Question

I have been researching diffraction for the purposes of implementation into a ray tracer (based on the geometrical theory of diffraction). However, I found a few statements that seem conflicting to me with regards to the phase jumps at a diffraction location. The Kirchhoff diffraction formula, for example, can be solved to get a phase jump of 90° at an aperture, the same goes for the Rayleigh-Sommerfeld diffraction integral. On the other hand, with the GTD or the UTD, the diffraction coefficients include phase jumps of usually 45° (see e.g. the original paper by Keller ). In all cases, the fundamental physical conditions seem to be the same to me, a wave is propagated to an aperture/half-plane and then creates diffracted waves. So where do these different phase jumps come from? Also, are these dependent on whether or not the aperture is bound by a perfect conductor?

Answer 1

The variations in phase jumps (such as 90° in Kirchhoff/Rayleigh-Sommerfeld formulas and 45° in GTD/UTD) arise due to different assumptions and approximations made in modeling diffraction phenomena. These differences are not solely contingent on whether the aperture is bounded by a perfect conductor.

Step-by-step explanation:

The disparities in phase jumps between diffraction models stem from their distinct theoretical foundations and approximations used in their derivation. The Kirchhoff and Rayleigh-Sommerfeld diffraction formulations consider a continuous aperture illuminated by a wave and typically yield a phase change of 90° at the aperture edge. This result is based on assumptions related to scalar wave theory and the integration of the Huygens-Fresnel principle, considering all points on the aperture as coherent sources.

On the other hand, Geometrical Theory of Diffraction (GTD) and Uniform Theory of Diffraction (UTD) incorporate a physical optics approach where diffraction arises due to the interaction of incident and diffracted rays at edges or discontinuities. In these cases, the phase jump of 45° generally emerges from the approximation that assumes a single reflection or diffraction process occurring at the edge or discontinuity. These models rely on high-frequency approximations and assume a limited number of interactions between rays, leading to a different phase change estimation.

While the presence of a perfect conductor at the aperture's boundary can affect the diffraction pattern by altering the boundary conditions, the discrepancies in phase jumps primarily arise from the underlying assumptions and mathematical formalisms employed in the specific diffraction models rather than being solely dependent on the boundary conditions.