The Chapman-Kolmogorov equation states^{25}Display Formula
$P(r\u2192d,r\u2192s)=\u222bP(r\u2192d,r\u2192)P(r\u2192,r\u2192s)d3r\u2192.$(14)
In the case of transitions to and from different side bands we must sum over the relevant permutations: Display Formula$P0n(r\u2192d,r\u2192s)=\u222b\u2211k=0\u221ePkn(r\u2192d,r\u2192)P0k(r\u2192,r\u2192s)d3r\u2192.$(15)
We assume that contributions to the $(n+1)$’th side-band are negligible and thus express the detected exitance at the zeroth side-band, Display Formula$W0(r\u2192d)=I0(r\u2192s)\u222bP00(r\u2192d,r\u2192)P00(r\u2192,r\u2192s)d3r\u2192,$(16)
where by optical reciprocity $P00(r\u2192d,r\u2192)=P00(r\u2192,r\u2192d)$. The detected exitance at the first sideband is given by, Display Formula$W1(r\u2192d)=I0(r\u2192s)\u222bP01(r\u2192d,r\u2192)P00(r\u2192,r\u2192s)+P11(r\u2192d,r\u2192)P01(r\u2192,r\u2192s)d3r\u2192.$(17)
We find $P01(r\u2192d,r\u2192)$ by optical reciprocity, and make the approximation $P11(r\u2192d,r\u2192)\u2248P00(r\u2192d,r\u2192)$. Thus, the first side-band modulation depth [$MD(1)$] detected at a point on the boundary is given by Display Formula$MD(1)\u2248\u222bP01(r\u2192d,r\u2192)P00(r\u2192,r\u2192s)+P00(r\u2192d,r\u2192)P01(r\u2192,r\u2192s)d3r\u2192\u222bP00(r\u2192d,r\u2192)P00(r\u2192,r\u2192s)d3r\u2192.$(18)