  The behavior of electromagnetic waves in a medium modulated in time and
space, largely investigated decades ago, has attracted emerging current
interest. Here, we address one of the key questions of this research: can an
optical wave be amplified by a slowly modulated realistic photonic circuit? To
solve this problem, we investigate the propagation of light with an initial
phase velocity $v_0$ and frequency $\omega_0$ propagating through an optical
waveguide and resonator modulated by a traveling wave with phase velocity $v_p$
and low frequency $\omega_p << \omega_0$. Close to the synchronous condition,
$v_p =v_0$, the modulation effects can significantly grow. Consequently, the
bandwidth of the optical frequency comb spectrum can be dramatically enhanced
by a synchronous traveling wave with $v_p \approx v_0$ compared to the
bandwidth generated by an instantaneous wave with $v_p=\infty$. However, for
realistic optical waveguides, the light amplification remains small due to its
essentially broadband character. In contrast, we show that the modulation of a
racetrack optical resonator by a traveling wave with a small phase velocity
$v_p <<v_0$ (rather than with synchronous $v_p\approx v_0$) can significantly
amplify light remaining within a relatively small bandwidth.
