Fig. 2. Geometry at an open-ended waveguide The PO currents Jp0 on the walls are defined as

in which ft^, is the normal to the wall directed toward \f,„. These currents radiate at any point P in free space the PO field (E^, H^).

The Kirchhoff-Kottler approximation assumes equivalent currents J, = z x H , ka mod , , v n (40)

on the aperture, that radiate a field (Eka, at any point P in the free space. The following theorem has been demostrated [Maci et al., 1996b ]

i.e., the AI and the PO fields are exactly the same outside the open ended waveguide; inside the same waveguide the sum of the AI field plus the modal field is equal to the PO field.

Note that, outside the OEW (in V^.), the PO field is a radiated field; to evidence this, it can be denoted by (Ep"d, \tp"d)- Inside the OEW (in V,H) the PO field is the sum

po mod po j rtf po where (Emoj, I?,»«/) is the incident field and {Epf, bfpf) is the field reflected back from the

po mod po end. This latter contains all the eigen-modes. Then, we can interpret the theorem (41), (42) as follows

This means that the AI field in the Kirchhoff-Kottler approach exactly equals the PO radiated and reflected fields.

It is worth noting that the assumption of perfectly conducting walls, is not at all necessary. This theorem is applicable to any arbitrary either boundary conditions or configurations of its walls, provided that in this special case the incident dominant waveguide mode of the original, canonical infinite waveguide may be assumed entirely confined within the waveguide itself. Also, it is applicable for any arbitrary modal field configuration which is compatible with the canonical infinite waveguide.

The above theorem may be extended to deal with dominant modes in waveguides smoothly expanded into horns, provided that in this special case it is assumed that the incident waveguide mode adiabatically transforms into the lowest order eigen-mode of the horn without reflection and excitation of other modes. The fields (Ep^, H/«"™) and (Epo , IIpf) are found by the Kirchhoff-Kottler aperture integration.

Finally, it is suggested that applying this theorem allows to more easily conceive an asymptotic reduction of AI into line integration, based on a rigorous high-frequency approximation of the PO integration along the semi-infinite generatrix of the OEW structure.

An important consequence of the above theorem within the framework of incremental theories is that of allowing to complement either AI or PO [.Michaeli, 1995; Capolino et al., 1995] by the same fringe contributions. We just note that the fringe field that can be deduced from ITD, as shown in Sect. 3, does not depend on any choice of the surface where the equivalence principle is applied. Thus, it is rather apparent that the ITD formulation is congruent with the above theorem.

A brief overview has been presented of well-established high-frequency methods form treating antenna and scattering problems. The attention has been focused on incremental techniques, that may provide useful extensions of either ray field or integration techniques. Recent developments within this framework have been for P e V, for P e V,

discussed in some details. A relationship between PO and AI has been addressed, which may be useful to provide an effective representation of the total field, after conveniently introducing either diffracted or fringe incremental contributions.

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