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arXiv:2409.12775 · Physical Review Research

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

M. Sumetsky

Published Sep 26, 2025• 3 citations
arXivHTMLPDF

Abstract

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 ω 0 be amplified solely pumped by a traveling wave with a much lower frequency ω p ≪ ω 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 by a traveling wave having the frequency ω p and phase velocity v p much smaller than the frequency ω 0 and phase velocity v 0 of light can result in large light amplification accompanied by conversion to multiple comb lines within a relatively small frequency band.

DOI
10.1103/bqyj-kggh

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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 ω 0 be amplified solely pumped by a traveling wave with a much lower frequency ω p ≪ ω 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 by a traveling wave having the frequency ω p and phase velocity v p much smaller than the frequency ω 0 and phase velocity v 0 of light can result in large light amplification accompanied by conversion to multiple comb lines within a relatively small frequency band.
Light propagating in an optical waveguide can gain or lose power through interaction with a travelling acoustic wave or radio-frequency modulation of permittivity. Here, we model this propagation by considering an optical wave interacting with a weak travelling-wave permittivity perturbation whose frequency is much smaller than the optical frequency and whose amplitude decays exponentially along the propagation direction. Four modulation cases are analyzed: instantaneous modulation, synchronous modulation, and the Stokes and anti-Stokes resonances. For these cases, simple expressions are obtained for the carrier and sideband transmission and reflection powers as well as for the total gain and loss powers. Although the achievable total gain remains small for realistic modulation and waveguide parameters, the anti-Stokes resonance is identified as the most promising condition for observing modulation-induced light amplification.
Light propagating in an optical waveguide can gain or lose power through interaction with a travelling acoustic wave or radio-frequency modulation of permittivity. Here, we model this propagation by considering an optical wave interacting with a weak travelling-wave permittivity perturbation whose frequency is much smaller than the optical frequency and whose amplitude decays exponentially along the propagation direction. Four modulation cases are analyzed: instantaneous modulation, synchronous modulation, and the Stokes and anti-Stokes resonances. For these cases, simple expressions are obtained for the carrier and sideband transmission and reflection powers as well as for the total gain and loss powers. Although the achievable total gain remains small for realistic modulation and waveguide parameters, the anti-Stokes resonance is identified as the most promising condition for observing modulation-induced light amplification.
A traveling-wave amplifier pumped by a femtosecond light pulse is numerically simulated. Essential output pulse characteristics, such as pulse duration, energy, and the spectrum, are investigated for various experimentally relevant situations. In particular, the influence of the dynamics associated with the amplifying transition and the sources and the role of phase modulation are discussed, as is as the influence of the group-velocity dispersion. Conclusions are drawn for achieving the shortest possible output pulses at high-energy conversion from the pump pulse.
The new possibilities of almost ideal amplification of optical solitons during the incoherent interaction of light pulses with a resonantly amplifying medium are considered. The mechanism of two-photon amplification of optical solitons with an optimal frequency-modulation law is proposed. It is shown that the entirely ideal amplification of solitons cannot be achieved because the law of phase modulation of radiation differs from a parabolic law. The possibility of using the phase cross modulation to produce the required initial phase of amplified solitons is studied.
The propagation of a strong coherent "pump" wave traveling through a resonant medium is discussed. The medium is assumed to consist of identical stationary atoms, and the pump-wave frequency is assumed to coincide exactly with the resonance frequency for transitions between a particular pair of atomic states. A previously developed theory of the response of a strongly driven atom to a weak nearly resonant "signal" field is used to deduce the existence of a coupling between any two waves travelling parallel to the pump wave whose (nearly equal) frequencies sum to twice the pump-wave frequency. In the limit of very intense (highly saturating) pump fields, the coupling between the two waves of nearby frequencies leads to amplification of both waves if their frequencies lie within an interval about the pump-wave frequency equal to the Rabi frequency of population inversion. The basic process, which is roughly described by travelling-wave parametric-amplifier equations of motion, consists of the absorption of two pump-field photons followed by their emission at different nearby frequencies, and thus implies a frequency instability of the initially coherent pump wave.
Using semiclassical radiation theory, a formalism similar to that used by Lamb in his Theory of an Optical Maser is developed for studying the amplification of vector traveling waves in a laser-type medium. The effect of the medium on the waves is given in terms of space (or time) dependent field amplitudes and phases and a nonlinear index of refraction. With particular emphasis on typical gaseous media, the effects of Doppler broadening are treated in detail for arbitrary ratios of natural to Doppler linewidths. Polarization and propagation vectors in various directions are considered, and the nonlinear effects are found to make an isotropic medium effectively anisotropic. Lowest order nonlinear effects (due to a polarization cubic in the field amplitudes) are studied extensively, and the frequency dependence of several of these processes is presented in graphical form. In particular, the introduction of fields at new frequencies and polarization effects are considered. The characteristics of these nonlinear processes peculiar to Doppler broadened lines are discussed, and the processes are interpreted in terms of saturation and coherent modulation of the population inversion density. Strong nonlinear effects are considered in a more approximate way and are found to consist of saturation of the various linear and nonlinear processes previously considered. These strong nonlinear effects should occur at low enough intensities to be easily observed in practice on a CW basis. With the present formalism, the analytical results of Gordon, White and Rigden regarding gain saturation in laser amplifiers are obtained, and the extension is made to include frequencies away from line center and the effects of multiple spectral components. Again, the introduction of fields at new frequencies is considered in detail. These results are also discussed in terms of saturation and coherent modulation of the populations and hole burning.