## 1. Introduction

[2] As it is well known, ray diffraction theories, such as the geometrical theory of diffraction (GTD) [*Keller*, 1962] and its uniform version (UTD) [*Kouyoumjian and Pathak*, 1974], give a description of the diffracted field in terms of rays emanating from diffraction points located on the scatterers. The diffraction point locations are calculated using ray tracing based on the minimum path Fermat principle. A field contribution is associated with each ray by resorting to a canonical problem that fits the local geometry at the diffraction point. Conversely, edge wave theories such as the incremental length diffraction coefficients (ILDC) [*Mitzner*, 1974; *Michaeli*, 1984; *Knott*, 1985; *Shore and Yaghjian*, 1988], the physical theory of diffraction (PTD) [*Ufimtsev*, 1991] and incremental theory of diffraction (ITD) [*Tiberio et al.*, 2004], build up the diffracted field by integral superposition of incremental field contributions emanating from every point of the edge. At each incremental point, the contribution is again calculated by invoking the high-frequency localization principle, and by resorting to a canonical problem that locally fits the actual geometry. The superposition of incremental contributions is performed via a numerical integration; this integration substitutes the ray tracing operation and overcomes the GTD/UTD impairment associated with curved edge caustic. When performing the analytical continuation of the UTD for complex point source (CPS) [*Green et al.*, 1979; *Suedan and Jull*, 1989, 1991; *Heyman and Ianconescu*, 1995], the diffracted field description requires a more general “complex ray tracing”; this consists of finding complex diffraction points whose determination and geometrical interpretation are no longer straightforward. This feature may be a strong drawback when implementing ray-tracing algorithms, where the estimate of diffraction points positions must be performed at first. Conversely, this problem does not appear using incremental theories. ILDC, PTD and ITD contributions are readily calculated by analytical continuation of the source coordinates into complex space. The main feature of the CPS is that its field satisfies the Helmholtz wave equation everywhere in space [*Deschamps*, 1971; *Shin and Felsen*, 1974; *Felsen*, 1976; *Heyman and Felsen*, 2001]. An appropriate CPS expansion can match more complicated wavefield in localized region of space; furthermore its paraxial approximation leads to a Gaussian beam field, whose effectiveness in treating electromagnetic scattering problem has been widely demonstrated in the literature [*Maciel and Felsen*, 1989; *Chou and Pathak*, 1997; *Chou et al.*, 2001; *Chou and Pathak*, 2004]. In this paper, we formulate the generalization of the ITD to CPS. In particular, section 2 describes the model and the properties of the vectorial CPS used in the formulation. Section 3 briefly summarizes the ITD formulation for (real) point source. In section 4 the ITD generalization to complex source is formulated. In order to gain physical insight, the canonical problem of an infinite straight wedge is considered in section 5 by reconstructing the canonical response using ITD integration. There it is shown that the present formulation leads to a different definition of the shadow boundaries of the geometrical optics (GO) field, which agrees with *Albani et al.* [2005a, 2005b]. Section 6 presents the derivation of the GO in an incremental form to simplify the implementation and shows the accuracy of the present formulation by a comparison with method of moments results.