Homework for Chapter 4 4.1 We wish to fabricate a planar ...

Homework for Chapter 4
4.1 We wish to fabricate a planar waveguide for light of wavelength λ0 = 1.15 μm that will operate in
the single (fundamental) mode. If we use the proton bombardment,
carrier-concentration-reduction method to form a 3μm thick waveguide in GaAs, what are the
minimum and maximum allowable carrier concentrations in the substrate? (Calculate for the two
cases of p-type or n-type substrate material if that will result in different answers.)
4.2 If light of λ0 = 1.06 μm is used, what are the answers to Problem 4.1?
4.3 Compare the results of Problems 4.1 and 4.2 with those of Problems 2.1 and 2.2, note the unique
wavelength dependence of the characteristics of the carrier-concentration-reduction type
waveguide.
4.4 We wish to fabricate a planar waveguide for light of λ0 = 0.9 μm in Ga(1−x)AlxAs. It will be a
double layer structure on a GaAs substrate. The top (waveguiding) layer will be 3.0 μm thick
with the composition Ga0.9Al0.1As. The lower (confining) layer will be 10 μm thick and have the
composition Ga0.17Al0.83As. How many modes will this structure be capable of waveguiding?
4.5 What is the answer to Problem 4.4 if the wavelength is λ0 = 1.15 μm?
4.6 A planar asymmetric waveguide is fabricated by depositing a 2.0 μm thick layer of Ta2O5 (n =
2.09) onto a quartz substrate (n = 1.5).
(a) How many modes can this waveguide support for light of 632.8 nm (vacuum wavelength)?
(b) Approximately what angle does the ray representing the highest-order mode make with the
surface of the waveguide?
4.7 A three-layer planar waveguide structure consists of a 5 μm thick waveguiding layer of GaAs
with electron concentration N = 1× 10 14 cm −3 , covered on top with a confining layer of
Ga0.75Al0.25As with N = 1× 10 18 cm −3 , and on bottom with a confining layer of GaAs with N = 1×
10 18 cm −3 . If this waveguide is used to guide light of λ0 = 1.06 μm, how many modes can it
support?
4.8 For an asymmetric planar waveguide like that of Fig. 3.1 with n1 = 1.0, n2 = 1.65, n3 = 1.52, tg =
1.18 μm, and λ0 = 0.63 μm, find the number of allowed TE modes.
Fig. 3.1 Basic three-layer planar waveguide structure
4.9 If the waveguide of Prob. 4.4 were to be modified so as to support one additional mode, would it
be practical to produce the required change in n by using the electro-optic effect? (Assume a
uniform electric field to be applied over the thickness of the waveguide in the direction normal to
the surface, with r41 = 1.4× 10 −12 m/V).
4.10 Explain why the cutoff condition for a carrier-concentration-reduction type waveguide is

independent of wavelength.
4.11 We desire an epitaxially-grown double-heterojunction In(1-x)GaxAs(1-y)Py waveguide that will have
a bandgap Eg = 1.1 eV in the waveguiding layer at room temperature.
(a) If lattice-matched layers are desired, what should be the relative concentrations of the
constituent elements? i.e. What are x and y?
(b) What is the shortest wavelength that can be guided in this waveguide without excessive
interband absorption?
4.12 We wish to make an epitaxially grown waveguide in In(1-x)GaxAs(1-y)Py that has a bandgap of 0.9
eV at liquid nitrogen temperature (77 K). Find the values of x and y that are required.