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Physical Review A

Relevance of rotational diffusion on the grating picture of four-wave mixing

J. P. Lavoine, A. A. Villaeys

Published Feb 1, 1989• 1 citation
Publisher

Abstract

A theoretical study of the nonlinear polarization in a molecular medium is presented. It enables us to stress the importance and to evaluate the contribution of the population and orientational gratings in four-wave mixing depending on the pump- and probe-beam polarizations. The medium is described as an ensemble of two-level systems and is subject to rotational diffusion. The model is valid far or near resonance, as well. It is only assumed that time variation of the field envelopes and rotational diffusion constant are much smaller than the transverse decay rate ${\ensuremath{\Gamma}}_{2}$. The nonlinear polarization is expressed as a function of the field polarizations. The results are applied to a four-wave-mixing experiment to discuss the influence of the rotational diffusion. Finally, by including propagation effects, we use this formalism to calculate a phase-conjugated signal. This enables us to stress the influence of rotational diffusion to achieve time reversal of the envelope of a light pulse.

DOI
10.1103/physreva.39.1922

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Rotational Diffusion Effect On Time Reversal In Phase Conjugation Spectroscopy

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Envelope time reversal can be achieved in a phase conjugation experiment. In this work, we study the influence of the various microscopic parameters of the nonlinear medium as well as the effects of rotational diffusion of the dipole moments on the envelope reversal. To this end, we consider a backward degenerate four-wave mixing experiment with pulsed fields and assume that any characteristic times of the envelopes or of the rotational diffusion are larger than the transverse decay time. Propagation and field polarizations effects are introduced. From the propagation equations solved in the slowly varying envelope approximation, we calculate the amplitude of the conjugated wave. Its expression depends on the field polarization as well as on the various overlap integrals of the pump and probe envelopes. In order to obtain a complete time reversal between the amplitudes of the conjugated wave E4(0,t) and probe field E3(0,t), supplementary contributions to E4(0,t) which do not satisfy the proportionality relation between E4(0,t) and E3*(0,-t) must wash-out. Some of these contributions cancel by taking a convenient choice of beam polarizations. It is shown here, that if the dipole moments undergo a rapid rotational diffusion, then complete time reversal is observed.

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Effects of permanent dipole moments in degenerate four-wave-mixing processes

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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
A theoretical study of the nonlinear polarization in a molecular medium is presented. It enables us to stress the importance and to evaluate the contribution of the population and orientational gratings in four-wave mixing depending on the pump- and probe-beam polarizations. The medium is described as an ensemble of two-level systems and is subject to rotational diffusion. The model is valid far or near resonance, as well. It is only assumed that time variation of the field envelopes and rotational diffusion constant are much smaller than the transverse decay rate \ensuremathΓ2{\ensuremath{\Gamma}}_{2}\ensuremathΓ2​. The nonlinear polarization is expressed as a function of the field polarizations. The results are applied to a four-wave-mixing experiment to discuss the influence of the rotational diffusion. Finally, by including propagation effects, we use this formalism to calculate a phase-conjugated signal. This enables us to stress the influence of rotational diffusion to achieve time reversal of the envelope of a light pulse.
Envelope time reversal can be achieved in a phase conjugation experiment. In this work, we study the influence of the various microscopic parameters of the nonlinear medium as well as the effects of rotational diffusion of the dipole moments on the envelope reversal. To this end, we consider a backward degenerate four-wave mixing experiment with pulsed fields and assume that any characteristic times of the envelopes or of the rotational diffusion are larger than the transverse decay time. Propagation and field polarizations effects are introduced. From the propagation equations solved in the slowly varying envelope approximation, we calculate the amplitude of the conjugated wave. Its expression depends on the field polarization as well as on the various overlap integrals of the pump and probe envelopes. In order to obtain a complete time reversal between the amplitudes of the conjugated wave E4(0,t) and probe field E3(0,t), supplementary contributions to E4(0,t) which do not satisfy the proportionality relation between E4(0,t) and E3*(0,-t) must wash-out. Some of these contributions cancel by taking a convenient choice of beam polarizations. It is shown here, that if the dipole moments undergo a rapid rotational diffusion, then complete time reversal is observed.
The main goal of this paper is to analyze a two-photon degenerate four-wave-mixing process that occurs in an isotropic media modeled by homogeneously broadened two-level systems having unequal permanent dipole moments and subject to rotational diffusion. Our theoretical treatment is based upon a third-order perturbative solution of the density-matrix equations and does not require the rotating-wave approximation. The medium being excited by three stationary fields, we give an analytical expression of the macroscopic polarization up to the third order, including permanent dipole moments, orientational diffusion, and field polarizations. We apply these general results to the description of a degenerate four-wave-mixing process by selecting, in the expressions, the contributions in the direction ${\mathbf{k}}_{\mathrm{\ensuremath{\alpha}}}$-${\mathbf{k}}_{\mathrm{\ensuremath{\beta}}}$+${\mathbf{k}}_{\ensuremath{\gamma}}$, ${\mathbf{k}}_{\mathrm{\ensuremath{\alpha}}}$, ${\mathbf{k}}_{\mathrm{\ensuremath{\beta}}}$, and ${\mathbf{k}}_{\ensuremath{\gamma}}$ being, respectively, the wave vectors of the exciting fields. We show that the two-photon degenerate four-wave mixing is dependent on the permanent dipole moments and is not present if these dipole moments are equal or zero. We analyze the influence of rotational diffusion. Different cases of field polarizations are also considered, and we discuss a comparison of the well-known one-photon degenerate four-wave-mixing process and this two-photon process. This process is also discussed in the particular case of phase conjugation.
We have derived the polarization dependent coupling constants for transmission and reflection gratings in degenerate four-wave mixing, measured phase-conjugate signals for various polarization conditions, and determined the relative contribution of the two gratings to the signal. The results obtained with cryptocyanine dissolved in methanol as the nonlinear medium show that the signal from transmission grating is 1.53–2.14 times larger than that from a reflection grating depending on the angle between the wave vectors of the forward pump beam and the probe beam.
We present the theory of resonant backward degenerate four-wave mixing in gaseous media for Doppler-broadened degenerate two-level systems. The degeneracy of the atomic levels as well as the influence of the polarization of the incident fields are taken into account within the framework of a tensorial formalism. The amplitude, line shape, and polarization of the reemitted phase-conjugate field are calculated up to the third order in the incident fields. The velocity average is performed in the approximation of large Doppler linewidths. One considers in detail the influence of depolarizing relaxation processes and the effects of pump-probe angular separation. In particular, the importance of the residual Doppler effect (shortening of the lifetime of the optically induced gratings by the atomic motion) associated with the pump-probe angle is emphasized. We show that the polarization of the reemitted fields is governed by very simple laws as soon as the residual Doppler effect overcomes the lifetimes of the atomic levels. Phase-conjugate mirrors can be simply characterized by a linear dichroism (parallel pump polarizations) or a birefringence (cross-polarized pumps). Vectorial phase conjugation is also analyzed, in the case of counter-rotating circular polarizations of the pumps. Finally, the case of two-photon transition is studied and compared with the case of a resonant transition.
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.
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.