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Optics Letters

Narrow-band optical filter through phase conjugation by nondegenerate four-wave mixing in sodium vapor

Joseph Nilsen, Natalie S. Gluck, Amnon Yariv

Published Aug 1, 1981• 30 citations
Publisher

Abstract

An ultrahigh-Q, tunable optical filter with a FWHM bandwidth of 41 MHz is demonstrated. The filtering is produced by nondegenerate phase conjugation through four-wave mixing in atomic-sodium vapor. The filter is observed to have a maximum quantum efficiency of 4 x 10(-3). However, degenerate phase-conjugation experiments in sodium suggest that a quantum efficiency of greater than unity can be attained on a cw basis.

DOI
10.1364/ol.6.000380

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IEEE Journal of Quantum Electronics•Nov 1, 1982•16 citations
An ultrahigh-Q, tunable optical filter with a FWHM bandwidth of 41 MHz is demonstrated. The filtering is produced by nondegenerate phase conjugation through four-wave mixing in atomic-sodium vapor. The filter is observed to have a maximum quantum efficiency of 4 x 10(-3). However, degenerate phase-conjugation experiments in sodium suggest that a quantum efficiency of greater than unity can be attained on a cw basis.
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.
Summary form only given. Previously, we have used four-wave mixing via coherent population trapping to achieve optical phase conjugation (OPC) with high gain, using two dye lasers and a sodium vapor cell. The /spl Lambda/ transition used involved the two hyperfine ground states and one excited state, so that two different optical frequencies were needed to satisfy the two-photon resonant condition between the probe and one of the pump beams. Later, we demonstrated in rubidium that a single hyperfine transition with Zeeman sublevels is enough to achieve OPC with equally high gain, thus eliminating the need for high-frequency sideband generation. However, in order to guarantee that the diffraction from the grating is primarily dispersive, the backward pump beam had to be detuned by about 60 MHz from the probe and the forward pump. One of the key disadvantages of such a nondegenerate scheme is that applications such as phase-conjugate mirrors are hard to implement. Here, we report the observation of optical phase conjugation in rubidium with equally high gain, but all the frequencies are degenerate.
The formulation of phase conjugation via nondegenerate four-wave mixing in a homogeneously broadened resonant medium is generalized to include saturating pump waves. An ultrahigh-Qoptical filter is predicted, showing both gain and a bandwidth less than the homogeneous linewidth of the atom. The dependence of the filter function on the ac Stark effect and on the use of asymmetric pump wave intensities is discussed. In addition, several new phenomena are shown which manifest themselves as a dip in the degenerate reflectivity at line center.