Problem 1:

A GaAlAs laser operating at 830 nm is pulsed and produces 10 ns long pulses.  It oscillates in 3 modes that are separated in frequency by c/2nl, where n = 3.5 and l = 200 mm.  Estimate the difference in pulse envelop speed for the highest and lowest frequency in the 11 km fiber for which the dispersion diagram is shown below.  If the pulse spreading is to be kept below 20%, what is the maximum fiber length?

Refer to the Web material.  As light traverses the optical fiber, different group of wavelengths travel at different speeds, which leads to chromatic dispersion. 
Chromatic dispersion is defined as the differential of the group velocity with respect to the wavelength l = l_free.
d(1/v_g)/dl = (d(1/v_g)/dw)(dw/dl) = (d²b/dw²)(dw/dl) = (d²b/dw²)(w/l)] which is usually given in units of ps/(nm-km) or ns/(nm-km).  
The dispersion diagram shows (d²b/dw²)(w/l)d in units of ps/nm with d = 11 km.
Note: (d²b/dw²)Dw = (d²b/dw²)w(Dw/w) = (d²b/dw²)w(Dl/l) = (d²b/dw²)(w/l)Dl.

Problem 2:

Investigate the properties of the gradient index rod (grinrod.len), an example "lens" included with OSLO LT.

Gradient index materials are specified in OSLO LT a variety of ways.  If the gradient is in the direction of the optical axis, it is an axial gradient.  If the gradient is perpendicular to the axis, it is a radial gradient.  If the gradient is spherically symmetrical about some point, it is a spherical gradient.  OSLO contains ray trace routines for handling ten different forms of gradient.  In each case, the gradient is given by a power-series expansion, similar to an aspheric surface.  If the standard gradient types are not sufficient, it is possible to specify a user-defined gradient.  The dispersion of gradient index materials is specified by giving values for the gradient coefficients at each of the defined wavelengths.

The gradient index rod is more like a fiber than a lens.  It is, however, an imaginary system, designed using OSLO without regard for actual refractive indices that can be manufactured.

The drawing above shows both on-axis and off axis beams.  Note that the off-axis beam for both the upper and lower rim rays is truncated by the edge of the rod.  In order for this to happen, you must turn on the Aperture check all GRIN segs general operating condition.  If you don’t, the rays will be free to propagate outside the boundary defined by the aperture on the surface for with the GRIN material is defined.

(a) Produce some graphs that display the properties of this lens more clearly than the default graph.

(b) Construct a rod with the same dimensions but made from glass with an index of refraction of 1.5 at the center of the rod and a parabolic index profile, n(r) = n₀ + nr1 r², with nr1 having the same value (nr1 = - 0.0999) as for the example lens.  Produce some graphs that display the properties of this lens.

(c) Shorten the rod to produce a 0.25 pitch and a 0.29 pitch GRIN lens.  Produce graphs for these lenses.

Problem 3:

Produce a model of a weakly-guiding optical fiber using OSLO LT.  Use the Clad Light Pipe (lpipe.len) as an example.  Produce a graph that shows rays for which the TIR condition is satisfied and rays for which it is not satisfied.