To employ the path of the excitation light to calculate the fluorescence $y(p)$, the fluorescence emission from state $p\u2032$ to $p$ must be transformed into the scattering of excitation photons. We use the following transformation to decouple the excitation-to-emission process of the fluorescence $p\u2032\u2192p$ from the special kernel describing the generation of fluorescence photons: Display Formula
$Kxm(p\u2032\u2192p,\mu af)=\eta T(r\u2192\u2032\u2192r\u2192\u2223s^\u2032,\mu sex,\mu aex+\mu af)Cxm(s^\u2032\u2192s^|r\u2192,\mu af)=\eta T(r\u2192\u2032\u2192r\u2192\u2223s^\u2032,\mu sex,\mu aex)\Gamma (p\u2032\u2192p,\mu af)\xd7\mu tex(r\u2192)+\mu af(r\u2192)\mu tex(r\u2192)Cxm(s^\u2032\u2192s^|r\u2192,\mu af)=K(p\u2032\u2192p,\mu sex,\mu aex,gex)\xd7\Gamma (p\u2032\u2192p,\mu af)\eta \mu af(r\u2192)\mu sex(r\u2192)PI(s^\u2032\xb7s^)PA(s^\u2032\xb7s^,gex(r\u2192\u2032)),$(5)
where $\Gamma (p\u2032\u2192p,\mu )=exp[\u2212\u222b0|r\u2192\u2212r\u2192\u2032|\mu (r\u2192\u2032+ls^)dl]$. From Eq. (5), we can see that the fluorescence photons of the state $p$ excited by the excitation photons of state $p\u2032$ are proportional to the excitation photons of state $p$ scattered from the state $p\u2032$. The transformation can maintain energy conservation. This means the excited fluorescence photons can be equivalent to the scattered excitation photons by the equivalent transformation. In addition, the transport of the fluorescence photons from $p\u2032$ to $p$ can be changed into that of excitation light, yielding the following transformation: Display Formula$K(p\u2032\u2192p,\mu sem,\mu aem,gem)=T(r\u2192\u2032\u2192r\u2192\u2223s^\u2032,\mu sem,\mu aem)C(s^\u2032\u2192s^|r\u2192,\mu sem,\mu aem,gem)=T(r\u2192\u2032\u2192r\u2192\u2223s^\u2032,\mu sex,\mu aex)\Gamma (p\u2032\u2192p,\mu tem\u2212\mu tex)\xd7C(s^\u2032\u2192s^|r\u2192,\mu sex,\mu aex,gex)\mu sem(r\u2192)\mu sex(r\u2192)=K(p\u2032\u2192p,\mu sex,\mu aex,gex)\xd7\mu sem(r\u2192)\mu sex(r\u2192)PA(s^\u2032\xb7s^,gem(r\u2192\u2032))PA(s^\u2032\xb7s^,gex(r\u2192\u2032))\Gamma (p\u2032\u2192p,\mu tem\u2212\mu tex).$(6)