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Radiometry of Rough SurfacesM. Nieto-Vesperinas aa Instituto de Optica, Serrano 121, Madrid-6, SpainPublished online: 03 Dec 2010.

To cite this article: M. Nieto-Vesperinas (1982) Radiometry of Rough Surfaces, Optica Acta:International Journal of Optics, 29:7, 961-971

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OPTICA ACTA, 1982, VOL . 29, NO. 7, 961-971

Radiometry of rough surfaces

M. NIETO-VESPERINASInstituto de Optica, Serrano 121, Madrid-6, Spain

(Received 3 December 1981)

Abstract . This paper discusses the radiometric properties of perfectly con-ductive, slightly rough random surfaces. The Kirchhoff approximation, so farused in this kind of study, is analysed . In particular, its inability to constructmodels of lambertian rough surfaces is pointed out . The small perturbationmethod, recently put forward, allows the scope and limitations of this approxi-mation to be established. This method also yields exact expressions for theradiant intensity, for either polarized or unpolarized incident radiation and allowsthe construction of surfaces which produce a lambertian distribution of intensity .

1 . IntroductionStudies of the connection between radiometry and coherence of fields [1-6] have

increased interest in the relationship between the structural statistics and theradiometric properties of random scatterers [7-9] (or as introduced in scatteringtheory [10-12]) . This is relevant to solving the inverse problem, i .e. to find thestatistical features of a scatterer which, after being illuminated, produces a certaindistribution of mean radiant intensity .

In the case of rough surfaces or phase screens [7-9], only the scalar Kirchhoffapproximation (KA) under low scattering angles has been used . However, it is wellknown that scattering from rough surfaces is highly polarization-dependent at largeangles . Also connected with this is the fact that any attempt based on the above modelto construct a lambertian random surface has so far failed [7-9] . This is notsurprising since an isotropic distribution of radiant intensity following Lambert'slaw must necessarily be in conflict with the small-angle approximation .

This paper deals with radiation scattered by perfectly conducting, slightly roughsurfaces, for which an exact scalar and vector formulation for the mean scatteredintensity has recently been suggested [13, 14] . This has been done by means of asmall perturbation method (SPM) based on the extinction theorem . This allows acomparison to be made with the mean radiant intensity yielded by the KA, thusestablishing its limitations, as well as the range of validity of a scalar theorydepending on whether the incident radiation is polarized or unpolarized . This is ofspecial importance in the solution of inverse problems . In particular, it is shown thatthe SPM allows the construction of slightly rough lambertian surfaces .

In § 2 the Kirchhoff approximation for the mean radiant intensity is revised,placing special emphasis on the inclination factor (about which there is currentlymuch controversy) . The inverse problem is also discussed and, specifically, thefailure of constructing lambertian rough surfaces. This problem is solved in §3,where the scalar and vector SPMs for the mean radiant intensity are discussed .

0030-3909/82/2907 0961 504 . 00 c~ 1982 Taylor & Francis Ltd

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2. The radiant intensity in the Kirchhoff approximation2.1 . Scalar theory

Let Z=D(R) be the random heights of a rough surface contained in the OXYplane (see the figure ), r=(R, Z), illuminated by a plane monochromatic wave ofwave-vector k o =k(s°, -s°) . The field tfi(rs), generated by reflection at a point P ofthe far zone given by the position vector rs=r(s l , sZ ) ( s being a unit vector), is givenby [15]

/J(rs) = - 2itik cos 9A(sl)exp (ikr)

k = ~,

(1)

where A(sl) is the angular spectrum of I(rs) . For example, for a planar source A(sl)is the two-dimensional Fourier transform of the field distribution across the plane ofthe source . For a perfectly conducting rough surface, the scalar KA yields

A(sl)=- F 2

s d2Rexp[-ik(s l -s°) • R]exp[-ik(s°+s-)D(R)],

(2)(2~)

the integral being over the illuminated area S . The geometric F factor is [16, 17]

F - 1 +cos00cos0 - sin 0 0 sin 0 sin 4cos 0(cos 00 + cos 0)

(3)

Unless explicitly stated, we shall consider normal incidence (0 0 =0) from now on .F then becomes

F=1/cos 0 . (4)

Equations (1)-(4) give the reflected field I(rs) in the far zone . Since, however, theinclination factor F is sometimes improperly considered, and there is considerabledebate in the literature about it, we derive equations (2) and (3) in the Appendix .

S°=(S1,-S=) , S=(S1 ,S )iS1= sin 0(sinO , -cos 0, 0)

S Z = `11 -Si = cos0

Scattering geometry .

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The mean radiant intensity <I(s)>, which represents the energy reflected perunit solid angle around the s direction is

<I(s)>=r2<I f(rs)I2>,

(5)

where the angular brackets denote an ensemble average . Substituting equations (1),(2) and (4) into (5), one obtains for normal incidence

<I(s)> (2n) 2 ,1 s ,/ sd2Rd2R'exp[-iksl • (R-R')]

x <exp {-ik(1 +s.) [D(R)-D(R')]}) . (6)

Setting R-R'=p, equation (6) becomes2

<I(s)>=(2n)2 d2pexp (-iks1 • p)W_

(p, s1),

(7)

where the surface-averaged correlation W(p, s1) of the effective source is

W(p,s 1)=J

d2R<exp{-ik(1+s-)[D(R)-D(R+p)]}>s

=J

d2R W(R, p, s1) . (8)S

Here W(R, p, s1 ) is the effective source correlation

W(R, p, s1)=<exp {-ik(1 +s-) [D(R) -D(R+p)]}>

(9)

Equation (7), together with equations (8) and (9), constitute the fundamentalradiometric relation for a random rough surface . It is interesting to compare thisexpression with the corresponding one for a planar source . Two important factsshould be observed in this connection. First, in contrast with the case of planarsources, equation (7) does not contain in front of the integral the inclination factorscost 0 or (1 + cos 0) 2 , which correspond respectively to the Rayleigh-Sommerfelddiffraction integral [5, 7] or the Helmholtz-Kirchhoff integral [18, 19] (see also [20]) .This is a very important point concerning 0 dependence when the inverse problem istackled, and it is a consequence of the random orientation of the surface local normal .

Secondly, W(R, p, s1) given by equation (9) is a function of the scattering angle 0 ;hence equation (7) does not constitute a two-dimensional Fourier transform, unlikethe case of planar sources . This makes the inversion of the transform given byequation (7) quite complicated .

There is, however, an instance, important in several situations with opticalsurfaces (a point made implicitly in [18], [21] and [22]), corresponding to thosesituations in which s 1 is very small so that

s=_s=° .

(10)

In this case k(s°+s=)D(R) may be approximated by 2ks°D(R), so that W(R, p, s1) inequation (9) becomes 0-independent :

W(R, p) = <exp { - 2ik[D(R) - D(R + p)] }>,

(11)

so that equation (7) becomes a two-dimensional Fourier transform .

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As an example of equation (7), let us consider a gaussian, statistically homoge-neous and isotropic random rough surface, for which IX(R, p, s) is given by [23]

i%V(p, s1)=Sexp { -g[1 - C(P)]},

( 12)where C(p) is the correlation function of D(R) . The parameter g is given by

g=k 2v 2(1 +cos 8) 2 ,

( 13)

where u is the variance of the random heights D(R) . By expanding the exponential ofequation (12) and assuming that the width of C(p) is much smaller than thedimensions of the surface, equations (7) and (9) lead to the mean radiant intensity[23]

k2

Xg'"

°°

<I(0)>=2-Sexp(-g) E M!

dppJo(kp sin0) [C(P)] m ,

( 14)m=o

o

where J0 is the zero-order Bessel function of the first kind . Equation (14) shows thedifficulty of solving the inverse problem, i .e. determining C(p) from <1(0)> .

However, for a slightly rough surface (a<<A), one can retain only the two firstterms of the expansion (14), so that if we consider just the diffuse component, weobtain

<I(8)>=9

k4a2(1 +cos0) 2

dppJo(kpsin8)C(p),

(15)J 0

which is easily inverted to give

27r

1 <,(S-L)>C(P)=Sk4a2fo ds1s1Jo(kpsl) [1 + J(1 - si)] 2

(16)

which constitutes the low-frequency part of the surface correlation function . This isalready known [3-5], since the high-frequency part of C(p) (namely, that partcorresponding to s1 > 1) is not directly retrievable by inverting equation (15) becauseit corresponds to evanescent waves which do not carry energy to the far zone .

2 .2 . Planar facet modelIn the model of planar facets as studied in [7], the random surface is described in

terms of the probability distribution P(m) of slopes m of planar facets of averagediameter ~ . Since this corresponds to the case in which the local normal n variesrandomly, equation (7) should be considered an adequate radiometric relation forthis model within the range of validity of the scalar Kirchhoff approximation .

For a distribution which is statistically homogeneous and isotropic, equation (7)leads, after a few calculations (analogous to equations (3 .4)-(3 .9) in [7]), to thefollowing expression for the mean radiant intensity :

k2<I(8) > _ (2~)2 P(k~ sin 8) .

(17)

This differs by the factor cos2 0 from the expression for the mean radiant intensitygiven by equation (3 .9) of [7], as expected . This difference is probably one of themain reasons why, in the experiment in which light is reflected by a sandpapersurface, consisting of small crystallites of random orientation [8], it is the functionP(kc sin 0) and not cost BP(k~ sin 8) that seems to match the experimental curves .

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It is, however, important to note that in the derivation of equation (17) (or,likewise, in that of equation (3 .9) of [9]), the random phase introduced by the surfaceis given by

k(1 +sZ)D(R) ocmR

,

(18)

which does not depend on sl , and thus conveys implicitly the assumption expressedin equation (10) .

2 .3 . Comparison with the vector Kirchhoff theoryFor planar sources, the main difference between the radiometric relation derived

from the scalar approximation, and that derived from the vector Kirchhoffapproximation, is that, whereas the former contains the inclination factors cos 2 0 or(1 +cos0) 2 , as stated above, the latter has the factor (1 +cos t 0)/2 when unpolarizedlight is incident on the source [20,:24] . (In the case of a three-dimensional volumesource, however, unpolarized light gives the same result as the scalar theory [25] .)

For perfectly conducting rough surfaces, the vector Kirchhoff theory does notdistinguish between s- and p-polarization of the incident field, and does not yielddepolarization, at least up to the first-order expansion in the slope, thus giving thesame result as the scalar Kirchhoff theory [26, 27] .

2.4 . Lambertian scatterersThere have been several unsuccessful attempts to construct a theoretical model of

a lambertian random rough surface starting from the scalar Kirchhoff approximationwith the assumption of equation (10) [7, 9] . From equation (15) it can be seen that, asin the vector treatment of planar sources [24], since (1 + cos 0) 2 can be approximatedby 4 cos 0 for 0 < 0 < 30 ° , a b-correlated rough surface, i.e . one with a correlationfunction C(p)=6(p)/p, would behave as a lambertian scatterer in the above rangeof 0 .

However, it is a well-established fact (see, for example, [13]) that the Kirchhoffapproximation works well only under low scattering angles 0, i .e . in surfaces withbroad C(p), or correlation lengths T such that T > 2 . Hence, the author considers itunlikely that such an approximation can constitute an appropriate framework inwhich to describe scatterers that produce a wide distribution of mean radiantintensity, such as the lambertian . Certainly, nothing else could be more against alambertian intensity distribution than the hypothesis of low values of sl =sin0,which implies equation (10) . Because of this, although equation (17) would yield theprobability distribution

P(M)=JC1_ I2~2 /, m,<k~

(m=k~sin0)

(19)

for the model of planar facets (which, unlike that of equation (7.4) of [7], is a well-behaved function), we-consider even equation (19) to-have little possibility of beingadequate, and that all this explains the failure to construct lambertian rough surfaces[9] .

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3 . The radiant intensity in the small perturbation method

3.1 . Scalar theoryThe angular spectrum obtained by using Green's theorem is given by [13]

A(sl)=-1f

d2 RF(R) exp (-iksl • R) exp [-ikszD(R)],

(20)ksZ s

where the `source function' F(R), which depends on D(R), can be associated with anelectric current density .

According to equations (1), (5) and (20), the mean radiant intensity (I(s)> will be

<I(s)>=(21r)2 Jd2p exp (- iksl - P)J(P, s1),

(21)

where the surface-averaged correlation of the effective source is now

W(p, sl)=Js

d2 R <F(R) exp [-iksZD(R)]F*(R+p) exp [iks2D(R+ p)] >

=J5 d2RW,P,5±);(R(22)

here W(R, p, s1 ) is the effective source correlation

W(R,p,sl)=(F(R)exp[-iksZD(R)]F*(R+p)exp[iksZD(R-+p)]> .

(23)

F(R) appears in equation (23) coupled to the surface heights D(R) . As it might appearthe inversion of equation (21) to obtain statistical features of the surface from <I(s)>is indeed a very involved task .

However, for surfaces with a«)., the SPM based on the extinction theorem [13]yields, by means of an expansion in powers of a, the diffuse component of the meanradiant intensity as

(I(0)>= 2Sk4a2 cos2 0

dppJo(kpsin0)C(p) .

(24)R

0

Apart form a constant, equation (24) differs from its equivalent in the KA (equation(15)) by the obliquity factor in front of the integral . For small angles (1 +cos 0) 2 ^_-4and cos2 0^_ 1, both expressions being identical as expected .

Equation (24) may now be inverted, yielding the low-frequency part of thesurface correlation function as

i L)~

C(P) = 2Sk~4a2

dsl s1J0(kPsI(s

l) 1-s 2f0

1(25)

It should be noted that surfaces with larger a will require higher-order terms of theexpansion of <I(s)> in powers of a in order to obtain convergence [13] . This willinclude higher-order correlation functions, <D(R1) D(R2 ) . . . D(RN)> . Thus, unlessthe surface is assumed to be statistically gaussian, the inverse problem would involvemore than just the second-order correlation function C(p) .

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3 .2. Vector theoryFor slightly rough surfaces, the vector SPM gives the diffuse component of the

mean radiant intensity as

<I(0,4')>=<Il(0,4')>+<III(9,4')>, (26)

j'(0, 4')> and <III(0, 0)>being respectively the intensities corresponding to s- andp-scattered waves, which under normal incidence become [14]

<11(0 4')> = 2S k4a2 cost 8u l sin o-u 2 cos 0 2

n

u

Here u =(u1 , u 2 ) is the polarization vector of the linearly polarized incident wave, u 1and u 2 being constants denoting respectively the components of u along OX and OY .

From equations (26)-(28), it is seen that, under normal incidence, measurementsof the reflected radiation in a plane given by sl perpendicular to u yields the sameresult as equation (24) of the scalar theory .

For incident umpolarized radiation, one must average over all incident polariz-ation directions . Let a be the angle that u makes with the OX axis (u is obviouslycontained in a plane parallel to OXY); then

u1 =u cos a,

u2 =u sin a .

(29)

But, since one has the averages :

<cos2 a) = <sin 2 a> = 2,

<cos a sin a>=0,

(30)

one obtains from equations (26)-(30), for unpolarized incident radiation,

2S

1 +cost B

°°<I(0)>= k4Q2(

2

/

dppJo(kpsin0)C(p) .

(31)7r

f 0

This expression for the mean radiant intensity differs from the corresponding one inthe scalar theory, equation (24), by the obliquity factor in front of the integral . Thisdifference is similar to the one between the scalar and vector theories for planarsources [20, 24] . Of course, at very low scattering angles both factors are similar .This would require C(p) to be sufficiently broad (T>> A) and constitutes the essenceof the validity of the scalar SPM for unpolarized radiation and the KA . Also, for0 0 30° we have that (1 +cost 0)/2 ^_cos 0 and (1 +cos 0)2 .4 cos 0 . Hence, thecomparison of equations (31) and (15) shows that, for 0 within the above interval, theKA is adequate to describe reflection of unpolarized incident radiation .

x f dppJo(kpsin0)C(p), (27)0

u 2 sin4'I 0 4 )> = 2S k4~2

C

u1cos4'2

<I

lII

7r

u

x

dp pJo(kp sin 0)C(p) . (28)J 0

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The inversion of equation (31) yields the low-frequency part of the surfacecorrelation function as

= i

1

U(Sl)>C(P) 2Sk46 2

f

1o dsl s1Jo(kPsl) 2 _S2

(32)

3 .3 . Lambertian scatterersThe scalar SPM, equations (24) or (25), yields the low-frequency part of the

correlation function of a slightly rough surface which produces a lambertiandistribution of mean radiant intensity Kcos0 in a plane normal to the polarizationdirection of the incident radiation as

Kit sin kpC(P)= 2Sk4Q 2 kp

(33)

This result is analogous to the one in the theory of planar sources [13] and givescorrelation lengths T of the order of ) .

For unpolarized incident radiation, however, equations (31) or (32) are ap-propriate to obtain the low-frequency correlation function of the surface whichproduces a lambertian distribution of radiant intensity . These equations are similarto those corresponding to the vector theory of planar sources, and the correspondingintegral has been evaluated by Pask [24] as

C(P)aC1-4)-ln~l(-1)°+ i(2n-1)!1~kP)1

(34)

I „ being the spherical Bessel functions .Again, for 0<,O<,30', the correlation function C(p)=6(p)1p yields an

approximately lambertian distribution for unpolarized radiation according toequation (31), this coincides also with the result,of the KA, equation (15), althoughthe KA lacks rigour for such a correlation function .

4 . ConclusionsThe radiometric characteristics of perfectly conductive, slightly rough random

surfaces have been discussed for normally incident radiation . The scope of the KAhas been studied by means of a comparison with the exact results given by the SPM .

It has been pointed out that the low-angle approximation is inconsistent with alambertian distribution of radiant intensity, and that this probably explains thefailure of previous attempts to construct lambertian rough surfaces using the KA andthe hypothesis of small reflection angles . Nevertheless, for 00530° the KAcorrectly describes the reflection of unpolarized incident radiation . On the otherhand, the difference between the vector and scalar SPM is similar to the theory ofplanar sources. Lambertian slighty rough surfaces have correlation distances of theorder of ), although within the range 050530° , a 6-correlated rough surfaceproduces an almost lambertian radiant intensity .

The inverse problem has been established only for the low-frequency part of thesurface correlation function C(p) . In principle; the high-frequency part could beobtained by exploiting the analytic properties of <I(s) >, as noted in [5] . However, theextrapolation of <I(s)> beyond the accessible- interval marked by the evanescentwaves will be highly unstable given a certain level of noise, i.e . this problem is ill-posed in the sense of Hadamard (see, for example, [28] for a review of this kind ofproblem) .

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AcknowledgmentsPart of this work was carried out while the author was with the Optics Group,

Physikalisches Institut der Universitat Erlangen-Nurnberg, F .R. Germany .

Appendix

Derivation of equation (2)The Kirchhoff approximation is obtained from the Helmholtz integral [29]

~(r)=J0(r)- 1 f d2S Gaanr )- 0(r)an '

(A 1 )J s'

where t 0(r) is the incident field and G the free-space Green's function

exp (iklr-r'I)G(r- r') =(A 2)

Ir-r IG is expanded into plane waves (Weyl representation ; [30, 31]) as

G(r-r')=2 f

d2slks

exp[iksl •(R-R')]exp(iksZlz-z'l) .

(A 3)

The boundary condition on the surface S' means that 0(r') is zero in the integral(A 1) . This, and the insertion of (A 3) into (A 1), yields

~(r) =tyo(r)- ~~ d2s i 1

d2 S'1 8n2

s-

00x exp [iksl • (R-R')] exp (iksZlz-z'l

8n',

(A4)

where z'= D(R') and the local normal n to D(R) is :

_ V[z - D(R)]

(A5)n.,/[I +(VD)2]'

For z > z', equation (A 4) gives00

i(r) =io(r) +J

A(s1) exp (iksl • R) exp (iksZz) d2s1 ,

(A 6)

where A(s1) is the angular spectrum given by

A(s1)= - 2J

d2S' exp (-iksl • R') exp (-iksZz')8

(A 7)87r ks1 s-

8n

Here d2s' may be replaced by dR'/n'z, (n'z being the z component of n') .The Kirchhoff approximation consists in replacing 8O/8n' in the integral (A7) by

28ro/8n' . For an incident field,

io(r')=exp [ik(sl • R'-s°z')],

(A8)

one has

2 an° 2ako n' exp [ik(s° • R'-s°z')] .

(A 9)

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Then

A(s1)ZS f sd2R' {s°+V'[D(R')] s°}-

zx exp [-ik(sl-si) • R'] exp [-ik(s°+sz)D(R')] .

(A 10)

Integrating (A 10) by parts, one obtains :

A(s1)= - 12 ~~-s°+(slo s~)•sl~ f d2R' exp [-ik(sl-s°)

• R'(2 n) sZ

sZ + s2

s'x exp [-ik(s°+s,)D(R')]

s°t+sZ f Yy exp [-ik(sl-so)• R'] exp [-ik(s°+sz)D(R')]IXxdy

- oZS° f X exp [-ik(sl-so) • R'] exp [-ik(s°+s,-)D(R')] I Yydx

(A 11)sZ +s. -x

The second and third terms of equation (A 11) represent an `edge effect' and arenegligible when compared with the first term for S>> t2, which expressed in terms ofthe incident and scattered polar and azimuthal angles becomes equal to equation (2) .

L'article presente une discussion des proprietes radiometriques de surfaces parfaitementconductrices a faible rugosite aleatoire . L'approximation de Kirchhoff, jusqu'ici utilisee danscette sorte d'etude, est analysee . En particulier, son inaptitude a construire des modeles desurfaces a rugosite lambertienne est mise en evidence . La ththode des petites perturbations,recemment mise au point, permet d'etablir la portee et les limitations de cette approximation .Cette methode conduit aussi a des expressions exactes pour 1'intensite du rayonnement enlumiere incidente polarisee ou non, et permet la construction de surfaces qui produiserit unerepartition lambertienne d'intensite .

Diese Arbeit diskutiert die radiometrischen Eigenschaften ideal leitender and leichtstochastisch rauher Oberflachen . Die fur these Art von Untersuchungen bisher benutzteKirchhoff-Naherung wird analysiert . Im einzelnen wird deren Unzulanglichkeit fur dieKonstruktion von Lambertstreuern ausgef ihrt . Die kurzlich zur Geltung gebrachte Methodekleiner Storungen erlaubt Moglichkeiten and Grenzen dieser Naherung anzugeben . DieseMethodeliefert auch exakte Ausdriicke fur die Strahlungsstarke entweder fur polarisierte oderfur unpolarisierte einfallende Strahlung and erlaubt die Konstruktion von Oberflachen,welche eine Lambertverteilung der Intensitat liefern .

References[1] WALTHER, A ., 1968, J. opt . Soc . Am., 58, 1256 .[2] MARCHAND, E . W ., and WOLF, E ., 1974, J . opt . Soc. Am ., 64, 1219 .[3] CARTER, W . H., and WOLF, E ., 1975, J. opt . Soc . Am., 65, 1067 .[4] WOLF, E ., 1978, J . opt . Soc. Am ., 68, 6 .[5] WOLF, E ., 1978, J . opt . Soc. Am., 68, 1597 .[6] BALTES, H . P ., GEIST, J ., and WALTHER, A ., 1978, Inverse Source Problems in Optics,

edited by H . P. Baltes, Topics in Current Physics Vol . 9 (Berlin, Heidelberg, NewYork: Springer-Verlag), Chap . 5 .

[7] HOENDERS, B . J ., JAKEMAN, E ., BALTES, H . P ., and STEINLE, B ., 1979, Optica Acta, 26,1307 .

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in Current Physics Vol . 20 (Berlin, Heidelberg, New York: Springer-Verlag), Chap . 1 .[10] Ross, G., 1977, Optica Acta, 25, 57 .[11] Ross, G., and NIETO-VESPERINAS, M., 1981, Optica Acta, 28, 77 .[12] Ross, G., FIDDY, M. A ., and NIETO-VESPERINAS, M., 1980, Inverse Scattering Problems in

Optics edited by H . P. Baltes, Topics in Current Physics Vol. 20 (Berlin, Heidelberg,New York: Springer-Verlag), Chap . 2 .

[13] NIETO-VESPERINAS, M., and GARCIA, N ., 1981, Optica Acta, 28, 1651 .[14] NIETO-VESPERINAS, M., 1982, J. opt . Soc. Am ., 72, 539 ; Opt. Lett ., 7, 165 . (See also,

for example, VALENZUELA, G . R., 1967, I.E.E.E. Trans . Antennas Propag . 15, 552 ;ToIGO, F ., MARVIN, A ., CELLI, V., and HILL, N . R., 1977, Phys . Rev . B, 15, 5618 ;AGARWAL, G . S., 1977, Ibid, 15, 2371 .)

[15] MIYAMOTO, K., and WOLF, E., 1962, J. opt . Soc. Am ., 52, 615 .[16] GARIBALDI, U., LEVI, A . C ., SPADACINI, R., and TOMMEI, G . E., 1975, Surf. Sci., 48,649 .[17] HILL, N . R., and, CELLI, V., 1978, Surf. Sci ., 75, 577 .[18] WELFORD, W. T., 1971, Optics quant . Electron ., 9, 269 .[19] JAKEMAN, E ., and PUSEY, P . N ., 1975, J. Phys . A, 8, 369 .[20] JACKSON, J . D ., 1975, Classical Electrodynamics (New York : J . Wiley), §§9.8-9 .12 .[21] CHANDLEY, P. J ., and WELFORD, W. T., 1975, Optics quant . Electron ., 7, 393 .[22] CHANDLEY, P. J ., 1976, Optics quant . Electron ., 8, 329 .[23] BECKMANN, P ., and SPIZZICHINO, A., 1963, The Scattering ofElectromagnetic Waves from

Rough Surfaces (New York: Pergamon Press), Chap . 5 .[24] PASK, C., 1977, Optica Acta, 24, 235 .[25] CARTER, W. H., 1980, J. opt . Soc. Am ., 70, 1067 .[26] BECKMANN, P., 1968, The Depolarization of Electromagnetic Waves (Boulder, Colorado :

Golem Press) .[27] LEADER, J . C ., 1971, J. appl. Phys ., 42, 4808 .[28] BERTERO, M ., DE MOL, C ., and VIAND, G . A., 1980, Inverse Scattering Problems in Optics

edited by H . P . Baltes, Topics in Current Physics Vol. 20 (Berlin, Heidelberg, NewYork: Springer-Verlag), Chap . 5 .

[29] BORN, M., and WOLF, E., 1975, Principles of Optics, fifth edition (New York : PergamonPress), p. 377 .

[30] MORSE, P . M ., and FESHBACH, H ., 1953, Methods of Theoretical Physics, Vol. 1 (NewYork: McGraw-Hill) .

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