Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Spectrum of three-wave mixing for multimode fields

Y. B. Band, Donald F. Heller, Jerzy S. Krasiński

Published Oct 1, 1989• 15 citations
Publisher

Abstract

Analytic solutions are developed for the time-dependent intensity that results from the three-wave mixing of multimode fields. These differ dramatically from the solutions for single-mode (time-independent) fields. New frequencies in the output spectrum are created as input intensities increase. Under conditions of strong nonlinear coupling the output spectrum broadens, ultimately reaching the phase-match bandwidth limit, and the dynamics can become chaotic. A new technique for enhancing mixing efficiencies is presented.

DOI
10.1103/physreva.40.4400

Related papers

Spectra of second harmonic generation and sum frequency generation for multimode fields

Jerzy S. Krasiński, Donald F. Heller, Jay R. Ackerhalt +1 more

The theory of three wave mixing was formulated in a classic paper by Armstrong, Bloembergen, Ducuing, and Pershan over 25 years ago.1 A theoretical description of three wave mixing for multimode time-dependent input fields (as opposed to cw single frequency time independent input fields) has recently been presented.2 Analytic solutions were developed for die time-dependent intensities of the output fields. These solutions differ dramatically from those for single-mode fields. New frequencies in the sum-frequency output spectrum and in the input-field output spectrum are created and grow in magnitude as the input intensities are increased. The output spectrum broadens under conditions of strong nonlinear coupling, ultimately reaching the phasematched bandwidth limit. Here, we review the results presented in Ref. 2 for sum frequency generation (SFG) and explicidy compare these results with those for second harmonic generation (SHG). We show that SHG cannot be described as a limit of the SFG formalism for multimode fields. The SFG formulation for the interaction of photons close to frequency ω1 with photons close to frequency ω2 producing photons close to frequency ω 3= ω 1+ ω 2 does not allow ω1(ω2) photons to interact with each other to produce photons close to frequency 2ω1(2ω2). This self interaction is crucial for SHG and must be incorporated. Therefore, a separate formulation is used to describe SHG with multimode input fields.

Nonlinear Dynamics in Optical Systems•Jan 1, 1990•0 citations

Theory of Optical Frequency Mixing Using Resonant Phenomena

Ali Javan, A. Szöke

The nonlinear interaction of two intense optical waves with three energy levels of an atomic system is analyzed theoretically. The frequencies of the applied fields are assumed to be on or near two of the resonances involving the three levels. Particular attention is given to the generation of power at the difference frequency of the two applied fields where their frequency separation lies in the far-infrared range of the spectrum. In the limit of large input fields where atomic transitions are driven to saturation, the generated difference-frequency power is no longer proportional to the product of the two input powers. It is shown that, in this limit, the total rate of emission of photons at the difference frequency may, in principle, approach a value given by $\frac{\mathrm{rN}}{T}$, where $N$ is the total number of atoms, $T$ is the average time of coherent interaction of atoms and the radiation field, and $r$ is a factor which in favorable cases approaches unity at low temperatures. Also, a number of additional effects are encountered due to coherent double-quantum transitions among various levels which may be ignored only for weak input powers. These effects are highly dependent on the modes of relaxations of atomic levels and the details of the line shape. The results are applied to several cases, in which the resonance condition yields large efficiency in nonlinear frequency mixing.

Physical Review•Jan 18, 1965•15 citations

Phase-modulation effects on the spectrum of third-harmonic generation

Y. B. Band

We calculate the spectrum of third-harmonic generation (THG) for multimode fundamental input fields. We consider input fields from (1) a laser with gain on several cavity modes, (2) pulses with short temporal duration, and (3) frequency-modulated input fields. The effects of phase modulation in the nonlinear medium on the spectrum of the THG field are considered. Using the estimates by Miles and Harris [IEEE J. Quantum Electron. QE-9, 470 (1973)] for the nonlinear susceptibilities ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-3\ensuremath{\omega};\ensuremath{\omega},\ensuremath{\omega},\ensuremath{\omega}) and ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-\ensuremath{\omega};\ensuremath{\omega},-\ensuremath{\omega},\ensuremath{\omega}) in alkali-metal vapors, we find that phase modulation can profoundly affect the efficiency and the spectrum of the THG. Generally, the self-phase modulation of the fundamental will tend to reduce the phase matching of an intense fundamental field and the third harmonic and thereby lower the conversion efficiency. New frequencies not contained in the set of frequencies ${\mathrm{\ensuremath{\omega}}}_{1\mathit{i}}$+${\mathrm{\ensuremath{\omega}}}_{1\mathit{j}}$+${\mathrm{\ensuremath{\omega}}}_{1\mathit{k}}$, where {${\mathrm{\ensuremath{\omega}}}_{1\mathit{i}}$} are frequencies in the fundamental laser spectrum, are created even at relatively low conversion due to phase modulation. Effects of cross phase modulation are also studied.

Physical Review A•Nov 1, 1990•11 citations

Intensity instability and correlation in amplified multimode wave mixing

Haechan An, Hal Owens, Hamza Ather +2 more

The dynamics of optical nonlinearity in the presence of gain and feedback can be complex leading to chaos in certain regimes. Temporal, spectral, spatial, or polarization instability of optical fields can emerge from chaotic response of an optical [Formula: see text] or [Formula: see text] nonlinear medium placed between two cavity mirrors or before a single feedback mirror. The complex mode dynamics, high-order correlations, and transition to instability in these systems are not well known. We consider a [Formula: see text] medium with amplified four-wave mixing process and study noise and correlation between multiple optical modes. Although individual modes show intensity instability, we observe relative intensity noise reduction close to the standard quantum noise, limited by the camera speed. We observe a relative noise reduction of more than 20 dB and fourth-order intensity correlation between four spatial modes. More than 100 distinct correlated quadruple modes can be generated using this process.

Scientific Reports•Aug 30, 2022•2 citations

Role of dressed states in resonant four-wave mixing with large Doppler broadening

Sergei A Babin, Evgeniy V. Podivilov, David A. Shapiro

Experiments on resonant four-wave mixing demonstrated high- efficiency CW down-conversion of frequency in gas (up to 25%). Meanwhile, the high intensities make the perturbation theory invalid to understand the behavior of output power measured as a function of input powers. The nonlinear susceptibility of optically thin medium is found for difference-sum scheme (omega) 4 equals (omega) 1 - (omega) 2 + (omega) 3, where fields one and three are strong and two and four are weak. In symmetric case (kappa) 2 equals (kappa) 4, the integral over velocity can be calculated analytically for Doppler limits in collinear geometry. The compact explicit formula obtained yields the intensity and frequency dependence. The peak of mixing coefficient as a function of intensity is found around equal Rabi frequencies of fields one and three. The effect is shown to base on resonance between two closed cycles via sublevels of dressed states. The number of peaks in a spectrum varies from three to eight. Explicit formula allows interpreting the measurements and predicting the optimal relation between parameters for high conversion efficiency in gas with large Doppler broadening.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jan 28, 1999•0 citations

Fully resonant three-wave mixing in isotropic media under induced transparency conditions

Vasily G. Arkhipkin, V. Yu. Apanovich

A theoretical investigation is reported of three-wave frequency mixing under simultaneous single-photon and two-photon resonance conditions when one of the fields being mixed is weak and the other is strong. A study is made of the influence of the field splitting on the nonlinear susceptibility in the case of homogeneously and Doppler-broadened transitions. An analysis is made of the role played by interference between quantum transitions via quasilevels in absorption and generation, and of its influence on mixing. It is shown that significant enhancement of the nonlinear generation efficiency is possible in optically dense media.

Quantum Electronics•Apr 30, 1997•0 citations
Analytic solutions are developed for the time-dependent intensity that results from the three-wave mixing of multimode fields. These differ dramatically from the solutions for single-mode (time-independent) fields. New frequencies in the output spectrum are created as input intensities increase. Under conditions of strong nonlinear coupling the output spectrum broadens, ultimately reaching the phase-match bandwidth limit, and the dynamics can become chaotic. A new technique for enhancing mixing efficiencies is presented.
The theory of three wave mixing was formulated in a classic paper by Armstrong, Bloembergen, Ducuing, and Pershan over 25 years ago.1 A theoretical description of three wave mixing for multimode time-dependent input fields (as opposed to cw single frequency time independent input fields) has recently been presented.2 Analytic solutions were developed for die time-dependent intensities of the output fields. These solutions differ dramatically from those for single-mode fields. New frequencies in the sum-frequency output spectrum and in the input-field output spectrum are created and grow in magnitude as the input intensities are increased. The output spectrum broadens under conditions of strong nonlinear coupling, ultimately reaching the phasematched bandwidth limit. Here, we review the results presented in Ref. 2 for sum frequency generation (SFG) and explicidy compare these results with those for second harmonic generation (SHG). We show that SHG cannot be described as a limit of the SFG formalism for multimode fields. The SFG formulation for the interaction of photons close to frequency ω1 with photons close to frequency ω2 producing photons close to frequency ω 3= ω 1+ ω 2 does not allow ω1(ω2) photons to interact with each other to produce photons close to frequency 2ω1(2ω2). This self interaction is crucial for SHG and must be incorporated. Therefore, a separate formulation is used to describe SHG with multimode input fields.
The nonlinear interaction of two intense optical waves with three energy levels of an atomic system is analyzed theoretically. The frequencies of the applied fields are assumed to be on or near two of the resonances involving the three levels. Particular attention is given to the generation of power at the difference frequency of the two applied fields where their frequency separation lies in the far-infrared range of the spectrum. In the limit of large input fields where atomic transitions are driven to saturation, the generated difference-frequency power is no longer proportional to the product of the two input powers. It is shown that, in this limit, the total rate of emission of photons at the difference frequency may, in principle, approach a value given by rNT\frac{\mathrm{rN}}{T}TrN​, where NNN is the total number of atoms, TTT is the average time of coherent interaction of atoms and the radiation field, and rrr is a factor which in favorable cases approaches unity at low temperatures. Also, a number of additional effects are encountered due to coherent double-quantum transitions among various levels which may be ignored only for weak input powers. These effects are highly dependent on the modes of relaxations of atomic levels and the details of the line shape. The results are applied to several cases, in which the resonance condition yields large efficiency in nonlinear frequency mixing.
We calculate the spectrum of third-harmonic generation (THG) for multimode fundamental input fields. We consider input fields from (1) a laser with gain on several cavity modes, (2) pulses with short temporal duration, and (3) frequency-modulated input fields. The effects of phase modulation in the nonlinear medium on the spectrum of the THG field are considered. Using the estimates by Miles and Harris [IEEE J. Quantum Electron. QE-9, 470 (1973)] for the nonlinear susceptibilities ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-3\ensuremath{\omega};\ensuremath{\omega},\ensuremath{\omega},\ensuremath{\omega}) and ${\mathrm{\ensuremath{\chi}}}^{(3)}$(-\ensuremath{\omega};\ensuremath{\omega},-\ensuremath{\omega},\ensuremath{\omega}) in alkali-metal vapors, we find that phase modulation can profoundly affect the efficiency and the spectrum of the THG. Generally, the self-phase modulation of the fundamental will tend to reduce the phase matching of an intense fundamental field and the third harmonic and thereby lower the conversion efficiency. New frequencies not contained in the set of frequencies ${\mathrm{\ensuremath{\omega}}}_{1\mathit{i}}$+${\mathrm{\ensuremath{\omega}}}_{1\mathit{j}}$+${\mathrm{\ensuremath{\omega}}}_{1\mathit{k}}$, where {${\mathrm{\ensuremath{\omega}}}_{1\mathit{i}}$} are frequencies in the fundamental laser spectrum, are created even at relatively low conversion due to phase modulation. Effects of cross phase modulation are also studied.
The dynamics of optical nonlinearity in the presence of gain and feedback can be complex leading to chaos in certain regimes. Temporal, spectral, spatial, or polarization instability of optical fields can emerge from chaotic response of an optical [Formula: see text] or [Formula: see text] nonlinear medium placed between two cavity mirrors or before a single feedback mirror. The complex mode dynamics, high-order correlations, and transition to instability in these systems are not well known. We consider a [Formula: see text] medium with amplified four-wave mixing process and study noise and correlation between multiple optical modes. Although individual modes show intensity instability, we observe relative intensity noise reduction close to the standard quantum noise, limited by the camera speed. We observe a relative noise reduction of more than 20 dB and fourth-order intensity correlation between four spatial modes. More than 100 distinct correlated quadruple modes can be generated using this process.
Experiments on resonant four-wave mixing demonstrated high- efficiency CW down-conversion of frequency in gas (up to 25%). Meanwhile, the high intensities make the perturbation theory invalid to understand the behavior of output power measured as a function of input powers. The nonlinear susceptibility of optically thin medium is found for difference-sum scheme (omega) 4 equals (omega) 1 - (omega) 2 + (omega) 3, where fields one and three are strong and two and four are weak. In symmetric case (kappa) 2 equals (kappa) 4, the integral over velocity can be calculated analytically for Doppler limits in collinear geometry. The compact explicit formula obtained yields the intensity and frequency dependence. The peak of mixing coefficient as a function of intensity is found around equal Rabi frequencies of fields one and three. The effect is shown to base on resonance between two closed cycles via sublevels of dressed states. The number of peaks in a spectrum varies from three to eight. Explicit formula allows interpreting the measurements and predicting the optimal relation between parameters for high conversion efficiency in gas with large Doppler broadening.
A theoretical investigation is reported of three-wave frequency mixing under simultaneous single-photon and two-photon resonance conditions when one of the fields being mixed is weak and the other is strong. A study is made of the influence of the field splitting on the nonlinear susceptibility in the case of homogeneously and Doppler-broadened transitions. An analysis is made of the role played by interference between quantum transitions via quasilevels in absorption and generation, and of its influence on mixing. It is shown that significant enhancement of the nonlinear generation efficiency is possible in optically dense media.