[Paper Review] Derivation of Nonlinear Evolution Equations for Coupled and Single Fields in a Quadratic Medium

Jeffrey Moses, Frank W. Wise|ArXiv.org|Apr 20, 2006

Differential Equations and Numerical Methods3 citations

TL;DR

This paper derives a generalized nonlinear evolution equation for ultrashort pulses in quadratic media, extending the slowly evolving wave approximation to model degenerate three-wave mixing. It derives an approximate single-field equation for the fundamental field by perturbatively collapsing coupled equations, yielding a higher-order nonlinear Schrödinger-type equation that includes dispersion, self-phase modulation, and third-order nonlinear terms due to group-velocity dispersion and pulse duration effects.

ABSTRACT

We derive coupled propagation equations for ultrashort pulses in a degenerate three-wave mixing process in quadratic media, using approximations consistent with the slowly evolving wave approximation [T. Brabec and F. Krausz, Phys. Rev. Lett. 78, 3282 (1997)]. From these we derive an approximate single-field equation for the fundamental field. This document expands upon mathematics used for work submitted by the same authors to Physical Review Letters.

Motivation & Objective

• To develop a more general framework for modeling ultrashort pulse propagation in quadratic nonlinear media beyond the standard slowly varying envelope approximation.
• To address limitations in pulse duration and spectral bandwidth by using the slowly evolving wave approximation (SEWA), which allows shorter pulses than SVEA.
• To derive a single-field evolution equation for the fundamental frequency by perturbatively eliminating the second-harmonic field from the coupled equations.
• To include higher-order nonlinear and dispersive terms arising from group-velocity dispersion and pulse duration effects in the final single-field equation.

Proposed method

• Formulate Maxwell’s equations in the Fourier domain with linear and quadratic polarization contributions.
• Decompose the electric field into fundamental and second-harmonic components using complex envelopes and apply the slowly evolving wave approximation.
• Expand wavevectors in Taylor series around central frequencies and assume instantaneous nonlinearity to derive coupled propagation equations in time domain.
• Apply multiple-scale perturbation analysis using slow variables for envelope evolution and dispersion, separating orders in the expansion parameter $\beta$.
• Eliminate the second-harmonic field order-by-order via asymptotic expansion, substituting $\hat{a}_{20}$ and $\hat{a}_{21}$ into the fundamental field equation.
• Derive a final single-field equation for the fundamental field that includes terms up to $1/\beta^2$, capturing higher-order dispersion and nonlinearities.

Experimental results

Research questions

• RQ1 How can the coupled propagation of fundamental and second-harmonic fields in quadratic media be modeled beyond the slowly varying envelope approximation?
• RQ2 What are the dominant higher-order nonlinear and dispersive effects that emerge in ultrashort pulse propagation in quadratic media?
• RQ3 How can the second-harmonic field be systematically eliminated from the coupled system to yield an effective single-field equation for the fundamental field?
• RQ4 What role does group-velocity dispersion play in shaping the nonlinear evolution of ultrashort pulses in quadratic media?
• RQ5 What is the structure of the resulting single-field nonlinear evolution equation, and which higher-order terms are significant for accurate modeling?

Key findings

• The derived single-field equation for the fundamental field includes a self-phase modulation term proportional to $|a_1|^2 a_1$ at order $1/\beta$, capturing intensity-dependent frequency shifts.
• A significant third-order nonlinear term appears at order $1/\beta^2$, proportional to $|a_1|^2 \partial a_1 / \partial s$, arising from group-velocity dispersion and pulse duration effects.
• A linear correction term involving $\partial |a_1|^2 / \partial s$ appears at order $1/\beta^2$, indicating amplitude-gradient coupling in the pulse envelope.
• The equation contains a complex combination of second- and third-order dispersion terms, with coefficients depending on $\alpha_1$, $\alpha_2'$, and $\omega_0\tau_0$, reflecting material and pulse characteristics.
• The final equation is a higher-order nonlinear Schrödinger-type equation with terms up to $O(1/\beta^3)$, but only a few terms—particularly those at $1/\beta$ and $1/\beta^2$—are dominant.
• The perturbative elimination of the second-harmonic field results in a consistent, closed-form evolution equation for the fundamental field that retains essential nonlinear and dispersive physics.