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

Journal of the Optical Society of America B

Phase conjugation by degenerate four-wave mixing via autoionizing states

S. Cohen, A. Lyras

Published Mar 1, 1998• 1 citation
Publisher

Abstract

The degenerate four-wave-mixing process in the vicinity of an autoionizing state is studied theoretically in the context of perturbation theory. The autoionizing state is represented either by an isolated Fano profile or by the appropriate expression of the multichannel quantum-defect theory for the corresponding Rydberg series of autoionizing states. Analytic expressions for the third-order susceptibility are given together with realistic numerical estimates based on atomic parameters extracted from accurate spectroscopic data. It is demonstrated that the observation of the effect is feasible in extensively studied atomic systems (e.g., alkaline earths) with existing coherent light sources. Constraints on the atomic parameters that will facilitate the experimental observation of the effect are identified and shown to be frequently met in real atomic systems.

DOI
10.1364/josab.15.001069

Related papers

Phase conjugation through autoionizing states: a density matrix approach

S. Cohen, A. Lyras

We present a theoretical study of phase conjugation via degenerate four wave mixing through an autoionizing state (AIS), formulated in terms of a density matrix approach. Within this quite general and flexible formulation the probe field is handled as a perturbation, while the coupling of the pump fields to the atomic system is treated exactly. The AIS is described as an isolated Fano resonance decaying into a single continuum. In the weak-field, steady-state limit it is possible to analytically recover earlier perturbative results for the corresponding nonlinear susceptibility. However, the advantage of this approach is best revealed when applied to the detailed study of realistic examples drawn from such diverse atomic systems as Xe and Mg. Both spectral and temporal features of the produced phase-conjugated (PC) signal are studied quantitatively through the numerical solution of the density matrix equations. Moreover, time-integrated quantities, such as the reflectivity, that are usually employed to measure the PC response, are also calculated. We find that the PC response is determined by the asymmetry of the autoionizing lineshape, the peak intensity of the pump pulses and the interplay between the pump pulse duration and the autoionization lifetime. Moreover, the role of the pump–probe delay in restoring reflectivity to high values under otherwise unfavourable conditions is highlighted. Finally, the impact of pump-induced ionization on the PC response is also studied and conditions to counter its deleterious effect on the reflectivity are determined.

Journal of Physics B Atomic Molecular and Optical Physics•Feb 17, 2004•0 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

Polarization properties of optical phase conjugation by two-photon resonant degenerate four-wave mixing

Martti Olavi Kauranen, Daniel J. Gauthier, Michelle S. Malcuit +1 more

We develop a semiclassical theory of the polarization properties of phase conjugation by two-photon resonant degenerate four-wave mixing. The theory includes the effects of saturation by the pump waves. We solve the density-matrix equations of motion in steady state for a nonlinear medium consisting of stationary atoms with a ground and excited state connected by two-photon transitions. As an illustration of the general results, we consider an ${S}_{0}$\ensuremath{\rightarrow}${S}_{0}$ two-photon transition, which is known to lead to perfect polarization conjugation in the limit of third-order theory. We show that the fidelity of the polarization-conjugation process is degraded for excessively large pump intensities. The degradation can occur both due to transfer of population to the excited state and due to nonresonant Stark shifts. Theoretical results are compared to those of a recent experiment [Malcuit, Gauthier, and Boyd, Opt. Lett. 13, 663 (1988)].

Physical Review A•Aug 1, 1989•13 citations

Role of atomic coherence effects in four-wave mixing using autoionizing resonances

M. Anscombe, Rebeca de Nalda, İbrahim Küçükkara +1 more

The generation of tunable, nanosecond-pulsed extreme-ultraviolet (EUV) light $(84.92\mathrm{nm}$ to $84.97\mathrm{nm}$ $\ensuremath{\sim}14.6\mathrm{eV})$ using four-wave mixing in optically deep krypton gas was investigated experimentally and theoretically. The sum-frequency mixing scheme generated light from two overlapping autoionizing resonances, ${17s}^{\ensuremath{'}}$ and ${15d}^{\ensuremath{'}}$. We investigated whether the laser-induced atomic coherence effect of electromagnetically induced transparency (EIT) could enhance the intensity of the EUV light generated. The maximum EUV output was seen from the region where the ${15d}^{\ensuremath{'}}$ autoionizing resonance has a transparency window due to Fano-type quantum interference. An analytical model using appropriate atomic parameters explains how background absorption can prevent EIT enhancement of EUV generation; and shows how a background Raman-like coupling term in the nonlinear susceptibility can well describe the measured EUV generation spectrum.

Physical Review A•Oct 8, 2003•2 citations

Theory of vector phase conjugation by two-photon-resonant degenerate four-wave mixing

Martti Olavi Kauranen, Daniel J. Gauthier, Michelle S. Malcuit +1 more

We develop a semiclassical theory of phase conjugation by two-photon-resonant degenerate four-wave mixing. Our theory treats both the tensor nature of the interaction and the effects of saturation of the atomic system. Previous theories that include saturation effects have treated only the scalar case,1 while theories that include polarization properties have treated the problem only in lowest-order perturbation theory.2 We consider an S → S two-photon transition. This interaction is of Interest because it has been predicted theoretically2 and shown experimentally3 that in the limit of nonsaturating pump waves this interaction leads to the simultaneous conjugation of the wavefront and polarization of the incident probe wave (vector phase conjugation).

Quantum Electronics and Laser Science Conference•Apr 24, 1989•0 citations

Polarisation theory of phase conjugation in resonant non-degenerate four-wave mixing

G. G. Adonts, D G Akopyan

The theory of polarised light phase conjugation (PC) in non-degenerate four-wave mixing in a resonant medium has been developed in the irreducible tensorial formalism. The theory is applied to a wave with any polarisation in a resonant medium with levels of arbitrary momentum in the absence of pump-induced saturation effects. The formula for the non-linear susceptibility of the medium has been found and it is shown that the change of the polarisation at PC is essentially connected with the relaxation of the atomic multipole momenta (population, orientation and alignment). Wave equations were investigated for different pump-wave polarisations and three different physical mechanisms of the PC are discussed. The influence of the different relaxation rates of the atomic multipole momenta on the PC reflectivity has been investigated. The spectral dependence of the reflectivity and the spectral ranges which depend on the wave polarisation are found. For motionless atoms with small momenta (J 1 = 1 / 2 to J 2 = 1 / 2 , J 1 =1 to J 2 =0), exact formulae for the non-linear susceptibility of the medium as a function of pump intensity were found. The spontaneous decay of higher level to the lower one was taken into account. Because of optical pumping, saturation effects can be shown at considerably smaller intensities than in the usual two-level system. The exact expressions for the PC reflectivity at different polarisations of the pump waves are given.

Journal of Physics B Atomic and Molecular Physics•Aug 28, 1985•10 citations
The degenerate four-wave-mixing process in the vicinity of an autoionizing state is studied theoretically in the context of perturbation theory. The autoionizing state is represented either by an isolated Fano profile or by the appropriate expression of the multichannel quantum-defect theory for the corresponding Rydberg series of autoionizing states. Analytic expressions for the third-order susceptibility are given together with realistic numerical estimates based on atomic parameters extracted from accurate spectroscopic data. It is demonstrated that the observation of the effect is feasible in extensively studied atomic systems (e.g., alkaline earths) with existing coherent light sources. Constraints on the atomic parameters that will facilitate the experimental observation of the effect are identified and shown to be frequently met in real atomic systems.
We present a theoretical study of phase conjugation via degenerate four wave mixing through an autoionizing state (AIS), formulated in terms of a density matrix approach. Within this quite general and flexible formulation the probe field is handled as a perturbation, while the coupling of the pump fields to the atomic system is treated exactly. The AIS is described as an isolated Fano resonance decaying into a single continuum. In the weak-field, steady-state limit it is possible to analytically recover earlier perturbative results for the corresponding nonlinear susceptibility. However, the advantage of this approach is best revealed when applied to the detailed study of realistic examples drawn from such diverse atomic systems as Xe and Mg. Both spectral and temporal features of the produced phase-conjugated (PC) signal are studied quantitatively through the numerical solution of the density matrix equations. Moreover, time-integrated quantities, such as the reflectivity, that are usually employed to measure the PC response, are also calculated. We find that the PC response is determined by the asymmetry of the autoionizing lineshape, the peak intensity of the pump pulses and the interplay between the pump pulse duration and the autoionization lifetime. Moreover, the role of the pump–probe delay in restoring reflectivity to high values under otherwise unfavourable conditions is highlighted. Finally, the impact of pump-induced ionization on the PC response is also studied and conditions to counter its deleterious effect on the reflectivity are determined.
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 develop a semiclassical theory of the polarization properties of phase conjugation by two-photon resonant degenerate four-wave mixing. The theory includes the effects of saturation by the pump waves. We solve the density-matrix equations of motion in steady state for a nonlinear medium consisting of stationary atoms with a ground and excited state connected by two-photon transitions. As an illustration of the general results, we consider an S0{S}_{0}S0​\ensuremath{\rightarrow}S0{S}_{0}S0​ two-photon transition, which is known to lead to perfect polarization conjugation in the limit of third-order theory. We show that the fidelity of the polarization-conjugation process is degraded for excessively large pump intensities. The degradation can occur both due to transfer of population to the excited state and due to nonresonant Stark shifts. Theoretical results are compared to those of a recent experiment [Malcuit, Gauthier, and Boyd, Opt. Lett. 13, 663 (1988)].
The generation of tunable, nanosecond-pulsed extreme-ultraviolet (EUV) light (84.92nm(84.92\mathrm{nm}(84.92nm to 84.97nm84.97\mathrm{nm}84.97nm $\ensuremath{\sim}14.6\mathrm{eV})$ using four-wave mixing in optically deep krypton gas was investigated experimentally and theoretically. The sum-frequency mixing scheme generated light from two overlapping autoionizing resonances, ${17s}^{\ensuremath{'}}$ and ${15d}^{\ensuremath{'}}$. We investigated whether the laser-induced atomic coherence effect of electromagnetically induced transparency (EIT) could enhance the intensity of the EUV light generated. The maximum EUV output was seen from the region where the ${15d}^{\ensuremath{'}}$ autoionizing resonance has a transparency window due to Fano-type quantum interference. An analytical model using appropriate atomic parameters explains how background absorption can prevent EIT enhancement of EUV generation; and shows how a background Raman-like coupling term in the nonlinear susceptibility can well describe the measured EUV generation spectrum.
We develop a semiclassical theory of phase conjugation by two-photon-resonant degenerate four-wave mixing. Our theory treats both the tensor nature of the interaction and the effects of saturation of the atomic system. Previous theories that include saturation effects have treated only the scalar case,1 while theories that include polarization properties have treated the problem only in lowest-order perturbation theory.2 We consider an S → S two-photon transition. This interaction is of Interest because it has been predicted theoretically2 and shown experimentally3 that in the limit of nonsaturating pump waves this interaction leads to the simultaneous conjugation of the wavefront and polarization of the incident probe wave (vector phase conjugation).
The theory of polarised light phase conjugation (PC) in non-degenerate four-wave mixing in a resonant medium has been developed in the irreducible tensorial formalism. The theory is applied to a wave with any polarisation in a resonant medium with levels of arbitrary momentum in the absence of pump-induced saturation effects. The formula for the non-linear susceptibility of the medium has been found and it is shown that the change of the polarisation at PC is essentially connected with the relaxation of the atomic multipole momenta (population, orientation and alignment). Wave equations were investigated for different pump-wave polarisations and three different physical mechanisms of the PC are discussed. The influence of the different relaxation rates of the atomic multipole momenta on the PC reflectivity has been investigated. The spectral dependence of the reflectivity and the spectral ranges which depend on the wave polarisation are found. For motionless atoms with small momenta (J 1 = 1 / 2 to J 2 = 1 / 2 , J 1 =1 to J 2 =0), exact formulae for the non-linear susceptibility of the medium as a function of pump intensity were found. The spontaneous decay of higher level to the lower one was taken into account. Because of optical pumping, saturation effects can be shown at considerably smaller intensities than in the usual two-level system. The exact expressions for the PC reflectivity at different polarisations of the pump waves are given.