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

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 nR/r is solved analytically. The solution gives a ray equation r(phi) that is closely related to Rutherford scattering trajectories; we therefore refer to this refraction process as photonic Rutherford scattering. It is shown that not only do the classical limits correspond but also the wave-mechanical pictures coincide-the time-independent Schrodingier equation and the Helmholtz equation permit the same mapping between the scattering of massive particles and optical scalar waves. Scattering of narrow beams of light finally recovers the classical trajectories. The analysis suggests that photothermal single-particle microscopy measures photonic Rutherford scattering in specific limits and allows for an individual single-scatterer probing. A macroscopic experiment is demonstrated to directly measure the scattering angle to impact parameter relation, which is otherwise accessible only indirectly in Rutherford-scattering experiments. (C) 2013 American Association of Physics Teachers.