EXACT solutions to boundary-value problems for a

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Exact Penetration, Radiation and Scattering for a Slotted Semielliptical Channel Filled with

Isorefractive Material

Piergiorgio L. E. Uslenghi, Fellow, IEEE

Abstract— A semielliptical channel flush-mounted under a metal plane and slotted along the interfocal distance of its cross-section is considered. The channel is filled with a material isorefractive to the medium that occupies the half- space above the metal plane. The boundary-value problem is solved exactly in terms of Mathieu functions, when the primary source is a plane wave with arbitrary direction of incidence and polarization, or an electric or magnetic line source parallel to the channel and located either above the channel or inside it.

Keywords— Electromagnetic radiation, electromagnetic scattering, complex media, cavities.

I. Introduction

E

XACT solutions to boundary-value problems for a complicated structure containing a cavity, sharp metallic edges, and two different penetrable media are presented. The geometry consists of a metallic channel of semielliptical cross-section slotted along the interfocal distance and flush-mounted under a metallic infinite plane.

The channel is filled with a material that is isorefractive to the medium occupying the half-space above the channel (e.g., air); in particular, the material filling the channel may be identical to that filling the half-space above it.

The solutions are found for a primary field that may be a plane wave or a line source. The plane wave has arbitrary polarization and direction of incidence; the electric or magnetic line source is parallel to the channel. While the plane wave is incident on the channel from the half-space above the metal plane, the line may be located inside the channel, in the half space above it, or on the interfocal slot separating the channel cavity from the half space. The solutions are determined by expanding the primary and secondary fields in infinite series of eigenfunctions involving the products of radial and angular Mathieu functions (in the notation of Stratton [1], [2]), and by imposing the boundary and radiation conditions. This process leads to the analytical determination of all modal coefficients in the eigenfunction expansions. Thus, the obtained solutions are exact and constitute new canonical solutions of boundary- value problems. A solution for E-polarized plane wave incidence and the same medium inside and outside the channel was given previously by the author [3], who also recently presented some of the results obtained below [4].

The author is with the Department of Electrical and Computer Engineering, University of Illinois at Chicago, 851 South Morgan St., Chicago, Illinois 60607, USA. E-mail: [email protected]. This research was supported by the U.S. Department of Defense under MURI grant F49620-01-1-0436.

The geometry of the problem is presented in Section II, and the solutions for plane wave incidence and line sources are obtained in Sections III and IV, respectively. Some considerations on numerical results, low and high frequency approximations are developed in section V, and some useful properties of the Mathieu functions are collected in Ap- pendix. The analysis is performed in the frequency do- main, and the time-dependence factor exp(+jωt) is omit- ted throughout.

The studies carried out in this paper are important for at least two reasons. First, they enrich the dictio- nary of known canonical solutions for boundary-value problems. Second, the problems whose exact solutions are given herein are of sufficient complexity to constitute challenging validation cases for general computer codes developed for studying penetration, radiation and scattering for complicated structures.

It should be noted that the method applied herein does not work for the more general case in which the media inside and outside the channel are not isorefractive: one- on-one mode matching cannot occur because of the different wavenumbers inside and outside the channel, and an explicit analytical determination of modal coefficients is precluded.

II. Geometry of the problem

With reference to Cartesian coordinates (x, y, z), the physical structure is invariant in the z direction and is symmetric with respect to the plane x = 0. A cross- section of the structure in a plane z = constant is shown in Fig. 1. The metallic plane y = 0 has the strip (|x| ≤ d/2, y = 0, −∞ < z < ∞) cut away; thus, the strip AF is an aperture in the plane BG that connects that half-space y > 0 above the plane to the semielliptical chan- nel CDE below it. The points A and F in Fig. 1 represent the positions of the edges of the infinitesimally thin metallic baffles AC and FE. The segment CE is the major axis of the semiellipse CDE, and A and F are the foci, whose interfocal distance AF=d is the width of the aperture. The half-space y > 0 is filled with a linear, homogeneous and isotropic material of permittivity ε1 and permeability µ1, whereas the semielliptical channel is filled with a linear, ho- mogeneous and isotropic materials of permittivity ε2 and permeability µ2. The two materials are in contact along the surface of the strip AF, and are isorefractive to each other:

ε1µ1= ε2µ2 (1)

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Foci

Semiellipse ξ=ξ₁ v=π/2

v=0 y

v=π

A v=π

B F G

D

C E v=2π

Medium 2 Medium 1

0

d/2 d/2

x

(u=u1)

Fig. 1. Geometry of the problem.

i.e., they have the same propagation constant k = ω√

εhµh, (h = 1, 2) (2) but, in general, different intrinsic impedances

Zh=p

µh/εh, (h = 1, 2). (3) It is expedient to introduce the elliptic coordinates (u, v, z) related to the Cartesian coordinates (x, y, z) by

x = d

2cosh u cos v, y = d

2sinh u sin v, z = z (4) where 0 ≤ u < ∞ and 0 ≤ v ≤ 2π. For simplicity, we also use

ξ = cosh u, η = cos v (5) and the dimensionless parameter

c = kd

2 . (6)

The surface of the slot is u = 0 (or ξ = 1); the half-planes AB and FG are given by v = π and v = 0, respectively;

the lower surfaces of the baffles CA and FE correspond to v = π and v = 2π, respectively; the metallic surface CDE of the channel is the semiellipse u = u1(or ξ = ξ1). The limit case u1 = 0 (or ξ1 = 1) corresponds to an infinite metal plane with no channel, whereas the limit case u1= ∞ (or ξ1= ∞) corresponds to a slotted metallic screen separating two isorefractive half-spaces.

III. Plane wave incidence

Only the case of an incident wave propagating in a direction perpendicular to the z axis is considered. Results for oblique incidence with respect to the axis of the two- dimensional structure can be obtained from the results at normal incidence [5] (this is always true if the structure is composed of perfect conductors and isorefractive materials, but not true if penetrable materials with different refractive indexes are present).

A. E-polarization

Consider an incident plane wave whose direction of prop- agation forms the angle ϕ0with the negative x axis and the

angle π/2 − ϕ0 with the negative y axis (0 < ϕ0 ≤ π/2), and whose electric field is given by:

Eⁱ= ˆzE_1zⁱ = ˆz exp{jk(x cos ϕ0+ y sin ϕ0)}. (7) The incident electric field may be expanded in a series of elliptic-cylinder wave functions:

E_1zⁱ =√ 8π

X∞ m=0

j^m

· 1 Nm^(e)

Re⁽¹⁾_m (c, ξ)Sem(c, η)Sem(c, cos ϕ0)

+ 1

Nm^(o)

Ro⁽¹⁾_m (c, ξ)Som(c, η)Som(c, cos ϕ0)

¸ (8)

where Re⁽¹⁾_m and Ro⁽¹⁾_m are even and odd radial Mathieu functions of the first kind, Sem and Som are even and odd angular functions, and Nm^(e),(o) are normalization coefficients [1], [2].

The total electric field in the half-space y ≥ 0 may be written as the sum of three terms:

E1z= E_1zⁱ + E_1z^r + E_1z^d , (9) where E_1z^r is the field that would be reflected by the metal plane y = 0 if there were no slot (i.e., ξ1 = 1), whereas E_1z^d is the diffracted field introduced by the presence of the cavity-backed slot and must satisfy the two-dimensional radiation condition. By symmetry considerations, it is easily seen that

E_1z^r = −√ 8π

X∞ m=0

j^m

· 1 Nm^(e)

Re⁽¹⁾_m (c, ξ)Sem(c, η)Sem(c, cos ϕ0)

− 1 Nm^(o)

Ro⁽¹⁾_m (c, ξ)Som(c, η)Som(c, cos ϕ0)

¸

, (10)

so that

E_1zⁱ + E_1z^r = 2√

8π X∞ m=0

j^m Nm^(o)

Ro⁽¹⁾_m (c, ξ)Som(c, η)Som(c, cos ϕ0) (11)

contains only odd eigenfunctions. The boundary condition of zero total electric field at v = 0 and v = π requires that also the diffracted field contain only odd eigenfunctions:

E_1z^d =

√8π X∞ m=0

j^m am

Nm^(o)

Ro⁽⁴⁾_m (c, ξ)Som(c, η)Som(c, cos ϕ0), (12)

where am are expansion (or modal) coefficients, and the radial Mathieu functions of the fourth kind ensure that the radiation condition is satisfied.

The total electric field inside the cavity (1 ≤ ξ ≤ ξ1, π ≤ v ≤ 2π, |z| < ∞) is parallel to the z axis and given by:

E2z=√ 8π

X∞ m=0

j^m Nm^(o)

cm

h

Ro⁽¹⁾_m (c, ξ)−

Ro⁽¹⁾_m (c, ξ1)

Ro⁽⁴⁾_m (c, ξ1)Ro⁽⁴⁾_m (c, ξ)

#

Som(c, η)Som(c, cos ϕ0), (13)

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where cm are modal coefficients and the boundary condi- tions of zero electric field at ξ = ξ1, v = π and v = 2π are satisfied.

The unknown coefficients am and cm are determined by imposing the continuity of the total tangential electric and magnetic fields across the interface ξ = 1, yielding:

am= Ro⁽¹⁾_m (c, ξ1) Ro⁽⁴⁾_m (c, ξ1)cm

= 2

ζRo⁽⁴⁾_m (c, ξ1)

Ro⁽¹⁾_m (c, ξ1)− (1 + ζ)Ro⁽⁴⁾_m ⁰(c, 1) Ro⁽¹⁾_m ⁰(c, 1)

, (14)

where

ζ = Z1

Z2

(15) and the prime means the derivative with respect to ξ.

The normalized bistatic radar cross section due to the slot is:

σE(ϕ) λ =8π

¯¯

¯ X∞ m=0

(−1)^m Nm^(o)

amSom(c, cos ϕ)Som(c, cos ϕ0)

¯¯

¯

2

,

(0 < ϕ < π) (16)

where λ is the wavelength. The normalized total scattering cross section of the slot is, when c is real:

σT,E

λ = 4 X∞ m=0

Re(−am) Nm^(o)

[Som(c, cos ϕ0)]², (17)

where, in the particular case of Z1and Z2real:

Re(−am) =

2



1 +

"

(1 + ζ)Ro⁽²⁾_m⁰(c, 1)

Ro⁽¹⁾_m⁰(c, 1) − ζRo⁽²⁾_m (c, ξ1) Ro⁽¹⁾_m (c, ξ1)

#₂





−1

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with Ro⁽²⁾_m as the odd radial Mathieu function of the second kind.

At the metal surface, the current density is (see Fig. 1):

J = −Hξz, on AC and FGˆ

= Hξz, on AB and FEˆ

= −Hvz, on CDE.ˆ (19) The magnetic field is given by:

H = j

cZp ξ²− η²

µ∂Ez

∂v u −ˆ ∂Ez

∂u vˆ

¶

, (20)

hence, in particular:

H1ξ|_v=0,π = j cZ1

r 8π ξ²− 1

X∞ m=0

(±j)^m Nm^(o)

h

2Ro⁽¹⁾_m (c, ξ)+

amRo⁽⁴⁾_m (c, ξ) i

Som(c, cos ϕ0), (21)

H2ξ|_v=π,2π= j cZ2

r 8π ξ²− 1

X∞ m=0

(∓j)^m Nm^(o)

cm

h

Ro⁽¹⁾_m (c, ξ)−

Ro⁽¹⁾_m (c, ξ1)

Ro⁽⁴⁾_m (c, ξ1)Ro⁽⁴⁾_m (c, ξ)

#

Som(c, cos ϕ0), (22)

H2v|_ξ=ξ₁ = 1 cZ2

s 8π ξ₁²− η² X∞

m=0

j^m Nm^(o)

cm

Ro⁽⁴⁾_m (c, ξ1)Som(c, η)Som(c, cos ϕ0). (23) The limit case of a slot in a screen separating two half- spaces of intrinsic impedances Z1 and Z2 is obtained from the above formulas by letting ξ1→ ∞ while Im c < 0. In particular,

E2z|_ξ

1→∞,Im c<0∼2√ 8π 1 + ζ

X∞ m=0

j^m Nm^(o)

Ro⁽¹⁾_m⁰(c, 1) Ro⁽⁴⁾_m⁰(c, 1)· Ro⁽⁴⁾_m (c, ξ)Som(c, η)Som(c, cos ϕ0), (π ≤ v ≤ 2π) (24) and E_1z^d is given by (24) multiplied by minus one, obviously for 0 ≤ v ≤ π.

B. H-polarization

Since the analysis is similar to that for E-polarization, only the results are given. For an incident magnetic field

Hⁱ = ˆzH_1zⁱ = ˆz exp{jk(x cos ϕ₀+ y sin ϕ₀)}, (25) the total magnetic field in the half-space y ≥ 0 is:

H1z= H_1zⁱ + H_1z^r + H_1z^d

=√ 8π

X∞ m=0

j^m Nm^(e)

h

2Re⁽¹⁾_m (c, ξ) + bmRe⁽⁴⁾_m (c, ξ) i

·

Sem(c, η)Sem(c, cos ϕ0) (26) and inside the channel is:

H2z=√ 8π

X∞ m=0

j^m Nm^(e)

dm

"

Re⁽¹⁾_m (c, ξ) −Re⁽¹⁾_m⁰(c, ξ1)

Re⁽⁴⁾_m⁰(c, ξ1)Re⁽⁴⁾_m (c, ξ)

#

·

Sem(c, η)Sem(c, cos ϕ0), (27) where the modal coefficients are:

b_m= Re⁽¹⁾_m ⁰(c, ξ1) ζRe⁽⁴⁾_m ⁰(c, ξ1)d_m

= 2

ζRe⁽⁴⁾_m⁰(c, ξ1)

Re⁽¹⁾_m⁰(c, ξ1)− (1 + ζ)Re⁽⁴⁾_m (c, 1) Re⁽¹⁾_m (c, 1)

. (28)

The normalized bistatic radar cross section of the slot is:

σH(ϕ)

λ =8π

¯¯

¯ X∞ m=0

(−1)^m Nm^(e)

bmSem(c, cos ϕ)Sem(c, cos ϕ0)

¯¯

¯

2

,

(0 < ϕ < π) (29)

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and the normalized total scattering cross section for c real is:

σT,H

λ = 4 X∞ m=0

Re(−bm) Nm^(e)

[Sem(c, cos ϕ0)]²; (30) in the case of Z1 and Z2 real:

Re(−bm) =

2



1 +

"

(1 + ζ)Re⁽²⁾_m (c, 1)

Re⁽¹⁾_m (c, 1)− ζRe⁽²⁾_m ⁰(c, ξ1) Re⁽¹⁾_m ⁰(c, ξ₁)

#₂





−1

, (31)

where Re⁽²⁾_m is the even radial Mathieu function of the second kind.

The electric field is given by:

E = jZ

cp ξ²− η²

µ

−∂Hz

∂v u +ˆ ∂Hz

∂u vˆ

¶

. (32) At the metal surface, the current density is (see Fig. 1):

J = Hzu, on AC and FGˆ

= −Hzu, on AB and FEˆ

= Hzˆv, on CDE. (33)

It is found that:

H1z|_v=0,π=√ 8π

X∞ m=0

(±j)^m Nm^(e)

h

2Re⁽¹⁾_m (c, ξ)+

bmRe⁽⁴⁾_m (c, ξ)i

Sem(c, cos ϕ0), (34)

H2z|_v=π,2π =√ 8π

X∞ m=0

(∓j)^m Nm^(e)

dm

h

Re⁽¹⁾_m (c, ξ)−

Re⁽¹⁾_m⁰(c, ξ1)

Re⁽⁴⁾_m⁰(c, ξ₁)Re⁽⁴⁾_m (c, ξ)

#

Sem(c, cos ϕ0), (35)

H2z|_ξ=ξ

1 = −j s

8π ξ₁²− 1· X∞

m=0

j^mdm

Nm^(e)Re⁽⁴⁾_m⁰(c, ξ1)Sem(c, η)Sem(c, cos ϕ0). (36) In the limit case of a slot in a screen separating two-half- spaces,

H_1z^d¯

¯ξ1→∞,Im c<0∼ −2√ 8π 1 + ζ

X∞ m=0

j^m Nm^(e)

Re⁽¹⁾_m (c, 1) Re⁽⁴⁾_m (c, 1)· Re⁽⁴⁾_m (c, ξ)Sem(c, η)Sem(c, cos ϕ0), (37) whereas H2z is given by (37) multiplied by −ζ.

IV. Line source incidence A. E-polarization

Consider an electric line source parallel to the z axis and located at (x0, y0) ≡ (ξ0, η0), whose primary electric field is

Eⁱ = ˆzEⁱ_1z= ˆzH₀⁽²⁾(kR) (38)

where

R =p

(x − x0)²+ (y − y0)² (39) is the distance of the observation point from the source.

The primary field may be rewritten as the series of Mathieu functions [2]:

E_1zⁱ = H₀⁽²⁾(kR) = 4

X∞ m=0

· 1 Nm^(e)

Re⁽¹⁾_m (c, ξ<)Re⁽⁴⁾_m (c, ξ>)Sem(c, η0)Sem(c, η)+

1 Nm^(o)

Ro⁽¹⁾_m (c, ξ<)Ro⁽⁴⁾_m (c, ξ>)Som(c, η0)Som(c, η)

¸

(40)

where ξ< (ξ>) is the smaller (larger) between ξ and ξ0. Let us first examine the case of a line source in the half- space above the channel (i.e., 0 < v0< π). The total field in the half-space y ≥ 0 is:

E1z= E_1zⁱ + E_1z^r + E_1z^d , (41) where E_1z^r is the field reflected by the infinite metal plane y = 0 when no channel is present, and E_z^d is the diffracted field due to the presence of the cavity-backed slot. By considering the image of the line source into the ground plane, it is found that

E_1zⁱ + E_1z^r = H₀⁽²⁾(kR) − H₀⁽²⁾(k ˜R) (42) where

R =˜ p

(x − x0)²+ (y + y0)² (43) is the distance between the observation point and the source’s image; the field (42) may be rewritten as

E_1zⁱ + E_1z^r = 8

X∞ m=0

1 Nm^(o)

Ro⁽¹⁾_m (c, ξ<)Ro⁽⁴⁾_m (c, ξ>)Som(c, η0)Som(c, η).

(44) The diffracted field E^d_1zin y ≥ 0 and the total field E2zin- side the cavity are given by, on consideration of the boundary conditions:

E_1z^d = 4

X∞ m=0

am

Nm^(o)

Ro⁽⁴⁾_m (c, ξ0)Ro⁽⁴⁾_m (c, ξ)Som(c, η0)Som(c, η), (45)

E2z= 4 X∞ m=0

cm

Nm^(o)

Ro⁽⁴⁾_m (c, ξ0) h

Ro⁽¹⁾_m (c, ξ)−

Ro⁽¹⁾_m (c, ξ1)

Ro⁽⁴⁾_m (c, ξ1)Ro⁽⁴⁾_m (c, ξ)

#

Som(c, η0)Som(c, η), (46)

with amand cmgiven by (14).

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The total far field in the half-space y > 0 is:

E1z|ξ→∞,Im c<0∼ e^−jkρ

√kρ4e^j^π⁴ X∞ m=0

j^m Nm^(o)

h

2Ro⁽¹⁾_m (c, ξ0)+

amRo⁽⁴⁾_m (c, ξ0)i

Som(c, η0)Som(c, cos ϕ). (47) In the limit when the line source is located at the interface between the two media:

E1z|_ξ₀₌₁= E_1z^d ¯

¯ξ0=1=

4 X∞ m=0

am

Nm^(o)

Ro⁽⁴⁾_m (c, 1)Ro⁽⁴⁾_m (c, ξ)Som(c, η0)Som(c, η). (48)

The current density on the metal surfaces of the structure is still given by (19), where now:

H1ξ|_v=0,π= 4j cZ1

pξ²− 1 X∞ m=0

(±1)^m Nm^(o)

h

2Ro⁽¹⁾_m (c, ξ<) +

amRo⁽⁴⁾_m (c, ξ<) i

Ro⁽⁴⁾_m (c, ξ>)Som(c, η0), (49)

H2ξ|_v=π,2π = −4j cZ2

pξ²− 1 X∞ m=0

(∓1)^m Nm^(o)

am

h

Ro⁽⁴⁾_m (c, ξ) −

Ro⁽⁴⁾_m (c, ξ1)

Ro⁽¹⁾_m (c, ξ₁)Ro⁽¹⁾_m (c, ξ)

#

Ro⁽⁴⁾_m (c, ξ0)Som(c, η0). (50)

H2v|_ξ=ξ₁ = 4 cZ2

pξ₁²− η²· X∞

m=0

am

Nm^(o)

Ro⁽⁴⁾_m (c, ξ0)

Ro⁽¹⁾_m (c, ξ1)Som(c, η0)Som(c, η). (51) The limit case of a slot in a screen separating two isorefractive half-spaces yields:

E2z|_ξ

1→∞,Im c<0∼ 8 1 + ζ

X∞ m=0

Ro⁽¹⁾_m⁰(c, 1) Nm^(o)Ro⁽⁴⁾_m⁰(c, 1)· Ro⁽⁴⁾_m (c, ξ0)Ro⁽⁴⁾_m (c, ξ)Som(c, η0)Som(c, η),

(π ≤ v ≤ 2π) (52)

and E_1z^d is given by the same expression multiplied by mi- nus one, obviously with 0 ≤ v ≤ π.

Let us now consider the case of a line source located inside the channel (u0 < u1, π < v0 < 2π). The total electric field inside the channel is

E2z= E_2zⁱ + E_2z^r + E^s_2z (53) where E_2zⁱ is the incident field given by (40), E_2z^r is the reflected field when no slot is present and the semielliptical channel is recessed to infinity (ξ1 → ∞), and E_2z^s is the perturbation that accounts for the slot at ξ = 1 and the metal wall at ξ = ξ1. Consequently, Eⁱ_2z+ E_2z^r is still given by (44), whereas the infinite series for E_2z^s contains

a linear combination of Ro⁽¹⁾_m (c, ξ) and Ro⁽⁴⁾_m (c, ξ). The infinite series for the total field E1zin y > 0 must satisfy the radiation condition, hence contains only Ro⁽⁴⁾_m (c, ξ). The fields are:

E2z= 4 X∞ m=0

1 Nm^(o)

h

2Ro⁽¹⁾_m (c, ξ<)Ro⁽⁴⁾_m (c, ξ>)+

˜amRo⁽¹⁾_m (c, ξ) + ˜cmRo⁽⁴⁾_m (c, ξ)i

Som(c, η0)Som(c, η), (54)

E1z= −4 X∞ m=0

˜cm

Nm^(o)

Ro⁽⁴⁾_m (c, ξ)Som(c, η0)Som(c, η), (55)

where, by imposing the boundary conditions:

˜am= −2Ro⁽⁴⁾_m (c, ξ1)·

(1 + ζ⁻¹)Ro⁽¹⁾_m (c, ξ0)Ro⁽⁴⁾_m⁰(c, 1) − Ro⁽¹⁾_m ⁰(c, 1)Ro⁽⁴⁾_m (c, ξ0) (1 + ζ⁻¹)Ro⁽¹⁾_m (c, ξ1)Ro⁽⁴⁾_m⁰(c, 1) − Ro⁽¹⁾_m ⁰(c, 1)Ro⁽⁴⁾_m (c, ξ1),

(56)

˜cm= −2Ro⁽¹⁾_m ⁰(c, 1)·

Ro⁽¹⁾_m (c, ξ1)Ro⁽⁴⁾_m (c, ξ0) − Ro⁽¹⁾_m (c, ξ0)Ro⁽⁴⁾_m (c, ξ1) (1 + ζ⁻¹)Ro⁽¹⁾_m (c, ξ1)Ro⁽⁴⁾_m⁰(c, 1) − Ro⁽¹⁾_m ⁰(c, 1)Ro⁽⁴⁾_m (c, ξ1).

(57) The far field in the half-space y > 0 is:

E1z|ξ→∞,Im c<0∼

−e^−jkρ

√kρ 4e^j^π⁴ X∞ m=0

j^m Nm^(o)

˜cmSom(c, η0)Som(c, cos ϕ). (58)

Using (19), the current density on the metal surfaces is found from:

H1ξ|_v=0,π =

−4j cZ1

pξ²− 1 X∞ m=0

(±1)^m Nm^(o)

˜cmRo⁽⁴⁾_m (c, ξ)Som(c, η0), (59)

H2ξ|_v=π,2π = 4j cZ2

pξ²− 1 X∞ m=0

(∓1)^m Nm^(o)

h

2Ro⁽¹⁾_m (c, ξ<)Ro⁽⁴⁾_m (c, ξ>)

+ ˜amRo⁽¹⁾_m (c, ξ) + ˜cmRo⁽⁴⁾_m (c, ξ) i

Som(c, η0), (60)

H2v|_ξ=ξ

1 =

−4j cZ2

s ξ²₁− 1 ξ₁²− η²

X∞ m=0

1 Nm^(o)

n

˜amRo⁽¹⁾_m ⁰(c, ξ1) + [˜cm+

2Ro⁽¹⁾_m (c, ξ0)i

Ro⁽⁴⁾_m ⁰(c, ξ1)o

Som(c, η0)Som(c, η). (61) In the limit case of a slot in an infinite screen separating two isorefractive half-spaces, E1z is given by (52) with ζ replaced by ζ⁻¹ and 0 ≤ v ≤ π; this is also the expression for E_2z^d , aside from a change in sign, when π ≤ v ≤ 2π.

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B. H-polarization

The derivation proceeds similarly to the case of E- polarization; thus, only the main results are given. For a magnetic line source

H_zⁱ = ˆzH₀⁽²⁾(kR) (62) located at (ξ0, η0) in y ≥ 0, the total fields in the half-space y ≥ 0 and inside the channel are, respectively:

H1z=4 X∞ m=0

1 Nm^(e)

h

2Re⁽¹⁾_m (c, ξ<)Re⁽⁴⁾_m (c, ξ>)+

bmRe⁽⁴⁾_m (c, ξ0)Re⁽⁴⁾_m (c, ξ)i

Sem(c, η0)Sem(c, η), (63)

H2z=4 X∞ m=0

dm

Nm^(e)

Re⁽⁴⁾_m (c, ξ0) h

Re⁽¹⁾_m (c, ξ)−

Re⁽¹⁾_m ⁰(c, ξ1)

Re⁽⁴⁾_m ⁰(c, ξ1)Re⁽⁴⁾_m (c, ξ)

#

where bmand dmare given by (28).

The total far field in the half-space y > 0 is:

H_1z|ξ→∞,Im c<0∼ e^−jkρ

√kρ4e^j^π⁴ X∞ m=0

j^m Nm^(e)

h

2Re⁽¹⁾_m (c, ξ0)+

bmRe⁽⁴⁾_m (c, ξ0) i

Sem(c, η0)Sem(c, cos ϕ). (65) The current density on the metal surfaces is given by (33) with:

H1z|_v=0,π= 4 X∞ m=0

(±1)^m Nm^(e)

h

2Re⁽¹⁾_m (c, ξ<)+

bmRe⁽⁴⁾_m (c, ξ<) i

Re⁽⁴⁾_m (c, ξ>)Sem(c, η0), (66)

H2z|_v=π,2π = 4 X∞ m=0

(∓1)^m Nm^(e)

dm

h

Re⁽¹⁾_m (c, ξ)−

Re⁽¹⁾_m⁰(c, ξ1)

Re⁽⁴⁾_m⁰(c, ξ1)Re⁽⁴⁾_m (c, ξ)

#

Re⁽⁴⁾_m (c, ξ₀)Se_m(c, η₀), (67)

H2z|_ξ=ξ₁ = p−4j

ξ₁²− 1 X∞ m=0

dm

Nm^(e)

Re⁽⁴⁾_m (c, ξ0)

Re⁽⁴⁾_m ⁰(c, ξ1)Sem(c, η0)Sem(c, η). (68) For a line source inside the channel (u0 < u1, π < v0 <

2π), the total magnetic field inside the channel is:

H2z= H_2zⁱ + H_2z^r + H_2z^s = 4

X∞ m=0

1 Nm^(e)

h

2Re⁽¹⁾_m (c, ξ<)Re⁽⁴⁾_m (c, ξ>) + ˜bmRe⁽¹⁾_m (c, ξ)−

d˜mRe⁽⁴⁾_m (c, ξ) i

and in the half-space y > 0 is:

H1z=4 ζ

X∞ m=0

d˜m

Nm^(e)

Re⁽⁴⁾_m (c, ξ)Sem(c, η0)Sem(c, η), (70)

where

˜bm= −2Re⁽⁴⁾_m⁰(c, ξ1)·

(1 + ζ⁻¹)Re⁽¹⁾_m (c, ξ0)Re⁽⁴⁾_m (c, 1) − Re⁽¹⁾_m (c, 1)Re⁽⁴⁾_m (c, ξ0) (1 + ζ⁻¹)Re⁽¹⁾_m⁰(c, ξ1)Re⁽⁴⁾_m (c, 1) − Re⁽¹⁾_m (c, 1)Re⁽⁴⁾_m ⁰(c, ξ1),

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d˜m= −2Re⁽¹⁾_m (c, 1)·

Re⁽¹⁾_m (c, ξ₀)Re⁽⁴⁾_m⁰(c, ξ₁) − Re⁽¹⁾_m⁰(c, ξ₁)Re⁽⁴⁾_m (c, ξ₀) (1 + ζ⁻¹)Re⁽¹⁾_m⁰(c, ξ1)Re⁽⁴⁾_m (c, 1) − Re⁽¹⁾_m (c, 1)Re⁽⁴⁾_m ⁰(c, ξ1).

(72) The far field in the half-space y > 0 is:

H1z|ξ→∞,Im c<0∼ e^−jkρ

√kρ4e^j^π⁴ ζ

X∞ m=0

j^m Nm^(e)

d˜mSem(c, η0)Sem(c, cos ϕ). (73)

The current density on the metal surfaces is given by (33) with:

H1z|_v=0,π =4 ζ

X∞ m=0

(±1)^m Nm^(e)

d˜mRe⁽⁴⁾_m (c, ξ)Sem(c, η0), (74)

H2z|_v=π,2π = 4 X∞ m=0

(∓1)^m Nm^(e)

h

2Re⁽¹⁾_m (c, ξ<)Re⁽⁴⁾_m (c, ξ>)+

˜bmRe⁽¹⁾_m (c, ξ) − ˜dmRe⁽⁴⁾_m (c, ξ) i

Sem(c, η0), (75)

H_2z|_ξ=ξ

1 = 4 X∞ m=0

1 Nm^(e)

nh

2Re⁽¹⁾_m (c, ξ₀) − ˜d_mi

Re⁽⁴⁾_m (c, ξ₁)+

˜bmRe⁽¹⁾_m (c, ξ1) o

Sem(c, η0)Sem(c, η). (76) V. Discussion and conclusion

The theoretical results obtained in the previous sections should be computed and compared to numerical results obtained directly, e.g. from the solution of an integral equation. Preliminary calculations based on the formulas for plane wave incidence given in Section III have been con- ducted [6], and integral equation solutions have also been shown [7]; a comprehensive numerical work in which the eigenfunction solutions for both plane wave incidence and line sources are compared to integral equation solutions has been recently completed [8],[9].

It is to be expected that the infinite series of Mathieu functions representing exact solutions are numerically well

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behaved, with the possible exception of the neighborhood of the sharp edges of the slot, where the convergence of some series may be poor. This is not surprising, since some field components are singular at the edge. This situation is akin to that occurring in the scattering by a metallic wedge, where the exact solution is expressed as an infinite series obtained by MacDonald [10].

Low-frequency expansions of the exact results obtained herein could be compared with predictions based on quasi- static techniques applied to cavities. Specifically, an effort is presently under way to obtain low-frequency (c ¿ 1) expansions for the Mathieu functions in the Stratton-Chu normalization, in order to compare low-frequency approximations of the exact solutions in this paper to the general formulas for low-frequency solutions of two-dimensional channels obtained by Hansen and Yaghjian [11]. When available, those results will be reported separately.

At high frequencies, a GTD approach would entail that rays diffracted at either of the slot edges would, after re- flection at the boundary of the channel, converge on the other edge as a concave cylindrical wave imploding on the edge itself. If uniform asymptotic formulas were available for Mathieu functions, it might be possible to obtain the diffraction coefficient for a cylindrical wave converging on the edge of a metallic half-plane.

Exact results can also be obtained for an arbitrarily ori- ented and located Hertzian dipole, by extending the tech- nique used for the isorefractive elliptic cylinder [12] to the more complicated geometry of Fig. 1. However, the ex- pressions for the Green functions are rather cumbersome, and are not given here.

The exact results obtained in this work represent a canonical solution to a rather complicated boundary-value problem involving a cavity, two diffracting edges, and two different materials. This exact solution is possible because the boundaries are either complete coordinate surfaces, or portions of coordinate surfaces whose continuation is em- bedded in metal. For example, this is not the case for a semicircular cavity either not covered [13] or partially covered [14] by metal, or for a semielliptical cavity slotted along the entire axis (no metal covering) [15], where an exact solution is precluded and one must resort to numerical techniques based on either coupling of eigenfunction expansions (inversion of truncated matrices), or integral equation formulation.

Finally, according to a communication by Prof. Boris Pavlov to the author [16], in the mid 1970’s the late Dr. Vladimir Lazutkin of St. Petersburg, Russia had sug- gested the possibility of solving the problem of Fig. 1 exactly, for the same medium inside and outside the channel.

However, Prof. Pavlov is unaware of any written record on this subject.

VI. Acknowledgement

The author is grateful to Dr. Danilo Erricolo for assis- tance with the manuscript, and to the Reviewers for useful suggestions.

Appendix

SOME PROPERTIES OF MATHIEU FUNCTIONS The notation used in this paper is that of Stratton [1]

[2]. The even and odd angular functions are Sem(c, η) and Som(c, η), respectively; note that these functions are even and odd with respect to the variable v, not with respect to η = cos v. The even and odd radial functions are Re^(h)_m (c, ξ) and Ro^(h)_m (c, ξ), respectively, where h = 1 or 4.

A. Wronskian relation

Re, o⁽¹⁾_m (c, ξ) ∂

∂uRe, o⁽⁴⁾_m (c, ξ)

− Re, o⁽⁴⁾_m (c, ξ) ∂

∂uRe, o⁽¹⁾_m (c, ξ) = −j (77)

B. Special values

Sem(c, 1) = 1 (78)

Sem(c, −1) = (−1)^m (79)

Som(c, ±1) = 0 (80)

Sem(c, η)|_v=π±v₀ = (−1)^mSem(c, η)|_v=v₀ (81) So_m(c, η)|_v=π±v

0 = ±(−1)^mSo_m(c, η)|_v=v

0 (82)

Sem(c, η)|_v=2π−v_o = Sem(c, η)|_v=v₀ (83) Som(c, η)|_v=2π−v

o = − Som(c, η)|_v=v

0 (84)

∂

∂v Sem(c, η)|_v=0,π= 0 (85)

∂

∂v So_m(c, η)|_v=0= 1 (86)

∂

∂v Som(c, η)|_v=π= (−1)^m (87)

∂

∂u Re⁽¹⁾_m (c, ξ)

¯¯

¯u=0= 0 (88)

Ro⁽¹⁾_m (c, 1) = 0. (89)

C. Asymptotic expansions

Re, o⁽⁴⁾_m (c, ξ)

¯¯

¯ξ→∞,Im c<0∼ j^m

√cξe^{−jcξ+jπ/4} ∼ j^m

√kρe^{−jkρ+jπ/4} (90)

where ρ = p

x²+ y². In the same limit, the functions Re, o⁽¹⁾m (c, ξ) tend to infinity.