Temporal Paraxial Optics under Adiabatic Modulations

This paper presents a temporal paraxial formulation for the propagation of ultrashort optical pulses in time-modulated media with slowly varying refractive index. By deriving the paraxial wave equation directly in the time domain from the Helmholtz equation under an adiabatic approximation, the model remains analytically tractable while extending paraxial optics beyond time-invariant backgrounds commonly treated by frequency-domain expansions. The resulting equation preserves a Schrödinger-like structure in the presence of explicit temporal modulation and admits closed-form solutions for ultrashort Gaussian pulses. The framework supports a Green’s-function description and an operator-based Hamiltonian formalism, from which an ABCD matrix representation for temporal propagation in time-varying media is obtained. The results demonstrate that temporal modulation provides an active means to control ultrashort pulse dynamics, enabling tailored evolution of pulse characteristics such as temporal width and chirp, with potential applications in ultrafast pulse shaping and a direct connection to temporal wave-packet dynamics.

💡 Research Summary

The manuscript introduces a comprehensive temporal‑paraxial framework for the propagation of ultrashort optical pulses in media whose refractive index varies slowly in time. Starting from Maxwell’s equations, the authors derive a scalar wave equation for the electric field and separate a fast carrier e^{iω₀t−ik₀z} from a slowly varying envelope ψ(t,z). By invoking the paraxial approximation (|∂²ψ/∂z²|≪k₀|∂ψ/∂z|) and an adiabatic condition (the modulation frequency ω_m is far smaller than the carrier ω₀), they neglect higher‑order spatial derivatives and the time derivatives of ε(t) and μ(t). This leads to a first‑order evolution equation in the propagation coordinate z:

∂ψ/∂z ≈ i a(τ) ∂²ψ/∂τ² + i b(τ) ψ,

where τ = t – z c/v² is a retarded time moving with the pulse, v(t)=c/n(t) is the instantaneous phase velocity, a(τ)=1/(2k₀v²(τ)) plays the role of an effective mass, and b(τ)=ω₀²c