Photonic Rutherford scattering: A classical and quantum mechanical analogy in ray and wave optics

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) = n0 + ∆nR/r, is solved analytically. The solution, i.e. the ray-equation r (φ), 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 Schrödingier 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.

Comment on “Optimal detection angle in sub-diffraction resolution photothermal microscopy: application for high sensitivity imaging of biological tissues”

“…Similar to the quantum mechanical treatment of the Coulomb scattering problem, the scattering cross-section dσ /dΩ = | f k (θ ) | 2 = R 2 Δn 2 sin −4 (θ /2) /4n 2 0 is found to be identical to the classical one and should thus be wavelength independent. This is indeed what was found before [4,9,14,20].…”

Diffraction of electromagnetic waves in the gravitational field of the Sun

“…The complex-valued function f describes the perturbation of the incoming plane wave and transforms (18) into a solvable differential equation for f [45]. The same equation appears in other problems of modern physics: for instance, in problems describing photothermal single-particle Rutherford scattering microscopy that involves the scattering of waves by a 1/r refractive index profile formed by the presence of a point-like heat source in a homogeneous medium (e.g., [48][49][50]). …”

The use of the stationary phase method as a mathematical tool to determine the path of optical beams

“…[12,13] The possibility of linking Maxwell's equations for photon propagation in the presence of dielectric blocks with the quantum-mechanical equations for the propagation of electrons in the presence of potential steps [14][15][16] allows one to determine, in a simple and intuitive way, the reflection and transmission coefficients at the dielectric block interfaces. [17][18][19][20] As the second step in our analysis (Sec. III), we obtain the reflection and transmission coefficients for s-polarized waves transmitted through the dielectric system of Fig.…”