Minimizing light reflection from dielectrictextured surfaces
Alexei Deinega,1,* Ilya Valuev,2 Boris Potapkin,1 and Yurii Lozovik3
1Kinetic Technologies Ltd., Kurchatov Sq. 1, Moscow 123182, Russia2Joint Institute for High Temperatures RAS, Izhorskaya, 13, Bld. 2, Moscow 125412, Russia3Institute for Spectroscopy RAS, Fizicheskaya, 5, Troitsk, Moscow Region 142190, Russia
*Corresponding author: firstname.lastname@example.org
Received January 4, 2011; revised February 11, 2011; accepted February 15, 2011;posted February 18, 2011 (Doc. ID 140551); published April 12, 2011
In this paper, we consider antireflective properties of textured surfaces for all texture size-to-wavelength ratios.Existence and location of the global reflection minimum with respect to geometrical parameters of the texture is asubject of our study. We also investigate asymptotic behavior of the reflection with the change of the texturegeometry for the long and short wavelength limits. As a particular example, we consider silicon-textured surfacesused in solar cells technology. Most of our results are obtained with the help of the finite-difference time-domain(FDTD) method. We also use effective medium theory and geometric optics approximation for the limiting cases.The FDTD results for these limits are in agreement with the corresponding approximations. 2011 OpticalSociety of America
OCIS codes: 310.1210, 050.2770.
1. INTRODUCTIONElimination of undesired reflection from optical surfaces isimportant for many technologies, such as solar cell, light-sensitive detectors, lenses, displays, etc. To reduce reflection,one can use single-layer quarter-wave coatings , but theirapplications are limited by a small range of wavelengthsand incidence angles. To extend this range, multilayer or gra-dient index coatings  can be used, however, their manufac-turing encounters problems with thermal mismatch, adhesion,and stability of thin-film stacks .
An alternative method for the reflection reduction consistsof texturing the surface with three-dimensional pyramids ortwo-dimensional grooves (gratings; Fig. 1) [4,5]. Many reportson successful fabrication of the antireflective textured sur-faces have been published [3,69]. Their use in solar technol-ogy leads to significant enhancement of solar cell efficiencyby reducing reflection from the surface of the cell by 1 or 2orders of magnitude.
In the limiting cases of long and short wavelengths, thisreduction in reflection can be explained theoretically withthe help of effective medium and geometric optics approxima-tions. According to the first approximation, if the wavelengthis greater than the texture size, the texture behaves like agradient index film with reduced reflection [10,11]. In theshort wavelength limiting case, reflection reduction can beillustrated geometrically: rays should be reflected many timesuntil being reverted back . At the same time, transmittedrays deviate from the incident direction that leads to the lighttrapping effect used in solar cells . It was also shownnumerically that using texture reduces reflection for wave-lengths comparable with its size. In particular, reflectiondecreases when texture is made higher and dielectric contrastbetween the texture and incident medium is lower [14,15].
In our previous work, we studied antireflective propertiesof dielectric textured surfaces at the whole wavelength range,
including long and short wavelength limits . We foundoptimal parameters for the period of the texture and thepyramids height. It was found that the key factor influencingthe optimal pyramid size is the character of substrate tiling bythe pyramid bases.
The present paper extends our previous work with newdetails and results. We closely analyze the limiting casesdue to their importance for finding global reflection minimumat the whole wavelength range. It is demonstrated that theconclusions of this work are applicable to real silicon-textured surfaces used in solar cell technology. In order tocheck the relevance of our calculations, we make a compar-ison with the experimental data.
As a main numerical tool, we use the finite-difference time-domain (FDTD) method . This method implements directnumerical solution of Maxwells equations, so it is valid for alltexture size-to-wavelength ratios, including long and shortwavelength limits. For accurate reflection calculation, it isimportant to reduce undesired numerical reflection fromthe artificial absorbing perfectly matched layer . For thispurpose, we use the additional backing absorbing layers tech-nique that we described in . We apply effective mediumtheory and ray tracing techniques for long and short wave-length limits as well. All calculations are performed usingthe free electromagnetic simulation package ElectromagneticTemplate Library . It incorporates all three simulationmethods (FDTD, effective medium theory, and ray tracing)and allows one to use them for the same geometry of thestudied structure.
In the following, we consider surfaces coated by a periodicpyramid-type texture with height d. Pyramids bases have theshape of triangles, squares, hexagons, and circles (in the lastcase, the pyramid is, in fact, a cone) with the distance betweenthe base side and its center L (Fig. 1). The pyramids are clo-sely packed on the substrate in the triangular or square lattice
770 J. Opt. Soc. Am. A / Vol. 28, No. 5 / May 2011 Deinega et al.
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with the period . In the following, we specially distinguishtwo cases (Fig. 1): complete tiling case when pyramids basestouch each other at every point of their perimeters (this corre-sponds to the polygon base pyramids in our study) and incom-plete tiling case when there are gaps between bases (thiscorresponds to cones). Most of the results presented hereare calculated for the normal light incidence case. However,as shown below, generalization to oblique incidence ispossible.
The remaining paper is organized as follows. In Sections 2and 3, we estimate asymptotic behavior of the reflection withthe increasing height-to-base size ratio for the pyramids forlong and short wavelength limits. In Section 4, we considerantireflective properties of the textured surfaces at the wholerange of their sizes and find the location of the global reflec-tion minimum with respect to pyramid geometrical param-eters. In these three sections, we study the nondispersive caseconsidering glass-textured surfaces (the refractive indexn 1:5). In Section 5, we present results for silicon-texturedsurfaces used in solar cell technology. In Section 6, we sum-marize our results.
2. LONG WAVELENGTH LIMITIn the long wavelength limit , electromagnetic wavespropagate in a heterogeneous structure as in an anisotropicmedium of some effective (or homogenized) dielectric permit-tivity. Its value is determined by the shapes and the relativefractions of the structure components . In our case, tex-tured surface can be treated as a layer with gradually changingdielectric permittivity tensor z . Here, the z direction isaligned along the pyramid axis (see Fig. 2), with z 0 at thepyramid tops and z d at the pyramid bases. The z compo-nent of z is the average 
zz f zs 1 f zi; 1
where s is the pyramids permittivity, i is the incidentmedium permittivity, and f z is the filling fraction occupiedby the pyramid at z, which is equal to the ratio between thecross-sectional area of the pyramid and the area of the unitcell of the lattice. Other components of z depend on thepyramid base shape. For example, xz yz for somesymmetrical cases.
For a circle base, the MaxwellGarnett [20,21] expressioncan be used:
x y i 2f is i
s i f s i: 2
For a square shape, Brauer and Bryngdahl  proposed thefollowing empirical expression:
x y n 2n 2n=52; 3
n 1 f 1=2i f 1=2s ; 4
n2 1 f 1=2i f 1=2f 1=2
s 1 f
1=n2 1 f
f 1=2s 1 f 1=2i: 6
As was shown previously, an increase in the pyramid height dand a decrease in the optical contrast between the incidentmedium and the texture reduces the reflection [10,11]. Somespecial profiles f z were proposed to reduce the reflectionas well . In this section, we first estimate the asymptoticbehavior of the reflection coefficient with the increasing d andshow how it depends on the tiling of the pyramid bases.
In the case of complete tiling, the filling fraction at the topof the pyramid is f 0 0 and f d 1 at the base of thepyramid; therefore, 0 i and d s. For the normalincidence case, if the reflectivity is low, the expressionderived by Franceschetti is applicable :
where ~nz 1=2z Fi; s; f z is the effective refrac-tive index, F is the function defined by the pyramid base shapeand the incident wave polarization [e.g., see Eqs. (2) and (3)].If xz yz, F does not depend on the incident wavepolarization.
Fig. 1. (Color online) Antireflective textured surface: front andside view.
Fig. 2. In a long wavelength limit, textured surface can be treated asa layer with gradually changing dielectric permittivity tensor z.
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