Table 2, where we now distinguish between $A_y$ and $A_z$. While we remain general for now and include the phase shifts, with the choices of functions in Table 2 it is possible to fix the phase shifts to be zero due to the aforementioned symmetry. In addition, adding phase shifts allows us to consider row 2 in Table 1 as a special case of row 1. Finally, since the $k$’s are actually used as arguments for the functions in columns 4-6 of Table 1 rather than the $\lambda$’s, we use them to characterize Table 2. With these modifications, though we have introduced three new parameters, we have reduced the number of possible forms for $\mathbf{A}$ to 3.\
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We are almost able to put forth a set of basis functions for $\mathbf{A}$. However, we must first multiply $A_y$ and $A_z$ by an appropriate scale factor, determined by the fact that $\mathbf{B}$ must have zero curl and divergence. A set of basis functions for $\mathbf{A}$ is presented below, with the proper scale factors.

$$\mathbf{A} =
\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 & 