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Iterative corrections to the Kirchhoff approximation in rough periodic surface scattering
Volume 41, number 3
OPTICS COMMUNICATIONS
l April 1982
ITERATIVE CORRECTIONS TO THE KIRCHHOFF APPROXIMATION IN ROUGH PERIODIC SURFACE SCATTERING A. WlRGIN Laboratoire de Mdcanique Th~orique, Universit~ Paris 6, 75230 Paris Cedex 05, France
Received 23 October 1981
It is shown, contrary to the opinion of Beckmann and Spizzichino, that the Kirchhoff approximation generally gives a less satisfactory description of rough periodic surface scattering than the Rayleigh theory. The errors inherent in the Kirchhoff approximation can, to a large extent, be corrected by an iteration process generated within the framework of the Rayleigh theory. These results hold for periodic opaque scattering surfaces exposed to E-polarized plane waves, but are easily extended to arbitrarily-polarizedwaves incident on non-periodic rough surfaces.
1. Introduction The most direct and rigorous method for treating the problem o f scattering o f waves from rough opaque surfaces makes use o f Green's theorem and leads, quite easily [ 1,2] to the H e l m h o l t z - K i r c h h o f f integral (HK|) relating the scattered field outside the scatterer to field components on the very surface of the scatterer. Some of these field components are known in virtue o f the b o u n d a r y conditions resulting from Maxwell's equations, but others are essentially unknown and must therefore be determined from an integral equation which is generated by evaluation of the HK! on the boundary of the scatterer [ 3 - 1 0 ] . In practice, this task is usually so formidable (and costly) that it is preferable to first apply a simpler approximation procedure. The Kirchhoff approximation (KA) [3,6,11 ], also known as the physical optics [8,12,13], tangent plane [14], semi-classical [15] or eikonal [16] approximation, is one of the best-known and most often used procedures o f this type. The association of the name o f Kirchhoff with the KA comes from the latter's contribution to the mathematical formulation of the H u y g h e n s - F r e s n e l principle (HFP) [1 ]. Loosely speaking [17], the KA assumes the incident wave W to proceed unperturbed to each point P of the scattering surface S; this means that all points P of S are reached by 154
W without traversing the (opaque) underlying medium (no shadowing condition) and that radiation scattered from other parts o f the surface cannot reach P (no multiple scattering condition). The only opaque surface S for which this is rigorously true is a fiat mirror of infinite extent, so that the KA makes the approximation that the field at P o f S is essentially what it would have been if S had been a fiat mirror with a slope equal to that of S at P. The introduction of the KA into the HKI then enables the determination of the field at any point exterior to the scatterer. This procedure was first applied, in the extensive form that we have just outlined, by Brekhovskikh [18], but all those who, starting with Rowland [19] used the HFP for the study of rough surface scattering, implicitly assumed at least a restricted form o f the KA [17,20]. Much has been written [6,11-14,18,21] about the domain of validity of the KA, but agreement is obtained only for the condition r >> X, with r the minimal radius o f curvature of S and X the wavelength of the incident radiation. Naturally, this condition is rather imprecise (how much is "much larger"?) and gives no indication o f the influence of the incident angle and scattering angle which are known to be of some importance [17]. For this and other reasons (related to the fact that some of the reported conditions of validity [6,11,21 ] are so restrictive that they preclude use of the KA in
0 0 3 0 - 4 0 1 8 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 02.75 © 1982 North-Holland
Volume 41, number 3
OPTICS COMMUNICATIONS
all but the most uninteresting scattering configurations), in practice one often adopts a "try and see" attitude concerning the opportunity of use of the KA. The results are sometimes gratifying [6,14,15,22,23], and often very discouraging [5,6,8,12,24]. The most dramatic index of failure of the KA is furnished by large-scale violations of the law of conservation of energy [8,12, 24]. Therefore, KA results which have not been submitted to the energy balance test [14,18,21,25] should be considered to be of questionable accuracy. It is interesting to note that energy non-conservation was one of the main factors which led to the almost unanimous rejection of the well-known Rayleigh theory (RT) of gratings in favor of the rigorous integral equation method [9,26]. For probably other (although rather obscure) reasons Beckmann and Spizzichino [14] suggest that the RT "usually yields a worse approximation than the Kirchhoff solution". Recent publications [27,28] have shown that the RT can actually be made to function quite well for gratings whose slopes are as much as five times larger than the slope beyond which Petit and Cadilhac showed the RT to be mathematically unrigorous [26]. Considering the importance of Beckmann and Spizzichino's book for all those who are confronted with rough surface scattering problems, it should be useful to confront the above-cited opinion with facts. As is shown herein, the latter indicate, on the contrary, the superiority of the RT over the KA. Moreover, it will be shown that the KA actually constitutes the lowestorder approximation of an iterative process generated within the very framework of the RT and that carrying out the iterations beyond the KA can enable a very substantial correction of the errors inherent in the latter. The method developed herein arose from several previous works. Meecham [3] showed that the KA constituted the lowest-order approximation in an iterarive process generated in the integral equation method (which was numerically implemented by Pavageau and Bousquet [4] ). We showed [10] that the KA could be identified with either the zeroth or first order solution arising from a perturbation scheme based on the smallness of ?t/d, with d the correlation length or period of S. Lynch [13] demonstrated that a useful correction to the KA could be obtained by a variational method. Hagfors [29], on searching for a way to improve the KA, suggested that the fields at S should
1 April 1982
contain, in addition to the specular component associated with the KA, an angular spectrum of plane waves traveling both toward and away from S. Here the angular spectrum of Hagfors is the one inherent in the Rayleigh theory [30], and we shall assume (as did Rayleigh) the scattering surface to be an opaque grating (although our method is easily extended to non-periodic surfaces by use of the Marsh [10,31,32] extension of the RT).
2. Preliminaries A periodic surface S, describable by y = f(x) = f(x + d), Vx C R, Vz E R, w i t h f a continuous and f = df/dx a piecewise continuous function of x, separates an infinitely conducting (totally opaque) medium M 2 from a lossless dielectric medium M l within which propagates a monochromatic plane wave W towards S with its propagation vector in the x - y plane. The time factor (implicit in the sequel) is e x p ( - i w t ) , with co the angular frequency related to ~, the wavelength in M 1 by co = kc = 27rc/X, wherein c designates the speed of light in M 1 . The total field in M 1 is of two-dimensional nature (z the ignorable coordinate) because of the symmetry of S and W. The field associated with W is assumed to be polarized in the E mode (i.e., only the z component of the electric field, designated herein by u, is non-zero). W is given by ui(x,y ) = exp [ik(six ciY)], with s i = sin 0i, c i = cos 0i, 0 i the incident angle measured counterclockwise from the positive y axis, and gives rise to the scattered field Us(X,y ) which is related to the total field u(x,y) in M 1 by u = u i + u s. The HKI and the boundary condition (u'-= u(x,f(x)) = 0, Vx E [0, d] ) enables the following rigourous result to be obtained [27,28] :
Us(X,y ) = ~
R n
OnUn;
y > Max(f(x)),
(1)
n =-oo
with R
= (i/2o,~',en ,o-'3,
(2)
and
on = exp(iksnx), cn = (1
o2~1/2. --
an)
Un = exp(ikcnY)'
Re(cn)/> 0, Im(cn)/> 0,
U'n = exp [ikcnf(X)] ,
(3) 155
Volume 4 i. number 3
OPTICS COMMUNICATIONS
The difficulty in this problem arises from the fact that although we have used all available information on u to obtain relations (1)-(3), we know nothing about ~'. The Kirchhoff approximation assumes
"~= k - l ( ~ O / O x - Ot8y) u ( x , f ( x ) ) = "~i + Vs' d (~, Q) = d -1 f 7 i x ) ~ ( x ) dx. 0
The problem is to determine the amplitudes R n of the plane waves composing the plane wave spectrum representation of the scattered field. These amplitudes are related to the so-called grating efficiencies [9] ~n by (4)
C n = IR n 12 Cn/Ci,
the latter satisfying, if all goes well, the conservation of energy relation oo
11 = - - ~
(5)
6:Re(n__~_ 6 n ) : l -
ij tl t li
1.5
I April 1982
= 2 v~
on s,
(6)
so that R m is approximated by (7)
R m ~-- Km = (i/°n U~nCn' ~i)"
An example of this type of calculation, for a sinusoidal grating f(x) = h cos 2~x/d, is given in fig. 1. Although both the Lynch condition [11,13] (kr cos30 >> 1, with r = d2/4n2h and 0 = 0 i in the present case) and the Bass and Fuks [13,21] condition ((2h/d) 2 ,~ 1 and k d ~ 1) for the validity of the KA are satisfied over a large portion of the range of k covered in fig. 1, it is seen that the KA leads to very serious energy conservation violations over this same range even though thle no shadowing condition is satisfied. These results give ample support to Chan and Fung's view [6] that further studies are needed to better define the conditions of validity of the KA.
A
3. The c o n v e n t i o n a l Rayleigh t h e o r y
li
The hypothesis underlying the RT is that some linear combination of the waves appearing in (1) is adequate to describe the field near and on S [27,28,30] Taking the first 2Q + 1 of these waves leads to the following expression of the boundary condition
I iI tL
I1 i L A AI \
I
\
AI
\
~A,,
a
Q
A
\',
A
"if" ~
A A a"*
\\
1
&& \
\
\
\ \ \ \\ \ \ \\ \
I I
1
I! ! t
0
,
,
w
,
i
1
,
r
,
k
,
,
,
2
Fig. 1. Total energy (~) versus wavenumber (k) for sinusoidal grating f(x)=h cos 2 ~rx/d. d = 13.6,h = 1, 0i= 40 °. - . . . . KA (zeroth order iterate). A A A A second order iterate. 156
R m o m "ffm '
V x G [O,d].
(8)
\\ \\
.5
~
m =-Q
\
The standard procedure in the RT is to project this relation on 2Q + l basis functions {(3n;n = Q ..... - 1 , 0, 1 .... Q} and solve the resulting linear system for increasing values of Q until (what is hoped) convergence of successive approximations o f R m to the true R m is achieved. We have shown elsewhere [27,28] that this method is entirely successful for gratings of the type and having the slope in fig. 1 provided/3 n is chosen to be 1/o n. Here we proceed somewhat differently by choosing
~,~ = % i
-- c n)/c,, o ~ .
Integrating by parts then permits one to obtain
(9)
Volume 41, number 3
OPTICS COMMUNICATIONS
1 April 1982
O Inl <~ Q,
R n =K n + ~ lnmR m m=-a
(lO) .9
IRol
with K n (as in (7)) given by I1 - SnS 0 + CnCo]
~.(°n ;Co-) J (°°' 1/~°~n)'
Kn : -
(11)
.T
and
[1--SnSm Cn__O on(% - %)
.5
J
X (0 m U'~m, l/On'fin),
.3
sm 4= ±sn ,
~,
[~m =
(Sn/Cn)(O m , / / o n ) ,
sm = --Sn,
O,
Sm = Sn .
(12)
Rather than solve (9) by matrix inversion, we here generate an iteration process by first neglecting the sum, then introducing R (0) = K n into the sum, and so forth. This leads to R(nO) = K n, In l < Q (the zeroth order approximation
in the KA),
Q R~)=Kn+
~
m =-a
IIII
.1
,
i
1
2
k
F i g . 2. IRol versus k f o r s a m e g r a t i n g as in fig. 1. o o o o L a Casce and Tamarkin experiment, c o n v e n t i o n a l R T reference result. - ..... K A ( z e r o t h o r d e r i t e r a t e ) , za A A A second order iterate.
(13) lnmR(i-1) ,
i : 1 , 2 ..... Inl~
with the optimal choice of Q being governed by indications given in [28]. It will be recognized that (13) is analogous to Meecham's integral equation iteration solution [3].
.7
IR_ll
&
.5
4. An unconventional Rayleigh method Another consequence of the Rayleigh hypothesis is that
.3
a m =-Q
R m i ( f s m - Cm)O m U""m .
(14)
By introducing this into the rigorous relation (2) we find exactly the same matrix equation ((9)-(11)) as in the previous section. Relation (14) can be recognized as the implementation of Hagfor's hint [29] for improving the KA.
\
\
,
o
,
i
.
.
,
,
,
1
,
i
2 k
F i g . 3. IR_ l I v e r s u s k. S a m e g r a t i n g a n d s a m e n o t a t i o n s as in fig. 2.
157
Volume 41, number 3
OPTICS COMMUNICATIONS
5. C o n f r o n t a t i o n o f iterated solutions w i t h e x p e r i m e n t and quasi-rigorous reference results The e x p e r i m e n t s are those o f La Casce and Tamarkin [33] and the quasi-rigorous reference results are those o b t a i n e d f r o m the conventional R T with ~n = 1/o n [ 2 7 , 2 8 ] . The iterative process was f o u n d to converge very q u i c k l y to the reference solution except in the vicinity o f the Rayleigh wavelengths (for which Isnl = 1). The second-order iterate is t h e r e f o r e close to the higher-order iterates and is f o u n d in fig. 1 to give a better energy balance than the K A whereas in figs. 2 and 3 it is f o u n d to be n m c h closer than the K A to exp e r i m e n t a l and reference results as concerns [R0[ and IR_ 1 I. Naturally, higher-order iterates give better results, especially with respect to the energy balance.
6. Conclusions It has been shown that n o t o n l y can the K i r c h h o f f a p p r o x i m a t i o n constitute a bad a p p r o x i m a t i o n in a case for w h i c h B e c k m a n n and Spizzichino w o u l d agree t h a t it should be a good a p p r o x i m a t i o n , b u t it also can be m u c h less satisfactory t h a n either the c o n v e n t i o n a l Rayleigh t h e o r y or u n c o n v e n t i o n a l R a y l e i g h m e t h o d solutions. The latter are o b t a i n e d w i t h little effort by an iterative process which constitutes a m e a n s for correcting the K i r c h h o f f a p p r o x i m a t i o n in a systematic manner.
References [1 ] M. Born and E. Wolf, Principles of optics (Pergamon, London, 1959) p. 374. [2] A. Wirgin, Rev. Opt. 43 (1964) 449; 44 (1965) 20. [3] W.C. Meecham, J. Rat. Mech. Anal. 5 (1956) 323.
158
[4] [5] [6] [7]
1 April 1982
J. Pavageau and J. Bousquet, Optica Acta 17 ~1970) 469. R.R. Lentz, Radio Sci. 9 (1974) 1139. H.L. Chan and A.K. Fung, Radio Sci. 13 (1978) 811. R.M. Axline and A.K. Fung, IEEE Trans. Anten. Prop. AP-26 (1978) 482. [8] K.A. Zaki and A.R. Neureuther, IEEE Trans. Anten. Prop. AP-19 (1971) 208. [9] R. Petit, Theis, Facultd des Sciences d'Orsay, 1966. [10] A. Wirgin, Thesis, Facultd des Sciences, d'Orsay, 1967. [11 ] A.K. Fung and HJ. Eom, Radio Sci. 16 (1981) 299. [12] P.J. Lynch and R.J. Wagner, J. Acoust. Soc. Am. 47 (1970) 816. [13] P.J. Lynch, J. Acoust. Soc. Am. 47 (1970) 804. [14] P. Beckmann and A. Spizzichino, The scattering of electromagnetic waves from rough surfaces (Pergamon, London, 1963) p. 19,29,42. [15] R.I. Masel, R.P. Merrill and W.H. Miller, J. Chem. Phys. 65 (1976) 2690. [16] U. Garibaldi, A.C. Levi, R. Spadacini and G.E. Tommei, Surf. Sci. 48 (1975) 649. [17] A. Sommerfeld, Optics (Academic, New York, 1967) p. 229. [18] L.M. Brekhovskikh, Zh. Eksp. i Yeor. Fiz. 23 (1952) 275; 289. [19] H.A. Rowland, Phil. Mag. 35 (1893) 397. [20] C. Janot and A. Hadni, J. Phys. Rad. 23 (1962) 152. [21] F.G. Bass and I.M. Fuks, Wave scattering from statistically rough surfaces (Pergamon, Oxford, 1979). [22] V.V. Liepa, IEEE Trans. Anten. Prop. AP-16 (1968) 273. [23] C. Chiroli and A.C. Levi, Surf. Sci. 59 (1976) 325. [24] G. Whitman, IEEE Trans. Anten. Prop. AP-25 (1977) 869. [25] M.A. Isakovich, Zh. Eksp. i Teor. Fiz. 23 (1952) 305. [26] R. Petit and M. Cadilhac, C.R. Acad. Sci. Paris, B262 (1966) 468. [27] A. Wirgin, J. Acoust. Soc. Am. 68 (1980) 692. [28] A. Wirgin, Optica Acta 27 (1980) 1671. [29] T. Hagfors, in: Electromagnetic wave theory, Pt. 2, cd. J. Brown (Pergamon, Oxford, 1967) p. 997. [30[ L. Rayleigh, Tile theory of sound, Vol. 2 (Dover, New York, 1945) p. 89. [31] H.W. Marsh, J. Acoust. Soc. Am. 33 (1961) 330. [32] P.M. Van den Berg and J.T. Fokkema, Radio Sci. 15 (1980) 723. [33] E.O. La Casce and P. Tamarkin, J. Appl. Phys. 27 (1957) 138.
OPTICS COMMUNICATIONS
l April 1982
ITERATIVE CORRECTIONS TO THE KIRCHHOFF APPROXIMATION IN ROUGH PERIODIC SURFACE SCATTERING A. WlRGIN Laboratoire de Mdcanique Th~orique, Universit~ Paris 6, 75230 Paris Cedex 05, France
Received 23 October 1981
It is shown, contrary to the opinion of Beckmann and Spizzichino, that the Kirchhoff approximation generally gives a less satisfactory description of rough periodic surface scattering than the Rayleigh theory. The errors inherent in the Kirchhoff approximation can, to a large extent, be corrected by an iteration process generated within the framework of the Rayleigh theory. These results hold for periodic opaque scattering surfaces exposed to E-polarized plane waves, but are easily extended to arbitrarily-polarizedwaves incident on non-periodic rough surfaces.
1. Introduction The most direct and rigorous method for treating the problem o f scattering o f waves from rough opaque surfaces makes use o f Green's theorem and leads, quite easily [ 1,2] to the H e l m h o l t z - K i r c h h o f f integral (HK|) relating the scattered field outside the scatterer to field components on the very surface of the scatterer. Some of these field components are known in virtue o f the b o u n d a r y conditions resulting from Maxwell's equations, but others are essentially unknown and must therefore be determined from an integral equation which is generated by evaluation of the HK! on the boundary of the scatterer [ 3 - 1 0 ] . In practice, this task is usually so formidable (and costly) that it is preferable to first apply a simpler approximation procedure. The Kirchhoff approximation (KA) [3,6,11 ], also known as the physical optics [8,12,13], tangent plane [14], semi-classical [15] or eikonal [16] approximation, is one of the best-known and most often used procedures o f this type. The association of the name o f Kirchhoff with the KA comes from the latter's contribution to the mathematical formulation of the H u y g h e n s - F r e s n e l principle (HFP) [1 ]. Loosely speaking [17], the KA assumes the incident wave W to proceed unperturbed to each point P of the scattering surface S; this means that all points P of S are reached by 154
W without traversing the (opaque) underlying medium (no shadowing condition) and that radiation scattered from other parts o f the surface cannot reach P (no multiple scattering condition). The only opaque surface S for which this is rigorously true is a fiat mirror of infinite extent, so that the KA makes the approximation that the field at P o f S is essentially what it would have been if S had been a fiat mirror with a slope equal to that of S at P. The introduction of the KA into the HKI then enables the determination of the field at any point exterior to the scatterer. This procedure was first applied, in the extensive form that we have just outlined, by Brekhovskikh [18], but all those who, starting with Rowland [19] used the HFP for the study of rough surface scattering, implicitly assumed at least a restricted form o f the KA [17,20]. Much has been written [6,11-14,18,21] about the domain of validity of the KA, but agreement is obtained only for the condition r >> X, with r the minimal radius o f curvature of S and X the wavelength of the incident radiation. Naturally, this condition is rather imprecise (how much is "much larger"?) and gives no indication o f the influence of the incident angle and scattering angle which are known to be of some importance [17]. For this and other reasons (related to the fact that some of the reported conditions of validity [6,11,21 ] are so restrictive that they preclude use of the KA in
0 0 3 0 - 4 0 1 8 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 02.75 © 1982 North-Holland
Volume 41, number 3
OPTICS COMMUNICATIONS
all but the most uninteresting scattering configurations), in practice one often adopts a "try and see" attitude concerning the opportunity of use of the KA. The results are sometimes gratifying [6,14,15,22,23], and often very discouraging [5,6,8,12,24]. The most dramatic index of failure of the KA is furnished by large-scale violations of the law of conservation of energy [8,12, 24]. Therefore, KA results which have not been submitted to the energy balance test [14,18,21,25] should be considered to be of questionable accuracy. It is interesting to note that energy non-conservation was one of the main factors which led to the almost unanimous rejection of the well-known Rayleigh theory (RT) of gratings in favor of the rigorous integral equation method [9,26]. For probably other (although rather obscure) reasons Beckmann and Spizzichino [14] suggest that the RT "usually yields a worse approximation than the Kirchhoff solution". Recent publications [27,28] have shown that the RT can actually be made to function quite well for gratings whose slopes are as much as five times larger than the slope beyond which Petit and Cadilhac showed the RT to be mathematically unrigorous [26]. Considering the importance of Beckmann and Spizzichino's book for all those who are confronted with rough surface scattering problems, it should be useful to confront the above-cited opinion with facts. As is shown herein, the latter indicate, on the contrary, the superiority of the RT over the KA. Moreover, it will be shown that the KA actually constitutes the lowestorder approximation of an iterative process generated within the very framework of the RT and that carrying out the iterations beyond the KA can enable a very substantial correction of the errors inherent in the latter. The method developed herein arose from several previous works. Meecham [3] showed that the KA constituted the lowest-order approximation in an iterarive process generated in the integral equation method (which was numerically implemented by Pavageau and Bousquet [4] ). We showed [10] that the KA could be identified with either the zeroth or first order solution arising from a perturbation scheme based on the smallness of ?t/d, with d the correlation length or period of S. Lynch [13] demonstrated that a useful correction to the KA could be obtained by a variational method. Hagfors [29], on searching for a way to improve the KA, suggested that the fields at S should
1 April 1982
contain, in addition to the specular component associated with the KA, an angular spectrum of plane waves traveling both toward and away from S. Here the angular spectrum of Hagfors is the one inherent in the Rayleigh theory [30], and we shall assume (as did Rayleigh) the scattering surface to be an opaque grating (although our method is easily extended to non-periodic surfaces by use of the Marsh [10,31,32] extension of the RT).
2. Preliminaries A periodic surface S, describable by y = f(x) = f(x + d), Vx C R, Vz E R, w i t h f a continuous and f = df/dx a piecewise continuous function of x, separates an infinitely conducting (totally opaque) medium M 2 from a lossless dielectric medium M l within which propagates a monochromatic plane wave W towards S with its propagation vector in the x - y plane. The time factor (implicit in the sequel) is e x p ( - i w t ) , with co the angular frequency related to ~, the wavelength in M 1 by co = kc = 27rc/X, wherein c designates the speed of light in M 1 . The total field in M 1 is of two-dimensional nature (z the ignorable coordinate) because of the symmetry of S and W. The field associated with W is assumed to be polarized in the E mode (i.e., only the z component of the electric field, designated herein by u, is non-zero). W is given by ui(x,y ) = exp [ik(six ciY)], with s i = sin 0i, c i = cos 0i, 0 i the incident angle measured counterclockwise from the positive y axis, and gives rise to the scattered field Us(X,y ) which is related to the total field u(x,y) in M 1 by u = u i + u s. The HKI and the boundary condition (u'-= u(x,f(x)) = 0, Vx E [0, d] ) enables the following rigourous result to be obtained [27,28] :
Us(X,y ) = ~
R n
OnUn;
y > Max(f(x)),
(1)
n =-oo
with R
= (i/2o,~',en ,o-'3,
(2)
and
on = exp(iksnx), cn = (1
o2~1/2. --
an)
Un = exp(ikcnY)'
Re(cn)/> 0, Im(cn)/> 0,
U'n = exp [ikcnf(X)] ,
(3) 155
Volume 4 i. number 3
OPTICS COMMUNICATIONS
The difficulty in this problem arises from the fact that although we have used all available information on u to obtain relations (1)-(3), we know nothing about ~'. The Kirchhoff approximation assumes
"~= k - l ( ~ O / O x - Ot8y) u ( x , f ( x ) ) = "~i + Vs' d (~, Q) = d -1 f 7 i x ) ~ ( x ) dx. 0
The problem is to determine the amplitudes R n of the plane waves composing the plane wave spectrum representation of the scattered field. These amplitudes are related to the so-called grating efficiencies [9] ~n by (4)
C n = IR n 12 Cn/Ci,
the latter satisfying, if all goes well, the conservation of energy relation oo
11 = - - ~
(5)
6:Re(n__~_ 6 n ) : l -
ij tl t li
1.5
I April 1982
= 2 v~
on s,
(6)
so that R m is approximated by (7)
R m ~-- Km = (i/°n U~nCn' ~i)"
An example of this type of calculation, for a sinusoidal grating f(x) = h cos 2~x/d, is given in fig. 1. Although both the Lynch condition [11,13] (kr cos30 >> 1, with r = d2/4n2h and 0 = 0 i in the present case) and the Bass and Fuks [13,21] condition ((2h/d) 2 ,~ 1 and k d ~ 1) for the validity of the KA are satisfied over a large portion of the range of k covered in fig. 1, it is seen that the KA leads to very serious energy conservation violations over this same range even though thle no shadowing condition is satisfied. These results give ample support to Chan and Fung's view [6] that further studies are needed to better define the conditions of validity of the KA.
A
3. The c o n v e n t i o n a l Rayleigh t h e o r y
li
The hypothesis underlying the RT is that some linear combination of the waves appearing in (1) is adequate to describe the field near and on S [27,28,30] Taking the first 2Q + 1 of these waves leads to the following expression of the boundary condition
I iI tL
I1 i L A AI \
I
\
AI
\
~A,,
a
Q
A
\',
A
"if" ~
A A a"*
\\
1
&& \
\
\
\ \ \ \\ \ \ \\ \
I I
1
I! ! t
0
,
,
w
,
i
1
,
r
,
k
,
,
,
2
Fig. 1. Total energy (~) versus wavenumber (k) for sinusoidal grating f(x)=h cos 2 ~rx/d. d = 13.6,h = 1, 0i= 40 °. - . . . . KA (zeroth order iterate). A A A A second order iterate. 156
R m o m "ffm '
V x G [O,d].
(8)
\\ \\
.5
~
m =-Q
\
The standard procedure in the RT is to project this relation on 2Q + l basis functions {(3n;n = Q ..... - 1 , 0, 1 .... Q} and solve the resulting linear system for increasing values of Q until (what is hoped) convergence of successive approximations o f R m to the true R m is achieved. We have shown elsewhere [27,28] that this method is entirely successful for gratings of the type and having the slope in fig. 1 provided/3 n is chosen to be 1/o n. Here we proceed somewhat differently by choosing
~,~ = % i
-- c n)/c,, o ~ .
Integrating by parts then permits one to obtain
(9)
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1 April 1982
O Inl <~ Q,
R n =K n + ~ lnmR m m=-a
(lO) .9
IRol
with K n (as in (7)) given by I1 - SnS 0 + CnCo]
~.(°n ;Co-) J (°°' 1/~°~n)'
Kn : -
(11)
.T
and
[1--SnSm Cn__O on(% - %)
.5
J
X (0 m U'~m, l/On'fin),
.3
sm 4= ±sn ,
~,
[~m =
(Sn/Cn)(O m , / / o n ) ,
sm = --Sn,
O,
Sm = Sn .
(12)
Rather than solve (9) by matrix inversion, we here generate an iteration process by first neglecting the sum, then introducing R (0) = K n into the sum, and so forth. This leads to R(nO) = K n, In l < Q (the zeroth order approximation
in the KA),
Q R~)=Kn+
~
m =-a
IIII
.1
,
i
1
2
k
F i g . 2. IRol versus k f o r s a m e g r a t i n g as in fig. 1. o o o o L a Casce and Tamarkin experiment, c o n v e n t i o n a l R T reference result. - ..... K A ( z e r o t h o r d e r i t e r a t e ) , za A A A second order iterate.
(13) lnmR(i-1) ,
i : 1 , 2 ..... Inl~
with the optimal choice of Q being governed by indications given in [28]. It will be recognized that (13) is analogous to Meecham's integral equation iteration solution [3].
.7
IR_ll
&
.5
4. An unconventional Rayleigh method Another consequence of the Rayleigh hypothesis is that
.3
a m =-Q
R m i ( f s m - Cm)O m U""m .
(14)
By introducing this into the rigorous relation (2) we find exactly the same matrix equation ((9)-(11)) as in the previous section. Relation (14) can be recognized as the implementation of Hagfor's hint [29] for improving the KA.
\
\
,
o
,
i
.
.
,
,
,
1
,
i
2 k
F i g . 3. IR_ l I v e r s u s k. S a m e g r a t i n g a n d s a m e n o t a t i o n s as in fig. 2.
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OPTICS COMMUNICATIONS
5. C o n f r o n t a t i o n o f iterated solutions w i t h e x p e r i m e n t and quasi-rigorous reference results The e x p e r i m e n t s are those o f La Casce and Tamarkin [33] and the quasi-rigorous reference results are those o b t a i n e d f r o m the conventional R T with ~n = 1/o n [ 2 7 , 2 8 ] . The iterative process was f o u n d to converge very q u i c k l y to the reference solution except in the vicinity o f the Rayleigh wavelengths (for which Isnl = 1). The second-order iterate is t h e r e f o r e close to the higher-order iterates and is f o u n d in fig. 1 to give a better energy balance than the K A whereas in figs. 2 and 3 it is f o u n d to be n m c h closer than the K A to exp e r i m e n t a l and reference results as concerns [R0[ and IR_ 1 I. Naturally, higher-order iterates give better results, especially with respect to the energy balance.
6. Conclusions It has been shown that n o t o n l y can the K i r c h h o f f a p p r o x i m a t i o n constitute a bad a p p r o x i m a t i o n in a case for w h i c h B e c k m a n n and Spizzichino w o u l d agree t h a t it should be a good a p p r o x i m a t i o n , b u t it also can be m u c h less satisfactory t h a n either the c o n v e n t i o n a l Rayleigh t h e o r y or u n c o n v e n t i o n a l R a y l e i g h m e t h o d solutions. The latter are o b t a i n e d w i t h little effort by an iterative process which constitutes a m e a n s for correcting the K i r c h h o f f a p p r o x i m a t i o n in a systematic manner.
References [1 ] M. Born and E. Wolf, Principles of optics (Pergamon, London, 1959) p. 374. [2] A. Wirgin, Rev. Opt. 43 (1964) 449; 44 (1965) 20. [3] W.C. Meecham, J. Rat. Mech. Anal. 5 (1956) 323.
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[4] [5] [6] [7]
1 April 1982
J. Pavageau and J. Bousquet, Optica Acta 17 ~1970) 469. R.R. Lentz, Radio Sci. 9 (1974) 1139. H.L. Chan and A.K. Fung, Radio Sci. 13 (1978) 811. R.M. Axline and A.K. Fung, IEEE Trans. Anten. Prop. AP-26 (1978) 482. [8] K.A. Zaki and A.R. Neureuther, IEEE Trans. Anten. Prop. AP-19 (1971) 208. [9] R. Petit, Theis, Facultd des Sciences d'Orsay, 1966. [10] A. Wirgin, Thesis, Facultd des Sciences, d'Orsay, 1967. [11 ] A.K. Fung and HJ. Eom, Radio Sci. 16 (1981) 299. [12] P.J. Lynch and R.J. Wagner, J. Acoust. Soc. Am. 47 (1970) 816. [13] P.J. Lynch, J. Acoust. Soc. Am. 47 (1970) 804. [14] P. Beckmann and A. Spizzichino, The scattering of electromagnetic waves from rough surfaces (Pergamon, London, 1963) p. 19,29,42. [15] R.I. Masel, R.P. Merrill and W.H. Miller, J. Chem. Phys. 65 (1976) 2690. [16] U. Garibaldi, A.C. Levi, R. Spadacini and G.E. Tommei, Surf. Sci. 48 (1975) 649. [17] A. Sommerfeld, Optics (Academic, New York, 1967) p. 229. [18] L.M. Brekhovskikh, Zh. Eksp. i Yeor. Fiz. 23 (1952) 275; 289. [19] H.A. Rowland, Phil. Mag. 35 (1893) 397. [20] C. Janot and A. Hadni, J. Phys. Rad. 23 (1962) 152. [21] F.G. Bass and I.M. Fuks, Wave scattering from statistically rough surfaces (Pergamon, Oxford, 1979). [22] V.V. Liepa, IEEE Trans. Anten. Prop. AP-16 (1968) 273. [23] C. Chiroli and A.C. Levi, Surf. Sci. 59 (1976) 325. [24] G. Whitman, IEEE Trans. Anten. Prop. AP-25 (1977) 869. [25] M.A. Isakovich, Zh. Eksp. i Teor. Fiz. 23 (1952) 305. [26] R. Petit and M. Cadilhac, C.R. Acad. Sci. Paris, B262 (1966) 468. [27] A. Wirgin, J. Acoust. Soc. Am. 68 (1980) 692. [28] A. Wirgin, Optica Acta 27 (1980) 1671. [29] T. Hagfors, in: Electromagnetic wave theory, Pt. 2, cd. J. Brown (Pergamon, Oxford, 1967) p. 997. [30[ L. Rayleigh, Tile theory of sound, Vol. 2 (Dover, New York, 1945) p. 89. [31] H.W. Marsh, J. Acoust. Soc. Am. 33 (1961) 330. [32] P.M. Van den Berg and J.T. Fokkema, Radio Sci. 15 (1980) 723. [33] E.O. La Casce and P. Tamarkin, J. Appl. Phys. 27 (1957) 138.