1.

Introduction

The Graduate School of Engineering and Management at the Air Force Institute of Technology (AFIT) educates military and civilian students from the U.S. Air and Space Forces (USAF/USSF), Department of Defense, other government agencies, international partners, and government contractors, providing academic programs with a defense-related focus and research on high-priority defense problems.¹ There is a clear need for optical expertise in USAF applications, ranging from active systems, such as laser range-finding and radar, to passive systems, such as imaging-, spectral-, and polarimetric-remote sensing. This paper introduces an experiment, which is centered on radiometry and radiometric instrumentation, from the laboratory portion of our optics curriculum.

Radiometry is the measurement of a radiometric quantity, e.g., irradiance or incident power/unit area on a surface, often with the goal of determining another radiometric quantity of a distant source, e.g., its intensity (power/unit solid angle into which that radiation flows) or its radiance (radiating power/unit area of a surface/unit solid angle into which the radiation flows), which is not otherwise accessible. Broadly, a radiometer is an apparatus for quantifying some property of incident radiation. The receiving aperture of a radiometer is used as the reference position for the incident radiation, with the assumption that the radiation would affect any other instrument similarly placed. Calculation of the desired radiometric quantity at the remote location using the values determined at the radiometer’s aperture requires additional information, such as knowledge of the intervening atmosphere or the distance to the source, etc. The challenge of radiometry is in understanding how these all affect the radiometer’s output.²

Radiometric theory appears to be simply geometric and phenomenological, but Wolf established the relationship between it and the modern theories of radiation, such as Maxwell’s electromagnetic theory and the quantum theory of radiation.³ When the optics student assumes radiometry is simply geometric, its nuances often become deceptively difficult for them to understand; e.g., the basic quantity in radiometry is radiance, $L$

At AFIT, we offer a three-course sequence, essentially in applied radiometry. “Optical radiometry and detection” develops the terminology of radiation transfer and measurement, as well as an understanding of optical detection. “Infrared (IR) Technology” presents the principles required for the analysis of electro-optic systems, with emphasis on those systems operating in the IR, and is often where the AFIT student finally masters the radiometric concepts. And “Electro-Optical Systems Lab” is a laboratory-and-lecture course that introduces laboratory techniques for the measurement of optical observables, i.e., emissions and reflections of optical radiation from targets.⁴ This paper will focus on a particular radiometry experiment developed to reassert and clarify the radiometric concepts and approximations for the students in our Electro-Optical-Systems-Lab course.

The bidirectional reflectance distribution function (BRDF) radiometrically describes reflections from a surface as the ratio of the radiance ($L_r$) reflected from a surface into a small solid angle in a given direction in spherical coordinates with respect to the surface normal $({\theta }_r,{\varphi }_r)$, to the incident irradiance ($E_i$) from a given direction $({\theta }_i,{\varphi }_i)$⁵

Although the BRDF is introduced in our optical-radiometry-and-detection text,⁶ we develop it much further in our IR-Technology course. Just as Wolf tied the phenomenological radiometry to modern optical theories, Greffet and Nieto-Vesperinas also derived the radiometric BRDF in terms of modern electromagnetic theory.⁷

We also introduce the scene rendering equation, often used in computer graphics and which uses the BRDF, in our IR-Technology course^8,9

We also introduce the scenario shown in Fig. 1 in our IR-Technology course as a tractable case in which the signature contributions from both self-emission and reflection of an observed surface, here at the center of the spherical coordinate system shown, may be computed using the rendering equation. The surface is illuminated by the entire hemisphere above the surface, e.g., it could be the sky with declination angles, $0\le {\theta }_{\text{sky}}\le \pi /2$, and which we assume is azimuthally symmetric to keep the problem tractable for the student. The surface is also illuminated by a source that subtends a small solid angle, such as the sun, at position (${\theta }_{\text{sun}},{\varphi }_{\text{sun}}=0$). The surface is then observed from position (${\theta }_{\mathrm{obs}},{\varphi }_{\mathrm{obs}}$), but since we set $({\varphi }_{\text{sun}}=0,{\varphi }_{\text{sun}}={\varphi }_{\mathrm{obs}}-\mathrm{\Delta }\varphi )$, and we only consider isotropic samples, again to keep the problem tractable for the student, the observation position becomes (${\theta }_{\mathrm{obs}},\mathrm{\Delta }\varphi$). We introduce scenarios for computation where the hemispherical illumination may be simple functions of ${\theta }_{\text{sky}}$, and higher-fidelity sky radiances may be incorporated using atmospheric models, such as Spectral Sciences, Inc.’s MODTRAN^® (MODerate resolution atmospheric TRANsmission)¹¹ or the AFIT’s LEEDR (Laser Environmental Effects Definition and Reference).¹² The “small-angle” source illumination may be a function of ${\theta }_{\text{sun}}$ (as with the zenith angle of the sun due to atmospheric absorption). The observed surface is then described by various BRDF’s ranging from perfectly diffuse (i.e., Lambertian) to perfectly specular (i.e., mirror-like), and where the spherical coordinate system defined by its surface normal $({\theta }_i,{\varphi }_i)$ may or may not align with the environmental coordinate systems $({\theta }_{\text{sky}},{\varphi }_{\text{sky}})$, $({\theta }_{\text{sun}},{\varphi }_{\text{sun}})$.¹³ All this allows the student to use the radiometry and understand the influence of each of these pieces on the observed radiance.

Fig. 1

Classroom remote-sensing scenario introduced in AFIT’s IR-Technology course. An observed surface is located at the center of the spherical coordinate system shown. It is illuminated by the entire hemisphere above it with declination angles, $0\le {\theta }_{\text{sky}}\le \pi /2$, and assumed to be symmetric azimuthally. It is illuminated by a source that subtends a small solid angle at position (${\theta }_{\text{sun}},{\varphi }_{\text{sun}}=0$). It is observed from position is $({\theta }_{\mathrm{obs}},\mathrm{\Delta }\varphi )$.

Note that both the self-emitted- and reflected-radiance terms of Eq. (5) are source quantities, i.e., the IR signature at zero range or as attenuated by a vacuum. In a remote-sensing scenario, apparent spectral radiance measurements, i.e., the IR signature of an object attenuated by the atmosphere over the range of the scenario, are made by a correctly calibrated instrument. Atmospheric attenuation may then be “calibrated out” by either calibrating at range to effectively provide source measurements or by correcting apparent spectral measurements to source spectral radiance using the aforementioned atmospheric models, MODTRAN^® or LEEDR. Once source spectral radiance is known, the optical characteristics of the unknown object may be determined using scene-rendering software, such as the Rochester Institute of Technology’s DIRSIG™ (Digital Imaging and Remote Sensing Image Generation),¹⁴ or ATA Engineering, Inc.’s SPIRITS (SPectral and In-band Radiometric Imaging of Targets and Scenes),¹⁵ which uses radiometry as its basis, and illumination sources, such as the sun and sky, which are known positionally, spectrally, etc., due to time of day, temperature, and other meteorological conditions.

The IR-Technology scenario described for Fig. 1 closely aligns with the radiometric data-collection scenario we developed for our Electro-Optical-Systems-Lab course (see Fig. 2). Here, again to keep the problem tractable as a learning experience for the student, the hemispheric illumination on the unknown sample is assumed to be spatially constant (isotropic) at room temperature, $T_{\text{Background}}$, and the cavity blackbody and the Fourier-transform IR spectrometer (FTS) are positioned in-plane relative to the sample’s surface normal to set $\mathrm{\Delta }\varphi =\pi$. The cavity blackbody and FTS may be positioned at a variety of specular angles relative to the sample’s surface normal, and the sample may then be rotated to many other $({\theta }_{\mathrm{BB}},{\theta }_r)$ angle pairs away from specular to provide a rich dataset from which the optical characteristics of the unknown sample are to be determined. The cavity blackbody at temperature, TBB, takes the place of the sun in the Fig. 1 scenario. The FTS is assumed to be correctly calibrated so that the measured data is the source spectral radiance, and the solid angle subtended by the FTS aperture is assumed to be small

Fig. 2

An experiment is AFIT’s Electro-Optical Systems Laboratory to provide the student with a data set from which the optical characteristics of an unknown sample are to be determined. Isotropic hemispheric illumination of the sample at $T_{\text{Background}}$ is assumed. The cavity blackbody and the FTS are positioned in-plane relative to the sample’s surface, and they may be positioned at a variety of specular angles relative to the sample’s surface normal. The sample may then be rotated to other $({\theta }_{\mathrm{BB}},{\theta }_r)$ angles away from specular.

With these simplifying assumptions, the self-emitted spectral radiance of Eq. (5) becomes,

Fig. 3

Planck’s radiation law [Eq. (8)]. (a) Linear plot and (b) log-linear plot with $T=500°\mathrm{C}$, 136°C, and 20.5°C (top to bottom) and peak wavelengths at ${\lambda }_{\text{peak}}=4.3$, 7.1, and $9.9\mu \mathrm{m}$, respectively.

The reflected spectral radiance of Eq. (5) [the BRDF integral of Eq. (5)] becomes

Butler says microfacet BRDF models are generally written as¹⁷

The spectral radiance leaving the surface in Eq. (5) becomes the measured spectral radiance here

With this background, Sec. 2 will cover the methods of instrument calibration, data collection and analysis, and a description of the two samples chosen for use in this experiment in our Electro-Optical-Systems-Lab course. Section 3 will show example measurement results and analyses and demonstrate through this experiment what radiometry can reveal to the student about a measured sample’s characteristics. Section 4 concludes the paper.

2.

Methods

Prior to the start of data collection for this experiment, the student is encouraged to explore the possibilities of Eq. (15) to understand the spectral data they will collect and how to begin to interpret/analyze it. Representing an empirically observed pattern mathematically is one of the scientific abilities the ISLE methodology encourages. As will be shown in Sec. 3, Eq. (15) fully represents the observed trends, and elicits analyses of just how well it agrees with the data.²⁰

Figure 4 shows a case where the sample is spectrally constant and perfectly diffuse (Lambertian), so $f_{\text{surf}}({\theta }_{BB},{\theta }_r,\mathrm{\Delta }\varphi =\pi )=0$ may be set and ${\rho }_{DDR}({\theta }_r)$ essentially becomes ${\rho }_{\mathrm{HDR}}({\theta }_r)$, with $0\le {\rho }_{\mathrm{HDR}}\le 1$; it uses $T_{\text{Background}}=20.5°\mathrm{C}$ and ${\theta }_{\mathrm{BB}}=45°$. Figure 4(a) uses $T_{\mathrm{BB}}=400°\mathrm{C}$ and shows four curves, from bottom to top, a Planckian at $T_{\text{Background}}$ and three plots of Eq. (15) with ${\rho }_{\mathrm{DDR}}=0,0.5$, and 1, respectively. It tells the student that if they collect data that appears Planckian at $T_{\text{Background}}$, they are likely measuring a diffuse sample and should begin data analysis appropriately. Figure 4(b) uses ${\rho }_{\mathrm{DDR}}=0.5$ and shows six curves, again from bottom to top, a Planckian at $T_{\text{Background}}$ and five curves with $T_{\mathrm{BB}}=100$, 300, 500, 700, and 900°C, respectively. It tells the student that for diffuse samples, the “small angle” illumination must be very bright before it pulls the spectrum away from that of the background Planckian, and then that that difference is largest at shorter wavelengths where the Planckian of the hotter source peaks.

Fig. 4

Plots of Eq. (15) for a diffuse (Lambertian) sample. $f_{\mathrm{surf}}({\theta }_{\mathrm{BB}},{\theta }_r,\mathrm{\Delta }\varphi =\pi )=0$, $T_{\text{Background}}=20.5°\mathrm{C}$, and ${\theta }_{\mathrm{BB}}=45\mathrm{deg}$. In both plots, the lowest curve is a Planckian at $T_{\text{Background}}$. (a) $T_{\mathrm{BB}}=400°\mathrm{C}$ and from the bottom, ${\rho }_{\mathrm{DDR}}=0$, 0.5, and 1, respectively. (b) ${\rho }_{\mathrm{DDR}}=0.5$ and from the bottom, $T_{\mathrm{BB}}=100°\mathrm{C}$, 300°C, 500°C, 700°C, and 900°C, respectively.

Figure 5 is a second example, where the sample is again spectrally constant but now perfectly specular. ${\rho }_{\mathrm{DDR}}({\theta }_r)=0$ may be set in Eq. (15) and the specular BRDF, $f_{\mathrm{surf}}({\theta }_{\mathrm{BB}},{\theta }_r,\mathrm{\Delta }\varphi =\pi )$, may be studied. It again uses $T_{\text{Background}}=20.5°\mathrm{C}$, $T_{\mathrm{BB}}=400°\mathrm{C}$, and ${\theta }_{\mathrm{BB}}=45°$. Figure 5(a) shows six plots, again from bottom to top, a Planckian at $T_{\text{Background}}$ and five plots of Eq. (15) with $f_{\mathrm{surf}}({\theta }_{\mathrm{BB}},{\theta }_r,\mathrm{\Delta }\varphi =\pi )=5$, 10, 20, 30, and $40{\mathrm{Sr}}^{-1}$, respectively. It tells the student that the more the spectrum moves away from the $T_{\text{Background}}$ Planckian, the more specular the sample likely is. Figure 5(b) shows four plots, this time from top to bottom, a Planckian at $T_{\mathrm{BB}}$ and three plots of Eq. (15) with $f_{\mathrm{surf}}({\theta }_{\mathrm{BB}},{\theta }_r,\mathrm{\Delta }\varphi =\pi )=300$, 200, and $100{\mathrm{Sr}}^{-1}$, respectively. It tells the student they are essentially observing the virtual image of the “small-angle source” through a specular “mirror,” and as the sample reflectance increases, the $T_{\mathrm{BB}}$ Planckian dominates the $T_{\text{Background}}$ Planckian, as expected since $T_{\mathrm{BB}}\gg T_{\text{Background}}$.

Fig. 5

Plots of Eq. (15) for a perfectly specular sample. $T_{\text{Background}}=20.5°\mathrm{C}$, $T_{\mathrm{BB}}=400°\mathrm{C}$ and ${\theta }_{\mathrm{BB}}=45°$. In both plots, the lowest curve is a Planckian at $T_{\text{Background}}$. (a) Bottom-most plot is a Planckian at $T_{\text{Background}}$. Moving upward, $f_{\mathrm{surf}}({\theta }_{\mathrm{BB}},{\theta }_r,\mathrm{\Delta }\varphi =\pi )=5$, 10, 20, 30 and $40{\mathrm{Sr}}^{-1}$, respectively. (b) Top-most plot is a Planckian at $T_{\mathrm{BB}}$. Moving downward, $f_{\mathrm{surf}}(\lambda ,{\theta }_{\mathrm{BB}},{\theta }_r,\mathrm{\Delta }\varphi =\pi )=300$, 200, and $100{\mathrm{Sr}}^{-1}$ respectively.

Again, referring to the ISLE methodology, the concept is that the best learning experience for the student is that they design experiments for themselves to find phenomenological patterns in the data, devise hypotheses the explain the phenomena, design experiments to test their hypotheses, and even rule out hypotheses based on those experimental results.¹⁶ Although that aspect of the process is not what is presented here, it is a paradigm we try to follow in our Electro-Optical-Systems-Lab course since this course is considered preparation for the student’s upcoming thesis/dissertation experience. e.g., one student lab group proposed doing this radiometry experiment with an imager rather than a spectrometer. As shown in Figs. 4 and 5, the useful patterns in this experiment are spectral. The strength of an imager is that it provides spatial information, of which there is very little useful in this experiment, generally at the expense of spectral information, which has been integrated out over the spectral band of the detector. This may be considered as a simplified version of the well-known temperature-emissivity-separation problem in remote sensing.²¹ With some work, Eq. (13) could be in a form like Eqs. (7) and (14), such that its spectral integral is ${\int }_{\mathrm{\Delta }\lambda }{\rho }_{\mathrm{eff}}(\lambda )L_{\mathrm{BB}}(\lambda ,T)\mathrm{d}\lambda$. Then, even when the emissivity or reflectance are constant, such as the cases shown in Fig. 6 for the mid-wave IR (MWIR, $3.3-5.6\mu \mathrm{m}$ here) and long-wave IR (LWIR, $5.6-12.5\mu \mathrm{m}$ here), separate identification of ${\rho }_{\mathrm{eff}}$ and $T$ is not possible. These students limited their success by “choosing” to integrate out important information. Although every student in our Electro-Optical-Systems-Lab course does not have the opportunity to propose and rule out hypotheses such as this, just in the interest of time in a 10-week course, the redacted laboratory report from this failed experiment is made available to the students working the spectral version for their information.

Fig. 6

Constant band-integrated radiance, ${\rho }_{\mathrm{eff}}{\int }_{\mathrm{\Delta }\lambda }{L}_{BB}(\lambda ,T)\mathrm{d}\lambda$, makes solution of $({\rho }_{\mathrm{eff}},T)$ nonunique. (a) MWIR, $3.3-5.6\mu \mathrm{m}$, $L({\rho }_{\mathrm{eff}},T)=310\mu \mathrm{W}{\mathrm{cm}}^{-2}{\mathrm{Sr}}^{-1}$. (b) LWIR, 5.6 to $12.5\mu \mathrm{m}$, $L({\rho }_{\mathrm{eff}},T)=5.3\mathrm{mW}{\mathrm{cm}}^{-2}{\mathrm{Sr}}^{-1}$.

In our Electro-Optical-Systems-Lab course, one of the instruments used is a Bomem MR 154 FTS with a liquid-nitrogen (${\mathrm{LN}}_2$)-cooled mercury-cadmium-telluride (MCT) photodiode for the 800 to $1800\text{-}{\mathrm{cm}}^{-1}$ ($5.6-12.5\text{-}\mu \mathrm{m}$) band and an ${\mathrm{LN}}_2$-cooled indium-antimonide photodiode for the 1800 to $3000{\mathrm{cm}}^{-1}$ (3.3 to $5.6\text{-}\mu \mathrm{m}$) band. The dataset presented henceforth was collected using a medium-angle telescope, and the detectors were calibrated for spectral radiance ($\mathrm{W}{\mathrm{cm}}^{-2}{\mathrm{Sr}}^{-1}\mathrm{cm}$) with $1.93\text{-}{\mathrm{cm}}^{-1}$ resolution, smoothed to $13.5\text{-}{\mathrm{cm}}^{-1}$ resolution, at the 123-cm collection distance using an Electro-Optical Industries, Inc. wide-area 2″ blackbody for temperatures of 20 to 200°C and an Electro-Optical Industries, Inc. cavity blackbody for temperatures of 200 to 410°C, and converted to ($\mathrm{W}{\mathrm{cm}}^{-2}{\mathrm{Sr}}^{-1}\mu {\mathrm{m}}^{-1}$) spectral radiance in post-processing. Both blackbody apertures overfilled the FTS FOV.

The reflectance samples used were polished aluminum (Al), henceforth referred to as the “specular” sample, and flamed-sprayed Al, henceforth referred to as the “diffuse” sample, both with gold (Au) deposited on them (Fig. 7). These are great initial samples for the student because they are near the extremes of specular/diffuse scatter that the student explored using Eq. (15); the specular sample is very mirror-like and the diffuse sample approaches Lambertian, especially at near-normal incidence angles. The Au deposition makes their reflectance rather constant spectrally, again, simplifying the application of Eq. (15) as a starting point by removing the spectral dependence of the reflectance terms.

Fig. 7

Reflectance samples used. (a) Polished Al, aka the “specular” sample, and (b) flamed-sprayed Al, aka the “diffuse” sample, both with Au deposited on them.

To quantify these samples’ characteristics, they were measured using a Surface Optics Corp. SOC-100 Hemispherical Directional Reflectometer²² with a Nicolet iS50 FT-IR FTS with a DTGS pyroelectric detector over the spectral band studied here. The ${\rho }_{\mathrm{HDR}}(\lambda ,{\theta }_r)$, ${\rho }_{\mathrm{DDR}}(\lambda ,{\theta }_r)$ and spectral specular-directional reflectance, ${\rho }_{\mathrm{SDR}}(\lambda ,{\theta }_r)$, are shown in Fig. 8. Details are included in the figure caption.

Fig. 8

At $40\mathrm{deg}\le {\theta }_r\le 50\mathrm{deg}$, (a) for the polished sample, ${\rho }_{\mathrm{HDR}}=0.98\pm 0.004$, ${\rho }_{\mathrm{DDR}}=0.024\pm 0.008$, and ${\rho }_{\mathrm{SDR}}=0.95\pm 0.01$, across the 3.3- to $12.5\text{-}\mu \mathrm{m}$ band, and (b) for the flame-sprayed sample, ${\rho }_{\mathrm{HDR}}=0.81\pm 0.02$, ${\rho }_{\mathrm{DDR}}=0.75\pm 0.02$, and ${\rho }_{\mathrm{SDR}}=0.06\pm 0.006$, across this band.

3.

Results and Analysis

Data should be collected by the student at several initial ${\theta }_{\mathrm{BB}}={\theta }_r$ specular angles (again, at any one of which, the sample may then be rotated away from specular) to give the student a rich data set to study how the angle-dependent reflectance impacts the radiometry. Two scientific abilities the ISLE methodology seeks to develop in the student are designing an experiment that yields interesting patterns for the investigation of some phenomenon and identifying those patterns in the data collected.²⁰ The data are excellent for that.

3.1.

Specular Sample

Data from initial specular angles of ${\theta }_{\mathrm{BB}}={\theta }_r=11°$, 21°, 30°, and 62° are used here and the sample was rotated by $\pm 1°$, 2°, and 3° in each case. The virtual image of the cavity blackbody aperture viewed through the sample “mirror” subtends a planar angle of

Eq. (16)

$$\theta =\frac{D_{\mathrm{BB}}}{R_{\mathrm{BB}-\mathrm{FTS}}}=\frac{2.5\mathrm{cm}}{(40+123)\mathrm{cm}}=15\mathrm{mrad}=0.88°,$$

so little interesting data change was expected beyond the $\pm 3°$. Figure 9 shows examples of the data collected. Figure 9(a) is a measurement at ${\theta }_{\mathrm{BB}}={\theta }_r=21°$ and (b) is off specular at $({\theta }_{\mathrm{BB}}=20°,{\theta }_r=22°)$. Note the noise in the spectra; between the low responsivity of the MCT detector and the low signal in the water-vapor band of $\sim 5.6$ to $8\mu \mathrm{m}$, this was not uncommon, and at times, made curve-fitting difficult. Although the student can see this is a mirror-like sample as they set up the experiment, the assumption is that even without this knowledge, they should recognize from their own preparation leading to plots like those of Figs. 4 and 5 and the behavior of this data that this is a specular sample and proceed with analysis accordingly.

Fig. 9

(Blue) Measurement data for specular sample and (red) best fit of Eq. (15). $T_{\text{Background}}=20.5\pm 0.5°\mathrm{C}$ and $T_{\mathrm{BB}}=400°\mathrm{C}$. ${\rho }_{\mathrm{DDR}}({\theta }_r)=0$ and $f_{\mathrm{surf}}({\theta }_{\mathrm{BB}},{\theta }_r,\mathrm{\Delta }\varphi =\pi )$ is the fitting parameter are set for the fits. (a) ${\theta }_{\mathrm{BB}}={\theta }_r=21\mathrm{deg}$ and $f_{\mathrm{surf}}=303{\mathrm{Sr}}^{-1}$. (b) ${\theta }_{\mathrm{BB}}=20\mathrm{deg}$ and ${\theta }_r=22\mathrm{deg}$ and $f_{\mathrm{surf}}=200{\mathrm{Sr}}^{-1}$.

The red lines in each plot of Fig. 9 are the best fit of Eq. (15) to the data, using $T_{\text{Background}}=20.5°\mathrm{C}$ and $T_{\mathrm{BB}}=400°\mathrm{C}$, and making the assumption that illumination sources are reasonably well known, as they are in many remote-sensing scenarios and that the sample is at $T_{\text{Background}}$ here. (Note that $T_{\text{Background}}=20.5\pm 0.5°\mathrm{C}$ across all the data-collection and instrument-calibration days, and the Electro-Optical Industries, Inc. cavity blackbody was set to $T_{\mathrm{BB}}=400°\mathrm{C}$.) Knowledge of $({\theta }_{\mathrm{BB}},{\theta }_r)$ is also assumed, again, as might be true in many remote-sensing scenarios. Following the argument from Fig. 5, ${\rho }_{\mathrm{DDR}}(\lambda ,{\theta }_r)=0$ is set (which works out well here since $\mathrm{DDR}=0.024\pm 0.008$ for the specular sample), leaving $f_{\mathrm{surf}}({\theta }_{\mathrm{BB}},{\theta }_r,\mathrm{\Delta }\varphi =\pi )$ as the only fitting parameter.

Table 1 shows results of all the $f_{\mathrm{surf}}$ fits. The rows are the incident angles, ${\theta }_{\mathrm{BB}}$, and the columns are the reflectance (observation) angles, ${\theta }_r$, making the table diagonal to the specular case, ${\theta }_{\mathrm{BB}}={\theta }_r$. Moving down the diagonal of the table from upper left to lower right, reflectance increases as incident angle, ${\theta }_{\mathrm{BB}}$, increases, as expected. Looking at the off-specular results for any of the three-element specular-datasets, reflectance is always greater on the forward-scatter side of specular (${\theta }_r>{\theta }_{\mathrm{BB}}$) than it is on the back-scatter side (${\theta }_{\mathrm{BB}}>{\theta }_r$), again as expected. The ISLE methodology also suggests the adequate experiment for developing the scientific ability of identifying relevant patterns in data allows those trends to be described in words.²⁰ This does that. These are concepts that were introduced to the student in our IR-Technology course but are likely not retained without sufficient repetition/use. Although this experiment is well designed to do that, we said earlier that the FTS is assumed to be well aligned/well calibrated, and in our Electro-Optical-Systems-Lab course, this is not always the case. Hence, the collected data may differ from the examples shown here and lead to confusion in the analyses. This can be a good thing! The course description says this lab introduces laboratory techniques for the measurement of radiometric quantities. The synergy between data collection and theory is more important than ever with modern computational power.²³ Encouragement by the instructor to double check alignment/calibration is not only part of learning the laboratory techniques but is also an introduction to that synergy. Back to the Table 1 data, the spectral shape of the measured data for angles greater than $\pm 1°$ off specular, as well as the noise in that data, prohibited good fitting results for those angles.

Table 1

Results of fitting fsurf(θBB,θr,Δϕ=π) (units of Sr−1) of Eq. (14) to the specular-sample data.

θr(deg)
θBB(deg)	10	11	12	20	21	22	29	30	31	61	62	63
10			235
11		261
12	122
20						200
21					303
22				160
29									195
30								315
31							245
61												300
62											625
63										410

3.2.

Diffuse Sample

For the diffuse sample, initial specular angles of ${\theta }_{\mathrm{BB}}={\theta }_r=11°$, 21°, 30°, 45°, and 62° were again used, but the sample was rotated away from specular by much larger angles, $\le |\pm 33°|$. Figure 10 shows examples of the data collected. Figure 10(a) is a measurement at $({\theta }_{\mathrm{BB}}=23°,{\theta }_r=19°)$, (b) at $({\theta }_{\mathrm{BB}}=53°,{\theta }_r=7°)$, and (c) at $({\theta }_{\mathrm{BB}}={\theta }_r=45°)$. Again, although the student can see this is a rough-surfaced sample as they set up the experiment, the assumption is that even without this knowledge, they should recognize from their own preparation leading to plots like those of Figs. 4 and 5 and the behavior of this data that this is a diffuse sample and proceed with analysis accordingly.

Fig. 10

(Blue) Measurement data for diffuse sample. The green lines are the best fits of Eq. (15) to the data, using $T_{\text{Background}}=20.5°\mathrm{C}$ and $T_{\mathrm{BB}}=400°\mathrm{C}$, and that the sample is at $T_{\text{Background}}$. The red lines are Plankians at $T_{\text{Background}}$. (a) $({\theta }_{\mathrm{BB}}=23\mathrm{deg},{\theta }_r=19\mathrm{deg})$, best-fit results $(f_{\mathrm{surf}}=0,{\rho }_{\mathrm{DDR}}=0.82)$. (b) $({\theta }_{\mathrm{BB}}=73\mathrm{deg},{\theta }_r=7\mathrm{deg})$, best-fit results $(f_{\mathrm{surf}}=0,{\rho }_{\mathrm{DDR}}=0)$, (best fit is a Planckian at $T_{\text{Background}}$). (c) $({\theta }_{\mathrm{BB}}=45\mathrm{deg},{\theta }_r=45\mathrm{deg})$, best-fit results $(f_{\mathrm{surf}}=1.6{\mathrm{Sr}}^{-1},{\rho }_{\mathrm{DDR}}\sim 1)$, (the $f_{\mathrm{surf}}$ component models the forward scattering of this non-Lambertian sample).

The green lines in each plot of Fig. 10 are the again best fits of Eq. (15) to the data, using $T_{\text{Background}}=20.5°\mathrm{C}$ and $T_{\mathrm{BB}}=400°\mathrm{C}$, and again making the assumptions that the sample is at $T_{\text{Background}}$ and that ${\theta }_{\mathrm{BB}}$ and ${\theta }_r$ are known. The red lines are Plankians at $T_{\text{Background}}$. Following the argument from Fig. 4, $f_{\mathrm{surf}}({\theta }_{\mathrm{BB}},{\theta }_r,\mathrm{\Delta }\varphi =\pi )=0$ should be set, leaving ${\rho }_{\mathrm{DDR}}({\theta }_r)$ as the fitting parameter. This works well as an initial guess, however, as will be shown by the lower plot of Fig. 10, this sample is diffuse but not Lambertian, so a simultaneous fit of $f_{\mathrm{surf}}$ and ${\rho }_{\mathrm{DDR}}$ in Eq. (15) is a better choice. Assuming the student can do, or at least follow, the argument of Eqs. (12)–(15), this is a very reasonable and computationally tractable stretch. Note that the fitting constraint used here is ${\rho }_{\mathrm{DDR}}\le 1$, which is obviously physical.

Figure 10(a) shows the $({\theta }_{\mathrm{BB}}=23°,{\theta }_r=19°)$ case, where the fit did produce $f_{\mathrm{surf}}=0$, with ${\rho }_{\mathrm{DDR}}=0.82$ (recall that the measured $\mathrm{DDR}=0.75\pm 0.02$ for the diffuse sample). As in Fig. 4(a), this data resembles the $T_{\text{Background}}$ Planckian but lies above it, so ${\rho }_{\mathrm{DDR}}>0$ gave a good fit. Figure 10(b) shows the $({\theta }_{\mathrm{BB}}=73°,{\theta }_r=7°)$ case; like much of the data well away from the specular angle, the best fit to the data was $(f_{\mathrm{surf}}=0,{\rho }_{\mathrm{DDR}}=0)$, indicating there is little appreciable contribution from the cavity blackbody and the best fit to the data is a $T_{\text{Background}}$ Planckian. Figure 10(c) shows the $({\theta }_{BB}=45°,{\theta }_r=45°)$ case, where the fit produced $(f_{\mathrm{surf}}=1.6{\mathrm{Sr}}^{-1},{\rho }_{\mathrm{DDR}}\sim 1)$. ${\rho }_{\mathrm{DDR}}$ being constrained to 1 was the case for many of the $30°\le {\theta }_{\mathrm{BB}}\le 62°$ data due to the forward-scattering nature of this non-Lambertian sample. Referencing Fig. 4, such data cannot be modeled by diffuse scatter alone and requires the $f_{\mathrm{surf}}$ component, but again, this is still a tractable stretch for the student. Recall that the measured $\mathrm{SDR}=0.06\pm 0.006$ for the diffuse sample; it is difficult to draw the link between $\mathrm{BRDF}({\mathrm{\Omega }}_i,{\mathrm{\Omega }}_r)$ and ${\rho }_{\mathrm{HDR}}({\mathrm{\Omega }}_r)$ or ${\rho }_{\mathrm{SDR}}({\mathrm{\Omega }}_r)$ without much higher resolution ${\mathrm{\Omega }}_r$ measurements.

Figure 11 shows results of all the $(f_{\mathrm{surf}},{\rho }_{\mathrm{DDR}})$ fitting, shown as $\mathrm{BRDF}({\theta }_{\mathrm{BB}},{\theta }_r)=f_{\mathrm{surf}}({\theta }_{\mathrm{BB}},{\theta }_r)+{\rho }_{\mathrm{DDR}}({\theta }_{\mathrm{BB}},{\theta }_r)/\pi$, and represents an estimated (and reasonable) BRDF plot for this diffuse sample. The specular $({\theta }_{\mathrm{BB}}={\theta }_r)$ diagonal runs lower left to upper right but is not very prominent in a diffuse sample. The BRDF increases with increasing incident angle, ${\theta }_{\mathrm{BB}}$, as expected for a non-Lambertian sample, and increases on the upper left of the $({\theta }_{\mathrm{BB}}={\theta }_r)$ diagonal compared to the lower right since the BRDF is typically greater on the forward-scatter side of specular than it is on the backscatter side. Again, these are concepts that were introduced to the student in our IR-Technology course but are likely not retained without sufficient repetition/use. And again, the ISLE methodology suggests the adequate experiment for developing the scientific ability of identifying relevant patterns in data also allows those trends to be described in words.²⁰ This experiment is well designed to do that.

Fig. 11

Results of fitting $\mathrm{BRDF}({\theta }_{\mathrm{BB}},{\theta }_r)=f_{\mathrm{surf}}({\theta }_{\mathrm{BB}},{\theta }_r)+{\rho }_{\mathrm{DDR}}({\theta }_{\mathrm{BB}},{\theta }_r)/\pi$ (units of ${\mathrm{Sr}}^{-1}$) of Eq. (15) to the diffuse-sample data.

3.3.

Findings

This BRDF experiment in our Electro-Optical-Systems-Lab course was devised to help the student develop a better understanding of radiometry and radiometric instrumentation, and how they can be used to gain an understanding of a material’s optical properties in a remote-sensing scenario, which is another concept introduced in our IR-Technology course. As an initial, tractable problem for the student, it allows them success in the differentiation between a specular sample and a diffuse one, and after that differentiation, success in their initial quantification of a sample’s reflectance properties of $f_{\mathrm{surf}}({\theta }_{\mathrm{BB}},{\theta }_r,\mathrm{\Delta }\varphi =\pi )$ and ${\rho }_{\mathrm{DDR}}({\theta }_r)$. It provides useful experience upon which much more difficult remote-sensing analyses may be built. Finally, all students present their findings and procedures to the rest of the class in our Electro-Optical-Systems-Lab course, another experience encouraged by the ISLE methodology.²⁰

4.

Conclusions

AFIT offers a three-course optics sequence to educate students in radiometry and radiometric instrumentation, and their applications. This paper described a BRDF experiment developed for our Electro-Optical-Systems-Lab course to help students gain a better understanding of radiometry and radiometric instrumentation. The groundwork for the BRDF and its analysis is laid in our earlier (IR)-Technology course. This experiment uses a Fourier-transform IR spectrometer (FTS), a cavity blackbody, and samples whose reflectance properties range from specular to diffuse. We find this experiment to be useful in helping the student not only to understand radiometry and radiometric instrumentation but also how they can be used to gain an understanding of a material’s optical properties in a remote-sensing scenario. It also provides success in an initial radiometry experience upon which more difficult remote-sensing analyses may be built.

Limitations of the initial experiment described here include sample measurements being limited to in-plane, i.e., both illumination source and detector were positioned in-plane relative to the sample’s surface normal. The generalization of this obviously includes out-of-plane measurements, but those results are likely not much different than the off-specular results reported here. Another limitation is that the reflectance properties of the initial samples presented here were limited to the simple cases of being either very specular or very diffuse. Furthermore, these samples were gold-coated metal, which is rather constant spectrally in the IR region studied here. Generalization of these includes the extension to samples whose specularity is somewhere between these two extremes (as most real-world materials are) and are dielectric, such that their reflectance might have some spectral content and volume scatter would complicate the BRDF model required for their analyses.

Even with these simplifying constraints, we believe this is a good radiometry-laboratory experiment to advance the student’s optics education.