Calculation of Diffraction Field Using Incremental Theory of Diffraction for Observation Points Outside the Keller Cone
Article information
Abstract
This letter presents an enhanced diffracted electromagnetic field calculation approach that addresses the critical limitations of conventional methods. Since traditional techniques, such as geometrical theory of diffraction, uniform theory of diffraction, and physical theory of diffraction, are constrained by infinite wedge assumptions, they fails to correctly calculate the electromagnetic field at observation points outside the Keller cone. In contrast, we implement the incremental theory of diffraction (ITD) for perfectly electrically conducting structures to accurately compute the diffraction field at arbitrary observation points through incremental integration along finite edges. The proposed hybrid methodology combines shooting-and-bouncing rays (SBR) with ITD for both multiple reflections and edge diffraction. Numerical simulations demonstrate significant performance improvements over conventional methods, particularly outside the Keller cone, where traditional techniques fail completely.
I. Introduction
Conventional diffraction modeling approaches, including the geometrical theory of diffraction (GTD) [1], uniform theory of diffraction (UTD) [2], and physical theory of diffraction (PTD) [3], rely on infinite wedge assumptions and are fundamentally limited to analyzing observation points within the Keller cone. These constraints lead to errors in the analysis of practical scenarios involving complex structures and arbitrary observation angles. To overcome these limitations, the incremental theory of diffraction (ITD) [4, 5] was developed as an extension of UTD, enabling accurate field predictions by incrementally integrating the diffraction contributions along finite edges [6, 7].
In this letter, we implement ITD methodology for perfect electric conductor (PEC) targets to resolve the critical shortcomings of conventional diffraction models. We demonstrate that ITD allows for accurate and stable diffraction field calculations at arbitrary observation points. Comparative analyses with UTD- and PTD-based approaches highlight the improved superior performance of ITD in computing diffracted fields, particularly in geometrically complex environments where conventional methods exhibit significant errors outside the Keller cone. Although our implementation focuses only on PEC structures and does not account for double diffraction effects due to the inherent limitations of ITD formulation, the proposed methodology demonstrates improved accuracy in diffraction field prediction for PEC structures.
II. Analysis Method
ITD is an advanced diffraction calculation method derived from UTD that involves incrementally calculating the diffraction fields over finite length elements and then superposing these contributions to determine the total diffraction field. The implementation of this method entails dividing the edge into micro-length elements, each associated with a locally tangent canonical wedge. For each element, the diffracted field contribution is calculated individually using locally tangent canonical problems, following which the total diffracted field is obtained through spatial integration along the edge. Effectively, this methodology eliminates the Keller cone constraint caused by infinite wedge assumptions, thereby enabling accurate diffraction calculations at arbitrary observation points, including those in regions inaccessible to conventional techniques. A high-frequency incremental field contributions can be expressed as follows [6]:
where D11 and D22 is the diffraction coefficient, which depends on incident angles βi and φi, observation angles βs and φs, and the wedge exterior angle n, as shown in Fig. 1.
The concept of incremental field contribution by the edge of an arbitrary perfect electric conductor.
To obtain the total electromagnetic field, both the reflected and diffracted waves must be calculated. Notably, the shooting-and-bouncing rays (SBR) method accounts for multiple reflections using physical optics (PO) for scattered field calculations. By combining ITD with SBR, accurate predictions that incorporate the effects of multiple reflections and diffraction can be achieved. Notably, since SBR employs PO, integrating ITD with SBR requires combining the PO and ITD results. This differs from PTD, which involves directly calculates the fringe field—the difference between the actual scattered field and the PO-predicted field—thus making it compatible with PO. Since, ITD does not directly produce fringe fields, it needs to be extracted by removing the PO diffraction component from the total ITD calculations. In ITD formulation, the PO component corresponds to the diffraction field when n = 1. The resulting fringe field can be expressed as follows [6]:
where
III. Numerical Results
For validation, the proposed method was compared with UTD, PTD, and full-wave solvers at 10 GHz UTD was implemented via FEKO, while PTD was analyzed using HFSS SBR. SBR was employed because it can handle multiple reflections a capability PO lacks. To ensure computational efficiency, this SBR ray-tracing engine was accelerated using NVIDIA OptiX and integrated with ITD diffraction calculations developed in C++.
The proposed method was verified using a simple model. For canonical cube geometry (Fig. 2(a)), observation points were positioned at r = 5 m over an angular range of θ = 270°–360° (ϕ = 0°). This configuration ensured some points were outside Keller cones, enabling evaluation where conventional techniques fail. Fig. 2(a) compares field predictions from UTD, PTD, and the proposed method against the multi-level fast multipole method (MLFMM) results. It is observed that UTD completely fails within θ = 270°–306°. In contrast, ITD combined with PO calculates fields accurately by converting the object into equivalent sources, thereby allowing diffracted waves to reach observation points. As a result, the proposed method achieves approximately 15% NRMSE (normalized root mean square error), closely matching the reference and validating its effectiveness outside the Keller cone. Meanwhile, PTD achieves partial predictions with approximately 25% NRMSE. Notably, the accuracy is limited as double diffraction and object thickness were not considered. However, the results confirm ITD achieves higher accuracy than conventional methods.
Comparison of diffraction field predictions made by conventional methods and the proposed method: (a) canonical cube model and (b) simplified tank model.
To further assess the proposed method under realistic conditions, we simulated a simplified tank structure (Fig. 2(b)). Observation points were placed across wide angular ranges. Notably, given the electrically large size of this structure (approximately 317λ × 117λ × 77λ), full-wave methods, such as MLFMM, were computationally prohibitive and failed to provide reference solutions. Fig. 2(b) shows that GO+UTD failed to calculate the fields in the 60°–80° range, which can be attributed to two fundamental limitations. First, UTD rays are incident on the front face of the model, which restricts diffraction calculations to only the front-facing edges. For these edges, observation points at 60°–80° lie outside the Keller cone, where UTD cannot be used to compute diffraction fields due to its inherent angular constraints. Second, since geometrical optics (GO) rays are predominantly reflected from front surfaces, they cannot reach the 60°–80° observation region, thus preventing GO from calculating reflected field contributions within this angular range. Furthermore, SBR+PTD achieved the same results as SBR without diffraction calculations, since PTD also cannot compute diffraction fields outside the Keller cone. The proposed method, however, maintains accurate field predictions throughout the entire domain, regardless of local Keller cone coverage. These results demonstrate that the proposed approach effectively overcomes the limitations of conventional diffraction models, enabling accurate field estimation in geometrically complex scenarios.
IV. Conclusion
This letter presents a hybrid SBR-ITD framework that overcomes the Keller cone limitation of conventional diffraction methods. By integrating ITD with incremental field calculations along finite edges, the proposed approach helps achieve accurate diffraction predictions for arbitrary observation points, including those in regions inaccessible to UTD and PTD techniques.
Numerical validation conducted using canonical geometries demonstrates the method’s effectiveness, confirming that SBR+ITD can calculate diffraction fields where conventional methods fail. Furthermore, successful validation against MLFMM solutions establishes the approach’s reliability in dealing with complex PEC diffraction problems.
Notes
This work was supported by the Laboratory of Computational Electromagnetics for Large-Scale Stealth Platform (UD200047JD).