Canonical quantization of envelope function equations in a PBG material

Suresh Pereira and J.E. Sipe
Department of Physics, University of Toronto, Toronto, ON M5S 1A7, Canada

We construct a canonically quantized Hamiltonian for slowly-varying pulses near a photonic bandgap (PBG) in a one dimensional, Kerr nonlinear, periodic medium. We start with an exact Hamiltonian density defined in terms of the Bloch modes of the dual field [1]. We then specialize to slowly varying pulses near the PBG by considering envelope functions that modulate Bloch functions at the upper and lower portions
of the centre or edge of the Brillouin zone. We use the method of multiple scales to separate the Bloch functions from the envelope functions, and to include the nonlinearity at the appropriate level of approximation [2]. The restricted Hamiltonian, written in terms of these envelope functions, is used to determine the appropriate equations of motion for the envelope function by applying a set of canonical commutation relations. This gives us a general procedure for deriving canonically quantized coupled mode equations in a PBG material. Use of the associated Lagrangian allows investigation of the conserved quantities of the coupled mode equations. Via a similar procedure we derive a nonlinear Schrodinger equation, also in terms of canonically quantized envelope functions. Generalizations of the method to 2 and 3 dimensions, and their usefulness, will be discussed.

[1] Peter D. Drummond, „Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics, Phys Rev A, V42 #11, 6845 (1990).

[2] C. Martijn de Sterke, David G. Salinas and J. E. Sipe, Coupled-mode theory for light propagation through deep nonlinear
gratings, Phys Rev E, V54 #2, 1969 (1996).