Half-plane diffraction in a chiral medium

Wave Motion 32 (2000) 157–200

Half-plane diffraction in a chiral medium S. Prze´zdziecki∗ ´ etokrzyska 21, Warsaw, Poland Institute of Fundamental Technological Research, Polish Academy of Sciences, Swi˛ Received 4 November 1998; received in revised form 11 May 1999; accepted 3 February 2000

Abstract An exact, closed-form solution is presented for the problem of diffraction of an electromagnetic plane wave by a perfectly conducting half-plane embedded in a chiral (optically active) medium. The direction of incidence is arbitrary (3D case). The fundamental feature of the problem is its two-mode character. A chiral medium is isotropic but handed and supports two modal fields propagating with distinct velocities. The modes couple at the screen and its edge. Arising phenomena are exemplified by the two cones of diffracted rays and the excitation of lateral waves at the half-plane. The problem is a generalization of the previously solved 2D case of normal incidence and this motivates the adopted approach via the modal Hertz potentials. A subtle role in this context is played by the modal ghost potentials, i.e. peculiar potentials corresponding to the zero electromagnetic field. The mathematical core structure underlying the problem is defined by a boundary value problem for two scalar functions satisfying distinct Helmholtz equations and subject to a pair of boundary conditions at the half-plane. Required solutions to this problem are obtained with the aid of the Wiener–Hopf technique. The main result of the paper consists in showing how to generate, via relevant operators, the 3D electromagnetic solution from three appropriate solutions to the core problem. These solutions are, in essence, available from the 2D case. Basic properties of the solution are briefly discussed. Definitions of the diffraction coefficients indicate the canonical role of the problem for a geometrical theory of diffraction in chiral media. © 2000 Elsevier Science B.V. All rights reserved.

1. Introduction The Sommerfeld, electromagnetic, half-plane diffraction problem [1] is extended in this paper by the assumption of a new environment: a chiral medium is taken to surround a perfectly conducting half-plane which scatters an incident electromagnetic plane wave. The direction of incidence is arbitrary. An exact, closed-form solution is presented for this new problem. A chiral medium is also referred to as optically active. In this paper a chiral medium is defined by assuming the following set of modified Maxwell’s equations for the time-harmonic electric and magnetic fields E , H : H = −iωεE E, (∇ × −ωχII )H

(1.1a)

E = iωµH H, (∇ × −ωχII )E

(1.1b)

∗

Fax: +48-22-26-98-15. E-mail address: [email protected] (S. Prze´zdziecki) 0165-2125/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 2 1 2 5 ( 0 0 ) 0 0 0 3 7 - 8

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where the parameters ε=ε(ω), µ=µ(ω), χ=χ(ω) characterize the medium. I denotes the unit tensor. The parameter χ , modifying the standard Maxwell system, is responsible for new, remarkable properties of the medium which motivate its Greek name. The harmonic time dependence is given via exp (−iωt). Alternatively, the system (1.1a) and (1.1b) can be rewritten in terms of the four vectors E , D ; H , B related by E + iχH H, D = εE

(1.2a)

H − iχE E. B = µH

(1.2b)

Relations (1.2a) and (1.2b) have the form of constitutive equations [2,3] but in such a context a physical reinterpretation of the fields D and B is required. Our only physical assumption about the fields in (1.1a) and (1.1b) is that the tangential component of E vanishes at a perfectly conducting surface. The basic property of an isotropic medium described by (1.1a) and (1.1b) is its handedness (chirality). This is most simply exhibited by plane wave solutions to (1.1a) and (1.1b). For any direction there exist two canonical plane waves of right- and left-hand circular polarizations and the medium distinguishes between these waves by propagating them with distinct speeds. In general terms: any electromagnetic field satisfying (1.1a) and (1.1b) splits into two handed modal fields obeying (1.1a) and (1.1b) and propagating with different velocities (Appendix A). In the present problem the modal fields interact and couple at the half-plane and its edge. This leads to the fundamental feature of the problem: its two-mode character. In contrast, the classical Sommerfeld electromagnetic problem separates into two independent one-mode (scalar) problems. The problem is a generalization to arbitrary incidence of the same problem considered previously in its 2D version of incidence normal to the edge of the half-plane [4]. This small step is not straightforward and motivates the adopted approach via the modal Hertz potentials (Appendix A). In both cases the mathematical core structure underlying the problems is identical and defined by the following boundary value problem for two functions u, v: 1. Equations to be satisfied are two distinct Helmholtz equations (∇ 2 + k12 )u = 0,

(1.3a)

(∇ 2 + k22 )v = 0.

(1.3b)

2. Boundary conditions to be fulfiled at the half-plane are ∂ (u + v) = −cw, ∂z

(1.3c)

k1 u − k2 ν = −dw,

(1.3d)

where z is a Cartesian coordinate perpendicular to the half-plane, w an exponential function determined by the incident wave, and c, d are some constants. Moreover, in order to assure uniqueness, u and v are required to behave properly at the edge of the half-plane and at infinity. Due to the simplicity of the boundary conditions (1.3c) and (1.3d) (of Neumann and Dirichlet type for some linear combinations of u, v) the core problem is an easy object for the Wiener–Hopf method [5]. After the decomposition of the problem into its even and odd parts with respect to z=0, a straightforward application of the technique yields a solution. It was presented in [4]. Since, however, the present paper is intended as independent of [4] we repeat the Wiener–Hopf constructions of the required solutions to (1.3a)–(1.3d). The arising Wiener–Hopf kernel function was encountered in earlier diffraction problems and can be factored effectively [6–8]. This leads to solutions given in quadratures. Yet, we should note that the Wiener–Hopf aspect is of purely technical character in this paper whose basic concerns, as explained below, are concentrated elsewhere. The Wiener–Hopf method is applied within its algorithmic framework and the reader is assumed to be familiar with the technique. On the other hand, no preliminary knowledge is required on the electrodynamics of chiral media.

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An electromagnetic problem closely related to [4] and thus to (1.3a)–(1.3d) (two distinct Helmholtz equations, more complicated boundary conditions) was solved in [9]. In the case of elastic waves a similar problem was treated in [10]. In [11], the authors attempted to present a solution alternative to [4]. An error, made in passing from (10a) and (11) to (12) in their paper, invalidates the presented results. The 2D electromagnetic solution in [4] is given directly by an appropriate solution to (1.3a)–(1.3d). The main result of this paper consists in showing how to generate, via relevant operators, the 3D electromagnetic solution from three appropriate solutions to (1.3a)–(1.3d) which are, in essence, available from the 2D case. The crucial tool for our construction of the solution is provided by the representation theorem of Section 2 which introduces the scalar Hertz potentials to represent compactly the electromagnetic field (cf. (2.3a)–(2.5b)). It is with their help that the vectorial, electromagnetic problem is translated into a scalar boundary value problem whose structure is determined by (1.3a)–(1.3d) (Sections 4–6). Then three appropriate solutions to (1.3a)–(1.3d) are specified that define three auxiliary electromagnetic fields to be used as constituents in a synthesis of the solution. It is in this context that the notion of an electromagnetic pre-solution emerges in a natural way. It is defined as an electromagnetic field that satisfies all the uniqueness conditions of the problem with the exception of the conditions on the behaviour at the edge of the screen where the pre-solution is allowed to have a singularity stronger than that admissible by the edge uniqueness conditions. Subsequently, two corrective singular fields are introduced into play. These fields satisfy at the half-plane homogeneous boundary conditions and represent outgoing waves at infinity. Thus they exist exclusively due to their strong, radiating singularities at the edge without any incident or reflected electromagnetic waves. The required electromagnetic solution is synthesized by a superposition of the pre-solution and the two corrective, singular fields. The inadmissible edge singularities are required to cancel and this determines uniquely the solution. A characteristic aspect of the paper is brought about by the way of construction of the two singular, corrective fields. These constituents of the solution are found with the help of the so-called modal ghost potentials, i.e. peculiar potentials corresponding to the zero electromagnetic field. For a chiral medium it is in this paper that they appear for the first time. Basic ideas of the approach adopted in this paper come from [12,13]. A detailed investigation and interpretation of the solution is beyond the limits of this long paper. In the light of the work done in [6–8,10,14–16], this is rather a matter of relevant adaptations. In Section 17, we only outline some of the basic features of the solution. In order to emphasize the two-mode nature of the problem we call attention to the phenomena of modal interaction exemplified by: 1. two cones of diffracted rays: diffraction coupling, 2. excitation of lateral waves: boundary interaction, 3. geometrical optics solution: coupling on reflection. Finally, the definitions of the diffraction coefficients given in Section 17 indicate that the problem may serve as canonical for an extension of Keller’s geometrical theory of diffraction to chiral media.

2. Modal fields and modal Hertz potentials for chiral media The fundamental, propagation property of electromagnetic fields in chiral media can be expressed in the form of the following theorem: Splitting theorem. Any electromagnetic field E , H given in a region of a chiral medium and satisfying therein the system (1.1a) and (1.1b) can be split in this region into two fields E (2) , H (2) ), E , H ) = (E E (1) , H (1) ) + (E (E so that each constituent field obeys (1.1a) and (1.1b) and fulfils the following defining equations:

(2.1)

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  E (1) 0 (∇ × −k1I ) = , 0 H (1)  (2)    E 0 = , (∇ × +k2I ) 0 H (2)

√ √ where k1 = ω εµ + χ , k2 = ω εµ − χ , and 0 denotes the zero vector.

(2.2a)

(2.2b)

By virtue of (1.1a) and (1.1b) the defining Eqs. (2.2a) and (2.2b) can be given the following equivalent remarkable forms: H (1) , E (1) = iZH H (2) , E (2) = −iZH

r Z=

(2.2c) µ . ε

(2.2d)

The constituent fields E (1) , H (1) ; E (2) , H (2) will be referred to as the modal fields or the modes and the relation (2.1) will be called the modal splitting. Eqs. (2.2a) and (2.2b) will be referred to as the modal equations. We show further on that the constituent fields propagate with distinct velocities and are handed, i.e. a characteristic (left or right) handedness can be ascribed to each of them. A simple proof of the theorem is given in Appendix A. The decomposition (2.1), (2.2a)–(2.2d) has been indicated in [17]. An analysis of the structure of the modal fields (see Appendix A) leads to the following theorem underlying our approach to the solution of the problem: Representation theorem (local). For an arbitrary electromagnetic field E , H given in a region V of a chiral E (2) , H (2) ) E (1) , H (1) ), (E medium and satisfying therein the system (1.1a) and (1.1b), each of its modal constituents (E can be represented in the vicinity of any point in v in terms of one scalar function in the following way: H (1) = ∇ × ∇ × uττ + k1 ∇ × uττ = K 1 u, E (1) = iZH

(2.3a)

H (2) = ∇ × ∇ × vττ + k2 ∇ × vττ = K 2 v E (2) = −iZH

(2.3b)

with u and v obeying in this vicinity the Helmholtz equations (∇ 2 + k12 )u = 0,

(2.4a)

(∇ 2 + k22 )v = 0,

(2.4b)

where τ is a unit vector whose direction may be chosen arbitrarily and K 1 = (∇ × ∇ ×

+k1 ∇×)τ,

K 2 = (∇ × ∇ ×

−k2 ∇×)τ.

The representation of the total field is given by E = ∇ × ∇ × (u + v)ττ + ∇ × (k1 u − k2 v)ττ = K 1 u + K 2 v,

(2.5a)

H = ∇ × ∇ × (u − v)ττ + ∇ × (k1 u + k2 v)ττ = K 1 u − K 2 v. iZH

(2.5b)

The auxiliary scalar functions u, v will be called the modal Hertz potentials. Straightforward calculations confirm that if the functions u, v satisfy their Helmholtz equations (2.4a) and (2.4b) in an arbitrary region of a chiral medium, then the fields generated via (2.3a), (2.3b), (2.5a) and (2.5b) obey in this region the system (1.1a) and (1.1b). This simple fact is sufficient for our ends in this paper. Within this narrower

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context of the theorem the formulas (2.5a) and (2.5b) constitute the basis for our considerations. The more subtle questions of the local completeness of (2.3a), (2.3b), (2.5a) and (2.5b) and their global validity in the entire region v can be treated similarly as in [18,19] and will be discussed in [20]. In Appendix A, we show how to arrive at the operators K 1 , K 2 . Let us now take a closer look at the basic properties of the modal fields. They follow easily from the splitting theorem and may be formulated as corollaries from it: 1. Taking ∇× of (2.2a) and (2.2b) yields the vectorial Helmholtz equations  (1)    E 0 = , (2.6a) (∇ 2 + k12 ) 0 H (1)  (2)    E 0 2 2 = , (2.6b) (∇ + k2 ) (2) 0 H which indicate that the modal fields propagate with distinct velocities determined by the wave numbers k1 , k2 . 2. As shown in Appendix A the specific form of the modal equations (2.2a) and (2.2b) allows to ascribe to each E (1) , H (1) )-mode as of the modal fields its characteristic handedness and accordingly we shall refer to the (E (2) (2) E , H )-mode as right-handed. left-handed and to the (E 3. From (2.1), (2.2c) and (2.2d) the modal fields are determined explicitly as H (1) = 21 (E E + iZH H ), E (1) = iZH

(2.7a)

H (2) = 21 (E E − iZH H ). E (2) = −iZH

(2.7b)

In what follows, the terms “a modal field” or “a mode” are slightly extended to denote any solution to (1.1a) and (1.1b) that fulfils (2.2a) or (2.2c), respectively, (2.2b) or (2.2d). Also a shorthand description of modal fields as the k1 - and the k2 -mode will be used. The physical property of a medium that distinguishes between handed modal fields by propagating them with distinct velocities is called chirality. The representation theorem may also be treated as a consequence of the modal splitting. The theorem results from the simple structure of the modal fields induced by the modal equations (2.2a) and (2.2b), (see Appendix A). Yet its proof goes far beyond a corollary from the splitting theorem. Two particular aspects of the representation theorem deserve explicit attention. The first involves a degree of freedom in (2.3a), (2.3b), (2.5a) and (2.5b), where the vector τ can be chosen arbitrarily (in the local case). The second, more subtle and surprising, arises due to the nonuniqueness of the representations (2.3a), (2.3b), (2.5a) and (2.5b), and leads to the modal potentials corresponding to the zero electromagnetic field. Obviously, such potentials u, ˜ v˜ must simultaneously satisfy two equations: 0 = K 1 u˜ and (2.4a) for the k1 -mode; 0 = K 2 v˜ and (2.4b) for the k2 -mode. As shown in Appendix A there exist quite wide and interesting classes of such solutions. Consider a Cartesian system ξ , η, τ with the τ -axis directed along τ , then for the k1 -mode we get u˜ 1 = f1 (ξ + iη)e−ik1 τ , u˜ 2 = f2 (ξ − iη)e

ik1 τ

(2.8a)

,

(2.8b)

v˜1 = f3 (ξ + iη)eik2 τ ,

(2.9a)

v˜2 = f4 (ξ − iη)e−ik2 τ ,

(2.9b)

and for the k2 -mode

where fn (·), n=1, 2, 3, 4, are analytic functions of a complex argument (·). The solutions u˜ n , v˜n will be called the modal ghost potentials.

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At first sight the ghost potentials may just look like a mathematical curious detail. It turns out, however, that they play a key role in our construction of the solution. This role consists in that they serve to good purpose in constructing electromagnetic fields that arise exclusively due to a strong, radiating singularity at the edge of the half-plane with no incident and reflected electromagnetic waves at infinity. As noted in Section 1, it is first in this paper that the ghost potentials manifest themselves for a chiral medium. Their existence was first noticed in [12] and later they have appeared several times [8,13,14].

3. Formulation of the problem We choose a right-handed system of Cartesian coordinates x, y, z, so that the diffracting half-plane is determined by z=0, x≥0 (see Fig. 1). By x , y , z we denote the unit vectors directed along the√respective axes, then r =xxx +yyy +zzz . We also introduce a system of cylindrical coordinates ρ, ϕ, y defined by ρ = x 2 + z2 , x = ρ cos ϕ, z = ρ sin ϕ. About the chiral medium defined by (1.1a) and (1.1b) we assume that , µ are real positive, χ is real and √ |χ | < εµ. Thus the medium is lossless and supports two plane waves propagating in the same direction with opposite circular polarizations (Appendix A). To aid in some considerations we may temporarily admit small losses by supplementing ε, µ with small imaginary parts iε0 , iµ0 (ε0 > 0, µ0 > 0), but immediately afterwards we return to the lossless case. The incident plane wave can be of either mode; for k1 -mode it is given by (Appendix A) (1) H (1) a + iss 1 × a )eik1s 1 ·rr , E i = iZH i = a(a

(3.1a)

and for k2 -mode by (2) H (2) b − iss 2 × b )eik2s 2 ·rr , E i = −iZH i = b(b

(3.1b)

where s 1 , s 2 are real unit vectors defining the directions of incidence; a , b are real unit vectors such that a ·ss 1 =0, b ·ss 2 =0; a, b are arbitrary (complex) constants. In what follows we shall consider simultaneously both incident waves and for this technical expediency we assume (3.1a) and (3.1b) to depend on x and y via the same exponential factors. Thus (1) E i = a(aa + iss 1 × a )ei(α0 x+βy+y10 z) = a(aa + iss 1 × a )eiβy eiκ1 ρcos(ϕ−ϕ10 ) ,

(3.1c)

(2) E i = b(bb − iss 2 × b )ei(α0 x+βy+y20 z) = b(bb − iss 2 × b )eiβy eiκ2 ρcos(ϕ−ϕ20 ) ,

(3.1d)

Fig. 1. System of coordinates, the diffracting half-plane and the incident wave.

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where kns n =α 0x +βyy +γ n0z . With no restriction on the results, β is taken to be positive; q moreover, we assume p 2 2 β0. We denote (1) (2) Ei = Ei + Ei ,

(3.2a)

(1) (2) Hi = Hi +Hi .

(3.2b)

The total field E (t) , H (t) , defining the solution, will be represented as E (t) = E s + E i , H

(t)

(3.3a)

= H s + H i,

(3.3b)

where the fields E s , H s are called the scattered fields and it is these fields that we shall be looking for to determine the solution. H g the total field determined by the laws of geometrical optics (see (4.12)). We denote by E g ,H In order to specify the field E s , H s we require it to obey: (1) Equations: the system (1.1a) and (1.1b).

(3.4)

(2) Boundary conditions: Ei × z E s × z = −E

for z = 0, x ≥ 0.

(3.5)

(3) Edge conditions: E s = ρ −1/2p e (ϕ, y) + E (r) s (ρ, ϕ, y),

(3.6a)

H s = ρ −1/2p m (ϕ, y) + H (r) s (ρ, ϕ, y),

(3.6b)

(r) p m =0 and p e , p m , E (r) p e =0, y ·p where y ·p s , H s are continuous for 0≤ϕ≤2π, ρ≥0, −∞
(4) Outgoing wave conditions: E s , H s can be extended analytically to account for small losses and then the diffracted field E s + E i) − E g, (E

(3.7a)

H s + H i) − H g (H

(3.7b)

decays exponentially, uniformly with respect to ϕ, for ρ→∞. The adopted form of the edge conditions (3.6a) and (3.6b) is motivated by our further needs. We note explicitly (r) that the nonsingular fields E (r) s , H s are required to be continuous at ρ=0. The edge behaviour anticipated by (3.6a) and (3.6b) is the same as for ordinary isotropic media. This may be confronted with the fact that the formulas interpreted for ordinary isotropic media as energy balance equations hold with no change in form for chiral media. In view of the boundary condition (3.5) which is periodic along the y-axis, the solution E s , H s must be invariant with respect to appropriate translations along y and this together with (1.1a) and (1.1b) implies that E s , H s depend on y via exp (iβy).

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The uniqueness of the solution of the problem can be proven with the aid of the following formula which holds for an electromagnetic field E , H obeying (1.1a) and (1.1b) in a region v of a chiral medium: ZZZ ZZ E ×H H ∗ ] · n dσ = −ω Re[E (ε0E · E ∗ + µ0H · H ∗ ) dV , (3.8) ∂V

V

where ∂V is the boundary of v, n the outward unit normal to ∂V, and dσ an element of ∂V. The star denotes a conjugate quantity. The formula (3.8) is identical with the real part of the energy balance equation for an ordinary, isotropic lossy medium. Consider a region v s whose boundary is determined by: ρ=ρ 1 , ρ=ρ 2 (ρ 1 <ρ 2 ), y=y1 , y=y1 +2π /β, and includes the portion of the screen: ρ 1
4. Formulation in terms of the potentials The problem of the preceding section will now be translated into the language of the modal Hertz potentials. We assume that the field E s , H s in the region outside the half-plane can be represented via (2.5a) and (2.5b) in terms of the corresponding potentials. Thereby the vectorial, electromagnetic problem will be mapped into an equivalent diffraction problem for two scalar functions. The key step in the translation is determined by the following choice in (2.5a) and (2.5b): τ = z.

(4.1)

It will be seen that due to (4.1) we obtain for the potentials a particularly simple form of the boundary conditions at the half-plane. The incident wave (3.1a) and (3.1c) is given via (2.5a) and (2.5b) by u=ui0 , v=0, where ui0 = a0 ei(α0 x+βy+y10 z) = a0 eiβy eiκ1 ρ cos(ϕ−ϕ10 ) = a0 eik1s 1 ·rr , a0 =

q1 a, ik12

q1 =

(4.2a)

i . a · [zz + i(ss 1 × z )]

Similarly for (3.1b) and (3.1d) u=0, v=v i0 , where vi0 = b0 ei(α0 x+βy+γ20 z) = b0 eiβy eiκ2 ρ cos(ϕ−ϕ20 ) = b0 eik2s 2 ·rr , b0 =

q2 b, −ik22

q2 =

(4.2b)

−i . b · [zz − i(ss 2 × z )]

The waves (3.1a)–(3.1d) can now be written as (1) 2 H (1) I + iss 1 ×](ss 1 × z ) eik1s 1 ·rr , E i = iZH i = K 1 ui0 = ik1 a0 [I

(4.3a)

(2) 2 H (2) I − iss 2 ×](ss 2 × z ) eik2s 2 ·rr . E i = −iZH i = K 2 vi0 = −ik2 b0 [I

(4.3b)

Expressions for a0 , q1 ; b0 , q2 follow from the equalities (3.1a)=(4.3a), (3.1b)=(4.3b), multiplied, respectively, by a and b . For s 1 =ss 2 =zz the waves (3.1a)–(3.1d) can be obtained from (4.3a) and (4.3b) as limiting cases: a0 →0, β→0, but q1 (ss 1 ×zz )→aa , q2 (ss 2 ×zz )→bb . However, within the adopted approach, via potentials, this case would require a special treatment. Nevertheless, in Section 15 it is shown that the present solution reduces correctly to that obtained directly for β=0, α 0 =0.

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As indicated in Section 3, both waves (4.2a) and (4.2b) will be treated simultaneously and only in the final results we assume a0 =0 or b0 =0. Our assumption about the global representation of E s , H s in terms of the potentials is now supplemented with the postulate that the potentials depend on y via the same exponential factor as the fields, i.e. via exp(iβy). This postulate reduces the class of the ghost potentials to the following four solutions: For the k1 -mode ◦

u1 = a1 eβ(x+iy) e−ik1 z ,

(4.4a)

◦

u2 = a2 e−β(x−iy) eik1 z ,

(4.4b)

and for the k2 -mode ◦

v 1 = b1 eβ(x+iy) eik2 z ,

(4.4c)

◦

v 2 = b2 e−β(x−iy) e−ik2 z .

(4.4d)

The potentials corresponding via (2.5a) and (2.5b) to E s , H s can now be represented in general as ◦

◦

◦

◦

us + u1 + u2 ,

(4.5a)

vs + v 1 + v 2 ,

(4.5b)

where us , v s denote any pair of the potentials defining E s , H s . In mapping the vectorial, electromagnetic problem into that for the scalar potentials we need for us ,v s conditions that are sufficient to guarantee the uniqueness of the electromagnetic solution and do not make the problem overdetermined so as to destroy the existence of the solution. Such conditions are easy to arrive at by inspection from formulas (2.5a) and (2.5b). Nevertheless, for the sake of generality, i.e. to fully clarify the relation between the fields and the potentials, we investigate what conditions on the potentials follow necessarily from the uniqueness conditions of Section 3 imposed on the field. Thus we arrive at the following diffraction problem for us , v s . (1) Equations: (2.4a) and (2.4b), respectively.

(4.6)

(2) Boundary condition:   ∂ (us + vs ) − z × ∇t (k1 us − k2 vs ) = −∇t c0 ei(α0 x+βy)+z × ∇t d0 ei(α0 x+βy) ∇t ∂z

(4.7)

on both sides of the half-plane z = 0+ and z = 0− , x ≥ 0; ∇t = ∇ − z ∂/∂z, c0 = iγ10 a0 + iγ20 b0 , d0 = k1 a0 − k2 b0 (3) Edge conditions: us = eiβy [ρ 3/2 p1 (ϕ) + u(r) s ],

(4.8a)

vs = eiβy [ρ 3/2 p2 (ϕ) + vs(r) ],

(4.8b) (r)

(r)

where ∂ 2 p1 (ϕ)/∂ϕ 2 , ∂ 2 p2 (ϕ)/∂ϕ 2 and K 1 us , K 2 vs are continuous for 0 ≤ ϕ ≤ 2π, ρ ≥ 0. Relations between (4.8a), (4.8b) and (3.6a), (3.6b) are discussed further on. (4) Outgoing wave conditions: us , v s depend analytically on k1 , k2 and for small losses K 1 (us + ui0 ) − K 1 ug0 ,

(4.9a)

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K 2 (vs + vi0 ) − K 2 vg0

(4.9b)

decay exponentially, uniformly with respect to ϕ, for ρ→∞. Here, ug0 , v g0 is the geometrical optics solution for the incident waves (4.2a) and (4.2b) with the boundary conditions to be satisfied at the screen given by (4.7). By a direct calculation using the laws of geometrical optics, we find  for 0 < ϕ < ϕ10 , 0 for ϕ10 < ϕ < 2π − ϕ10 , (4.10a) ug0 = ui0  ui0 + ur0 for 2π − ϕ10 < ϕ < 2π,  for 0 < ϕ < ϕ20 , 0 for ϕ20 < ϕ < 2π − ϕ20 , (4.10b) vg0 = vi0  vi0 + vr0 for 2π − ϕ20 < ϕ < 2π, where vr0 = br0 eiβy ei(α0 x−γ20 z) , ur0 = ar0 eiβy ei(α0 x−γ10 z) ,     a0 ar0 =R , br0 b0     1 2k2 γ20 k2 γ10 − k1 γ20 R11 R12 = . R= R21 R22 2k1 γ10 k1 γ20 − k2 γ10 k1 γ20 + k2 γ10

(4.11a) (4.11b)

The geometrical optics field E g , H g is now given by E g = K 1 ug0 + K 2 vg0 ,

(4.12a)

H g = K 1 ug0 − K 2 vg0 . iZH

(4.12b)

The conditions (4.9a) and (4.9b) arise as linear combinations of (3.7a) and (3.7b) expressed in terms of us , v s and ug0 , v g0 . The boundary condition (4.7) is a simple literal translation of (3.5), but for the sake of generality is required to hold separately at each side of the half-plane. In the next section the second-order vectorial condition (4.7) will be transformed into two independent (uncoupled) boundary conditions of the form (1.3c) and (1.3d). The translation of (3.6a) and (3.6b) into (4.8a) and (4.8b) proceeds as follows: from (2.5a) and (2.5b), we have 1 2 (Esz

+ iZHsz ) =

∂ 2 us + k12 us , ∂z2

(4.13a)

1 2 (Esz

− iZHsz ) =

∂ 2 vs + k22 vs . ∂z2

(4.13b)

Taking into account (2.4a) and (2.4b) we obtain 1 2 (Esz

+ iZHsz ) = −

∂ 2 us + β 2 us , ∂x 2

(4.14a)

1 2 (Esz

− iZHsz ) = −

∂ 2 vs + β 2 vs . ∂x 2

(4.14b)

Expansions of the explicit solutions to (4.13a), (4.13b) and (4.14a), (4.14b) [12,18,19] indicate that the lowest order terms in ρ defining the behaviour of us , v s at the edge cannot be lower than ρ 3/2 . From the same solutions we find that us , v s are continuous for 0≤ϕ≤2π , ρ≥0. Thus us , v s may be assumed in the forms (4.8a) and (4.8b).

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167

Substituting (3.6a), (3.6b), (4.8a), (4.8b) into (4.13a), (4.13b), (4.14a), (4.14b), respectively, we determine the regularity of p1 (ϕ), p2 (ϕ) and establish the class of regularity for us (r) , vs (r) . Consequently, conditions (4.8a) and (4.8b) are both sufficient and necessary for E s , H s to fulfil (3.6a) and (3.6b). Conditions (4.9a) and (4.9b) are both sufficient and necessary for (3.7) to be fulfiled but, strictly speaking, they cannot be recognized as outgoing wave conditions for the potentials since they fail to specify explicitly the behaviour of the potentials at infinity. The explicit behaviour will be determined further on. Finally, let us observe that the conditions imposed above on the potentials us , v s determine the fields E s , H s uniquely. Consequently, by virtue of the representation theorem, the potentials are determined uniquely modulo the ghost potentials (4.4a)–(4.4d).

5. Reformulation of the problem for the potentials First we shall find a pair of scalar boundary conditions equivalent to the vectorial condition (4.7) (decoupling). Then we reformulate the problem of Section 4 so as to introduce into play the ghost potentials (4.4a)–(4.4d). Let us take ∇ t · of (4.7) and then of (4.7) multiplied vectorially by z . We get   ∂ (us + vs ) = −∇t 2 c0 ei(α0 x+βy) , z = 0+ , z = 0− , x ≥ 0, (5.1) ∇t 2 ∂z ∇t 2 (k1 us − k2 vs ) = −∇t 2 d0 ei(α0 x+βy) ,

z = 0+ , z = 0− , x ≥ 0.

(5.2)

where ∇t = ∇t · ∇t . At the half-plane ∇t 2 = d2 /dx 2 − β 2 and integration of Eqs. (5.1) and (5.2) gives 2

∂ (us + vs ) = −c0 ei(α0 x+βy) − c1 + eβ(x+iy) − c2 + e−β(x−iy) , ∂z

z = 0+ , x ≥ 0,

(5.3a)

∂ (us + vs ) = −c0 ei(α0 x+βy) − c1 − eβ(x+iy) − c2 − e−β(x−iy) , ∂z

z = 0− , x ≥ 0.

(5.3b)

k1 us − k2 vs = −d0 ei(α0 x+βy) − d1 + eβ(x+iy) − d2 + e−β(x−iy) ,

z = 0+ , x ≥ 0,

(5.4a)

k1 us − k2 vs = −d0 ei(α0 x+βy) − d1 − eβ(x+iy) − d2 − e−β(x−iy) ,

z = 0− , x ≥ 0,

(5.4b)

c1 ± , c2 ± ; d1 ± , d2 ±

are some unknown constants. where Relations (5.3a)–(5.4b) are equivalent to (5.1) and (5.2), however, to satisfy the basic condition (4.7) we must have c1 ± = −id1 ± ,

(5.5a)

c2 ± = id2 ± .

(5.5b)

By virtue of the edge conditions (4.8a) and (4.8b) us , v s ; ∂us /∂ z , ∂v s /∂ z are continuous at ρ=0, consequently d1 + + d2 + = d1 − + d2 − ,

(5.6a)

c1 + + c2 + = c1 − + c2 − .

(5.6b)

However, in view of (5.5a) and (5.5b), relation (5.6a) gives c1 + − c2 + = c1 − − c2 − ,

(5.7)

hence c1 + = c1 − = c1 ,

c2 + = c2 − = c2 ,

d1 + = d1 − = d1 = ic1 ,

d2 + = d2 − = d2 = −ic2 .

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Thus for both sides of the half-plane we obtain the same conditions which for (5.3a) and (5.3b) may be summarized as one condition ∂ (us + vs ) = −c0 ei(α0 x+βy) − c1 eβ(x+iy) − c2 e−β(x−iy) , ∂z

z = 0, x ≥ 0,

(5.8a)

and similarly for (5.4a) and (5.4b), k1 us − k2 vs = −d0 ei(α0 x+βy) − ic1 eβ(x+iy) + ic2 e−β(x−iy) ,

z = 0, x > 0.

(5.8b)

We conclude that conditions (5.8a) and (5.8b) are equivalent to the basic boundary condition (4.7) and note that they involve two unknown constants c1 , c2 to be determined in the course of the solution. We are now in a position to pinpoint the crucial feature of the diffraction problem for the potentials us , v s . It consists in the connection between the type of restrictions imposed by the edge conditions (4.8a) and (4.8b) and the undetermined form of the boundary conditions (5.8a) and (5.8b). As will be seen later on, it is only due to the two unknown constants c1 , c2 in (5.8a) and (5.8b), that there exists a possibility to fulfil the edge conditions (4.8a) and (4.8b). For c1 =c2 =0 a solution for the potentials could only exist if the conditions (4.8a) and (4.8b) had been relaxed (cf. Section 6). This is why conditions (4.8a) and (4.8b) will be referred to as strong (restrictive) edge conditions. The form of the boundary conditions (5.8a) and (5.8b) confronted with that of the ghost potentials (4.4a)–(4.4d), suggests a natural reformulation of the problem. We introduce the ghost potentials (4.4a)–(4.4d) as incident waves and then arrive at the boundary conditions (5.8a) and (5.8b), in a way standard for diffraction problems. Thus we assume ◦

◦

◦

◦

ui = ui0 + u1 + u2 , vi = vi0 + v 1 + v 2 ,

(5.9a) (5.9b)

and replace (5.8a) and (5.8b) by ∂ ∂ (us + vs ) = − (ui + vi ), ∂z ∂z

(5.10a)

k1 us − k2 vs = −(k1 ui − k2 vi ),

(5.10b)

for z=0, x≥0. The unknown constants c1 , c2 are now given by c1 = −ik1 a1 + ik2 b1 ,

(5.11a)

c2 = ik1 a2 − ik2 b2 .

(5.11b)

Technically, this reformulation only transfers the unknown constants from the boundary conditions (5.8a) and (5.8b) to the unknown amplitudes of the incident waves (5.9a) and (5.9b). Let us note that while c1 , c2 , as will be seen, are uniquely determined, the redundant number of amplitudes results from the symmetry of the problem. The same scattered field may correspond to two different incident waves falling at symmetric angles. Thus only one pair of ghost potentials could be sufficient but such a pair must include waves linearly independent for z=0, i.e. one decaying and one growing for x→∞. For the sake of symmetry we preserve all waves (4.4a)–(4.4d) with possible reduction in the final results. Within this reformulation we still preserve the implicit form of the outgoing wave conditions (4.9a) and 4.9b), and its role will be clarified in Sections 10–12 and 17. Here we may only add that (4.9a) and (4.9b) allow us , v s to comprise, asymptotically for ρ→∞, ghost-potential constituents in relevant space sectors. Summarizing, the diffraction problem for us , v s is now specified by (1) Incident waves given by (5.9a), (5.9b).

(5.12a)

(2) Equations to be satisfied (2.4a) and (2.4b).

(5.12b)

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169

(3) Boundary conditions given by (5.10a) and (5.10b).

(5.12c)

(4) Edge conditions determined by (4.8a) and (4.8b).

(5.12d)

(5) Outgoing wave conditions determined by (4.9a) and (4.9b).

(5.12e)

6. Potentials uss 0 , vss 0 for the pre-solution Ess 0 , H ss 0 From the problem re-defined by the list at the end of the last section we now single out an auxiliary subproblem that constitutes the core structure of our problem. This subproblem for the modal potentials us0 , v s0 is uniquely specified by the following conditions: (1) Incident waves are ui0 ,vi0 given by (4.2a) and (4.2b).

(6.1)

(2) Equations to be satisfied are (2.4a) and (2.4b), respectively.

(6.2)

(3) Boundary conditions at z=0, x≥0 are ∂ ∂ (us0 + vs0 ) = − (ui0 + vi0 ), ∂z ∂z

(6.3a)

k1 us0 − k2 vs0 = −(k1 ui0 − k2 vi0 ).

(6.3b)

(4) Edge conditions: us0 = O(1),

→0

(6.4a)

vs0 = O(1),

→0

(6.4b)

|∇us0 | = O(ρ −1/2 ),

ρ → 0,

(6.4c)

|∇vs0 | = O(ρ −1/2 ),

ρ → 0.

(6.4d)

(5) Outgoing wave conditions: us0 , v s0 depend analytically on k1 , k2 and for small losses (us0 + ui0 ) − ug0 ,

(6.5a)

(vs0 + vi0 ) − vg0

(6.5b)

decay exponentially for ρ→∞, uniformly with respect to ϕ. This subproblem differs from (5.12a)–(5.12e) in that it takes from (5.9a) and (5.9b) only the incident nonghost waves and that it relaxes the strong (restrictive) edge conditions (4.8a) and (4.8b). Uniqueness of the solution of this subproblem is of fundamental importance and it can be proven with the aid of the following formula: ZZZ [k1 ∇u · ∇u∗ + k2 ∇v · ∇v ∗ − k1 ∗ (k1 k1 ∗ )uu∗ − k2 ∗ (k2 k2 ∗ )vv ∗ ] dV V ZZ n dσ, [k1 k2 (u + v)∇(u∗ + v ∗ ) + (k1 u − k2 v)∇(k1 u∗ − k2 v ∗ )]n (6.6) = (k1 + k2 )−1 ∂V

which holds for u,v satisfying Eqs. (2.4a) and (2.4b) in a region v with a sufficiently smooth boundary ∂v; n is the outward unit normal to ∂v. The formula (6.6) follows from the asymmetric integral green formula where we first substitute k1 (u+v), k2 (u∗ +v ∗ ), next (k1 u−k2 v), (k1 u∗ −k2 v ∗ ), and add the results.

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By applying (6.6) to the difference of two solutions of the problem in a region v s as defined at the end of Section 3, and considering the limits ρ 1 →0, ρ 2 →∞, we conclude that for small losses the right-hand side limit is zero by virtue of Eq. (6.3a)–(6.5b). Consequently, the imaginary part of the left-hand side limit must also be zero and this implies that the difference of the solutions is identically zero. Formula (6.6) indicates that the boundary value problem for two Helmholtz equations with boundary conditions (6.3a) and (6.3b) on ∂v is, in its own way, an instructive example of a generalization of the Dirichlet and Neumann problems for a single Helmholtz equation. We may also conclude that when c1 , c2 in (5.8a) and (5.8b) are determined then the solution to (5.12a)–(5.12e) is unique. The electromagnetic field E s0 , H s0 corresponding to us0 , v s0 is defined in the following way: E s0 = E s0 (1) + E s0 (2) ,

(6.7a)

H s0 = H s0 (1) + H s0 (2) ,

(6.7b)

H s0 (1) = K 1 us0 , E s0 (1) = iZH

(6.8a)

H s0 (2) = K 2 vs0 . E s0 (2) = −iZH

(6.8b)

Conditions (6.1)–(6.5b) confronted with (3.4)–(3.7b) via (4.6)–(4.9b) indicate that the fields E s0 , H s0 fulfil all the uniqueness conditions of Section 3 with the exception of (3.6a) and (3.6b) on the behaviour at the edge, where they are too singular. It is in view of these properties that E s0 , H s0 is called a pre-solution. In the subsequent sections we shall construct appropriate corrective fields that will cancel the inadmissible singularities of E s0 , H s0 and will not harm any of the remaining uniqueness conditions of Section 3.

7. Even and odd parts of the solution for uss 0 , vss 0 The boundary conditions (6.3a) and (6.3b) are of particularly simple form being of the Neumann or Dirichlet type for the respective linear combinations of us0 , v s0 . It is well known that for such conditions and the considered geometry of the problem it is possible to obtain simple boundary conditions at the two half-planes: x<0, z=0, and x≥0, z=0, for the even and odd parts of these linear combinations. Let us denote us0 = u0s0 + u00s0 ,

(7.1a)

0 00 + vs0 , vs0 = vs0

(7.1b)

where u0s0 (x, y, z) = u0s0 (x, y, −z),

(7.2a)

u00s0 (x, y, z) = −u00s0 (x, y, −z),

(7.2b)

0 , v 00 . and similarly for vs0 s0 For the even parts of the potentials we obtain in the aperture x<0, z=0,

∂ 0 0 (u + vs0 ) = 0, ∂z s0

(7.3a)

∂ 0 (k1 u0s0 − k2 vs0 ) = 0, ∂z

(7.3b)

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171

while at the diffracting half-plane x<0, z=0, we have ∂ 0 0 (u + vs0 ) = 0, ∂z s0

(7.4a)

0 = −d0 ei(α0 x+βy) . k1 u0s0 − k2 vs0

(7.4b)

For the odd parts of the potentials we get in the aperture x<0, z=0, 00 = 0, u00s0 + vs0

(7.5a)

00 = 0, k1 u00s0 − k2 vs0

(7.5b)

and at the half-plane x≥0, z=0, ∂ 00 00 (u + vs0 ) = −c0 ei(α0 x+βy) , ∂z s0

(7.6a)

00 = 0. k1 u00s0 − k2 vs0

(7.6b)

The boundary conditions for the even parts can now be rewritten in the following way: ∂ 0 0 )=0 (u + vs0 ∂z s0

in the entire plane z = 0,

∂ 0 )=0 (k1 u0s0 − k2 vs0 ∂z

in the aperture x < 0, z = 0,

0 = −d0 ei(α0 x+βy) k1 u0s0 − k2 vs0

at the half-plane x ≥ 0, z = 0.

(7.7a) (7.7b) (7.7c)

For the odd parts we have 00 =0 k1 u00s0 − vs0 00 =0 u00s0 + vs0

in the entire plane z = 0,

(7.8a)

in the aperture x < 0, z = 0,

(7.8b)

∂ 00 00 ) = −c0 ei(α0 x+βy) (u + vs0 ∂z s0

at the half-plane x ≥ 0, z = 0.

(7.8c)

0 , v 00 are required to obey (2.4a) and (2.4b), respectively. Decompositions of the Obviously both u0s0 , u00s0 and vs0 s0 incident waves (4.2a) and (4.2b), edge conditions (6.4a) and (6.4b), and the geometrical optics solution ug0 , v go are straightforward. The uniqueness for the even and odd parts results from the uniqueness of us0 , v s0 .

0 8. Solution for u0s0 , vs0 0 will be obtained with the aid of a variant of the Wiener–Hopf technique [5]. A solution for u0s0 , vs0 We shall consider the half-space z≥0 and assume there the following plane wave spectral representations for 0 : u0s0 , vs0 Z u0s0 = eiβy A00 (α) ei(αx+γ1 z) dα, (8.1a) 0 = eiβy vs0

Z

Q

Q

B00 (α) ei(αx+γ2 z) dα.

(8.1b)

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S. Prze´zdziecki / Wave Motion 32 (2000) 157–200

Fig. 2. The complex α-plane with the contour Q, the branch points at α =±κ 1 ,α =±κ 2 , the branch cuts 01+ , 02+ , 01− , 02− , and the incident-wave pole at α =α 0 . The poles at α =±iβ of the ghost potentials and the modification of the contour Q into Qβ are also indicated (χ >0, k1 >k2 ).

p Integration in (8.1a) and (8.1b) is in the complex α-plane, γ (α) = γ = κ1 2 − α 2 , γ2 (α) = γ2 = 1 1 p 2 2 κ2 − α . The roots for ␥1 , ␥2 are chosen so that for α=0, γ 1 =κ 1 , γ 2 =κ 2 . The contour Q is oriented along and follows the real axis except for appropriate indentations such that −κ 1 ,−κ 2 are below Q while κ 1 , κ 2 , α 0 are above Q as shown in Fig. 2, where also the branch cuts 01 + , 01 − ; 02 + , 02 − , outgoing from ±κ 1 and ±κ 2 , respectively, are indicated. We denote by U the region above Q, and by L that below Q. A00 (α) = A00 , B00 (α) = B00 are the unknown amplitude functions to be found. To this end they are assumed to be defined in the entire α-plane cut by the indicated branch cuts. Some combinations of A00 ,B00 are required to 0 to fulfil the stipulated have certain analytical properties and a specific behaviour for |α|→∞ sufficient for u0s0 , vs0 boundary and edge conditions. This leads to a functional equation which will be solved via the Wiener–Hopf technique. The validity of the representations (8.1a) and (8.1b) depends on the convergence of the integrals. This will be verified after the solution has been obtained. Since for a slightly lossy medium γ 1 , γ 2 have small positive imaginary parts, it will be seen that the outgoing wave conditions are fulfiled. The condition (7.7a) will be satisfied if in the entire α-plane γ1 A00 + γ2 B00 = 0.

(8.2a)

The fulfilment of (7.7b) follows from the following relation: k1 γ1 A00 − k2 γ2 B00 = (k1 + k2 )L00 (α),

(8.2b)

where L00 (α) = L00 is required 1. to be analytic in L and continuous in L +Q; 2. to behave for |α|→∞ in L so that integrating L00 (α) eiαx (x < 0) along a semi-circle closing the contour in L , we get zero for the semi-circle’s radius becoming infinite. The factor (k1 +k2 ) is introduced for the sake of convenience. The relation (8.2b) admits a class of functions for L00 . However from (8.1a), (8.1b), (7.7b) and (8.2b) we obtain the following explicit representation: Z 1 ∞∂ 0 (k1 u0s0 − k2 vs0 ) z=0 e−iαx dx, α ∈ L (8.3a) (k1 + k2 )L00 (α) = e−iβy 2π 0 ∂z From the edge behaviour (6.4a)–(6.4d) we get, according to [5] (p. 36), the following additional restriction on L00 :

S. Prze´zdziecki / Wave Motion 32 (2000) 157–200

L00 (α) = O(α −1/2 )

as |α| → ∞ in L .

173

(8.3b)

For (7.7c) to hold we are led to the following relation: k1 A00 − k2 B00 = U00 (α) − F00 (α).

(8.3c)

where F00 (α) = F00 = d0 /2π i(α − α0 ) and U00 (α) = U00 fulfils in U +Q the same requirements as L00 in L +Q. An additional restriction on U00 follows from the explicit representation resulting from (7.7c), (8.1a), (8.1b) and (8.2c), k1 A00 − k2 B00 = e−iβy

1 2π

Z

d0 0 (k1 u0s0 − k2 vs0 ) z=0 e−iαx dx − 2π −∞ 0

Z

∞

eiα0 x e−iαx dx

(8.4a)

0

where the second integral is convergent in L and is equal there, along with its limiting value on Q, to F00 . Consequently, in view of (8.2c), the first integral defines U00 (α) in U . From the edge behaviour it follows [5] (p. 36), that U00 (α) = O(α −1 )

as |α| → ∞ in U .

(8.4b)

By eliminating from (8.2a)–(8.2c) the amplitudes A00 ,B00 , we obtain the following Wiener–Hopf functional equation: G(α)L00 (α) = U00 (α) − F00 (α),

(8.5)

where G(α) = G =

g(α) , γ1 (α)γ2 (α)

g(α) = g = k1 γ2 (α) + k2 γ1 (α).

Let us observe that for the chosen roots of γ 1 ,γ 2 , we have g(α)6=0 in the whole ␣-plane cut by 01 + , 01 − , 02 + , 02 − . In order to solve (8.5) we factor g(α) in the Wiener–Hopf sense g(α) = gU (α)gL (α),

(8.6)

where gU (α)=gU is analytic in U and gL (α)=gL is analytic in L . According to [5], (p. 15, 16, 42) we have for Im α>0,   Z ∞ 1 logg(t) dt , Im t = 0. (8.7) gU (α) = exp 2πi −∞ t − α The integral in (8.7) is understood in the sense of a principal value at infinity. Since g(t)=g(−t) we have for Im α<0, gL (α) = gU (−α).

(8.8)

Effective factorization of a function identical in form with g(α) has been carried out by Ament [6] and applied in [7,8]. This constructive and crucial step is postponed to a later paper concerned with the analysis and interpretation of the solution. Factorization of γ 1 (α)γ 2 (α) is straightforward p p γ1 (α)γ2 (α) = (κ1 + α)(κ2 + α) (κ1 − α)(κ2 − α), (8.9) and consequently, G(α) = GU (α)GL (α),

(8.10)

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S. Prze´zdziecki / Wave Motion 32 (2000) 157–200

where gU (α) , (κ1 + α)(κ2 + α)

(8.11a)

gL (α) , GL (α) = √ (κ1 − α)(κ2 − α)

(8.11b)

GU (α) = √

with GU (α) analytic in U and GL (α) analytic in Ω L . In what follows we shall need asymptotic estimates of gU (α) and GU (α) for |α|→∞ in U . These can be obtained in the following way (J. Boersma, private commun.). Rewrite the Cauchy integral in (8.7) as Z ∞ Z ∞ log g(t) logg(t) α 1 dt, Im α > 0. dt = 2πi −∞ t − α 2πi −∞ t 2 − α 2 Replace log g(t) bylog g(t) = log[i(k1 + k2 )] + log|t| + R(t) where R(t) = O(t −2 ) as t → ±∞. Then the Cauchy integral can be estimated by Z Z ∞ Z ∞ α α α log |t| R(t) log[i(k1 + k2 )] ∞ dt + dt + dt 2 − α2 2 − α2 2 − α2 2πi 2πi 2π i t t t −∞ −∞ −∞   1 iπ +O as |α| → ∞, Im α > 0. = 21 log[i(k1 + k2 )] + 21 log α − 4 α On using this result we obtain from (8.7) and (8.11a), gU (α) = (k1 + k2 )1/2 α 1/2 [1 + O(α −1 )], GU (α) = (k1 + k2 )1/2 α −1/2 [1 + O(α −1 )]

(8.12a) as |α| → ∞ in U + Q.

(8.12b)

By virtue of (8.10), (8.5) can be given the following form: GL (α)L00 (α) = GU −1 (α)[U00 (α) − F00 (α)],

(8.13)

Let us decompose GU −1 (α)F00 (α) in the following way: GU −1 (α)F00 (α) = [GU −1 (α) − GU −1 (α0 )]F00 (α) + GU −1 (α0 )F00 (α),

(8.14)

and write (8.13) as GL (α)L00 (α) + GU −1 (α0 )F00 (α) = GU −1 (α)U00 (α) − [GU −1 (α) − GU −1 (α0 )]F00 (α).

(8.15)

The left-hand side of (8.15) considered in L constitutes there an analytic function, while the right-hand side considered in U is analytic there. Thus by virtue of (8.15) considered at Q, these sides are analytic continuations of each other and together they form an entire function. From (8.3b), (8.4b) and (8.12b) we find that for |α|→∞ this entire function is O(α −1 ) in L and O(α −1/2 ) in U . Consequently, it tends to zero as |α|→∞ and by Liouville’s theorem vanishes identically. Thus each of the sides of (8.15) is equal to zero in the entire ␣-plane and we obtain L00 (α) = −GL −1 (α)GU −1 (α0 )F00 (α),

(8.16a)

U00 (α) = [1 − GU (α)GU −1 (α0 )]F00 (α).

(8.16b)

Either of the results (8.16a) and (8.16b) can be used to find A00 and B00 . For example from (8.2a), (8.2b) and (8.16a), we get s 0 (α) L (κ2 − α)(κ2 + α0 ) −1 =− gL (α)gU −1 (α0 )γ1 (α0 )F00 (α), (8.17a) A00 (α) = 0 γ1 (α) (κ1 + α)(κ1 − α0 )

S. Prze´zdziecki / Wave Motion 32 (2000) 157–200

B00 (α) =

−L00 (α) = γ2 (α)

s

(κ1 − α)(κ1 + α0 ) −1 gL (α)gU −1 (α0 )γ2 (α0 )F00 (α). (κ2 + α)(κ2 − α0 )

175

(8.17b)

It is seen from (8.12b) and (8.16a) that the integrals in (8.1a) and (8.1b) are convergent. Let us record for further needs that L00 (α) =

d0 (k1 + k2 )−1/2 GU −1 (α0 )(−α)−1/2 [1 + O(α −1 )] 2πi

as |α| → ∞ in L + Q.

(8.18)

00 9. Solution for u00s0 , vs0

As in the preceding section we consider the half-space z≥0 and assume there the following representations for 00 : u00s0 , vs0 Z (9.1a) u00s0 = eiβy A00s0 (α)ei(αx+γ1 z) dα 00 = eiβy vs0

Z

Q

Q

B000 (α)ei(αx+γ2 z) dα,

(9.1b)

where A000 (α) = A000 , B000 (α) = B000 are the amplitude functions to be found. The remaining notation and remarks are the same as in Section 8. From (7.8a) we get k1 A000 − k2 B000 = 0.

(9.2a)

The condition (7.8b) will be fulfiled if A000 + B000 = −i(k1 + k2 )L000 (α),

(9.2b)

where L000 (α) = L000 is required to be analytic in L and behave correctly for |α|→∞. Similarly as in Section 8, from an explicit expression for L00 (α) and the edge behaviour (6.4a)–(6.4d) we find L000 (α) = O(α −1 )

as |α| → ∞ in L .

(9.3)

The condition (7.8c) will be satisfied if iγ1 A000 + iγ2 B000 = U000 (α) − F000 (α),

(9.2c)

where F000 (α) = F000 = c0 /2π i(α − α0 ). U000 (α) = U000 is required to be analytic in U and behave correctly for |α|→∞. Similarly as in Section 8, from an explicit representation and the edge behaviour (6.4a)–(6.4d) we get U000 (α) = O(α −1/2 )

as |α| → ∞ in U .

(9.4)

By eliminating the amplitudes A000 , B000 from (9.2a)–(9.2c) we obtain the Wiener–Hopf equation g(α)L000 (α) = U000 (α) − F000 (α).

(9.5)

In a way identical to that of Section 8, we get L000 (α) = −gL −1 (α)gU −1 (α0 )F000 (α),

(9.6a)

U000 (α) = [1 − gU (α)gU −1 (α0 )]F000 (α).

(9.6b)

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and A000 (α) = −ik2 L000 (α) = ik2 gL −1 (α)gU −1 (α0 )F000 (α),

(9.7a)

B000 (α) = −ik1 L000 (α) = ik1 gL −1 (α)gU −1 (α0 )F000 (α).

(9.7b)

For further ends we record c0 (k1 + k2 )1/2 gU −1 (α0 )α −1/2 [1 + O(α −1/2 )] U000 (α) = − 2πi

as |α| → ∞ in U + Q.

(9.8)

10. Potentials uss 0 , vss 0 and the pre-solution Ess 0 , H ss 0 We assume for us0 , v s0 the following representations: Z us0 = eiβy A0 ± (α) ei(αx±γ1 z) dα, Z vs0 = eiβy

Q

Q

B0 ± (α) ei(αx±γ2 z) dα,

(10.1a) (10.1b)

where the upper signs are for z>0 and the lower for z<0. From (7.1a), (7.1b), (8.1a), (8.1b), (9.1a) and (9.1b) we have A0 + (α) = A0 + = A00 + A000 ,

(10.2a)

A0 − (α) = A0 − = A00 − A000 ,

(10.2b)

B0 + (α) = B0 + = B00 + B000 ,

(10.2c)

B0 − (α) = B0 − = B00 − B000 .

(10.2d)

Consequently, from (8.17a), (8.17b) (9.7a) and (9.7b) we obtain r r r  κ2 − α −1 κ2 + α0 κ1 + α ± −1 0 00 γ1 (α0 )F0 (α) ∓ gL (α)gU (α0 ) ik2 F0 (α) , A0 (α) = − κ1 + α κ1 − α0 κ2 − α r r r  κ1 − α −1 κ1 + α0 κ2 + α γ2 (α0 )F00 (α) ± gL (α)gU −1 (α0 ) ik1 F000 (α) . B0 ± (α) = κ2 + α κ2 − α0 κ1 − α

(10.3a) (10.3b)

For z=0, x<0, no choice of the sign in (10.3a) and (10.3b) are involved, since Z gL −1 (α)F000 (α) eiαx dα = 0 for x < 0. Q

For x>0, limiting cases z=0+ and z=0− should be considered. The required edge behaviour of us0 , v s0 in the plane z=0 is confirmed by the asymptotic behaviour of U00 , U000 and L00 , L000 for |α|→∞. As shown in Section 17, this is sufficient to fulfil the stipulations (6.4a)–(6.4d). Also the outgoing wave conditions are discussed in Section 17. Formulas (10.1) could be used for extensive investigation and interpretation of us0 , v s0 and this, as will be seen, could serve as a basis for a discussion of the solution us , v s . This however is beyond our space limits and only basic properties of us0 , v s0 are indicated in Section 17. Remark. Let us note for further ends that us0 , v s0 are analytic functions of α 0 in the α 0 -plane with cuts outgoing from ±κ 1 ,±κ 2 as in the α-plane (Fig. 2).

S. Prze´zdziecki / Wave Motion 32 (2000) 157–200

177

For Im α 0 >0 this is seen from (10.1a), (10.1b), (10.3a) and (10.3a), since gU (α 0 ) is analytic there and gU (α 0 )6=0. Furthermore, the factor functions gU (α), gL (α), introduced in (8.6), are defined in the entire α-plane and their explicit analytic continuations into L and U cut by 0n − and 0n + are gU (α) =

k1 γ2 + k2 γ1 gL (α)

for Im α < 0,

gL (α) =

k1 γ2 + k2 γ1 gU (α)

for Im α > 0.

Then by an appropriate modification of the contour Q, so that α 0 is above the contour, us 0 , v s 0 are analytically continued for Im α<0. The electromagnetic pre-solution E s0 , H s0 is given by (6.7a)–(6.8b). As anticipated it fulfils the uniqueness conditions (3.4), (3.5), (3.7a) and (3.7b), and fails to behave properly only at the edge. Its inadmissible terms in the edge behaviour in the aperture are determined by the asymptotic behaviour of U00 (α), U 00 (α), for |α|→∞. We get for x→0− , z=0: ∂ ∂ 00 00 ) + finite terms = O(|x|−3/2 ), (10.4a) (u + vs0 Es0x = ∂x ∂z s0 Es0y = − Es0z =

∂ ∂ 0 00 (k1 u0s0 − k2 vs0 ) + iβ (u00s0 + vs0 ) = O(|x|−1/2 ), ∂x ∂z

∂ ∂ 0 0 (u + vs0 ) + finite terms = O(|x|−3/2 ). ∂z ∂z s0

(10.4b) (10.4c)

Formulas (6.7a)–(6.8b) exhibit the modal structure of the pre-solution. Insofar as its symmetries with respect to z=0 are concerned, they reduce only to a decomposition into fields with even and odd components Es0z , Hs0z , which 0 and u00 , v 00 , respectively. are determined via (2.5a) and (2.5b) by u0s0 , vs0 s0 s0 For the spectral representations of the pre-solution we get Z H s0 (1) = eiβy A 0 ± (α) ei(αx±γ1 z) dα, (10.5a) E s0 (1) = iZH Q

H s0 (2) = eiβy E s0 (2) = −iZH

Z Q

B 0 ± (α) ei(αx±γ2 z) dα,

(10.5b)

where A 0 ± (α) = S ± (α)A0 ± (α),

(10.6a)

B 0 ± (α) = T ± (α)B0 ± (α),

(10.6b)

S ± (α) = S ± = (∓αγ1 + ik1 β)xx + (∓βγ1 − ik1 α)yy + (α 2 + β 2 )zz ,

(10.7a)

T ± (α) = T ± = (∓αγ2 − ik2 β)xx + (∓βγ2 + ik2 α)yy + (α 2 + β 2 )zz .

(10.7b)

As earlier the upper signs are for z>0 and the lower for z<0. For z=0, the integrals (10.5a) and (10.5b) are divergent since the vectorial amplitudes (10.6a) and (10.6b) behave like O(α 1/2 ) for |a|→∞ . Thus formulas (6.8a) and (6.8b) should be used there unless a distributional interpretation of (10.5a) and (10.5b) is introduced. The choice of the signs is then immaterial. Finally, let us observe explicitly that the pre-solution arises due to a two-fold excitation: 1. by the inhomogeneous boundary condition (3.5) impressed by the incident waves, 2. by the strong radiating singularity at the edge.

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Obviously it is only the first excitation that is physically justified and must remain intact in the final solution while the inadmissible strong singularity must be removed. Fortunately, with the help of the ghost potentials, we are able to construct auxiliary electromagnetic fields that satisfy at the half-plane homogeneous boundary conditions and exist exclusively due to their strong, radiating singularities at the edge. These fields serve to single out from the pre-solution its inadmissible part and to cancel it. 11. Potentials uss 1 , vss 1 for the singular field E −ββ , H −ββ ◦

◦

From the problem (5.12a)–(5.12e) we now choose as incident waves the potentials u1 , ν 1 and denote ◦

ui1 = u1 = a1 eβ(x+iy) e−ik1 z ,

(11.1a)

◦

vi1 = v 1 = b1 eβ(x+iy) eik2 z .

(11.1b)

We shall now look for a solution us1 , v s1 , which satisfies the boundary conditions (6.3a) and (6.3b) and the edge conditions (6.4a)–(6.4d) with the index 0 substituted by 1, whereas its behaviour at infinity (ρ→∞) is not yet specified. As in Section 7 we use the even–odd decomposition with respect to z=0 and denote us1 = u0s1 + u00s1 ,

(11.2a)

0 00 + vs1 . vs1 = vs1

(11.2b)

0 By virtue of the remark in Section 10 about the analytic dependence of us0 , v s0 on α 0 , the solutions for u0s1 , vs1 00 00 0 0 00 00 and us1 , vs1 can be obtained from the spectral representations of us0 , vs0 and us0 , vs0 by substituting in their spectral quantities: α0 = −iβ, F00 (α) = F10 (α) = ic1 /2πi(α + iβ), F000 (α) = F100 (α) = c1 /2πi(α + iβ). In the spectral quantities arrived at in this way the index 0 is changed to 1 and we get for z>0, ! ! Z A01 (α) u0s1 iβy i(αx+γ1 z) =e dα, (11.3a) 00 (α) e u00s1 A Qβ 1 ! ! Z 0 B10 (α) vs1 iβy ei(αx+γ2 z) dα. =e (11.3b) 00 vs1 B100 (α) Qβ

Here, the contour Qβ denotes a modification of the contour Q so that −iβ is above Qβ ; the respective loop of Qβ is shown in Fig. 2 by a dashed line. Moreover, A01 (α) = (8.17a),

B10 (α) = (8.17b)

L01 (α) = (8.16a),

U10 (α) = (8.16b)

with α0 = −iβ, F00 = F10 . Similarly A001 (α) = (9.7a),

B100 (α) = (9.7b),

with α0 = −iβ, F00 = F10 . Then the solution us 1 , v s 1 is given by Z iβy A1 ± (α) ei(αx±γ1 z) dα, us1 = e Z vs1 = e

Qβ

iβy Qβ

B1 ± (α) ei(αx±γ2 z) dα,

L001 (α) = (9.6a),

U100 (α) = (9.6b)

(11.4a) (11.4b)

S. Prze´zdziecki / Wave Motion 32 (2000) 157–200

179

where A1 ± (α) = A01 (α) ± A001 (α), ±

B1 (α) =

(11.5a)

B10 (α) ± B100 (α).

(11.5b)

The convention about the signs is the same as in (10.1a) and (10.1b). For further needs we record c1 (k1 + k2 )−1/2 GU −1 (−iβ)(−α)−1/2 [1 + O(α −1 )], (11.6a) L01 (α) = 2π −c1 −1 (k1 + k2 )1/2 gU (−iβ)α −1/2 [1 + O(α −1/2 )], (11.6b) U100 (α) = 2πi for |α|→∞. It will be seen (Section 17) that the solution us1 , v s1 is suitable for our purposes to find a corrective field since: 1. asymptotically for ρ→∞ the plane wave constituents in us1 , v s1 consist exclusively of ghost potentials. Thus H −β .; no plane wave constituents will appear in the electromagnetic field E −β ,H 2. the remaining constituents of us1 , v s1 decay exponentially for ρ→∞, due to the appropriate choice of the roots γ 1 (α), γ 2 (α). Let us add that it can be shown (J. Boersma, private commun.; see Section 17) that the shadow boundaries for the incident and reflected waves in the asymptotics of us1 , v s1 coincide with the shadow boundaries determined by the geometrical construction from [21]. Thus we may conclude that the analytic continuation used to determine us1 , v s1 , correctly defines the geometrical optics solution. Consequently, the diffraction problem for us1 , v s1 could be formulated in exactly the same way as that for us0 , v s0 . The electromagnetic field E −β , H −β corresponding to us1 , v s1 is given by (1) (2) E −β = E −β + E −β ,

(11.7a)

(1) (2) H −β = H −β + H −β ,

(11.7b)

where (1) H (1) E −β = iZH −β = K 1 us1 ,

(11.8a)

(2) H (2) E −β = −iZH −β = K 2 vs1 ,

(11.8b)

and the spectral representations are Z (1) iβy i(αx±γ1 z) H (1) = e A± dα, E −β = iZH −β 1 (α) e Q

(2) iβy H (2) E −β = −iZH −β = e

Z Q

i(αx±γ2 z) B± dα, 1 (α) e

(11.9a) (11.9b)

where ± ± A± 1 (α) = S (α)A1 (α),

± ± B± 1 (α) = T (α)B1 (α).

Since Res A ± 1 (α) = 0,

(11.10a)

Res B ± 1 (α) = 0,

(11.10b)

α=−iβ α=−iβ

we are allowed to return in (11.9a) and (11.9b) to the original contour Q.

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± By means of (10.7a) and (10.7b) we note that in the amplitudes A ± 1 , B 1 , the poles at α=−iβ are cancelled identically for the z-components and transformed into removable singularities for the x- and y-components. For z=0, the sense of the integrals (11.9a) and (11.9b) is the same as in (10.5a) and (10.5b). Three basic properties of E −β , H −β deserve to be mentioned explicitly. 1. No plane wave constituents appear in the asymptotics of E −β , H −β for ρ→∞, since in the asymptotics of the corresponding potentials only ghost potentials are present. This is equivalent to the explicit relations (11.10a) and (11.10b). 2. From (11.9a) and (11.9b) we find that E −β satisfies at the half-plane the homogeneous boundary condition

E −β × z = 0.

(11.11)

3. In the aperture the edge behaviour of E −β is the same as that of the pre-solution E s0 , given by (10.4a)–(10.4c). Thus we conclude that E −β , H −β is excited exclusively by its strong, radiating singularity at the edge and consequently may be useful in compensating the inadmissible singularity in the pre-solution. 12. Potentials uss 2 , vss 2 for the singular field Eββ , H ββ The incident waves singled out from (5.12a) are now ◦

ui2 = u2 = a2 e−β(x−iy) eik1 z , ◦

vi2 = v 2 = b2 e−β(x−iy) e−ik2 z .

(12.1a) (12.1b)

The formulation and solution of the diffraction problem for (12.1a) and (12.1b) is the same as in the preceding section. The solution is denoted by us2 , v s2 , and its even–odd decomposition is given by us2 = u0s2 + u00s2 ,

(12.2a)

0 00 + vs2 . vs2 = vs2

(12.2b)

0 and u00 , v 00 arise by substituting in the spectral quantities of Sections The spectral quantities defining u0s2 , vs2 s2 s2 0 0 8 and 9: α 0 =iβ, F0 (α) = F2 (α) = −ic2 /2π i(α − iβ), F000 (α) = F200 (α) = c2 /2πi(α − iβ). We obtain for z>0, ! ! Z u0s2 A02 (α) iβy ei(αx+γ1 z) dα, (12.3a) =e 00 u00s2 Q A2 (α) ! ! Z 0 B20 (α) vs2 iβy i(αx+γ2 z) dα, (12.3b) =e 00 00 (α) e vs2 B Q 2

where A02 (α) = (8.17a), B20 (α) = (8.17b), L02 (α) = (8.16a), U20 (α) = (8.16b), with α0 = iβ, F00 = F20 ; similarly A002 (α) = (9.7a), B200 (α) = (9.7b), L002 (α) = (9.6a), U200 (α) = (9.6b), with α0 = iβ, F000 = F200 . Then the solution us 2 , v s 2 is given by Z i(αx±γ1 z) dα, (12.4a) us2 = eiβy A± 2 (α) e Z vs2 = eiβy

Q

Q

B2± (α) ei(αx±γ2 z) dα,

(12.4b)

where 0 00 A± 2 (α) = A2 (α) ± A2 (α),

(12.5a)

S. Prze´zdziecki / Wave Motion 32 (2000) 157–200

B2± (α) = B20 (α) ± B200 (α).

181

(12.5b)

For further needs we record c2 −1/2 [1 + O(α −1 )], L02 (α) = − (k1 + k2 )−1/2 G−1 U (iβ)(−α) 2π −c2 −1 (iβ)α −1/2 [1 + O(α −1/2 )], (k1 + k2 )1/2 gU U200 (α) = 2πi

(12.6a) (12.6b)

for |α|→∞. As in the preceding section we find that the plane wave constituents in the asymptotics (ρ→∞) of us2 , v s2 consist exclusively of ghost potentials. The electromagnetic field E β , H β corresponding to us2 , v s2 is given by (1) (2) Eβ = Eβ + Eβ ,

(12.7a)

(1) (2) Hβ = Hβ +Hβ ,

(12.7b)

(1) H (1) E β = iZH β = K 1 us2 ,

(12.8a)

(2) H (2) E β = −iZH β = K 2 vs2 ,

(12.8b)

and the spectral representations are Z (1) iβy i(αx±γ1 z) A± dα, Eβ = e 2 (α) e Q

(2) Eβ

Z =e

iβy Q

i(αx±γ2 z) B± dα, 2 (α) e

(12.9a) (12.9b)

where ± ± A± 2 (α) = S (α)A2 (α),

± ± B± 2 (α) = T (α)B2 (α)

and Res A ± 2 (α) = 0,

(12.10a)

Res B ± 2 (α) = 0.

(12.10b)

α=iβ α=iβ

Basic properties of E β , H β are identical with those described for E −β , H −β at the end of the preceding section. 13. Synthesis of the solution uss , vss We shall now construct the solution us , v s by representing it as us = us0 + us1 + us2 ,

(13.1a)

vs = vs0 + vs1 + vs2 .

(13.1b)

Then the even/odd parts of us , v s are given by u0s = u0s0 + u0s1 + u0s2 ,

(13.2a)

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S. Prze´zdziecki / Wave Motion 32 (2000) 157–200 0 0 0 vs0 = vs0 + vs1 + vs2 ,

(13.2b)

u00s = u00s0 + u00s1 + u00s2 ,

(13.2c)

00 00 00 + vs1 + vs2 . vs00 = vs0

(13.2d)

By virtue of the uniqueness conditions satisfied by the constituent potentials in (13.1a)–(13.2d), the only uniqueness conditions not yet satisfied by us , v s are the edge conditions (4.8a) and (4.8b), i.e. (5.12d). Fortunately, the unknown constants c1 , c2 in the constituent potentials allow to remedy this defect. In order to find these constants we shall exploit the Abelian connection between the asymptotic behaviour at infinity of the functions L00 , L01 , L02 and U000 , U100 , U200 , and the asymptotic behaviour for x → 0± , z = 0 of the corresponding combinations of the potentials. Consider first the even parts of us , v s . We have ∂ ∂ ∂ ∂ 0 0 0 ) + (k1 u0s1 − k2 vs1 ) + (k1 u0s2 − k2 vs2 ). (k1 u0s − k2 vs0 ) = (k1 u0s0 − k2 vs0 ∂z ∂z ∂z ∂z

(13.3a)

By virtue of (4.8a) and (4.8b), the left-hand side of (13.3a) is required to behave like O(x 1/2 ) for x → 0+ , z = 0. However, in view of (6.4), each of the terms of the right-hand side behaves like O(x −1/2 ). The asymptotic behaviour of the constituent expressions in (13.3a) is, by virtue of (8.3a), determined by L00 , L01 , L02 , and consequently, from the Abelian connection we must have L00 + L01 + L02 = O(α −3/2 )

for |α| → ∞ in L .

(13.4a)

Thus from (8.18), (11.6a), (12.6a) we obtain −1 −1 iG−1 U (−iβ)c1 − iGU (iβ)c2 = −GU (α0 )d0 .

(13.5a)

Similarly, for the odd parts of us , v s we have ∂ 00 ∂ ∂ ∂ 00 00 00 00 ) + (u00s1 + vs1 ) + (u00s2 + vs2 ). (u + vs00 ) = (u + vs0 ∂z s ∂z s0 ∂z ∂z

(13.3b)

By virtue of (4.8a) and (4.8b), the left-hand side of (13.3b) is required to behave like O(x0 )=O(1) for x→0− , z=0. In view of (6.4a)–(6.4d), each of the terms of the right-hand side behaves like O(|x|−1/2 ). The asymptotic behaviour of the constituent expressions in (13.3b) is, by virtue of (9.2c), determined by U000 , U100 , U200 , and consequently from the Abelian connection we must have U000 + U100 + U200 = O(α −1 )

for |α| → ∞ in U .

(13.4b)

Thus from (9.8), (11.6b), (12.6b) we obtain −1 −1 −1 (−iβ)c1 + gU (iβ)c2 = −gU (α0 )c0 . gU

(13.5b)

Obviously, (13.5a) and (13.5b) are obtained by virtue of necessary conditions on (13.3a) and (13.b) which hold for z=0. In Section 17, we show they are also sufficient for the required behaviour of the potentials in the whole vicinity of the edge. From (13.5a) and (13.5b) we get c1 =

11 , 1

(13.6a)

c2 =

12 , 1

(13.6b)

S. Prze´zdziecki / Wave Motion 32 (2000) 157–200

183

where 1=

i 2k1 k2

p p ( (κ1 − iβ)(κ2 − iβ) + (κ1 + iβ)(κ2 + iβ)),

(13.7a)

11 =

p p −1 [ (κ1 + α0 )(κ2 + α0 )d0 + i (κ1 + iβ)(κ2 + iβ)c0 ], gU (α0 )gU (iβ)

(13.7b)

12 =

p p −1 [ (κ1 + α0 )(κ2 + α0 )d0 − i (κ1 − iβ)(κ2 − iβ)c0 ]. gU (α0 )gU (−iβ)

(13.7c)

Determination of the constants c1 , c2 given by (13.6a) and (13.6b), constitutes the final step in our construction of the solution Zfor the potentials us , v s , which are now given by A± (α) ei(αx±γ1 z) dα,

(13.8a)

B ± (α) ei(αx±γ2 z) dα,

(13.8b)

± ± A± (α) = A± = A± 0 + A1 + A2 ,

(13.9a)

B ± (α) = B ± = B0± + B1± + B2± ,

(13.9b)

us = eiβy Z vs = e

Qβ

iβy Qβ

where

u0s , vs0 , u00s , vs00

are given by the integrals of the form (8.1a), (8.1b), (9.1a) and (9.1b) with the

A0 (α) = A0 = A00 + A01 + A02 ,

(13.10a)

B 0 (α) = B 0 = B00 + B10 + B20 ,

(13.10b)

A00 (α) = A00 = A000 + A001 + A002 ,

(13.10c)

B 00 (α) = B 00 = B000 + B100 + B200 .

(13.10d)

Representations for amplitudes

Properties of us , v s follow from those of the constituent solutions and are briefly discussed in Section 17. 14. Electromagnetic solution The solution for the scattered electromagnetic field E s , H s is given by (2) E s = E (1) s + Es ,

(14.1a)

(2) H s = H (1) s +Hs ,

(14.1b)

where H (1) E (1) s = iZH s = K 1 us ,

(14.2a)

H (2) E (2) s = −iZH s = K 2 vs ,

(14.2b)

and the spectral representations are Z (1) iβy H = iZH = e A ± (α) ei(αx±γ1 z) dα, E (1) s s Q

iβy H (2) E (2) s = −iZH s =e

Z Q

B ± (α) ei(αx±γ2 z) dα,

(14.3a) (14.3b)

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where A ± (α) = S ± (α)A± (α),

B ± (α) = T ± (α)B ± (α).

The convention about the signs in (14.3a) and (14.3b) is the same as in (10.1a) and (10.1b). We note however that, in contrast to (10.5a) and (10.5b), the integrals in (14.3a) and (14.3b) are convergent for z=0, due to (13.5a) and (13.5b). Representations (14.1a)–(14.3b) exhibit the natural modal structure of the solution. Its structure in terms of the auxiliary, singular constituent fields is given by E s = E s0 + E −β + E β ,

(14.4a)

H s = H s0 + H −β + H β .

(14.4b)

Insofar as the symmetries of E s , H s with respect to z=0 are concerned, they reduce only to a decomposition into fields with even and odd components Esz , Hsz , which are determined via (2.5a), (2.5b) by u0s vs0 and u00s , vs00 , respectively. Formulas (14.3a) and (14.3b) constitute the final result of the paper and can be used for extensive investigations and interpretations of the solution. Due to space limits this is postponed to a later paper. In Section 17 only the basic properties of the solution are discussed.

15. Reduction to normal incidence We now show that the present solution for β=0 reduces correctly to that of [4]. The solution in [4] was given in terms of the modal constituents of the y-component of the scattered field ˆ Eˆ ys = uˆ + v.

(15.1)

We put carets over the symbols from [4]. For the present solution assuming β=0 we get Esy = −k1

∂us ∂vs + k2 . ∂x ∂x

(15.2)

This relation determines the modal constituents of Esy , which in turn, as shown in [4], define all remaining components of the electromagnetic field. Thus to identify both solutions it is sufficient to show that uˆ = −k1 vˆ = k2

∂us , ∂x

∂vs . ∂x

(15.3a) (15.3b)

In order that the incident waves be the same for both problems we take αˆ 10 = αˆ 20 = α0 ,

(15.4a)

aˆ = −ik1 α0 a0 ,

(15.4b)

bˆ = ik2 α0 b0 .

(15.4c)

Relations (15.3a) and (15.3b) will hold provided for β=0, Aˆ (e) = −iαk1 A0 ,

(15.5a)

Bˆ (e) = iαk2 B 0 ,

(15.5b)

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185

Aˆ (0) = −iαk1 A00 ,

(15.5c)

Bˆ (0) = iαk2 B 00 .

(15.5d)

From (8.17a) and its counterparts for α 0 =±iβ=0 we find s s i(c1 − c2 ) k − α k2 − α −1 2 −1 −1 g (α)GU (α0 )F00 [(α − α0 ) + α0 ] + ik1 g (α)G−1 , (15.6) −iαk1 A0 = ik1 U (0) k1 + α L k1 + α L 2π i hence

s

−iαk1 A0 =ik1 α0

s    d0 ic1 − ic2 k2 − α −1 k2 − α −1 −1 −1 −1 0 g (α)GU (α0 )F0 +ik1 g (α) GU (α0 ) + GU (0) . k1 + α L k1 + α L 2π i 2π i (15.7)

The second term in (15.7) vanishes by virtue of (13.5a) and consequently s ik1 α0 a0 − ik2 α0 b0 k2 − α −1 g (α)G−1 , −iαk1 A0 = k1 U (α0 ) k1 + α L 2π i(α − α0 )

(15.8)

which coincides with (5.14a) in [4] when (15.4a)–(15.4c) and accounted for Since A0 = −(γ2 /γ1 )B 0 and Aˆ (e) = (k1 /k2 )(γ2 /γ1 )Bˆ (e) , (15.5a) implies (15.5b). For A00 we get from (9.7a) and its counterparts for α 0 =±iβ=0, −1 −1 −1 (α0 )F000 + k1 k2 gL−1 (α)[gU (α0 )c0 + gU (0)(c1 + c2 )] −iαk1 A00 = k1 k2 α0 gL−1 (α)gU

The second term vanishes by virtue of (13.5b) and consequently # " k2 γ10 aˆ − k1 γ20 bˆ −1 −1 00 , −iαk1 A = −gL (α)gU (α0 ) 2πi(α − α0 )

1 . 2π i

(15.9)

(15.10)

which coincides with (6.7a) in [4]. Since A00 = (k2 /k1 )B 00 and Aˆ (0) = −Bˆ (0) , (15.5c) implies (15.5d).

16. Reduction to the ordinary medium For χ =0 we get √ k1 = k2 = k = ω εµ, q κ1 = κ2 = κ = k 2 − β 2 , p γ1 = γ2 = γ = κ 2 − α 2 , q γ10 = γ20 = γ0 = κ 2 − α02 ,

(16.2b)

g = 2kγ ,

(16.3a)

2k , γ

(16.3b)

G=

(16.1a) (16.1b) (16.2a)

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p gU (α) = 2k(κ + α), p gL (α) = 2k(κ − α), r 2k , GU (α) = κ +α r 2k , GL (α) = κ −α A0n (α)

=

−Bn0 (α)

1 =− 2k

A00n (α) = Bn00 (α) =

(16.4a) (16.4b) (16.4c)

(16.4d) 

κ + αn κ +α

1/2

Fn0 ,

i F 00 , 2((κ − α)(κ + αn ))1/2 n

(16.5a) n = 0, 1, 2, α1 = −iβ, α2 = iβ,

(16.5b)

u0s = −vs0 ,

(16.6a)

u00s = −vs00 ,

(16.6b)

The system (13.5a) and (13.5b) takes the form p p √ i κ − iβ c1 − i κ + iβ c2 = − κ + α0 d0 , c2 c0 c1 . +√ = −√ √ (κ − iβ) κ + iβ (κ + α0 )

(16.7a) (16.7b)

Taking into account relations (16.1a)–(16.7b) we arrive from (13.8a), (13.8b), (14.3a) and (14.3b) at the correct solutions for us , v s and E s , H s for an ordinary medium. This means that they coincide with the solutions obtained directly for the ordinary medium but constructed in terms of the handed modal potentials determined by the choice τ =zz in (2.5a) and (2.5b). The structures of these solutions are difficult to compare with the conventional solutions which are usually given in terms of the y-components of the fields. As is well known, in ordinary media all field components can be expressed in terms of the y-components. For the considered diffraction problem two independent (uncoupled) scalar problems are obtained for these components. Let us recall these problems. Assume the incident electromagnetic field to have the following y-components: Eˆ iy = aˆ e ei(α0 x+βy+y0 z) ,

(16.8a)

Hˆ iy = aˆ m ei(α0 x+βy+y0 z) .

(16.8b)

We put carets over the symbols of the direct solutions and denote the y-components of the scattered field by Eˆ sy , Hˆ sy . The boundary conditions at the half-plane are Eˆ sy = −Eˆ iy ,

(16.9a)

∂ Hˆ iy ∂ Hˆ sy =− . ∂z ∂z

(16.9b)

We apply the even–odd decompositions 0 00 + Eˆ sy , Eˆ sy = Eˆ sy

(16.10a)

0 00 + Hˆ sy , Hˆ sy = Hˆ sy

(16.10b)

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and find that 00 = 0, Eˆ sy

(16.11a)

0 = 0. Hˆ sy

(16.11b)

0 , Hˆ 00 (z≥0): We assume the following representations forEˆ sy sy Z 0 = eiβy Aˆ e (α) ei(αx+γ z) dα, Eˆ sy Q

00 = eiβy Hˆ sy

Z

Q

Aˆ m (α) ei(αx+γ z) dα.

Via the standard Wiener–Hopf technique we obtain r aˆ e κ + α0 ˆ , Ae (α) = − κ + α 2πi(α − α0 ) r aˆ m κ − α0 . Aˆ m (α) = − κ − α 2πi(α − α0 )

(16.12a) (16.12b)

(16.13a) (16.13b)

In order that the incident waves be the same for both problems we take aˆ e = −βγ0 (a0 + b0 ) − iα0 k(a0 − b0 ),

(16.14a)

iZ aˆ m = −βγ0 (a0 − b0 ) − iα0 k(a0 + b0 ).

(16.14b)

Let us consider the y-components of our solution (14.1a)–(14.3b) in the case of χ =0. By virtue of (16.6a) and (16.6b), we get 0 = 2iβ Esy

∂u00s ∂u0 − 2k s , ∂z ∂x

(16.15a)

00 = 0, Esy

(16.15b)

iZH0sy = 0,

(16.16a)

iZH00sy = 2iβ

∂u00 ∂u0s − 2k s , ∂z ∂x

(16.16b)

As indicated above, all field components are expressible in terms of Esy , Hsy . Thus to identify both solutions it is sufficient to show that Aˆ e (α) = −2(βγ A00 + iαkA0 ),

(16.17a)

iZ Aˆ m (α) = −2(βγ A0 + iαkA00 ).

(16.17b)

From (16.5a) and (16.5b) with n=0 we get −2(βγ A000

√ −iβ[(κ + α) − (κ + α0 ) + (κ + α0 )] 00 i κ + α0 (α − α0 + α0 ) 0 F0 F0 + √ √ κ +α κ + α)(κ + α0 ) r √
κ + α0 iβc0 id0 κ + α0 iβF000 − iα0 F00 − =− . + √ √ κ +α 2π i κ + α 2π i (κ + α)(κ + α0 )

+ iαkA00 ) =

(16.18)

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Similarly from (16.5a) and (16.5b) with n=1, 2, we get r r iβc1 κ − iβ iβ(c1 − c1 ) κ − iβ c1 00 0 − , − −2(βγ A1 + iαkA1 ) = − √ κ + α 2πi(α + iβ) 2π i (κ + α)(κ − iβ) κ + α 2π i r r iβc2 κ − iβ iβ(c2 − c2 ) κ + iβ c2 − . + −2(βγ A002 + iαkA02 ) = − √ κ + α 2πi(α + iβ) 2π i (κ + α)(κ − iβ) κ + α 2π i Hence, summing up we obtain r   κ + α0 −βγ0 (a0 + b0 ) − iα0 k(a0 − b0 ) , −2(βγ A00 + iαkA0 ) = − κ +α 2π i(α − α0 )

(16.19)

(16.20)

(16.21)

since all other terms in (16.18)–(16.20) vanish either identically or by virtue of (16.7a) and (16.7b). Thus (16.21) coincides with (16.13a) when (16.14a) is accounted for. Similarly β(κ + α0 )1/2 [(κ−α)−(κ − α0 ) + (κ − α0 )] 0 k(α − α0 + α0 ) F0 + √ F 00 √ k κ−α (κ − α)(κ + α0 ) 0 r   r βγ0 0 kα0 00 1 κ − α0 κ+α0 βd0 kc0 − F − +√ ,(16.22) =− F − κ−α k 0 γ0 0 κ−α k2π i (κ − α)(κ+α0 ) 2π i r r 1 kc1 κ + iβ iβ(−c1 + c1 ) κ − iβ iβc1 0 00 − +√ , (16.23) −2(βγ A1 + iαkA1 ) = − κ − α 2πi(α + iβ) κ − α k2π i (κ − α)(κ − iβ) 2π i r r 1 kc2 κ − iβ iβ(c2 − c2 ) κ + iβ iβc2 0 00 + +√ . (16.24) −2(βγ A2 + iαkA2 ) = − κ − α 2πi(α − iβ) κ − α k2π i (κ − α)(κ + iβ) 2π i −2(βγ A00 + iαkA000 ) =

Summing up, by virtue of (16.7a) and (16.7b) we obtain r   κ − α0 −βγ0 (a0 − b0 ) − iα0 k(a0 + b0 ) 0 00 −2(βγ A + iαkA ) = − κ −α 2π i(α − α0 )

(16.25)

and this coincides with iZ Aˆ m from (16.13b) when (16.14b) is accounted for. 17. Properties of the solution An analysis and interpretation of the solution concern its various properties and involve several view-points. Fortunately, most of this task reduces to results obtained for earlier diffraction problems. The decisive step that simplifies the problem consists in an effective factorization of g(α) [6–8]. A fundamental insight stems from an asymptotic analysis for ω→∞ and a ray interpretation of the waves singled out thereby [7,8,10,14,16]. Significant features of these waves follow from uniform asymptotic expansions in the vicinities of the shadow boundaries, including those of lateral waves [7,14]. A detailed discussion of the solution is postponed to a later paper; here we merely present an outline of its basic properties. 17.1. Geometrical optics solutions As indicated in Sections 10–12, only the potentials us0 , v s0 defining the pre-solution E s0 , H s0 contribute asymptotically for ρ→∞, plane wave constituents to the asymptotics of the electromagnetic solution E s , H s .

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189

We shall now show that the asymptotics of us0 +ui0 , v s0 +v i0 for ρ→∞ leads correctly to the geometrical optics solution ug0 , v g0 given by (4.10a) and (4.10b) and employed in (6.5a) and (6.5b). To this end let us determine the ± residues of A± 0 (α), B0 (α) at α=α 0 , which define the amplitudes of the relevant plane wave constituents. From (10.3a) and (10.3b) we get 2πi Res A+ 0 (α) = −a0 ,

(17.1a)

2πi Res B0+ (α) = −b0 ,

(17.1b)

2π i Res A− 0 (α) = R11 a0 + R12 b0 ,

(17.1c)

2πi Res B0− (α) = R21 a0 + R22 b0 ,

(17.1d)

α=α0 α=α0 α=α0 α=α0

where Rmn , m, n=1,2, are given in (4.11b). In the spectral integral representation (10.1a) we apply the transformation α = κ1 cos θ (with real κ 1 >0) leading to Z κ 1 A± z> (17.2) us0 = eiβy < 0, 0 (κ1 cos θ) sin θ exp[iκ1 ρ cos(θ ∓ ϕ)]dθ, Q(1)

where Q(1) is the integration contour in the θ -plane as shown in Fig. 3. Q(1) passes above the transformed pole at θ=ϕ 10 and above the “alien” (lateral wave) branch point at θ =ϕ h1 =arccos(κ 2 /κ 1 ) remaining in the amplitudes A± 0 (κ1 cos θ ) (θ=ϕ h1 corresponds to the branch point α=κ 2 ; cf. relations (8.17a), (9.7a) and (10.2a), (10.2b), where the lower functions L00 (α), L000 (α) have branch points at α=κ 1 and α=κ 2 ). The integrand in (17.2) contains an exponential factor which has a saddle point at θ =ϕ (if z>0, 0<ϕ<π) and at θ =2π −ϕ (if z<0, π<ϕ<2π). Next, Q(1) is deformed into the steepest descent path SDP± through the saddle point, which is possibly interrupted by a loop on both sides of a cut outgoing from θ =ϕ h1 . SDP± is described by Re cos(θ ∓ ϕ) = 1. In the deformation of Q(1) the pole at θ=ϕ 10 is intercepted if 0<ϕ<ϕ 10 and if 2π−ϕ 10 <ϕ<2π . According to (17.1a) and (17.1c), the corresponding residue contributions are given by −ui0 and ur0 , and consequently we are led to (4.10a) which was obtained directly as the geometrical optics solution.

Fig. 3. The complex θ -plane with the integration contour Q(1) , the steepest descent paths SDP± , the incident-wave pole at θ =ϕ 10 , and the lateral-wave branch point at θ =ϕ h1 .

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S. Prze´zdziecki / Wave Motion 32 (2000) 157–200

By applying the transformation α = κ2 cos θ in (10.1b) and proceeding in the same manner, we recover (4.10b) (the lateral-wave branch point θ=ϕ h2 =arccos(κ 1 /κ 2 ) is in this case purely imaginary). By evaluating for (17.2) the saddle point contribution and the asymptotic contribution (ρ→∞) from the loop around the cut outgoing from θ=ϕ h1 , it is confirmed [7,8,14] that these contributions decay exponentially for ρ→∞, when small losses are introduced (Im κ1 > 0). The first terms (under appropriate restrictions) of these contributions are given in the next two sections. Similar results hold for v s0 and we thus conclude that the outgoing wave conditions (6.5a) and (6.5b) are fulfiled. We now turn our attention to the plane wave constituents in the asymptotics (ρ→∞) of the potentials us1 , v s1 . ± The amplitudes of these waves are determined by the residues of A± 1 (α), B1 (α) at α=−iβ. We get from (11.5a) and (11.5b), 2π i Res A+ 1 (α) = 0,

(17.3a)

α=−iβ

2πi Res B1+ (α) = −b1 +

k1 a1 , k2

(17.3b)

2πi Res A− 1 (α) = −a1 +

k2 b1 , k1

(17.3c)

α=−iβ

α=−iβ

2πi Res B1− (α) = 0.

(17.3d)

α=−iβ

In the spectral integral (11.4a) we apply the transformation α = κ1 cos θ , leading to Z κ1 A± z> us1 = eiβy < 0. 1 (κ1 cos θ)sin θexp[iκ1 ρ cos(θ ∓ ϕ)] dθ, (1)

Qβ

◦

(17.4)

◦

(1) The pole location α=−iβ is transformed into θ = π/2 + iϕ 1 , where κ1 sinhϕ 1 = β. The integration contour Qβ , shown in Fig. 4, has a loop above the transformed pole and passes above the lateral-wave branch point at θ =ϕ h1 . The saddle points in the integrand of (17.4) and the corresponding steepest descent paths SDP± are the same as for

± ϕ◦ Fig. 4. The complex θ -plane with the integration contour Q(1) β , the steepest descent paths SDP , the incident-wave pole at θ = π/2 + i 1 , and the lateral-wave branch point at θ =ϕ h1 .

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191 ◦

ϕ (17.2). Since Resα=−iβ A+ 1 (α) = 0, it is sufficient to determine whether the pole at θ = π/2 + i 1 is intercepted ◦

(1)

in the deformation of Qβ into SDP− . We note that SDP− passes through this pole if Re cos(π/2 + iϕ 1 + ϕ) = ◦

cos(π/2 + ϕ)coshϕ 1 = (k1 /κ1 ) cos(π/2 + ϕ) = 1, subject to 2π−ϕ>π/2, that is, if ϕ = π + arcsin(κ1 /k1 ). Let us introduce the notation   κn , n = 1, 2. (17.5) ϕgn = π − arcsin kn ◦

Then the pole at θ = π/2 + iϕ 1 is intercepted if 2π −ϕ g1 <ϕ<2π . This corresponds to the occurrence of a shadow boundary at ϕ=2π−ϕ g1 for us1 . Identical considerations hold for v s1 with the transformation α = κ2 cos θ applied in (11.4b). In the deformation of the integration contour the relevant pole is intercepted if 0<ϕ<ϕ g2 . This corresponds to the occurrence of a shadow boundary at ϕ=ϕ g 2 for v s1 . As already indicated in Section 11, the shadow boundaries thus found coincide with those determined by the geometrical construction from [21]. Hence, the plane wave constituents in the asymptotics of us1 +ui1 , v s1 +v i1 , may be identified with the geometrical optics solution ug1 , v g1 . The interpretation of this solution is simplified if we denote ug1 = ug1a + ug1b ,

(17.6a)

vg1 = vg1a + vg1b ,

(17.6b)

where ug1a , v g1a are for b1 =0, and ug1b , v g1b for a1 =0. Then we find  ui1 for 0 < ϕ < 2π − ϕg1 , ugla = 0 for 2π − ϕg1 < ϕ < 2π,  vr1 for 0 < ϕ < ϕg2 , vg1a = 0 for ϕg2 < ϕ < 2π, with vr1 = (k1 /k2 )a1 eβ(x+iy) eik2 z ;  0 for 0 < ϕ < 2π − ϕg1 , ug1b = ur1 for 2π − ϕg1 < ϕ < 2π,  0 for 0 < ϕ < ϕg2 , vg1b = ur1 for ϕg2 < ϕ < 2π,

(17.7a) (17.7b)

(17.8a) (17.8b)

with ur1 = (k2 /k1 )b1 eβ(x+iy) e−ik1 z . These results also show that the plane wave constituents in the asymptotics of us1 , v s1 consist exclusively of ghost potentials, as anticipated in Section 11. Note that for the pairs ui1 , v i1 ; ur1 , v r1 , we have γ 10 =k1 , γ 20 =k2 in the reflection matrix (4.11b), whereupon   ◦ 0 k2 /k1 . (17.9) R = R (1) = k1 /k2 0 This shows that the incident ghost potential of k1 (k2 )-mode is reflected as the ghost potential of k2 (k1 )-mode. Just like for us0 , v s0 , the remaining contributions to the asymptotics of us1 , v s1 decay exponentially for ρ→∞, when small losses are accounted for. The previous considerations can be repeated for us2 , v s2 , with identical conclusions. Consequently, the geometrical optics field E g , H g is given by (4.12a) and (4.12b), and the outgoing wave conditions (3.7a) and (3.7b) are fulfiled.

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17.2. Diffracted far field By means of the analysis of the preceding subsection the spectral representations (13.8a) and (13.8b) for us , v s can be reduced to integrals along steepest descent paths through saddle points and to integrals along appropriate loops around the cuts outgoing from the lateral-wave branch points θ =ϕ h1 , θ =ϕ h2 . Asymptotic evaluation (for ρ→∞) of the integrals along the steepest descent paths by the saddle point method provides the diffracted far field, to be denoted by uf , v f . Assuming that the observation parameter α = κn cos ϕ (where ϕ is the polar angle of the observation point) is well separated from the characteristic points in the α-plane, we find for ρ→∞, uf =

√ ei(κ1 ρ+βy) 2πe−iπ/4 κ1 A± (κ1 cos ϕ)|sin ϕ| + O(ρ −3/2 ), (κ1 ρ)1/2

(17.10a)

vf =

√ ei(κ2 ρ+βy) 2πe−iπ/4 κ2 B ± (κ2 cos ϕ)|sin ϕ| + O(ρ −3/2 ). (κ2 ρ)1/2

(17.10b)

Assuming the incident wave of k1 -mode (b0 =0), we have uf =uf1 , v f =v f1 , where uf1 = D11 (ϕ)a0

eik1ν 1 ·r + O(ρ −3/2 ), (ρ)1/2

(17.11a)

vf1 = D21 (ϕ)a0

eik2ν 2 ·r + O(ρ −3/2 ). (ρ)1/2

(17.11b)

Here we introduced the notation knv n (ϕ)=knv n =κ nρ 0 +βyy , n=1,2, ρ 0 =∇ρ. The values of D11 (ϕ), D21 (ϕ) are gathered from (17.10a) and (17.10b), whereby A± (κ1 cos ϕ), B ± (κ2 cos ϕ) are determined from (13.9a) and (13.9b) with b0 =0. The coefficients D11 (ϕ), D21 (ϕ) may be interpreted as Keller’s diffraction coefficients within a geometrical theory of diffraction for the modal Hertz potentials. Similar coefficients D12 (ϕ), D12 (ϕ) can be defined for the incident wave of k2 -mode. The electromagnetic far fields are given by eik1ν 1 ·r + O(ρ −3/2 ), (ρ)1/2 ! k22 eik2 ν 2 ·r = q1 a[II − ivv 2 ×](vv 2 × z ) − 2 D21 (ϕ) + O(ρ −3/2 ). (ρ)1/2 k1

H (1) I + ivv 1 ×](vv 1 × z )D11 (ϕ) E (1) f1 = iZH f1 = K 1 uf1 = q1 a[I

(17.12a)

(2) H (2) E f1 = −iZH f1 = K 2 vf1

(17.12b)

Confronting (17.12a) and (17.12b) with (4.3a) we see that the diffraction coefficients for the potentials may serve to good purpose in a vectorial version of Keller’s geometrical theory of diffraction (GTD) in chiral media. Alternatively, the tensorial diffraction coefficients can be defined by D 11a , [II + ivv 1 ×][vv 1 × z ]q1 D11 (ϕ) = [II + ivv 1 ×]D " # k2 D 21a . [II − ivv 2 ×][vv 2 × z ] − 22 q1 D21 (ϕ) = [II − ivv 2 ×]D k1

(17.13a) (17.13b)

A GTD interpretation of (17.10a)–(17.12b) exhibits the two cones of diffracted rays (Fig. 5). For (17.12a) and (17.12b) these rays support circularly polarized electromagnetic waves. Corresponding results hold for the incident wave of k2 -mode. A complete verification of the conditions (3.7a) and (3.7b) would require uniform asymptotic expansions in the vicinities of the shadow boundaries of the geometrical optics solution E g , H g and of the shadow boundaries of the excited lateral waves. The diffraction coefficients introduced in (17.11a) and (17.11b) are singular at these boundaries.

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193

Fig. 5. The incident k1 -ray and the two cones of diffracted rays (χ >0, k1 >k2 ).

17.3. Lateral waves (n)

While deforming the integration contours Q(n ) , Qβ , n=1,2, in the θ -plane we encounter, for certain sectors of the observation angle ϕ, the “alien” branch cuts outgoing from θ =ϕ h1 for us and from θ =ϕ h2 for v s [7,8,14]. The asymptotic contributions (ρ→∞) from the integrals along loops around these cuts determine the lateral (head) waves supported at each side of the half-plane in the relevant sectors of ϕ. Consider the case b0 =0 and assume χ>0, i.e. k1 >k2 . In the half-space z>0, the first terms in the asymptotic evaluations of the integrals along the loops around the indicated branch cuts are u+ h1 =

λ+ + iβy iκ1 ρ cos(ϕ−ϕh1 ) 11 a0 eiβy ei(κ2 x+γh z) =λ+ e 11 a0 ah e [κ1 ρ sin(ϕh1 − ϕ)]3/2 +

+ ik1l 1 ·rr = λ+ 11 a0 ah e

+ = vh1

for 0 <ϕ < ϕh1 ,

(17.14a)

λ+ + iβy iκ2 ρ cos(ϕ−ϕh2 ) 21 a0 eiβy ei(κ1 x+iγh z) = λ+ e 21 a0 bh e [κ2 ρ sin(ϕh2 − ϕ)]3/2 +

+ ik2l 2 ·r = λ+ 21 a0 bh e

for 0 <ϕ < arcsinγh /κ1 = ϕh1 ,

(17.14b)

+ from asymptotic evaluations; γ h is determined by κ22 + γh2 = κ12 or by where λ+ 11 , λ21 are coefficients resulting √ κ12 − γh2 = κ22 , hence γh = 2ω[χ εµ]1/2 ; κ1 cos ϕh1 = κ2 , κ1 sin ϕh1 = γh , κ2 cos ϕh2 = κ1 , κ2 sin ϕh2 = iγh , −1 y + γhz ), l+ 1 = k1 (κ2x + βy −1 y + iγhz ), , l+ 2 = k2 (κ1x + βy

ah+ = [κ1 ρ sin(ϕh1 − ϕ)]−3/2 , bh+ = [κ2 ρ sin(ϕh2 − ϕ)]−3/2 .

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The electromagnetic lateral waves are given by +

+ + + + ik1 l 1 ·rr H (1)+ I + i l+ E (1)+ , 1 ×][ll 1 × z ]ah e h1 = iZH h1 = K 1 uh1 = λ11 q1 a[I (2)+ + H (2)+ E h1 = −iZH h1 = K 2 vh1 = −

k22

λ+ q1 a[II k12 21

(17.15a) +

+ + ik2l 2 ·rr − i l+ . 2 ×][ll 2 × z ]bh e

(17.15b)

Similar expressions for the potentials and the fields can be obtained for z<0. The case of the incident wave of k2 -mode is strictly analogous. The geometrical optics interpretation of (17.14a) and (17.15a) is straightforward: their common eiconal is real and the rays are parallel to l + 1 , while the amplitudes are constant along these rays and are determined by the initial values at the half-plane. For (17.14b) and (17.15b) the rays determined by l + 2 are complex and the interpretation goes beyond the limits of this paper. It would be interesting to find out whether the lateral waves could be detected experimentally, e.g. by observing some shadows from their rays. 17.4. Behaviour at the edge From the construction used in Section 13 we conclude only that us , v s fulfil the edge conditions (4.8a) and (4.8b) in the plane z=0, and this follows from the asymptotic behaviour for x→0± of the combinations given by the left-hand sides of (13.3a) and (13.3b). Below we show that the edge conditions (4.8a) and (4.8b) are satisfied in a full vicinity of the edge. In view of (2.4a) and (2.4b) and the edge behaviour of us0 , v s0 in the plane z=0, we can write h ϕ ϕ i −1/2 (17.16a) us0 + ui0 = eiβy C00 sin + C000 cos κ1 J1/2 (κ1 ρ) + ur0 , 2 2 h ϕ ϕ i −1/2 (17.16b) vs0 + vi0 = eiβy C00 sin + C000 cos κ2 J1/2 (κ2 ρ) + ν0r , 2 2 where J1/2 (·) is the Bessel function of the first kind of order 1/2, and ∇ur0 , ∇v0r are continuous for 0≤ϕ≤2π, ρ≥0, −∞
(17.17a)

(C000 + C100 + C200 ) + (C¯ 000 + C¯ 100 + C¯ 200 ) = 0.

(17.17b)

By virtue of the boundary conditions (5.8a) and (5.8b) at the half-plane we get Cn0 + C¯ n0 = 0,

n = 0, 1, 2,

k1 Cn00 − k2 C¯ n00 = 0,

n = 0, 1, 2,

(17.18a) (17.18b)

and consequently C00 + C10 + C20 = 0,

(17.19a)

C000 + C100 + C200 = 0,

(17.19b)

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195

C¯ 00 + C¯ 10 + C¯ 20 = 0,

(17.19c)

C¯ 000 + C¯ 100 + C¯ 200 = 0.

(17.19d)

Thus from (17.16a), (17.16b) (17.19a)–(17.19d) it is seen that the expansions of us , v s do not contain terms with ρ 1/2 . Considering now the behaviour of us , v s in the plane z=0, we conclude that their expansions analogous to (17.16a) and (17.16b) begin with terms determined by J3/2 (·) and consequently we arrive at (4.8a) and (4.8b). Hence, also the edge conditions (3.6a) and (3.6b) for the electromagnetic fields are fulfiled. Thus, as anticipated, the fields behave at the edge in the same way as in an ordinary isotropic medium.

18. Conclusions and remarks The set of exactly solved half-plane diffraction problems (Sommerfeld class) has been enlarged by a new member. Among electromagnetic problems it represents the line of generalizations concerned with new properties of the environment. Similar earlier problems dealt with anisotropic media [8,9,12,13], or a medial interface [6–8,14,16]. Alternative generalizations account for modified properties of the half-plane. Symbolic of the paper are the two wave numbers k1 , k2 and the two ghost potential parameters ±iβ. The wave numbers depict the two-mode character of the problem illustrated by the phenomena of the two cones of diffracted rays and the excitation of the lateral waves. In the complex α-plane the wave numbers define the relative positions of the characteristic branch points ±κ 1 , ±κ 2 , which enter the spectral quantities of both the fields and the potentials. The parameters ±iβ symbolize the adopted approach and the crucial role of the ghost potentials. They appear as supplementary poles in the spectral quantities of the potentials. In the spectral amplitudes of the fields these poles are either cancelled or transformed into removable singularities. The parameters ±iβ also highlight the relation between the 2D and 3D versions of the problem. For the sake of curiosity let us notice that the 2D/3D relation for electromagnetic half-plane diffraction problems is indeed intriguing: it reaches from trivial (ordinary medium) through quite puzzling, as in the present case and in [8,12–14], up to no relation at all [9]. It is in this context that the ghost potentials deserve an explicit credit: for a considerable group of problems [8,12–14] they provide an effective tool for the 2D/3D generalizations. Obviously the ghost potentials constitute a purely mathematical aspect in the analysis of physically relevant fields. Within the adopted approach some properties of the solution may be interpreted via their correspondence with the ghost potentials. In a different approach these properties may be expected to find appropriate equivalent contexts. (1) (2) A suggestive alternative approach is to use directly the Esz -component. By assuming Esz = Esz + Esz , we are immediately led to the core structure (1.3a)–(1.3d). But the full construction of the solution looks more complicated than that given in this paper. A particular but interesting contribution of the potentials consists in revealing the diffractional structure of the singular fields E −β , H −β ; E β , H β . The incident waves exciting these fields can be identified only at the level of the potentials. Moreover, the potentials are not to be blamed for the rather lengthy discussions of Sections 4 and 5. The discussions were intended to show what follows necessarily for the potentials from the uniqueness conditions postulated for the fields, i.e., to establish the necessary character of the boundary, edge and outgoing wave conditions imposed on the potentials. They could be entirely avoided by postulating just a set of sufficient conditions which are easy to arrive at by inspection. In retrospect, the fundamental simplicity in the construction of the electromagnetic solution appears very clearly. Complete data for the construction are provided by just one solution to a conventional, scalar diffraction problem for two Helmholtz equations (cf. (1.3a)–(1.3d) and Section 6). This solution determines the so-called electromagnetic pre-solution. Next, two simple analytic extensions of the scalar solution, from the incidence parameter α 0 to the ghost potential parameters ±iβ, lead to two auxiliary electromagnetic fields singular at the edge of the half-plane. By canceling the inadmissible edge singularities in the sum of these three vectorial constituents we arrive at an

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electromagnetic solution to the formulated problem. In this context, the role of the core structure (1.3a)–(1.3d), to underlie the electromagnetic problem, is evident. Finally, the problem presented in the paper is the first and simplest, exactly solved case of edge diffraction in a chiral medium (it includes [4]). Hopefully, it will prove helpful for more general diffraction results in the electromagnetics of chiral media. Occasional note: Centenary of the Sommerfeld half-plane diffraction problem The paper is dedicated to be part in a tribute to Arnold Sommerfeld on the occasion of the centenary of his landmark work [1]. The centenary was commemorated by the Sommerfeld ’96 Workshop “Modern Mathematical Methods in Diffraction Theory and its Applications in Engineering” organized by Professor E. Meister in Freudenstadt, Germany, September–October 1996 [22]. Basic results of the paper were presented at a Plenary Session of the Workshop. The author also wishes to recall the memory of his teacher Adalbert Rubinowicz whose book [23] bears the dedication “Dem Andenken an meinen unvergesslichen Lehrer dem Schöpfer der exakten Beugungstheorie ARNOLD SOMMERFELD gewidmet”. Acknowledgements The author wishes to thank Professor J. Boersma for his critical reading of the first two versions of this paper. His extensive and incisive comments helped directly to simplify some of the results and implicitly induced significant changes in the presentation of the material. Appendix A. Modes, potentials and plane waves for chiral media A.1. Splitting theorem A standard, first move in dealing with the system (1.1a) and (1.1b) is to eliminate from it the field H or E . Thus we get       E E 0 2 2 I I I I = (∇ × −k1 )(∇ × +k2 ) = , (A.1) [(∇ × −ωχI ) − ω εµI ] H H 0

√ √ where k1 = ω εµ + χ , k2 = ω εµ − χ and 0 denotes the zero vector. It is the factorization in (A.1) that suggests the splitting theorem with the defining equations given by (2.2a) and (2.2b). The factorization serves also to good purpose for a proof of the theorem. Consider the field E , then from (A.1) we get E= 8, =8 (∇ × +k2I )E

(A.2a)

8= 0. (∇ × −k1I )8 Thus E can be represented

(A.2b) E (2) +9, as E =E

where E (2)

fulfils the equation

E (2)= 0, (∇ × +k2I )E

(A.3a)

and 9 is a particular solution to 9 =8 . (∇ × +k2I )9

(A.4) )−1

since, by virtue of (A.2b), equation (A.4) reduces to Such a solution is given by 9=q8 with q=(k1 +k2 E (1) , we get q(k1 +k2 )8=8. Consequently, by denoting 9=q8=E E = E (1) + E (2) ,

(A.5a)

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197

and, by virtue of (A.2b), E (1)= 0. (∇ × −k1I )E

(A.3b)

E (2) and, by defining the constituent magnetic fields as iZH H (1) =E E (1) , iZH H (2) = H =E (1) −E From (1.1b) we obtain iZH (2) E , we arrive at −E H = H (1) + H (2) .

(A.5b) E (1) , (E

H (1) );

E (2) , (E

H (2) )

obey the system (1.1a) and (1.1b), and in view of We now easily verify that the fields (A.5a) and (A.5b), this implies the splitting theorem. Let us note that in the splitting of a given electromagnetic field obeying (1.1a) and (1.1b) into its modal constituents, (2.2a) equations (2.2b), and (2.2c) equations (2.2d) are equivalent. However, in order to construct a modal solution to (1.1a) and (1.1b), say of the k1 -mode, we have to use one of the equations (2.2a) for E (1) or H (1) and then determine H (1) or E (1) from (2.2c) (cf. Section A.3). A.2. Modal handedness The modal equations (2.2a) and (2.2b) induce in the fields E (n ) , n=1,2, a specific, local geometrical structure: E (n ) are collinear, and the distinction between the modal fields E (1) and E (2) is determined by the and ∇×E signof the scalar product

E (n )

E (n)∗ · ∇ × E (n) ,

n = 1, 2.

(A.6)

From (2.2a) and (2.2b), for every point in the region considered in the splitting theorem, we obtain E (1)∗ · ∇ × E (1) = k1E (1)∗ · E (1) > 0,

(A.7a)

E (2)∗ · ∇ × E (2) = −k2E (2)∗ · E (2) < 0.

(A.7b)

The signs of (A.6) are used below to define the characteristic modal handedness of the fields E (n ) . C , then the sign of the product C ·∇×C C characterizes a local Let C be a real vector field collinear with ∇×C geometrical structure of C that, in view of the definition of ∇×, can be described as the characteristic handedness. Consequently, this sign is taken to define the right or left handedness of C . For the modal fields E (n ) =Re E (n ) +i Im E (n ) obeying (2.2a) and (2.2b), where kn are real, the handedness of Re E (1) and Im E (1) is the same and opposite to that of Re E (2) and Im E (2) . Hence, the sign of (A.6), for n=1, is taken to define the characteristic handedness of E(1) , and for n=2, the handedness of E (2) . For circularly polarized fields (3.1a)–(3.1d), the handedness of polarization is determined by the equations (2.2a) and (2.2b), respectively, and consequently, can be defined by the signs of (A.6). Since, according to the generally adopted convention from optics, the polarization of the waves (3.1a) and (3.1c) of the k1 -mode is left-handed and right-handed for the waves (3.1b) and (3.1d) of the k2 -mode, we adopt, to avoid confusion, the same terminology for the characteristic modal handedness.Thus, we shall refer to the characteristic handedness of the fields of the k1 -mode as left-handed and to that of the fields of the k2 -mode as right-handed. For the modal physical fields E (n) e−iωt ) = (ReE E (n) ) cos ωt + (Im E (n) )sin ωt, E (n) (rr , t) = Re(E

n = 1, 2,

by virtue of (2.2a) and (2.2b), we get for every point in the region considered in the splitting theorem E (1) (rr , t) · ∇ × E (1) (rr , t) > 0,

(A.8a)

E (2) (rr , t) · ∇ × E (2) (rr , t) < 0,

(A.8b)

and thus, the adopted definition of the characteristic handedness is also valid for these basic fields.

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A.3. Representation theorem By exhibiting the specific structure of the modal fields following from the modal equations (2.2a) and (2.2b), we show how to generate modal fields from scalar functions satisfying appropriate Helmholtz equations. Thereby we find the forms of the operators K 1 , K 2 of Section 2. Let us denote (n) E (n) = E t + Eτ(n)τ ,

Eτ(n) = E (n) · τ ,

∇t = ∇ − τ

∂ , ∂τ

n = 1, 2,

and τ is the Cartesian coordinate directed along τ . Let us consider the k1 -mode, then from (2.2a) we get (1)

Et ∇ · (E

(1)

× τ ) = k1 Et ,

(A.9a)

(1)

∂Eτ , ∂τ

(A.9b)

(∇ 2 + k12 )Eτ(1) = 0.

(A.9c)

(1)

∇ · Et

=−

and from (2.6a)

It is the integration of the system (A.9a)–(A.9c) in terms of one scalar function that underlies the representation theorem. As follows by inspection, from an arbitrary function ψ (1) of class C2 we can generate a field E(1) that obeys (A.9a) and (A.9b), and is given by E (1) t = ∇t

∂ψ (1) − k1τ × ∇t ψ (1) , ∂τ

Eτ(1) = −∇t2 ψ (1) .

(A.10a) (A.10b)

In order to fulfil also (A.9c) we simply demand that (∇ 2 + k12 )ψ (1) = 0.

(A.11a)

Eqs. (A.10a) and (A.10b) can be cast into the form E (1) = K 1 ψ (1)

(A.12a)

E (1) we easily verify that this field obeys (1.1a), (1.1b) and (2.2a). H (1) =E and by assuming iZH Similarly for the k2 -mode H (2) = E (2) = K 2 ψ (2) , −iZH

(A.12b)

(∇ 2 + k22 )ψ (2) = 0.

(A.11b)

The formulas (A.12a) and (A.12b) show that in an arbitrary region of a chiral medium a wide class of electromagnetic fields can be generated via these formulas from all solutions to the Helmholtz equations (A.11a) and (A.11b) in this region. The representation problem is concerned with the question whether this class exhausts the set of all solutions to (1.1a) and (1.1b). In the local case the positive answer is given by the theorem of Section 2. Globally some restrictions on the region are necessary [19]. For a proof of the theorem it is necessary to show that for an arbitrary modal field E (1) it is possible to solve simultaneously both (A.10a) and (A.10b) and satisfy (A.11a) and similarly for the k2 -mode.

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A.4. Modal ghost potentials Let us consider the k1 -mode and the Cartesian coordinate system ξ , η, τ introduced in Section 2. A modal ghost potential u˜ is defined by ˜ 0 = K 1 u,

(A.13a)

(∇ 2 + k12 )u˜ = 0.

(A.13b)

Hence ∇t2 u˜ = 0,   2 ∂ 2 + k ˜ = 0. 1 u ∂τ 2

(A.14a) (A.14b)

The general solution to (A.14a) and (A.14b) can be written as u˜ = [f1 (ξ + iη) + h2 (ξ − iη)]e−ik1 τ + [h1 (ξ + iη) + f2 (ξ − iη)]eik1 τ ,

(A.15)

where fn (·), hn (·), n=1, 2, are analytic functions of a complex argument (·). In order to satisfy the equation (A.13a) by (A.15), we must assume hn (·)=0 and then we arrive at (2.8a) and (2.8b). In the same way we obtain (2.9a) and (2.9b) for the k2 -mode. A.5. Plane wave solutions Finding plane wave solutions to (1.1a) and (1.1b) may be simplified by making use of the modal splitting. Let us assume the plane waves to propagate in the direction specified by a unit vector s and let s =ss ·rr . Then the modal equations (2.2a) and (2.2b) can be written as s×

E (1) dE − k1E (1) = 0, ds

(A.16a)

s×

E (2) dE + k2E (2) = 0. ds

(A.16b)

E (2) =0, consequently the plane waves are transverse with respect to E (1) =0, s ·E From (A.16a) and (A.16b) we get s ·E the direction of propagation. Let us now assume s =ss 1 for (A.16a) and s =ss 2 for (A.16b), then the relevant solutions are given by (3.1a) and (3.1b). Formulas (3.1a) and (3.1b) show that the plane waves are circularly polarized and their left- or right-hand polarization coincides with their modal handedness defined by (A.6).The phenomenon of chirality is now clearly exhibited by the property of the medium to distinguish electromagnetically between the right and left orientations by propagating the plane waves of opposite circular polarizations with distinct velocities. References [1] [2] [3] [4] [5] [6]

A. Sommerfeld, Mathematische Theorie der Diffraction, Math. Ann. 47 (1896) 317–374. Ll.G. Chambers, Propagation in a gyrational medium, Quart. J. Mech. Appl. Math. 9 (1956) 360–370. F.I. Fedorov, On the theory of optical activity in crystals, Opt. Spectrosc. 6 (1959) 49–53. S. Prze´zdziecki, Diffraction by a half-plane in a chiral medium (normal incidence), Acta Phys. Polon. A 83 (1993) 739–750. B. Noble, Methods Based on the Wiener–Hopf Technique, Pergamon Press, London, 1958. W.S. Ament, Application of a Wiener–Hopf technique to certain diffraction problems, US Naval Res. Lab. Rep. No 4334, Washington, 1954.

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