Temporal Paraxial Optics under Adiabatic Modulations

TL;DR

The paper develops a temporal paraxial framework for ultrashort pulses propagating in slowly time-modulated media, deriving a time-domain paraxial equation under an adiabatic approximation. A Schrödinger-like evolution along the propagation axis $z$ is obtained with a time-dependent Hamiltonian $\hat{H} = -a(\tau) \partial^2/\partial \tau^2 - b(\tau)$ and a retarded time $\tau$, enabling explicit incorporation of $v(\tau)$ and $a(\tau), b(\tau)$. The analysis yields a Green's function $G_\chi(\tau,z;\tau_0)$ and a Gaussian-pulse solution $\chi(\tau,z) = \sqrt{\frac{q_0}{q(\tau,z)}} \exp\left(-\frac{(\tau-\tau_c)^2}{2 q(\tau,z)}\right)$ with $q(\tau,z)= q_0 + 2 i a(\tau) z$, plus an ABCD matrix formalism with $A=1$, $B=2 a(\tau) z$, $C=0$, $D=1$. These results imply that temporal modulation acts as an active degree of freedom to tailor pulse duration and chirp, offering pathways for ultrafast pulse shaping and time-modulated photonics.

Abstract

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.

Figures (2)

• Figure 1: Propagation and diffraction of temporal pulse across a time-invariant (left) and a time-modulated (right) medium. The time-modulated scenario offers a dynamical control on the pulse characteristics (duration, chirp, etc.) as the medium parameters change in time. On the other hand, the time-invariant medium shows the same response independently of the chosen time instant.
• Figure 2: Evolution of five identical pulses launched into a time-varying optical medium at five different time instants $\tau_\mathrm{c} = 1, 3, 5, 7, 9\,$ ps. (Top panel) Initial pulses at the spatial origin $z = 0$. (Bottom panel) Propagated and diffracted pulses at $z = 100\, \mu$m. System parameters: $q_{0} = 10^{-26}\, \mathrm{s}^2$ , $\mu_{\text{r}}(t) = 1$, $\varepsilon_{\text{r}}(\tau) = \alpha \tau + 1.5$, $\alpha = 4\cdot 10^{11}\,\mathrm{s}^{-1}$. This permittivity definition gives $\varepsilon_{\text{r}}(1\,\mathrm{ps}) = 1.9$, $\varepsilon_{\text{r}}(3\,\mathrm{ps}) = 2.7$, $\varepsilon_{\text{r}}(5\,\mathrm{ps}) = 3.5$, $\varepsilon_{\text{r}}(7\,\mathrm{ps}) = 4.3$, $\varepsilon_{\text{r}}(9\,\mathrm{ps}) = 5.1\,$. Carrier frequency $f_\text{c} = 10\,$THz.