Condition III: ${\bf n}\cdot ({\bf D}_1 - {\bf D}_2 )$

For the boundary condition in Eq.40

$$\tilde{\epsilon_1} \epsilon_0 ( E_{z} + E_{z}' ) = \tilde{\epsilon_2} \epsilon_0 E_{z}'' + \sigma_{12}$$ (78)

$$\tilde{\epsilon_1} \epsilon_0 ( -E_p \sin\theta - R_p \sin\theta) e^{i(k\sin\theta x - \omega t)} = \tilde{\epsilon_2} \epsilon_0 (-T_p \sin\theta'' ) e^{i(k'' \sin\theta'' x - \omega t)} + \sigma_{12}({\bf r}, t)$$

$$\sigma_{12} ({\bf r}, t) = \delta (z) \int_{-\infty}^{\infty} dk \int_{-\infty}^{\infty} d\omega \sigma_{12} (k, \omega) e^{i(kx-\omega t)}$$ (79)

If we apply $(1/2\pi)^2 \int \int dx dt e^{-i(k'x - \omega' t)}$, then we use $(1/4\pi^2 ) \int\int dx dt e^{i(k\sin \theta x - k' x)}e^{-i(\omega - \omega')t} = \delta(k\sin \theta - k') \delta(\omega - \omega' )$ and we keep in maind that the above equations are held at $z= 0$

$$\tilde{\epsilon_1} \epsilon_0 ( -E_p \sin\theta - R_p \sin\theta) \delta(k\sin \theta - k') \delta(\omega - \omega' ) = -\tilde{\epsilon_2} \epsilon_0 T_p \sin\theta'' \delta(k'' \sin \theta'' - k') \delta(\omega - \omega' )$$

$$+ \int\int dk''' d\omega''' \sigma_{12}(k''', \omega''') \delta(k''' - k' ) \delta (\omega''' - \omega)$$ (80)

$$\tilde{n_1}^2 (E_p + R_p ) \sin\theta = \tilde{n_2}^2 T_p \sin\theta'' + \frac{\sigma_{12} (k\sin\theta, \omega)}{\epsilon _0}$$ (81)

$${\rm From \ Snell's \ law}$$ (82)

$$\tilde{n_1} (E_p + R_p ) = \tilde{n_2} T_p + \frac{\sigma_{12}(k\sin\theta, \omega)}{\epsilon_0 \tilde{n_1} \sin\theta }$$ (83)

[The following discussion may be useless because the surface charge is in the $\omega=0$ limit.] For electrochemical system $\sigma_{12}(k\sin\theta=0,\omega=0)$ is order of 10 $\mu$C$/$cm$^2$= 0.1 C$/$m$^2$ (MPA fully disociated surface in $\sqrt{3}\times\sqrt{3}$ structure = 0.74 C$/$m$^2$). $\sigma_{12}/\epsilon_0 = (1\sim 8) \times 10^{10}$ V$/$m. Laser light(10 mW He-Ne, focused to 20 $\mu$m, $10^{26}$ photons$/$s$/$m $^2 = I/(\hbar\omega)$ $I= c\epsilon_0 E_0^2/2$ (J/m$^2$/s = W$/$m$^2$) c=299792458 m, $E_0 = 1.5 \times 10^{5}$ V$/$m ) Bright sunlight (average 480nm, 10$^{18}$ photons$/$s$/$m$^2$ $E_0 = 18$V$/$m.)