Propagation characteristics of planar segmented waveguides with parabolic index segments

Abstract

We have analyzed the linear propagation characteristics of planar periodic segmented waveguides with infinitely extended parabolic refractive-index segments in the transverse direction, using the matrix formulation of Gaussian beam propagation. Modal propagation and the variation of effective index as a function of duty cycle and the period of segmentation have been studied. Conditions for the existence of guided modes in such a segmented waveguide are presented.

Unexpected light behaviour in periodic segmented waveguides

“…The duty-cycle (DC) of a PSW is commonly defined as the ratio of the length d of the high index segment over the segmentation period K, DC ¼ d/K. A step index profile is assumed along the propagation direction z as it has been done in previous works 21,25,26 waveguides realized in LiNbO 3 , wave chaos development in PSW is not based on a specific waveguide fabrication process and may be applied to many materials.…”

“…The analysis has been developed using a 2D structure, and for simplicity, a TE (transverse electric) polarization has also been assumed for the fields. Using the classical assumption of the slowly envelope variation and neglecting backward propagating waves, 15,21,25,26 the parabolic wave equation representing the evolution of the field in a PSW is given by…”

Ray dispersion strongly modified by a periodic index segmentation

“…The PSW presents a transverse gaussian index profile in the high index segments that is, for example, naturally encountered with the proton exchange technique waveguide fabrication on LiNbO 3 substrate [16]. The dutycycle of a PSW is usually defined as DC =d/Λ and a step index profile is assumed along the propagation direction z as it has been done in previous works [8][9][10]. It can be mentioned that a smoother profile will not change qualitatively the results.…”

“…4 is a slice at θ = 0 of the dispersion map for the same waveguide and for the corresponding equivalent continuous waveguide. A PSW can be represented by a continuous equivalent waveguide with an index n eq (x) = n 2 + δn(x) × DC [9,10,19]. The incident angle is fixed and initial conditions are taken such as only the position of the incident ray varies.…”

“…More recently, it has been shown that multimode PSW can be a practical support for classical chaos investigation in the geometrical limit [8]. The ray dynamics in the simple and ideal 2D case of a PSW characterized by a transverse gaussian index profile is responsible for the emergence of a genuine chaotic behavior of the ray dynamics which is much more complex than that encountered with the usually considered parabolic index profile [9][10][11]. Such a complex behavior is actually comparable to what has already been studied in a periodically perturbed waveguide [12,13] or more generally, the analysis shows a dynamic that can be found in usual periodically forced nonlinear pendulum.…”