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Embedding formula for an electromagnetic diffraction problem
A. V. Shanin November 29, 2004
Abstract Emb edding formulae is a p owerful to ol enabling one to reduce the dimension of the space of variables for a diffraction problem. Let the scatterer b e finite, planar and p erfectly conducting. The idea of the metho d is to substitute the initial problem of diffraction of a plane wave by finding an edge Green's function, i.e. to solve a problem with a source lo cated near the edge of a scatterer. Emb edding formula is an integral relation connecting the solution of the initial plane wave incidence problem with the edge Green's function. Earlier, the emb edding formulae have b een derived for acoustic and elasticity problems. Here we derive en emb edding formula for an electromagnetic problem.

1

Introduction

Emb edding formulae b elong to a rather new typ e of the relations in diffraction theory. For the first time (up to our knowledge) the emb edding formula was intro duced by M. H. Williams [1]. The idea was the following. The 2D scalar problem of scattering of a plane wave by a thin strip has b een considered. The main ob jective of the research was to find the diffraction co efficient f (, 0 ) dep ending on the angle of incidence and the angle of scattering. M. H. Williams intro duced a set of auxiliary functions, who were the diffraction co efficients calculated for the case of a fixed incidence angle 0 , namely for the grazing incidence 0 = ± /2. Since the incidence angle was fixed, the auxiliary functions dep ended only on a single variable. By manipulation with the integral equation M. H. Williams managed to express the diffraction co efficient f (, 0 ) in terms of the auxiliary functions f1,2 () in rather simple manner. Since the diffraction co efficient dep ends on two variables and each of the auxiliary functions dep ends on a single variable, we can say that diffraction co efficient was factorized in some sense. For the next time, the emb edding formula app eared in the pap er by P. A. Martin and G. R. Wickham [2]. They studied the problem of of a plane wave scattering by a p enny-shap ed flat crack in a bulk of a solid. As the result of tedious calculations, the authors obtained a formula expressing the solution for an arbitrary incidence angle in terms of the solutions related to the grazing incidence. 1


After this, the emb edding formulae were forgotten for a long time. Recently, the emb edding formulae have b een revived Biggs et. al. [3, 4, 5]. They obtained and checked the emb edding formulae for the problems of scattering by several thin strips, by strips of finite thickness and by the walls of a p erforated duct. In all cases the way to derive the emb edding formulae and the form of the formulae themselves remained rather complicated. As the result, such a bright result of diffraction theory remained almost unknown to most of the sp ecialists. The author of this pap er in collab oration with Dr. R. V. Craster develop ed an easier way to derive the emb edding formula for some diffraction problems [6, 7]. Earlier R. V. Craster applied the emb edding ideas to solving the ordinary differential equations of Heun's typ e [8]. In our approach we cho ose the auxiliary functions as follows. Instead of the grazing incident plane wave we use a p oint source lo cated close to one of the edges of the scatterer. The solution corresp onding to this source is called an edge Green's function. Formally, the pro cess of constructing the edge Green's function is describ ed by a limiting pro cedure, since one cannot place a source directly at the edge. However, the approach based on the edge Green's functions seems to b e simple and physically transparent. The purp ose of the current work is to derive the emb edding formula for the case of an electromagnetic wave diffraction by a scatterer containing the edges. For this we use our standard approach and mo dify it according to the vector nature of the electromagnetic field.

2

Problem formulation

We study the problem of diffraction of a plane electromagnetic wave by an ideal (conducting) plane scatterer S of zero thickness lo cated in the plane (x, y ). The edge of the scatterer is a curve , which is smo oth enough. A co ordinate l is defined on ; here l is the length along the curve counted from some starting p oint.

Figure 1: Geometry of the problem At each p oint of construct a unit internal normal vector in the plane (x, y ). The angle b etween this vector and the x-axis will b e denoted as (l) (see Figure 1). Intro duce the lo cal cylindrical co ordinates near each p oint of the edge. One of these 2


co ordinates is l, two other ones are and (see Figure 2). For the correctness we can consider these co ordinates as curvilinear ones making the edge b e describ ed by the relation = 0.

Figure 2: Lo cal cylindrical co ordinates near the edge of the scatterer The Maxwell equations are assumed to b e valid in the 3D space for the vectors of electric and magnetic fields: â â · · E H E H = = = = ik0 H, -ik0 E, 0, 0, (1) (2) (3) (4)

where k0 = /c0 , is the circular frequency (the time dep endence has the form of e-it and it is omitted henceforth), c0 is the sp eed of light. The b oundary conditions on the surfaces of the scatterer can b e written using the unit vector n normal to the surface as E â n = 0, (5)

i.e. the tangential comp onent of E should b e equal to zero. A correct formulation of the diffraction problem includes the Meixner's edge condition. The edge condition denotes the fact that the total energy concentrated near a finite fragment of the edge is finite. This means that the combination E 2 + H 2 grows slower than for 0, where > -2. Formulate the radiation conditions for our problem. The total field should b e represented as a sum of incident wave and the outgoing spherical wave. The form of the spherical wave is given b elow. Intro duce the notation for the directions of propagation and for the p olarizations of the plane electromagnetic waves as follows. Denote the directions of incidence and scattering by the p oints on a unit sphere (or the unit vectors, which is the same) 0 and (see Figure 3). The p oints are identified by their spherical co ordinates ( , ), 0 (0 , 0 ). At each p oint of the sphere define a tangential plane. Let V( ) b e the 2D vector space in this plane. Indicate the amplitude and p olarization of the plane electromagnetic wave by the vector E b elonging to V( ). The vector H for the outgoing wave can b e expressed by the well-known identity H = â E. 3 (6)


Figure 3: Directions of incidence and scattering The incident plane wave in these notations have the form: E = E0 exp{-ik0 (x sin 0 cos 0 + y sin 0 sin 0 + z cos 0 )}, (7)

E0 is the amplitude. Our purp ose here will b e to establish the emb edding relations for the tensor function f ( , 0 ) having its values in V( ) V(0 ) and describing the spherical part of the scattered field: Esc (r, ) = 2 e k0 r
ik0 r

f ( , 0 )E0 + O (e

ik0 r

(k0 r )-2 )

for

r ,

(8)

where r is the distance from the origin. To make the last formula clear, intro duce some orthonormal co ordinates in the i spaces V( ) and V(0 ). Let Esc -- b e the comp onents of the p olarization of the j scattered wave pro jected onto V( ), and let E0 b e the comp onents of the vector of p olarization of the incident wave. We are lo oking for such a tensor f ij , that
i Esc (r, ) =

2 e k0 r

ik0 r j

j f ij ( , 0 )E0 + O (e

ik0 r

(k0 r )-2 ).

(9)

2.1

Overview of the pro cedure of deriving the emb edding formula

Let us give a sketch of the pro cedure leading to the emb edding formula. On the first (auxiliary) step we find the form of the edge asymptotics of our solution ob eying the edge (Meixner's) conditions. These asymptotics can b e obtained from the solution of the classical Sommerfeld half-plane problem. Also we define the edge Green's functions by placing p oint sources near the edge of the scatterer. Obviously, the edge Green's function violates the edge conditions due to the presence of the sources. However, the degree of growth (the exp onent of -1 ) is only by 1 higher than that of the solution ob eying the edge conditions. We shall say that such a function is slightly oversingular. 4


On the second step we apply to the total field (b oth electric and magnetic part of it) the differential op erator Px = + ik0 sin 0 cos 0 . x (10)

Note that this op erator kills the incident wave. The result Px [E, H] will b e interpreted as a new electromagnetic field. This field ob eys Maxwell equations and the radiation condition. Moreover, a simple analysis shows that this field is slightly oversingular. On the third step we prove the Lemma stating that any slightly oversingular solution of the Maxwell equation ob eying the radiation conditions can b e represented as a linear combination of the edge Green's functions. The sources are unknown but they are connected with the main term of edge asymptotics of the field. Therefore Pz [E, H] can b e represented as a convolution-form integral of the edge Green's function over the co ordinate of the source with the unknown density of the source. This representation is a weak form of the emb edding formula. On the fourth step we express the density of the source through the edge Green's function. The recipro city theorem is used for this.

3

Edge asymptotics of the field

The edge asymptotics of the field can b e found from the solution of the Sommerfeld problem. Namely, the singular part of the field can b e split into two mo des: in one of them the electric comp onent is parallel to the edge, and in another the magnetic comp onent is parallel to the edge: E H = EE HE + EH HH , (11)

where the fields are determined by their asymptotics as 0 EE = HH = 2CE (x)
1/2

sin

+ O (), 2

HE =

â EE , ik0

(12)

2CH (x) 1/2 â HH EH = - . (13) cos + O (), 2 ik0 Here is the unit vectors tangential to and directed in the p ositive l direction. The functions CE (l) and CH (l) are unknown co efficients playing an imp ortant role b elow.

4

Edge Green's functions
EE HE EH HH

Define a pair of edge Green's functions GE (x, y , z ; ) and GH (x, y , z ; ): GE = , 5 GH = (14)


Define the field GE approximations to EE ^ equations for EE and

as a result of the following limiting pro cedure. Denote the ^ ^ and HE by EE and HE . Consider the inhomogeneous Maxwell ^ HE : ^ ^ â EE - ik0 H ^ ^ â HE + ik0 E ^ ·E ^ ·H
E E E E

= 0, 4 ^ = j, c0 = 0, = 0,

(15) (16) (17) (18)

where

1/2 ^ = (l - ) ( - ) ( - ), j 3/2

(19)

i.e. ^ is the element of current lo cated at the distance from the p oint l = of the j edge (see Figure 4). The amplitude of the current is chosen to b e such that there exists a finite non-zero limit of the field as 0. Equations (15)­(18) are considered with the edge conditions, radiation conditions and the b oundary conditions.

Figure 4: Lo cation of the source for the edge Green's function We define EE and HE as follows: ^ EE = lim EE ,
0

^ HE = lim HE .
0

(20)

Consider the edge asymptotics of the edge Green function GE . Due to the source lo cated near the edge, the field has a complicated structure near the edge along the l co ordinate. Avoid considering this structure by constructing the convolution-typ e integral


I(x, y , z ) =
-

h(x)GE (x, y , z ; )d ,

where h( ) is an arbitrary smo oth enough density function, which is non-zero on a small segment of the edge. Consider the vicinity of the p oint l = of the edge. Return to the limiting pro cedure and find the asymptotics of the solution for some small but finite in a small area near this p oint. 6


Lo cally the inhomogeneous Maxwell equations can b e approximately reduced to the 2D inhomogeneous Poisson equation E = -h( ) 4 c0 ( - ) ( - ), (21)

where is the Laplacian in the plane l = . Equation (21) can b e solved using the conformal mappings metho d. Taking into account the b oundary conditions we find: E - 2h( ) c0 Re [log( - ) - log( + )], (22)

where = (i sin - cos ). Calculating the outer asymptotics of this solution as 0, obtain the following representation for the electric field in I: 4 h(l) -1/2 E= sin + O () for 0. (23) c0 2 The representation for the magnetic field can b e obtained by using the equation (15). Analogously, the comp onents EH and HH of GH are approximated by the func^ ^ tions EH and HH , for which the inhomogeneous Maxwell equations are written: ^ ^ â EH - ik0 H ^ ^ â HH + ik0 E ·E ·H where
H H H H

=

4 ^ k, c0 = 0, = 0, = 0,

(24) (25) (26) (27)

1/2 ^ k = 3/2 (l - ) ( - ) ( - )

(28)

is the unphysical "magnetic current". Taking the limit 0 we obtain the comp onents EH and HH . The H-comp onent of the integral of the form


I(x, y , z ) =
-

h( )GH (x, y , z ; )d ,

has the edge asymptotics as 0 lo oking like 4 h(l) -1/2 H = - cos + O (). c0 2 The E-comp onent is describ ed by equation (25). 7

(29)


5

Application of the operator Px to the field

Apply the op erator Px defined according to (10) to the total field. The result will b e treated as a new electromagnetic field (obviously, Px [E] is its electrical comp onent, and Px [H] is the magnetic one). This interpretation is p ossible b ecause Helmholtz equations are invariant with resp ect to translations, i.e. they admit differentiations with resp ect to the co ordinates. Obviously, the new field ob eys the same b oundary conditions as the old one. Also it ob eys the radiation condition, i.e. it contains no comp onents coming from infinity. It could b e necessarily equal to zero due to the theorem of uniqueness, however the differentiation with resp ect to x makes the edge singularity stronger, i.e. the edge (Meixner) conditions b ecome broken. Physically it means that the new field is generated by the sources lo cated near the edge. The main term of the asymptotics as 0 can b e written as follows: Px [EE ] = - Px [HH ] = CE (l) cos (l)
-1/2

sin

+ O (), 2

Px [HE ] = Px [EH ] = -

â Px [EE ] , ik0

(30)

CH (l) cos (l)

-1/2

cos

+ O (), 2

â Px [HH ] . (31) ik0

The main terms of these asymptotics violate the edge conditions. The co eficients at these terms are prop ortional to the strengths of the linear source of electric and magnetic typ e lo cated near the edge.

6

Weak form of the embedding formula

Let the theorem of uniqueness is valid for the chosen scatterer, i.e. the following statement is true: if the field (E, H) obeys homogeneous Maxwel l equations, radiation conditions, boundary conditions on the scatterer and Meixner edge conditions, then it is identical ly equal to zero. We are not going to prove this theorem, however, we are sure that it is true at least in the simplest cases (S is compact, its b oundary is smo oth enough). In worse case if b elongs to the discrete sp ectrum of the problem and has finite degeneration, the metho d can b e mo dified. Now we are ready to formulate the following Lemma: Lemma 1 Let F= E H

be a solution of the homogeneous Maxwel l equations obeying the radiation conditions and the ideal conducting boundary conditions on the surfaces of the scatterer introduced above, but violating the Meixner edge conditions. Let the edge behaviour of the solution is given by the asymptotics as 0: E H 1 =
-1/2

DE (l) sin(/2) DH (l) cos(/2) 8

+ Meixner terms.

(32)


Then F(x, y , z ) =

c0 4



(DE (l)GE (x, y , z ; l) - DH (l)GH (x, y , z ; l)) dl.

(33)

To prove the Lemma one can subtract the right-hand side of (33) from F. The result ob eys all the conditions of the theorem of uniqueness, therefore it is zero. Note that the asymptotics (30), (31) has the form ob eying the conditions of the Lemma. Therefore if U is the solution of the initial plane-wave incidence problem, then c0 Pz [U(x, y , z )] = - cos (l) {CE (l)GE (x, y , z ; l) + CH (l)GH (x, y , z ; l)} dl. 4 (34) where CE ,H are the co efficients intro duced in (12), (13). We can rewrite this expression in a slightly different way by intro ducing the directivities of the edge Green's functions. Namely define fE ,H by the following asymptotic relation as r : E
E ,H

(r, ; l) =

2 e k0 r

ik0 r

f

E ,H

( ; l) + O (e

ik0 r

(k0 r )-2 ),

(35)

where E1,2 is the electrical vector of G E ,H directivity as follows:
P

E ,H

. Note that the op erator Px acts on the (36)

x f ( , 0 ) - ik0 (sin cos + sin 0 cos 0 )f ( , 0 ).

Substituting the fields far from the source in (34) by their directivities we obtain ik0 (sin cos + sin 0 cos 0 )f ( , 0 )E0 = c0 cos (l) {CE (l)fE ( ; l) + CH (l)fH ( ; l)} dl, (37) 4 We call the last expression the weak form of the emb edding formula b ecause it contains unknown co efficients CE and CH . In the next section we shall express these co efficients in terms of the edge Green's functions. Here we just note that these co efficients dep end on the direction of incidence 0 and the incident field p olarization. -

7

Application of the reciprocity principle and obtaining the "strong" embedding formula

We formulate the recipro city principle as follows. Let (E1 , H1 ) and (E2 , H2 ) b e two fields ob eying the inhomogeneous Maxwell equations âE
1,2

- ik0 H + ik0 E ·E ·H

1,2

=

âH

1,2

1,2 1,2 1,2

4 k1,2 , c0 4 = j1,2 , c0 = 0, = 0.

(38) (39) (40) (41)

9


We remind that k denotes the unphysical magnetic currents. Both fields satisfy b oundary, edge and radiation conditions common for b oth fields. Then [(j1 · E2 - j2 · E1 ) + (k1 · H2 - k2 · H1 )] dx dy dz = 0 (42)

where the integral is taken over the whole space (or its part containing the sources). The relation (42) is the recipro city principle. The pro cedure of its derivation is rather standard. Equation (38) is multiplied by H2,1 , equation (39) is multiplied by E2,1 , then the equation are summed and the divergent part is taken out. Let us apply the relation (42) to transform the emb edding formula (37). Consider, say, the directivity fE (0 ; l) of the edge Green's function GE . By definition, we can find this directivity as follows. Take the source j1 =
1/2 3/2

(l - ) ( - ) ( - )

near the edge of the scatterer and calculate the field E1 at the p oint (R, 0 ) pro duced by this source (R is large). The directivity will b e equal to the limit fE (0 ; l) =
0,R

lim

k0 R e 2

-ik0 R

E1 (R, 0 ).

(43)

Now consider the field E2 pro duced by scattering of the incident plane wave (7). Note that this plane wave can b e approximately replaced by a spherical wave pro duced by a p oint source lo cated far enough. Namely, take the lo calized source with the amplitude c0 R -ik0 R j2 = e E0 (44) ik0 lo cated at the p oint (R, 0 ) with R large enough. The -comp onent of the field E2 pro duced by this source at the p oint (l = , = , = ) is equal to (E2 ) = 2CE ( )
1/2

+ o(

1/2

)

Applying formula (42) and taking the limits CE (l) =

c0 f (0 ; l) · E0 . 2E ik0 c0 f 2 ik0

0, R , we obtain the formula

Rep eating the same pro cedure, we obtain a more general formula C
E ,H

=

E ,H

· E0 .

(45)

So, the co efficients CE ,H are now expressed in terms of the directivities of the edge Green's functions and the p olarization of the incident field. Substituting (45) into (37) we obtain the strong form of the emb edding formula: f ( , 0 ) = c2 0 â 3 4k0 (sin cos + sin 0 cos 0 ) 10




{fE ( ; l) fE (0 ; l) + fH ( ; l) fH (0 ; l)} dl.

(46)

Here we use the symb ol in the sense on Kronecker pro duct, i.e. in the co ordinate reprsentation f ij ( , 0 ) = c2 0 â 3 4k0 (sin cos + sin 0 cos 0 )
j j i i fE ( ; l)fE (0 ; l) + fH ( ; l)fH (0 ; l) dl,

(47)



i where fE ,H are the comp onents fE ,H in the co oresp onding co ordinates in V( ) and V(0 ). The usage of the edge Green's function can b e preferable from different p oint of views. First, the functions through which we expressed the diffraction co efficient, namely fE ,H ( ; l) dep end on three scalar variables , and l while the initial diffraction co efficient dep end on four scalar variables , , 0 and 0 . It means that numerical tabulation of the new functions gives some gain in memory and time. Second, fE ,H ( ; l) are physically measured values. Such a measurement gives the information what area is resp onsible for the maximums of the diffraction co efficient. The work is supp orted by RFBR grant 03-02-16889 and NSh grant 1575.003.02. The work was partly held within the visits to UK under the LMS and Royal So ciety funding. Author is grateful to Prof. V. P. Smyshlyaev, Prof. V. M. Babich and R. V. Craster for helpful discussions.

References
[1] M. H. Williams, Diffraction by a finite strip, Quart. Jl. Mech. Appl. Math., 35 (1982) pp.103­124. [2] P. A. Martin and G. R. Wickham, Diffraction of elastic waves by a pennyshaped crack: analytical and numerical results, Pro c. R. So c. Lond. A, 390 (1983), pp. 91­129. [3] N. R. T. Biggs, D. Porter, and D.S.G. Stirling, Wave diffraction through a perforated breakwater, Quart. Jl. Mech. Appl. Math., 53 (2000), pp. 375­391. [4] Biggs, N.R.T., Porter, D., Wave diffraction through a perforated barrier of non-zero thickness, Quart. Jl. Mech. Appl. Math. 54 (2001) 523­547. [5] Biggs, N.R.T., Porter, D., Wave scattering by a perforated duct, Quart. Jl. Mech. Appl. Math. 55 (2002) 249­272. [6] A. V. Shanin and R. V. Craster, Removing false singular points as a method of solving ordinary differential equations, Euro. Jl. Appl. Math., 13 (2002), pp. 617­639. 11


[7] R. V. Craster, A. V. Shanin, and E.M.Doubravsky, Embedding formulae in diffraction theory, Pro c. Roy. So c. Lond. A, 459, 2475-2496, (2003). [8] R. V. Craster and V. H. Hoang, Application of Fuchsian differential equations to free boundary problems, Pro c. Roy. So c. Lond. A. 453 (1998), pp. 1241­1252. [9] A. V. Shanin, Diffraction of a plane wave by two ideal strips, Quart. Jl. Mech. Appl. Math., 56 (2003), pp. 187­215 [10] A. V. Shanin, A generalization of the separation of variables method for some 2D diffraction problems, Wave Motion, 37 (2003) pp. 241­256. [11] A. V. Shanin, Three theorems concerning diffraction by a strip or a slit, Quart. Jl. Mech. Appl. Math., 54 (2001), pp.107­137.

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