POLARIZATION PROPERTIES OF LIGHT SCATTERED BY IRREGULAR SUR-

FACES

I. D e g r e e of P o l a r i z a t i o n

A. P. Ivanov and P. B. Boiko

Zhurna l P r i k l a d n o i Spek t ro skop i i , Vol. 8, No. 5, pp. 816-820 , 1968

UDC 535.51

E l e c t r o m a g n e t i c r a d i a t i o n s c a t t e r e d by ob jec t s with an i r r e g u l a r s u r f a c e s t r u c t u r e i s i n t e r e s t i n g to s p e - c i a l i s t s in v a r i o u s b r a n c h e s of s c i ence : s p e c t r o s c o p y , hyd roop t i c s , r a d i o p h y s i c s , a s t r o p h y s i c s , t h e r m a l p h y s i c s , l ight technology, and a c o u s t i c s . A l a r g e n u m - b e r of t h e o r e t i c a l and e x p e r i m e n t a l p a p e r s [1-23] have been pub l i shed on this p r o b l e m .

In the i nves t iga t ion of the r e g u l a r f e a t u r e s of the r e f l e c t i o n of r a d i a t i o n f r o m an i r r e g u l a r s u r f a c e , u se is often made of a hypo thes i s a c c o r d i n g to which the s u r f a c e is r e p r e s e n t e d as c o n s i s t i n g of a l a r g e num- b e r of uni t a r e a s , o r i e n t e d in a p a r t i c u l a r f ash ion and r e f l e c t i n g r a d i a t i o n a c c o r d i n g to the laws of g e o m e t r i c op t i c s . The foundat ions of t he se t h e o r i e s we re l a id by Bouguer [1] and w e r e subsequen t ly deve loped in [2 - 12]. This a p p r o a c h is f a i r l y s i m p l e , and in many s i t - ua t ions i t is f ru i t fu l and r e a d i l y v i s u a l i z e d . I t is not app l i cab le , however , to the phenomena of d i f f rac t ion , i n t e r f e r e n c e , and the shad ing of s o m e a r e a s by o t h e r s .

Another a p p r o a c h [13-20] invo lves the m a t h e m a t - i c a l s tudy of the d i f f r ac t ion of p lane waves a t the e l e - m e n t s of a rough su r f ace ; th is a p p r o a c h was o r i g i n a t e d by Ray le igh [13]. Although in th is c a se the o r i g i n a l p r o b l e m is f o r m u l a t e d m o r e r i g o r o u s l y , subsequen t ly , in o r d e r to ob ta in f in i te a n a l y t i c a l e x p r e s s i o n s , s i g - n i f i can t a s s u m p t i o n s a r e made r e g a r d i n g the na tu r e of the s u r f a c e , the condi t ions of i l l umina t ion and o b s e r - vat ion, and the r e l a t i o n s h i p be tween the wavelength of the r a d i a t i o n and the d i m e n s i o n s of the i r r e g u l a r i t i e s of the s u r f a c e . Even with t hese s i m p l i f i c a t i o n s , how- e v e r , the equat ions a r e of ten so c o m p l i c a t e d and d i f - f i cu l t to v i s u a l i z e p h y s i c a l l y tha t t he i r p r a c t i c a l p r a c - t i c a l u t i l i z a t i on is d i f f icul t .

E x p e r i m e n t a l i nves t i ga t i on of the c h a r a c t e r i s t i c f e a t u r e s of the s c a t t e r i n g of r a d i a t i o n on rough s u r - f a c e s is t h e r e f o r e d e s i r a b l e ; on the one hand, this wi l l give a l a r g e amount of m a t e r i a l r e l a t i n g the p r o p e r t i e s of an ob j ec t to the r a d i a t i o n s c a t t e r e d by it , and, on the o the r , i t wi l l make i t p o s s i b l e to ob ta in m o r e a c - c u r a t e i n fo rma t ion on the m e c h a n i s m of s c a t t e r i n g in given c a s e s . In th is work , among the l a r g e number of p r o b l e m s a s s o c i a t e d with th is ques t ion , we s tud ied the dependence of the d e g r e e of p o l a r i z a t i o n of the l ight s c a t t e r e d by i r r e g u l a r s u r f a c e s on the condi t ions of i l l umina t ion and o b s e r v a t i o n , and on the p r o p e r t i e s of the ob j ec t s .

The ob jec t s s tud ied w e r e b l a c k g l a s s , made m a t t with c o a r s e and f ine a b r a s i v e s , a l a y e r of c a r b o n b l a c k on a p l a t e , depos i t ed on a g l a s s s u b s t r a t e and a f l a t m e t a l p la te , cove red with type 1519 b l a c k m a t t ename l . Since a l l the ob jec t s a b s o r b e d l ight s t rong ly , we e f -

f ec t ive ly examined mode l s in which the r a d i a t i o n did not p a s s through the t h i ckness of the m a t e r i a l and s c a t t e r i n g was a s s o c i a t e d only with the i n t e r a c t i o n of the r ad i a t i on and the s u r f a c e of the med ium. M e a - s u r e m e n t s by m i c r o s c o p e showed that the a v e r a g e d i m e n s i o n s of the i r r e g u l a r i t i e s were 3 - 4 p and 3 0 - 40 ~ fo r the ma t t g l a s s e s , 1 p fo r the ca rbon b lack , and 8 - 1 0 # for the b l a c k enamel . The c h a r a c t e r i s t i c cu rves fo r the r e f l e c t i o n of these s p e c i m e n s a r e given in Fig . 1. These c h a r a c t e r i s t i c cu rves were obta ined by i l l umina t ion with n o n p o l a r i z e d l ight a t an angle of 7 ~ to the n o r m a l to the s u r f a c e of the s p e c i m e n s . F o r c l a r i t y , they a r e r e f e r r e d to uni ty a t an angle of r e - f l e c t i on 0 = 0 ~ Al l m e a s u r e m e n t s we re made on the s e t - u p d e s c r i b e d in [23], which m a d e i t p o s s i b l e to ach ieve angles of inc idence r of l ight on the s p e c i m e n f r o m 0 to 70 ~ with a 30' a p e r t u r e angle of d ive rgence . The r e f l e c t e d l ight could be m e a s u r e d in any d i r ec t ion , c h a r a c t e r i z e d by the r a d i a l angle 0 and the az imu tha l angle r where 0 is the angle be tween the n o r m a l to the s u r f a c e of the s p e c i m e n and the d i r e c t i o n of p r o p a - gat ion of the s c a t t e r e d r a d i a t i on , ~ i s the angle be tween the p r o j e c t i o n s of the inc ident and r e f l e c t e d b e a m s on the su r f a c e of the spec imen . The angu la r r e s o l u t i o n o f the r e c e i v i n g unit was 1 ~ . The m e a s u r e m e n t s were made at a wavelength of 546 nm. The i l l umin a t i ng a s - s e m b l y was f i t t ed with a p o l a r i z e r , which p e r m i t t e d i l l umina t ion of the s p e c i m e n with l ight p o l a r i z e d in the p lane of inc idence and p e r p e n d i c u l a r to it. The use of an a n a l y z e r in the r e c e i v i n g unit made i t p o s s i b l e to m e a s u r e the d e g r e e of p o l a r i z a t i o n of the s c a t t e r e d

I

7,0.

30 50 '~"

Fig. 1. Characteristic curves for reflection from the objects studied (the data in the range O = 70-90 ~ were obtained by extrapolation): 1) layer of carbon black deposited on a support; 2) glass with coarse matt surface; 3) glass with fine matt surface; 4) plate covered with

1519 matt enamel.

492

~/~ _ 5

80 ~0 0 KEg a*

A

T

o , __ !

j i J i ] I i ' d

80 qO 0 140 8" B

Fig . 2. Dependence of the d e g r e e of p o l a r i z a t i o n on the angle of ob - s e r v a t i o n 0 (v ib ra t ions of the v e c t o r a r e p e r p e n d i c u l a r (A) to the p lane of inc idence and p a r a l l e l to the p lane (B) for va lues of r equal to (a) 0 - 1 8 0 ~ (b) 4 5 - 2 2 5 ~ and (c) 90 -270 ~ and d i f f e ren t va lues ofq~:

1) q~ = 0~ 2) 20~ 3) 45~ ~i) 60~ 5) 70 ~ .

r ad i a t i on , which was unde r s tood to mean the r a t i o P = = (I1 - I2)/(I1 + I2), whe re Is and I2 r e p r e s e n t r e s p e c - t ive ly , the m a x i m u m and m i n i m u m r e a d i n g s on the r e c o r d i n g a s s e m b l y on ro t a t i on of the a n a l y z e r about an axis co inc iden t with the r e f l e c t e d b e a m .

F i g u r e 2A gives the dependence of the d e g r e e of p o l a r i z a t i o n of the l ight on the r a d i a l angle of o b s e r - vat ion 0 fo r d i f f e r en t angles ~ and r The da ta w e r e ob ta ined for f ine ly m a t t g l a s s i l l umina t ed with r a d i a - t ion p o l a r i z e d in the p lane p e r p e n d i c u l a r to the p lane of inc idence . Here and in s u b s e q u e n t f i g u r e s the angles of inc idence c o r r e s p o n d to the l e f t -hand quadran t s of the coo rd ina t e s y s t e m . In F ig . 2A(a) the l e f t -hand quad ran t c o r r e s p o n d s to r -- 0 ~ and the r i g h t - h a n d quad ran t to ~ = 180 ~ S i m i l a r l y , fo r F ig . 2A(b), r = 45 and 225 ~ , and for F ig . 2A(c), r = 90 and 270 ~ .

When the p l anes of i nc idence and o b s e r v a t i o n co in - c ide (Fig . 2A(a)), i t is found tha t for ang les of i n c i - dence f rom 0 to 45 ~ the P fo r the r e f l e c t e d l ight r e m a i n s c l o s e to 100%. With f u r t h e r i n c r e a s e in ~, a t e n d e n c y t owards a s l igh t d e c r e a s e in the d e g r e e of p o l a r i z a t i o n is o b s e r v e d only in the zone of s m a l l ang les 6. Thus, the t heo ry r e g a r d i n g the p r e s e r v a t i o n of the p o l a r i z a - t ion of the " e x t e r n a l component" [7, 8] in the e x p e r i - m e n t a l s c h e m e be ing c o n s i d e r e d is d i r e c t l y c o n f i r m e d in th is case . When ~ d i f f e r s f r o m 0 and 180 ~ however , the zone c o r r e s p o n d i n g to 100% p o l a r i z a t i o n of the r e - f l ec t ed l ight d e c r e a s e s s h a r p l y .

With i n c r e a s i n g ~b and ~p, as a r e s u l t of the i n t e r - ac t ion of the r a d i a t i o n with the subs t ance , the d e g r e e of d e p o l a r i z a t i o n of the s c a t t e r e d l ight beg ins to i n - c r e a s e . This is p a r t i c u l a r l y m a r k e d in the p lane p e r - p e n d i c u l a r to the p lane of inc idence (r equal to 90 and 270 ~ fo r va lues of 0 and ~ g r e a t e r than 60 ~ A c h a r - a c t e r i s t i c f e a t u r e is that if the p lane of o b s e r v a t i o n co inc ides with the p lane of inc idence o r is p e r p e n d i c u l a r

to it , the cu rves P = f ( O ) a r e s y m m e t r i c with r e s p e c t to the axis of o rd ina t e s . In the o t h e r c a s e s , however , the d e g r e e of p o l a r i z a t i o n is s l i gh t ly g r e a t e r in " f o r - w a r d " d i r e c t i o n s than in "back" d i r e c t i o n s (F ig . 2A(b)).

Thus in m o s t s i tua t ions which a r e r e a l i z e d , the p o l a r i z e d l ight is p a r t l y d e p o l a r i z e d a f te r r e f l e c t i on f r o m a rough s u r f a c e . This should not be the ca se within the f r a m e w o r k of g e o m e t r i c op t ics . The p h e - nomenon can be d e s c r i b e d qua l i t a t ive ly , however , on the b a s i s of the ideas deve loped by Rvachev and P o i y a n - sk i t [21, 22], a c c o r d i n g to which each m i e r o a r e a r e f l e c t s not in m i r r o r fash ion but in a c e r t a i n d i f f r a c - t ion cone. The s c a t t e r i n g in the d i r e c t i o n of o b s e r v a - t ion is then due to a l a r g e number of m i c r o a r e a s r e f l e c t i n g noncoheren t l ight b e a m s ; the n o r m a l s l ie within an e f fec t ive cone a s s o c i a t e d with the given d i - r e c t i o n of o b s e r v a t i o n . The combina t ion of t he se l ight b e a m s l eads to p a r t i a l d e p o l a r i z a t i o n of the r e f l e c t e d r ad i a t i on . Unfor tunate ly , the ca lcu la t ions given in [22], b e c a u s e of a number of a s s u m p t i o n s and the l ack of knowledge of v a r i o u s c h a r a c t e r i s t i c s of the rough s u r - face , cannot be used for quant i ta t ive c o m p a r i s o n with the r e s u l t s of the e x p e r i m e n t be ing c o n s i d e r e d .

Let us c o n s i d e r the ca se in which the v i b r a t i o n s of the l ight v e c t o r in the inc iden t beam take p l a c e in the p lane ~of inc idence . The a n a l y s i s of the d e g r e e of p o l a r i z a t i o n of the s c a t t e r e d r a d i a t i o n can be c a r r i e d out f r o m Fig. 2B, in which the s i tua t ion examined is the s a m e as that in F ig . 2A. F i g u r e 2B shows that when the plane of o b s e r v a t i o n is not p e r p e n d i c u l a r to the p lane of inc idence , the P = f ( O ) curves a r e a s y m - m e t r i c with r e s p e c t to the o rd ina t e ax is . As a ru le , the d e g r e e of p o l a r i z a t i o n is s m a l l e r in the " fo rward" d i r e c t i o n ( r i gh t -hand quadrant) than in the "back" d i r e c t i on . F o r r equal to 0 and 45 ~ the p o l a r i z a t i o n of the r ad i a t i on d e c r e a s e s smoo th ly with i n c r e a s e in 0,

493

i F - - - " Z ~ ~ ' V i r - / - - : - - a 1

J : / : . I "-o K'r / . ~ OUl \ \ \ \

. l l ~ . a \ " " ".~. < ~ 6

. j ' / z~ 1 . , , , , , , ~ .

-~O qO 0 4's 8 ~

Fig. 3. Dependence of the degree of polarization on the angle of o b s e r v a t i o n 0 for d i f f e r en t s p e c i - m e n s (v ib ra t ions of the v e c t o r p e r p e n d i c u l a r to the p lane of inc idence) : 1, 2) f inely m a t t g l a s s ; 3, 4) c o a r s e l y m a t t g l a s s ; 5, 6) l a y e r of ca rbon b l a c k depos i t ed on a suppor t ; 7, 8) p la t e cove re d with 1519 m a t t ename l , a) r = 0-180~ b) r = 9 0 -

270~ ~ = 4 5 ~ .

w h e r e a s fo r ~ equal to 180and 225 ~ the P = f(O) c u r v e s show a m i n i m u m in m o s t s i tua t ions . Th is m i n i m u m is d i s p l a c e d t owards low va lues of 0 with an i n c r e a s e in the angle of inc idence 4o.

A n a l y s i s of the r e s u l t s shows that a p o l a r i z a t i o n m i n i m u m is r e a l i z e d when the angle be tween the i n c i - dent and r e f l e c t e d b e a m s is a p p r o x i m a t e l y twice the B r e w s t e r angle . In th is c a s e the r e g u l a r p o l a r i z e d componen t f r o m the m i e r o a r e a s r e f l e c t i n g l ight in m i r r o r fash ion at the B r e w s t e r angle is a m i n i m u m , and the o t h e r a r e a s , t aken toge the r , as noted above, r e f l e c t l ight which i s d e p o l a r i z e d to a c o n s i d e r a b l e extent . However , this exp lana t ion is only qua l i t a t ive . M e a s u r e m e n t s of the dependence of P on 0, c a r r i e d out fo r d i f f e r e n t v a l u e s of ~, show tha t the magni tude of the m i n i m u m on the P = f(O) c u r v e s at f i r s t d e - c r e a s e s s h a r p l y with i n c r e a s e in the a z i m u t h a l angle , up to ~b = 200 ~ At h igher v a l u e s of r the d e g r e e of p o l a r i z a t i o n for a l l va lue s of 0 beg ins to i n c r e a s e , a l though it s t i l l r e m a i n s s m a l l e r than in the p lane co inc iden t with the p lane of inc idence of the l ight . These phenomena cannot be exp la ined f r o m the d i f f r ac t ion v iewpoin t e i t he r .

C o m p a r i s o n of F i g s . 2A and 2B shows that the g r e a t e s t d e p o l a r i z a t i o n and changes in i t s va lue with change i n the condi t ions of i l l umina t ion and o b s e r v a - t ion a r e found fo r the ca se where the v i b r a t i o n s of the l ight v e c t o r of the inc iden t r a d i a t i o n take p l ace in the p lane of i nc idence . He re , the p o l a r i z a t i o n c h a r a c t e r - i s t i c s show a s e n s i t i v e dependence on the p r o p e r t i e s of the s u r f a c e and the m a t e r i a l . To obta in i n fo rma t ion on the o r i en t a t i on of the rough s u r f a c e in space r e l a - t ive to the angles of i nc idence and o b s e r v a t i o n , and a l so to ana lyze the t ex tu re of the su r f ace , i t is t h e r e - f o r e m o r e convenien t to i l l umina t e the ob jec t u s i n g t h e v i b r a t i o n s of the l ight v e c t o r in the p lane of inc idence . On the o ther hand, if a f t e r the i n t e r a c t i o n of the e l e c - t r o m a g n e t i c r a d i a t i o n with the i r r e g u l a r s u r f a c e i t is n e c e s s a r y , as f a r as p o s s i b l e , to p r e s e r v e unchanged the p o l a r i z a t i o n c h a r a c t e r i s t i c s of the s c a t t e r e d r a d i -

a t ion (e. g., in the p ropaga t i on of r a d i o waves c a r r y i n g in fo rma t ion ove r e a r t h and w a t e r su r f a c e s ) , i t i s b e s t to have the p l anes of inc idence and o b s e r v a t i o n c o i n - c ident , with the o r i g i n a l e l e c t r i c v e c t o r v i b r a t i n g p e r - pe nd i c u l a r to the p lane of inc idence .

Unti l now we have c o n s i d e r e d the d e g r e e of p o l a r i - za t ion of l ight r e f l e c t e d f r o m one type of rough s u r f a c e (f inely ma t t g l a s s ) . It i s of i n t e r e s t to inves t iga t e the inf luence of the s u r f a c e on the p o l a r i z a t i o n c h a r a c t e r - i s t i c s . Of the l a r g e number of r e l a t i o n s h i p s examined fo r s p e c i m e n s with d i f f e ren t s u r f a c e s , Fig . 3 shows g r a p h i c a l l y the dependence of P on 0 f o r f o u r s u r f a c e s . We cons ide red i l l umina t ion of the ob jec t a t an angle of 45 ~ with the v i b r a t i o n s of the l ight v e c t o r p e r p e n d i c u - l a r to the p lane of inc idence . The so l id l ines c o r r e - spond to o b s e r v a t i o n of the s c a t t e r i n g in the p lane of inc idence , and the dashed l ines to that in the p lane p e r p e n d i c u l a r to it . A n a l y s i s of the f igure shows that i t i s not p o s s i b l e to e s t a b l i s h an unambiguous r e l a t i o n - ship be tween the d e g r e e of p o l a r i z a t i o n of the s c a t t e r e d l ight on the one hand and the m a c r o i r r e g u l a r i t i e s o r c h a r a c t e r i s t i c curve of the r e f l e c t i o n (see Fig . 1) on the o ther . The s i tua t ion is c o m p l i c a t e d by the inf lu- ence of the m a t e r i a l and the m i e r o t e x t u r e of the s u r - face , which a r e d i f f e r en t fo r d i f fe ren t s p e c i m e n s . In sp i t e of the quant i ta t ive d i f f e r ences , however , the qua l i t a t ive p i c t u r e is the s a m e for d i f fe ren t s p e c i m e n s , if the d ime ns ions of the i r r e g u l a r i t i e s a r e much g r e a t e r than the wavelength of the inc iden t r a d i a t i o n and the m a t e r i a l has the p r o p e r t i e s of a d i e l e c t r i c .

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18. E. L. Feinberg, Propagation of Radiowaves Over the Ear th ' s Surface [in Russian], Izd. AN SSSR, Moscow, 1961.

19. B. I. Semenov, Radiotekhnika i elektronika, 11, 8, 1966.

20. F. G. Bass and V. G. Bocharov, Radiotekhnika i elektronika, 3, 2, 1958.

21. V. K. Polyanskii and V. P. Rvachev, Opt. i spektr., 20, no. 4, 1966.

22. V. K. Polyanskii and V. P. Rvachev, Opt. i spektr., 22, no. 2, 1967.

23. G. K. II ' ich and P. B. Boiko, Vestsi AN BSSR, seryya f iz . -mat , navuk, no. 3, 1966.

23 November 1967

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