182
4 Light Sources for Fiber Links
spontaneous emission into the lasing mode (which is much smaller than the total
spontaneous-emission rate), τ ph is the photon lifetime, τ sp is the spontaneousrecombination lifetime, and J is the injection-current density.
Equations (4.30) and (4.31) may be balanced by considering all the factors that
affect the number of carriers in the laser cavity. The first term in Eq. (4.30) is a source
of photons resulting from stimulated emission. The second term, describing the
number of photons produced by spontaneous emission, is relatively small compared
with the first term. The third term in Eq. (4.30) indicates the decay in the number
of photons caused by loss mechanisms in the lasing cavity. In Eq. (4.31), the first
term represents the increase in the electron concentration in the conduction band as
current flows into the device. The second and third terms give the number of electrons
lost from the conduction band owing to spontaneous and stimulated transitions,
respectively.
Solving these two equations for a steady-state condition will yield an expression
for the output power. The steady state is characterized when the left-hand sides of
Eq. (4.30) and Eq. (4.31) are equal to zero. First, from Eq. (4.30), assuming R sp is
negligible and noting that d/dt must be positive when is small, it follows that
Cn −
1
τ ph
≥ 0
(4.32)
This shows that n must exceed a threshold value n th in order for to increase.
Using Eq. (4.31), this threshold value can be expressed in terms of the threshold
current J th needed to maintain an inversion level n = n th in the steady state when the
number of photons = 0:
n th
τ sp
=
J th
qd
(4.33)
This expression defines the current required to sustain an excess electron density
in the laser when spontaneous emission is the only decay mechanism.
Next, consider the photon and electron rate equations in the steady-state condition
at the lasing threshold. Respectively, Eqs. (4.30) and (4.31) become
0 = Cn th s + R sp −
s
τ ph
= stimulated emission + spontaneous emission + photon loss
(4.34)
and
0 =
J
qd
−
n th
τ sp
− Cn th s
= injection + spontaneous recombination + stimulated emission
(4.35)
4 Light Sources for Fiber Links
spontaneous emission into the lasing mode (which is much smaller than the total
spontaneous-emission rate), τ ph is the photon lifetime, τ sp is the spontaneousrecombination lifetime, and J is the injection-current density.
Equations (4.30) and (4.31) may be balanced by considering all the factors that
affect the number of carriers in the laser cavity. The first term in Eq. (4.30) is a source
of photons resulting from stimulated emission. The second term, describing the
number of photons produced by spontaneous emission, is relatively small compared
with the first term. The third term in Eq. (4.30) indicates the decay in the number
of photons caused by loss mechanisms in the lasing cavity. In Eq. (4.31), the first
term represents the increase in the electron concentration in the conduction band as
current flows into the device. The second and third terms give the number of electrons
lost from the conduction band owing to spontaneous and stimulated transitions,
respectively.
Solving these two equations for a steady-state condition will yield an expression
for the output power. The steady state is characterized when the left-hand sides of
Eq. (4.30) and Eq. (4.31) are equal to zero. First, from Eq. (4.30), assuming R sp is
negligible and noting that d/dt must be positive when is small, it follows that
Cn −
1
τ ph
≥ 0
(4.32)
This shows that n must exceed a threshold value n th in order for to increase.
Using Eq. (4.31), this threshold value can be expressed in terms of the threshold
current J th needed to maintain an inversion level n = n th in the steady state when the
number of photons = 0:
n th
τ sp
=
J th
qd
(4.33)
This expression defines the current required to sustain an excess electron density
in the laser when spontaneous emission is the only decay mechanism.
Next, consider the photon and electron rate equations in the steady-state condition
at the lasing threshold. Respectively, Eqs. (4.30) and (4.31) become
0 = Cn th s + R sp −
s
τ ph
= stimulated emission + spontaneous emission + photon loss
(4.34)
and
0 =
J
qd
−
n th
τ sp
− Cn th s
= injection + spontaneous recombination + stimulated emission
(4.35)