Transformation and amplification of light modulated by a traveling wave with a relatively low frequency

The behavior of electromagnetic waves in media modulated in both time and space, extensively studied decades ago, has recently attracted renewed attention. In this work, we address a central question of this research: can light at an initial frequency $\omega_{0}$ be amplified solely by pumping with a traveling wave of much lower frequency $\omega_{p} \ll \omega_{0}$? In general, the bandwidth of the modulation-induced optical frequency-comb spectrum increases substantially when the phase velocity of the traveling wave, $v_{p}$, approaches the phase velocity of light, $v_{0}$. However, in realistic photonic waveguides, the resulting amplification remains negligible due to the unfeasible modulation strengths and waveguide parameters required. In contrast, we demonstrate that modulating an optical resonator with a traveling wave of frequency $\omega_{p}$ and phase velocity $v_{p}$ much smaller than the frequency $\omega_{0}$ and phase velocity $v_{0}$ of light can produce strong amplification. This effect is accompanied by conversion into multiple comb lines within a relatively narrow frequency band.