[Paper Review] An alternate way to obtain the aberration expansion in Helmholtz Optics

Sameen Ahmed Khan|arXiv (Cornell University)|Sep 30, 2002

Photonic Crystal and Fiber Optics34 references21 citations

TL;DR

This paper presents an alternative derivation of the aberration expansion in Helmholtz optics by leveraging the algebraic similarity between the Helmholtz and Klein-Gordon equations, using the Feschbach-Villars linearization and Foldy-Wouthuysen iterative diagonalization. The method yields a wavelength-dependent optical Hamiltonian that modifies traditional aberration coefficients, with explicit third-order corrections derived for axially symmetric graded-index fibers.

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.

Motivation & Objective

• To develop an alternative formalism for deriving the aberration expansion in Helmholtz optics by exploiting algebraic parallels with the Klein-Gordon equation.
• To incorporate wavelength-dependent corrections into the optical Hamiltonian that are absent in traditional quasiparaxial treatments.
• To demonstrate the formalism's applicability to axially symmetric graded-index media, where it modifies standard aberration coefficients.
• To unify the treatment of light optics and charged-particle beam optics through shared algebraic structures, particularly in the matrix formulation.
• To recover the classical (traditional) limit in the small-wavelength ($\lambda \to 0$) regime, analogous to the $\hbar \to 0$ limit in quantum mechanics.

Proposed method

• Linearize the Helmholtz equation using the Feschbach-Villars procedure, analogous to the method used for the Klein-Gordon equation in relativistic quantum mechanics.
• Apply the Foldy-Wouthuysen iterative diagonalization technique to the linearized Hamiltonian to obtain a quasiparaxial optical Hamiltonian with explicit wavelength dependence.
• Derive a matrix-form optical Hamiltonian that includes polarization-like terms analogous to spin terms in Dirac theory.
• Use the resulting Hamiltonian to compute aberration coefficients up to third order in the paraxial limit for axially symmetric systems.
• Construct sixth- and eighth-order Hamiltonians for the graded-index fiber to capture higher-order aberrations.
• Compare the derived Hamiltonians with traditional prescriptions, showing convergence in the $\lambda \to 0$ limit.

Experimental results

Research questions

• RQ1 Can the aberration expansion in Helmholtz optics be derived via a quantum-like formalism based on algebraic analogies with the Klein-Gordon equation?
• RQ2 What are the wavelength-dependent corrections to the standard aberration coefficients in a graded-index medium?
• RQ3 How do the polarization-like terms in the matrix Hamiltonian affect the paraxial behavior of light beams?
• RQ4 In what way does the new formalism generalize the traditional Fermat’s principle-based approach?
• RQ5 How is the classical limit ($\lambda \to 0$) recovered in this quantum-inspired optical formalism?

Key findings

• The formalism produces a Hamiltonian that includes wavelength-dependent terms not present in the traditional quasiparaxial approximation.
• For axially symmetric graded-index fibers, the standard six aberration coefficients are modified by explicit wavelength-dependent contributions.
• Explicit third-order expressions for the modified aberration coefficients are derived, showing the influence of the refractive index gradient and wavelength.
• Sixth- and eighth-order Hamiltonians are constructed, enabling higher-order aberration analysis beyond the standard paraxial expansion.
• The method reproduces the traditional beam-optical results in the $\lambda \to 0$ limit, confirming consistency with classical optics.
• The matrix structure of the Hamiltonian reveals a deep analogy with Dirac and Klein-Gordon theories, particularly in the emergence of polarization-like terms.