A semielliptical channel flush-mounted under a metal plane and slotted along the interfocal distance of its cross section is separated from the half-space above by a multilayer diaphragm. The cavity, the diaphragm, and the half-space are all isorefractive to each other. Both the cavity and the multilayer diaphragm are filled with materials isorefractive to the medium in the half-space above. This is a two-dimensional geometry where the source is invariant with respect to the axial variable. The resulting electromagnetic boundary-value problem is solved exactly in terms of Mathieu functions, when the excitation source is either a plane wave or a line source. For plane wave incidence, the polarization is with either the electric or the magnetic field parallel to the axis of the structure and the direction of incidence is arbitrary in a plane perpendicular to the axis. For line source excitation, the polarization is with either the electric or the magnetic field parallel to the axis of the structure and the source is arbitrarily located. Numerical results are also provided.
 The exact analytical solution to a new canonical problem is provided. Few exact analytical solutions exist for electromagnetic scattering problems, but they are very important because they provide useful insight into the interpretation of the behavior of the electromagnetic field. This new solution is possible due to the choice of isorefractive materials and of a particular geometry, which allows for the application of the boundary conditions using mode matching, thus leading to closed form solutions expressed as series. Additionally, the exact solution of this complicated problem, which involves sharp curved metallic edges, a cavity, and a curved surface separating different penetrable media provides a challenging test for the validation of frequency-domain computational software.
 The exact solution to a two-dimensional electromagnetic boundary-value problem involving a channel of semielliptical cross section with metallic walls flush-mounted under a metallic ground plane and coupled to the half-space above via a slot is considered. The half-space above the ground plane is separated from the material inside the channel by a multilayer diaphragm. The multilayer diaphragm, the material filling the channel, and the half-space are isorefractive to each other. The coupling slot occupies the interfocal distance in the cross section of the structure. The diaphragm is composed of N − 2 semielliptical shells confocal to the cavity wall and with their flat surfaces flush to the ground plane.
 This geometry is considered because it allows to model the penetration of electromagnetic radiation into a cavity or the radiation that escapes from an aperture of a cavity that contains an electromagnetic source. As an example, in a problem of electromagnetic compatibility, this geometry may model a wire inside a channel. The additional complication of the diaphragm models the presence of a mechanical cover that protects the cavity, similar to the function of a radome for a radar antenna.
 This is a two-dimensional geometry where the excitation is invariant with respect to the axial variable. The excitation is either a plane wave or a line source. For a plane wave source, the field is polarized with either or parallel to the axis of the structure and the direction of incidence is arbitrary in the plane of the cross section. For a line source, the or field is polarized parallel to the axis of the structure and the source is arbitrarily located in any of the N regions, but never exactly on any boundary.
 The exact solution is expressed in the form of series expansions involving Mathieu functions. The expansion coefficients in the series are determined analytically by imposing the boundary conditions, thereby leading to a canonical solution of the boundary-value problem. The notation for the Mathieu function is essentially the one of Stratton , with the difference that the argument of the angular Mathieu function is the angle, rather than the cosine of the angle, as explained in Blanch  and Valentino and Erricolo .
 This work is an extension of the results by Uslenghi [2004a], where no diaphragm is considered, and Valentino and Erricolo , where the diaphragm consists of only two layers. The technique used in the solution is similar to the one applied in Erricolo and Uslenghi [2005b], for which numerical results were given by Erricolo et al. [2005b, 2005a]. Prior related works consider two-dimensional cavities flush-mounted under a conducting ground plane with, however, the following differences: first, there is no diaphragm; second, the materials inside the cavity are usually the same as outside; third, the solution is not expressed in analytic closed form. For example, these prior works include: Hinders and Yaghjian , who consider a semicircular channel with a dual series eigenfunction approach; Lockard and Butler [2004, 2006], who examine an arbitrarily shaped cavity with an integral equation approach; and Scharstein et al. [2007, 2008], who analyze rectangular trough with a variational approach.
 Numerical results based on the evaluation of the series of Mathieu functions are provided for the fields inside the channel, inside the diaphragm and in the open space above the structure. Issues such as the evaluation of the fields and surface currents on the metallic boundaries are analyzed in detail. The geometry of the problem is presented in section 2, and the solutions for plane wave incidence and line source excitation are shown in sections 3 and 4, respectively. Some numerical results for the different media are reported in section 5. The time-dependence factor exp(+iωt) is omitted throughout.
2. Geometry of the Problem
 Geometry 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 Figure 1.
 The various boundaries correspond to coordinate surfaces of the elliptic cylinder orthogonal system, with foci located at points A and B, whose distance is the interfocal distance d. The elliptic cylinder coordinates (ξ, η, z) are a right-handed system related to Cartesian coordinates by:
where 0 ≤ u < ∞, 0 ≤ v ≤ 2π, −∞ < z < +∞. It is also convenient to introduce the variables (ξ, η) defined as ξ = cosh u, η = cos v, with 1 ≤ ξ and −1 ≤ η ≤ 1. The inverse transformation is reported in Erricolo et al. [2005b].
 The semielliptical channel is limited by a metallic boundary given by u = uN. The coordinate surface ξ = ξN corresponds to an elliptic surface with interfocal distance d, major axis a = dξN, minor axis b = d, and eccentricity e = d/a = 1/ξN. In the limit when the interfocal distance d is zero, the elliptic cylinder coordinates reduce to the circular cylinder coordinates; on the other hand, when d is finite, the coordinate surfaces u = constant become circular as u approaches infinity:
where ρ and ϕ are the radial and angular circular cylinder coordinates, respectively. The metallic plane y = 0 has the strip (∣x∣ ≤ d/2, y = 0, −∞ < z < ∞) cut away; thus the slot AB is an aperture in the metallic plane that connects the half-space y > 0 to the semielliptical channel below it. The points A and B also represent the edges of the infinitesimally thin metallic baffles AC and BD.
 Across the aperture is located a diaphragm, which is made of N − 2 different isorefractive materials and limited by the surfaces u = u1 and u = uN−1 inside and outside the channel, respectively. The inner surfaces of the multilayer diaphragm are called respectively u = un with n = 2, …, N − 2. All the semielliptical surfaces are confocal to each other. Both the slot u = 0 and the diaphragm provide coupling between the cavity and the unbounded medium. L is the index of the medium whose lower boundary is u = 0.
N different media are considered in this problem, as shown in Figure 1: the unbounded medium 1 with electric permittivity ε1 and magnetic permeability μ1; the diaphragm that is made of materials with parameters εn, μn for n = 2, …, N − 1; and the cavity with parameters εN and μN. All media are isorefractive to each other, which means they satisfy the condition
This condition implies that the wavevector does not change from one medium to the other, since
but the intrinsic impedance of the material is usually different, because
 We consider a plane wave with arbitrary direction of arrival in the plane perpendicular to the z axis. For this direction of arrival, we consider two orthogonal polarizations with or parallel to the z axis. All other polarizations for the same direction of incidence may be obtained by superposition of the results for E and H polarization. Results for oblique incidence with respect to the axis of the two dimensional structure can be obtained from the studies at normal incidence, as shown by Uslenghi : this is always true only if the structure is composed of perfect conductors and isorefractive materials.
 As first case, an E-polarized incident plane wave, whose direction of propagation forms an arbitrary angle ϕ0 with the negative x axis, and an angle π/2 − ϕ0 with the negative y axis (0 < ϕ0 < π/2) is considered. The incident electric field is given by:
The magnetic field may be computed everywhere by applying Maxwell's equations in elliptic cylinder coordinates:
where Y is the intrinsic admittance of the medium. The incident electric field may be expanded in a series of elliptic-cylinder wave functions:
E1zr is the field that would be reflected by the metal plane y = 0 if there were no slot (i.e., u1 = 0) and in absence of the diaphragm. E1zd is the diffracted field introduced by the presence of the cavity-backed slot, the multilayer diaphragm and must satisfy the two-dimensional radiation condition. By applying the image theory and by symmetry considerations it is trivial to determine the reflected component of the total field when there is no channel and no diaphragm: the reflected wave is a plane wave that forms an angle ϕ0 with the negative x axis and an angle π/2 − ϕ0 with the positive y axis. The expression for E1zr can be computed applying the properties of the angular Mathieu functions reported in the appendix of Uslenghi [2004a], to obtain
The superposition of the incident and reflected field is the geometrical optics total field in medium 1, which only contains odd radial Mathieu functions
In order to compute the total field in the unbounded medium, the contribution of the field E1zd must be added to (14), according to (11). The expression for the diffracted field is
where the presence of the Mathieu radial function of the fourth kind guarantees the satisfaction of the radiation condition, whereas the modal coefficient 1m(e) is introduced to account for the effect of the presence of the cavity and the diaphragm. The analytical expression for the total field E1z is:
The general expression for the total magnetic field in the other media is given by a linear combination of odd radial functions of the first and fourth kind (the subscript h = 2, …, N − 1 identifies the medium):
All the modal coefficients account for the presence of both the cavity and the diaphragm; they are determined imposing a vanishing total tangential electric field on the metallic surfaces, and the continuity of the total tangential electric and magnetic field across the surfaces separating different penetrable media. Because all interfaces are coordinate surfaces of the elliptic cylinder system and because of the use of isorefractive materials, the application of the boundary conditions leads to a matching of the modal coefficients of each term of the pertinent series. Hence, imposing the boundary conditions leads to a system of 2(N − 1) linear equations that is solved using Gaussian elimination for small N or Thomas' algorithm for block tridiagonal matrices for larger N. The matrix expression of the linear system is
The following parameters and functions are adopted:
The main diagonal blocks have the form
The lower diagonal block Bn are all equal to
The upper blocks take the form
 The unknown vectors are:
while the parameter terms are
The expression for the normalized bistatic radar cross section is computed according to
The current density vector = × on the metallic surfaces is obtained using the following expressions
The explicit expression for the induced current density vector is not reported here; it should also be stressed that the previous equations are valid for any type of E-polarized source.
 The analytical procedure for an H-polarized plane wave illuminating the semielliptical cavity is very similar to the one outlined in the previous section. A unit-amplitude, H-polarized plane wave is assumed to be propagating in a direction perpendicular to the z axis forming an angle ϕ0 with the negative x axis, and an angle π/2 − ϕ0 with the negative y axis (0 < ϕ0 < π/2). Its mathematical expression is formally equivalent to (7)
and its expansion in terms of Mathieu functions is given by (10). The total magnetic field in medium 1 is given by the sum of three components:
By applying image theory and by symmetry considerations, the reflected field is obtained as
Since the diffracted field H1zd must satisfy the two-dimensional radiation condition, the boundary condition on the metallic surface v = 0, π, and the boundary condition along u = uN, its expression is similar to (15) and the total field in the unbounded medium can be written as follows:
In the previous expression, only even modes are present and the modal coefficient 1m(m) is introduced to account for the presence of the cavity and the diaphragm. The electric field components are obtained from the magnetic field Hz according to
The total magnetic field inside the diaphragm is identified by the subscript h = 2, …, N − 1:
whereas in the material filling the cavity:
2(N − 1) boundary conditions are required to compute all the modal coefficients. It is also useful to define the auxiliary function:
The linear system to compute the coefficients is
 The main diagonal blocks are of the type
The lower diagonal block Bn are all equal to
The upper blocks take the form
The unknown vectors are:
while the parameter terms are
The normalized bistatic radar cross section is thus given for (0 < ϕ < π) by:
The induced current density on the metallic surfaces may be computed using the following expressions
4. Line Source Incidence
 The excitation of the cavity in Figure 1 may be provided by an ideal, infinitely long electric or magnetic line source parallel to the z axis and arbitrarily located in one of the N media. The most physically meaningful cases are represented by a line source located either in medium 1 or N, since the diaphragm is mainly meant to model mechanical protection to the interior of the cavity; however analytical solutions are presented for a line source located in all N media. Issues such as the determination of the far field expression in the unbounded medium are also addressed.
4.1. Electric Line Source
 An electric line source parallel to the z axis and located at (u0, v0) generates a primary electric field given by the Hankel function of the second kind and zero order:
where in rectangular coordinates (x, y, z)
is the distance of the observation point (x, y) from the source (x0, y0). According to Bowman et al. , the incident field may be rewritten as series expansion of Mathieu functions:
where u< (u>) is the smaller (larger) between u and u0.
4.1.1. Electric Line in the Unbounded Medium
 When the source is located in the unbounded medium, the general expression for the total field in medium 1 is still given by (11), where E1zi is given by (48), E1zr is the reflected field by the infinite metal plane y = 0 when there is no channel and no diaphragm, and E1zd is the diffracted field due to the presence of the cavity-backed slot and of the diaphragm.
 As shown in Figure 2, and by considering image theory, the effect of an electric line horizontally located above a ground plane is equivalent to the sum of the field due to the original line and the field due to another line located symmetrically with respect to the ground plane in absence of the ground plane. Therefore the total geometrical optics field is:
where the distance R′ is given by:
The series expansion for the reflected electric field E1zr can be computed recalling some properties of the angular radial Mathieu functions:
The total geometrical optics field (49) in the unbounded medium is given by:
The expression for the diffracted field E1zd is written so that the radiation condition is satisfied and accounting for the presence of the cavity and of the diaphragm:
Therefore the expression for the total electric field in the unbounded medium is computed according to (11), thus yielding:
The general expression of the total electric field inside the diaphragm is written as a linear combination of Mathieu radial functions of the first and fourth kind:
 In the material filling the cavity, the requirement for a vanishing tangential total electric field yields
The procedure for the determination of the seven modal coefficients is identical to the one for plane wave incidence. Moreover, it is evident that the linear system originating from the imposition of the boundary conditions yields a solution for the modal coefficients that is formally identical to the one shown in section 3.1 (equations (19) through (25)) derived for an E-polarized incident plane wave. Given this equivalence, in every medium the relative energy distribution across the modes of the scattered field is independent from the radiation pattern of the illuminating source.
 The far field expression for the total electric field is obtained by recalling the asymptotic expressions for the radial Mathieu function of the fourth kind in passive media, and that when u → ∞, c cosh u → kρ and v → ϕ:
4.1.2. Electric Line in Medium N
 For an infinitely long electric line source located in the material filling the cavity, the total electric field in medium N is given by the sum of the geometrical optics field in (52) plus a perturbation component ENzs that must satisfy the boundary conditions along the border.
 Specifically, the diffracted field ENzs contains a linear combination of odd radial Mathieu functions of the first and fourth kind. The resulting expression for ENz is
where the modal coefficients Nm(e) and Nm(e) are introduced.
 The total field inside the isorefractive diaphragm is formally equivalent to (17), except for some constants
The total field in medium 1 is the diffracted field due to the presence of the cavity and the diaphragm; the radiation condition must be met, thus yielding:
The boundary conditions, together with the Maxwell's equation necessary to obtain the magnetic field once the total electric field is known, are identical to the ones used in section 4.1.1, for an electric line source located in the unbounded medium. The solution is very similar to the one previously determined, with the following exception:
The far field in the passive medium 1 is given by:
4.1.3. Electric Line in Medium 2 ≤ M ≤ N − 1
 For an electric line source located in medium M and parallel to the z axis, the expression of the total electric field therein is:
in the unbounded medium, the electric field is the scattered contribution that satisfies the radiation condition
In media 2 ≤ n ≤ N − 1, n ≠ M the total field is a linear combination of radial Mathieu functions of the first and fourth kind:
similarly, inside the cavity, the electric field is given by
 The solution is the same as for the linear polarization incidence, with the following differences. For M ≤ L:
For M = L + 1
For L + 1 < M ≤ N − 1
4.2. Magnetic Line Source
 The derivation of the analytical solution of the boundary-value problem, when the primary source is represented by an H-polarized line source located at (u0, v0), proceeds similarly to the case of an electric line source. The primary field is:
and its series expansion in elliptic cylinder eigenfunctions is still given by (48).
4.2.1. Magnetic Line in the Unbounded Medium
 The total magnetic field in medium 1 is still given by (30). The total geometrical optics field H1zi + H1zr is computed applying image theory to the case of a magnetic line source
with R and R′ given by (47) and (50), respectively. The total field in the unbounded medium also accounts for the diffracted component:
Inside the diaphragm, the total z component of the magnetic field is
whereas in medium N:
The application of the boundary conditions yields a linear system of equations whose solution is formally identical to the equations derived for an H-polarized incident plane wave. The expression of the total magnetic field at large distance from the cavity is:
4.2.2. Magnetic Line in Medium N
 For a line source in the material filling the cavity, the total magnetic field in all the media is given by the following expressions
4.2.3. Magnetic Line in Medium 2 ≤ M ≤ N − 1
 The total magnetic field in M-th portion of the isorefractive diaphragm in Figure 1, when the source is located therein, is given by:
Inside the cavity, only the scattered component of the magnetic field is present, and its most general expression may be written as a linear combination of Mathieu radial functions of the first and fourth kind
whereas in the unbounded medium the total field is required to satisfy the radiation condition, thus yielding:
In all other media, the expression of the total field is formally identical to (81)
 The solution is the same as for the linear polarization incidence, with the following differenced. For M ≤ L:
For M = L + 1
For L + 1 < M ≤ N − 1
5. Numerical Results
 The numerical evaluation of the fields was performed using some of the Fortran subroutines that implement Mathieu radial and angular functions reported by Zhang and Jin . However, since these subroutines apply the Goldstein-Ince normalization [see Goldstein, 1927; Ince, 1932], also used by Abramovitz and Stegun , they were modified to account for the Stratton-Chu normalization adopted in this work. Also, the expansion coefficients of the Mathieu functions were computed according to the algorithm of Blanch . Details of the computation of the expansion coefficients are described by Erricolo .
 All the computations of series expansions of the fields were carried out by applying Shanks transform [see Singh et al., 1990] to the imaginary part only, following what is described by Erricolo . For all the series involved in the following results, convergence was achieved within the first 25 terms. Whenever line source coefficients are involved for a line source located in a medium that has the slot as a boundary, then the series is unstable and only 15 terms are sufficient before encountering numerical misbehaviors. Each curve representing an electric or magnetic field was evaluated in an arbitrary plane z = costant at 200 points along the y axis located between the metallic cavity identified by (ξ1, −1) to the point on the y axis of elliptic coordinates (2ξ1, 1). The computation time for each of the following figures, except for the contour plot, was less than 5 s. All simulations were run on a Turion X2 64 with a clock frequency of 1.60 GHz with 1 Gibyte of RAM.
 This geometry of six isorefractive layers was chosen because it corresponds to the minimum number of media to cover all the possible different cases presented in our analysis.
 The evaluation of the fields was performed in all the cases for which an exact solution was presented, except for an electric or magnetic line source inside the isorefractive diaphragm. The fields were evaluated in all the interesting regions along the y axis, because the y axis penetrates the cavity, the diaphragm, and the unbounded medium.
 Moreover, the evaluation of the analytical formulas was carried out for several values of the most meaningful parameters, which are: (1) the dimensionless parameter c = kd/2, which has the physical meaning of the ratio between the aperture size and the wavelength; specifically, when c < π (c > π) the aperture is smaller (larger) than the wavelength; (2) the location of the N confocal semielliptical surfaces, such as the metallic cavity wall ξ1 or the the lower boundary ξN of the diaphragm; (3) the material properties, through the impedance ratios ζn,n+1 (see (20a)); and (4) the incidence angle ϕ0 for plane wave illumination and the location (ξ0, η0) for line source illumination.
 To reduce the complexity of the possible combinations of all parameters, a particular cavity was selected, such that the size of the major semiaxes of the ellipses, in order, is: 4λ, 3λ, d/2, 2λ, 3.5λ, 4λ; the remaining parameters and the impedance ratios ζ are varied from case to case. Additionally, we verified that in the special case ζn,n+1 = 1, i.e. all materials are the same, the numerical results are in agreement with those obtained by Uslenghi [2004a], where the diaphragm is not present, and Valentino , where the diaphragm consists of only two layers.
5.1. Plane Wave Incidence
 For an E-polarized plane wave illuminating the cavity, the analytical results were presented in section 3.1. Figure 3 shows the magnitude of the z component of the total electric field ∣Ez∣x=0 along the y axis, computed using equations (16) through (18), for different sets of impedance ratios. One may observe that (1) at the intersection between the y axis and the metallic channel, the boundary condition for the electric field is satisfied; (2) the continuity of the electric field is respected at all interfaces; (3) all the maxima and minima occur at the same locations because the value of c is the same for all three curves; and, (4) within the cavity and the multilayer diaphragm the differences among the three curves are more evident because of the differences in the material properties, whereas in medium 1 the three curves approach each other.
Figure 4 shows the magnitude of the z component of the total magnetic field ∣Hz∣x=0 along the y axis, computed using equations (32), (35) and (36) for different values of c and for a specific choice of the impedance ratios. One may observe that (1) the derivatives of the curves correspond to the normal derivative of the tangential magnetic field and they satisfy the boundary condition at the intersection of the y axis with the channel; (2) additionally, the number of oscillations increases as c becomes larger, which is consistent with what one would expect when the aperture appears wider compared to the wavelength; (3) moreover, the choice of c = π/2, π, 2π corresponds to λ/d = 2, 1, 1/2 respectively.
5.2. Line Source Incidence
 For an infinitely long electric line source located at (u0 = 1.1u1, v0 = π/2), i.e. parallel to the z axis in the unbounded medium, the analytical derivation was outlined in section 4. A contour plot of the magnitude of the electric field Ez is provided in Figure 5. Also, as theoretically expected, the contour plot is symmetric with respect to the y axis. The intensity of the electric field is stronger for y > 0 because the source is located in the unbounded medium. The electric field satisfies the boundary conditions at every interface and close to the metallic boundaries, where its amplitude vanishes. When an electric line source is located inside the cavity at (u0 = (uN−1 + uN)/2, v0 = 3π/2), the evaluation of the total electric field is represented in Figure 6 for c = 2π, ζn,n+1 = and the usual geometry. It appears that a resonance condition is satisfied, since the electric field is mostly confined within the cavity notwithstanding the fact that an aperture exists. When a magnetic line source is located in the diaphragm at (u0 = (u3 + u4)/2, v0 = 3π/2), the relevant expressions for the total magnetic field ∣Hz∣ outlined in section 4.2.1, and explicitly reported in equations (72) through (74), are reported in Figure 7. Similar to the case of Figure 6 it appears that a resonance condition is also satisfied with the difference that in this case the source is located closer to the aperture and therefore more field appears to be radiated towards the unbounded medium through the aperture.
 Exact analytical results and evaluation for the electromagnetic behavior of a semielliptical cavity with sharp curved edges, a diaphragm and four isorefractive media were presented. The new canonical electromagnetic boundary-value problem solved herein enrich the catalog of exact solutions while providing a good test for the validation of frequency-domain codes. This validation is especially useful for testing the behavior of these codes around critical points such as close to edges.
 The exact formulas derived in this work could be used to extract low-frequency expansions to make comparisons with quasi static techniques applied to cavities, such as those obtained by Hansen and Yaghjian . This should be possible by making use of the low-frequency expansions of angular and radial Mathieu functions, using the normalization of Stratton, recently obtained by T. Larsen et al. (Expansions of Mathieu functions into products of Bessel functions, submitted to Mathematics and Computation, 2006).
 Numerical examples are herein provided; furthermore, in the specific case of a diaphragm of only two layers, our solution agrees with the one shown in Valentino , while when the diaphragm is removed, the solution agrees with the one in Uslenghi [2004a].
 This work was supported in part by the Aileen S. Andrew Foundation, by the U.S. Department of Defense, and by the U.S. Air Force Office of Scientific Research under MURI grant FA9550-05-1-0443, and by a grant of computer time from the DOD High Performance Computing Modernization Program at ASC. The authors are thankful to the reviewers for their comments and to Giuseppe Carluccio for his help with the figures.