Editing Nonlinear optics (section)

====Classical picture ====

In ''classical Maxwell electrodynamics'' a phase-conjugating mirror performs reversal of the [[Poynting vector]]:

:<math>\mathbf{S}_\text{out}(\mathbf{r},t) = -\mathbf{S}_\text{in}(\mathbf{r},t),</math>

("in" means incident field, "out" means reflected field) where

:<math>\mathbf{S}(\mathbf{r},t) = \epsilon_0 c^2 \mathbf{E}(\mathbf{r},t) \times \mathbf{B}(\mathbf{r},t),</math>
 
which is a linear momentum density of electromagnetic field.<ref name="Okulov2008">{{cite journal |first=A. Yu. |last=Okulov |title=Angular momentum of photons and phase conjugation |journal=J. Phys. B: At. Mol. Opt. Phys. |volume=41 |issue=10 |pages=101001 |date=2008 |doi=10.1088/0953-4075/41/10/101001 |arxiv=0801.2675}}</ref>
In the same way a phase-conjugated wave has an opposite angular momentum density vector <math> \mathbf{L}(\mathbf{r},t) = \mathbf{r} \times \mathbf{S}(\mathbf{r},t) </math> 
with respect to incident field:<ref name="Okulov2008J">{{cite journal |first=A. Yu. |last=Okulov |title=Optical and Sound Helical structures in a Mandelstam–Brillouin mirror |journal=JETP Lett. |volume=88 |issue=8 |pages=561–566 |date=2008 |doi=10.1134/S0021364008200046 }}</ref>

:<math>\mathbf{L}_\text{out}(\mathbf{r},t) = -\mathbf{L}_\text{in}(\mathbf{r},t).</math>

The above identities are valid ''locally'', i.e. in each space point <math>\mathbf{r}</math> in a given moment <math>t</math> for an ''ideal phase-conjugating mirror''.