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Optical Engineering

Optical Beam Phase Conjugation By Four-Wave Mixing In A Waveguide

R. W. Hellwarth

Published Apr 1, 1982• 6 citations
Publisher

Abstract

Theory and experiments are reviewed which suggest that optical-beam phase conjugation by four-wave mixing in a waveguide has the following advantages over four-wave mixing of unguided waves: (1 ) pump power requirements are reduced; (2) the pump beams can contain many transverse modes, both outside and inside the waveguide; (3) pump beam alignment is not critical; (4) the image-wave, being conjugated, can serve simultaneously as its own pump beam; (5) many frequencies can be conjugated simultaneously without intermixing of signals; and (6) frequency- and angle-response characteristics of the conjugator as an optical filter are extended. Frequency depen-dent resonances in the phase-conjugate signal may be employed in spectroscopy of small samples.

DOI
10.1117/12.7972892

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Theory of phase-conjugation by four-wave mixing in a waveguide

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We show that one can generate the time-reversed replica of an "input" monochromatic, image-bearing beam by coupling it into a waveguide where it interacts with counterpropagating multimode "pump" waves of the same frequency. The nonlinear electric polarization density that is third order in the propagating electric fields in the guide medium generates the replica by the process of "four-wave mixing." We show also that the input beam can serve simultaneously as its own pump beam. If the frequency ν of the backward pump beam is different from the frequency ω to of the input beam, then the "phase-conjugate" to the input is generated at the entrance plane to the guide, and this radiates a replica of the input field, magnified by\omega/\nu, back along the input beam. The pump power required per resolution element to phase-conjugate a beam in a waveguide is orders-of-magnitude less than for the corresponding process with free (unguided) waves interacting in an infinite medium. Unlike the requirements for free waves, the pump waves do not need to be well aligned or single-mode to produce high fidelity in the replication process. Neither does the guiding structure or enclosed medium have to be precise in dimension or uniformity; the main requirements on the guiding structure being that it not attenuate the waves too heavily. Formulas are derived for the replication efficiency and fidelity in the various guided configurations. We also show how the process can be used: 1) to make a narrowband optical filter with a large acceptance solid angle; 2) to perform image-frequency conversion; 3) to obtain Raman and two-photon spectra of small samples; and 4) to achieve broadband optical amplification. We examine the conditions under which phase conjugation and these applications can be performed at several frequencies simultaneously. Limitations placed by the power-dependence of propagation constants are derived.

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Theory and experiments are reviewed which suggest that optical-beam phase conjugation by four-wave mixing in a waveguide has the following advantages over four-wave mixing of unguided waves: (1 ) pump power requirements are reduced; (2) the pump beams can contain many transverse modes, both outside and inside the waveguide; (3) pump beam alignment is not critical; (4) the image-wave, being conjugated, can serve simultaneously as its own pump beam; (5) many frequencies can be conjugated simultaneously without intermixing of signals; and (6) frequency- and angle-response characteristics of the conjugator as an optical filter are extended. Frequency depen-dent resonances in the phase-conjugate signal may be employed in spectroscopy of small samples.
We show that one can generate the time-reversed replica of an "input" monochromatic, image-bearing beam by coupling it into a waveguide where it interacts with counterpropagating multimode "pump" waves of the same frequency. The nonlinear electric polarization density that is third order in the propagating electric fields in the guide medium generates the replica by the process of "four-wave mixing." We show also that the input beam can serve simultaneously as its own pump beam. If the frequency ν of the backward pump beam is different from the frequency ω to of the input beam, then the "phase-conjugate" to the input is generated at the entrance plane to the guide, and this radiates a replica of the input field, magnified by\omega/\nu, back along the input beam. The pump power required per resolution element to phase-conjugate a beam in a waveguide is orders-of-magnitude less than for the corresponding process with free (unguided) waves interacting in an infinite medium. Unlike the requirements for free waves, the pump waves do not need to be well aligned or single-mode to produce high fidelity in the replication process. Neither does the guiding structure or enclosed medium have to be precise in dimension or uniformity; the main requirements on the guiding structure being that it not attenuate the waves too heavily. Formulas are derived for the replication efficiency and fidelity in the various guided configurations. We also show how the process can be used: 1) to make a narrowband optical filter with a large acceptance solid angle; 2) to perform image-frequency conversion; 3) to obtain Raman and two-photon spectra of small samples; and 4) to achieve broadband optical amplification. We examine the conditions under which phase conjugation and these applications can be performed at several frequencies simultaneously. Limitations placed by the power-dependence of propagation constants are derived.
We analyze the frequency-shifted conjugate beam generated by nearly degenerate four-wave mixing in an array of single-mode waveguides. We show that if the array sufficiently samples the signal beam, a high-quality conjugate will be generated when the pump beams are counterpropagating. The ability of a waveguide mode to couple to incoming beams over a range of incident directions relaxes the strict phase-matching requirement present in bulkmedia four-wave mixing. A shift in the propagation direction of one of the pump beams changes the propagation direction of the generated beam while maintaining the spatial phase coherence.
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.
The technique of polarization-decoupled Brillouin-enhanced four-wave mixing is known for its high-reflectivity, high-quality phase conjugation. The treatment accounts for the transverse structure of the pump and signal beams within a waveguide in the paraxial approximation, assuming a steady-state acoustic wave and assuming that many transverse modes of the waveguide are excited by the four mixing waves. The zeroth-order equations are obtained for the polarization states of the signal and conjugate beams by dropping all phase-mismatch terms. Under the stated assumptions, the phase-mismatched terms are found to make only a small contribution. A first-order solution accounts for phase-mismatch terms and determines the loss of conjugation fidelity as a function of the angular spread of the pump laser beam, the pump-signal angle, the pump intensities, and the pump mutual fidelity.>
Part I of this thesis describes the theoretical and experimental aspects of a new area in coherent optics known as phase conjugate optics. By using nonlinear optical mixing techniques, the complex conjugate of an arbitrary incident monochromatic wave can be generated. In particular, the theory of the degenerate four-wave mixing technique is presented. Experimental observation of phase conjugate wave front generation in a liquid crystal material (MBBA) is described. The ability of the conjugate wave front to correct for phase aberration is demonstrated. The quantum mechanical origin of the nonlinear third order susceptibility responsible for the four-wave mixing process is illustrated in an atomic system. Because of a finite memory time of such an atomic system, the electromagnetic waves taking part in the mixing process need not occur simultaneously. The applications of the phase conjugation process to real-time information processing and the characteristics of a phase conjugate resonator are discussed in detail. The ability of an optical fiber to confine the propagation of electromagnetic waves over a long distance and within a small cross sectional area implies that large optical intensities can be maintained over a long " interaction length even at moderate incident optical power. Part II of this thesis describes some third order nonlinear optical phenomena that have been observed under this condition. Especially, phase conjugation by degenerate four-wave mixing in a fiber has been shown to require significantly less pumping power. The theoretical and experimental studies of four-wave mixing in a liquid-filled fiber are presented. Extensive experimental efforts have been devoted over the last few years to the study of stimulated Raman scattering and the oscillation of the Stokes radiation inside fibers. A detailed theoretical treatment will be given here to describe these effects. The analysis shows that pump depletion due to the nonlinear process is very important in describing the stimulated Raman scattering in low loss fibers and is the dominant saturation effect in a fiber Raman oscillator. Finally, the implications of nonlinear optical phenomena to fiber optic communication will be mentioned.
It has previously been shown that Brillouin enhanced four-wave mixing can lead to large phase conjugate reflectivities for moderate laser powers.1,2 However, in the geometry considered by previous workers the pumps have equal frequency and hence the conjugate signal is shifted in frequency with respect to the probe. We have conducted an experimental and theoretical study of Brillouin-enhanced four-wave mixing for the different case in which the two pump waves differ in frequency by twice the Brillouin frequency of the four-wave mixing medium, and in which the probe wave frequency is halfway between those of the pump waves. In this case, the phase conjugate wave has the same frequency as the probe. Reflectivities > 100 % have been measured. Experimental results are compared to those of a theory that includes the effects of phase mismatch, pump depletion, and the finite response time of the medium.