The double phase-conjugate mirror (DPCM) is a stimulated four-wave mix
ing effect, mediated by either a chi(3) mechanism or a photorefractive
effect. The type of grafting that operates in the DPCM can be a trans
mission or reflection grating, depending on the wave-mixing configurat
ion. Any combination of nonlinearity and grating geometry is possible,
so that four general DPCM classes emerge. An analytic solution of the
last unsolved class, the photorefractive reflection DPCM, is presente
d. With these results, general observations concerning the four catego
ries of double phase conjugation are made. An important conclusion is
that the type of grating geometry and its associated boundary conditio
ns dictate many of the basic features of the DPCM, regardless of the n
onlinearity involved.