In Chapter 4 we have derived perturbation theory for Maxwell's equations to find corrections to the electromagnetic state eigenfrequency ω due to small changes in the material dielectric constant. Applied to the case of systems incorporating absorbing materials, we have concluded that absorption introduces an imaginary part to the modal frequency, thus resulting in decay of the modal power in time. While this result is intuitive for resonator states localized in all spatial directions, it is somewhat not straightforward to interpret for the case of waveguides in which energy travels freely along the waveguide direction. In the case of waveguides, a more natural description of the phenomenon of energy dissipation would be in terms of a characteristic modal decay length, or, in other words, in terms of the imaginary contribution to the modal propagation constant. The Hamiltonian formulation of Maxwell's equations in the form (2.70) is an eigenvalue problem with respect to ω2, thus perturbation theory formalism based on this Hamiltonian form is most naturally applicable for finding frequency corrections. In the following sections we develop the Hamiltonian formulation of Maxwell's equations in terms of the modal propagation constant, which allows, naturally, perturbative formulation with respect to the modal propagation constant.

Eigenstates of a waveguide in Hamiltonian formulation

In what follows, we introduce the Hamiltonian formulation of Maxwell's equations for waveguides, [1] which is an eigenvalue problem with respect to the modal propagation constant β. A waveguide is considered to possess continuous translational symmetry in the longitudinal ẑ direction.