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.