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Experimental data demonstrates that continuous-wave (CW) theories of non-linear optics do not accurately model the optical turbulence (chaos) arising from delayed discrete feedback in nonlinear optical ring cavities (Ikeda, 1979). To accurately predict this onset of optical turbulence within a ring cavity, this study implements a high-performance computational architecture simulating the complex-valued, discrete-time Ikeda map. This map, in comparison, accurately models the behavior within a nonlinear ring cavity. By utilizing a decoupled two-dimensional real-valued mapping, the heavy complex-number computational overhead was eliminated, thus optimizing the algorithm to scan high-dimensional parameter spaces with minimal latency. The current implementation successfully plotted the structural transition from stable periodic orbits to persistent chaos via a period-doubling bifurcation cascade. To produce even better latency benchmarks for larger parameter sweeps, the rendering engine and numerical solvers are actively being translated into modern C++.
After generating the bifurcation data, the chaotic regime was analyzed. Using Jacobian matrix differentiation to compute the Lyapunov exponent for varying values of the reflection coefficient $B$, the exponential trajectory divergence at precise bifurcation thresholds was confirmed mathematically. Additionally, a box-counting algorithm extracted a non-integer fractal dimension of $D\approx1.714$, confirming the self-similar geometry of the resulting strange attractor. Ultimately, the modelling of the discrete map revealed that the period-1 stability limit in the route to optical turbulence was $B_{1}\approx0.322$, which disagrees with R. W. Boyd's theoretical CW threshold of $B_{CW}\approx0.0624$ (Boyd, 2020), as expected. The numerical precision of the iterative architecture was orthogonally validated by calculating the ratio of successive bifurcation intervals, which converged toward the universal Feigenbaum scaling constant with an 84.03% relative agreement.
[1] K. Ikeda, "Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system," Opt. Commun., vol. 30, no. 2, pp. 257-261, 1979.
[2] R. W. Boyd, Nonlinear Optics, 4th ed. Academic Press, 2020.