Minimizing light reflection from dielectric textured surfacesfdtd.kintechlab.com/_media/deinega_-_minimizing_light_reflection... · Minimizing light reflection from dielectric textured

Minimizing light reflection from dielectrictextured surfaces

Alexei Deinega1 Ilya Valuev2 Boris Potapkin1 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 poblizostikintechru

Received January 4 2011 revised February 11 2011 accepted February 15 2011posted 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 ratiosExistence 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 casesThe FDTD results for these limits are in agreement with the corresponding approximations copy 2011 OpticalSociety of America

OCIS codes 3101210 0502770

1 INTRODUCTIONElimination of undesired reflection from optical surfaces isimportant for many technologies such as solar cell light-sensitive detectors lenses displays etc To reduce reflectionone can use single-layer quarter-wave coatings [1] but theirapplications are limited by a small range of wavelengthsand incidence angles To extend this range multilayer or gra-dient index coatings [2] can be used however their manufac-turing encounters problems with thermal mismatch adhesionand stability of thin-film stacks [3]

An alternative method for the reflection reduction consistsof texturing the surface with three-dimensional pyramids ortwo-dimensional grooves (gratings Fig 1) [45] Many reportson successful fabrication of the antireflective textured sur-faces have been published [36ndash9] 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 [1011] In theshort wavelength limiting case reflection reduction can beillustrated geometrically rays should be reflected many timesuntil being reverted back [12] At the same time transmittedrays deviate from the incident direction that leads to the lighttrapping effect used in solar cells [13] 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 [1415]

In our previous work we studied antireflective propertiesof dielectric textured surfaces at the whole wavelength range

including long and short wavelength limits [16] We foundoptimal parameters for the period of the texture and thepyramidsrsquo 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 [17] This method implements directnumerical solution of Maxwellrsquos 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 [17] For thispurpose we use the additional backing absorbing layers tech-nique that we described in [18] 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 [19] 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

1084-752911050770-08$15000 copy 2011 Optical Society of America

with the period Λ 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 Howeveras 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 frac14 15) 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 Λ ≪ λ 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 [20] In our case tex-tured surface can be treated as a layer with gradually changingdielectric permittivity tensor εethzTHORN [11] Here the z direction isaligned along the pyramid axis (see Fig 2) with z frac14 0 at thepyramid tops and z frac14 d at the pyramid bases The z compo-nent of εethzTHORN is the average [11]

εzethzTHORN frac14 f ethzTHORNεs thorn eth1 minus f ethzTHORNTHORNεi eth1THORN

where εs is the pyramidrsquos permittivity εi is the incidentmedium permittivity and f ethzTHORN 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 εethzTHORN depend on thepyramid base shape For example εxethzTHORN frac14 εyethzTHORN for somesymmetrical cases

For a circle base the MaxwellndashGarnett [2021] expressioncan be used

εx frac14 εy frac14 εi thorn 2f εiεs minus εi

εs thorn εi minus f ethεs minus εiTHORN eth2THORN

For a square shape Brauer and Bryngdahl [22] proposed thefollowing empirical expression

εx frac14 εy frac14 ethfrac12nthorn 2nthorn 2n=5THORN2 eth3THORN

where

n frac14 eth1 minus f THORNε1=2i thorn f ε1=2s eth4THORN

n2 frac14 eth1 minus f 1=2THORNεi thorn f 1=2f 1=2

εsthorn 1 minus f 1=2

εi

minus1 eth5THORN

1=n2

frac14 eth1 minus f 1=2THORNεi

thorn f 1=2

f 1=2εs thorn eth1 minus f 1=2THORNεi eth6THORN

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 [1011] Somespecial profiles f ethzTHORN were proposed to reduce the reflectionas well [10] 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 eth0THORN frac14 0 and f ethdTHORN frac14 1 at the base of thepyramid therefore εeth0THORN frac14 εi and εethdTHORN frac14 εs For the normalincidence case if the reflectivity ρ is low the expressionderived by Franceschetti is applicable [23]

ρ frac14 minus

Zd

0

12 ~n

d ~ndz

exp

minusi

4πλ

Zz

0~nethz0THORNdz0

dz eth7THORN

where ~nethzTHORN frac14 ε1=2ethzTHORN frac14 Fethεi εs f ethzTHORNTHORN is the effective refrac-tive index F is the function defined by the pyramid base shapeand the incident wave polarization [eg see Eqs (2) and (3)]If εxethzTHORN frac14 εyethzTHORN 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 εethzTHORN

Deinega et al Vol 28 No 5 May 2011 J Opt Soc Am A 771

Let us introduce a new integration variable g frac14 4πRz0 ~nethz0THORNdz0 Under the assumption of infinite differentiabilityof the function ~n and correspondingly h frac14 1

2 ~nd ~ndz

dzdg we can in-

tegrate Eq (7) by parts

ρ frac14 minus

ZgethdTHORN

0heminusig=λdg frac14

Xkfrac141

ethminusiTHORNkλkhethkminus1THORNeminusig=λgethdTHORN

0 eth8THORN

One can see from Eq (8) that the polynomial degree of thedependence of the reflectance R frac14 jρj2 on λ is determined bythe index k0 of the first nonzero item in Eq (8) as R sim λ2k0 Notethat for the effective medium limit the only scaling parameterof the system is d=λ so R sim ethd=λTHORNminus2k0 at different d and λ Forthe linear profile case f ethzTHORN frac14 z=d the first term in Eq (8) isnonzero due to h sim d ~n

dz simdfdz ne 0 therefore R sim ethd=λTHORNminus2

One can influence the rate of the reflectance reduction withthe growth of d=λ using a special profile f ethzTHORN For examplef ethzTHORN can be chosen as polynomial of the degree of eth2k0 minus 1THORNand f ethiTHORNeth0THORN frac14 f ethiTHORNethdTHORN frac14 0 for i lt k0 where f ethiTHORN is the i derivativeof f This leads to hethiTHORNeth0THORN frac14 hethiTHORNethdTHORN frac14 0 for i lt k0 minus 1 thereforethe first k0 minus 1 items in Eq (8) are zero and R sim ethd=λTHORNminus2k0 In particular for the profiles proposed by Southwell [10]f ethzTHORN frac14 3z2 minus 2z3 and f ethzTHORN frac14 10z3 minus 15z4 thorn 6z5 (in theseexpressions d frac14 1) we obtain R sim ethd=λTHORNminus4 and R sim ethd=λTHORNminus6correspondingly

Let us find a profile characterized by zero derivatives of allorders at the points 0 and d f ethiTHORNeth0THORN frac14 f ethiTHORNethdTHORN frac14 0 foralli gt 0Without any restriction assume that d frac14 1 Consider firstthe infinitely differentiable function eminusz

minus1eth1minuszTHORNminus1 which is zerowith all its derivatives at z frac14 0 and z frac14 1 After its integrationwe get a monotone function increasing from 0 to 1 f ethzTHORN frac14CRzo eminusζ

minus1eth1minusζTHORNminus1dζ where the value of C is chosen to ensuref eth1THORN frac14 1 Using this ldquointegralrdquo profile leads to the exponentialdecrease of the reflectance with the growth of d=λ due tof ethiTHORNeth0THORN frac14 f ethiTHORNethdTHORN frac14 0 foralli gt 0 The shapes of the discussed pro-files are presented in Fig 3

We calculated dependence of the reflectance on d=λ forgradient index layers corresponding to closely packed squarepyramids with flat-sided (f ethzTHORN frac14 z2 k0 frac14 1 because for flat-sided pyramids the width depends linearly on the heightand the filling fraction is proportional to the width squared)cubic (f ethzTHORN frac14 3z2 minus 2z3 k0 frac14 2) and quintic (f ethzTHORN frac14 10z3minus15z4 thorn 6z5 k0 frac14 3) profiles We used Eq (3) for the effectiverefractive index of square pyramids Calculations wereperformed using the 2 times 2 matrix technique described in [1]We obtained R sim ethd=λTHORNminus2 R sim ethd=λTHORNminus4 and R sim ethd=λTHORNminus6 forthe cases considered here (Fig 4 left)

We calculated the reflectance for the gradient index filmcorresponding to a single periodic grating with the integralprofile One can see that using this profile leads to an expo-nential decrease of the reflection with the growth of d=λ(Fig 4 right)

At the incomplete tiling case there are gaps betweenpyramid bases 0 lt f ethdTHORN lt 1 (see cones in Fig 1) causing adiscontinuity of the permittivity εethzTHORN at the pyramid basesz frac14 d εethdTHORN ne εi Because of this fact by increasing d=λ thereflectance tends to a constant value equal to the reflectancebetween the media with εethdTHORN and εi To demonstrate this wecalculated the reflectance for the gradient index film corre-sponding to cones closely packed in the triangular lattice(Fig 4 right) We used the MaxwellndashGarnett expression[Eq (2)] for the effective refractive index of cones

We performed FDTD calculations for pyramidally texturedsurfaces corresponding to the gradient index films consideredhere at a wavelength larger than the pyramid base size4 lt λ lt infin Λ frac14 1 d frac14 16 (FDTD mesh step δx frac14 001) Calcu-lations were performed using the subpixel smoothing method[2425] which allowed us to increase the accuracy comparedto the usual staircase model of the pyramids One can see fromFig 4 that the FDTD results for pyramids and the results fora layer with gradually changing refractive index are in goodagreement However a deviation of the FDTD results fromthe effective medium theory is still seen at large d=λ forpyramids with quintic profiles (k0 frac14 3) and gratings withthe integral profile It may be explained by the insufficientmesh resolution to represent the pointed pyramid tips char-acterizing the selected profiles

3 SHORT WAVELENGTH LIMITIn the short wavelength limit Λ ≫ λ the optical properties oftextured surfaces do not depend on the wavelength λ and aredefined by geometry only The ray tracing technique (eg see[14]) is widely used for their numerical modeling

We calculated the reflection for closely packed triangularhexagonal and square pyramids (complete tiling) and cones(incomplete tiling) for different values of d=Λ using thistechnique (Fig 5) We obtained exponential decrease of the

Fig 3 (Color online) Profiles f ethzTHORN frac14 z f ethzTHORN frac14 3z2 minus 2z3 f ethzTHORN frac1410z3 minus 15z4 thorn 6z5 and f ethzTHORN frac14 C

Rzo eminusζ

minus1eth1minusζTHORNminus1dζ

Fig 4 (Color online) Curves are the reflectance from a graded indexfilm with the optical properties corresponding to square pyramidswith flat-sided cubic and quintic profiles closely packed in thesquare lattice (left side) grating with the integral profile f ethzTHORN frac14CRzo eminusζ

minus1ethdminusζTHORNminus1dζ f ethdTHORN frac14 1 TE case cones closely packed in thetriangular lattice (right side) For the effective dielectric permittivityof square pyramids and cones Eqs (2) and (3) were used The FDTDcalculations (points) were performed for the corresponding pyramidswith Λ frac14 1 d frac14 16 and 4 lt λ lt infin (FDTD mesh step δx frac14 001)


reflection with the growth d=Λ for the complete tiling caseFor the incomplete tiling case the reflection tends to a con-stant value passing a local minimum while d=Λ increases Inthe following we give our explanation of these results

According to [26] we introduce the following ray classifi-cation (a) incident rays (b) reflected rays formed by incidentrays after their reflection from the texture which revert backinto the incident medium after some numberM of consecutivereflections (c) refracted rays formed by incident or reflectedrays after they get into the texture and (d) secondary raysformed by refracted rays if they leave the texture (Fig 6) Onlyreflected and secondary rays make contributions to the totalreflection R frac14 Rrefl thorn Rsec Here the partial reflections are de-fined as the total reflected intensity of the corresponding raytype divided by the incident light intensity Consider these par-tial contributions for the complete tiling case Each incidentray consists of two subrays with parallel or perpendicularpolarization relative to the plane of incidence (this plane isdefined by the ray and the normal to the pyramid side atthe point the ray moves onto it)

At the complete tiling case bases touch each other at everypoint of their perimeters Therefore the incidence plane doesnot change while a ray consecutively reflects between neigh-bouring sides [Fig 7(a)] Thus the total reflection is Rrefl frac14ethRrefl∥ thorn RreflperpTHORN=2 We assume that the rays with paralleland perpendicular polarizations make equal contributionsto the total incident light intensity (which is achieved in caseof uniform illumination)

Let us estimate Rreflperp As was shown in [26] during eachmth reflection the ray comes onto the pyramid side at theangle jπ=2 minus eth2m minus 1THORNβj where β is half of the angle betweensides of the neighboring pyramids (Fig 1) Therefore after mreflections the ray intensity is multiplied by Rperpethjπ=2minuseth2m minus 1THORNβjTHORN where RperpethϕTHORN is Fresnel reflection coefficientfor the incidence angle ϕ and perpendicular polarizationAccording to [26] the number of reflections required forthe ray to finally revert back into the incident medium is equalto M or M thorn 1 where

M le π=eth2βTHORN le M thorn 1 eth9THORN

Therefore

Rreflperp asympYMmfrac141

Rperpethjπ=2 minus eth2m minus 1THORNβjTHORN eth10THORN

Taking the logarithm of this expression and assuming that β issmall and M is large we obtain

lnRreflperp asympXMmfrac141

lnRperpethjπ=2 minus eth2m minus 1THORNβjTHORN asymp

asymp2Mπ

Z π=2

0lnRperpethϕTHORNdϕ asymp

1β

Z π=2

0lnRperpethϕTHORNdϕ

frac14 minusCβminus1 eth11THORNwhere C gt 0 since RperpethϕTHORN le 1 and lnRperpethϕTHORN le 0 Since β issmall β frac14 arctan L

d asympLd where L is the distance between the

base side and its center (Fig 1) As a result we obtain

Rreflperp asymp exp

minusC

dL

eth12THORN

R∥ethϕTHORN le RperpethϕTHORN for any ϕ therefore Rrefl∥ le Rreflperp (for somevalues of β Rreflperp can be zero since after some mthreflection a ray can move onto the pyramid side under theBrewsterrsquos angle) Therefore

Rrefl asymp exp

minusC

dL

eth13THORN

According to our calculations secondary rays make a smallcontribution to the reflection Rsec asymp Rrefl which can beexplained by the following considerations First pyramidsdeflect secondary rays downward since ns gt ni preventingthem from reverting back to the incident medium [Fig 7(b)]

Fig 5 (Color online) Curves are reflectance from different closelypacked structures in the geometric optics limit FDTD calculations(points) are performed for Λ=λ frac14 15

Fig 6 (Color online) Rays propagated in the texture (the ray tracingsimulation) Gray color intensity corresponds to the intensity ofthe rays

Fig 7 (Color online) Rays propagated in the texture


Second refracted rays can transmit to the substrate directly[Fig 7(c)] or move onto the inner pyramid side under the totalinternal reflection angle [Fig 7(d)] This prevents them fromforming secondary rays Therefore

R asymp expminusC

dL

eth14THORN

in agreement with the numerical results in Fig 5 since Ldepends linearly on Λ (Fig 1)

Note that results for hexagonal and square pyramids arealmost identical since their L is equal at fixed Λ (Fig 1)The curve tangent for square and triangular pyramids in Fig 5is approximately different by a factor of

ffiffiffi3

p This is because L

for a triangular pyramid isffiffiffi3

ptimes smaller than for a square

pyramid at fixed Λ (Fig 1)In the case of incomplete tiling (cones) the reflectance

tends to the constant value with the growth of d=Λ passingover a local minimum It can be explained by the followingconsiderations While d=Λ rarr infin normal rays remain almostparallel to the scatterer surface after the first reflection andsome of them go to the gap between the bases not reachingthe neighboring scatterer Afterward they are directly re-flected back into the incident medium [Fig 7(e)] For thecones case almost all incident rays behave in this way there-fore while d=Λ rarr infin the reflectance tends to the substratereflectance value

We performed FDTD calculations for the structures dis-cussed here at Λ=λ frac14 15 Since the FDTD mesh step shouldbe at least 10 times smaller than the wavelength calculationsrequired large FDTD mesh sizes and were performed on aparallel computer (ie calculations with the mesh 150 times 150 times300 were performed using 160 processing cores) FDTD andray tracing results are in good agreement with each other(Fig 5) The difference between them appearing by increasingd=Λ is due to the fact that the value of Λ=λ used in FDTDbecomes insufficiently large for light diffraction effects tobe neglected

4 FDTD RESULTS FOR ALL TEXTURESIZE-TO-WAVELENGTH RATIOSIn this section we perform a numerical parameter sweep inorder to find the global reflection minimum with respect totexture geometrical parameters We perform calculationsusing the FDTD method which is valid for all texture size-to-wavelength ratios including effective medium and geo-metric optics limits

We use a standard scheme of the FDTD calculation inwhich propagation of a plane wave impulse through the struc-ture is modeled During the numerical experiment the ampli-tude of the reflected wave is recorded and then it istransformed to the frequency representation and normalizedby the incident spectrum Finally we obtain the reflection atsome wavelength range corresponding to the simulated tem-poral width of the incident impulse w

We perform calculations for a fixed texture period valueΛ frac14 1 and a set of height values 0 lt d lt 5 We perform threenumerical experiments with different temporal widths of theincident impulse w frac14 001 01 1 for each geometry whichallows us to cover a large total wavelength span becausedifferent values ofw cover different wavelength ranges Since

in our nondispersive case Maxwellrsquos equations are scalinginvariant we rescale the results for the corresponding d=Λand frequency as if they were obtained at variable Λ=λ

Consider first the complete tiling case using the results forsquare pyramids (Fig 8) Let us fix some value d=Λ and followthe reflectance R behavior while Λ=λ increases (see corre-sponding black curves at Fig 8) By the increase of Λ=λ from0 to 1 the reflectance decreases achieving local minimum atΛ=λ sim 1 This can be explained by the fact that the effectivedielectric permittivity does not depend on the Λ=λ at zerothapproximation and the reflectance of corresponding gradientindex film decreases while d=λ frac14 ethd=ΛTHORNethΛ=λTHORN increases AtΛ=λ ge 1 the reflection decreases further passing local minimacorresponding to the values of Λ=λ at which the next diffrac-tion orders appear However these reflectance oscillationsbecome smaller at greater Λ=λ while the curve approachesthe geometric optics limit

Note that by greater fixed d=Λ the curve reaches this limitat greaterΛ since there are more reflections required for a rayto leave the texture according to Eq (9) Most of these reflec-tions are near pyramid bases and the path covered by a raybetween consecutive reflections becomes small If Λ=λ isinsufficiently large then this path is of the order of thewavelength This makes the geometric optics approximationinvalid and increases the effective Λ values for which diffrac-tion effects are still important In the case of incomplete tilingthe minimal reflectance is achieved at Λ of the order of the

Fig 8 Reflectance from closely packed pyramids with square basesas a function of Λ=λ and d=Λ (FDTD results)

Fig 9 Reflectance from cones closely packed in the triangularlattice as a function of Λ=λ and d=Λ (FDTD results)


wavelength (see results for cones closely packed in thetriangular lattice in Fig 9)

We come to a conclusion that the optimal texture size isdetermined by the character of the pyramid tiling In the com-plete tiling case the lowest reflection is reached by Λ=λ rarr infin

for a fixed ratio d=Λ This is in agreement with the fact that therate of the reflection reduction with increasing d and fixed Λand λ is exponential in the geometrical optics limit and is thushigher than the polynomial rate in the effective medium limitAt the same time small reflectance can be achieved at Λ ofthe order of the wavelength In the incomplete tiling case theoptimal value Λ is of the wavelength order This is also inagreement with the limiting approximations consideredabove

Note that high-precision fabrication of completely tiledmactotextured surfaces is a complicated technological taskFabrication of textured surfaces with periodicity of the orderof some hundred nanometers by lithographic technique oretching [7] seems to be more advantageous A possibility ofachieving very small reflection values for texture sizes ofthe wavelength order is the evidence of this texture efficiencyfor the visible range

5 OPTIMIZATION OF ANTIREFLECTIVESURFACE OF SOLAR CELLSIn this section we consider silicon antireflective textured sur-faces used for solar cell efficiency enhancement As followsfrom Section 4 at the incomplete tiling case the optimal scat-terer size should be of the order of the wavelength to ensurethe minimal reflectance for pure dielectrics In this section wedemonstrate the validity of this conclusion numerically forsilicon-textured surfaces also taking into account the disper-sion in the dielectric permittivity of silicon

Experimental data on the silicon dielectric permittivity εethωTHORNwere taken from [27] Usually the frequency dependence ofthe dielectric permittivity can be assigned in FDTD by consid-

ering a DrudendashLorentz approximation We find an approxima-tion of the silicon permittivity by three Lorentz terms

εethωTHORN frac14 εinfin thornX3pfrac141

Δεpω2p

ω2p minus 2iωγp minus ω2 eth15THORN

with the following parameter values (ωp and γp are in 1=μm)εinfin frac14 1 Δε1 frac14 8 Δε2 frac14 285 Δε3 frac14 minus0107 ω1 frac14 364 ω2 frac14276 ω3 frac14 173 γ1 frac14 0 γ2 frac14 0063 and γ3 frac14 25 There is ex-cellent agreement between tabular frequency dependence andour approximation in the visible range (Fig 10)

We have simulated the behavior of a silicon surface coveredby coneswith diameterΛ and height d To check the relevanceof this geometry to describe experimental structures theexperimental data taken from [3] (Λ frac14 015 μm d frac14 035 μmtriangular lattice) and [6] (Λ frac14 02 μm d frac14 052 μm squarelattice) were comparedwith the results calculated numericallyusing FDTD In these two works the fabrication of gratingsformed by cone-shaped nanoscale silicon pillars is considered(see images inFigs 11and12)and their reflectivity ismeasuredIn both works the filling fraction at the base of the cones f ethdTHORNdefining the gap between them is not specified and should beimplied from the images We calculated the reflectance of thecorresponding structures with different filling fraction valuesand the best agreement with the experimental data was foundto be f ethdTHORN frac14 09 for the first structure and f ethdTHORN frac14 055 for thesecond one (Figs 11 and 12) The FDTD results presented hereas well as all FDTD results in this section were obtained usingmesh steps δx frac14 10nm (using smaller mesh steps does notinfluence the FDTD results)

Now we switch to the numerical results obtained for coneswith different values of height d and diameter Λ We assumethat the cone bases are closely packed in the triangular lattice(filling fraction is f ethdTHORN frac14 π=eth2 ffiffiffi

3p THORN)

Fig 10 (Color online) Real and imaginary components of the silicondielectric permittivity comparison of the experimental data (dots)with approximation by three Lorentz terms (curves)

Fig 11 (Color online) Image of the textured surface investigated in[3] (left) Comparison of the corresponding experimental data withFDTD calculations (right)


Fig 13 (Color online) Reflectance for cones with diameter Λ frac1403 μm for different heights d (left) Reflectance for cones with heightd frac14 05 μm for different diametersΛ (right) Cones are arranged in theclosely packed triangular lattice One can see that the wavelengthrange in which the reflection reaches a minimum is shifted to the leftwith decreasing Λ The results are obtained by FDTD


We found that the reflectance R decreases with the growthof the cone height d (see the FDTD results for cones of diam-eter Λ frac14 03 μm in Fig 13) We also found that the minimalreflectance at fixed d is achieved at diameter Λ of the orderof the average wavelength between the free space and silicon(Fig 13) The wavelength in silicon is by a factor n frac14 Reeth ffiffiffiεp THORNless than the wavelength in free space (one can see fromFig 10 that this ratio is 4 for the visible range) Thereforethe optimal size Λ for the visible range (380ndash760nm) is aboutseveral hundred nanometers

Finally the minimal reflectance R is achievable at the diam-eter size of the order of the wavelength and when the heightis made as large as possible At the same time d asymp 05 μm isalready sufficient for the low reflectance value (sim1)

This conclusion remains valid for the oblique incidencecase as well (Fig 14) In this case the reflectance increaseswith the growth of the incidence angle θ When a p-polarizedwave is incident at an angle close to the Brewster angle thereflection of the substrate appears lower than of the texturedsurfaces It can be explained by the fact that in the latter casethe wave ceases to be p-polarized with respect to the sidecone surface However the reflection coefficient integratedover all angles is still smaller for the textured surface thanthe one of the substrate

6 CONCLUSIONIn this paper we consider antireflective properties of texturedsurfaces at the whole range of their size relative to the wave-length including long and short wavelength limits We use theFDTD method for direct numerical solution of Maxwellrsquosequations Also we apply effective medium and geometric op-tics approximations for the long and short wavelength limitscorrespondingly

We found asymptotic behavior of the reflectance with thechange of geometric parameters of scatterers composing thetexture for both limiting cases We established that key factorinfluencing the optimal scatterer size is the character of thetiling substrate of the scatterers In particular we showed thatthe textured surfaces with the periodicity of the order ofseveral hundred nanometers are highly efficient for the visiblerange In future work we plan to extend our study to theoblique light incidence with the help of the iterative FDTDtechnique [28]

ACKNOWLEDGMENTSThis work is supported by the Programs of Fundamental Re-search of the Presidium of RAS number 13 (coord Savin GI)and the Russian Federal Agency for Science and Innovations(contract 02523113014) Calculations were performedpartially on the cluster MVS-100K (JSCC RAS)

REFERENCES1 M Born and E Wolf Principles of Optics (Pergamon 1980)2 H A Macleod Thin Film Optical Filter (McGraw-Hill 1989)3 Y Kanamori M Sasaki and K Hane ldquoBroadband antireflection

gratings fabricated upon silicon substratesrdquo Opt Lett 241422ndash1424 (1999)

4 C G Bernhard ldquoStructural and functional adaptation in a visualsystemrdquo Endeavour 26 79ndash84 (1967)

5 P B Clapham and M C Hutley ldquoReduction of lens reflexion bythe lsquoMoth Eyersquo principlerdquo Nature 244 281ndash282 (1973)

6 Z Yu H Gao W Wu H Ge and S Y Chou ldquoFabrication of largearea subwavelength anti-reflection structures on Si using tri-layer resist nanoimprint lithography and lift-offrdquo J Vac SciTechnol B 21 2874ndash2877 (2003)

7 Y F Huang S Chattopadhyay Y J Jen C Y Peng T A LiuY K Hsu C L Pan H C Lo C H Hsu Y H Chang C S LeeK H Chen and L C Chen ldquoImproved broadband and quasi-omnidirectional anti-reflection properties with biomimetic sili-con nanostructuresrdquo Nat Nanotechnol 2 770ndash774 (2007)

8 H L Chen S Y Chuang C H Lin and Y H Lin ldquoUsing colloidallithography to fabricate and optimize sub-wavelength pyramidaland honeycomb structures in solar cellsrdquo Opt Express 1514793ndash14803 (2007)

9 Y M Song S Y Bae J S Yu and Y T Lee ldquoClosely packed andaspect-ratio-controlled antireflection subwavelength gratings onGaAs using a lenslike shape transferrdquo Opt Lett 34 1702ndash1704(2009)

10 W H Southwell ldquoPyramid-array surface-relief structures produ-cing antireflection index matching on optical surfacesrdquo J OptSoc Am A 8 549ndash553 (1991)

11 D H Raguin and G M Morris ldquoAnalysis of antireflection-structured surfaces with continuous one-dimensional surfaceprofilesrdquo Appl Opt 32 2582ndash2598 (1993)

12 B L Sopori and R A Pryor ldquoDesign of antireflection coatingsfor textured silicon solar cellsrdquo Sol Cells 8 249ndash261 (1983)

13 P Campbell and M A Green ldquoLight trapping properties of pyr-amidally textured surfacesrdquo J Appl Phys 62 243ndash249 (1987)

14 A A Abouelsaood S A El-Naggar and M Y Ghannam ldquoShapeand size dependence of the anti-reflective and light-trappingaction of periodic groovesrdquo Prog Photovolt Res Appl 10513ndash526 (2002)

15 F Llopis and I Tobias ldquoTexture profile and aspect ratio influ-ence on the front reflectance of solar cellsrdquo J Appl Phys 100124504 (2006)

16 A Deinega I Valuev B Potapkin and Y Lozovik ldquoAntireflec-tive properties of pyramidally textured surfacesrdquo Opt Lett 35106ndash108 (2010)

17 A Taflove and S H Hagness Computational Electrodynamicsthe Finite Difference Time-Domain Method (Artech House2005)

18 A Deinega and I Valuev ldquoLong-time behavior of PML absorbingboundaries for layered periodic structuresrdquo Comput PhysCommun 182 149ndash151 (2011)

19 Electromagnetic Template Library httpfdtd kintechlab com20 T C Choy Effective Medium Theory Principles and Applica-

tions (Clarendon 1999)21 F Wu and K W Whites ldquoComputation of static effective permit-

tivity for a multiphase lattice of cylindersrdquo Electromagnetics 2197ndash114 (2001)

22 R Braumluer and O Bryngdahl ldquoDesign of antireflection gratingswith approximate and rigorous methodsrdquo Appl Opt 337875ndash7882 (1994)

23 G Franceschetti ldquoScattering from plane layered mediardquo IEEETrans Antennas Propag 12 754ndash763 (1964)

24 A Farjadpour D Roundy A Rodriguez M Ibanescu P BermelJ D Joannopoulos S G Johnson and G Burr ldquoImproving

Fig 14 (Color online) Reflectance for bare substrate and cones withΛ frac14 03 μm d frac14 05 μm arranged in the closely packed triangular lat-tice for different angles of incidence and the wavelength λ frac14 05 μmResults were calculated by FDTD For oblique incidence calculationthe iterative FDTD technique [28] is applied


accuracy by subpixel smoothing in FDTDrdquo Opt Lett 31 2972ndash2974 (2006)

25 A Deinega and I Valuev ldquoSubpixel smoothing for conductiveand dispersive media in the FDTD methodrdquo Opt Lett 323429ndash3431 (2007)

26 O Bucci and G Franceschetti ldquoScattering from wedge-taperedabsorbersrdquo IEEE Trans Antennas Propag 19 96ndash104 (1971)

27 M A Green and M Keevers ldquoOptical properties ofintrinsic silicon at 300Krdquo Prog Photovoltaics 3 189ndash192(1995)

28 I Valuev A Deinega and S Belousov ldquoIterative technique foranalysis of periodic structures at oblique incidence in the finite-difference time-domain methodrdquo Opt Lett 33 1491ndash1493(2008)


with the period Λ 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 Howeveras 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 frac14 15) 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 Λ ≪ λ 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 [20] In our case tex-tured surface can be treated as a layer with gradually changingdielectric permittivity tensor εethzTHORN [11] Here the z direction isaligned along the pyramid axis (see Fig 2) with z frac14 0 at thepyramid tops and z frac14 d at the pyramid bases The z compo-nent of εethzTHORN is the average [11]

εzethzTHORN frac14 f ethzTHORNεs thorn eth1 minus f ethzTHORNTHORNεi eth1THORN

where εs is the pyramidrsquos permittivity εi is the incidentmedium permittivity and f ethzTHORN 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 εethzTHORN depend on thepyramid base shape For example εxethzTHORN frac14 εyethzTHORN for somesymmetrical cases

For a circle base the MaxwellndashGarnett [2021] expressioncan be used

εx frac14 εy frac14 εi thorn 2f εiεs minus εi

εs thorn εi minus f ethεs minus εiTHORN eth2THORN

For a square shape Brauer and Bryngdahl [22] proposed thefollowing empirical expression

εx frac14 εy frac14 ethfrac12nthorn 2nthorn 2n=5THORN2 eth3THORN

where

n frac14 eth1 minus f THORNε1=2i thorn f ε1=2s eth4THORN

n2 frac14 eth1 minus f 1=2THORNεi thorn f 1=2f 1=2

εsthorn 1 minus f 1=2

εi

minus1 eth5THORN

1=n2

frac14 eth1 minus f 1=2THORNεi

thorn f 1=2

f 1=2εs thorn eth1 minus f 1=2THORNεi eth6THORN

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 [1011] Somespecial profiles f ethzTHORN were proposed to reduce the reflectionas well [10] 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 eth0THORN frac14 0 and f ethdTHORN frac14 1 at the base of thepyramid therefore εeth0THORN frac14 εi and εethdTHORN frac14 εs For the normalincidence case if the reflectivity ρ is low the expressionderived by Franceschetti is applicable [23]

ρ frac14 minus

Zd

0

12 ~n

d ~ndz

exp

minusi

4πλ

Zz

0~nethz0THORNdz0

dz eth7THORN

where ~nethzTHORN frac14 ε1=2ethzTHORN frac14 Fethεi εs f ethzTHORNTHORN is the effective refrac-tive index F is the function defined by the pyramid base shapeand the incident wave polarization [eg see Eqs (2) and (3)]If εxethzTHORN frac14 εyethzTHORN 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 εethzTHORN



2 ~nd ~ndz

dzdg we can in-


ρ frac14 minus

ZgethdTHORN


Xkfrac141


0 eth8THORN














Rzo eminusζ










Therefore






asymp2Mπ

Z π=2


1β

Z π=2





Rreflperp asymp exp

minusC

dL

eth12THORN


Rrefl asymp exp

minusC

dL

eth13THORN







R asymp expminusC

dL

eth14THORN



ffiffiffi3

p This is because L























Δεpω2p





3p THORN)












































2 ~nd ~ndz

dzdg we can in-


ρ frac14 minus

ZgethdTHORN


Xkfrac141


0 eth8THORN














Rzo eminusζ










Therefore






asymp2Mπ

Z π=2


1β

Z π=2





Rreflperp asymp exp

minusC

dL

eth12THORN


Rrefl asymp exp

minusC

dL

eth13THORN







R asymp expminusC

dL

eth14THORN



ffiffiffi3

p This is because L























Δεpω2p





3p THORN)
















































Therefore






asymp2Mπ

Z π=2


1β

Z π=2





Rreflperp asymp exp

minusC

dL

eth12THORN


Rrefl asymp exp

minusC

dL

eth13THORN







R asymp expminusC

dL

eth14THORN



ffiffiffi3

p This is because L























Δεpω2p





3p THORN)












































R asymp expminusC

dL

eth14THORN



ffiffiffi3

p This is because L























Δεpω2p





3p THORN)



















































Δεpω2p





3p THORN)






















































































Documents

Minimizing light reflection from dielectric textured surfacesfdtd.kintechlab.com/_media/deinega_-_minimizing_light_reflection... · Minimizing light reflection from dielectric textured