Page 193 - Photonics Essentials an introduction with experiments

Direct Modulation of Laser Diodes

Direct Modulation of Laser Diodes 187

Now insert the expression for N in this equation:
dn 1 dn 1 1 dn 1
1
N + = – N +
dt B 21 fan dt r B 21 fan dt
– B 21 fK th N
d 2 1 d 1
2
2
N + N + B 21 f aK th n N + = 0 (8.17)
dt 2 r dt r
This is a second order differential equation that describes a damped
oscillation with an angular frequency:
2
2 n
R = B 21 f aK th (8.18)
and a decay time of 2 r .
The solution will be of the form:

N (t) ~ e –t/2 r sin( R t + ) (8.19)
We have obtained some results that we would like to use to direct
modulation of semiconductor lasers in communications applications:

1. The relaxation oscillation dies out in a time ~ 2 r . This would put a
limit on the bit rate, which must be low enough to allow the optical
output power to come to steady state. A typical value for the free
carrier recombination time in GaAs is about 2 nanoseconds. If the
relaxation oscillation dies out in 5 nanoseconds, the corresponding
–9
modulation bandwidth would be estimated at f = 1/(5 × 10 ) =
60 MHz. However, modulation rates of 10 GHz in semiconductor
lasers can be obtained experimentally. This would imply a much
shorter carrier lifetime, on the order of 30 picoseconds. Such a com-
parison suggests that the recombination time is not constant, but
in fact depends strongly on the injection rate. This is an under-
standable result. Photon emission must be balanced with the
pumping rate. So, the recombination rate must increase in order to
maintain equilibrium at high carrier injection rates.
2. In order to reach higher modulation rates you would want to push
the relaxation oscillation frequency well above the modulation
rate. The relaxation oscillation frequency will depend on the stimu-
lated emission rate B 21 and the band gap of the material (E g = hf).
Whereas the band gap will not change, there is no physical reason
why the stimulated emission rate could not increase as the photon

density increases. The frequency also depends through n and K th
on the amount by which the laser is driven beyond threshold in or-
der to send a “1.”

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