Condition IV:

$\displaystyle H_x + H'_x$	$\textstyle =$	$\displaystyle H''_x + (J_s)_x$	(84)
$\displaystyle H_y + H'_y$	$\textstyle =$	$\displaystyle H''_y + (J_s)_y$	(85)
$\displaystyle B_x$	$\textstyle =$	$\displaystyle \mu\mu_0 H_x, ......$	(86)
$\displaystyle \left[ J_s ({\bf r}, t) \right] _x$	$\textstyle =$	$\displaystyle \int\int dk d\omega [J_s (k, \omega)]_x e^{i(kx - \omega t)}, ..$	(87)
$\displaystyle \left( -\frac{kE_s \cos\theta}{\mu_1 \mu_0 \omega} + \frac{kR_s \cos\theta}{\mu_1 \mu_0 \omega}\right) e^{i(k \sin\theta x - \omega t)}$	$\textstyle =$	$\displaystyle -\frac{k'' T_s \cos\theta'' }{\mu_2 \mu_0 \omega}e^{i(k'' \sin\theta'' - \omega t)} + [J_s ({\bf r}, t)]_x$
$\displaystyle \frac{\tilde{n_1} \cos\theta}{ \mu_1 } (E_s - R_s )$	$\textstyle =$	$\displaystyle \frac{\tilde{n_2} \cos\theta''}{ \mu_2 } T_s + c \mu_0 [J_s (k\sin\theta, \omega)]_x$	(88)
$\displaystyle \left( \frac{kE_p}{\mu_1 \mu_0 \omega} + \frac{kR_p}{\mu_1 \mu_0 \omega} \right) e^{i(k\sin\theta x -\omega t)}$	$\textstyle =$	$\displaystyle \frac{k'' T_p}{\mu_2 \mu_0 \omega}e^{i(k'' \sin\theta'' x -\omega t)} + [J_s ({\bf r}, t)]_y$	(89)
$\displaystyle \frac{\tilde{n_1} }{ \mu_1 }( E_p + R_p )$	$\textstyle =$	$\displaystyle \frac{\tilde{n_2}}{ \mu_2 }T_p + c \mu_0 [J_s (k\sin\theta, \omega)]_y$	(90)