Delay distortion in generalized lens-like media

Explicit expressions are derived for the phase constant, the specific-group-delay constant, and the rms width of the impulse response for two-dimensional or square media having a transverse variation of index of refraction according to n = n
1
(1 − 1/2a
u
x
^u
− 1/2a
r
x
^r
), in which x is the transverse dimension, a
u
and a
r
are constants with |a
r
| ≪ a
u
, and (n-n
1
) ≪ 1. Use is made of an approximation which the author has previously shown yields significant results. The results are applied to fibers with graded-index variation, clad by an additional medium of index n = n
1
(1 − δ). The ideal index gradient, a near-parabolic profile, gives delay distortion orders of magnitude less than for the conventional fiber with a step-change in index at the core-cladding boundary. However, it is shown that several forms of 5-percent error in the ideal gradient yield improvement of the order of 50 compared with the conventional clad fiber. The delay distortion is shown to be very sensitive to the exact index distribution in the vicinity of the ideal distribution but increasingly insensitive to perturbations in the index distribution as that distribution departs more and more from the ideal.