Accurate and independent spectral
response scale based on silicon trap
detectors and spectrally invariant detectors
Ver. 4.2
Jarle Gran
Faculty of Mathematics and Natural Sciences
University of Oslo
2005
Preface
This thesis is a product of the work for the degree of Dr. Scient at the Department of Physics,
Faculty of Mathematics and Natural Sciences, University of Oslo (UiO). Most of the work has
been carried out at Justervesenet (JV) in the period from 2001 to 2005 and I wish to thank the
director and employer, Helge Kildal, for providing me the opportunity to make this degree. I
also wish to thank the University Graduate Centre (UniK) and its director, Ivar Jardar Aasen
for the scholarship, the working place and facilities provided at UniK, from which I am most
grateful.
I am most thankful to my supervisor professor Aasmund Sudbø for his guidance,
encouragement and support during these years of research. I would also like to use the
opportunity to thank my colleagues and friends at Justervesenet and UniK for being an
important social and professional environment enabling the work of this thesis.
At last I would like to thank my friends and family for their support, and especially my
children Markus and Maren, who mean so much to me.
Jarle Gran
Kjeller, Mars 2005
1
List of abbreviations
IQD
Internal quantum deficiency
QE
Quantum efficiency
CR
Cryogenic radiometer
ESR
Electrical substitution radiometer
NMI
National Metrology Institutes
UV
Ultraviolet
e-h
Electron – hole pair
CPD
Cavity pyroelectric detector
IR
Infrared
SNR
Signal to noise ratio
NICe
Nordic Innovations Centre
MOSFET
Metal oxide semiconductor field effect transistor
DVM
Digital voltmeter
NPL
National Physical Laboratory, UK
JV
Justervesenet – Norwegian Metrology Service, Kjeller
UiO
University of Oslo, Oslo
UniK
University Graduate Centre, Kjeller
NIST
National Institute of Standards and Technology, USA
NIR
Near infrared
FIR
Finite impulse response
2
List of symbols
h
Planck constant = 6.62607·10-34 Js
c
Speed of light = 299 792 458 m/s
k
Boltzmann constant = 1.38065·10-23 J/K
T
Thermodynamic temperature
e
Elementary charge = 1.60218·10-19 C
λ
Wavelength of the radiation
ρ(λ)
Spectral reflectance
δ(λ)
Spectrally dependent internal quantum deficiency
R(λ)
Spectral responsivity
d
Oxide thickness
a = (df,T, D, dr, b, R)
Parameters in the internal quantum deficiency model
α(λ)
Spectral absorption coefficient
V
Vector of measured values
σ2
Variance
k
Scaling constant, also used as fitting parameter a1
F(λ)
Fit function, equals R(λ)/k
χ2
Generalised least squares equation
u(y)
Covariance matrix of quantity y
f(a)
Matrix of sensitivity coefficients for the fit function
r(a)
Matrix of sensitivity coefficients for the responsivity
fb , fs
Frequency band, sampling frequency
W
Filter matrix
y(m),Y(m)
Relative response of detector m
y (m)
Filtered value of y(m)
F(m,n)
Responsivity ratio between detector m and n
K
Scaling constant
A(m)
Vector of absolute calibration values of detector m
Ψ
Sensitivity vector of scaling constant K with respect on filtered
values
α
Sensitivity vector of scaling constant K with respect on absolute
calibration values
3
Δ(λ)
Difference in IQD between an unbiased and a biased detector
d(b)
Matrix of sensitivity coefficients for Δ(λ) with respect on fitted
parameters b = (D, dr)
4
Contents
Preface
1
List of abbreviations
2
List of symbols
3
Contents
5
1. Introduction
6
1.1 Background
1.2 Progress of this work
2. Summary of research
6
10
13
2.1 Goal
13
2.2 Independent spectral response scale
13
2.3 Establishing a spectral response scale in the infrared
20
3. Summary, conclusion and future directions
23
References
26
Publications
28
5
1. Introduction
1.1 Background
Lightning and equipment used to measure different quantities by optical methods plays an
important role in our society. Within the health sector optically based equipment is used for
instance to analyse blood samples, measure the oxygen contents in patients blood and treat
skin cancer. Our infrastructure depends on products with good optical properties as proper
signalling light for air, road and marine traffic, including pilot displays and traffic light. In
addition, proper car lightning and road traffic lightning helps reducing accidents in the traffic.
Optical radiation might also harm the human body. Classification of laser products and UV
irradiance measurements in sun loungers are introduced to help protect the consumer against
radiation damage. Equipment used to measure all these and other quantities has to be
regularly calibrated in an unbroken chain back to a primary standard to maintain its accuracy.
A primary method yields the value of a primary standard and is defined as an independent
realisation relating the measurand to an independent physical quantity, preferably a
fundamental constant or another SI – unit [1-3]. In optical radiometry there are a number of
primary methods designed to measure specific quantities as spectral irradiance or spectral
responsivity. There are basically two different approaches for the primary method, either
defined by a primary source or a primary detector [4, 5].
The most common standard source is the blackbody radiator, which obeys Planck’s radiaton
law. This law determines the spectral radiance of a blackbody radiator expressed by Planck
constant h, the speed of light c, Boltzmann constant k and the thermodynamic temperature T
[6]. The 1948 definition of the SI – unit candela was based on a blackbody defined by the
temperature of the freezing point of platinum under standard atmospheric pressure [7].
More recently, storage rings and synchrotron radiation, generated from accelerated charges,
have been used as a calculable source determined from the measurement of a few parameters
as magnetic field, electron energy, electron current while it is applied in a defined geometry
[8]. The enormous cost in establishing and operating this facility, makes it unavailable to most
laboratories.
6
During the last 25 years detectors have been used as primary standards. One of the reasons for
this approach is the 1979 definition of the candela where the photometric quantities are
related to the radiometric quantities through the responsivity of a standard human eye and a
defined optical power value at a certain frequency [9, 10]. This definition enabled the
establishment of measurement standards for all radiometric and photometric quantities by
linking them to the one common unit, the watt.
In 1980, Geist and Zalewski proposed to use silicon photodiodes as a primary standard [11].
The principles were based on the fact that the responsivity of an ideal photodiode is expressed
by fundamental constants and the wavelength, and that the deviation from ideal performance
was measured separately. Deviations were caused by two loss mechanisms; reflectance from
the surface of the diode and the non-ideal conversion of photons into measurable electron
current. The reflection losses could be reduced to negligible levels [12] or measured in a
purely relative measurement. The internal losses were eliminated through two bias
techniques; reverse bias for the losses in the deeper layers of the diode and front oxide bias for
the losses in the shallow part of the diode. In fig. 1 a simple drawing of the diode and the
principles, as it was presented in [11], is displayed.
Figure 1. A drawing of the diode and the bias technique as presented in [11] is shown.
The oxide is biased with a transparent electrolyte and the reverse bias was done by standard
techniques. Because one could measure the increased responsivity by biasing the detectors
with the two techniques, the responsivity change was a direct measure of the internal quantum
deficiency (IQD), defined as one minus the quantum efficiency, and the method was called
the “self-calibration procedure”.
7
Figure 2 shows the results of a self-calibration procedure. When the applied bias of the oxide
(a) and the rear of the detector (b) was increased, an increase in responsivity was observed
until a saturation level was reached. When the responsivity reached the saturation level, it was
assumed that the quantum deficiency was eliminated. The difference between saturated and
unbiased response was assumed to be the measure of the quantum deficiency directly. The
clear advantage of this method was the simplicity in determining the optical power with
silicon detectors.
(a)
(b)
Figure 2. The figure shows the typical results of the self - calibration procedure. In (a) the
responsivity change with applied oxide bias technique is given and in (b) the responsivity
change given by the reverse bias technique. (Figures are taken from [17]).
Intensive work was done on the self - calibration technique in the next decade [13-21] and
many national metrology institutes (NMI) used this technique for their primary realisation
[17]. Unfortunately it turned out that the detectors changed their quantum efficiency after the
application of the oxide bias technique, which made them less suitable as a primary standard
[22]. This was a large disadvantage. To avoid the problem, inversion layer n on p 100 %
quantum efficient (QE) detectors were developed and the later research was based on these
detectors. It turned out that these detectors had 100 % QE only over a short spectral range and
methods to predict the QE at longer wavelengths from the shorter ones, where the detectors
have 100 % QE, were developed [23].
8
The research in the latter half of the 1980s was mainly based on the inversion layer detectors,
but in parallel with the development of the self-calibration procedures a new primary
instrument was developed. The cryogenic radiometer (CR) [24, 25] was a further
development of the previously known electrical substitution radiometer (ESR). The principles
of these two techniques are similar. The radiation is directed onto an absorbing active element
resulting in a temperature rise of that element. Then electrical power is used to heat the active
element to the same temperature as generated from the optical power, thus providing a link
from electrical power to optical power. Even if ESRs have been known for more than a
hundred years, only the last 20 - 25 years the CR was developed. The improved performance
of the CR is simply an effect of cooling. The CR can measure the optical power with an
uncertainty of 0.004% at discrete wavelengths [26].
A number of comparisons between the self-calibration procedure and the CR were performed
in the 1980’s with good equivalence between the two methods [17, 23, 27]. Despite this good
agreement between CR and self-calibration technique, the development of inversion layer trap
detectors and the fact that silicon based self-calibration detectors is a more cost efficient
realisation, the self-calibration technique lost interest among the NMIs as a primary method of
measuring optical power. One of the reasons might be that the uncertainty was not properly
analysed, as expressed in [23], and that the responsivity was therefore not linked to
fundamental constants with a sufficiently known accuracy.
Because CRs performs best only at discrete laser wavelengths, one needed another standard to
transfer the accuracy to users. Polarisation insensitive silicon trap detectors based on
Hamamatsu S1337 detectors are the most common detectors used nowadays for this purpose
[28], but several alternative configurations have been developed [29, 30]. To get a continuous
wavelength scale from the discrete calibration points, interpolation models of silicon trap
detectors were developed. These models were based upon the same philosophy as that given
from the self-calibration procedure where the loss mechanisms, reflectance and IQD were
treated separately [18,31,32]. The model developed by Gentile et al [32] is used by many
NMIs to disseminate a continuous spectral responsivity scale [33-35]. The model is based on
the assumption that the recombination probability for electron hole (e-h) pairs is given by the
position where they were created. By adding two more parameters Werner et al extended the
model beyond 1015 nm [33]. Kübarsepp et al developed a model of the quantum yield in the
UV region with two free parameters, extending the model to 250 nm [34].
9
In the last decade a new primary technique for photon counters and detectors for ultra low
optical powers (10^-12 W to 10^-18 W) was realised through “parametric down-conversion”,
also named “correlated photons” [36]. The principles consists of pumping a nonlinear crystal
with a laser where the photons of the pump laser split into pairs of photons under the
constraints of power and momentum maintenance. Measuring their individual and
coincidental appearance on the detection channels one can calculate the detector quantum
efficiency from the ratio of number of counts on the coincidental channel and a single
channel. This principle is elegant but not yet practical to use in calibration applications. Still,
research is going on but uncertainties below 0.1 % might be hard to achieve [37]. Counting
photons directly will not work as a primary method, because the detector used has an
uncertainty in its QE, which is determined in the correlated photons technique.
1.2 Progress of this work
Since the development of the self-calibration procedure there has been some large
technological advances affecting the research and applications within the field. Comparing the
reported quantum deficiency of detectors from publications made in 1980 – 1985 to those
made on Hamamatsu detectors 1991 – 2004, the quantum deficiency is reduced by an order of
magnitude [18, 28]. In addition, the fast technological advances of integrated circuit
technology and computers have improved the availability of computational resources
significantly, enabling more accurate calculations of the detectors and opened the way for
improved models [32, 38, 39]. In principle, it is these two improvements that are exploited in
this new approach to the self-calibration procedure, which replaces the previous procedure
based on oxide bias.
Publication I. Absolute calibration of silicon photodiodes by purely relative measurements
[Metrologia, 2004, 41, 204-212]
This publication exploits the properties of the developed interpolation models to quantify the
internal quantum deficiency of a silicon trap detector. This can be done because the
recombination probability for electron – hole (e-h) pairs created in the depletion region is
zero. By measuring the spectral shape of the trap’s responsivity one could estimate the
internal quantum deficiency by fitting the model to the purely relative measurement. The
intrinsic property of the recombination probability function at the depletion region ensures
10
correct scaling of the relative measurement. The major advances with this article is that one
can estimate the quantum deficiency through a purely relative measurement to a spectrally
invariant detector and through that eliminate the oxide bias experiment. Another very
important result from this article is the full covariance analysis of the results ensuring the link
of the spectral response to fundamental constants with a known accuracy. The uncertainty is
propagated from the observed variance of the measured data and is used to estimate the
covariance of the responsivity values, where the responsivity function is a sub-function of the
fit function itself. The model of the deficiency, rather than the efficiency, is developed and
used in this article. A correction caused by the diffusely reflected light is developed. The
author of this thesis has done all the work including the idea, measurement, analysis and
uncertainty evaluation.
Publication II. Establishing a high accuracy spectral response scale in the near infrared with
digital filters [To be published Applied Optics, May 2005]
The publication shows how one can calibrate a group of InGaAs detectors by measuring their
relative spectral response and then scale the relative measurements by calibrating them
directly to a silicon trap detector. The major contributions of this article are that the group as a
whole is regarded as the standard in the infrared. In addition, we suppress the influence of the
noise on the relative measurements by introducing a dedicated digital filter with a designed
cut off frequency. This is the first introduction of digital filters in radiometry to my
knowledge. The scaling constant is calculated by comparing the relative measurement of the
InGaAs to the absolute measurement against a trap detector at the same wavelengths. The
uncertainty was different for different wavelengths and we selected the wavelength range that
gave the lowest uncertainty in the scaling constant. A full covariance analysis of the
responsivity values caused by the digital filter and the scaling constant is given. The author of
this thesis has made all the measurements and made the analysis and had the basic idea to the
principles.
Publication III. Calibration of silicon trap detectors with a hybrid self-calibration method
[Submitted Metrologia, 2005]
This publication combines the original self-calibration procedure and the relative method
introduced in publication I. The purpose is to reduce the uncertainty and extend the spectral
range compared to that given in publ. I. The extension was done by including two more
parameters in the quantum deficiency model. The major contributions of this article are the
11
combination of different measurements in the estimation of the detectors responsivity. Each
measurement estimates two parameters in the quantum deficiency model with a low
uncertainty. Due to the low QD of the trap detector and the high stability of the source used in
the monochromator, we achieved very low uncertainties in the responsivity. A more dense
spacing between the measurements of the relative response enabled us to lower the
uncertainties in the responsivity based on these experiments as well. The calculated
uncertainty given from the method is at best comparable to that given at laser wavelength
cryogenic radiometers. In this article the covariance of the responsivity is calculated from the
observed covariance of the parameters from the individual experiments. The accuracy is
limited by the properties of the detector. The author of this thesis had the idea to this method
and has designed and done the experiments, performed the analysis of the results and their
uncertainty.
12
2. Summary of Research
2.1 Goal
We aimed to establish an independent high accuracy spectral response scale over a broad
spectral range based on standard laboratory equipment at moderate cost. This had to be done
by a primary method, where the responsivity of the detector is linked to fundamental
constants.
2.2 Independent spectral response scale
2.2.1 Model of the responsivity
A silicon detector can be manufactured with very high quality and their response can be
modelled and described by a few parameters. An ideal photodiode has the necessary
properties of a primary standard because its responsivity is expressed by fundamental
constants and the wavelength of the light. A silicon photodiode can be modelled as a modified
ideal quantum detector, where the two loss mechanisms, reflectance from the surface and
internal losses, are taken into account. The responsivity is modelled as
R ( ) 
e
 (1   ( ))  (1   ( )) ,
hc
(1)
where e is the elementary charge, h is Planck’s constant, c is the speed of light in vacuum, and
λ is the vacuum wavelength of the radiation. These quantities form the ideal term of a
quantum detector. The spectrally dependent reflectance is given by ρ(λ), and the quantum
deficiency is given by δ(λ). Our approach to get a primary realisation is to estimate the
reflectance and the internal quantum deficiency (IQD) separately. Both loss mechanisms,
reflectance and IQD, can be modelled and described by a few parameters.
2.2.2 Spectral reflectance
Fresnel’s equations allow us to calculate the spectral reflectance over the whole spectrum with
an uncertainty of less than 1% from a single measurement, when we know the spectrally
dependent refractive indices [40-42], polarisation, angle of incidence and oxide thickness. A
polarisation insensitive silicon trap detector, shown in fig. 3, can be calculated as
13
 trap ( , d )   ( ,0, d )  s2 ( ,45 , d )  2p ( ,45 , d ) ,
(2)
where index s and p indicates polarization direction and d is the oxide thickness.
Figure 3. A schematic drawing of a reflection trap detector is shown in the figure. The figure
is taken from [43].
A reflection trap detector is constructed so that light reflected from the first detector is
collected by the second detector and then again by the third. The first two detectors gets
irradiated at a 45 degrees angle with opposite polarisation, while the third detector was
illuminated at zero degrees implying that the radiation is retro reflected to the second and then
the first detector. The radiation is therefore reflected five times before the residual radiation
exits the detector, reducing the reflectance by two orders of magnitude [12, 44].
The main result here is that by measuring the reflectance, at one wavelength only, one can
find the oxide thickness and from that calculate the full spectral reflectance according to (2).
2.2.3 Internal quantum deficiency
The IQD model δ(λ) is based on the assumption that the recombination probability for a
generated electron hole pair (e-h) depends only upon the depth, x, into the detector where it is
created. The quantum deficiency is found by integrating over the depth of the detector. The
spectral dependence comes from the normalised distribution of photons into the detector
given from the highly spectral dependent absorption coefficient. These values are given by the
14
extinction coefficient of the refractive index of silicon. The recombination probability
function used in our model is shown in fig. 4.
R
df
dr
T
D
b
x
Figure 4. The recombination probability as a function of generation depth of electron – hole
pairs is shown.
The model of the IQD achieved from the recombination probability given in fig. 4 is
 (  , d f , T , D, d r , R , b ) 

df
T ( )
exp( T ( ))  T ( )  1
dr
D ( )  T ( )  1 exp(  ( ) D)  exp   ( )T 
( D  T ) ( )
 d r exp   ( ) D  exp   ( )b
 1  R  R  d r  exp   ( )b .
(3)
Here df is the recombination probability of (e-h) pairs generated at the SiO2/Si interface.
Further into the detector, the recombination probability decreases linearly to zero at position
T, the position of the pn junction of the diode. From position T we have a linear increase in
recombination probability until the bulk region of the detector starting at position D with
recombination probability dr. b is the depth of the detector and R is the reflectance at the back
of the detector, whereas α(λ) is the spectrally dependent absorption coefficient.
15
To establish an independent spectral response scale we have to determine the parameters in
the IQD model given in (3). The important property of this model is that at the pn junction the
recombination probability of (e-h) pairs created there is zero. In the presented work this
property is exploited by two different methods. In publ. I we estimated the IQD in the spectral
range from 400 to 920 nm, where the last term in (3) can be omitted, by fitting the parameters
in (3) and a necessary scaling constant to the relative spectral response of the silicon trap
detector to establish an independent spectral response scale. The property of the model
ensures correct scaling. In these measurements we used a cavity pyroelectric detector (CPD)
as a spectrally invariant detector.
In publ. III we used and combined the original self-calibration procedure and the relative
method in publ I to estimate the parameters in the IQD model. The classical bias experiment
was used to find the parameters dr and D by measuring the increased response with an applied
bias of 25 V and fitting those parameters to describe the change in response. The relative
method was used to find the parameters that describe the losses in the front (df, T) and at the
back of the detector (R, b) in two different but similar experiments. The IQD is found by
putting the parameters into (3). The uncertainty of this method is lower than the purely
relative method, but it requires 3 separate experiments each determining 2 parameters in the
model.
2.2.4 Uncertainty evaluation
The uncertainty in both publications is calculated from the observed variance in the input
measurement and is brought through the calculations to the output covariance in the
responsivity. These calculations are valid for random variables and are based upon the
observed variance in these random variables, which requires a number of observations of
those variables. We define u(x,y) to represent the covariance of two random variables x and y.
In the general case we get a covariance matrix u(x,y) for vector variables x and y containing
the matrix elements um,n(x,y). Each matrix element um,n(x,y) is defined as the covariance
between the two random variables xm and yn. The uncertainty calculation follows the guidance
given in [45], but the calculations are done in matrix form as
u ( Rm , Rn ) 
a  Rn Rm
Rm a
R

 u (Vi , V j ) 


 u (a , a  )  n ,
a Vi
V j a 
a
a 
(4)
16
where we use the convention of summation over equal indices. Here Rm is the responsivity at
wavelength λm. a is the vector of fitting parameters given in fig. 4 describing the
recombination probability as a = (df, T, .. b, R). The scaling constant is also a parameter when
it is used in the fit function. Greek index, α and β, is used to identify the index of the fitting
parameters while roman indices represents the wavelength λm,n,i or j. V is the vector of relative
measurements. u(Vi,Vj) is the covariance of the measured responsivities at different
wavelengths. We assume that the measurement noise at one wavelength is independent from
the measurement noise at any other wavelength. Therefore the covariance matrix u(Vi,Vj) is
diagonal, where u(Vi,Vj) = 0 unless i = j, and we can use the observed measurement noise to
set up statistical estimators for the diagonal elements of the covariance matrix.
In publ I (the relative method) the covariance in the responsivity values shows some
interesting characteristics. In this publication the responsivity function R(λ) is a sub-function
of the fit function F(λ) itself with the relation
F ( ) 
R ( )
,
k
(5)
where the scaling constant k is one of the parameters. Comparing the correlation coefficients
of the fit function and the responsivity, as done in fig. 5, shows that the correlation
coefficients of the fit function show a diagonal like characteristics, while the correlation
coefficients of the responsivity has a much broader form, caused by the uncertainty in the
scaling constant.
17
(a)
(b)
Figure 5. The correlation coefficient for the fit function values (a) and the responsivity values
(b) is compared in the figure.
The correlation matrix is calculated by dividing the covariance matrix with the type A
uncertainty in each measurement point involved as r(Yi,Yj) = u(Yi,Yj)/ u(Yi)u(Yj), where u(Yi)
equals the positive square root of u(Yi,Yi).
In publication III the covariance in the parameters of the individual measurements were
calculated and combined to find the covariance of the responsivity values. In this article we
aimed to extend the spectral range and reduce the uncertainty. The resulting correlation matrix
for the responsivity values by this method is shown in fig. 6.
Figure 6. Correlation coefficients of the responsivity calculated from the hybrid self calibration method is shown.
In the hybrid self - calibration method the low uncertainty in the fitted parameters gives
strongly correlated responsivity values. The diagonal elements in the covariance matrix give
the uncertainty in the responsivity. The calculated uncertainty in the responsivity given by the
method itself is shown fig. 7. Other uncertainty components are given by the properties of the
18
detectors used, like uniformity, linearity, temperature dependence and CPD absorbance. The
combined uncertainty in the reflectance and IQD is below 70 ppm between 820 and 940 nm at
the 1 σ level of confidence, which is close to the best measurements done by the cryogenic
radiometer.
0.2
Unc IQD (fit only)
0.18
Unc refl (2 nm)
Relative uncertainty / %
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
350
450
550
650
750
850
950
1050
Wavelength / nm
Figure 7. The uncertainty components given by the method of the hybrid self - calibration
procedure is shown in the figure.
2.2.5 Diffusely reflected light
The diffusely reflected light is not accounted for in the model given in (1). But the uniformity
and the diffusely reflected light limit the accuracy in the optical power measurements. The
source of diffusely reflected light is localised dust spots on the surface of the detector. By
measuring the uniformity of the detector with a small laser spot we can estimate the diffusely
reflected component. The detector has its maximum signal where no dust is present and
through numerical integration of uniformity values, over an area matching the spotsize in the
application, the difference between maximum signal and integrated value in the central part of
the detector gives the diffusely reflected component. The detector used in the hybrid self calibration procedure was used for a long time with no protection cap covering the surface.
During a period of 6 months the detector was intensively used and therefore collected some
dust, which limited the accuracy. The estimated diffuse components were 0.06 % in the blue
part of the spectrum and 0.04 % in the red part of the spectrum.
19
2.2.6 Summary
An independent spectral response scale is established with silicon reflection trap detectors
with the use of a spectrally invariant CPD. The loss mechanisms are estimated separately. The
basis for following this procedure is that the responsivity of an ideal photodiode can be
expressed in terms of fundamental constants and we measure the deviations from its ideal
performance. This is equivalent to a change in zero point value. Instead of measuring the
responsivity from the usual zero response we measure the responsivity as the deviation from
an ideal response. Because the responsivity of trap detectors is close to the responsivity of an
ideal photodiode the estimated uncertainty in the responsivity values are low with this
method, but their values are strongly correlated. The internal constraint that the recombination
probability for (e-h) pairs created at the pn junction is zero enables us to follow this
procedure. Please note that this is the only constraint needed. It is not necessary to use the
IQD function given in (3), but a IQD function with the same internal constraint is needed.
Likewise, the relative spectral response does not need to be measured with a CPD, but any
spectrally invariant detector would do, for instance a thermopile. In that sense the principles
presented are general and the methods used here is one way of realising an independent
spectral response scale.
2.3 Establishing a spectral response scale in the infrared
In publication II a method to establish a spectral response scale in the infrared from the
known responsivity of a silicon trap detector is presented.
2.3.1 Basic principles
The manufacturing of InGaAs detectors has improved the last decade and they have become
the most popular working standards in the near infrared (IR). A description of a calibration
procedure of infrared standard detectors are given by several groups [46-49]. The principles
are based upon a relative calibration of the InGaAs detector to a spectrally invariant thermal
detector and then an absolute calibration at a few wavelengths in order to scale the relative
measurement to absolute values. As pointed out in these references, the disadvantage of
thermal detectors is their signal to noise ratio (SNR) when they are used in a narrow
20
bandwidth double monochromator set-up at room temperature and limits the accuracy
achievable by this method.
We applied a digital filter, designed for the purpose, to reduce the influence of noise in the
relative measurements. We chose to use a Chebychev low pass filter on the data. This filter is
a symmetric lowpass FIR (Finite Impulse Response) filter with a flat monotonically
decreasing passband and equiripple stopband. The resulting filter response and its taps are
given in fig. 8. The application of this filter will reduce the standard deviation by 30 %. In
addition, our algorithm takes advantage of the known responsivity ratio within a group of
several InGaAs detectors in order to get the best estimate of the responsivity from the group.
(a)
(b)
Figure 8. The filter response in terms of sample frequency (a) and the according filter weights
(b) is shown in the figure.
In our analysis we used a scaling constant that gives the lowest uncertainty in the output
responsivity. The basic argument is the more overlapping points between the absolute and
relative calibration of the InGaAs detector the better estimate we get for the scaling constant.
But the SNR at the shortest wavelengths is so much poorer than for slightly longer
wavelengths, and using them in the calculation of the scaling constant actually increases the
uncertainty in the scaling constant compared to the result achieved when omitting them.
Therefore, the number of samples can be optimised and lies in the spectral range of
approximately 900 to 1000 nm in our set-up.
21
2.3.2 Uncertainty evaluation
The calculation of the uncertainty follows the same principle as shown in publ I and III. We
differentiate the analyzed result (the responsivity of the InGaAs detectors) with respect to the
parameters and functions used to calculate the responsivity. We made a full covariance
analysis calculated in matrix form. The correlation matrix achieved is shown below in fig. 9.
One important observation here is the strong correlation obtained by the uncertainty in the
scaling constant.
Figure 9. The correlation matrix of the responsivity values of the InGaAs detector is given in
the figure.
When the uncertainty in the scaling constant contributes significantly to the total uncertainty
this appears as a correlation between the responsivity values of the different wavelengths. The
more dominant the uncertainty in the scaling constant becomes, the stronger becomes the
correlation between the responsivity values. The reason is clear. When the scaling constant
changes its value all responsivity values changes accordingly and if this is the major source of
uncertainty the correlation approaches 1. The ridge along the diagonal shows the correlation
caused by the digital filter.
22
3. Summary, conclusion and future directions
In this thesis it has been demonstrated that an independent spectral response scale from the
visual to the IR based on simple relative measurements can be established. The accuracy
obtained by the hybrid self-calibration method demonstrates that state of the art accuracy is
obtained with self-calibration principles. A calculable silicon trap detector with low internal
losses over a wide spectral range is needed to establish the scale, in addition to a linear,
spectrally independent detector with a good signal to noise ratio. By fitting the parameters in
the responsivity model to a purely relative measurement we express the spectral response in
terms of fundamental constants with a known uncertainty. This is therefore a primary method.
By applying a digital filter on the relative measurements of the InGaAs detectors in the
infrared reduces the standard deviation by 30 %. In addition, by optimising the necessary
scaling constant converting the relative calibration to absolute values, we have managed to
establish an accurate and cost efficient spectral response scale in the IR. The full covariance
analysis, which takes into account the correlation in the absolute values of the silicon detector,
the correlation caused by the filter and the scaling constant, shows that the spectral response
scale established in the infrared with InGaAs detectors is done with high accuracy. A similar
procedure can be used in the UV, though it has not been demonstrated here.
In fig. 10 the responsivities of the detectors (a) and their associated uncertainties (b) at the 1 σ
level of confidence is compared for the three publications. We see that the responsivity of the
trap detector is very close to the responisivity of an ideal detector over most of its spectral
range. The uncertainties given in (b) are very low uncertainties for the spectral response scales
in the visual and infrared. The improvements of using the hybrid self – calibration method is
clearly demonstrated, though longer integration time and more measurement series in the
purely relative method is expected to reduce the uncertainty in that method as well.
23
1.6
Ideal
Trap
InGaAs
1.4
Response [A/W]
1.2
1
0.8
0.6
0.4
0.2
0
350
550
750
950
1150
1350
1550
1750
Wavelength [nm]
(a)
Combined uncertainty [%]
0.6
Comb Unc Hybrid
Comb unc InGaAs
Relative method
0.5
0.4
0.3
0.2
0.1
0
350
550
750
950
1150
1350
1550
1750
Wavelength [nm]
(b)
Figure 10. The responsivity of the trap detector and an InGaAs detector is compared to the
response of an ideal photodiode in (a). The calculated combined uncertainty of the spectral
response scale from 390 to 1650 nm at the 1 σ level of confidence is shown in (b) as it is
calculated in the three publications.
The suggested methods presented in this thesis would improve if better spectrally invariant
detectors were developed. Designing spectrally invariant detectors to be chilled with liquid
24
nitrogen, but without all the facilities needed for a CR, would reduce the noise of such
detectors. This “CR-light” should preferably be small and the silicon detector should be
placed behind the needed window so that window effects are minimised and full advantage of
the method is obtained. The disadvantages by this method are that the system is more
complex and requires vacuum. In addition, the temperature differences will cause
condensation problems, which also have to be handled.
The uncertainties are obtained with a power levels in the order of 1 μW per nm. If we had
access to a smoothly varying spectrally selective system with higher throughput, we expect to
reduce the uncertainties accordingly. This could be a continuous laser source or a
monochromator system and source with higher brightness. On the other hand, the cost of such
a system would be rather high, so the main advantage with the presented methods would
therefore be reduced.
The evolution of self-calibration is going further. Geist et al has suggested to cool the silicon
detectors down to cryogenic temperatures in order to achieve an ultra high accuracy primary
standard below the 1 ppm level [50]. Gran has initiated a NICe (Nordic Innovations Centre)
funded project with custom designed silicon detectors where half of the surface is covered
with a conducting, semitransparent layer of gold. The detector would then be more stable and
reproducible than the detectors used by the original self-calibration procedure and would have
a lot of similarities with the MOSFET transistor. The response change with bias under the
gold layer would be a direct measure of the quantum deficiency of the uncoated part of the
detector. We have called the suggested method fractional self – calibration, and the detectors
would work nicely in cryogenic environment as well. The reason for this is that the detectors
are deposited a semitransparent metal layer which is expected to be connected to the surface
without substantial changes in its properties at liquid nitrogen temperatures.
25
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Publications
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