An Efﬁcient Formulation for Calculating the Modal Coupling for Open-Ended Waveguide Problems

— A double line integral representation of the mutual coupling between open-ended waveguides of arbitrary cross section is presented, which is useful to speed up calculations inside the framework of a Galerkin Abstract— This letter presents a novel design of triangular microstrip antenna with dual-frequency operation. In this design the microstrip patch is short-circuited using a shorting pin and fed by a single probe feed. By varying the shorting-pin position in the microstrip patch, such a design can provide a large tunable frequency ratio of about 2.5–4.9 for the two operating frequencies. Experimental results are presented and discussed.

the integration end-points are determined by the condition z s = 0 and described by the unit step function in (5).
It is worth noting that for large apertures in terms of a wavelength, F po (`0) is a rapidly oscillating function of`0. Consequently, the integration in (3) can be asymptotically evaluated by its stationary phase point contributions, thus, leading to a UTD-type ray-field representation. However, this latter fails in describing the field close to and at the axial caustic and it has also been found less accurate with respect to the present numerical line integration for moderate sized apertures.

II. NUMERICAL RESULTS
Numerical results from AI (continuous line) have been compared with those from LI (dashed line) for the case of a circular OEW with radius a. In particular, Fig. 2(a) shows results for a waveguide with radius a = 0:5, excited by the TE 11 mode. The component of the electric field in the H plane is plotted at a distance r = 1:5 and r = 0:7, respectively. Both curves are normalized with respect to the maximum value obtained in the case r = 0:7; furthermore, the field is calculated in the region external to the waveguide; i.e., < 130 for r = 0:7 and < 160 for r = 1:5. Normalized near field patterns for TM 11 -mode excitation are presented in Fig. 2 The component of the electric field in the H plane is plotted for the two cases a = 0:65, r = 1:5, and a = , r = 2, respectively. The curves corresponding to this latter case are shifted 10 dB down to render the figure more readable.
In spite of the moderate size of the apertures, the agreement between the AI and its corresponding LI has been found quite satisfactory over the total 40-dB dynamic range. The small glitches arise from the fact that the IGCO integration has been turned off when L does not intersect the edge [see Fig. 1(c)]. The result presented here also suggests an effective method to speed-up practical calculations of the interaction between modes [6].

I. INTRODUCTION AND FORMULATION
The theorem demonstrated in [1] establishes a rigorous equivalence between the field predicted by the Kirchhoff-type aperture integration (AI) and that radiated by the physical optics (PO) wall current. By using this equivalence, a formulation has been presented [2] to asymptotically reduce the AI into a line integration (LI) along the waveguide edge.
In this paper, the equivalence between PO and AI [1] is applied to modal coupling between OEW's of arbitrary cross-sections. This allows for the derivation of a convenient double integral expression for the modal coupling to be applied in the framework of a method of moments (MoM) for arrays of OEW's and horns. Note that a MoM Galerkin mutual impedance is generally given in terms of a quadruple integral in the space domain and the reduction to a double line integral is possible only for rectangular coplanar apertures [3]. When closed-form Fourier transform representations of modal coplanar distributions are available, one can resort to a spectraldomain approach; however, the resulting double spectral integrals are improper and slowly convergent. The method presented here is independent from the waveguide cross sections and, although it is developed here only for coplanar apertures, it can be easily extended to the noncoplanar case.
Let us consider two open-ended waveguides OEW1 and OEW2 of arbitrary cross sections. For the sake of simplicity, but without loss of generality, we will assume the two axes of the OEW's to be parallel. Two reference systems (x 0 ; y 0 ; z 0 ) and (x 00 ; y 00 ; z 00 ) are introduced in which z 0 < 0 and z 00 < 0 are the semi-axes of OEW1 and OEW2, respectively ( Fig. 1). Let us denote by where A i is the aperture of OEWi. It is worth pointing out that calculating C nm 12 as in (1) where the surface S1 is now the wall of OEW1. By inspection, it is apparent that (1) and (3) where S2 denotes the wall of OEW2, G E J (jr2 0r1j) is the pertinent free-space dyadic Green's function andr i is the position vector of dSi.
The double integral is asymptotically dominated by the contributions from three two-dimensional critical points, i.e., one double end-point (EP) at (z 0 ; z 00 ) = (0;0) and two EP-stationary phase points (SPP's) at (z 0 ; z 00 ) (z 1 ; 0) and at (z 0 ; z 00 ) (0; z 2 ). The integration along`0 of all the contributions at (z 1 ; 0) asymptotically reconstructs the field radiated by the modal current of waveguide 1 when the latter is supposed to be infinite. Since these modal currents are exact for the infinite structure, their radiation rigorously produces a zero field outside the waveguide. This means that practically we need to calulate only the contribution c 00 (`0;`0 0 ) from the double EP.
The transition function in parentheses in (8) reduces to one for large kR(0; 0) and far from the shadow boundary cones of each strip [2]. In this case, the contribution c 00 (`0;`0 0 ) can be interpreted as the coupling coefficient between two elementary electric dipoles distributed on the two rims and oriented as the local modal currents. In practical applications, the presence of the factor into parentheses of (8) becomes practically relevant close to the cut-off condition of one of the two coupled modes, where the associated ray-mode angle approaches 90 .
In Fig. 2, the real part of the coupling coefficient C nm 12 calculated via (1) (quadruple integral, dashed line) is compared to the one calculated with the present method (double integral, dashed line) for two equal circular OEW's with various radii a. The coupling coefficient has been calculated as a function of the minimum distance between the aperture rims for a pair of TE11 [ Fig. 2(a)] and TM 11 [ Fig. 2(b)] mode distributions (shown in the insets). The calculation time for the double line integration is reduced by a factor 34 with respect to the corresponding quadruple integral. In spite of the moderate size of the apertures, the agreement has been found quite satisfactory even for small distances between the OEW's.
In conclusion, it should be noted that owing to the exponential factor exp[0jkR(0; 0)], the incremental coupling contribution c00(`0;`0 0 )i (8) is a rapidly oscillating function of both`0 and`0 0 . Consequently, the integration in (5) could be asymptotically evaluated by its stationary phase point contributions for larger d, thus leading to a complete closed-form representation.

I. INTRODUCTION
Although dual-frequency operations of microstrip patch antennas have received much attention and many related designs have been demonstrated, most of the designs provide only a moderate frequency ratio (FR) between the two operating frequencies (about 1.3-2). The reports on dual-frequency operation with FR greater than two are relatively limited. Typical designs that use a single-layer, singlepatch structure and provide a large frequency ratio include loading the rectangular microstrip patch with two varactor diodes [1], shortcircuiting the rectangular microstrip patch with a shorting pin [2], etc. The former design can provide a dual-frequency operation with FR about 5.0 but it needs external circuitry to supply the bias voltage for the diodes, which makes the structure relatively complicated. As for the latter, the structure is much simpler and requires only proper selection of the shorting-pin position in the microstrip patch. Moreover, the impedance matching for operating at the two frequencies can be achieved using a single probe feed. The reported FR for such a design using a rectangular patch is tunable in the range 2.0-3.2 [2] and from the study on compact triangular microstrip antennas that use a similar shorting-pin technique [3], which shows a greater resonant frequency reduction than the rectangular or circular patches, it is expected that the tunable FR range can also be greater for a triangular patch than for a rectangular one. This motivates the present study. The design for a dual-frequency triangular microstrip antenna with a shorting pin is described and typical experimental results are presented.

II. ANTENNA DESIGN AND EXPERIMENTAL RESULTS
The antenna geometry is given in Fig. 1. The equilateral triangular patch of side length d is considered. For comparison with the dual-frequency rectangular microstrip antenna in [2], the same FR4 substrate ("r = 4:4; h = 1:6 mm) of 7.5 cm 2 7.5 cm in size was used in this study and the triangular patch was designed to resonate at 1.9 GHz, which gives a side length of 50 mm. The shorting pin of radius (rs = 0:32 mm) and probe feed of radius (rp = 0:63 mm) are placed along the line segment between the triangle tip and the bottom side of the patch, as shown in Fig. 1. By varying the shorting-pin position (ds), a strong dependence of the first two resonant frequencies on d s is observed. The results are shown in