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Optical and Quantum Sensing and Precision Metrology

Investigations on phase conjugation based on nondegenerate four wave mixing in a Rb vapor cell

Jie Zhao, Zhifan Zhou, Paul D. Lett

Published Mar 4, 2021• 0 citations
Publisher

Abstract

Phase conjugation is an nonlinear effect that results from the four-wave-mixing process. This peculiar effect, in analogy to a mirror, reflects an incident beam in a way that the reflection constitutes the time-reversal of the original beam. By combining the phase conjugate mirror with ordinary mirrors, a special resonator, called phase conjugate resonator (PCR), is constructed. The phase conjugate resonator has been used in a wide range of applications, including phase aberration correction, image transmission, holography, and phase sensing using interferometers. Here, we explore theoretically the properties of phase conjugate mirror by investigating its action upon a variety of quantum states. In addition, we analyze the input-output relations of a phase conjugate resonator. Experimental results are reported that faithfully verify PCR's enhanced stability as compared to normal resonators. Furthermore, we observe a linewidth narrowing effect that is only present in the phase conjugate resonator.

DOI
10.1117/12.2586674

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Phase conjugate optics : on the theory, observation, and utilization of temporallly-reversed wavefronts as generated via nonlinear optical parametric interactions

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This thesis describes theoretical and experimental aspects as well as potential applications in the field of coherent optics known as Phase Conjugate Optics (PCO). By utilizing nonlinear optical techniques, real-time or wavefront reversal of an arbitrary incident electromagnetic field can be realized. The nonlinear optical interaction gives rise to what is referred to as the phase conjugate (of the incident monochromatic wave) by performing the operation of complex conjugation upon the incident wave's complex spatial amplitude in real time. This conjugate wave, which is also designated as being a wavefront, has the property of exactly retracing the path of the incident field. The ability of the conjugate wave to correct for inhomogeneous linear and nonlinear (intensity-dependent) aberrations as well as polarization distortions is proved. In particular, the theory of a degenerate four-wave nonlinear optical interaction as providing for the conjugator is presented. The effects of linear and nonlinear losses upon this interaction are discussed. The quantum mechanical origin of the third order nonlinear optical susceptibility responsible for the four-wave mixing process is analyzed for both single and two-photon allowed transitions in an atomic (or molecular) system. The analogies of four-wave mixing with that of real-time holography are discussed. The theory of conjugation via four-wave mixing in optical waveguides is presented. Several of the above characteristics of conjugate fields are verified experimentally where conjugate fields via degenerate four-wave mixing were observed both in the bulk and in waveguide geometries, using carbon disulfide as the nonlinear medium. Amplified time-reversed wavefront generation as well as a mirrorless optical parametric mode of oscillation have been observed, both in agreement with theoretical predictions. Potential applications of PCO are discussed in three different regimes: spatial-frequency, temporal-frequency, and spatial/temporal frequency domains. In the first category, the ability of PCO to correct for image modal dispersion in optical waveguides as well as the use of PCO to perform real-time coherent image processing and nonlinear microscopy is discussed. Temporal-frequency domain applications of PCO to be analyzed include the use of a nearly degenerate four-wave mixing process as a narrowband, wide field-of-view optical filter, capable of an amplified bandpass. The ability of a PCO interaction to renarrow (transform limited) optical pulses which have been temporally spread due to propagation through (group velocity) dispersive channels is analyzed. A potential application of PCO in the field of nonlinear laser spectroscopy is presented. Specifically, the scattering of a probe photon off a two-photon coherent state (created in a three-level atomic system) is shown to yield a conjugate replica. This conjugate replica is capable of providing sub-Doppler width resolution of the two-photon resonance. Further, in the transient regime, the optical free-induction decay of the conjugate wave is capable of yielding detailed spectral features of the three-level system such as the anharmonic contribution to a (nearly) harmonic potential. This technique, performed in the time domain, is known as [alpha]-beat spectroscopy. Finally, a detailed theoretical and experimental study is presented of a laser resonator in which one (or both) of the mirror(s) comprising the optical cavity is replaced by a conjugate mirror (PCM). This novel resonator, which is termed a conjugate resonator (PCR), combines many of the spatial- and temporal-frequency aspects of PCO interactions discussed above. The stability criterion, transverse and longitudinal mode spectra, and the PCR output energy, as well as the frequency locking features of the laser modes to the PCM are discussed.

•Mar 1, 1980•2 citations

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CRC Press eBooks•Oct 3, 2018•0 citations

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It is often desirable that a phase conjugate mirror be able to correct for polarization distortions in addition to correcting for wavefront aberrations. Such a device is called a vector phase conjugate mirror. It has been predicted within the context of the third-order theory that vector phase conjugation can be achieved by degenerate four-wave mixing using atomic two-photon transitions between S-states.1 We recently developed a theory of the polarization properties of two-photon resonantly enhanced degenerate four-wave mixing which can treat saturation effects due to pump waves.2 Our results show that for the case of linear and parallel pump wave polarizations, vector phase conjugation occurs with high quality and high reflectivity for two-photon transitions for which the nonresonant Stark shift is very small. We have also found that for the case of circular and counterrotating pump wave polarizations, perfect vector phase conjugation occurs for equal pump wave intensities. We have studied experimentally the effects of saturation and Stark shifts for S → S two-photon transitions of atomic sodium. We have found that high fidelity vector phase conjugation can be obtained.

Annual Meeting Optical Society of America•Jan 1, 1989•0 citations

Theory of phase-conjugate oscillators I

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We have developed a theory for nondegenerate oscillations in optical resonators containing an intracavity phase-conjugate element. The phase-conjugate element consists of a nonlinear transparent medium that is pumped externally by a pair of counterpropagating laser beams of the same frequency and intensity. Phase conjugation of an input beam of slightly different frequency occurs because of nondegenerate four-wave mixing. The theory takes into account linear absorption (or gain) in the medium and is applied to study the threshold behavior of phase-conjugate oscillators. For the special case of no conventional mirrors, the phase-conjugate oscillator reduces to an ordinary phase-conjugate mirror, and our general formulation yields the results of previous studies. Our analysis shows that the parametric gain required for oscillation increases (or decreases) as a result of linear absorption (or gain) in the medium, and oscillation can occur at a frequency different from that of the pump beams in the presence of large linear gain (or loss). The effects of linear absorption (or gain) on the filter operation are also examined.

Journal of the Optical Society of America B•Aug 1, 1990•5 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 is an nonlinear effect that results from the four-wave-mixing process. This peculiar effect, in analogy to a mirror, reflects an incident beam in a way that the reflection constitutes the time-reversal of the original beam. By combining the phase conjugate mirror with ordinary mirrors, a special resonator, called phase conjugate resonator (PCR), is constructed. The phase conjugate resonator has been used in a wide range of applications, including phase aberration correction, image transmission, holography, and phase sensing using interferometers. Here, we explore theoretically the properties of phase conjugate mirror by investigating its action upon a variety of quantum states. In addition, we analyze the input-output relations of a phase conjugate resonator. Experimental results are reported that faithfully verify PCR's enhanced stability as compared to normal resonators. Furthermore, we observe a linewidth narrowing effect that is only present in the phase conjugate resonator.
This thesis describes theoretical and experimental aspects as well as potential applications in the field of coherent optics known as Phase Conjugate Optics (PCO). By utilizing nonlinear optical techniques, real-time or wavefront reversal of an arbitrary incident electromagnetic field can be realized. The nonlinear optical interaction gives rise to what is referred to as the phase conjugate (of the incident monochromatic wave) by performing the operation of complex conjugation upon the incident wave's complex spatial amplitude in real time. This conjugate wave, which is also designated as being a wavefront, has the property of exactly retracing the path of the incident field. The ability of the conjugate wave to correct for inhomogeneous linear and nonlinear (intensity-dependent) aberrations as well as polarization distortions is proved. In particular, the theory of a degenerate four-wave nonlinear optical interaction as providing for the conjugator is presented. The effects of linear and nonlinear losses upon this interaction are discussed. The quantum mechanical origin of the third order nonlinear optical susceptibility responsible for the four-wave mixing process is analyzed for both single and two-photon allowed transitions in an atomic (or molecular) system. The analogies of four-wave mixing with that of real-time holography are discussed. The theory of conjugation via four-wave mixing in optical waveguides is presented. Several of the above characteristics of conjugate fields are verified experimentally where conjugate fields via degenerate four-wave mixing were observed both in the bulk and in waveguide geometries, using carbon disulfide as the nonlinear medium. Amplified time-reversed wavefront generation as well as a mirrorless optical parametric mode of oscillation have been observed, both in agreement with theoretical predictions. Potential applications of PCO are discussed in three different regimes: spatial-frequency, temporal-frequency, and spatial/temporal frequency domains. In the first category, the ability of PCO to correct for image modal dispersion in optical waveguides as well as the use of PCO to perform real-time coherent image processing and nonlinear microscopy is discussed. Temporal-frequency domain applications of PCO to be analyzed include the use of a nearly degenerate four-wave mixing process as a narrowband, wide field-of-view optical filter, capable of an amplified bandpass. The ability of a PCO interaction to renarrow (transform limited) optical pulses which have been temporally spread due to propagation through (group velocity) dispersive channels is analyzed. A potential application of PCO in the field of nonlinear laser spectroscopy is presented. Specifically, the scattering of a probe photon off a two-photon coherent state (created in a three-level atomic system) is shown to yield a conjugate replica. This conjugate replica is capable of providing sub-Doppler width resolution of the two-photon resonance. Further, in the transient regime, the optical free-induction decay of the conjugate wave is capable of yielding detailed spectral features of the three-level system such as the anharmonic contribution to a (nearly) harmonic potential. This technique, performed in the time domain, is known as [alpha]-beat spectroscopy. Finally, a detailed theoretical and experimental study is presented of a laser resonator in which one (or both) of the mirror(s) comprising the optical cavity is replaced by a conjugate mirror (PCM). This novel resonator, which is termed a conjugate resonator (PCR), combines many of the spatial- and temporal-frequency aspects of PCO interactions discussed above. The stability criterion, transverse and longitudinal mode spectra, and the PCR output energy, as well as the frequency locking features of the laser modes to the PCM are discussed.
Four wave mixing, a third-order optical parametric process, is the nonlinear optical process in which four waves interact with each other through the third-order optical nonlinearity. In this process, three waves form a nonlinear polarization at the frequency of the fourth wave. An important application of degenerate four wave mixing is the generation of phase conjugate wave or simply the phase conjugation. The phase conjugate wave has the same wavefront as the incident wave; however, it propagates backward. The phase conjugation is equivalent to the time reversal. Straightforward application of the time reversal is the correction of phase aberration. Suppose that, the incident wave propagates in the nonuniform medium, such as a laser amplifier, where the temperature is fluctuated by intense pumping. Then, the incident wave suffers severe phase aberration during the propagation and the transmitted wave is then reflected by the phase conjugate mirror and goes back into the same nonuniform medium.
It is often desirable that a phase conjugate mirror be able to correct for polarization distortions in addition to correcting for wavefront aberrations. Such a device is called a vector phase conjugate mirror. It has been predicted within the context of the third-order theory that vector phase conjugation can be achieved by degenerate four-wave mixing using atomic two-photon transitions between S-states.1 We recently developed a theory of the polarization properties of two-photon resonantly enhanced degenerate four-wave mixing which can treat saturation effects due to pump waves.2 Our results show that for the case of linear and parallel pump wave polarizations, vector phase conjugation occurs with high quality and high reflectivity for two-photon transitions for which the nonresonant Stark shift is very small. We have also found that for the case of circular and counterrotating pump wave polarizations, perfect vector phase conjugation occurs for equal pump wave intensities. We have studied experimentally the effects of saturation and Stark shifts for S → S two-photon transitions of atomic sodium. We have found that high fidelity vector phase conjugation can be obtained.
We have developed a theory for nondegenerate oscillations in optical resonators containing an intracavity phase-conjugate element. The phase-conjugate element consists of a nonlinear transparent medium that is pumped externally by a pair of counterpropagating laser beams of the same frequency and intensity. Phase conjugation of an input beam of slightly different frequency occurs because of nondegenerate four-wave mixing. The theory takes into account linear absorption (or gain) in the medium and is applied to study the threshold behavior of phase-conjugate oscillators. For the special case of no conventional mirrors, the phase-conjugate oscillator reduces to an ordinary phase-conjugate mirror, and our general formulation yields the results of previous studies. Our analysis shows that the parametric gain required for oscillation increases (or decreases) as a result of linear absorption (or gain) in the medium, and oscillation can occur at a frequency different from that of the pump beams in the presence of large linear gain (or loss). The effects of linear absorption (or gain) on the filter operation are also examined.
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.