Spring 2011 Dr. Andrew W. Poon
ELEC 509 Advanced Photonics Technologies
Department of Electronic and Computer Engineering, HKUST
Problem Set 5 (due May 18 Wed 5pm, room 2529)
1.
Explain why in an indirect-gap semiconductor both direct absorption and indirect absorption are possible but direct recombination is highly improbable. Under what condition does direct absorption take place in an indirect-gap semiconductor? What is the difference between a direct-absorption process that takes place in an indirect-gap semiconductor and one that takes place in a directgap semiconductor?
2.
Use the relation for the joint density of states

(

) and that for the absorption coefficient in thermal equilibrium

0
(

) to find and plot the absorption coefficient of intrinsic GaAs in thermal equilibrium at 300 K as a function of photon energy over a range of photon energies from 1.424 to 1.8 eV, corresponding to optical wavelengths from 871 to 689 nm. Take spontaneous time constant
 sp
= 500 ps and refractive index n = 3.65 over the entire range considered here.
3.
In this problem, we consider the gain bandwidth of a GaAs sample that is injected with excess electron-hole pairs of a concentration of N = 2.83

10
24
m
-3
at 300 K.
Take spontaneous time constant
 sp
= 500 ps for GaAs at 300 K. For simplicity, ignore wavelength-dependent variations of the refractive index by taking n = 3.65 over the wavelength range of interest. (see example 13.4 of Photonic Devices,
Jia-Ming Liu) a.
Find the spectral range for which the gain coefficient (cm
-1
) g > 0 by finding the two photon energies for which g = 0 at the two ends of this range. What is the gain bandwidth? What is the wavelength range covered by the gain spectrum? b.
Find and plot the gain coefficient g(

) as a function of photon energy over the gain spectral range found in (a). At what photon energy and corresponding optical wavelength does the gain peak occur? c.
What is the gain cross section at the gain peak?
4.
a. What are the two most important considerations in lowering the threshold and in improving the efficiency of a semiconductor laser? What are the basic strategies used in the structural design of semiconductor lasers to address these two issues? (see discussion in Photonic Devices, Jia-Ming Liu, Chapter 13) b. Why do most edge-emitting LEDs and semiconductor lasers have stripe geometry for their lateral structures? Why is index-guiding stripe geometry used for almost all practical semiconductor lasers? When is gain-guiding stripe geometry used for a semiconductor laser? (see discussion in Photonic Devices,
Jia-Ming Liu, Chapter 13)
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c. Why are most LEDs surface-emitting devices? In what situation are edgeemitting LEDs used? (see discussion in Photonic Devices, Jia-Ming Liu, Chapter
13)
5. An AlGaInP LED with a flat surface is to be encapsulated in a plastic epoxy that has a refractive index of 1.5. Take the refractive index of AlGaInP to be 3.4 and approximate the transmittance T in all cases with that of normal incidence. The critical angle, the solid angle
 esc
of the escape cone, and the escape efficiency
 esc
before encapsulation when the LED surface is exposed to the air are already found in Example 13.11 of Photonic Devices, Jia-Ming Liu’s book. a. After the LED is encapsulated in the plastic epoxy, what is the critical angle between the LED and the epoxy? Show that if the epoxy has a slab geometry with a flat epoxy/air interface parallel to the LED/epoxy interface, there is no improvement on
 esc
for the emission to enter the air with or without the epoxy encapsulation. However, there is still some improvement on
 esc
. How much is this improvement? b. If the epoxy/air interface has a spherical dome shape, what is
 esc the LED to the air through the epoxy encapsulation? Find
 esc
from inside
in this case by taking into account reflections at all interfaces.
6. Find the following performance parameters for the VCSEL considered in
Example 13.17 of Photonic Devices, Jia-Ming Liu, by using the results obtained in Examples 13.18-13.21 when it operates at three times the threshold current with I = 3I th
: a. the output power P out
, b. the efficiencies
 e
(external quantum efficiency),
 c
(power conversion efficiency) and c. the relaxation rates
 rate),
 n
 s
(slope efficiency), c
(cavity decay rate),
 s
(spontaneous carrier relaxation
(differential carrier relaxation rate) and
 p
(nonlinear carrier relaxation rate), d. the parameters
 r
(total carrier relaxation rate), f r
(relaxation resonance frequency), K (K factor), f pk
(resonance peak frequency) and f
3dB
(3-dB modulation bandwidth) for its frequency response.
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