Photonic Rutherford Scattering: A Classical and Quantum Mechanical Analogy in Ray- and Wave-Optics

TL;DR

This paper establishes an analogy between ray and wave optics in gradient index lenses and Rutherford scattering, revealing that optical wave behavior can be mapped to quantum mechanical scattering, with implications for microscopy.

Contribution

It analytically solves ray trajectories in a specific gradient index lens and demonstrates their equivalence to Rutherford scattering, linking classical, wave, and quantum optics.

Findings

Ray trajectories are analytically related to Rutherford scattering.

Wave-mechanical models coincide with classical scattering predictions.

Photothermal microscopy may measure photonic Rutherford scattering.

Abstract

Using Fermat's least optical path principle the family of ray-trajectories through a special but common type of a gradient refractive index lens, n(r)=n_0+\Delta n R/r, is solved analytically. The solution, i.e. the ray-equation r(phi), is shown to be closely related to the famous Rutherford scattering and therefore termed photonic Rutherford scattering. It is shown that not only do these classical limits correspond, but also the wave-mechanical pictures coincide: The time-independent Schr\"odingier equation and the inhomogeneous Helmholz equation permit the same mapping between massive particle scattering and diffracted optical scalar waves. Scattering of narrow wave-packets finally recovers the classical trajectories. The analysis suggests that photothermal single particle microscopy infact measures photonic Rutherford scattering in specific limits.