- #1
EmilyRuck
- 136
- 6
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 [itex]r = a[/itex].
For example, the most general solution (pp. 18-19) for [itex]E_z[/itex] is
[itex]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][/itex]
In page 21 (17.70) though, just the sine variation is kept, putting (I think, arbitrarily) [itex]K_1 = 0[/itex] and [itex]K_2 = 1[/itex].
- Why should [itex]K_2[/itex] equal unity while [itex]A[/itex] 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 [itex]\phi[/itex] in both electric and magnatic fields can violate the continuity of the tangential components across the surface [itex]r = a[/itex]?
Moreover, in page 22 it is pointed out that:
« In this case, where we have assumed the [itex]sin(\nu \phi)[/itex] for the electric field, we must have the [itex]cos(\nu \phi)[/itex] variation for the [itex]H_z[/itex] field to allow matching of the tangential fields (which include both [itex]z[/itex] and [itex]\phi[/itex] components) at the core/cladding boundary ».
- How can this be proved?
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 [itex]r = a[/itex].
For example, the most general solution (pp. 18-19) for [itex]E_z[/itex] is
[itex]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][/itex]
In page 21 (17.70) though, just the sine variation is kept, putting (I think, arbitrarily) [itex]K_1 = 0[/itex] and [itex]K_2 = 1[/itex].
- Why should [itex]K_2[/itex] equal unity while [itex]A[/itex] 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 [itex]\phi[/itex] in both electric and magnatic fields can violate the continuity of the tangential components across the surface [itex]r = a[/itex]?
Moreover, in page 22 it is pointed out that:
« In this case, where we have assumed the [itex]sin(\nu \phi)[/itex] for the electric field, we must have the [itex]cos(\nu \phi)[/itex] variation for the [itex]H_z[/itex] field to allow matching of the tangential fields (which include both [itex]z[/itex] and [itex]\phi[/itex] components) at the core/cladding boundary ».
- How can this be proved?