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Optics Letters

Theory of two-photon phase conjugation

Tao-Yi Fu, Murray Sargent

Published Oct 1, 1980• 59 citations
Publisher

Abstract

The theory of degenerate four-wave mixing using two-photon transitions is formulated. Two kinds of mechanisms compete in generating conjugate waves leading to two phenomena that do not occur in two-level media: (1) a double-peaked reflection spectrum and (2) coupled-mode oscillation in absorbing media. In addition, because of the dynamic Stark-shifted line center, the reflection coefficient approaches a constant value at large intensities, unlike the two-level coefficient, which bleaches to zero.

DOI
10.1364/ol.5.000433

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The theory of degenerate four-wave mixing using two-photon transitions is formulated. Two kinds of mechanisms compete in generating conjugate waves leading to two phenomena that do not occur in two-level media: (1) a double-peaked reflection spectrum and (2) coupled-mode oscillation in absorbing media. In addition, because of the dynamic Stark-shifted line center, the reflection coefficient approaches a constant value at large intensities, unlike the two-level coefficient, which bleaches to zero.
Our semiclassical theory of two-photon multiwave mixing1 is a generalization of the degenerate four-wave mixing case2 to treat nondegenerate pump/signal/conjugate frequencies. We apply that theory to four-wave mixing in two-photon media. We present explicit formulas for the nondegenerate signal/conjugate absorption and coupling coefficients for cases for which the two-photon Stark shifts are sufficiently small compared to the homogeneous linewidths. More general cases require that the spatial hole burning averages be performed numerically. We find intricate dependencies of the transmission and reflection spectra on the pump intensity and on dynamic Stark shifts. In particular, the reflection coefficients do not necessarily bleach to zero as the pump intensity is increased indefinitely, and interferences occur between pure index and mixed absorption/index contributions
Two-photon transitions have provided the basis for numerous useful nonlinear interactions with radiation. In this paper we calculate the absorption and coupling coefficients for one-, two-, and three-wave mixing in two-photon homogeneously broadened media. The resulting field equations form the basis for understanding saturation spectroscopy, phase conjugation,1 and single-mode instabilities2 in lasers and optical bistability in two-photon media. We assume that the wave frequencies are not resonant with any intermediate state j and that the probe and conjugate waves cannot saturate the two-photon transition. This allows us to use the two-photon two-level model and to transfer much of our knowledge of the one-photon two-level systems to the two-photon case. Two major differences occur between the two models. First, dynamic Stark shifts of the active levels play an important role in generating asymmetric spectra and nonzero dispersion (≃ω s T1) at line center. Second, high pump intensities do not bleach the absorption coefficient at the Rabi sidebands.
The theory of resonant four‐wave mixing using one and two‐photon transitions is described in terms of saturation spectroscopy. Detuning the signal wave from the pump wave in the one‐photon case leads to a narrow‐band reflection spectrum, with width given by 1/T1, where T1 is the lifetime of the population difference. In the corresponding time‐domain picture (the fourth dimension), this produces a reflected pulse at least as long as T1, regardless of the signal. This time‐domain picture also leads to a novel kind of coherent transient spectroscopy. In the two‐photon case, an infinite number of levels are involved in principle, i.e., phase conjugation in infinite dimensions! In this case, two kinds of mechanisms compete in generating conjugate waves leading to two phenomena that do not occur in two‐level media: 1) a double‐peaked reflection spectrum and 2) coupled‐mode oscillation in absorbing media. In addition, due to the dynamic‐Stark shifted line center, the reflection coefficient approaches a constant value at large intensities, unlike the two‐level coefficient which bleaches to zero.
Multiphoton transitions are of great interest in laser spectroscopy because they permit the study of new physical phenomena and because of their potential applications. Malcuit et al.1 studied two-photon transitions in atomic sodium and showed that for appropriate phase-matching four-wave mixing can overcome the usual spontaneous decay processes. We have recently developed a quantum theory2 to treat such multiwave mixing processes in two-photon media. Because of the greater complexity of two-photon transitions, many effects arise that are absent in the one-photon case, some of which have important consequences in the generation of squeezed states. For the two-photon two-level model, the field modes have frequencies approximately one-half the frequency difference between the levels. We find that our coefficients differ from the corresponding one-photon theory in that dynamic Stark shifts have a major effect and that the pump-induced coherence between the two levels has a new term in the conjugate coupling terms, which leads to improved squeezing.
Steady-state interactions of one, two, and three waves with two-photon two-level media are treated semiclassically. Dynamic Stark shifts as well as dynamic Stark splittings are included in the theory. The intermediate levels are assumed to be nonresonant, and one wave is allowed to have arbitrarily large intensity. The theory parallels the one-photon theory of multiwave mixing couched in terms of population pulsations. We find the one-, two-, and three-wave two-photon absorption coefficients and the three-wave coupling coefficients. We find similarities and significant differences between the one- and two-photon polarizations of the medium. The work treats two-photon saturation spectroscopy, laser- and optical-bistability instabilities, and phase conjugation with a unified formalism.
Considers phase conjugation through degenerate four-wave mixing in a Λ-type three-level system under certain simplifying assumptions, the most restrictive among them being the assumption of symmetrical detuning of the pump from the two lower lying sublevels. When compared to an absorbing resonant two-level system, it is found that significantly higher values of the phase-conjugate reflectivity are possible at pump intensities well below saturation provided the sublevel splitting is less than a homogeneous linewidth. The physical mechanism responsible for phase conjugation is coherent population trapping arising from two-photon-induced atomic coherence between the two sublevels.