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Abstract
Light curve analysis is often used to discern information about satellites in geosynchronous orbits. Solar panels, comprising a large part of the satellite’s body, contribute significantly to these light curves. Historically, theoretical bidirectional reflectance distribution functions (BRDFs) have failed to capture key features in the scattered light from solar panels. In recently published work, a new solar cell BRDF was developed by combining specular microfacet and “two-slit” diffraction terms to capture specular and periodic/array scattering, respectively. This BRDF was experimentally motivated and predicted many features of the solar cell scattered irradiance. However, the experiments that informed the BRDF were limited to a single laser wavelength, single beam size, and single solar cell sample. In addition, the BRDF was not physics based and therefore, physical insight into what causes certain features in the scattered irradiance was not evident. In this work, we examine solar cell scattering from first principles and derive a simple physics-based expression for the scattered irradiance. We analyze this expression and physically link terms to important scattering features, e.g., out-of-plane phenomena. In addition, we compare our model with experimental data and find good agreement in the locations and behaviors of these features. Our new model, being more predictive by nature, will allow for greater flexibility and accuracy when modeling reflection from solar cells in both real-world and experimental situations.
1. Introduction
Remote sensing is a critical technique for studying faraway objects such as geosynchronous satellites. For such objects, ground-based imaging is virtually impossible. Instead, tools like light curve analysis are used to understand satellite characteristics and activities under low resolution conditions [1–3]. A light curve displays an object’s intensity over time, and light curve analysis uses radiometry to understand the propagation of light through a scene from source to satellite to detector in order to infer information about orbiting satellites. At the satellite, there are two principal factors that contribute to a light curve: (1) the satellite’s state, which includes its shape, location, and orientation, and (2) the satellite’s material composition and properties, which includes directional reflectance properties defined by bidirectional reflectance distribution functions (BRDFs). Historically, inaccuracies within satellite material BRDF models have led to well-documented errors between light curve observations and simulations [3–7]. In particular, solar cells often comprise a relatively high percentage of a satellite’s surface area, so errors in solar cell BRDFs have the potential to greatly affect light curve analysis and simulation.
Commonly used solar cell BRDF models often employ relatively simple isotropic microfacet formulations which neglect diffraction [5,8,9]. Microfacet BRDFs are a stochastic approach to modeling a rough surface with the typical benefit of computational efficiency compared to a full physical optics approach. Microfacet models are useful when exact surface characteristics are unknown, but are ultimately a mathematical approximation with their own associated errors. Recent experiments have shown that satellite solar cells can exhibit significant anisotropic diffraction in their reflected light due to the periodic spacing of metal grids atop the photovoltaic layers, which BRDFs commonly fail to accurately represent [10,11]. In 2020, a microfacet-based “two-slit" BRDF was published to account for newly measured and identified anisotropic diffraction effects, such as those in Fig. 1 [10].
Fig. 1. Laboratory image of scattering from a solar cell illuminated by a HeNe laser with the wire grid orientation parallel to the plane of incidence. This view is from behind the solar cell.
The two-slit model was a combination of a specular microfacet term, which used an isotropic Beckmann microfacet distribution function [12], and a closed-form far-field solution of the two-slit aperture diffraction problem [10]. The model’s development was based on an experimental setup, which used a single wavelength ($\lambda =632.8$ nm), a single illumination spot size, and a single solar cell sample. Nevertheless, it successfully matched important features in the data, including the magnitude and width of the central peak, as well as the magnitudes, spacing, and directionally dependent orientation of the diffraction peaks. In particular, when the metal grids were oriented perpendicular to the plane of incidence, the diffraction pattern extended completely within the plane of incidence (in-plane), and the diffraction was planar. Conversely, when the grids were oriented at least partly parallel to the plane of incidence, the diffraction extended outside the plane of incidence (out-of-plane) and demonstrated a curved behavior as shown in Fig. 1. Although not explicitly identified or explained in previous work, this out-of-plane curvature originates from the phenomenon known as conical diffraction [13].
The two-slit model improved solar cell simulations by capturing important anisotropic behavior, but did not include the flexibility to scale up illumination spot size to capture “multi-slit" features in the observed reflection from the metal grids. Furthermore, it modeled the photovoltaic layer with a specular microfacet BRDF, which could not capture the periodic nature of the structure under general illumination conditions. Finally, the out-of-plane observed conical diffraction was accounted for in the two-slit model parametrically and did not have a direct physical link to the mathematical representation. To address these limitations, electromagnetic theory is applied to develop a more physically based scattering model for solar cells.
Section 2 presents the general theory for the electromagnetics approach used to determine scatter from a solar cell. Here, we derive the far-zone scattered irradiance and physically link terms in the expression to important scattering features. Section 3 briefly describes the experimental setup and compares measurements to predictions from our improved solar cell model. Lastly, we conclude with a summary and discussion of future work.
2. Theory
2.1 Geometry and assumptions
Figure 2 shows a three-dimensional view of the geometry of the coordinate system and the description of the solar cell. Coordinates are chosen with the solar cell fixed to the $x$-$y$ plane, the metal grids fixed specifically in the $y$ direction, and the surface normal fixed in the $z$ direction. The spacing $D$ is the center-to-center distance between neighboring wire or grid elements, $d_{M}$ is the metal wire width, and $d_{I}=D-d_{M}$ is the width of the photovoltaic indium gallium phosphide (InGaP) segments. The incident field propagates along the $\bar {k}_i$ vector, with the incident direction set by the spherical coordinates ($\theta _i, \phi _i$), which govern the incident angle and the structure’s orientation relative to the plane of incidence. Incident and scatter azimuthal angles ($\phi _i$ and $\phi _s$) are measured from the positive $x$ axis, and the incident and scatter zenith angles ($\theta _i$ and $\theta _s$) are measured from the positive $z$ axis. Here, $\bar {E_i}$ is assumed to lie in a plane parallel to the surface of the structure and therefore, perpendicularly polarized. Considering that the structure’s features are several times the wavelength of the incident field, the primary features of the scattered diffraction pattern will be generally polarization agnostic. Since the goal of this paper is to physically describe the features of the scattered irradiance pattern, with less emphasis devoted to radiometry and polarization state, we do not consider parallel polarization. Nevertheless, if needed, including polarization effects is a relatively straight forward modification of the theory presented below.
Fig. 2. (left) 3D view of solar panel geometry used for theory development under plane wave illumination. The geometry of the solar cell’s orientation is fixed, while the incident beam can rotate by changing $\theta _i$ and $\phi _i$. (right) Cross-sectional illustration of solar cell surface structure with dimensions. The gold rectangles represent the solar cell’s metal grids (aluminum wires), which are sitting atop an InGaP substrate. At the visible wavelength used in this work, the light penetrating beyond the top InGaP layer is negligible. The drawing is not to scale.
The exact electric field $\bar {E}_s$ at observation point $\bar {r}$ is found by
Equations (1) and (2) can be used to find the exact scattered field anywhere in space; however, doing so is computationally intensive. Even in cases where the scatterer is much larger than $\lambda$ (the realm of Fourier, or wave optics) and $\bar {J}$ and $\bar {M}$ can be approximated using just the incident fields, the evaluation of the convolution integrals in Eq. (2) is difficult due to the form of the free-space Green’s function. The problem becomes tractable and extremely physical when observation is restricted to the far zone. For the satellite-light-curve scenario motivating this work, the far-zone scattered field is appropriate.
2.2 Physical optics approximation
As alluded to above, when observation is restricted to the far zone, Eqs. (1) and (2) simplify considerably. Neglecting the details, which can be found in [14], the far-zone scattered electric field is described by
The electric and magnetic current densities are approximated in terms of the perpendicular polarization Fresnel reflection coefficient $r_{\perp }$ as [15]
In Eq. (5), the incident field $\bar {E_i}$ is a monochromatic plane wave incident from $\theta _i$ and $\phi _i$, namely,
For aluminum, $r_{\perp }\approx -1$ and the contribution from $\bar {M}$ is negligible. For InGaP at $\lambda = 500 \text { nm}$,
The scattered irradiance is proportional to
The far-zone potential is $\bar {N} = \bar {N}_{I} + \bar {N}_{M}$ via superposition, where $\bar {N}_{I}$ and $\bar {N}_{M}$ are the contributions to $\bar {N}$ from the InGaP and aluminum parts of the solar cell, respectively. As a result,
The last term in Eq. (9) models the interference of the fields scattered from the aluminum and InGaP parts of the structure. This term is weak compared to the “self” terms because of the size difference between those features (see Fig. 2). In addition to this physical argument, the effects of the interference term are not apparent in the measured data (shown in Section 3.2). For these reasons, we neglect it hereafter.
Referring to the solar cell geometry depicted in Figs. 2 and 3, the potential $\bar {N} = \bar {N}_{I} + \bar {N}_{M}$ can be represented as
Recall that this expression was derived assuming perpendicular polarization and by neglecting the interference between the aluminum and InGaP segments.
Fig. 3. (left) 2D view of the solar panel with alternating sections of thin aluminum wires and large InGaP sections. (right) Cross-sectional mathematical view of the solar panel.
2.2.1 Discussion
Physical features of the diffraction pattern arise from the second line of Eq. (11). The final term on that line is immediately recognized as the multi-slit diffraction contribution. The zeros of its denominator, namely,
The first two terms comprise the far-zone scattered pattern from a single rectangular element, either an aluminum wire or InGaP segment. The widths of these $\operatorname {sinc}^2$ functions generally constrain the scattered light to the specular direction. Therefore, to examine the behavior of these terms more closely, it makes physical sense to expand the observation angles $\left (\theta,\phi \right )$ around the specular scattered direction, i.e., $\theta \approx \theta _i + \Delta \theta$ and $\phi \approx \phi _i - \pi + \Delta \phi$. Expanding the arguments of the $\operatorname {sinc}^2$ functions in Taylor series and keeping second-order terms produces
The maxima of the $\operatorname {sinc}^2$ functions occur when the above quadratic equations in $\Delta \phi$ equal zero. The roots can easily be found using the quadratic equation. Note that a similar set of quadratic equations in $\Delta \theta$ can also be derived. To gain insight into the behaviors of Eq. (14), let us choose a convenient incident azimuth angle (such as $\phi _i = 270^{\circ }$ or $\phi _i = 180^{\circ }$) such that some of the terms in Eq. (14) are zero. Selecting $\phi _i = 270^{\circ }$ and setting the resulting expressions equal to zero produces
Finally, solving for $\Delta \phi$, assuming small $\Delta \theta$, yields $\Delta \phi = 0$ and $\Delta \phi \approx \pm \sqrt {2 \Delta \theta \cot \theta _i}$, respectively.
The former physically means that the $\hat {x}$ $\operatorname {sinc}^2$ function, at $\phi _i = 270^\circ$, obtains its maximum only in the specular direction. On the other hand, the argument of the $\hat {y}$ $\operatorname {sinc}^2$ function generally has two zeros and therefore, the function obtains its maxima at two different $\Delta \phi$ locations. Physically, this manifests as a bright parabolic arc that opens in the positive $\Delta \theta$ direction. The width of the arc is given approximately by $\sqrt {2\cot \theta _i}$. Thus, the $\hat {y}$ $\operatorname {sinc}^2$ term in Eq. (12) is responsible for the out-of-specular-plane feature pictured in Fig. 1. Note that the expressions in Eq. (15) flip if $\phi _i = 180$°; however, the arc, although present, is dim and not generally observed because of the small width $d_M$ of the aluminum wires and relatively weak reflection from the InGaP segments.
Before proceeding, we note that there is a height difference between the aluminum wires and InGaP segments (see Fig. 2), which we have neglected. In our model this effect would be captured in the interference term in Eq. (9). As stated previously, we do not observe this interference in the measured data. Therefore, we conclude that the height difference between the aluminum and InGaP elements does not contribute significantly to the scattered irradiance. Be that as it may, our model does not account for the possibility that an InGaP segment is shadowed by an aluminum wire. From Fig. 2, we can assess when shadowing will be significant using simple geometry. Any $\theta _i > 0^{\circ }$ decreases the illuminated area of an InGaP segment; however, for the geometry of this solar cell, a 10% reduction in illuminated area requires $\theta _i \geq 82^{\circ }$. For solar cells with larger wire height to photovoltaic width ratios, shadowing will be significant at smaller incident angles. This is the subject of on going efforts.
3. Experiments
3.1 Setup
The Complete Angle Scatter Instrument (CASI) is a measurement tool used to characterize the reflected and transmitted light from material samples. The device uses a laser to interrogate the sample under test. The CASI has an optics box to control the path and focus of the incident beam. A graphical user interface (GUI) controls the goniometer, mechanically changing and measuring the angle of the sample and detector. The CASI’s detector is a charge-coupled device (CCD) array on the goniometer arm shown in Fig. 4. The CCD array allows each pixel to act as an individual detector and measure the reflected light in a unique scatter direction [11,16–18]. The CCD captures out-of-plane scatter, but only in a limited range of angles around the observation direction. This range of angles depends on the distance from the sample to the CCD and is somewhat adjustable. However, even for the largest range of possible observation angles, only a small portion of the large out-of-plane arc photographed in Fig. 1 will be captured.
For the experiments whose results are presented below, BRDF data was collected from the solar cell at $\left (\theta _i, \phi _i\right )=\left (70^{\circ }, 270^{\circ }\right )$ using a green HeNe laser ($\lambda$ = 543 nm). The solar cell used in this work is a commercially available triple junction cell and is shown in Fig. 5. Data was collected at different exposure times and integrated to improve the dynamic range of the CCD.
3.2 Results
Figure 6(a) shows the measured BRDF (represented on a log color scale) from the solar cell. We converted the digital CCD output to BRDF units by subtracting the background light from the raw CCD data and dividing each pixel by the incident light. For more details on this calibration procedure, see Ref. [19]. There are two distinct features visible in Fig. 6(a): the out-of-plane parabolic arc and a sharp specular peak. The latter, which is not included in our model, is caused by reflection from a protective glass coating that covers the wire and InGaP parts of the solar cell. The specular coverglass reflection is slightly offset from the expected specular direction due to a slight misalignment between the glass and wire/photovoltaic layer. Including the coverglass in our solar cell scattering model is a work progress.
Fig. 6. (a) Measured BRDF data from the solar cell in Fig. 5 at $(\theta _i, \phi _i)=(70^{\circ }, 270^{\circ })$ illuminated by a HeNe laser ($\lambda = 543 \text { nm}$). (b) Theoretical scattered irradiance [Eq. (12)] corresponding to the geometry of the measurement in (a). (c) Measured BRDF data overlayed with the derived relation for the out-of-plane parabolic arc. (d) $\Delta \phi = 0$ (in-plane) slices through (a) and (b). (e) $\Delta \phi = -\sqrt {2 \Delta \theta \cot \theta _i}$ slices through (a) and (b).
Figure 6(b) is the theoretical prediction of the scattered irradiance from the solar cell [see Eq. (12)] assuming $L_y = 1 \text { mm}$, $I = 1$, $M = 2$, and $\left |E_o\right | = 1$. The other solar cell parameters are given in Fig. 2. Like in Fig. 6(a), we observe two primary scattering features in Fig. 6(b). The first is the out-of-plane parabolic arc discussed in more detail below and an in-plane feature, which is the $\operatorname {sinc}^2$ diffraction pattern from a single InGaP segment. Naturally, this diffraction pattern has a maximum at the specular angle. The off-specular maxima along the $\Delta \phi = 0$ (in-plane) slice in Fig. 6(b) are the “sidelobes” of the $\operatorname {sinc}^2$. Many are visible in this image because the scattered irradiance is plotted on a log color scale. These maxima do not appear in the measured result [Fig. 6(a)] due to the Gaussian-shaped laser illumination on the solar cell. Figure 6(d) shows the $\Delta \phi = 0$ (in-plane) slices through Figs. 6(a) and (b) (blue and red traces, respectively). Although our model does not account for beam shape (assumes uniform illumination over, at least, part of the solar cell) and therefore, incorrectly predicts the overall shape of this feature, the experimental and theoretical widths of the feature are in good agreement. This proves that beam shape is the cause of this discrepancy.
Lastly, Fig. 6(c) overlays the predicted equation for the out-of-plane arc (red-dotted line) from Section 2.2.1 onto Fig. 6(a). It is important to note that this red-dotted curve is not fitted. It is derived from the $\hat {y}$ $\operatorname {sinc}^2$ term and results directly from the geometry of the solar cell. One key feature of the arc that is visible in the theoretical result and not in the measurement is multi-slit (in this case, two-slit because of beam size) interference. This does not appear in the experimental result due to a combination of beam shape, surface roughness, and noise. Both the measurement in Fig. 6(a) and theory in Fig. 6(b) share symmetric “nulls” in the arc located at approximately $\Delta \phi = \pm 3.5^\circ$. These nulls are caused by the $\operatorname {sinc}^2$ function physically modeling diffraction from a single aluminum wire. Figure 6(e) shows the $\Delta \phi = -\sqrt {2 \Delta \theta \cot \theta _i}$ slices through Figs. 6(a) and (b), i.e., along the $-\Delta \phi$ branches of the arcs in (a) and (b). Upon closer inspection of the arcs, we observe that experimental (solid blue trace) and theoretical (dotted gold trace) nulls are slightly displaced. This discrepancy is completely explained by uncertainty in the width of an aluminum wire. To show this, we included another $I_s$ (solid red trace), this time with $d_M = 12 \text { } \mu \text {m}$, $d_I = 740 \text { } \mu \text {m}$, and $D = 752 \text { } \mu \text {m}$. The locations of the nulls are now in much better agreement.
In summary, the quality of the results in Fig. 6 validates our model.
4. Conclusion
Solar panels account for a large portion of a satellite’s overall structure and thus contribute significantly to their observed brightness. Light curve analysis, which considers brightness over time, is often used in low resolution scenarios to discern satellite features, characteristics, and maneuvers for remote sensing purposes. Historically, radiometric simulators have relied on isotropic microfacet BRDF models to account for solar panel light scattering. However, microfacet models assume geometrical optics and typically neglect wave effects, such as interference and diffraction.
In recent work, a new experimental BRDF was developed to predict solar panel light scattering. The new BRDF consisted of a two-slit far-field diffractive contribution plus an isotropic microfacet term to account for interference and specular features in the measured scattered irradiance. Subsequent experiments revealed limitations with this “two-slit” BRDF as it failed to account for incident beam size and material parameters. In addition, being experimentally informed, insight into what caused certain scattering features was not readily apparent.
In this work, we derived an expression for the far-zone scattered irradiance from a solar cell using electromagnetic theory. Our model overcame the limitations of the two-slit BRDF and was extremely physical, i.e., terms in the mathematical expression were linked directly to the observed scattering phenomena. We validated our theoretical expression by comparing its predictions to BRDF solar cell measurements; the results were in very good agreement.
Our work is an important step toward predicting and generating BRDFs for solar cells with the ultimate goal being to reduce light curve error in satellite assessment. Although our model successfully predicts features in the scattered irradiance from solar cells, there are several aspects where it can be improved and thus more accurate. First, our model is scalar and neglects polarization. Second, while it can account for beam size, it does not account for shape (assumes uniform illumination). Third, we do not include shadowing nor any protective coating over the photovoltaic layer. Lastly, we do not account for incident light coherence. All of these factors can be incorporated into our development and are the subject of future work.
Appendix
The scattered irradiance in terms of incident and observation angles can be found from Eq. (12) by making the following substitutions:
In addition, it can be converted to a BRDF in $1/\text {sr}$ using
Funding
Air Force Office of Scientific Research (F4FGA09014J002).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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