Field form in the optic fibers from Maxwell's equations

[EmilyRuck]

Hello!
In this document a solution of Maxwell's equations in cylindrical coordinates is provided, in order to determine the electric and magnetic fields inside an optic fiber with a step-index variation. The interface between core and cladding is the cylindrical surface $r = a$.
For example, the most general solution (pp. 18-19) for $E_z$ is

$E_z (r, \phi) = \Phi(\phi) R(r) = \left[ K_1 \cos (\nu \phi) + K_2 \sin (\nu \phi) \right] \left[ A J_{\nu}(k_c r) + CY_{\nu}(k_c r) \right]$

In page 21 (17.70) though, just the sine variation is kept, putting (I think, arbitrarily) $K_1 = 0$ and $K_2 = 1$.

- Why should $K_2$ equal unity while $A$ is left unchanged?
- Is this a standard choice? Why is the cosine variation excluded? Is this just for convenience, or the contemporary presence of sine and cosine variation with respect to $\phi$ in both electric and magnatic fields can violate the continuity of the tangential components across the surface $r = a$?

Moreover, in page 22 it is pointed out that:

« In this case, where we have assumed the $sin(\nu \phi)$ for the electric field, we must have the $cos(\nu \phi)$ variation for the $H_z$ field to allow matching of the tangential fields (which include both $z$ and $\phi$ components) at the core/cladding boundary ».

- How can this be proved?

 

[EmilyRuck]

Do you have any further information, come to any new conclusions or is it possible to reword the post?

Not new conclusions, but: is there a textbook where similar computations are showed? Or another pdf from the web, with the same subject matter?
I need the derivation of the longitudinal fields for a step-index optic fiber, with the considerations about the choice of the $(r,\phi)$ dependence: when $\sin (\nu \phi)$ is to be chosen, or the $\cos (\nu \phi)$, etc.
Even if you can't suggest the proofs I'm looking for, could you provide some (maybe) useful link?
Thank you anyway,

Emily