126 CHAPTER 3. OPTICAL TRANSMITTERS
is used to hold the laser and fiber in place. Coupling stability in this case depends
on how epoxy changes with aging of the transmitter. In the lens-coupling scheme,
laser welding is used to hold various parts of the assembly together. The laser package
becomes a part of the transmitter package, which includes other electrical components
associated with the driving circuit. The choice of transmitter package depends on the
type of application; a dual-in-line package or a butterfly housing with multiple pins is
typically used.
Testing and packaging of optical transmitters are two important parts of the manu-
facturing process [149], and both of them add considerably to the cost of a transmitter.
The development of low-cost packaged transmitters is necessary, especially for local-
area and local-loop applications.
Problems
3.1 Show that the external quantum efficiency of a planar LED is given approx-
imately byηext=n−^1 (n+ 1 )−^2 , wherenis the refractive index of the semi-
conductor–air interface. Consider Fresnel reflection and total internal reflection
at the output facet. Assume that the internal radiation is uniform in all directions.
3.2 Prove that the 3-dB optical bandwidth of a LED is related to the 3-dB electrical
bandwidth by the relationf3dB(optical)=√
3 f3dB(electrical).
3.3 Find the composition of the quaternary alloy InGaAsP for making semiconductor
lasers operating at 1.3- and 1.55-μm wavelengths.
3.4 The active region of a 1.3-μm InGaAsP laser is 250μm long. Find the active-
region gain required for the laser to reach threshold. Assume that the internal
loss is 30 cm−^1 , the mode index is 3.3, and the confinement factor is 0.4.
3.5 Derive the eigenvalue equation for the transverse-electric (TE) modes of a pla-
nar waveguide of thicknessdand refractive indexn 1 sandwiched between two
cladding layers of refractive indexn 2 .(Hint: Follow the method of Section 2.2.2
using Cartesian coordinates.)
3.6 Use the result of Problem 3.5 to find the single-mode condition. Use this condi-
tion to find the maximum allowed thickness of the active layer for a 1.3-μm semi-
conductor laser. How does this value change if the laser operates at 1.55μm?
Assumen 1 = 3 .5 andn 2 = 3 .2.
3.7 Solve the rate equations in the steady state and obtain the analytic expressions for
PandNas a function of the injection currentI. Neglect spontaneous emission
for simplicity.
3.8 A semiconductor laser is operating continuously at a certain current. Its output
power changes slightly because of a transient current fluctuation. Show that the
laser power will attain its original value through an oscillatory approach. Obtain
the frequency and the damping time of such relaxation oscillations.
3.9 A 250-μm-long InGaAsP laser has an internal loss of 40 cm−^1. It operates in
a single mode with the modal index 3.3 and the group index 3.4. Calculate the