= Phase-conjugate ray retracing
Consider an <optical point source> at $(z,x)=(0,0)$, a uniform transparent medium, and a flat return surface $z=L$. A <light ray> with slope $s$ reaches $(L,sL)$. A <phase-conjugate mirror> reverses its full direction and returns it to $(0,0)$. An ordinary mirror reverses only its normal direction; its returning <light ray> reaches $x=2sL$ at $z=0$. Thus rays with $s=-1/4,0,1/4$ at $L=3$ return to $x=-3/2,0,3/2$ with ordinary <specular reflection>, but all return to $x=0$ under ideal <phase conjugation>. These are stipulated geometric coordinates, not laboratory measurements.
\Image[/phase-conjugate-wavefront-ray-retracing.png]
{title=Ordinary reflection and ideal <phase conjugation>: curved <wavefronts> and returning <light rays>}
{id=phase-conjugate-wavefront-ray-retracing}
Solid blue curves mark outward <wavefronts>; dashed curves mark returning <wavefronts>. In the <phase-conjugate mirror> panel the returning curves coincide with the outward ones, and the arrows reverse on the same <light rays>. The diagram uses equal spatial scales. In <geometrical optics>, reversing each <light ray> in an unchanged medium with <wave reciprocity> retraces its route. At the <optical point source> itself, the ray construction becomes singular and <diffraction> sets the physical focal size.
A wave-level proof outline applies to a stationary scalar medium with real <refractive index> $n(\mathbf r)$ and no <wave absorption>. Writing $c$ for the <vacuum> <speed of light>, its <Helmholtz equation> is
$$
\nabla^2U+k_0^2n(\mathbf r)^2U=0,\qquad k_0=\omega/c.
$$
All coefficients are real, so <complex conjugation> gives the same <Helmholtz equation> for $U^*$. The conjugated outgoing boundary data select the incoming solution. This explains the connection between <phase conjugation> and retracing, provided the required incoming field is supplied over the necessary <aperture>. It also exposes the limits: <wave reciprocity> alone does not imply absence of <wave absorption>; conjugation of complex material coefficients would describe a different medium. Loss, an incomplete <aperture>, a changing medium or failure of <wave reciprocity> can prevent exact reconstruction. Intensity-dependent material changes need a separate analysis and are not covered by this linear proof.
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