Accurate and independent spectral response scale based on silicon trap detectors and spectrally invariant detectors
Faculty of Mathematics and Natural Sciences University of Oslo 2005
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
List of abbreviations
IQD QE CR ESR NMI UV e-h CPD IR SNR NICe MOSFET DVM NPL JV UiO UniK NIST NIR FIR Internal quantum deficiency Quantum efficiency Cryogenic radiometer Electrical substitution radiometer National Metrology Institutes Ultraviolet Electron – hole pair Cavity pyroelectric detector Infrared Signal to noise ratio Nordic Innovations Centre Metal oxide semiconductor field effect transistor Digital voltmeter National Physical Laboratory, UK Justervesenet – Norwegian Metrology Service, Kjeller University of Oslo, Oslo University Graduate Centre, Kjeller National Institute of Standards and Technology, USA Near infrared Finite impulse response
List of symbols
h c k T e λ ρ(λ) δ(λ) R(λ) d a = (df,T, D, dr, b, R) α(λ) V σ2 k F(λ) χ
Planck constant = 6.62607·10-34 Js Speed of light = 299 792 458 m/s Boltzmann constant = 1.38065·10-23 J/K Thermodynamic temperature Elementary charge = 1.60218·10-19 C Wavelength of the radiation Spectral reflectance Spectrally dependent internal quantum deficiency Spectral responsivity Oxide thickness Parameters in the internal quantum deficiency model Spectral absorption coefficient Vector of measured values Variance Scaling constant, also used as fitting parameter a1 Fit function, equals R(λ)/k Generalised least squares equation Covariance matrix of quantity y Matrix of sensitivity coefficients for the fit function Matrix of sensitivity coefficients for the responsivity Frequency band, sampling frequency Filter matrix Relative response of detector m Filtered value of y(m) Responsivity ratio between detector m and n Scaling constant Vector of absolute calibration values of detector m Sensitivity vector of scaling constant K with respect on filtered values
f(a) r(a) fb , fs W y(m),Y(m)
F(m,n) K A(m) Ψ α
Sensitivity vector of scaling constant K with respect on absolute calibration values
Difference in IQD between an unbiased and a biased detector Matrix of sensitivity coefficients for Δ(λ) with respect on fitted parameters b = (D, dr)
Preface List of abbreviations List of symbols Contents 1. Introduction 1.1 Background 1.2 Progress of this work 2. Summary of research 2.1 Goal 2.2 Independent spectral response scale 2.3 Establishing a spectral response scale in the infrared 3. Summary, conclusion and future directions References Publications 1 2 3 5 6 6 10 13 13 13 20 23 26 28
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 . 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 .
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 . The enormous cost in establishing and operating this facility, makes it unavailable to most laboratories.
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 . 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  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 , is displayed.
Figure 1. A drawing of the diode and the bias technique as presented in  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.
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 ).
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 . 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 . 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 .
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 .
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 , 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 , 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  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 . Kübarsepp et al developed a model of the quantum yield in the UV region with two free parameters, extending the model to 250 nm . 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” . 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 . 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
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.
2. Summary of Research
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
e (1 ( )) (1 ( )) , hc
R ( )
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
trap ( , d ) ( ,0, d ) s2 ( ,45, d ) 2 ( ,45, d ) , p
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 .
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
extinction coefficient of the refractive index of silicon. The recombination probability function used in our model is shown in fig. 4.
R df dr
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 .
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.
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 , 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
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
R ( ) , k
F ( )
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.
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
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 0.18 0.16 Unc IQD (fit only) Unc refl (2 nm)
Relative uncertainty / %
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.
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
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) is shown in the figure.
Figure 8. The filter response in terms of sample frequency (a) and the according filter weights
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.
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.
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.
1.6 1.4 1.2 Ideal Trap InGaAs
1 0.8 0.6 0.4 0.2 0 350 550 750 950 1150 1350 1550 1750 Wavelength [nm]
0.6 0.5 0.4 0.3 0.2 0.1 0 350 550 750 950 1150 1350 1550 1750 Wavelength [nm]
Combined uncertainty [%]
Comb Unc Hybrid Comb unc InGaAs Relative method
(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 . 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.
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