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

The behavior of electromagnetic waves in a medium modulated in time and space, largely investigated decades ago, has recently attracted renewed interest. Here, we solve an intriguing problem of this research: can light with an initial frequency ${\omega }_0$ be amplified in a realistic photonic circuit solely pumped by a traveling wave with a much lower frequency ${\omega }_p\ll {\omega }_0$? Generally, the bandwidth of the modulation-induced optical frequency comb spectrum can be substantially broadened when the phase velocity of the traveling wave, $v_p$, approaches the phase velocity of light, $v_0$. However, in realistic photonic waveguides, the amplification effect remains small due to the unfeasible modulation and waveguide parameters required. In contrast, we demonstrate that modulating an optical resonator with a traveling wave that has a small phase velocity $v_p\ll v_0$ (rather than a synchronous $v_p\approx v_0$) can result in narrow-band light amplification, which is dramatically enhanced near the Brillouin phase-matching condition ${\omega }_p\mathrm{/}{v}_p\approx 2{\omega }_0\mathrm{/}{v}_0$. Our calculations show that the proposed amplifier of light can be realized in a lithium niobate racetrack resonator with millimeter-scale perimeter modulated by a surface acoustic wave with surprisingly small and practically achievable amplitude.