Empty Sym: 0-by-1 for Solve

Hi there,

I'm not sure why my code returns an empty Sym:

syms beta
beta = sym('beta', 'real');

%Constants
d = 2e-6;
n_s = 1.5;
n_c = 1;
es = n_s^2;
ec = n_c^2;
c = 3e8;
lambda = 1.5e-6;
f = c/lambda;
w = 2*pi*f;

kappa = sqrt((n_s^2*w^2/c^2)-beta^2);
gamma = sqrt(beta^2-(n_s^2*w^2/c^2));

eqn = tan(kappa*d) == -kappa/gamma;

S = vpasolve(eqn, beta)

 
S =
 
Empty sym: 0-by-1
 

Any advice would be greatly appreciated.

Answer 1

[Walter Roberson]

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.

[Tom Field]

Hmm, interesting. Thank you very much