- #1
EmilyRuck
- 134
- 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 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?
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?