Hamilton-Jacobi equation in momentum space

Area of Science:

• Optics and Photonics
• Mathematical Physics
• Geometrical Optics

Background:

• The Hamilton-Jacobi equation is fundamental in classical mechanics and optics.
• In geometrical optics, it simplifies to the eikonal equation, defining ray paths.
• Symmetries in Hamiltonian formulations offer deeper insights into physical systems.

Purpose of the Study:

• To explore the dual Hamilton-Jacobi equation in momentum space for optical systems.
• To analyze the relationship between coordinate and momentum space wavefronts.
• To investigate optical systems with spherical symmetry and exchanged ray trajectories.

Main Methods:

• Application of the Hamilton-Jacobi equation to isotropic optical materials.
• Exploitation of coordinate-momentum symmetry in Hamiltonian geometrical optics.
• Analysis of dual eikonal equations in spherical symmetric refractive index distributions.

Main Results:

• Derivation of a dual Hamilton-Jacobi equation for wavefronts in momentum space.
• Demonstration that this dual equation is also an eikonal equation under spherical symmetry.
• Identification of specific refractive index distributions (e.g., Maxwell fish-eye, Luneburg lens) where coordinate and momentum space ray trajectories are exchanged.

Conclusions:

• The study establishes a formal duality between coordinate and momentum space descriptions of light propagation.
• This duality provides new perspectives on optical systems, particularly those with spherical symmetry.
• The findings offer a framework for designing novel optical elements with tailored ray trajectories.