Spatial coherence of sunlight and its implications for light management in photovoltaics

New technologies are emerging that capture and redirect sunlight inside solar cells. This type of light management can significantly increase efficiency, but the device behavior will fundamentally depend on the spatial coherence of the incoming light. This dependence calls for a complete characterization of the spatial coherence of sunlight. Here, we present the first spectral measurements of the spatial degree of coherence of direct, diffuse, and simulated sunlight. An expression is derived for both the cross-spectral density and the spatial degree of coherence in an arbitrarily oriented device plane, including the effects of overcast skies. Implications of the present work are discussed and may lead to a better understanding of light-managing components in solar cells as well as a new class of solar simulator that provides both the same spectrum and spatial coherence as direct sunlight. © 2015 Optical Society of America

Light management can increase efficiency in single-and multi-junction cells, but the device behavior will depend on both the spatial coherence and the bandwidth of the incoming light [13].The behavior of rectenna solar cells has similar dependencies [14,15].It is therefore necessary to fully characterize the spectral variation of the spatial coherence of sunlight.We refer to devices whose behavior depends on spatial coherence as coherent photovoltaics or co-PVs.
The spatial coherence of direct terrestrial sunlight (sunshine) was first considered in the 19th century by Verdet [16].Much more recently the machinery of modern coherence theory was applied to the calculation of the spectral degree of spatial coherence expected from sunshine by Agarwal et al. [17] and found to essentially match Verdet's expectation.Indeed, the mutual intensity of sunshine, a broadband quantity that is related to the spectral degree of coherence [18], was subsequently measured by Mashaal et al. [19] and found to substantially agree with the expectation [20].However, the calculations of Agarwal et al. are based on a uniform sphere model and therefore ignore real optical effects such as solar limb darkening.
While the previous measurements demonstrate agreement with the theory in the broadband case, the spectral degree of coherence, which considers each spectral component separately, has not been measured.Further, if co-PVs are to be used in real applications, it is necessary to understand the spatial coherence of sunlight under non-ideal weather conditions as well as to establish a spatial coherence standard for device testing.Here, we present the first interferometric measurements of the transverse spectral degree of coherence of both direct sunshine and diffuse terrestrial sunlight, where the Sun is obscured by clouds or fog.In this context, transverse indicates that the coherence function is measured in a plane whose normal vector points directly toward the center of the Sun.
Our measurements are performed using the method illustrated in Fig. 1.The method involves non-parallel slits and its working principles have been described previously [21].The measurements are precise enough to distinguish between the expectations of uniform and limb-darkened incoherent spheres.To demonstrate the precision of the measurements, we compare our findings with those expected by theory as applied to hyperspectral solar limb darkening measurements made by Neckel and Labs [22].We also present a measurement of the spectral degree of coherence of light emitted by a solar simulator, establishing the need for a spatial coherence standard in co-PV device testing.
To provide context to the measurements and their practical use, we present the theory of the spectral degree of coherence of sunlight beyond the transverse case and/or under a uniformly cloudy sky.Unless sophisticated tracking methods are used, a photovoltaic cell is generally never in a plane whose normal vector points directly toward the center of the Sun.Consequently, a description of how the coherence properties of sunlight vary with the position of the Sun in the sky relative to a fixed device plane is important to predict the behavior of co-PVs at different times in the day.We provide an expression that derives any non-transverse spectral degree of coherence function in terms of the measured transverse function.Having established these results, we discuss how to apply them to the simulation of co-PV devices.

PROBLEM STATEMENT
Spatial coherence is a measure for the correlation between optical fields in different locations of space.More specifically, we are interested in measuring the degree of spatial coherence, η ⃗ r 1 ; ⃗ r 2 ; ω, which gives a scalar description of the statistical correlation between the fields with frequency ω at locations ⃗ r 1 and ⃗ r 2 .In the following we will refer to η ⃗ r 1 ; ⃗ r 2 ; ω as the spectral degree of coherence [23].In general, a degree of coherence can be described between any pair of field components for any pair of points in space.A rigorous description of second-order coherence is given by a 3 × 3 covariance matrix known as the cross-spectral density (CSD) matrix W ⃗ r 1 ; ⃗ r 2 ; ω (see Mandel and Wolf [24], Sect.6.6.1).The CSD matrix is directly applicable to the numerical simulation of light propagating in a solar cell (see Implications).However, the spectral degree of coherence can be more easily measured and provides evidence for the structure of a theoretical CSD matrix.
The spectral degree of coherence can be defined in terms of the CSD matrix as [23] where Tr denotes the matrix trace.The two-slit apparatus illustrated in Fig. 1 provides a simple and robust measurement of η as defined by Eq. (1) (see Appendix A).We note that alternative definitions of the spectral degree of coherence exist that take all elements of W into account [25].However, the use of Eq. ( 1) is justified by the facts that direct and cloud-diffused sunlight are unpolarized [26] and therefore Eq. ( 1) represents the essential elements of W. Partially polarized skylight (blue sky) is discussed as a separate case in Appendix E.
For scalar fields, the van Cittert-Zernike theorem has been traditionally used to calculate the CSD of sunshine (see Born and Wolf [27], Sect.10.4.2).For electromagnetic fields, the general case can be calculated using an angular spectrum of plane waves propagating within a solid angle σ subtended by the source at the device plane.Under the assumption of angularly uncorrelated and unpolarized plane waves the CSD matrix is given by [28,29] where ⃗ E is the random electric field column vector; û − sin θ cos ϕx − sin θ sin ϕŷ − cos θẑ is the unit vector that specifies the propagation direction of a plane wave (θ and ϕ represent polar and azimuthal spherical coordinates, respectively); the vectors ĵj x; y; z are the unit column vectors along the coordinate axes (see Fig. 2(a)); dΩ≔ sin θdθdϕ is the differential solid angle; k≔ω∕c is the wavenumber; ⃗ r≔ ⃗ r 2 − ⃗ r 1 ; U 3 is the 3 × 3 identity matrix; aû; ω is, up to a constant, the average power spectral density per unit solid angle of waves traveling with frequency ω in the û direction; indicates complex conjugate; ⊺ indicates vector transpose; and the angle brackets indicate an ensemble average over monochromatic field realizations.Equation (2) immediately implies, under the given assumptions, that W ⃗ r 1 ; ⃗ r 1 ; ω W ⃗ r 2 ; ⃗ r 2 ; ω and that W ⃗ r 1 ; ⃗ r 2 ; ω is a function of ⃗ r alone such that we can write W ⃗ r; ω and η ⃗ r; ω in their respective places.For a very distant source, the function aû; ω is given by the power spectral density distribution of the source that is apparent from the origin (see Novotny and Hecht [30], Sect.3.4).
In general, the device plane shown in Fig. 2(a) can have an arbitrary orientation with respect to the source.We first consider the special case of a device plane that is transverse to a circularly symmetric source, that is, z in Fig. 2(a) passes through the center of the source.For this case we denote the spectral degree of coherence as η ⊥ , or the transverse spectral degree of coherence.Due to rotational symmetry, η ⊥ ⃗ r; ω is a function of r≔j ⃗ rj alone and can be written η ⊥ r; ω.Later, we will consider an arbitrarily oriented device plane and express η in terms of η ⊥ and orientational angles.
In the following we will distinguish between two different types of sources, direct sunshine and diffuse sunlight.The latter corresponds to a uniformly bright, cloudy sky.We model this situation with a spatially incoherent half-shell source whose radius is large, surrounding the device plane (see Fig. 2(b)).The origin of the device plane is placed at the center of the half-shell.The spectral degree of coherence in this case is given by (see Appendix B) [23,31] This result indicates that, for a spatially incoherent half-shell source, the spatial coherence length along any direction at the center of the shell is on the order of one wavelength of the light.Light illuminating the device plane is in general a mixture of direct and diffuse sunlight.We model this situation as the combination of a disk and a half-shell source, as shown in Fig. 2(b).Consequently, the spectral degree of coherence is given by (see Appendix C) where s d and s s represent the spectral densities in the device plane due to the disk and shell, η s is given by Eq. ( 3) and η d is the spectral degree of coherence in the disk-only case as defined by Eqs. ( 1) and ( 2).Introducing the parameter γ≔s d ∕s d s s , we can rewrite the equation above as In the next section we will use a simple experimental procedure to measure η in the transverse plane and compare the results with the theoretical model, Eq. ( 5).

EXPERIMENTAL RESULTS
We have fabricated a pair of non-parallel slits to measure the transverse spectral degree of coherence of i) direct sunshine, ii) light emitted by a solar simulator, and iii) diffuse sunlight (see Appendix A for materials and methods).i) Direct sunshine: Figure 3 shows the measured and calculated transverse spectral degree of coherence η ⊥ r; ω for direct sunshine with wavelength λ between points separated by a distance r.Both the amplitude and phase of η ⊥ are shown.The theoretical maps were calculated by applying Eqs. ( 1) and ( 2) to hyperspectral measurements of solar limb darkening [22] (see Appendix D).The excellent agreement of the two amplitudes is made clear by the cross sections, given in Fig. 4(a), along the lines indicated in Figs.3(a) and 3(c) corresponding to a wavelength of 510 nm.For the phase we obtain similar good agreement (Fig. 4(b)), aside from inconsequential offsets of integer multiples of 2π.
We emphasize that the measured and calculated amplitudes do not strictly agree with previous theoretical estimates based  on the uniform incoherent sphere model [17].The coherence amplitude function expected from a spatially uniform incoherent sphere source is shown in Fig. 4(a).The disagreement is minor; the estimate has an error of at most 10% for separations below 80λ and gets better for longer wavelengths.This result shows that solar limb darkening only weakly affects the coherence properties of sunlight at visible frequencies.Our measurements of η ⊥ given in Fig. 3 can be used to derive a measure for the coherence length.For example, defining the coherence length L ⊥ as the slit separation for which the amplitude of η ⊥ decays to 1∕2, we find that L ⊥ 80λ for direct sunshine.
ii) Solar simulator: The transverse spectral degree of coherence was also measured with a solar simulator source.Typically, such a simulator is used to test the performance of photovoltaic devices that are placed within a test area (device plane) where the power spectral density of the light approximately matches that of direct sunshine.It is evident that the measured η ⊥ of simulated sunlight, shown in Fig. 4(a), does not match the sunshine case.This serves to highlight the fact that the power spectral density at the device plane should not be the only consideration in solar simulator design.For solar cells whose performance depends on the coherence properties of light, a simulator should be used that renders both the correct power spectral density and the correct spatial coherence.
The requirements on such a simulator follow from Eq. ( 2): the source should not only have the same spectrum as the Sun but its size and shape should also be matched to render the same spatial coherence as sunshine.A hyperspectral set of images recorded from the point of view of the test area would appear identical to a hyperspectral set of images of the Sun, except perhaps for a global intensity scaling factor.As previously discussed, a uniform disk/sphere source approximates the spatial distribution of the Sun fairly well.Consequently, wavelength-dependent limb darkening can be safely dismissed in solar simulator design unless a very high degree of accuracy is required in measuring the device performance relative to that under direct sunshine.
iii) Diffuse sunlight: The measurements of diffuse sunlight are given in Fig. 5.In each case, the sky was completely and uniformly covered by clouds or fog in such a way that the position of the Sun was still discernible.Notably, the positions of the amplitude minima in the spectral degree of coherence remain unchanged under each measured weather condition while other aspects of the function change significantly.This fact is also borne out in the phase information as given in Fig. 4(b), where the steps in phase occur invariably at the same distances.Of interest is the behavior of the spectral degree of coherence relative to the direct sunshine case.From the inset in Fig. 5 it is noticeable that, in each diffuse case, the degree of coherence appears to be simply offset vertically from the direct sunshine case.This implies that the coherence functions in the diffuse cases can be described as fractions of the direct function, as implied by Eq. ( 5).
Indeed, the measurements in the diffuse case are well fitted by Eq. ( 5).The second term on the right-hand side of the equation falls off quickly and is negligible for r > 5λ, where The inset in (a) shows jη ⊥ j on a logarithmic scale.The measured and calculated curves relating to sunshine in both (a) and (b) correspond to cross sections along the respective lines given in Figs.3(a-d).Fig. 5. Amplitude of the transverse spectral degree of coherence (jη ⊥ j) at 510 nm for different weather conditions.The inset shows jη ⊥ j on a logarithmic sale.The direct sunshine case corresponds to the measured curve given in Fig. 4(a).
r j ⃗ r 2 − ⃗ r 1 j is the distance between points in the transverse plane.Then, ignoring the second term, Eq. ( 5) can be approximated by γη ⊥ r; ω.Thus, the simple relationship between the diffuse cases and the direct function becomes clear.This approximation was used with a value γ 0.45 in fitting the first fog case.The fit and measurement overlap significantly.
The second fog measurement was taken two minutes after the first using a slightly different system (see Appendix A).In this case, a peak-like feature related to the second term on the right-hand side of Eq. ( 5) becomes visible in the lowest slit separation region.This feature is significantly wider than expected by the second term.The discrepancy between the second fog measurement and the second term in Eq. ( 5) is due simply to the fact that the diffused light was not truly uniform.In reality, the brightness of the fog fell off for positions away from the Sun.This factor effectively narrowed the spatial distribution of the source and thus widened the part of the coherence function due to diffused light.However, even given this discrepancy, it is clear that the spatial coherence length is greatly reduced in the diffuse case as compared to the direct sunshine case.For example, using the same definition for the coherence length L ⊥ as before, we find for the second fog case L ⊥ 7λ as given by Fig. 5.This length is expected by Eq. ( 5) to be as low as λ∕3 in the extreme (γ 0) case.

ARBITRARILY ORIENTED DEVICE PLANE
We proceed by describing how, in the solar disk-only case, the cross spectral density W d ⃗ r 1 ; ⃗ r 2 ; ω and spectral degree of coherence η d ⃗ r 1 ; ⃗ r 2 ; ω between points in an arbitrarily oriented device plane can be given in terms of the transverse functions W ⊥ ⃗ r; ω and η ⊥ r; ω, where ⃗ r≔ ⃗ r 2 − ⃗ r 1 and r≔j ⃗ rj.We assume a spatially incoherent, broadband source that appears circularly symmetric (but not necessarily uniform) from the origin.We define the set of rectangular coordinate axes x; y; z such that ⃗ r 1 and ⃗ r 2 lie in the x-y plane and the center of the source is located at position ⃗ r 0 , with polar angle θ 0 and azimuthal angle ϕ 0 (see Fig. 2(b)).We assume that the source is far from the device plane such that j ⃗ r 0 j is much larger than r.Now, by applying the appropriate rotation matrix to the cross-spectral density in the transverse case, we can find the cross-spectral density and spectral degree of coherence between any two points in the device plane.Let Mθ 0 ; ϕ 0 ≔ R z ϕ 0 R y θ 0 where R i φ is a rotation about the i axis by angle φ as given by the right-hand rule.Then, as shown explicitly in Appendix D, W d and η can be expressed as and where p≔ sin θ 0 cosϕ 0 − ϑ, ϑ≔ arctan ⃗ r • ŷ; ⃗ r • x, and arctany; x is the four-quadrant inverse tangent.The functions W ⊥ ⃗ r; ω and η ⊥ r; ω describe, respectively, the cross-spectral density and spectral degree of coherence in the transverse plane (the θ 0 0, sunshine-only case), the latter of which is given in Fig. 3.We note that the degree of coherence between points in the transverse plane is a function of only ω and the distance r between the points; fields of this type are known as Schellmodel fields [32].
Coherence functions in the general case: We now consider a device plane that is oriented obliquely to the transverse plane and illuminated by both diffuse and direct sunlight.As before, we model this situation as the combination of a disk and a half-shell (see Fig. 2(b)).
By the additive nature of the cross-spectral density for mutually-incoherent sources, and by Eqs. ( 3), ( 5), (6), and (7) we find and where W s is the cross-spectral density in the device plane due to the uniform half-shell (explicit expressions for W ⊥ and W s are given in Appendix D).Equations ( 8) and ( 9) are the main results of this section.They state that the cross-spectral density and spectral degree of coherence between two points in a device plane can be described by their relative distance (r), their orientation with respect to the Sun (θ 0 ; ϕ 0 ; ϑ), the transverse coherence functions of direct sunshine (W ⊥ , η ⊥ ), the frequency (ω), and the ratio (γ) between the spectral densities in the device plane due to the Sun and a uniform cloud cover or fog.Using Eq. ( 9) together with our experimental results for η ⊥ we can derive the spectral degree of coherence in an arbitrarily oriented device plane.

IMPLICATIONS FOR THE SIMULATION OF COHERENT PHOTOVOLTAICS
In general, the electromagnetic field impinging on a device is not a plane wave but rather it fluctuates randomly in time and space.The average response of a device is thereby an ensemble average of responses to random field realizations.In simulating devices, it is not enough to consider the response under plane wave illumination; a rigorous treatment requires that they be simulated under realizations of the full field.The cross-spectral density matrix W is a particular measure for the statistics of these realizations.In fact, W as given by Eq. ( 8) can be used to directly generate realizations of the field [33].Another method for generating realizations, as implied by Eq. ( 2), is to use a Monte Carlo simulation in which plane waves with random polarization, phase, and wave vector are coherently superposed.
For devices with a linear response, the requirements for realistic simulations become more relaxed.For some ensemble averaged observable Bω (absorption, external quantum efficiency, etc.), we can write [34] Bω Z σ aû; ωχû; ωdΩ; (10) where a, σ, and dΩ are as before (see Problem Statement and Appendix D). χ is the generalized linear susceptibility that corresponds to the observable B and describes the response of the system to unpolarized plane waves traveling in the û direction with frequency ω.Simply, the device can be simulated under illumination by plane waves of different wave vectors and polarizations and the responses to these waves can be linearly summed with the proper weight to determine the overall response.Implementation of Eq. ( 10) is therefore compatible with simulation modalities that consider plane wave illumination.
It is also possible, from knowledge of the dielectric susceptibility of the scattering medium and the CSD matrix W of the incoming light, to determine the power spectral density distribution of light within a device.Such a calculation would involve propagating W into the device [13,35] and using the basic equations for scattering in random media (see Mandel and Wolf [24], Sect.7.6) to determine the power spectrum.

CONCLUSIONS
We have used a double slit aperture, in which the slit separation slowly varies, to measure the transverse spectral degree of coherence of direct sunshine, sunlight diffused by clouds or fog, and light emitted by a solar simulator.The results in the direct case were compared to those expected by theory as applied to hyperspectral image data and found to agree.The results are precise enough to distinguish the true transverse spectral degree of coherence of direct sunlight from the uniform disk approximation.It was shown that, in general, the uniform disk approximation is very close to the measured function.
We have shown that the spectral degree of coherence in a plane that is arbitrarily oriented with respect to a circularly symmetric source, such as the Sun, can be derived given the transverse spectral degree of coherence.A simple model was also developed to describe the spectral degree of coherence under overcast skies.This model was shown to accurately fit the diffuse sunlight measurements in the transverse case.Further, an intuitive set of requirements was provided for the design of a solar simulator with spatial coherence properties similar to those of the Sun.
We have also described a method by which simulations of co-PVs can include the effects of partial coherence.For linear devices, an ensemble average of plane wave responses is sufficient.For non-linear devices, a rigorous treatment includes an ensemble average of the device response to random realizations of the field.We have also shown that, for devices with effective dimensions on the order of a few wavelengths or less, direct sunlight can be approximated as spatially coherent.However, this is not true for overcast skies where the coherence function falls off quickly within these dimensions.This implies that future co-PV designs should take into account the spatial coherence of sunlight, especially if the components are expected to be used under cloud cover.Additionally, co-PVs should be tested under a spatial coherence standard such that fair comparisons may be made between different devices.

APPENDIX A: MATERIALS AND METHODS
Measurement apparatus: The measurement procedure has been described previously in Ref. [21].The measurements were performed using a stock Nikon DX-format camera sensor placed at a distance of 6.19 cm behind non-parallel slits, as shown in Fig. 1.The slit separation varies from 10 to 300 μm.6.19 cm is sufficient to be in the far-field of the two slits together but still in the near-field of the long dimension of each slit.Consequently, diffraction can be approximated as taking place only along the direction that connects the two slits and the resulting interference pattern describes an ensemble of simultaneous Thompson-Wolf experiments [36].The spectral degree of coherence is recovered by combining information from the interference pattern with power spectra as measured by the same device.Technically, Ref. [21] provides a method to measure the spectral degree of coherence μ ⃗ r 1 ; ⃗ r 2 ; ω of a scalar field.However, under comparison with Ref. [23], it is clear that the same method measures the spectral degree of coherence η ⃗ r 1 ; ⃗ r 2 ; ω of an electromagnetic vector field.Direct sunlight: Our measurement of direct sunlight was performed by recording only two images, one to measure the spectrum and one to measure the spectral degree of coherence.The first image, to measure the spectrum, is of an interference pattern created by illuminating the slits with sunshine reflected from a small, distant mirror.The second image is of an interference pattern created behind the aperture by direct sunshine.By using the first image to normalize the second, as described in [21], the transverse spectral degree of coherence can be measured.The measurement was made on 22 March 2013 in Zurich, Switzerland.A solar semi-diameter of 16.1 minutes, corresponding to 15 March 2013 [37], was used in the calculation of the theoretical maps.The uniform disk source calculation was also made assuming a 16.1 minute semi-diameter.
Solar simulator: The simulator consists of a xenon arclamp with air-mass 1.5 optical filters.Our measurement method for the solar simulator source is similar to that of the direct sunshine case.The coherence image was recorded at the device plane and the spectral image was recorded at a distance of 2 m from the source.Further, since the simulator source subtended a significant solid angle at the device plane, an extra post-processing step was required beyond those described in [21]: the spectral image was deconvolved from the coherence function image.This allowed for the retrieval of the pointspread function that, when projected along one dimension and convolved again with the spectrum image, gives the result that would be expected in the limit of very long slits and a Schell-model coherence function.The lowest spatial frequency components of the point-spread function were obscured in the particular images that were recorded.These components were recovered by extrapolation using a polynomial fit of higher frequency components.
Diffuse sunlight: For diffuse sunlight we chose a simpler measurement procedure.The transverse spectral degree of coherence corresponding to a wavelength of 510 nm was obtained by recording only a single image.The slits were placed behind a narrow-band-pass optical filter centered at 510 nm and the equal time degree of coherence j r was found using the method described in [21] and directly equated to the spectral degree of coherence via an equation first derived by Wolf [38]: μr; ω 0 j r, where ω 0 is the center frequency of the optical bandpass filter.This method was used, as opposed to that used for measuring direct sunshine, because it requires only a single image to be recorded and avoids the complication of a time-varying spectral density of the source.In the second fog measurement, a 4x telescope was used to decrease the apparent size of the Sun from the point of view of the slits.The use of a telescope in this fashion is equivalent to a reduction in the slit-to-slit separation by a factor of 4. Consequently the minimum available separation was reduced from 10 to 2.5 μm.The rotation of a vector field ⃗ F ⃗ r using a rotation matrix R is given by R ⃗ F R ⊺ ⃗ r.Since W ⃗ r 1 ; ⃗ r 2 ; ω h ⃗ E ⃗ r 1 ; ω ⃗ E ⊺ ⃗ r 2 ; ωi, the general solar disk case W d is found by applying Mθ 0 ; ϕ 0 to Eq. (D4), after noting that the

Fig. 1 .
Fig. 1.Illustration of the experimental measurement procedure.Broadband light from the source passes through a pair of non-parallel slits creating an interference pattern on a detector.

Fig. 2 .
Fig. 2. Illustration of the theoretical model.(a) An plane wave propagating along the û direction is emitted from the corresponding position on a distant source σ.Positions ⃗ r 1 and ⃗ r 2 lie in the device x; y plane.(b) A cutaway illustration of a combined disk-shell source.The device plane is illuminated by diffuse light (half-shell source) with spectral density a s û; ω and by direct sunlight (disk source centered at ⃗ r 0 ) with spectral density a d û; ω.The front portion of the half-shell has been removed to reveal the origin.

Fig. 3 .
Fig. 3. Measured and calculated transverse spectral degrees of coherence η ⊥ of direct sunshine.Amplitude (a) and phase (b) of the transverse spectral degree of coherence measured by the double slit apparatus.Amplitude (c) and phase (d) of the spectral degree of coherence calculated by Eqs.(1) and (2) as applied to hyperspectral images of the Sun.

Fig. 4 .
Fig. 4. Transverse spectral degree of coherence η ⊥ at 510 nm for various sources.The amplitude of η ⊥ is shown in (a) and the phase in (b).The inset in (a) shows jη ⊥ j on a logarithmic scale.The measured and calculated curves relating to sunshine in both (a) and (b) correspond to cross sections along the respective lines given in Figs.3(a-d).