Nonlinear Optics
Phase Conjugation
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Optical Phase Conjugation

Optical phase conjugation (OPC) is used as a generic term for a multitude of nonlinear optical processes. The common feature is that all these processes are capable of reversing both the direction of propagation and the phase factor for each plane wave component of an arbitrary incoming beam of light. This means that a phase conjugator can be considered as a kind of mirror with very unusual reflection properties. Unlike a conventional mirror, where a ray is redirected according to the ordinary law of reflection, a phase conjugate mirror (PCM) retroreflects all incoming rays back to their origin. Thus, any optical beam that is reflected by a PCM will retrace its original path. When a ray is reflected by a conventional mirror only the wave vector component normal to the surface of the mirror changes sign. The tangential components are unchanged. This means that the propagation direction of the reflected ray depends on the angle between the surface normal and the incident ray. This is just a complicated way of describing our every-day experience that a beam of light can be precisely redirected by a mirror just by tilting it. A PCM, on the other hand, changes the sign of the complete wave vector so that the reflected ray is always antiparallel to the incident ray, independent of the orientation of the mirror surface.


Four Wave Mixing

Four wave mixing can be divided into degenerate four wave mixing (DFWM) and nondegenerate four wave mixing. DFWM is one of the most important processes used for OPC. It is a third-order nonlinear optical process involving the mixing of four separate optical waves, all with the same frequency. The three input waves consist of two antiparallel, high power pump or reference waves and a weaker probe wave. Two of the three input waves interfere and form either a spatially or temporally modulated grating; the third input wave is scattered by the grating to yield the output wave. In terms of photons DFWM can be described as a parametric process where the energy from two pump photons, one from each wave, is converted into one probe and one output signal photon. The energy conservation requirement is obviously fulfilled in this case since all photons have the same frequency. To have an efficient energy transfer from the pumps to the probe and signal we must also have momentum conservation.  With k1, k2, kp and k3 as the wave vectors of the two pumps, the probe, and the signal wave, respectively, this so called phase-matching condition states that k1 + k2 = kp + k3. Since the two pump photons are antiparallel, with zero total momentum ( k1 + k2 = 0), this means that the signal wave must be antiparallel to the probe wave. Momentum conservation is the basis of the phase-conjugate nature of the output signal in DFWM. Here the signal is proportional to the complex conjugate of the probe field. Since the output is directly related to the nonlinear response of the medium DFWM is often used to measure third-order nonlinearities.

In DFWM, the conjugated output wave is essentially at the same wavelength as the probe wave. To achieve phase conjugation with large frequency conversion we have to use nondegenerate four wave mixing. The phase matching scheme is shown above. In this case the two pump photons carry a nonzero total momentum in the z direction that is tuned to match the frequency difference of the probe and signal waves. As a consequence of this, exact phase matching can be achieved only for one direction of propagation of the probe wave.
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