= Phase conjugation
{title2=$U_{\mathrm c}(\mathbf r)=\gamma U(\mathbf r)^*$}
{wiki}
= Optical phase conjugation
{synonym}
<Phase conjugation> creates a returning <monochromatic wave> whose spatial <complex amplitude> is proportional to the <complex conjugate> of the incident field. Fix the time convention throughout:
$$
E(\mathbf r,t)=\operatorname{Re}\!\left\{U(\mathbf r)e^{-i\omega t}\right\},
\qquad U(\mathbf r)=A(\mathbf r)e^{i\phi(\mathbf r)},
\qquad U_{\mathrm c}(\mathbf r)=\gamma A(\mathbf r)e^{-i\phi(\mathbf r)}.
$$
Here $E$ is one scalar component of the <electric field>, $A\geq0$ is the <wave amplitude>, $\omega>0$ is the <angular frequency>, and $\gamma$ is a spatially constant <complex number> setting the return strength and reference <wave phase>. The <complex conjugation> acts on $U$, including its spatial carrier, while the time factor remains $e^{-i\omega t}$. Conjugating both factors would merely give another representation of the same real field. With $\gamma=1$, the new real field equals $E(\mathbf r,-t)$ in this ideal <monochromatic wave> model; that identity does not assert reversal of an arbitrary pulse or of the apparatus producing it.
The spatial <wave phase> becomes $-\phi+\arg\gamma$, so the local <wavevector> changes from $\nabla\phi$ to $-\nabla\phi$. For example, $U=Ae^{i(k_xx+k_zz)}$ with $k_z>0$ becomes $\gamma Ae^{-i(k_xx+k_zz)}$: both components of the <wavevector> reverse. In a transparent stationary medium with <isotropy> and positive <refractive index>, <light rays> follow the <wavevector>, giving counterpropagation. The same <wavefronts> now have opposite propagation normals. This is why <phase conjugation> can return a distorted beam toward its origin: it reverses the detailed spatial <wave phase>, rather than requiring that <wave phase> to be flat.
The attractive application is correction of <optical aberrations> accumulated on an outward trip. The return uses the distortion itself as the prescription for correction, as the original <thin phase-plate cancellation by phase conjugation> example makes explicit. An experimentally accessible implementation measures the spatial <wave phase> and uses a <spatial light modulator> to generate its negative; <digital optical phase conjugation> describes an inspected 2010 demonstration of recovery through <multiple scattering>.
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