BARKER
MASSACHUSETTS
OF TECHNOLOGY
INSTITUTE
Soliton Squeezing in Optical Fibers
APR 2 420
by
LIBRARIES
Charles Xiao Yu
Submitted to the Department of Electrical Engineering
and Computer Science in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Electrical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
November, 2000
© Massachusetts Institute of Technology, 2000. All Rights Reserved.
A uthor ..............................
.......................
Department of Electrical EngineeriAg and Computer Science
November 14, 2000
Certified by .....
..................
Hermann A. Haus
Institute Professor
Thesi§ Supervisor
Accepted by .........
Arthur C. Smith
Chairman, Departmental Committee on Graduate Students
2
Soliton Squeezing in Optical Fibers
by
Charles Xiao Yu
Submitted to the Department of Electrical Engineering and Computer Science on November 14, 2000, in partial fulfillment of the
requirements for the degree of Doctor of Philosophy in Electrical
Engineering
Abstract
The use of squeezed light can overcome the standard quantum limit in phase sensitive
optical measurements. This thesis is a theoretical and experimental investigation of soliton
squeezing at 1.55gm. Theoretically, the effects of the continuum on squeezing have been
investigated. Experimentally, modelocked fiber laser sources at 1.55gm have been developed for both schemes and their noise properties have been investigated. The timing jitter
of these lasers are found to be quantum limited. Squeezing has been observed with two
schemes, both using the optical fiber as the nonlinear medium. As much as 4.4dB of noise
reduction has been detected.
Thesis Supervisor: Hermann A. Haus
Title: Institute Professor
Acknowledgments
I want to thank my advisor, Prof. Haus. Prof. Haus matches my idea of a great
scholar pretty well, except for the fact of not having white hair. I'm very lucky to have
him.
I want to thank my co-advisor, Prof. Ippen. Prof. Ippen first served as my undergraduate academic advisor, then as my thesis co-advisor. He always reminds me of something Sam Ulman said about youth, that youth is not a time of life, but a state of mind. "As
long as your aerials are up, to catch the wave of optimism, there is hope you may die
young at eighty." It's a pity that I won't have his "continued supervision" at Crawford Hill
after graduation.
I want to acknowledge Prof. Fujimoto. It's nice to see that an Asian American can
rise to the top in a still very white world. It's also nice to know that all that work and
tuition at MIT will pay off eventually/sometimes.
I want to thank Prof. Shapiro. Prof. Shapiro is sharp. He read this thesis and gave
lots of advice.
I want to thank the people in and around our optics group: Cindy, Donna, Jalal,
Krist, Farhan, Pat, Mike, Farzana, John, Jerry, Stu, Will, Shu, Moti, Dan, Juliet, Matt and
lots of others whose names escaped me at this moment. It's been great.
Finally I'd like to thank my parents, for always giving me lots of freedom and leading me to question the orthodox. I dedicate this thesis to them.
Table of Contents
1 Introduction................................................................................................................11
1.1
M otivations ..................................................................................................
11
1.2
H istorical Background on Soliton Squeezing ...............................................
12
1.3
Thesis Content ..............................................................................................
17
2 Soliton Squeezing in Optical Fibers, Theory ........................................................
19
2.1
Quantized Nonlinear Schr6dinger Equation and its Linearization ..............
20
2.2
Renorm alization of the Soliton Operators ...................................................
24
2.3
The Continuum ............................................................................................
29
2.4
Soliton Squeezing in a Fiber ........................................................................
31
2.5
Continuum Contribution to Squeezing ............................................................
38
2.6
Sum m ary .....................................................................................................
42
3 Squeezing with a short piece of fiber....................................................................
3.1
3.2
Laser and its N oise......................................................................................
45
45
3.1.1
Introdution..........................................................................................
45
3.1.2
Theoretical Overview ..........................................................................
47
3.1.3
Experim ental Setup and Procedure ...................................................
52
3.1.4
Experim ental Results ..........................................................................
55
3.1.5
Com parison between Theory and Experim ent....................................
66
Cross Phase M odulation(XPM ) Squeezing .................................................
4 Squeezing with a Sagnac loop and l-GHz ps Pulses.............................................
68
75
4.1
GHz Laser Source ........................................................................................
75
4.2
N oise of the GHz laser .................................................................................
79
4.3
The Double Clad Fiber Am plifier...............................................................
86
4.4
The Balanced Detector.................................................................................
88
4.5
Sagnac Loop Squeezing...............................................................................
90
5 Conclusions and Future W ork ..............................................................................
97
5.1
Conclusions.................................................................................................
97
5.2
Future W ork .................................................................................................
98
5
Appendix A Continuum M atrix Elements ...................................................................
101
Appendix B Direct Measurement of Self-Phase Shift due to Fiber Nonlinearity........105
B ibliograp h y ...............................................................................................................
6
113
List of Figures
Figure 1.1: A sample GAWBS spectrum. Inset: Low frequency GAWBS....................14
Figure 1.2: Effect of loss is analogous to going through a beamsplitter.......................16
Figure 2.1: The deformation of uncertainty ellipse. ......................................................
32
Figure 2.2: The squeezing set-up...................................................................................
33
Figure 2.3: The squeezing and anti-squeezing (the minor and major axes of the squeezing
ellipse) as functions of 2 ) ............................................................................................
37
Figure 2.4: The root mean square fluctuations as function of the phase angle with respect
to local oscillator: 20 = 2, 4, 8 ....................................................................................
37
Figure 2.5: The minimum and maximum fluctuations of the soliton alone as detected by local oscillator of secant hyperbolic shape. Comparison with ideal local oscillator use......40
Figure 2.6: The fluctuations of the soliton alone as detected by local oscillator of secant hyperbolic shape as function of phase angle with respect to local oscillator, (D = 1......40
Figure 2.7: The matrix elements (a) 22 (cont,cont) and (b)
22(sol,cont) as a function of phase
D . N ote the "beats." ...........................................................................................................
43
Figure 2.8: The minimum fluctuations detected by a local oscillator which is orthogonal to
continuum, and by a secant hyperbolic local oscillator with and without continuum.......44
Figure 3.1: Stretched pulse laser schem atic...................................................................
46
Figure 3.2: Experim ental setup .....................................................................................
53
Figure 3.3: A typical sampling scope trace(20ps/div)...................................................53
Figure 3.4: (a) Local oscillator of R3265 with RBW=lOHz. (b) Local oscillator of HP
8560E with RBW =1 Hz, the peak is ~15dB above the maximum on the graph. ......
54
Figure 3.5: Hybrid state (a) Optical Spectrum. (b) Harmonic 1. (c) Harmonic 21........57
Figure 3.6: Energy fluctuation at very low frequencies. Note that in this case the noise
structure is buried under the local oscillator noise........................................................58
Figure 3.7: Broadband pulse energy noise: (a)without pedestal. (b) with pedestal. RBW =
7
K H z ..............................................................................................................................................
59
Figure 3.8: (a)The optical spectrum, and (b) the autocorrelation of the compressed pulse corresponding to the case of GVD=0.0169ps 2 and highest output power. The dashed line in (a) is a theoretical gaussian fit of the spectrum. ........................................................................................
62
Figure 3.9: Jitter fitting for harmonic 21, measurement time = 0.09 second. Dashed=data, Solid
= fitted jitter. ...................................................................................................................................
62
Figure 3.10: (a) Timing jitter due to white noise, jitter=13ppm. (b) Timing jitter as a function of
output power. The theoretical curve is the quantum jitter using the following parameters: t0 =5 Ifs,
Gain bandwidth =40 nm, D=0.5 * 0.0169 ps 2 , a=0.2, g=0.8, TR=1 8 ns, O= 3.2. Measurement time
= 0.0 9 secon d . .................................................................................................................................
63
Figure 3.11: Data taken using the HP 8560E, RBW= 1Hz, Measurement time= 1.92 seconds:
(a)Harmonic 1, (b) Harmonic 21. Dashed curve is the experimental data. Solid curve is the sum of
harmonic 1 and the theoretically predicted lorentzian-shaped jitter. The signals peaks are 30 dBm
above the maximum shown in the graph. ......................................................................................
64
Figure 3.12: (a) The optical spectrum and (b) the autocorrelation of the compressed pulse corresponding to the case of GVD<0.005ps 2 . The dashed line in (a) is a theoretical gaussian fit of the
sp ectru m .........................................................................................................................................
65
Figure 3.13: RF spectrum, Solid=Harmonic 1, Dashed = Harmonic 21. ..................................
66
Figure 3.14: Quantum jitter vs. gain bandwidth for different O's. Pulse energy(wo) = 0.42 nJ. Pulsew id th (t) = 5 1 fs. .............................................................................................................................
68
Figure 3.15: Schematic description of cross phase modulation squeezing...............................
69
Figure 3.16: Experimental Setup ..............................................................................................
70
Figure 3.17: Pulse Spectra before(dashed) and after(solid) SMF fiber, Power of 35mW...... 70
Figure 3.18: Autocorrelation of pulses after SMF with power of 35mW. Dashed line is the sech
fit ....................................................................................................................................................
71
Figure 3.19: RF spectrum of the homodyne detector. Shot noise is obtained by blocking the vacuum p o rt ...........................................................................................................................................
72
Figure 3.20: Dependence of noise on the relative local oscillator phase...................................73
Figure 4.1: Laser Schematic, HWP: Half Wave Plate; QWP: Quarter Wave Plate. D: Phase Delay
8
line, G:RF amplifier, F: 1GHz Filter, PD: Photodetector...............................................76
Figure 4.2: Output of the laser: a) RF spectrum b) optical spectrum when the modulator is
on, c) RF spectrum d) optical spectrum when the modulator is off. The RBW of the RF
spectrum is 100KHz, and the RBW of the optical spectrum is 0.5nm.The suppression in (a)
is >60dB. The spectral width is 5.6 nm in (b) and (d). e) Autocorrelation of the laser output
when the modulator is on. The pulsewidth is 480 fs......................................................78
Figure 4.3: Harmonic 1 (a) Broadband amplitude noise, RBW=IKHz. (b) Narrowband
structure, RB W = I0H z. .................................................................................................
83
Figure 4.4: (a) Quadratic dependence of the jitter on frequency, (b) Harmonic 7,
RBW=lOHz, Dashed line: experimental; Solid line, theoretical quantum fitting. ........ 85
Figure 4.5: (a) Schematic of the EYDFA (b) Cross section of the double clad fiber........87
Figure 4.6: Amplifier input-output curve......................................................................
88
Figure 4.7: Detector power spectrum with one detector and with both detectors when illum inated with ASE from the EYDFA............................................................................
90
Figure 4.8: Propagation of a soliton state input through the nonlinear Mach-Zehnder interfero m eter ............................................................................................................................
91
Figure 4.9: Setup for Sagnac loop squeezing.................................................................
93
Figure 4.10: Unaveraged spectrum at 8MHz, RBW= 100KHz. Piezo stack is driven with a
S aw too th ............................................................................................................................
94
Figure 4.11: Averaged spectrum at 8MHz, RBW=IOOKHz, Piezo stack driven with a Sinuso id .....................................................................................................................................
95
Figure 4.12: Loss vs. Noise Reduction for Different Squeezing ...................................
96
Figure B. 1: (a) $NL(t=0) vs. $NL(W=O) for P2 =-3.6ps 2 /km. (b) $NL(t=O) vS. PNL(w=O) for
P2 =0.5ps 2/km. The two phases are equal along the straight line.....................................107
Figure B.2: Experimental Setup, Zero dispersion wavelength of DSF is ~ 1547nm. The inset is the autocorrelation of the signal-reference pair after the PM fiber...........109
Figure B.3: (a) Spectra before and after DSF, 20nm span (b) Spectra before and after DSF,
1 nm span for different powers. The center wavelength=1574nm. (c) Spectra before and after DSF, 20nm span (d) Spectra before and after DSF, 1 nm span for different powers. The
9
center w avelength= 1541 nm ...................................................................................
........
110
Figure B.4: (a) Spectral bandwidth for Xo=1574nm and Xo=1543nm. (b)Nonlinear Phase Shift vs.
Input Power, Xo=1574nm and Xo=1543nm............................................................................111
10
Chapter 1
Introduction
1.1 Motivations
Highly accurate optical phase measurement instruments such as the laser gyroscope[l]-[3] are increasingly being deployed in many applications that require inertial
navigation and platform stabilization. These sensors use an interferometric arrangement to
sense movements. The accuracy of such high precision phase sensitive interferometric
measurements is fast approaching the limits set by the shot noise, which originates from
fluctuations of the vacuum field. The necessary existence of this noise is a direct consequence of Heisenberg's Uncertainty Principle, which forms a cornerstone of modem quantum mechanics. Heisenberg's Uncertainty Principle states that a lower bound exists for the
product of the variances of two observables that do not commute with each other. In any
measurements this lower bound dictates the achievable signal to noise ratio(SNR). However if the measurement only involves one of the observables, then one can reduce the
uncertainty of that observable at the expense of the other observable, and still preserve
Heisenberg's Uncertainty Principle. This kind of noise reduction is termed "squeezing"
because in the phase space the initial uncertainty ellipse is further elongated and thus
looks "squeezed." It has been shown that an interferometric phase measurement, such as
the interferometric fiber optic gyroscope(IFOG)[4]-[8], can achieve higher SNR than the
standard quantum limit by using squeezed vacuum. In these measurements the two
observables are the in-phase and the quadrature components of the optical field. Since the
measurement is phase sensitive, only one of the two components is necessary.
In order to achieve squeezing some nonlinear medium must be used to alter the statistics of the optical field[9]-[14]. The X3 nonlinearity, more specifically the self-phase
modulation(SPM) of the optical fiber is used in the experiments done at MIT[13],[14].
11
The MIT scheme uses a balanced fiber Sagnac loop that separates the squeezed vacuum
from the pump pulse. Homodyne detection[15] is used to observe the effects. In the experiment done at 1.3gm, as much as 5dB of squeezing has been obtained. At that wavelength
the fiber has zero dispersion and theory indicates that, taking into account the losses, 5dB
of squeezing is the achievable limit for a Gaussian pulse propagating in a zero dispersion
fiber. This is because the different temporal portions of the Gaussian pulse do not interact
with each other. Each portion experiences different nonlinearity depending on its amplitude and thus different amounts of squeezing. The overall squeezing is an average over the
entire pulse. Because of the pulse wings the overall squeezing cannot go lower.
To overcome this limitation we perform squeezing at 1.55gm. At 1.55gm the fiber
has negative dispersion and the pulses form solitons[17]-[25]. The different temporal portions of an ideal soliton are correlated with each other and experience the same squeezing.
Thus no limit due to the pulse shape exists for soliton squeezing. Theoretical analysis[19][22] indicates that even for sub-100fs pulses greater than 20dB squeezing is achievable
before the Raman effect[23],[24] places a floor on the observable squeezing. Numerical
simulation[25] indicates that even when the pump pulse deviates from an ideal soliton and
the local oscillator is non-optimal, the achievable squeezing is still not limited by the pulse
shape when the fiber has negative dispersion. This is fortunate since the wavelength
1.55gm corresponds to the lowest loss for the optical fiber and is becoming increasingly
important for optical telecommunication. As a result, high quality, inexpensive optical
components are abundant at 1.55gm.
1.2 Historical Background on Soliton Squeezing
It is well established that soliton propagation in optical fiber can generate squeezed
light. Analytical study via the soliton perturbation theorem[19] and numerical investigation via the generalized- P representation[20] both predict quantitatively the degree of
achievable squeezing given the nonlinear phase shift experienced by the soliton. It was
later shown that the two formalisms lead to the same results. Numerical simulations via
the split-step algorithm[25] have also been done for an input pulse that is not an ideal soli-
12
ton. The results indicate that squeezing is still achievable and does not degrade significantly under these conditions. The Raman effect on squeezing has also been
analyzed [23],[24]. It was found that Raman noise a floor on the amount of observable
squeezing. Though this floor exists even in the case of continunous-wave squeezing[25], it
is greater than 20dB. Thus the Raman effect will not become significant except for sub100fs pulses and large phase shifts. The remaining theoretical concern is the effect of continuum on squeezing. Because the local oscillator used for the detection of squeezing is
usually not matched to the soliton perturbations, it does not discriminate against the continuum completely. We will extend the soliton perturbation theory to cover the continuum.
The projection functions required are readily available[26]. So this extension is straightforward.
Though many theoretical papers have been published on soliton squeezing, the
reported experimental work on this subject is not nearly as numerous. The first experiment
on soliton quadrature squeezing was done by Shelby et. al. at IBM[17]. Only 1.7dB of
noise reduction below shot noise was reported and the fiber had to be cooled to liquid
nitrogen temperature. A later effort by Doerr et. al.[18] achieved 1.9dB at room temperature. Recently progress has been made on soliton amplitude squeezing. Using external filtering 3.2dB of noise reduction has been reported. Further efforts by two separate groups
using an asymmetric Mach-Zehnder interferometer has led to 3.9dB and 5.7dB of squeezing, respectively. Amplitude squeezing presumes that the laser light is shot noise limited.
The drawback of this requirement is that squeezing can only be detected at higher frequencies since the laser light is rarely shot noise limited at low frequencies. Furthermore, if a
robust and compact, but low power, fiber laser or semiconductor laser is used as the
source, then an amplifier must follow the laser source to reach the soliton energy. Since it's
difficult for amplified light to be shot noise limited even at higher frequencies, amplitude
squeezing may necessarily require high power solid state lasers to avoid the use of amplifiers. Using bulky lasers is a severe limitation for many potential applications of squeezed
light. Furthermore, squeezed vacuum, not amplitude squeezed light, is what is needed for
high precision measurements. Despite these issues, these soliton amplitude squeezing
13
experiments still show that better results should be obtainable for quadrature squeezing as
well. Therefore we have decided to take a second look at soliton quadrature squeezing and
to resolve the remaining experimental obstacles.
There are several experimental difficulties when one tries to realize quadrature
squeezing, they are listed below:
1) Guided Acoustic Wave Brillouin Scattering(GAWBS).
2) Noise of the laser source.
3) Pump leakage into the vacuum port.
4) Raman noise.
5) Optical loss in the system.
-65
-70
-75
'0-80
-75
-85
-90
-Q!;-
0
40
80
MHz
120
160
-851
-.95'
)
j1 'm
4
500
V4
MHz
1000
1500
Figure 1.1: A sample GAWBS spectrum. Inset: Low frequency GAWBS
The first one of these: GAWBS, is a phase noise inherent to the optical fiber that
counteracts squeezing of optical waves[28]. Extensive measurements have been performed, both cw and pulsed, by the IBM group to characterize the effects [27],[28].
GAWBS is produced by the thermal fluctuations of the index in a fiber via the acoustic
14
modes of the fiber. The acoustic wavelength must be long so as to nearly phase match the
scattered radiation to the incident radiation. This means that only modes near the acoustic
cutoff frequency are GAWBS active. The IBM experiments have shown that the main contribution to the polarized GAWBS noise is due to the compressional waves. Fig. 1.1 shows
a sample GAWBS spectrum under cw excitation. The inset is the low frequency portion of
the same spectrum. The spectrum is a series of discrete lines starting from ~35MHz so that
squeezing can be observed between these lines via a laser source that has the appropriate
repetition rate. The magnitude of GAWBS increases with the fiber length and usually rolls
off for frequencies higher than 1 GHz. Thus to avoid GAWBS, one either uses a very short
piece of fiber where GAWBS noise is below shot noise, or uses a 1 GHz repetition-rate
pulsed source and looks for squeezing at frequencies between GAWBS peaks.
The noise of the pump laser is experimentally significant for two reasons. In order
to see quantum noise reduction, one has to make sure that the classical noise is insignificant. Though a balanced detector is used and ideally all of the classical noise can be cancelled, in practice 40dB of cancellation can be expected from such a device at the best.
Thus the classical noise has to be within 40dB of the quantum noise. The second effect is
more severe: usually some of the pump leaks into the vacuum port. The vacuum port output is the signal and will not be cancelled by the balanced detector. Thus any classical
noise associated with the leaked pump laser will mask squeezing directly.
The amount of pump leakage depends on the specific setup. If a Sagnac loop is
used, then the leakage depends on how well-balanced the Sagnac loop is. Since the loop
uses a fiber coupler whose coupling ratio changes as a function of wavelength, the total
leakage is directly proportional to the bandwidth of the input pulse. A narrower bandwidth
leads to acceptable leakage. It also implies that the input pulsewidth cannot be very short.
Since squeezing is directly proportional to the nonlinear phase shift the pump pulse experiences in the Sagnac loop, a long pump pulse implies that the loop length has to be long.
For example, for a 1 ps pulse, the Sagnac loop has to be on the order of 100 meters to
achieve 15dB of squeezing.
15
As mentioned earlier, Raman noise places a floor on the observable squeezing
when the pump pulses are sub-100fs. To alleviate this problem one can either use a short
piece of fiber or use picosecond pulses.
The power loss experienced by the squeezed vacuum can be modeled as injection
of unsqueezed vacuum via a beamsplitter with the correct splitting ratio, as illustrated in
Fig. 1.2. Suppose that the squeezed signal mode is described by the annihilation operator
a. Let
a4 be the same signal after it experienced power losses of 1-1. then
(1.1)
a4 = la, - JO - )a2
where the mode a2 is in vacuum. The noise reduction ratios for the quadratures of a^ and
a4 are therefore related by
R4 =
1R 1 + 1 -1
(1.2)
Vacuum a2
Loss of
Squeezing a4
Squeezed State a
Figure 1.2: Effect of loss is analogous to going through a beamsplitter
We see that if 1=0.9 and R1 =O(infinite squeezing), the noise reduction ratio
increases to R4 =0.1(lOdB squeezing) because of loss. The problem of optical loss can be
16
partially solved by using a low loss nonlinear medium such as the optical fiber, which is
the approach in this thesis.
1.3 Thesis Content
Chapter Two of this thesis addresses the theoretical background of soliton squeezing. A linearized approach of Haus and Lai is presented. We have examined the effects of
an unmatched local oscillator for the detection of squeezing. Specifically, a sech local
oscillator is used as an example to demonstrate the effects of continuum.
Chapter Three describes a squeezing experiment using a short piece of fiber. By
using a short piece of fiber, GAWBS is not an issue and a low rep-rate modelocked
stretched pulse laser is used. The stretched pulse laser was invented at MIT. Though most
of its operation is well understood, study on its noise behavior does not yet exist. We have
characterized the laser noise thoroughly and compared the experiments with theory.
Squeezing results, which utilize cross phase modulation of the optical fiber, are then presented.
Chapter Four describes a second squeezing experiment employing a long Sagnac
loop and 1 GHz laser source. The 1 GHz laser source combined both passive modelocking
and active modelocking mechanisms to achieve short pulsewidth at high rep-rate. Record
pulsewidth reduction from the Kuizenga-Siegman pulsewidth is also reported. The noise
of this laser is again characterized and compared with theory. Experimental results from
the Sagnac loop squeezing are then presented. The two approaches described in Chapters
Three and Four both address the experimental difficulties mentioned in Section 1.2.
Finally in Chapter Five we summarize the work presented in this thesis and propose some future research topics.
17
18
Chapter 2
Soliton Squeezing in Optical Fibers, Theory
Soliton squeezing has been extensively analyzed analytically[ 2 9- 4 1] and demonstrated experimentally[ 17,18,42-44]. Soliton squeezing was first numerically investigated
via stochastic differential equations by Drummond et. al. [20]. The stochastic approach introduced noise sources into the equations of propagation. Haus and Lai[19] analyzed the
problem of squeezed soliton vacuum generation in a balanced Sagnac loop using linearization of the Nonlinear Schridinger Equation(NLSE). Since this equation is derivable from
a Hamiltonian it conserves commutator brackets and thus does not require the introduction
of noise sources[ 3 9 ,4 0]. It was shown the two formalisms lead to the same results[32]. The
approach based on the linearized NLSE leads to linear differential equations for the operators. Since the solutions of linear operator equations do not involve commutators, they are
identical in form with solutions of differential equations of classical c-number variables.
This correspondence with classical evolution allows for simple interpretations of the quantum behavior.
The linearized equations were solved[19] by expressing the solution as a superposition of the four soliton perturbations, photon number, phase, position and momentum and
the continuum. Generally, squeezing is described by the evolution of the in-phase and
quadrature components of the electric field. These have the same dimensions, contrary to
the operators of photon number and phase. To bring out more closely the correspondence
with the conventional approach to squeezing via the second order nonlinearity we renormalize the pertinent operators. One welcome consequence of the renormalization is that it
establishes symmetry in the equations of evolution between pairs of the four soliton perturbation parameters; in-phase and quadrature amplitudes; position and momentum.
Optimal detection of squeezing soliton vacuum requires a local oscillator that is not
a simple secant hyperbolic. Coupling to the continuum is thereby avoided. In practice it is
19
much easier to use the secant hyperbolic of the squeezing "pump" for the local oscillator.
Under these conditions coupling to continuum is unavoidable. The question then arises as
to the effect of this coupling, which has been partially addressed in Refs. [41] and [37].The
coupling to the continuum leads to an oscillatory dependence of shot noise reduction upon
the Kerr induced phase shift. In fact, the coupling to the continuum can provide a (small)
improvement in the shot noise reduction by virtue of the fact that the noise in the continuum
is correlated with the soliton fluctuations. The work on amplitude squeezing[42-44] has
also dealt with the contribution of continuum[46, 4 7]. Oscillatory behavior has been found
in the amplitude squeezing experiments in an asymmetric Sagnac loop[4 6 ,4 7 ]. In our case
the variation is due to two beating effects: (a) the self-beating of the continuum, (b) the
beating between the continuum and the soliton.
In Section 2.1 we review soliton perturbation theory of Ref. [19]. Section 2.2 renormalizes the operators so as to cast soliton squeezing in the linearized approximation into a
standard Bogolyubov transformation. Section 2.3 reviews the orthogonality properties of
the continuum. Section 2.4 analyzes the detection of a squeezed soliton with the choice of
local oscillator that rejects the continuum in preparation to the main thrust of this paper,
namely the detection of the squeezed soliton with a secant hyperbolic local oscillator pulse
as is usually done in practice, because then the local oscillator is directly available from the
squeezing pump. This pulse couples to the continuum which may add to, or subtract from,
the noise level.
2.1 Quantized Nonlinear Schr6dinger Equation and its Linearization
The unnormalized Heisenberg equation of motion is
2
a3t-
Si
-
2
d (o a
2
2
2 (x) + iKat (x)a(x)-a(x)
(2.1)
This is the quantized nonlinear Schr~dinger equation. a(x) is an annihilation
operator annihilating a photon at the position x, at(x) is a creation operator creating a photon
at the position x. The classical analogs of a(x) and at(x) are the complex amplitude of the
20
pulse envelope, and its complex conjugate; do /d
2
is the second derivative with respect
to the propagation constant o(p); K is the Kerr parameter. The quantized form of the
Nonlinear Schr6dinger equation was solved rigorously using the Bethe ansatz[2]. An
approach that leads to simple analytic expressions that permits physical insight is based on
the linearization approximation[3]. One sets for the operator a(x)
a(x) = a (x) + A ^(x)
(2.2)
where the first term is a c-number, and the second is an operator that takes over the
commutation relation of a(x). Thus,
[A^(x),Alt(x')] = 8(x-x')
(2.3)
The replacement (2.2) is rigorous, and by itself does not imply any approximation.
Approximations are made when ao(x) is made to obey the nonlinear Schr6dinger equation
and the equation is linearized in terms of Aa(x). Thus, ao(x) obeys the equation
2
Ca ao
.3ao
- a =+
d2
*
Ka
a a ,C =
(2.4)
The solution is:
-/ 2
KA02t
a (t,x) = A
exp i
-
C
-2V
2
t+p
x+O
sech
X-X-Cp
t
(2.5)
with the constraint
Ao 2 2= C/K.
(2.6)
The solution has four arbitrary integration constants, Ao, po, Oo, and xo. These have
been chosen on account of their interpretation as average amplitude, momentum, phase and
position.
The average photon number no is given by f dx ao* (x) ao (x):
Ja(t,
0
2
atx
x)j2dx =
JA 2sech(
-x-Cp 0 t)2
2A0 2
dx = 20)2,=n
J
21
Id
(2.7)
In the subsequent analysis, we shall set po = Oo =
xo
= 0, which simply means that
we have chosen a coordinate system whose origin is at the pulse center, we have set the
phase equal to 0, and picked a momentum (or carrier frequency) that coincides with the
nominal carrier frequency oo.
When the ansatz (2.2) is introduced into the nonlinear Schrodinger equation, and
terms of order higher than first in Aa and Aat are dropped, one obtains a linear equation of
motion for these two operators:
-i
2
S C a 2_
Aa =
Aa + 2K Ia
2
0 2a+Ka
a
(2.8)
The equation couples Aa and Aat in a way characteristic of a parametric process.
Linear equations of motion of an operator are in one-to-one correspondence with linear
equations of motion of the classical evolution equation. In the integration of such equations
one does not encounter products of operators, for the inclusion of which one would have to
use the commutation relations. Hence, the integration can proceed "classically," as if the
operators were c-numbers.
We note that Aa consists of two parts: a part AaSe that describes the change of the
soliton parameters, i.e. a part that is associated with the soliton, and a part Aaon
1 that is not
associated with the soliton, the continuum part:
Aa = Aasol+
acont
(2.9)
The soliton perturbation is with respect to the four degrees of freedom of the
soliton: the photon number, the phase, the momentum, and the position. These
perturbations are all operators. They are functions of x. A solution of (2.8) has been
obtained[2] through separation of variables, using the solutions of the classical form of the
nonlinear Schr6dinger equation as a guide. The perturbation is written as a superposition
of operators with associated functions of x. The operators for photon number and phase,
An , and AO have the usual interpretation. The operator of the position, Ax^, is associated
with the displaced position xo, the operator for momentum with the shift from the carrier
22
frequency po. A carrier frequency shift Ap corresponds to a change of propagation constant
AP, hP being the momentum. It is important to note that the change of momentum of a wave
packet with the average number of photons no is equal to no Ap. Hence, it is natural to write
the perturbation in the form
A as
[A~n~x + A(t)f g(x) + AX(t)fx(x) +n,Ap(t)f (x)]exp
iK A 2
2
.
(2.10)
The functions fi(x) are the derivatives of the soliton evaluated at t = 0:
sech
tanh
fn (X)
f 0 (x) = iA 0 sech(
(2.11)
(2.12)
A
f (x)
sech
tanh
=
xsech (X)
fP(x) =
(2.13)
(2.14)
where we have fixed the phase by defining Ao real and positive. When the ansatz (2.10) is
introduced into the linearized nonlinear Schr6dinger equation one finds that no new
functions are generated by the derivatives with respect to x. Equating the coefficients of the
functions fQ(x),
Q
= n, 0, x, p, one finds equations of motion for the operator soliton
perturbations:
d
n = 0
dt
(2.15)
-AO$ = K Ah
dt
2
(2.16)
d Axi
C
- AP
23
(2.17)
dA^ = 0
dt
(2.18)
This is a review of the approach presented in Ref. [19]. Squeezing via a second
order nonlinearity operates on the uncertainty ellipse of the in-phase and quadrature
components of the electric field. Vacuum fluctuations, a stationary process, is represented
by an uncertainty circle. Squeezing is observed when the circle deforms into an ellipse of
the same area. This process is described by a Bogolyubov transformation. In order to
represent soliton squeezing by a third order nonlinearity in terms of a Bogolyubov
transformation it is necessary to renormalize the photon number and phase operators into
in-phase and quadrature component operators. There is an analogous squeezing process
that operates on the uncertainty ellipse of momentum and position. It is equally convenient
to express this process as a Bogolyubov transformation and thus introduce renormalized
momentum and position that have the same dimension.
2.2 Renormalization of the Soliton Operators
The perturbation operator Aa(x) has the commutator [Aa(x), Aat (x')] = 6 (x - x'),
and thus has dimensions of inverse length to the half-power. The photon number perturbation A n is given by A n = A (2AO 2 ) = 4A0 A A0 4 + 2AO2 A 4 = 2A0 A A 0
where we have
used the area theorem to relate the pulsewidth change to the pulse amplitude change. Next,
consider a cw wave of amplitude A0 and its associated photon number no = A 2. The change
in photon number is A n = 2Ao A Ao. When quantized, the perturbation A Ao would be replaced by the in-phase operator, A A 0 -+- A Al. This fact, and the dimensions of the Aa(x)
operator suggest that the soliton perturbation A Ao
AA
0
J4-AA1
is to be replaced by
(2.19)
Its associated expansion function is changed by the renormalization from the expansion
function of the photon number perturbation (2.11):
24
-
f l(x) =
sech
tanh
(2.20)
The same approach suggests the definition of the quadrature component as
A AA
-+ AA
2
(2.21)
We find for the expansion function
(2.22)
sech
f 2 (x) =
A similar renormalization is possible for the perturbation operators of position and
momentum. As we shall see, it is convenient to change the commutation relation by a factor
of 1/2. This is accomplished by the identification of the new operators Ak = AOAx/J
and AP = n Ap F4/2AO. The commutator is now:
[AAA] = 1
2
(2.23)
The respective perturbation functions become
fX(x)
=
tanh
sech
(2.24)
and
fP(x) =
xsech
(2.25)
The expansion (2.10) of the pulse is now in the form:
AaSol = [AA (t)fI (x) + AA(t)f
2
A2 t
iKA1
2
(2.26)
(x) +A AX(t)fx(x) + AP(t)fp(x)]exp
The adjoint functions are defined by
Re { f dxf,* (x)f, (x)} = mn
for m, n = 1, 2, P, Q and cont. It is easy to show that the equation adjoint to (2.8) is:
25
(2.27)
i
a
Aa=
C a2
Aa + 2K a
2
-Ka
2
_
Aa(2.28)
Note the change of sign in the last term. There is a simple physical reason for the
form of the adjoint: The physical process is parametric pumping. Such pumping can
produce growing and decaying excitations. Energy is not conserved in growth or decay.
However, cross-energy can be conserved between a growing and a decaying solution. The
growing and decaying solutions are shifted in phase with respect to the pump. This is the
origin of the phase change between the equation and its adjoint.
The solutions of the adjoint equation, properly normalized are
f1 (x)
=
-
t2(x) =
1
-sech
(2.29)
(
1J
fLX(x)
(2.30)
tanh ()]sech
X)
xsech (
=
(2.31)
and
f_,(x)
=
tanh
sech
(2.32)
The commutator of the in-phase and quadrature components is
[AAAA2 ]= if dxf 1 * (x) J dx'f 2 * (x') [Aa ' (x), A^(2) (x,)]
= -1/2 f dxf1 * (x) f dx'f 2 * (x') 6 (x - x') = i/2
(2.33)
and has the expected value.
It is clear from the preceding discussions that the expansion of a pulse excitation
into a soliton part and a continuum part is an expansion in a complete set of modes. These
modes are phase dependent, the components in phase with the pulse ao(t, x) are different
from those in quadrature. They form a complete set into which any excitation can be decomposed and whose amplitudes are quantized. Of course, the decomposition makes phys-
26
ical sense only when the expansion represents perturbations of a secant hyperbolic pulse.
But, the pulse need not be a soliton; e.g. it could be the secant hyperbolic pulse produced
in the output of a beam splitter with a soliton impingent on one of its input ports.
It is of interest to determine the mean square fluctuations of the soliton perturbation
parameters, if the background is zero-point-fluctuations. In this way one finds
<I(AA 1)21>= f dxf, * (x) f dx' fl (x') { Aa() (x Aa(1) (x)
= 1/4 Jdxfi* (x) fdx'f1 (x') 6 (x- x')
= 1/4 f dx! 1 * (x)12 = 1/2
(2.34)
The mean square fluctuations are twice the minimum uncertainty value for equal inphase and quadrature fluctuations. The remaining three fluctuations can be computed
analogously. It is clear that they involve the values of the integrals
2 = 2,(1.214), x2/6, 2/3; Q = 1, 2,X,P
fdx!fQ (x)1
(2.35)
The uncertainty products are
2
<I(AAi) ><(AA 2) 1> = 2.43/16
(2.36)
and
<l AX I2
P 21>= 1.09/16
(2.37)
The in-phase and quadrature fluctuations are uncorrelated
<LA 1I AA2 + AA2 A
11>=
0
(2.38)
In order to appreciate better the significance of the in-phase fluctuations, we return
to (2.19) and take note of the fact that the photon number fluctuations are given by
Ah=2A 0 A AO-2AO4AA1
(2.39)
Thus, the mean square photon number fluctuations are
(Mi 2)= 2 AO 2
A 2 >=<n>
(2.40)
They have the Poisson value. Hence, the in-phase fluctuations of a soliton with
twice the minimum value are, in fact, the fluctuations associated with a Poisson distribution
27
of photons. The renormalization has changed the uncertainty ellipse. In the photonnumber-phase description, the photon number fluctuations were at the Poisson value; the
phase fluctuations were larger than the minimum uncertainty[19]. In the in-phase and
quadrature description, the amplitude fluctuations are excessive, whereas the quadrature
fluctuations are close to the minimum value. This shows that the description of squeezing
is dependent upon the representation. In fact, the minimum uncertainty ellipse of the
momentum and position of the particle is plotted along axes of different dimensions and
thus the shape of the ellipse is not an indication of "squeezing". It is only when the
noncommuting variables are of the same dimensions and of the same character, such as the
in-phase and quadrature components of the electric field, when squeezing can be identified.
The stationary character of zero-point fluctuations guarantees equal in-phase and
quadrature mean square fluctuations of each mode of magnitude 1/4 each, giving an uncertainty circle of radius 1/2 and area it /4 (defining the circle as the locus of points of probability e-1). The projection of zero-point fluctuations into the soliton converts the circle
into an ellipse of area greater than r /4. The fluctuations along the major and minor axes of
the ellipse are uncorrelated. The position and momentum operators do not obey the standard commutator relation, but a new one, in one-to-one correspondence with the commutator of the in-phase and quadrature components. The renormalization leads to new
equations of motion for the operators. The derivation is parallel to that of (2.15)-(2.18) and
leads to the same form of the equations, but with new coefficients:
d
d
dt A
2
AA 1
2
=C
-AX
(.2
(2.42)
d
AP
-tAX=C-2
dt
2
d
-AP = 0
28
(2.43)
(2.44)
where we have used the area theorem KA20 = C/Y 2. The initial conditions for the equations
of motion for the operators are evaluated by projection of the excitations Aa(x) and Ait (x)
at t = 0. The renormalization has led to a welcome symmetrization between the pairs of
equations coupling in-phase to quadrature amplitudes, and renormalized position to
momentum observables.
2.3 The Continuum
The four soliton perturbation operators describe only part of the evolution of the
field, the continuum describes the rest. Gordon[20] derived the orthogonal functions in
terms of which the continuum can be expressed. They obey equation (2.8). Outside the time
slot occupied by the soliton, they are of the simple form exp (-iQ x) and have the same amplitude on both sides of the time slot. In the interval of overlap they change their amplitude
and phase:
fcs=
c[Q2 - 2 i Q tanh (x) - tanh2 (x)] exp - i(K x + Q2 t/2)
+c* sech2 (x) e(i t) exp i(Ki x + Q2t/2)
(2.45)
where
c = 1 for the in-phase component
c = i for the quadrature component
We have set C = K = 1 as we shall do henceforth in order to keep the notation
simple. There are two types of excitations
(a) In phase with the soliton denoted by the subscript c (reminiscent of cosine).
These excitations change the depth of the well since they affect the net intensity.
(b) In quadrature to the soliton denoted by subscript s (reminiscent of sine). These
excitations do not change the depth of the well and are the well known solutions of linear
scattering from a secant hyperbolic well[45].
The adjoint functions satisfy the adjoint equation (2.28). They are:
29
fC's= c[Q 2 -2 i
2
tanh (x) - tanh 2 (x)] exp - i(Q x + Q t/2)
- C* sech2 (x) e(i t) exp i(Q x + Q2t/2)
(2.46)
where
C = 1/(1 + Q2) for the in-phase component
c = i/(1 + Q 2) for the quadrature component
The continuum functions are orthogonal to the soliton perturbation functions in the
sense
Re { J dxfQ (x)fcs* (x, t = 0)}= 0;Q= 1, 2, X, P
(2.47)
The functionsfc(Q,x,t) are orthogonal tof(Q, x, t), and they obey the orthonormality
condition:
Re { J dxfc* (Qx, t)fc (Q',x, t)} = Re { f dxfs* (a x, t)f, (Q',x, t)} = 2n8(Q - Q') (2.48)
The soliton perturbation functionsfQ (x),
Q=
1, 2, X, P and thefc's (Q, x, t = 0) form
a complete set of functions in terms of which any excitation can be expanded. The crossorthogonality with their adjoints allows evaluation of the operator coefficients for any
given initial conditions. In this way one can express the continuum excitation at t, x:
Aacont
J dQ/2n
{c (Q)fc (x, " t) + Ps (Q)fs (x, a t)}
(2.49)
where the operators F, s (Q) are obtained by projection from the initial condition Aa (x, t =
0) and Aat (x, t = 0)
F, (Q) = 1/2 J dx {Aa(x)f*c,, (x, " t) + Aat(x)fes (x, , t)}
(2.50)
An initial Gaussian pulse would have a continuum contribution. In the case of
soliton propagation, the evolution of the basis functions is simple. The Gaussian pulse
would evolve into a secant hyperbolic pulse, the remainder would radiate away as
continuum. If the propagation is not along a guide characterized by the nonlinear
30
Schr6dinger equation, the evolution of this basis set would be complicated and use of the
expansion may not be useful. However, for the passage of the pulse and the continuum
through a beam splitter, the description in terms of the orthonormal set of modes is
convenient.
2.4 Soliton Squeezing in a Fiber
Propagation of a soliton along a lossless dispersive fiber leads to squeezing of the
soliton fluctuations. Squeezing occurs due to the coupling between operators. As pointed
out in Sect. 2.3, the uncertainty locus of the in-phase and quadrature components of a soliton in a zero-point fluctuation background is an ellipse, not a circle. The major and minor
axes of the ellipse are aligned with the in-phase and quadrature axes and the in-phase and
quadrature fluctuations are uncorrelated. This initial ellipse is distorted by the squeezing
process due to the coupling of the in-phase component to the quadrature component. Within the linearization approximation, the ellipse is stretched in the direction of the quadrature
component as shown in Fig. 2.1. Its area remains constant. The in-phase and quadrature
fluctuations become correlated. The fluctuations along the major and minor axes are uncorrelated. This we now proceed to show.
The solutions of (2.41) and (2.42) are
AA 1 (t) =AA 1 (0)
(2.51)
AZ2 (t) = AA 2 (0) + 2 D (t) AA1 (0)
(2.52)
and
where (D (t) =1/2 KA
2
t = 1/2 (CI, 2) t is the classical soliton phase shift.
31
AAA
dAAA1
Figure 2.1: The deformation of uncertainty ellipse.
These two equations describe the evolution of the uncertainty ellipse in the plane of the inphase and quadrature components. The mean square deviations along the in-phase and
quadrature directions are respectively
2
2
<[AA (t)]2>=
<[A A1 (0)] >
(2.53)
and
A 2
2
2 <
<[AA 2 (t)] > = <AA2 (0)] > + 40<I
A
with the cross-correlation
32
-
1 0 ]2 >(
(0)]>
. 4
(2.54)
1/2<AA 1 (0) AA 2 (t) + AA 2 (0) AA I (t)>= 2 ( <A2 (0)>
(2.55)
The process of squeezing can be treated in a formal way by establishing the correspondence
of the solutions (2.51) and (2.52) with the Bogolyubov transformation. We have
AA(t) = AAI (t) + i
2 (t) =
(t)AA (0) + v (t)AA (0)
(2.56)
with
[t = 1 + i 20 (t) and v = i 20 (t)
input
(2.57)
3 dBY
A~
85/ 15 coupler
coupler
local oscillator
Vacuum
squeezed
vacuum
fiber
(nonlinear)
-----
pulse
transformer
balanced
/ detector
output
Figure 2.2: The squeezing set-up.
The perturbation (2.56) accompanies the soliton pulse ao (t,x). We are now ready to
analyze the generation of squeezed "soliton vacuum" by the set-up illustrated schematically
in Fig. 2.2. The transform-limited secant hyperbolic pulse is incident upon one of the ports
of the Sagnac loop acting as a nonlinear Mach-Zehnder interferometer. A Sagnac loop is
chosen so that index fluctuations of the fiber, slow compared with the transit time, do not
lead to imbalance of the interferometer. The single coupler doubles as both input- and
output-coupler. The collision of the two pulses at the symmetry point has negligible effect
since the Kerr effect is weak and nonlinear phase shifts are accumulated only over fibers
33
that are many meters long. The energy of the secant hyperbolic pulse is adjusted so that the
two pulses in the two arms propagate as solitons. The zero-point fluctuations
accompanying each of the two pulses are produced by the superposition of the vacuum
incident from port (b) and the vacuum accompanying the pulse in input port (a). The
fluctuations accompanying each pulse are incoherent with each other. (The situation is
analogous to the splitting of thermal power by a beam splitter, each port of which has equal
thermal excitations.) The fluctuations are squeezed as indicated by equation (2.56). Upon
their return to the coupler, the classical (c-number), excitation exits into port (a) and the
squeezed zero-point fluctuations superimpose incoherently into port (b). The classical
excitation is used as the local oscillator in a balanced detector after an (optional) reshaping
in the pulse transformer. The noise accompanying the local oscillator pulse cancels. We
assume that the reshaping produces the local oscillator (L.O.) waveform:
ifL (x) = 1/2 {cos
wf1
(x) + sin Ngf
2
(x)} e(i1)
(2.58)
which can be put into the ideal form for the purpose of projecting out a linear combination
of AAi (t) and AA2 (t). In a real experiment the L.O. will of course produce gain G. For
convenience (2.58) is normalized for G = 1. The squeezed vacuum fluctuations emerging
from output port (b) are projected out in the balanced detector resulting in the net charge
operator[45]
A Q Iq= if dx {fL (x) Aa^0 - fL (x) Aai ts,,
=
{cos4 AA1 (t) + sinN AA 2 (t)} = 1/2 {e& AA(t) + elN A At(t)}
=
1/2 {e"
[g AA(O) + v AAt(0)] + elf [p* AAt(O) + v* AA(0)]}
(2.59)
where q is the electrical charge. The mean square fluctuations of the detector current follow
from (2.59). Equation (2.59) expresses the normalized difference charge in two ways: (i)
as the projection of a vector with components AA, (t), AA 2 (t) onto an axis inclined at an
angle V with respect to the (1) axis, and (ii) as the sum of the phase-shifted squeezed input
excitations g AA(O) + v AAt (0). The two representations are equivalent, but in particular
34
applications one may be more convenient than the other. We shall determine the degree of
squeezing and antisqueezing from representation (i). The mean square fluctuations of the
charge are
(JA
= Cos2
2
A 21 (t) 1> + sin 212<IAA 2 (t) 1>
+ sin (2W) 1/2 <1 AAI (t) A A2 (t) + AZ 2 (t) AA, (t)l>
If the projection ( A Q2 (t)I)
1/2
(2.60)
Iq is plotted in the (1)-(2) plane as a function of the
orientation angle V, an ellipse is traced out, the locus of the root mean square deviation of
the Gaussian distribution of (IAQ 2(t)I) /q 2 . According to (2.52) the component in direction
(1) remains unchanged, whereas the component in direction (2) shifts proportionally to
[<IAA12 (0)1>]1/2 . The mean square fluctuations along the two axes are
(2.61)
<1 AAi (0)1>, <AA22(0)1> + <AA1 2 (0)1>4(D2
The crosscorrelation is
1/2 <1 A Al A A2 + AA2
1 I>= 2 D < AA
(2.62)
1>
The probability distribution of the normalized difference charge in the (1)-(2) plane with
coordinates
, and 2 is a Gaussian given by
( 1,
2, t) oc exp -1/2 ( 421/0T
(t) + 4 2 2 1a 2 2 (t) + 24, 42/
where
35
12 (t)
)
(2.63)
a 1 1(t) ( 1 2 (t)
(2.64)
U21(t)(522(t)
(AA
2(t))
( AA I(t)AA 2 (t) + AA
I!
2
AA 22))
( A1 (t)AAl2 (t) + AAl1(t)A'2(t)J)
( AA
^2
(0))
2
(t)AA 2 (t)
I
222(t)
_20(t) il + 4(D (t)
and
= <1 A 22 2(0)1>/<1
A1 2(0) I> = 0.607. The Fourier transform of the probability
distribution expressed in k-space, the characteristic function, is of the form
C(k1 , k2, t) c-< exp - 1/2 (ayi (t) k21 + (22 (t) k2 2+ 2Y12 (t) k, k 2 )
The quadratic form in the exponent of the characteristic
(2.65)
function can be
diagonalized by a reorientation of the axes. A coordinate transformation into new
orthogonal coordinates ki' and k2' finds the mutually orthogonal directions along which the
fluctuations are uncorrelated. These are the major and minor axes of an ellipse. The
transformation is a unitary transformation of the matrix which leaves the eigenvalues of the
matrix (2.64) invariant. The eigenvalues are
2i
± = <I AA, (0)I>{ (1 +r + 4D 2 )/2±[(( 1 +,q + 42 )/2)
2
/
1/ 2
}
(.6
(2.66)
The product of the eigenvalues is
'2
+X-=i<1 AA1 (0) I>
2
(2.67)
and is constant, independent of the degree of squeezing. The squeezing and antisqueezing
is illustrated in Fig. 2.3. With zero phase shift, the fluctuations in the (1) direction are shot
noise fluctuations. These are equal to twice the zero point fluctuations of 1/4. In the
orthogonal direction, the fluctuations are less, but they are still larger than 1/4. As the
36
nonlinear phase shift increases, the branch that represents shot noise at D = 0 shows
monotonically increasing fluctuations, whereas the orthogonal direction decreases and
reaches zero asymptotically. Figure 2.4 shows the fluctuations as a function of phase angle
V for different degrees of squeezing and antisqueezing. This figure shows that the regime
of phase angle within which a large degree of squeezing is observed becomes narrower and
15 10 5
dB
1
2
3
4
5
-5
-10
20D
-0
Figure 2.3: The squeezing and anti-squeezing (the minor and major axes of the squeezing
ellipse) as functions of 20.
8
4
20 10 3g
2
dB
0.5
1
1.5
-10 -1
2
2.5
ow
Figure 2.4: The root mean square fluctuations as function of the phaseangle with respect
to local oscillator: 2D = 2, 4, 8
37
narrower as the degree of squeezing is increased. The greater the degree of squeezing, the
harder it is to find the squeezing angle and stabilize the system at that angle.
2.5 Continuum Contribution to Squeezing
The projection with a local oscillator orthogonal to the continuum permitted us to
evaluate the perturbation of the soliton without consideration of the continuum. In practice
it is inconvenient to reshape the pulse for use in the local oscillator. Any reshaping introduces phase delays that may undergo fluctuations and prevent optimization of the relative
phase between LO and the squeezed vacuum. Use of the secant hyperbolic pulse as the local
oscillator pulse leads to detection of the continuum contribution. We shall now show that
the penalty incurred when a secant hyperbolic LO pulse is used is not severe and, in fact,
at some phase shifts the noise is less than for the case of a local oscillator pulse orthogonal
to the continuum.
The first order perturbation of the pulse is composed of the soliton perturbation and the continuum:
Ad = Adsol + Adcont
(2.68)
When the local oscillator pulse is a secant hyperbolic,
ifL(x,t) = (1/2) sech (x/4) e(it/2) e(iV) = (1/2)f 1 (x,t) e(1V)
(2.69)
The function f1 (x,t) is the adjoint projection that evaluates AA 1 (t) from the perturbation
Aa(x,t). In addition, the local oscillator waveform contains the phase factor exp(iV). We
first look at the contribution to the difference current of the soliton part of the squeezed
radiation. Equation (2.65) must now be modified to take into account that the quadrature
component is projected out with this particular function, namely sin x f 1(x,t), rather than
sin AVf
2
(x,t). We find that (2.59) changes to:
AQS0 /q= i f dx{ f*L (x,t) A ^ 0i -fL (x,t) A solI} ={cosipAA 1(t)+2 sinNfAA 2 (t)}
The mean square fluctuations due to the soliton part are
38
(2.70)
(t)) soJ/q 2==<CO s2
(2A IAQc(ts
<
AA 1 2 (t) 1> + 4 sin 2 1y <1
(t)l>
AA2 2 ()I
(2.71)
+ sin (21g) <1 AAI (t) AA 2 (t) + AA 2 (t) A AI (t)I>
with the mean square fluctuations of the in-phase and quadrature components at t = 0 the
same as before. The matrix
.sol'sol)
analogous to (2.64) is
(AA^2 (O)[ 1
GII(t) C 12 (t) solsol
(0 )(2.72)
=
22]
_721 (t) G22(t)
4D(t)
2
1I
40(t) 4[il + 4(D2(t)]_
where T1 = 0.607. It is of interest to determine the major and minor axes of the squeezing
ellipse due to the soliton noise alone, as affected by the nonideal local oscillator waveform.
This is shown in Fig. 2.5. As the squeezing increases, the minor axes of the ellipse turns
more and more closely into the (1) direction. In this direction, the sech local oscillator is
the projection function orthogonal to the continuum. For this reason we see that the
squeezing and antisqueezing approaches more and more the values (2.66) with increasing
phase shift D. However, an undesirable consequence of the increased coupling to AA 2 (t)
is the narrowing of the range of angles over which squeezing is observed as shown in Fig.
2.6 where we compare the squeezing as a function of angle as evaluated from (6.5) and
(2.64) for D = 1. This is due to the significant increase in the value of the anti-squeezing
39
eigenvalue.
20 15 -
dB
10 5
.
deal LO
3
2
-5
sech LO
4
5
-10 -
20D
Figure 2.5: The minimum and maximum fluctuations of the soliton alone as detected by
local oscillator of secant hyperbolic shape. Comparison with ideal local oscillator use.
15
sech LO
10
dB
ideal LO
5
0.5
-5
1
1.5
2
2.5
3
-
Figure 2.6: The fluctuations of the soliton alone as detected by local oscillator of secant
hyperbolic shape as function of phase angle with respect to local oscillator, (D = 1.
The secant hyperbolic local oscillator waveform couples to the continuum as well.
We encounter a new ingredient of the detection process: beat terms of the continuum with
the soliton noise are found. This is the consequence of the fact that the noise within the
40
soliton does not commute with the noise of the continuum:
[Aado 0 , Aatcont] + [Adcont, Aatso;]#O
(2.73)
The partial coherence between the noise in the continuum and the soliton noise is
the consequence of the fact that the linearized NLSE is not self-adjoint. We construct a
matrix of the self- and cross-correlations of the terms mutliplying cos Xy and sin Ag
respectively as in (2.64). To systematize this step we construct separately aij(sol,
sol)
G (cont, cont), and a j(sol, cont), three matrices, from the sum of which we may construct the
major and minor axes of the squeezing ellipse. The matrix aijfso'
Sol)
was given in (2.64).
Note that the only nonzero components of the matrices ay (cont, cont), and a (sol, cont) are the
22 components, by virtue of the fact that A Qcont has no contribution to the cos A' term. The
22 (cont, cont) and a 2 2 (sol,cont) is carried out in Appendix A.
evaluation of
The results are:
CY(cont,cont) - i/8 J dM sech2 (
22 (sol,cont)= 1/4
(contcont)
and
J dQ
C 2 2(contsoO
IQ/2) cos 2 [( + Q2 )t/2]
(2.74)
1/(1 + Q2){2/3 + 7r2/6 Q tanh(tQ/2) + t2 g3/6 tanh(Q/2 )}
sech2(Q/2) cos [(1 + Q2)t/2]
(2.75)
are plotted in Figs. 2.7a and 2.7b. Both quantities oscillate
depending on the phase between the local oscillator and the continuum. As the phase (D
increases, this contribution gets smaller and smaller. Collecting the three matrices, we may
evaluate the eigenvalues of the sum matrix that give the squares of the major and minor
axes of the squeezing ellipse. Figure 2.8 shows the amount of squeezing (a) in the case of
a L.O. pulse orthogonal to the continuum, (b) without continuum and a sech L.O., and (c)
with the contribution of continuum. We find that the continuum improves or reduces the
amount of squeezing relative to both (a) and (b), depending on the phase between the
continuum and the soliton. However, as before, we find that at large degrees of squeezing,
all three curves approach each other. In this limit, inclusion of the continuum has negligible
41
effect on the anti-squeezing branch, which is already a large number.
2.6 Summary
This chapter renormalized the four soliton perturbation parameters in a way that, pair-wise,
the equations for the noncommuting observables became identical in form. The dimensionless expansion functions permit the quantization of any secant hyperbolic pulse in terms a
complete set of expansion functions, including a pulse that is not a soliton, such as the pulse
emitted into a linear medium from the end of a fiber. Of course, the evolution equations of
the operator coefficients must be changed to accommodate the change of the medium of
propagation. The renormalization cast the squeezing formalism in terms of a Bogolyubov
transformation. If the initial excitation is described by a soliton in a zero point fluctuation
background the in-phase operator does not have the minimum uncertainty value, unlike the
photon perturbation operator which exhibits Poissonian fluctuations. The local oscillator
pulse shape in the balanced detector orthogonal to the continuum is a superposition of the
functions sech (x/4) and i[l - (x/4) tanh (x/4)] sech (x/4). If a secant hyperbolic pulse is used
for convenience, coupling occurs to the continuum. The squeezing of the continuum is partially coherent with that of the soliton. The squeezing of the continuum is oscillatory. In
fact, the continuum contribution can improve the squeezing beyond that achievable with an
LO whose shape is adjusted to be orthogonal to the continuum.
42
(cont,cont)
22
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0C
5
10
I
I
15
20
25
(a)
(cont,sol)
a 22
0.8
0.6
0.4
-
-
0.2
0
-7
-0.2
-0.4
-0. 6
0
5
10
15
20
25
(b)
Figure 2.7: The matrix elements (a) 022(cont,cont) and (b)
(D. Note the "beats."
43
G22(sol,cont) as
a function of phase
51
0 -
m[4
N
without continuum
.
with continuum
~--
ideal
= -10-
Co
-20
0
0.5
1
1.5
2
2.5
3
3.5
4
Figure 2.8: The minimum fluctuations detected by a local oscillator which is orthogonal
to continuum, and by a secant hyperbolic local oscillator with and without continuum.
44
Chapter 3
Squeezing with a short piece of fiber
The squeezing scheme in this chapter uses femtosecond pulses and a short piece of
fiber. By using a short piece of fiber we avoid problems caused by GAWBS and Raman
noise. Sub- IOOfs pulses are needed to achieve significant nonlinear phase shift over such a
short fiber length. A stretched-pulse laser has been built to fulfill this purpose. The large
bandwidth(>50nm) generated by such a short pulse laser implies that a broadband splitter
must be used to separate the pump and the squeezed vacuum. If the splitter deviates from
the stringent accuracy required of squeezing experiments within the broad bandwidth then
excess pump light will leak into the squeezed port and mask squeezing. Very few optical
splitters can meet this broadband requirement but a polarization beamsplitter(PBS) is one
of them. Thus for this experiment cross phase modulation of the fiber is used as the dominant nonlinearity for the generation of squeezed vacuum and a PBS follows the fiber to
achieve the separation of pump light and squeezed signal. Thus the most challenging part
of this experiment is building a low noise, high power laser. In the first section of this
chapter we present a detailed noise characterization of our pump laser: the stretched pulse
fiber laser.
3.1 Laser and its Noise
3.1.1 Introduction
The stretched pulse laser[52-55], first invented by K. Tamura et al., uses sections
of fiber with large positive and negative group velocity dispersion(GVD). The pulse is
temporally stretched and compressed in one round trip. This stretching serves to lower the
peak power of the pulse and thus reduces the overall nonlinearity the pulse experiences.The modelocking mechanism is Additive Pulse Modelocking(APM)[56]-[62]. APM
45
converts the reactive Kerr nonlinearity into fast saturable absorber action. By coherently
interfering two self-phase modulated(SPM) versions of the mode, it converts phase modulation into amplitude modulation. A schematic of the all fiber APM stretched pulse ring
laser is shown in Figure 3.1. The laser consists only of erbium doped fiber(EDF), regular
SMF-28 fiber, a polarization sensitive isolator, and polarization controllers.The pumping
is done via a WDM coupler, and the output is taken from a 10%-90% coupler. The polarization controllers and the polarizer produce APM action by transforming linear polarization into elliptic polarization, followed by SPM in the fiber which rotates the ellipse. The
polarizer transforms the rotation of the ellipse into amplitude modulation.The isolator
forces the unidirectional operation; and a second polarization controller compensates the
spurious polarization transformation in the non-polarization-maintaining fiber. Since the
excited states of the erbium ions have very long relaxation times, the ring laser is not sensitive to the pump noise and gain fluctuation except at very low frequencies, and is particularly well-suited for noise study.
Polariz at ion
Cont rollers
90/10
Coupler
Isolator
with
Polarizer
Pulse Output
PositiveNegative
ispersion
Erbium Doped
Pump Input
Fib-er
980/1550
WDM Coupler
Figure 3.1: Stretched pulse laser schematic
46
The stretched pulse laser has pulse energy as large as 3nJ per pulse[53], and pulse
widths as short as 64 fs[57]. It already has found applications as the source in pulse amplification[63], WDM communication, second harmonic generation, and supercontinuum
generation[64], and is commercially available[65]. The laser is also attractive as a source
for pump-probe diagnostics, oscillator synchronization, and fiber squeezing experiments[66]-[68] and could be the successor to the modelocked color-center laser. Because
of the potentially wide range applications of the stretched pulse laser, its noise characteristics are of great importance. In a companion paper a theoretical model has been proposed
to treat the noise of the stretched pulse laser[5 1]. This model employs perturbation theory
and is similar to the Haus-Mecozzi noise theory for modelocked lasers in the soliton
regime.
In Section 3.1.2 of this chapter we review the noise theory in [51]. Only the first
order effects are included in this review. The equations of motion for different perturbation
parameters are presented. Section 3.1.3 contains the experimental setup. The power spectrum technique is used in this study and only the amplitude fluctuations and timing jitter of
the laser are measured. Section 3.1.4 contains the experimental results. Some necessary
conditions for quiet laser operation are discussed. Section 3.1.5 compares the first order
theory in Section 3.1.2 with the experimental results in Section 3.1.4. In addition to qualitative comparisons, the amplitude noise and timing jitter due to amplified spontaneous
emission(quantum) noise are numerically estimated. The timing jitter is shown to be
nearly quantum-limited when the measurement time is on the order of 0.1 seconds.
3.1.2 Theoretical Overview[51]
The output from the stretched pulse laser can be approximated by a gaussian with
pulsewidth(ro) and chirp parameter(),
47
t2
a
A0
1
2,r 21 - j$
1
e
(3.1)
A pulse has six degrees of freedom: height, width, phase, chirp, carrier frequency, and timing. It can be shown that adiabatic (slow) changes of the pulse leave the chirp parameter
invariant and change the height and width so as to leave the product of the peak intensity
and pulse-width to the fourth power fixed (a form of an area theorem for the stretched
pulse laser). Hence, there remain four parameters of the pulse that undergo changes under
the influence of noise excitation: energy(Aw), timing(At), frequency(Ap, where p stands
for the deviation of the angular carrier frequency o from a nominal value o, p=o 0 -o),
and phase(Ae). The phase fluctuations do not couple to the remaining perturbations and
hence we require equations of motion for only three parameters. They are obtained from
the modelocking equation, supplemented by a noise source S(t,T), by projection via
adjoint functions:
TR
~ ~
+ TRSW
(3.2)
p
TR
At = -4|DIAp - 4
g Ap + TR St( T)
(3.4)
g
where t, and rP are relaxation constants. Energy fluctuation relaxes mainly due to gain
saturation and frequency fluctuation relaxes mainly due to filtering(- = IT P
TR
2 IT2
g o
Here, TR is the round trip time; T is a time variable on the scale of many cavity round-trip
times; g is the incremental gain per pass,
g is the gain bandwidth; D is the net group
velocity dispersion(GVD) due to an imbalance between positive and negative dispersion
contributions of the two fiber segments; ro is the minimum pulse width during each round
48
trip; the noise sources Si(T) are defined by projecting the respective orthonormal functions
onto the noise source S(t,T). Equation (3.4) indicates that because of GVD and chirp, a
change in pulse frequency deviation changes the group velocity.
Equations (3.2) -(3.4) can be easily solved using Fourier transform. The energy
fluctuations spectrum is
<IS (1
2>
<IA w(Q)l > =
(3.5)
(22+
/
W2
the frequency fluctuations spectrum is
(Q)I12>_
-<ISa
2
(3.6)
(Q2 +
2
the timing jitter spectrum is:
2
p
2
<IAt(T)I >
=
2
( 2 +
/
2
Ist ()I 2>
+
+
-2Re
(3.7)
4 D+P -
TRg J<IS
2 P(Q)S t* L00
TR
Q 2 (jQ+ 1/t )
The asymptotic behavior for large T is determined by the leading singularity, which has a
spectrum of the form 1/Q2. This corresponds to random walk, of which the r.m.s. value is
of the form: <IAt(T + T 0 ) - At(T 0 ) 2 > = Dt T , where Dt is a diffusion constant. This
random walk is the reason that all free running passively modelocked lasers have large
49
timing jitter at low frequencies. Thus the removal of this leading singularity via active
retiming of the pulse train[88] is a very effective method to reduce the timing jitter in passively modelocked lasers. Note that the cross correlation terms in Equation (3.7) are small
for small chirp.
The spontaneous emission noise(quantum noise) can be modeled by a white noise
source[80], [81],[85]. The sources Si's that are given in Equations (3.3) and (3.4) and drive
the frequency and timing deviations have the correlations:
<S.(T)S .(VT)> = D..(T- T')
1
J
ii
with "diffusion constants" D
D
= (1 + 0 2 )213
2 T
wt T R
(frequency-timing
gh-o
2w
D tt =
PP
2 5/2
= (1+3 2)5
(3.8)
(frequency deviation),
cross-correlation),
and
TR
2
2 3/2to 2g
Th
0
(1+1)
W
(timing deviation), where 0 is the enhancement factor
TR
due to incomplete inversion of the medium, and wo is the steady state pulse energy. Timing jitter induced by spontaneous emission can be calculated by solving (3.4) using (3.8)
as the noise source. The r.m.s value of the quantum limited timing jitter can be obtained by
Fourier transforming Equation (3.7). In the limit
T >> rP and large net GVD it has the
form of a random walk as mentioned before and is approximately
<lAt(T + T 0 ) - At(T 0 ) 2> = BT
(3.9)
where T is the measurement time and
)2
16
g
8L1
+D
g
TR 2
2
+D
8g 2
Q
9
2
-
D t +D
pt p
tt
R
TR
When the pulse train impinges on a detector, the power spectrum of the detector cur-
50
rent is [85]:
<2ik
jkO
> = A+
<2'w(
) >+k2 Q 0<lAt(Q)l > +
(3.10)
[<At* (Q)Aw(Q)>-<At(Q)Aw*(Q)>]
where A is a scaling constant, and De=2K/TR. Equation 3.10 reduces to the expression
derived by D. von der Linde[86] by neglecting the cross-correlation terms. The amplitude
noise has a contribution independent of harmonic number, and the timing jitter grows quadratically with the harmonic k.
In reality, an RF spectrum analyzer has a finite integration time so that the timing jitter power spectrum does not have a singularity at Q=0. This can be accounted for by solving the corresponding transient problem for 3.4. Assuming "white" noise source, Q << 1/
T, and T >> ,p, we obtain the following expressions for the power spectrum corresponding
to the timing jitter:
for K <> 0,
2
B
<lAt(O)l 2> = 2
and for Q=0,
(3.11)
2
<lAt(Q) 2 > = B T-
6
(3.12)
Note that Equation (3.11) is the same as (3.7) when the noise source is white and Q << 1/
TP. Equation (3.12) predicts that the singularity no longer exists if T is not infinite. The
noise power spectrum increases quadratically with T at Q=0. So if the noise source is
white, Q << /tU, and T>>tP, the timing jitter structure does not deviate noticeably from
the infinite measurement time limit except near Q=0[51],[87].
51
3.1.3 Experimental Setup and Procedure
The laser shown in Figure 1 consists of SMF-28 fiber(P 2 =-23ps 2 /km), Coming Flexcor 1060 fiber(
2=-7ps
2 /km),
erbium fiber 1 from AT&T( 2=75ps 2/km), and erbium fiber
2 from INO NOI(P 2 =14.5ps 2/km). A Master-Oscillator Power Amplifier(MOPA) from
SDL. Inc. is used as the pump source. A quiet pump such as the MOPA is essential for
noise studies. The pump power is monitored using an HP power sensor module(HP
81533B and HP 81525A) via a 1%-99% coupler. The power meter reading of the 1% port
reveals that the low frequency pump noise is less than 1% of the power coupled into the
WDM coupler. The laser is first biased at relatively large net GVD(p 2 =0.0169ps 2 ), where
the jitter structure is more pronounced. The length of the ring in this case is 3.6 meters,
which leads to a 55.62 MHz repetition rate. The cavity's net GVD is then decreased by
adjusting the length ratio of the erbium doped fiber to SMF-28 fiber in the laser cavity. The
experimental setup is shown in Figure 3.2. Through flip mirrors, the collimated laser output is either sent into an HP Optical Spectrum Analyzer(OSA) to measure its optical spectrum, or an autocorrelator, or the HP power meter to measure its average power, or a fast
photodetector. Since the chirp of the output is negative, a concatenated SMF-dispersioncompensating-fiber(DCF) is used to externally recompress the pulse. Autocorrelation is
then taken for the compressed pulse. A 45 GHz photodetector from New Focus is used to
obtain the scope traces and the sampling scope (25 ps rise time) traces. A typical scope
sampling scope trace for the laser output is shown in Figure 3.3. Single pulse operation is
ascertained through the sampling scope and the autocorrelator. A detector with a 1.2 GHz
bandwidth is used for noise measurements because of its superior dynamic range. The
power spectrum is obtained by feeding the photodetector output to an RF spectrum analyzer. Two RF spectrum analyzers, an Advantest R3265 and an HP 8560E are used. Their
noise sidebands are estimated from their DC components with their inputs terminated.
Unless specified, the Advantest R3562 with a measurement time of 0.09 second[36] at
52
RBW=10 Hz(digital-IF mode) is used because of its measured local oscillator noise sidebands are smaller(Fig 3.4a). The HP 8560E is used when better resolution(l Hz) and
Laser Output
OpPulse
Fiber
Lase
Co
1%
Detector
R
Len
Scope
9
I Autocorrelator
MoP
Powermeter
I
Figure 3.2: Experimental setup
Figure 3.3: A typical sampling scope trace(20ps/div)
53
ter
-20
-40
-60
CI
lit
. 101
LMis
-80
-100
-1
-0.5
0
KHz
(a)
0.5
1
-20
-40
-60
0
-80
-100
-1
0
-0.5
0.5
1
KHz
(b)
Figure 3.4: (a) Local oscillator of R3265 with RBW=IOHz. (b) Local oscillator of HP
8560E with RBW =1 Hz, the peak is -15dBm above the maximum shown in the graph.
54
longer measurement time(l.92 seconds at RBW = 1Hz, digital IF mode) are needed[68].
Its measured noise sidebands are shown in Figure 3.4b. The noise structure of the
stretched pulse laser is buried under the local oscillator noise of the HP 8560E when RBW
is 10Hz, as will be shown later. Note the measurement taken with the HP spectrum analyzer looks similar to those taken with an analog analyzer because of the "rosenfell" algorithm used[70]. The measured noise sidebands match well with typical data provided by
the two companies[68],[69]. By blocking the optical beam, we verified that the measurements are not limited by the noise floor of the measurement system. No effort has been
made to stabilize the laser externally except shielding it with foam to minimize length and
refractive index fluctuations due to thermal effects.
3.1.4. Experimental Results.
A. Noise of the stretched pulse laser when GVD=0.0169ps 2
Unlike the soliton laser, the stretched pulse laser is not tunable. Moreover, it has
very little continuum generation[7 1]-[74]. The GVD of the laser cavity can not be measured during pulsing[75],[76]. In this case, the measurement of the laser's GVD is carried
out in cw operation. By adjusting the polarization controllers, we can excite a narrowband
cw radiation, and change its center lasing wavelength. The center lasing wavelength is
observed using the OSA(minimum RBW = 0.1 nm). Because this cw light consists of multiple axial modes, the beats of these modes can be observed on an electrical spectrum analyzer(e. g. Figure 3.5) after an optoelectronic detector. Changing the center wavelength
leads to a frequency shift of these beats, and the cavity's GVD can be calculated from the
shift. The measured value is 0.0169ps 2 . The GVD is also calculated based on the fiber
parameters and is found to be 0.01 84ps 2 . The small discrepancy could be due to the uncertainties in the specification of the fiber geometry or pump induced change in refractive
index of the erbium fiber[77],[78]. The measured value is used in the subsequent calculations.
The co-existence of cw and pulse(hybrid state)[55] may lead to relaxation oscillations[82]. Moreover, cw adds its own noise. Figure 3.5 shows the optical spectrum and RF
55
spectra of a hybrid state. The two structures corresponding to the pulse and the cw can be
easily identified. The cw structure overwhelms the pulse noise. Noise measurements were
made in the absence of a cw component which was eliminated by lowering the pump
power or by enhancing the APM action by adjustment of the polarization controllers.
i) Energy fluctuations
Two types of energy fluctuations have been observed. One occurs at very low frequencies.
The source of this energy noise is classical: i.e. pump noise, thermal noise, electrical
power line noise, etc[87]. Because of the long relaxation time of the erbium ions, this type
of noise is low pass filtered with a cutoff around 1 KHz. However, for the stretched pulse
laser reported in this paper this narrowband amplitude noise is buried under the low frequency noise sidebands of the Advantest spectrum analyzer(Figure 3.4(a) and Figure 3.6).
Small disturbances such as the cooling fan on the MOPA can add AM resonance sidebands 10dB above the noise floor[83]. The data presented in this paper are taken while the
cooling fan is temporarily disconnected. The other amplitude noise structure extends from
-500KHz to 500KHz(Figure 3.7a). This noise structure represents energy fluctuation equal
to 0.05% of the pulse energy.
-20
-30
*-
-40
-50
AN-
.
.
-
-
-
.
.
.
.
-60
-70
-
-
-oil-
-
-80
-90
14 50
1500
1550
1600
Wavelength, nm
(a)
56
1650
-20
I
H Armol i C. 1
-40
S
-60
0
-80
-100
-20
-10
0
10
20
KHz
(b)
-40
-60
-80
CW
0-100
-120
-20
-10
0
10
20
KHz
(c)
Figure 3.5: Hybrid state (a) Optical Spectrum. (b) Harmonic 1. (c) Harmonic 21.
57
At higher pump power, the amplitude fluctuation structure exhibits pedestals(Figure 3.7b),
which may be due to gain relaxation. This energy fluctuation pedestal can be avoided by
reducing the pump power.
ii) Timing Jitter
Only one timing jitter structure has been discovered. It spans the range [-1 KHz, 1
KHz]. The structure has the shape of a 1/Q2. This shape is induced by white noise, as predicted by Equations (3.7), (3.11) & (3.12). The very low frequency white noise jitter is
buried under the signal. To include this portion of the jitter, Equations. (3.11) and (3.12)
I
I
I
I
I
I
I
I
I
I
I
I
I
I
Har nonic
-40
I
1
-60
----------
.-80
S-100
- - -
-
-
-
- - -
----
-
-
- -
-
-
-
-120
-1
-0.5
0
KHz
0.5
1
Figure 3.6: Energy fluctuation at very low frequencies. Note that in this case the noise
structure is buried under the local oscillator noise.
58
-40
-60
'N
0
-80
-100
-120
-400
0
-200
200
400
KHz
(a)
-40
armonic
-60
PP
-80
E
-100
-120
-400
-200
0
KHz
(b)
200
400
Figure 3.7: Broadband Energy Noise: (a)without pedestal. (b) with pedestal. RBW =
1KHz
59
are used to fit the data based on the l/22 wings at higher frequencies, with the noise
sources as fitting parameters. This fitting is then numerically integrated and normalized to
the peak value of the signal. The result of the integration gives the r.m.s. value of the noise,
according to Parseval's theorem. Repeating this procedure for several harmonics, and taking into account the resolution bandwidth of the spectrum analyzer, we use the following
relationship to do a least square fit of the jitter data[86]:
jpj(E)dQ
JQ(= P
AQ
(2i)k)
res o
2(At)2
T
\T)
(3.13)
where Pj(Q) is the noise spectrum, and PO is the peak power of the signal, and (27rk) 2
comes from von der Linde's result that the jitter increases quadratically with k.
We then vary the output power by varying the pump power alone. The maximum
output power without inducing the energy noise pedestal(Fig. 3.7b) is 2.33mW. The optical spectrum and the autocorrelation of the unchirped pulse for this output power are
shown in Figure 3.8. Note that the optical spectrum can be fitted reasonably well by a
Gaussian(the dashed line), as predicted by Equation (3.1). The two broad humps near the
center frequency are typical in systems with SPM and positive GVD[84]. The cause of the
sidebands near the wings of the spectrum is still unknown, and could be due to continuum
generation[71]-[74]. Figure 3.9 shows Harmonic 21 of the laser output and its assumed
Lorentzian-shaped jitter structure. The fit is reasonable. Using (13) and fitting the jitter
data, we find that the jitter is l3ppm of the round trip time, as shown in Figure 3.10a. The
timing jitter increases as the output pulse energy decreases. Calculated rms jitter values at
different output power levels are plotted in Figure 3.1Gb. Though these values are within a
small range, the higher harmonics(k>15) at large output power consistently have larger jitter structures than their counterparts at smaller output power.
The timing jitter also increases as the measurement time increases. When operating
at 1 Hz resolution bandwidth, the HP 8560E's measurement time is 1.92 seconds[68]. Figure 3.11 shows that the timing jitter is significantly larger than in the previous case with
measurement time = 0.09 seconds. Even though Equations. (3.11) and (3.12) still fit rea-
60
sonably well, we believe that some of the jitter is due to the well known 1/f noise. The jitter is estimated to be 1 l6ppm. The HP unit is able to clearly resolve spurs 60Hz away
from the carrier for all the harmonics. These spurs are due to electrical power supply of the
MOPA pump. Note that the height of these two spurs has increased dramatically in Harmonic 21. This can be due to AM-FM conversion.
B. Laser noise when the GVD < 0.005 ps2
Decreasing the net GVD by adding SMF-28 fiber leads to smaller pulse energy
because of enhanced soliton effects[56]. In this case the lengths of both the erbium fiber
and SMF-28 are adjusted to achieve an average output power of 1.26mW, a power that is
sufficient to overcome the detector noise. The repetition rate is 49.98 MHz. The GVD is
estimated to be smaller than 0.005ps 2 . The optical spectrum and the autocorrelation of the
compressed pulse are shown in Figure 3.12. Again the optical spectrum is well approximated by a Gaussian, as indicated by the dashed line. The two energy fluctuation structures
-20
-30
. .
. . .
. .
. ..
-40
E
-50
-60
-70
-
-
-
-
-
-
-
-80
-90
14 50
1500
1550
1600
Wavelength, nm
(a)
61
1650
0.25
0.2
F HM
0.15
0.1
0.05
0
-600-400-200
200 400
0
600
fs
(b)
Figure 3.8: (a)The optical spectrum, and (b) the autocorrelation of the compressed pulse
corresponding to the case of GVD=0.0169ps 2 and highest output power. The dashed line
in (a) is a theoretical gaussian fit of the spectrum.
Harmonic 21
-40
-A
-60
S
-iJ
-80
-
-100
I
g
-I
-120
e
-1
4
0
-0.5
0.5
'
1
KHz
Figure 3.9: Jitter fitting for harmonic 21, measurement time = 0.09 second. Dashed=data,
Solid =fitted jitter.
62
4
10-6 -
3.5 10-
Measurement Time(T) = 0.09 Sec
Jitter = 13 ppm
3 10~6
0
2.5 10-6
0
2 10~6
z
-0
1.5 101 10-6
5 10~7
I
I
15
20
U
10
5
Order of Harmonics
(a)
0. 5
0.45
I
-
0
Experiment
I
Theory
-
0.4
0.35
0.3
0.25
0.2
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Intracavity Energy [nJ]
(b)
Figure 3.10: (a) Timing jitter due to white noise, jitter= 13ppm. (b) Timing jitter as a function of output power. The theoretical curve is the quantum jitter using the following
parameters: co=51 fs, Gain bandwidth =40 nm, D=0.5 * 0.0169 ps2 , (x=0.2, g=0.8, TR=1 8
63
ns, 0= 3.2. Measurement time =0.09 second.
-60
-80
-100
0
-120
-140
-1
-0.5
0
0.5
1
KHz
Harmonic 21
Dashed = Data=ftfer--S
-80
+ Ha rmonic 1
Il
-100
"U5K
ii.'ri
-120
-
0
-
-
r
- II-
-140
-160
_-
Iii ii~
Ii.
._7-171!I
-1000
I
I
I
I
LIj~
I--
i
-
~I~I
'il
r
iI
1
-500
-
0
i Ia...
III.
,I.
500
~
L
-
I
1000
Hz
(b)
Figure 3.11: Data taken using the HP 8560E, RBW=lHz, Measurement time=1.92 seconds: (a)Harmonic 1, (b) Harmonic 21. Dashed curve is the experimental data. Solid curve
is the sum of harmonic 1 and the theoretically predicted lorentzian-shaped jitter. The signals peaks are 30 dBm above the maximum shown in the graph.
64
-10
-20
-30
E
P
rj
.
. . .
. .. .
.
-40
.
-50
-60
-70
-80
1450
1600
1550
1500
1650
Wavelength, nm
(a)
0.04
0.035
0.03
0.025
101 fs
0.02
0.015
0.01
0.005
0
-600
-400
0
fs
-200
200
400
600
(b)
Figure 3.12: (a) The optical spectrum and (b) the autocorrelation of the compressed pulse
corresponding to the case of GVD<0.005ps 2 . The dashed line in (a) is a theoretical gaussian fit of the spectrum.
65
Solid=
-Dashd
-40
armonic
1Iarmoni 21
-60
~-80
-100
I'
-I -- - -
-120
-1
0
-0.5
0.5
1
KHz
Figure 3.13: RF spectrum, Solid=Harmonic 1, Dashed = Harmonic 21.
remain the same. However, the timing jitter structure decreases as the net GVD decreases,
as shown in Figure 3.13. The jitter is mostly buried under the local oscillator noise sidebands of the Advantest. Its upper bound has been estimated from the 21st harmonic to be 4
ppm(80 fs). Note that the 60 Hz spurs have been greatly reduced by using a better power
supply.
3.1.5 Comparison between Theory and Experiment
The quantum limited jitter can be estimated from Equations (3.8) & (3.9). The round
trip time, net GVD and the pulse width have all been measured fairly accurately. By adding all the linear losses in the cavity, g is estimated to be 0.8. The average pulse energy is
estimated for g=0.8 and used in subsequent computations. The chirp(p) of the pulse is a
function of dispersion, gain bandwidth and the ratio of APM action to SPM action:z. For
66
P-APM, a does not vary much and a=0.2 is used for subsequent calculations[51]. So the
only uncertainties left are the incompleteness of population inversion(0) and the gain
bandwidth(Qg) of the erbium fiber. Though ideally the inversion enhancement factor is 2
at 980nm, it can be significantly higher for high concentration erbium fiber at larger signal
and pump powers[79],[80]. The gain bandwidth is usually taken as 35-40nm. Figure 3.14
shows the calculated quantum jitter corresponding to the laser configuration in Figure
3.8.(GVD=0.0169ps 2 , highest output energy) as a function of gain bandwidth for O's. The
measured jitter is on the same order of magnitude as this theoretical prediction. We believe
that the stretched pulse laser reported in this paper has quantum limited jitter for three reasons: 1) only quantum noise is white, and the jitter spectra are clearly induced by a white
noise source 2) jitter due to classical noise is usually orders of magnitude above the quantum limit, 3) the measured jitter increases as the pulse energy decreases. Though other
laser parameters can also vary slightly as the pulse energy changes, this energy dependence is most readily explained by the quantum origin of the jitter: Since the quantum
noise source is fixed, the perturbation experienced by each photon in the pulse decreases
as the number of photons in the pulse increases.
As we increase the measurement time from 0.09 sec to 1.92 sec, the jitter increases
as predicted by Equation (3.9), but the increase is two times the theoretically predicted
square root dependence. This is partly due to the 1/f noise, whose presence is inevitable
whenever the measurement time is large.
The GVD dependence of the timing jitter has been experimentally confirmed. The
timing jitter decreases even though the pulse energy decreases. The calculated quantum
jitter in this case is 62 fs, and the experimentally measured upper bound is 80 fs. Eventually, the jitter is due to higher order effects such as third order dispersion, chirp, or AMFM conversion via the Kramers-Kronig relation.
Finally, energy fluctuations are larger than those predicted from quantum effects.
Though it is difficult to estimate rw in Equation (3.2), it is usually on the order of 0.5 Rs
for an APM fiber laser. The quantum noise induced amplitude fluctuation is then calculated to be about 10 dB below the measured noise.
67
At (psec)
0.5
0
= 12,10,8, 6,
2
4,
0.4
0.3
0.2
Measured At
0.1
20
30
40
50
60
Gain Spectral FWHM AXg (nm)
Figure 3.14: Quantum jitter vs. gain bandwidth for different Os. Pulse energy(wo) = 0.42
nJ. Pulsewidth(r) = 51 fs.
3.2 Cross Phase Modulation(XPM) Squeezing
The squeezing setup is schematically shown in Fig. 3.15. An incoming pulse linearly
polarized along the x axis, is coupled into a single mode fiber. The X
nonlinearity of the
fiber couples the creation and annihilation operators of the y-polarized mode. As a result
of the cross phase modulation, the quantum state of the y-polarized mode, which was initially a vacuum state, is transformed into squeezed vacuum state at the output of the
fiber[91,93-95]. In order to detect the squeezed vacuum, the pump and the vacuum are
68
separated by a polarization beam splitter. The pump polarization is rotated to match the
polarization of the squeezed vacuum, and used as the local oscillator in a homodyne detection scheme. The main limitation on the attainable squeezing in this scheme arises from
the residual birefringence of the single mode fiber. The birefringence will cause a phase
walk-off between the pump and the squeezed vacuum, limiting the maximum attainable
squeezing. In our system we used standard telecommunication fiber with a beat length of
-4m. In order to minimize the phase and/or polarization walk-off the length of the fiber
used was 20cm. By using pulses with a soliton period comparable to the fiber length a
nonlinear phase shift of about 1 Rad can be obtained. This amount of nonlinear phase shift
should result in 9dB of noise reduction for an ideal soliton squeezing scheme. Another
advantage of using a short fiber is the reduction of GAWBS, which scales linearly with
fiber length.
t
pump
Y
single mode fiber
0
-~
vacuum
0
squeezed
vacuum
Figure 3.15: Schematic description of cross phase modulation squeezing
Figure 3.16 shows the experimental setup. To obtain high power, an air gap is added
to the laser cavity. A half-waveplate and a PBS are used as a variable coupler to extract as
much power from the laser as possible. Waveplates are used in lieu of fiber polarizer controllers to improve the laser stability. The average power from our stretched pulse laser
was 100mW and the repetition rate was 30MHz. The corresponding pulse energy was 3nJ.
The pulses were compressed to 150fs using a double pass through a silicon prism pair, and
injected into an SMF-28 fiber. Soliton effects in the fiber further compressed the pulses.
The soliton effects are evident in Figure 3.17 which shows the pulse spectra before and
69
after the squeezing fiber.
nJ, 150 fs
pulses
I
piezo controlled
mirror
of
J2
1/
771
7
;V2
PBS
SMF fiber
poliarlzer
1/
double pass
prism compressor
/
BS
balanced
detector
stretched pulse laser
Figure 3.16: Experimental Setup
-30
-40
-50
'3)
E
0.
-60
-70
-80
1450
1475
1500
1550
1525
1575
1600
1625
wavelength [nm]
Figure 3.17: Pulse Spectra before(dashed) and after(solid) SMF fiber, Power of 35mW
As a result of the soliton pulse compression the spectrum develops wings which fall
70
off exponentially as expected from a soliton, rather than the Gaussian tail which is typical
of stretched pulse lasers. It should be noted that the spectrum of the pulse doesnot broaden
since the pulse was initially chirped and the compression is counteracting the initial chirp.
The maximum power achieved in the fiber was 3mW which resulted in 85fs pulses at
FWHM, shown in Figure 3.18.
0.15
0.10
0
C.
C')
0.05
0.00
-4 30
-300
-200
0
-100
100
200
300
400
time [fsI
Figure 3.18: Autocorrelation of pulses after SMF with power of 35mW. Dashed line is the
sech fit
For this pulse width and for the dispersion of the SMF-28(-22ps 2/km) the soliton
period is
-17cm.
To detect the quadrature squeezing a polarizing beam-splitter separated
the pump from the squeezed vacuum. The pump was attenuated with a polarizer and its
polarization was rotated to match the vacuum polarization. The pump and vacuum were
then detected in a balanced homodyne detector. The amplitude noise of the stretched pulse
laser, which was 10dB above the shot noise at 2.5MHz, was cancelled by the balanced
detector scheme which had a common mode suppression of 20dB at up to 5MHz.
In Figure 3.19 we show the shot noise level which was obtained by blocking the
71
squeezed vacuum port, and the noise level obtained with the squeezed vacuum. To obtain
the minimum noise level the delay between the pump and vacuum has to be carefully
adjusted to obtain the relative phase for maximum squeezing. At other angles anti-squeezing, i.e. noise levels above shot noise, were observed as shown in Figure 3.20.
-70
-75
E
..- 80 V
o
-85
Shot noise
fl~
\4
fi
v.,yJoli
-90
-95
-100
2.45
2.50
2.55
2.60
2.65
frequency [MHz]
Figure 3.19: RF spectrum of the homodyne detector. Shot noise is obtained by blocking
the vacuum port
The results shown in Figure 3.20 were obtained by modulating the piezo displacer
shown in Figure 3.16 at 20Hz and synchronizing the scan of the RF spectrum analyzer to
the modulation. Since both the local oscillator pulse and the squeezed vacuum emerge
from the same fiber the interferometric stability necessary for the homodyne detection can
be obtained without active stabilization. Due to the short fiber length the noise associated
with GAWBS spectrum was not evident in our system. The optical power was attenuated
with the polarizer to 8mW and the photocurrent was 4mA through each of the detectors.
The RF noise was measured using an HP 8560E RF spectrum analyzer, with a resolution
of 10KHz, and an averaging of 30 times. The calculated noise level based on the detector
photocurrent, resolution bandwidth and transimpedance gain(-54dB) was -84dBm. The
72
electronics noise floor was limited by the noise of the transimpendance amplifier and was
20dB below the measured optical shot noise. The linear dependence of the noise level and
the detector current on the optical power were verified up to 15mW of the incident optical
power. The average noise reduction from Figure 3.19 was 3dB. Accounting for the quantum efficiency of the detector and an estimated 1dB of optical losses, the actual squeezing
is 5dB. The expected noise reduction for this scheme is 6dB which was obtained from
simulations similar to those described in Ref. 25. The bandwidth of the squeezed radiation, as in other schemes employing Kerr nonlinearity and short pulses, is defined by the
pulse bandwidth and the response time of the nonlinear mechanism. In our case the measurement of the squeezed radiation bandwidth was limited by the 5MHz bandwidth of our
balanced detector.
-70E
-~75-
CO
shot noise level
0
CD,
0-95
2nt
0
LO phase [Rad]
Figure 3.20: Dependence of noise on the relative local oscillator phase
73
74
Chapter 4
Squeezing with a Sagnac loop and 1-GHz ps Pulses
The squeezing scheme in this chapter uses picosecond pulses and a balanced Sagnac loop to separate the squeezed vacuum from the pump. By using picosecond pulse
Raman noise is negligible. An added advantage of picosecond pulses is their narrow spectrum, which allows the use of fiber devices without incurring excessive pump leakage.
Self-phase modulation(SPM), which is more effective than XPM in Chapter 3, can then be
the nonlinearity producing squeezing. If GAWBS can be suppressed with a 1 GHz rep-rate
laser source, then very long fiber lengths can be used. Long fiber lengths are crucial to
achieving large squeezing. For example, if the pump is a lps soliton in standard singlemode fiber, then -100m of fiber is required to achieve >10dB of squeezing. On the other
hand, the required fiber length scales with the square of the pulsewidth. If a 5ps soliton is
used then the fiber length increases to 2.5km, which has >10% propagation loss. Therefore
the soliton pulsewidth should not be much wider than 1 ps.
4.1 GHz Laser Source
Constructing modelocked lasers with 1 GHz rep-rate and pulsewidth -Ips is not an
easy task. Actively modelocked lasers can provide high rep-rate, good noise performance
and can be easily locked to external clocks through their intracavity modulator[96],[97].
However even with soliton pulse shortening these lasers will only produce pulses close to
5ps when operating at 1 GHz. Passively modelocked lasers can provide short pulses
directly[99] but at a low rep-rate. Though passive harmonic modelocking[98] or external
time-division-multiplexing(TDM)[99]
can increase the rep-rate of such lasers, they
require extensive stabilization and/or suffer from poor timing jitter and poor supermode
suppression. By incorporating an amplitude modulator and passive modelocking mechanisms in an Er-doped fiber laser -ips pulses have been generated at 500 MHz with good
supermode suppression[97]. Since the effectiveness of the modulator decreases with
75
shorter pulses, the modulator in [97] was driven with short electrical pulses. In contrast to
amplitude modulation, phase modulation changes the carrier frequency of the pulse and
affects the pulse timing via the group velocity dispersion of the laser. Thus a phase modulator[100],[101] in a laser with large anomalous group velocity dispersion may provide
more effective timing restoration and stabilize shorter pulses.
F ISO
HWP QWP
HWP QWP
SMF + DSF
PC
Phase Modulator
10 % 0
30 %
D
G
F PD
710%
Laser
Output
Figure 4.1: Laser Schematic, HWP: Half Wave Plate; QWP: Quarter Wave Plate. D:
Phase Delay line, G:RF amplifier, F: 1GHz Filter, PD: Photodetector.
The laser schematic is shown in Fig. 4.1. The ring laser is pumped by 400mW at
977nm and the fundamental rep-rate is 15.2 MHz. The waveplates, the polarization controller(PC) and the polarization-sensitive isolator provides the modelocking mechanism,
76
i.e. Polarization Additive Pulse Modelocking(P-APM)[52]. The phase modulator in the
laser cavity is polarizing and a polarization controller is placed in front of the modulator to
make the input light linearly polarized. The laser consists of SMF-28, dispersion shifted
fiber(DSF) and erbium fiber. The average cavity dispersion is calculated to be -8ps/nm/
km. A 10nm filter is inserted to suppress the erbium gain peak at 1530nm. A 10% coupler
is used as the output. A 30/70 splitter is placed at the output and 30% of the light is
detected with a homemade 2GHz detector. The electrical signal is filtered with a 1 -GHz
RF filter, amplified to 23 dBm and phase-delayed to regeneratively drive the phase modulator in the laser cavity[101]. The laser generates multiple pulses in each round trip. The
output light from the 70% port has a power of 7.2mW. When the laser is first turned on
without applying RF modulation, the pulsing self-starts. Applying modulation leads to an
RF suppression of the sidemodes by 25-30dB from the 1 GHz harmonic. Further adjustment of the waveplates leads to improved RF suppression.The RF spectrum and the optical spectrum of the laser output when the modulator is turned on are shown in Figs. 4.2a
and 4.2b. The supermode RF suppression is >60dB and the optical spectral width is
5.6nm. Since the RF suppression deteriorates to 37dB with one pulse dropout per round
trip, this laser is free of pulse dropouts. This RF suppression is sensitive to the polarization
bias, indicating that Additive Pulse Limiting(APL)[102] is an important mechanism for
pulse equalization in this laser. The center wavelength of the laser can be tuned by adjusting the optical filter in the laser cavity and the electrical delay line in the feedback loop.
Long-range autocorrelation reveals an echo pulse 23ps away, which accounts for the small
ripples on the optical spectrum. This echo, due to the reflections caused by the interference
filter, can be eliminated with a non-normal incidence filter. Figs. 4.2c and 4.2d show the
RF spectrum and the optical spectrum when the modulator is off. The absence of regenerative feedback leaves the pulses in random time slots, as can be inferred from Fig. 4.2c.
However the envelope of the optical spectrum is the same as that in Fig. 4.2b for over
20dB. The wings of the spectrum fall off slightly more slowly than the one in Fig. 4.2b,
indicating that the modulator provides some pulse shaping at the temporal center of the
pulses. Although there may be other stabilization mechanisms, Fig. 4.2 shows conclu-
77
sively that the role of the phase modulator is that of synchronization, not of modelocking,
and that the pulse shaping is mainly due to P-APM. Fig. 4.2e shows the autocorrelation of
the laser output when the modulator is on. The pulsewidth is 480fs. This gives a time
bandwidth product of 0.336.
-
0
m
-20 -
-20-
-40-
-40
-60
-60III.'.
-80
-80
-100
800
900
-100
1000 1100 1200
iii IJi
I
II'I
800
900
1000 1100 1200
Freq, MHz
Freq, MHz
(a)
(b)
-10
-10
m -30
m -30
r-n
-70-
1525
-502
1545
-70
1525
1565
1545
Wavelength, nm
Wavelength, nm
(c)
(d)
1565
1
480fs
0.80.60.40.2
-10
0
-1000 -500
500
1000
1500
(e)
Figure 4.2: Output of the laser: a) RF spectrum b) optical spectrum when the modulator is
on, c) RF spectrum d) optical spectrum when the modulator is off. The RBW of the RF
spectrum is 100KHz, and the RBW of the optical spectrum is 0.5nm.The suppression in
(a) is >60dB. The spectral width is 5.6 nm in (b) and (d). e) Autocorrelation of the laser
output when the modulator is on. The pulsewidth is 480 fs.
78
The Kuizenga-Siegman pulsewidth for this laser is 9.6ps[100]. With solitonic shortening
the pulsewidth can only be reduced by about a factor of 4.4[103]. The pulse from our laser
is much shorter than both due to P-APM. If the dispersion of fiber pigtail of the output
coupler is taken into account then the pulse is transform limited at the coupler. By adjusting the waveplates it is possible to obtain even shorter pulses at the expense of RF suppression ratio. The broadest bandwidth is 10nm, a limitation imposed by the intracavity 10nm
interference filter. Thus even shorter pulses with good supermode suppression might be
produced by this scheme if higher intracavity power and a more effective modulator were
available.
4.2 Noise of the GHz laser
Since the noise of a regenerative PM scheme reported in 4.1 has not been measured, characterizing the noise of such a laser is important. There is another reason to
study the noise of this regeneratively driven laser. Previous studies have compared regenerative amplitude modulation(AM) schemes with actively modelocked lasers driven by
external oscillators[104]-[106]. Even though the overall jitter for the regenerative case can
be smaller partly because no external electronic oscillator adds noise to the laser, the low
frequency jitter spectrum increases dramatically for the regenerative laser compared to an
actively modelocked laser[105],[107]. In the case of a single pulse per round trip this is
well understood. The regenerative laser behaves like a free running oscillator whose jitter
undergoes a random walk. A random walk means large energy at low frequencies. However with harmonic modelocking multiple pulses act on the modulator. In our case the
number of pulses(N) in the cavity is 66. Their effect on the modulator is an averaged one
and can be less than that due to a single pulse. Thus regenerative harmonic modelocking
may resemble active modelocking and its low frequency jitter can be smaller than that of
passive modelocking.
Our laser operates in the soliton regime and a soliton has four degrees of freedom:
energy(Aw), timing(At), frequency(Ap, where p stands for the deviation of the angular carrier frequency o from a nominal value oo, p=o o -o), and phase(AO). In this section we are
79
interested in only the amplitude and timing noise. The relevant equations of motion [110][111] for these perturbations are:
- Aw=-+
Ap
Ap
_
-
S (T)
Aw
(4.1)
W
2
+
MO) mAt
T
TR
+
+S
p
(4.2)
(T)
aAtS-2IDIAp
=
TJ~ + St(T)
aJT
(4.3)
TR
1
where t, and rP are relaxation constants and 1
P
4
T
1
2T. Eqns. (4.1)-(4.3) are
3T R f2
obtained from the modelocking equation, supplemented by a noise source S(t,T), by projection via adjoint functions. Here, TR is the round trip time of the fundamental cavity; T
is a time variable on the scale of many cavity round-trip times; Qf is the filter bandwidth;
D is the net group velocity dispersion(GVD); t is the sech pulse width; M is the modulation depth of the modulator and orm is the modulation frequency; the noise sources Si(T)
are defined by projecting the respective orthonormal functions onto the noise source
S(t,T). Eqn (4.1) predicts that energy deviations decay for a stable oscillator. Eqn (4.2)
shows that both filtering and phase modulation stabilizes frequency fluctuations. Eqn.
(4.3) indicates that because of GVD, a change in pulse frequency deviation changes the
group velocity. This is the well-known Gordon-Haus effect. Through the interplay
between Eqns. (4.2) and (4.3), the phase modulator first affects the laser carrier frequency,
then the laser timing. Thus timing restoration with phase modulation does not degrade
with shorter pulsewidth.
Equations (4.1) -(4.3) can be easily solved. The energy fluctuations spectrum is
22
<IAw(Q)I >
(4.4)
=
2+ 1/W 2)
80
For a soliton laser the dominant timing jitter is due to the Gordon-Haus effect and the contribution from St(T) is negligible when compared with the contribution from Sp(T). The
timing jitter spectrum is approximately:
4D 2
2
<2At(Q)I > =
<Isa (Q2)I>
2
TR
R
<
Q2K2 +
Q2
1
22
(4.5)
22PM 2P4+QPM
IP
where QPM
2DIMCOm 2/TR2 accounts for the effect of the phase modulator.
For the case of passive modelocking, Qpm= 0 . The laser becomes a free running
oscillator. The energy fluctuations stay the same. The jitter spectrum when Qpm= 0 is:
4D 2 <'Sp(Q)'
2
<IAt(Q)2 > =
2
T R2Q2
2
)
(4.6)
2 + 12
The singularity at Q=O is typical of a free running oscillator undergoing random walk.
Comparing Eqns. (5) and (6), we see that the phase modulator, when driven by a
noise-free external oscillator, provides pulse retiming and stabilization which removes the
singularity at Q=O in (6) and reduces the jitter at low frequencies. For the laser reported
here, 1/tp is on the order of MHz and Qpm is on the order of KHz. Thus the effect of the
modulator can only be observed at very low frequencies.
The spontaneous emission noise(quantum noise) can be modeled by a white noise
source[85], [108]. The sources Si that are in (4.1)-(4.3) and drive the frequency and timing
deviations have the correlations
<S (T)S .(T)> = D .8(T - T')
81
(4.7)
with "diffusion constants" D
ww,qn
deviation), and D
2
pp,qn = 3wt
= 4w
0
2g_
2g
hu= <1S (Q)l22 > (ASE amplitude
TR
2
g-h=
2 0 TR
<IS (Q)l 2 > (ASE frequency deviation),
P
0
where 0 is the enhancement factor due to incomplete inversion of the medium, g is the
incremental gain per pass, and wo is the steady state pulse energy. Amplitude noise and
timing jitter induced by spontaneous emission can be calculated using (4.4)-(4.5) with
(4.6) as the noise source.
The noise characterization of this laser is measured in the frequency domain with a
12GHz detector and an 8-GHz RF spectrum analyzer(Advantest R3265). Fig. 4.3 shows
Harmonic 1. One of the amplitude noise structures(Fig. 4.3a) extends to 300KHz and is
constant around each harmonic. Its overall magnitude is 0.04% via numerical integration.
There is another narrowband noise structure(Fig. 4.3b) which extends to 250Hz. Part of
this structure is due to the local oscillator noise of the spectrum analyzer. However there
can be some pump noise. The upper limit for this noise structure is 0.1%. This amplitude
noise is lower than actively modelocked lasers[104]-[106] but comparable to that of passively modelocked fiber lasers reported in [87].
Fig. 4.4a shows the jitter's quadratic dependence on harmonics/frequency. Taking
into account the resolution bandwidth(AQres) of the spectrum analyzer, we use the following relationship to fit the jitter data:
f Pj(Q)dQ = (2rk)2
resPo
At
2
(4.8)
TRm
where Pj(Q) is the timing jitter noise spectrum, and P0 is the peak power of the signal, k is
the order of harmonic, and TRm is the temporal spacing between consecutive pulses and is
different from the fundamental round trip time TR in Eqns. (4.1)-(4.3). We'd like to point
out that the noise contribution due to the jitter is proportion to (At/TRm) 2 . In addition to the
well-known fact that the noise structure of the higher harmonics is dominated by jitter
82
0
-20F
-40F
-60F
.1
I
-80 F
1004.8
1004.6
04.4
MHz
1005.2
1005
1005.4
0r-
-20-
-40-
m
V
-60
-
-80
-
-
1
-
-1003
-3
-.
-2
JA.
-1
0
Freq, KHz
1
JA
2
3
Figure 4.3: Harmonic 1 (a) Broadband amplitude noise, RBW=1KHz. (b) Narrowband
structure, RBW=10Hz.
83
contribution, this jitter contribution is larger for smaller TRm. Thus for high rep-rate lasers
one does not need to examine very high harmonics before jitter can be accurately measured, as shown in Fig. 4.4a. In fact, Ref. [104] reported that in the case of a 10GHz laser
the jitter dominates the low frequency noise structure even for its first harmonic. The jitter
is 35fs for frequencies above 100Hz and is lower than the 90fs[104] and 260fs[105] previously reported for regeneratively modelocked fiber lasers. However, it's higher than the
IOfs[106] reported for an actively modelocked fiber laser.
Fig. 4.4b shows Harmonic 7(dotted line), which is dominated by the jitter. The jitter is low frequency, narrowband and mainly due to white noise. The solid line is the theoretical quantum jitter spectrum calculated assuming passive modelocking. The singularity
in (6) disappears because of the finite integration time of the spectrum analyzer[108],[87].
The dash-dotted line is the theoretical quantum jitter calculated assuming active modelocking. Here t=480/1.76=273 fs, D=-0.12ps 2 , g=0.79, TR= 65.8ns, M=iT, (om=27*1GHz,
w0 =130pJ and rp=0.2 gs. Both curves match well with the experiment for frequencies
above 400Hz. For frequencies below 400Hz the active modelocking model underestimates
1.8 x 101.61.4cd 1.2-
0.83
0.60.4--
2
3
4
5
Order of Harmonics
84
6
7
-20
-30
-40
E
M -50
-60
-L70-0
-80
-90-100-110
-120
-2
---
-
-1.5
-1
-0.5
0
0.5
1
1.5
2
Freq, KHz
Figure 4.4: (a) Quadratic dependence of the jitter on frequency, (b) Harmonic 7,
RBW=IOHz, Dashed line: experimental; Solid line, theoretical quantum fitting.
the jitter significantly. The jitter spectrum follows the passive modelocking model much
more closely. At very low frequencies(<200Hz) the jitter is buried under the local oscillator of the RF spectrum analyzer and can be smaller than that predicted by the passive modelocking model because the modulator is affected by multiple pulses. Though the local
oscillator at very low frequencies is noisy as shown in Fig. 4.3b and the noise theory for
passive modelocking may not apply to our laser at frequencies <200Hz, the good agreement in the wings of the jitter spectra shows that the jitter induced by the white noise is
quantum limited. Small disturbances such as the cooling fan on the pump laser can add
sidebands significantly above the quantum noise[ 12]. The data presented in this Letter
are taken while the cooling fan is temporarily disconnected.
The data presented in 4.1 and 4.2 show that our laser has noise characteristics
superior to and pulsewidth shorter than harmonic modelocking by amplitude modulation.
85
The pulsewidth is comparable to that obtained by passive modelocking but its rep-rate is
much higher.
4.3 The Double Clad Fiber Amplifier
For a ips soliton in a standard fiber the soliton energy amounts to an average
power of ~150mW at 1GHz rep-rate. This power level is pushing the limits of conventional Erbium-doped fiber amplifiers. Thus the output of the laser is amplified by a homemade double-clad Er/Yb co-doped fiber amplifer(EYDFA). Fig. 4.5a shows the amplifier
setup and Fig. 4.5b shows a cross section of the gain fiber. The double-clad Er/Yb fiber has
a cladding numerical aperture(NA) of 0.45. The cladding is hexagonally shaped and the
diameter is 130gm. It is backward pumped with a 3-Watt 975nm diode(Polaroid 4000975-TO3). The output beam is shaped by a cascade of circular and cylindrical lenses and
the coupling efficiency to the fiber is 70%. The amplifier is 11 meters long. It has -30dB
gain and its saturation power is 760mW. The input-output curve when the amplifier input
is a CW laser at 1549nm is shown in Figure 4.6. No filter is used at the output of the
amplifier. Thus at low input powers the output power is equivalent to the total backward
ASE power, which is at 540mW. The noise figure at 1549nm is measured using an OSA to
be 5.7dB. Recently we have replaced the 3-Watt diode with a multimode-fiber-pigtailed
diode bar from Optopower Inc. Up to 10 Watts pump power can be obtained with this
pump module. Since the diode bar is already fiber pigtailed, coupling into the gain fiber
only requires a telescope. Coupling efficiency is -60%. Greater than 1.5 watt of output
power can be obtained. The output power is limited by the isolators in the amplifiers. The
isolators fail at high powers and insufficient isolation leads to lasing by the amplifier at the
Yb wavelength. If high power isolators were used then the output power could exceed 2
watts. The dispersion of the double-clad Er/Yb fiber is calculated to be -27.5 ps 2 /km and
86
the pulses are dispersion compensated with 4.5 meters of Furukawa dispersion compensating fiber(DCF) prior to the amplifier. A polarization controller(PC) before the amplifier
Output
Polarization
Sensitive Isolator
X/2
Dichroic
Beam Shaper
Pump Laser
Input
Gain Fiber
00
(a)
rrT
DCF
(b)
Figure 4.5: (a) Schematic of the EYDFA (b) Cross section of the double clad fiber
87
800
750700 -E
0
60055q500
0
1
2
3
Input, mW
4
5
Figure 4.6: Amplifier input-output curve
and waveplates after the amplifier compensate for the fiber birefringence. In our case -Ips
pulses are being amplified. Depending on the input polarization, significant nonlinear
polarization rotation can be accumulated in the EYDFA because of the high peak power of
these pulses. After passing though the polarization-dependent isolator the output spectrum
can have sharp dips. The input polarization is carefully controlled to avoid these sharp
dips.
4.4 The Balanced Detector
Because the laser source is followed by an amplifier, its noise is expected to
degrade significantly after amplification. Thus the balanced detector used must be able to
88
cancel large amounts of noise to ensure that shot noise has been reached. Two photodiodes(Epitaxx ETX 1000) are tied together and their difference current is amplified by an
Op-amp(Analog Devices AD829JN). The diodes are linear for input powers up to 8mW.
The resistor values are 500Q and 4.9kg before and after the Op-amp, and the capacitor
values are 0.lpF and 3.3pF before and after the Op-amp, respectively. Since the fundamental rep-rate of our GHz laser is 15.2MHz, the detector bandwidth is chosen to be
-10MHz so that imperfect suppression of the fundamental harmonic will not saturate the
detector. The shot noise level and the detector linearity are measured with flashlights. The
shot noise level is at least 10dB above the noise floor of our measurement system. To test
noise cancellation, we illuminate the detector with the ASE coming from our EYDFA.
Fig. 4.7 shows the noise spectrum with one port blocked and with both ports open when
detector currents are 8mA at each photodiode. The noise cancellation is 38.7dB. When
both diodes are illuminated the noise spectrum falls to the shot noise level. Thus if the
ASE source were noisier, more than 38.7dB of noise can be cancelled. Since the fiber laser
source amplified by the EYDFA will not be noisier than the ASE alone, our balanced
detector can reach the shot noise level for the frequencies of interest.
89
-20 -
-
-35
E
- 5 0 -.
....
.
..........
-..
Both detectors
65
-80
1
2
3
4
5
6
7
8
9
10
MHz
Figure 4.7: Detector power spectrum with one detector and with both detectors when illuminated with ASE from the EYDFA.
4.5 Sagnac Loop Squeezing
As first proposed by Shirasaki and Haus[66], the squeezed vacuum is separated
from the pump if a nonlinear interferometer is used. The scheme has the further advantage
that the pump power is saved and can be reused in full. A qualitative description of the
setup is shown in Fig. 4.8. A soliton state as represented in the phasor diagram enters one
port of the first 50/50 beamsplitter, while zero point fluctuations enter into the second port.
The beamsplitter divides the input fields into coherent soliton states of half power, and the
noise associated with each input port splits incoherently. The fluctuations entering the two
separate ports are uncorrelated with each other and maintain incoherence following the
90
power division by the beamsplitter. Following the beamsplitter, the two phasors propagate
through the same fiber Kerr medium in the Mach-Zehnder arms and experience the same
nonlinear propagation. The phasors thus rotate and the circles are distorted into ellipses.
At the second beamsplitter the phasors add coherently at one port and they subtract at the
I
II
Fiber -
50/50
505K
ICD
I
Figure 4.8: Propagation of a soliton state input through the nonlinear Mach-Zehnder
interferometer
second. As before, no correlation exists between the squeezed noises entering the two separate ports of the second beamsplitter. The uncorrelated fluctuations add incoherently at
both output ports and as a result, only squeezed vacuum fluctuations remain in the subtraction port while essentially all the pump power exits through the addition port.
Fig. 4.9 shows the experimental setup. The nonlinear Mach-Zehnder interferometer in Fig. 4.8 is implemented with a Sagnac loop. A Sagnac loop replaces the two arms of
91
the interferometer with a common path, which is inherently stable against environmentally
fluctuations. The laser output is passed through a dielectric filter to obtain pulsewidth
closer to ips and to have less nonlinearity in the EDFA. The commercial filters have very
steep cut-offs in frequency and a factory reject is used in our case to ensure less energy in
the pulse wings. The pulses are amplified and passed into the squeezer. The Sagnac loop is
balanced and the input is reflected back towards the amplifier. The leakage due to the
imperfect balancing at each detector is 25gW. This leakage level is much less than the LO
pump at each detector, which is 7.5mW. The circulator, consists of a polarization beam
splitter and a Faraday rotator, separates it so that it can be reused as the LO for homodyne
detection after being attenuated to 15mW. The squeezed vacuum exits the vacuum port
and meets the LO at the 50/50 beamsplitter. The two beams are carefully matched in
space, time and polarization. A piezo stack is attached to one of the mirrors to allow fine
tuning of the displacement. A balanced detector is used to cancel the classical noise of the
LO. The noise of the amplified pulses is 12dB above the shot noise for 4-10MHz. Since
our balanced detector can cancel 38dB of noise, our detector output is shot-noise limited
for those frequencies. These shot noise levels are confirmed when we comparing them
with flashlight calibrated results.
92
a+EDF
+
*
R
2
Filter
ZT
3dB
80m
5050 X2Fiber
Figure 4.9: Setup for Sagnac loop squeezing
Unlike the XPM scheme reported in Chapter 3, the LO and the signal propagate
significant distances before meeting at the 50/50 beamsplitter for homodyne detection.
This is because the lengths of the two fiber pigtails of the Sagnac loop cannot be matched
to within submicron precision and a delay stage is necessary for this task. This large distance, especially the fiber portion, makes the setup in Fig. 4.9 very susceptible to environmental fluctuations. Since squeezing is phase sensitive, submicron displacement in path
length is sufficient to make squeezing detection difficult.
Figure 4.10 shows measured shot noise level and the noise spectrum at 8MHz
when the piezo stack is driven with a 20Hz sawtooth. The spectrum analyzer is operated in
93
the "zero span" mode and the resolution bandwidth is set at 100KHz. Thus the horizontal
axis is the phase difference between the squeezed vacuum and the local oscillator. The
noise spectrum is not averaged and is noisy. However, the noise level is clearly phase sensitive and dips below shot noise for some phase differences. To measure the exact squeezing and anti-squeezing levels many averages are needed. This is difficult since the path
length difference between the signal and the LO may have shifted within the measurement
period as mentioned in the previous paragraph. Figure 4.11 shows the averaged noise
spectrum. The squeezing is 4.4dB and the anti-squeezing is almost 18dB.
-20
-30
-
.
.
.
ii
.
.
...
......
.-
-40
-50
E -60
mO
-70
-80
....................................
.-.
.............................
-90
-100
-110
C
100
200
300
400
500
Phase Difference, a.u.
600
700
Figure 4.10: Unaveraged spectrum at 8MHz, RBW=100KHz. Piezo stack is driven with a
Sawtooth
94
17 .5 ..... ... .
10-
52.50
-2.5
-
20
40
120
60
80
100
Phase Difference, a. u.
140
160
Figure 4.11: Averaged spectrum at 8MHz, RBW=100KHz, Piezo Driven with a Sinusoid
The experiment is designed to achieve 13dB of squeezing. Only 4.4dB of noise reduction has been observed. This large discrepancy can be partly accounted for when loss in
our system is considered. The total loss in our system is 23%. The loss due to the detector
quantum efficiency is 12%. At the fiber tip 4% is lost and the polarizer insertion loss is
2%. The remaining 5% are due to the AR coatings, the spatial beam mismatch between the
signal and the LO, and the fact that the 50/50 beamsplitter used for balanced detection is
imperfect. Figure 4.12 shows loss vs. noise reduction for different squeezing levels. With
26% system loss, the observable noise reduction for 13dB squeezing is only 5.6dB. Thus
the difference between the theoretical value and the experimental value is only 1.2dB. This
95
difference may be due to the slow path lengths drifts which makes observed minimum
non-optimal.
-2
-3
-4
0
....
.. .
-.....
-5
0
...
. ..
. . .. . .
.. . .
I...it
00 .
-6
-7
. .. . ... . . . ..
-0
.... .........
-
-8
z
-9
-
18
1
0.2
0.3
Loss
0.4
Figure 4.12: Loss vs. Noise Reduction for Different Squeezing
96
0.5
Chapter 5
Conclusions and Future Work
5.1 Conclusions
In this thesis we investigated, both theoretically and experimentally, generation of
squeezed vacuum via soliton propagation. In chapter two we renormalized the four soliton
perturbation parameters in a way that, pair-wise, the equations for the noncommuting observables became identical in form. The dimensionless expansion functions permit the
quantization of any secant hyperbolic pulse in terms of a complete set of expansion functions, including a pulse that is not a soliton, such as the pulse emitted into a linear medium
from the end of a fiber. Of course, the evolution equations of the operator coefficients must
be changed to accommodate the change of the medium of propagation. The renormalization
cast the squeezing formalism in terms of a Bogolyubov transformation. If the initial excitation is described by a soliton in a zero point fluctuation background the in-phase operator
does not have the minimum uncertainty value, unlike the photon perturbation operator
which exhibits Poissonian fluctuations. The local oscillator pulse shape in the balanced detector orthogonal to the continuum is a superposition of the functions sech (x/4) and i[ 1 (x/4) tanh (x/4)] sech (x/4). If a secant hyperbolic pulse is used for convenience, coupling
occurs to the continuum. The squeezing of the continuum is partially coherent with that of
the soliton. The squeezing of the continuum is oscillatory. In fact, the continuum contribution can improve the squeezing beyond that achievable with an LO whose shape is adjusted
to be orthogonal to the continuum.
As early as in Chapter one we outlined the experimental obstacles to achieving
soliton squeezing. In Chapters Three and Four we report two approaches to solve these
obstacles. In Chapter Three we used a short piece of fiber and sub-100fs pulses. We provide a thorough noise characterization of the stretched pulse modelocked fiber laser,
97
which is used as the pump laser. The noise theory for this laser has been reviewed. The
experimental results agree well with the theoretical predictions. The energy fluctuation is
0.05% of the pulse energy and the jitter is quantum limited for frequencies >60Hz. Moreover, the timing jitter can be reduced by adjusting the net GVD. The stretched pulse laser
is the quietest passively modelocked laser reported to date. We then used XPM to generate
squeezed vacuum. As much as 3dB of noise reduction has been detected and is highest
reported for an XPM scheme. Using a fiber with higher nonlinearity or a fiber with less
birefringence should lead to even higher squeezing.
In Chapter Four we describe squeezing with a 1-GHz laser and a Sagnac loop. The
GHz laser was discussed in some detail. we demonstrated, for the first time, that by regeneratively synchronizing multiple pulses from an P-APM laser it is possible to obtain RF
suppression comparable to that achieved by active modelocking, and with shorter pulses.
The pulses are self-starting and a pulsewidth reduction from the Kuizenga-Siegman pulsewidth by a record factor of 20 is reported. We then reviewed the noise theory for this laser
and characterized its amplitude noise and timing jitter. The jitter due to the white noise is
quantum-limited and the energy noise is 0.04% of the pulse energy. The timing jitter is
35fs for frequencies greater than 100Hz. This low noise source is then amplified to reach
the soliton energy for a lps pulse in a standard SMF fiber. A double-clad Er/Yb fiber
amplifier has been developed for this purpose. A Sagnac loop is used to separate the
squeezed vacuum from the pump and the pump is reused as the local oscillator for detection. The resulting squeezing is 4.4dB. This is partly due to phase sensitive nature of the
experiment. With interferometric stabilization the result may be better.
5.2 Future Work
Soliton quadrature squeezing is well understood and is, as this thesis has shown,
experimentally realizable. Thus it's time to engineer the squeezing apparatus so that
squeezed light can move out of the research labs and into the real world. There are still
challenges that need to be resolved. As mentioned previously, using squeezed light
98
requires interferometric stabilization, which in itself can be a mini-thesis. This is even
more difficult because in a real application free space delay line may not be desirable and
matching the path lengths of the LO and the signal may have to be done with stretching
fiber alone. Compactness is another issue. In Chapter 3 we used XPM to generate squeezing. Though less efficient comparing to SPM, XPM is a very promising approach because
the pump and the vacuum travel a common path and their separation is relatively easy.
Recently high nonlinearity fiber is starting to become available. These are air-silica microstructured fiber that have very small core. Furthermore, the Cr:YAG laser can generate
pulses as short as 43fs, which is significantly shorter than the stretched pulse laser. If high
nonlinearity fiber and/or the Cr:YAG were used then XPM may generate large squeezing
over a short distance. Finally, optical loss is a third concern. The system must be very
carefully engineered to avoid loss since even 10% loss can affect the resulting noise reduction significantly. Ultimately loss is limited by the detector quantum efficiency, which is
constantly improving. Detectors with quantum efficiency as high as 95% at 1550nm has
been reported amongst products made by Epitaxx Inc. when one is allowed to test and
hand-select the individual devices[42].
Though soliton squeezing via the photon-number-phase pair has been thoroughly
investigated, squeezing via the frequency-timing pair has not received much attention.
This pair truly distinguishes a soliton from a CW wave because it's derived from the "particle nature" of the soliton. For example, a pulse made of a superposition of CW waves
traveling in a zero dispersion medium will maintain its shape classically. However, quantum mechanically this pulse is not treated differently from a CW wave and can be completely described by the photon number and phase of its individual components. A soliton,
on the other hand, has four degrees of freedom. In addition to photon number and phase, a
soliton also has freedom in timing(position) and frequency(momentum). The latter pair
also obeys standard commutation relations and thus has standard minimum uncertainties.
Recently efforts in measurement techniques have arisen which can benefit from
less uncertainty in these two operators. Timing jitter is important for the photonic analogto-digital applications. Signals can be sampled with greater accuracy when the laser jitter
99
is less. In this application the squeezing of position is analogous to amplitude squeezing:
the aim is to reduce the jitter at the expense of carrier frequency fluctuations. However, no
squeezed vacuum is extracted. Position squeezed solitons can be used directly for sampling. To benefit from jitter squeezing the laser must operate near the uncertainty limit. Jitter measurements reported in Chapter 3 and Chapter 4 have shown the jitter can be
attributed to amplified spontaneous emission. It's well known that ASE in a laser far above
threshold produces amplitude fluctuations consistent with a Poissonian process. Thus a
system with fluctuations determined by ASE can operate at the quantum uncertainty limit
and its jitter or frequency fluctuations can be reduced below the quantum limit. Of course
an actively modelocked laser is needed to clamp the random walk nature of the laser jitter.
Amplitude modulation introduces loss and with it noise. Phase modulation is a Hamiltonian process and does not introduce noise of its own. However, if a steady state is to be
achieved, phase modulation calls for filtering which introduces loss as well, and with it
quantum noise. Thus studies should be done to examine which scheme generates states
closest to minimum uncertainty states. Once this has been accomplished the product of the
mean square jitter and frequency uncertainty is fixed. By changing the operating conditions one trades the decreased fluctuations of one against the increased fluctuations of the
other.
In addition to the photonic A/D sampling, there is also great interest in the generation of very precise combs of frequencies using the output of modelocked lasers. These
combs of frequencies are intended for use in precision spectral measurements. Frequency
jitter is an obvious issue in these measurements. Frequency uncertainty from the source
laser determines the ultimate accuracy of such a frequency comb. The statistics of both
frequency and timing uncertainties are important not only for scientific but also for engineering reasons and investigation of their squeezing is tantalizing indeed.
100
Appendix A
Continuum Matrix Elements
In this appendix we derive the continuum matrix elements stated in Equations (2.74)
and (2.75) of Chapter 2 of this thesis. The contribution of the continuum to the normalized
difference current is
Ao
cont
=-{
dx sech(x)
q
+
+
zf
,[exp(-iO - it/2)f,5 (x, Q, t)
C's f2
2
+ it/2)f*,(x, Q, t)]
exp(i
1
= 2 cos
dQ
f 2E
'Qw'
7 ,,s dx[f*(xLt)fe(x
i
dQ
2
-sin
7
C, S
'
(A.1)
,t)+
f1 (xt)fc*s(xQt)]
dx [f*(x, t)fc, (x, Q,t) - 11(x, t)fc*,(X, Qt)
CS0
The term multiplying cos 4 is zero, since it is the standard projection with the adjoint
function f 1 (x, t) that is orthogonal to the continuum. Hence, the operator AQcont can be
expressed more simply
cos
q27r
()Ic(Q,
t) + sin O.s (Q)I,(Q, t)
(A.2)
where
Ic(Q, t) = Re [
dx{ffI(x, t)fc*(x, Q, t)}]
(A.3a)
Is (Q, t) = Im [I dx{f 1 (x, t)f*(x, Q, t)}]
(A.3b)
and
The integrals contain hyperbolic functions with the mutliplier exp(±iQx). Therefore, the integrals involved in (A.3) can be reduced to simple Fourier transforms of hyperbolic functions.
Using these transforms one finds:
(A.4a)
IC(At) = 0
and
Is(Q,
For the evaluation of
t) = 7r(1 + Q2 ) sech
a2ontcont)
(2Q)
cos[(1 + Q2 )t/2]
(A.4b)
we need four correlation functions (|1,8 (Q)Fc,5 (Q')|).
101
Consider first the correlation function (IFC()C(Q')|)
(c
c(Q)Fc
(Q')|
dx{A&(x)f*(x,Q,O) +A&t(x)f (xQ,O)}
dx'{A&(x')f*(x', Q', 0) + Alit (x')f (x', Q', 0) })
-
1
dxJ dx'(IA&(x) Aat(x') )f*(x, Q, 0)f(x', Q', 0)
{ dxf((x,
1
0)f0(x,
0) }
(A.5)
,, f',
2wT
In the same way one finds for (IF,(Q)F,(Q')I):
(1F5(Q)F5(I')I)
dxf*(x,Q, 0)f_(x, Q', 0)
=
(A.6)
2Q
1
4 [1+
Q 2 ]2 6 ~
'
Finally, the cross-correlation gives
1(
2
PC(Q)S (Q') + PS (Q')PC(Q))
8K-
1 dx{A&(x)fjc(x, Q, 0) + Aht(x)f(x, Q,0)}
dx'{\Ae(x')f*(x', Q', 0) + Adt(x')f_(x', Q', 0)}) + h.c.
(A.7)
dxf dx'(I A&(x)A et(x')I)f*(x, Q, 0)f(x', Q', 0) + h.c.
8
I dxf*(x, Q, 0)f(x Q, 0) +
( t
,c
tins
dxf*(x,
Q',0) f(x,Q,0)}
Now, the functions fLc(x, Q, 0) and f .(x, Q, 0) differ only by the factor i. Hence the crosscor-
102
relation (A.7) is zero. We find for the autocorrelation of the continuum:
(cont,cont)
d
1 1
42w
8
f
Q
2 FP(Q)I,
s(I
(, t)
-_
0722
Q
2
(
(Qt)
(A.8)
+Q2]2
][1
dQ sech 2 (
cos2 (1 + Q 2 )t/2]
Next we turn to the cross-correlation matrix. Again, only the 22-element of the matrix is
nonzero:
sol,cont) = (|2AA 2 (t)
(Q')I (Q',t)|)
(A.9)
JQ
Since AA 2 (t) = AA 2 (0)+ 4<D (t),AA 1 (0), we must compute the cross-correlations between
AA 1 (0) and AA 2 (0) on one hand, and F, on the other hand. Consider first the crosscorrelation (IAzA 1 (0) F (Q) 1)
(IAA1(O)Ps(Q)I)
I(~Ifdx{zA6(x)f(x)
+ A& (x)f,(x)}
J dx'{A&(x')f*(x', Q, 0) + A&t(x')fS(x', Q, 0)}I)
1 Jdxfdx'(A&(x)Aet (x')1)L*(x)f (x', Q, 0)
dx
1
j
f
dx'6(x
-
(A.10)
x')f*(x)f (x' , 0)
dxf*(x)L8 (x,Q,0)
and thus
(1AA 1 (0)Fs(Q) + Fs(Q)AA1 (0)1)
{Jdxf*(x)f_(x,Q, 0) +
CS()
103
dxf,(x)f*(x,Q, 0)}
(A.11)
In the same way one finds:
2(IAA 2 (0)Fs(Q) + F(Q)AZA 2 (0)1)
={Jdxf*(x)f(x, , 0) +
dxf
2
(x)f*(x, Q, 0)}
(A.12)
C 2s(Q)
Again, one may evaluate the integrals by simple Fourier transforms given in the Table. One
finds:
{)
4(1+
Q tanh ( Q) + 63
+
tanh (2Q)
} sech
(A.13)
where C 1, = 0. Thus, we find for the cross-correlation matrix
(sol,cont)
=2 f
4
+
I
[C2 8 (Q) + 2(D(t)Cs (Q)]Is(Q,t) = 21
dQ
2 Q3
t
7F2
2+ r -Q
±
6
(I1+Q2)} 3
tanh (
2
Q) COS
1
tanh (2-Q)
sech
104
C2 s (Q)Is (Q, t)
(A.14)
[(1 +22)t]
Appendix B
Direct Measurement of Self-Phase Shift due to Fiber Nonlinearity
One of the most important nonlinear effects experienced by a pulse propagating through
the fiber is the accumulation of a nonlinear phase shift. This nonlinear phase shift is a
good measure of the fiber nonlinearity. Knowledge of the nonlinear phase is very helpful
for the diagnosis of our squeezer. In this appendix we use spectral interferometry(SI) [116]
to measure the nonlinear phase ($NL) by measuring the shift of the reference-signal interference fringes[ 117]. SI measures the sum of a reference and a signal pulse that are coherent and separated in time by t. It has been used previously to measure the phase variation
within a pulse[118]. However, if a long fiber forms one arm of the Mach-Zehnder interferometer the fluctuations of the length and the refractive index of the fiber shift the signal
phase randomly. To surmount this obstacle we propagate both reference and signal
through the fiber[ 119] and attenuate the reference so that it only experiences linear effects
in the fiber. The spectrum before propagation(S(o)) is:
S(o) = Ss(o)+S r(o)+2 SS(O)Sr(o)cos (or + $ + H(o)),
(B.1)
and after propagation(S'(o)) is:
S'(o) = S's(o)
+ Sr(o)
+ 2 S'(-)S(o)Scos
(or + $ + $NL
s
r
s
roN
L0 + H'(o))
(B.2)
Here Ss(o) and S s(o) are the signal spectra before and after propagation; Sr(o), the reference spectrum, is unchanged by propagation; t is the-time-delay-between the signal and
the reference; $NL is the nonlinear phase shift experienced by the center frequency(defined
as o=O) of the signal spectrum; $0 is the initial phase of the beat spectrum at o=0; and
H(w) and H'(o) includes linear and nonlinear chirp before and after fiber propagation.
Thus when H'(oi) is small compared to osT, $NL manifests itself as a shift of the interference pattern resulting from the signal-reference beating. Note that the phase that charac-
105
terizes nonlinearity is the time domain phase shift. Specifically, for a pulse that
symmetrically decreases in magnitude from its peak in time(defined as t=O), the phase
shift at t=O is the relevant quantity in most nonlinear experiments[ 113]-[114]. This phase
is not a function of time and techniques that only characterize dynamic phase within the
pulse, including FROG[115], inherently leave the time-independent nonlinear phase
ambiguous. In general the phase at t=O and the phase at o=O are not the same and in this
letter we use ONL(t=O) and $NL((O= 0 ) to differentiate the two. This technique was first
applied in a pump-probe setup where nonlinearity dominated[ 117]. In our case fiber dispersion makes the results less easy to interpret.
To gain insight into the relationship between $NL(o=O) and PNL(t=O) we consider
two cases. In the first case the signal is a soliton. H(o) consists of the linear chirp seen by
the reference. It changes the period of the interference fringes but doesn't contribute to the
phase shift. $NL(O =O) = ONL(t=O) in the soliton case. In the second case the nonlinearity
is small so that nonlinear and linear effects decouple. The linear effects, e.g. dispersion,
cancel completely. Moreover small nonlinearity implies that the nonlinear phase near the
peak of the pulse can be approximated by a parabola in time whose peak is at t=O. Fourier
transform of the quadratic phase doesn't contribute to $NL(o=O). Thus $NL(O=O)
$NL(t=O) when nonlinearity is small.
To check whether $NL(CO= 0 ) =PNL(t=O) in the presence of large dispersion and
large nonlinearity we use numerical simulations. The simulation parameters correspond to
our experimental conditions. We will present the experimental results in the following section. In the simulation a reference pulse and a signal pulse are propagated through 1.7 km
of lossless dispersion-shift fiber(DSF) using the split-step algorithm. Both pulses are gaussian with initial bandwidth equal to 5 nm. The reference pulse power is 3% of the signal
pulse power. To simulate DSF we use Aeff=50gm 2 , f3=-O. 1 1ps 3 /km, and
f2=-3.6ps2 /km
or P2 =0.5ps2/km to simulate different pulse center wavelength. We also study the case
where the initial pulse has a linear chirp. We quantify this chirp 8 by defining the initial
106
12
pulse to be:
Ae
0
,
2r2(
where A is the pulse amplitude and t is the transform limited
pulsewidth. The phases of both reference and signal are computed. We confirm that the
reference pulse experiences negligible nonlinear effects.
S=3
3
0, 5=1.7
3
13=-0.11 ps /km,6=1 .7
3
p3=0,8=0
£1=30.11
ps3 /km,8=-1.7
II __j
(a)
0
1
4
3
2
5
6
7
8
ONL(t=O)
S
S133 =0,8=O
0
v
3
A
p3=-0.1 1 ps /km,s=0
v
M
133=-0.11 ps 3/km,5=1.7
133=-0.1 1 ps3 /km,S=-1.7
1
2
4
3
5
6
(b)
7
7
8
$NL(t=O)
Figure B.1: 1. (a) $NLtO) VS- ONL(O=O) for 02=- 3 .6 PS2 /km. (b) ONL(tO) Vs. ONL
for P2=0.5ps2/km. The two phases are equa I along the straight line.
107
Fig. B. la shows $NL(t=O) vs. PNL(O=O) for $ 2=-3.6ps 2/km. The initial pulsewidth
varies from 0.7 ps to 1.3 ps depending on the chirp. The input power is varied between
50gW to 250gW to achieve a range of phase shifts. The reference pulse has pulsewidth
around 25ps at the end of propagation because of large dispersion. We see that the two
phases match each other very well when they are small. At large phase shifts they deviate,
but not by more than 30% regardless of initial conditions. Fig. B. la also shows that third
order dispersion doesn't have significant effects on the phase mismatch when P2 is large.
Fig. B. lb shows $NL(t=O) vs. ONL(o=0) for P2 =0.5ps 2 /km. The dispersion is small
so that the reference pulse at the end of propagation has pulsewidth around 3-4 ps depending on the initial pulsewidth. The input power is varied between 30gW to 300gW. Here
the two phases match each other very well even at large phase shifts and the deviation is
no more than 10%.
To demonstrate the feasibility of this technique we use the experimental setup
shown in Fig. B.2. Chirped pulses from a stretched pulse laser go through an interference
filter and are coupled into -20m PM fiber to generate a delay of -42ps between reference
and signal, as shown by the autocorrelation(inset). The reference pulse energy is -3% of
the signal energy. The temporal profile of the signal is already different from the reference
because the signal is affected by the PM fiber nonlinearity. The long delay and weak reference make signal-reference interaction negligible. The half-wave plate controls the ratio
between signal and reference. The pulses then pass through a PBS and an HP variable
attenuator before propagating through 1.7 km of dispersion shifted fiber(DSF). We vary
the attenuator to achieve a large range of nonlinear phase shift. The zero dispersion wavelength of the DSF is
-
1547nm. The pulse powers and spectra before and after the DSF are
monitored with a power meter(HP 81525A) and an HP optical spectrum analyzer, respectively.
108
Dispersion-Shifted
Fiber(DSF) Spool
Ref.
APBS
Stretched
Laser
----
PM
r
-
Variabie
Filter
AJ2
Att.
Plate
_ _
Inset
A utoco rrelatio n
- _-_
1E-3
-5;
.40
.22
0
0
-0
0
10
Power
20
30
40
OSA
so
PS
Figure B.2: Experimental Setup, Zero dispersion wavelength of DSF is ~ 1547nm. The
inset is the autocorrelation of the signal-reference pair after the PM fiber.
Fig. B.3a shows the optical spectrum of the pulse pair before and after the DSF
when dispersion is negative (pulse center wavelength XO= 1574nm). Soliton effects induce
large spectral changes at high input powers. Fig. B.3b shows the centers of the same spectra on an expanded 1- nm scale. The fringes are uniform even with large spectral changes.
The period of the fringes implies a delay of -42ps between signal and reference. Figs.
B.3c & B.3d show the spectra for the positive dispersion case(Xo = 1541nm) with different
spans. The fringes are again uniform and shifts can be easily deduced.
Fig. B.4a shows the output spectral bandwidths of the X0=1574nm and XO=1541nm
pulses as a function of power. The spectrum of the 1574nm pulse narrows as the total
power increases because of soliton effects. When the total output power is above 90gW
the pulse bandwidth stays constant at 1.6nm. Since the soliton energy for a 1.6nm bandwidth pulse in DSF corresponds to an average power of about 90gW, we believe that the
pulse has evolved into a soliton at these powers. An increase of the input power beyond the
soliton limit generates continuum and does not contribute to soliton energy, and thus does
109
not further alter the output bandwidth. We have also numerically integrated the optical
spectra at different powers to separate the pulses from the continuum[ 120]. The result confirms that the pulse energy indeed stays approximately constant for input powers exceeding the soliton condition. For the 1541nm pulse the spectrum broadens as expected
because of SPM.
9
-20
-25
W
-30
-30
-40
0)
Input
-50
-
-35
Op
26pt
Input
-40
-45
-50
-60
-55
-70
-60
-80,
1575
1570
1565
1532 1534 1536 1538 1540 1542 1544 1546 15481550 1552
1585
1580
Wavelength, nm
Wavelength, nm
(c)
(a)
-28
-24
E
0
A
-29
-26
-31
-25
-26
-27
-28
0' -32
02
60RW
~
57.5
sm
73&
a
74.0
574.?
15i'4
1 .0
15s 2
1574A
(V
52RW
-33
-34-
-36
-37
In put
-38
1540.8 1540.9 1541.0 1541.1 1541.21541.3 1541.4 1541.5 1541.6
-25
1573.6
1574.0
1573.8
1574.2
1574.4
Wavelength, nm
Wavelength, nm
(d)
(b)
Figure B.3: (a) Spectra before and after DSF, 20nm span (b) Spectra before and after DSF,
1 nm span for different powers. The center wavelength=1574nm. (c) Spectra before and
after DSF, 20nm span (d) Spectra before and after DSF, 1 nm span for different powers.
The center wavelength= 1541nm.
Fig. B.4b plots the phase shift vs. power for the X0=1574nm and kO=154 1nm pulses.
For XO= 1 574nm the nonlinear phase shift is clamped for higher powers because the soliton
condition is reached. This result is consistent with the previous observation that the pulse
bandwidth doesn't vary once the soliton condition is reached. For a 1.6nm bandwidth
110
pulse with average power = 90pW, soliton propagation leads to a 3.1 radian phase shift for
1.7km of DSF. This calculated result is lower than the measured values which range from
3.4 to 3.8. We believe that this inconsistency arises because the input bandwidth is 5 nm.
Before it evolves into a soliton the pulse experiences a larger phase shift. Below the
(a)
7
Na
6
"
1574nm pulse
1541nm pulse
A
-o
4 -
ct
3A
A A
2
A
A
ALAA
A A A AA
1
20
40
60
80
100
A
A
120
140
Total Power, pW
(b)
4 -
A
AAAAA
A
A
A A A
A
C
3
A
.
A
A
U
A
u)
2 -A
CDA
u
l1574nm Pulse
aa
1
1541 nm Pulse
A
A
20
40
60
80
100
120
140
Total Power, pW
Figure BA4: (a) Spectral bandwidth for k0 =1574nm and ko=1 543nm. (b)Nonlinear Phase
Shift vs. Input Power, o= 1 574nm and ko= I543nm.
111
soliton condition the phase decreases rapidly as the power decreases and becomes negligible at very low input powers. This is because large negative fiber dispersion causes the
pulse peak to disperse rapidly. For X0=1541nm the dispersion is positive and a nearly linear dependence is observed once SPM dominates.
In conclusion, we have directly measured a constant phase shift despite the chirp
of the pulse and fiber fluctuations due to thermal effects. The measured results agree with
theoretical calculations. We also demonstrate that the measured phase at 0=O doesn't
deviated significantly from the phase at t=O. Thus the results give us a good measure of the
total nonlinearity. We finally note that the setup in Fig. B.2 will also enable the TADPOLE
technique[121] to measure arbitrary phase profiles and thus characterize completely the
nonlinearity of the fiber.
112
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