Classical Electrodynamics · Problem 8.14

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · graded-index-optical-fiber-ray-tracing

Problem

Consider an optical fiber with a graded index of refraction for rays confined to the

x

z

plane,

n(x) = n(0)\operatorname{sech}(\alpha x)

. The fiber has large enough transverse dimensions (

x

) to contain all rays of interest, which are evidently symmetric about

x = 0

. The invariant

\bar{n} = n(x_{\max}) = n(0)\cos\theta(0)

. (a) Solve the eikonal equations for the transverse coordinate

x(z)

of the ray and show that

\alpha x = \sinh^{-1}[\sinh(\alpha x_{\max})\sin(\alpha z)]

where the origin in

z

is chosen when the ray has

x = 0

. Sketch rays over one half-period for ``launch angles''

\theta(0) = \pi/6

\pi/4

, and

\pi/3

. (b) Find the half-period

Z

of the ray. Does it depend on

\bar{n}

? (c) Show that the optical path length for a half period,

L_{\text{opt}} = \int n(x)\, ds

L_{\text{opt}} = n(0)Z

. Comment on the effectiveness of this particular grading of the index. Hint: In the computation of

L_{\text{opt}}

, a useful change of variables is

\sinh(\alpha x) = \sinh(\alpha x_{\max})\sin t

. The resulting integral can be done by contour integration:

\int_0^{\pi/2}\frac{d\theta}{1 + a^2\sin^2\theta} = \frac{\pi}{2\sqrt{1+a^2}}