RayleighSommerfeld

Rayleigh–Sommerfeld theory refers to a set of mathematical formulations used in wave optics and acoustics to calculate how waves propagate, diffract, and interfere when encountered with openings or obstacles. Named after Lord Rayleigh (John William Strutt) and the physicist Arnold Sommerfeld, the theory provides precise models for the field distributions resulting from wave diffraction, especially in the near-field region.

The Rayleigh–Sommerfeld diffraction equations are integral solutions derived from the wave equation, describing how a wavefront passes through an aperture and propagates beyond it. These formulations are rooted in the Huygens–Fresnel principle, but with specific boundary conditions that improve accuracy for certain application ranges. They are often contrasted with the Kirchhoff diffraction approximation, offering more rigorous results without some of Kirchhoff’s simplifying assumptions.

In practical applications, Rayleigh–Sommerfeld integrals are utilized in optical system design, holography, electron microscopy, and acoustic modeling to predict diffraction patterns and focus fields with high fidelity. They are particularly useful in situations involving complex geometries and require high-precision calculations of wave behavior.

The formulations come in two main variants: the first type, which considers only forward diffraction, and the second, which accounts for backscattering effects. Numerical methods, such as boundary element techniques, are employed to evaluate the integrals efficiently for real-world problems.

Overall, Rayleigh–Sommerfeld theory plays a crucial role in understanding wave propagation phenomena and is fundamental to advances in optical engineering, acoustics, and electromagnetic wave analysis.