The Contributions of Prof. Roberto Tiberio to the Incremental Theory of Diffraction for Electromagnetic

In this paper a brief overview of the incremental theory of diffraction has been presented, of which Prof. Tiberio was the principal author and the main source of inspiration of our contributions. This activity covers only a very small part of his overall significant research contributions in high-frequency electromagnetic, to which he has indefatigably dedicated until he passed away. However, in this short paper, it has not been possible to describe also the excellent human qualities of Roberto. He was a unique individual and we also know we will all keep his memory alive


Introduction
This paper provides a brief review of the fundamental contributions of Prof. Roberto Tiberio to the area of high-frequency (HF) incremental techniques for antennas and scattering problems. It is focused, in particular, to the main features of the Incremental Theory of Diffraction (ITD), originally introduced by Tiberio and S. Maci in the early nineties, and to development of which he has dedicated with great passion until he passed away on May 2, 2005 at the age of 59.
It was during the late 1980s, that Tiberio's research interest was attracted by new emerging techniques for the prediction of the radar cross section of complex objects. In particular he was fascinated by the early work of P.Y. Ufimtsev on the Physical Theory of Diffraction (PTD), which has the ability to overcome the UTD impairment at caustics. In those years, P.Y. Ufimtsev improved his Edge Wave theory within the framework of PTD, so that it remained valid at any observation aspect outside the Keller cone of edge diffraction. Tiberio extended the PTD edge-wave theory to impedance surfaces [1]. In order to overcome some limitations of the PTD, Tiberio started thinking of a new general formulation for treating edge waves. The original idea was to resort to a description of the scattered field directly in terms of fields rather than in terms of currents. The incremental field contribution of the ITD was initially derived from the problem of the infinite wedge under plane wave illumination, on the basis of a simple Fourier-transform pair between the canonical field and the incremental contribution. The final representation, given in terms of a double integral, was uniformly evaluated, thus providing a closed form solution. It was formally presented for the first time by Tiberio at the 1992 IEEE Antennas and Propagation Society Symposium in Chicago [2] and then published in its scalar version on IEEE Trans. on Ant. and Propagat. in 1994 [3]. Later, Tiberio often interacted with Prof. Ufimtsev and Prof. Yaghjian; their discussions centered on ways of interpreting the two incremental theories based on PTD and ITD, respectively. In 1995-96, Tiberio and his research group extended the original formulation to electromagnetic problems [4] and to treat the case when both the source and the observation point are at finite distance form the edge [5].
In 1996-97 Tiberio, with Maci and Toccafondi, introduced a generalization of the ITD localization process to treat any canonical problem with a uniform cylindrical configuration [6]. This generalized ITD is a unified, self-consistent framework for the description of a wide class of practical scattering phenomena. It introduced a localization process based on constructing a suitable spatial integral representation of the 3D canonical Green's function (GF) through a 2D GF factorization. This method was successfully applied, in a unified framework, to both locally-tangent perfectly conducting wedges and moderately sized circular cylinders. In both these cases this theory provided incremental contributions which explicitly satisfied reciprocity and naturally blends into the UTD when the ray field regime is established. This formulation was also successfully applied to the scattering of electromagnetic fields by plane angular sector [7], and extended to the case of the time domain incremental diffraction by a generally curved edge illuminated by an impulsive source [8].
However, in practical applications the high-frequency ITD expressions were affected by some impairments, especially at incidence and observation aspects in the paraxial region of the local canonical configurations, from which the incremental fields were deduced. To overcome these difficulties, Tiberio and his group developed an improved asymptotic analysis for the incremental field contributions. The above formulation of the new improved version of ITD was presented in various conferences and finally it also appeared as a journal article [9]. The new high frequency incremental field contributions still retained the property of reciprocity, but, more importantly they now provided a well-behaved and valid description of the field at any incidence and observation aspects, including the paraxial region. Using the improved ITD concept, expressions were also recently obtained for moderately sized circular cylinders [10] and edges in a planar configuration with surface impedance boundary conditions [11]. Most recently Tiberio was working on the extension of the ITD for single-edge diffraction to a pair of skewed separated wedges, and to a pair of joined wedges with a common face. This research activity has been continued by his research group because of his untimely demise and presented last year [12].

The ITD localization process
The general procedure for defining ITD incremental contributions is based on a suitable interpretation of the exact solution for the scattering in the near zone from a canonical cylindrical structure, when it is illuminated by an elementary source. In particular, the spectral integral formulation of the canonical solution is represented as a spatial integral convolution along the longitudinal coordinate axis of the canonical structure. Then, its integrand is used to directly define the local incremental field contribution. For the sake of simplicity in the explanation, the formulation here refers to the scalar case. The more general electromagnetic problem is addressed in published papers where dyadic high-frequency expressions are presented. Let us consider a local canonical cylindrical configuration depicted in Fig.1a or 1b. As well known [7], an exact spectral integral representation of the Green's function of an object with a uniform cylindrical canonical configuration along the z axis may be obtained as where is a linear operator. Using (2) in (1) (3) It is now rather apparent that the argument of L in (3) represents the inverse Fourier transform (FT) of the product of two spectrum functions. As a consequence, by FT analysis it is seen that the integral expression in (3) represents the spatial convolution product of two functions 1 ( ) ( " , ) ( ' ", ') " 2 c P L u z z u z z dz in which ( , )  (2)-(4), is referred to as the ITD FT-convolution process. It provides a general, systematic procedure for obtaining incremental field contributions from the Green's function of infinite, uniform cylindrical local canonical problems.

Application to actual structures
In order to provide practical engineering tools, the four fold spectral integral representation for the incremental field contributions obtained in (5), needs to be asymptotically approximated for large r and r'. For this purpose, a specific asymptotic analysis has been conceived and presented in [9].
These results have been used to obtain high-frequency, dyadic, closed form expressions for the ITD incremental field contributions for both wedge shaped (Fig 1a) and moderately sized circular cylinder configuration (Fig. 1b) where is the electromagnetic incident field at . Detailed expressions of the diffraction coefficients ( , , ) e h D υ φ φ for several canonical structures have been obtained and reported in [9]- [11]. In (6), υ is the value of a spectral variable which satisfies the ITD stationary phase condition [9], and is defined by ' ' ' 1 cos cos cos cos cos sin sin sin sin sin j Finally, in order to provide an estimate of the singly diffracted or scattered field by an actual structure the incremental field contribution (6) is distributed and then integrated along the actual edge of the structure itself l . (1) It has been found that this formulation explicitly satisfies reciprocity and leads to a wellbehaved description of the field at any incident and observation aspect, including those close to and at the longitudinal coordinate axis of the local canonical configuration. It also may be successfully applied for either complementing ray field methods or providing fringe field augmentations to the PO approximation.
Numerical results obtained applying this technique for estimating the electromagnetic field scattered or radiated by complex structures, will be shown during the oral presentation.

Conclusion
In this paper a brief overview of the Incremental Theory of Diffraction has been presented, of which Prof. Tiberio was the principal author and the main source of inspiration of our contributions. This activity covers only a very small part of his overall significant research contributions in high-frequency electromagnetic, to which he has indefatigably dedicated until he passed away. However, in this short paper, it has not been possible to describe also the excellent human qualities of Roberto. He was a unique individual and we also know we will all keep his memory alive.