Table 1.

Frequency-domain Green’s functions for Equation (12) in several homogeneous geometries subject to the extrapolated-zero boundary condition (Equation (19)). r_s is the position of a normalized isotropic point source. With the exception of the infinite case, cylindrical coordinates are used explicitly to specify position, i.e. r = (ρ, z). Notation is defined in the lower part of the Table. In practice, the infinite sums are truncated after a desired accuracy has been reached.

Case	Green’s function (frequency-domain)
Infinite	$G_0(\mathbf{r},{\mathbf{r}}_{\mathbf{s}})=\frac{1}{4\pi |\mathbf{r}-{\mathbf{r}}_{\mathbf{s}}|}\text{exp}(-k|\mathbf{r}-{\mathbf{r}}_{\mathbf{s}}|)$
Semi-infinite	$G_0([\rho ,z],[{\rho }_s=0,z_s={\ell }_{\mathit{\text{tr}}}])=\frac{1}{4\pi }[\frac{\text{exp}(-k{r}_1)}{r_1}-\frac{\text{exp}(-k{r}_b)}{r_b}]$
Infinite Slab	$G_0([\rho ,z],[{\rho }_s=0,z_s={\ell }_{\mathit{\text{tr}}}])=\frac{1}{4\pi }{\sum }_{m=-\infty }^{\infty }(\frac{\text{exp}[-k{r}_{+,m}]}{r_{+·m}}-\frac{\text{exp}[-k{r}_{-·m}])}{r_{-,m}})$
Infinite Cylindrical	$G_0([\rho ,z],[{\rho }_s,z_s])=\frac{1}{2\pi a_b^2}{\sum }_{m=-\infty }^{\infty }\text{cos}m\phi {\sum }_{{\beta }_m}\frac{e^{-|z-z_s|}/\sqrt{k^2+{\beta }_m^2}}{\sqrt{k^2+{\beta }_m^2}}\frac{J_m({\beta }_m\rho )J_m({\beta }_m{\rho }_s)}{{[J_m^{\prime }({\beta }_m{a}_b)]}^2}$
$k\equiv \sqrt{({\mu }_a\upsilon -i\omega )/D}$	Jm(z) mth order Bessel function, 1st kind
$r_{\pm ,m}\equiv \sqrt{{\rho }^2+{(z-z_{\pm ,m})}^2}$	a, cylinder radius
z+,m ≡ 2m(d + 2zb) + ℓtr	ab = a + zb, i.e. extrapolated-zero boundary (cylinder)
z−,m ≡ 2m(d + 2zb) − 2zb − ℓtr	βm, a positive root of Jm(βmab) = 0
$k_m\equiv \sqrt{k^2+m^2{\pi }^2/h_b^2}$	ρ, radial cylindrical coordinate
$r_1\equiv \sqrt{{(z-{\ell }_{\mathit{\text{tr}}})}^2+{\rho }^2}$	$r_b\equiv \sqrt{{(z+2z_b+{\ell }_{\mathit{\text{tr}}})}^2+{\rho }^2}$
$R_{\mathit{\text{eff}}}\equiv \frac{R_{\varphi }+R_J}{2-R_{\varphi }+R_j}$	$z_b=2{\ell }_{\mathit{\text{tr}}}\frac{1+R_{\mathit{\text{eff}}}}{3(1-R_{\mathit{\text{eff}}})}$
$R_{\varphi }\equiv {\int }_0^{\pi /2}\text{Sin}(2\vartheta )R_{\text{Fresnel}}(\vartheta )\mathrm{d}\vartheta$	$R_J\equiv {\int }_0^{\pi /2}3\text{Sin}\vartheta {\text{cos}}^2\vartheta R_{\text{Fresnel}}(\vartheta )\mathrm{d}\vartheta$
m, an integer	d, slab thickness (Figure 4)
cos ϑ = Ω̂ · n̂ Figure 4)	RFresnel(ϑ), Fresnel reflection coefficient
ϑ, angle of incidence in RFresnel(ϑ)	φ, angle between input/output light beams (cylinder)