1. Introduction

Surface texturing of optical sheets is used for reflectance reduction, absorptance or transmittance enhancement or for light redirection [1–6]. In the special case of silicon solar cells, surface texturing significantly reduces the number of photons lost due to surface reflection and further enhances absorption because of improved light trapping properties. Depending on the feature size, shape and arrangement of the texture, the coherence of light must be taken into account in some cases and can be neglected in other cases. Consequently, a large variety of theoretical optical methods exists that are tailored to specific surface morphologies. Textures that can be described by geometrical optics, such as pyramidal textures or isotextures, are modelled using ray tracing [7–10]. Scattering textures are described using analytical models [11, 12] or by statistical methods [13] and grating textures are described by rigorous wave optical methods such as finite difference time domain (FDTD) [14, 15] or rigorous coupled wave analysis (RCWA) [16]. Each of these methods is specific as to which optical properties it can accommodate and which texture dimensions it can describe efficiently. Modeling of other texture dimensions can be very tedious or even impossible with another method. Therefore, it is inherently difficult to model sheets that feature textures operating in different optical regimes with one simulation method – a task which is becoming increasingly important in the context of high-efficiency photovoltaics. One example is a silicon wafer solar cell with a pyramidal front surface texture and a rear side scattering texture or a rear side grating. Such a system can in principal be simulated using wave-optical methods. This is computationally very demanding in two dimensions and unfeasible in three dimensions [17].

To solve this problem Rothemund et al. [18] presented a method that uses a combination of RCWA and ray-tracing. A bidirectional reflectance function for one surface, containing a diffractive rear side grating, is calculated via RCWA. Absorption within the solar cell with front side pyramidal texture is afterwards calculated via ray-tracing. Their method has so far only been used for pyramidal front side textures and binary rear gratings. The introduction of a different kind of surface texture or a variation of other parameters like the sheet thickness can still be very time consuming, as a complete re-calculation is required. A. Mellor et al. proposed a matrix based formalism in which the absorption of a planar silicon sheet with rear side grating is described [19, 20]. The matrix entries are calculated via RCWA and represent a reflection distribution function. Calculation of the total absorption of the system is done via matrix multiplications, which describes the non-coherent light propagation inside the optical sheet. However, this method is generally restricted to systems with a finite number of diffraction orders. This is a correct description either for one surface with a grating and one planar surface or for two gratings with one period being a multiple of the other one.

To overcome the mentioned limitations to certain combinations of textures, we introduce the OPTOS (Optical Properties of Textured Optical Sheets) formalism for calculating the optical properties of an absorber with two arbitrary surface textures with comparatively low computational effort. Arbitrary light path directions that occur during the simulation are binned to a previously defined angle discretization. Surface interactions as well as light propagation inside the textured optical sheet are simulated only for a discrete set of angles. This allows a description of every surface by means of a redistribution matrix that is specific to a surface and needs to be calculated only once for each surface. These matrices describe the redistribution of light between the pre-assigned discrete angles. The results for front and rear side redistribution are then coupled non-coherently using a propagation matrix that accounts for the light propagating through the absorber. Thereby, the total absorption and a one-dimensional absorption profile of a sheet with arbitrary surface textures can be obtained via matrix multiplications.

The main focus of the first part (section 2) of this work is a thorough description of the OPTOS formalism. Afterwards exemplary calculations of redistribution matrices for different surface textures are presented (section 3). The calculated redistribution matrices are used to evaluate the formalism and compare OPTOS to other simulation techniques and optical measurements (section 4). After this validation the same matrices are used for the simulation of a silicon sheet with surface textures of different length scales at front and rear side (section 5). To further demonstrate the versatility of the formalism a thickness variation of this sheet is carried out. This can be done with low computational resources as for different combinations of front and rear surfaces and also for different cell thicknesses the same redistribution matrices are used.

2. Mathematical formulation of matrix based absorption calculation

2.1 Light field description via angular power distribution vector

The optical systems discussed in this paper consist of a thick sheet with an arbitrary front and rear surface texture as depicted in Fig. 1. When light impinges the front surface, it is redirected, either by single- or multiple reflection(s), diffraction or scattering. The redistribution of power, which happens as well for all further surface interactions, is in general given by a continuous function over the polar- and azimuth angle.

 

Fig. 1 Light propagation in a continuous medium with textured surfaces on both sides. Incoming light is divided into different channels as depicted in (a). The power fractions in these channels are described by a vector $v{\text{'}}_0$. Before the reflection (and redistribution) at the rear side it is called v₁, after reflection $v{\text{'}}_1$ and so on. The reflectance and redistribution at the surfaces is described by the matrices (B) and (C). The bulk propagation, where no redistribution but only absorption occurs, is described by the propagation matrix (D).

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In our formalism, the angle space is divided into discrete angle channels. The power distribution inside the textured sheet can therefore be represented by a vector with discrete elements. Each element contains the power fraction distributed into a certain solid angle. In the case of a grating, a convenient angle discretization could be defined by the different discrete orders of diffraction as done by Mellor et al [19, 20]. For genuinely continuous distributions, as for example provided by scattering surfaces, the angle discretization is a more elaborate process and will be discussed in more detail in section 2.7.

Assuming a discretization into n polar angles θ_i and m azimuth angles φ_i, there are in total $n\cdot m$channels in which light is allowed to propagate. The distribution of power in all channels of the system can then be described by the vector v consisting of $n\cdot m$entries while every entry contains the fraction p(θ_i, φ_j) of incoming power. Note that in principle the number of azimuth angles could be different for each polar angle. For the sake of simplicity we use the same set of azimuth angles here for each polar angle.

2.2 Light propagation via propagation matrix

The optical systems discussed in this paper consist of an optically thick sheet, i.e. light propagation in the sheet is treated non-coherently. Propagation of light in a homogeneous absorbing medium leads to an exponential decrease of the power in the different channels according to Lambert-Beer’s law. The multiplication of every vector entry with the corresponding exponential function can be represented by a multiplication of vector v with diagonal matrix

In this matrix, α is the absorption coefficient, d the thickness of the optical sheet and θ_i the polar angle of the respective channel. The dimension of the matrix is $(n\cdot m)\times (n\cdot m)$. For each of the n polar angles the m azimuth angle entries of the matrix are identical.

2.3 Surface interaction via redistribution matrices

A surface interaction of the light leads to redistribution of power between the different channels and possibly also to a power loss, e. g. via transmission or absorption in an extended three dimensional surface structure. The redistribution itself depends on the actual surface under consideration but it can again be represented by a matrix multiplication with a $(n\cdot m)\times (n\cdot m)$-matrix. As this matrix describes the redistribution of light at a surface, we will refer to it as redistribution matrix. The entries of the redistribution matrix describe the fraction of light being transferred from one angle channel $({\theta }_i,{\phi }_j)$ to another channel $({\theta }_k,{\phi }_l)$. In Eq. (3) the scheme of the redistribution matrices B and C, for front and rear side respectively, is shown.

The use of redistribution matrices is very similar to the concept of reflectance distribution functions in ray tracing formalisms. As power can, in general, be redistributed from any channel to any other channel, these matrices could be fully occupied. Due to energy conservation, the sum of each column has to be smaller than or equal to one. If it is equal to one the matrix is stochastic. This means that light is neither coupled out nor absorbed by the surface texture but everything is reflected.

2.4 Simulation procedure

When light impinges the front surfaces of the system the transmitted power is divided into the different channels as depicted in Fig. 1 (a). The vector $v{\text{'}}_0$ in Fig. 1 (b) contains the information about the power distribution after this first surface interaction and is used as starting point for the formalism.

As described above the propagation through the optical sheet including the angle dependent absorption can be obtained by a multiplication of vector v₀ with the propagation matrix D. The power distribution at this point is described by the vector v₁. Similarly the rear side surface interaction, the backward propagation and the subsequent front side surface interaction can be described by matrix multiplications with matrices C, D and B, respectively. This leads to the power distribution in vector $v{\text{'}}_2$. The primed vector signs always indicate, that they describe the system directly after a surface interaction.

In this notation all vectors v can be calculated by the following matrix products starting from vector v₀. The remaining total power fraction P_i can be obtained by summation over all vector entries. Note, that this is not the sum over the index i, which is related to the path number.

The surface interaction and propagation with absorption can be repeated via multiplication of the corresponding matrices until no significant fraction of the incoming power is left in any channel. At this point all relevant information is contained in power distribution vectors v_i and $v{\text{'}}_i$.

2.5 Determination of total absorptance

A quantity of special interest is the total absorptance inside the optical sheet. It can be calculated by an iterative procedure based on all the v_i-vectors described above. The absorbed fraction of incoming power Abs is equal to the sum of the power loss during the transition of the optical sheet from rear to front surface, Abs_up, and from front to rear surface, Abs_down. The absorbed power fraction during such a transition is equal to the difference between the total power fraction at the beginning (directly after a surface interaction) and the total power fraction at the end of the path (directly before a surface interaction). The summation of all of these components leads to the total absorptance Abs:

This iterative approach requires the calculation of the vectors v_i up to a certain path number i_max. The required number i_max depends on the strength of absorption and escape losses. In weakly absorbing media that trap the light effectively, e. g. a cavity, i_max becomes very large. In this case, the iterative calculation can be replaced by a mathematically more appropriate and elegant approach using the Neumann series (geometric series for matrices). Using the Neumann series allows the calculation of the total absorption as described by the following derivation, where I stands for the unity matrix.

Conservation of energy leads to the fact that $\Vert B\Vert \le 1$ and $\Vert C\Vert \le 1$ with respect to the column-sum norm, whereas for an absorption coefficient α > 0 $\Vert D\Vert <1$ . This means, for a matrix T, which is the result of multiplications between matrices D, B and C, that the Neumann series ${\displaystyle {\sum }_{i=0}^{\infty }{(T)}^i}$ converges and the matrices $(I-T)$ are invertible [21]. For this geometric series approach only four matrices have to be calculated and inverted. The result is exact and corresponds to the iterative approach for the limit of $i_{\max }\to \infty$.

In a similar manner to the calculation of the total absorption of power also the total power loss can be obtained for each surface. The result includes the absorption inside the surface texture as well as the transmission to the surrounding medium.

2.6 Determination of absorption profile

Based on the already determined vectors v_i a one-dimensional cumulative absorption profile Abs(z) is calculated. The cumulative absorption contains the information how much light is absorbed up to a depth z. In addition to the v_i, we also need the z-dependent propagation matrix D(z) and a z-dependent absorption matrix A_down(z):

The absorption for upwards and downwards propagating light is again separated and is also depending on z. For the upwards propagating light, the path length starts at the rear side, so the correct exponent is not $z/\cos {\theta }_i$ but $(d-z)/\cos {\theta }_i(d-z)/\cos {\theta }_i$. An additional absorption matrix A_up(z) has to be defined for upwards propagating light:

In the context of photovoltaics a generation profile that is directly related to the absorption profile might be of interest. A generation profile contains the information how many electron hole pairs are created between a certain depth $z$ and $z+dz$. It may be calculated either by considering the absorbed amount of light according to the cumulative absorption profile in finite steps or by differentiating an interpolated cumulative absorption profile.

2.7 Angle discretization and symmetry of matrices

Before actually calculating redistribution matrices one has to define an angle discretization of the half space. After the angular discretization all discrete angles (described by θ and φ) are subsequently used as incoming angles and the corresponding outgoing (reflected) angles are calculated. The direction of the reflected light is attributed to the closest channel angle value and the reflection efficiency written to the corresponding matrix entry. In this work, we only present systems that can be described in two dimensions, so only a discretization of the polar angle θ is needed.

For different surface textures, different angle discretizations could be preferable. However, for the absorption calculation via matrix multiplication with arbitrary surface textures, one discretization has to be chosen that can be used for all textures. We believe that an equidistant spacing of sin(θ) is generally a good choice for systems in which only the polar angle is considered. This choice corresponds to an equidistant discretization of the momentum-vector of light, projected onto the surface plane, and leads to convergence of the resulting total absorption with a low number of angle channels and low computation time.

Looking at a diffractive grating, a change of sin(θ_in) in the incoming angle leads to an equally large change of sin(θ_out) for the diffracted orders. By the choice of a discretization with an equidistant sin(θ)-spacing all rays within one incoming angle channel would, without binning, result in diffracted orders that belong to the same outgoing angle channel. Thereby, this discretization results in a symmetrical matrix. The symmetry can, thus, be used as a verification criterion for the redistribution matrix calculations.

The redistribution matrix being symmetrical can be attributed to the reciprocity of light paths. Please note that the redistribution matrix will not be symmetrical for arbitrary choices of the angle discretization. However, this does not mean that the reciprocity of light paths is violated or that further simulation results based on such a matrix are necessarily incorrect.

2.8 Polarization effects

The formalism described so far does not consider polarization effects. For many real systems, these play an important role to describe the optical properties correctly. The OPTOS formalism can easily be adapted to consider polarization effects by converting every vector entry into two entries, one for each independent direction of polarization, e.g. s- and p-polarization. Accordingly, every matrix element has to be converted into a 2x2-matrix that describes the redistribution between s- and p-polarization. Hence, all vector lengths and matrix dimensions increase by a factor of two. This, of course leads, to longer calculation times especially for the initial determination of the redistribution matrices, but allows for a correct description of polarization effects.

3. Calculating redistribution matrices

The initial calculation of the redistribution matrices is a crucial step for the OPTOS formalism. As described above, one fundamental advantage is that each surface structure can be treated with the most suitable simulation technique. In this section, a planar surface and three different surface textures will be examined using analytical solutions as well as ray- and wave-optical methods.

Figure 2 and 3 show examples for redistribution matrices for different systems represented by density plots at a wavelength of 1100 nm. All systems can be described in two dimensions and with a mirror symmetry regarding the surface normal. It is therefore sufficient to look at angles between 0° and 90° which correspond to sin(θ)-values between 0 and 1. The x-axis represents the incoming angle sin(θ_in) between 0 and 1, while the y-axis is the outgoing angle sin(θ_out) with the same range. Based on convergence tests we used an angle discretization with 100 values of sin(θ) for all matrices (a short description of this discretization procedure can be found in the appendix). For each incoming angle θ_in the reflected light paths are depicted while the colour scheme represents the corresponding reflection efficiency.

 

Fig. 2 Redistribution matrices for two different structures at a wavelength of 1100 nm using sin(θ) binning and averaged polarization. The colour scales are a measure of the diffraction or scattering intensity into the respective channels. (a) shows a planar silicon-air interface. Due to the specular reflection only diagonal entries appear. For all angles outside the loss cone the values are one. (b) shows V-grooves with dimensions that allow a treatment with ray tracing.

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Fig. 3 Redistribution matrices for two different structures at a wavelength of 1100 nm using sin(θ) binning and averaged polarization. (a) shows a linear grating at a silicon-air interface. (b) shows a 2D representation of a 3D Lambertian scatterer with 100% reflection. For all incoming angles a Lambertian light distribution is created. The values increase for increasing values of θ_out because of integration over the azimuth angles φ. In a fully 3D-description with appropriate angle discretization the representation of the Lambertian matrix would be single-valued.

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The most trivial investigated surface is a planar surface, as for specular reflectance the absolute value of the polar angle is not changed, and hence, the redistribution matrix has only diagonal entries. If we consider a perfect rear side mirror, the redistribution matrix is the unity matrix. If we consider a non-perfect mirror, the diagonal elements can also be smaller than 1. Figure 2(a) shows the density plot of the redistribution matrix for a silicon-air interface. For incoming angles inside the escape cone, transmission leads to matrix elements smaller than 1, whereas total internal reflection leads to a value of 1 for larger angles.

The second structure presented here are two-dimensional V-grooves. This structure is typically realized with a period in the range of 10 µm, which allows the use of ray-optical simulation methods as a good approximation. The redistribution matrices discussed here are calculated with the non-sequential forward ray-tracing software OptiCAD®. The system under consideration is depicted in Fig. 2(b). The opening angle at the top of the structure is 70.52°.

The redistribution matrix representing a line grating can be calculated by wave optical methods, for example rigorous coupled wave analysis (RCWA). We used a RCWA code by Hugonin and Lalanne [22, 23]. Because we discuss the line gratings here in terms of light trapping in silicon solar cells, we show a structure period in the range of 1000 nm [17].

In Fig. 3 (a) we show a redistribution matrix for a line grating with binary profile, a period of 990 nm, a fill factor of 0.5 and a grating depth of 160 nm. As the structure will be on the rear side of the silicon sheet, only light in a wavelength region, where the absorption of silicon can be neglected for the diffraction calculation, reaches the texture. Therefore, our simulations assume that the light impinges onto the silicon air boundary with the grating from a half-infinite non-absorbing silicon surrounding. The distinct nature of diffracted orders is clearly visible due to the distinct entries of the matrix that differ from zero.

The redistribution matrix corresponding to a Lambertian scatterer can be calculated analytically. We chose the same angle θ-discretization as for the other systems and integrated over all azimuth angles φ (for a detailed description of this calculation see Appendix). The resulting matrix shown in Fig. 3(b) is not symmetric because of the azimuth integration. If we chose a discretization of the azimuth angles with equally large projections of the half-sphere surface to the x-y-plane, the Lambertian matrix would show identical values for all entries.

4. Validation and comparison to existing methods and measurements

In this section, we validate the presented method by comparison of the obtained results to those of other simulation methods and measurement results. Therefore, we have selected four simple exemplary systems which can also be calculated with alternative methods. The four systems are a planar silicon slab (reference), a slab with planar front side and Lambertian rear side scatterer, a slab with planar front side and line grating rear side and a slab with V-grooves at the front side and a planar rear side.

4.1 Planar front side – line grating rear side

For a system consisting of a planar front side and a rear side line grating we validated the formalism by comparison to other simulation results taken from literature [9] and to measure-ments. In the work of Mellor [24] an exemplary system with a line grating and different grating parameters is considered. It consists of a 40 µm thick silicon sheet with a 114 nm thick SiO₂-anti-reflection coating on the front and a binary grating at the rear side. Between the silicon grating and an ideal dielectric reflector (refractive index n = 100000), there is an additional dielectric buffer layer (DBL) made of SiO₂ with a thickness of 1 µm. The grating parameters are in the first case a period of 990 nm and a depth of 160 nm and in the second case a period of 350 nm and a depth of 180 nm. The volume fill factor of the grating is in both cases 0.5. Note, that the thickness of the structure in the following always refers to the thickness of the bulk material in between the surface structures. In the current version of the formalism any extension of the surface structures does not contribute to the thickness.

The simulation results are shown in Fig. 4. As a reference the case of planar front and rear side surface was added. It was calculated via the transfer matrix formalism [25]. In all three cases the OPTOS results show an excellent agreement to the previously established simulation techniques.

 

Fig. 4 Calculated absorption for a silicon wafer with a thickness of 40 µm. The black line shows the absorption of a planar reference wafer calculated by a transfer matrix method (TMM) corresponding to [25]. The orange triangles show the results for the same system obtained by the OPTOS method. The orange circles and rhombs are OPTOS calculations for systems with planar front and line grating rear side. Grating 1 has a period of 990 nm and a grating depth of 160 nm, grating 2 a period of 350 nm and a depth of 180 nm. The results are compared to calculations of Mellor, where the diffraction orders of the gratings are directly used as angle channels [24] (blue and green line). The OPTOS formalism shows excellent agreement with the results obtained using other simulation techniques.

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Figure 5 shows a comparison of the OPTOS simulation with measured absorptance data of a 250 µm thick silicon substrate with a linear rear side grating. The sample was fabricated via interference lithography and subsequent plasma etching [26, 27] and characterized via Fourier spectrometry. The grating period was in this case 410 nm, the grating depth 120 nm and the fill factor 0.5. The measured absorption is in accordance with the OPTOS simulation results. The slightly higher absorption of the measured sample could result from enhanced scattering due to surface imperfections or deviations from the binary grating profile.

 

Fig. 5 Calculated and measured absorption for a silicon wafer with a thickness of 250 µm and a binary line grating on the rear side. The measured absorption is in accordance to the absorption calculated using OPTOS.

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4.2 Planar front side – Lambertian rear side scatterer

The second system investigated for validation is a 100 µm thick wafer with planar front side and a Lambertian rear side scatterer. The used matrices are the planar and Lambertian matrix presented in Fig. 2(a) and 3(b). This structure is equivalent to the situation Yablonovitch described in his work [28]. In Fig. 6 a good agreement between the results of the OPTOS formalism and the Yablonovitch limit, calculated analytically according to [28], is shown.

 

Fig. 6 Calculated absorption for a 100 µm thick silicon wafer with planar front and Lambertian rear. The simulation result of the OPTOS formalism (orange) and the Yablonovitch limit (green) agree very well. For comparison the absorption of a planar-planar reference wafer is shown (black).

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4.3 V-groove front side – planar rear side

While the first two examples had a planar front side and a textured rear side, described in the first case by RCWA and in the second case analytically, we apply a front side texture in form of V-shaped grooves in this third validation system. As described in section 3, these V-grooves can be simulated by ray-tracing. For a system with V-grooves at the front, a silicon sheet with 100 µm thickness and planar rear we conducted calculations with OPTOS and comparison simulations based on the wafer ray tracer available at PV Lighthouse (input parameters: regular V-shaped grooves, characteristic angle 54.74°, height 3.536 µm, substrate thickness 100 µm, rear reflector R = 1, total number of rays 1000000). Figure 7 shows an excellent agreement of both calculations, which again validates the presented OPTOS matrix formalism. Note, that the current version of the PV Lighthouse wafer ray tracer places the grooves on top of the 100 µm thick substrate and hence the total thickness of the complete structure is larger than 100 µm by the height of the grooves (3.536 µm). Accounting for this in OPTOS, we chose a substrate thickness of 102 µm, which corresponds approximately to the net thickness of the complete structure. The small differences in the wavelength range between 1000 and 1100 nm may be due to a different treatment of polarization. The PV Lighthouse ray tracer calculates with both TE and TM polarization, but it averages these results after each interaction. In contrast, OPTOS keeps track of the polarization throughout the simulation and only averages the final result.

 

Fig. 7 Calculated absorption for a 100 µm thick silicon wafer with additional V-groove front (groove height 3.536 µm) and planar rear. The lines with symbols are the results of the OPTOS formalism for TE polarization (orange), TM polarization (green) and the averaged values (blue). The latter agree very well with results of a ray tracing simulation with the same texture and sheet parameters that was run based on the PV-Lighthouse wafer ray tracer (black).

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By using a redistribution matrix, absorptance within the surface structure does not contribute to the calculated absorptance of the sheet itself. This is a valid approximation for bulk thicknesses large compared to the surface structure and weak absorption within the surface structure. For very thin sheets with large V-grooves or strong absorption however, the absorption in the grooves must not be ignored. In the case of overall strong absorption all light is absorbed before reaching the rear side, thus the absorptance can be calculated via $Abs=1-R$ . For other cases one option is an adapted sheet thickness. A redistribution matrix that differentiates between transmission out of the sheet and absorption is beyond the scope of this paper.

5. Combined simulation of front and rear side textured sheets

The OPTOS formalism allows not only the combined simulation of a textured with a planar surface, as described above. Its original purpose is the combined simulation of different textured surfaces, especially with feature sizes of different length scales. This case cannot easily be covered with standard existing techniques like ray-tracing or wave optical methods. With the functionality of the matrix formalism validated in section 4, all of the described surface textures can be combined with each other. As an example, we conducted calculations of a wafer that is textured with V-grooves on the front side and a line grating rear side, both individually described above. With this simulation we want to demonstrate the capabilities of the method, even if we do not propose this exact approach for solar cell applications.

5.1 V-grooves front side - line grating rear side

In addition to the absorption calculations presented in section 4, where we had reference methods or values from literature, we present here a textured sheet with an optically large scale structure (V-grooves) at the front side and an optically small scale structure (grating) at the rear side. The same matrices for V-grooves and line gratings calculated before were used for these simulations. In Fig. 8 the absorption for a sheet with V-groove front and line grating rear side is compared to the sheet with V-grooves front and planar rear, that is the same curve as in Fig. 7. A significant absorption enhancement in the near infrared can be seen. With respect to standard silicon wafer solar cells with pyramidal texture at the front side, this is a hint, that additional redistribution of light at the rear surface can further improve the overall light trapping properties. Note however, that the presented case is only two-dimensional, and no additional rear reflector, which is common in solar cells, has been considered. Both of these aspects will be covered in further studies, which focus for example on simulations of a three-dimensional pyramidal texture at the front side and a crossed grating at the rear side.

 

Fig. 8 Absorption in a 100 µm thick silicon wafer with V-groove front and planar rear (black) and with V-groove front and line grating rear (orange). Although the presented case is only two-dimensional and no additional rear reflector has been considered, the result is a hint, that additional redistribution of light at the rear surface can further improve the overall light trapping properties also for three-dimensional, pyramidal front side textures.

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Simulations with a textured rear surface, that needs wave-optical treatment, can in principle also be incorporated in a ray-tracing formalism via bidirectional reflectance functions. This was demonstrated by Rothemund et al. [18]. With this kind of technique it is, however, necessary to start completely new simulations when different surface textures are combined with each other. Especially in the three-dimensional case, the calculation of the redistribution matrices for a specific surface can be numerically demanding and time consuming. In the OPTOS formalism, the matrices only need to be calculated once and can then be reused for different simulations. This efficiency in computation is one of the benefits of our new formalism.

5.2 Variation of the sheet thickness

The possibility to re-use obtained simulation results in form of redistribution matrices furthermore allows a fast variation of the sheet thickness. The propagation matrix changes but can be easily calculated. The final absorption calculation requires only matrix multiplications or matrix inversions. Especially for only partly occupied matrices there are very efficient numerical methods for matrix operations and hence the absorption calculation is numerically not very demanding. In Fig. 9 we present the results of such a sheet thickness variation for a system with the same surface properties as described in section 5.1. (i.e. V-grooves front side and line grating rear side). The whole simulation time was a matter of minutes on a single core CPU which is much less computationally intensive than the original calculation of the redistribution matrices. The result shows the expected absorption increase for larger sheet thicknesses. Similar to the comparison between a sheet with planar rear side and one with rear side grating, the three-dimensional systems will be of more practical relevance, e.g. for solar cell applications. Nevertheless, the fact that OPTOS allows fast optimizations of the sheet thickness also for complex surface textures is of high importance.

 

Fig. 9 Sheet thickness variation of an OPTOS calculation for a silicon wafer with V-grooves front and line grating rear. The simulation results show the expected absorption increase for larger sheet thicknesses. The computation is remarkably efficient, requiring only several minutes for this thickness variation.

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6. Summary and outlook

We presented a novel matrix formalism called OPTOS (Optical Properties of Textured Optical Sheets) that allows the calculation of the optical properties in optically thick sheets with arbitrarily textured front- and rear surface. This method is of particular interest to investigate advanced optical concepts for silicon wafer solar cells.

In this formalism, the angle distribution of the light is mapped to discrete channels. The complete set of channels constitutes a disjunct partition of the half-space within the absorber. Each channel is defined by a suitable discretization of polar- and azimuth angle. This discretization enables the description of arbitrary textured surfaces by discrete redistribution matrices that contain the information how much light is redirected into each channel by an interaction with the corresponding surface. The redistribution matrix of any front or rear surface texture can be calculated independently by the most suitable method. This approach allows combining front- and rear surface textures with fundamentally different optical properties in one absorption calculation. The propagation of light within the absorber sheet is calculated incoherently by a propagation matrix that considers absorption according to Lambert-Beer’s law. The absorption is then calculated either by subsequent multiplication of the matrices in the right order, and considering the absorbed fraction at every path, or by a geometric series of matrices. The formalism works in two or three dimensions and it can rigorously consider polarization.

To demonstrate the functionality of the formalism and for its validation, we presented redistribution matrices for planar surfaces, grating structures, a Lambertian scatterer and a V-groove texture. All these matrices were calculated with different methods, adapted to the respective structure sizes. The comparison of absorption data calculated with the new OPTOS formalism with results obtained by established methods or from literature, shows very good agreement for all considered examples, which were sheets featuring a planar front and either a Lambertian scatterer or a line grating at the rear and also for sheets with V-grooves at the front and a planar rear. For the case with planar front and line grating rear the simulation results were validated against experimental data. Also here, good agreement was found. By calculating the absorption for sheets with V-grooves at the front and a line grating rear for different bulk thicknesses, we highlighted the capability of OPTOS to calculate different combinations of front and rear side textures and use the obtained matrices for different calculations. This was demonstrated by a variation of the sheet thickness, which was performed with great computational speed.

Further studies will focus on the simulation of three-dimensional surface textures. Especially the case of a pyramidal front side texture in combination with a diffractive rear side is very relevant for silicon solar cells. As it is possible to calculate not only the total absorption but also an absorption and charge carrier generation profile, the results of the OPTOS formalism can be used for simulations of the electrical solar cell properties. This combination allows a complete solar cell simulation with two textured surfaces and will also include rear side mirrors as they are used in many realistic cell architectures.

For a description of the optical properties of a whole module, an extension of the formalism to a multiple textured sheet system is required. Furthermore, OPTOS might be suitable for applications beyond photovoltaics, which was the focus of the examples presented in this paper. For instance brightness enhancement films used in displays feature a front and rear side texture similar to the V-groove and Lambertian texture presented here. Also further development of the formalism regarding absorption within surface structures or scattering within the sheet is possible.