Hamiltonian preserving nonlinear optics

Generalized Hamiltonian Formalism in Nonlinear Optics

The basic mathematical apparatus of nonlinear optics consists of an array of nonlinear PDEs for the complex amplitudes of an envelope of interacting wave trains. In the general case, these equations include linear and nonlinear dissipative terms. However, in many important cases, they are small and can be neglected: therefore the equations are conservative, and the medium is transparent. Accord...

Hamiltonian-preserving schemes for the Liouville equation of geometrical optics with discontinuous local wave speeds

In this paper, we construct two classes of Hamiltonian-preserving numerical schemes for a Liouville equation with discontinuous local wave speed. This equation arises in the phase space description of geometrical optics, and has been the foundation of the recently developed level set methods for multivalued solution in geometrical optics. We extend our previous work in [S. Jin, X. Wen, Hamilton...

A Hamiltonian-preserving scheme for the Liouville equation of geometrical optics with transmissions and reflections

We construct a class of Hamiltonian-preserving numerical schemes for a Liouville equation of geometrical optics, with transmissions and reflections. This equation arises in the high frequency limit of the linear wave equation, with discontinuous local wave speed. In our previous work [23], we introduced the Hamiltonian-preserving schemes for the same equation when only complete transmissions or...

A Hamiltonian-Preserving Scheme for the Liouville Equation of Geometrical Optics with Partial Transmissions and Reflections

We construct a class of Hamiltonian-preserving numerical schemes for the Liouville equation of geometrical optics, with partial transmissions and reflections. This equation arises in the high frequency limit of the linear wave equation, with a discontinuous index of refraction. In our previous work [Hamiltonian-preserving schemes for the Liouville equation of geometrical optics with discontinuo...

Nonlinear Optics Basics: Kramers-Krönig Relations in Nonlinear Optics