OPTICAL PROPERTIES OF THIN FILMS ON ROUGH SURFACES

(1)

HAL Id: jpa-00223531

https://hal.archives-ouvertes.fr/jpa-00223531

Submitted on 1 Jan 1983

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

OPTICAL PROPERTIES OF THIN FILMS ON ROUGH SURFACES

J. Vlieger, M. Wind

To cite this version:

J. Vlieger, M. Wind. OPTICAL PROPERTIES OF THIN FILMS ON ROUGH SURFACES. Jour-

nal de Physique Colloques, 1983, 44 (C10), pp.C10-363-C10-366. �10.1051/jphyscol:19831073�. �jpa-

00223531�

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JOURNAL

DE PHYSIQUE

Colloque CIO, supplCrnent au n012, Tome 44, d k e m b r e 1983 page

C

10-363

OPTICAL PROPERTIES OF THIN FILMS ON ROUGH SURFACES

J. V l i e g e r and M.M. Wind

Institute-Lorentz, University of Leiden, Nieuwsteeg 1 8 , 2311 SB Leiden, The Uether Zands

R6suni6

-

On dBveloppe une t h 6 o r i e g s n d r a l e , du deuxibrne o r d r e par r a p p o r t 2 m i s s e u r e t 5 l a r u g o s i t 6 s u p e r f i c i e l l e , d e s p r o p r i d t 6 s o p t i q u e s d'un f i l m mince c o n t i n u . Ces p r o p r i E t S s s o n t d s c r i t e s par un p e t i t nombre d e c o e f f i c i e n t s c o n s t i t u t i f s 6 l e c t r o m a g n l t i q u e s . On d t a b l i t d e s r e l a t i o n s e n t r e c e s c o e f f i c i e n t s , l e s f o n c t i o n s de c o r r B l a t i o n d e s h a u t e u r s d e s s u r - f a c e s s u p s r i e u r e e t i n f d r i e u r e du f i l m , e t son Bpaisseur moyenne. La r d f l e c - t i v i t d , l a t r a n s r n i t t i v i t E e t l e s c o e f f i c i e n t s e l l i p s o m 6 t r i q u e s s o n t exprimes en f o n c t i o n des c o e f f i c i e n t s c o n s t i t u t i f s , pour d e s a n g l e s d ' i n c i d e n c e a r b i - t r a i r e s . Les r 6 s u l t a t s d e ce t r a v a i l s o n t a u s s i Btudi6s dans l a l i m i t e d e s grandes longueurs d'onde de l a r u g o s i t d s u p e r f i c i e l l e , e t compar6s aux r e s u l t a t s d r O h l i d a l e t c o l l a b o r a t e u r s , obtenus prEci5demment.

A b s t r a c t

-

A g e n e r a l theory, of second o r d e r i n t h e f i l m t h i c k n e s s and sur- f a c e roughness, i s developed of t h e o p t i c a l p r o p e r t i e s of a t h i n continuous f i l m . These p r o p e r t i e s a r e d e s c r i b e d by a small number of e l e c t r o m a g n e t i c c o n s t i t u t i v e c o e f f i c i e n t s . Formulae f o r t h e s e c o e f f i c i e n t s a r e d e r i v e d i n terms of t h e h e i g h t - h e i g h t c o r r e l a t i o n f u n c t i o n s of t h e upper and lower sur- f aces of t h e f i l m , and i t s average t h i c k n e s s . The r e f l e c t a n c e , transmittance and e l l i p s o m e t r i c c o e f f i c i e n t a r e expressed i n terms of t h e c o n s t i t u t i v e co- e f f i c i e n t s , f o r a r b i t r a r y a n g l e s of i n c i d e n c e . The r e s u l t s of t h i s work a r e a l s o s t u d i e d i n t h e long wave l e n g t h l i m i t of s u r f a c e roughness and compared w i t h r e s u l t s , obtained e a r l i e r by O h l i d a l and co-workers.

1 . INTRODUCTION

-

O p t i c a l p r o p e r t i e s of s u r f a c e s a r e s t r o n g l y i n f l u e n c e d by rough- n e s s and t h e p r e s e n c e of t h i n f i l m s . We s h a l l s t u d y t h e s e i n f l u e n c e s by c o n s i d e r i n g

a t h i n continuous i s o t r o p i c f i l m ( l i q u i d o r s o l i d , average t h i c k n e s s d<<wave-length A), w i t h d i e l e c t r i c c o n s t a n t E , covering t h e rough s u r f a c e of a s u b s t r a t e ( l i q u i d o r s o l i d ) w i t h d i e l e c t r i c c o n s t a n t E'. The i n s t a n t a n e o u s p o s i t i o n of t h i s s u r f a c e i s g i v e n by z=f+(Pll ,t ) , where $ ( l = ( x , Y , ~ ) . The upper s u r f a c e of the f i l m i s a l s o rough ( z = f - ( z U , t ) ) . We assume t h a t ^E and E+ a r e r e a l , b u t t h e theory can a l s o b e extended t o complex E and ( o r ) E + . The d i e l e c t r i c c o n s t a n t of t h e ambient E--1 u s u a l l y . The f u n c t i o n 3 z=fC(TU , t ) a r e assumed t o be s i n g l e valued and d i f f e r e n t i a b l e w i t h l o c a l normals v'(?~~ ,t) and c o n s t a n t averages. (We assume <f->SO and <f+>=d). The boundary c o n d i t i o n s f o r t h e electro-magnetic f i e l d s

3

and and t h e displacement f i e l d s

d

and

Tf

a t t h e i n t e r f a c e s a r e g i v e n by

The f i e l d s w i t h s u p e r s c r i p t s

+

o r

-

a r e t h e f i e l d s i n t h e s u b s t r a t e and ambient r e s - p e c t i v e l y , whereas f i e l d s w i t h o u t s u p e r s c r i p t s a r e w i t h i n t h e f i l m . From t h e Maxwell e q u a t i o n s , t o g e t h e r w i t h t h e boundary c o n d i t i o n s ( I ) and t h e s t a t i s t i c a l p r o p e r t i e s

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19831073

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ClO-364 JOURNAL

DE

PHYSIQUE

of t h e rough s u r f a c e s , i t i s p o s s i b l e t o c a l c u l a t e t h e o p t i c a l p r o p e r t i e s of t h e system. This w i l l be done i n t h e n e x t s e c t i o n s .

2. DESCRIPTION WITH FLUCTUATING SURFACE SUSCEPTIBILITIES

-

The system d e s c r i b e d i n s e c t i o n 1 can b e r e p l a c e d by t h e f o l l o w i n g system: a f l a t s u b s t r a t e w i t h d i e l e c t r i c c o n s t a n t &+, f i l l i n g t h e h a l f space z>0, an ambient w i t h d i e l e c t r i c c o n s t a n t E-, f i l - l i n z<O, and s i n g u l a r p o l a r i z a t i o n and m a g n e t i z a t i o n d e n s i t i e s

5

, t ) G ( z ) i n z=0. These d e n s i t i e s , which can be c a l c u l a t e d from t h e e x c e s s pola- t)G(z) and r i z a t i o n and magnetization i n t h e f i l m / I / , produce t h e same f i e l d s i n t h e r e g i o n s where z>f

+(Sn ,

t ) and z<f-(Pll

,

t )

,

as were p r e s e n t i n t h e o r i g i n a l s y s tern, d e s c r i b e d

i n se$tion 1 ( c . f . a l s o r e f 121). The s u r f a c e p o l a r i z a t i o n and magnetization d e n s i - t i ~ s pS and ;fs can b e e r e s s e d i n terms of t h e e x t r a p o l a t e d f i e l d s / I /

8 ; ~

(e;,et7dt) and i$,'Z(hi,z,b$) on b o t h s i d e s of z=O. Up t o second o r d e r i n f'one

f i n d s Y / 3 / Y

where v = + o r

- .

The f l u c t u a t i n g s u r f a c e s u s c e p t i b i l i t i e s a r e given by -+

+ ( I ) v

cb

_{( r l l}-+ , t ) = 1 ) ( € - ~ ~ ) f ~ ( z ~ ~ , t ) (z%+?)+v(~-e") (EEV) -1 f

v

( r , + ,t)'22

,

+

t ~ 2 ) v ~ ~ l l

,

^t):v(c-sv) kv(;,,

,

t ) r {(sV)-'(

aja:,,

)2-&-'2(a/aZIl

)} ^,

-f

2 ~ 3 ) v ( ~ l ~ ,

a

Z;V(E-E~) (Ev1-l ( a ~ a : ~ , ) ~ - E - ' ~ ( a / a :

)}

Jfv(;ll, t)}'

, .

-f

( 4 ) ~ ^-+

i

¹¹

¹

Sb

( r l l , t )

-

~$v(c-Ev){fv(Zl,, t)}'

( 3 - 3 % ) ,

⁽³⁾

with %=(I , 0 , 0 ) , 7 = ( 0 , 1 , 0 ) , 2=(0,0,1) and a / a r , , = ( a / a x , a / a y , o ) . +

3. SURFACE CONSTITUTIVE COEFFICIENTS

-

These a r e fouzd by e x g r e s s i n g t h e average s u r f a c e p o l a r i z 2 t i o n and ma n e t p a t i o n d e n s i t i e s FS:S-<~S> and M ~ Z & > i n terms of t h e average f i e l d s and $m%<n,,,>. One o b t a i n s , up t o second o r d e r i n t h e s u r f a c e roughness and f i l m t h i c k n e s s / 3 /

2 A A

w i t h

3

Here < $ i ) y ~ i = l ,2,3,4) a r e averages of f l u c t u a t i n g s u s c e p t i b i l i t i e s , eq. ( 3 ) , whereas -+

'f(C)v i s an e x p r e s s i o n i n terms of t h e height-height c o r r e l a t i o n f u n c t i o n s of t h e

:zu,

s u r f a c e s :

where A f V f f v - < f v > add v , v l =

+

1

.

The s u r f 3 c e s u s c e p t i b i l i t i e s , eq. (5), a r e d i a gonal i n wave vector-frequency r e p r e s e n t a t i o n (kll,w). The d i a g o n a l elements a r e given by

i t

G I / 1 , 1 .

The s u b s c r i p t s

R

and tr s t a n d f o r l o n g i t u d i n a l and t r a n s -

(4)

v e r s a l . For yRL) one f i n d s

v + v v

-

³

yR (kll ,w) = ZV(E-E ) < f > + i n i P ~ V V ' (E-E') (E-Ev')

I ^v

x / d f i d o ' S v ' v ( f , - ~ i , w - u l )

1

c ~ s ~ @ q ; q ; ( ~ + ~ > - q + ) - ~

I

+

^{- 1}

+

sin2m(w1

1.1

(ql + P;)

}

^E^2v(E-EV)^<^{f v}^>^+y([)' ^{(TIl ,u)}

,

where

sVv'

(XI, ,w)

"sVV'

( z I l ) 2 n 6 ( ~ ) i s t h e F o u r i e r transform of eq. (6)

, qy

^E

and

0

i s t h e a n g l e between

2,

and

2; .

S i m i l a r e x p r e s s i o n s c a n b e d e r i v e d f o r yYr, f3'

,

6'

, nV

and 'rV and a r e given i n r e f . 1 3 1 . The con- s t i t u t i v e c o e f f i c i e n t s can i n p r i n c i p l e b e e v a l u a t e d , i f t h e 4 c o r r e l a t i o n f u n c t i o n s

+ -

SVV' a r e known, t o g e t h e r w i t h d, E

,

^E and E.

4. REFLECTANCE, TRANSMITTANCE AND ELLIPSOMETRIC COEFFICIENT

-

Using t h e c o n s t i t u - t i v e e q u a t i o n s (4) and t h e boundary c o n d i t i o n s f o r t h e f i e l d s i n z=0 ( s e e e.g. / 4 / ) , one can e x p r e s s t h e r e f l e c t a n c e R, t r a n s m i t t a n c e T and e l l i p s o m e t r i c c o e f f i c i e n t r i n terms of t h e s u r f a c e c o n s t i t u t i v e c o e f f i c i e n t s , u s i n g e q . ( 7 ) . One t h e n ob- t a i n s f o r s- and p-polarized l i g h t :

w i t h t h e amplitudes, v a l i d up t o second o r d e r i n t h e s u r f a c e roughness and f i l m

and analogous e x p r e s s i o n s f o r tp and r p m / 3 /

.

I n t h e above e q u a t i o n s O and a r e t h e a n g l e s of i n c i d e n c e and t ~ a n s m i s s ~ o n , t:fd) and r O (d) t h e t r a n s m i s s i o n O t and r e f l e

teen

amplitudes f o r h p l a n e p a r a l l e l f i l m w i t h S t h i c k n e s s d

,

n'

~d

^E',

whereas E

5

^y(:) and 'rtc'E Z T ( d v d e s c r i b e t h e i n f l u e n c e of s u r f ace

roughness, ( c f . eq. (8))

.'

5. LONG WAVE LENGTH LIMIT OF SURFACE ROUGHNESS; COMPARISON WITH EARLIER WORK

-

We s h a l l assume t h a t i n t h i s l i m i t t h e c o r r e l a t i o n f u n c t i o n s S V ~ ' ( P ~ , ) m$l be approx- imated by 6-functions i n e q . ( 8 ) and corresponding e q u a t i o n s f o r ~ t r t

.BV1

6'9 qV and

xV ,

s o t h a t i n t e g r a t i o n s c a n e a s i l y b e performed. For normally xncxdent l i g h t we t h e n o b t a i n /5/ t h e same r e s u l t f o r R as O h l i d a l , Navr'ltil and ~ u k e g 161.

I n o r d e r t o c a l c u l a t e t h e e l l i p s o m e t r i c c o e f f i c i e n t r

,

we have t o make t h e b e t t e r approximation

i n eq. (8) and corresponding e q u a t i o n s f o r yYr,

B',

e t c . Otherwise we would f i n d no i n f l u e n c e of s u r f a c e roughness. We assume h e r e i d e n t i c a l f i l m s , i . e . a l l 4 cor- r e l a t i o n f u n c t i o n s

sVv'

a r e equal w i t h same < ( ~ f ' ) ~ > r < ( ~ f ) ~ > and same c o r r e l a t i o n l e n g t h R

(R >>A

i n considered l i m i t ) . The symbol 6" denotes second d e r i v a t i v e of t h e & f u n c t i o n . With e q . ( l l ) i t i s a g a i n p o s s i b l e t o e v a l u a t e

yx ^,

e t c . 151. One t h e n o b t a i n s , u s i n g eqs.(9) and ( l o ) , t h e f o l l o w i n g second o r d e r e x p r e s s i o n f o r r :

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C10-366 JOURNAL DE PHYSIQUE

where r i ( d ) / r z ( d ) i s t h e e l l i p s o m e t r i c c o e f f i c i e n t of t h e p l a n e p a r a l l e l f i l m w i t h t h i c k n e s s d

,

wher_eas+the l a s t term d e s c r i b e s t h e e f f e c t of s u r -

f a c e roughness. The f u n c t i o n F(O,E , E ) i s r a t h e r c mplicate+d-- d given e x p l i c i t - l y i n r e f .

.

For the Brewster angle 0: 5 a r c r o s rn-(E-+E )

7

one g e t s This r e s u l t i s d i f f e r e n t from t h a t o b t a i n e d by O h l i d a l and Lukes, u s i n g t h e Helm- holtz-Kirchhoff i n t e g r a l / 7 / , b u t a l s o from t h e r e s u l t o b t a i n e d by t h e same a u t h o r s , u s i n g t h e Stratton-Chu-Silver i n t e g r a l / 8 / . I n t h e f i r s t case t h e s e a u t h o r s o b t a i n a c o n t r i b u t i o n t o r from s u r f a c e roughness, which does n o t v a n i s h f o r G-0, a s i t should do f o r i s o t r o p i c s u r f a c e s . So we o n l y have t o compare our r e s u l t w i t h t h a t of r e f e r e n c e / 8 / . I t can b e shown /5/ t h a t t h e d i f f e r e n c e i n r e s u l t s i s due t o a n i n c o n s i s t e n c y i n t h e assumptions made by O h l i d a l and ~ u k e s " /7,8/ i n o r d e r t o calcu- l a t e t h e l o c a l electro-magnetic f i e l d on a roughness s u r f a c e . We show / 5 / , by a d i r e c t c a l c u l a t i o n of t h i s f i e l d , up t o second o r d e r i n t h e s u r f a c e roughness, t h a t

i t d i f f e r s from t h e l o c a l f i e l d , p o s t u l a t e d by O h l i d a l and ~ u k e g /7,8/. Repeating t h e i r c a l c u l a t i o n s w i t h our f i e l d s , we f i n d t h a t b o t h methods (H.K. and S.C.S.) l e a d t o our r e s u l t , e q s . ( l 2 ) and (13). We furthermore prove i n r e f . / 5 / t h a t t h e l o c a l f i e l d on a rough s u r f ace, p o s t u l a t e d by O h l i d a l and Lu$es' / 7 , 8 / , can be ob- t a i n e d by n e g l e c t i n g a l l terms w i t h second d e r i v a t i v e s of f - w i t h r e s p e c t t o x and y i n o u r c a l c u l a t i o n . T h i s i s i n agreement w i t h t h e f a c t t h a t t h e s e a u t h o r s n e g l e c t l o c a l c u r v a t u r e of t h e rough s u r f a c e s . We prove, however, i n r e f . / 5 / , t h a t i t i s i n c o n s i s t e n t t o make t h i s approximation i n a second o r d e r c a l c u l a t i o n of t h e f i e l d s . It seems t o be a p u r e coincidence t h a t f o r t h e numerical example of t h e S i

-

S i 0 system, given by OhlZdal and ~ u k e g 2 / 7 / , our formulae (12) and (13) g i v e , w i t h i n a few p e r c e n t s , t h e same r e s u l t s a s t h e i r s .

REFERENCES

1 Albano, A.M., Bedeaux, D. and V l i e g e r , J . , P h y s i c a s (1980) 105 / 2 / KrEger, E. and Kretschmann, E . , 2. Physik

237

(1970) 1

/ 3 / Wind, M.M. and V l i e g e r , J . , Physica A , t o be published

/ 4 / Bedeaux, D. and V l i e g e r , J., Physica

67

(1973) 55

/5/ Wind, M.M. and V l i e g e r , J., P h y s i c a A, t o b e p u b l i s h e d

/6/ O h l i d a l , I., N a v r ' i t i l , K . , and Luke;, F., J.Opt.Soc.Am.

61

(1971) 1630 / 7 / O h l i d a l , I. a n d ~ u k e z , F., O p t . A c t a E ( 1 9 7 2 ) 817

/8/ O h l i d a l , I . and Lukeg, F., 0pt.Commun.

1

(1973) 76

OPTICAL PROPERTIES OF THIN FILMS ON ROUGH SURFACES

HAL Id: jpa-00223531

https://hal.archives-ouvertes.fr/jpa-00223531

Submitted on 1 Jan 1983

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.