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Correction of Spatial Nonuniformity in Spectroradiometer Field-of-View Using a Concentric-Circles Method

Correction of Spatial Nonuniformity in Spectroradiometer Field-of-View Using a Concentric-Circles... hv photonics Article Correction of Spatial Nonuniformity in Spectroradiometer Field-of-View Using a Concentric-Circles Method Zhaoqiang Jiao , Yiwen Li *, Ge Chen, Yao Li, Shijie Chai and Puyousen Zhang Aviation Engineering School, Air Force Engineering University, Xi’an 710043, China; jiao_zq@163.com (Z.J.); 18189144368@163.com (G.C.); liyao_0927@163.com (Y.L.); chaishijie@sohu.com (S.C.); z374699183@163.com (P.Z.) * Correspondence: lee_yiwen@163.com Abstract: Spectroradiometers exhibit the smallest aberration and the optimum response at the field- of-view (FOV) center. The aberration increases and the response deteriorates at positions further away from the FOV center, which leads to nonuniformity in the spectroradiometer FOV. In this study, a concentric-circles method for correcting the spectroradiometer FOV nonuniformity was developed. The calibration experiment for FOV nonuniformity was conducted by establishing the experimental platform. The nonuniformity correction coefficients were obtained and then used to fit the correction function curve within the whole FOV, allowing for correction of measurement targets with an arbitrary shape. The radiation intensity of the blackbody at different temperatures was obtained by measurement, and the nonuniformity coefficient was used to correct it. After correction, the error was within 1.84% for the spectrally integrated radiant intensity in the non-absorption band. Using this correction method, efficient calibration of spectroradiometer nonuniformity can be achieved, thereby enhancing the measurement accuracy of the spectroradiometer. Keywords: spectroradiometer; aberration; field-of-view (FOV) nonuniformity; concentric-circles correction Citation: Jiao, Z.; Li, Y.; Chen, G.; Li, Y.; Chai, S.; Zhang, P. Correction of 1. Introduction Spatial Nonuniformity in Fourier-transform infrared (FTIR) spectroscopy has found increasingly extensive ap- Spectroradiometer Field-of-View plications in environment monitoring, pollution prevention and control [1–3], infrared Using a Concentric-Circles Method. target detection for the military [4–7], atmospheric transmittance measurement, and other Photonics 2022, 9, 56. fields [8–11]. A Fourier infrared spectroradiometer can obtain the spectral radiation charac- https://doi.org/10.3390/ teristics of a source, but its measurement results generally differ considerably from those photonics9020056 calculated under ideal conditions. The causes of the errors include issues with the repeata- Received: 11 December 2021 bility of spectroradiometer measurements, detector nonlinearity, interference from infrared Accepted: 17 January 2022 background radiation, atmospheric transmission attenuation, and human errors in exper- Published: 21 January 2022 imental apparatus testing. The nonuniformity of the spectroradiometer ’s field-of-view (FOV) response caused by off-axis aberration also significantly impacts measurements. At Publisher’s Note: MDPI stays neutral the center of the spectroradiometer FOV, the aberration is the smallest and the best response with regard to jurisdictional claims in can be obtained. At long distances from the FOV center, the aberration increases and the published maps and institutional affil- iations. response deteriorates. Therefore, when the target to be measured deviates from the FOV center or occupies a major part of the spectroradiometer FOV, the radiation measurement results contain considerable errors compared to the theoretical values. The spectroradiometer consists of four parts: an optical system, a detection system, Copyright: © 2022 by the authors. a signal processing module, and a computer module [12,13]. The optical system receives Licensee MDPI, Basel, Switzerland. and collects the energy of the target radiation source. The detection system then transforms This article is an open access article the collected energy into physical quantities, such as voltage and resistance. The signal distributed under the terms and processing module amplifies the physical quantities, which are ultimately transmitted to conditions of the Creative Commons the computer module for data visualization by supporting software. Attribution (CC BY) license (https:// Among the modules in the spectroradiometer, the off-axis parabolic mirror in the creativecommons.org/licenses/by/ optical system constitutes the fundamental cause of FOV nonuniformity. For the off-axis 4.0/). Photonics 2022, 9, 56. https://doi.org/10.3390/photonics9020056 https://www.mdpi.com/journal/photonics Photonics 2022, 9, 56 2 of 14 parabolic mirror, the aberration can be ignored on its optical axis, but it increases rapidly when the deviation from the optical axis exceeds a certain value. Therefore, the target light can be recovered well at positions close to the optical axis. However, at positions further away from the optical axis, aberration may result in different responses from the spectroradiometer for the same target at different FOV positions. The greater the deviation from the optical axis, the worse the response of the spectroradiometer is. Furthermore, aberration is a complex function related to the structure of an optical system that cannot be directly expressed as a specific function. References [14,15] studied the response nonuniformity of the spectral testing appara- tus in the theoretical measurement regions. They evaluated the practical spectroradiometer FOV range and the responses at different positions within the FOV range and obtained the directional response function, which is of far-reaching significance for accurate spectrora- diometer measurements. In 2015, Huang, W.; Ji, H.H.; Si, R. [16] corrected the nonuniformity in the results measured by an FTIR spectroradiometer. By studying the effects of FOV and field area on spectral radiant intensity, they concluded that measurements for the same target varied with the relative target position in the FOV. Furthermore, by integrating the theoretical spectral radiant intensities in the band from 3.5~4.0 m and comparing the integral with the test value, a correction coefficient was obtained, which was then used for uniformity correction. With this approach, the error between the corrected test result and the spectral radiant intensity calculated under ideal conditions was reduced. However, this method requires the acquisition of the target radiation source’s test and theoretical radiation values, which are then used to obtain the correction coefficient. Furthermore, it does not explain the specific law of nonuniformity. In 2018, Wang, X.X.; Yang, H.R.; Yu, B. et al. [17] corrected the nonuniformity of the spectroradiometer FOV using equal-solid-angle calibration. First, by studying the nonuniformity of spectroradiometer FOV, the voltage responses at different FOV positions were obtained, revealing that the response at the edge was approximately 50% lower than that at the center. For the nonuniformity of the FOV, an equal-solid-angle calibration method was proposed. As the distance between the spectroradiometer and the blackbody was set reasonably, the solid angle when the spectroradiometer measured the target was the same as the solid angle when the blackbody was measured. In this way, the optical paths of the spectroradiometer when measuring the blackbody and the target to be measured were the same, eliminating the influence of the nonuniformity of the FOV. The final results revealed that the measurement error was less than 2%. This method requires that the blackbody and the target have substantially the same shape. The spectroradiometer has the same optical path when measuring both objects. However, it is difficult to find blackbodies with similar shapes for equal-solid-angle calibration for irregular target radiation sources. There are certain limitations present. There have been in-depth reports about the suppression techniques that can be used for background radiation and the spectroradiometer response function, with the aim of improving the measurement accuracy of the spectroradiometer. However, few studies have focused on correcting the spatial nonuniformity of the spectroradiometer FOV. With the existing methods, the inversion is performed with the help of the theoretical value of the radiation intensity of the target radiation source. However, the theoretical value of the radiation intensity of the target radiation source is generally difficult to obtain in practical tests. Alternatively, it is necessary to use a blackbody with a similar shape as the target radiation source to achieve better operation. However, this method cannot be applied when the target radiation source has an irregular shape; for example, when measuring an engine tail jet. In general, the existing processing methods still have limitations. Therefore, a concentric-circles method for correcting the nonuniformity of spectroradiometer FOV was developed in this study. It can be applied to target radiation sources of any shape and with unknown theoretical radiation intensities. Photonics 2022, 9, x FOR PEER REVIEW 3 of 14 Therefore, a concentric-circles method for correcting the nonuniformity of spectroradiom- eter FOV was developed in this study. It can be applied to target radiation sources of any Photonics 2022, 9, 56 3 of 14 shape and with unknown theoretical radiation intensities. 2. Calibration Scheme for FOV Nonuniformity 2. Calibration Scheme for FOV Nonuniformity 2.1. Correction using Concentric-Circles Method 2.1. Correction Using Concentric-Circles Method Spectroradiometers exhibit minor aberration and an optimum response at the FOV Spectroradiometers exhibit minor aberration and an optimum response at the FOV center. The aberration increases and the response deteriorates further away from the FOV center. The aberration increases and the response deteriorates further away from the FOV center, which leads to phase nonuniformity in the spectroradiometer FOV. The axisym- center, which leads to phase nonuniformity in the spectroradiometer FOV. The axisymmet- metric aberration of the optical system, which causes the nonuniform response of the FOV, ric aberration of the optical system, which causes the nonuniform response of the FOV, is is circularly symmetric and gradually increases along the FOV, taking the center of the circularly symmetric and gradually increases along the FOV, taking the center of the FOV FOV spectroradiometer as the center of the circle. This indicates a centrosymmetric distri- spectroradiometer as the center of the circle. This indicates a centrosymmetric distribution. bution. Therefore, a correction scheme using a concentric-circles method for the nonuni- Therefore, a correction scheme using a concentric-circles method for the nonuniformity of formity of the FOV of the spectroradiometer can be proposed. the FOV of the spectroradiometer can be proposed. The center of the FOV of the spectroradiometer was taken as the concentric center of The center of the FOV of the spectroradiometer was taken as the concentric center the circle. Moreover, the FOV was divided into concentric rings. The nonuniformity of the of the circle. Moreover, the FOV was divided into concentric rings. The nonuniformity FOV along the same ring was the same. The blackbody to be calibrated moved on different of the FOV along the same ring was the same. The blackbody to be calibrated moved on rings along the red line, as shown in Figure 1. For the entirety of the moving process, the different rings along the red line, as shown in Figure 1. For the entirety of the moving blackbody was always completely located in the FOV of the spectroradiometer. At this process, the blackbody was always completely located in the FOV of the spectroradiometer. time, the radiation value received by the spectroradiometer did not change, and the theo- At this time, the radiation value received by the spectroradiometer did not change, and the theor retical output response re etical output response main remained ed unch unchanged. anged. Howe However ver, due , due to th to e nonuniform the nonuniformity ity of th of e the FOV, th FOV,e output the output response response value o valuefof the the sp spectr ectror oradiometer adiometer chan changed ged when the when the b blackbody lackbody was was in in di difffe ferre ent nt p positions. ositions.Using Using the the test test r res esults ultfor s for spectr spectror oradiometers adiometers with with blackbodies blackbodies in dif in di fer ff ent eren rings, t rings, thethe non nonuniformity uniformit of y of the th FOV e FOV of of the the spectr spectro oradiometers radiometers was wa calibrated. s calibrated. Figure 1. Schematic diagram of the concentric-circles method. Figure 1. Schematic diagram of the concentric-circles method. In the calibration process, in order to reduce the measurement error, two measure- In the calibration process, in order to reduce the measurement error, two measurements ments were made on the left and right sides of the same ring. The measurement numbers were made on the left and right sides of the same ring. The measurement numbers on on the left were marked as −1, −2, …, −i, …, −n, and the measurement numbers on the right the left were marked as 1, 2, . . . , i, . . . , n, and the measurement numbers on the were marked as 1, 2, …, i, …, n. The average of the two measurements of i and -i was right were marked as 1, 2, . . . , i, . . . , n. The average of the two measurements of i and -i calculated in order to replace the measured response value for the entire ring. When the was calculated in order to replace the measured response value for the entire ring. When the number of ring divisions increases infinitely, a correction coefficient curve can be obtained. In this way, a more accurate response value at each position in the FOV of the spectroradiometer can be obtained, and the nonuniformity coefficient at all positions of the FOV can be calibrated. Photonics 2022, 9, 56 4 of 14 2.2. Experimental Scheme The experiment setup included a spectroradiometer, an electrically controlled mobile platform, and standard-surface blackbodies. The spectroradiometer was a model MR170 produced by ABB, with a spectral range of 2~15 m and optional lenses of 75 mrad, 28 mrad, and 4.9 mrad. Two types of HT2M and B-500HE-20 blackbodies produced by DEMEI and LR Tech respectively, with uniform and stable surface temperature distributions, were used: blackbody I, with a diameter of 100 mm, and blackbody II, with a size of 200 mm 200 mm and blackbody emissivity of 0.95. The electronically controlled mobile platform was a model FZVAC1200 produced by Fuzhou Vacuum Electromechanical Equipment, and the moving accuracy was 0.05 mm. Driven by the electrically controlled mobile platform, the blackbody moved the given distance along the direction perpendicular to the optical axis of the spectroradiometer. Thus, the blackbody radiance at each given position on the mobile platform could be measured. According to the characteristics and purpose of this experiment, and considering the need to reduce the influence of the signal-to-noise ratio, when the output of the spectroradiometer was unsaturated, the blackbody temperature was maximized. After fully considering the temperature range of the blackbody and the responsiveness of the spectroradiometer, the following experiment scheme was configured. The spectroradiometer had to be calibrated before use so as to facilitate convenient test operations while guaranteeing the accuracy requirements. As this experiment was conducted in the laboratory, the measurement distance was short, and the temperature variations of the target radiation source were small. Under these conditions, the response function of the spectroradiometer was considered to be linear, so the two-point calibration method was selected [18–20]. Specifically, two different temperatures were configured for the blackbody to calibrate the spectroradiometer. To eliminate the influence of background radiation and improve the calibration accuracy, the blackbody filled the spectroradiometer FOV during calibration. The theoretical formulae for the two-point calibration method are: V(, T ) = R()L(, T ) + O() (1) H H V(, T ) = R()L(, T ) + O() (2) C C where T is high temperature, T is low temperature, V(,T ) is the output voltage of the H C H spectroradiometer when testing the high-temperature blackbody, L(,T ) is the radiance of the high-temperature blackbody, V(,T ) is the output voltage of the spectroradiometer when testing the low-temperature blackbody, L(,T ) is the radiance of the low-temperature blackbody, R() is the response of the spectroradiometer, and O() is the error of the radia- tion measurement, which does not change when the temperature of the blackbody changes. By combining Equations (1) and (2), the response R() and radiation measurement error O() can be obtained as follows: V(, T ) V(, T ) H C R() = (3) L(, T ) L(, T ) H C V(, T )L(, T ) V(, T )L(, T ) C H H C O() = (4) L(, T ) L(, T ) H C After two-point calibration, the response and measurement error of the spectrora- diometer could be determined, and the linear relationship between the voltage and the spectral radiance could be obtained as follows: V(, T) = R()L(, T) + O() (5) Thus, the voltage measured by the spectroradiometer was matched with the spectral radiance of the target, and the response function between the two was obtained. The schematic diagram of the experimental setup for measuring the spatial nonunifor- mity of the spectroradiometer FOV is shown in Figure 2. The blackbody was placed on the Photonics 2022, 9, x FOR PEER REVIEW 5 of 14 After two-point calibration, the response and measurement error of the spectroradi- ometer could be determined, and the linear relationship between the voltage and the spec- tral radiance could be obtained as follows: V(λ= ,T) R()λ ⋅L(λ,T)+O()λ (5) Thus, the voltage measured by the spectroradiometer was matched with the spectral radiance of the target, and the response function between the two was obtained. Photonics 2022, 9, 56 5 of 14 The schematic diagram of the experimental setup for measuring the spatial nonuni- formity of the spectroradiometer FOV is shown in Figure 2. The blackbody was placed on the electronically controlled mobile platform and in the center of the spectroradiometer electronically controlled mobile platform and in the center of the spectroradiometer FOV. FOV. It could be moved to the left and right under the control of the electronically con- It could be moved to the left and right under the control of the electronically controlled trolled mobile platform. The blackbody moving path is shown in the figure. A 75 mrad mobile platform. The blackbody moving path is shown in the figure. A 75 mrad lens was lens was selected, and the test distance D1 was set to 5.33 m. The temperature of the black- selected, and the test distance D was set to 5.33 m. The temperature of the blackbody body was set to 533 K. In the specific test, the FOV nonuniformity calibration was per- was set to 533 K. In the specific test, the FOV nonuniformity calibration was performed by formed by first measuring the spectrum radiance of blackbody I at the central axial posi- first measuring the spectrum radiance of blackbody I at the central axial position of the tion of the spectroradiometer FOV. Then, starting from the center measurement point, the spectroradiometer FOV. Then, starting from the center measurement point, the blackbody blackbody was moved to the left of the center position 1 cm at a time to perform each was moved to the left of the center position 1 cm at a time to perform each measurement, measurement, with 15 measurements taken in total across 15 cm. The same measurements with 15 measurements taken in total across 15 cm. The same measurements were recorded were recorded on the right side of the center of the FOV, and a total of 31 measurements on the right side of the center of the FOV, and a total of 31 measurements were performed. were performed. Figure 2. Schematic diagram of the experimental setup for measuring the spatial nonuniformity of Figure 2. Schematic diagram of the experimental setup for measuring the spatial nonuniformity of the spectroradiometer FOV. the spectroradiometer FOV. Afterward, as shown in Figure 3, the FOV nonuniformity calibration result was Afterward, as shown in Figure 3, the FOV nonuniformity calibration result was ver- verified by placing blackbody II at the spectroradiometer FOV center and measuring its ified by placing blackbody II at the spectroradiometer FOV center and measuring its spec- spectral radiation. tral radiation. The measurement scheme is shown in Table 1. A total of two sets of tests were carried out. When the distance D between the blackbody and the spectroradiometer was set reasonably, the blackbody accounted for 50.11% of the spectroradiometer ’s FOV. Here, D was 4.25 m; the blackbody temperatures were set as 547 K and 557 K, respectively. Photonics 2022, 9, 56 6 of 14 Photonics 2022, 9, x FOR PEER REVIEW 6 of 14 Figure 3. Schematic diagram of experiment to verify the effect of the spatial nonuniformity correc- Figure 3. Schematic diagram of experiment to verify the effect of the spatial nonuniformity correction tion of the spectrometer FOV. of the spectrometer FOV. The measurement scheme is shown in Table 1. A total of two sets of tests were carried Table 1. FOV nonuniformity correction verification scheme. out. When the distance D2 between the blackbody and the spectroradiometer was set rea- sonably, the blackbody accounted for 50.11% of the spectroradiometer’s FOV. Here, D2 Percentage in the FOV of the D (m) Temperature (K) Spectroradiometer (%) was 4.25 m; the blackbody temperatures were set as 547 K and 557 K, respectively. 4.25 547 50.11 Table 1. FOV nonuniformity correction verification scheme. 4.25 557 50.11 Temperature Percentage in the FOV of the L (m) 3. Results (K) Spectroradiometer (%) 3.1. FOV Nonuniformity Correction Coefficient 4.25 547 50.11 The distance i between each measurement point and the FOV center was divided by 4.25 557 50.11 the FOV radius for normalization and the ratio was denoted as . In this study, the FOV radius Q of the spectroradiometer was 200 mm. 3. Results 3.1. FOV Nonuniformity Correction Coefficient = (6) The distance i between each measurement point and the FOV center was divided by the FOV radius for normalization and the ratio was denoted as β. In this study, the FOV The radiance measured by the spectroradiometer was transformed into the radiant in- radius Q of the spectroradiometer was 200 mm. tensity at different wavelengths by averaging the spectral test data from two measurements on the same ring. Meanwhile, the theoretical radiant intensity value of the blackbody was β = (6) obtained by the following Equations [21]. " a The radiance measured L by the sp = ectroradiometer was td ran  sformed into the radian (7) t (  ) 1 2 5 a /T (e 1) intensity at different wavelengths by averaging the spectral test data from two measure- ments on the same ring. Meanwhile, the theoretical radiant intensity value of the black- I = L A (8) (  ) (  ) 1 2 1 2 body was obtained by the following Equations. [21]. Here, L is the radiance within the   band; I is the radiant intensity (  ) 1 2 (  ) 1 2 1 2 within the   band; A is the effective radiation area of the target; " is the emissivity 1 2 Photonics 2022, 9, x FOR PEER REVIEW 7 of 14 ε 2 a Ld =⋅λ (7) () λ−λ 5 a/λT π 1λ− (e 1) IL = ⋅A (8) () λ−λ (λ −λ) 12 1 2 Here, L is the radiance within the λλ − band; I is the radiant inten- () λ−λ 12 (- λλ ) 12 12 sity within the λλ − band; A is the effective radiation area of the target; ε is the emissiv- ity of the blackbody; a and a are radiation constants, with values of 3.7415 ± 0.0003 × 1 2 Photonics 2022, 9, 56 7 of 14 8 4 2 4 10 (W·μm /m ) and 1.43879 ± 0.00019 × 10 (μm·K), respectively; and T is the temperature of the blackbody. of the Th blackbody; e measure a and d d aaar ta e we radiation re pr constants, ocessed with acco values rding of 3.7415 to Equ 0.0003 ations ( 107) and (8). First, the rela- 1 2 4 2 4 (Wm /m ) and 1.43879  0.00019  10 (mK), respectively; and T is the temperature tionship curve between the wavelength and the radiant intensity at different measure- of the blackbody. ment positions was obtained for blackbody I, as shown in Figure 4. The measured data were processed according to Equations (7) and (8). First, the rela- tionship curve between the wavelength and the radiant intensity at different measurement positions was obtained for blackbody I, as shown in Figure 4. Figure 4. Theoretical and measured values of spectral radiant intensity at T = 533 K. Figure 4. Theoretical and measured values of spectral radiant intensity at T = 533 K. The variation curve of the radiant intensity with the wavelength measured at different positions and the theoretical radiant intensity curves at the two temperatures are given in The variation curve of the radiant intensity with the wavelength measured at differ- Figure 4. As can be seen, the spectral radiant intensity curves of 3~5 m obtained for the same target varied at different positions. Basically, the smaller was—i.e., the closer it was ent positions and the theoretical radiant intensity curves at the two temperatures are given to the FOV center—the greater the measured spectral radiant intensity was and the closer in Figure 4. As can be seen, the spectral radiant intensity curves of 3~5 μm obtained for to the theoretical value. Furthermore, as can be observed in Figure 4, the practical spectral radiant intensity the same target varied at different positions. Basically, the smaller β was—i.e., the closer curve fluctuated obviously at wavelengths of 3~3.4 m,4.2~4.4 m, and 4.5~5 m. Specifi- it was to the FOV center—the greater the measured spectral radiant intensity was and the cally, the curve decreased rapidly at approximately 4.2 m and then rose at around 4.4 m. closer These to changes the theoretical v were due to energy alue. attenuation during atmospheric transmission. The wave- lengths of the infrared absorption bands for the main atmospheric components at 3~5 m Furthermore, as can be observed in Figure 4, the practical spectral radiant intensity are shown in Table 2, in which CO and H O exhibited the highest absorptions. Therefore, 2 2 curve fluctuated obviously at wavelengths of 3~3.4 μm,4.2~4.4 μm, and 4.5~5 μm. Specif- these two components should be the focus when discussing atmospheric transmittance in the range of 3~5 m. The fluctuations near 3.2 m were due to the influence of H O in the ically, the curve decreased rapidly at approximately 4.2 μm and then rose at around 4.4 atmosphere, and the significant fluctuations at 4.2~4.4 m were due to atmospheric CO , μm. These changes were due to energy attenuation during atmospheric transmission [9]. which had a strong absorption band at 4.3 m, causing an evident drop in the curve [21]. The wavelengths of the infrared absorption bands for the main atmospheric components at 3~5 μm are shown in Table 2, in which CO2 and H2O exhibited the highest absorptions. Therefore, these two components should be the focus when discussing atmospheric trans- mittance in the range of 3~5 μm. The fluctuations near 3.2 μm were due to the influence of H2O in the atmosphere, and the significant fluctuations at 4.2~4.4 μm were due to at- mospheric CO2, which had a strong absorption band at 4.3 μm, causing an evident drop in the curve [21]. Photonics 2022, 9, x FOR PEER REVIEW 8 of 14 Table 2. Center wavelengths of infrared absorption bands for main atmospheric components at 3~5 μm. Composition Center Wavelength of Absorption Band (µm) CO2 4.3, 4.8 H2O 3.2 CO 4.7 CH4 3.3 Photonics 2022, 9, 56 8 of 14 O3 4.8 Overall, the influence of atmospheric absorption was minor within the range from Table 2. Center wavelengths of infrared absorption bands for main atmospheric components at 3~5 m. 3.4~4.15 μm, so the curve was generally smooth, exhibiting a trend similar to that of the theoretical curve. To eliminate the influence of atmospheric transmission attenuation on Composition Center Wavelength of Absorption Band (m) CO 4.3, 4.8 the experiment when correcting the FOV nonuniformity, the band from 3.5~4.15 μm was H O 3.2 selected for calibration in the data processing. CO 4.7 CH 3.3 The spectral radiation data measured at the FOV center was closest to the theoretical O 4.8 value. After obtaining the total radiant intensity at 3.5~4.15 μm for each measurement point, the correction coefficient α was calculated by taking the radiant intensity at β = 0 as Overall, the influence of atmospheric absorption was minor within the range from 3.4~4.15 m, so the curve was generally smooth, exhibiting a trend similar to that of the the reference value. theoretical curve. To eliminate the influence of atmospheric transmission attenuation on the experiment when correcting the FOV nonuniformity, the band from 3.5~4.15 m was selected for calibration in the data processing. i α = i = 0, 1, 2 … 15. (9) The spectral radiation data measured at the FOV center was closest to the theoretical value. After obtaining the total radiant intensity at 3.5~4.15 m for each measurement point, the correction coefficient was calculated by taking the radiant intensity at = 0 as the reference value. where I is the 3.5~4.15 μm radiant intensity at βi and I is the 3.5~4.15 μm radiant in- i 0 = i = 0, 1, 2 . . . 15. (9) tensity at β0. where I is the 3.5~4.15 m radiant intensity at and I is the 3.5~4.15 m radiant intensity i i Figure 5 presents the correction coefficient α at different values of β at 533 K and the at . spectrally integrated radiant intensity in the 3.5~4.15 μm band. Specifically, the correction coef- Figure 5 presents the correction coefficient at different values of at 533 K and the spectrally integrated radiant intensity in the 3.5~4.15 m band. Specifically, the correc- ficient changed gently near the FOV center. The correction coefficient dropped signifi- tion coefficient changed gently near the FOV center. The correction coefficient dropped cantly when β became larger than 0.35. significantly when became larger than 0.35. Figure 5. The correction coefficient at different values of and the spectrally integrated radiant intensity in the 3.5~4.15 m band. Figure 5. The correction coefficient α at different values of β and the spectrally integrated radiant intensity in the 3.5~4.15 μm band. The correction coefficient α at different values of β was obtained using the above test. The distance β from the center of the FOV of the spectroradiometer was taken as the ab- scissa and the nonuniformity correction coefficient α as the ordinate. The quartic polyno- mial function was used for fitting, and the correction function α = f(β) was obtained, as shown in the corresponding curve in Figure 6a. Photonics 2022, 9, x FOR PEER REVIEW 9 of 14 The FOV of the spectroradiometer was represented in the X-Y coordinate system, with the center of the FOV of the spectroradiometer as the origin, the normalized horizon- tal distance as X, and the normalized vertical distance as Y. Photonics 2022, 9, 56 The distance from any point (xi,yi) in the coordinate system to the center of 9 th of 14 e FOV 22 22 xy + α=+ f( x y ) was , so the correction coefficient at (xi,yi) was , and it ii ii i The correction coefficient at different values of was obtained using the above was recorded as α = f (x, y) . From this, the correction coefficient α at any position of ii i test. The distance from the center of the FOV of the spectroradiometer was taken as the FOV of the spectroradiometer could be obtained. The correction coefficient cloud im- the abscissa and the nonuniformity correction coefficient as the ordinate. The quartic age for the whole FOV is shown in Figure 6b. polynomial function was used for fitting, and the correction function = f( ) was obtained, as shown in the corresponding curve in Figure 6a. Figure 6. (a) The FOV nonuniformity correction coefficient fitting curve; (b) the correction coefficient Figure 6. (a) The FOV nonuniformity correction coefficient fitting curve; (b) the correction coeffi- cloud image. cient cloud image. The FOV of the spectroradiometer was represented in the X-Y coordinate system, with As can be seen in Figure 6, the nonuniformity reached 0.55 at the edge of the FOV of the center of the FOV of the spectroradiometer as the origin, the normalized horizontal the spectroradiometer. When the target occupied a large area in the spectroradiometer, distance as X, and the normalized vertical distance as Y. the effect of the spatial nonuniformity of the spectroradiometer FOV was very large and The distance from any point (x , y ) in the coordinate system to the center of the FOV i i q q needed to be corrected. 2 2 2 2 was x + y , so the correction coefficient at (x , y ) was = f( x + y ), and it was i i i i i i i recorded as = f(x , y ). From this, the correction coefficient at any position of the FOV i i 3.2. Correction of FOV Nonuniformity of the spectroradiometer could be obtained. The correction coefficient cloud image for the whole FOV is shown in Figure 6b. When testing the target radiation source with the spectroradiometer, the distance As can be seen in Figure 6, the nonuniformity reached 0.55 at the edge of the FOV from the edge of the FOV of the spectroradiometer to the center of the FOV was regarded of the spectroradiometer. When the target occupied a large area in the spectroradiometer, as 1 to normalize the area of the target radiation source. The area was denoted as S. When the effect of the spatial nonuniformity of the spectroradiometer FOV was very large and correcting the target radiation source, there was a correction coefficient for each concentric needed to be corrected. ring in the FOV of the spectroradiometer, and the correction coefficients were different for different rings. 3.2. Correction of FOV Nonuniformity When testing the target radiation source with the spectroradiometer, the distance from For the i-th circle, the correction coefficient at the distance xy + = βi from the ii the edge of the FOV of the spectroradiometer to the center of the FOV was regarded as center of the FOV of the spectroradiometer was αi. As shown in the Figure 7, the target 1 to normalize the area of the target radiation source. The area was denoted as S. When radiation source was in this red area. The correction coefficient was αi. correcting the target radiation source, there was a correction coefficient for each concentric ring in the FOV of the spectroradiometer, and the correction coefficients were different for different rings. 2 2 For the i-th circle, the correction coefficient at the distance x + y = from the i i center of the FOV of the spectroradiometer was . As shown in the Figure 7, the target radiation source was in this red area. The correction coefficient was . Photonics 2022, 9, x FOR PEER REVIEW 10 of 14 Photonics 2022, 9, 56 10 of 14 Figure 7. Correction coefficient for the target radiation source in the FOV. Figure 7. Correction coefficient for the target radiation source in the FOV. From this, it can be seen that every position of the target radiation source in the FOV From this, it can be seen that every position of the target radiation source in the FOV had a certain correction coefficient. For the calculation of radiation intensity correction, had a certain correction coefficient. For the calculation of radiation intensity correction, the method of splitting, approximating, summing, and taking the limit was used to derive the method of splitting, approximating, summing, and taking the limit was used to derive the formula. The area of the target radiation source was arbitrarily divided into n area elements. the formula. For the FOV area with position coordinates ("  ), the area element of this area was D j, j j The area of the target radiation source was arbitrarily divided into n area elements. and the radiation intensity was: For the FOV area with position coordinates (εj, μj), the area element of this area was Δσj and the radiation intensity was:j DI  LD (10) f(" , ) q j Δ≈IL⋅Δσ ⋅ 2 2 (10) where f(" , ) = f( " +  ) and L is the measured radiance of the target radiation source. j j j j f(ε ,μ ) jj The total radiation intensity of the target radiation source was: 22 n where f(ε ,μ ) = f ( ε +μ ) and L is th j e measured radiance of the target radiation jj j j I = DI (11) j=1 source. When The to the tal r lara gest diat ar ion ea inten in all s niar ty of ea elements the targ tends et ra to di 0, at the ion limit source w can be expr as: essed as a double integral, namely: n x II =Δ 1 1 (11) I = lim LD = L d (12) å j j1 = !0 f(" , ) f(x, y) j j j=1 When the largest area η in all n area elements tends to 0, the limit can be expressed I = L dxdy (13) as a double integral, namely: f(x, y) where S is the normalized total area of the target radiation source. Considering the response of the spectroradiometer and the experimental test condi- I=⋅ lim LΔσ⋅ = L⋅ dσ (12)  η→0 f(εμ , ) f (x,y) tions, blackbody II was selected to verify the correction results. The measurements were j1 = jj S performed when the blackbody temperatures were 547 K and 557 K. In accordance with the shape characteristic of the blackbody square, the square area located in the 0– range 1 4 I=⋅ L dxdy of the first quadrant was taken for integral calculation. Then, the radiation was multiply by (13)  f (x, y) eight, so that Equation (13) could be derived as Equation (14). Z Z b x where S is the normalized total area of the target radiation source. I = 8L dx p dy (14) 2 2 0 0 f( x + y ) Considering the response of the spectroradiometer and the experimental test condi- tions, blackbody II was selected to verify the correction results. The measurements were performed when the blackbody temperatures were 547 K and 557 K. In accordance with the shape characteristic of the blackbody square, the square area located in the 0– range of the first quadrant was taken for integral calculation. Then, the radiation was multiply by eight, so that Equation (13) could be derived as Equation (14). bx I8=⋅L dx dy (14)  f( x + y ) Photonics 2022, 9, x FOR PEER REVIEW 11 of 14 Photonics 2022, 9, x FOR PEER REVIEW 11 of 14 where b is the normalized maximum distance of the target radiation source in the hori- zontal direction, which was 0.6274 in this verification. where b is the normalized maximum distance of the target radiation source in the hori- Photonics 2022, 9, 56 11 of 14 These data can also be converted to polar coordinates as follows: zontal direction, which was 0.6274 in this verification. These data can also be converted to polar coordinates as follows: π b 4cos θ where b is the normalized maximum distance of the target radiation source in the horizontal I8=⋅L dθ rdr (15) π b  direction, which was 0.6274 in this verification. f(r) 4cos θ I8=⋅L dθ rdr (15)  These data can also be converted to polar00 coordinates as follows: f(r) Here, the value of r equals the distance β from any point to the center of the FOV, f(r) Z  Z b 4 cos 1 = f(β). Here, the value of r equals the distance β from any point to the center of the FOV, f(r) I = 8L d rdr (15) f(r) 0 0 The radiation intensity curves of the corrected value, the measured value, and the = f(β). theoretical value in the 3.5~4.15 μm band are shown in Figure 8. It can be seen from the The radiation intensity curves of the corrected value, the measured value, and the Here, the value of r equals the distance from any point to the center of the FOV, figure that the spectral radiant intensity curve after correction of the FOV nonuniformity f(r) = f( ). theoretical value in the 3.5~4.15 μm band are shown in Figure 8. It can be seen from the was, as a The radiatio whole, n intensity closer to curves the th of eoreti the corr calected value. In value,tthe he 3.5~4 measur .1ed 5 μ value, m band, the theoretical and the the- figure that the spectral radiant intensity curve after correction of the FOV nonuniformity oretical value in the 3.5~4.15 m band are shown in Figure 8. It can be seen from the figure radiant intensity curve and the corrected radiant intensity curve almost overlapped, indi- was, as a whole, closer to the theoretical value. In the 3.5~4.15 μm band, the theoretical that the spectral radiant intensity curve after correction of the FOV nonuniformity was, as cating that the error was small. This shows the excellent effect of nonuniformity correc- radiant intensity curve and the corrected radiant intensity curve almost overlapped, indi- a whole, closer to the theoretical value. In the 3.5~4.15 m band, the theoretical radiant tion. cating that the error was small. This shows the excellent effect of nonuniformity correc- intensity curve and the corrected radiant intensity curve almost overlapped, indicating that tion. the error was small. This shows the excellent effect of nonuniformity correction. Figure 8. The theoretical curve, the experimental measurement curve, and the corrected curve of the spectral radiation intensity of the blackbody in the 3.5~4.15 μm band with different test tempera- Figure 8. The theoretical curve, the experimental measurement curve, and the corrected curve of the Figure 8. The theoretical curve, the experimental measurement curve, and the corrected curve of the tures. (a) L = 4.25 m, T = 547 K; (b) L = 4.25 m, T = 557 K. spectral radiation intensity of the blackbody in the 3.5~4.15 m band with different test temperatures. spectral radiation intensity of the blackbody in the 3.5~4.15 μm band with different test tempera- (a) L = 4.25 m, T = 547 K; (b) L = 4.25 m, T = 557 K. tures. (a) L = 4.25 m, T = 547 K; (b) L = 4.25 m, T = 557 K. The spectrally integrated radiant intensity in the 3.5~4.15 μm band was calculated, The spectrally integrated radiant intensity in the 3.5~4.15 m band was calculated, and the results are given in Figure 9. The maximum error compared to the theoretical The spectrally integrated radiant intensity in the 3.5~4.15 μm band was calculated, and the results are given in Figure 9. The maximum error compared to the theoretical value value within the 3.5~4.15 μm band was 1.84% after correction, exhibiting improved meas- and the results are given in Figure 9. The maximum error compared to the theoretical within the 3.5~4.15 m band was 1.84% after correction, exhibiting improved measurement urement accuracy and verifying the effectiveness of the correction method. value within the 3.5~4.15 μm band was 1.84% after correction, exhibiting improved meas- accuracy and verifying the effectiveness of the correction method. urement accuracy and verifying the effectiveness of the correction method. Figure 9. Theoretical, measured, and corrected spectrally integrated radiation intensity values and Figure 9. Theoretical, measured, and corrected spectrally integrated radiation intensity values and corrected errors in the 3.5~4.15 m band. corrected errors in the 3.5~4.15 μm band. Figure 9. Theoretical, measured, and corrected spectrally integrated radiation intensity values and corrected errors in the 3.5~4.15 μm band. Photonics 2022, 9, 56 12 of 14 3.3. Uncertainty Analysis The uncertainty of the spectrally integrated radiant intensity in the non-absorption band existed across three aspects: the measurement instrument, the object, and the condi- tions [22–24]. 1. Measurement instrument According to the type A evaluation of measurement uncertainty, the error N intro- duced by issues with measurement repeatability was approximately 1.0%. The uncertainty component N due to inaccurate spectroradiometer measurements was approximately 0.3%. 2. Measurement object Following calibration, the blackbody temperature stability was 0.5 C. The uncer- tainty caused by the inaccurate temperature of the blackbody was 0.5 C. Assuming that it followed a normal distribution, the confidence probability was 0.95, and the k was 2 [25]. According to the type B evaluation of measurement uncertainty, the uncertainty of the blackbody was highest at 50 C, so N was: 0.5 N =  100% = 0.5% 2 50 Following calibration, when the temperature was below 673 K, the emissivity of the blackbody was 0.950  0.005. Assuming that it followed a normal distribution, the confidence probability was 0.95, and the k was 2. According to the type B evaluation of measurement uncertainty, the blackbody emissivity causing the uncertainty N was: 0.005 N =  100% = 0.26% 2 0.95 3. Measurement conditions According to the type B evaluation of measurement uncertainty, the uncertainty owing to the inaccurate distance and angle between the spectroradiometer and the blackbody (N ) was approximately 0.1%. The change in ambient temperature was less than 2 K. According to the type B evalua- tion of measurement uncertainty, the influence resulted in an uncertainty (N ) of approxi- mately 0.2%. In acquiring the correction coefficient, the spectral radiant intensity measurements on both sides of the same ring were averaged. Errors accrued at this point. Similarly, in the fitting of the calibration curve, the use of different fitting functions also led to errors. Moreover, in the part of the method where was less than 0.75, the obtained fitting coefficients were more accurate. In summary, the uncertainty of the correction factor N was about 2.5%. The above uncertainty components were independent of each other, so the combined uncertainty N was: 2 2 2 2 2 2 2 N = N + N + N + N + N + N + N = 2.78% 1 2 3 4 5 6 7 4. Conclusions In an experiment examining practical spectral radiation characteristics, a concentric- circles method was used to obtain the nonuniformity fitting function. A correction formula was used to correct the measured results of the spectroradiometer. In this context, it is useful to attend to the influence of the spatial phase nonuniformity of the spectroradiometer FOV on the actual measurement. After correcting the spatial nonuniformity of the spectroradiometer FOV using the concentric-circles method, for the blackbody occupying 50.11% of the spectroradiometer FOV, the corrected spectrally integrated radiation in the 3.5~4.15 m band was close to Photonics 2022, 9, 56 13 of 14 the theoretical value, with an error less than 1.84%, demonstrating an improved FOV nonuniformity and verifying the effectiveness of the correction method. Author Contributions: Conceptualization, Z.J. and G.C.; methodology, Z.J. and Y.L.(Yiwen Li); software, Z.J., Y.L. (Yao Li) and P.Z.; validation, P.Z.; formal analysis, Z.J. and S.C.; investigation, Z.J.; resources, G.C.; data curation, Z.J. and Y.L.(Yiwen Li); writing—original draft preparation, Z.J.; writing—review and editing, Z.J., G.C. and Y.L. (Yao Li); visualization, Z.J. and P.Z.; supervision, G.C. and S.C.; project administration, Y.L.(Yiwen Li); funding acquisition, Y.L. (Yao Li) and Y.L.(Yiwen Li) All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Science and Technology Major Project of China (Grant No. J2019-V-0008-0102) and the China Postdoctoral Science Foundation (No. BX2021370). Conflicts of Interest: The authors declare no conflict of interest. References 1. Che, K.; Liu, Y.; Cai, Z.N.; Yang, D.X.; Wang, H.B.; Zhu, S.H. Review of Atmospheric Greenhouse Gas Observation and Application based on Portable Fourier Transform Infrared Spectrometer. Remote Sens. Technol. Appl. 2021, 36, 44–54. 2. Veerasingam, S.; Ranjani, M.; Venkatachalapathy, R.; Bagaev, A.; Mukhanov, V.; Litvinyuk, D.; Mugilarasan, M.; Gurumoorthi, K.; Guganathan, L.; Aboobacker, V.M. 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Reasons and Probability-Theoretical Bases of the Revision. Meas. Tech. 2017, 60, 317–324. [CrossRef] 24. Zhao, C.S. The Theory and Practice Design of The Indetermination Degree of Measure. Master ’s Thesis, Changchun University of Science and Technology, Changchun, China, 2007. 25. Zou, M.S. The Sources and Analysis of Common Probability Distribution Coverage Factor k Value in Evaluation and Expression for the Uncertainty of Measurement. Metrol. Meas. Tech. 2014, 41, 84–86. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Photonics Multidisciplinary Digital Publishing Institute

Correction of Spatial Nonuniformity in Spectroradiometer Field-of-View Using a Concentric-Circles Method

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hv photonics Article Correction of Spatial Nonuniformity in Spectroradiometer Field-of-View Using a Concentric-Circles Method Zhaoqiang Jiao , Yiwen Li *, Ge Chen, Yao Li, Shijie Chai and Puyousen Zhang Aviation Engineering School, Air Force Engineering University, Xi’an 710043, China; jiao_zq@163.com (Z.J.); 18189144368@163.com (G.C.); liyao_0927@163.com (Y.L.); chaishijie@sohu.com (S.C.); z374699183@163.com (P.Z.) * Correspondence: lee_yiwen@163.com Abstract: Spectroradiometers exhibit the smallest aberration and the optimum response at the field- of-view (FOV) center. The aberration increases and the response deteriorates at positions further away from the FOV center, which leads to nonuniformity in the spectroradiometer FOV. In this study, a concentric-circles method for correcting the spectroradiometer FOV nonuniformity was developed. The calibration experiment for FOV nonuniformity was conducted by establishing the experimental platform. The nonuniformity correction coefficients were obtained and then used to fit the correction function curve within the whole FOV, allowing for correction of measurement targets with an arbitrary shape. The radiation intensity of the blackbody at different temperatures was obtained by measurement, and the nonuniformity coefficient was used to correct it. After correction, the error was within 1.84% for the spectrally integrated radiant intensity in the non-absorption band. Using this correction method, efficient calibration of spectroradiometer nonuniformity can be achieved, thereby enhancing the measurement accuracy of the spectroradiometer. Keywords: spectroradiometer; aberration; field-of-view (FOV) nonuniformity; concentric-circles correction Citation: Jiao, Z.; Li, Y.; Chen, G.; Li, Y.; Chai, S.; Zhang, P. Correction of 1. Introduction Spatial Nonuniformity in Fourier-transform infrared (FTIR) spectroscopy has found increasingly extensive ap- Spectroradiometer Field-of-View plications in environment monitoring, pollution prevention and control [1–3], infrared Using a Concentric-Circles Method. target detection for the military [4–7], atmospheric transmittance measurement, and other Photonics 2022, 9, 56. fields [8–11]. A Fourier infrared spectroradiometer can obtain the spectral radiation charac- https://doi.org/10.3390/ teristics of a source, but its measurement results generally differ considerably from those photonics9020056 calculated under ideal conditions. The causes of the errors include issues with the repeata- Received: 11 December 2021 bility of spectroradiometer measurements, detector nonlinearity, interference from infrared Accepted: 17 January 2022 background radiation, atmospheric transmission attenuation, and human errors in exper- Published: 21 January 2022 imental apparatus testing. The nonuniformity of the spectroradiometer ’s field-of-view (FOV) response caused by off-axis aberration also significantly impacts measurements. At Publisher’s Note: MDPI stays neutral the center of the spectroradiometer FOV, the aberration is the smallest and the best response with regard to jurisdictional claims in can be obtained. At long distances from the FOV center, the aberration increases and the published maps and institutional affil- iations. response deteriorates. Therefore, when the target to be measured deviates from the FOV center or occupies a major part of the spectroradiometer FOV, the radiation measurement results contain considerable errors compared to the theoretical values. The spectroradiometer consists of four parts: an optical system, a detection system, Copyright: © 2022 by the authors. a signal processing module, and a computer module [12,13]. The optical system receives Licensee MDPI, Basel, Switzerland. and collects the energy of the target radiation source. The detection system then transforms This article is an open access article the collected energy into physical quantities, such as voltage and resistance. The signal distributed under the terms and processing module amplifies the physical quantities, which are ultimately transmitted to conditions of the Creative Commons the computer module for data visualization by supporting software. Attribution (CC BY) license (https:// Among the modules in the spectroradiometer, the off-axis parabolic mirror in the creativecommons.org/licenses/by/ optical system constitutes the fundamental cause of FOV nonuniformity. For the off-axis 4.0/). Photonics 2022, 9, 56. https://doi.org/10.3390/photonics9020056 https://www.mdpi.com/journal/photonics Photonics 2022, 9, 56 2 of 14 parabolic mirror, the aberration can be ignored on its optical axis, but it increases rapidly when the deviation from the optical axis exceeds a certain value. Therefore, the target light can be recovered well at positions close to the optical axis. However, at positions further away from the optical axis, aberration may result in different responses from the spectroradiometer for the same target at different FOV positions. The greater the deviation from the optical axis, the worse the response of the spectroradiometer is. Furthermore, aberration is a complex function related to the structure of an optical system that cannot be directly expressed as a specific function. References [14,15] studied the response nonuniformity of the spectral testing appara- tus in the theoretical measurement regions. They evaluated the practical spectroradiometer FOV range and the responses at different positions within the FOV range and obtained the directional response function, which is of far-reaching significance for accurate spectrora- diometer measurements. In 2015, Huang, W.; Ji, H.H.; Si, R. [16] corrected the nonuniformity in the results measured by an FTIR spectroradiometer. By studying the effects of FOV and field area on spectral radiant intensity, they concluded that measurements for the same target varied with the relative target position in the FOV. Furthermore, by integrating the theoretical spectral radiant intensities in the band from 3.5~4.0 m and comparing the integral with the test value, a correction coefficient was obtained, which was then used for uniformity correction. With this approach, the error between the corrected test result and the spectral radiant intensity calculated under ideal conditions was reduced. However, this method requires the acquisition of the target radiation source’s test and theoretical radiation values, which are then used to obtain the correction coefficient. Furthermore, it does not explain the specific law of nonuniformity. In 2018, Wang, X.X.; Yang, H.R.; Yu, B. et al. [17] corrected the nonuniformity of the spectroradiometer FOV using equal-solid-angle calibration. First, by studying the nonuniformity of spectroradiometer FOV, the voltage responses at different FOV positions were obtained, revealing that the response at the edge was approximately 50% lower than that at the center. For the nonuniformity of the FOV, an equal-solid-angle calibration method was proposed. As the distance between the spectroradiometer and the blackbody was set reasonably, the solid angle when the spectroradiometer measured the target was the same as the solid angle when the blackbody was measured. In this way, the optical paths of the spectroradiometer when measuring the blackbody and the target to be measured were the same, eliminating the influence of the nonuniformity of the FOV. The final results revealed that the measurement error was less than 2%. This method requires that the blackbody and the target have substantially the same shape. The spectroradiometer has the same optical path when measuring both objects. However, it is difficult to find blackbodies with similar shapes for equal-solid-angle calibration for irregular target radiation sources. There are certain limitations present. There have been in-depth reports about the suppression techniques that can be used for background radiation and the spectroradiometer response function, with the aim of improving the measurement accuracy of the spectroradiometer. However, few studies have focused on correcting the spatial nonuniformity of the spectroradiometer FOV. With the existing methods, the inversion is performed with the help of the theoretical value of the radiation intensity of the target radiation source. However, the theoretical value of the radiation intensity of the target radiation source is generally difficult to obtain in practical tests. Alternatively, it is necessary to use a blackbody with a similar shape as the target radiation source to achieve better operation. However, this method cannot be applied when the target radiation source has an irregular shape; for example, when measuring an engine tail jet. In general, the existing processing methods still have limitations. Therefore, a concentric-circles method for correcting the nonuniformity of spectroradiometer FOV was developed in this study. It can be applied to target radiation sources of any shape and with unknown theoretical radiation intensities. Photonics 2022, 9, x FOR PEER REVIEW 3 of 14 Therefore, a concentric-circles method for correcting the nonuniformity of spectroradiom- eter FOV was developed in this study. It can be applied to target radiation sources of any Photonics 2022, 9, 56 3 of 14 shape and with unknown theoretical radiation intensities. 2. Calibration Scheme for FOV Nonuniformity 2. Calibration Scheme for FOV Nonuniformity 2.1. Correction using Concentric-Circles Method 2.1. Correction Using Concentric-Circles Method Spectroradiometers exhibit minor aberration and an optimum response at the FOV Spectroradiometers exhibit minor aberration and an optimum response at the FOV center. The aberration increases and the response deteriorates further away from the FOV center. The aberration increases and the response deteriorates further away from the FOV center, which leads to phase nonuniformity in the spectroradiometer FOV. The axisym- center, which leads to phase nonuniformity in the spectroradiometer FOV. The axisymmet- metric aberration of the optical system, which causes the nonuniform response of the FOV, ric aberration of the optical system, which causes the nonuniform response of the FOV, is is circularly symmetric and gradually increases along the FOV, taking the center of the circularly symmetric and gradually increases along the FOV, taking the center of the FOV FOV spectroradiometer as the center of the circle. This indicates a centrosymmetric distri- spectroradiometer as the center of the circle. This indicates a centrosymmetric distribution. bution. Therefore, a correction scheme using a concentric-circles method for the nonuni- Therefore, a correction scheme using a concentric-circles method for the nonuniformity of formity of the FOV of the spectroradiometer can be proposed. the FOV of the spectroradiometer can be proposed. The center of the FOV of the spectroradiometer was taken as the concentric center of The center of the FOV of the spectroradiometer was taken as the concentric center the circle. Moreover, the FOV was divided into concentric rings. The nonuniformity of the of the circle. Moreover, the FOV was divided into concentric rings. The nonuniformity FOV along the same ring was the same. The blackbody to be calibrated moved on different of the FOV along the same ring was the same. The blackbody to be calibrated moved on rings along the red line, as shown in Figure 1. For the entirety of the moving process, the different rings along the red line, as shown in Figure 1. For the entirety of the moving blackbody was always completely located in the FOV of the spectroradiometer. At this process, the blackbody was always completely located in the FOV of the spectroradiometer. time, the radiation value received by the spectroradiometer did not change, and the theo- At this time, the radiation value received by the spectroradiometer did not change, and the theor retical output response re etical output response main remained ed unch unchanged. anged. Howe However ver, due , due to th to e nonuniform the nonuniformity ity of th of e the FOV, th FOV,e output the output response response value o valuefof the the sp spectr ectror oradiometer adiometer chan changed ged when the when the b blackbody lackbody was was in in di difffe ferre ent nt p positions. ositions.Using Using the the test test r res esults ultfor s for spectr spectror oradiometers adiometers with with blackbodies blackbodies in dif in di fer ff ent eren rings, t rings, thethe non nonuniformity uniformit of y of the th FOV e FOV of of the the spectr spectro oradiometers radiometers was wa calibrated. s calibrated. Figure 1. Schematic diagram of the concentric-circles method. Figure 1. Schematic diagram of the concentric-circles method. In the calibration process, in order to reduce the measurement error, two measure- In the calibration process, in order to reduce the measurement error, two measurements ments were made on the left and right sides of the same ring. The measurement numbers were made on the left and right sides of the same ring. The measurement numbers on on the left were marked as −1, −2, …, −i, …, −n, and the measurement numbers on the right the left were marked as 1, 2, . . . , i, . . . , n, and the measurement numbers on the were marked as 1, 2, …, i, …, n. The average of the two measurements of i and -i was right were marked as 1, 2, . . . , i, . . . , n. The average of the two measurements of i and -i calculated in order to replace the measured response value for the entire ring. When the was calculated in order to replace the measured response value for the entire ring. When the number of ring divisions increases infinitely, a correction coefficient curve can be obtained. In this way, a more accurate response value at each position in the FOV of the spectroradiometer can be obtained, and the nonuniformity coefficient at all positions of the FOV can be calibrated. Photonics 2022, 9, 56 4 of 14 2.2. Experimental Scheme The experiment setup included a spectroradiometer, an electrically controlled mobile platform, and standard-surface blackbodies. The spectroradiometer was a model MR170 produced by ABB, with a spectral range of 2~15 m and optional lenses of 75 mrad, 28 mrad, and 4.9 mrad. Two types of HT2M and B-500HE-20 blackbodies produced by DEMEI and LR Tech respectively, with uniform and stable surface temperature distributions, were used: blackbody I, with a diameter of 100 mm, and blackbody II, with a size of 200 mm 200 mm and blackbody emissivity of 0.95. The electronically controlled mobile platform was a model FZVAC1200 produced by Fuzhou Vacuum Electromechanical Equipment, and the moving accuracy was 0.05 mm. Driven by the electrically controlled mobile platform, the blackbody moved the given distance along the direction perpendicular to the optical axis of the spectroradiometer. Thus, the blackbody radiance at each given position on the mobile platform could be measured. According to the characteristics and purpose of this experiment, and considering the need to reduce the influence of the signal-to-noise ratio, when the output of the spectroradiometer was unsaturated, the blackbody temperature was maximized. After fully considering the temperature range of the blackbody and the responsiveness of the spectroradiometer, the following experiment scheme was configured. The spectroradiometer had to be calibrated before use so as to facilitate convenient test operations while guaranteeing the accuracy requirements. As this experiment was conducted in the laboratory, the measurement distance was short, and the temperature variations of the target radiation source were small. Under these conditions, the response function of the spectroradiometer was considered to be linear, so the two-point calibration method was selected [18–20]. Specifically, two different temperatures were configured for the blackbody to calibrate the spectroradiometer. To eliminate the influence of background radiation and improve the calibration accuracy, the blackbody filled the spectroradiometer FOV during calibration. The theoretical formulae for the two-point calibration method are: V(, T ) = R()L(, T ) + O() (1) H H V(, T ) = R()L(, T ) + O() (2) C C where T is high temperature, T is low temperature, V(,T ) is the output voltage of the H C H spectroradiometer when testing the high-temperature blackbody, L(,T ) is the radiance of the high-temperature blackbody, V(,T ) is the output voltage of the spectroradiometer when testing the low-temperature blackbody, L(,T ) is the radiance of the low-temperature blackbody, R() is the response of the spectroradiometer, and O() is the error of the radia- tion measurement, which does not change when the temperature of the blackbody changes. By combining Equations (1) and (2), the response R() and radiation measurement error O() can be obtained as follows: V(, T ) V(, T ) H C R() = (3) L(, T ) L(, T ) H C V(, T )L(, T ) V(, T )L(, T ) C H H C O() = (4) L(, T ) L(, T ) H C After two-point calibration, the response and measurement error of the spectrora- diometer could be determined, and the linear relationship between the voltage and the spectral radiance could be obtained as follows: V(, T) = R()L(, T) + O() (5) Thus, the voltage measured by the spectroradiometer was matched with the spectral radiance of the target, and the response function between the two was obtained. The schematic diagram of the experimental setup for measuring the spatial nonunifor- mity of the spectroradiometer FOV is shown in Figure 2. The blackbody was placed on the Photonics 2022, 9, x FOR PEER REVIEW 5 of 14 After two-point calibration, the response and measurement error of the spectroradi- ometer could be determined, and the linear relationship between the voltage and the spec- tral radiance could be obtained as follows: V(λ= ,T) R()λ ⋅L(λ,T)+O()λ (5) Thus, the voltage measured by the spectroradiometer was matched with the spectral radiance of the target, and the response function between the two was obtained. Photonics 2022, 9, 56 5 of 14 The schematic diagram of the experimental setup for measuring the spatial nonuni- formity of the spectroradiometer FOV is shown in Figure 2. The blackbody was placed on the electronically controlled mobile platform and in the center of the spectroradiometer electronically controlled mobile platform and in the center of the spectroradiometer FOV. FOV. It could be moved to the left and right under the control of the electronically con- It could be moved to the left and right under the control of the electronically controlled trolled mobile platform. The blackbody moving path is shown in the figure. A 75 mrad mobile platform. The blackbody moving path is shown in the figure. A 75 mrad lens was lens was selected, and the test distance D1 was set to 5.33 m. The temperature of the black- selected, and the test distance D was set to 5.33 m. The temperature of the blackbody body was set to 533 K. In the specific test, the FOV nonuniformity calibration was per- was set to 533 K. In the specific test, the FOV nonuniformity calibration was performed by formed by first measuring the spectrum radiance of blackbody I at the central axial posi- first measuring the spectrum radiance of blackbody I at the central axial position of the tion of the spectroradiometer FOV. Then, starting from the center measurement point, the spectroradiometer FOV. Then, starting from the center measurement point, the blackbody blackbody was moved to the left of the center position 1 cm at a time to perform each was moved to the left of the center position 1 cm at a time to perform each measurement, measurement, with 15 measurements taken in total across 15 cm. The same measurements with 15 measurements taken in total across 15 cm. The same measurements were recorded were recorded on the right side of the center of the FOV, and a total of 31 measurements on the right side of the center of the FOV, and a total of 31 measurements were performed. were performed. Figure 2. Schematic diagram of the experimental setup for measuring the spatial nonuniformity of Figure 2. Schematic diagram of the experimental setup for measuring the spatial nonuniformity of the spectroradiometer FOV. the spectroradiometer FOV. Afterward, as shown in Figure 3, the FOV nonuniformity calibration result was Afterward, as shown in Figure 3, the FOV nonuniformity calibration result was ver- verified by placing blackbody II at the spectroradiometer FOV center and measuring its ified by placing blackbody II at the spectroradiometer FOV center and measuring its spec- spectral radiation. tral radiation. The measurement scheme is shown in Table 1. A total of two sets of tests were carried out. When the distance D between the blackbody and the spectroradiometer was set reasonably, the blackbody accounted for 50.11% of the spectroradiometer ’s FOV. Here, D was 4.25 m; the blackbody temperatures were set as 547 K and 557 K, respectively. Photonics 2022, 9, 56 6 of 14 Photonics 2022, 9, x FOR PEER REVIEW 6 of 14 Figure 3. Schematic diagram of experiment to verify the effect of the spatial nonuniformity correc- Figure 3. Schematic diagram of experiment to verify the effect of the spatial nonuniformity correction tion of the spectrometer FOV. of the spectrometer FOV. The measurement scheme is shown in Table 1. A total of two sets of tests were carried Table 1. FOV nonuniformity correction verification scheme. out. When the distance D2 between the blackbody and the spectroradiometer was set rea- sonably, the blackbody accounted for 50.11% of the spectroradiometer’s FOV. Here, D2 Percentage in the FOV of the D (m) Temperature (K) Spectroradiometer (%) was 4.25 m; the blackbody temperatures were set as 547 K and 557 K, respectively. 4.25 547 50.11 Table 1. FOV nonuniformity correction verification scheme. 4.25 557 50.11 Temperature Percentage in the FOV of the L (m) 3. Results (K) Spectroradiometer (%) 3.1. FOV Nonuniformity Correction Coefficient 4.25 547 50.11 The distance i between each measurement point and the FOV center was divided by 4.25 557 50.11 the FOV radius for normalization and the ratio was denoted as . In this study, the FOV radius Q of the spectroradiometer was 200 mm. 3. Results 3.1. FOV Nonuniformity Correction Coefficient = (6) The distance i between each measurement point and the FOV center was divided by the FOV radius for normalization and the ratio was denoted as β. In this study, the FOV The radiance measured by the spectroradiometer was transformed into the radiant in- radius Q of the spectroradiometer was 200 mm. tensity at different wavelengths by averaging the spectral test data from two measurements on the same ring. Meanwhile, the theoretical radiant intensity value of the blackbody was β = (6) obtained by the following Equations [21]. " a The radiance measured L by the sp = ectroradiometer was td ran  sformed into the radian (7) t (  ) 1 2 5 a /T (e 1) intensity at different wavelengths by averaging the spectral test data from two measure- ments on the same ring. Meanwhile, the theoretical radiant intensity value of the black- I = L A (8) (  ) (  ) 1 2 1 2 body was obtained by the following Equations. [21]. Here, L is the radiance within the   band; I is the radiant intensity (  ) 1 2 (  ) 1 2 1 2 within the   band; A is the effective radiation area of the target; " is the emissivity 1 2 Photonics 2022, 9, x FOR PEER REVIEW 7 of 14 ε 2 a Ld =⋅λ (7) () λ−λ 5 a/λT π 1λ− (e 1) IL = ⋅A (8) () λ−λ (λ −λ) 12 1 2 Here, L is the radiance within the λλ − band; I is the radiant inten- () λ−λ 12 (- λλ ) 12 12 sity within the λλ − band; A is the effective radiation area of the target; ε is the emissiv- ity of the blackbody; a and a are radiation constants, with values of 3.7415 ± 0.0003 × 1 2 Photonics 2022, 9, 56 7 of 14 8 4 2 4 10 (W·μm /m ) and 1.43879 ± 0.00019 × 10 (μm·K), respectively; and T is the temperature of the blackbody. of the Th blackbody; e measure a and d d aaar ta e we radiation re pr constants, ocessed with acco values rding of 3.7415 to Equ 0.0003 ations ( 107) and (8). First, the rela- 1 2 4 2 4 (Wm /m ) and 1.43879  0.00019  10 (mK), respectively; and T is the temperature tionship curve between the wavelength and the radiant intensity at different measure- of the blackbody. ment positions was obtained for blackbody I, as shown in Figure 4. The measured data were processed according to Equations (7) and (8). First, the rela- tionship curve between the wavelength and the radiant intensity at different measurement positions was obtained for blackbody I, as shown in Figure 4. Figure 4. Theoretical and measured values of spectral radiant intensity at T = 533 K. Figure 4. Theoretical and measured values of spectral radiant intensity at T = 533 K. The variation curve of the radiant intensity with the wavelength measured at different positions and the theoretical radiant intensity curves at the two temperatures are given in The variation curve of the radiant intensity with the wavelength measured at differ- Figure 4. As can be seen, the spectral radiant intensity curves of 3~5 m obtained for the same target varied at different positions. Basically, the smaller was—i.e., the closer it was ent positions and the theoretical radiant intensity curves at the two temperatures are given to the FOV center—the greater the measured spectral radiant intensity was and the closer in Figure 4. As can be seen, the spectral radiant intensity curves of 3~5 μm obtained for to the theoretical value. Furthermore, as can be observed in Figure 4, the practical spectral radiant intensity the same target varied at different positions. Basically, the smaller β was—i.e., the closer curve fluctuated obviously at wavelengths of 3~3.4 m,4.2~4.4 m, and 4.5~5 m. Specifi- it was to the FOV center—the greater the measured spectral radiant intensity was and the cally, the curve decreased rapidly at approximately 4.2 m and then rose at around 4.4 m. closer These to changes the theoretical v were due to energy alue. attenuation during atmospheric transmission. The wave- lengths of the infrared absorption bands for the main atmospheric components at 3~5 m Furthermore, as can be observed in Figure 4, the practical spectral radiant intensity are shown in Table 2, in which CO and H O exhibited the highest absorptions. Therefore, 2 2 curve fluctuated obviously at wavelengths of 3~3.4 μm,4.2~4.4 μm, and 4.5~5 μm. Specif- these two components should be the focus when discussing atmospheric transmittance in the range of 3~5 m. The fluctuations near 3.2 m were due to the influence of H O in the ically, the curve decreased rapidly at approximately 4.2 μm and then rose at around 4.4 atmosphere, and the significant fluctuations at 4.2~4.4 m were due to atmospheric CO , μm. These changes were due to energy attenuation during atmospheric transmission [9]. which had a strong absorption band at 4.3 m, causing an evident drop in the curve [21]. The wavelengths of the infrared absorption bands for the main atmospheric components at 3~5 μm are shown in Table 2, in which CO2 and H2O exhibited the highest absorptions. Therefore, these two components should be the focus when discussing atmospheric trans- mittance in the range of 3~5 μm. The fluctuations near 3.2 μm were due to the influence of H2O in the atmosphere, and the significant fluctuations at 4.2~4.4 μm were due to at- mospheric CO2, which had a strong absorption band at 4.3 μm, causing an evident drop in the curve [21]. Photonics 2022, 9, x FOR PEER REVIEW 8 of 14 Table 2. Center wavelengths of infrared absorption bands for main atmospheric components at 3~5 μm. Composition Center Wavelength of Absorption Band (µm) CO2 4.3, 4.8 H2O 3.2 CO 4.7 CH4 3.3 Photonics 2022, 9, 56 8 of 14 O3 4.8 Overall, the influence of atmospheric absorption was minor within the range from Table 2. Center wavelengths of infrared absorption bands for main atmospheric components at 3~5 m. 3.4~4.15 μm, so the curve was generally smooth, exhibiting a trend similar to that of the theoretical curve. To eliminate the influence of atmospheric transmission attenuation on Composition Center Wavelength of Absorption Band (m) CO 4.3, 4.8 the experiment when correcting the FOV nonuniformity, the band from 3.5~4.15 μm was H O 3.2 selected for calibration in the data processing. CO 4.7 CH 3.3 The spectral radiation data measured at the FOV center was closest to the theoretical O 4.8 value. After obtaining the total radiant intensity at 3.5~4.15 μm for each measurement point, the correction coefficient α was calculated by taking the radiant intensity at β = 0 as Overall, the influence of atmospheric absorption was minor within the range from 3.4~4.15 m, so the curve was generally smooth, exhibiting a trend similar to that of the the reference value. theoretical curve. To eliminate the influence of atmospheric transmission attenuation on the experiment when correcting the FOV nonuniformity, the band from 3.5~4.15 m was selected for calibration in the data processing. i α = i = 0, 1, 2 … 15. (9) The spectral radiation data measured at the FOV center was closest to the theoretical value. After obtaining the total radiant intensity at 3.5~4.15 m for each measurement point, the correction coefficient was calculated by taking the radiant intensity at = 0 as the reference value. where I is the 3.5~4.15 μm radiant intensity at βi and I is the 3.5~4.15 μm radiant in- i 0 = i = 0, 1, 2 . . . 15. (9) tensity at β0. where I is the 3.5~4.15 m radiant intensity at and I is the 3.5~4.15 m radiant intensity i i Figure 5 presents the correction coefficient α at different values of β at 533 K and the at . spectrally integrated radiant intensity in the 3.5~4.15 μm band. Specifically, the correction coef- Figure 5 presents the correction coefficient at different values of at 533 K and the spectrally integrated radiant intensity in the 3.5~4.15 m band. Specifically, the correc- ficient changed gently near the FOV center. The correction coefficient dropped signifi- tion coefficient changed gently near the FOV center. The correction coefficient dropped cantly when β became larger than 0.35. significantly when became larger than 0.35. Figure 5. The correction coefficient at different values of and the spectrally integrated radiant intensity in the 3.5~4.15 m band. Figure 5. The correction coefficient α at different values of β and the spectrally integrated radiant intensity in the 3.5~4.15 μm band. The correction coefficient α at different values of β was obtained using the above test. The distance β from the center of the FOV of the spectroradiometer was taken as the ab- scissa and the nonuniformity correction coefficient α as the ordinate. The quartic polyno- mial function was used for fitting, and the correction function α = f(β) was obtained, as shown in the corresponding curve in Figure 6a. Photonics 2022, 9, x FOR PEER REVIEW 9 of 14 The FOV of the spectroradiometer was represented in the X-Y coordinate system, with the center of the FOV of the spectroradiometer as the origin, the normalized horizon- tal distance as X, and the normalized vertical distance as Y. Photonics 2022, 9, 56 The distance from any point (xi,yi) in the coordinate system to the center of 9 th of 14 e FOV 22 22 xy + α=+ f( x y ) was , so the correction coefficient at (xi,yi) was , and it ii ii i The correction coefficient at different values of was obtained using the above was recorded as α = f (x, y) . From this, the correction coefficient α at any position of ii i test. The distance from the center of the FOV of the spectroradiometer was taken as the FOV of the spectroradiometer could be obtained. The correction coefficient cloud im- the abscissa and the nonuniformity correction coefficient as the ordinate. The quartic age for the whole FOV is shown in Figure 6b. polynomial function was used for fitting, and the correction function = f( ) was obtained, as shown in the corresponding curve in Figure 6a. Figure 6. (a) The FOV nonuniformity correction coefficient fitting curve; (b) the correction coefficient Figure 6. (a) The FOV nonuniformity correction coefficient fitting curve; (b) the correction coeffi- cloud image. cient cloud image. The FOV of the spectroradiometer was represented in the X-Y coordinate system, with As can be seen in Figure 6, the nonuniformity reached 0.55 at the edge of the FOV of the center of the FOV of the spectroradiometer as the origin, the normalized horizontal the spectroradiometer. When the target occupied a large area in the spectroradiometer, distance as X, and the normalized vertical distance as Y. the effect of the spatial nonuniformity of the spectroradiometer FOV was very large and The distance from any point (x , y ) in the coordinate system to the center of the FOV i i q q needed to be corrected. 2 2 2 2 was x + y , so the correction coefficient at (x , y ) was = f( x + y ), and it was i i i i i i i recorded as = f(x , y ). From this, the correction coefficient at any position of the FOV i i 3.2. Correction of FOV Nonuniformity of the spectroradiometer could be obtained. The correction coefficient cloud image for the whole FOV is shown in Figure 6b. When testing the target radiation source with the spectroradiometer, the distance As can be seen in Figure 6, the nonuniformity reached 0.55 at the edge of the FOV from the edge of the FOV of the spectroradiometer to the center of the FOV was regarded of the spectroradiometer. When the target occupied a large area in the spectroradiometer, as 1 to normalize the area of the target radiation source. The area was denoted as S. When the effect of the spatial nonuniformity of the spectroradiometer FOV was very large and correcting the target radiation source, there was a correction coefficient for each concentric needed to be corrected. ring in the FOV of the spectroradiometer, and the correction coefficients were different for different rings. 3.2. Correction of FOV Nonuniformity When testing the target radiation source with the spectroradiometer, the distance from For the i-th circle, the correction coefficient at the distance xy + = βi from the ii the edge of the FOV of the spectroradiometer to the center of the FOV was regarded as center of the FOV of the spectroradiometer was αi. As shown in the Figure 7, the target 1 to normalize the area of the target radiation source. The area was denoted as S. When radiation source was in this red area. The correction coefficient was αi. correcting the target radiation source, there was a correction coefficient for each concentric ring in the FOV of the spectroradiometer, and the correction coefficients were different for different rings. 2 2 For the i-th circle, the correction coefficient at the distance x + y = from the i i center of the FOV of the spectroradiometer was . As shown in the Figure 7, the target radiation source was in this red area. The correction coefficient was . Photonics 2022, 9, x FOR PEER REVIEW 10 of 14 Photonics 2022, 9, 56 10 of 14 Figure 7. Correction coefficient for the target radiation source in the FOV. Figure 7. Correction coefficient for the target radiation source in the FOV. From this, it can be seen that every position of the target radiation source in the FOV From this, it can be seen that every position of the target radiation source in the FOV had a certain correction coefficient. For the calculation of radiation intensity correction, had a certain correction coefficient. For the calculation of radiation intensity correction, the method of splitting, approximating, summing, and taking the limit was used to derive the method of splitting, approximating, summing, and taking the limit was used to derive the formula. The area of the target radiation source was arbitrarily divided into n area elements. the formula. For the FOV area with position coordinates ("  ), the area element of this area was D j, j j The area of the target radiation source was arbitrarily divided into n area elements. and the radiation intensity was: For the FOV area with position coordinates (εj, μj), the area element of this area was Δσj and the radiation intensity was:j DI  LD (10) f(" , ) q j Δ≈IL⋅Δσ ⋅ 2 2 (10) where f(" , ) = f( " +  ) and L is the measured radiance of the target radiation source. j j j j f(ε ,μ ) jj The total radiation intensity of the target radiation source was: 22 n where f(ε ,μ ) = f ( ε +μ ) and L is th j e measured radiance of the target radiation jj j j I = DI (11) j=1 source. When The to the tal r lara gest diat ar ion ea inten in all s niar ty of ea elements the targ tends et ra to di 0, at the ion limit source w can be expr as: essed as a double integral, namely: n x II =Δ 1 1 (11) I = lim LD = L d (12) å j j1 = !0 f(" , ) f(x, y) j j j=1 When the largest area η in all n area elements tends to 0, the limit can be expressed I = L dxdy (13) as a double integral, namely: f(x, y) where S is the normalized total area of the target radiation source. Considering the response of the spectroradiometer and the experimental test condi- I=⋅ lim LΔσ⋅ = L⋅ dσ (12)  η→0 f(εμ , ) f (x,y) tions, blackbody II was selected to verify the correction results. The measurements were j1 = jj S performed when the blackbody temperatures were 547 K and 557 K. In accordance with the shape characteristic of the blackbody square, the square area located in the 0– range 1 4 I=⋅ L dxdy of the first quadrant was taken for integral calculation. Then, the radiation was multiply by (13)  f (x, y) eight, so that Equation (13) could be derived as Equation (14). Z Z b x where S is the normalized total area of the target radiation source. I = 8L dx p dy (14) 2 2 0 0 f( x + y ) Considering the response of the spectroradiometer and the experimental test condi- tions, blackbody II was selected to verify the correction results. The measurements were performed when the blackbody temperatures were 547 K and 557 K. In accordance with the shape characteristic of the blackbody square, the square area located in the 0– range of the first quadrant was taken for integral calculation. Then, the radiation was multiply by eight, so that Equation (13) could be derived as Equation (14). bx I8=⋅L dx dy (14)  f( x + y ) Photonics 2022, 9, x FOR PEER REVIEW 11 of 14 Photonics 2022, 9, x FOR PEER REVIEW 11 of 14 where b is the normalized maximum distance of the target radiation source in the hori- zontal direction, which was 0.6274 in this verification. where b is the normalized maximum distance of the target radiation source in the hori- Photonics 2022, 9, 56 11 of 14 These data can also be converted to polar coordinates as follows: zontal direction, which was 0.6274 in this verification. These data can also be converted to polar coordinates as follows: π b 4cos θ where b is the normalized maximum distance of the target radiation source in the horizontal I8=⋅L dθ rdr (15) π b  direction, which was 0.6274 in this verification. f(r) 4cos θ I8=⋅L dθ rdr (15)  These data can also be converted to polar00 coordinates as follows: f(r) Here, the value of r equals the distance β from any point to the center of the FOV, f(r) Z  Z b 4 cos 1 = f(β). Here, the value of r equals the distance β from any point to the center of the FOV, f(r) I = 8L d rdr (15) f(r) 0 0 The radiation intensity curves of the corrected value, the measured value, and the = f(β). theoretical value in the 3.5~4.15 μm band are shown in Figure 8. It can be seen from the The radiation intensity curves of the corrected value, the measured value, and the Here, the value of r equals the distance from any point to the center of the FOV, figure that the spectral radiant intensity curve after correction of the FOV nonuniformity f(r) = f( ). theoretical value in the 3.5~4.15 μm band are shown in Figure 8. It can be seen from the was, as a The radiatio whole, n intensity closer to curves the th of eoreti the corr calected value. In value,tthe he 3.5~4 measur .1ed 5 μ value, m band, the theoretical and the the- figure that the spectral radiant intensity curve after correction of the FOV nonuniformity oretical value in the 3.5~4.15 m band are shown in Figure 8. It can be seen from the figure radiant intensity curve and the corrected radiant intensity curve almost overlapped, indi- was, as a whole, closer to the theoretical value. In the 3.5~4.15 μm band, the theoretical that the spectral radiant intensity curve after correction of the FOV nonuniformity was, as cating that the error was small. This shows the excellent effect of nonuniformity correc- radiant intensity curve and the corrected radiant intensity curve almost overlapped, indi- a whole, closer to the theoretical value. In the 3.5~4.15 m band, the theoretical radiant tion. cating that the error was small. This shows the excellent effect of nonuniformity correc- intensity curve and the corrected radiant intensity curve almost overlapped, indicating that tion. the error was small. This shows the excellent effect of nonuniformity correction. Figure 8. The theoretical curve, the experimental measurement curve, and the corrected curve of the spectral radiation intensity of the blackbody in the 3.5~4.15 μm band with different test tempera- Figure 8. The theoretical curve, the experimental measurement curve, and the corrected curve of the Figure 8. The theoretical curve, the experimental measurement curve, and the corrected curve of the tures. (a) L = 4.25 m, T = 547 K; (b) L = 4.25 m, T = 557 K. spectral radiation intensity of the blackbody in the 3.5~4.15 m band with different test temperatures. spectral radiation intensity of the blackbody in the 3.5~4.15 μm band with different test tempera- (a) L = 4.25 m, T = 547 K; (b) L = 4.25 m, T = 557 K. tures. (a) L = 4.25 m, T = 547 K; (b) L = 4.25 m, T = 557 K. The spectrally integrated radiant intensity in the 3.5~4.15 μm band was calculated, The spectrally integrated radiant intensity in the 3.5~4.15 m band was calculated, and the results are given in Figure 9. The maximum error compared to the theoretical The spectrally integrated radiant intensity in the 3.5~4.15 μm band was calculated, and the results are given in Figure 9. The maximum error compared to the theoretical value value within the 3.5~4.15 μm band was 1.84% after correction, exhibiting improved meas- and the results are given in Figure 9. The maximum error compared to the theoretical within the 3.5~4.15 m band was 1.84% after correction, exhibiting improved measurement urement accuracy and verifying the effectiveness of the correction method. value within the 3.5~4.15 μm band was 1.84% after correction, exhibiting improved meas- accuracy and verifying the effectiveness of the correction method. urement accuracy and verifying the effectiveness of the correction method. Figure 9. Theoretical, measured, and corrected spectrally integrated radiation intensity values and Figure 9. Theoretical, measured, and corrected spectrally integrated radiation intensity values and corrected errors in the 3.5~4.15 m band. corrected errors in the 3.5~4.15 μm band. Figure 9. Theoretical, measured, and corrected spectrally integrated radiation intensity values and corrected errors in the 3.5~4.15 μm band. Photonics 2022, 9, 56 12 of 14 3.3. Uncertainty Analysis The uncertainty of the spectrally integrated radiant intensity in the non-absorption band existed across three aspects: the measurement instrument, the object, and the condi- tions [22–24]. 1. Measurement instrument According to the type A evaluation of measurement uncertainty, the error N intro- duced by issues with measurement repeatability was approximately 1.0%. The uncertainty component N due to inaccurate spectroradiometer measurements was approximately 0.3%. 2. Measurement object Following calibration, the blackbody temperature stability was 0.5 C. The uncer- tainty caused by the inaccurate temperature of the blackbody was 0.5 C. Assuming that it followed a normal distribution, the confidence probability was 0.95, and the k was 2 [25]. According to the type B evaluation of measurement uncertainty, the uncertainty of the blackbody was highest at 50 C, so N was: 0.5 N =  100% = 0.5% 2 50 Following calibration, when the temperature was below 673 K, the emissivity of the blackbody was 0.950  0.005. Assuming that it followed a normal distribution, the confidence probability was 0.95, and the k was 2. According to the type B evaluation of measurement uncertainty, the blackbody emissivity causing the uncertainty N was: 0.005 N =  100% = 0.26% 2 0.95 3. Measurement conditions According to the type B evaluation of measurement uncertainty, the uncertainty owing to the inaccurate distance and angle between the spectroradiometer and the blackbody (N ) was approximately 0.1%. The change in ambient temperature was less than 2 K. According to the type B evalua- tion of measurement uncertainty, the influence resulted in an uncertainty (N ) of approxi- mately 0.2%. In acquiring the correction coefficient, the spectral radiant intensity measurements on both sides of the same ring were averaged. Errors accrued at this point. Similarly, in the fitting of the calibration curve, the use of different fitting functions also led to errors. Moreover, in the part of the method where was less than 0.75, the obtained fitting coefficients were more accurate. In summary, the uncertainty of the correction factor N was about 2.5%. The above uncertainty components were independent of each other, so the combined uncertainty N was: 2 2 2 2 2 2 2 N = N + N + N + N + N + N + N = 2.78% 1 2 3 4 5 6 7 4. Conclusions In an experiment examining practical spectral radiation characteristics, a concentric- circles method was used to obtain the nonuniformity fitting function. A correction formula was used to correct the measured results of the spectroradiometer. In this context, it is useful to attend to the influence of the spatial phase nonuniformity of the spectroradiometer FOV on the actual measurement. After correcting the spatial nonuniformity of the spectroradiometer FOV using the concentric-circles method, for the blackbody occupying 50.11% of the spectroradiometer FOV, the corrected spectrally integrated radiation in the 3.5~4.15 m band was close to Photonics 2022, 9, 56 13 of 14 the theoretical value, with an error less than 1.84%, demonstrating an improved FOV nonuniformity and verifying the effectiveness of the correction method. Author Contributions: Conceptualization, Z.J. and G.C.; methodology, Z.J. and Y.L.(Yiwen Li); software, Z.J., Y.L. (Yao Li) and P.Z.; validation, P.Z.; formal analysis, Z.J. and S.C.; investigation, Z.J.; resources, G.C.; data curation, Z.J. and Y.L.(Yiwen Li); writing—original draft preparation, Z.J.; writing—review and editing, Z.J., G.C. and Y.L. 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Journal

Photonics – Multidisciplinary Digital Publishing Institute

Published: Jan 21, 2022

Keywords: spectroradiometer; aberration; field-of-view (FOV) nonuniformity; concentric-circles correction

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