Green's function analysis of an ideal hard surface rectangular waveguide

Authors

  • Wei Huang,

    1. Center for Applied Electromagnetic Systems Research, Department of Electrical Engineering, University of Mississippi, University, Mississippi, USA
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  • Alexander B. Yakovlev,

    1. Center for Applied Electromagnetic Systems Research, Department of Electrical Engineering, University of Mississippi, University, Mississippi, USA
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  • Ahmed A. Kishk,

    1. Center for Applied Electromagnetic Systems Research, Department of Electrical Engineering, University of Mississippi, University, Mississippi, USA
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  • Allen W. Glisson,

    1. Center for Applied Electromagnetic Systems Research, Department of Electrical Engineering, University of Mississippi, University, Mississippi, USA
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  • Islam A. Eshrah

    1. Center for Applied Electromagnetic Systems Research, Department of Electrical Engineering, University of Mississippi, University, Mississippi, USA
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Abstract

[1] Green's function analysis of an ideal hard surface rectangular waveguide is proposed for characterization of the modal spectrum of the structure. A decomposition of the hard surface waveguide into perfect electric conductor and perfect magnetic conductor waveguides allows the representation of dyadic Green's function as a superposition of transverse magnetic (TM) and transverse electric (TE) waveguide modes, respectively. In addition, a term corresponding to a transverse electromagnetic (TEM) mode is included in the eigenmode expansion of the Green's dyadic. It is shown that the TEM mode solution can be obtained by solving vector Helmholtz's equation in the zero cutoff limit with the corresponding boundary conditions of electric field on the ideal hard surface. The electric field distribution due to an arbitrarily oriented electric dipole source is illustrated for a few representative TM, TE, and TEM modes propagating in the ideal hard surface rectangular waveguide. The proposed model is verified by analyzing a realistic hard surface square waveguide using the Ansoft High-Frequency Structure Simulator (HFSS).

1. Introduction

[2] The concept of artificial hard surfaces, commonly used in acoustic applications, has been introduced in electromagnetics in the last two decades [Kildal, 1990]. The artificial hard surface represents a surface with anisotropic impedance. For a planar artificial hard surface, the relation between the tangential electric field and tangential magnetic field is given by, Etequation image + Elequation image = ZtHlequation imageZlHtequation image, where Zt and Zl are the transverse and longitudinal anisotropic surface impedances, respectively, with respect to the propagating direction [Ruvio et al., 2003]. This results in the characterization of the ideal hard surface by imposing conditions on the surface impedances such that, Zl = 0 and ∣Zt∣ → ∞. Because of this property, ideal hard surfaces can be created by introducing narrow longitudinal alternating perfect electric conductor (PEC) and perfect magnetic conductor (PMC) strips along the propagating direction and allowing the strip width to approach zero. However, in practice, there are no microwave materials which behave similarly to a PMC. One way to realize an artificial PMC is to use a grounded dielectric slab of certain thickness. Another realization is the corrugated surface with dielectric-filled longitudinal corrugations along the propagating direction. Compared with the strip-loaded dielectric slab, the corrugated surface has more weight and higher manufacturing cost [Sipus et al., 1997].

[3] Hard surfaces can also be implemented in guided-wave structures, for example, rectangular or circular waveguides. Modal characteristics of an ideal hard surface circular waveguide have been recently investigated by Klymko et al. [2005]. In this paper, we study a rectangular waveguide with an ideal hard surface boundary conditions, which is modeled by alternating longitudinal PEC and PMC strips with vanishing widths. The geometry is depicted in Figure 1, where l is the longitudinal direction of wave propagation, n is the inward normal direction to the waveguide boundary, and t is the direction tangential to the waveguide boundary and transverse to the longitudinal strips. The longitudinal PEC strips enforce ElS = 0, so that Zl satisfies the longitudinal impedance condition, Zl = 0. The longitudinal PMC strips require HlS = 0, resulting in the transverse impedance condition, ∣Zt∣ → ∞ [Ruvio et al., 2003]. When the hard surface behaves like a PEC conductor, Et and the normal derivative of En are zero on the waveguide surface Se, that is, equation image and equation image, and when the hard surface behaves like a PMC conductor, En and the normal derivative of Et are zero on the waveguide surface Sm, that is, equation image and equation image.

Figure 1.

PEC/PMC strip model for an ideal hard surface rectangular waveguide. The hard surface behaves like a PEC waveguide for TM modes and as a PMC waveguide for TE modes. A TEM wave is obtained as the solution of vector Helmholtz's equation in the zero cutoff limit subject to the boundary conditions of electric field (shown in the figure) on the ideal hard surface.

[4] Compared to PEC and PMC rectangular waveguides, which can support only transverse magnetic (TM) and transverse electric (TE) modes, the most important feature of this “ideal hard surface rectangular waveguide” is that it allows a propagation of the arbitrarily polarized uniform transverse electromagnetic (TEM) mode with a zero cutoff frequency, and can be used as a miniaturized feeding network in dual-polarized antenna arrays. This waveguide will support only the TEM mode if the frequency is below the cutoff of any of TE and TM modes. Whereas the TM and TE modes “see” the waveguide as a PEC and PMC waveguide, respectively, the TEM mode “sees” the ideal hard surface waveguide as a combination of PEC and PMC waveguides modeled by alternating PEC/PMC strips with vanishing widths, and it satisfies the boundary conditions on the ideal hard surface partially contributed by PEC and PMC waveguides, that is, ∂En/∂nS = 0 and ∂Et/∂nS = 0 (as shown in Figure 1).

[5] Different analytical and numerical techniques have been used for the analysis of hard surfaces. A Green's function approach has been used to model different realizations of open boundary planar hard surfaces, including ideal PEC/PMC strip models, corrugated surfaces, and strip-loaded grounded slabs [Sipus et al., 1997]. Image theory has been applied to study hard surfaces with a time-harmonic source [Lindell, 1995]. Another effective and simple approach to analyze hard surfaces is based on the asymptotic boundary condition, which has been used to analyze a corrugated surface [Kishk et al., 1998], grids of metal strips [Kishk and Kildal, 1997], circular waveguide with strip-loaded walls [Kishk and Morgan, 2001], and hard surfaces of two-dimensional structures [Kishk, 2003].

[6] The most common use of artificial hard surfaces is the hard horn antenna, which can support a TEM wave at the design frequency. The hard boundary results in a uniform field distribution over the horn aperture with zero cross polarization [Kildal, 1988]. The advantage of using a hard horn antenna is that it can increase the directivity and decrease the beam width for given aperture dimensions [Lier and Kildal, 1988]. Also, an infinite planar array of open-ended rectangular waveguides, whose E plane walls are dielectric-loaded hard walls, can achieve better matching of the array to free space and increase the aperture and gain to remove the scan blindness [Skobelev and Kildal, 1998]. The performance of an open-ended circular waveguide with strip-loaded dielectric hard walls used as the element of an antenna array is also studied by Skobelev and Kildal [2000].

[7] Recently, a rectangular waveguide with sidewalls covered by a printed dipole frequency selective surface (FSS) has been designed to obtain the quasi-TEM mode [Cucini et al., 2004; Maci et al., 2005]. Also, an electromagnetic band gap (EBG) structure implemented on the sidewalls of a rectangular waveguide has been used to create the TEM mode propagation [Yang et al., 1999]. Miniaturized dielectric-loaded quasi-TEM rectangular waveguides have been proposed for dual-band array applications [Kehn et al., 2004; Kehn and Kildal, 2005].

[8] The purpose of the present paper is to develop an electric dyadic Green's function for the modal analysis of an ideal hard surface rectangular waveguide excited by an arbitrarily oriented electric current source. A procedure for deriving the Green's function in terms of solenoidal and irrotational parts is presented, wherein the solenoidal part of the Green's function is obtained in the eigenmode expansion form as a superposition of three terms associated with the TM, TE, and TEM modes of the ideal hard surface waveguide. A term corresponding to the TEM mode is obtained analytically as the solution of vector Helmholtz's equation in the zero cutoff limit subject to the boundary conditions for the electric field on the ideal hard surface. Numerical results for field distributions are demonstrated for the TEM mode and a few representative TM and TE modes propagating in a rectangular waveguide with ideal hard surface boundary conditions due to an arbitrarily oriented electric dipole source. A uniform field distribution of the TEM mode is verified by modelling a realistic hard surface square waveguide using High-Frequency Structure Simulator (HFSS) commercial software [Ansoft, 2004].

2. Dyadic Green's Function for Ideal Hard Surface Waveguide

[9] Consider an ideal hard surface rectangular waveguide with the impressed electric current source as shown in Figure 2. The electric field at an interior point of the waveguide (including the source region) is expressed in the integral form

equation image

where equation image(r, r′) is the electric-type dyadic Green's function for the ideal hard surface rectangular waveguide obtained as the solution of a dyadic wave equation

equation image

subject to boundary conditions on the waveguide surface. In (2), k0 = ωequation image.

Figure 2.

Geometry of an infinite ideal hard surface rectangular waveguide with the impressed electric current source. A slice pillbox principal volume associated with the source plane singularity of the Green's function at z = z′ is shown in the source region.

[10] The Green's function is obtained in the eigenmode expansion form as a superposition of three parts associated with the TM modes of the PEC waveguide, the TE modes of the PMC waveguide, and the TEM mode of the ideal hard surface:

equation image

This corresponds to the decomposition of the boundary value problem for the Green's function into three problems. Thus the equation imageTM(r, r′) part of the Green's function is obtained in terms of the TM modes of the PEC waveguide subject to the first-kind boundary conditions on Se,

equation image

where equation image is the inward normal to Se; equation imageTE(r, r′) part is expressed in terms of the TE modes of the PMC waveguide subject to the second-kind boundary conditions on Sm:

equation image

The TEM part of the Green's function satisfies mixed boundary conditions, which are analogous to those for the normal and tangential components of the electric field, ∂En/∂nS = 0 and ∂Et/∂nS = 0 (shown in Figure 1), which may be expressed as

equation image

The TEM part of the Green's function is obtained by solving the vector Helmholtz's equation in the zero cutoff limit subject to the boundary conditions for the electric field components on the ideal hard surface.

[11] It should be noted that the procedure for obtaining the Green's function for the ideal hard surface waveguide is based on the eigenmode expansion originally introduced for cylindrical PEC waveguides [Collin, 1991; Pathak, 1983]. Because of the decomposition of the ideal hard surface waveguide into PEC and PMC waveguides, the Green's function is expressed in terms of TM modes of the PEC waveguide and TE modes of the PMC waveguide, with an additional term corresponding to the TEM mode.

[12] The electric Green's dyadic of the ideal hard surface waveguide consists of solenoidal and irrotational parts [Collin, 1991; Johnson et al., 1979], that is,

equation image

where the solenoidal part (first term) is obtained in the eigenmode expansion form in terms of TM, TE, and TEM modes of the hard surface waveguide and it is understood in the principal value (PV) sense [Yaghjian, 1982; Chew, 1989; Eshrah et al., 2004]. The irrotational part (second term) includes a depolarizing dyadic equation image, which is associated with a specific principal exclusion volume [Yaghjian, 1980]. For the eigenmode expansion used in this formulation, it is natural to choose a slice pillbox principal volume with the normal in the propagating direction (z direction, Figure 2) [Eshrah et al., 2004; Yaghjian, 1980; Wang, 1982; Viola and Nyquist, 1988], wherein equation image = equation imageequation image. It should be noted that the irrotational part in (7) contains a delta function singularity, which does not represent a main singularity of the Green's function [Johnson et al., 1979]. The irrotational part contributes to the field in the source region as an integral over the entire source region. The main singularity of the Green's function is contained in the solenoidal part and represents a source plane singularity at z = z′. It is understood as an improper integral with the slice pillbox principal volume. The Green's function represented by (7) is uniquely defined for the ideal hard surface waveguide as a superposition of solenoidal and irrotational parts. However, in general, the solenoidal and irrotational parts are not necessarily uniquely defined. Their representation depends on the shape of the principal volume.

[13] The electric field inside the waveguide given by (1) is uniquely defined as the sum of the improper integral of the solenoidal part understood in the principal value sense and the integral of the irrotational part over the entire volume of the impressed current source as

equation image

[14] Below we summarize the procedure for obtaining electric and magnetic fields of TM and TE eigenmodes of PEC and PMC waveguides, respectively. The electric fields of TM and TE eigenmodes propagating in the positive (plus) and negative (minus) z direction, EmnTM±(r) and EmnTE±(r), are expressed in terms of electric vector wave functions [Collin, 1991] as

equation image

where γmn = equation image, kxm = mπ/a, kyn = nπ/b. The corresponding magnetic fields of the TE and TM eigenmodes are obtained in terms of magnetic vector wave functions

equation image

The transverse and longitudinal vector wave functions of the TM modes normalized by power in the PEC waveguide are obtained as

equation image

where kcmn = equation image and Zmne = γmn/jωɛ0. Similarly, the transverse and longitudinal vector wave functions of TE modes normalized by power in the PMC waveguide are obtained as

equation image

where Zmnh = jωμ0mn.

[15] Substituting (9) together with (11) and (12) into the eigenmode expansion (7), we obtain the representation for the nine components of the electric dyadic Green's function of the ideal hard surface rectangular waveguide:

equation image

Below it is shown that the TEM part of the Green's function consists of GxxTEM and GyyTEM components only.

[16] The vector Helmholtz's equation for the transverse electric field is reduced to Laplace's equation for the transverse electric vector wave functions in the zero cutoff limit, kc → 0, so

equation image

subject to the boundary conditions on the ideal hard surface of the waveguide a × b (Figure 1) given by

equation image

The solution of the boundary value problem (14), (15) is obtained by separation of variables as

equation image

where A and B are constants. The corresponding magnetic vector wave functions are related to ex and ey through the intrinsic impedance of free space, η0, as

equation image

The two orthogonally polarized fields (ex, hy) and (ey, hx) are normalized by power resulting in

equation image

Finally, the TEM term of the Green's function is obtained as

equation image

3. Numerical Results

[17] The electric dyadic Green's function derived in the previous section has been used to obtain the electric field of a few representative TM, TE, and TEM modes excited in the ideal hard surface rectangular waveguide by an electric dipole source. Figures 3, 4, 5, 6, and 7demonstrate the transverse electric field of the TM11, TE11, and TEM modes at a distance of z = λg/2 (where λg is the guided wavelength of the corresponding mode) from the source plane at z0 = 0. The modes propagate in the ideal hard surface rectangular waveguide of cross section a × b = 22.86 mm × 10.16 mm at a frequency of 17 GHz. The electric dipole is positioned at (x0, y0, z0) = (a/4, b/4, 0).

Figure 3.

Transverse electric field of the TM11 mode due to the electric dipole J(r) = equation imageδ(xa/4)δ(yb/4)δ(z).

Figure 4.

Transverse electric field of the TE11 mode due to the electric dipole J(r) = equation imageδ(xa/4)δ(yb/4)δ(z). The behavior of the electric field is similar to that of the magnetic field of the TM11 mode propagating in a PEC waveguide.

Figure 5.

Transverse electric field of the TEM mode due to the electric dipole J(r) = equation imageδ(xa/4)δ(yb/4)δ(z).

Figure 6.

Transverse electric field of the TEM mode due to the electric dipole J(r) = equation imageδ(xa/4)δ(yb/4)δ(z).

Figure 7.

Transverse electric field of the TEM mode due to the electric dipole J(r) = (equation image cos(π/3) + equation image sin(π/3))δ(xa/4)δ(yb/4)δ(z).

[18] Figure 3 shows the transverse electric field of the TM11 mode due to an x-directed electric dipole source. Obviously, the electric field distribution of this and, in general, all TMmn modes is the same as in the PEC waveguide.

[19] The transverse electric field of the TE11 mode due to an x-directed electric dipole is shown in Figure 4. It should be noted that the electric field of TEmn modes propagating in the ideal hard surface waveguide (as in the PMC waveguide) is similar to the distribution of the magnetic field of the corresponding TMmn modes (propagating in a PEC waveguide). This results in the conclusion that the TEm0 and TE0n modes cannot propagate in the ideal hard surface waveguide.

[20] Figures 5, 6, and 7 demonstrate the transverse electric field of the TEM mode due to an x-directed, y-directed, and 60°-directed electric dipole, respectively. It can be seen that the field is uniform over the waveguide cross section, and the polarization of the field depends on the polarization of the source. Clearly, the ideal hard surface rectangular waveguide supports a uniform TEM mode polarized according to the orientation of the source. This waveguide can be used for the miniaturization of feeding networks for dual-polarized antenna applications.

[21] To validate the proposed modal analysis of the ideal hard surface waveguide model, a realistic hard surface square waveguide is simulated using commercial High-Frequency Structure Simulator (HFSS) software [Ansoft, 2004]. The structure consists of a strip-loaded dielectric-filled square waveguide (with the waveguide cross section shown in Figure 8) in order to create a hard surface boundary condition on the interface of a dielectric slab of thickness d = λ0/4equation image [Kildal, 1990], where λ0 is the free space wavelength at the operation frequency and ɛr is the relative dielectric permittivity of the slab. This hard surface square waveguide is designed at 10 GHz with the following geometrical and material parameters: outside waveguide dimensions are 23.74 mm × 23.74 mm, inside waveguide dimensions are 9.5 mm × 9.5 mm, waveguide length is 60 mm, slab thickness is 6.12 mm and permittivity is 2.5, strip width is 0.5 mm and gap width is 1.5 mm (w/p = 0.75). The number of strips on each side is chosen to satisfy a hard surface condition, typically 10∼20 strips per guide wavelength.

Figure 8.

Cross section of a realistic hard surface square waveguide model.

[22] The results shown in Figures 9, 10, and 11 are obtained for the electric field distribution and field magnitude of the modes supported by the realistic hard surface square waveguide. It can be seen that this waveguide supports an arbitrarily polarized uniform field distribution in the waveguide inner region (similar to the TEM mode propagating in the ideal hard surface waveguide). Strip-loaded dielectric slabs create a hard surface condition, which allows to obtain a uniform field in the region bounded by the hard surface. It should be noted that the field magnitude level shown for all three polarizations (Figures 9–11) is normalized to the case of 45° polarization (Figure 11), which is obtained as a superposition of horizontally and vertically polarized fields shown in Figures 9 and 10. The presented analysis clearly demonstrates that the TEM modal behavior of the ideal hard surface waveguide can adequately model a uniform field distribution of the realistic hard surface waveguide at the design frequency. A TEM field distribution obtained by the Green's function analysis for the ideal hard surface square waveguide with dimensions 9.5 mm × 9.5 mm due to a 45°-polarized electric dipole source is shown in Figure 12. It can be seen that the field in Figure 12 is similar to that of the realistic hard surface square waveguide (inner region bounded by the hard surface) shown in Figure 11. Other cases of vertically and horizontally polarized fields for ideal hard surface square waveguide produce similar results to those shown in Figures 9 and 10 for the case of the realistic waveguide and omitted here for brevity.

Figure 9.

(top) Electric field distribution and (bottom) electric field magnitude with a horizontal polarization in the realistic hard surface square waveguide at 10 GHz.

Figure 10.

(top) Electric field distribution and (bottom) electric field magnitude with a vertical polarization in the realistic hard surface square waveguide at 10 GHz.

Figure 11.

(top) Electric field distribution and (bottom) electric field magnitude with a 45° polarization in the realistic hard surface square waveguide at 10 GHz.

Figure 12.

Transverse electric field of the TEM mode due to the electric dipole J(r) = (equation image cos(π/4) + equation image sin(π/4))δ(xa/2)δ(yb/2)δ(z).

[23] The good agreement between the field behavior obtained using the theoretical analysis of the ideal structure and the simulations of the realistic structure implies that the constructed Green's function can be used to predict the response of the practical hard surface waveguide within the frequency range where the hard boundary conditions are satisfied.

4. Conclusion

[24] The Green's function analysis of the ideal hard surface rectangular waveguide is based on the decomposition of the structure into PEC and PMC waveguides. The electric Green's dyadic due to an arbitrarily oriented electric dipole source is obtained in terms of solenoidal and irrotational parts. The solenoidal part is expressed by the eigenmode expansion of TM and TE modes of PEC and PMC waveguides, respectively, with an additional term associated with the TEM mode. The TEM term of the Green's dyadic is obtained as the solution of the vector Helmholtz's equation for the electric field in the zero cutoff limit subject to the appropriate boundary conditions on the ideal hard surface. It is shown that the TEM mode of the ideal hard surface waveguide has uniform field distribution over the waveguide cross section, and the polarization of the mode necessarily depends on the polarization of the electric dipole source. A uniform field distribution of the TEM mode is verified by modeling a realistic hard surface square waveguide using HFSS. It is shown that the arbitrarily polarized uniform field obtained in the realistic hard surface waveguide at the design frequency is similar to the TEM mode distribution of the ideal hard surface waveguide.

Acknowledgments

[25] This work was partially supported by the National Science Foundation under grant ECS-0220218 and by NASA under EPSCoR Cooperative Agreement NCC5-574. The authors are thankful to Dejan Filipovic and Michael Buck from the University of Colorado at Boulder and Richard Remski from Ansoft Corporation for helpful discussions on modeling hard surface waveguides using HFSS.

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