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

Optics Letters

Configuration dependence of optical filtering characteristics in backward nearly degenerate four-wave mixing

S. Saikan, H. Wakata

Published Jun 1, 1981• 12 citations
Publisher

Abstract

Backward nearly degenerate four-wave mixing close to the atomic resonance line has been investigated experimentally. It has been found that phase mismatch provides a window factor to the reflectivity that masks the frequency spectrum of the medium and that the line profile depends on the geometrical configuration of the three interacting beams. The capability of obtaining an optical bandpass filter with nearly degenerate four-wave mixing has also been demonstrated experimentally.

DOI
10.1364/ol.6.000281

Related papers

Polarization and angular-momentum dependences of degenerate backward four-wave mixing in atomic lines

S. Saikan

The third-order nonlinear susceptibilities responsible for degenerate backward four-wave mixing in single-photon resonance are calculated for atomic lines with arbitrary values of total angular momentum J. All possible combinations for the polarization of three interacting beams have been considered.

Journal of the Optical Society of America•Apr 1, 1982•18 citations

Phase Conjugation via Four-Wave Mixing in a Resonant Medium

Joseph Nilsen

This thesis describes the theoretical solution and experimental verification of phase conjugation via nondegenerate four-wave mixing in resonant media. The theoretical work models the resonant medium as a two-level atomic system with the lower state of the system being the ground state of the atom. Working initially with an ensemble of stationary atoms, the density matrix equations are solved by third-order perturbation theory in the presence of the four applied electro-magnetic fields which are assumed to be nearly resonant with the atomic transition. Two of the applied fields are assumed to be non-depleted counterpropagating pump waves while the third wave is an incident signal wave. The fourth wave is the phase conjugate wave which is generated by the interaction of the three previous waves with the nonlinear medium. The solution of the density matrix equations gives the local polarization of the atom. The polarization is used in Maxwell's equations as a source term to solve for the propagation and generation of the signal wave and phase conjugate wave through the nonlinear medium. Studying the dependence of the phase conjugate signal on the various parameters such as frequency, we show how an ultrahigh-Q isotropically sensitive optical filter can be constructed using the phase conjugation process. In many cases the pump waves may saturate the resonant medium so we also present another solution to the density matrix equations which is correct to all orders in the amplitude of the pump waves since the third-order solution is correct only to first-order in each of the field amplitudes. In the saturated regime, we predict several new phenomena associated with degenerate four-wave mixing and also describe the ac Stark effect and how it modifies the frequency response of the filtering process. We also show how a narrow bandwidth optical filter with an efficiency greater than unity can be constructed. In many atomic systems the atoms are moving at significant velocities such that the Doppler linewidth of the system is larger than the homogeneous linewidth. The latter linewidth dominates the response of the ensemble of stationary atoms. To better understand this case the density matrix equations are solved to third-order by perturbation theory for an atom of velocity v. The solution for the polarization is then integrated over the velocity distribution of the macroscopic system which is assumed to be a gaussian distribution of velocities since that is an excellent model of many real systems. Using the Doppler broadened system, we explain how a tunable optical filter can be constructed whose bandwidth is limited by the homogeneous linewidth of the atom while the tuning range of the filter extends over the entire Doppler profile. Since it is a resonant system, sodium vapor is used as the nonlinear medium in our experiments. The relevant properties of sodium are discussed in great detail. In particular, the wavefunctions of the 3S and 3P states are analyzed and a discussion of how the 3S-3P transition models a two-level system is given. Using sodium as the nonlinear medium we demonstrate an ultrahigh-Q optical filter using phase conjugation via nondegenerate four-wave mixing as the filtering process. The filter has a FWHM bandwidth of 41 MHz and a maximum efficiency of 4 x 10-3. However, our theoretical work and other experimental work with sodium suggest that an efficient filter with both gain and a narrower bandwidth should be quite feasible.

•Jan 1, 1982•0 citations

Theory of degenerate four-wave mixing in resonant Doppler-broadened systems - I. Angular dependence of intensity and lineshape of phase-conjugate emission

M. Ducloy, Daniel Bloch

We present in this paper a theoretical study of the angular dependence of phase-conjugate emission via resonant degenerate four-wave mixing (FWM) in Doppler-broadened gas media. The nonlinear optical polarization is calculated through an expansion of the atomic density matrix up to the third order in the incident fields. One finds that, when the incident frequency is resonant for a single-photon transition, both strength and lineshape of the phase-conjugate emission strongly depend on the angle 0 between standing pump wave and probe wave. The emission intensity, maximum at grazing incidences, decreases rapidly by several orders of magnitude when 0 increases. Simultaneously the emission linewidth increases from homogeneous to Doppler width. Similar effects are predicted for nearly-degenerate FWM (different pump and probe frequencies). In particular, one shows that the bandwidth of the equivalent optical reflection filter undergoes a Doppler-broadening linearly increasing with sin θ. On the opposite, none of the above effects is predicted in two-photon-resonant FWM which does not exhibit any angular dependence.

Journal de physique•Jan 1, 1981•74 citations

Multiwave-mixing-mediated optical bandpass filter with a thin Kerr medium

Rose W. Wang, Iam Choon Khoo

A theory for a phase-conjugate bandpass filter based on multiwave-mixing effects in a thin Kerr medium is presented. As a result of the presence of side diffractions, several new features are observed in the power reflectivity, the bandpass, and the central frequency. Aside from a strong dependence on the wave-mixing angle, it is shown that components of different frequencies are generated at different reflected angles; at small input angles the filter can have a wide frequency bandpass with amplified output, and the central frequency is equal to the pump frequency regardless of the ratio of the pump beams.

Journal of the Optical Society of America B•Nov 1, 1992•2 citations

Nearly degenerate four-wave mixing enhanced by the ac Stark effect

Donald J. Harter, Robert W. Boyd

A model is developed to describe nondegenerate four-wave mixing in a medium characterized by a single strong atomic resonance. The nonlinear response is calculated using the density-matrix formalism, and propagation effects are included by solving coupled-wave equations. This model predicts large resonant responses for signals detuned from the pump frequency by approximately the generalized Rabi frequency. Potential applications of the effects predicted by this model include construction of tunable bandpass filters and four-wave parametric oscillators.

IEEE Journal of Quantum Electronics•Oct 1, 1980•85 citations

Detailed velocity-dependent line shapes for degenerate four-wave mixing spectra in a two-level atomic system

Binh Do, Daniel S. Elliott

We study phase-conjugate degenerate four-wave mixing in an optically pumped, two-level atomic system (the ${3s}^{2}{S}_{1/2},$ $F=2,$ ${m}_{\mathrm{F}}=\stackrel{\ensuremath{\rightarrow}}{2}{3p}^{2}{P}_{3/2},$ $F=3,$ ${m}_{\mathrm{F}}=3$ transition in a diffuse, collision-free, thermal beam of sodium). In this investigation, we study the detailed line shape and strength of the four-wave mixing signal as a function of the intensity of the cw pump waves, and of the velocity of the atoms. Our technique allows us to examine the four-wave mixing interaction under conditions in which the Doppler shift, the natural linewidth of the transition, and the Rabi frequency for the interaction are each comparable with one another. From our experimental measurements and computational results, we show that the line shapes are very complex in this regime, and that they depend sensitively upon the atomic velocity, especially in the case when the intensity of the forward and backward pump beams exceeds the saturation intensity for the transition.

Physical Review A•Jun 19, 2003•3 citations
Backward nearly degenerate four-wave mixing close to the atomic resonance line has been investigated experimentally. It has been found that phase mismatch provides a window factor to the reflectivity that masks the frequency spectrum of the medium and that the line profile depends on the geometrical configuration of the three interacting beams. The capability of obtaining an optical bandpass filter with nearly degenerate four-wave mixing has also been demonstrated experimentally.
This thesis describes the theoretical solution and experimental verification of phase conjugation via nondegenerate four-wave mixing in resonant media. The theoretical work models the resonant medium as a two-level atomic system with the lower state of the system being the ground state of the atom. Working initially with an ensemble of stationary atoms, the density matrix equations are solved by third-order perturbation theory in the presence of the four applied electro-magnetic fields which are assumed to be nearly resonant with the atomic transition. Two of the applied fields are assumed to be non-depleted counterpropagating pump waves while the third wave is an incident signal wave. The fourth wave is the phase conjugate wave which is generated by the interaction of the three previous waves with the nonlinear medium. The solution of the density matrix equations gives the local polarization of the atom. The polarization is used in Maxwell's equations as a source term to solve for the propagation and generation of the signal wave and phase conjugate wave through the nonlinear medium. Studying the dependence of the phase conjugate signal on the various parameters such as frequency, we show how an ultrahigh-Q isotropically sensitive optical filter can be constructed using the phase conjugation process. In many cases the pump waves may saturate the resonant medium so we also present another solution to the density matrix equations which is correct to all orders in the amplitude of the pump waves since the third-order solution is correct only to first-order in each of the field amplitudes. In the saturated regime, we predict several new phenomena associated with degenerate four-wave mixing and also describe the ac Stark effect and how it modifies the frequency response of the filtering process. We also show how a narrow bandwidth optical filter with an efficiency greater than unity can be constructed. In many atomic systems the atoms are moving at significant velocities such that the Doppler linewidth of the system is larger than the homogeneous linewidth. The latter linewidth dominates the response of the ensemble of stationary atoms. To better understand this case the density matrix equations are solved to third-order by perturbation theory for an atom of velocity v. The solution for the polarization is then integrated over the velocity distribution of the macroscopic system which is assumed to be a gaussian distribution of velocities since that is an excellent model of many real systems. Using the Doppler broadened system, we explain how a tunable optical filter can be constructed whose bandwidth is limited by the homogeneous linewidth of the atom while the tuning range of the filter extends over the entire Doppler profile. Since it is a resonant system, sodium vapor is used as the nonlinear medium in our experiments. The relevant properties of sodium are discussed in great detail. In particular, the wavefunctions of the 3S and 3P states are analyzed and a discussion of how the 3S-3P transition models a two-level system is given. Using sodium as the nonlinear medium we demonstrate an ultrahigh-Q optical filter using phase conjugation via nondegenerate four-wave mixing as the filtering process. The filter has a FWHM bandwidth of 41 MHz and a maximum efficiency of 4 x 10-3. However, our theoretical work and other experimental work with sodium suggest that an efficient filter with both gain and a narrower bandwidth should be quite feasible.
We present in this paper a theoretical study of the angular dependence of phase-conjugate emission via resonant degenerate four-wave mixing (FWM) in Doppler-broadened gas media. The nonlinear optical polarization is calculated through an expansion of the atomic density matrix up to the third order in the incident fields. One finds that, when the incident frequency is resonant for a single-photon transition, both strength and lineshape of the phase-conjugate emission strongly depend on the angle 0 between standing pump wave and probe wave. The emission intensity, maximum at grazing incidences, decreases rapidly by several orders of magnitude when 0 increases. Simultaneously the emission linewidth increases from homogeneous to Doppler width. Similar effects are predicted for nearly-degenerate FWM (different pump and probe frequencies). In particular, one shows that the bandwidth of the equivalent optical reflection filter undergoes a Doppler-broadening linearly increasing with sin θ. On the opposite, none of the above effects is predicted in two-photon-resonant FWM which does not exhibit any angular dependence.
A theory for a phase-conjugate bandpass filter based on multiwave-mixing effects in a thin Kerr medium is presented. As a result of the presence of side diffractions, several new features are observed in the power reflectivity, the bandpass, and the central frequency. Aside from a strong dependence on the wave-mixing angle, it is shown that components of different frequencies are generated at different reflected angles; at small input angles the filter can have a wide frequency bandpass with amplified output, and the central frequency is equal to the pump frequency regardless of the ratio of the pump beams.
A model is developed to describe nondegenerate four-wave mixing in a medium characterized by a single strong atomic resonance. The nonlinear response is calculated using the density-matrix formalism, and propagation effects are included by solving coupled-wave equations. This model predicts large resonant responses for signals detuned from the pump frequency by approximately the generalized Rabi frequency. Potential applications of the effects predicted by this model include construction of tunable bandpass filters and four-wave parametric oscillators.
We study phase-conjugate degenerate four-wave mixing in an optically pumped, two-level atomic system (the 3s2S1/2,{3s}^{2}{S}_{1/2},3s2S1/2​, F=2,F=2,F=2, ${m}_{\mathrm{F}}=\stackrel{\ensuremath{\rightarrow}}{2}{3p}^{2}{P}_{3/2},$ F=3,F=3,F=3, mF=3{m}_{\mathrm{F}}=3mF​=3 transition in a diffuse, collision-free, thermal beam of sodium). In this investigation, we study the detailed line shape and strength of the four-wave mixing signal as a function of the intensity of the cw pump waves, and of the velocity of the atoms. Our technique allows us to examine the four-wave mixing interaction under conditions in which the Doppler shift, the natural linewidth of the transition, and the Rabi frequency for the interaction are each comparable with one another. From our experimental measurements and computational results, we show that the line shapes are very complex in this regime, and that they depend sensitively upon the atomic velocity, especially in the case when the intensity of the forward and backward pump beams exceeds the saturation intensity for the transition.