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THEORETICAL AND MATHEMATICAL PHYSICS

Reflection and scattering of light from a three-layer structure with two rough interfaces

A. V. Fedorov and Yu. V. RozhdestvenskiT

All-Russian Science Center at the S. L Vavilov Institute, St. Petersburg

(Submitted April 17, 1992)

Zh. Tekh. Fiz, 63(3),1-13 (March 1993)

The s",lall-perturbation metho~ is used to solve the boundary-value problem describing the scattering of light waves m a s~stem of three medIa separated by rough surfaces. Explicit expressions are obtained for the average and fluctuatmg fields. The elhpsometric parameters ~ and IlJ are constructed. The possibility of using these results to determine optically the statistical parameters of rough surfaces is discussed.

INTRODUCTION

The extensive use of layered structures and superlattices lllk'iliding quantum superlattices) in microwave and optical ,'ic'(tronics. and also the trend toward microminiaturization \1~lIifican[ly increase the requirements on the quality of the ,cclll11<::tric characteristics of interfaces between media. Indeed, (PI' (omplex multilayer elements with characteristic linear dinlensions of order 100 A, inter,facial roughness significantly ,!il(cts the electrophysical and optical properties of the de­\1I.:t..:.

It is well known that the roughness of a surface ,can vary ",id~l: as a result of etching, oxidation, epitaxy, and so on.

I!<:ncc it is necessary to be able to quickly and accurately ,k:t~rmine the statistical parameters of rough surfaces such as :IIC ~ranclard deviation 0' of tile heights of the irregularities "nd thc correlation length L.

TI;ere are several methods of determining the geometri­cal characteristics of surfaces. Optical methods are the most promising in our opinion, because they allow nondestructive

mal scattering problem was solved using perturbation theory and rules for the calculation of rhe terms of the Born series to arbitrary finite order were given. In the opposite limiting case, when min(O', L)'k » I, one uses the Kirchhoff meth­od, which is the quasiclassical approximation for the scatter­ing problem, The two-scale model' is used when the surfaces have large-scale irregularities modulated by small "ripples".

Previous papers have considered the scattering of light on a single rough surface. This assumption significantly limits the applicability of the results, since it describes the interac­

lion of light only in the case of \ery pure metallic or large, strongly absorbing objects, where the effect of the back sur­

face can be neglected. But thin organic or oxide tilms form naturally on the surfaces of materials, and therefore :lny scattering object has at least two rougll surfaces wllich'make comparable contributions to the photoresponse, Multilaver structures, such as insulator - semiconductor structures, wh'ich are essential elements in charge-coupled devices and field-effect transistors, and metal- insulator - semiconductor struc­

icsting, ,·\n opticaLmethod.....llJUch.......is [eliltiyeJ.~ __slmp-le_ and. __tl~r~s a~e also of great interest. highly sensitive is the ellipsometric method, in \vhich the phasc rclations between retlected (scattered) light waves of dirr~rciH polarizations are analyzed. An ellipsometric measur· in)! system. together with computer programs 10 analyze the cxpcrimcntal data. provides a unique capability of monitoring (cchilolo)!ic~tl rrocesses in re:li rime, However the results o!ltaiil<:d wili be relevant to the re:ll situation onl\' when the C'~I'l'cr or tile inrcrt':lcial rou~ill1\::ss nn Ihe rctlecti~n (scatter· lilg) 01' light is laken into ace-ount.

The lntcrae:tilll1 of different types 01' waves wirll rough ,mr;\ces oc'curs III :nanv fields 01' phvsies slldl ;IS ;\Coustics. :',:diopl1\"lc", \lptics, 11H;lecular !lhv.sic~, ~lnc! so Oil [II s!Jite of ;[s ml'I'C ::~;lll 'r)() \ear hlSlOf\', '111e :JrOh\cl11 i~ [':lr irom a ::lLd '\'iUllO;} i'i,c :;letilOds :,,'cd 10 ;;u,\ck ,11l~ probiem C;lIl

"'c' c::i"i!:','\\ ;1,:1' ';l~e..: gmuLJs' ;;~c sm;til'perturi),lIlon l11elllOd. ',il\;i1hll:' , ':lerIIPd. ;1I1d :i,c :1\,'-se;1Ic modei,' ri1~ sm:lll· ,:',':':,,(1):11:(1;1 ::,CII\l'd I~ I:le L)~~llllSllricd 111;\l11CI11'1ti-:all\, It !S ''::\L'(: . H1 :..'\i.'::!l~:\)!~ ,l! ',;~c.: :--c!~c..;.•..'tet.i I :-"c3ll~rcd I '.\ a\\?s ill ~1

"ic' '.',\"() l):Lr~lJ1ier<:rs "'/' <-::< ! JIHi ,f',' « !. , ,,\,' C:-::('~ :,-: ,:1';lii-Dcrturh:\llllI1 !'1eli1l1d

:,,::", "'llrl-\CU \\l:i :.~ i~l:'(\ '-_ '"'Il'O:.'rc :i'll: j"clr-

In the present paper we conSider tile inter:lction of light waves with two statistically independent and statistically uniform rough surfaces. Explicit expressions are obtained for the e1lipsometric parameters for an isotropic autocorrelation function of arbitrary form. For appropriate values of the parameters the results go o\'er to the known expressions for a single surface, The c!epen(leilee of rhe ellipsometric parame­ters on tile angle \)r incidellce (3 ;Ind on the sl:lnc!:lrc! deviation <J of the heights (Ind the correlation kngtil L is studied (Figs. ~·8), ,,\s a com[larison with previous resulrs. calculations are done for a Gaussian :lutocorrelation function \n rhe case of a single surface. The practical :lpplic:ltion of our metilod to the determination of surraee fOugllness ant! its extension to starts­ilcallv :lilisotropie :Ind correl:ltcd Illtcrt';lces arc also discussec!,

LIGIIT SC\ TTERI"iG I.\Y ROlGIl SL'RL\CES

We c\)llsitler ~n.:: sc::tterillg ut' a 111allc !IH)nochromatic '1~11t \\'a\'e oI (reuucncy 0J Incident :It all angle l-) on a system I Fig, i) o( three ;ned!:I with complex dielectriC cOlIst:lnts Gm

(IIi = I, ~. ,~l. I'he sur(ac~s bell\'een tile n1cdia are on a\'er­:::!-c oaral!ei 10 onc another :Inti .scnarat<:'t! bv a disrance d. The

'063,7842193,03 016i,05 S~OCO ".;) , 994 Amenr.an Institute of Prwslcs 16'

z

FIG. I. Scallering of light waves by an object with two rough surfaces: (J;

i~ the standard deviation of the heights of the irregularities. L; is Ihe

correlation length (i = t, 2), {;/11 is the dielectric constant of the medium

(11/ = I, 2, 3), e is the angle of incidence of Ihe light wave, d is the

effective thickness of the second medium.

surfaces are described by the random normal functions !,,(x). It is well known 6 that fn(x) is completely described by its average value (f,,(x»), its variance a} = (fn2(x»), and its auto­correlation function Wn(x Ix2) = an - 2(f,,(x 1}fn(X2»)' Here and below the angle brackets denote a statistical average of a ranuol1l quantity with respect to tile appropriate probability dc:nsitv. 1

To calculate the electromagnetic fields in the system it is necessary to solve a boundary-value problem. We assume that

(1)

where c is the speed of light in vacuum and L" is the smallest of tile correlation lengths for the autocorrelation function. It follows from (I) that we can use the small-perturbation meth­od. Til.: c'kctromagnetic tields E"I' H,,, in the mtll'mediul1l are wrillen as sums of average em, 1tm and tluctuating eill • hll/ cOl1lponents i.2

£ .., = (En,). 1t,,, = (H on ), (em) = (hm) = 0,

where. in view of the assumed statistical uniformity of the surfaces. tile average tields £"" 1tm obey Snell's law and have ~'iC same form as the fields in the case of plane bound-

I,or

J.751

j

I lor- =

,~()

9

. 62 Technical PhYSICS 38131. Marcn ; 993

aries. Unlike the latter case, however, here the reflection Rill and transmission T,1l coefficients of the average tield are not given by the simple Fresnel formulas,7 but are unknown quantities to be determined as part of the solution of the boundary-value problem.

Using the representation (2), the linearity of Maxwell's equations, and the abov8 inequalities, we expand the bound­ary conditions in the small parameters and obtain a closed system of equations for the retlection and transmission coeffi­cients of the average fields and the amplitudes of the fluctuat­ing fields. In the coordinate system defined with respect to the plane of incidence (Fig. I) these equations have the fol­lowing form to the lowest nonvanishing order in the small parameters, assuming that the first and second surfaces are statistically independent:

{Oi-t (Z)(£i.o(X, z) - £i-l.Q(X, z))

a + (fi-t(X)az(ei.Q(x,z) - Ci_l.Q(X,z»

+ri-l.o(X)(Ci,=(X,z) - ei_l.=(X,z»)) }I===._. =0,

{lii_dz)(1ti.o(X, z) -1ti-l.o(X, z) + (fi-l(X) :;/hi.O(X, z)

-hi-l.o(x. z))) ]1-- = 0, (3) ¥- ... -t

and

{Ci.O(X, z) - ei-t.o(X. z) + A,_t.o(x. =) }I===._. = 0..

{hi.O(x, z) - h i - 1,o(X,,;;) + Bi-l.O(x,,;;)} I===,_. = 0,

div(em(x,=) = 0, (4)

where

'y' (x) _ afi(x) I.y - ay ,

a=x.y; i=2.1; m = 1. 2, J; -- ----- -_._~ ._----------­

and the functions Ai,,,,(x. :::) and Bi.",(x. :::) are

iJ .-li.,,(x. =) = fi(x) u= (£,.;.t.,,(x. =) - £,.,,(x. =)) + "Ii,o(X)

x (£,+I.=(X. =) - £,Ax. =)).

iJ . 't..I)1lJi",(x.:) = f,(x)-:-I_l1t, ... :.,,(x. =) - fLi,o(X.,;; . (5), ­

fiG. 2. Der;;:'IH.ieon ...'C: nl" till:' L'lIif':-\~Jlleln~' p;tr:tJlld~r:-; ~. 0/ lin lh~ ;m:k of Hh.:u.lenl.:(.· B Itlf ;lO lIn.icl.:t \\.Hh lIne: ;..t1rf~h.:C;-;: I}! = 0.1. I.! = J .0. C1 =

IU - /'[\.0 1:1Ir). C' = l:i ~ti - '·(1. iO.. lSi). Ii",' = 1')5 c\'.

162A. V. Fedorov and Yu. V. RozhaestvensKIi

1'­

I I !

..'~

II

ot

'n Ie

's )­

d i­t­

o

e

~'. ".

,1~.6l I \ tJ

I \ O.It I \/ r~ 0.2 // ,t

!-/ 2/ J _--- ,/ I'"--- --.,...._ ........

0.0 F-:;.............~.=-=::.=-..::;;~""""-';.:=-'=...==:;......---::': ....\'oo:-=!~--:_~~-55 55 75 \ ,... ---85' 8

'- ......

FIG. J. Chang~ in th~ dlipsomelric param~l~r !l uue to roughn~ss (otl =

j - ~\l' wh~r~ !lo corresponus 10 the abs~nc~ of roughn~ss) as a function

"I' the angle of inciuence e for aj = 0'.1. ClIrv~ I: L 1 = 1.0, cnrv~ 2: L 1

c.O. cllr\'~ 3: L 1

= 4.0. The other paramel~rs are the same as in Fig.

The retlection and transmISSIOn coefficients of the aver­

;If:e fields and the amplitudes of the fluctuating fields are

d<:terhlined from (3) and (4) as follows. First the system of

l'quation (4) is solved, treating Ai.,)x, z) and Bi.c.(x, z) as

known quantities. This is conveniently done in the wave­l'l'Clor representation, which is related to the coordinate rep­

resentation by the two-dimensional Fourier transform

+'>0

iiJ2 iqxe2(x . .:-) == (2~F Jdq [ei'(q)e;iJ2= +e;'(q)e =] . e ,

-'>0

+?O

J dqfJe·\(x . .:-) == --eJ(q)e' ,=+'qx {J I . -- (k2i_ q2

)1/2 1{211')2 '

-'X>

+00 iqx

.·l,.,,(x,.:-) == J(:~2 A;,,,(q)c ,

-?O

+00

fI,,,(x,z) == J('2d:)2 Bi.o(q)e,qx. (6)

-00

In writing down the tluctuating fields in the first and

third media we have used the Raykigh hypothesis. As was

(" I " .......

0.2 I ""

~ I "" ~1~ 1( 2 "

I I f----...--.. "­~ J ........ " \ I 50' :'D' "-.Sl.I&-----,1.0! . _ . ------~ M ~ :-------__ /1

·).1k --- /

I:.. ,/

i-I{; -t. ~ ;;.10°_';: in ill,,' ::.::pSIIIIl:-:n.: par:lllld~r '-it ;.)y ~-: IV 'V".' '.\ ih:rc

·i·:·.· ... pl:nu' :,l ::!:: ::" n,.:~,: ,'i ;".·u-;lll1c:::-. .... \ ~l=" ,I :lll~,:(I()li I>t tll~ ;ln~'k ll(

.~\,.·IJ;,.·IL;,: (.) :.',[ .;. :-·:1:: ::l;;'7")~'Cr' i.lh~lln~ In;: ~~lr\~" "'·,'lr'-· ... pnnJ iu

tl:(,,;" ":;\1111: ,:orrd:Hh)n i·.·r.·:-':!":..: ;1" ir. F=:-; . .;.

0.6 8Jl I~ I \ ",I II II / I ~

/ f ~

0."

0.2 / / t l\ 1//a/J\\ _ ---:::::/ I

-- _- ./ \1 8,5'----=:;:-.;::::;--.----,..- ­ ,1 ,

FIG. 5. Chang<\ in Ih~ ellipsom~tric parameter .'l (oil = :l - ~) as a

function of ch<\ angl~ of inciu~nc~ e for LI = 1.0. Curve I: a l = o. I,

curv~ 2: a\ = 0.08, Curve 3: a\ = 0.05.

shown in Ref. 8, this is a good approximation if 'Yi.",(x) Z

anlLn ~ 0.4. In our case this inequality is satisfied, since

anlLn is a small parameter of the theory [Eq. (1)]. The solu­

tion of (4) in the wave-vector representation is conveniently

written in the form

eliJ(q) == L [C1iJ.i"Ai,o(q) + DIfJ.ioBi,o(q)] , itO

e2iJ(q) == L [Ct.o,."Ai.O(q) + DiO.;"Bi.,,(ql] , '.0

i,o

(7)i == 1,'2; Q == x,y; 13 == x,y,Z.

The coefficients C"f3,i'" , D"f3,i", are given In Appendix A. We next substitute expressions (6) and (7) for the fluctuating

fields into system (3), which becomes closed by virtue of (5).

The equations were solved for s- and p-polarized incident

light waves. The results in the general case are quite compli ­cated, and we therefore assume statistically isotropic surfaces

and present the explicit expressions only for the average

fields E~~), 1t~~), in the first medium, since Ihey are need­

ed to construct the ellipsometric parameters '¥ and ~. Be­

cause of the statistical isotropy of the surface, for ail s-polar­

ized incident wave the nonzero components of Ihe average

tields In the tirst medium are

E~y(x, z) == 'R.IJEoexp[ik1rx - ik1:z],

1t;:(x . .:-) =E;)x . .:-)cos0. 1t;jx. .:-) =E;y(x . .:-)sin0. (8)

and f0r a p-polarizcd wa\'e the Iwnzcro componelHs are

1tfy(x.:) == 'R. 1;,/16'pxp[iklr I - ik l ::],

£·;·,.(x.:) == - (0" (-)1ify(x . .:-). k lx == ko sin 0.

E;·:(x . .:-) == -1ify(x. .:-). sin 0. kl : == (kl - kij/~, (9)

kn = ..; i c.

where Eo' and fir{' are the amplitude of the incident waves of

!x'iarizaiions s ;~nd p ;lnu 'R. I ,. R,ll' ;m;: the retlection

,'Oerri~ients t'or tile :J\'eragc tielcis.

Tile ellirsollletri..: paramerers 'Jr ,mel ~ can be written in

:e~lll~ ,11 'R.l.,. 'R.;; ill the usual \I'av

,:,.. V. F·o10rov and Yu. V. Rnzhdeslvenskll 153

-------­--------­-----­ .,/ -­

0.2

0.1

-0.1 -----/ ./

FIG. 6. Change in the ellipsometric parameter 'it (o'it = 'it - 'ito) as a function of the angle of incidence e for L1 = 1.0. Curves \·3 are for the

sam" values of al as in Fig. 5.

l:he ref1ection coefficients 'R.1• and 'R.1p have the form

Q =s,p (11)

because of the multiple-wave interference effect. Because of

the surface roughness, the ret1ection coefficients R"JIl CI. of a single surface are not antisymmetric under permutation of the indices /I and 111:

R:2 = r:z(1 - 2aik1:(kh + Mlr )),

R~I = rid 1 - 2aik2:(k l : - M lr )),

R~3 = ri3(1- 2aik2:(k3: - M2r )),

R:2 Ril - Tt'2T;1 = - [I - 2ai(kl: - k 2:)M1r ] ,

p p • 2 • [ kz: - krr N k k2z

P kL 'vil~ ] ,RI2 =T 1Z - 2(71S12kl: kf 1 - 2 Ir 1 + k~ 1

p p • 2 [ kr r k kL 'vi ]RZI = r ZI - 2a l s Zl k2: k1: + k~ :VI - 2 -lrkl:PI - kfl I~ ,

Rp ? .) 2 k. [I.' + kr x \' 2k k p kL M ]'!3 = r23 -"a2 sZ3 2: 3: k~' 2 - Ir 3: 2 - k~ 2~,

TizTft - RjzR~l = 1+ 2(7;.<12 (~IZ: + :22:) (kixNI + klzk2:MI~)' 1 2 (12)

In (12) we have introduced the notation

k?l k~1l (k~,: - k;) kH : - k m :

[kn.:k~l + k~'l:k~P k,..: + k m :'I

0.81\

I " O·lt-If--. ....... ~ -- ---. ---

2

-..._--- --­O. 0 --,-'-,---------o---:---------LC------'-:.==--==-=::JI-I.:=:.=:.=:..=-~-==

I 0.1 C.:-_ ----------Zt­

-O·~r, ,J/-- -­-(1,8 ~ /

/

II I

-'!.ltV.

, \

J\ \ \ O.J

'\ '\.

"........ .......- --­2\ \

- - - -- - - 0.8 __ 0.1 -:1_0.2 O.¥­

FIG. 8. Change in the ellipsometric parameter 'it due to roughness (o'it 'it - 'ito) as a function of the correlation length L 1 for a, = 10- 2 The

angleS of incidence and the other parameters ar" the sam" as in Fig. 7.

kn:"-';'l - A'm::J..:?l {Il.m} = 1.2.3. (13) A.'m:: A:;! + kll ::k;1l .

The quantities MnCl.' Nn, P" (/I = 1, 2; Ct x, y) are two-dimensional integrals of the form

{.\1.",. S". p,.} (14)

-i-'X­

= J(.~-q zll",(CJr - k1r . qy) {.I/",,(q). S,,(q), P,,(q)}.~" )

where WIl(qx - k l". q,.) is the Fourier transform of the ailto­correlation function or' the rirsl (11 = 1) or second (/I = 2)

surface. and explicit expressions for the functions .HIlCl.(q).

Nn(q). PIl(q) are gi\'en in Appendix B. It follolVs from (II) that when the variances of the

heights of the tirst and second interfaces are set e:qual to

zero, the retlection coenicienrs 'R. 1" reduce to tile results (see Ret'. 7, for example) for multilayer interference: in the absence of roughness. As in the case of plane boundaries

there are three different ways of obtaining the re:tlection coefticients for a single interfacial surface from (II). For C)

= c2' Eq. (II) describes the ret1ection of light from the boundary between the tirst and second media. and for ci =

c2' Eq. (11) describes the retlection of light from the: bound­ary between the second and third media. As in the absence of roughness. the ret1ection coefficients in the second case have the additional pllase factor exp(i2kl~ because the: boundary lies a distance cI from lh~ origin of th~ coordinate: system.

The third \\'ay of obt:1ining the rcllection cOefticieIHs for a single surface from (l i) IS (0 set the thickness of the second mediul11 cI equal ro zero and to equate the variances of the heights and aUlocorrel:Hion functions of tile t'irst and se:cond surfaces, Th~n Eq. (II) will lkscrihe: rhe relkction llf light

from the bouIH..ian' het\leen the rirs! and third me:dia. The reflection codtici~iHs ,'or a )ingk surface obtained by the ;tbOl't' mctllOds ;lgree Illth rhe \":nolln results (s~e R~r', J for

e\<lmflkl if rhe\' are \lTi!len In lh~ ~<lme notation.

J)[SCLSSIO~ OF THE RESLLTS

'l'l) nroceed fUrl her Ille e:q)licil r'l1 rm of the allillcorrela­:,on r'llncnon 11111~1 be sDecllie:d, \\·c as)llmc tbt the: <lutocor­,~I;lIl11n ,",Incill'ns ot [;',e ;irst and ,econcl in:crf;lCl<l1 )urr'aces

164~ 'J. Fecof0v and Yu. v. ~.::zndeStverS<ll

..... 1 I

,­ ar~ single-parameter Gaussians of the form

n = 1,2. (15)

This form is chosen because it is often used in models of randomly rough surfaces in both theoretical calculations and in the interpretation of experimental data. 9 In addition. when (15) is put into (14) of the integrations can be performed explicitly, which greatly simplifies the numerical calculation of the rellection coefficients and the ellipsometric parameters.

In the numerical calculations the parameters a and L are measured in units of ko = w/c. Figures 2-8 show the numeri­cal results for the ellipsometric parameters Cl. '1' and the cllanges in these parameters oCl = Cl - Lio• 0'1' = '1' - '1'0 (~o. '1'0 are the ellipsometric parameters in the absence of roughness) .as functions of the angle of incidence 9. the standard de~iation a of the heights. and the correlation length L for a single rough surface e2 = eJ. A single surface was assumed so that our results could be compared to previous r~sults2-4 and also to more easily demonstrate the main quali­

tative features of the rellection (scattering) of light from a

rough surface. The dependence of Li and '1' on the angle of incidence 9

of the light wave is shown in Fig. 2. We see that the pres­~nce of roughness is equivalent to additional extinction result­Ing from light-scattering by the surface irregularities. [n the case· of absorbing media surface roughness does not change the qualitative forms of Li(e) and; '1'(9). But for transparent media surface roughness leads to a deviation of Cl(9) from a st~p function in the angle of incidence e, and min '1'(e) ;z; O.

Tile changes in the ellipsometric parameters oLi, 0'1' due to the presence of surface roughness are shown in Figs. 3-6. As would be expected. oLi and 0'1' vary most strongly with e n~:lr the effective Brewster angle 7 The absolute values of oCl, a'It decrease with increasing correlation length L (Figs. 3 and 4) or with decreasing standard deviation a of the heights (Figs. 5 and 6). The dependence of 0Li, 0'1' on a is obvious :1Ild the dependence on L is connected with the fact that, in the limit L -+ co. Eq.-ttl) beeotfteS;--------­

and therefore from (10) the ellipsometric parameters Cl and 'It do not depend on the roughness parameters. This is seen more clearly in Figs. 7 and 8. \vllich sho\\' tile dependence of

;)~. o'lt on the correlation length L .

. CONCLLSION

Let us summarize the In:lin results of this paper. Light sClll~ring (rom a D:lir of rough surfaces h:lS been considered '1:',[ ng the small-I)erturo:ltion method. [,pressions ha \ e been lli)l:lined !'l1f the J\ ~rage ;\nd ,luL'tuJting rields ;\nd aiso :Of

the rerlectlon and transnllssion coe(ricients for .1- ;\Ild ()-polar­:/-.:d lnc\ck:n \\;\\~s ,I:> (unctions u( the roughne~s l)aramet~rs.

t!\e et'r'<:ctl\e thiCKness or' rile I·iim. ;\nd ih~ <Ingle l)r' Incidence 01' tile ii~i1i t,)r ,·(1::1pie.\ dielce.:tric constallts. V,mous phvsic;\l

L!.lldlltl~il..:S "lh:il J~ :ht.: ~C~lt[C:::lg. ilil..!lC,1Irl\ .1I1d ;ilC S(OKeS ;\!r~\ilh:~cr~ \.·.ii~ .~~ ·':)~~~!ned ;'il'n~ ::h~'c ;..·.\prc.)slon~. in :1<) n il..:­l:i~H. ',\~ \::~k'Ul~:[C'U ,:':~ :...·'ilp~uinctf1..: .\~r;~lnC\l:r~ j. ,.:iU +

which can be used to solve the inverse scattering problell1 lo , II

to determine the statistical parameters of the surface irregu­larities. A computer program has been written to automate the ellipsometric system for this task.

A generalization of the approach considered here is to solve the direct scattering problem in the case of a large number of rough interfacial surfaces with possible correIa, tions between the irregularities on di fferent surfaces. Another

generalization is to remove the restriction T·k « I. [t can be shown 12 that the only necessary condition for the smail-per­

;""

turbation method is T/L « I. Therefore one can attempt to solve the direct scattering problem for arbitrary values of the ratio of the irregularity heights to the irregularity heights to the wavelength. The solution to this problem would reduce to the solutions obtained by Kirchhoff's method and the usual small-perturbation method in the appropriate limiting cases.

APPENDIX A

The coefficients determining tile tluctuating fields (7) are

oo C(+) C(-)C1...,,;0-= Ji...,n+ 2-y,io+ 2--Y,iQl

':.'I·

(3,C,:.io = qrC'r,io + qyC1y,io.

/)lDlz,io = qrD1r,io +qyD\y,io.

('1-,.io = UJ-I (q){02i O..,0 + [;2(q)C~~.),o + U2-I(q)C~~\,] •

Dhio = UJ-1(q) [U2(q)D~~\0 + U2-1(q)D~~!iO]'

IhCJ:. io = -qrCJr,io - qyCJy,io,

IhDJ:. io = -qrDJr,io -.qyDJy,io.

,62C~~,)iO = 1= [qrC~;?io + qyC~;.)iO] ,

!J2D~~,)io = 1= [qrD~;!io +qyD~~,)iO]'

C',,(q)=exp[;(U,,], 11=2.3; i""I,2: O,,=x.y;

__!J;,') = q~~~I(q). VI") =U2'(q)jf!. n =O. l.2;

[ )(+) - _[)'+I __ I.: \.(O)D,ll D(-l = -D(-) = I.: ~'(llD(I) :.!r.lr - .!!j.ly - 0 ry 13' 2r.lr 1y.2y 0 Iy 31'

[)~~), ... = -Di;.l,y = -l.:nV~~)D\;l. D~~\ ... = -D~~,\y = ~·o\-;~lD\;l.

[)~;\v = -kuV(O)O\;l(rL

D'-) = k ~-'ll D(ll(x).2",,2yO.)1

[)\-) = 11: \",) d'l(L) D(-l - _I.: V(21 d 21 ( )1:.ly 0 31 I 2y,l.r - 0 .11 y,

-.(-\ _ .. \-) _ "1 1)(,(1)(,~:i _ = ('\:1 = \ ·.i~IC~ I l. ,,;.1, ",I Y _._ ,J ( :~ ..!y - ( .!:J.!..r - v: y -.11 •

('1+ 1 _ .,(+1 = _\.111(.") C·\-i =C~-) = _V i 'lC(2),"!;J,:.!.c - l ".!.r:.!y ~-tJ 13' .!I.II .x.IV ry 31

(');\" = _~'iO)C\~I((}),

C\.,.\ = \-1.11(,:2'(0), ........ 0 .1

. &::5 Tec~ntc(lt Pl;vsrcs 3813:. r,,1J'~'" , ?93 A. V. Fedo(ov ana Yu. V RozhdeSlvenskll 165

APPENDIX B

The explicit forms of the functions Mna(q) , N/l(q) , P/l(q)

in (14) are, in the notation of the text and Appendix A:

31 - 130 [ z'\/I,,(q) = :-n j3 I lh<[aJ + TJz(3)b1Uz (q)]

+(I,~ r~[U1 - T1,(iJ)hUi(Q)] - 2j3z(f3 1 + r3J)TJz(£1)[1 - TdJ3lJl!i(q)J]

.\/z .... (q) = )1; liz [rhI3z<[UI +- Td.2i )b I Ui(q)J] + q; [<[al

-rn())bIUi(q)]- 2/3z(/3 1 + jiJ)[I - TJZ(,I1)]Tn(j3)Ui(q)J] ,

, f3 1 - r3z z [ z ]'\dq) =-n-q <aJ - TJ2(/3)b JUz (q),

. ' ,1, - i3z 0 ,z '\2(q) = -n-q-<[al - Tdi3)b IUZ(q)],

Pdq) = <~qr raJ - T1Z(£1)TJ2(£1)b Jui(q)],

Pz(q) = <~qr[al - TIZ(;3)TJZ(;3)b I Ui(q)].

IF. G. Bass and I. M. Fuchs, Scallerillg vf WlII'es 011 St(l{ixtimlly Irregu­

lar Sill/aces [in Russian], Nauka, Moscow (1972).

2N. P. Zhuk, Opt. Spektrosk. 61. 560 (1986) [Opt. Spectrv.\l·. (USSR) 6l.

351 (1986)].

3G. Y. RozMOv, Zh. Eksp. TeoI'. Fh. 94(2), 50 (1988) [SOl'. Phy.,. JETP

67,240 (1988)].

4N. P. Zhuk, O. A. Trel'yakov, and A. G. Yarovoi, Zh. Ek.,p. Tl!or. Fiz.

98,1520 (1990) [Sov. Phys. JETP 71. 850 (1990)1.

5B. F. Kur'yanov, Aku;·t. Zh. 8, 325 (1962) ISo,'. Phyx. AWllst. 8,252

( 1963)J.

6y. S. Pugachev, TheOly of Ralldoll/ FllllctiOlls alld its Al'plic(I{ioll ((J

Problell/s of AU/olI/mic COli/rot [in Russianl, Fizmatgiz. Muscow (1962).

7M. Born and E, Wolf, Prillciples of Optics. 4th cd" Pergamon Press,

Oxford (1970).

8A. Isimaru, Wave Propag(l{ioll alld ScallerillS ill RIIIIllolII Medi(l: Mulli ­

pie Scnllerillg Turbulellce, Rough Stll/aces, '1IIId Relllote Sen'illg, Yol. 2,'

Acauemic Press, New York (1978).

9E. Marx anu T. Y. Yorhurger, Appt. Opt. 29,3613 (1990),

IOH. P, Baltes (cd.), IlIl'ene Problelll;' il/ Optin, Springer-Yerlag (1980),

110. S. Dron', Yu. Y. Rozhdeslvenskii, Y. A. Tolmachev. and A. V.

Feuoruv, Abstr, XlV til/em. COllI 011 Coherellt IIlId NOli Ii Ileal' Oplic'\

[in Russian], Leningrau (1991), Yol. 2, pp. 105-106.

12A. G. Yoronovich. Zh. Eksp. Tevr. Fi~. 89, 116 (1985) ISuv. Phyx"

JETP 62. 65 (1985)].

Translated by J. D. Parsons

Calculating the properties of heterogeneous mixtures with the helpof-·a-phenomenological'function-

V. B. TsvitsinskiT

(Submitteu April 17, 1992)

Zh. Tekh. Fiz, 63(3), 14--22 (March 1993)

A method is proposed for calculating the properties of heterogeneous mixtures with the help of a phenomenological function which makes it possIble to incorporate the structure of the material. i.e., the spatial distribution of the components. The corresponding function contains two coefficients. one of which detennines t~e type of problem (a site problem or a bond problem), while the second has a power-law relationship with the threshold concentration of inclusions. These expressions can describe the properties of heterogeneous mix.lures over their entire range of existence, from a series connection of the components to a rarallel one.

J)CV'C!OpIllCIltS in science and technology are making ;~,':,--r(, .. ,'ilC()US J'l:11eri:lls illl[Jonant \1n a progrcsslv'ciy !1roader

,,-,lie 111 ,IlIl1C diverse fields Ilt technology: cicctrotechnologv. l'\l\vdcr ::lctallurgv. :le:H engineering. ~·~c. r!le cesearch on :i;-: :)~\)I1C.rtICS \\1 :hC5C j11<uerials ~iale5 hack nl\lre .~han iUO " ,·:\r5.\ :,caIISC ov hmes ' 'cn; \la\weii. (lne c!laDler of \1,1111..:11 '.\ .!~ dC\"Ol('U :\.) ;11~ :\~!\(lt~\.·~~\·!t\· \)l :letertJgcncl1l1S

mixtures. can be regarded chis direction. The further Lorentz. Rayleigh, Vicker. ,tIld Lifshitz :lll1ong many

as the first fundamental work in development of this research by Bri·lggeman. Odelevskii. Landau,

others laid the f(1~Indation for the ',ileory (.1f heterogeneous mixtures. Tllat theory continues to ,leveloD lodav.

It should he recognized that the methods ·.,hid have

':)63·78~2.93 03 O~55·05 S1000 .") I 994 American Instil ute of PhYSICS 166