A comprehensive study on beam dynamics inside symmetrically chirped waveguide array mimicking the graded index media

TL;DR

This work analyzes beam dynamics in symmetrically chirped nonlinear waveguide arrays by combining a continuous approximation of the discrete nonlinear Schrödinger equation with semi-analytical variational techniques. It develops two chirping schemes (linear and quadratic) and demonstrates that the varying inter-waveguide coupling acts as an effective graded-index medium, enabling oscillatory Gaussian-beam propagation in the linear regime and discrete soliton formation in the nonlinear regime. The study provides explicit stationary solutions, linear stability analysis, and soliton dynamics, validated against full DNLSE simulations, and derives closed-form expressions for soliton trajectories and beaming characteristics in both chirp configurations. Overall, the results establish chirped waveguide arrays as versatile photonic lattices for precise light routing and soliton control, bridging discrete lattices and graded-index analogies.

Abstract

In this article, we explore the beam dynamics within symmetrically chirped nonlinear waveguide arrays, focusing on linear and quadratic chirping schemes. We propose a practical structure for these arrays that enhances control over light propagation. By employing a continuous approximation of the discrete nonlinear Schrödinger equation (DNLSE), we utilize a semi-analytical variational method to analyze beam behavior under waveguide chirping. Our findings indicate that the symmetrically chirped waveguide arrays behave similarly to graded index systems, with varying coupling coefficients analogous to the refractive index in continuous media. We derive a steady-state solution and validate it against numerical simulations, alongside conducting a linear stability analysis to assess the robustness of these solutions. The results reveal that input Gaussian beams in such waveguide arrays follow an oscillatory trajectory akin to that in parabolic index media. Notably, under nonlinear conditions, these beams evolve as discrete solitons. Our rigorous investigation of the propagation characteristics in both linear and nonlinear regimes highlights the intricate dynamics of optical beams within the engineered chirped waveguide arrays, supported by comparisons to comprehensive numerical simulations.

Figures (11)

• Figure 1: (a) Schematic representation of the waveguide arrangement under a symmetric linear chirp, along with the variation in refractive index in the transverse direction. (b) The variation in the inter-waveguide separation $d_n$ (solid black line) and the corresponding normalized coupling coefficient $\eta$ (dashed red line) as a function of $n$, for typical parameters $\delta = 30~\text{nm}$ and $d_{\text{ref}} = 30~\mu \text{m}$. (c) Variation of the coupling coefficient $\eta$ along the transverse coordinate $n$ for the chirping strength $\delta$ ranging from 10 nm to 40 nm (shaded region). In the right inset we plot $\alpha_l$ as a function of $g_c$.
• Figure 2: (a) Schematic representation of the waveguide arrangement under a symmetric quadratic chirp, along with the variation in refractive index in the transverse direction. (b) The variation of $d_n$, the separation between adjacent waveguides, (solid black line) and the corresponding normalized coupling coefficient profile $\eta$ (dashed red line) for $\delta = 1.5$ nm along the transverse coordinate $n$. (c) Variation of the coupling coefficient $\eta$ along the transverse coordinate $n$ for the chirping strength $\delta$ ranging from 0.05 nm to 0.25 nm (shaded region). In the right inset we plot $\alpha_q$ as a function of $g_c$.
• Figure 3: Comparison of a soliton evolving as per the DNLSE and the approximate NLSE under (i) linear and (ii) quadratic chirping. (a) and (b) are the evolution in $n$ space along with the corresponding evolution in the wavenumber ($\kappa$) space (e) and (f) as per the DNLSE and NLSE, respectively. The evolution of the parameters as per the DNLSE (solid red dots) and NLSE (solid black line) are given in (c) $n_0$, (d) $\psi_0$, (g) $\kappa_0$, and (h) $\phi$.
• Figure 4: Evolution of (a) a Gaussian beam as an input of Eq. \ref{['eq:norm_dnlse']} and (b) the numerical solution for the DNLSE in a linearly chirped WA under low power conditions ($\mathcal{N}=0$) for $\delta = 10 ~\text{nm}$ and $\alpha_l = 3 \times 10^{-3}$ which makes $n_{sol}= 6.65$. In plot (c) we superimpose the Gaussian beam shape with the numerical stationary solution of Eq. \ref{['eq:fsolve']}. In upper inset we demonstrate how much these two shapes differ. Plot (d)-(g) depict the evolution of beam parameters where the solid line represents the variation predictions. The circle and solid dots represent the corresponding values obtained form the numerical stationary solution and Gaussian feed in respectively.
• Figure 5: Evolution of (a) A Gaussian beam as an input of Eq. \ref{['eq:norm_dnlse']} and (b) the numerical solution for the DNLSEs in a quadratic chirped WA under low power conditions ($\mathcal{N}=0$) for $\delta = 1 ~\text{nm}$ and $\alpha_q = 3\times 10^{-4}$ which makes $n_{sol}=6.4$. In plot (c) we superimpose the Gaussian beam shape with numerical stationary solution of Eq. \ref{['eq:fsolve']}. In upper inset we demonstrate how much these two shapes differ. Plot (d)-(g) depict the evolution of beam parameters where the solid line represents the variation predictions. The circle and solid dots represent the corresponding value obtain from the numerical stationary solution and Gaussian feeding, respectively.