Parametric processes in nonlinear structures with reflections: a transfer matrix method approach

TL;DR

This work extends the Transfer Matrix Method to nonlinear optics in layered structures with reflections, enabling accurate simulation of energy transfer and phase matching in processes such as $ω$-conserving difference-frequency generation, spontaneous parametric down-conversion, and four-wave mixing. A nonlinear propagation transfer matrix is introduced to capture bidirectional fields and arbitrary material profiles, allowing seamless stacking of multilayer systems. The approach is validated against analytic DFG solutions and experimental SPDC and FWM in Fabry–Pérot and Bragg cavities, including counter-propagating generation, and it preserves bosonic commutation via a Bogoliubov scattering matrix. The resulting framework provides a practical, extensible tool for quantum photonics device design and nonlinear optics simulations, with open-source code available for integration into existing workflows.

Abstract

The Transfer Matrix Method is a powerful numerical tool for simulating wave propagation in layered media. It has been widely applied in many fields, although its use is typically restricted to passive media. In this paper, we develop the transfer method to simulate optical nonlinear generation and give examples for processes such as difference frequency generation, spontaneous parametric down conversion, and four-wave mixing including generation of counter propagating waves. This framework enables accurate simulation of complex multilayer structures and resonant cavities, providing a versatile tool for designing nonlinear photonic devices.

Figures (7)

• Figure 1: Schematic of nonlinear propagation single cell
• Figure 2: TMM simulaton of DFG in straight waveguide with phase matching and quasi-phase matching, compared to the analytical solution. The signal input fields is 1 V/m and the idler input field is 0 V/m and the pump field is $10^7$ V/m.
• Figure 3: Fabry-Pérot TMM spectral results of the photon pairs in forward-forward emission (Eq. \ref{['eq:P_ff']}) and backward-backward emission (Eq. \ref{['eq:P_bb']}).
• Figure 4: Schematic of the electric field nonlinear generation for counter-propagating idler and signal.
• Figure 5: (a) Coincidence rate at different lengths of the crystal. (b) Spectral response for the forward-forward emitted probability, backward-backward emitted probability and forward-backward emitted probability, for a crystal length of 1 mm.