An alternate way to obtain the aberration expansion in Helmholtz Optics

TL;DR

This paper introduces a novel method for deriving the aberration expansion in Helmholtz optics by leveraging analogies with the Klein-Gordon equation, resulting in wavelength-dependent modifications to optical Hamiltonians.

Contribution

It presents an alternative approach using the Feschbach-Villars and Foldy-Wouthuysen techniques to obtain wavelength-dependent aberration coefficients in Helmholtz optics.

Findings

Derived explicit expressions for third-order aberration coefficients.

Obtained sixth and eighth order Hamiltonians for graded index fibers.

Showed wavelength dependence modifies traditional paraxial and aberration terms.

Abstract

Exploiting the similarities between the Helmholtz wave equation and the Klein-Gordon equation, the former is linearized using the Feschbach-Villars procedure used for linearizing the Klein-Gordon equation. Then the Foldy-Wouthuysen iterative diagonalization technique is applied to obtain a Hamiltonian description for a system with varying refractive index. Besides reproducing all the traditional quasiparaxial terms, this method leads to additional terms, which are dependent on the wavelength, in the optical Hamiltonian. This alternate prescription to obtain the aberration expansion is applied to the axially symmetric graded index fiber. This results in the wavelength-dependent modification of the paraxial behaviour and the aberration coefficients. Explicit expression for the modified coefficients of the aberration to third-order are presented. Sixth and eighth order Hamiltonians are derived for this system.