# Decay rates of a dipole near a planar multilayer stack

Following (Le Ru, Etchegoin, p. 571), and (Novotny, Hecht pp. 335–360), the enhancement factor for the total decay rate for a dipole perpendicular to the interface is

\[ M^\perp_\text{tot} = 1 + \frac 3 2 \int_0^\infty \Re\left\{ \frac{q^3}{\sqrt{1 - q^2}} r^p (q) \exp\left(2 i k_1 d\sqrt{1 - q^2} \right)\right\}\text{d}q \]

The integrand diverges as \(q\to 1\), it is therefore advantageous to perform the substitution \(u:=\sqrt{1 - q^2}\). In order to maintain a real path of integration, the integral is first split into a radiative region (\(0\leq q\leq 1\), \(u:=\sqrt{1 - q^2}\geq 0\)), and an evanescent region (\(1\leq q\leq \infty\), \(-i u:=\sqrt{q^2 - 1}\geq 0\)). After some algebraic manipulation, we obtain, \[ M^\perp_\text{tot} = 1 + \frac 3 2 \left(I_1 + I_2\right) \] where \[ \begin{split} I_1 + I_2 =& \int_0^1 \left[1 - u^2\right]\cdot\Re\left\{ r^p (\sqrt{1 - u^2}) \exp\left(2 id k_1 u\right)\right\}\text{d}u \ & +\int_0^\infty \left[1 + u^2\right]\cdot\exp\left(-2d k_1 u\right)\cdot\Im\left\{ r^p (\sqrt{1 + u^2}) \right\}\text{d}u \end{split} \] Similarly, for the parallel dipole \[ M^\parallel_\text{tot} = 1 + \frac 3 4 \int_0^\infty \Re\left\{ \left[ \frac{r^s(q)}{\sqrt{1 - q^2}} - r^p (q) \sqrt{1 - q^2}\right] \cdot q\cdot\exp\left(2 i k_1 d\sqrt{1 - q^2} \right)\right\}\text{d}q \label{eq:Mstot1} \] which can be rewritten as, \[ M^\parallel_\text{tot} = 1 + \frac 3 4 \left(I^\parallel_1 + I^\parallel_2\right) \] where \[ \begin{split} I^\parallel_1 + I^\parallel_2 =& \int_0^1 \Re\left\{\left[r^s (\sqrt{1 - u^2}) - u^2\cdot r^p (\sqrt{1 - u^2}) \right] \exp\left(2 id k_1 u\right)\right\}\text{d}u \ & +\int_0^\infty \exp\left(-2d k_1 u\right)\cdot \Im\left\{r^s (\sqrt{1 + u^2}) + u^2\cdot r^p (\sqrt{1 + u^2}) \right\}\text{d}u \end{split} \]