Thus, we are expressing the z dependence of $\mathbf{A}$ (and thus $\mathbf{B}$) as a sum of sine’s and cosine’s. Though we may have imposed this restriction on $\lambda_x$ or $\lambda_y$ it is useful to choose $\lambda_z$ for LGB fields, since qualitatively the dependence of $\mathbf{B}$ on z looks (co)sinusoidal (see Figures 1(b) and 2(b)).

Through an additional simplification, we may combine the cases in which $\lambda_x = 0$ and $\lambda_y = 0$ with the cases in which $\lambda_x< 0$ and $\lambda_y > 0$. In the cases in which separation constants are zero, the ODE’s in equation (2) yield solutions of the form $ax + b$. First, we need not consider $b$, since its presence does not affect the curl of $\mathbf{A}$. Furthermore, using the approximation that $\sinh(ku) \approx ku$ for small $k$, we may effectively consider linear solutions to the ODE’s as special case of hyperbolic trigonometric solutions. This a valid simplification because if we ever require such a linear solution, in the process of fitting the model to our data we may achieve arbitrary accuracy by choosing an arbitrarily small value for $k$. In this instance, only being arbitrarily accurate rather than exact is allowable, since we will be 