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.
Highlights
The application of coupled-mode theory to the problem of propagation in an optical waveguide is well known [1], and is useful in the description of gratings and other periodic optical systems where the strength of the perturbation is weak
The same formalism has been applied in solid state physics to the description of electrons in a weak periodic potential [2]
The optical structures that can be described using the tight binding approximation are those that consist of isolated structural elements weakly coupled to one another; the propagating eigenmodes of the overall system are closely related to the eigenmodes of the individual elements, rather than the free-space eigenmodes as in coupled-mode theory
Summary
The application of coupled-mode theory to the problem of propagation in an optical waveguide is well known [1], and is useful in the description of gratings and other periodic optical systems where the strength of the perturbation (relative to the free-space equations) is weak. Recent experiments in the microwave regime have demonstrated the validity of the tight binding approximation in such a structure [6] Another application of this framework is in the description of superstructure Bragg gratings (SSGs), called optical superlattices, which are fiber or semiconductor gratings with parameters that vary periodically as a function of position [7]. Whereas shallow SSGs can be described by the standard coupled-mode theory, deep SSGs require the complementary approach of the tight-binding approximation [8] In both these physical realizations of the tight binding approximation, the analysis has so far been restricted to the propagation of monochromatic waves at the eigenfrequencies of the structure. We describe the propagation of pulses with a nonzero spread of wave vectors in a one-dimensional structure described by the tight binding approximation and comment on certain limits in which a simplified analysis is justifiable
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