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Optics Letters

Conjugate generation and beam steering by nearly degenerate four-wave mixing in waveguide arrays

Thomas B. Simpson, J. M. Liu

Published Dec 15, 1989• 1 citation
Publisher

Abstract

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.

DOI
10.1364/ol.14.001383

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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.
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.
In this paper, nearly degenerate four-wave mixing created by pump-probe structure with small frequency spacing and enough high power is demonstrated and discussed by introducing single optical beam into single-mode Fabry-Pérot laser diode, in which the injected beam is called the probe wave, whose frequency detuning is controlled within ±50 GHz, and lasing resonance mode such as dominant mode or arbitrarily residual side mode is selected as the pump wave. In the case of dominant mode like the pump wave, conjugate signals around the pump-probe wave are easily observed for positive and negative probe detuning by judiciously tuning injected probe power. While external injection mode is moved towards the side mode selected as the pump wave, newly generated signals only occurred with small positive probe detuning. As the probe wave with positive frequency detuning is close to the pump wave, higher order conjugate signals are gradually grown. Under the condition of side mode as the pump wave, the strongly nonlinear mixing of the pump-probe wave also gives rise to new frequency signals on both sides of the dominant mode with enough high peak level. Against the change of correspondingly initial power and frequency detuning of the probe wave, outcome spectrum, output power, and conversion efficiency of conjugate signals are analyzed in detail, while dominant or one side mode is, respectively, selected as the pump wave.
Nondegenerate four-wave mixing in fluorescein-doped glass was used to demonstrate frequency- dependent beam steering and tilt/focus phase control. An argon-ion laser at 457.9 nm provided the probe beam and one of the pump beams. An argon-pumped dye laser, tunable from 560 to 615 nm, provided the second pump beam. The conjugate emerged at the same wavelength as the dye laser and at a frequency tunable, phase matching steering angle. It was shown conclusively that this beam was due to four-wave mixing.
Phase-conjugate beam fidelity is studied in degenerate four-wave mixing with spatially varying pump beams. The theory includes long-cell-length (or high-gain) effects, diffraction, and nonlinear phase variation, focusing of the probe beam, and, finally, losses. Relatively simple algebraic expressions for the reflectivity are given that cover the cases of collinear or near-collinear beam geometries. An easily calculated parameter is defined that characterizes phase-conjugate fidelity and is applied to study focusing effects. It is shown that focusing effects may be useful in greatly reducing the principal distortion of the phase-conjugate beam over a considerable range of variation of beam parameters.