Editing Nonlinear optics (section)

====Four-wave mixing technique====
For the four-wave mixing technique, we can describe four beams (''j'' = 1, 2, 3, 4) with electric fields:

:<math>\Xi_j(\mathbf{x},t) = \frac{1}{2} E_j(\mathbf{x}) e^{i \left(\omega_j t - \mathbf{k} \cdot \mathbf{x}\right)} + \text{c.c.},</math>

where ''E<sub>j</sub>'' are the electric field amplitudes. Ξ<sub>1</sub> and Ξ<sub>2</sub> are known as the two pump waves, with Ξ<sub>3</sub> being the signal wave, and Ξ<sub>4</sub> being the generated conjugate wave.

If the pump waves and the signal wave are superimposed in a medium with a non-zero χ<sup>(3)</sup>, this produces a nonlinear polarization field:

:<math>P_\text{NL} = \varepsilon_0 \chi^{(3)} (\Xi_1 + \Xi_2 + \Xi_3)^3,</math>

resulting in generation of waves with frequencies given by ω = ±ω<sub>1</sub> ± ω<sub>2</sub> ± ω<sub>3</sub> in addition to third-harmonic generation waves with ω = 3ω<sub>1</sub>, 3ω<sub>2</sub>, 3ω<sub>3</sub>.

As above, the phase-matching condition determines which of these waves is the dominant. By choosing conditions such that ω = ω<sub>1</sub>  + ω<sub>2</sub> − ω<sub>3</sub> and '''k''' = '''k'''<sub>1</sub> + '''k'''<sub>2</sub> − '''k'''<sub>3</sub>, this gives a polarization field:

:<math>P_\omega = \frac{1}{2} \chi^{(3)} \varepsilon_0 E_1 E_2 E_3^* e^{i(\omega t - \mathbf{k} \cdot \mathbf{x})} + \text{c.c.}</math>

This is the generating field for the phase-conjugate beam, Ξ<sub>4</sub>. Its direction is given by '''k'''<sub>4</sub> = '''k'''<sub>1</sub> + '''k'''<sub>2</sub> − '''k'''<sub>3</sub>, and so if the two pump beams are counterpropagating ('''k'''<sub>1</sub> = −'''k'''<sub>2</sub>), then the conjugate and signal beams propagate in opposite directions ('''k'''<sub>4</sub> = −'''k'''<sub>3</sub>). This results in the retroreflecting property of the effect.

Further, it can be shown that for a medium with refractive index ''n'' and a beam interaction length ''l'', the electric field amplitude of the conjugate beam is approximated by

:<math>E_4 =  \frac{i \omega l}{2 n c} \chi^{(3)} E_1 E_2 E_3^*, </math>

where ''c'' is the speed of light. If the pump beams ''E''<sub>1</sub> and ''E''<sub>2</sub> are plane (counterpropagating) waves, then

:<math>E_4(\mathbf{x}) \propto E_3^*(\mathbf{x}),</math>

that is, the generated beam amplitude is the complex conjugate of the signal beam amplitude. Since the imaginary part of the amplitude contains the phase of the beam, this results in the reversal of phase property of the effect.

Note that the constant of proportionality between the signal and conjugate beams can be greater than 1. This is effectively a mirror with a reflection coefficient greater than 100%, producing an amplified reflection. The power for this comes from the two pump beams, which are depleted by the process.

The frequency of the conjugate wave can be different from that of the signal wave. If the pump waves are of frequency ω<sub>1</sub> = ω<sub>2</sub> = ω, and the signal wave is higher in frequency such that ω<sub>3</sub> = ω + Δω, then the conjugate wave is of frequency ω<sub>4</sub> = ω&nbsp;− Δω. This is known as ''frequency flipping''.