APARTIALLY covered trench located at the corner of

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1 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, Penetration, Radiation and Scattering for a Cavity-backed Gap in a Corner Danio Erricoo, Senior Member, IEEE, Piergiorgio L. E. Usenghi, Feow, IEEE Abstract A partiay covered cavity, or trench, ocated aong the edge of two intersecting metaic was perpendicuar to each other is considered. The cross section of the cavity is a quarter eipse, and is sotted aong the interfoca distance. The cavity is fied with a materia isorefractive to that occupying the quarter-space between the was. This two-dimensiona boundary-vaue probem is soved exacty, when the primary fied is either a pane wave of arbitrary poarization and direction of incidence, or an eectric or magnetic ine source. Numerica resuts are given and discussed. Keywords Eectromagnetic radiation, eectromagnetic scattering, compex media, cavities, isorefractive media, Mathieu functions. I. Introduction APARTIALLY covered trench ocated at the corner of two metaic was perpendicuar to each other is considered. The cross section of the trench is a quarter eipse, and it is sotted from the focus to the center of the eipse; the trench is partiay covered by a thin meta strip extending outward from the foca ine, as part of the meta wa under which the trench is fush mounted. The trench is fied with a materia that is isorefractive to the medium (e.g. air) fiing the quarter space between the was. This boundary-vaue probem is soved exacty in the frequency domain, when the primary source is either a pane wave of arbitrary direction of incidence and poarization, or an eectric or magnetic ine source parae to the was and ocated either outside or inside the trench. The fied components are expanded in infinite series of eigenfunctions that are products of radia and anguar Mathieu functions, in the Stratton-Chu notation,. The moda coefficients in the expansion of the secondary fieds are anayticay determined by imposing the boundary conditions. The technique utiized in obtaining this new canonica soution is akin to that used for a sotted semieiptica channe 3, that ed to numerica resuts in exceent agreement with integra equation approaches 456. Some preiminary resuts for E-poarized pane wave incidence were reported previousy in 7. Cavity-fied probems arise, for exampe, when antennas are ocated inside a cavity at the surface of a patform such as an airpane, a missie or a sateite. The smoothness of the patform surface is dictated by aerodynamica or other considerations, and is achieved by fiing the cavity, either The authors are with the Department of Eectrica and Computer Engineering, University of Iinois at Chicago, 85 South Morgan St., Chicago, Iinois 667, USA. E-mai: derrico@ece.uic.edu, usenghi@uic.edu. This research was supported by the U.S. Department of Defense under MURI grant F Additionay, this work was supported in part by a grant of computer time from the DoD High Performance Computing Modernization Program at ASC. totay or partiay (as in the case of a radome), with a materia that is transparent to eectromagnetic radiation. A numerica study of this type of boundary-vaue probems may encounter difficuties, such as estimating the portion of patform surface surrounding the cavity structure that needs to be taken into account in the numerica modeing, or evauating fieds in the vicinity of geometric singuarities. The canonica soutions presented herein are the ony exact, anaytica soutions avaiabe for a trench ocated at the corner of two was. Aside from the intrinsic importance of soving exacty a compicated boundary-vaue probem invoving a cavity, a sharp edge and two different penetrabe materias, these canonica soutions constitute a benchmark for the vaidation of frequency-domain computer codes. The geometry of the probem is iustrated in section II, and the exact soutions for pane wave incidence and ine sources are obtained in sections III and IV, respectivey. Some numerica resuts are presented and discussed in Section V. The time-dependence factor exp(+jωt) is omitted throughout. II. Geometry of the probem A cross-sectiona view of the structure in a pane z = constant is shown in Fig.. Fig.. Geometry of the probem. The metaic was OA(x = ) and OE(y = ) are perpendicuar to each other. The trench OBC is fushmounted under the horizonta wa OE, and its cross section is a quarter eipse with semi-major axis OC and semi-

2 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, minor axis OB. The wa OE is sotted aong the sit of width OD equa to haf the interfoca distance d of the eiptica trench. The trench is partiay covered by the thin meta baffe DC. The rectanguar coordinates (x, y, z) are reated to the eiptic cyinder coordinates (u, v, z) by: x = d cosh u cos v, () y = d sinh u sin v, () z = z (3) where u <, v π, < z <. expedient to introduce the coordinates It is = cosh u, η = cos v (4) with < and η. Curves with constant are eipses and curves with η constant are hyperboas. The haf-pane DE corresponds to v =, the surface DC of the baffe inside the trench corresponds to v = π, whereas the portions OA and OB of the vertica wa outside and inside the trench are given by v = π/ and v = 3π/, respectivey; finay, the curved surface BC of the trench corresponds to =, and the surface OD of the sit to = (see Fig. ). The semi-interfoca distance OD = d/, the semi-major axis OC = d /, and the semi-minor axis OB = (d/). The medium fiing the quarter space (x >, y > ) between the was has an eectric permittivity ε and a magnetic permeabiity µ, whereas the materia fiing the trench has permittivity ε and permeabiity µ. The two media are isorefractive, i.e. so that the propagation constant ε µ = ε µ (5) k = ω ε h µ h, (h =, ) (6) is the same for both media, whist the intrinsic impedances Z h = µ h /ε h, (h =, ). (7) are, in genera, different from each other. It is important to understand that condition (5) is necessary for one-to-one mode matching of the fieds inside and outside the trench, and the consequent anaytica determination of the moda coefficients. If (5) were not satisfied, an anaytica soution woud be precuded and one woud have to dea with the inversion of an infinite matrix for the determination of the moda coefficients. Finay, in the foowing it is convenient to introduce the dimensioness parameter c = kd = π d λ. (8) where λ is the waveength. The geometry studied herein and shown in Fig. may be considered a particuar case of the sotted channe examined in 3, by inserting a PEC wa in the pane x =. As such, the resuts obtained in this paper may be derived by combining the formuas in 3 with the method of images, and this is essentiay the technique utiized in sections III and IV beow. However, the derivations are not trivia; in particuar, the foowing two formuas for the anguar Mathieu functions are needed, in addition to a the formuas isted in the Appendix of 3: v Se m(c, η) = (9) v=π/,v=3π/ So m (c, η) v=π/ = So m (c, η) v=3π/ = () III. Pane wave incidence Any incident pane wave in the quarter space (x >, y > ) may be considered as a superposition of an E- poarized wave (eectric fied parae to the z-axis) and an H-poarized wave (magnetic fied parae to the z-axis). The two poarization cases are treated separatey. A. E-poarization Consider an incident pane wave whose direction of propagation forms the ange ϕ with the negative x axis and the ange π/ ϕ with the negative y axis ( < ϕ < π/, see Fig. ), and whose eectric fied is given by: E i = ẑe i z = ẑe jk(x cos ϕ +y sin ϕ ), ( < ϕ < π/), () where ẑ is a unit vector in the positive z-direction. The incident fied () may be expanded in a series of eipticcyinder wave functions : E i z = 8π j m Re () N m (e) m (c, )Se m (c, η)se m (c, cos ϕ ) Ro () m (c, )So m (c, η)so m (c, cos ϕ ), () m= + jm N m (o) where Re () m and Ro () m are even and odd radia Mathieu functions of the first kind, Se m and So m are even and odd anguar functions, and N m (e),(o) are normaization coefficients,. The tota eectric fied in the quarter-space (x, y ) may be written as the sum of a geometric-optics fied E go z due to the corner refector without the trench and a diffracted fied Ez d due to the presence of the trench: E z = E go z + Ed z. (3) The geometric-optics fied is the sum of four terms (see Fig. ): E go z = Ei z + Ez OE + Ez OA + Ez OA,OE (4) where the fied Ez OE corresponds to a wave with incidence ange π ϕ mutipied by a refection coefficient -, the fied Ez OA corresponds to a wave with incidence ange π ϕ mutipied by a refection coefficient -, and the fied Ez OA,OE corresponds to a douby-refected wave, i.e. a wave with incidence ange π + ϕ. When the even and

3 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, 3 Of particuar interest are the surface current densities J on the metaic boundaries of the structure of Fig.. With E-poarization, these currents are parae to the z-axis and given by: where: J z = H on DE(v = ), = H on DC(v = π), = H on OA(v = π/), = H on OB(v = 3π/), = H v on BC( = ), () Fig.. Geometric-optics contribution for pane-wave incidence. H v= = j8 π cz ( ) Ro () (c, )+ a (c, ) So (c, cos ϕ ), () odd functions of ϕ are separated out in the various fied components, it is found that the overa geometric-optics fied is: E go z =8 π ( ) Ro () (c, )So (c, η)so (c, cos ϕ ). (5) As a consequence of the boundary conditions on the two was, ony the odd functions of even order remain in the geometric-optics contribution. The diffracted fied must satisfy the boundary conditions on the was and the twodimensiona radiation condition, hence it must be of the type: E d z =8 π ( ) a (c, )So (c, η)so (c, cos ϕ ), (6) where is the radia Mathieu function of the fourth kind, and the moda coefficients a are to be determined. The tota eectric fied inside the trench may be written as E z = 8 π where a = ζro () with ( ) a (c, ) Ro () (c, ) Ro() (c, ) (c, ) So (c, η)so (c, cos ϕ ), (7) Ro () (c, )Ro() (c, ) (c, )Ro(4) (c, ) ( + ζ) (c, )Ro() (c, ), (8) ζ = Z /Z (9) and the prime indicates the derivative with respect to. H v=π = j8 π cz ( ) a (c, ) Ro () (c, ) Ro () (c, ) Ro(4) (c, ) So (c, cos ϕ ), () H v= π = j8 π cz ( ) So (c, cos ϕ ) H v=3π/ = j8 π cz Ro () (c, ) + a (c, ) v So (c, η) ( ) (c, ) So (c, cos ϕ ) H v = = 8 π cz η v= π a (c, ) Ro () v So (c, η) ( ), (3) (c, ) Ro() (c, ), (4) v=3π/ a Ro () (c, ) So (c, η)so (c, cos ϕ ). (5) Within the quadrant (x >, y > ), the bistatic radar cross section (RCS) σ E (ϕ) of the partiay covered trench is: σ E (ϕ) λ =8π a So (c, cos ϕ)so (c, cos ϕ ), < ϕ, ϕ < π/. (6) In the imit when the trench recedes to infinity ( with Im c < ), the sit = connects two isorefractive quarter spaces. In this case, a,im c< = + ζ Ro () (c, ) (c, ), (7)

4 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, 4 and the diffracted fied in the quarter space (x, y ) becomes: Ez d = 8 π,im c< + ζ ( ) Ro () (c, ) (c, ) (c, )So (c, η)so (c, cos ϕ ) (8) and is vaid for v π/. The diffracted fied Ez d in the quarter space (x, y ) is,im c< given by (8) mutipied by minus one, and is vaid for 3π/ v π. B. H-poarization For an incident pane wave with the magnetic fied parae to the z-axis, H i = ẑh i z = ẑe jk(x cos ϕ +y sin ϕ ), ( < ϕ < π/), (9) the derivations are simiar to those for E-poarization, hence ony the resuts are given. The tota magnetic fied H z in the quarter space (x, y ) is: H z = H go z + Hd z, (3) where the geometric-optics fied H go z (i.e., the fied in the absence of the trench) is the sum of four pane waves: H go z =8 π ( ) Re () (c, )Se (c, η)se (c, cos ϕ ); (3) ony the even Mathieu functions of even order appear, because of the boundary conditions on the two metaic was. The diffracted fied due to the sotted trench is: H d z =8 π ( ) The fied inside the trench is H z = 8 πζ ( ) (c, )Se (c, η)se (c, cos ϕ ). (c, ) (c, ) Re() (c, ) Re () (3) (c, ) Se (c, η)se (c, cos ϕ ), (33) where ζ is given by (9), the prime means the derivative with respect to, and = Re () (c, )Re() (c, ) ζre () (c, )Re(4) (c, ) ( + ζ) (c, )Re() (c, ). (34) The surface current density J on the metaic boundaries of Fig. is given by: where: J = ˆH z on DE(v = ), = ˆH z on DC(v = π), = ˆH z on OA(v = π/), = ˆH z on OB(v = 3π/), = ˆvH z on BC( = ), (35) H z v= =8 π H z v=π = 8 πζ H z v=π/ =8 π ( ) Re () (c, )+ (c, ) Se (c, cos ϕ ), (36) ( ) (c, ) (c, ) Re() (c, ) Re () (c, ) Se (c, cos ϕ ), (37) ( ) H z v=3π/ = 8 πζ H z = = j8ζ Re () (c, ) + (c, ) Se (c, cos ϕ ) Se (c, η) v=π/, (38) ( ) (c, ) (c, ) Re() (c, ) Re () (c, ) Se (c, cos ϕ ) Se (c, η) v=3π/ ; (39) π ( ) Re () (c, ) Se (c, cos ϕ )Se (c, η); (4) note that in (38) and (39), Se (c, η) has the same vaue at v = π/ and v = 3π/. The bistatic radar cross section σ H (ϕ) of the partiay covered trench is: σ H (ϕ) λ =8π Se (c, cos ϕ)se (c, cos ϕ ), < ϕ, ϕ < π/. (4) In the imit when the trench recedes to infinity ( with Im c < ):,Im c< = Re () (c, ) + ζ (c, ) (4)

5 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, 5 and the diffracted fied in the quarter space (x, y ) becomes: Hz d,im c< = 8 π + ζ ( ) Re () (c, ) (c, ) (c, )Se (c, cos ϕ )Se (c, η) (43) and is vaid for v π/. The diffracted fied H z,im c< in the quarter space (x, y ) is given by (43) mutipied by ζ, and is vaid in the range 3π/ v π. A. E-poarization IV. Line source incidence Consider an eectric ine source parae to the z-axis and ocated outside the trench at (x, y ) (u, v ), where x >, y >, whose primary eectric fied is E i = ẑez i = ẑh () (kr) (44) Fig. 3. Geometric-optics contribution for ine-source excitation where R = (x x ) + (y y ) (45) is the distance between the ine source and the observation point (x, y) (u, v). The incident fied may be expanded in a series of eiptic-cyinder functions : Ez i = H () (kr) = 4 Re () m= N m (e) m (c, < ) m (c, > )Se m (c, η )Se m (c, η)+ Ro () m (c, < ) m (c, > )So m (c, η )So m (c, η) (46) m where < ( > ) is the smaer (arger) between and. In the quarter-space (x, y ) outside the trench, the tota eectric fied E z may be written as the sum of two terms, as in (3). The geometrica-optics fied E go z is the tota fied that woud be present in the absence of the trench, and is the sum of the fieds due to four ine sources, i.e. the primary ine S and its three images S, S and S 3 (see Fig. 3): E go z = H() where: (kr) H() (kr ) H () (kr ) + H () (kr 3), (47) R = (x x ) + (y + y ), (48) R = (x + x ) + (y y ), (49) R 3 = (x + x ) + (y + y ). (5) The image ine S is ocated at (x, y ) (u, π v ), the image ine S at ( x, y ) (u, π vo), and the image ine S 3 at ( x, y ) (u, π + v ). By expanding the Hanke functions in (47) into series of the type (46) and utiizing properties of the anguar Mathieu functions, it is found that E go z = 6 Ro () (c, <) (c, >)So (c, η )So (c, η), (5) which invoves ony odd Mathieu functions of even order. The diffracted portion Ez d of the fied in the quadrant (x, y ), due to the presence of the trench, and the tota fied E z inside the trench, are simiary given by infinite series of odd Mathieu functions of even order, whose moda coefficients are determined by imposing the boundary conditions. It is found that Ez d = 6 a E z = 6 (c, ) (c, )So (c, η )So (c, η), (5) a (c, ) (c, ) Ro () (c, ) Ro() (c, ) (c, ) So (c, η )So (c, η) (53) where a is given by (8). The surface current densities on the meta boundaries are sti given by () with: 6j H v= = cz a (c, <) Ro () (c, <)+ (c, >)So (c, η ), (54)

6 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, 6 6j H v=π = cz (c, ) Ro () (c, ) Ro() H v=π/ = 6j cz a (c, ) Ro(4) (c, ) (c, >)So (c, η ) H v=3π/ = 6j cz (c, ) So (c, η ), (55) Ro () (c, <) + a (c, <) v So (c, η) a (c, ) Ro () (c, ) Ro(4) (c, ) So (c, η ) 6 H v = = cz η, (56) v=π/ (c, ) Ro () (c, ) v So (c, η), v=3π/ (57) a (c, ) Ro () (c, ) So (c, η )So (c, η). (58) If the trench recedes to infinity ( with Im c < ), then the sit = connects two isorefractive quarter spaces. Use of (7) in (5) and (53) yieds: Ez d,im c< = 6 + ζ Ro () (c, ) (c, ) (c, ) (c, )So (c, η )So (c, η), (59) vaid for v π/. The fied E z,im c< is given by (59) mutipied by minus one, and is vaid for 3π/ v π. Let us now consider the case when the eectric ine source (44) is ocated inside the trench, i.e. < and 3π/ < v < π. The tota eectric fied inside the trench is: where E go z E z = E go z + Ed z, (6) is the geometric-optics fied due to the ine source in the presence of a 9 metaic corner refector, and is sti given by the right-hand side of (5), whereas the diffracted fied component Ez d accounts for the sot at = as we as the eiptic meta wa at =, and may be written as: E d z = 6 ã Ro () (c, ) + c (c, ) So (c, η )So (c, η). (6) The tota fied E z in the quadrant (x, y ) is: E z = 6 c (c, )So (c, η )So (c, η), (6) and the moda coefficients ã and c are expicity determined by imposing the boundary conditions: where ã = Ro(4) (c, ) (o) Ro () ( + ζ ) (c, )Ro(4) (c, ) (c, )Ro() (c, ), (63) c = Ro() (c, ) Ro () (o) (c, ) (c, ) Ro () (c, ) (c, ), (64) (o) =Ro () (c, )Ro(4) (c, ) ( + ζ ) (c, )Ro() (c, ). (65) The surface current densities on the meta boundaries are given by () with: 6j H v= = cz 6j H v=π = cz c (c, )So (c, η ), (66) Ro () (c, <) (c, >) +ã Ro () (c, ) + c (c, ) So (c, η ), (67) H v=π/ = 6j cz H v=3π/ = 6j cz c (c, )So (c, η ) v So (c, η) ã Ro () (c, ) + c (c, ) So (c, η ), (68) v=π/ Ro () (c, <) (c, >)+ v So (c, η), v=3π/ (69) H v = = 6j {ã cz η Ro () (c, )+ } Ro () (c, ) + c (c, ) So (c, η )So (c, η). B. H-poarization (7) The derivations are simiar to those for E-poarization, hence ony the resuts are given. For a magnetic ine source

7 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, 7 parae to the z-axis and ocated outside the trench at (x, y o), whose primary magnetic fied is H i = ẑhz i = ẑh () (kr) (7) with R given by (45), the tota magnetic fied H z in the quarter-space (x, y ) is given by (3), where, with reference to Fig. 3: H go z = H() 6 (kr) + H() (kr ) + H () (kr ) + H () )kr 3) = Re () (c, <) (c, >)Se (c, η )Se (c, η). (7) The diffracted fied H d z in (x, y ) and the tota fied H z inside the trench are given by: H d z = 6 H z = 6ζ (c, ) (c, )Se (c, η )Se (c, η), (c, ) (73) (c, ) (c, ) Re() (c, ) Re () (c, ) Se (c, η )Se (c, η), (74) where is given by (34). The surface current density on the metaic boundaries is given by (35) with H z v= = 6 H z v=π/ = 6 Re () (c, <) + (c, <) (c, >)Se (c, η ), (75) Re () (c, <) + (c, <) (c, >)Se (c, η ) Se (c, η) v=π/, (76) If the magnetic ine source (7) is ocated inside the trench, then the tota magnetic fied inside the trench is H z = H go z + Hd z (8) and outside the trench (i.e., x, y ) is H z, where is given by the right-hand side of (7) and H go z H d z = 6 H z = 6ζ where with b Re () (c, ) + d (c, ) d b = Re(4) (c, ) (e) Re () Se (c, η )Se (c, η), (8) (c, )Se (c, η )Se (c, η), (c, )Re(4) (c, ) (8) ( + ζ ) (c, )Re() (c, ), (83) d = Re() (c, ) Re () (e) (c, ) (c, ) Re () (c, ) (c, ), (84) (e) =Re () (c, )Re(4) (c, ) ( + ζ ) (c, )Re() (c, ). (85) The surface current density on the meta boundaries is given by (35) with H z v= = 6ζ d (c, )Se (c, η ), (86) H z v=3π/ = 6ζ (c, ) (c, ) Re () (c, ) Re () (c, ) Re(4) (c, ) Se (c, η ) Se (c, η) v=3π/, (77) H z v=π =6ζ H z = = 6jζ (c, ) (c, ) Re () (c, ) Re () (c, ) Re(4) (c, ) Se (c, η ), (78) (c, ) Re () (c, ) Se (c, η )Se (c, η). (79) H z v=π/ = 6ζ H z v=3π/ = 6 d (c, )Se (c, η ) Se (c, η) v=π/, (87) Re () (c, <) (c, >)+ b Re () (c, ) + d (c, ) Se (c, η ) Se (c, η) v=3π/, H z v=π = 6 Re () (c, <) (c, >)+ (88) b Re () (c, ) + d (c, ) Se (c, η ), (89)

8 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, 8 σ E (φ)/λ σ H (φ)/λ φ Fig. 4. Bistatic scattering cross section for a pane wave E-poarized incident at an ange ϕ = π/4 when ζ =. The resuts shown correspond to c = (dotted ine), c = π (soid ine), and c = (dash-dot ine). H z = = 6 ( Re () (c, ) + d ) 4 (c, ) V. Numerica resuts b Re () (c, )+ Se (c, η )Se (c, η). (9) As exampes of the computation of the expressions derived in the previous two sections, we provide resuts for bistatic radar cross sections, currents induced aong the metaic was, and radiated fied. The computation of the Mathieu functions was performed using some of the Fortran subroutines provided in 8. However, these subroutines are based on the Gostein-Ince normaization 9,, σ E (φ)/λ φ Fig. 5. Bistatic scattering cross section for a pane wave E-poarized incident at an ange ϕ = π/4 when ζ = /. The resuts shown correspond to c = (dotted ine), c = π (soid ine), and c = (dash-dot ine) φ Fig. 6. Bistatic scattering cross section for a pane wave H-poarized incident at an ange ϕ = π/4 when ζ =. The resuts shown correspond to c = (dotted ine), c = π (soid ine), and c = (dash-dot ine). so they were appropriatey modified to account for the normaization used in this work. In particuar, these routines were transated to Fortran 9 so that verification of computations at quadrupe precision coud be performed. In a the figures that foow, we refer to a cavity with size =. Resuts are presented for three vaues of the dimensioness parameter c given in (8):, π, to account for different cases for the size of the aperture OD of Fig. with respect to the waveength λ. The first resut, shown in Fig. 4, is the radar cross section (6) for the E-poarized pane wave () incident at an ange ϕ = π/4 when ζ =. When c = (dotted ine) the aperture is smaer than the waveength and the RCS does not vary as much as it does for the two other cases where the aperture is equa to the waveength, c = π (soidine), or greater than it, c = (dash-dot ine). When the materia properties are changed, the shape of the RCS curves remains quaitativey the same but the magnitudes change, as it is shown in Fig. 5 that anayzes the RCS when ζ = /. The RCS for the H-poarized pane wave (9) incident at an ange ϕ = π/4 when ζ = is given in Fig. 6. The currents induced on the metaic was of the structure of Fig. are of interest because they provide a way to compare resuts with other numerica methods such as discussed in 5, 6. Fig. 7 shows the currents J z induced by the E-poarized pane wave (), incident at an ange ϕ = π/4 when ζ =. These resuts were computed using eqs. ()-(5) with the hep of the Shanks transform acceeration method appied ony to the imaginary part of the terms of the series invoved, simiar to what is described in. Resuts for the eectrica ine source (44) ocated outside the trench at ( =.5, v = π/6) when ζ = are given in Fig. 8. Note that there is no singuar behavior of J z

9 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, 9 J z J z b..8 a J z x 3 a b.6.4. J z J z x 3 c 6 4 J z x 3 c x 3 d x 3 d J z 4 J z e v 3 x x 3 v e J z J z Fig. 7. Currents induced on the metaic was of the structure of Fig. due to an unit ampitude pane wave eectricay poarized and incident at an ange ϕ = π/4. Current J z aong DE is shown in (a); current J z aong OA is shown in (b); current J z aong OB is shown in (c); current J z aong BC is shown in (d); and current J z aong DC is shown in (e). The resuts shown in (a-e) correspond to c = (dotted ine), c = π (soid ine), and c = (dash-dot ine) Fig. 8. Currents induced on the metaic was of the structure of Fig. due to an eectric ine source ocated at ( =.5, v = π/6) when ζ =. Current J z aong DE is shown in (a); current J z aong OA is shown in (b); current J z aong OB is shown in (c); current J z aong BC is shown in (d); and current J z aong DC is shown in (e). The resuts shown in (a-e) correspond to c = (dotted ine), c = π (soid ine), and c = (dash-dot ine). in part c) around =. These resuts were computed using eqs. (54)-(58). In particuar, in addition to the use of the aforementioned acceeration technique, the geometrica optics contributions were computed using the direct expression in terms of Hanke functions instead of using its expansion with Mathieu functions. Resuts for an eectric ine ocated inside the trench at ( =.5, v = 5π/) when ζ = / are reported in Fig. 9. These resuts were computed using eqs. (66)-(7). As before, Shanks acceeration technique was used and the geometrica optics terms were evauated using the direct expressions in terms of Hanke functions. Fig. gives J z induced by a ine source ocated inside the trench at ( =.5, v = 5π/) when ζ =. These resuts were computed using eqs. (86)-(9) and were com- puted using Shanks acceeration technique as we as with a direct evauation of the geometrica optics contribution using Hanke functions. A figures 7- have in common that a more compex behavior of J z is observed for arger vaues of c (dash-dot ine, c = ). Another common feature among a figures 7- is the continuity of J z whie moving aong the metaic surfaces around points B and C of Fig.. In fact, since there are no ocaized currents at such points, the continuity around B is proven by the behavior at = of part c) and at v = 3π/ of part d), whereas the continuity around C is guaranteed by the behavior around v = π of part d) and = of part e).

10 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, 5 x 4 a a 4.8 J z 3 J.6.4 J z J z J z J z x x x Fig. 9. Currents induced on the metaic was of the structure of Fig. due to an eectric ine source ocated at ( =.5, v = 5π/) when ζ = /. Current J z aong DE is shown in (a); current J z aong OA is shown in (b); current J z aong OB is shown in (c); current J z aong BC is shown in (d); and current J z aong DC is shown in (e). The resuts shown in (a-e) correspond to c = (dotted ine), c = π (soid ine), and c = (dash-dot ine). v b c d e J J J v J b c d v e Fig.. Currents induced on the metaic was of the structure of Fig. due to a magnetic ine source ocated at ( =.5, v = 5π/) when ζ =. Current J aong DE is shown in (a); current J aong OA is shown in (b); current J aong OB is shown in (c); current J v aong BC is shown in (d); and current J aong DC is shown in (e). The resuts shown in (a-e) correspond to c = (dotted ine), c = π (soid ine), and c = (dash-dot ine). A parts a) and e) of the figures reated to E-poarized sources, i.e. Fig. 7-9, show singuar behaviors at =. These singuarities occur near the edge of the baffe DC at point D of Fig. and are due to the factor / that appears in () and () for pane wave incidence, (54) and (55) for a ine source outside the trench, (66) and (67) for a ine source inside the trench. Note that this singuar behavior is not observed in Fig. that refers to a magnetic ine source. As a ast exampe, Fig. shows the magnitude of the eectric fied due to an eectric ine ocated inside the trench at ( =.5, v = 5π/). The magnitude Ez is evauated aong the hyperboa with η = / when ζ = /. This geometry may be used to examine the fied radiated by a wire that is ocated inside the trench of Fig.. In addition, the fied computed for this situation may be compared with the fied computed by other methods, simiar to what is shown in 4. As before, Shanks acceeration technique was used and the geometrica optics terms were evauated using the direct expressions in terms of Hanke functions. The strongest fied contribution is observed when c = π, i.e. d/λ =, which may suggest the existence of a resonance condition that causes a better couping with the fied outside the trench. One shoud aso observe the change of sope at y/d =, due to the transition from the interior of the cavity to the exterior, as we as to the change of materia.

11 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, E z J. A. Stratton, Eectromagnetic Theory, McGraw-Hi, New York, 94. J. J. Bowman, T. B. A. Senior, and P. L. E. Usenghi, Eectromagnetic and Acoustic Scattering by Simpe Shapes, Hemisphere Pubishing Corporation, New York, P.L.E. Usenghi, Exact penetration, radiation and scattering for a sotted semieiptica channe fied with isorefractive materia, IEEE Trans. Antennas Propagat., vo. 5, no. 6, pp , June 4. 4 D. Erricoo, M. D. Lockard, C. M. Buter, and P. L. E. Usenghi, Numerica anaysis of penetration, radiation, and scattering for a D sotted semieiptica channe fied with isorefractive materia, PIER, vo. 53, pp , 5. 5 D. Erricoo, M.D. Lockard, C.M. Buter, and P.L.E. Usenghi, Comparison among currents on surfaces inside and near a semieiptica channe fied with isorefractive materia that backs a sotted pane: currents computed by anaytica formuas and by integra equation methods, in Proc. Int. Conf. on Eectromagnetics in Advanced Appications (ICEAA 3), Torino, Itay, Sept 3, pp D. Erricoo, M.D. Lockard, C.M. Buter, and P.L.E. Usenghi, Currents on conducting surfaces of a semieiptica-channebacked sotted screen in an isorefractive environment, IEEE Trans. Antennas Propagat., 3, accepted. 7 P.L.E. Usenghi, Exact eectromagnetic penetration througha gap in a corner, in Proc. 999 IEEE-APS/URSI Int. Symp, Orando, FL, USA, Juy 999, p S. Zhang and J.-M. Jin, Computation of Specia Functions, Wiey, New York, S. Godstein, Mathieu functions, Camb. Phi. Soc. Trans., vo. 3, pp , 97. E. L. Ince, Tabes of eiptic cyinder functions, Roy. Soc. Edin. Proc., vo. 5, pp , 93. M. Abramovitz and I. A. Stegun, Handbook of Mathematica Functions, Dover Pubications, Inc, New York, 97. D. Erricoo, Acceeration of the convergence of series containing Mathieu functions using Shanks transformation, IEEE Antennas and Wireess Propagation Letters, vo., pp. 58 6, y/d Fig.. Magnitude of the eectric fied aong the hyperboa η = / due to an eectric ine source ocated at ( =.5, v = 5π/) when ζ = /. The resuts correspond to c = (dotted ine), c = π (soid ine), and c = (dash-dot ine). VI. Concusion Exact anaytica soutions have been derived for pane wave and ine source incidence on a sotted quartereiptica trench in a corner. These new canonica soutions are important not ony per se, but aso because they provide benchmarks for the vaidation of frequency-domain codes invoving geometries with trenches, sharp edges, and different penetrabe materias. Numerica resuts based on these exact soutions have been obtained and discussed. It woud be interesting to investigate the approximate forms that these exact soutions take in the ow-frequency (c ) and high-frequency (c ) imits. However, these studies woud require a detaied knowedge of the behavior of the Mathieu functions in the Stratton-Chu notation for sma and arge vaues of the parameter c, and wi constitute the topic of future investigations. Danio Erricoo (S 97 - M 99 - SM 3) For a biography and photo, see the September 4 issue of this TRANSACTIONS. Piergiorgio L. E. Usenghi (SM 7 - F 9) For a biography and photo, see the June 4 issue of this TRANSACTIONS. References