Page 171 - Photonics Essentials an introduction with experiments

Lasers

Lasers 165

Substituting (7.19) into (7.18),
2 2
8 n f
k th
N th = · (7.19)
2
A 21 c g(f)
The spontaneous emission rate A 21 is determined by the inverse of
the spontaneous emission lifetime 21 . This lifetime can be measured
by exciting the laser material with a light pulse from an external
laser emitting photons with energy above the band gap of the semi-
conductor. The semiconductor will emit photoluminescence that dies
out with the spontaneous emission lifetime. Typical values for 21 are
several nanoseconds. Although this may seem short, it is easily meas-
ured with conventional equipment.
The threshold current density can be expressed by combining Eqs.
7.15, 7.16, and 7.19:
2 2
qtN th k th n f
21
J = = qt A-cm –2 (7.20)
th
2
r r c g(f)
You would prefer to have a lower threshold current. There are some
variables in this expression that are under the control of the laser de-
signer. The thickness of the recombination region can be reduced
physically. This was first done by making a heterostructure, and has
been developed into the currently used quantum well design, where
the recombination is restricted to a potential well of thickness compa-
rable to the de Broglie wavelength; that is, about 10 nm. The quan-
tum well laser design has a second equally important effect of lower-
ing the threshold current. This structure narrows the gain spectrum,
increasing g(f).
In a semiconductor laser having a band structure similar to that
shown in Fig. 7.6, the gain function can be adequately represented by
a Gaussian distribution. The value of the distribution at its maximum
value can be expressed in terms of its full width at half maximum
(usually abbreviated FWHM).
2
g(f max ) = (Gaussian gain distribution) (7.21)
f

The exact form of the gain distribution function is almost never
known. It can be adequately approximated by

1
g(f max ) (7.22)
f
The fundamental nature of stimulated emission dictates that the
laser will want to emit light whose frequency lies as close as possible

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