Page 192 - Photonics Essentials an introduction with experiments

Direct Modulation of Laser Diodes

186 Advanced Topics

threshold is to set in motion this cycle, which leads quite naturally to
oscillations in both the carrier density and the photon density. In this
section, we will make an estimate of the two important parameters
that define this dynamic process: the frequency of the self-pulsations
or relaxation oscillations, and the decay time of these oscillations.
The approach we will use will be to decouple the equations to the
extent that we obtain a single equation that shows how the nonequi-
librium carrier concentration changes with time. To do this we will
have to make some approximations in order to discard some terms
that are smaller than others. Since the objective is not to solve for the
time dependence of the carrier concentrations, the errors introduced
by these approximations do not play an important role in the result
we are seeking.
Start with the photon equation:

dN N – spont N A 21
= +B 21 N · f·K th – 2 (8.11)
dt
We presume that the fraction of spontaneous emission in the laser
mode is so small it can be neglected.
Next we separate the photon density into a constant term plus a
small change:

N = n + N
Finally, we parametrize the gain coefficient :

K th = a N (8.12)
With these conditions, we can write
d n
N B 21 · f·a · N · n + (8.13)
dt
where the remaining terms are small by comparison and will be neg-
lected. This equation is solved for N:
1 d 1
N = N + (8.14)
B 21 an dt
The equation for the carrier density is

dN J N
= – B 21 N · f·K th – (8.15)
dt qd r
By substituting N = n + N and N = n + N , this equation becomes
d N
N = –B 21 fK th N – (8.16)
dt r

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)
Copyright © 2004 The McGraw-Hill Companies. All rights reserved.
Any use is subject to the Terms of Use as given at the website.