# Field form in the optic fibers from Maxwell's equations

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1. Jan 8, 2016

### 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?

2. Jan 13, 2016

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Jan 20, 2016

### EmilyRuck

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