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 address an intriguing problem of this research: can light with an initial frequency <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>ω</a:mi><a:mn>0</a:mn></a:msub></a:math> be amplified solely pumped by a traveling wave with a much lower frequency <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msub><b:mi>ω</b:mi><b:mi>p</b:mi></b:msub><b:mo>≪</b:mo><b:msub><b:mi>ω</b:mi><b:mn>0</b:mn></b:msub></b:mrow></b:math>? Generally, the bandwidth of the modulation-induced optical frequency comb spectrum can be substantially broadened when the phase velocity of the traveling wave, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>v</c:mi><c:mi>p</c:mi></c:msub></c:math>, approaches the phase velocity of light, <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi>v</d:mi><d:mn>0</d:mn></d:msub></d:math>. 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 by a traveling wave having the frequency <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msub><e:mi>ω</e:mi><e:mi>p</e:mi></e:msub></e:math> and phase velocity <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msub><f:mi>v</f:mi><f:mi>p</f:mi></f:msub></f:math> much smaller than the frequency <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msub><g:mi>ω</g:mi><g:mn>0</g:mn></g:msub></g:math> and phase velocity <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:msub><h:mi>v</h:mi><h:mn>0</h:mn></h:msub></h:math> of light can result in large light amplification accompanied by conversion to multiple comb lines within a relatively small frequency band.