Page 170 - Photonics Essentials an introduction with experiments

Lasers

164 Photonic Devices

7.6 Threshold—Going Over the Edge
You are sitting at the lab bench. The laser is mounted in a test socket,
and you are ready to increase the forward current. The question you
would really like to answer before beginning the test is, “How much
current will I have to supply in order for the laser to reach threshold?”
The answer is that the threshold current is attained when the num-
ber of electrons per second being injected into the diode is equal to the
threshold population density, taking into account that some of the
electrons will be lost to recombination before a suitable population in-
version is built up.
The number of electrons injected per second per square cm into the
diode is just the current density divided by the electronic charge: J/q.
If we consider the rate of electrons per second in the recombination re-
gion, we need to divide this expression by the thickness t (= w in
Figs. 7.8 and 7.9) of the recombination region:
J
Pumping rate = (electrons-sec -cm ) (7.15)
–3
–1
qt
What goes in must come out, so to speak, and so the recombination
rate must equal the pumping rate. This is the optoelectronic equiva-
lent of the principle that absorption must be equal to emission. The
recombination rate is the population inversion necessary to achieve
threshold divided by the recombination rate:
N th
Recombination rate = (electrons-sec -cm ) (7.16)
–1
–3
r
We have already developed an expression for the population inver-
sion in Eq. 7.10.
nhfg(f)
k th = (N 2 – N 1 ) th = B 21
c
Therefore,
c
k th
N th = (N 2 – N 1 ) th = · (7.17)
nhfg(f)
B 21
where B 21 is the stimulated emission coefficient. In Eq. 7.4, we related
B 21 to the spontaneous recombination rate. This is a useful relation-
ship to know because you can measure this rate directly:
3
8 n hf 3
A 21 = · B 21
c 3
c 3
B 21 = A 21 (7.18)
3
8 n hf 3

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