We describe a procedure to estimate the calibration function for an experimental setup that contains a transmitting diffuser whose bidirectional transmittance distribution function for light transmitted in normal direction is nearly independent of the incidence angle. This type of diffusing screen may be used in experimental setups to measure the irradiance distribution in the focal plane of concentrator optics that have large angles of incidence in the focal plane. It is shown that the influence of this screen and of the remaining components on the irradiance distribution may be described empirically by a Gaussian function and thus may be corrected for. Furthermore, it is demonstrated that the correction is necessary to avoid an underestimation of the concentrator optics’ ability to concentrate the incoming radiation. © 2015 Optical Society of America OCIS codes: (100.0100) Image processing; (100.3010) Image reconstruction techniques; (220.1770) Concentrators; (220.4840) Testing; (230.1980) Diffusers. http://dx.doi.org/10.1364/AO.54.000927

1. Introduction

In concentrator photovoltaics (CPV) the incoming sunlight is focused via large primary concentrator optics onto small high-efficiency solar cells or the entry aperture of a secondary optical concentrator placed on the solar cell. The irradiance distribution (unit: W∕m2 ) on this target plane generated by the primary concentrator influences the conversion efficiency from sunlight to electrical energy [1]. Therefore a detailed knowledge of it is of high interest for the development of cost-effective CPV systems. Ideally, the irradiance distribution can be calculated by raytracing simulations neglecting, e.g., shape deviations, surface errors, and diffraction effects. In order for a CPV system to be cost-effective, concentrator optics are usually produced by low-cost mass manufacturing processes. Therefore, the simplifications commonly 1559-128X/15/040927-07$15.00/0 © 2015 Optical Society of America

made in the ray-tracing simulations are not necessarily justified. On the other hand, a quick and reproducible measurement of the irradiance distribution is not yet state of the art. Descriptions of experimental setups investigating the irradiance distribution in the focal plane may be found in [2–9]. Several of the authors [4–9] use a transmitting diffuser at the position of the target plane. The diffuser scatters and diffusely transmits the incoming light and may thus be imaged from the rear side by a camera. The camera signal may be processed to obtain an estimate of the irradiance distribution. The disadvantage of this technique is that the measured data are biased especially by the diffuser’s influence. We presented results obtained with this experimental method in [8] and described a model to obtain a calibration function that will give satisfactory results if the dynamic range of the experimental setup does not allow us to resolve a signal more than two orders of magnitude below the maximum concentration. With improved accuracy and 1 February 2015 / Vol. 54, No. 4 / APPLIED OPTICS

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dynamic range of the experimental setup, a more precise calibration function needs to be determined. We present a method to determine a calibration function that is adjusted to the expected angles of incidence on the diffuser. The method provides the possibility for an in-depth analysis of the measurement uncertainty. An example of this analysis is presented. The paper is organized as follows: in Section 2 we review the experimental procedure for the characterization of the concentrator optics via a diffuser and the problems inherent to it. In Section 3 we describe our experimental approach and an analysis of the experimental uncertainties. In Section 4 the experimental results and the determination of the calibration function are presented together with a brief discussion of the findings’ implications. 2. Experimental Concept or the Measurement of the Focal-Plane Irradiance Distribution Using a Transparent Diffuser

As an example of an experimental setup for the measurement of the focal-plane irradiance distribution using a transparent diffuser, we briefly review the one we presented in [8]. A schematic drawing is shown in Fig. 1. The light source is monochromatic (λ 622 nm) and has a sun-like divergence half-angle (4.58 mrad step-function); more details may be found in [10]. The diffuser is irradiated on one side by light reflected off the sample mirror and imaged by a CCD camera from the other side resulting in a matrix of gray values (measurement image, M ij ). A second image (reference image, Rij ) is taken in which the sample is replaced by a flat mirror (not shown in Fig. 1), which is slightly tilted so that it simply redirects the incoming light to the diffuser without focusing it. By dividing the recorded gray values of the measurement image by the ones of the reference image, one obtains a so-called concentration matrix (Cij , dimensionless). With this concentration matrix it is possible to calculate the irradiance distribution in the target plane simply by multiplying the concentration matrix’s values by the irradiance at the entrance aperture of the concentrator optics.

Fig. 1. Schematic representation of the experimental setup to measure the irradiance distribution in the focal plane of a parabolic mirror. The incoming and reflected light (solid, red arrows) is scattered by the diffuser (dashed, red arrows), which is imaged by an objective lens onto a CCD array. An aperture diaphragm may be used to analyze a section of the mirror only. 928

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The choice of an appropriate transmitting diffuser for this application is not trivial. Transmitting diffusers are usually used to produce an even illumination or as rear projection screens [11]. When used as a rear projection screen (similar to what is done here) the light usually hits the diffuser almost vertically, see e.g., [12] for commercially available rear projection screen technology. On the contrary, the target plane of primary concentrator optics may be irradiated under incidence angles of up to φ 45°. This demands that the bidirectional transmittance distribution function (BTDF) for light transmitted in normal direction (i.e., toward the objective lens) should be independent of the incidence angle φ. This may be achieved by a thick volume diffusor. In that case the drawback is the reduction in spatial resolution that is addressed in this paper. We chose a Zenith Polymer White Diffuser, which is an approximately 0.1 mm thick volume diffuser consisting of a compressed and sintered polymer composition, mainly Polytetrafluoroethylene [13]. In Fig. 2 its behavior under irradiation is shown schematically. For angles of incidence φ of 0° and 25°, the radiation transmitted by the diffuser shows an almost Lambertian distribution. The values of the cosine corrected BTDF at 20° exit angle are reduced by about 5% relative to the BTDF values at 0° exit angle for both angles of incidence. Ideally, the diffuser scatters the incoming light so that the BTDF for light transmitted in normal direction does not depend on the incidence angle. The radiant emittance (unit: W∕m2 ) at the top should be identical to the irradiance (W∕m2 ) at the bottom. In reality the total emitted radiant flux (W) at the top is smaller than the total incoming radiant flux (W) at the bottom due to the nonzero diffuse reflectance of the diffuser. The reflected light has to be taken into account as extraneous light in the experimental setup, which may distort the measurement. Furthermore, in reality, the BTDF for light transmitted in normal direction decreases slightly with an increasing angle of incidence (up to φ 45°). We measured this behavior separately with a photo-goniometer by varying only the incidence angle φ and keeping the

Fig. 2. Schematic representation of the diffuser’s behavior. Left, ideal behavior. Incoming ray bundles (arrows from below) are completely transmitted independently of their position and angle of incidence φ, scattered and transmitted according to Lambert’s emission law. Right, real behavior. Due to material inhomogeneity (thickness variation, partial crystallinity, etc.) the BTDF depends on the position and angle of incidence φ. The ray bundles are transmitted differently depending on φ and additionally broadened.

Fig. 3. Diffuser’s normalized BTDF in normal direction versus incidence angle φ. The BTDF is normalized to its value at φ 0°. The measurement uncertainty is estimated to be 1%.

detector normally aligned to the diffuser surface (see Fig. 3). In this measurement we ensured that the opening angle of the detector was similar to that of the objective lens (approximately 4°). In this study the largest incidence angle at the diffuser is φ 20°. For this incidence angle the value of the BTDF decreases by about 2% relative to the BTDF value for φ 0°. This shows that in our case this deviation from the desired behavior has little influence but becomes more important for larger angles of incidence. Nevertheless, we took these data into account when performing ray-tracing simulations to find the calibration function that is described in Sections 3 and 4. Each simulated ray was weighted according to the measured relative decrease of the BTDF depending on its incidence angle φ on the diffuser. Furthermore, in reality, the radiant emittance at the top of the diffuser has a larger spatial extension due to the scattering inside the diffuser (called ray broadening from here on). From the ray broadening it follows that the diameter of a focal point appears larger in the measurement than it actually is. In the following we describe how we quantify this effect to correct the measurements.

ray-broadening effect is likely to depend on the incidence angle (see Fig. 2), a slight dependence of k on the distribution of incidence angles is expected. Apart from this, the calibration function k is assumed to be independent of the concentrator under test. The calibration function k also contains the point-spread function of the objective lens. The measurement setup for the determination of the calibration function k is shown in Fig. 4. The light source, the objective lens, and the camera are the same as the ones used for the measurement of concentrating mirrors (see Fig. 1). Thereby, it is ensured that the results are comparable. The incoming light is focused by the lens instead of a mirror. This is done first because it is easier to find off-the-shelf point focusing lenses with the desired f-number (about 1.4 in our case) that have a sufficiently high surface quality. Second, this setup is easier to align because it does not contain the tilted mirror. A. Apertures and Collecting Lens

In this study the dimension of the aperture as well as the focal length of the collecting lens are chosen so that the distribution of incidence angles approximately matches that of the concentrating mirrors to be tested. This prevents systematic errors due to a possible dependence of k on the distributions of incidence angles. In our case the aperture is square because most CPV systems use several small concentrator optics placed on a single, large tracking unit to follow the sun during the day. To avoid gaps between the optical concentrators their apertures are usually square instead of circular, saving costly area on the tracking unit. The largest incidence angle on the diffuser is about 20°, and the mean value of the incidence angles across the full aperture is about 13°. The opaque disk centered over the collecting lens masks

3. Determination of the Diffuser’s Ray-Broadening Effect and Measurement Uncertainties

To estimate the diffuser’s ray-broadening effect, we measured the concentration matrix (Cmeas ) of a well-known optical concentrator (collecting lens, see below). Additionally, we generated a concentration matrix by a ray-tracing simulation (Cid ). It is assumed that a two-dimensional convolution of this simulated concentration matrix Cid with a suitable calibration function k will approximate the measured concentration matrix. To obtain the calibration function k, the relation Cmeas ≈ k ⊗ Cid ;

(1)

is solved via a least square fit using a suitable function k with a set of fit parameters. Since the

Fig. 4. Schematic representation of our experimental setup. In the small figure on the right the ray-broadening effect due to the scattering inside the diffuser is schematically shown. The z-position is the distance between the apex of the lens pointing toward the diffuser and the surface of the diffuser. 1 February 2015 / Vol. 54, No. 4 / APPLIED OPTICS

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small incidence angles in the same way the setup shown in Fig. 1 does. It is fixed on an antireflective coated cover glass (transmittance T g 0.981 0.005). Any surface error or shape deviation of the lens may lead to an overestimation of the raybroadening effect. Thus, one has to use a lens whose surface errors and shape deviations alter the expected focal-plane irradiance distribution to an amount that is small compared to that caused by the ray broadening of the diffusing screen itself. Therefore we chose a commercially available, grinded and polished biconvex lens made from optical crown glass. Inspection of the lens’ surface by a light-optical microscope did not uncover any defects that would alter the irradiance distribution significantly. B.

Data Acquisition and Correction

Concentration matrices are measured for six different, approximately equally spaced z-positions of the diffusor in the range 1% of the focal length (see Fig. 4) to estimate the sensitivity of k on the exact z-position. Every concentration matrix is composed of three images with different exposure times in order to increase the gray values from the low irradiated parts of the target plane to the linear range of the camera. The reference image is taken by removing the collecting lens and the opaque disk centered over the collecting lens. An exemplary result for a concentration matrix is shown in Fig. 5. For easier comparison the two-dimensional concentration matrices are azimuthally averaged, resulting in a radial distribution, according to Crk

1 · Nrk

X

rk −Δr 2≤

Cij p 2 2

:

C.

Analysis of Measurement Uncertainties

An analysis of the uncertainties is necessary to assess the quality of the fit of the ray-tracing simulations to the measured data. The aim is to calculate realistic values for the radial distribution’s error bars, which can be used in the fitting procedure presented in Section 4. As explained in Section 2, the measurement equation can be stated as Cij M ij ∕Rij ;

(3)

with variables explained in Section 2. The following sources of uncertainties were found to have a significant influence. The light source’s irradiance varies according to Kiefel et al. [10] by about 1% relative to the mean irradiance across the light source

(2)

xi yj 1.7 mm some minor extraneous light is observable. From the comparison between measurement and simulation, it is clear that the qualification of concentrator optics using this kind of transparent diffuser will lead to an underestimation of the optics’ ability to concentrate the light because if the ray broadening is not corrected for, the area of the focal spot is enlarged. This result is likely to be valid at least to some extent for any transparent diffuser, and one has to check whether the ray-broadening effect reported here is negligible or not. In the example shown in Fig. 6, the maximum concentration (at r 0 mm) would be underestimated by approximately 20%. The radius of a circle around the origin that intercepts 90% of the total detected radiant energy would be overestimated by about 17%. Both are calculated relative to the values obtained by raytracing simulations. Different model functions, including the model we presented in [8], were fitted by a least-square-fit to the data, according to Eq. (1), to find the calibration 932

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function k. To assess the quality of the fit, the radial distributions were compared. The best fitting model function found in this investigation is a twodimensional, rotationally symmetric Gaussian function where σ (unit: mm) is the free parameter of the fit 1 1 x2 y2 kσ · exp − : (6) 2πσ 2 σ2 The result using the data shown in Fig. 6 is σ 0.105 0.001 mm. We performed the same procedure for six different z-positions and obtained, by averaging the six resulting values, σ 0.106 0.003 mm showing a good agreement among the fit results. The low deviation among the results for the six z-positions indicates that kσ may be used to correct measurements of mass manufactured concentrator optics, even if their focal plane’s position cannot be accurately determined because of possible shape deviations. As the resulting function kσ is a calibration function (including contributions from effects described in Section 3) and not a model of the physical mechanism of light scattering inside the diffuser, we refrained from performing an χ 2 -test of the fit. The fit describes the measured data accurately enough to correct any measurement of unknown concentrator optics generating a similar distribution of incidence angles. The determination of the calibration function presented here is fast, simpler than a direct deconvolution approach, takes into account the systematic errors as well as noise in the measured data, and gives sufficiently accurate results. However, since we know that the calibration function is a Gaussian, the correction of measured data from concentrator optics with unknown shape deviations may be done numerically by a deconvolution approach [14]. If one would rather seek to obtain an analytical expression for the influence of the shape deviations on the irradiance distribution, one may extend Eq. (1) by a function g, which describes the influence of the shape deviations on the irradiance distribution in the focal plane Cmeas ≈ kσ ⊗ Cid ⊗ g:

(7)

Cid is found by a ray-tracing simulation assuming error free optics, and kσ is the calibration function found here. The function g may depend on a suitable set of fit parameters, which may be found in exactly the same way as the parameter σ. This approach seems sufficiently general and repeatable and has already been presented in [8] and similarly in [15]. Yet, it should be noted that the calibration function and the fit parameter σ only apply to the specific diffuser and concentrator optics investigated here. It should be repeated for other diffusers and other distributions of incidence angles from different concentrators. 5. Summary

The ray-broadening effect of a transparent diffuser has been assessed. In principle, this effect occurs

in any experiment using a transparent screen for spatially resolved data acquisition. We have shown experimentally that the transparent diffuser placed at the target area of concentrator optics changes the measured irradiance (concentration) distribution in a way that will lead to an underestimation of the concentrator optics’ performance if this ray-broadening effect is not accounted for. In the presented investigation the ray-broadening effect reduced the maximum concentration by about 20% and increased the radius of the circle that intercepts 90% of the total detected radiant energy by 17%. We conducted an experiment using a lens of high surface quality and presented a data-evaluation procedure to estimate the influence of the transparent diffuser under a given distribution of incidence angles. We performed an uncertainty analysis and showed that a convolution of the simulated concentration distribution and a Gaussian calibration function kσ can describe the measured data accurately enough. The resulting value of the fit parameter found for our particular setup is σ 0.106 0.003 mm. The basic approach presented here can be generally used and may be applied to other setups and similar cases, e.g., a point focusing Fresnel lens with different angles of incidence on the diffuser. It may even be helpful in other applications where diffusing screens are used for spatially resolved measurements of irradiance. References 1. P. Benítez and J. C. Miñano, “Concentrator optics for the nextgeneration photovoltaics,” in Next Generation Photovoltaics, A. Martí and A. Luque, eds. (IOP, 2004). 2. P. Nitz, A. Heller, and W. J. Platzer, “Indoor characterisation of Fresnel type concentrator lenses,” in Proceedings of the 4th International Conference on Solar Concentrators for the Generation of Electricity or Hydrogen, El Escorial, Spain, 2007. 3. R. Herrero, D. C. Miller, S. R. Kurz, I. Antón, and G. Sala, “A novel scanning lens instrument for evaluating Fresnel lens

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