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 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.
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.
Four wave mixing, a third-order optical parametric process, is the nonlinear optical process in which four waves interact with each other through the third-order optical nonlinearity. In this process, three waves form a nonlinear polarization at the frequency of the fourth wave. An important application of degenerate four wave mixing is the generation of phase conjugate wave or simply the phase conjugation. The phase conjugate wave has the same wavefront as the incident wave; however, it propagates backward. The phase conjugation is equivalent to the time reversal. Straightforward application of the time reversal is the correction of phase aberration. Suppose that, the incident wave propagates in the nonuniform medium, such as a laser amplifier, where the temperature is fluctuated by intense pumping. Then, the incident wave suffers severe phase aberration during the propagation and the transmitted wave is then reflected by the phase conjugate mirror and goes back into the same nonuniform medium.
This thesis describes theoretical and experimental aspects as well as potential applications in the field of coherent optics known as Phase Conjugate Optics (PCO). By utilizing nonlinear optical techniques, real-time or wavefront reversal of an arbitrary incident electromagnetic field can be realized. The nonlinear optical interaction gives rise to what is referred to as the phase conjugate (of the incident monochromatic wave) by performing the operation of complex conjugation upon the incident wave's complex spatial amplitude in real time. This conjugate wave, which is also designated as being a wavefront, has the property of exactly retracing the path of the incident field. The ability of the conjugate wave to correct for inhomogeneous linear and nonlinear (intensity-dependent) aberrations as well as polarization distortions is proved. In particular, the theory of a degenerate four-wave nonlinear optical interaction as providing for the conjugator is presented. The effects of linear and nonlinear losses upon this interaction are discussed. The quantum mechanical origin of the third order nonlinear optical susceptibility responsible for the four-wave mixing process is analyzed for both single and two-photon allowed transitions in an atomic (or molecular) system. The analogies of four-wave mixing with that of real-time holography are discussed. The theory of conjugation via four-wave mixing in optical waveguides is presented.
Several of the above characteristics of conjugate fields are verified experimentally where conjugate fields via degenerate four-wave mixing were observed both in the bulk and in waveguide geometries, using carbon disulfide as the nonlinear medium. Amplified time-reversed wavefront generation as well as a mirrorless optical parametric mode of oscillation have been observed, both in agreement with theoretical predictions.
Potential applications of PCO are discussed in three different regimes: spatial-frequency, temporal-frequency, and spatial/temporal frequency domains. In the first category, the ability of PCO to correct for image modal dispersion in optical waveguides as well as the use of PCO to perform real-time coherent image processing and nonlinear microscopy is discussed. Temporal-frequency domain applications of PCO to be analyzed include the use of a nearly degenerate four-wave mixing process as a narrowband, wide field-of-view optical filter, capable of an amplified bandpass. The ability of a PCO interaction to renarrow (transform limited) optical pulses which have been temporally spread due to propagation through (group velocity) dispersive channels is analyzed. A potential application of PCO in the field of nonlinear laser spectroscopy is presented. Specifically, the scattering of a probe photon off a two-photon coherent state (created in a three-level atomic system) is shown to yield a conjugate replica. This conjugate replica is capable of providing sub-Doppler width resolution of the two-photon resonance. Further, in the transient regime, the optical free-induction decay of the conjugate wave is capable of yielding detailed spectral features of the three-level system such as the anharmonic contribution to a (nearly) harmonic potential. This technique, performed in the time domain, is known as [alpha]-beat spectroscopy.
Finally, a detailed theoretical and experimental study is presented of a laser resonator in which one (or both) of the mirror(s) comprising the optical cavity is replaced by a conjugate mirror (PCM). This novel resonator, which is termed a conjugate resonator (PCR), combines many of the spatial- and temporal-frequency aspects of PCO interactions discussed above. The stability criterion, transverse and longitudinal mode spectra, and the PCR output energy, as well as the frequency locking features of the laser modes to the PCM are discussed.
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.
We consider the backward optical wave stimulated by a multimode, monochromatic, incident optical wave in a waveguide filled with a transparent nonlinear medium, when the incident wave is negligibly perturbed by the nonlinear processes. We derive the conditions on guide length, area, mode number, and Stokes shift in order that a given high percentage of the power in the backscattered field be the “phase conjugate” of the incident field, i.e., be proportional to its complex conjugate in the entrance plane of the waveguide.
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.