Anisotropic protective coating forBrewster angle Windows

Gregory Surdutovich, Ritta Vitlina, and Vitor Baranauskas

We propose what we believe is a new type of dielectric anisotropic coating of arbitrary thickness that canprotect Brewster angle windows without degrading their optical quality. Such a coating may be fabri-cated as a multilayer two-component structure. The parameters of the structure, i.e., the dielectricpermittivities of the components and their concentrations, are calculated. For ZnSe windows twoexamples of anisotropic coatings are presented. The optical quality of the multilayer films does notdepend on their precise thickness, which makes them less sensitive to surface damage. © 1999 OpticalSociety of America

OCIS codes: 240.0310, 310.1210, 310.1620.

1. Introduction

Advanced technological applications are increasingthe demands for optical coatings of different types.Protective coatings are used to avoid degradation ofthe optical elements, to impede wear, and to resistscratching. As a rule, any protective films worsen tosome extent the qualitative characteristics of a sam-ple. This rule is also related to Brewster angle win-dows in high-power lasers, since the null reflectivityat the Brewster angle is an attribute of a flat interfacebetween two lossless homogeneous dielectrics.Therefore it seems impossible to retain the sub-strate’s Brewster angle quality for any nonideal-surface or film-covered sample. For a rough surfacethe shift of the Brewster angle was first discovered inRef. 1 in the course of numerical simulation and thenexplained in terms of the destructive interference ef-fect between the reflected and the coherently scat-tered fields.2,3 In these studies it was shown that fora low-relief rough surface the minimum of the reflec-tivity never reaches zero and occurs at a smaller an-gle of incidence, i.e., shifts the substrate’s Brewsterangle uB in the negative direction. In Ref. 4 Surdu-

The authors are with the Departamento de Semicondutores,Instrumentos e Fotonica, Faculdade de Engenharia Eletrica e deComputacao, Unicamp, CEP 13083-970, Campinas, Sao Paulo,Brazil. G. Surdutovich’s e-mail address is gregory@dsif.fee.unicamp.br.

Received 4 December 1998; revised manuscript received 22March 1999.

0003-6935y99y194172-05$15.00y0© 1999 Optical Society of America

4172 APPLIED OPTICS y Vol. 38, No. 19 y 1 July 1999

tovich et al. considered the effective anisotropy of asample produced by a thin isotropic film and ex-plained the shift of the Brewster angle in terms of theeffective anisotropy of a sample that is due to a low-relief surface roughness. Although an isotropic filmcannot retain the substrate’s Brewster angle quality,it is possible to design special coatings for which boththe position of the Brewster angle and a true zero ofthe reflection factor remain unchanged. To achievethis it is sufficient to use a uniaxial optical anisotropyof the coating film. Our aim in this paper is to showthat a uniaxial anisotropy, as an additional film pa-rameter, allows us to make both reflection coefficientszero at the ambient–film and the film–substrate in-terfaces and thus to attain zero reflectivity for a film-protected Brewster angle windows. An isotropicfilm does not present such a possibility and, as weshow in Section 2, always produces a shift from thesubstrate’s Brewster-angle position.

2. Thin Isotropic Film on an Isotropic Substrate

Let us consider a thin film on an isotropic substrateas a starting point for the treatment of thick films ormultilayer stacks. Any thin isotropic film with di-electric constant ε and thickness d ~much less thanthe wavelength! on a substrate with dielectric con-stant ε may be matched with an effective positive ornegative uniaxial anisotropic crystal. By postulat-ing the equality of the reflectances at the normalincidence and equating the pseudo-Brewster angle ofa real film-coated sample with the Brewster angle uB

e

of the effective anisotropic homogeneous crystal, oneobtains the parameters—the horizontal, εxx

e, and thevertical, εzz

e, components of the effective dielectric

itgvpitiD

ecdv

r

Twoame

εw

permittivity tensor, as well as the effective anisot-ropy, be 5 ~εzz

eyεxxe! 2 1—of such a semi-space-

equivalent anisotropic crystal. An anisotropy doesnot destroy the Brewster angle but only shifts itsposition with regard to an isotropic case. For ananisotropic crystal the position of the Brewster angleuB

e is given by ~see, for example, Refs. 4 and 5!

tan uBe 5 Fεxx

e~εxxe 2 1!~be 1 1!

εxxe~be 1 1! 2 1 G1y2

, (1)

so that the derivative

S]uB

]beDbe50

5 2Îε

2~ε2 2 1!(2)

s always negative. Therefore the negative ~posi-ive! anisotropy shifts, within itself, the Brewster an-le of the equivalent crystal to the right ~left!. Theertical inhomogeneity of a sample that is due to theresence of a thin film reveals itself both in the an-sotropy of the equivalent crystal and in the change ofhe horizontal component of the dielectric permittiv-ty tensor. Their contributions to the total shiftu 5 uB

e 2 uB of the Brewster angle are proportionalto the difference ε 2 ε and have opposite signs. Inprinciple they might compensate each other. How-ever, for the isotropic film the shift in the Brewsterangle that is due to the change dεxx 5 εxx 2 ε of thehorizontal component of the permittivity tensor dom-inates so that a total shift has the form4

Du 5k2

4ε3y2

~ε 1 1!2 ~ε 2 ε!, k 54pd

l, (3)

which always differs from zero. This is one reasonwhy it is impossible to guarantee the invariability ofthe Brewster angle position by use of isotropic coat-ings.

3. Method

The amplitude reflection coefficients r01 and r12 forthe p-polarized light on the interfaces ambient ~0!–uniaxial film ~1! and the uniaxial film–isotropic sub-strate ~2! are given by6

r01~u! 5~εxxεzz!

1y2 cos u 2 ~εzz 2 sin2 u!1y2

~εxxεzz!1y2cos u 1 ~εzz 2 sin2 u!1y2 ,

r12~u! 52 @εxxεzz~ε 2 sin2 u!#1y2 1 ε~εzz 2 sin2 u!1y2

@εxxεzz~ε 2 sin2 u!#1y2 1 ε~εzz 2 sin2 u!1y2 .

(4)

Here the optical axis z is perpendicular to the inter-faces, εzz Þ εxx are the components of the film’s di-lectric permittivity tensor, ε is the dielectriconstant of the substrate, and u is the angle of inci-ence in a vacuum ambient ε0 5 1. One can readilyerify this at the Brewster angle, uB, of a substrate,

where tan uB 5 =ε and Eqs. ~4! become identical; i.e.,

01~uB! 5 2r12~uB!. It is clear that, to reduce thereflection coefficient of the film-covered sample

r01 1 r12 exp@2id~d!#

1 1 r01r12 exp@2id~d!#

to zero for any film of thickness d, it is necessary toset both terms of Eqs. ~4! to zero. This requirementmay be satisfied at u 5 uB under the following uniqueconstraint on the horizontal and the vertical compo-nents of the dielectric permittivity tensor of a film:

εzz 5ε

ε 1 1 2 εxx. (5)

herefore it is sufficient to arrange a uniaxial filmith such a relation between tensor components tobtain a protective coating that keeps the positionnd a true zero of the Brewster angle. Equation ~5!ay be rewritten in terms of the anisotropy param-

ter b 5 ~εzzyεxx! 2 1:

b 5 2~ε 2 εxx!~εxx 2 1!

εxx~ε 1 1 2 εxx!. (6)

Condition ~5! is satisfied in two regions: under theconstraint ε , εxx , ε 1 1, i.e., positive anistropy b .0, or when εxx , ε and b is negative. Loci ε 5 constin the plane ~εxx, εzz! are shown in Fig. 1. For thechosen substrate, ε, the minimum achievable anisot-ropy bmin of the films is given by the formula

bmin 5 2~ε 2 1yε 1 1!2. (7)

Fig. 1. Curves ε 5 const and dashed lines b 5 bmin for each curve. At the curve ε 5 5.76 points A, B, and C correspond to filmsith the anistropy b 5 20.024 ~A!, 20.13 ~B!, 20.496 ~C!.

1 July 1999 y Vol. 38, No. 19 y APPLIED OPTICS 4173

wo

ptqfib

T

nt

p

tto

Ii

da

caE2

~

l

bt

4

From Fig. 1 it is clear that for any negative b . bminthere are two solutions in terms of εxx~b! and εzz~b!:

εzz~b! 5 1y2$~b 1 1!~ε 1 1! 6 @~b 1 1!2~ε 1 1!2

2 4ε~b 1 1!#1y2%, εxx~b! 5 @εzz~b!yb 1 1#, (8)

hereas for positive anisotropy there is always onlyne solution.Unfortunately, Eq. ~5! as a formal solution of the

roblem has only a limited value. The question ishis: How can one fabricate films with the re-uired anisotropy? Although an anisotropic thin-lm fabrication technique by oblique deposition,elieved to be novel, was developed recently,7,8 one

may hope to find or fabricate suitable materialswith the exactly prescribed properties, i.e., the val-ues of εxx and εzz, only occasionally. To overcomethis difficulty, one can, however, design a thin-layered structure with the needed anisotropy or ef-fective dielectric tensor ε.

4. Multilayer Structure

Any thin-layered structure ~assuming that each layeris much thinner than an optical wavelength! isknown to be equivalent to an effective homogeneousuniaxial anisotropic medium.9–11 This is true bothfor periodically and for stochastically distributed lay-ers.11 Moreover, all layered structures have a neg-ative effective anisotropy ~see, for example, Ref. 4!.

herefore it is possible to satisfy Eq. ~5! in the regionεxx , ε by use of a two-component composite. For

onabsorbing isotropic coating materials with dielec-ric constants ε1 and ε2 ~assume ε1 , ε2!, one can

apply the well-known thin laminar structure rela-tions4,9–11:

εxx 5 cε1 1 ~1 2 c!ε2, εzz21 5

cε1

1~1 2 c!

ε2, (9)

where c represents the volume fraction of the firstmaterial. By substitution of Eqs. ~9! into Eq. ~5! oneobtains an expression for c in terms of ε1 and ε2:

c 5 ε1

~ε 2 ε2!~ε2 2 1!

~ε2 2 ε1!~ε 2 ε1ε2!. (10)

Since this relationship has a physical sense only for0 # c # 1, then for a vacuum ambient Eq. ~10! has

hysical solutions only in the region ε1 # ε # ε2~where ε1ε2 . ε!. In the case of an arbitrary incidentmedium ~ε0 . 1! one can arrange a situation in whichε1 , ε0. Hence an additional physical region whereboth ε1 and ε2 are less than ε ~i.e., ε1ε2 , ε! appears.In Fig. 2 the first ~A! and the second ~B! regions areshown for a substrate with ε 5 5.76, the characteris-ic value of ZnSe in the IR. However, after substi-ution of Eq. ~10! into the first term of Eqs. ~9! webtain the relation

εxx 5 ε 2~ε 2 ε1!~ε2 2 ε!

ε1ε2 2 ε. (11)

174 APPLIED OPTICS y Vol. 38, No. 19 y 1 July 1999

n the physical region the second term is always pos-tive, so that εxx , ε, and therefore b , 0 ~see Fig. 2!.

There are two possible situations of practical inter-est:

~i! For fabrication of the needed coating from twochosen materials with dielectric permittivities ε1 andε2 one needs only to determine the concentration cfrom Eq. ~10!. In Fig. 3 the loci c are shown for two

ifferent substrates ε 5 2.25 and ε 5 5.76. Throughny point ~ε1, ε2! in the region 1 # ε1 # ε and ε2 $ ε

there passes only one locus; i.e., there is a uniquesolution for c. The maxima of all loci are along theline ε1 5 =ε. In Fig. 3~b! examples of the protectiveoatings on a Brewster angle window with ε 5 5.76re shown. Choosing ε1 5 4.93 and ε2 5 6.76 fromqs. ~10! and ~9!, one finds c 5 56.29% and b 50.024 ~point A1!. For the components ε1 5 4 and

ε2 5 9 one has c 5 68.57% and εxx 5 5.571, b 5 20.13point B1!.

~ii! If for a given substrate one needs to fabricate aayered coating with the prescribed anisotropy b,

then it is possible to do so only for b . bmin. Thevalue ε2 is

ε2 5εzz~b!

b 1 1 F1 1bε1

ε1 2 εzz~b!G , (12)

where εzz~b! is given by Eq. ~8!. This equation hasa solution only for ε1 , εzz. The curves b 5 const inFig. 3 corresponds to each point of loci ε 5 const inFig. 1. Each such curve has two branches thatcorrespond to the greater and the lesser solutions ofEq. ~8!. In Fig. 3~b! we drop the lesser solution

ranches of loci b as impractical, since they assumehe adoption of the components with ε1 close to 1.

The intersection of the greater solution branches of

Fig. 2. Regions ~shaded! of the physical solutions of Eq. ~10! for~A! a vacuum ambient, ε0 5 1, and ~B! an additional region for ageneral ambient, ε0 . 1.

tp

bema

ws

psSvcs~roav

t

curves b with loci c determines the parameters ε1and ε2 of the components and their proportion; i.e.,here is a one-to-one correspondence between theairs c, b and ε1, ε2, respectively. The sole excep-

tion is the point ~ε1 5 1, ε2 5 ε!, where all loci c andintersect. This point corresponds to the neutral

quilibrium of the vacuum-substrate’s materialixture, when Eq. ~10! is satisfied identically. For

ny pair of parameters ε1, ε2 one should determinethe concentration c @the intersection of curve ~12!

Fig. 3. Loci of the concentration c ~solid curves! in terms of ε1 andε2 starting from c 5 0.1 through the interval 0.1: ~a! ε 5 2.25, ~b!ε 5 5.76. Dashed curves b 5 const correspond to the greatersolution of Eq. ~8!; their asymptotes ε1 5 εzz equal ~A! 5.59, ~B! 4.84,and ~C! 1.704, respectively. Similar curves for the lesser solutionof Eq. ~8! are located in a narrow strip on the left-hand side fromthe line b 5 bmin and are not shown.

ith loci ~9!#. In Fig. 3~b! the curves of Eq. ~12! arehown by the dashed curves.

5. Accuracy of Preparation

During fabrication of the coatings the optical thick-ness rather than the geometric thickness is moni-tored,12 so that the concentration ce turns out to be alittle different from the concentration c calculatedwith Eq. ~10!. To estimate the influence of this effecton the Brewster angle reflectance, we expand thisreflectance R~c! in the parameter ce 2 c. The result

R~c! , ~ε1 1 ε2 2 ε 2 1!~ce 2 c!2 (13)

depends on the distance of a given point ~ε1, ε2! froma straight line ε1 1 ε2 5 ε 1 1 in Fig. 2. The common

oint ~ε1 5 1, ε2 5 ε! of the regions A and B corre-ponds to the neutral equilibrium point in Fig. 3.uch an imaginary fine-layer “substrate material–acuum sandwich” with arbitrary proportions of theomponents will retain zero reflectivity at the Brew-ter angle. The practical conclusion from relation13! is that to obtain the limiting insensitivity to er-ors in the materials’ concentration under fabricationf multilayer structures we should choose from allccessible components the material with a minimumalue of ε1 1 ε2, i.e., the nearest point to the line ε1 1

ε2 5 ε 1 1 in Fig. 2.Under practical employment of the multilayer two-

component stack there are two sources of error: theinherent model error that is due to the effective sub-stitution of the homogeneous anisotropic film by astack of layers with thicknesses ;Dd and the tech-nological limitation that is due to inaccurate fabrica-tion of the layers. The model error is connected withthe fact that all the effective tensor calculations maybe performed only in the electrostatic approximationwith the precision up to the first order in the param-eter Ddyl.13 However, for a given experimental ac-curacy ~relative error! x of each layer’s fabrication therelative error in the total thickness ~concentration! ofeach of the material’s components ~Dcyc) ; (xy=N!.This error decreases with an increase in the numberof the stack’s layers but, of course, there is no sense intrying to do it less than the inherent model error.Therefore the optimal choice of N will correspond tohe estimate ~xy=N! ; ~Ddyl! ; ~dylN!. Assuming

the stack’s total thickness D as ~Dyl! ' ~1y4! and x ;0.03, we obtain for Nopt ; ~Dylx!2 the value of ;102

layers. With such a stack the relative error in con-centration will be a few tenths of 1%.

6. Summary

Multilayered anisotropic coatings may be useful inapplications of Brewster angle windows within pow-erful lasers where the windows must be protectedfrom a harsh environment. A two-component com-posite is sufficient for preparation of the needed fine-layered structure. The adoption of three or agreater number of components opens more possibili-ties for fabrication of the structures with the pre-scribed anisotropy.

1 July 1999 y Vol. 38, No. 19 y APPLIED OPTICS 4175

A

6. R. M. Azzam and N. M. Bashara, “Ellipsometry and polarized

4

We are grateful to the referee for a valuable sugges-tion. This study was supported by the Fundacao de

mparo a Pesquisa do Estado de Sao Paulo, Brazil.

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