kappa = sqrt((n_s^2*w^2/c^2)-beta^2);
gamma = sqrt(beta^2-(n_s^2*w^2/c^2));
the expressions within the sqrt() are the negative of each other
eqn = tan(kappa*d) == -kappa/gamma;
When beta^2 > (n_s^2*w^2/c^2) then -kappa/gamma --> -1i . When beta^2 < (n_s^2*w^2/c^2) then -kappa/gamma --> 1i . When beta^2 == (n_s^2*w^2/c^2) then you have a singularity.
So you are attempting to find a beta such that tan(kappa*d) = 1i or tan(kappa*d) = -1i depending on the range of beta.
If you proceed by way of taking arctan of both sides, then providing you stay within the primary range, you get
kappa*d == atan(-1i)
%or
kappa*d == atan(1i)
but the arctan of 1i and -1i are complex infinities.
The only way to get a complex infinity out of kappa is for beta to be infinite. However, if you substitute everything in then for the left hand side, tan(kappa*d) for infinite beta, you get 1i and for the right hand side of -kappa/gamma you get -1i .
Therefore there is no solution.
