Physical Review A

Third-order nonlinear optical response resulting from optical pumping: Effects of atomic motion

M. Pinard, Robert W. Boyd, G. Grynberg

Published Feb 1, 1994• 10 citations

Abstract

The formalism of Maker, Terhune, and Savage [Phys. Rev. Lett. 12, 507 (1964)] for third-order nonlinear susceptibilities is extended to account for nonlocal response. In the case of counterpropagating beams, we show that five constants instead of two are required in general to describe the nonlinearity. In the particular case of optical pumping nonlinearities, which we study extensively, we show that four constants, which describe the magnetization and electric-quadrupole moment of the medium induced by the forward and backward beams, are sufficient. We evaluate these constants for various values of the angular momenta of the atomic levels connected by the incident field. We finally show how this formalism can be applied to several problems such as induced focusing, four-wave-mixing generation, optical instability, and polarization properties of phase-conjugate and phase-contrast mirrors.

DOI: 10.1103/physreva.49.1326

Related papers

Theory of optical phase conjugation in Kerr media

Henk F. Arnoldus, Thomas F. George

Optical phase conjugation by four-wave mixing in a Kerr medium is considered. Two strong counterpropagating lasers pump a nonlinear medium in which they excite the third-order polarization. Mixing with a weak incident field then generates the phase-conjugated image of that field. It is shown how the polarization of the pumps and the tensorial nature of the nonlinear interaction can be accounted for by a geometrical polarization tensor. The electric field is shown to obey two coupled wave equations, which couple a positive- and negative-frequency component of the field. These wave equations allow plane-wave solutions, and we derive the dispersion relation for the wave vectors of these modes. By matching the field of such a mode across the boundaries of the medium to an external incident field, we are able to obtain analytically the Fresnel coefficients for reflection and transmission. Our expressions reduce to earlier results in the appropriate limits.

Physical Review AMay 1, 199510 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, 19820 citations

Modeling the nonlinear refractive index in atomic gases

Christian Kohler, Roland Guichard, Emmanuel Lorin +4 more

Accurate modeling of optical nonlinearities is crucial to describe macroscopic laser propagation in a medium, including sum frequency generation, spectral broadening due to self-phase modulation, various ionization processes and soliton formation. For incident laser light the response of the medium is given by the induced polarization of the microscopic system. The polarization is usually expanded in a Taylor series for the electric field amplitude, which is truncated after the first non-linear term being of third order for isotropic media. A third-order nonlinearity leads to the well-known optical Kerr effect, where the refractive index of the medium becomes intensity dependent via n = n0+ n2I. This leads to an inherent problem when modeling laser propagation in two or more spatial dimensions, linked to the formal divergence (n2> 0) of the refractive index for increasing intensity.

Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)May 1, 20130 citations

Polarimetric Study of the Phase Conjugate Wave in an Isotropic Medium

Santosh Kumar Saha

We derive expressions for the third-order nonlinearities responsible for the optical phase conjugation via backward degenerate four-wave mixing using various combinations of left and right circularly polarized beams in isotropic media. The effects of polarization conjugation and the nulling effect of phase conjugate waves are also discussed in an isotropic medium.

Journal of Modern OpticsSep 1, 19922 citations

Introduction

P. D. Drummond, Mark S. Hillery

Nonlinear optics is the study of the response of dielectric media to strong optical fields. The fields are sufficiently strong that the response of the medium is, as its name implies, nonlinear. That is, the polarization, which is the dipole moment per unit volume in the medium, is not a linear function of the applied electric field. In the equation for the polarization, there is a linear term, but, in addition, there are terms containing higher powers of the electric field. This leads to significant new types of behavior, one of the most notable being that frequencies different from that of the incident electromagnetic wave, such as harmonics or subharmonics, can be generated. Linear media do not change the frequency of light incident upon them. The first observation of a nonlinear optical effect was, in fact, second-harmonic generation – a laser beam entering a nonlinear medium produced a second beam at twice the frequency of the original. Another type of behavior that becomes possible in nonlinear media is that the index of refraction, rather than being a constant, is a function of the intensity of the light. For a light beam with a nonuniform intensity profile, this can lead to self-focusing of the beam. Most nonlinear optical effects can be described using classical electromagnetic fields, and, in fact, the initial theory of nonlinear optics was formulated assuming the fields were classical. When the fields are quantized, however, a number of new effects emerge. Quantized fields are necessary if we want to describe fields that originate from spontaneous emission. For example, in a process known as spontaneous parametric down-conversion, a beam of light at one frequency, the pump, produces a beam at half the original frequency, the signal. This second beam is a result of spontaneous emission.

Cambridge University Press eBooksMar 27, 20140 citations

Atomic coherence effects in resonant degenerate four-wave mixing

Jenny Lam, Duncan G. Steel, R. A. McFarlane +1 more

We present a study of atomic coherence effects in resonant degenerate four-wave mixing. We find for energy levels having magnetic degeneracies, the third-order nonlinear response resulting from two counterpropagating pump waves orthogonally polarized to an incident probe wave generates a signal whose polarization is phase conjugate to that of the incident probe wave. This effect is due to the coupling of different magnetic states resulting from coherent excitation giving rise to a net electric quadrupole. Experimental studies in Na on the D2 line at 5890 Å confirm the analysis.

Applied Physics LettersJun 15, 198139 citations