Current Fluctuations in Semiconductor Devices
by
Peter Mayer
Submitted to the Department of Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degrees of
Master of Engineering in Electrical Engineering
BARKER
and
MA SSACHUSETTS INSTITUTE
OF TECHNOLOGY
Bachelor of Science in Physics
at the
JUL 3 1 2002
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
LIBRARIES
Feb 2002
@ Peter Mayer, MMII. All rights reserved.
The author hereby grants to MIT permission to reproduce and distribute publicly
paper and electronic copies of this thesis document in whole or in part.
IAuthor .....................................
Department of Electrical hngineering and CdTnAputer Science
February 10, 2002
Certified by..............................-.
Accepted by ..........
.................................
.
ajeev J. Ram
ssociate Professor
Thesis Supervisor
..........
Arthur C. Smith
Students
Graduate
on
Chairman, Department Committee
2
Current Fluctuations in Semiconductor Devices
by
Peter Mayer
Submitted to the Department of Electrical Engineering and Computer Science
on February 10, 2002, in partial fulfillment of the
requirements for the degrees of
Master of Engineering in Electrical Engineering
and
Bachelor of Science in Physics
Abstract
Current fluctuations in semiconductor devices are important for both practical and fundamental reasons. Measurements of the current noise in devices can establish fundamental
limits on the attainable signal-to-noise ratio in communication links and can also provide
insight into the basic physics of the device's operation. This work presents a suite of current noise measurement techniques useful for studying a range of devices. These techniques
are applied to investigate the extent to which the photon noise from lasers biased in the
same circuit is correlated due to the current noise in the shared bias currents. The first
measurements of the circuit-induced photon noise correlations in semiconductor lasers are
presented. A calibrated measurement of the photon noise of a single laser as a function of
its bias current is also presented.
Thesis Supervisor: Rajeev J. Ram
Title: Associate Professor
3
Acknowledgments
There are several people without whom this thesis would not have been possible.
My
advisor Professor Rajeev Ram has been a constant source of support. His unique gift of
understanding and explaining complicated phenomena with simple clarity was often called
upon, and his contagious enthusiasm for research was invaluable. Mr. Fahan Rana was
my principal collaborator on this work, and is responsible for the theory of noise in lasers
which was used in this thesis. Fortunately for me, his incredibly deep knowledge of physics
was matched by his patience in explaining things, and I leaned on his knowledge frequently.
Mr. Harry Lee has been involved in my research in one way or another since I joined this
research group. Harry has the dubious distinction of being the firefighter, the guy who I
can go to to fix anything. I cannot list all of the ways in which I have depended on his
expertise.
I have confined myself in these acknowledgements to only addressing my debts to people
which are most immediately related to the content of this thesis. More valuable to me than
any of the work I have done are the people I have worked with. Some of these friendships
are very old, and some are relatively new, but each means more to me than I feel capable
of expressing in words.
I must break this rule to thank some people who did not have make technical contributions to this thesis, but who at times entirely sustained me. My mother and father have
been a firm, unwavering source of support throughout my life. They taught me everything
I really need to know. Their honest, simple approach to life is my ideal. In a thesis about
electrical measurements, an analogy seems appropriate: they are my ground.
There is one more person who should certainly be thanked here, but I can't think of
how to do it. With luck, I will have the rest of my life to thank her properly.
4
Contents
1
2
15
Introduction
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1.2
Measuring and Characterizing Noise . . . . . . . . . . . . . . . . . . . . . .
17
1.3
Noise in a Fiber Communication Link
. . . . . . . . . . . . . . . . . . . . .
22
1.4
Link Slope Efficiency and the SNR . . . . . . . . . . . . . . . . . . . . . . .
25
1.5
Thermal Noise
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
1.6
Thesis Outline
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
31
Theory of Electrical System Noise Modeling
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
. . . . . . . . . . . . . . . . . . . . . .
32
2.2.1
Resistors, Capacitors, and Inductors . . . . . . . . . . . . . . . . . .
32
2.2.2
D iodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
Two-port Equivalent Noise Models . . . . . . . . . . . . . . . . . . . . . . .
38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
. . . . . . . . . . . . . . . . . . . . . .
41
2.4.1
One-port Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . .
42
2.4.2
Generalized Two-port Noise Models
. . . . . . . . . . . . . . . . . .
42
2.4.3
Multiple Amplifier Stages (Friss's Formula)
. . . . . . . . . . . . . .
48
2.4.4
Optimum Noise Resistance
. . . . . . . . . . . . . . . . . . . . . . .
49
2.1
O verview
2.2
One-port Equivalent Circuit Models
2.3
2.4
2.3.1
Transistors
2.3.2
Transformers
Theory of One-ports and Two-ports
5
6
3
CONTENTS
2.5
External Low Frequency Sources of Noise
2.6
Summary
51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
Current Noise Measurements
59
3.1
Summary of Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.2
Measurement Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
3.2.1
Low Noise Current Preamplifier . . . . . . . . . . . . . . . . . . . . .
60
3.2.2
Low Noise Voltage Preamplifier . . . . . . . . . . . . . . . . . . . . .
63
3.2.3
Low Noise Transformer Preamplifier . . . . . . . . . . . . . . . . . .
64
3.2.4
Data Aquisition System . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.3
Johnson Noise Measurements
. . . . . . . . . . . . . . . . . . . . . . . . . .
70
3.3.1
Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
3.3.2
Noise Thermometer
76
. . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
High Impedance Measurement
. . . . . . . . . . . . . . . . . . . . . . . . .
79
3.5
Low Impedance Measurement . . . . . . . . . . . . . . . . . . . . . . . . . .
83
3.5.1
New Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
3.5.2
Two Solutions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
3.5.3
Transformer-Coupled Measurement . . . . . . . . . . . . . . . . . . .
86
3.5.4
Calibration
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.5.5
Measurement
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
3.6
4
. . . . . . . . . . . . . . . . . . .
Circuit-Induced Laser Noise Correlations
93
4.1
Semiconductor Laser Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
4.1.1
Theories of Diode Noise . . . . . . . . . . . . . . . . . . . . . . . . .
95
4.1.2
High-Impedance Supression of Noise . . . . . . . . . . . . . . . . . .
96
4.1.3
External Current Correlations . . . . . . . . . . . . . . . . . . . . . .
97
Correlation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
4.2.1
104
4.2
The Transimpedance Preamplifiers . . . . . . . . . . . . . . . . . . .
7
CONTENTS
4.3
4.4
4.5
5
4.2.2
Low Frequency Sensitivity vs. Microwave Measurement Sensitivity .
105
4.2.3
The Photodetectors.....
. . . . . . . . . . . . . . . . . . . . . .
10 6
4.2.4
The Lasers
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 7
Measured Single Laser Fano Factor . . . . . . . . . . . . . . . . . . . . . . .
10 9
4.3.1
Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2
Measurement and Results
. . . . . . . . . . . . . . . . . . . . . .
1 12
Correlation Measurement......
. . . . . . . . . . . . . . . . . . . . . .
1 14
4.4.1
Preliminary Measurements
. . . . . . . . . . . . . . . . . . . . . .
116
4.4.2
Correlation Measurement
. . . . . . . . . . . . . . . . . . . . . .
1 19
Summary
121
. . . . . . . . . . . . . . .
123
Conclusions and Future Directions
5.1
5.2
Summary and Conclusions . . . . . . . . . . . . . . . .
. . . . . .
123
. . . . . . . . . . . . . . . . . . . . .
. . . . . .
123
. . . .
. . . . . .
124
. . . . . .
126
. . . . . .
126
5.1.1
M odeling
5.1.2
Current Noise Measurement Techniques
5.1.3
Photon Correlations in Circuit-Coupled Lasers
Directions for Future Work
. . . . . . . . . . . . . . .
131
A Current Noise in a Resonant Tunneling Diode
[1]
A.0.1
Tunnel Junctions at DC
. . . . . . . . . . . . . . . . . . . . . .
132
A.0.2
Mesoscopic Noise [2] . . . . . . . . . . . . . . . . . . . . . . . . . .
135
A .0.3
N oise in RTD s
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139
143
B Basics of Semiconductor Lasers
. . . . . . . . . . . . . . . . . . . . . . . .
143
. . . . . . . . . . . . . . . .
145
. . . . . . . . . . . . . . . . . . . . . . . . .
146
. . . . . . . . . . . .
147
B.1
Semiconductor Laser Structure
B.2
Carrier Recombination and Light Generation
B.2.1
B.3
1 10
Laser Rate Equations
Solution of the Rate Equations for Low Frequencies
C Matlab Code
151
8
CONTENTS
List of Figures
1-1
The definition of the relative intensity noise (RIN). . . . . . . . . . . . . . .
20
1-2
The two alternate circuit representations of a resistor's thermal noise.
. . .
21
1-3
A digital optical fiber link .. . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
1-4
The conditional PDFs of the output voltage for a "1" input and a "0" input.
25
1-5
L-R circuit with Langevin voltage noise source. . . . . . . . . . . . . . . . .
28
2-1
One-port noise model of a resistor. . . . . . . . . . . . . . . . . . . . . . . .
32
2-2
One-port noise model of a capacitor. . . . . . . . . . . . . . . . . . . . . . .
34
2-3
One-port noise model of an inductor. . . . . . . . . . . . . . . . . . . . . . .
34
2-4
One-port noise model of a diode. . . . . . . . . . . . . . . . . . . . . . . . .
35
2-5
Two-port noise model of a bipolar junction transistor.
. . . . . . . . . . . .
38
2-6
Two-port noise model of a field effect transistor.
. . . . . . . . . . . . . . .
39
2-7
Two-port noise model of a transformer.
. . . . . . . . . . . . . . . . . . . .
41
2-8
Two-port noiseless model of an operational amplifier..
. . . . . . . . . . . .
43
2-9
General noiseless model of a two-port network.....
. . . . . . . . . . . .
43
2-10 General two-port network with voltage noise sources at the input and output. 44
. . . . . . . . . . . . .
45
. . . . . .
46
2-13 Measurement with noisy voltage amplifier. . . . . . . . . . . . . . . . . . . .
49
2-14 Capacitive coupling of noise into a measurement. . . . . . . . . . . . . . . .
52
. . . . . . . . . .
53
2-11 General two-port network with input referred noise.
2-12 Balanced detection scheme for the measurement of sma 11 signals.
2-15 Capacitive coupling of noise into a current measuremen t .
9
LIST OF FIGURES
10
2-16 Correct and incorrect shielding of a sensitive measurement.
. . . . . . . . .
55
2-17 Inductive coupling of noise into a measurement. . . . . . . . . . . . . . . . .
55
2-18 Microphonic coupling of noise into a measurement. . . . . . . . . . . . . . .
56
3-1
Two-port model of a current amplifier. . . . . . . . . . . . . . . . . . . . . .
62
3-2
Two-port model of a voltage amplifier. . . . . . . . . . . . . . . . . . . . . .
64
3-3
Resolution and bandwidth of analog-to-digital converters (1999).
. . . . . .
66
3-4
Transfer function of ideal DAQ. . . . . . . . . . . . . . . . . . . . . . . . . .
67
3-5
Original, quantized, and error signals for 3-bit and 6-bit quantization.
. . .
68
3-6
Experimental setup for Johnson noise measurements. . . . . . . . . . . . . .
70
3-7
Experimental setup for room temperature Johnson noise measurements.
. .
71
3-8
Transfer function of amplifier chain.
. . . . . . . . . . . . . . . . . . . . . .
73
3-9
Measured noise with different source resistances, and theoretical contributions to the noise obtained with a curve fit.
. . . . . . . . . . . . . . . . . .
3-10 Noise power measured as a function of temperature.
. . . . . . . . . . . . .
75
78
3-11 Noise measurement and calibration scheme for photodetector optical noise
m easurem ent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
3-12 Noise measurement and calibration scheme for photodetector optical noise
measurement, written to emphasize symmetry of the signal generators. . . .
3-13 DUT for calibration circuit, with parasitic capacitance.
the Thevenin equivalent source and impedance values.
81
Also included are
. . . . . . . . . . . .
82
3-14 M easured Fano factor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
3-15 inferred fundamental charge over valid frequencies.
. . . . . . . . . . . . . .
84
3-16 Low-frequency noise model of a diode. . . . . . . . . . . . . . . . . . . . . .
85
3-17 Transformer-coupled noise setup. . . . . . . . . . . . . . . . . . . . . . . . .
87
3-18 Results of parameter extraction from I-V curve. . . . . . . . . . . . . . . . .
90
3-19 Transfer function of the transformer-coupled measurement with diode DUT.
91
3-20 Measurement of diode shot noise using transformer-coupled measurement. .
92
LIST OF FIGURES
11
4-1
One-port noise model of a diode. . . . . . . . . . . . . . . . . . . . . . . . .
94
4-2
Pump supression in a diode laser. . . . . . . . . . . . . . . . . . . . . . . . .
96
4-3
Four simple bias circuit topologies. . . . . . . . . . . . . . . . . . . . . . . .
98
4-4
The correlated photon noise measurement setup. . . . . . . . . . . . . . . .
102
4-5
Laser characterization curves. . . . . . . . . . . . . . . . . . . . . . . . . . .
108
4-6
The calibration measurement setup at high frequencies.
. . . . . . . . . . .
109
4-7
Measured transfer functions needed for calibrated photon noise measurements. 112
4-8
Power spectrum of the measured voltage noise, taken at Ibias = 81 mA.
4-9
Power spectrum of the photodetector current noise, taken at
bias =
.
81 mA.
113
114
4-10 Photodetector current and incident light Fano factors as a function of laser
b ia s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 15
. . . . . . . . . . . . . . .
116
4-12 Spurious correlation between two lasers voltage biased in separate circuits. .
117
4-13 Spurious correlations measured with one laser off. . . . . . . . . . . . . . . .
118
4-14 Noise correlation measured for lasers in 4 different circuits.
120
4-11 The correlated photon noise measurement setup.
. . . . . . . . .
4-15 Possible mechanism for spurious correlation in circuits (B) and (C).
. . . .
121
5-1
The four measured bias circuit topologies. . . . . . . . . . . . . . . . . . . .
127
5-2
One-port noise model of a diode. . . . . . . . . . . . . . . . . . . . . . . . .
128
A-1
A practical realization of an RTD, and a schematic representation of the
structure [3].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
132
A-2 A schematic energy level diagram for an RTD in the unbiased and biased
case, showing the well density of states.
. . . . . . . . . . . . . . . . . . . .
133
A-3 A typical I-V curve of an RTD. . . . . . . . . . . . . . . . . . . . . . . . . .
134
A-4 A 1-D channel connecting two reservoirs, showing orthogonal occupied and
unoccupied transmission states. . . . . . . . . . . . . . . . . . . . . . . . . .
136
A-5 A 1-D channel connecting two reservoirs at low temperature and with an
applied voltage bias. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
138
12
LIST OF FIGURES
B-1
Basic structure and operation of a laser. . . . . . . . . . . . . . . . . . . . .
B-2 Important radiative processes in a semiconductor laser.
. . . . . . . . . . .
143
145
List of Tables
3.1
Performance of the SR570 low noise transimpedance amplifier.
. . . . . . .
61
3.2
Performance of the SR554 transformer amplifier.
. . . . . . . . . . . . . . .
65
3.3
Best fit param eters.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
3.4
Best fit parameters for the measurement of Chen and Kuan
4.1
Key specifications of the Analog Devices OP-27 operational amplifier used
[4].
. . . . . . .
76
for the m easurem ent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
104
4.2
Key specifications of the photodetectors used for the measurement. . . . . .
107
4.3
Key specifications of the lasers used for the measurement.
. . . . . . . . . .
108
4.4
Average measured spurious correlation with one laser off.
. . . . . . . . . .
119
4.5
Average noise correlations. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
13
14
LIST OF TABLES
Chapter 1
Introduction
1.1
Introduction
Understanding current fluctuations (also called current noise) in semiconductors is important for practical and fundamental reasons. The most important practical reason is that
the fluctuations often fundamentally limit the device's signal-to-noise ratio. In most systems in which information must be extracted from an analog electrical signal, current noise
plays a part in setting the specifications of the devices [5], [6], [7], [8]. With the continued
shrinking of the MOS transistor and the resulting smaller signal levels, noise will become
more important in the digital realm as well. Understanding noise's origins, how it can be
modeled, and how it propagates through a complex system is essential to modern device
design. Noise has also proven valuable as a test of quality and reliability in many devices,
since it is highly sensitive to impurities and defects.
Aside from practical considerations, the measurement of current fluctuations in semiconductors provides a way to study interesting physics in the device. The easiest access to the
physics of a device is provided by DC response curves such as the I-V (current vs. voltage)
curve for electron devices and the L-I (light intensity vs. current) curve for light-emitting
structures.
However, at the price of a slightly more challenging measurement, studying
current fluctuations often yields more useful information about the underlying physics of a
15
CHAPTER 1.
16
INTRODUCTION
device than a DC measurement [9]. There are several reasons for this.
In any device, there is always some 'white' noise superimposed on a DC biased device
(noise is called white if it has equal power over a wide range of frequencies, in analogy with
white light).
The presence of some of this noise in any dissipative system is a necessary
consequence of the fluctuation-dissipation theorem [10].
Shot noise, noise resulting from
the discreteness of charge in a biased device, also contributes white noise. A measurement
of these simple noise sources generally would not yield more physical information than
DC measurements.
However, because the white noise is filtered by the dynamics of a
particular device, measuring the device's current noise is a way to examine the microscopic
dynamics of the device. A practical example of this technique is the determination of the
relaxation-oscillation frequency of a laser diode, a measure of the rate at which the laser's
light can be modulated with an AC current [11].
This measurement can be performed
by simply examining the current noise spectrum of the device with a DC bias. It is not
necessary to modulate the current driving the laser, or detect the emitted laser light. This
is a particularly useful measurement for lasers emitting at wavelengths for which fast or
sensitive detectors do not exist.
Current fluctuation measurements can also provide information about mesoscopic charge
transport, since interactions between charge carriers either through Coulomb repulsion or
through the Pauli exclusion principle result in correlated motion of the electrons, altering
the noise of the device [12]. Such many-body effects are generally not reflected in the DC
response curve of a device. In addition, noise is a very sensitive probe of many forms of
scattering. In many devices the noise at low frequencies (called 1/f noise for the shape of its
power spectrum) has been found to be strongly correlated with the impurity concentration
of the device. Thus noise is often a measure of the electrical quality of a fabricated device.
This thesis develops a suite of techniques for measuring low-frequency current fluctuations in semiconductor devices and reviews the theory necessary for designing these mneasurements. In general, noise measurements can be divided into frequency and time domain
methods. The former is common in the literature [13] [14] [15] [16], and relies on a frequency
1.2. MEASURING AND CHARACTERIZING NOISE
17
measurement of the noise using a microwave spectrum analyzer and a low noise amplifier.
A time-domain setup relies on sampling and quantization of a time signal to measure lowfrequency noise (<1 MHz).
In many cases, this method allows for greater accuracy and
flexibility than the standard RF techniques.
In this thesis, the focus is on time domain
measurements. Several setups are designed to allow the measurement of noise from devices
of varying impedance. These methods are then applied to make the core measurements of
this thesis. First, the photon noise from a heterostructure semiconductor laser diode is measured as a function of bias. Second, the correlations introduced into the photon streams of
series and parallel electrically coupled laser arrays are measured. Both results are analyzed
in the context of current theory.
In general, the challenges in designing these measurements fall into three basic categories.
First, the noise from the measurement apparatus which inevitably pollutes the measurement
of device noise must be minimized. Second, the exact contribution of the test device's noise
to the total measured noise must be measured, and the other unrelated noise subtracted
out. Finally, a measurement and calibration method must be found to match both low
and high impedance devices. The noise measurement methods described in this thesis will
be discussed here in a broad context so as to aid readers in the custom design of noise
measurement systems.
In the remainder of this chapter, a general introduction to the subject of current noise
in semiconductor devices is given. The various measures used to describe noise are defined,
and the basic classes of noise measurement techniques are described.
To motivate the
practical importance of noise considerations and to demonstrate the analysis of a system's
noise budget, the noise in a fiber communication link is discussed. Finally, the Langevin
formalism is introduced to derive the spectral density of the thermal noise of a resistor.
1.2
Measuring and Characterizing Noise
The fluctuations in the current measured through a device generally contain contributions
from the motion of large numbers of interacting charge carriers.
Modeling such a sys-
CHAPTER 1.
18
INTRODUCTION
tem deterministically is impossible, so current fluctuations are treated mathematically as
stochastic processes. All information about a general stochastic process is contained in the
joint probability distributions (PDFs) for the process at all possible times. There is no easy
way to measure these PDFs in general, and in most practically interesting cases there is no
need for such an exhaustive measurement. Many physically interesting random processes
are characterized entirely or chiefly by the first and second order moments of their PDFs,
so that the others may be neglected. The measurable quantities of these are the mean, the
variance, and the autocorrelation function.
A simple way to obtain information about the first and second moments of an unknown
process is to repeatedly measure the fluctuating signal at small time intervals. Such a measurement is called a time-domain technique. The samples of the random process should
be taken with time separations much shorter than the time span for which the autocorrelation function shows interesting structure. The mean, variance, and autocorrelation of
the process are then easily computed. Often performed with a sampling oscilloscope, the
measurement is attractive for its simplicity and its general applicability, but is limited by
the sampling speed of the measuring instrument. Many physical processes of interest occur
over time scales shorter than a nanosecond, which is too fast to be accurately measured by
modern time-domain sampling techniques.
Other measurements characterizing the first and second moments of the random process
are performed in the frequency domain. To connect the time and frequency domains for a
deterministic signal, one typically relies on the Fourier or Laplace transform
[17]. However,
the Fourier and Laplace transform of a stochastic processes are not well defined
[18]. A
different route to the frequency domain is through the Wiener-Khinchine theorem, which
states that the autocorrelation of a time signal and the power spectral density (PSD) of the
signal are related through the Fourier transform. The PSD can alternately be viewed as the
result of a measurement of the RMS power in small frequency bins at every frequency. This
is exactly the job of a spectrum analyzer, the work-horse of frequency domain measurements.
Spectrum analyzers which can measure signals in the tens of gigahertz range are readily
1.2. MEASURING AND CHARACTERIZING NOISE
19
available today. However, because microwave spectrum analyzers are usually operated in
an environment matched to 50 Q, there is a best-case noise floor set by the thermal noise
of a 50 Q load. Time domain measurments are free of this constraint, and can often be
optimized for better noise performance than a comparable microwave system, depending on
the specific details of the circuit. More will be said about this in Chapter 4.
The metrics used to describe noise vary. In theoretical discussions, noise is typically
modeled as an instance of some random process.
Starting with an understanding of the
basic probability distributions of the noise process, the noise's autocorrelation and power
spectral density can be derived. An example of this is found later in this chapter, when the
power spectral density of the noise in a 1-D quantum channel is derived. When describing
unwanted noise in practical analog systems, the relevant quantity is generally the signal to
noise ratio (SNR), defined as the RMS power of the signal divided by the RMS power of
the noise:
2
SNR - -V(Psignai(t) )
(Pnoise (t)2 )
For zero mean signals, this reduces to the ratio of the variance of the signal to the noise
process. The SNR is more commonly expressed in decibels:
SNRdB = 10log1 0 (SNR)
One point of possible confusion should be clarified.
(1.2)
When discussing electrical current
signals, the SNR is a ratio of mean square electrical currents. Equivalently, the SNR for the
electrical signal is a ratio of electrical powers. When dealing with light, the SNR is a ratio
of mean square optical powers. This is because in light-wave systems, signals are typically
measured in power, whereas in a circuit, current or voltage are generally the signals.
When describing noise in a digital communication channel, the preferred metric for noise
is the bit error rate (BER). The BER is the ratio of the average number of bits transmitted
incorrectly per unit time to the average total number of bits transmitted per unit time. In
practical digital fiber systems, bit error rates of 10-9-10--"
are commonly required [19].
CHAPTER 1.
20
INTRODUCTION
For more specific applications, there are other noise metrics. The relative intensity noise
(RIN) is of interest because it is the accepted way to describe noise in lasers. It is defined as
the mean square noise power divided by the mean square average power level. In general,
the noise is white within the communication band so that the noise can be characterized
with a certain power per 1 Hz of bandwidth. Expressed in decibels, a typical value for the
RIN in a modern communication system is between -130 dBm and -140 dBm
[19].
Fig. 1-1
[11] illustrates a noisy analog light signal with a mean power level Pave, a signal power
and some rms noise power
Pmod,
a. The SNR and the RIN for this signal are given by:
P(t)-PaveP modSin((ot)
P(t)
Pmod
P
ave
t
Figure 1-1: The definition of the relative intensity noise (RIN).
SNR _((Pmod sin(Wt)) 2 )
mod
(1.3)
o2
RIN = -"
Pave
Given the modulation index m
(1.4)
Pmod/Pave, the RIN and the SNR can therefore be easily
related. Later in this section, a relation between the RIN and the BER is derived for a
simple communication channel.
1.2. MEASURING AND CHARACTERIZING NOISE
21
When modeling noise in electrical circuits at low frequencies, it is customary to use
noise generators
[20].
These are idealized current or voltage sources whose output is a
stochastic process in time, generally with a mean value of zero, a Gaussian distribution,
and a white spectrum. Because the variance of true white noise is not well-defined (due to
the finite signal power over an infinite bandwidth), these generators are typically specified
with a value in units of Volts/v' Hz or Amps/v Hz. In electronic circuits, the noise power
across any two ports can be represented as a noise current (voltage) generator in parallel
(series) with the Thevenin resistance of the network defined by the two terminals, as shown
in Fig. 1-2. A more careful description of the noise models for circuit elements is given in
Chap. 2.
VTh =Th RT
RT
+
Th
RT
-0
tw
a
Figure 1-2: The two alternate circuit representations of a resistor's thermal noise.
In microwave and RF circuits, a simplification of the low frequency description given
above is possible due to presence of a universal input and output line impedance (typically
50 Q). In this case, a single number, such as the SNR, is used to characterize the noise at
a node in the circuit. Likewise, a single number known as the noise figure (NF), is used to
characterize the change in SNR between the input and the output of a device. The NF of
a device is given by:
SNR~
NF = 10 log1 o SNRin
SNROut
(1.5)
When the NF is given as a specification of a RF or microwave device, the noise at the input
CHAPTER 1. INTRODUCTION
22
of the device is assumed to be the thermal noise associated with a 50 Q resistor. Typical
noise figures for low-noise room temperature RF amplifiers are between 1 and 3 dB. Note
that when describing the noise performance of a device in a circuit not impedance matched
to 50 Q, the impedance looking out from the device's input must be specified for the NF to
have meaning. More will be said about these issues in Chapter 2.
1.3
Noise in a Fiber Communication Link
To understand how noise enters and affects a practical system, consider the direct detection
of a fiber optic signal. Noise present in the incident light beam and noise introduced in the
detection process combine to set a fundamental limit on the accuracy of the received signal,
quantified here by the bit error rate at the receiver.
A direct detection receiver typical of those used in fiber optic communication links [8] is
shown in Fig. 1-3. Digital data is transmitted using light of frequency v through the fiber
link using a simple on-off keying format. The light signal intensity L,(t) is assumed to take
the values Lo and L, with equal probability. Superimposed on the signal is some noise with
intensity L,(t). This light signal is transduced into current using a PIN diode photodetector
with a quantum efficiency r, an area A, and a bandwidth Av. The quantum efficiency is the
average number of electrons generated for each incident photon. The resulting current signal
is gained up by an amplifier with a transimpedance of R and, for simplicity, a bandwidth
equal to that of the detector. The amplifier will always contribute some noise to the input
signal, modeled here as a noise current generator Ian (t) placed across the input of the
amplifier. Methods of modeling noise from amplifiers will be discussed in more detail in
chapter 2. The output voltage from the signal alone is V(t), and the output voltage from the
noise is V,(t). The output signal voltage V1(t) can take on values of Vo or V corresponding
to the input light signal levels of Lo and L 1 . Finally, the output voltage (the sum of the
output signal voltage and the output noise voltage) is fed into a comparator which interprets
the signal as a one or zero depending on whether it is greater or less than some threshold
voltage.
1.3. NOISE IN A FIBER COMMUNICATION LINK
23
R
L-
--
F_
optical
---
+
an
+
Figure 1-3: A digital optical fiber link.
For simplicity only amplitude noise is considered in this simple model, and uncertainties
in the timing of the pulses are neglected. If a photon in the incoming stream of light has an
arrival time independent of the others (so-called Poissonian statistics), then two principal
contributions to the output noise voltage V(t) can be expected: the noise present in the
light L, and the current noise Ian (t) from the amplifier. The noise present in the incident
light signal L, is shot noise, due to the signal's Poisson statistics. This is generally a good
assumption in practical communication systems, especially if the link between the receiver
and the source is lossy. The noise L,(t) is transduced by the photodiode into current noise
In(t)
-
L,(t) Aq.
Here
Ln(t)A
is simply the number of noise photons incident on the detector
per unit time. The signal L,(t) is similarly transduced into a current 1 (t) = L,
r/qq. This
current signal is then tranduced by the amplifier into a voltage V(t) = Is(t)R. The two
current noise sources at the amplifier's input are similarly transduced into output noise
voltage noises V1, (t) from the incident light and Van (t) from the amplifier noise.
Both V 1,(t)
and Van(t) can be modeled as random processes. Because it was assumed
that each photon arrival was independent of other photon arrivals, In (t) is a Poisson process
with mean I(t). Its variance is therefore:
2
or,=21, (t)qAv(16
(1.6)
CHAPTER 1.
24
INTRODUCTION
At the output, this results in a voltage noise:
= 21(t)RqAv = 2V(t)qAv
a
(1.7)
The variance of Ian(t) is determined by the internal details of the amplifier (see Chapter 2),
and is typically given by some a
a, giving rise to an output voltage noise variance of:
2
UVa
= R2202 a
_
(1.8)
Since the light and amplifier noise sources are statistically independent of one another, their
variances can be summed to calculate the total variance of the output noise voltage:
2 = o a + o7
(1.9)
The plot in Fig. 1-4 depicts the PDFs of the total output signal V(t) + V"(t) for the
case when a zero and a one are transmitted. In general the variance of the PDF given that
a '0' was transmitted can be different from the variance of the PDF given that a '1' was
ao and a1, respectively. The output signal is sampled at discrete
transmitted. Call these
times, once per bit. To optimize the performance of the receiver, a threshold voltage V'h
must be determined.
If a voltage above V'h is sampled, the signal is recorded as a one;
otherwise, the signal is recorded as a zero.
The optimum cutoff voltage minimizes the
probability of error, which is equal to the sum of the two shaded areas in Fig. 1-4 (with
each area weighted by the probability of receiving a one or zero).
A reasonable (but not
quite optimal) choice for a cutoff voltage is the intersection of the two Gaussian curves.
This voltage is given by:
a
VinIf
Q(X)
=
exp(
4) dy,
1
0o 1
+ uV
Oo + or11 0
(1.10)
the shaded area under the Gaussian can be calculated to
find the bit error rate:
V1 - V
Oro + (T
1.4. LINK SLOPE EFFICIENCY AND THE SNR
probability density
function if a 'O' was
transmitted
25
probability density
function if a '1' was
transmitted
probability
of error
V0
V
V
V
Figure 1-4: The conditional PDFs of the output voltage for a "1" input and a "0" input.
In the preceeding discussion, the BER of a system and the statistics of a system's
fundamental noise sources were connected. The basic method can be extended to account
for more complicated communication protocols or detection schemes [6],[21],[22].
1.4
Link Slope Efficiency and the SNR
During the discussion of the noise in a fiber link given in the previous section, the noise in
the photons incident on the photodetector was assumed to be shot noise. This was done
mainly to avoid involving a more detailed noise model for the photons emitted from the
laser into the discussion; in a real link, the noise can be many times shot noise, and may or
may not even be set by the laser's intrinsic noise. But in any case, the noise which degrades
the signal to noise ratio of an optical link does not typically scale with the modulation
CHAPTER 1. INTRODUCTION
26
power of the signal, but rather with the average output of the laser, or with some other
unrelated source (e.g. thermal noise in the modulation circuit). In such a sitiiation, one
way to improve the SNR of the link is to increase the link slope efficiency.
In a microwave analog link, the link slope efficiency is the small signal gain between the
modulated current driving the laser (or external modulator) and the small signal output
current at the receiving photodiode. If we confine the discussion to links whose components
are impedance matched to 50 Ohms, the link slope efficiency for a simple link of the type
discussed in the previous section is simply the product of the laser slope efficiency and
the photodetector slope efficiency, where the photodetector slope efficiency is assumed to
include any optical loss in the link. Clearly, the best possible link slope efficiency for such a
setup is 1, and this is often difficult to achieve. This limit is physically set by the simple fact
that for every electron which is injected into the laser as part of the modulation current,
at most one photon is emitted; at the photodetector, at best one electron of modulation
current results from this incident photon.
Several schemes have been proposed for improving the link slope efficiency [23].
One
proposal is to use a series cascade of lasers. In this scheme, a single electron is capable of
producing multiple photons (one photon from each laser). Using a series array of discrete
lasers, efficiencies of greater than
1 have been attained [24]. By epitaxially growing the
lasers together and coupling them using tunnel junctions, one can overcome many of the
bandwidth-limiting parasitic issues associated with the series discrete lasers, and still see
enhanced slope efficiencies. The bipolar cascade laser, a working prototype of this concept
useful for fiber links, has recently been demonstrated [25].
A pressing theoretical and
experimental question associated with this approach is to what extent the added link slope
efficiency improves the current-to-current link SNR.
To calculate the theoretical SNR in a series cascade of N lasers, one might reason that
total signal could be found by adding the magnitudes of each of the N individual laser signals
in phase. To find the total noise from the N lasers one would add the variances of the N
individual noise signals, assuming each laser's noise is independent of the other lasers' noise.
THERMAL NOISE
1.5.
This would give a
27
= vN improvement in the SNR compared to a single laser. According
to this simple reasoning, by adding more series laser stages one can achieve arbitrarily high
SNR. Of course the real world is not so kind, and there are several ways in which this
analysis fails to hold. One fundamental error of the calculation is the assumption of totally
independent noise sources. In this thesis, this assumption is investigated experimentally by
measuring the correlation between the light of series and parallel coupled lasers. To the
extent that this light is correlated, the noise contributed by each must not be considered
independent.
1.5
Thermal Noise
One fundamental source of noise in practical systems is thermal noise.
is present in every dissipative system.
Thermal noise
In their seminal paper, Callen and Welton
{10
showed that fluctuation and dissipation are inextricably linked together on the quantum
level. Dissipation of energy in a system occurs through a coupling into some reservoir. This
coupling between the states of the system and the bath of reservoir states causes information
about the state of the system to be leak out; this loss of information manifests itself as noise
in the system.
The most important source of dissipation from a circuit persepctive is the resistor.
Nyquist [26] originally derived an expression for the thermal noise of a resistor with a very
clever thermodynamical argument involving a resistor attached to a matched transmission
line. Here the thermal noise of a resistor is derived in a different way using the Langevin
method. The Langevin method is a very powerful and general tool for analyzing noise in
linear systems.
Consider the simple L-R circuit shown in Fig. 1.5, along with a white thermal voltage
noise generator whose spectral density we wish to determine. A differential equation relating
the voltage source v(t) to the current in the circuit i(t) can be written:
L
ditt)
+ R i(t) = v(t)
dt
(1.12)
CHAPTER1. INTRODUCTION
28
+
v(t)
L
1 i(t)
R
Figure 1-5: L-R circuit with Langevin voltage noise source.
Fourier transforming this equation, we can write:
LdjwI(w) + RI(w) = V(w)
(1.13)
To find the power spectral densities of the current and voltage, we multiply by the complex
conjugate:
(LjwI(w) + RI(w)) (-LjwI*(w) + RI*(w)) = V(w)V*(w)
(1.14)
Multiplying out and rearranging:
,P)
S~(Ll) =R
2
Sv
+
w2 L 2
(1.15)
It will be convenient to have our spectral densities in terms of frequency f rather than
angular frequency w, so that the result of the calculation will be in a recognizable form.
Si(f) =
With the use of a trigonometric identity,
Sv
f)
R2 + (2bf)2L2
(1.16)
Si can be easily integrated to find the total current
1.5.
THERMAL NOISE
29
power:
(z2) (i=
*
f*X
10
Si(f) df = S
df
R 2 + (2rf)2L 2
Jo
_
-
_
_
"O
4RL
(1.17)
We can assume that the inductor and the resistor are in thermal equilibrium. From the
equipartition theorem of statistical mechanics, it is known that the stored energy in the
circuit is given by:
1
2
1
2
-L(2)
-kT
(1.18)
Inserting this result into Eqn. 1.17, we immediately obtain:
Sv = 4kTR
(1.19)
The Langevin approach combined with the equipartition theorem of statistical mechanics
has allowed the rapid calculation of the thermal noise in a resistor.
Unfortunately, like
Nyquist's original derivation, the Langevin method does not really provide much insight
into the microscopic motions of the electrons which give rise to thermal noise.
If the derivation is taken a step further, the voltage noise spectral density can be inserted
into Eqn. 1.15 to calculate the current power spectral density.
4kTR
R
2
+ W2 L
2
(1.20)
The current power spectral density is not white, but is low-pass filtered by the inductor
in the circuit. This makes it clear how systems can display complex noise behavior even
though the fundamental sources of noise in the system may be very simple. The power of
the Langevin formalism, barely used in this simple example, lies in its ability to propagate
simple sources of noise through complex system models.
CHAPTER 1.
30
1.6
INTRODUCTION
Thesis Outline
The basic outline of this thesis is as follows. Chapter 2 investigates the basic sources of noise
in high sensitivity measurement systems. Models for the noise in important practical devices
are given, and a general framework for dealing with noise in electronic circuits is presented.
Less fundamental sources of noise important in the design of low noise measurements are
discussed. In Chapter 3, the measurement instruments built in this thesis are introduced.
The challenges involved in performing the specific measurements of this thesis are outlined,
and solutions are chosen. Measurements calibrating and testing the setups are described
and results of these preliminary measurements are presented.
The goal of Chapter 4 is
twofold. First, the circuit model for laser noise is used to calculate correlations in the light
of circuit coupled lasers. In the second part of the Chapter, measurements made on lasers
are described and the results are presented. Results are compared to theoretical calculations
when appropriate. Finally, Chapter 5 discusses possible improvements to the measurements,
and indicates directions for future work in the field.
Chapter 2
Theory of Electrical System Noise
Modeling
2.1
Overview
It is a fundamental tenet of experimental physics that a measurement always disturbs, to
some degree, the system which is measured. When possible, one designs the measurement
such that this disturbance is small. In the case of noise measurements, this is not often
practically feasible. It is necessary to understand the noise sources present in each of the
basic building blocks of a measurement apparatus, so that their effects may be properly
accounted for during an analysis of the measured results.
The purpose of this chapter is to list some useful equivalent circuit noise models and
describe how these models can be used to design measurements. Noise circuit models for
circuit elements used for this thesis are presented. The goal here is not to review how all of
the models are derived, but rather to present the models with enough physical motivation
to aid an experimenter attempting measurements similar to the ones undertaken in this
thesis. References to more detailed treatments of the models are given.
Along with the intrinsic noise in circuit elements, other important external sources of
spurious noise are reviewed, along with practical methods for avoiding them.
31
CHAPTER 2. THEORY OF ELECTRICAL SYSTEM NOISE MODELING
32
2.2
2.2.1
One-port Equivalent Circuit Models
Resistors, Capacitors, and Inductors
Resistors, capacitors, and inductors are the simplest one port electronic devices. Of the
three, only resistors contribute significant noise to the system, due to the fact that they
are inherently dissipative. The noise model for a resistor shown in Fig. 2-1 consists of the
resistor in parallel with a current source which supplies the noise signal. Ideal elements are
shown with a dotted box around them.
R
R
Figure 2-1: One-port noise model of a resistor.
The current noise signal Ith is treated as an instance of a Gaussian stochastic process
with a white frequency spectrum. The magnitude of the current noise is most conveniently
described by giving the noise current power per 1 Hz of bandwidth. This spectral density
was shown by Nyquist [26] to be:
(2.1)
S1,th =4kT
R
In addition to the thermal noise (also called Johnson noise) current in parallel with the
resistor, there is also a 1/f noise source. This noise also has zero-mean Gaussian statistics,
and is well represented by a spectral density:
S,1/f (
=
C
(2.2)
2.2. ONE-PORT EQUIVALENT CIRCUIT MODELS
33
In this expression IDC is the DC current flowing through the resistor and a is a constant
whose value depends on the specific type of resistor. Of the four common types of resistances, carbon-composition have the highest
a. Carbon-film resitors typically have a about
1/4 the value of the carbon-composition type. Metal-film resistors are next, with a about
1/4 the value of the carbon-film resistors. Best are wire-wound resistors, with a roughly
1/4 the value of the metal-film resistors [27]. Of course these numbers are intended as a
rough practical guide for a circuit designer, not as a physical theorem of accurate or universal validity. Identifying the physical mechanisms responsible for 1/f noise is an unsolved
problem, in the sense that there exists no accepted explanation for why it is found in so
many different physical systems, from the tides of the ocean to the voltage flucuations in
the gate of a FET. For a review of the more popular theories for 1/f noise and a guide to
the extensive literature on the subject, see [28].
Because ideal capacitors and inductors do not dissipate energy, they do not contribute
noise to the system. However, to the extent that real devices are lossy (non-zero leakage
currents in capacitors and non-zero parasitic resistances in inductors) this is violated. With
some care in selecting high-quality components, these unwanted noise sources can be made
negligible for a practical experimental setup. Also, while inductors and capacitors may be
intrinsically noiseless, their environment is not. Inductors can be particularly effective in
picking up stray magnetic fields from power lines or other electronic equipment and coupling
unwanted inductive currents into measurement. Stray electric fields can capacitively couple
unwanted voltages into the circuit, but this problem is can be somewhat alleviated by proper
electrostatic shielding, as will be discussed later.
The noise model for a capacitor is shown in 2-2. Note that the lead and leakage resistances are not ideal, and therefore can contribute thermal and 1/f noise according to the
resistor noise model given above. For economy of notation, the noise generators for resistors
will not be explicitly drawn in this thesis, unless their presence deserves special emphasis.
Resistors in noise models can be assumed to possess thermal and 1/f noise unless a dotted
box is drawn around the resistor, indicating that it is ideal. In capacitors, the resistors
34
CHAPTER 2. THEORY OF ELECTRICAL SYSTEM NOISE MODELING
I
I
Rlead
Rleak
C
Figure 2-2: One-port noise model of a capacitor.
and their noise sources can usually be neglected.
A notable exception is for electrolytic
capacitors, which have comparatively large leakage currents and should be used with care
in sensitive circuits.
Also, polar electrolytic capacitors can emit large amounts of burst
noise for hours if they experience even a momentary reverse (incorrect) bias.
The noise model for an inductor is shown in 2-3. The model includes a non-ideal resistor
R series
Li
Figure 2-3: One-port noise model of an inductor.
in series with an ideal inductance to model the series resitance of the windings and any
2.2. ONE-PORT EQUIVALENT CIRCUIT MODELS
35
magnetic losses in the core of the inductor. In contrast to the capacitor, this resistance is
generally noticable. For a 100 mH inductor, a typical Rseries is about 100 Q.
2.2.2
Diodes
The basic current noise model of a diode, laser or otherwise, is shown in Fig. 2-4.
R series
r
In(ALI11/f
Figure 2-4: One-port noise model of a diode.
The resistor Rseries is a parasitic resistance representing the losses in the Ohmic contacts
of the diode. For the laser diodes measured for this thesis, this value is in the 2-10 Q range.
Because it is a real resistance with actual power loss, it contributes noise as specified by the
resistor noise model given above. In series are the parallel combination of the differential
resistance rd, a 1/f noise source Il/f, and a current noise source I,.
The differential resistance rd is not a real resistance in the sense that it is capable of
dissipating power, but is rather a ratio of the small change in voltage to the corresponding
change in current across the junction; it is therefore depicted as ideal. The current through
CHAPTER 2. THEORY OF ELECTRICAL SYSTEM NOISE MODELING
36
a diode is typically approximated by an exponential law of the form
Vi
I = Is (e nkT
-
1)
(2.3)
where I is the current through the diode, I, is the saturation current of the diode, V is the
voltage across the diode, q is the charge of an electron, k is Boltzmann's constant, T is the
temperature of the diode, and n is an empirical fitting constant of order 1 which can have
a weak bias dependence. The differential resistance is found as:
dV
ddI
nkT
-
Vq
qIenk
nkT
qI
(2.4)
(
For higher frequency modeling, one includes diffusion and depletion capacitance in parallel
with Rd.
The precise origins of the Ii/f noise source are still poorly understood, but it typically
takes the form:
SI,/f(f) = 0I C
(2.5)
Typically 3 and -y are of order 1, and a varies widely depending on the diode. A helpful
parameter is the 1/f noise corner frequency, which is the frequency at which the 1/f current
noise power SI,i/f equals the noise current power Sin. This typically occurs at around 10
kHz, but can range between 1 kHz and 1 MHz.
The noise current source I1 in the diode model is taken to be an instance of a Gaussian
stochastic process with white spectral density. This is a reasonable approximation in the
low frequency limit in which one is interested in frequencies less than the inverse of the
characteristic carrier scattering and transport times of the diode, as is the case in this
thesis. The magnitude of this noise, or more precisely the noise power per unit frequency,
is dependent on the detailed physics of the diode, and sometimes the bias current.
For
example, in a simple tunnel junction, each carrier's transport across the junction is largely
2.2. ONE-PORT EQUIVALENT CIRCUIT MODELS
37
independent of the other carriers. This results in a current spectral density of:
qV
SItj = 2qlDCcoth( k)
2kT
(2.6)
Here IDC is the DC bias current through the diode, and V is the voltage across the junction
of the diode. For most other types of diodes the situation is more complicated. Even in
a simple p-n homojunction, which is known to (roughly) display shot noise behavior, the
physics of what really goes on and where the noise comes from is quite subtle [28]
[29].
It is
very common even in recent papers to attribute the noise in a forward biased semiconductor
homojunction laser to the noise of two terminal currents in the laser, one in the forward
Vq
direction of magnitude If = Ienkr and one in the reverse direction of magnitude I,
Is.
With some fuzzy thinking, one could decide that the carriers in these currents cross the
junction of the device in a independent manner, resulting in Poisson statistics and shot noise.
For low frequencies (compared to the transport times of the diode) this model happens to
give the correct noise spectral density:
(2.7)
SI,hid = 2q(If + Ir)
Unfortunately, this simple explanation, while appropriate for some devices (like a vacuum
triode operated in the thermally-limited current regime), is simply wrong for a typical
semiconductor diode.
The noise model of the semiconductor diode is discussed more in
Chapter 4, but it is sufficient to observe here that the carrier fluxes which actually cross
the depletion layer of a diode are much larger than the currents measured at the terminals
[28], invalidating the 'independent transport' explanation. Unfortunately, this erroneous
explanation is still common in published work and textbooks on electronic devices.
In general, in trying to assess the noise added to a measurement from a diode, it is
reasonable to assume a simple shot noise model SI, =
2
qIDC as an order of magnitude
estimate. But to take a Parthian shot at even this rough of a generalization, a heterojunction
laser biased just above the threshold of lasing can display SI, at a thousand times the shot
CHAPTER 2. THEORY OF ELECTRICAL SYSTEM NOISE MODELING
38
noise level.
2.3
2.3.1
Two-port Equivalent Noise Models
Transistors
The most important part of most sensitive electronic measurements is the amplifier which
boosts the signal to a level useful for data aquisition. To understand the noise in amplifiers,
it is necessary to first understand the noise in their constituent transistors.
The noise
models presented for these devices will be in the common-emitter hybrid-pi model for bipolar
junction transistors (BJTs), and the analogous common-source small signal model for fieldeffect transistors (FETs). While these models are very useful, like the diode model they
should not be taken as gospel truth, but rather as good approximations which may differ
from reality for different materials and geometries.
The noise model for a BJT is shown in Fig. 2-5. The resistance Tb is the base parasitic
b
rb/ 2
p
'1/f
rb/ 2
~
C
rbc
-
...
A/LJ%...........
1b,sh
be
ce
9mvs
Iec,sh
Figure 2-5: Two-port noise model of a bipolar junction transistor.
resistance and is engineered to be as small as possible, generally between 10 and 200 Ohms.
The resistance rbc is the parasitic resistance between the base and the collector and is usually
large enough to neglect. The two ideal resistances are Tbe and rce. They are the differential
resistances of the forward biased base-emitter junction and the differential resistance of the
reverse biased base-collector junction, respectively.
b,sh is due to shot noise in the base
2.3.
TWO-PORT EQUIVALENT NOISE MODELS
39
current and has the power spectral density
S, ,sh =
2
(2.8)
qIB
where IB is the DC current flowing into the base. Likewise Ic,sh is shot noise from the
collector current with spectral density
Sl,c,sh =
2
(2.9)
qIc
where Ic is the DC collector current. Also shown in Fig. 2-5 is the 1/f noise generator
I/f accompanying the base current. As in the diode noise model, this noise has a spectral
density
SIi/f (f)
DC
(2.10)
This noise generator has been placed in the middle of the base resistance rb in an effort
to accurately model empirical results
[30].
Finally, it must be remembered that the non-
ideal resistances in the model carry noise according to the resistor noise model. Practically
speaking, Tbc is large enough so as to render its noise sources unimportant, and the base
resistance rb generally does not display significant 1/f noise.
The noise model for a general FET device is shown in Fig. 2-6.
For a typical FET,
d
Zgd
r0
p
Fiur2-:Topr
osemdlo
il
efc
rnitr
40
CHAPTER 2. THEORY OF ELECTRICAL SYSTEM NOISE MODELING
R(z,,) and R(zgd) are very small, and the impedances can be taken to be pure capacitors.
Physically speaking, the real parts of these impedances are vanishingly small because in
modern processes the gate oxide in a MOS structure which insulates the gate from the other
two terminals can be made with very few defects. At low frequencies, the only remaining
impedance is the drain to source resistance, which is the effective resistance seen across the
channel from the output of the transistor. The resistance is modelled as ideal even though it
is dissipative and therefore contributes thermal noise. The thermal noise from the channel
is added in separately later.
I/f
is the usual 1/f noise generator for the drain current with spectral density
(2.11)
SI,1/f (f )=
Ith is related to the thermal noise of the channel, although its exact calculation is somewhat
complicated, and involves an integral over the length of the channel [29] [28]. Below saturation, when the channel of the FET is not pinched off, the value of the spectral density is
the usual thermal noise:
St
~ 4kTgm
(2.12)
Above saturation, when the FET is pinched off (normal operating condition), the spectral
density is different:
S1th ~
-4kTgmn
3
(2.13)
The noise source I9 represents shot noise of the gate leakage current, and has the spectral
density
Sl,g =- 2qIG
(2.14)
where IG is the DC gate leakage current [29]. For most modern FETs, this leakage current
is very small, and this noise source is unimportant compared to the thermal noise.
The noise models for BJTs and FETs described above are important for deciding what
devices to use for a particular measurement.
41
2.4. THEORY OF ONE-PORTS AND TWO-PORTS
2.3.2
Transformers
In the context of this thesis, transformers were used to impedance match noise sources with
measuring amplifiers. Fig. 2-7 shows the noise model of a 1:N transformer with the '1' side
chosen as the input. The resistances rp and r, are primary and secondary coil resistances,
......
.........
r
rCr
1
: N
L
1I N
C
Figure 2-7: Two-port noise model of a transformer.
respectively. They obey the usual resistor noise model; thermal noise from these devices
typically dominates the noise from the transformer. The resistance rec represents magnetic
core losses, and like any dissipative process, contributes thermal noise.
The inductance
LP is the primary inductance, which sets the low frequency limit of the transformer. The
capacitance C, models the distributed parasitic capacitance of the secondary coil, and often
sets the high frequency limit of the transformer.
2.4
Theory of One-ports and Two-ports
Because the noise generated in the circuit models given above is typically a small signal, the
formalism of linear circuit theory can be applied to predict a system's noise behavior even
when nonlinear elements are involved, as long as the system maintains a stable bias. This
formalism allows one to abstract away from the particular circuit elements to generalized
one-port and two-port devices. Some results of this theory which are central to this thesis
are presented here.
CHAPTER 2. THEORY OF ELECTRICAL SYSTEM NOISE MODELING
42
2.4.1
One-port Thermal Noise
It is a standard result of linear circuit theory that an arbitrary network of resistors, capacitors, and inductors with two nodes of the network designated as the measurement port can
be written in terms of a Thevenin or Norton equivalent circuit [31]. A Thevenin equivalent
circuit consists of a voltage source in series with a general complex impedance ZT, and
the Norton equivalent circuit consists of the parallel combination of a current source and
a general complex impedance ZN. To calculate the thermal noise exhibited at the output
port due to resistive elements within this network, one could insert the noise models for
each of the elements earlier (neglecting the 1/f noise sources) and turn the crank of linear
circuit theory for each independent source of noise, summing all of the variances (because
they are uncorrelated) for all of the noise sources to obtain the total noise.
There is a much faster way to obtain the noise if the Thevenin impedance ZT is known
[30]: simply take the real part of the impedance and calculate its thermal noise. This can
be seen to be true from a simple thermodynamic argument. If a resistor with value
R(ZT is
connected across the ports of the Thevenin network, thermal equilibrium guarantees that
at each frequency there is no net power transfer between the network and the terminating
resistor. This means that the thermal noise emitted by the network must be equal to the
thermal noise emitted by the resistors.
2.4.2
Generalized Two-port Noise Models
Is is often useful to regard some portion of a circuit as a noiseless two-port device. For
example, the BJTs and FETs discussed above are best regarded as two port devices. Although they contain three terminals, one is designated as a shared terminal common to the
both the input and the output; in Fig. 2-5 and Fig. 2-6 the common emitter and common
source configurations were given, respectively. More complicated circuits such as an operational amplifiers can often be well described by simplified two port models, as in Fig. 2-8
[32]. Note that this paricular noise model, by choosing a side of the two port to place the
voltage dependent voltage source and a side of the port to measure the voltage upon which
2.4. THEORY OF ONE-PORTS AND TWO-PORTS
43
+
Vin
Zj-
Figure 2-8: Two-port noiseless model of an operational amplifier.
the source depends, has implicitly designated one port as the input and one port as the
output. The model was not designed to work in reverse; it does not predict the response
at the input of the circuit to a disturbance at the output. Also note that not only the
gain, but also the input and output impedance of the two-port have a non-trivial frequency
dependence. While this is the standard noiseless amplifier model, a more complete model
is possible. The input and output voltages can be related to one another with complete
generality using four complex frequency dependent parameters:
out
(2.15)
z 11 Iin + k 12 Vout
in
=k 2 1 Vin
+ Z22 Iout
This is shown below in Fig. 2-9. This model uses voltage controlled voltage sources to drive
'in
.........................................................................................
1out
z+
zil
in
z2 2
+
k 12Vout
-
V
+
ut
-k21Vi
B-ut
Figure 2-9: General noiseless model of a two-port network.
44
CHAPTER 2. THEORY OF ELECTRICAL SYSTEM NOISE MODELING
signals through the two-port. It is possible to formulate the two-port model in terms of any
combination of current or voltage controlled current or voltage sources.
The standard theory of noiseless two ports can be easily extended to handle noise sources
within the circuit which the two-port represents
[30].
By successively treating each port of
the two-port as a one-port (with the other port left open circuited), the noise at each port
can be calculated using the principle of superposition from ordinary linear circuit theory,
and, where appropriate, the one-port thermal noise technique discussed above. The noisy
network can then be replaced with a noiseless two-port model with voltage generators at the
input and the output, as in Fig. 2-10. These generators will still have zero-mean Gaussian
I
Iout
1+
+I
in
+
k
o'ut
_1
VN,out
22
V'
V'
VN,in
out
+
-out
_ills
Figure 2-10: General two-port network with voltage noise sources at the input and output.
statistics (assuming all of the circuit noise generators had zero-mean Gaussian statistics),
but they can have spectral densities with non-trivial frequency dependence, and they can
have some non-trivial correlation with one another.
To make it easier to compare the noise contributed by a two port network to a signal
applied at its input it is helpful to express all of the two port's noise at its input. To do this
one starts with the general two-port input and output relations with voltage noise sources
at both the input and output, shown in Fig. 2-10. By inspection, one can write:
k 12 (Vout + VN,out)
Vin
=
Z111In +
Vout
=
k2 1 (Vin + VN,in) + z22I 1out
-VN,in
-
VN,out
(2.16)
2.4. THEORY OF ONE-PORTS AND TWO-PORTS
45
Now consider an identical noiseless two-port network with current and voltage noise sources
at the input, as depicted in Fig. 2-11. The input and output voltage relations can be written
...............................................
lout
II-
Z2 2
Z
VN
In
ot
ut
+,
kV
{N
1-
Ut
k2 1V
-k-
---
o
+
Figure 2-11: General two-port network with input referred noise.
by inspection (note that V..t = V0ut/ for this model):
+IN ) +k12Vout
Vin
=
z11(Iin
Vout
=
k21(Vin+VN)+z22Iout
(2.17)
-VN
Referring to Eqn. 2.16 and Eqn. 2.17, the noise appearing at the input (Vin) and output
(Vout) can be made equal by setting the values of I
and V, appropriately.
VN
VN,in
IN
VN,out(1 ± k 12 )
-
(2.18)
VN,out
k21
z11k21
Because this can always be done (as long as k 2 1
/
0 and z 11
4 0), all practical noisy
two-port circuits can be expressed in this form. If the parameters k 2 1 or z 11 become equal
to zero, one or both of the noise sources in Eqn. 2.18 will diverge. That these constraints
apply is not surprising; if either k 21 or zii equals zero the output portion of the circuit in
Fig. 2-11 becomes totally decoupled from the input noise sources, and the input referred
model is incapable of reproducing the output noise VN,Out in Fig. 2-10.
A common caution in the literature on this subject [30] [28] is that the input referred
46
CHAPTER 2. THEORY OF ELECTRICAL SYSTEM NOISE MODELING
noise model is only valid in the forward direction, and may not be used to calculate the
noise coupled out of the input. This is not true; as formulated above, the general noise
model for a two-port is totally symmetrical with respect to input and output, and can
be used to calculate the noise from either port. However it must be remembered that at a
given frequency, the two-port is characterized by 12 real parameters: the real and imaginary
parts of z11 , k 12 , k 21 , and z22 , as well as the power of each noise source, and the real and
imaginary part of the cross-correlation between the noise sources.
The cross-correlation
is often ignored in practical calculations, and it is seldom important when calculating the
effects of noise at the output of a real amplifier. This is a result of the specific positions
of the noise sources in the FETs and BJTs which make up the amplifier. However, when
calculating the noise out of the input port of an amplifier, the correlation typically becomes
very important and must not be forgotten. One application for which this is important is
the balanced detection scheme, shown in Fig. 2-12. The idea behind balanced detection is
F VI +
cross-
nV
DUT
F ----
-
-
- -
correlator
ZV,
ZSig
+
V
+
V
-
A2
In,2
Figure 2-12: Balanced detection scheme for the measurement of small signals.
to measure a small signal Voi
from the DUT using two amplifying circuits in parallel, and
then cross-correlating the measured signals (VI and V2) to remove the effects of the noise.
2.4. THEORY OF ONE-PORTS AND TWO-PORTS
47
More quantitatively,
Here the V
eoise
V
A 1 (Vig + VnIolise + VIo2ijse)
V2
A2
(Vsig + V
(2.19)
+o2 V1ise)
represents the noise signal at the '1' amplifier due to the noise generators
on the input of the '1' amplifier, V' o2se represents the noise signal at the '1' amplifier due to
the noise generators on the input of the '2' amplifier, and so on. These noise signals contain
the combined effects of the voltage and current input noise sources of each amplifier. The
cross-correlation of V and V2 can be taken:
Rv 1 v
2
=
(T)
AIA
2
(Vs-g(t)Vsig(t - T) +Vsig(t)V 7 2se(t
-
VSig(t)V2 se(t
---
V~oise ft)V2ise(t
...
V112se(t)Vsig(t
Vs(t)
$,jise
..
+
-
)+
-
(2.20)
(t)Vg(t - T) + ---
) +Vaise(t)VJise(t
-T)+Vro2se(t)Vise(t
(t - T)
T)
-
-T)
)
+±--
++---
All of the uncorrelated terms (signal with noise, or noise from amplifier '1' with noise from
amplifier '2') are zero, and the cross-correlation can be written:
Rv1 v2
(7)
=
AIA
2
(Vsig(t)Vsig(t - T) +V
'lise(t)Vraise(t- 7)+
(2.21)
...02(t)V22se(tT)
The only terms which remain are the desired autocorrelation of the signal and the undesired
cross correlation terms resulting from the coupling of noise from the '1' (or '2') amplifier
into the '2' (or '1') channel. The size of these noise sources can be determined using the
input referred noise model discussed above. For most amplifiers, they can be shown to be
small, and for FET input amplifiers they are virtually zero, allowing the balanced detection
CHAPTER 2. THEORY OF ELECTRICAL SYSTEM NOISE MODELING
48
method to be used with confidence. A detailed discussion of this technique can be found in
[33].
2.4.3
Multiple Amplifier Stages (Friss's Formula)
Often when measuring small signals, two or more cascaded amplifier gain stages must be
used. As amplifers frequently have different gains and contribute different amounts of noise
to a measurement, the ordering of the amplifiers can be important. This was first noted
by Friss
[34] for impedance matched radio circuits. The result given here is for unmatched
amplifiers which are not significantly loaded by the impedance of the measured signal source,
relevant for all of the measurements in this thesis.
Suppose that one has two voltage amplifiers, one with gain A 1 and input noise voltage
N 1, and the other with gain A 2 and input noise voltage N 2 . Suppose further that the
measured signal is a pure voltage source of voltage S. If the amplifiers are arranged with
'1' first and '2' second, the output signal is:
MA = A 2 (A 1 S + N1 ) + N2
For amplifier '2' first, and '1' second, the signal is:
MB
= A, (A 2 S + N 2 ) + N 1
In both cases, the desired signal is A 1 A 2 S, and the rest is noise. Examining the noise terms,
it is clear that amplifier '1' should be used in the measurement chain if
A1- 1
A2 - 1
N1
N2
(2.22)
Otherwise amplifier '2' should be used. This criteria explains why designers of low noise
measurement systems spend most of their effort on reducing the noise of the first stage. As
long as the designer has some flexibility in setting the first stage gain (generally the case),
2.4. THEORY OF ONE-PORTS AND TWO-PORTS
49
the gain can almost always be made large enough to render the noise of the second stage
unimportant.
2.4.4
Optimum Noise Resistance
Another important result which follows naturally from the two-port input-referred noise
model discussed above is that of the optimum noise resistance. Consider a voltage amplifier
with independent input referred noise sources V and I, used to measure a Thevenin voltage
source with resistance R.
Note that R, expresses thermal noise (Fig. 2-13). The voltage
S+
+
|
n
Vth
n
R
Vsg
sig
DUT
Figure 2-13: Measurement with noisy voltage amplifier.
signal at the input is simply
Vin = Vsig + Vh + Vn + In RS
(2.23)
The goal of an experimenter is to measure Voi 9 with as much accuracy is possible. The
thermal noise from R, which corrupts the signal will always be present, and represents an
ultimate limit to how well the signal can be measured.
This observation motivates the
definition of the noise factor F (recognizable as the noise figure defined in Eqn. 1.5, but this
CHAPTER 2. THEORY OF ELECTRICAL SYSTEM NOISE MODELING
50
time not expressed in dB):
F
total noise power
ni
noise power from thermal noise in source
For our present scenario, we can write
F =v
S_
+ Sv_ + RS
Sth
=
1
+
S
+R\Si
4kTRs
(2.24)
where Sth is the PSD of the thermal noise of R5, and SI and Sv are the PSDs of the
current and voltage noise sources, respectively. The noise factor is a useful figure of merit
for an amplifier because it recognizes the inevitability of thermal noise at the source, and
quantifies how much more noise is added by the amplifier. A perfect amplifier has F = 1,
and real amplifiers have F > 1.
It is clear from Eqn. 2.24 that for large R, the noise factor F will be large due to the
contribution of the R2S 1 term in the numerator. Similarly, for small R, the denominator
approaches zero, and F becomes large. There is an optimum value of R. for which the F is
minimized. Using simple calculus, this can be shown to be:
Ro =
Sv =nVn
_L
(2.25)
This important result should be interpreted as a guide for the selection of an amplifier
based on the Thevenin equivalent resistance of the device being measured. There are two
points of common confusion which should be clarified. First, it never makes sense to place a
resistor in series or parallel with a device in order to achieve the optimum source resistance;
thermal noise from the added resistor will only make things worse. Second, the concept
of optimum noise resistance is seldom very useful if the DUT has a significant reactance
!(Z,)
at the measurement frequencies. One can derive a relation between the current and
voltage noise generators and the real and imaginary part of Z. in a manner analagous to
2.5.
EXTERNAL LOW FREQUENCY SOURCES OF NOISE
51
the method above:
SV
S1
-
(Z)2
(2.26)
Unfortunately, because the reactance of any practical DUT tends to be frequency dependent
(ZIapacitor
jI= '
Zinductor =
JwL), the optimum ratio between the current and voltage
noise depends on frequency as well.
From a design perspective, one rarely has enough
control over the current and voltage noise of the measuring amplifier to match this frequency
dependence over any significant bandwidth.
2.5
External Low Frequency Sources of Noise
There are a number of other sources of noise which find their origin in the environment
outside of the measurement, but are coupled in through vulnerable components. They can
usually be avoided by following certain rules in laying out the measurement. However, one
often finds that some of the rules offered must be broken, for convenience or by necessity.
Whether or not breaking a particular rule will result in a significant degradation in signal
quality is usually most easily determined by just trying it.
Capacitive Coupling, Shielding, and Ground Loops
Unless one enjoys the luxury of working in an electrostatically shielded room, there are
many sources of stray electrical fields in a room which can corrupt a measurement. A room
is essentially a large capacitor, floor and ceiling defining the sides. Large conductors in
the room (people, optical tables, etc.)
distort the electrical fields and the corresponding
electrostatic potential. Room lighting, electrical conduits, and electronic equipment provide
the origins and terminations of the stray field lines. In engineering terms, every conductive
object in the room has a capacitance with every other metal object in the room, including
the conductors in the measurement apparatus. The problem which can arise from this is
shown in Fig. 2-14.
The measured device (DUT) is taken to consist of a Thevenin equivalent resistance R
52
CHAPTER 2. THEORY OF ELECTRICAL SYSTEM NOISE MODELING
/C
//,
%4
+___
VorI
+,R
Rm
s
>
measurement
device
DUT
Figure 2-14: Capacitive coupling of noise into a measurement.
and a voltage signal source V, and the measured quantity is taken to be a voltage or current measurement performed across the resistor Rm. For voltage measurement (e.g. voltage
amplifier) Rm is taken to be large, and for a current measurement (e.g. transimpedance amplifier) Rm is very small. The capacitor C drawn in dotted line is some parasitic capacitance
between the signal wire of a measurement and some other metal object in the room with a
fluctuating potential V.
If one supposes a C=2 pF coupling capacitance, R=10000 Q, and
a voltage measurement, a voltage of 15 [W is measured due to the parasitic capacitance,
enough to swamp any reasonably small signal. Note that the larger the value of the parallel
combination of R and Rm, the more vulnerable the circuit is to capacitive voltage coupling.
Thus for voltage measurements (large Rm) across high Thevenin equivalent impedance devices (high R) one is particularly vulnerable to this form of noise. This is one benefit of
measuring a large impedance device with a transimpedance amplifier. However, even this
may not be enough. Fig. 2-15 depicts an unshielded DC current measurement of a weak
luminescent source. The large transient spikes at the beginning of the measurement were
caused by the experimenter walking away from the measurement apparatus, disturbing the
pattern of electrostatic fields in the room (the large glitch at roughly 350 seconds is also
from movement). A more general solution to the problem of capacitive coupling is shielding.
The intuitive idea behind shielding is simply to prevent the random spurious electric
2.5. EXTERNAL LOW FREQUENCY SOURCES OF NOISE
53
10
987-
605-
E 30
2
0101
0
100
200
300
400
time (seconds)
500
600
700
Figure 2-15: Capacitive coupling of noise into a current measurement.
fields in the room from talking to sensitive nodes of the measurement circuit.
Shielding
around an amplifier can also reduce unwanted capacitive feedback that could potentially
degrade the bandwidth or stablity of the amplifier. A detailed and clear discussion of shielding and related issues are presented in [35]. Here we simply note that for an electrostatic
shield to be effective, three conditions must be satisfied.
First, the shield must be a conductor which completely encloses the sensitive portion of
the measurement circuit. For low frequencies (<10 MHz) and for moderately sized shields
(< 1 m 3 ), alumininum foil is perfectly sufficient, as the electric fields may be treated quasistatically, and travelling-wave effects ignored. For proper shielding at higher frequencies
(10 MHz to 100 GHz), the skin depth of the shielding material must be considered.
Second, the shield should be connected to the zero reference of potential in the circuit.
Connecting the shield to wall ground if the measurement circuitry inside is floating relative
to ground is generally a bad idea, and may even make for more noise pickup than having
no shield at all. Coaxial cables, which carry the signal inside of a cylindrical grounded case,
can then be conveniently used to exit the shielded enclosure and obtain or deliver signals.
CHAPTER 2. THEORY OF ELECTRICAL SYSTEM NOISE MODELING
rX
U4
If the transmitted signals are extremely small (pA to fA range), triaxial cables are often
used in place of coaxial cables. These feature a third shell in between the signal and the
ground, which is kept at the same potential as the signal using a buffer amplifier at the
signal source. This prevents the radial leakage currents which would otherwise be present
from the signal to the ground, and also eliminates much of the mutual capacitance in the
cable between the signal and the ground.
Lastly, the shield should connect at one point only to the ground of the measurement
apparatus. If the signal source which is being measured is connected to earth ground, efforts
should be made to have the shield connect to earth ground at the same physical location
as the signal-earth ground connection.
This eliminates the possibility that current flow
in the shield (necessary to perform its function) can couple into the signal ground of the
measurement. If shield current flows through the small but finite resistance of the signal
ground path, it can cause spurious voltage fluctuations to appear in the measured voltage.
This problem is known as a 'ground loop'. For similar reasons, it is always a good idea to
wall-plug any necessary electrical measurement equipment into the same electrical main,
preferably through a power supply regulator which quiets the wall power. Noisy equipment
(arc lamps, fluorescent lighting, computers) is ideally plugged in on a separate circuit.
There are some basic difficulties with the above description. One is getting power into
the shield to supply electronic equipment. There exist well developed techniques for doing
this using shielded transformers
solution.
[35]. Relying on battery powered devices is an even simpler
Another issue which can arise is the measurement of two independent signals
within the shielded enclosure.
Eliminating capacitive crosstalk between these signals by
following the above prescriptions requires the construction of two separate shielded areas,
one for each signal. Where not practical, simply spatially separating the signals within the
shield can be helpful.
Fig. 2-16 shows a shielded single ended amplifier reading a signal from a DUT represented
by a Thevenin equivalent impedance. Note that the shield connects with the signal ground
at the point where the signal ground is connected to earth ground.
Also shown is an
2.5. EXTERNAL LOW FREQUENCY SOURCES OF NOISE
55
incorrect signal connection (with an 'x' through the wire), which would result in a ground
loop and a corresponding deterioration in the measured signal quality.
Vsig
+
..
i/I
mproper
onnectin
RThe.
different
ground
groundVose
Figure 2-16: Correct and incorrect shielding of a sensitive measurement.
Inductive Coupling
Inductive coupling is the magnetic analog to capacitive coupling. An example of this is
sketched in Fig. 2-17.
An AC source of current outside of the measurement apparatus
generates a time varying magnetic flux which passes through a loop in the measurement
circuit. By Faraday's law, this induces a voltage into the measurement circuit. In effect,
induced H-field
or I
InV
-DU
measurement
device
Figure 2-17: Inductive coupling of noise into a measurement.
there is a transformer whose primary is the noise source coil (Ia) and whose secondary is
56
CHAPTER 2. THEORY OF ELECTRICAL SYSTEM NOISE MODELING
the measurement circuit. Like capacitive coupling noise, the problem is worst for voltage
measurements on large DUT impedances.
Unlike capacitive coupling noise, there is no
simple way to shield measurements from magnetic fields.
In principle, P--metal (metal
with a high magnetic permeability) can be used to enclose an experiment. This channels
magnetic fields around the measurement and away from the loops. However, because pmetal is expensive, difficult to shape, and susceptible to shock-induced damage, it is almost
always easier to eliminate the loops which are causing the coupling. Twisting wires and
even coaxial cables will often solve the problem. Twinax cable (two twisted wires within a
concentric guarding shield) is often a good solution to the combined problem of capacitive
and inductive coupling. Sometimes simply changing the orientation of equipment is helpful.
Microphonics,
dC
--
noise
Noise can also be mechanically introduced into a circuit. This is known as microphonic, or
more generally
noise. an example of the basic mechanism is depicted in Fig. 2-18. In
CV
mechanical
eetoee
eetotr
UTvibration
amplifier
Figure 2-18: Microphonic coupling of noise into a measurement.
some way, the environment causes a time dependent change in the value of a capacitance
in the circuit (C). This is sometimes through the mechanical vibration of a coaxial cable
which has not been properly secured. That this causes a spurious signal can be seen by
differentiating the standard voltage-charge relation of a capacitor:
d
d(CV)
Q=->I
dt
dt
dQ
dt
dV
dC
C dt +V dt
(2.27)
2.2
2.6. SUMMARY
57
The !dt term gives rise to a spurious current if there is a voltage V across the terminals.
Often the offset voltage of an op amp is sufficient to see this effect. In particular, the electrometer op amps which are typically used to measure small currents (e.g. Analog Devices
AD546) often have relatively large (~
mV) offset voltages at their inputs. While micro-
phonic noise is typically much smaller than the capacitive, inductive, and ground related
noise discussed earlier, it can become important when looking at very small current signals,
or when the voltage across a high impedance load is measured. Tying down loose cables
and wires and using low-noise cabling are the best ways of dealing with microphonic noise.
There are more exotic spurious environmental sources of noise which can potentially
plague the experimenter.
For a more thorough investigation, the reader is referred to
Chapter 3 of [30].
2.6
Summary
In this chapter, the modeling of noise in electrical systems is discussed from an experimental
perspective. The goal of the chapter is to demystify the myriad sources of noise which are
present in any sensitive measurement and help in the design and optimization of customized
measurement systems. The key results of this chapter are summarized here.
First, noise circuit models for various elements common to experimental setups are
presented. These are divided into one-port and two-port models; the one-port models are
resistors, capacitors, inductors, and diodes, and the two-port models are BJTs, FETs, and
transformers.
In each device, there are typically several sources of noise, although one
source is often dominant.
Once the noise models are available, the machinery of linear circuit theory can be applied
to model the noise of arbitrary systems. In particular, the abstractions of one-port and twoport models, traditional fare of introductory circuits classes, are generalized here to describe
networks with arbitrary noise. The thermal noise from an arbitrary one-port network is
shown to be equal to the thermal noise of a resistor whose value is the real part of the
Thevenin equivalent impedance of the one-port.
The noise in a two port is referenced
CHAPTER 2. THEORY OF ELECTRICAL SYSTEM NOISE MODELING
58
Q
back to the input and expressed in terms of a noise voltage source and a noise current
source, allowing the easy comparison of signal levels and noise levels for a measurement
system. This model is used throughout this thesis and is standard in the electronic circuit
community.
Having provided the necessary tools for modeling noise in electronic circuits, two important principles of low noise circuit design are presented. The first stage of an amplifier
chain is shown be the main contributer to the noise factor (and noise figure) of the chain.
This suggests that the low noise circuit designer focus his or her efforts on the careful design
of the preamplification stage. This can often be done by matching the optimum noise resistance of the measurement apparatus to the DUT's Thevenin resistance. For the important
special case of a resistive DUT, the optimum noise resistance was shown to be
, where
V,, and I, are the voltage and current noise generators of the measurement apparatus.
Finally, some other sources of spurious noise external to the measurement circuit are
discussed. Capacitive coupling noise is by far the biggest offender of these, but it is also
the easiest to prevent. A careful description of electrostatic shielding shielding is given, and
the next most common gremlin of low noise measurements, the ground loop, is discussed.
Inductive coupling noise is touched upon, as well as microphonic 4- noise.
Chapter 3
Current Noise Measurements
3.1
Summary of Instruments
For this thesis a set of time domain measurement systems were developed to make noise
measurements in a range of devices under various operating conditions. When reviewing
these systems, one should remember the three key difficulties in making measurements of
current noise.
The first difficulty stems from the nature of the noise signal.
Physically interesting
current noise is invariably a small signal of wide bandwidth. Both high sensitivity and wide
bandwidth are generally important goals. With high gain and high bandwidth issues of
stability and sensitivity to parasitics often arise. Small perturbations in the topology of the
circuit can result in large changes in the behavior of the system.
The second difficulty is related to the many other signals in the environment of the
measurement along with the noise of interest. It should be clear from the discussion in
Chapter 2 that for a given measurement system, there are many uninteresting sources
of spurious noise present along with the interesting noise which is to be studied.
The
philosophy adopted here is that if unwanted noise cannot be turned off, it must be minimized
by design. If it cannot be minimized, it must be carefully subtracted out. In every case,
it must be understood. Dealing with spurious sources of noise can necessitate a detailed
59
CHAPTER 3. CURRENT NOISE MEASUREMENTS
60
understanding of things which initially do not seem relevant to the measurement.
The third and greatest difficulty in noise measurements is in calibrating the measurement. Because of the constraints imposed by the first difficulty (high sensitivity and high
bandwidth), the noise measurement apparatus is often highly optimized and very sensitive
to small changes, even parasitic changes, in its circuitry and in the device under test. This
means that the transfer function describing the action of the device on the measured signal
is nontrivial and unstable with respect to small changes in the measurement. This means
that a method of calibration is required, and further, the calibration must take pains not
to perturb the circuit from its behavior during the measurement. Interesting noise has a
spectral and amplitude character very different than the sinusoidal signal sources which are
known with confidence and used for calibration. Therefore further contortions on the part
of the experimentalist are required to ensure that the calibration with a known signal is
really providing information which pertains to the unknown noise.
The measurement systems described in this chapter address these issues.
3.2
Measurement Equipment
The most important intrumentation used for the time domain measurements were two commercial low-noise amplifiers, a transformer, and an analog-to-digital converter. In what follows, a brief description of each is given, and other possible choices of intrumentation are discussed. Also used for the measurements in Chapter 4 were two home-made transmpedance
amplifiers. These are discussed in more detail in Chapter 4.
3.2.1
Low Noise Current Preamplifier
The key element of a noise measurement system is typically the first device or amplifier
in the measurement chain which provides signal gain. As shown in Section 2.4.3 the noise
floor (and the sensitivity) of the measurement is usually determined by this stage. This
first stage is often a solid state amplifier. For a particular device under test (DUT), the
optimum first-stage amplifier for noise measurement depends on the Thevenin equivalent
3.2. MEASUREMENT EQUIPMENT
61
impedance of the device seen by the measurement system. In this thesis, two complimentary
techniques are developed to deal with high and low DUT impedances.
For higher impedance measurements, a current preamplifier is chosen for the first stage.
The job of the current preamplifier (also called a transimpedance amplifier) is to transform a current signal at the amplifier input to a voltage signal at the output. The gain
(transimpedance) of the amplifier is therefore specified in Ohms. The transimpedance amplifier used in the time-domain measurements of this thesis was the SR570 low noise current
preamplifier from Stanford Research Systems. Table 3.1 depicts the range of settings possible on the SR570, as specified by the manufacturer. The current preamplifier contributes
Transimpedance
Input Current Noise
Bandwidth
Input Impedance
I kQ
10 kQ
100 kQ
150 pA/ Hz
100 pA/ IHz
60 pA/ Hz
1 MHz
1 MHz
800 kHz
1Q
IQ
100 Q
1 MQ
2 pA/ IHz
600 fA/ Hz
200 kHz
100 Q
20 kHz
2 kHz
200 Hz
100 Hz
20 Hz
10 Hz
10 kQ
10 kQ
1 MQ
1 MQ
1 MQ
I MQ
10 MQ
100 MQ
1 GQ
10 GQ
100 GQ
1 TQ
100 fA/ Iz
60 fA/ FHz
10 fA/ Hz
10 fA/ \Hz
5 fA/f Hz
Table 3.1: Performance of the SR570 low noise transimpedance amplifier.
unwanted noise to the measurement, which can be modeled at the amplifier's input with
a current noise source I, and a voltage noise source V,. Because a current preamplifier
is typically used when the Thevenin equivalent resistance of the DUT is large, the input
current noise of the amplifier typically dominates over the input voltage noise, and therefore only the current noise is specified for a commercial amplifier. "Large", in this context,
means large enough so that parasitic capacitance at the input of the amplifier becomes
troublesome, either by degrading the bandwidth of the amplifier or by capacitively coupling
spurious signals into the measurement. In this thesis, the equivalent resistance of the DUT
was not always large enough to ignore the voltage noise. Fig. 3-1 depicts the two-port model
CHAPTER, 3.
62
CURRENT NOISE MEASUREMENTS
of a transimpedance amplifier with both noise sources.
In
Zzout
zin
Ilin
V nI
n
V=RGInl
Figure 3-1: Two-port model of a current amplifier.
While the SR570 is a well-regarded commercial low noise amplifier [36]
[4],
the best
custom designed amplifiers can allow measurement of signals which are 10 to 100 times
smaller [37] [36] [38]. This is due mainly to the fact that a custom designed amplifier can
be highly optimized for the specific measurement for which the amplifier is intended. Other
techniques which can be employed to reduce the noise of the amplifier include cooling the
input stage of the amplifier stage to reduce the thermal noise from the channels of the
input JFETs [39], measuring the DUT with two identical systems to allow uncorrelated
noise to be subtracted [33], and designing very careful electromagnetic shielding [40]. The
latter two techniques are explored later in this chapter. Even with these improvements, a
severe reduction in the dynamic range (the range of input signal amplitudes which can be
measured) often must be accepted to improve significantly over the SR570 [36].
The advantage of using a commercial transimpedance amplifier over a custom built
amplifier is the ease with which measurement bandwidth, noise performance, sensitivity,
and impedance matching can be traded off (using the wide range of programmable settings),
while still retaining reasonable overall performance. This allows for a measurement system
to be developed which can be easily adapted to various DUTs. It is the opinion of this author
that an experimenter's time is better spent developing a careful method of calibration for
a good commercial preamplifier rather than attempting to custom design a different low
noise amplifier for every DUT. As will be seen, with careful calibration, voltage or current
3.2. MEASUREMENT EQUIPMENT
63
signal to noise ratios of 'TO can be tolerated while still measuring signals to better than 5%
accuracy.
A promising method for measuring small currents not investigated in this thesis is
contact-free detection.
This method relies upon an inductive mesurement of the mag-
netic field created by the DUT current [41]. Further improvements in sensitivity might be
attained with the use of a SQUID (superconducting quantum interference device) as the
sensing element. There exists an extensive literature on high sensitivity DC current measurements (for example, [42]) and commercial DC current sensing SQUIDs are available
nowadays.
3.2.2
Low Noise Voltage Preamplifier
To measure noise from lower impedance devices, a low noise voltage preamplifier is used
for the first stage of amplification. A voltage amplifier is also useful as a second stage amplifier for high impedance device measurements. For these applications, the SR560 voltage
preamplifier was selected. The amplifier allows the user to select from gains between 10 and
50000, while maintaining a bandwidth of greater than 1 MHz, an input impedance of 100
MQ + 25 pF, and an input referred voltage noise of < 4 nVvlHz. While the voltage amplifier's noise performance can again be characterized by current and voltage noise sources
at the amplifier input, only the voltage noise is typically specified, since the amplifier is
usually used to measure DUTs with Thevenin equivalent impedances much smaller than
the optimal noise resistance of the voltage amplifier. Fig. 3-2 depicts the two-port model of
a voltage amplifier.
As was true for the transimpedance amplifier, the best custom designed voltage amplifiers achieve better noise performance than the SR560 by roughly a factor of 10-100. The
most popular technique is to design a low-gain first stage of the amplifier using very low
noise (and often cryogenically cooled) discrete JFETs. This stage is then followed by a
commercial low noise amplifier (e.g.
OP17, OP27) second stage. There are several exam-
ples of this in the literature [43] [44] [45] [46]. In general, these amplifiers suffer from the
CHAPTER 3. CURRENT NOISE MEASUREMENTS
64
Zout
+
V=AVl
Figure 3-2: Two-port model of a voltage amplifier.
same drawbacks as the custom designed current preamplifiers discussed above, with two
differences. Custom designed voltage preamplifiers are typically not as sensitive to changes
in the DUT's resistance, making measurements on a variety of devices easier. This is due
to the high input impedance seen while looking into the gate of the FET which typically
composes the first stage of the amplifier. However, custom designed voltage preamplifiers
are typically much more sensitive than current preamplifiers to variations in the capacitance
seen at the input of the amplifier. An amplifier whose performance has been optimized for
a given DUT may become unstable if slightly more parasitic capacitance appears across the
amplifier input. Even a small change in the coaxial cable length between the DUT and the
amplifier can change the capacitance seen at the input by a factor of two or more, upsetting
the stability of the amplifier. For this reason, a commercial amplifier was selected over a
custom design.
3.2.3
Low Noise Transformer Preamplifier
A transformer preamplifier can be helpful to measure signals from very low impedance
(< 50Q) sources. It is typically built with a low-noise transformer as an input stage and
a voltage amplifier as a second stage to provide additional voltage amplification. The two
port model for a transformer is discussed in Section 2.3.2.
The SR554 transformer preamplifier chosen for the measurement in this thesis consists
of a transformer with a 1:100 coil ratio connected in series with a low noise voltage amplifier.
3.2. MEASUREMENT EQUIPMENT
65
The internal amplifier of the SR554 was bypassed and the SR560 was used in its place. The
key performance specifications of the SR554 are summarized in Table 3.2, as specified by the
manufacturer. The noise floor of the SR554 is almost entirely determined by the Johnson
Gain
Input Voltage Noise
500
100 pV/
Hz
Available Bandwidth
Amplifier Input Impedance
<15 kHz
0.5 Q
Table 3.2: Performance of the SR554 transformer amplifier.
noise of the parasitic resistance of the primary and secondary coils of the transformer. The
low frequency performance of the device is set by the primary inductance and the DUT
resistance, and the high frequency performance is set by the shunt capacitance seen by
the output terminal and the DUT resistance.
Care must be taken to avoid loading the
transformer output with cable capacitance, as this will degrade the bandwidth.
3.2.4
Data Aquisition System
The final step in all of the time domain measurements is to sample and quantize the voltage
signal from the previous stages of the measurement chain with an analog-to-digital converter. This converter is typically packaged with some other electronincs such as variable
gain amplifiers and memory buffers on a single circuit board. The board is known as a
data aquisition system (DAQ). There are four specifications which are important for low
noise measurements: the number of bits of quantization, the maximum sampling rate, the
minimum full-scale range, and the input-referred voltage noise.
The input to a DAQ is a continous time signal taking on a continous range of voltages.
A computer can only input a finite number of data points per unit time and can retain only
finite precision voltage values in memory. The limitation imposed by the first contraint sets
the maximum sampling rate of the DAQ, and the limitation imposed by the second sets
the number of bits of quantization. A signal quantized with 16 bit quantization can take
on any one of 216 = 65536 different values at a given sampling instant. Due to constraints
imposed by high-speed analog design, sampling rate and resolution are traded off against
one another. This is evident in Fig. 3-3
[47],
which is a plot of the resolution and sampling
22
-
I1-
_
1
slope: -1 bit/octave [--_
18
20
_
MI-v
4
2
-
uhybrid
S8i
IC
L-.
C
..
_
-
-state-of-the-art---
E+4
1E+5
IE+7
IE+6
1E+8
IE+9
1E+10
E+11
Sample Rate (Sampess)
Figure 3-3: Resolution and bandwidth of analog-to-digital converters (1999).
rate of various commercial analog-to-digital converters available in 1999. The line overlaying
the data points depicts the state of the art A-to-D converters at that time.
The maximum sampling rate of a DAQ sets the bandwidth of the signal which can be
measured. This is given by the well-known Nyquist sampling theorem, which states that
sampling at a rate
f,
is sufficient to reconstruct a signal which is band-limited to frequencies
of less that f-. Frequencies
greater than f2 will be aliased down to lower frequencies. For
2
a discussion of these issues, see [171.
The full-scale range (FSR) of a DAQ is the maximum voltage which a DAQ can sample
correctly.
Signals above the FSR are "truncated", meaning that they are assigned the
maximum quantization level permitted irrespective of their actual value.
The minimum
FSR of the DAQ is the smallest FSR possible for the DAQ while still maintaining the
proper number of bits of quantization.
The minimum FSR is important because taken
together with the number of quantization bits B, it sets the smallest voltage interval which
can be measured with the DAQ. This voltage is known as the minimum step size and is
given by FSRgin. The transfer function of an ideal DAQ is sketched in Fig. 3-4 [48].
67
3.2. MEASUREMENT EQUIPMENT
vout
4
3
2
1
full scale range
(FSR)
--
1
-2
tep size
-3
Figure 3-4: Transfer function of ideal DAQ.
The sensitivity of a DAQ is theoretically set by the FSR and the number of quantization
bits B. This limit on the sensitivity can be expressed in terms of the circuit noise models
developed in Chapter 2. The key idea is to consider the difference between the real analog
signal and the quantized version of the signal to be a noise source. Of course this noise is an
artifical construct, physically unrelated to the thermal and shot noise discussed throughout
Chapter 2.
The noise is best understood as a convenient signal processing abstraction.
Figure 3-5 depicts an unquantized signal, a quantized signal, and the difference between the
quantized and unquantized signals for 4 bits and 16 bits of quantization. It is evident from
Fig 3-5 that as the number of bits of quantization increases, the noise injected by the DAQ
(the difference between the original and quantized signals) begins to look more like white
noise, in the sense that the noise signal at a time becomes less and less correlated with noise
a short time later. A good rule of thumb is that the error signal can be said to be white
noise as long as the unquantized signal changes by more than one quantization level between
consecutive sampling instants. This allows the noise to be conveniently represented as an
CHAPTER 3. CURRENT NOISE MEASUREMENTS
68
1
original and quantized signals
1
original and quantized signals
6-bit quantization
3-bit quantization
0.5
a
cci
0.5
ci)
0)
cci
01
0
0
0
-0.5
-0.5 1
-1'-
0
2
4
6
8
0
2
error signal
8
0.02
3-b1
0.1
uantization
0.015
0.05
0.01
0
0.005
a
01
0)
cci
0c -0.05
0
0
-0.1
-0.005
-0.15
-0.01
-0.2
6
error signal
0.15
0a
4
0
2
4
6
8
-0.015
6-bit Luan t iza t ion
0
2
4
6
8
Figure 3-5: Original, quantized, and error signals for 3-bit and 6-bit quantization.
additive voltage noise source in series with the input voltage signal of the DAQ. A more
detailed analysis [48] finds that the standard deviation of the quantization noise voltage
signal is:
1 FSR
UQN =
vT2
2B
(3-1)
where FSR is the full scale range in volts and B is the number of bits of quantization. This
noise is known as quantization noise; it is a fundamental result of the quantization process,
and is present even in an "ideal" DAQ. In addition to quantization noise, a real DAQ has
additional noise from the amplifiers that perform the sampling. This noise can dominate
the quantization noise when the DAQ is operated at small FSRs, but the quantization noise
is usually a good estimate of the noise performance of a DAQ. In either case, the noise can
3.2. MEASUREMENT EQUIPMENT
69
be treated as an input voltage noise source at the input of the DAQ.
Two DAQ systems were used in this thesis. The first DAQ used for measurements was
the National Instruments PCI-6052E. The NI DAQ performs 16-bit quantization, has a
maximum single-channel sampling rate of 333kHz, and a minimum FSR (for differential
signals) of 100mV. At the minimum FSR, the standard deviation of the DAQ's noise is
approximately 5/6 amplifier noise and 1/6 quantization noise. This translates to a signal-tonoise ration of approximately 90 dB. At the higher FSRs, the NI DAQ system is quantization
noise limited, with an SNR of about 107 DB. Because of the high SNR, the quatization
noise could be ignored.
The second DAQ was the Gage Compuscope 14100. The Gage DAQ performs 14-bit
quantization, has a maximum single-channel sampling rate of 100 MHz, and a minimum
FSR of 100mV. The dominant source of noise in the Gage DAQ was amplifier noise from
the sampling process. The SNR of the Gage DAQ at the speed and input range for which
it was used was over 60 dB, and therefore its noise could be ignored.
Another important signal processing issue pertaining to time sampled measurements is
aliasing. A complete discussion of aliasing can be found in [48]. Here it is sufficient to note
that sampling a continuous, wide-band signal with a finite sampling frequency Fs maps the
signal power present across the entire spectrum of the real signal to a finite frequency band
(-F,/2 to Fs/2). Input signals with frequency magnitudes greater than Fs/2 are mapped
onto this finite band in the output signal by the sampling process, and prevent the spectrum
of the output from being interpreted as a sampled representation of the spectrum of the
input. This problem is normally taken care of with an anti-aliasing filter, a filter before the
sampler which attempts to limit the spectrum of the input to frequencies less than IFs/2L.
In this thesis, a variety of filtering methods were employed to ensure that aliasing was not
a source of error.
CHAPTER 3.
70
3.3
CURRENT NOISE MEASUREMENTS
Johnson Noise Measurements
Two sets of Johnson (thermal) noise measurements were made. The goal of the first set
of measurements was to characterize the full noise behavior of the measurement chain,
allowing the accurate measurement of the noise of an arbitrary device. We use a method
similar to the one used in [4]. The goal of the second set of measurements was to confirm
the linear dependence of thermal noise power on temperature by measuring a resistor over
a range of temperatures using a liquid Helium cryostat. Both sets of measurements were
made using the experimental setup depicted in Fig. 3-6.
Equivalent Two-port
Amplifier
Cryostat
n
DUT
(resistor)
A resistor of impedance Z, was
+
V
OU
_
V
+
V
V=H(w)Iin
Computer
DAQ
Board
Figure 3-6: Experimental setup for Johnson noise measurements.
placed in the device under test (DUT) position across the input of an SR570 low noise
current preamplifier. The DUT was kept at some temperature T,. Signal gain was either
provided by the SR570 transimpedance amplifier, or by the cascaded combination of the
SR570 with the SR560 voltage amplifier. In Fig. 3-6, the amplifier(s) are represented with a
single equivalent two-port amplifier. The two-port is characterized by an input impedance
Zi,(w), a transimpedance H(w), and two equivalent input noise sources I(W) and V"(w).
As discussed in Section 2.4.3, if the gain of the first stage amplifier is sufficient, the noise
3.3. JOHNSON NOISE MEASUREMENTS
71
from the second stage is unimportant. This condition was easily met in this measurement.
In general all of the quantities which characterize the two-port model depend on frequency.
The calibration procedure described here can be used to determine these quantities at any
particular frequency.
The output signal from the voltage amplifier signal is sampled at 300kHz using the
DAQ and stored as a time series on a computer. Aliasing was not an issue because the
The MATLAB
input signal was band-limited by the bandwidth of the ampifier chain.
programming environment was used to perform any necessary digital signal processing and
curve-fitting.
3.3.1
Calibration
DUT
(1).
Equivalent Two-port Amplifier
in
(2~
Zcal
(3
Zout
+
+
VOut
ZS
+
vn
fg
|V=H(c>)Ii
(1) - calibration DUT
(2) - open circuit
subrmComputreoA
DAQ Board
(3)- measurement DU
Figure 3-7: Experimental setup for room temperature Johnson noise measurements.
Fig. 3-7 depicts the Johnson noise measurement setup with the DUTs used for room
temperature noise measurements.
Johnson noise from an impedance Z, is modeled as a
parallel current source I,. Also shown is a calibration circuit with a an impedance of Zai
and a voltage source Vfg.
The voltage V0 st at the output of the amplifier circuit can be
CHAPTER 3.
72
CURRENT NOISE MEASUREMENTS
calculated from the sources at the input. When the noise resistor (Z,) is the DUT (position
(3) in Fig. 3-7), the result is:
Z8 |2 (IP + I) + V2 + 2IV, - R(CIvZ*)
ount = |H (j)| 1 ZIjs+Zi12
Zs + Zin|
(3.2)
In the Johnson noise measurement calibration setup, there are seven unknown real
parameters which must be determined. First, the effective transimpedance H(w) of the
amplifier chain must be determined. The magnitude of both of the amplifier noise sources
I(w) and V(w) must also be measured. Both the real and imaginary parts of the AC
amplifier input impedance Zi,(w) are unknown and must be determined. As discussed in
Chapter 2, there can be a non-zero correlation CIV between the current and voltage noises.
In general, the real and imaginary parts of this correlation coefficient must be determined.
However, for the case of the JFET input of the SR570, this correlation can be safely ignored.
The reason for this is that for low frequencies the noise of a modern low-noise JFET is almost
entirely due to the thermal fluctuations in its channel. Referring to the FET noise model in
Chapter 2, this means that all of the noise is expressed as a single current noise generator
between the drain and the source (output), or alternatively a single voltage noise generator
between the gate and the source (input). Because the input referred current noise source for
the JFET is basically zero, we know that its correlation with the input referred voltage noise
source will be unimportant for this measurement. This leaves 5 unknown real parameters
in the expression for VIout:
Rs2(Is ± I+) + V 2
n 2 n
|Rt + Zin1
2)2
v2n = H (bj)2R
(3.3)
Because noise is a wide-band signal, it is useful to express Eqn. 3.3 as a relation between
the output voltage power density and the input power voltage and current densities:
2
Svot =|H()|
RH2(SIs + Sin) + SVn
sR2
(3.4)
3.3.
JOHNSON NOISE MEASUREMENTS
73
To determine the effective gain of the two-port, the calibration DUT (position (1) in
Fig. 3-7) was used. A function generator (Vfg) was applied in series with a Rcai =1 MQ
resistor, resulting in a 10nA sinusoidal current entering the input of the amplifier. Care was
taken in selecting a resistor Rai with a negligible parasistic capacitance Cpar. At frequencies
greater than
1
2irRcaiCpar,
the parasitic capacitance shunts the the resistor, resulting in a
greater current into the amplifier, and distorting the calibration of the amplifier gain. The
parasitic capacitance Cpar of the resistor was measured as 0.2 pF, resulting in a negligibly
small calibration error. Using the output voltage measured on the DAQ and sweeping the
frequency of the function generator's input current signal, the effective transimpedance of
the amplifier chain was measured and stored on the computer. Fig. 3-8 is the measured
transfer function.
Johnson Noise Transfer Function
X 10
0
C
_0
0-
E
Cn
C:
0)
2)
()
00
10
20
30
frequency (kHz)
40
50
60
Figure 3-8: Transfer function of amplifier chain.
Next, the DUT was removed (position (2) in Fig. 3-7), and the input of the transimpedance amplifier was left open (but shielded). This removed the effects of the voltage
noise on the output of the amplifier, leaving only the current noise Sin. A measurement of
CHAPTER 3. CURRENT NOISE MEASUREMENTS
74
the output voltage signal by the DAQ was stored on the computer. The expression for the
output voltage for this measurement simplifies to:
Svout(w) = |H(cj)j2Sj.(u)
(3.5)
The measured V0ut contains noise power in a wide spectrum. To single out a particular
frequency (5 kHz was chosen), Vst was digitally filtered by a narrow high order Bessel
filter, leaving only noise in a 500 Hz band around 5 kHz.
This narrow-band voltage noise at the output was then used to find the current noise
at the input in the same narrow bandwidth. The output voltage noise power per Hz of
bandwidth is related to the input current noise power per Hz by:
Svout(w) = G(w)12 SI"(w)
(3.6)
Here G(w) is the transfer function of the measurement apparatus cascaded with the transfer
function of the Bessel function digital filter. This transfer function is found in practice by
multiplying the measured transfer function of the amplifier with the transfer function of
the chosen Bessel filter. Because this G(w) is narrow band, Sv0 ut and S1n can be taken to
be independent of frequency in the band of interest, and Eqn. 3.6 can be integrated over
frequency to yield:
vout W = SIn fIG(w)2 dw
(3.7)
Here the integral of the output voltage noise power over frequency is recognized as the
variance of the Bessel filtered output voltage noise, which can be easily calculated from the
filtered data. The quantity
f
JG(w)12 dw is found using a numerical integration, allowing an
equivalent input current noise power density Smn to be calculated.
Determining the remaining three unknown real parameters Vn(w), !R(Zi,(w)), and a(Zjn(w))
required a curve fitting technique. Measurements of V0st(w) 2 were taken for several different
resistances R, using the resistor noise measurement setup ((position (3) in Fig. 3-7). The
noise was then filtered by the same high-order Bessel bandpass filter used earlier, and an
75
3.3. JOHNSON NOISE MEASUREMENTS
identical procedure was followed to find the equivalent input current power
-i== R(Ss + SIn) + SVn
S hn
|SRs
+ Zin2
from Eqn. 3.4. To obtain a good fit, the resistance values R, spanned several orders of
magnitude. Fig. 3-9 depicts the variance of the measured input current noise variance cri,
7
Measured Noise and Contributions
x 10-
6
a) 5
sum of noise
sources
amplifier
voltage noise
E4
)
x>
Cz
0
measured total noise
a)
thermal
noise
/amplifier
1
-
current noise
7
100
104
106
resistance at amplifier input (Ohms)
102
108
Figure 3-9: Measured noise with different source resistances, and theoretical contributions
to the noise obtained with a curve fit.
for the different DUT resistances. Also shown in Fig. 3-9 are the contributions of the three
noise sources
I, Vs, and I, to the output noise obtained with the curve fit using Svn,
WR(Zin), and
(Zin) as fitting parameters. In this calibration, SI, is a known parameter,
CHAPTER 3.
76
CURRENT NOISE MEASUREMENTS
and is calculated from the Johnson noise formula:
SIs =
AkTs
Rs
(3.8)
The values for the parameters for the best fit are shown in Table 3.3. The same parameters
were measured in
[4],
and are presented in Table 3.4. The agreement is reasonably good, and
the discrepency can most likely be explained by normal variation between the amplifiers.
Voltage Noise
Current Noise
R(Zi,)
a(Z,)1
3.05 nV/
628 fA/
630 Q
670 i
Hz
Hz
Table 3.3: Best fit parameters.
Voltage Noise
Current Noise
R(Zin)
'n(Z)
3 nV/v Hz
560 fA/v/Hz
500 Q
1000 Q
Table 3.4: Best fit parameters for the measurement of Chen and Kuan
[4].
In this measurement, the parameters were determined at a particular frequency, 5 kHz.
By simply iterating the procedure described above using different values for the center
frequency of the narrow Bessel bandpass filter, all of the parameters of interest can be
calculated at any given frequency. However, in practice, the 3-parameter nonlinear curve
fitting proved difficult to automate, making characterization at all frequencies very tedious.
For the rest of the measurements in this thesis, it was possible to develop calibration procedures to eliminate the added noise sources and effects of the amplifier input impedance
from the measured noise without the careful independent determination of the measurement
noise sources and amplifier impedance described above. However, it is important to note
that such a detailed calibration is possible, and may sometimes be necessary.
3.3.2
Noise Thermometer
In the calibration section discussed above, the variation of Johnson noise with resistance R,
was used to characterize the noise behavior of the two-port amplifier. However, it is clear
3.3. JOHNSON NOISE MEASUREMENTS
77
from the Johnson noise formula (Eqn. 3.8) that the noise scales not only with resistance Rs,
but also with temperature T. This observation is the basis for the field of Johnson noise
thermometry [49] [50] [51]. The idea is to use a resistor's Johnson noise as a measure of the
absolute temperature. In this thesis a simple Johnson noise thermometer was developed
and tested at temperatures between 5K to room temperature. The SR560 voltage amplifier
was used for the measurement because of its wider measurment bandwidth. An anti-aliasing
filter was present within the amplifier. As might be expected, using the voltage amplifier
rather than the transimpedance amplifier initially introduced a great deal of capacitive
coupling noise into the measurement, as discussed in Section 2.5. The metal walls of the
cryostat were employed as a shield, allowing the measurement to be made.
Fig. 3-6 (including the cryostat) depicts the Johnson noise thermometry setup. A liquid
Helium flow cryostat was used to cool a 4.75 kQ wire-wound resistor to temperatures ranging
from 5K and room temperature.
A wire-wound resistor is used because its resistance is
constant over the measurement temperature range. Measurements of the output voltage
noise V 0st at each temperature were recorded, and the power spectral density of the noise
was calculated. The values of this power spectrum were averaged in a frequency band of 6
kHz to 30 kHz at each temperature. The results are plotted in Fig. 3-10, using arbitrary
units for the power. The linearity of the total noise power with temperature is apparent.
This is the signature of Johnson noise. The deviations from linearity at low temperatures
can be explained by the difference in position of the thermocouple measuring temperature
and the DUT resistor. Because the resistor is located further away from the He reservoir
than the thermocouple, it does not cool all the way to 5K, resulting in spurious thermal
noise.
The amplifier noise sources V and I, simply add a constant background to the measured
noise. If necessary, the background noise could be obtained by fitting a line to Fig. 3-10
and finding the noise power at the intercept of the fitted line with the power axis. Since the
amplifier noise does not depend on temperature, this noise power can be subtracted from
the total voltage variance to yield the voltage variance due to Johnson noise VOtJN. This
CHAPTER 3. CURRENT NOISE MEASUREMENTS
78
Measured Noise Power with Temperature
2.62.4,2.2
2
0
I-
1.8
Z 1.6
1.4
1.2
S
1
10
50
100
150
200
Temperature (K)
250
300
Figure 3-10: Noise power measured as a function of temperature.
output voltage noise variance is the result of spectrally white thermal noise passing through
the amplifier with gain H(w), and can therefore be expressed as:
(V
2J=
(outJN)
2
4kTs
dw
Rs f c H(w)1
H()|
o
(3.9)
The integral on the right hand side of Eqn. 3.9 is taken over the frequency band of the
measurement, and can be easily calculated numerically from the data taken in Section 3.3.1.
Eqn. 3.9 can then be solved for temperature T .
This is an example of a measurement that does not require the independent calculation
of the amplifier's noise contributions and input characteristics described in Section 3.3.1;
3.4. HIGH IMPEDANCE MEASUREMENT
79
the effects of the amplifier on the output noise voltage are neatly removed by subracting all
of the noise which does not vary with DUT temperature.
3.4
High Impedance Measurement
The measurements discussed in Section 3.3 were made on resistors with mid-range impedance.
In this context mid-range means that both the current noise and the voltage noise of the
amplifier made important contributions to the measured signal. The measurements discussed in this section were made on a photodiode whose impedance was on the order of
tens of MQ. For most amplifers (inlcuding the SR570) this means that the voltage noise is
unimportant, and results in a substantial simplification in the measurement. Additionally,
the use of a transimpedance amplifier eliminates many of the effects of parasitic capacitance
at the amplifier's input (as discussed in Section 2.5) on the amplifier's performance. This
allows cicuits to be substituted in and out of the noise measurement as necessary while remaining confident that the overall system function which the measured noise sees does not
change due to changing parasitics. This freedom is very useful in developing a calibration
for the measurement.
To measure current noise from a photodiode detector, the SR570 was used. The setup
is shown in Fig. 3-11. This setup is similar to the Johnson noise setup described in Section 3.3.1; the DUT, a large-area Hamamatsu photodiode detector, is represented by an
equivalent impedance and a noise current source. Near the detector, a red LED was used
as a light source. A quiet source of current was necessary to bias the LED; a battery with
a 1000 Q resistor was the best solution. The photodiode was left unbiased to eliminate the
shot noise of the detector's dark current, and the input of the transimpedance amplifier
acted as a sink for the detector current. The transfer function H(w) of the amplifier was
measured using the calibration circuit in Fig. 3-12.
The key question for an experimenter to ask when making calibrated noise measurements
is whether the measurement circuit looks the same looking out from the known source and
from the noise producing device.
If the voltage source Vt est in Fig. 3-11 is really just a
CHAPTER 3. CURRENT NOISE MEASUREMENTS
80
In
(1)
Equivalent Two-port Amplifier
r--
Zca
------
- -
(2)
+
Zs
_
s
Is
s
test4
(1)
--- --
V=H(w)I-
- calibration circuit
(2) - DUT (photodiode)
Computer
(LabVIEW, MATLAB)
DAQ Board
Figure 3-11: Noise measurement and calibration scheme for photodetector optical noise
measurement.
simple voltage source, and does not vary for different frequencies, the setup can be redrawn
using a Norton equivalent circuit in a way which emphasizes the symmetrical placement
of the calibration source and the noise source (Fig. 3-12). Both the calibration source and
the noise source see the same circuit at their output ports, ensuring that the calibration is
valid.
To be sure that the Vtest had no frequency dependence, Zca included a voltage divider
and a medium size resistor, rather than a single large resistor. This provides the small
currents necessary for the calibration while pushing the pole introduced by the parasitic capacitance of the large resistor to frequencies much higher than the measurement bandwidth
(Fig. 3-13).
As in the Johnson noise calibration, the voltage source used in the transfer
function was a function generator (Vfg).
Next, two noise measurements were recorded using the DAQ and the computer. First
the detector noise with the red LED turned on was recorded, and the DC current was
measured. Next the background noise was measured with the LED turned off. The power
spectral densities of both measurements were caluculated using MATLAB, and the PSD
with the LED off was subtracted from the PSD with the LED on.
This eliminated the
3.4. HIGH IMPEDANCE MEASUREMENT
81
In
(1)
Equivalent Two-port Amplifier
(2)
Zcal
Z0 u
ZZ
Vn
test
V=H(o)Ili
itest
test/Z
al
(1) - calibration circuit
(2) - DUT (photodiode)
Computer
(LabVIEW, MATLAB)
DAQ Board
Figure 3-12: Noise measurement and calibration scheme for photodetector optical noise
measurement, written to emphasize symmetry of the signal generators.
amplifier noise. As with the Johnson noise measurement, there is an input output relation
which can be written:
Svoutw = H (w) I' Sh n ()(3.10)
Using Eqn. 3.10, the PSD of the input current noise is calculated. For the measurement
to work, two stringent conditions must be met. The first, that the signal source for the
calibration must see the same impedance at its port as the noise source, has been discussed
already. The second condition is that the input impedance seen by the amplifer be the same
during the calibration measurements, diring the measurement with the LED illuminating
the photodiode, and during the measurement with the LED off.
To help ensure that this is the case, both the calibration source and the photodiode
are both always connected, as shown in Fig. 3-11. It remains to verify that the impedance
seen by the amplifier does not change depending on whether the illuminating LED is off
or on. This is not a priori obvious, and would most likely have been a serious problem if
a high impedance (voltage amplifier) measurement had been made. Fortunately, the transimpedance setup was stable in this respect. The depletion capacitance of the photodetector
(on the order of pF) was always in parallel with cable capacitance on the order of 100 pF,
CHAPTER 3. CURRENT NOISE MEASUREMENTS
82
Zcal
416 Q
120 kQ
50.0 Q
Cpar
10.2 Q
+
-
+V
test
Vfg
10.2
-zca=.
v tes~-v f" 466
120
1+
kQ
jo) -120000 -Cpar
Figure 3-13: DUT for calibration circuit, with parasitic capacitance. Also included are the
Thevenin equivalent source and impedance values.
washing out the effects of any changes in its value. The differential resistance of the photodiode was always much larger than !R(Zcai), and therefore had little effect on the resistance
seen from the input of the amplifier.
Fig. 3-14 depicts the so-called Fano factor of the noise current as a function of frequency.
The Fano factor, a useful measure of non-equilibrium noise, is the ratio of the PSD of the
measured current noise divided by the PSD of perfect shot noise, which is simply 2q IDCFor most of the frequency range shown, the measured Fano factor was approximately 1,
consistant with perfect shot noise. This is not surprising, since the photon noise from the
LED was most likely of order shot noise (see Chapter 2), and the extremely low collection
efficiency of the setup strongly promotes Poisson photon statistics in the light striking the
detector.
Fig. 3-14 shows the Fano factor over the entire frequency band of the measurement, but
actually the measurement was only accurate over a portion of that band. Below 10 kHz the
DC blocking capacitor in the first stage of hampered the accuracy of the measurement, and
above 70 kHz there was excess noise due to aliasing in the DAQ. In the measurement band
10 kHz-70 kHz, the noise was measured to be shot noise to within 2%. Fig. 3-15 shows the
measured data over the actual valid frequency range. In this measurement, the shot noise
83
3.5. LOW IMPEDANCE MEASUREMENT
LED shot noise (3.6 pA DC photocurrent)
2.5
2
- 1.5
0
U0
C
p..
.
.
u-Z
0.5-
0
2
4
10
6
8
Frequency of Measurement (Hz)
12
14
x 104
Figure 3-14: Measured Fano factor.
formula is turned around, and the measurement is used to infer the value of the charge of
an electron.
3.5
3.5.1
Low Impedance Measurement
New Difficulties
Measuring noise from a low impedance source such as a forward biased diode presents
some new difficulties, and the techniques described above must be modified.
There are
two reasons for this. The main reason best discussed in terms of the concept of optimal
source resistance described in Section 2.4.4. For every amplifier, there is a unique source
resistance R, at which the noise factor of the measurement is minimized. In other words,
CHAPTER 3. CURRENT NOISE MEASUREMENTS
84
x
Inferred Electron Charge
10-19
1.9
1.8-1.7
0
1.6
1.5
1.41.31
1
2
3
4
5
Frequency of Measurement (Hz)
6
7
X 104
Figure 3-15: inferred fundamental charge over valid frequencies.
if the measured DUT has this characteristic resistance, the ratio between the DUT noise
and the intrinsic noise of the amplifier is maximized. It is generally a good idea to attempt
to construct a measurement amplifier so as to match R, to the impedance of the DUT of
interest. Typical values of R, for BJTs range between a few kQ and a few hundred kQ [20].
For FET devices, R, is even higher, ranging from hundreds of kQ (JFETs) to hundreds of
GQ (MOSFETs). Because a laser or a LED has a Thevenin equivalent resistance of about 5
, the mismatch between the DUT impedance and R, is a severe impediment to a sensitive
measurement.
A second problem associated with low-impedance measurements is specifically related
to measurements of diode structures.
The equivalent circuit for a forward biased diode
relevant for the frequency range of interest was discussed in Chapter 2, and is redrawn here
85
3.5. LOW IMPEDANCE MEASUREMENT
in Fig. 3-16.
There are two sources of noise in the circuit: the "shot" noise of the diode
R series
Figure 3-16: Low-frequency noise model of a diode.
(which we know may actually vary from the traditional shot noise limit somewhat depending
on the details of the specific device), and the thermal noise associated with the parasitic
resistance of the device. The latter is not of primary theoretical interest, and corrupts the
measurement of the former. If the diode is operated in its forward biased regime, things
become even more difficult.
The resistance rd becomes very small and shunts the shot
noise away from the measurement apparatus.
Even if a perfect current measurement is
made at the output of the diode (meaning that the Thevenin resistance of the measurement
apparatus is zero) the series parasitic resistance of the diode will cause a large fraction of
the shot noise to be shunted. The fraction of the shot noise current which actually makes
it into the measurement is given by
fmeas =
Srp
rp
+ Rseries
(3.11)
CHAPTER 3. CURRENT NOISE MEASUREMENTS
86
3.5.2
Two Solutions
There are two common solutions to the impedance matching problem between the DUT
and the R, of the amplifier. The first is to use several input devices (either BJTs or FETs,
depending on the desired R,) in parallel for the first amplifier stage [20].
If N paralleled
devices are used, the voltage equivalent input voltage noise E, is reduced by a factor of
/N, and the equivalent input current noise is increased by the same factor. The result is
that R, =,
is reduced by a factor of N. The price paid for this is reduced bandwidth,
since the gate-to-source and source-to-drain capacitances both scale with N.
A second solution to the impedance mismatch problem is to use a transformer as the
input stage of the noise measurement apparatus. As discussed in Chapter 2, a transformer
with a 1:N primary to secondary winding ratio reduces the impedance seen by the DUT
by a factor of N 2 . This results in a reduction of the optimum source resistance R, of the
amplifier following the transformer by a factor of N 2 . The price paid by the addition of a
transformer to the measurment chain is the added Johnson noise associated with the parasitics of the transformer, and also the bandwidth restrictions imposed by the transformer
(see Chapter 2).
The method attempted for this thesis was the transformer coupled method. The bandwidth was judged acceptable for the measurements of interest, and the technique provided
the best chance for achieving an adequate reduction of the R, of the SR560 voltage amplifier. It should be noted that the paralleled-device technique appears to be the preferred
method in the noise measurement literature
3.5.3
[44],
[45], [43].
Transformer-Coupled Measurement
A diagram of the transformer-coupled measurement is shown in Fig. 3-17. There are several signal sources at the input of the transformer. The DUT, taken here to be an ideal
diode, contributes shot noise from the junction, and Johnson noise from the parasitic series impedance. All of the resistors in the surrounding circuit contribute Johnson noise.
The function generator voltage source is used for calibration of the measurement, and a 6
87
3.5. LOW IMPEDANCE MEASUREMENT
_ SRS56O
300Q
10009i
DUT
diode)
+
VI
I
transformer
-
I-
Function
generator
Computer
Bias
F1 DAQ
Figure 3-17: Transformer-coupled noise setup.
volt lantern battery provides a quiet current bias for the diode. The the noise model of
the transformer and the diode were discussed in Chapter 2. They should be inserted into
Fig. 3-17 for a complete analysis of the measurement.
There are two key parameters of interest; the signal/noise ratio at the output, and
the bandwidth. The former can be calculated for a DUT whose parameters are known by
analyzing the circuit of Fig. 3-17 and comparing the signal and noise level at the input of the
second stage amplifier. The low end of the bandwidth is set by the source resistance and the
primary inductance of the transformer
(f
=
A), and the high end of the bandwidth is
A consequence of
set by the source resistance and the output capacitance (fhigh =
this high frequency limit is that one must be careful not to load the output of the transformer
with parasitic cable capacitance.
For example, 100 pF of cable capacitance, due to the
impedance stepping of the 1:100 transformer, looks like a 100pF * 1002
=
1pF capacitance
from the input, which can significantly affect bandwidth. Another important point is that
the coupling capacitor at the input of the transformer should be made as large as possible
to avoid decreasing the low frequency bandwidth limit of the measurement. For the setup
CHAPTER 3. CURRENT NOISE MEASUREMENTS
88
used in this thesis, the coupling capacitance was required to be larger than 10
pF or so.
This is an awkwardly large value for a non-electrolytic capacitor, but it was achieved with
a parallel combination of four 5 pF polypropylene capacitors. While nonpolar electrolytic
capacitors of the required capacitance are available, their intrinsic noise and relatively high
leakage currents make them a questionable choice for sensitive measurements.
On a final practical note, this setup proved extremely sensitive to both capacitive and
inductive parasitic noise. The former was minimized by placing every part of the measurement up and including the SR560 in a properly shielded box. The latter was a much more
vexing issue; even with perfectly sound wiring practices the coils of the transformer still
picked up significant noise. The magnetic coupling problem was eventually dealt with by
moving to an area as far away as possible from power lines and computers, and carefully
choosing the orientation of the transformer coils to minimize flux linkage.
3.5.4
Calibration
A new calibration of the instrument must be performed for every DUT which is measured.
Further, if the DUT is a nonlinear device, then a calibration must be performed for each
bias point.
This is tedious, but necessary, since the bandwidth of the measurement is
strongly dependent on the Thevenin equivalent resistance of the DUT, due to the bandwidth
limitations of the transformer discussed in Section 3.5.3.
The calibration was performed in the usual manner, by applying a known sinusoidal
signal in the function generator and sweeping the frequency, while recording the output on
the DAQ. The key to the calibration is knowing the value of all of the impedances at the
input of the transformer. The transfer functions of the signal of interest (shot noise in the
diode, for example) to the transformer input and the function generator to the transformer
input are different, and both must be computed in order for the calibration to be useful.
The only unknown impedances are those of the DUT, and measuring them is the first step
in the calibration.
A high-power silicon homojunction p-n diode was chosen to test the calibration and
3.5. LOW IMPEDANCE MEASUREMENT
89
measurement of the tranformer-coupled setup. The values of the parasitic and differential
resistance were obtained by carefully measuring the I-V curve of the device. A diode model
of the form
nkT
=(3.12)
was assumed for this device in the forward bias regime, where V is the voltage drop across
the junction, I, is the unknown saturation current, and n is an unknown ideality factor
modeling the deviation of the device from the ideal diode law. The contact resistance due
to the Ohmic contacts in the device can be assumed to be constant with bias
[521.
This is a
reasonable model for forward biased low-level injection, and the high-power device can be
easily kept within this regime. This model gives a differential resistance of
r = nkT
qI
(3.13)
Along with the differential resistance, a diode posseses a parasitic series resistance rp on the
order of Ohms. This becomes important after the the device turns on (when the differential
resistance becomes small). Using this model, a curve-fit was obtained to extract the values
of n and rp from the numerical derivative of the measured I-V curve. The data and the
curve fit are shown in Fig. 3-18. While more detailed models of a diode are available
[52],
this model is good enough to capture the effects of interest. Note that the impedance of
the diode is dependent on the bias current.
The next step in the calibration is to obtain the transfer function of the measurement
apparatus from the DUT shot noise source to the measured output by using the function
generator and sweeping a test signal across the required bandwidth. This is conceptually
the same as what was done in the other noise calibration techniques discussed earlier in the
chapter, although a factor of fmeas (from Eqn. 3.11) must be manually inserted to complete
the calculation. The resulting transfer function of the measurement is plotted in Fig. 3-19.
The total DUT impedance seen by the transformer input can be calculated from the I-V
CHAPTER 3. CURRENT NOISE MEASUREMENTS
90
Measured Resistance With Fit
20
18
1614
-12
0
U)IU
8
6
4
1
01
0
10
5
15
bias current (mA)
Figure 3-18: Results of parameter extraction from I-V curve.
curve. However, a more accurate method was found for this particular type of measurement.
Since the bandwidth of the measurement depends sensitively on the equivalent impedence
of the DUT, a small potentiometer can be used in place of the diode, and the value can
be changed until the transfer function of the measurement with the resistor is equal to
the transfer function of the measurement with the diode. Using this technique, the DUT
equivalent impedance could be determined to within 0.1 Q. However, the curve fit above is
still required to resolve the separate contributions of r, and rd to the total impedance.
3.5.5
Measurement
Once the calibration was done, two more measurements were necessary to find the shot noise
of the diode. The function generator was disconnected and replaced with a 50Q termination,
3.5. LOW IMPEDANCE MEASUREMENT
1
91
Transformer Transimpedance
X106
0
1.6
1.4
C)
CU
_0
a)
1.2
0-
E
0.8
0.6
0.4
0.2
0
10
20
30
frequency (kHz)
40
50
Figure 3-19: Transfer function of the transformer-coupled measurement with diode DUT.
accurately duplicating the function generator's impedance load on the measurement circuit.
The first measurement was taken with the diode in the DUT position. Next, the measurement was repeated with the matched potentiometer from the calibration as the DUT.
The power spectral densities of the amplified voltage signals of both measurements were
calculated. The measurement of the diode (Sd) included significant noise contributions from
the signal (the shot noise), the Johnson noise of the parasitic resistance of the diode, the
Johnson noise of the input of the transformer, and the input referred noise of the second
stage voltage amplifier. The measurement of the matched potentiometer (Spot) included
contributions from the Johnson noise of the potentiomenter (whose value is rd + r
~ rd),
the Johnson noise of the transformer input, and the second stage voltage amplifier noise.
The measured shot noise Seh can then be found by:
Sh(L
h)
=
Sd(w) - Spot (w)
This power spectral density can then be deconvolved using the transfer function from the
CHAPTER 3. CURRENT NOISE MEASUREMENTS
92
calibration measurements to yield the shot noise in the same manner as outlined in Section 3.4. This process, and the final result are illustrated in Fig. 3-20. The noise spectral
500450
400
350
.2
300
C
L250
200
150
100
50-
-...
....
0.
5
25
15
20
10
Frequency of Measurement (kHz)
30
Figure 3-20: Measurement of diode shot noise using transformer-coupled measurement.
density measured is consistent with a 1/f-type noise spectrum.
3.6
Conclusions
This chapter has described a set of general techniques for measuring the current noise
from devices of various impedances.
The basic equipment used for these measurements
included the SR560 voltage amplfier, the SR570 current preamplifier, the SR554 transformer
preamplifier, and two data acquisition systems. The philosophy of this work is that when
measuring current noise, understanding and compensating for the spurious sources of noise
which corrupt the measurement is more effective than attempting to eliminate all spurious
noise through heroic custom designed circuitry.
Measurements using each of the setups have been presented here.
These include a
measurement of the input-reffered noise sources of the SR570 amplifier, a cryogenic measurement of the Johnson noise of a resistor with temperature, a calibrated measurement of
the shot noise generated from a low-efficiency optical link, and a calibrated measurement
of the 1/f noise spectrum of a silicon diode using a transformer coupled setup.
Chapter 4
Circuit-Induced Laser Noise
Correlations
The correlation of the photon noise in two lasers which are biased in the same circuit, under
various biasing conditions and in different circuit topologies, is studied here. To appreciate
the motivation and results of these experiments, some background in semiconductor lasers
is helpful. This can be found in Appendix B. The first part of this chapter assumes this
background, and reviews current theories of noise in semiconductor lasers.
Some experimental and theoretical context is also necessary to appreciate the measurements of this thesis. The work which first sparked interest in the relationship of the photon
noise of a laser to its biasing circuit was the seminal paper of Yamamoto and Machida
[53] pointing out that a laser biased with a current source well above threshold can emit
amplitude squeezed light. Squeezed light is light with a noise spectral sensity less than that
of the so-called "standard quantum limit" of shot noise (2hvPDC). This sparked a series
of experiments [13] [16] [54] [55] attempting to achieve the maximum level of squeezing
permitted by [53].
This proved difficult, and a series of theoretical papers (e.g. [56] [57]
[58]) delved further into the basic physics at work within the laser to find the source of the
excess noise.
As the work relating to squeezing advanced, optical correlation measurements began to
93
94
CHAPTER 4. CIRCUIT-INDUCED LASER NOISE CORRELATIONS
appear in the literature. A method for squeezing was proposed taking advantage of the
intensity noise correlations expected from series coupled lasers [59].
At around the same
time, the photon noise correlations from series and parallel coupled LEDs were measured and
shown to display positive and negative correlations [60], consistant with the measurements
in this thesis on laser devices. The bias-induced correlation between two lasers was studied
theoretically [61].
In [62] the correlation in the intensity noise between two facets of a
semiconductor laser is measured using a hybrid-pi splitter and a spectrum analyzer. Using
a similar measurement technique, series and parallel coupled LEDs are investigated in [63],
and squeezing is observed. The correlation between the electrical and optical noise of a laser
is measured for the first time in [64]. Sadly, a clear description of the experimental apparatus
is missing from the latter, and no attempt could be made to repeat the measurement. To
the best of the author's knowledge, the measurement in this thesis is the first measurement
on the photon noise of series and parallel coupled laser diodes.
Consider the circuit noise model originally presented in Section 2.2.2, reproduced here
in Fig. 4-1. Recall that Rseries is the series parasitic resistance of the laser (mainly from
Rseries
d
n
Figure 4-1: One-port noise model of a diode.
1/f
4.1. SEMICONDUCTOR LASER DIODES
95
its Ohmic contacts) and rd is the diode differential resistance.
The resistance Rseries is
dissipative and therefore contributes thermal noise to the circuit. The two current noise
sources in parallel with the differential resistance model the laser noise and the 1/f noise
of the laser. For the rest of this discussion the 1/f noise source I[/f will be ignored; since
it appears in parallel with the laser noise I
it can be added back to the model anytime
if necessary. The noise model of Fig. 4-1 does not provide two key pieces of information.
First, in a heterostructure laser we do not know the value of the spectral density of the I
noise generator with precision, and even the differential resistance can be more complicated
than the simple expression in Eqn. 2.4. Second, the circuit model makes no mention of the
photon noise, or its relation to the the current noise. To address both of these questions,
an understanding of the basic processes at work within a diode laser is necessary.
4.1
4.1.1
Semiconductor Laser Diodes
Theories of Diode Noise
One accepted theory of homojunction P-N junction noise is outlined by Buckingham in his
book on electronic noise [28]. According to Buckingham's theory, noise in a homojunction
diode results from thermal fluctuations and generation-recombination noise in the minority
carriers within a diffusion length of the junction.
The mathematical approach for this
theory is to write an expression for a single current pulse due to a noise event a certain
distance away from the junction. At a given location these events are taken to be a Poisson
process independent of the noise events at other locations, and Carson's theorem (see [65])
can be used to write the noise power spectral density resulting from noise events at that
location. The contributions to the noise power spectral density can then be integrated over
the length of the quasineutral regions to obtain the full noise spectral density for the diode.
The end result for a homojunction P-N diode is that at low frequencies, the noise generator
I displays shot noise. Van der Ziel [29], using a different method for the noise calculation,
arrives at the same result.
CHAPTER 4. CIRCUIT-INDUCED LASER NOISE CORRELATIONS
96
The first theoretical attempt to derive the noise model for a heterostructure laser diode
was by Yamamoto and Machida [53] using an approach similar to that of Buckingham's.
A more detailed theory generalized to series and parallel laser arrays was recently given by
Rana and Ram [661. This theory relied upon a detailed model of the heterostructure to write
down a rate equation for all of the independedent transport and recombination processes
of importance.
The Langevin method introduced in Section 1.5 was used to model the
noise. This resulted in expressions for the current noise generator I,, and the photon noise
generator P,a describing the noise in the photon stream.
4.1.2
High-Impedance Supression of Noise
When the models of [53] or [66] are biased with an impedance which is large compared to
the differential impedance of the diode, a unique phenomenon is observed.
Consider the
circuit seen by the noise source I, shown in Fig. 4-2.
Rseries
Rlarge
+V
VBL
rd
n
Figure 4-2: Pump supression in a diode laser.
If
Rlarge is much larger than rd, then the current from the noise generator will be
4.1.
SEMICONDUCTOR LASER DIODES
97
shunted by rd and no noise current from the laser will flow through the external circuit. At
the same time, the thermal noise of the series resistance of the circuit R. = Rseries + Riarge
also drops. Therefore the noise in the external circuit can be surpressed well below shot
noise. The spectral density of the current noise in the external circuit can be written (at
low frequencies):
Sext=(
R
rd + RS
)2
Rs
+ (
Td + Rs
(4.1)
)2
The supressed external current noise also exerts a profound influence on the noise power
of the light emitted from the laser, which is correlated with the external current noise. It
can be shown that for high bias currents (relative to the threshold current of the laser), the
Fano factor of the intensity noise at low frequencies is given by [66]
F
where
=
(4.2)
- TL
L is the quantum efficiency of the laser. The TIL term is known as partition noise,
because it is introduced by the random partitioning of the charge carriers and light within
the laser. This effect is known as amplitude-squeezing, or high impedance pump supression.
The proposed aplications of highly squeezed light are numerous, and can be found in the
fields of communication, spectroscopy, and quantum computing, to name a few.
4.1.3
External Current Correlations
If two or more lasers are biased in the same circuit, the theory of [66] predicts that the
lasers can influence one another through their shared biased current. Because the external
current noise of a laser is intimately correlated with the photon noise, the light from lasers
sharing the same bias circuit can be correlated by the external current noise. Measuring
these correlations, and in so doing helping to confirm the theory of [66] is the goal these
measurements.
Consider the four two-laser circuit topologies shown in Fig. 4-3.
In each circuit, Rp
represents the series parasitic impedance of the laser, rd its differential resistance, and IT
CHAPTER 4. CIRCUIT-INDUCED LASER NOISE CORRELATIONS
98
d1ii
Rd1
(A)
(C
Rp2
~C'
R
+:
(B)C
I........
...
R
p2
:
.
p2
1
~
n1
n:n2
dd
n2
d2
Figure 4-3: Four simple bias circuit topologies.
its noise. The voltage sources V and the current sources I supply the bias for the lasers.
Each of the resistors R, also contribute thermal noise to the circuit, although the thermal
noise sources are not shown. Also shown in the figure are the "external" noise currents going
through each
laser's terminals. This current is important in determining the photon noise
of the lasers, as was mentioned in Section 4.1.2.
Note that in the series coupled circuits
((A) and (B)) the external noise passing through both
Circuit (A) biases the two diode
for the external
e
(
lasers in series with a voltage source. An expression
current noises in the circuit can be written down by inspection:
AA)
t)
(
lasers is the same.
)
In2Td2
R
d+
R1+rd2 +R2
Ith2Rp2
Rd 1 +lrd
+R2+-rd2
d2Inld
rd2 + RP2+1 +Rp1
+Ith1Rp1
R 2 +r
2
+RP1 + d1
99
4.1. SEMICONDUCTOR LASER DIODES
Because the noise current from each of the lasers flows through both lasers, and the photon
noise of each laser is correlated with its own external current noise, the photon noise of the
lasers due to the external current noise is expected to become positively correlated.
Circuit (B) has the two lasers in series and biased with a current source. This is exactly
the case where squeezing (discussed in Section 4.1.2) is expected. None of the noise current
from the diode noise sources (I41
and In2) or the thermal noise sources of the parasitic
resistances (Rpi and Rp2 ) travels through the external circuit, because the impedance presented by the biasing current source I is assumed to be very large. For the experiments
performed for this thesis (and for any current source built from a battery in series with a
large resistance), the current source I contributes negligable noise to the external current.
Therefore
I(B)
0
exti = I ext2 = 0(4)
(4.4)
In circuit (C), the lasers are placed in parallel and voltage biased. The current noise is
I~c)
-
RpInIrail
1 + rdl
+ Rp1
rlIth1Rpi
45
(4.5)
In
=
In2rd2
Ith2Rp2
(4.6)
exti
rd2
+ Rp2
Rp2 + Td2
In this case, the noise from the two lasers is completely decoupled; none of the noise current
from laser 1 passes through laser 2, and vice versa. Therefore correlations between the
noises of the two lasers are not expected.
The final circuit (D) places the two lasers in parallel with a voltage bias. The external
CHAPTER 4. CIRCUIT-INDUCED LASER NOISE CORRELATIONS
100
noise is
1
(D)
extJ1
Td2
+ Rp2 + rdl + Rp1
Rp2 + rd2 + Rp1 +
(4.7)
Vdl
In2rd2
Ith2Rp2
7d1 + Rp 1 + rd2 + Rp2
Rp1 + Tdl + Rp2 + rd2
I(D)
ext2
+thlRp1
Inlrdl
Tdl
In2rd2
+ Rp1 + rd2 + Rp2
Ith2 Rp2
Inlrdl
7
'd2 + Rp
+-
--
Rp1 + rd1 + Rp2 + rd2
(4.8)
IthI RpI
Rp2 + rd2 + Rp1 +
2 + rdl + Rp1
Vdl
Physically, what is happening in (D) is that the current noise from each laser is forced
through the other laser, becasue none of it can pass through the high impedance of the
current bias source I. In this circuit, as with circuit (A), correlations in the part of the
ouput photon noise due to the external current noise of the two lasers is expected.
The
expressions in this case look identical to those for the circuit (A), except for the sign
reversals. In this case, the magnitude of the correlation is expected to be the same as the
correlation in circuit (A), but the correlation is negative.
Lastly, we explicitly evaluate the cross-spectral densities of the photon noise in cases
(A) and (D) in terms of the power spectral densities of the thermal and laser noise sources.
We assume laser converts the current to light power with a factor
sources I1,
r7_, and that the noise
In2, Ithl, and 1th2 are independent. One finds:
C(A)
Vp
hv
72 + R2R2+±
2
q)
(q
h
q )
Rp1 + ri
2
A--
Rp2+
Td1
2
(1q
(hv
2
2
(rd1+ Rl1+r2+
2
=2
(Iq
(R
t
*
2
Rp2 +r7-2
1
Rp2
+ Rp2 +
2
Td2
A
+Rp1 +r7-c
Rd2 1
-.
+ lR1
2
(4.9)
4.1. SEMICONDUCTOR LASER DIODES
C(D)
( \i
I
) iv P1-,
2
.1
2
q)
(
101
rdl1+ Rp1+
Td2 rd2 + Rp2 )
+
Rp2
(hv
2
771 qIJ'th2 \Rp+rdl
hv
2
(Pq
Pitl
*..
(4.10)
2
/hv
r) 2
7
2
+r1
+ Rp)
Rp 2
+ R+ 2rd2J
2
R1p
2
(Rp2 + rd2 + RpI + rdli
It is clear that if the differential resistance of the lasers is very much smaller than the series
parasitic resistance, this model predicts that thermal noise will be the dominant souce of
laser correlation.
To get a more quanitative estimate for the extent to which thermal noise acts to correlate
the noise of the two lasers, we define a measure of the correlation equal to the ratio of the
cross spectral density C of the laser noise to the geometric mean of the power spectral
densities P of each laser:
C =
Assuming rdl = rd2
< Rp = Rpi
Paser1
(C(4.11)
X
Piaser 2
Rp2 and Paser i = Paser 2, we find for cases (A) and (D):
1 4kT
JCn
=
2 RP
PlaserI x Paser 2
(4.12)
To get a sense for how large this correlation is for typical lasers, we assume values of
Rp=3.1 Q, v=8809 nm'adm
, and T1 = 0.68.
These are the values for the lasers used in this thesis.
06.Teeaetevlefothlaesueintithi.
In Section 4.3 the lasers' noise is measured to be approximately 10 times the standard
quantum limit (shot noise).
Assuming each laser has a DC bias current of IDC the shot
noise is
P = 2hv(IDCl h)
q
(4.13)
With IDC=90 mA as it was for the correlation measurents later in this thesis, the correlation
CHAPTER 4. CIRCUIT-INDUCED LASER NOISE CORRELATIONS
102
in the photon noise due to thermal noise is:
(4.14)
Cn = 0.0065
This is a rather small correlation, and it is clear that if any significant correlation is measured
in circuits (A) and (D), it must be due to the intrinsic laser noise, and not the thermal noise
of the parasitic resistances.
4.2
Correlation Setup
The basic setup for the correlation measurement is shown in Fig. 4-4.
series
config.
Lb
Apr
LC
D+
L
SL
bt
Cbt
shunt:
_VB
otBBD
AA
--
f-.---
- -- -- -- - -- -- ----
- -- -- -- - -- -- -- -
Figure 4-4: The correlated photon noise measurement setup.
The setup is composed of two separate circuits, one which drives the lasers and one
which measures the noise current from the photodetectors. The bias current for the
laser was
4.2.
CORRELATION SETUP
103
provided by a 12 Volt lantern battery VBL. The current was controlled using a potentiometer
Rpot.
A 15 mH inductor was used for additional high frequency impedance (note that
throughout this thesis, the prefix 'in' stands for 'milli', contrary to some electronics part
labeling conventions). The inductor had a parasitic resistance of 15 Q, which can be added
for modeling purposes to the potentiometer resistance
Rpot. The voltage source Vat (a
function generator) and the resistor Rcal are used for calibration of the measurement. The
10 mF capacitor
Csah,,t
can be switched in or out of the circuit. The lasers L 1 and L 1 were
the diode lasers, specifications for which are given later.
The light from the lasers was directed at the PIN photodiode detectors Di and D 2.
Their specifications are also given later. The photodiodes were biased using a conventional
40 V power supply. They were isolated from the power supply and from each other at
high frequencies by using the identical inductors Lbt, which had an inductance of 100 mH
and a parasitic resistance of 60 Q.
The identical capacitors Cbt were each 20 mF. With
the inductors Lbt they formed two bias-Ts, allowing the the photodiodes to be voltage
biased at DC while passing AC signals through the capacitors into the transimpedance
preamplifiers (Apre with feedback impedance Zf). The specifications of the preamplifiers
are also described later. The two output channels were then amplified again with SR 560
low noise voltage amplifiers, whose specifications are given in Chapter 3.
The resulting
voltage signals were then sampled at 25 MHz using the Gage data acquisition card (also
discussed in Chapter 3) and stored on a computer.
To eliminate capacitive coupling noise, most of the setup (everything up to and including
the preamplifiers) was placed in a large metal box. The box was connected to the ground
of the photodiode circuit at the point where the connection to the earth was made, to avoid
ground loops.
The three most important devices for the correlation measurement are discussed next.
CHAPTER 4. CIRCUIT-INDUCED LASER NOISE CORRELATIONS
104
4.2.1
The Transimpedance Preamplifiers
The preamplifer chosen for this measurement (shown in Fig. 4-4) was the Analog Devices
OP-27 in the transimpedence configuration with feedback impedance Zf.
a low noise, high speed, precision operational BJT amplifier.
The OP-27 is
Its key specifications are
summarized in Table 4.1.
Gain Bandwidth
63 MHz
Voltage Noise VN
Slew Rate
17 V/ps
Current Noise IN
3nV/VHz
0.4 pA/
Common Mode Rejection
126 dB
Open Loop Gain
1.8 million
Hz
Table 4.1: Key specifications of the Analog Devices OP-27 operational amplifier used for
the measurement.
The transimpedance (ratio of output voltage to input current) of the preamplifers is
set by the feedback impedances Rf. The impedance was 10000 Q in parallel with a 18 pF
capacitor.
The 18 pF capacitor was necessary to stablize the amplifer.
The bandwidth
of a transimpedance amplifer is usually a nontrivial function involving the amplifer gain
characteristics, the feedback impedance, the impedance of the input, and various parasitic
capacitances. In this case, the most important pole is set by the R and C of the feedback
impedance alone, and occurs at about 880 kHz. The feedback capacitance was set small
enough to allow a small amount of gain peaking, pushing the 3 dB point out to around
1.1 MHz. The bandwidth of the second stage amplifier (SR 560) was also approximately 1
MHz.
There are three sources of noise in the transimpedance amplifier: the input referred
voltage noise, the input referred current noise, and the thermal noise of the feedback resistor.
Once these sources are identified, elementary circuit theory can be used to find the input
referred noise from each. The result is that the thermal noise of the resistor and the input
noise of the amplifier combined give an input referred current noise of approximately 1.4
pA/
Hz. The noise from the amplifier noise voltage source was on the order of the voltage
noise divided by the impedance of the photodetector
3nV/
100 MQ
= te
10-1nV/v
V/tHz
did
andwere
not make a significant contribution. The current noise signals from the photodetectors were
4.2. CORRELATION SETUP
105
always larger than the shot noise of the 35 mA photocurrent. This corresponds to a signal
of about 110 pA/v -Hz,more than 100 times that of the amplifier noise. Because in the final
results of the measurements is in terms of power and not amplitude, this corresponds to a
signal to noise ratio of 1102 ~ 10000.
A point should be made regarding the choice of the OP-27. Choosing a BJT amplifier over a FET amplifier appears slightly inconsitent with the noise models presented in
Chapter 2.
The noise performance of a FET amplifier is usually far superior to that of
a BJT amplifier when measuring high impedances like a reverse biased photodiode. For
example, the Analog Devices AD546 electrometer amplifier has an input current noise only
1.6 fA/vfHz. The reason for choosing the BJT amplifier over a FET amplifier was its better
was its high gain-bandwidth product, coupled with the fact that its noise performance was
more than good enough more the job at hand. The gain-bandwidth product of the AD546
amplifier has a gain-bandwidth product of only 1 MHz, by way of comparison.
4.2.2
Low Frequency Sensitivity vs. Microwave Measurement Sensitivity
Microwave measurements like the one in
[63] rely on a low noise amplifier and a spectrum
analyzer to characterize the measured noise. It is worthwhile to compare the sensitivity
achievable through microwave techniques with the techniques associated with the correlation
measurements here. Consider a photodetector AC coupled to a microwave amplifier and a
low frequency amplifier in turn.
For a typical microwave measurement, the impedance seen by the input of the amplifier
is 50 Q, because at high frequencies microwave components must be impedance matched
to avoid reflections.
The total current seen at the microwave amplifer is then the signal
from the photodetector I, the thermal noise of 50 Q, the noise from the dark current of
the photodiode Id, the current noise of the operational amplifier In, and the voltage noise
of the amplifier
Vn
50 Q*
Itot IttIs
Is +
0kT + I + In +
vn
(4.15)
An excellent noise figure for an LNA is 1.6 dB. This can be used with the known source
CHAPTER 4. CIRCUIT-INDUCEDLASER NOISE CORRELATIONS
106
impedance of 50 Q to calculate the root spectral density of the amplifier's noise current
I, +
" = 11.9 pA / vHz
50 Q
(4.16)
The total current at the input terminals of the low voltage amplifier is a similar sum, but
thermal noise of a 50 Q resistor need no longer be included. If the same expression is written
for the low frequency amplifier
Itt
(4.17)
= Is + Id + In + Vf
Td
where rd is the differential resistance of the photodetector under reverse bias. An AD546
electrometer op-amp in the transimpedance configuration can have
In +
(4.18)
r ~1.6 fA/v/Hz
rd
depending on the specifics of the setup. Therefore, in this case, the sensitivity of the low
frequency setup can be as much as 10000 times that of the microwave setup. When the
thermal noise which is usually present due to the 50 Q matching condition (the second term
on the RHS of Eqn. 4.15) is added, it only makes things worse for the LNA.
Of course, the price paid for this improved sensitivity is bandwidth; to operate the
AD546 at that sensitivity might mean having a bandwidth of only 10 Hz. Also, the benifits
of using low frequencies are best reaped in circuits with large impedances. The noise performance of microwave circuits and low frequency circuits for devices with low impedances
is much more comparable, with microwave circuits often having an edge.
4.2.3
The Photodetectors
The photodetectors used in this measurement were the Hamamatsu S3590-01 silicon PIN
photodiodes.
The detectors were chosen for their large area (1 cm 2 ).
This allowed the
photodetectors to be tilted at an angle relative to the incident laser while still collecting
4.2. CORRELATION SETUP
107
most of the laser's light. The photodetectors were tilted to avoid reflecting light back into
the laser and causing instabilities which might distort the noise. The key specifications of
the photodetectors are given in Table 4.2.
Photosensitivity
0.51 A/W
Terminal Capacitance
75 pF
Quantum Efficiency
78%
Dark Current
1.7 nA
Max. Power Dissipation
100 mW
Cut-off Frequency
35 MHz
Table 4.2: Key specifications of the photodetectors used for the measurement.
From the model in Chapter 2, we know that the dominant noise of the photodetector is
the shot noise of the dark current. This is much smaller than the currents to be measured,
and does not contribute significant noise to the measurement.
There are two important cautions pertaining to the photodiodes. First, if the photodiodes were not properly heat sunk, the large incident laser power caused them to overheat,
degrading their bandwidth and quantum efficiency noticably. This was avoided by heatsinking the detectors with a large block of aluminum, and blowing a fan on the aluminum block
to promote convective cooling. The second experimental pitfall associated with using these
photodetectors to detect lasers were spatial saturation effects due to the large incident photon flux. To avoid this, the photodetector was reverse biased at 40 V, the maximum allowed
according to the manufacturers specifications. The presence of either of these problems is
very apparent when the laser-to-detector transfer function is measured, and both problems
were confirmed absent when the measurements where made.
4.2.4
The Lasers
The lasers used for this measurement were SDL-5400-C GaAlAs cw laser diodes from JDS
Uniphase. The lasers were Fabry-Perot index guided structures, and were guaranteed to be
single mode at high bias and in the absence of temperature changes, drive current changes,
and optical feedback. The manufacturers measured the key specifications of each laser, and
the information is given in Table 4.3.
108
CHAPTER 4. CIRCUIT-INDUCED LASER NOISE CORRELATIONS
Thresh. Current
Slope Eff.
Quantum Eff.
Series Resistance
Wavelength
19.3 mA
1.04 A/W
68%
3.1 Q
809 nm
Table 4.3: Key specifications of the lasers used for the measurement.
Plots of the voltage device, optical output power, and slope of the optical output power
versus drive current are also provided by the manufacturer. They are reproduced here in
Fig. 4-5.
P vs I /
0
dP/dI
U vs I
/ P vs MPD-I
In
3
E
M
U
SN
Ir
D
3+
0
03
I
I-Q
L
0
ouptpwradth.
50
EM
+
+
+
+
+
+
E-
100
150
DRIVJE CURRENT,
200
P\
. 3
M
1
co
W
03
Li
G
+rn
+
L N3+
+
0
D
0
50
100
ISO
200
DRIVE CURRENT, mA
rnA
Figure
4-5: Laser
struture characterization curves.
ipemd
The lasers are popular in the quantum optics literature, and are often used to generate
single-mode squeezed light [55] [58] [67] [54] [68].
They were chosen here for their high
output power and their simple mode structure.
The lasers were temperature stablized using a hand-made Peltier cooling setup. Each
laser was mounted on a copper rail which was also the ground contact of the laser. The
laser package was allowed to overhang the edge of the copper by a small amount. This did
not compromise the thermal contact between the laser and the copper rail, but prevented
any light escaping through the back facet of the laser from reflecting back into the laser.
The copper rails were in thermal contact with a Peltier cooler, which dumped the heat from
the laser into a large aluminum block. While the copper rails were electrically insulated
from one another, heat sink grease was used to assure a good thermal contact between
them. The Peltier cooler was feedback controlled through a thermistor mounted in one of
109
4.3. MEASURED SINGLE LASER FANO FACTOR
the copper rails, and the temperature of the laser was kept at 16 degrees C throughout the
measurement.
It was important that the lasers be in a stable, single mode during the experiments.
Any mode hopping behavior would have dominated the noise properties of the device, and
washed out the relatively quiet stationary white noise spectrum from the driving current.
To confirm that the lasers were quiet enough to be useful, normalized measurements of the
laser light's Fano factor were made.
4.3
Measured Single Laser Fano Factor
To obtain to Fano factor of the photon stream the measurement apparatus must be characterized. To understand this, a simplified picture of the measurement is helpful. Fig. 4-6
shows the measurement setup of Fig. 4-4 at high frequencies and with only one channel in
use. Also shown in the figure are some of the transfer functions which will be useful in the
discussion of calibration. The circuit is shown below:
G=KH
K
H
Zf
Rcal
L2
D2
+,
Vcal
-Computer
DAQ
Figure 4-6: The calibration measurement setup at high frequencies.
The measured quantities in this experiment are the DC photocurrents from each of
the photodiodes, and the fluctuating voltage which is sampled by the DAQ. To obtain the
CHAPTER 4. CIRCUIT-INDUCED LASER NOISE CORRELATIONS
110
Fano factor of the light, all that is needed is the transfer function H characterizing the
conversion of an AC current in the photodetector to a voltage measured by the DAQ. Once
H is obtained, the Fano factor of the current noise in the photodetector can be found from
F 1 (f)
S1 (f)
2
qIDC
_
Sv(f)/H(f)
2
(4.19)
qIDC
Once the F1 is known, the Fano factor F(P) of the light can be obtained using the value
for the quantum efficiency of the photodetector, qd. For a perfect photodetector (Id = 1),
every incident photon is transduced to an electron of current, and F 1 = Fp. A photodetector
with
Wi
< 1 adds partition noise to the signal, as discussed in [18]. The relation between
the power spectrum of the current noise S, and the photon noise Sp can be written:
S 1 = riSP + (1 - qd)qdPDC
(4.20)
Here PDC is the DC laser power incident on the detector, given by:
PDC =
IDC/Id
(4.21)
Equations 4.19, 4.20, and 4.21 can be combined to give an expression for the the Fano
factor of the light in terms of measurable quantities.
SF
2hvPDC
_
Sv/H - (1 - rid)IDC
Td 2hvlIDC
(4.22)
In this expression, IDC and Sv can be directly measured, and H can be obtained using
calibration methods similar to those developed in Chapter 3.
4.3.1
Calibration
The transfer function H could be directly measured by inserting a known current source
into the measurement circuit in place of the photodetector. By varying the frequency of the
current source and recording the voltage at the output of the measurement, the transfer
4.3. MEASURED SINGLE LASER FANO FACTOR
ill
function H -- V/I could be measured. This was the method used for the measurement of
LED shot noise in Section 3.4.
The only problem with a direct measurement of H is that the current source inserted
into the calibration circuit inevitably has a different Thevenin equivalent impedance than
the real device, which can bias the results. For the measurement in Section 3.4 this was not
an issue, because the measurement was made with a less agressive bandwith (< 150 kHz)
and with a more well-behaved amplifier (SR 570).
Also, the parasitic capacitance of the
detector used for this measurment is considerably higher than that of the detector used in
Section 3.4.
To be sure to include all of the circuit parasitics, the calibration was done by modulating
the laser bias current. This allows the measurement of the bias-current-to-DAQ-voltage
transfer function G(f).
From this, the transfer function H was obtained by using the
known bias-current-to-detector-current transfer function K. K can be measured easily at
DC:
K =
0.58
IDC
Ibias - Ith
(4.23)
Here IDC is the DC photocurrent, Ibia, is the DC laser bias current, and Ith is the
threshold current of the laser. The measurement was made at various bias levels, and only
small variations in K with bias were seen.
From the measured G and K, H can be calculated from
H(f) =
G(f)
K
(4.24)
Here it has been assumed that K does not vary with frequency. This is consistant with the
specifications for the laser and the photodetector.
The measurement of G(f) was made using the the calibration circuit shown in Fig. 44. A 1 V oscillating signal from a function generator Vcal was applied through the 10000
Q resistor Rca, resulting in a 0.1 mA modulation of the laser current.
The results of a
CHAPTER 4. CIRCUIT-INDUCEDLASER NOISE CORRELATIONS
112
measurement of G, along with the resulting H, are shown in Fig. 4-7
12000
--- H
--- G
10000
8000
C:
CL
6000
E
4000
I2000
0.5
1
1.5
2
Frequency (MHz)
Figure 4-7: Measured transfer functions needed for calibrated photon noise measurements.
4.3.2
Measurement and Results
Armed with all of the necessary quantities, the measurement of the voltage noise at the
DAQ can be made and used to calculate the photon stream noise.
The measurements were performed in a dark room to eliminate spurious signals from
the fluorescent room lighting. It was also helpful to turn off the monitor of the computer
screen, which happens to produce noise within the measurement band.
The measurement was made at 16 different laser biases, ranging from just above the
threshold current of the laser to around 4.5 times the laser's threshold (90 mA). All of the
data were taken without the shunting capacitor; at high frequencies, the laser was current
biased, not voltage biased. For each measurement, the voltage at the DAQ was sampled
about 500000 times at 25 MHz. The calibration was repeated at each individual bias, to
capture and bias-dependent parasitic effects which might affect the transfer characteristics
of the amplifier.
4.3. MEASURED SINGLE LASER FANO FACTOR
113
Fig. 4-8 shows the measured power spectrum of a sample scan taken at a laser bias
current of 81 mA. The PSD function in MATLAB was used to obtain an estimate of the
4x1-11
10
3. 5
C')
30
a)
5
0.
a)
2
a)
5
0:
01. 1
5
0
0.5
1
1.5
2
Frequency of Measurement (MHz)
Figure 4-8: Power spectrum of the measured voltage noise, taken at Ibias = 81 mA.
power spectrum. The MATLAB algorithm relies upon the Welch's averaged periodogram
method, which windows the data and performs ensemble averaging. Care was taken to stay
well within acceptable bounds of the method's accuracy. The spikes in the power spectral
density near 1 MHz and below 100 kHz were caused by spurious coupling to the computer
needed for the data acquisition.
Using Eqn. 4.24, the power spectrum of the photodetector current noise was calculated.
Fig. 4-9 shows the results.
To obtain the Fano factor of the current noise Fr, the power spectra at various bias
points were averaged over a 100 kHz bandwidth centered at 550 kHz. The results for the
photodetector current Fano factor and the light Fano factor are plotted in Fig. 4-10.
The results of the measurement indicated that the noise of the laser was a strong function
of the bias current.
In particular, at bias currents near 40 mA, the noise in the laser is
114
CHAPTER 4. CIRCUIT-INDUCED LASER NOISE CORRELATIONS
x 1019
1.8
co)
C:
1.6
1.4
a)
0
Cl)
0
1.2
1
0.8
0.6
C:) 0.4
'U
0.2
0
0.5
1
1.5
2
Frequency of Measurement (MHz)
Figure 4-9: Power spectrum of the photodetector current noise, taken at Ibias = 81 mA.
especially pronounced. This excess noise is attributed to the unstable mode character of
the laser within that bias range. Measurements with an optical spectrum analyzer (OSA)
allowed the mode-hopping of the laser to be seen very clearly.
At higher biases (> 60
mA), the OSA indicated stable lasing without mode-hopping, and a corresponding drop in
the light's Fano factor was measured. At 90 mA of bias current, where the manufacturers
suggest operating the laser for optimal performance, the laser's Fano factor was measured
to be about 10. Based on the results of these measurements, a laser bias current of 90 mA
was chosen for the correlation measurements.
4.4
Correlation Measurement
The setup used for measuring the correlated photon noise of the lasers was shown in Fig. 4-4.
The high frequency equivalent circuit of the measurement is shown in Fig. 4.4. By putting
the lasers in parallel or in series and by closing or opening the switch, the laser circuit can
be made to match all of the four topologies shown in Fig. 4-3.
4.4. CORRELATION MEASUREMENT
115
104
-.
F,
...
FF,
103
0
C15
L 102
0
C
IL
101
9 3
Ith=1 . mA
100'
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Laser Bias (mA)
Figure 4-10: Photodetector current and incident light Fano factors as a function of laser
bias.
The correlation metric measured in this experiment is given by
-
R (CSD(si, 82))
YPSD(si) x PSD(s 2 )
where CSD(si, S2) is the cross spectral density of the two signals s, and S2. PSD(si) and
PSD(s 2 ) are the power spectral densities of s, and s2. A completely correlated signal results
in Cn=l, and a completely anticorrelated signal results in Cn=-1. Becasue of this metric,
the correlation measurement is much simpler than the measurement of the Fano factor;
knowledge of the transfer function H is not necessary, because any dependence on H in
R (CSD(si, S2)) is canceled out by the same dependence in VPSD(si) and VPSD(s 2 )Therefore the signals si and S2 which are used for the measurements are simply the noise
voltages measured in the DAQ.
116
CHAPTER 4. CIRCUIT-INDUCED LASER NOISE CORRELATIONS
series
config.
Ai-
Di
LI
+"
Zf
config.
L2D2
....
AV
AV
A
B
As discussed in Section 4.2.1, the noise contributions from the amplifier noise of the circuit
are very small. As small as they are, the are also uncorrelated, making them totally unimportant in setting the accuracy limits of the measurement. The two main sources of error in
the correlation measurement are spurious correlations introduced into the circuit through
capacitive or inductive coupling between the measurement circuits (spurious mutual correlation), and spurious correlations from some other environmental noise source which is
picked up by both
circuits (spurious environmental correlation). A great deal of care was
taken to properly shield the measurement, eliminate ground loops, and reduce the mag-
4.4. CORRELATION MEASUREMENT
117
netic flux linkage between the measurement circuits. Despite this care some preliminary
measurements were necessary to be sure that the measured correlation was that of the laser
noise, and not from some other spurious source.
First, the noise was measured when both lasers were voltage biased with separate batteries. Any correlation which appears as the result of such a measurement is spurious, and
is a potential source of error.
The results of this measurement are shown in Fig. 4.4.1.
The correlation is roughly zero until it starts increasing at around 1 MHz.
The flatness
0.5
0
-1
0
0.5
1
1.5
Frequency (MIHz)
2
Figure 4-12: Spurious correlation between two lasers voltage biased in separate circuits.
of the noise trace below 1 MHz lends support suggests that spurious correlations in the
measurement bandwidth are small. The bandwidth over which the measurements are to be
made is between 450 kHz and 650 kHz. Averaging the correlation trace of Fig. 4.4.1 over
that bandwidth results in a correlation of 0.03.
As an even more rigorous test for spurious correlation between the detector circuits, noise
measurements were made with one laser at normal (90 mA) bias and the other turned off.
This was repeated for both lasers and with both current and voltage bias. This measurement
is far more sensitive to spurious mutual correlations in the photodector circuits than the
a measurement with both sources on. The noise floor of the photodetection circuit was
shown earlier to be much smaller than the typical noise due to the laser, so that even the
CHAPTER 4. CIRCUIT-INDUCED LASER NOISE CORRELATIONS
118
smallest amounts of spurious signal picked up in the dark (laser off) photodetector circuit
can easily be the dominant signals in that circuit. Because the correlation metric used here
is normalized, the weakness of the spurious signal is not important, only the degree of its
correlation with the strong signal. Hence a small spurious signal which would be unnoticable
in the noise correlation with both lasers on can cause huge correlations when one laser is off.
This measurement does not rule out the possibility of mutual spurious noise between the
laser bias circuits, since one of the lasers was turned off. The results of these measurements
are shown in Fig. 4.4.1. Signs of mutual spurious correlation are evident in these traces.
I
0.51-
0.5
0
0
U
U-0.5 .4.
0
0.5
1
1.5
0.5
-1
D
2
re
1
Fre que ncy (M Hz)
Frequency (MHz)
1.5
2
I
+
0.5
0.5F
0.
0
U
-0.51
0.5
0
0.5
1
Frequency (MHz)
1.5
2
~0
0.5
1
1.5
Frequency (MHz)
Figure 4-13: Spurious correlations measured with one laser off.
Still, within the measurement bandwidth the correlation is remarkably small. The average
correlation of each of the measurements is given in Table 4.4 The small correlations measured
indicate that in the actual measurements, the correlations due to the spurious pickup of
noise between detector circuits is negligable. Iterating the measurements in Table 4.4 proved
4.4. CORRELATION MEASUREMENT
scenario
laser
laser
laser
laser
1
1
2
2
119
[correlation
voltage
current
voltage
current
biased,
biased,
biased,
biased,
laser
laser
laser
laser
2
2
1
1
off
off
off
off
0.00
0.04
-0.01
0.01
Table 4.4: Average measured spurious correlation with one laser off.
to be a very useful and effective way to diagnose spurious mutual coupling. From these
measuerements it is conluded that the correlation measured in Figure 4.4.1 is most likely
environmental in origin, or related to spurious coupling between the laser circuits.
4.4.2
Correlation Measurement
Four measurements of the correlated noise were taken, each one corresponding to one of
the circuits discussed in Section 4.1.3. The results are shown in Figure 4.4.2. The average
correlations over the band between 450 kHz and 650 kHz are given in Table 4.5.
circuit (A)
circuit (B)
circuit (C)
circuit (D)
0.31 +/- 0.01
0.05 +/- 0.01
-0.07 +/- 0.01
-0.31 +/- 0.01
Table 4.5: Average noise correlations.
The error bars indicated in the table reflect only the estimated random error in the
averaging process, and do not include any systematic errors due to spurious correlations.
A large positive correlation (0.31) is observed for the voltage biased, series coupled lasers.
A negative correlation of the same magnitude was measured for the current biased, parallel
coupled lasers, agreeing with the theory presented in 4.1.3. Smaller correlations are observed
with the series coupled, current biased circuit and the parallel coupled, voltage biased
circuit. The exact origin of these correlations is not understood. One theory is that these
correlations are spurious correlations due to inductive coupling between the lasers and some
other source of fluctuating magnetic field. Fig. 4.4.2 shows how inductive coupling might
give rise to the measured correlations for both laser circuits. In circuit (B), the inductive
coupling induces a voltage in the circuit driving a noise current Iimd through both of the
CHAPTER 4. CIRCUIT-INDUCED LASER NOISE CORRELATIONS
120
I
(A)
0.5
0
0.5
-zZ*
=(0).
-0.5
1-1 -
I
0.5
0
1
1.5
2
-1
Frequency (MHz)
1
1.5
Frequency (MHz)
2
(D).
+
0.5
0 -
0
0.5
0
2
I
(C)
-I 0
1.5
1
0.5
Frequency (MHz)
ir
0.5
0
-0.5
0.5
0.5
1
1
1.5
1.5
2
Frequency (MHz)
0
0.5
Figure 4-14: Noise correlation measured for lasers in 4 different circuits.
lasers in the same direction, resulting in a positive spurious correlation, consistent with
what was measured. In circuit (C), the same positive correlation could be induced by a
Ifnd.
However, a noise current I'
will also be induced in the circuit. This current flows in
the opposite direction through the lasers, and would act to negatively correlate the noises.
Becuase of the relatively large loop
(~
3 cm) formed by the leads to each of the lasers,
this latter loop is larger than the former, and therefore this negative correlation is expected
to dominate over the positive correlation from IAd, resulting in a net negative correlation.
This also matches with the measured data. Further tests must be made before this model
for the correlations in (B) and (C) is accepted, as the existence of magnetic fields capable
of inducing a coupling over a wide bandwidth would be surprising.
4.5. SUMMARY
121
(B)
(C)
ind
ind
"id
Figure 4-15: Possible mechanism for spurious correlation in circuits (B) and (C).
4.5
Summary
A model for the correlations in circuit coupled lasers has been presented and tested. Correlations in the light are theoretically expected due to the shared biasing circuitry of the
lasers. Noise from the laser and which flows through the external biasing circuitry can couple into the other laser, creating correlations. The circuit noise of the laser can be divided
into thermal noise due to the parasitic series resistance of the Ohmic contacts of the laser,
and more complicated noise originating from within the laser.
Four representative circuit topologies were examined theroretically and experimentally.
Circuit (A) biased the lasers in series using a voltage source bias.
Positive correlations
due to the thermal and laser noise from both lasers flowing through the external circuit
are expected.
Circuit (B) biased the lasers in series using a current source bias. Circuit
(C) biased the lasers in parallel using a voltage source. In both cases, no correlations were
theoretically expected, because the external circuits shunt the noise currents from each laser
away from the other laser. In circuit (D), the lasers are biased in parallel using a current
source. A negative correlation of the same magnitude as the positive correlation of circuit
(A) is predicted. Finally, assuming values for parameters consistent with those used in the
measurements, an expression for the photon correlation due to thermal noise is derived.
This is expected to be very small, and so any correlation observed by the measurement is
assumed to be from the intrinsic laser noise.
CHAPTER 4. CIRCUIT-INDUCED LASER NOISE CORRELATIONS
122
To test the predictions of the theory, two commercial SDL-5400 GaAlAs single transverse mode Fabry-Perot lasers were used, along with two Hamamatsu S3590-01 silicon PIN
photodetectors.
The photodetector signals were amplified using two custom made tran-
simpedance amplifiers, allowing a measurement bandwidth of approximately 1 MHz. To
measure the photon noise, each laser was directed at a detector, resulting in a 58% currentto-current quantum efficiency. Before measuring the photon correlations, a single laser was
current biased, and a normalized measurement of its photon noise was made. At bias currents of 90 mA (about 4.5 times the threshold current) the Fano factor of the photon noise
was meased to be approximately 10.
When operating in this bias regime, the laser was
free of any mode-hopping behavior which might wash out the correlations which were to be
measured.
Finally, the correlation meaurements were made for each of the 4 circuit topologies.
Several preliminary measurements are made to ensure that spurious correlations are not
present. Circuit (A) displayed a current correlation of 31%, and circuit (D) resulted in a
-31% correlation. The opposite signs and equal magnitudes of the correlations are in good
agreement with theory. The large degree of correlation indicate that the biasing circuit has a
very noticable effect on the noise of a laser. Also measured were the correlations in circuits
(B) and (C). While much smaller, the correlations observed in these measurements are
inconsitent with the theory. A possible mechanism for the spurious correlation, inductive
coupling, is discussed.
Chapter 5
Conclusions and Future Directions
5.1
Summary and Conclusions
This work in this thesis is directed towards three intertwined goals. The first, primarily
addressed in Chapter 2 is to present the models and the formalism needed to describe noise
in an arbitrary electronic circuit, as well as some simple low noise design principles. The
second goal is to provide a suite of techniques for measuring current noise in various devices.
These were presented in Chapter 3. The culmination of this work was the first measurement
of the circuit-induced correlations in the light of two lasers. The results of the measurement
are in agreement with a theoretical calculation of the correlation.
5.1.1
Modeling
To allow the experimenter to understand and predict the behavior of measurement circuits,
noise models for the most common circuit elements are presented.
These are resistors,
capacitors, inductors, diodes, BJTs, FETs, and transformers.
The models given are specific instances of one port and two port devices. Some useful results regarding the modeling of noise within idealized one and two port systems are
presented. First, with a simple thermodynamical argument it is shown that the thermal
noise from an arbitrary one port network is easily related to the real part of the one port's
123
CHAPTER 5. CONCLUSIONS AND FUTURE DIRECTIONS
124
Thevenin equivalent resistance. The two-port model from standard circuit theory is generalized to include input and output noise sources to model arbitrary noise within the two-port.
In this general case, the noise of the input and output generators can be both non-white
and correlated. To allow for easy comparison of the signal level at the input to the noise
contributed by the two port, the noise source at the output of the two port is referred back
to the input. This is consistant with the convention in commercial electronics specifications.
Contrary to what is often said in the literature, as long as certain basic conditions are met,
the input referred noise sources have all of the power and generality of the original noise
description in terms of noise sources at both ports. The sensitivity limits for the so-called
balanced detection scheme are presented in light of these results.
The concept of the optimal noise resistance is introduced. If an amplifier whose input
referred current and voltage noise generators are known is used to measure a small signal
from a device, there is a certain device impedance which minimizes the noise figure of the
setup. For a device whose impedance is resistive, this ratio is simply V"/I,
the ratio of
the input referred noise voltage generator to the input referred current noise generator. For
more general device impedances and noise behavior, the method of minimizing the noise
figure will lead to a different optimization condition. This method gives the designer of
low noise measurements a design constraint on an amplifier used to measure signals from a
particular device.
In short, this thesis has endeavored to present a practical, unified framework for modeling
noise in electronic circuits, along with some tips on low noise circuit design. The ideas and
methods of this chapter do not pertain only to measurements of current noise, but more
generally to any sensitive electronic measurement.
5.1.2
Current Noise Measurement Techniques
A set of current noise measurement techinques were described. The basic equipment used
for the preliminary measurements in this thesis was described and characterized.
These
included the SR560 voltage amplfier, the SR570 current preamplifier, the SR554 transformer
5.1.
SUMMARY AND CONCLUSIONS
125
preamplifier, and the data acquisition system. The two-port input referred noise formalism
developed in Chapter 2 is used to describe the noise contributions from the measurment
chain.
A method for characterizing the noise of an arbitrary amplifier was described and applied
to the SR570 amplifier.
By measuring the noise at the output for a variety of source
impedances, the results could be fit with very good agreement to a two-port input referred
noise model. Results agreed with a similar calibration in the literature. Using a series of
cryogenic measurements, the linear temperature dependence of thermal noise power from a
4.75 kQ resistor was confirmed.
A noise measurement technique for high impedance devices was developed. The measurement setup was used to perform a calibrated measurement of the shot noise of a photodetector's photocurrent. The photocurrent was produced from a weakly coupled optical
link with an LED. The principle of the calibration is to include a known signal source whose
input-to-output transfer function is identical to that of the noise source to be measured.
Knowledge of this transfer function can be used to deconvolve the measured power spectral
density of the noise at the output. By turning the LED off, a the noise due only to the
amplifier can be measured. This is subtracted from the measured spectral density of the
noise with the LED on to eliminate the noise from the amplifier. Using this method, the
standard quantum limit was measured to better than 2% accuracy.
A transformer coupled method of noise measurement was developed to handle devices
whose low impedances are poorly matched to the available optimal noise resistances of
FETs and BJTs. The current noise in a diode was measured at low frequencies and found
to follow a 1/f type spectrum.
In describing specific current noise measurement techniques, this thesis attempts to
highlight the main difficulties of noise measurements, and to suggest techniques by which
these difficulties can be overcome.
CHAPTER 5. CONCLUSIONS AND FUTURE DIRECTIONS
126
5.1.3
Photon Correlations in Circuit-Coupled Lasers
A model for the correlations in circuit coupled lasers has been presented and tested. Correlations in the photons are expected due to the shared biasing circuitry of the lasers; current
noise from one laser can flow through the external biasing circuitry and couple into the
other laser. The circuit noise of the laser can be divided into thermal noise of a parasitic
series resistance, and more complicated current noise source originating from within the
laser.
Two sets of measurements were made using SDL-5400 lasers. First, a calibrated measurement of the Fano factor of the laser at frequencies of less that 1 MHz was made for
several bias currents between threshold and 4.5 times threshold. A region of significantly
enhanced noise was measured near 2 times the laser's threshold. Measurements with an
optical spectrum analyzer confirmed that the enhanced noise was due to mode hopping
within the laser. A quieter region at 4.5 times the threshold current of the laser was chosen
for the correlation measurements.
The circuit induced current correlations in four simple circuit topologies have been measured (see Fig. 5-1), and the dominant source of error is attributed to spurious correlations
due to capacitive and inductive coupling of noise into the measurement circuit. The measured correlations agreed with the theoretically predicted correlations, except for the small
but non-zero correlations measured in the series coupled current biased lasers and the parallel coupled voltage biased lasers. A possible cause of this spurious correlation involving
inductive coupling is proposed, and is shown to be consistent with the measured data.
5.2
Directions for Future Work
There are several possible directions for future work building on the results and methods of
this thesis.
The most immediate direction is to continue with the photon correlation measurements.
The most striking discrepancy between theory and experiment was in the non-zero cor-
5.2. DIRECTIONS FOR FUTURE WORK
127
*II
Figure 5-1: The four measured bias circuit topologies.
relations measured from the series coupled current biased lasers and the parallel coupled
voltage bias lasers. Correlations on the order of +/- 5% were measured, a significant deviation from the expected uncorrelated noise. In Chapter 4, a mechanism for these correlations
in terms of spurious inductive coupling was proposed, but time and circumstance did not
allow this mechanism to be verified. In subsequent measurements, more attention should
be paid to reducing the area of current loops which may inductively couple noise into the
measurement, introducing spurious correlations. More care should probably also be taken
to physically separate the circuit biasing the laser from the measurement circuit. Despite
significant efforts which were already taken in these areas, it is possible that the importance
of inductive coupling was underestimated in the measurements of Chapter 4.
The correlation measurements have provided clear confirmation for the general predictions of the laser noise circuit model. However, further effort should be made to separate
out the contributions to the correlation from the thermal current noise source and the laser
current noise source in the model (see Fig. 5-2).
CHAPTER 5. CONCLUSIONS AND FUTURE DIRECTIONS
128
R series
rd
n
1/f
Figure 5-2: One-port noise model of a diode.
It was argued in Section 4.1.3 that for the laser used in this thesis, the correlation due
to the thermal noise of the lasers' parasitic resistances was too small to be observed. A chief
reason for this is the presence of noise in the photons which is not correlated with either
of the two current noise sources. While this excess noise is not unexpected, in a practical
measurement this excess noise tends to wash out the correlations due to the circuit noise.
A possible solution to this problem is cooling the lasers in a cryostat.
This lowers the
threshold current of the laser, and is expected to also lower some of the intrinsic noise
of the laser.
By cooling the laser to 15 K, the lasers can be operated at more than 30
times their threshold currents [54]. Operating the laser at high biases is known to further
decrease the intrinsic noise of the lasers
[541.
The likely mechanism for this is the further
suppresion of longitudinal side modes whose presence is responsible for much of the excess
noise in the laser [67] [56] [571. If the intrinsic noise of the laser is lowered, the portion of
the lasers' photon correlation due to thermal noise will be more easily seen. Of course the
thermal noise from the resistor will also fall at low temperatures, but it is hoped that the
advantages of lower laser noise will outweigh the loss of signal due to the reduced thermal
5.2. DIRECTIONS FOR FUTURE WORK
129
noise. An added benefit from cooling the setup can be realized if the photodetectors are
included in the cryostat. The photodetectors often limit the maximum light power which
can be measured due to saturation effects which drive down the bandwidth of the detectors.
Limiting the light power in the context of these correlation measurements means limiting
maximum bias of the laser; this in turn limits the noise reduction which can be attained by
biasing the laser well above threshold. If the detectors are cooled, significant improvements
in this bandwidth are expected. At 77 K, a five-fold improvement in bandwidth over room
temperature performance was reported using detectors similar to the ones used in this thesis
[691.
If a temperature regime and a bias current are found which reduce the laser's intrinsic
noise to an acceptable level, then a potentiometer can be used in series with the circuit
coupled lasers to test the circuit model of Figure 5-2. By taking correlation measurements
at diffrent values of the potentiometer, the functional dependence of the photon correlation
may be extracted, a further test of the laser circuit noise model.
In the discussion, so far, the focus has been on the relative contributions of the two circuit
noise sources to the photon correlation. The relative contribution of the noise sources can
also be probed in a different way, by varying the bias current. Because the thermal noise of
the parasitic resistance of the laser is independent of bias current, examining the correlation
as a function of bias is a probe of the lasers' current noise sources. The correlation of the
lasers should be measured as a function of bias to investigate how the current noise source
of the laser varies with bias. Comparing this measurement with theory requires a much
deeper investigation of the microscopic noise processes at work within the laser than was
given in this thesis [66].
A final direction for future study in this area is the direct measurement of the correlation
between the laser's current noise and the noise in its light. This measurement was actually
undertaken over the course of this thesis on two occasions, but without much success. The
difficulty lay with the measurement of the current noise of the laser, given the low impedance
of the laser at forward bias conditions ~ 5 Q. During the first attempt at measuring the
CHAPTER 5. CONCLUSIONS AND FUTURE DIRECTIONS
130
laser noise, the transformer coupled setup of Chapter 3 was used to help match the low
impedance of the laser circuit to the large optimal noise resistance of the SR 560 low noise
voltage amplifier. The measurement was successful, but unfortunately the bandwidth was
limited to under about 20 kHz, which turned out to be well before the 1/f noise corner
of the device.
Any of the laser noise of interest was swamped by the 1/f noise signal.
Another measurement was attempted with the SDL-5400 lasers using a transformer with
a smaller turn ratio in the hopes that they would display less 1/f noise, but the results
were inconclusive. A drawback of applying transformer coupled measurements to lasers is
the possiblity of inadvertant inductive current spikes which can destroy the lasers. A safer
method to obtain comparable levels of sensitivity to the transformer coupled measurement
might be to include a cooled JFET preamplifier in the cryostat along with the laser and the
detector. The noise in a JFET is almost entirely due to the thermal noise of the channel
(see Chapter 2), and can be therefore be decreased by going to lower temperatures. An
added advantage of a cooled JFET amplifier over the transformer coupled measurement
would be a much wider bandwidth, and a diminished sensitivity to parasitic elements in
the measurement circuit.
Appendix A
Current Noise in a Resonant
Tunneling Diode
The purpose of this appendix is to provide a detailed theoretical description of the fundamental origin of noise in the representative system of the resonant tunneling diode (RTD).
Aside from the theoretical benefits of understanding the fundamental processes responsible
for current noise, it is hoped that the reader will get a sense for some of the exciting physics
which is accessable to the experiementalist through noise measurements. Some knowledge
of elementary quantum mechanics is helpful to follow the discussion. It draws frequently
on insights from the excellent text by Datta [1] and a very thorough review of shot noise in
mesoscopic conductors by Blanter and Biittiker
[2].
A practical realization of the RTD structure using a GaAs quantum well and AlGaAs
tunnel barriers is shown in Fig. A-1.
[3].
Also sketched in Fig. A-1 is the energy of the
bottom of the conduction band. The valence band is not needed for a first-pass discussion
of transport because the RTD is a majority carrier device, due to the degenerate doping
in the emitter and collector. During normal operation, the Fermi-levels of the electrons on
either side of the device are far below the height of the barriers in the conduction band
(Vo). The electrons traverse the barriers by sequentially tunneling through each barrier, an
entirely quantum mechanical process.
131
APPENDIX A. CURRENT NOISE IN A RESONANT TUNNELING DIODE
132
V0
AIGaAs -- +E
barriers --
GaAs
GaAs well
GaAs
............
Z
substrate
4
Figure A-1: A practical realization of an RTD, and a schematic representation of the
structure [3].
A.0.1
Tunnel Junctions at DC [1]
The RTD is composed of two tunnel barriers as shown in Fig. A-1.
For a single barrier,
considered alone, Schr6dinger's equation is easily solved to calculate the transmission probability T(E) of an electron incident on the barrier. The key result of the calculation is that
there is no resonant behavior in the tunneling; the transmission probability monotonically
increases as the energy of the electron is increased towards the top of the potential barrier.
Even if the device is biased, the picture is qualitatively the same.
There is no resonant
behavior because the solution to Schr6dinger's equation in the barrier has no sinusoidal
character, and this is the case as long as the energy of the tunneling electron is less than
the minimum of the barrier potential.
Resonant behavior can be obtained by adding another identical barrier in series with the
first. If the tunneling probabilities for transmission through the barriers are small, as they
typically are for a real device, then the well region of the RTD can be said to possess quasibound states. The non-zero tunneling probability out of the well introduces a broadening in
the density of well energy states (and therefore a finite lifetime). This broadened density of
133
Vbias=O
E
well density
of states
EFr
EF(z
E
Vbias>O
well density
of states
EF
__ Eb
EFr
Figure A-2: A schematic energy level diagram for an RTD in the unbiased and biased case,
showing the well density of states.
states acts like a window for electron transport through the structure. Applying a voltage
bias to the structure raises or lowers this window relative to the reservoir of electrons on
either side of the RTD. This is sketched in Fig. A-2. When no bias is applied to the structure,
the RTD acts as a barrier, allowing no electrons to pass. As a bias is gradually added, the
Fermi-level of the emitter (EF on the left-hand side) eventually reaches the energy level of
the bound state (Eb). A channel for conduction is opened, and current begins to flow. The
current through the RTD continues to increase as the energy of the quasi-bound state is
swept across the Fermi sea of the emitter charge reservoir, and then decreases for a while
until other channels for conduction are opened (not shown in Fig. A-2).
The resulting
I-V curve for the RTD, with its characteristic regime of negative differential resistance, is
sketched in Fig. A-3.
In this discussion, several important approximations were made. A low enough temperature was assumed so that the spread of the Fermi level of the emitter was less than
APPENDIX A. CURRENT NOISE IN A RESONANT TUNNELING DIODE
134
region of negative
differential resistance
V
Figure A-3: A typical I-V curve of an RTD.
the quasi-bound state width. Increasing the temperature has a smearing effect on the I-V
curve. Likewise, scattering processes ensure that the current does not decrease all of the
way to zero in the valley of the I-V curve. Inelastic scattering current contributes to the
valley current because electrons in the emitter above the well quasi-bound state energy can
give up energy to the lattice and tunnel through the Eb bound state, or receive energy and
tunnel through one of the higher well bound states or over the barrier. Elastic scattering (electron-electron or interface) also effects transport. While elastic scattering does not
change the total energy of the electrons, it does redistribute the electrons' momentum in
k-space. Because only k, matters for tunneling through the well, this redistribution affects
transport.
In sketching the potential profile for forward bias, for simplicity it was assumed that the
emitter and collector are heavily doped, causing the potential to drop entirely across the well
and barriers. In reality, the conduction band in the emitter bends down next to the barrier,
which makes a small triangular quantum well. Also, it was assumed that there is no space
135
charge buildup in the device. This makes the potential drop occur evenly. A more careful
calculation would self-consistantly solve the Schr5dinger and Poisson equations, taking into
account the effects of the static charge distribution on the potential profile.
the simple assumptions made here capture the necessary features of the device.
extensive research and a variety of approaches
[70],
However,
Despite
modeling RTDs is still an active area
of research.
A.0.2
Mesoscopic Noise [2]
In a mesoscopic device (one in which the device dimensions are comparable or smaller than
the mean free path of the electron), current fluctuations through two fundamental sources:
thermal fluctuations and shot noise. At non-zero temperatures, all systems coupled to a
reservoir of states show thermal fluctuations. The size of these fluctuations can be related
to the dissipation in the system through the fluctuation-dissipation theorem [10].
This
makes some intuitive sense, as fluctuation and dissipation are in many respects two sides of
the same coin; the dissipation is a measure of the power flowing from the system into the
reservoir, and the size of the fluctuations are a measure of power flowing from the reservoir
to the system. Shot noise is best defined as noise resulting from the quantization of charge.
Unlike thermal noise, shot noise is not present in equilibrium. It is present only when there
is DC transport.
On a mesoscopic scale, the difference between shot noise and thermal
noise is a matter of definition; when calculating the noise in a structure, it is often possible
to obtain both types of noise as two different limits of one general expression [71].
A useful model for understanding the origins of thermal and shot noise is the onedimensional single energy channel. The channel consists of a single barrier transmitting an
incident electron of energy E with a probability T. For now, transport is examined only in
a range dE around a particular energy E0 . It is assumed that there are two identical reservoirs of electrons on either side of the conducting channel, and T is taken to be 1, meaning
that there is perfect transmission of incident electrons with energy E 0 . The energies of
the electrons in each reservoir are distributed according to identical Fermi-Dirac distribu-
136
APPENDIX A. CURRENT NOISE IN A RESONANT TUNNELING DIODE
1-D Channel
Emitter
reservoir
Collector
reservoir
Occupied (dark) or
unoccupied (light)
transmission states
Figure A-4: A 1-D channel connecting two reservoirs, showing orthogonal occupied and
unoccupied transmission states.
tions f(E), each at the same non-zero temperature, with the same Fermi-levels. Because
the channel only transmits at energy Eo, only electrons in the reservoir with energy E0
contribute to transport. The probability that the stream of transport states of energy E0
incident on the barrier from the left-most reservoir will be occupied by electrons is given by
the Fermi-Dirac distribution evaluated at E0 , f(Eo). The situation on the right side of the
barrier is identical, as the system is symmetric. This scenario is sketched in Fig. A-4. It is
a fundamental postulate of statistical mechanicals that a system is equally likely to be in
any of its accessable microstates. Therefore, in the stream of particles incident on one side
of the barrier, the occupation of a given state is statistically independent of the occupation
of every other state. This allows an easy calculation of the variance in occupation number
of the states incident on one side of the barrier:
(An 2 )
(n2)
-
(n)2
(A.1)
For Fermi-Dirac particles like electrons, the average occupation number (n) and the mean-
137
square occupation (n 2 ) are both simply f(Eo) (Poisson statistics), giving:
(An 2) =
f(Eo)(1
-
(A.2)
f(E0 ))
Already, it is clear that there will be fluctuations in the current between the two sides of this
device due to the fluctuation in occupation in the reservoir. This is the defining characteristic of thermal noise. To calculate the current fluctuations due to electrons incident on one
side of the channel, the number fluctuations above must be weighted by a proportionality
factor 4'dE Av [72].
Taking into account both of the statistically independent incident
electron streams gives an additional factor of 2. One finds:
6(AI 2 ) =
2e 2
h f(Eo)(1 - f(Eo))dE Av
(A.3)
The factor of f(Eo)(1 - f(E 0 )) is sharply peaked at the Fermi-level, indicating that only
electrons near the Fermi-level are important in conduction.
Equation A.3 can be easily
integrated over energy to find the total current fluctuations due to all of the reservoir
electrons. If the result is expressed in terms of the quantum of conductance G =
one
finds [73]:
(AI 2) = 4GkTAv
(A.4)
This expression for thermal noise also holds for macroscopic systems (eg. resistors) where
it is traditionally referred to as the Johnson or Nyquist formula.
To understand the origin of shot noise in the single energy channel model, consider low
temperatures so that thermal noise is not important. If a voltage bias Vapplied is applied
to this device, current will flow preferentially from the emitter to the collector due to the
difference in the Fermi-levels of the emitter and collector reservoirs. Only the carriers in the
emitter reservoir which are above the Fermi-level of the collector reservoir can participate
in transport (as sketched in Fig. A-5), since all of the states below the collector Fermilevel are filled in the low temperature limit. Restricting the discussion to emitter electron
energies capable of transport, the probability that an electron from the emitter reservoir
138
APPENDIX A.
CURRENT NOISE IN A RESONANT TUNNELING DIODE
1-D Channel
incident
reflected
Emitter
Collector
transmitted
E
E
EFg
0
00*
a0
at
Vapplied
Figure A-5: A 1-D channel connecting two reservoirs at low temperature and with an
applied voltage bias.
will be transmitted to the collector reservoir is simply T, and therefore the probability
of occupation of the transmitted current stream will also be T. Because the transmission
or reflection of an electron at the barrier is a quantum mechanically random process, each
transmission or reflection is independent of the others, and the stream of particles exiting the
device on the right exhibits Poisson statistics. The mean square occupation is therefore T.
The standard deviation of the occupation number fluctuations of the transmitted electron
stream is then
(An 2 )
=
(
_' (nT)2 = T(1 - T)
(A.5)
These fluctuations are known as partition noise because the scatterer partitions the ordered
stream of current into two disordered streams. This random scattering of certain electrons
but not others is the fundamental origin of shot noise. To convert from occupation number
fluctuations to current fluctuations, the factor of 2V dE Av from the previous discussion of
thermal noise is used again. In this case, dE = -fE(emitter) - E(collector)
f
=
eVpplied is the
width in energy of the conducting channel.
2e 2
6(A1 2 ) = 2 h T(1 - T)eVapplied Av
(A.6)
139
In the limiting case of small T and expressing the result in terms of the conductance of the
channel Gchan = T
- , this becomes:
6(AI
2
) = 2GchaneVapplied Av
(A.7)
Recognizing that Ichan = GchanVapplicd, the traditional shot noise formula is obtained:
6(AI 2 )
= 2 elchan AV
(A.8)
[74].
For larger values of T, it is clear
This is known as the Schottky limit for shot noise
from the discussion that the noise will be suppressed under the Schottky limit. The ratio
of the current fluctuations in a system to the Schottky limit is called the Fano factor.
A.0.3
Noise in RTDs
In the previous sections the fundamental microscopic origins of noise were discussed using
a coherent 1-D channel model. As primitive as the model was, it provides some insight into
the noise properties of the RTD.
In Section A.0.1 it was argued that there is a maximum in the DC current when the
quasi-bound state energy is centered within the energy reservoir of the emitter (Fig. A2).
The transmission T(E) through the device in this situation is simply proportional
to the density of states of the smeared out quasi-bound well state.
This can be simply
motivated with no mathematics by noting that the tunneling current into or out of the well
is proportional to three factors: the density of occupied states the electron leaves, the density
of unoccupied states the electron can enter, and some tranmission coefficient T(E). In the
case sketched in Fig. A-2, the emitter states are filled up to the Fermi level. The states which
the electron is tunneling into (on the right of Fig. A-2) are all empty. The transmission
T(E) due to just a single barrier is not strongly dependent on energy, and can be taken
as constant T for the energy range over which transport is possible. The only important
energy dependence in the tunneling current is that of the quasi-bound state which the
140
APPENDIX A. CURRENT NOISE IN A RESONANT TUNNELING DIODE
electron must pass through to cross the device. As is often the case with lifetime broadened
discrete states, the density of states g(E) (and therefore T(E)) is well approximated by a
Lorentzian distribution in energy. Knowing the form of T(E) for the RTD and applying
the result of Section A.0.2, the total shot noise can be calculated analytically by dividing
T(E) into narrow channels in energy and integrating the noise from each channel. It is
worth noting that T(E) can be quite large on resonance; for identical barriers T(Eb) = 1.
From the discussion in Section A.0.2, this large transmission will give rise to current noise
suppression in the device. In other words, the Fano factor should be less than one when
device is biased on resonance. In the ideal situation considered here, it can be shown
[1]
that the Fano factor is exactly 1/2. This result agrees well with the suppression of shot
noise observed in real devices
[75].
However, this agreement should be viewed with some
suspicion; in real devices at room temperature, electron-phonon scattering often destroys
the coherence of the electrons in the well before they have a chance to tunnel out. This
casts the entire procedure used above into doubt, as the 1-D channel model was completely
coherent (i.e. no scattering). Further, it is not even clear that the RTD's I-V curve will still
possess the same transmission maxima on resonance, since the discussion of the I-V curve
and T(E) was also based on a totally coherent model. Both of these concerns have been
addressed by developing a sequential tunneling model for the RTD, which treats transport
and noise in the device using a master equation. Is has been demonstrated that dephasing
does not alter the DC transport [1] or Fano factor [761 of the RTD significantly.
A measurement of the Fano factor in the RTD is sensitive to much of the important
physics of the device.
The shape of the quasi-bound states, the tunneling probability
through the double barrier structure, and the asymmetry of the device affect the noise
power measured through the Fano factor. Many of these properties are also reflected in the
DC transfer function of the device. However, so far only single electron transport has been
considered. It was assumed throughout the discussion that the electrons interact only with a
static potential, and not with each other. When effects on transport due to electron-electron
interaction are introduced, the DC I-V characteristic may not be significantly altered, but
141
the current noise may show dramatic changes. This can be seen by considering Fig. A-2
and the effect of a single electron's entrance into the RTD well.
After one electron tunnels into the well of the device, an electron immediately following
it sees a different potential in the quantum well due to the electrostatic repulsion of the
first charge, which is still in the well. This can be modeled as an increase in the potential
energy floor of the well, or equivalently as a capacitance between the well and the emitter.
If the change in potential is sufficient to influence the alignment of the quasi-bound state
with the emitter reservoir of states, then the transport of the two electrons can become
correlated. For example, suppose the device is biased so that the well quasi-bound state is
below the emitter Fermi level (in the negative differential resistance regime).
The added
well potential from the new electron pushes the well quasi-bound state up relative to the
emitter Fermi-level, and the next electron tunneling into the well from the emitter will have
an easier time. This introduces a positive correlation between the tunneling of electrons
into the well. The resulting 'bunching' in the tunneling electrons gives rise to shot noise
enhanced well above the standard quantum limit of 2eI. A Fano factor of 6.6 has been
attributed to this effect [18]. This mechanism can also give rise to shot noise suppression if
the device is biased so that the well quasi-bound state is just above the emitter Fermi level.
Whether this effect or the shot noise suppression mechanism discussed earlier (using the
1-D coherent channel model) is more important in determining the supressed Fano factor
depends on the details of the device.
142
APPENDIX A. CURRENT NOISE IN A RESONANT TUNNELING DIODE
Appendix B
Basics of Semiconductor Lasers
B.1
Semiconductor Laser Structure
A idealized semiconductor laser of the type used for the measurements in this thesis is
shown on the left side of Fig. B-1. The two terminals of the device are the metal layers on
electrons
photons
mow
S
substrate
Metal
S
E
holes
Figure B-1: Basic structure and operation of a laser.
the top and the bottom of the structure. Below the top metal contact is a p-doped region.
Above the botton metal contact is an n-doped region. So far this is similar to a normal
homojunction or heterojunction diodes. In between the two quantum wells is an undoped
region of a lesser-bandgap material known as the cladding. This region is lattice matched to
143
APPENDIX B. BASICS OF SEMICONDUCTOR LASERS
144
the n-doped and p-doped regions, meaning that the lattice describing the crystal structure
of the materials is ideally continuous throughout the device. Not shown in the figure is the
length and geometry of the laser in the direction coming out of the page. Lasers of the sort
used in this thesis are typically a few hundred microns long, and they are terminated on
each side by smooth reflective facets which serve as mirrors for the light generated within
the structure.
The smaller band-gap of the of the cladding material serves two purposes. First, because
the index of refraction of a material is inveresely proportional to its bandgap, light emitted
from inside the device is confined inside of the high index cladding. In the laser diagram of
Fig. B-1, the light is guided in the direction emerging from the page.
The other key benefit of the small band-gap of the cladding is to confine the electrons
and holes which enter the device from the n-doped and p-doped regions. This increases
the probability of electron-hole recombination, resulting in photon emission. This is one
big difference between an LED or laser structure as compared to a traditional diode or
transistor; for the laser and the LED, the device is engineered to maximized electron-hole
recombination in order to increase the radiative efficiency of the device, whereas electrical
devices are designed with as little electron-hole recombination as possible. Note also that
unlike the homojunction diode, the heterojunction diode is a majority carrier device, in the
sense that the electrons and holes responsible for the devices operation are injected from the
n-doped and p-doped sides, respectively. This process is shown in the schematic diagram
on the right side of Fig. B-1.
Also shown in the figure are multiple quantum wells (MQW). The quantum wells are
made from another material with an even smaller band-gap, again lattice matched to the
surrounding structure. The wells are small enough
(~
80
A) to offer quantum mechanical
confinement in the dimension of the well for the carriers in the well, which turns out to
enhance the performance of the laser. Carriers injected into the cladding from the n-doped
and p-doped regions relax into the lower energy of the quantum well before undergoing
recombination.
B.2. CARRIER RECOMBINATION AND LIGHT GENERATION
B.2
145
Carrier Recombination and Light Generation
A semiconductor laser is pumped applying a bias to the metal contacts, which injects holes
(from the p-doped side) and electrons (from the n-doped side) into the intrinsic cladding.
Once in the cladding, the carriers quickly relax into the quantum wells. There are several
processes by which carriers injected into the quantum wells recombine. They are divided
into radiative and non-radiative processes.
A good laser is designed to minimize non-
radiative recombination. The other important recombination occurs through the processes
of spontanous and stimulated emission, shown in Fig. B-2. During a spontaneous emission
E,
-
spontaneous
emission
absorption
stimulated
emission
Figure B-2: Important radiative processes in a semiconductor laser.
event, an electron recombines with a hole and emits a photon. The inverse of this process
is absoption, also shown in the figure; an incoming photon creates an electron hole pair and
is itself annihilated. This process is responsible for light emission in LEDs and in semiconductor lasers pumped with current below the threshold of lasing. The recombination rate
per unit volume in the laser scales with the density of carriers. During a stimulated emission event, an incoming photon induces another electron to recombine, producing another
identical photon. In this context, identical means that the excited photon is in the same
mode as the exciting electron. It is this process which allows for photon gain. The rate
of stimulated emission within the laser scales as the product of the electron and photon
densities, and therefore this process only becomes important at high photon densities. In
a properly engineered structure, at a certain bias current level this process becomes the
dominant form of recombination. At this point the device is said to begin lasing. To model
APPENDIX B. BASICS OF SEMICONDUCTOR LASERS
146
the processes at work within a laser more precisely, we rely on rate equations.
B.2.1
Laser Rate Equations
The physics of a semiconductor laser, including its noise, can be well modeled using rate
equations. To derive the rate equations the laser is divided the laser conceptually into two
reservoirs, one for the carriers injected into the active region of the laser and one for the
photons in the lasing mode of the laser. For multiple modes, more reservoirs can be used,
but here we focus on a laser which lases in a single mode. All of the different rates by which
particles can enter or leave each of the reservoirs can then be written down.
If the current intering the terminals of the laser is I, the amount of current entering the
active region of the laser is
T1I.
The injection efficiency
miis
a number between 0 and 1
modeling the internal efficiency of the laser, and it is designed to be as close to 1 as possible.
Once inside the carrier reservoir, the carriers can recombine nonradiatively, spontaneously,
or through stimulated emission, as discussed above.
A photon which is created through the radiative recombination processes has three
possible alternatives.
The photon might be absorbed in the active region and create an
electron-hole pair, it might be emitted through one of the end facets as useful light, or it
might be lost through free carrier absorption, absorption outside of the active region, or
through a scattering process.
Equations describing the time rate of change of the carrier and photon numbers subject
to all of the above processes can be written. The various rates are typically written in units
of
events
second x volume
dN
V -dt
Vp
=
=
-
dt
I
q
-
(R 8p(N) + Rnr (N) - (Rstim (N, Np) - Rabs(N, Np)) V
(Rstim(N, Np) - Rabs(Np))
-
+ R'P(N)V
+r
(B.1)
(B.2)
Here N is the carrier density and V is the active region volume, so that NV is the total
number of carriers in the carrier reservoir. Likewise Np is defined as the density of photons
B.3. SOLUTION OF THE RATE EQUATIONS FOR LOW FREQUENCIES
147
in the cavity, and V is the effective volume the photons occupy (related to the volume of the
cladding), so that their product NpVp is the number of photons in the cavity. The quantity
7
is simply the total number of carriers electrons
(or holes) injected into the active region.
The rates Rnr, RP, and Rstim are the nonradiative, spontaneous emission, and stimulated
emission rates; they have a negative sign because they act to decrease the carrier density.
The absorption rate Rabs acts to increase the carrier density and enters with a positive sign.
In the photon number rate equation, the Rstim and Rabs terms enter with the opposite
signs from the carrier number rate equation, as one would expect. The R,
term models
spontaneous emission into the lasing mode. Although we are discussing a single mode laser,
no laser is truly single mode, especially before the threshold of lasing. All of the rest of the
photon loss, including the useful output power from the facet, is in the term
P
The time
Tp
constant
Tp
is the known as the photon lifetime. The useful output power P is taken to be
a fraction % of the photon loss term, where %o is known as the optical efficiency.
P, = 71,hv
(B.3)
Tp
B.3
Solution of the Rate Equations for Low Frequencies
The rate equations are the basic tools for modeling a laser's operation and dynamics. With
the Langevin formalism introduced in Chapter 1, they can also be used to model a laser's
noise. In this thesis we are concerned with the noise properties of lasers at frequencies of
under 1 MHz. Because the dynamics modeled by the laser rate equations occurs on time
scales of nanoseconds or faster, the rate equations solved for the steady state contain all of
the information we need. Therefore the time derivatives on the left side of Eqn. B.1 can be
set to zero, and we can set about solving the equations for the steady state values of the
carrier and photon density.
Clearly much of the physics at work in a laser is hidden inside the dependence of the
various rate constants on the carrier density and the photon density. The rate constants
Rstim and Rabs should both be proportional to Np; one would intuively expect that the
APPENDIX B. BASICS OF SEMICONDUCTOR LASERS
148
number of photons absorbed per unit time and the number of carriers stimulated into
emitting per unit time should be proportional to the number of photons which initiate the
events. The photon density to be factored out of the rate constants, allowing the definition
of new rate constants depending only on N.
Np (stim(N) - rabs(N)) = Rstim(N, Np) - Rabs(N, Np)
(B.4)
The basic dependence of the rate constants rstim (N) and rabs (N) on the carrier density
N can also be deduced by thinking of the laser as a simplified two state system. In this
picture, an electron-hole pair is regarded as an electron excited from the ground state
(valence band) into the excited state (conduction band). Both bands have finite density
of electron states.
Clearly an absorption event in the laser depends on the presence of
an electron in the valence band. Just as important, however, is the presence of a vacant
space in the conduction band for the electron to be excited into.
The presence of this
space is not guaranteed because electrons obey the exclusion priciple; if all of the states
in the conduction band are occupied by electrons, no other electrons may be excited to
those states. For stimulated emission, the situation is reversed.
A stimulated emission
event depends on the presence of an electron in the conduction band, and the presence of
a vacant space in the valence band into which the electron can decay. The key insight is
that as the laser is pumped with more and more current, the conduction band is populated
with an increasing number of electrons (with density N) waiting to decay to the valence
band. As this happens, the rate constant rstim(N) increases and the rate constant rabs(N)
decreases, due to the changes in the available states for each process.
Eventually, the
difference (rstim(N)-rabs(N)) actually becomes positive. Physically this corresponds to net
photon gain in the medium.
With this information, the photon density rate equation (Eqn. B.1 can be solved for the
steady state photon density.
N
=s
S V/7p -
(rstim -
b
Tabs )v
(B.5)
B.3. SOLUTION OF THE RATE EQUATIONS FOR LOW FREQUENCIES
As the laser is pumped harder the quantity
(rstim - rabsV)
149
approaches the value of V/rp,
causing the photon density to diverge. This is exactly the threshold condition, at which
the gain a photon sees through one round trip through the cavity is exactly balanced by its
roundtrip loss. Of course, the photon density does not actually blow up; instead, (Tstim(N)
and rabsV)(N) are forced to clamp at their threshold values. Through this clamping of
the gain the carrier density N is also clamped at some value
Nth.
This has profound
consequences for the second steady state rate equation, which is solved here for the pumping
current
I.
I =
(Rp(Nh) + Rnr(Nth)) +
qV(rim(Nth) - Vabs(Nth))Np
(B.6)
Here all of the quantities which depended on N have been clamped at their threshold values.
As the pumping current I is increased beyond threshold, the only quantity on the right side
of Eqn. B.6 which can change is the photon density Np. All of the recombining current past
the threshold current Ith is taken up by the stimulated emission of the device. This allows
us to define
Ith =
qV
(Rsp(Nth) + Rnr(Nth))
(B.7)
h
and to write a simple equation for the laser output power above threshold:
Po =Toihv(I -
th) + Pp,
(B.8)
The term Pp is the power of the output light due to spontaneous emission. Its magnitide
is clamped at threshold, and at high pumping levels is negligable.
Below threshold, a
semiconductor behaves like a convential light emitting diode, which emits spontaneously
emitted light into all of the available modes. Above threshold, one mode of the laser wins
the competition between the modes, and any further electrical current pumping the laser
supports stimulated emission into the lasing mode, giving wise to an approximately linear
output power vs. input current relation.
150
APPENDIX B. BASICS OF SEMICONDUCTOR LASERS
Appendix C
Matlab Code
fano-vs-bias .m
% This code performs the calibrated noise measurement on the laser
% It calls the function meas-noisejfun
clear;
startfreq=2e4;
endfreq=2e6;
startfreqifor-avg=.5e6;
endfreqfor-avg=.6e6;
10
hnames={
%'h20.O-1.26'
'h20.0-1.26'
'h22.5-2.76'
'h22.5-2.76'
'h27.5-5.68'
'h35.2-10.1'
'h35.2-10.1'
'h35.2-10.1'
20
'h45.7-16.1'
'h45.7-16.1'
'h45.7-16.1'
'h51.2-19.15'
'h54.9-21.3'
'h70.8-30.2'
'h75.1-32.6'
'h80.9-35.8'
151
APPENDIX C. MATLAB CODE
152
'h89.2-40.4'
30
onnames={
%'d0.306'
'dl.26'
'd3.38'
'd5.96'
'd8.9'
'd10.2'
'dll.3'
'd12.3'
'd13.9'
'd14.7'
'd17.1'
'd19.2'
'd22.3'
'd28.32'
'd32.0'
'd36.6'
'd40.50'
40
50
I.offname=' dof f'
dc=[
%0.306e-3
1.26e-3
3.38e-3
5.96e-3
8.9e-3
10.2e-3
60
11.3e-3
12.3e-3
13.9e-3
14.7e-3
17.le-3
19.2e-3
22.3e-3
28.3e-3
32.0e-3
36.6e-3
40.5e-3
bias=[
I1;
70
153
for I=1:length(onnames)
[freq(:,1),fano-c(:,1),fano-l(:,1)] = measnoise_fun(hnames{1},...
onnames{1},offname,dc(l),startfreq,endfreq);
end
80
figure;
plot (freq,fano-c);
figure;
plot (freq,fanoi1);
for l=1:length(onnames)
min-found=0;
max-found=0;
min-ind=l;
90
max-ind=length(freq(:,1));
len=maxind;
for ind=1:1en
if ((freq(ind,1) >startfreqifor avg)&(min found==0))
min-ind=ind;
min-found=1;
end
if ((freq(ind,1)>endfreqifor-avg)&(max-found==0))
max-ind=ind-1;
max-found=1;
100
end
end
fano c_avg(1)=0;
fanoLavg(1)= 0;
for m=min-ind:max-ind
fano-c-avg(l)=fano-c-avg(l)+fano-c(m,1);
fano-l-avg(l)=fano-l-avg(l)+fanoil(m,1);
end
fano-c-avg(l)=fano-cavg(l)/(max-ind-min-ind+1);
fano-l-avg(l)=fano-l-avg(l)/(max-ind-min-ind+1);
110
end
bias=dc*1.78+18e-3;
figure;
semilogy(bias,fanocavg,' bo-');
hold on;
semilogy(biasfano-l-avg,'ro-');
xlabel('Laser Bias (mA)');
ylabel('Fano Factor');
set(gca,'linewidth',3);
set(gca,'fontsize',16);
120
APPENDIX C. MATLAB CODE
154
meas-noise-fun.m
function [freq,fano-c,fano-1]
= meas-noisefun (hnane,onnameoffname,...
dc,startfreq,endfreq)
% necessary constants
lambda=807e-9;
q=1.602e-19;
hbar=1.054e-34;
h-hbar*2*pi;
c=2.998e8;
slope-efficiency las=1.05; %W/A
%A/ W
photo sensitivity-det= 0.58
nu=c/lambda;
10
eta-l=slope-efficiencylas*q/h/nu;
etadd=photo-sensitivity-det*h*nu/q;
freq_s=2.5e7;
rawdata-file1=onname;
rawdata-file2=offname;
SRSfilter-file=hname;
% 'on' file
% 'off' file
% filter file
20
num=2^14;
DC-current=dc;
rawdatal=load(rawdata-file1);
rawdata2=load(rawdata-file2);
data1=rawdata1 -mean(rawdatal);
2
data2=rawdata2-mean(rawdata );
[psdl, f]=psd(data1,num,freqs);
[psd2, f]=psd(data2,numfreqs);
30
SRSfilter-raw= load(SRSfilter-file);
mninfound=0;
max-found=0;
nin-ind=1;
maxind= length(f);
ler=length(f);
for ind=l:len
if ((f(ind)>startfreq)&(min-found==0))
min-ind=ind;
min-found=1;
end
if ((f(ind)>endfreq)&(max-found==0))
max_ind=ind-1;
40
155
max-found=l;
end
end
50
f-trunc=f (min-ind:max-ind);
psdltrunc=psd1 (min-ind:max-ind);
psd2-trunc=psd2(min-ind:max-ind);
H=interp1 (SRSfilter raw(:, 1),SRSfilter-raw
SRSfilter= [f-trunc,H];
Hsqr=H.^2;
(:,2),f-trunc)/(eta-d*eta1);
freq=f trunc;
60
Ldc=D C _current;
psd-current _noise= (psdItrunc-psd2_trunc) ./Hsqr /freq-s*2;
psd-theory=2*1.602e- 19*I-dc;
fano-c=psd-current noise/psd-theory;
figure;
plot (fitrunc,fanoc,'b');
xlabel('Frequency of Measurement (Hz)');
ylabel('Fano Factor (ratio of noise to standard quantum limit) ');
70
title('current and photon noise');
hold on;
Pdc=Lidc/(etad*q/h/nu);
photon-fluxdc=P-dc/h/nu;
psd-post psd-current-noise/(q^2);
psd-pre=(psd-post -(1-eta-d)*eta-d*photon-flux-dc)/(eta-d^2);
psd_light=psdpre*((h*nu)^2);
psd_light_theory=2*h*nu*P_dc;
80
fanoI= psd-light/psd-light-theory;
plot (ftrunc,fano, ' g');
xcorr. m
% This code calculates the noise correlations
Fs=25e6;
num=2^13;
Fstart=250000;
Fend=550000;
APPENDIX C. MATLAB CODE
156
10
corrdata=load(' 11seriesl2v-sep');
psd1=psd(corrdata(:,1) ,num,Fs);
psd2=psd(corrdata(:,2),num,Fs);
[csd,freq] =csd(corrdata(:,1),corrdata(:,2),num,Fs);
cmetric=real(csd./ (sqrt (psd1.*psd2)));
%figure;
%subplot(2,1,1);
%semilogy(freq, abs(cmetric),'r');
%figure;
%semilogy(freq, cmetric,'b');
%hold off;
%subplot(2,1,2);
figure;
plot(freq/le6, cmetric);
set(gco, ' linewidth',3);
set(gca,' Fontsize',16);
set(gca, ' linewidth',3);
axis([O 2 -1 1]);
xlabel('Frequency (MHz) ');
20
30
ylabel('C-n (correlation)');
index=1;
while freq(index)<Fstart
index=index+1;
end
sind=index;
40
index=1;
while freq(index)<Fend
index=index+1;
end
eind=index-1;
sum=0;
for index=sind:eind
sum= cmetric(index) +sum;
end
avgcorr=sum/ (eind-sind+ 1)
50
Bibliography
[1] S. Datta, Electronic Transport in Mesoscopic Systems.
University Press, 1995.
Great Britain:
Cambridge
[2] Y. M. Blanter and M. Buttiker, "Shot noise in mesoscopic conductors," Physics Reports, vol. 336, pp. 1-166, 2000.
[3] F. Capasso and S. Datta, "Quantum electron devices," Physics Today, vol. 43, p. 74,
1990.
[4] C. Chen and C. Kuan, "Design and calibration of a noise measurement system," IEEE
Transactions on Intrumentation and Measurement, vol. 49, pp. 77-82, 2000.
[5] B. Mazhari and R. Sharan, "Photodetector-preamplifier for fiber optic systems," IETE
Journal of Education, vol. 38, p. 175, 1997.
[6] 0. K. Tonguz and L. G. Kazovsky, "Theory of direct-detection lightwave receivers using
optical amplifiers," Journal of Lightwave Technology, vol. 9, p. 174, 1991.
[7] W. M. Leach, "Fundamentals of low-noise analog circuit design," Proceedings of the
IEEE, vol. 82, p. 1515, 1994.
[8] R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective. San
Diego: Academic Press, 1998.
[9] R. Landauer, "The noise is the signal," Nature, vol. 392, p. 658, 1998.
[10] H. Callen and T. Welton, "Irreversibility and generalized noise," Physical Review,
vol. 83, p. 34, 1951.
[11] L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits.
Microwave and Optical Engineering, New York: John Wiley & Sons, Inc., 1995.
[12] M. Resnikov, M. Heiblum, H. Shtrikman, and D. Mahalu, "Correlation of electrons:
Suppression of shot noise in a ballistic quantum point contact," Phys. Rev. Lett., vol. 75,
p. 3340, 1995.
157
BIBLIOGRAPHY
158
[13] W. H. Richardson, S. Machida, and Y. Yamamoto, "Squeezed photon-number noise
and sub-poissonian electrical partition noise in a semiconductor laser," Phys. Rev.
Lett., vol. 66, pp. 2867-70, 1991.
[141 A. A. Saavedra, P.-J. Rigole, E. Goobar, R. Schatz, and S. Nilsson, "Relative intensity
noise and linewidth measurements of a widely tunable gcsr laser," IEEE Photonics
Technology Letters, vol. 10, pp. 481-3, 1998.
[15]
N. M. Hall and J. L. Carlsten, "Low-frequency intensity noise in semiconductor lasers,"
Applied Optics, vol. 35, pp. 6438-44, 1996.
[16] M. Kobayashi, Y. Kadoya, H. Yuji, R. Masuyama, and M. Yamanishi, "Squeezing
of photon-number fluctuations in the frequency range wider than 300 mhz in lightemitting diodes at room temperature," J. Opt. Soc. Am. B, vol. 17, pp. 1257-62, 2000.
[17] Oppenheim, Willsky, and Nawab, Signals and Systems. NJ: Prentice Hall, 1997.
[18] C. W. Gardiner, Handbook of Stochastic Methods. Berlin: Springer, 1997.
[19] M. Fukada, Optical Semiconductor Devices. NY: Wiley, 1999.
[20] C. D. Motchenbacher and J. A. Connelly, Low-Noise Electronic System Design. New
York: John Wiley and Sons, 1993.
[21] T. Darcie and G. Bodeep, "Lightwave subcarrier catv transmission systems," IEEE
Transactions on Microwave Theory and Techniques, vol. 38, p. 524, 1990.
[22
I. Garrett and G. Jacobsen, "Theoretical analysis of heterodyne optical receivers for
transmission systems using (semiconductor) lasers with nonnegligible linewidth," Journal of Lightwave Technology, vol. LT-4, p. 323, 1986.
[23] C. Cox, E. Ackerman, R. Helkey, and G. E. Betts, "Techniques and performance of
intensity-modulation direct-detection analog optical links," IEEE Transactions on Microwave Theory and Techniques, vol. 45, pp. 1375-83, 1997.
[24] C. Cox, H. V. Roussell, R. J. Ram, and R. J. Helkey, "Broadband, directly modulated
analog fiber link with positive intrinsic gain and reduced noise figure," in IEEE International Topical Meeting on Microwave Photonics, Technical Digest, (Piscataway,
NJ), pp. 157-60, 1998.
[25] S. G. Patterson, Bipolar Cascade Lasers. PhD dissertation, MIT, Sept. 2000.
[26] H. Nyquist, "Thermal agitation of electric charge in conductors," Physical Review,
vol. 32, pp. 110-113, 1928.
[27] P. Horowitz and W. Hill, The Art of Electronics. New York: Cambridge University
Press, 1989.
BIBLIOGRAPHY
159
[28] M. J. Buckingham, Noise in Electronic Devices and Systems. Ellis Horwood Series in
Electrical and Electronic Engineering, Chichester, England: Ellis Horwood Limited,
1983.
[29] A. van der Ziel, Noise in Solid State Devices and Circuits. New York: John Wiley and
Sons, 1986.
[30] P. J. Fish, Electronic Noise and Low Noise Design. New York: McGraw-Hill, Inc.,
1994.
[31] W. M. Siebert, Circuits, Signals and Systems. Cambridge, MA: The MIT Press, 1986.
[32] P. R. Gray and R. G. Meyer, Analysis and Design of Analog Integrated Circuits. New
York: John Wiley and Sons, 3rd ed., 1993.
[33] M. Sampietro, L. Fasoli, and G. Ferrari, "Spectrum analyzer with noise reduction by
cross-correlation technique on two channels," Rev. Sci. Instrum., vol. 70, pp. 2520-5,
1999.
[34] H. T. Friss, "Noise figure of radio receivers," Proceedings of the IRE, vol. 32, pp. 419422, 1944.
[35] R. Morrison, Grounding and Shielding Techniques in Instrumentation. New York: John
Wiley and Sons, 1967.
[36] R. M. Howard, "Ultralow noise high gain transimpedance amplifier for characterizing
the low frequency noise of infrared detectors," Rev. Sci. Instrum., vol. 70, pp. 1860-7,
1999.
[37] G. D. Geronimo, G. Bertuccio, and A. Longoni, "A low noise wide-band transimpedance amplifier for current noise spectra measurements," Rev. Sci. Intrum., vol. 7,
pp. 2643-7, 1996.
[38] C. Ciofi, G. Festa, R. Gianetti, and B. Neri, "Ultra low noise, current to voltage
converter with offset compensation independent of the source impedance," in IMTC/98
Conference Proceedings, (New York, NY), pp. 752-4, 1998.
[39] H. Birk, H. Oostveen, and C. Sch6nenberger, "Preamplifier for electric-current noise
measurements at low temperatures," Rev. Sci. Instrum., vol. 67, pp. 2977-80, 1996.
[40] J. Yao and J. Yoon, "Low-noise electrometer and its low-noise cryogenic probe with
completely guarded sample chamber," Rev. Sci. Instrum., vol. 71, pp. 1776-80, 2000.
[41] S. Hasiguchi, "Sensitivity of the dual-carrier scheme for the contact-free detection
of current noise," IEEE Transactions on Instrumentation and Measurement, vol. 38,
pp. 434-7, 1989.
160
BIBLIOGRAPHY
[42] J. Sese, G. Rietveld, A. Camon, C. Rillo, L. Vargas, G. Christian, S. Brons, M. G. H.
Hiddink, J. Flokstra, H. Rogalla, W. Jaszczuk, and H. Altenburg, "Design and realization of an optimal current sensitive ccc," IEEE Transactions on Instrumentation and
Measurement, vol. 48, pp. 370-4, 1999.
[43] B. Neri, B. Pellegrini, and R. Saletti, "Ultra low-noise preamplifier for low-frequency
noise measurements in electron devices," IEEE Transactions on Instrumentation and
Measurement, vol. 40, pp. 2-6, 1991.
[44]
R. B. Hallgren, "Paralleld transconductance ultralow-noise preamplifier," Rev. Sci.
Instrum., vol. 59, pp. 2070-3, 1988.
[45] E. R. Brown, "Ultra-low-noise high-impedance preamp for cryogenic detectors," Electronics Letters, vol. 21, pp. 417-8, 1985.
[46] H. P. Gush, M. Halpern, and S. Knotek, "Cooled low-noise preamplifier for a bolometer," Rev. Sci. Instrum., vol. 63, pp. 90-2, 1992.
[47] R. H. Walden, "Analog-to-digital converter survey and analysis," IEEE Journal on
Selected Areas in Communications, vol. 17, pp. 539-50, apr 1999.
[48] A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing.
Prentice Hall Signal Processing Series, Upper Saddle River, New Jersey: Prentice Hall,
2 ed., 1999.
[49] T. V. Blalock and R. L. Shepard, "Survey, applications, and prospects of johnson
noise thermometry," in Sixth International Conference on Noise in Physical Systems,
(Washington, DC), pp. 260-8, 1981.
[50] P. Kittel, W. R. Hackleman, and R. J. Donnelly, "Undergraduate experiment on noise
thermometry," Am. J. Phys., vol. 46, pp. 94-100, 1978.
[51] W. T. Vetterling and M. Andelman, "Comments on: Undergraduate experiment on
noise thermometry," Am. J. Phys., vol. 47, pp. 382-4, 1979.
[52] C. Fonstad, Microelectronic Devices and Circuits. McGraw-Hill series in electrical and
computer engineering, New York: McGraw-Hill, Inc., 1994.
[53] Y. Yamamoto and S. Machida, "High-impedance suppression of pump fluctuation and
amplitude squeezing in semiconductor lasers," Phys. Rev. A, vol. 35, pp. 5114-30, June
1987.
[54] M. J. Freeman, H. Wang, D. G. Steel, R. Craig, and D. R. Scifres, "Amplitude-squeezed
light from a quantum well lasers," Optics Lett., vol. 18, pp. 379-381, 1993.
[55] J. Kitching and A. Yariv, "Room temperature generation of amplitude squeezed light
from a semiconductor laser with weak optical feedback," Phys. Rev. Lett., vol. 74,
pp. 3372-3375, 1995.
BIBLIOGRAPHY
[56]
161
S. Lathi and Y. Yamamoto, "Influence of nonlinear gain and loss on the intensity noise
of a multimode semiconductor laser," Phys. Rev. A, vol. 59, pp. 819-25, 1999.
[57] A. Inoue, S. Lathi, and Y. Yamamoto, "Longitudinal-mode-partition noise and amplitude squeezing in semiconductor lasers," J. Opt. Soc. Am. B, vol. 14, pp. 2761-2766,
1997.
[58] F. Marin, A. Bramati, and E. Giacobino, "Squeezing and intermode correlations in
laser diodes," Phys. Rev. Lett., vol. 75, pp. 4606-4609, 1995.
[59] G. Bj6rk, "Generation of intensity-correlated twin beams using series-coupled semiconductor lasers," Phys. Rev. A, vol. 45, pp. 8259-8267, 1992.
[60] P. J. Edwards and G. H. Pollard, "Quantum noise-correlated operation of electrically
coupled semiconductor light emitters," Phys. Rev. Lett., vol. 69, pp. 1757-60, 1992.
[61] A. Z. Khoury, "Pump-induced correlation between two lasers," Phys. Rev. A, vol. 60,
pp. 1610-13, 1999.
[62] E. Goobar, R. J. Ram, R. Nagarajan, L. A. Coldren, and J. E. Bowers, "Intensity
noise and facet correlation in fabry-perot laser diodes with low facet reflectivities,"
Appl. Phys. Lett., vol. 66, pp. 3419-21, 1995.
[63] E. Goobar, A. Karlsson, G. Bj6rk, and P. J. Rigole, "Quantum correlated light beams
and sub-poissonian electrical partition noise in parallel and series coupled semiconductor light emitters," Phys. Rev. Lett., vol. 70, pp. 437-40, 1993.
[64] B. Orsal, P. Signoret, J. M. Peransin, K. Daulasim, and R. Alabedra, "Correlation
between electrical and optical photocurrent noises in semiconductor laser diodes.,"
IEEE Transactions on Electron Devices, vol. 41, pp. 2151-2161, 1994.
[65] A. Yariv, Optical Electronics in Modern Communications. The Oxford Series in Electrical Engineering, Oxford: Oxford University Press, 5th ed., 1997.
[66] F. Rana and R. Ram, "Current noise and photon noise in quantum cascade lasers,"
Phys. Rev. B, vol. 65, 2002. accepted for publication.
[67] J. Kitching, D. Provenzano, and A. Yariv, "Generation of amplitude-squeezed light
from a room-temperature fabry-perot semiconductor laser," Optics Lett., vol. 20,
pp. 2526-2528, 1995.
[68] H. Wang, M. J. Freeman, and D. G. Steel, "Squeezed light from injection-locked quantum well lasers," Phys. Rev. Lett., vol. 71, pp. 3951-3954, 1993.
[69] M. J. Freeman, D. C. Kilper, D. G. Steel, R. Craig, and D. R. Scifres, "Room temperature amplitude-squeezed light from an injection-locked quantum-well laser with a
time-varying drive current," Optics Lett., vol. 20, pp. 183-185, 1995.
BIBLIOGRAPHY
162
[70] J. P. Sun, G. I. Haddad, P. Mazumder, and J. L. Schulman, "Resonant tunneling
diodes: Models and properties," Proceedings of the IEEE, vol. 86, pp. 641-61, 1998.
[71] R. Landauer, "Solid state shot noise," Phys. Rev. B, vol. 47, pp. 16427-32, 1993.
[72] S. Kogan, Electronic Noise and Fluctuations in Solids.
University Press, 1996.
Great Britain:
Cambridge
[73] R. Landauer, "Johnson-nyquist noise derived from quantum mechanical transmission,"
Physica D, vol. 38, p. 226, 1989.
[74]
W. Schottky,
"Uber
spontane stromschwankungen
trizit atsleitern," Ann. Phys., vol. 57, p. 541, 1918.
in
verschiedenen
elek-
[75] G. lannaccone, G. Lombardi, M. Macucci, and B. Pellegrini, "Enhanced shot noise in
resonant tunneling: Theory and experiment," Phys. Rev. Lett., vol. 80, p. 1054, 1998.
[76] G. Iannaccone, M. Macucci, and B. Pellegrini, "Shot noise in resonant-tunneling structures," Phys. Rev. B, vol. 55, pp. 4539-50, 1997.