Room temperature dual-mode internal quantum deficiency measurement with propagated uncertainty to 0.03% (k = 2)

Metrologia PAPER • OPEN ACCESS Room temperature dual-mode internal quantum deficiency measurement with propagated uncertainty to 0.03% ( k = 2) To cite this article: Johanne Heitmann Solheim et al 2024 Metrologia 61 065007 View the article online for updates and enhancements. You may also like Predictable quantum efficient detector based on n-type silicon photodiodes Timo Dönsberg, Farshid Manoocheri, Meelis Sildoja et al. - Simulations of a predictable quantum efficient detector with PC1D Jarle Gran, Toomas Kübarsepp, Meelis Sildoja et al. - Evaluation of CMIP5 and CMIP6 simulations of historical surface air temperature extremes using proper evaluation methods Thordis L Thorarinsdottir, Jana Sillmann, Marion Haugen et al. - This content was downloaded from IP address 77.40.226.121 on 28/10/2024 at 15:10 Metrologia Metrologia 61 (2024) 065007 (12pp) https://doi.org/10.1088/1681-7575/ad7b24 Room temperature dual-mode internal quantum deficiency measurement with propagated uncertainty to 0.03% (k=2) Johanne Heitmann Solheim1,∗, Eivind Bardalen2and Jarle Gran1 1Justervesenet, Fetveien 99, 2007 Kjeller, Norway 2University of South-Eastern Norway, Borre, Norway E-mail: [email protected] Received 21 May 2024, revised 9 September 2024 Accepted for publication 12 September 2024 Published 28 October 2024 Abstract We present improvements in dual-mode calibration of predictable quantum efficient detectors and demonstrate the importance of calculating absolute uncertainties instead of relative uncertainties. We have implemented a new uncertainty component for the thermal fluctuations in the temperature signal which results in a propagated Type A uncertainty, matching the observed standard deviation. A new thermal drift correction method exploiting a monitor thermistor on the heat sink is relaxing the need for thermal stabilisation of the experimental set-up. With beam position uncertainty ±0.25 mm and background electrical power varying from 10 µW to 900 µW, the measured internal quantum deficiency (IQD) is in average 0.00% ±0.03% (k=2). The IQD exhibits clear systematic effects of beam position and background power, showing the need for improved design of dual-mode modules to further improve the uncertainty. Keywords: PQED, dual-mode, self-calibration, photodiodes, radiometry, optical power 1. Introduction Since the late 1980s, the cryogenic electrical substitution radiometer has served as a primary standard for measuring radiant power [1]. Based on initial work started in the late 1970s, the predictable quantum efficient detector (PQED) has emerged as an alternative to the cryogenic radiometer [2–9]. The PQED is composed of two low-loss induced-junction photodiodes arranged in a trap configuration to reduce reflectance losses to negligible levels. The internal quantum deficiency (IQD) of these photodiodes [8,10] are predicted by the use of numerical tools, facilitating accurate optical power measurements derived from the photocurrent and fundamental physical constants. Photodiodes presents numerous advantages over the ∗Author to whom any correspondence should be addressed. Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. cryogenic radiometer, such as speed, cost, user-friendliness, and the ability to operate at room temperature. Today, the mise en practique for the realisation of the candela includes both the cryogenic radiometer and the PQED as primary standards. In a conventional PQED, internal losses are determined using numerical methods. However, in 2014, White et al published an innovative approach by integrating the principles of the electrical substitution with the PQED, introducing a technique known as the dual-mode method [7]. This method extracts the internal losses of the PQED by comparing the optical power obtained in two modes of operation. With the current design, the photodiode serves both as the absorbing element and as the resistive heater by applying a forward bias. This experimental design is therefore limited to indirect bandgap photodiodes, but not specifically limited to PQEDs. However, due to the known and extremely low IQD of the PQED, it is beneficial to use it as a reference when developing a well-functioning dual-mode procedure. For direct bandgap photodiodes, an external heater could be mounted on the photodiode in order to perform the dual-mode measurements (not discussed in this article). 1 © 2024 The Author(s). Published on behalf of BIPM by IOP Publishing Ltd Metrologia 61 (2024) 065007 J H Solheim et al The incorporation of dual-mode operation not only enhances the versatility of the PQED but also represents a purely experimental method and a significant step forward in achieving more accurate and reliable measurements of other photodiodes. Introducing self-calibration into PQEDs opens for integrating self-calibration into instruments like power meters and spectrometers. In addition, this innovation facilitates traceability in possibly remote locations, such as space. The dual-mode detector used in this study was developed in the chipS·CALe project [11], which made significant advancements in PQED technology. The latest contribution in dual-mode measurements comes from the work of Ulset et al, whose study has successfully implemented an improved design for dual-mode PQED detectors as well as proposing novel algorithms for electrical substitution, resulting in more robust estimations of the IQD [9]. Their work has demonstrated reduced uncertainties associated with the IQD in dualmode operation, from percentage levels to below 0.05%, but increasing uncertainty at power levels below 500 µW. Further, they suggested an approach for the electrical substitution mode, where electrical heating is applied in addition to optical heating. This method holds the potential for shifting towards closed-loop operation. In this paper, we present the results on a dual-mode module studied at a wavelength of 633 nm and power level of 366 µW. We have experimentally studied the influence of the calculated IQD with the beam absorbed at different positions on the detector with various levels of background electrical heating during a dual-mode measurement. We have developed a new drift correction method exploiting a monitor thermistor to compensate for temperature fluctuations, relaxing the need for thermal stabilisation of the experimental set-up. The need for an improved absolute uncertainty evaluation is presented and a new temperature fluctuation component is implemented, yielding an agreement between absolute propagated uncertainty and observed standard deviation in the calculated IQD. The new component enables weighting of each measurement point giving an overall lower experimental uncertainty. 2. Theory and methods The goal behind developing a self-calibrated detector is to use it as a normal photodiode during operation, while regularly running the dual-mode measurement as a calibration procedure to detect changes in the response by a purely experimental method. In this way, the detector can in principle be used and re-calibrated on site without the need to take it to the lab for recalibration. The detailed method for extracting the IQD from the two experiments is presented in this section. 2.1. Estimating internal losses through dual-mode operation Dual-mode refers to the concept of estimating the incoming optical power which reach a photodiode via two distinct modes of operation; i.e. through a photocurrent measurement and through electrical substitution. Starting with the photocurrent measurements, we have that the photocurrent Iphoto from a photodiode is related to the incoming optical power from the source Φby Iphoto =R(λ)Φ (1) where Ris the photodiode responsivity. The responsivity takes into account the internal losses δ, the quantum yield yand the reflection losses ρ, and can be expressed as R(λ) = eλ hc (1−δ(λ))(1−ρ(λ))y(λ)(2) where λis the wavelength of the incoming radiation, his Planck’s constant, cis the speed of light in vacuum and eis the elementary charge. In the dual mode experiment, the photodiode acts as the absorbing element and generates the signal in both modes of operation. As a result, the reflection losses are the same and can therefore be neglected. This further implies that the dual-mode method is independent of polarization. The wavelength applied in this study will not produce a quantum yield above unity [12], but in general the quantum yield can be included as a part of the effective internal losses. By combining equations (1) and (2) we have Iphoto =eλ hc (1−δ(λ))Φ.(3) By recognizing the ideal loss-less diode response called Popt,pc as Popt,pc =Iphotohc eλ,(4) we can express the internal losses as δ(λ) = 1−Popt,pc Φ.(5) The absorbed optical power Φcan be measured through electrical substitution [1], as described in the next section. By using the power estimated from electrical substitution, Popt,es, as a reference for the optical power, i.e. Φ = Popt,es, we arrive at an expression for the internal losses of the photodiode based on the two experimental results δ(λ) = 1−Popt,pc Popt,es .(6) Once the internal losses, commonly referred to as the IQD, are known, the diode can be used as a primary standard for optical power measurements (given that we have eliminated or estimated the reflection losses by other means). In this case, the optical power Φis found through equation (1). In the following section, we describe how electrical substitution is applied in the estimation of Popt,es. 2 Metrologia 61 (2024) 065007 J H Solheim et al Figure 1. A typical thermal signal from the dual-mode detector is shown with R1and R2being the electrical low and high heating signals respectively and Ropt being the optically heated signal. The electrical low power heating last for the first 20 min, followed by 20 min of optical heating and 20 min of high electrical power. 2.2. Electrical substitution Electrical substitution is a well established method in radiometry for estimating optical power [13]. The fundamental principle of this measurement technique involves equating the absorbed radiant power at an absorber with a precisely defined electrical power, through the use of a resistive heater. The electrical and optical power levels are matched by monitoring the temperature on the absorber. The power levels can be exactly matched through temperature regulation, called closed-loop operation. Alternatively an unregulated open-loop approach can be employed, as done in this study. In open-loop operation, the electrical power level is usually set to an operating point slightly below and sightly above the optical power, denoted Pel,1and Pel,2, respectively. For the dual-mode detector used in this study, the electrical heating is achieved through forward biasing the photodiode. The temperature on the photodiode is monitored by use of a thermistor, such that the associated temperature values are represented by resistance. For the thermistor used in this study, an increasing temperature is associated with a decreasing resistance. The optical power Popt,el is then found by interpolation between the resistance levels R1and R2 corresponding to Pel,1and Pel,2; Popt,es =(Ropt −R1) (R2−R1)(Pel,2−Pel,1) + Pel,1−Pel,b(7) where Ropt is the resistance of the thermistor during optical heating, and R1and R2is the resistance during low and high electrical heating, respectively. The term Pel,bis the additional current heating that can be applied during the optical heating step by operating the photodiode in forward bias under illumination, as suggested by Ulset et al [9]. A typical resistance series is shown in figure 1, where the resistance levels for each step is marked. The principle of electrical substitution requires equivalence in radiant and resistive heating. The non-equivalence between electrical and optical heating results in an apparent IQD which is dependent on where the beam hits the detector. The heat equivalence of the module used in this study has a design based on COMSOL®simulations and detailed description can be found elsewhere [9]. COMSOL simulations are used in this study for comparing the measured IQD and the thermal non-equivalence. 2.3. Experimental setup 2.3.1. Dual-mode module. The photodiode is mounted on a copper trap at 45◦relative to the incoming radiation. After hitting the photodiode, the beam is reflected at a mirror mounted below the diode, as shown in figure 2(a). During electrical substitution, the copper structure serves as a heat sink. A picture of the module mounted in the copper trap is shown in figure 2(b). The dual-mode module used in this study is a p-type induced-junction photodiode developed and manufactured in the chipS·CALe project [8] designed for electrical substitution at room temperature. The active area measures 14 ×11mm−2. The photodiode is thermally connected on one side to a printed circuit board (PCB), through a silicon thermal diffuser. The PCB acts as a weak heat link between the diode and the heat sink. A Semitec 103FT1005A5P1 thermistor, that is used to measure the photodiode temperature during electrical substitution, is mounted on the PCB close to the thermal diffuser. This thermistor is referred to as the photodiode thermistor. An aerogel spacer is used to thermally isolate the diode from the heat sink, ensuring heat flow through the PCB. The time constant of the system depends on the dimensions of this heat link. The dual-mode module used in this study has a 4 mm heat link, leading to an approximate time constant of 173 s, which deviates significantly from the simulated 110 s time constant. During optical heating, the beam hits the center of the diode. During electrical heating, when the diode is forward biased, the heat flows from the edges where the connection rings are located. Differences in the heat profiles in the two modes of heating causes a thermal non-equivalence, and the dual-mode module is designed for minimizing this nonequivalence. Further details about the module, its wire connections and the thermal non-equivalence can be found in [9]. 2.3.2. Measurement setup. The measurement setup used in this study is based on that of Ulset et al [9]. In this study, all measurements were conducted in a climate controlled room, set to 22.3◦C. The copper trap with the photodiode is placed in a vacuum chamber (ideal vacuum cube from Ideal Vacuum Products), which is mounted on a vertical translation stage. The beam position on the photodiode was varied from the center (y=0, see figure 2(a)), to a displacement of y=± 1 mm. Positive yvalues refer to a position closer to the heat link. The vacuum cube is supplied with an optical antireflective (AR) coated window to reduce the potential influence of 3 Metrologia 61 (2024) 065007 J H Solheim et al Figure 2. (a) A sketch of the mounting of the dual-mode module in a three reflection trap configuration. A mirror is mounted horisontally beneath the dual-mode module reflecting the radiation back to the module and out. (b) The picture shows the full trap detector structure with peltier element (placed in the gap between the lower and upper part) and thermistors for thermal stabilisation and temperature monitoring. The thermistor used in temperature control is placed in the gap close to the peltier element, while the monitor thermistor is placed towards the top of the copper block. back reflectance of the beam. The vacuum chamber was evacuated using an Agilent Mini-Task AG81 pump to a pressure below 1 ×10−6mbar. The dimensions of the copper trap is described in [9], and is shown in figure 2(a). The copper trap is mounted on a peltier element with a pt1000 thermistor mounted at the bottom of the trap, close to the peltier element (referred to as the temperature control thermistor). A Thorlabs TED 350 temperature controller was used to stabilise the temperature of the copper trap slightly below room temperature exploiting the peltier element and temperature control thermistor (see figure 2(b)). A pt1000 thermistor used for monitoring the temperature at the heat sink was mounted on top of the copper trap close to the photodiode module, as shown in figure 2(b). The resistance was logged using a Keysight 3458A digital multimeter. This thermistor is referred to as the monitor thermistor, which signal is used for temperature drift correction. A simplified sketch of the electrical setup is shown in figure 3. A Datron 4700 calibrator was used as a voltage source, and the current in the circuit was measured using a calibrated Femto DLPCA-200 transimpedance amplifier (TIA), where the voltage was measured with a Keithley 2002 digital multimeter. Another multimeter of the same type was used to measure the voltage over the photodiode. The resistance of the Semitec 103FT1005A5P1 thermistor was recorded with a SIM921 AC resistance bridge (Stanford Research Systems). In the improved set-up, the current measurements are placed on the low side of the voltage source, which makes it unaffected by the switching between forward and reverse bias. All electrical wires are shielded twisted pairs, and all instrument chassis are connected to common ground. Using the same current measurement equipment in both electrical substitution and photocurrent mode gives a more relaxed demand for calibrating the absolute value of the TIA. The calibrated TIA showed a proportional gain deviation of 0.100% to an Figure 3. A simplified sketch of the circuits used in the dual-mode measurement set-up is shown. uncertainty of 5 ppm in both directions with an offset error of −5.68 ×10−5mA. Correction for the TIA offset and gain were implemented in the analysis. An illustration of the experimental set-up is given in figure 4. The laser used in the experiment is a red He–Ne laser with a nominal vacuum wavelength of 632.991 nm. The laser cavity is approximately 0.3 m long giving a free spectral range of 0.5 GHz. The beam is vertically polarised and is power stabilised to 366 µW with a laser power controller. The beam goes through a spatial filter and is expanded to a beam diameter of approximately 1.9 mm 1/e2. The spatial filter is located between the polarisation controller and the monitor detector in the power stabiliser. From the stabiliser, the beam enters the vacuum cube through its wedged AR coated window. 2.3.3. Measurement procedure. When performing a measurement, the series starts with a photocurrent measurement, followed by five electrical substitution cycles. Between each 4 Metrologia 61 (2024) 065007 J H Solheim et al Figure 4. Schematic overview of the experimental setup with the dual-mode detector mounted in the copper trap and placed inside the vacuum cube (marked DMD). electrical substitution cycle, a new photocurrent measurement is made. Photocurrent measurements: The photodiode was reverse biased with 10 V and dark current Idark is measured (shutter closed), before the photocurrent Ilight is measured (shutter open), followed by a new dark current measurement. As a large bias voltage is applied to ensure linearity in the photocurrent measurement, large heat dissipation and temperature rise is gained in the photodiode. Therefore, only the last dark current measurement is used in the analysis, ensuring that the temperature at the diode closely corresponds to that in the measurements with shutter open. All measurements last for 1.5 min, and values from the last 30 s to the last 5 s of the signal is averaged. The photocurrent is found by Iphoto =Ilight −Idark and used to calculate the neccessary application of Pel,1and Pel,2in the electrical substitution measurement. Electrical substitution measurements: The electrical substitution procedure starts with a temperature stabilisation step, where the photodiode is driven in forward bias at a power which matches the sum of the optical power Popt and the background electrical power Pel,b. The following sequence is then run for five rounds: 1. Low electrical heating (shutter closed): A forward bias voltage is applied, leading to a power dissipation (Pel,1) in the diode. This results in a corresponding thermistor reading R1. 2. Optical heating (shutter open): The optical power Popt is absorbed by the photodiode, resulting in a thermistor value Ropt. If a forward bias voltage is applied, an additional electrical heating at Pel,bis dissipated in the diode. This additional electrical power is measured and its component subtracted from the measured optical power (see equation (7)). 3. High electrical heating (shutter closed): A forward bias voltage is applied, leading to a power dissipation (Pel,2) Figure 5. Resistance values in a measurement series consisting of four photocurrent measurements and three electrical substitution cycles. The inserted figure shows an electrical substitution cycle used to estimate the optical power in step down (orange dots) and step up (green stars) is shown. The dissipated power difference between R1 and R2 signals are 10 µW. in the diode. This results in a corresponding thermistor reading R2. 4. Optical heating (shutter open): Identical to the first optical heating step as described in step 2. figure 5shows an example of a resistance signal from a measurement series consisting of four photocurrent measurements and three electrical substitution cycles. The electrical power levels in electrical substitution is found through the voltage and current signal. The average values from the last 120 s to the last 14 s of each step is used. The resistance values are estimated as explained in section 2.4.1. In this study, we have performed experiments where the difference between Pel,1and Pel,2was set to 10 µW and 20 µW. The background electrical power Pel,bwas varied and set to 10 µW, 100 µW and 900 µW, and the beam position was set to −1, 0 and 1 mm relative to the photodiode center. 2.4. Data processing In the following section, we describe how the data from the dual-mode measurements is used to calculate the IQD. To overcome the effect of drift in the ambient temperature, two different approaches for drift correction are presented. 2.4.1. Estimating resistance levels Ropt, R1and R2.In electrical substitution, the resistance levels Ropt,R1and R2need to be estimated from the resistance signal. Each Rvalue is equivalent to a temperature proportional to the dissipated heat inside the photodiode. There are different ways to extract the resistance from the time series. Under the assumption that the system reach steady state for each step (both electrical and optical heating), the resistance can be taken as the mean of a certain interval towards the end of the series, similar to what 5 Metrologia 61 (2024) 065007 J H Solheim et al was described for current and voltage signals. This method will be referred to as the average method. Alternatively, one can estimate the resistance by fitting the datapoints in each step to an exponential function given by r(c1,c2,c3,t) = c1e−c3t+c2(8) where c1,c2and c3are the parameters to be fitted. Here, c2represent the asymptotic value to which the function approaches for large time values t. The resistance value for each step is given by the function value and therefore given by c2. This method for estimating the resistance levels is referred to as the exponential fit method. When using the exponential fit method, it is not strictly necessary to let the system reach steady state. The steady state resistance value is estimated through c2, and the length of the interval used in the exponential fit is a parameter to be optimised. The exponential fit method may use all data points in the interval, whereas the average method only uses the data points where steady state is reached. Therefore, the exponential fit method is used further in this study. 2.4.2. Estimating the IQD. For each electrical substitution cycle we can give several individual estimates for Popt,es, using subsequent values for R2,Ropt and R1either when the power is increasing for each step (decreasing resistance; step down) or power is decreasing for each step (step up). The two methods are illustrated in figure 5, where points used in step down and step up are marked in orange and green, respectively. When calculating the IQD, Popt,es is paired with the optical power found through photocurrent mode Popt,pc. In order to account for potential drift in the photocurrent over time, Popt,pc is estimated on each side of the electrical substitution cycle, and interpolated to fit the sampling timepoint of the different Popt,es. In the data analysis presented in this paper, we use both step up and down for estimating Popt,es, resulting in ten values of IQD for each electrical substitution cycle. This method, described in the next section, differs from the one described in detail in [9], where a local average is used to compensate for temperature drift. The local average method from [9] is performed by first estimating an average R1from two successive low electrical heating steps, and an average Ropt from two optical heating steps. They are combined with the value R2for the high electrical heating step. The values for Ropt,R1and R2are further averaged by an additional set of Ropt,R1and R2, found by using the same method with the only difference being that this time it is two subsequent values for R2which are averaged, and not R1. Choosing the step up and down method leads to more transparent results than the local average method, at the expense of a greater dispersion in the IQD values. The transparency is particularly noticeable when the resistance signal in one of the steps deviates from the expected value. We picture a situation where R1appears lower than anticipated, resulting in a lower IQD than expected. With the local average method, the effect of a lower R1is masked by averaging the value effectively over two step down and two step up rounds. By keeping the step up and down estimates separate, the effect of a lower R1is apparent, which in turn makes it possible to assign this IQD value less confidence. The reason why this becomes possible relates to the uncertainty propagation, which will be explained in section 3.1.3. 2.4.3. Temperature drift correction. Although the ambient temperature was maintained at a stable value, variations in the thermistor signal were observed, which translates to temperature variations at the diode in the order of 80 mK. Both the average method and the exponential fit method are affected by temperature drift, although the exponential fit method exhibits a significantly greater sensitivity. This is because temperature drift may disturb the signals such that they no longer resemble exponential functions, which introduces an error of the fit. To mitigate temperature drift, a temperature controller was installed as explained in section 2.3.2. However, the thermal variations could not be removed completely, as seen in figure 6(a). To correct for the temperature drift, the following two methods were developed. The first method, the monitor temperature drift correction, require an additional temperature sensor to be mounted on the copper trap. The resistance signal from the monitor sensor tracks the background temperature drift in the copper trap, and can be subtracted from the diode resistance signal to remove the varying baseline. Before subtraction, the monitor resistance signal is shifted in time (215 s) relative to the diode signal, to account for the heat transfer time constant through the system. The time shift is derived from the observed unstabilised data fluctuations from the two thermistors, and is validated both through data analyses and by analysing the response from a pulsed thermal signal. The monitor signal is also scaled to account for differences in the response of the two thermistors. The second method is based on the drift correction described in [9], and is referred to as the piece-wise linear drift correction. This correction method can be performed on the diode resistor signal alone, and does not require a monitor sensor. The piece-wise linear drift correction originates from the assumption that the temperature drift is approximately linear during shorter time intervals. The optical heating steps are used as a reference, and resistance drift between these steps are corrected by removing a linear function from one Ropt to the next. The drift correction method employed here is similar to the approach outlined in [9] with the only difference being the choice of reference resistance. Where Ulset et al have selected the low electrical heating steps R1, we have chosen Ropt as the reference resistance. This results in a linear function that spans half the time interval compared to the method by Ulset et al, consequently resulting in a better resolution in the correction process. Figure 6(a) shows a thermistor signal from an electrical substitution measurement cycle where the raw signal is shown in blue. When correcting for the temperature drift, different baselines are subtracted from the raw signal. The baselines from the monitor drift correction, the piece-wise linear drift correction and the drift correction suggested by Ulset et al are plotted in green, orange and pink, respectively. The resulting drift corrected signals are shown in figure 6(b). It is evident 6 Metrologia 61 (2024) 065007 J H Solheim et al Figure 6. Comparison of three different temperature drift compensation methods in a measurement cycle are shown (a) shows the thermal signal and compensation functions used. (b) Shows the result after the compensation functions are applied. The colours correspond to the same method as in figure (a). that the monitor drift correction method is able to better resolve the background drift, leading to a superior drift correction. The piece-wise linear drift correction method and the method suggested by Ulset et al both resemble the monitor correction, although with a lower resolution. The differences are particularly evident in the start of the cycle. The new temperature correction method have demonstrated a significant improvement in observed standard deviation in the dual-mode measurements. Using one series as an example, the observed standard deviation in IQD values was 0.1520% using the uncorrected resistance series, 0.1295% using the method of Ulset et al and 0.0933% using the piece-wise linear drift correction proposed in this work. Using the monitor temperature drift correction, the observed standard deviation drops to 0.0285%. 3. Results and discussion 3.1. Propagation of uncertainty in dual-mode measurements 3.1.1. Estimating the uncertainty in resistance values. For all estimated parameters which are calculated as average values, i.e. current, voltage and resistance (using the average method), the uncertainty is taken to be the standard deviation of the corresponding measurement points. In the following, we describe how the uncertainty in the resistance levels R1,R2and Ropt are estimated when the exponential fit method is employed. We assume that the resistance in each interval can be described by r(c1,c2,c3)in equation (8). We denote the fitted parameters by vector c= [c1,c2,c3], with the 3 ×3 covariance matrix of the fitted values, U(c), given as U(c)=hF(c)TU(r)−1F(c)i−1 (9) where U(r)is the K×Kvariance matrix of the resistance sample values, with Kbeing the number of values. Fis the K×3 sensitivity matrix of the fit function with respect to each cjparameter; f(c)i,j=∂r(c,ti)/∂cj. The variance matrix for the sample values is a diagonal matrix where each diagonal element is given as U(r) i,i=qσ2 Ri,i+RMSE2,(10) where σ2 Ri,iis the variance in the resistance, estimated at steady state over a short time interval and represents the electrical noise in the resistance measurement. The RMSE is the root mean squared error obtained from the fitted function ˆ r(c1,c2,c3,t)to the measured resistance values and implements also an average true variation in the thermal signal as compared to the fit function. The RMSE is given as RMSE =v u u t 1 K K X i=1 (ri−ˆ ri)2(11) where riis the measured value andˆ riis the corresponding function value for measurement number i, and Kis the total amount of measurement points. Ui,jis zero when i=j. The resistance values and related uncertainties values are best estimated as the fit function value and associated uncertainty at the a large time point, given by the last diagonal value of the covariance matrix for the fit function values, derived as: U(r)=F(c)U(c)F(c)T.(12) The function value and associated uncertainty converges against the c2value and its uncertainty as found in equation (9) for the variance of c2. We have verified that the function value and associated uncertainty are similar as the c2value and associated uncertainty, and because of simplicity we have therefore used the c2value and related uncertainty in the estimates for the function value for each resistance step. 7 Metrologia 61 (2024) 065007 J H Solheim et al Table 1. Typical uncertainty budget used in sensitivity analysis. Xixiu(xi)ciui(y)Relative uncertainty R110 490 Ω0.05 Ω1.25 0.0625 µW R210 486 Ω0.05 Ω1.25 0.0625 µW Ropt 10 488 Ω0.05 Ω−2.50 0.125 µW Pel,1471 µW 0.0002 µW 0.50 0.0001 µW Pel,2461 µW 0.0002 µW 0.50 0.0001 µW Pel,b100 µW 0.0002 µW−1.00 0.0002 µW Popt,es 366 µW 0.153 µW 0.042% 3.1.2. Absolute uncertainty analysis. In this section we show why the uncertainty analysis in electrical substitution measurements should be performed by propagating the absolute uncertainties [14], as opposed to relative uncertainties as commonly used in the radiometric community. The central point we are emphasising here concerns how the uncertainties should be propagated. When propagating the square sum of the relative uncertainties, we may end up with a different answer than using the absolute uncertainties and their corresponding sensitivity factors. In the method where relative uncertainties are propagated, the final uncertainty only depends on the signal to noise ratio of the different input parameters. When we instead propagate absolute uncertainties, the sensitivity factors depend on how the input parameters relate to each other. Therefore, this method better reflects the measurement situation, and catches an effect that is also present in experimental data. We demonstrate that the position of the optical power level relative to the two electrical power levels significantly impacts the overall uncertainty in Popt,es. This dependence is inherent to the method, i.e. electrical substitution in general, however it only becomes evident when propagating absolute uncertainties. When the uncertainty is calculated, both absolute and relative uncertainties can be used for the measurand, without any practical difference. The following is a constructed example, however with realistic values for the uncertainties in the variables, which are stated in table 1. The uncertainty in Popt,es is found by propagating the absolute uncertainty through the sensitivity factors of each component in equation (7). The separation of Pel,1and Pel,2, called Pdiff, is varied in three levels; 5 µW, 10 µW and 20 µW, where table 1gives the uncertainty budget for the 10 µW separation. In this example, the background power Pel,bis set to 100 µW. In all calculations, the uncertainty in each component value is kept constant. From the initial value given in table 1, the optical power Popt is now varied (through varying Ropt) with respect to Pel,1 and Pel,2, whose midpoint is denoted Pel,mid. All other values in table 1maintain the same. Figure 7illustrates how the relative uncertainty in Popt,es is affected by the difference between Popt and Pel,mid. When Popt falls outside the range spanned by Pel,1 and Pel,2, a significant increase in Type A uncertainty u(Popt,es) is observed. We further observe that the sensitivity towards a relative change in Popt depends on the value of Pdiff, with a higher sensitivity for narrower separation of the two levels. Figure 7indicates that the optical power Popt should ideally fall Figure 7. The calculated relative uncertainty of Popt,es (solid lines), based on the numerical values from table 1. The x-axis gives the position of the optical power level relative to the electrical power levels. Three different values of the electrical power separations are used (indicated in dashed lines, and colour coded). The measured relative standard deviation with different experimental settings are shown as dots with colors matching the calculated example. exactly halfway between Pel,1and Pel,2, resulting in the lowest propagated uncertainty in Popt,es and with the same value independent of the value of Pdiff. The sensitivity calculations of uncertainty were experimentally confirmed in our set-up with Popt equal to a power 30 µW higher than Pel,mean with Pdiff values of 10 µW and 20 µW. The results are shown as dots in figure 7with corresponding color to calculated curves. The measured relative standard deviation shows even higher sensitivity to Popt position relative to Pel,mean than the calculations, with a lower relative standard deviation at the center position and higher relative standard deviation outside. Based on the preceding analysis, it becomes evident that utilizing absolute uncertainties in uncertainty propagation is crucial. When using relative uncertainties, the resulting uncertainty in Popt,es is 0.04%, regardless of where Popt is positioned relative to Pel,1and Pel,2. Without the sensitivity factors, the u(Popt,es)dependence of the optical power relative to the electrical ones will not be shown. Further, we draw the conclusion that choosing a wider range between Pel,1and Pel,2results in a more robust estimate for Popt,es. This makes the measurement 8