PHYSICS 415/416 SUPPLEMENTAL PROBLEMS
Part 11
11A. (Slab Waveguide)
Consider a symmetric, 3-layer slab waveguide. The core thickness is d and its index is n1. The clad indices
have the same value of n2.
(a) Prove that the effective index N of any of the guided modes obeys the relationship n2 < N < n1.
(b) Show that the dispersion equation for TE light can be written in the form
 b 

V 1  b  m  2 tan 1 

1

b


N 2  n22
.
n12  n22
(c) Plot b vs. V for the fundamental mode (m = 0). To do this, let b range from 0 to 1 and calculate the
corresponding V values. Then plot the values. You should get a curve like the one in the hand-out for m = 0
and a = 0.
where V  dk o n12  n22 and b 
11B. (Slab Waveguide)
Consider 632.8 nm light traveling in a plastic guide where the core is 0.5 m thick. The index of the plastic
core is 1.72 while that of the plastic clad is 1.62.
(a) Find the effective index and the zig-zag angle of the zeroth mode. (Use your dispersion plot from Problem
11A or the “Effective Index and Thickness Curves” worksheet that is posted on the course page.)
(b) Can this waveguide support higher modes? If not, explain why! If so, how many more modes?
(c) Find the effective thickness of this guide for the m=0 mode. How much wider is it than the actual
thickness?
(d) What would be the optimum thickness of this core so that the effective thickness is minimized? What is the
value of this minimum effective thickness? Find the effective index of the mode in this case.
(e) Compare the values of d, deff, and N found in (d) to the actual waveguide values.
2
11C. (Laser Waveguide and Fiber)
Consider a 500 m long semiconductor, quantum well laser which emits 867 nm light. The GaAs quantum well
is sandwiched between two layers of Al.3Ga.7As which act as the core of a waveguide. Each of these layers is
0.175 m thick. On either side of these layers are thick Al.55Ga.45As cladding layers. The refractive indices of
the core layers and clad layers are 3.42 and 3.28, respectively. The GaAs quantum well is so thin that it can be
ignored when analyzing the waveguide properties of the laser. The light leaves the end facet of the laser with a
wide divergence angle. We can assume that the waist of the beam is at the center of the laser and is a point
source. Measured from this point, the light leaves the laser as a cone with a full cone-angle of 66. The light is
to be coupled into a step-index optical fiber with a 40x microscope objective. The objective is placed 10 mm
from the output facet of the laser. The fiber has a core index of 1.481 and a clad index of 1.462 at 867 nm. The
core is 10 m in diameter.
(a)
(b)
(c)
(d)
Find the effective index of the laser structure.
How fast does the light travel in the laser structure?
Find the numerical aperture of the fiber.
How far from the objective lens should the fiber end be placed so that the light is focused to a “point” at the
center of the core face? Use the convention that mag m = (16 cm)/f for a microscope objective.
(e) Neglecting reflection loss, will all of the light in (d) be coupled into guided modes? Give evidence for your
answer.
(f) What is the largest distance that the objective lens can be placed from the laser facet so that all of the
focused light is coupled into a guided mode? Again, neglect reflection loss. How far should the fiber be
from the objective in this case if we want the light focused to a “point” at the center of the core face?
(g) How many modes will this fiber support?
11D. (Waveguiding in Semiconductor Lasers)
Another laser is made that is identical to the laser in Problem 11C except that each core layer is increased from
0.175 m to 0.275 m by mistake. (The guy who was growing the laser made a programming error.) A
waveguide analysis shows that this structure will support two modes at 867 nm.
(a) Find the effective indices of the two modes.
(b) Which mode will propagate faster along the laser cavity?
(c) Find the effective thicknesses of the waveguide for each mode. Which mode penetrates more into the clads?
(d) Plot the TE intensity profiles of the two modes on the same graph. Let the peak intensity in the core be
unity for each mode. Plot the profiles from the core center to 0.75 m into each cladding layer.
(e) Comment on the differences between the two mode profiles.
11E. (Pulse Broadening in Fiber)
Do Parts (a) and (b) of Problem 10-24. Assume that this is a single mode fiber so that modal distortion is
eliminated. Notice that in (b), the two wavelengths chosen are the ones that give zero material dispersion (1.27
m) or minimum absorption (1.55 m) in a silica fiber.
(c) If the 1.27 m, 1 nm linewidth source is used, then material dispersion is eliminated (see Fig. 10-13) and the
only pulse-broadening mechanism is waveguide dispersion. Find the total pulse broadening () that would
occur for such a system for 1 km of fiber and then find the corresponding maximum bit rate for 1 km.
(d) Let’s replace the source with the1.55 m, 1 nm linewidth source so that fiber absorption is minimized.
However, material dispersion is not eliminated now. Using Fig. 10-13, find the amount of material dispersion
present. For 1 km of fiber, find the pulse broadening due to both material dispersion (mat) and waveguide
dispersion (wg). The net pulse-broadening is given by   (mat ) 2  (wg ) 2 . Find this net broadening
and the corresponding maximum bit rate for 1 km of fiber.
(e) Compare the results. Does shifting to 1.55 m significantly affect the pulse-broadening and bit rate?