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Physical Review A

Quantum interference in phase conjugation by resonant, nearly degenerate four-wave mixing

O. K. Andersen, Daan Lenstra, Steven Stolte

Published Aug 1, 1999• 9 citations
Publisher

Abstract

We present a theoretical investigation of optical phase conjugation by use of nearly degenerate four-wave mixing in a two-level atomic system. The description is based on the semiclassical dressed-state approach, which offers a very transparent picture of the physics involved in the nonlinear process. The theory provides an explanation in terms of a quantum interference between two channels for the strong reduction of the optical phase-conjugate yield when the pump fields are tuned to the atomic resonance. In addition, the model predicts an efficient generation of the optical phase conjugate if the system is prepared in a pure dressed state.

DOI
10.1103/physreva.60.1672

Related papers

Quantum interference in optical phase-conjugation spectra of vapors

O. K. Andersen, Daan Lenstra, Steven Stolte

Summary form only given.The present summary reports about a dressed-state approach to optical phase conjugation (OPC) by resonant nearly degenerate four-wave mixing in vapors. The basic idea is to decouple the pump fields from the probe field by first treating the problem of the pump fields interacting with the atom and then consider the interaction with the probe field. From this model, the dip, observed in the OPC yield when the pump fields are tuned to an atomic resonance, can be interpreted as an interference effect between contributions to the polarization originating from two dressed-state transitions.

Conference on Lasers and Electro-Optics•Nov 27, 2002•0 citations

Degenerate four-wave mixing spectra in a two-level atomic system with phase-modulated time-delayed coherent fields

Arthur K. Mills, Daniel S. Elliott

We study phase-conjugate degenerate four-wave mixing (DFWM) in an optically pumped, two-level atomic system (the $3s\phantom{\rule{0.2em}{0ex}}^{2}S_{1∕2}$, $F=2$, ${m}_{F}=2\ensuremath{\rightarrow}3p\phantom{\rule{0.2em}{0ex}}^{2}P_{3∕2}$, $F=3$, ${m}_{F}=3$ transition in a diffuse, collision-free, thermal beam of sodium) using phase-modulated laser beams. In this investigation, we study the detailed line shape of the four-wave mixing signal as a function of the temporal delay between the probe field and the pump fields at different intensities of the cw pump waves and modulation frequencies. We show that phase modulation of the pump and probe beams leads to strong amplitude modulation of the phase-conjugate beam and report excellent agreement between our experimental observations of the DFWM spectra and the results of numerical calculations. Our numerical results suggest an enhancement of the ``local'' DFWM signal when the sidebands of the phase-modulated laser are resonant with dressed-state transitions, but that this resonance is less obvious in the total signal generated. We also show that the complex line shapes depend sensitively upon the delay of the probe beam, with the peak DFWM signal occurring when the probe beam is advanced compared to the pump beam.

Physical Review A•Aug 12, 2005•2 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

Phase Conjugation By Resonantly Enhanced Degenerate Four-Wave Mixing

The use of resonantly enhanced four-wave mixing to generate high reflectivity phase-conjugate waves is discussed. A physical understanding of the mechanisms underlying the nonlinear response for a simple two-level system is presented followed by a discussion of the complicating effects of pump absorption, pump depletion, unbalanced pumps, and thermal motion. Modifications of the theory to describe four-level behavior are presented and compared with experiments. The application of nearly degenerate four-wave mixing to narrow-band filtering complicated by the A. C. Stark effect is dis-cussed along with examples of experimental observations.

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Optical phase conjugation in saturable absorbers

Gilmore J. Dunning, Juan Francisco Lam

Current experimental studies of degenerate four-wave mixing in saturable absorbers1,2 have shown that the dependence of the reflectivity on the pump intensity exhibits a double maxima. This phenomenon can arise from the fact that the medium approximates a set of four-level quantum systems. We have carried out a detailed analysis of the nonlinear processes of degenerate four-wave mixing in a four-level system. Because resonant saturable absorbers are being considered for systems applications, e.g., optical phase conjugation and wide FOV filters, the results presented are relevant to various optical phase conjugation devices. There are two distinct regimes which are of interest. The first case is when the intramolecular relaxation is very weak or absent. The second is when the intramolecular relaxation is strong. For the weak case the medium behaves like a set of two distinct dipole radiation sources which can generate interference effects.

Conference on Lasers and Electro-Optics•Jan 1, 1981•0 citations

Degenerate phase-conjugate four-wave mixing in a nearly-Doppler-free two-level atomic medium

Binh Do, Jongwhan Cha, Daniel S. Elliott +1 more

We present experimental observations of phase-conjugate degenerate four-wave mixing in a two-level atomic system (the ${3s}^{2}{S}_{1/2}, F=2, {m}_{F}=\stackrel{\ensuremath{\rightarrow}}{2}{3p}^{2}{P}_{3/2}, F=3, {m}_{F}=3$ transition in a diffuse, collision-free, thermal beam of atomic sodium) driven by cw, narrow-band $(\ensuremath{\sim}250 \mathrm{kHz}),$ stabilized pump and probe fields. The primary measurements that we report are of the peak signal strength of the four-wave-mixing interaction and the bandwidth of the four-wave-mixing spectrum (as the laser frequency is tuned through the resonant frequency) as a function of the intensity of the pump field. We compare these measurements with the predictions of a theoretical analysis in which extension to include the effect of the small transverse velocity $(\ensuremath{\sim}3 \mathrm{m}/\mathrm{s}\mathrm{e}\mathrm{c})$ of the atomic beam is critical. These measurements provide a direct verification of the theory of this important nonlinear interaction.

Physical Review A•Oct 1, 1998•11 citations
We present a theoretical investigation of optical phase conjugation by use of nearly degenerate four-wave mixing in a two-level atomic system. The description is based on the semiclassical dressed-state approach, which offers a very transparent picture of the physics involved in the nonlinear process. The theory provides an explanation in terms of a quantum interference between two channels for the strong reduction of the optical phase-conjugate yield when the pump fields are tuned to the atomic resonance. In addition, the model predicts an efficient generation of the optical phase conjugate if the system is prepared in a pure dressed state.
Summary form only given.The present summary reports about a dressed-state approach to optical phase conjugation (OPC) by resonant nearly degenerate four-wave mixing in vapors. The basic idea is to decouple the pump fields from the probe field by first treating the problem of the pump fields interacting with the atom and then consider the interaction with the probe field. From this model, the dip, observed in the OPC yield when the pump fields are tuned to an atomic resonance, can be interpreted as an interference effect between contributions to the polarization originating from two dressed-state transitions.
We study phase-conjugate degenerate four-wave mixing (DFWM) in an optically pumped, two-level atomic system (the 3s2S1∕23s\phantom{\rule{0.2em}{0ex}}^{2}S_{1∕2}3s2S1∕2​, F=2F=2F=2, ${m}_{F}=2\ensuremath{\rightarrow}3p\phantom{\rule{0.2em}{0ex}}^{2}P_{3∕2}$, F=3F=3F=3, mF=3{m}_{F}=3mF​=3 transition in a diffuse, collision-free, thermal beam of sodium) using phase-modulated laser beams. In this investigation, we study the detailed line shape of the four-wave mixing signal as a function of the temporal delay between the probe field and the pump fields at different intensities of the cw pump waves and modulation frequencies. We show that phase modulation of the pump and probe beams leads to strong amplitude modulation of the phase-conjugate beam and report excellent agreement between our experimental observations of the DFWM spectra and the results of numerical calculations. Our numerical results suggest an enhancement of the ``local'' DFWM signal when the sidebands of the phase-modulated laser are resonant with dressed-state transitions, but that this resonance is less obvious in the total signal generated. We also show that the complex line shapes depend sensitively upon the delay of the probe beam, with the peak DFWM signal occurring when the probe beam is advanced compared to the pump beam.
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.
The use of resonantly enhanced four-wave mixing to generate high reflectivity phase-conjugate waves is discussed. A physical understanding of the mechanisms underlying the nonlinear response for a simple two-level system is presented followed by a discussion of the complicating effects of pump absorption, pump depletion, unbalanced pumps, and thermal motion. Modifications of the theory to describe four-level behavior are presented and compared with experiments. The application of nearly degenerate four-wave mixing to narrow-band filtering complicated by the A. C. Stark effect is dis-cussed along with examples of experimental observations.
Current experimental studies of degenerate four-wave mixing in saturable absorbers1,2 have shown that the dependence of the reflectivity on the pump intensity exhibits a double maxima. This phenomenon can arise from the fact that the medium approximates a set of four-level quantum systems. We have carried out a detailed analysis of the nonlinear processes of degenerate four-wave mixing in a four-level system. Because resonant saturable absorbers are being considered for systems applications, e.g., optical phase conjugation and wide FOV filters, the results presented are relevant to various optical phase conjugation devices. There are two distinct regimes which are of interest. The first case is when the intramolecular relaxation is very weak or absent. The second is when the intramolecular relaxation is strong. For the weak case the medium behaves like a set of two distinct dipole radiation sources which can generate interference effects.
We present experimental observations of phase-conjugate degenerate four-wave mixing in a two-level atomic system (the ${3s}^{2}{S}_{1/2}, F=2, {m}_{F}=\stackrel{\ensuremath{\rightarrow}}{2}{3p}^{2}{P}_{3/2}, F=3, {m}_{F}=3$ transition in a diffuse, collision-free, thermal beam of atomic sodium) driven by cw, narrow-band $(\ensuremath{\sim}250 \mathrm{kHz}),$ stabilized pump and probe fields. The primary measurements that we report are of the peak signal strength of the four-wave-mixing interaction and the bandwidth of the four-wave-mixing spectrum (as the laser frequency is tuned through the resonant frequency) as a function of the intensity of the pump field. We compare these measurements with the predictions of a theoretical analysis in which extension to include the effect of the small transverse velocity $(\ensuremath{\sim}3 \mathrm{m}/\mathrm{s}\mathrm{e}\mathrm{c})$ of the atomic beam is critical. These measurements provide a direct verification of the theory of this important nonlinear interaction.