Question: transcendental equation plot on complex plane

hello,everyone,
I have a problem to make a figure, commands are below:

restart;
1+(2*(S1*e2/(S2*e1)+S2*e1/(S1*e2)))*tanh(S2*k*d);
S1 = sqrt(beta^2-e1), S2 = sqrt(beta^2-e2);
e1 = 4, e2 = -19-.53*I;
subs(`%%`, %, k = 2*Pi/(0.633e-6), `%%%`);

I want to plot the relation between d and beta(real part named betar and imaginary part betai),but meet some trouble.
I got some points from the article is when d=400e-10, betar in the range (2.1,2.2) correspond to betai in the range(1e-3,1e-2) and another point is betar in the range (2.25,2.5) while betai in the range(1e-2,0.1). It means that there are double curves on the complex plane for beta.
d is a real number, its range is (0,1000e-10).
fsolve command can't give a right point. RootFinding[Analytic] command takes a long time so I stop it.

Any method or advice is appreciated.
thank you.