A simplified method to modulate colors on industrial multicrystalline silicon solar cells with reduced current losses

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ScienceDirect Solar Energy 103 (2014) 343–349 www.elsevier.com/locate/solener

A simplified method to modulate colors on industrial multicrystalline silicon solar cells with reduced current losses Libin Zeng a, Minghua Li a, Yifeng Chen a,b, Hui Shen a,⇑ a

School of Physics and Engineering, Institute for Solar Energy Systems, State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510275, China b State Key Lab of PV Science and Technology, Trina Solar Limited, No. 2 Trina Road, Trina PV Park, New District, Changzhou, Jiangsu 213031, China Received 3 July 2013; received in revised form 1 December 2013; accepted 4 February 2014 Available online 13 March 2014 Communicated by: Associate Editor Elias Stefanakos

Abstract This paper investigates the impact of SiOxNy/SiNx:H double layer antireflection (DLAR) coatings on color modulation and cell efficiencies of multicrystalline silicon (mc-Si) solar cells. We presented a three dimensional ellipsoid surface model to simulate the concave-like morphology formed by acidic texturization. Afterwards, reflectivities of these acidic textures coated by DLAR coatings over 300–1100 nm are calculated by Monte Carlo ray tracing method. Simulated results show that DLAR coatings can flexibly modulate the color with reduced current losses compared to single SiNx:H layer. SiOxNy was deposited by electron beam evaporation onto fabricated industrial solar cells to vary their colors. The busbars were sheltered by a mask to prevent the deposition of SiOxNy on them. This simplified method avoids adjustment of the standard fabricating process. High agreement between simulated and measured reflectance is achieved. The IV test results of colored solar cells are in good accord with the calculated results, which indicates the effectiveness of DLAR coatings in reducing the current losses. Ó 2014 Elsevier Ltd. All rights reserved.

Keywords: Ellipsoid model; Double layer antireflection coatings; Monte Carlo ray tracing; Colored mc-Si solar cells

1. Introduction Promoting the use of photovoltaics (PV) in buildings is one of the most important ways of reducing building energy consumption. Building-integrated and buildingattached photovoltaics (BIPV and BAPV) have therefore attracted a great deal of research interests (Yoo and Manz, 2011; Yoon et al., 2011; Santos and Ru¨ther, 2012). Two factors are hindering the widespread utilization of BIPV and BAPV: high cost and limited choices of aesthetics. The first factor has become less influential as module ⇑ Corresponding author. Tel.: +86 (20)3933 2863; fax: +86 (20)3933 2866. E-mail address: [email protected] (H. Shen).

http://dx.doi.org/10.1016/j.solener.2014.02.012 0038-092X/Ó 2014 Elsevier Ltd. All rights reserved.

price is undergoing a continuously downward trend (Ja¨ger-Waldau, 2012). The second factor is being increasingly critical as PV modules are being more visible in urban area. The mere colors of black and blue of conventional PV modules may easily cause incompatibility with the outer appearance of architectures. Hence PV modules with multiple colors are desirable. One approach is to vary the thickness of the antireflection (AR) coating and as a result a wide range of colors can be obtained. Tobias et al. (1999) reported the colored solar cells with single layer antireflection (SLAR) coating on random pyramid textures, but the drawback was the significant drop in short-circuit current (about 2–3 mA/cm2 less than optimum short-circuit current). To improve the optical performance, multilayer AR coatings had been studied by simulation and

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experiments (Spiegel et al., 1995; Selj et al., 2011). However, both references only presented the outcomes on polish surface. For pyramid texture surface, we showed that it is possible for colored cells with multilayer AR coatings to achieve equal Jsc with standard blue cell (Chen and Yang, 2012). Nevertheless, results on acidic texture for multicrystalline silicon (mc-Si) solar cells are absent. For acidic texture surface, we preliminarily showed that SiO2/SiNx:H double layer antireflection (DLAR) coatings can modulate the colors of solar cells (Li et al., 2013). However, since the refractive index of SiO2 coating is almost identical with that of ethylene vinyl acetate (EVA) sheet, encapsulating the solar cells into module would eliminate the effectiveness of SiO2 coating, turning the DLAR coatings into SLAR coating. Therefore, material with refractive index higher than SiO2 needs to be studied to address the issue of encapsulation. In this paper, we aimed at studying the impact of SiOxNy/SiNx:H DLAR coatings thickness upon the color and efficiency of mc-Si solar cells by numerical simulation and experiments. Typically, concave-like morphology is formed by acidic solutions of HF and HNO3. Spherical surface model was studied in Nishimoto et al. (1999) and Li et al. (2012), however, the assumption that the spherical surface is ideally smooth is not precise, because the surface is partially Lambertian. We therefore developed a three dimensional (3D) ellipsoid model to simulate the acidic textures of mc-Si wafers to include Lambertian reflectance. Monte Carlo ray tracing algorithm (Brendel, 1995; Brendel and Scholten, 1999) was adopted for the calculation of the reflectivity and absorptivity to yield colors and Jsc. On experiments, 156  156 mm2 industrial mc-Si solar cells were fabricated with SiNx:H SLAR coating. Then SiOxNy(n = 1.8) was deposited by the use of electron beam (e-beam) evaporation technique onto the solar cells to form SiOxNy/SiNx:H DLAR coatings. To prevent the deposition of SiOxNy onto the busbars from forming a dielectric layer and thus affecting the subsequent cell testing and soldering process, a mask was used to shelter the busbars. Therefore, only one additional step is necessary for colored cells fabrication. No additional requirement to match the front metallization process with the AR coating thicknesses is needed. Semi-sphere reflectance and I–V measurements were performed for comparison and analysis, showing that the DLAR coatings could effectively reduce current losses.

Fig. 1. Scanning electron microscope photographs showing the cross section of acidic textured mc-Si wafers (a) and 3D ellipsoid surface model symbolizing one unit pit of on the wafer surface (b).

The geometry of the ellipsoid surface can be defined by the following equation: x2 y 2 z2 þ þ  1 ¼ 0; r21 r22 r23

ðz < 0Þ

ð1Þ

where r1, r2, r3 are the geometric parameters for the ellipsoid surface. As shown in Fig. 1(b), point A with its coordinate (x0, y0, z0) randomly generated is the starting point of inci! dent ray AB , while point B(xi, yi, zi) is the point at which ! ray AB hitted the textured surface. The unit vector of ! AB is noted as (xd, yd, zd) which represents the direction ! ! of AB . For normal incident ray, AB = (0, 0, 1). The parametric equation of is defined as follows:

2. Optical simulation 2.1. Geometrical model The typical acidic texturing of mc-Si wafer is formed by etching the surface defects caused by saw damage. The formed surface morphology is concave-like pits, as shown in Fig. 1(a). To symbolize the surface, we chose a 3D ellipsoid surface model which is illustrated in Fig. 1(b), as the unit cell for the concave-like texture.

x ¼ x0 þ xd  t y ¼ y0 þ yd  t

ð2Þ

z ¼ z0 þ y d  t where t > 0 indicates that A is the starting point. By substituting Eq. (2) into Eq. (1), one can obtain at2 þ bt þ c ¼ 0 where

ð3Þ

L. Zeng et al. / Solar Energy 103 (2014) 343–349

a¼

x2d r21

y2

1

  at A ¼ expðaxÞ ¼ exp  ! ! N  T

z2

þ r2d þ rd2 2 3 h i y0 yd x0 xd b ¼ 2 r2 þ r2 þ z0r2zd 2

x2

y2

z2

1

2

3

ð4Þ

3

c ¼ r20 þ r20 þ r02  1 With the condition t > 0, we can deduce t by solving Eq. (3): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b þ b2  4ac t¼ ð5Þ 2a Hence the coordinate of B is expressed as ðxi ; y i ; zi Þ ¼ ðx0 þ xd  t; y 0 þ y d  t; z0 þ zd  tÞ

ð6Þ

Therefore knowing both the coordinate of the starting ! point A and the unit vector of AB , the coordinate of point B can be calculated. In the case of reflection, point B is the subsequent starting point, and similarly we can calculate point C by figuring out the unit vector of the reflected ! ray BC and applying the above calculations. 2.2. Front and rear surface reflection Two types of reflection are considered here: Lambertian reflection and specular reflection. Lambertian reflection refers to ideal diffuse reflection, which suggests that the intensity of the reflected light is uniformly distributed in all directions. In fact, all reflections are partial Lambertian reflection, so a factor f is introduced to address the possibility that the reflection is treated as Lambertian reflection. Unrelated to the direction of the incident light, the intensity of the reflected light is subject to the cosine distribution IðhÞ ¼ I 0 cos ðhÞ

ð7Þ

345

ð9Þ

where a is the absorption coefficient, and x is the ray path length in the wafer, t is the thickness of the wafer, and ~ ¼ ð0; 0; 1Þ is the unit normal vector to the wafer. N 2.4. Monte Carlo ray tracing Rays are traced starting from incidence and ending to reflection, absorption by the wafer or transmission through the wafer exclusively. Monte Carlo ray tracing algorithm is employed to simulate the probability of the above independent outcomes and hence the reflectance, transmission and absorptivity can be obtained. For example, the probability of a ray being reflected from the front surface is equal to its reflectance R. Total internal reflection is also taken into account in the simulation. For each wavelength, a large number of random rays are generated and traced. Therefore, the average reflectance, transmission and absorptance of incident light over a spectrum of 300–1100 nm are calculated. The entire simulation was performed by programming on MATLAB. The parameters r1, r2, r3 in Eq. (1) is determined by fitting the simulated reflectance curve to measured results of texture samples without any AR coating. Besides, Lambertian factors of front and rear sides are also adjusted to achieve high precision (f = 0.2 and 0.4 are the conventional values for front and rear sides respectively). The comparison of simulated and measured reflectance in Fig. 2 shows high agreement over 400–1100 nm, but a slight deviation below 400 nm can be observed. The deviation might be due to the imprecision of the ellipsoid assumption which neglects the irregular size distribution of concave pits. The same phenomenon occurred in Li et al. (2012), and further work is needed to investigate the reason.

where I0 is the radiance along the normal and h denotes the angle of reflection (Nayar et al., 1989). Therefore the unit ! vector of BC can be calculated by the distribution. When specular reflection occurs, the angle of reflection k AB and ~ k BC are is equal to that of incidence. Assume that ~ ! ! k BC is the unit vectors of AB and BC correspondingly, ~ determined by (see Appendix A) ~ n  ð~ n ~ k BC ¼ ~ k AB  2~ k AB Þ

ð8Þ

2.3. Bulk absorption The texturization also serves to enhance light trapping ! by oblique coupling of incident light. If ray AB refracts at B, the unit vector of the transmitted ray is expressed T (see Appendix A). as ~ Bulk absorption is described by attenuation factor A, written as

Fig. 2. Simulated and measured reflectance of industrial acidic textured mc-Si wafers depicted as a function of wavelength. Solid line and black triangle symbols denote the simulated and measured values respectively.

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2.5. Calculation of colors and short-circuit current density The absorption, reflectance and transmittance of DLAR coatings are calculated via the transfer-matrix-method (TMM) (Harbecke, 1986; Hiller et al., 2002; Katsidis and Siapkas, 2002). The bottom layer of SiNx:H was deposited by PECVD, with refractive index 2.05 at 633 nm and thickness of 80 nm. Then SiOxNy is introduced as the top layer since its refractive index can be changed continuously between 1.45 (SiO2) and 2.0 (Si3N4) and it has been proved to have high thermal stability and chemical inertness (Godinho et al., 2010). The three color matching functions x(k), y(k) and z(k) defined by the International Commission on Illumination (CIE) (CIE, 2006a, b) are multiplied with simulated reflectance and the standard D65 spectrum for color calculation. Furthermore, to better evaluate optical performance of the DLAR coatings stack, we calculated the short-circuit current density (Jsc) with the transmittance spectra: Z 1100 nm J sc ¼ q  Fluxphoton ðkÞ  IQEðkÞ  TRðkÞ dk ð10Þ 300 nm

where Fluxphoton ðkÞ is the wavelength dependent photon flux under AM 1.5 Global, q is the element charge, and IQE(k) is the internal quantum efficiency of conventional mc-Si solar cells, and TR(k) is the transmission spectra of the DLAR coatings. 3. Experimental 156  156 mm2 mc-Si solar cells were fabricated from ptype silicon wafers (1 X cm) using a standard industrial manufacturing process. After cleaning and acidic texturization, a standard POCl3 emitter diffusion in a quartz tube resulted in a sheet resistivity of 60 X/sq. The wafers were coated with SiNx:H(n = 2.05, 80 nm) in a PECVD (Roth&Rau) system, and metalized by screen printing

and firing. Then, SiOxNy was deposited onto mc-Si solar cells by e-beam evaporation. To avoid the deposition of SiOxNy on the front side of the busbars, a mask was utilized to shelter the busbars before the solar cell was placed into the chamber of the e-beam system (Shen et al., 2010). High purity SiNx (99.99%) granules were used as the source material for evaporation and the source-to-substrate distance was 30 cm. The substrates temperature was controlled at 200 °C. High purity oxygen (99.99%) was introduced into the chamber to maintain a pressure of 3.0  102 Pa and used as reactive gas during the deposition. The deposition rate was controlled using a quartz ˚/ crystal sensor placed near the substrate, and set as 2 A s. The refractive index of the SiOxNy coating on a fabricated solar cell was measured by a Sentech SE800 PV ellipsometer. According to Fig. 3(b), three different thicknesses of 90 nm, 110 nm and 210 nm were deliberately selected to fabricate brown, purple and green solar cells respectively. Ten solar cells were fabricated for each group. In order to research into the impact of SiOxNy layers upon efficiency, the I–V characteristics were measured with an Optosolar cell tester. The reflectance was measured with a Hitachi U-4100 spectrophotometer. 4. Results and discussions Fig. 3(a) shows the color and the Jsc of mc-Si solar cells as a function of the thickness of SiNx:H SLAR coating. The highest Jsc is achieved with thickness of 80 nm, but Jsc drops significantly as the thickness deviates from 80 nm, despite colors can be modulated. For comparison, the Jsc loss is greatly suppressed using DLAR coatings, as shown in Fig. 3(b), with SiNx:H (thickness = 80 nm, n = 2.05) as the bottom layer and SiOxNy as the top layer with varied thickness. The difference between the maximum and minimum Jsc is approximately 1.6 mA/cm2 for the DLAR coatings scenario, while in the SLAR coating scenario the difference is as high as 3.4 mA/cm2. The color

Fig. 3. Calculated perceived color and Jsc in dependence of AR coating thickness. (a) Thickness of SiNx:H (n = 2.05) SLAR coating is varied from 0 nm to 300 nm. (b) Thickness of top layer SiOxNy (n = 1.8) is varied from 0 nm to 300 nm, and the bottom layer SiNx:H (n = 2.05) is 80 nm. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 4. Photographs of (a) conventional solar cells with SLAR SiNx:H coating and colored solar cells with DLAR SiOxNy/SiNx:H coatings (b) brown, (c) purple, and (d) green. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. Measured and simulated reflectance spectra of blue (a), brown (b), purple (c) and green (d) mc-Si solar cells depicted as a function of wavelength. Solid line and black triangle symbols denote the simulated and measured values respectively.

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saturation of DLAR coatings decreases slightly due to reduced reflectance and consequently increased Jsc. Fig. 4 shows the reference SLAR coating blue cell and three types of colored mc-Si solar cells fabricated in this work. The ellipsometer test showed the refractive index of SiOxNy was 1.78, highly close to the desired refractive index. The measured reflectance of the fabricated solar cells are shown in Fig. 5, compared to the simulated reflectance. Reflectance peak can be obviously observed. Color modulation is basically fulfilled by modulating the wavelength position of reflectance peak, which contributes dominantly to the perceived color of solar cells. In general the simulated reflectance predicts the measured one. Especially in the range of 400–1000 nm the two spectra almost overlap. In the range of 300–400 nm, the simulated reflectance is lower than the measured one. This is due to the underestimation of the simulated reflectance of acidic textured mc-Si wafers in this range as shown in Fig. 2, and the difference is narrowed since the AR coatings reduced the reflectance in overall. In the range of 1000–1100 nm, the simulated reflectance is higher than the measured one. Because the aluminum layer of solar cells strongly absorbs long wavelength light, less light is reflected at the rear side. This effect is not yet considered in the simulation, and further work could improve the accuracy in this range. The CIE color space coordinates and the reflected photon flux in terms of current density (JR) of the fabricated AR coating are shown in Table 1, both calculated from the measured reflectance of the colored solar cells. The brown, purple and green cells with DLAR coatings show higher JR than blue cells with SLAR coating due to higher reflectance. But the reflected current density loss is notably suppressed, only 1–1.5 mA/cm2 higher than blue cells. The median IV test results of the colored mc-Si solar cells are shown in Table 2. While Voc and fill factor (FF) deviate slightly, the main difference that contributes to the variation of efficiency is Jsc. The colored solar cells show no decrease in FF compared to conventional blue solar cell, demonstrating that the busbars were effectively sheltered by the mask during the deposition of SiOxNy. Usually in order to achieve good electrode contact, the firing condition should be adjusted to suit different AR

coating thickness, which can be complicated in fabricating multiple colors of solar cells. This method avoids any change of the firing condition. All colored solar cells show lower Jsc than that of blue solar cells, but the decrease in Jsc is relatively insignificant compared to the SLAR coating scenario as indicated in Fig. 3(a). This result proves the validity of the DLAR coatings. 5. Conclusions A model was presented to investigate the impact of DLAR coatings on the color and Jsc of mc-Si solar cells with Monte Carlo ray tracing simulation. Compared to the SLAR coating, simulation showed that SiOxNy/SiNx DLAR coatings can produce effective reduction in current losses in color modulation. SiOxNy was deposited by ebeam evaporation onto industrial solar cells of which the busbars were sheltered by a mask. This simplified method avoided the adjustment of firing process while realizing equal FF, and reduced the current losses of colored solar cells. The simulated reflectance spectra fitted the measured ones well, as a demonstration of the validity of the optical simulation. Further work will study the possibility of encapsulating the colored solar cells with DLAR coatings into modules. Acknowledgement This work was supported by the National 863 Project (No. 2012AA050302).

Fig. A.1. Schematic of specular reflection. AB is incident light, BC is reflected light, and BD is normal at point B.

Table 1 JR and CIE coordinates of the fabricated AR coatings. Color 2

JR (mA/cm ) CIE coord. (x, y)

Blue

Brown

Purple

Green

3.9 (0.21, 0.21)

4.9 (0.40, 0.37)

5.4 (0.31, 0.21)

5.3 (0.30, 0.38)

Table 2 IV test results of conventional and colored solar cells. Color

Jsc (mA/cm2)

Voc (mV)

FF (%)

Eff (%)

Blue Brown Purple Green

34.6 34.2 33.8 33.6

612 612 611 613

77.6 77.4 77.8 77.1

16.4 16.2 16.1 15.9

Fig. A.2. Schematic of refraction. AB is incident light and BE is refracted light.

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Appendix A. Calculation of the unit vector of reflected and refracted light

References

! As shown in Fig. A.1, AB reflects at B, where the unit normal vector is      2x ;  2y ;  2z  @F ;  @F ;  @F @x @y @z r22 r21 r23   ¼   ~ ðA:1Þ n ¼    2y 2z  ;  @F ;  @F ;  ;    @F   2x  2 2 2 @x @y @z  r r r

Brendel, R., 1995. Coupling of light into mechanically textured silicon solar cells: a ray tracing study. Prog. Photovoltaics: Res. Appl. 3 (1), 25–38. Brendel, R., Scholten, D., 1999. Modeling light trapping and electronic transport of waffle-shaped crystalline thin-film Si solar cells. Appl. Phys. A: Mater. Sci. Process. 69 (2), 201–213. Chen, Y., Yang, Y., et al., 2012. Color modulation of c-Si solar cells without significant current-loss by means of a double-layer antireflective coating. In: 27th European Photovoltaic Solar Energy Conference and Exhibition, 2014–2016. CIE, 2006. ISO 11664–1:2008(E)/CIE S 014–1/E:2006: Joint ISO/CIE standard: CIE colorimetry—Part 1: Standard colorimetric observers. CIE, 2006. ISO 11664–2:2008(E)/CIE S 014–2/E:2006: Joint ISO/CIE standard: CIE colorimetry—Part 2: Standard illuminants for colorimetry. Godinho, V., Haro, M.C.J.d., et al., 2010. SiOxNy thin films with variable refraction index: microstructural, chemical and mechanical properties. Appl. Surf. Sci. 256 (14), 4548–4553. Harbecke, B., 1986. Coherent and incoherent reflection and transmission of multilayer structures. Appl. Phys. B: Lasers Opt. 39 (3), 165–170. Hiller, J.A., Mendelsohn, J.D., et al., 2002. Reversibly erasable nanoporous anti-reflection coatings from polyelectrolyte multilayers. Nat. Mater. 1 (1), 59–63. Ja¨ger-Waldau, A., 2012. PV Status Report 2012, Luxembourg: Office for Official Publications of the European Union, Brussels. Katsidis, C.C., Siapkas, D.I., 2002. General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference. Appl. Opt. 41 (19), 3978–3987. Li, M., Zeng, L., et al., 2013. Realization of colored multicrystalline silicon solar cells with SiO2/SiNx:H double layer antireflection coatings. Int. J. Photoenergy. Li, Y., Li, Z., et al., 2012. Modelling of light trapping in acidic-textured multicrystalline silicon wafers. Int. J. Photoenergy. Nayar, S.K., Ikeuchi, K., et al., 1989. Determining shape and reflectance of Lambertian, specular, and hybrid surfaces using extended sources. In: International Workshop on Industrial Applications of Machine Intelligence and Vision, pp. 169–175. Nishimoto, Y., Ishihara, T., et al., 1999. Investigation of acidic texturization for multicrystalline silicon solar cells. J. Electrochem. Soc. 146, 457. Santos, I´.P.d., Ru¨ther, R., 2012. The potential of building-integrated (BIPV) and building – applied photovoltaics (BAPV) in single-family, urban residences at low latitudes in Brazil. Energy Buildings 50, 290– 297. Selj, J., Mongstad, T., et al., 2011. Reduction of optical losses in colored solar cells with multilayer antireflection coatings. Solar Energy Mater Solar Cells 95 (9), 2576–2582. Shen, H., Chen, Y., et al., 2010. A structure to form colorful protective thin film via mask. PR China patent. CN101834230A. Spiegel, M., Willeke, G., et al., 1995. Colored solar cells with single, multiple and continuous layer antireflection coatings. In: Proceedings of the 13th European Photovoltaic Solar Energy Conference, pp. 417– 420. Tobias, I., El Moussaoni, A., et al., 1999. Colored solar cells with minimal current mismatch. IEEE Trans. Electron Dev. 46 (9), 1858–1865. Yoo, S.-H., Manz, H., 2011. Available remodeling simulation for a BIPV as a shading device. Solar Energy Mater. Solar Cells 95 (1), 394–397. Yoon, J.-H., Song, J., et al., 2011. Practical application of building integrated photovoltaic (BIPV) system using transparent amorphous silicon thin-film PV module. Solar Energy 85 (5), 723–733.

1

2

3

! Assume that ~ k AB and ~ k BC are the unit vectors of AB and ! BC correspondingly, according to parallelogram rule ! ! ! ~ n  j BD j k BC  j BC j ¼ ~ k AB  j AB j þ ~ ! ! Specular reflection means AB ¼ BC , therefore

ðA:2Þ

! j BD j ~ ~ n  ! k BC ¼ k AB þ ~ j AB j

ðA:3Þ

! j BD j n ~ k AB Þ ! ¼ 2 cos \ABD ¼ 2ð~ j AB j

ðA:4Þ

By substituting Eq. (A.4) into Eq. (A.3), one can obtain ~ n  ð~ n ~ k BC ¼ ~ k AB  2~ k AB Þ

ðA:5Þ

! As shown in Fig. A.2, AB refracts at B. h1 and h2 are the ! ! incident and refracted angle respectively. AB and BC can be written as ! AB ¼

~ k AB ~ jk AB  ~ nj

ðA:6Þ

! ! n þ bð AB þ~ nÞ BE ¼ ~

ðA:7Þ

where b is the ratio factor and can be defined as sin h

b¼

tan h2 cos h22 n1  cos h1 ¼ sin h1 ¼ tan h1 cos n2  cos h2 h

ðA:8Þ

1

where n1 and n2 are the refractive indexes of air and silicon nj and cos h2 can be deduced accordk AB  ~ wafer, cos h1 ¼ j~ ing to Snell’s law. ! Normalize BE and the unit vector of the refractive light can be obtained ! BE T ¼ ! j BE j !

ðA:9Þ