PHENOMENOLOGICAL DESCRIPTION OF SINGULARITIES OF WAVE DIFFRACTION IN THE ELECTRODYNAMICS OF PERIODIC AND INHOMOGENEOUS MEDIA Yu. M. Aivazyan and V. A. Sozinov

UDC 537.874

A phenomenological description of multiwave diffraction on periodic and inhomogeneous structures without losses is proposed, using the concept of a scattering matrix S in the form which permits one to distinguish explicitly the analytic singularities (radical branchings, poles, and nulls). It is shown that the Heitler K-matrix formalism makes possible a qualitative analysis of the superposition of threshold and resonance singularities in wave diffraction without the solution of boundary-value problems.

i. As is well known, the amplitudes and phases of electromagnetic or acoustic waves undergo sharp changes in certain fairly narrow spectral or angular intervals upon diffraction on periodic or inhomogeneous structures (diffraction gratings, stratified media, etc.). Observable singularities include a) threshold (Rayleigh) singularities corresponding to the formation or disappearance of spectra of uniform waves [I, 2] and b) resonance singularities associated with the rereflection of waves in elements of periodic structures under the conditions of multiple scattering [3-5]. Singularities of the second type can also be related to complex waves or surface plasmons generated near the surface of a structure [6]. Depending on the concrete conditions (the geometry of the structure, its dielectric properties, etc.), these singularities may either be separated in frequency or angle of incidence of the wave or nearly coincide. Threshold and resonance phenomena are associated with analytic singularities of the amplitudes and p h a s e s o f diffracted waves in the complex plane of wave frequency or angle of incidence, i.e., with radical branchings and simple poles. In a number of cases, nulls of complex amplitudes have a large influence on the observed phenomena. Superposition of the above-noted singularities can lead to more complicated behavior of the amplitudes and phases of diffracted waves than isolated threshold or resonance singularities, such as when one of the poles or nulls lies near a threshold point. In the present paper it is shown that the application of methods similar to those used in the theory of multichannel nuclear reactions permits a qualitatively correct description of all the singularities in wave diffraction without the solution of boundary-value problems. In such an approach one uses considerations of a general nature, such as the unitary and mutual nature and the analytic properties of the scattering matrix, the existence of threshold parameters, and the periodic and asymptotic properties of the solutions. Such a description turns out to be useful in the analysis of singularities of the complex amplitudes of waves scattered without losses by structures. For example, the use of phenomenological methods made it possible to predict the possibility of total reflection of energy in one of two spatial harmonics of the spectrum of an echelette grating without using an autocollimation regime [7]. 2. To describe diffraction phenomena we can represent the scattering matrix S in a form which enables us to distinguish explicitly the analytic singularities of matrix elements (radical branchings, poles, and nulls) [8-10]. For a system without losses, in which N waves are propagating (there are N open channels) under the given conditions (given values of the wave frequency or angle of incidence), the block of the matrix S associated with uniform waves is unitary and has the order N • N. It can be represented in the form

Scientific Production Combine of the All-Union Scientific-Research Institute of Physicotechnical and Radio-Engineering Measurements. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 32, No. 5, pp. 593-599, May, 1989. Original article submitted July 20, 1987. 446

0033-8443/89/3205-0446512.50

9 1989 Plenum Publishing Corporation

S =

~/2~-I: =

~ 1 - - i~i/2K~l/2) .(1 + i~/2K~i/2)-i,

(1)

where ~ is a real diagonal matrix of order N • N composed only of radical threshold parameters of the open channels. In the case of the diffraction of plane electromagnetic waves on a one-dimensional, reflective diffraction grating, for example, the diagonal elements of the matrix ~ have the form ~n = /k0 ~ - ~n = (
S~

K~

~

where the index ~ refers to open channels and y to closed channels, w i t h Im ~ = 0 a n d Cy = i [ ~ y [ . F o r N - 1 o p e n c h a n n e l s t h e S~ m a t r i x i s o f o r d e r (N - 1) x (N - 1 ) , S B a n d ~ a r e a n (N - 1 ) - d i m e n s i o n a l c o l u m n v e c t o r a n d row v e c t o r , a n d Sy i s a s c a l a r (the tilde sign above a vector denotes the transposition operation while a "+" sign denotes the operation of Hermitian conjugation). For L closed channels the S= matrix is of order (N - L) • (N - L), the $8 matrix is of order (N - L) • L, and the SX matrix is of order L • L. The same thing pertains to the orders of the ~ and K matrices. Since the unitarity relation is valid only for open channels (S=+S= = i), from (I) and (2) for S= we find

S, = _ ~ ~,~D. ~ [ I / ~ =

(1 _


,

(3)

where K r is the reduced K matrix for the case of N - 1 open channels. This K r matrix has the same order as the matrix K~ and is related to the total K matrix, which pertains to both the open and closed channels, as follows: K~ = K= - - i K ~ v (1 + i K v ~ ) - 1 ~ . The reduced K r matrix below the threshold of opening of the N-th K r = K s + KsI~yI(l - KyI~TI)-IKs, while above the threshold (~72 matrix K r does not have a continuous derivative at the threshold when ~X = 0, since in the transition through the threshold il~yl

(4) channel (~u < 0) is real, > 0) it is complex. The of opening of a new channel, changes into ~y.

From (I) and (3) it follows that the behavior of the S matrix above the threshold of opening of the N-th channel is determined by the matrix ~ (~ contains the radical threshold parameter ~7). Below the threshold the behavior of the matrix S is determined by the reduced K r matrix. Here the total K matrix is real, it is a function of ~X= over the entire real axis, and it does not have radical threshold singularities (the discontinuity at a branch cut is given only by the imaginary part of the matrix). Knowing the analytic behavior of the elements of the K matrix above the threshold of opening of the N-th channel, using Eqs. (3) and (4) and analytic continuation of the elements of the K matrix into the domain below the threshold one can find the values of the elements of the S matrix over the entire real axis. As follows from (i), all the matrix elements of S have the same branching points and the same poles [when det(l + i ~ / = K ~ / = ) = 0], but the nulls of different matrix elements of S can be different. Resonance singularities in a multichannel system associated with poles of the S matrix can be determined as poles of the total K matrix for certain real values of the wave frequency or angle of incidence, common for all elements of K, as well as poles of the reduced K r matrix, with additional poles appearing in the K r matrix [det (i + iKx~ Y) = 0], which are associated with the appearance of resonances in the closed channels ("virtual

447

resonances"). The mechanism of formation of such resonances is associated with complex waves or surface plasmons, generated near the surface of the structure, the field of which decreases exponentially as they propagate along the periodic structure because of energy losses to emission. The behavior of elements of the scattering matrix S near the threshold point ~7 = 0 can be found by expanding S with respect to the small branching quantity ~7. From Eq. (3) for the matrix S= we obtain S~ = S (~

- il~yIS(~

S~ ffi S (~

at the threshold,

S= = S (~

- ~S(~

where S (~

= (i -

threshold (for ~

before the threshold,

after the threshold,

i~=i/2K~a~/2)(l + i~a~/2Ka~=~/2)-~ = 0).

is the value of the matrix S at the

The matrix B has the form B = ~ -~/2~8(0) + ~8(0)~ -~/2, where

R8 (~ ffi-2iKs~e~/2(l + i~/2K~a ~/2) -x~a~/2 is a vector consisting of the coefficients of transformation of waves from open channels = to the newly opened channel with the index at the threshold. A difference of the matrix S(~ from zero is a sufficient condition for the appearance of radical singularities. When one of the poles of the scattering matrix S= is located close to the threshold point ~7 ffi 0, the character of the behavior of all the elements of the matrix of coefficients of wave transformation near the threshold of opening of the new channel is also radical. In the representation (I) the matrix K can be written in a more general form, K = ~tK,~, where the matrix K' is also real and symmetric, while a lffi0, • • ... The elements of the K matrix for each concrete case can be found, in principle, explicit form by solving the diffraction boundary-value problems.

in

3. We apply the representation (i) to the case of two open channels, i.e., to fourpole systems. For example, we consider the incidence of a plane, TE-polarized electromagnetic wave (the electric vector is perpendicular to the plane of incidence) on a uniform dielectric layer of thickness h with a permittivity e2, located between two half-spaces with permittivities E~ and ga. The scattering matrix S in (i) is a two-row matrix, while the matrices ~ and K for the wave incident from the medium with a permittivity ~ at an angle 8 i relative to the normal to the surface of the layer have the form

ffi(~ 0 ' K=(~2sin(;2h))-'(c~ 2h) 1 where ~i = Jk02r

-

~2

i = 1,

2,

3,

~ = k0e 1 s i n 8 i a r e

the

'

projections

(5)

o f t h e wave v e c t o r s

onto the plane interfaces between the media. In the case of TM polarization, ~i must be replaced by ~i/ei. It is seen from (5) that the matrix K is symmetric, is a function of ~a 2, and does not have a threshold singularity in the transition through the threshold of total internal reflection, when ~a = 0. The reduced matrix K r contains one element and has the form Kr =

;2 --/~stan(~2 h)

(6)

From (3) and (6) for the energy coefficient RTE of wave reflection from the layer before the threshold of total internal reflection (~a = 0) we find k

RTE= ]S~ [~= i

After the threshold the quantity ~a becomes imaginary and RTE = i. The dependence of RTE on sin 8 i for e I ffi 4, e 2 = 3, and e s = 2.8 is shown in Fig. i. Curves 1 and 2 correspond to the values h/X ffi 4 and h/X = 4.5. The behavior of RTE near the threshold point ~3 = 0 is radical, and the oscillating nature of the reflection coefficient is related to the poles and nulls of the scattering matrix S. The location of the poles of the matrix elements of S in the complex plane of the sine of the angle of incidence is determined by the roots of the equation ~2h = m~ + arctan (~i/i~a) + arctan (~s/i~2), and the location

448

~TF..~, tel. units

I

38 06 ,

0,4

1

5

0,2

0,78

O,B2

0,56

5 ~ 8i,rel. units Fig. 1 of the nulls, S=, for example, is determined by the roots of the equation ~2h = m~ - arctan (~I/i~2) + arctan (~3/i~2), m = 0, • • ..... When the null S= on the Re (sin 8 i) axis approaches a threshold point [ii], the reflection coefficient changes abruptly from zero to one near the threshold of total internal reflection. A graph of the dependence of RTE on sin 8 i for e I = 4, e 2 = 3, ~3 = 2.96, and h/% = 1.2 is given in Fig. i (curve 3 ) . 4. An example of a case of N-wave diffraction, permitting the solution of the boundaryvalue problem in closed form, is the scattering of plane, monochromatic electromagnes waves on a one-dimensional, perfectly conducting, reflecting grating with a rectangular line profile [3, 4]. The matrix of coefficients Rn, n of transformation of waves incident on the grating at angles sin On = sin 0 i + nk/d, n = 0, • • . . . . . into diffracted waves of order n' can be reduced to the form

R= (1 -- iK~) (1 + iK~)-~= -- l+2(~+iK')-~,

K'=~Kg.

(7)

The diagonal matrix ~ is constructed from the constants ~n'n = ~n6n'n of wave propagation normal to the grating plane, where ~n = /k02 - ~n 2 and [sin 0n[ ~ 1 for uniform waves and ~n = i[~nl and Isin 0nl > 1 for nonuniform waves. The elements of the matrix K' have the form

x'.,. =

A;.., c . a . . ,

(8)

ntmO

where

A~)

}

a\ ~• ' -cos (--1) "/2sin /~. |'-E--| - - ~(U.h) 2 '

m=~

\ z / x--tm~laj

I

~i (__ l)(m+l)/2 COS {Xna) xn cos (~mh) ~ 2 x~--(m~/a) 2 "

[ 2~(_ 1).,,2sin x a

(rn.~./a) s i n ( ~ m h )

m=2k+

1

m=2k

P

A,.,,= (--1)o'~+l~/2 cos

.

x~--(m~/a)'

9

m=2k+

1

k = 0, i, 2 . . . . , gm = Ck02 - (m~/a)2 for m k / 2 a ~ i and ~m = i[gml for m k / 2 a > i, and h and a are the depth and width, respectively, of the grating lines. The quantities C m are real and have the meaning of the impedance for the field in the grating lines:

449

0'+4"[ su~ 84,

/ ~

/ 0,40 ~ r~=-2 / . 2 "

/

0,696

0,704.

0,742

X/d,

f

Fig. 2

~,o ts~o ~

1,o

O,B ~=0

0,8

0,6

0

~

V

23~

..........

ts~,olZ

06L

'I

~

]

Oi= 24~

3 : ~ ~-~ ,Jr -. - ~ "- " OTC? 0704 ~/d

0,69+ 06~6 06~8 )~~d

4 0

"

r : .....

~

....

O,E ~ ~-'0~'"'~

~

~

":~

/

"'

A Z~~ / ~

9

_

,

n = 0

= , ~ . 7 _ ~

/

|

' '~

1 I

I

!

~ o4l

J 0,706

O,T08

0510 ,k /d

0,7'~5

Fig.

c,.

f--~m tan

(~mh)

O,7~e, 0.720 ,kid

3 -

-

TM polarization,

l

{ ~m Jot (~mh) -- TE polarization. The energy coefficients of reflection of a wave incident on the grating at an angle 8 i into diffracted waves of order n are determined by the squares of the matrix elements of S: ISn,01 = = (~n/~o)IRn,01 =. The matrices R, K', and ~ and the scattering matrix S associated with them are infinite-dimensional. The blocks S= and ~= of the scattering matrix S and the matrix ~, associated with uniform waves, have the order N • N, where N is the number of uniform diffracted waves. The elements of the matrix ~ are real, while the matrix S~ is unitary and satisfies the reciprocity condition. From Eq. (8) it follows that the matrix K' is real and does not have radical threshold singularities corresponding to the appearance or disappearance of diffracted waves. The location of the poles of the matrix S~ is determined by the equation d e t ( ~ + iKr') = 0, where K r' = K s' - K~'([~71 + KT')-lK~'.

450

In Fig. 2 we show the trajectory of motion of one of the poles of the matrix S (or order 2 • 2) in the (sin 9i, k/d) plane in the case when there are two uniform diffracted waves (n = -i, 0). The dashed line is the boundary of the region of propagation of the harmonic n = -2. The results of a calculation of the dependence of the reflection coefficients ISn,012 for spatial harmonics of orders n = -2, -i, and 0 on wavelength in the diffraction of a TM-polarized wave on a grating with the parameters h/d = 0.7 and a/d = 0.43 are given in Fig. 3. The values of the angles e i correspond to the location of points on the trajectory of motion of a pole of the matrix Se when the threshold of formation of the wave n = -2 approaches a resonance singularity. In the calculations the infinite system of equations (7) is replaced by a finite system (Inl ~ i0, m ~ i0). The accuracy of the solutions obtained is 10 -2 in this case. Allowance for nonuniform spatial harmonics of higher order does not lead to a significant increase in accuracy, since the transition to higher values of n and m is accompanied by a rapid decrease in the amplitudes of the spatial harmonics in the direction of the normal to the grating plane. The resonance behavior of the reflection coefficients, associated with the excitation of intense vibrations of the electromagnetic field in elements of the periodic structure considered as open resonators under the conditions of multiple scattering, is determined by the resonance behavior of the elements of the K matrix. Near the threshold of the appearance of the spatial harmonic n = -2 the width of the resonance is not constant, while the intensity of the radical threshold singularity increases sharply at resonance. This phenomenological description of threshold and resonance singularities of wave diffraction shows that the complex amplitudes of uniform waves have a radical behavior at the threshold of the appearance or disappearance of a wave of a new order near a resonance singularity as well. The examples given show that the elements of the K matrix do not have radical singularities at thresholds, which allows one to find the threshold be, havior of elements of the S matrix by assigning values of the phenomenological parameters to the matrix. Energy losses in the medium were ignored above in the analysis of the diffraction of electromagnetic waves on inhomogeneous and periodic structures. In the case of energy losses, the block of the scattering matrix S associated with uniform waves is not unitary, while the K matrix is Hermitian. Allowance for energy absorption in the medium (diffraction gratings with a finite conductivity, etc.) requires the introduction of new phenomenological parameters that do not have analogs in the theory of multichannel nuclear reactions. This question is of definite interest and will be considered in a separate paper. LITERATURE CITED i. 2. 3. 4. 5. 6. 7. 8. 9. i0. Ii.

B. M. Bolotovskii and A. N. Lebedev, Zh. ~ksp. Teor. Fiz., 53, No. i0, 1349 (1967). Yu. M. Aivazyan and B. M. Bolotovskii, Akust. Zh., 28, No. 2, 145 (1982). J. R. Andrewartha, J. R. Fox, and I. J. Wilson, Opt. Acta, 26, No. I, 69 (1979). J. R. Fox, Opt. Acta, 27, No. 3, 289 (1980). I. A. Urusovskii, Dokl. Akad. Nauk SSSR, 131, No. 4, 801 (1960). P. Sheng, R. S. Stepleman, and P. N. Sanda, Phys. Rev. B, 26, No. 6, 2907 (1982). D. Maystre, M. Cadilhac, and J. Chandezon, Opt. Acta, 28, No. 4, 457 (1981). W. Heitler, Proc. Cambridge Philos. Soc., 37, 291 (1941). E. P. Wigner, Phys. Rev., 70, 15 (1946). R. H. Dalitz, Strange PartiCles and Strong Interactions, Oxford Univ. Press, London (1962). Yu. M. Aivazyan and V. A. Sozinov, in: Abstracts of Papers of the 9th All-Union Symposium on Wave diffraction and Propagation [in Russian], Vol. i, Tbilisi (1985), p. 266.

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