High-frequency description of the Kirchhoff-type modal coupling between open ended waveguides

The Kirchhoff-type aperture integration (AI) is the simplest way to calculate the radiation from an open-ended waveguide (OEW). Recently, a rigorous equivalence between the field predicted by AI and that radiated by the physical optics (PO) wall-current was demonstrated, in which the PO currents are defined as that associated to the unperturbed mode. By using this equivalence, a method for asymptotically reducing the AI into a line integration (LI) of incremental diffraction coefficients along the waveguide edge was previously presented. A LI representation of the aperture field is well suited for introducing a fringe contribution as provided by the physical theory of diffraction (PTD) or by the incremental theory of diffraction (ITD). In this paper, the equivalence between PO and AI is extended to evaluate the coupling between two OEWs of arbitrary cross-section. Furthermore, a Kirchhoff-type coupling coefficient is derived in terms of a double line integration of incremental coupling coefficients. This may provide a useful tool when the mutual impedance of two modal distributions has to be calculated in the framework of a method of moments (MoM) procedure which is formulated in terms of mode-shaped basis functions.


INTRODUCTION
The Kirchhoff-type aperture integration (AI) is the simplest way to calculate the radiation from an open-ended waveguide (OEW).Recently, a rigorous equivalence between the field predicted by AI and that radiated by the Physical Optics (PO) wall-current was demonstrated [l], in which the PO currents are defined as that associated to the unperturbed mode.By using this equivalence, a method for asymptotically reducing the AI into a line integration (LI) of incremental diffraction coefficients along the waveguide edge was presented [2].A LI representation of the aperture field is well suited for introducing a fringe contribution as provided by the Physical Theory of Diffraction (PTD) [3] or by the Incremental Theory of Diffraction (ITD) 141, [5].In this paper, the equivalence between PO and AI [l] is extended to evaluate the coupling between two OEWs of arbitrary cross-section.Furthermore, a Kirchhoff-type coupling coefficient is derived in terms of a double line integration of incremental coupling coefficients.This may provide a useful tool when the mutual impedance of two modal distributions has to be calculated in the framework of a Method of Moments (MOM) procedure which is formulated in terms of mode-shaped basis functions.

RECIPROCITY OF THE EQUIVALENCE BETWEEN PO AND AI.
Let us consider a receiving open ended waveguide (OEW1) of arbitrary cross section, with its axis along the z axis of a reference system and its aperture on the s-y plane; suppose that OEWl extends for z<O and denote by e',", iln (n integer), an arbitrary normalized mode propagating into the waveguide toward negative z.The normdiaation constant may be chosen in such a way that the integration of Z,".h, " on the aperture is equal to 6, (Kronecker's delta).Fu_rthejmore, suppose that the waveguide is jlluqinate2 by an incident field (E,, H,) produced by an external source (J2, M,).By invoking the Kirchhoff approximation, the excitation coefficient Cn of the n-th mode onto the OEWl can be calculated as in which +,= Zlnxz*, Jln=z*xhl", and the total aperture field has been approximated by the incident field.The minus sign depends from the fact that the mode has been assumed as propagating toward negative t. Applying the reciprocity principle, yields 0-7803 -4 178-3/97/$10.000 199fIEEE where (El", H,") is the field pro$uc%d by the apertur: distribution (iln, ?ln) and Vz is a volume containing ( J ., Mz).Since E,", Hin), is equal to the field application of the reciprocity principle leads to produced by the radiation in free space of t Le PO currents [l], a further Cln=-I k2*j,E dS,

S l
where the surface S, denotes the waveguide wall and yic =CW x i," (where 7iWl is the internal normal to the wall).Equations ( 1) and (2) express the equivalence between AI and PO for a receiving OEW.

MODAL COUPLING
Let us now consider a second, arbitrarily shaped open-eGded waveguide (OEW2) which is fed by the normalized modal field Z2", h, " propagating toward the positive z axis (Fig. 1).An aperture field distribution +"=ZZm x 5, Yzm=i x hzm is associated to the unperturbed incident mode, that radiates the Kirchhoff-type field (E2", H2m).Using (l), the Kirchhoff-type excitation coefficient of the n-th mode into OEWl due to the Kirchhoff-type aperture radiation of OEW2 is

A1
This quantity involves a four folded integral.Although for circular and rectangular waveguides, it can be reduced to a single spectral integral [Bird], this cannot be done for general shape.
By invoking the equivalence between PO and AI, (E2m, Hzm) may also be though as produced by the currents j2'"=fiw2 x i 2 " .Applying eq.(3) leads to C,",m = -1 1 1 ( FJ * E:(/ F, -Pz/) .jzz (F2) dSi dS2 (5) where E(/ F2 -Pi/) is the free-space dyadic Green's function pertinent to the electric source and the electric field, S, and Fi are the wall-surfaces of OEWZ and the position vector on it, respectively.By integrating along strips parallel to the z axis (Fig. 1) belonging to the two waveguides, (2) is rearranged as where c(e :e") is an incremental coupling coefficient that represents the interaction between the two elementary strip for two circular TE,l modes versus the distance d between two circular OEWs with the same radius.The geometry and the polarisation of the modes are shown in the inset.Two different radii of the OEWs are considered; i.e., a=0.3X, and a=0.5X.Continues lines refer to the double LI solution (eq.6), and the dotted lines refer to the the double AI solution (eq. 4).In spite of the moderate size of the apertures, the agreement has been found quite satisfactory over all the dynamic range.