\text{if }  k_x<-|k_z|  
\end{cases}$$

[l]{} Form 1\
$A_x = 0$\
$A_y = -\frac{k_z}{k_x k_y}\sin(k_x x + \phi_x)\sinh(k_yy + \phi_y)\sin(k_zz + \phi_z)$\
$A_z = -\frac{1}{k_x}\sin(k_x x + \phi_x)\cosh(k_yy + \phi_y)\cos(k_zz + \phi_z)$\
with $k_y^2 = k_x^2 + k_z^2$\
\
Form 2\
$A_x = 0$\
$A_y = -\frac{k_z}{k_x k_y}\sinh(k_x x + \phi_x)\sinh(k_yy + \phi_y)\sin(k_zz + \phi_z)$\
$A_z = -\frac{1}{k_x}\sinh(k_x x + \phi_x)\cosh(k_yy + \phi_y)\cos(k_zz + \phi_z)$\
with $k_y^2 = k_z^2 - k_x^2$\
\
Form 3\
$A_x = 0$\
$A_y =-\frac{k_z}{k_x}\sinh(k_x x + \phi_x)\sin(k_yy + \phi_y)\sin(k_zz + \phi_z)$\
$A_z = -\frac{1}{k_x}\sinh(k_x x + \phi_x)\cos(k_yy + \phi_y)\cos(k_zz + \phi_z)$\
with $k_y^2 = k_x^2 - k_z^2$\

Finally, taking the curl of $\mathbf{A}$ gives us $\mathbf{B}$, which is presented below.

$$\mathbf{B} =
\begin{cases}
\text{Form } 1 & \text{if }  k_x>0       \\
\text{Form } 2 & \text{if }  -|k_z| \leq k_x \leq 0 \\
\text{Form } 3 & \text{if }  k_x<-|k_z|    
\end{cases}$$

[l]{} Form 1\
$B_x = -\frac{k_x}{k_y}\sin(k_x x + \phi_x)\sinh(k_yy + \phi_y)\cos(k_zz + \phi_z)$\
$B_y = \cos(k_x x + \phi_x)\cosh(k_yy + \phi_y)\cos(k_zz + \phi_z)$\
$B_z = -\frac{k_z}{k_y}\cos(k_x x + \phi_x)\sinh(k_yy + \phi_y)\sin(k_zz + \phi_z)$\
with $k_y^2 = k_x^2 + k_z^2$\
\
Form 2\
$B_x = \frac{k_x}{k_y}\sinh(k_x x + \phi_x)\sinh(k_yy + \phi_y)\cos(k_zz + \phi_z)$\
$B_y = \cosh(k_x 