Temporal Bragg Gratings: Broadband Reconfigurable Parametric Amplifiers

TL;DR

The paper addresses transforming passive Bragg gratings into tunable amplifiers by introducing temporal modulation of the refractive index in time, controlled by a modulation frequency near the Bragg frequency $\omega_B$ and analyzed through a spatio-temporal Floquet–Bloch framework. It develops a comprehensive theoretical model combining a quarter-wave Bragg architecture, multi-harmonic scattering, Maxwell's equations, and coupled-mode dynamics to reveal how energy transfers from the modulation pump to optical signals via parametric processes, with distinct regimes of operation. Key contributions include the demonstration of coherent amplification at sub- and supra-Bragg detunings when modulating high- or low-index layers, the spectral agility of gain controlled by the modulation frequency $\omega_m$, an asymmetry between sub-Bragg and supra-Bragg regimes, and the emergence of a broadband gain continuum at extreme sub-Bragg via multi-phase-matching. These findings establish temporal Bragg gratings as versatile, integrable platforms for active, reconfigurable photonic devices such as tunable amplifiers, frequency converters, and broadband light sources, with potential impact on on-chip photonics and quantum-optical applications.

Abstract

This paper introduces temporal Bragg gratings as a new class of broadband, reconfigurable parametric amplifiers. We present a comprehensive investigation of power amplification in these structures, where a spatially periodic refractive index profile is modulated in time at frequencies near the Bragg condition. Through systematic numerical simulations, we explore the impact of modulation location (high-index vs. low-index layers), modulation frequency relative to the Bragg frequency, and modulation amplitude on the gain spectrum and field dynamics. We demonstrate that both high-index and low-index layer modulations can produce significant parametric amplification, with high-index modulation yielding higher gain for comparable modulation depths. The amplification is frequency-agile, with gain peaks tunable across a broad spectral range, and exhibits strong asymmetry between sub-Bragg and supra-Bragg regimes, the former requires substantially stronger modulation for comparable gain. In the extreme sub-Bragg limit, the system transitions from discrete sideband amplification to a broadband gain continuum at high frequencies, explained by multi-phase-matching of parametric processes. These results provide a unified framework for designing dynamically reconfigurable optical amplifiers, tunable frequency converters, and broadband light sources using temporally modulated photonic crystals, offering new pathways toward active, agile, and integrable photonic devices.

Figures (8)

• Figure 1: Schematic of a temporal Bragg grating for through-port parametric amplification, composing alternating time-periodic high-index ($n_\text{H}(t)$) and low-index ($n_\text{L}(t)$) refractive indices.
• Figure 2: Amplification of $\omega_0$ arises from multiple reflections and transmissions of the main frequency $\omega_0$ and generated sidebands $\omega_{\pm1} = \omega_0 \pm \omega_\text{m}$ through time-modulated layers. While $\omega_{\pm1}$ are generated via parametric coupling at each interface, only $\omega_0$ experiences constructive interference across the structure, leading to net gain in transmission.
• Figure 3: Power amplification in a coherent temporal Bragg grating with a modulation frequency equal to the Bragg frequency, $\omega_{\rm m} = \omega_{\rm B} = 300$ THz. The structure parameters are $n_{\rm L}=1.5$, $n_{\rm H}=2.5$, $\lambda_{0}=1\ \mu\text{m}$, $\Lambda=20$, $\delta_{\rm H}=0$, and $\delta_{\rm L}=0.12$. (a)-(c) Normal electric field distribution at 150 THz ($=\omega_{\rm B}/2$), 300 THz ($=\omega_{\rm B}$), and 450 THz ($=1.5\omega_{\rm B}$), respectively. (d) Transmission gain for different time-periodic low-index modulations $\delta_{\rm L}>0$, with static high-index layers ($\delta_{\rm H}=0$).
• Figure 4: Time-averaged power flow and energy distribution in the Bragg grating in Fig. \ref{['Fig:Res1']}. (a)-(c) Power flow at frequencies 150, 300 and 450 THz, respectively. (d)-(f) Energy density at 150, 300 and 450 THz, respectively.
• Figure 5: Power amplification in a coherent temporal Bragg grating with modulation frequency $\omega_\text{m} = \omega_\text{B}$. Spectra are shown for the case where the low-index layers are static ($\delta_L = 0$) and the high-index layers are time-modulated with varying amplitudes $\delta_H$, along with one case of concurrent modulation ($\delta_L = 0.06, \delta_H = 0.036$). All other parameters match those in Fig. 2. The plot demonstrates strong parametric gain at $f = 150\ \text{THz} = \omega_\text{B}/2$ and $450\ \text{THz} = 1.5\omega_\text{B}$.