Optical pulse propagation in the tight-binding approximation

Abstract

Abstract:We formulate the equations describing pulse propagation in a one-dimensional optical structure described by the tight binding approximation, commonly used in solid-state physics to describe electrons levels in a periodic potential. The analysis is carried out in a way that highlights the correspondence with the analysis of pulse propagation in a conventional waveguide. Explicit expressions for the pulse in the waveguide are derived and discussed in the context of the sampling theorems of finite-energy space and time signals.

Coupled resonator optical waveguides

“…We can write an approximate dispersion relationship around the central wave vector of the pulse as (5) where is the group velocity of the pulse. Pulse propagation governed by (5) will be discussed in Section III-A, and as governed by the nonlinear relationship (3) in Section III-B.…”

“…A few algebraic manipulations (in the form of Fourier transform relationships [5]) show that (13) where the summation over indexes the resonators that comprise the CROW, and the summation over reflects the Born-von Karman quantization of the propagation constant in structures of finite length (2). Fig.…”

“…Since the allowed s are quantized in a structure of finite length, the dispersion relationship (5) implies that the allowed values are quantized. To avoid aliasing [25], the temporal interval between two samples must be greater than twice the temporal extent of the pulse envelope , where according to (5). Therefore which implies (14) Further, Fig.…”

“…In this approximation, (27) simplifies to (29) where we define (30) and the notation is a reminder of the presence of a dispersion relationship as in (5).…”

“…1) The extensive research on the properties of defects in photonic crystals [11], [19], [20] directly leads to both analytical [2], [5], [9] and numerical [2], [21] descriptions of the waveguide modes and the propagation of pulses.…”