23.4 Lasers
671
(a)
(b)
Fig. 23.45 a Threshold current density for (three-fold InGaAs/GaAs QD stack) laser (λ = 1150 nm) at 10 ◦ C with
different cavity length vs. the optical loss (∝ 1/L). The extrapolated transparency current density is 21.5 ±0.9 A/cm 2 .
b Inverse external quantum efficiency versus cavity length. The internal quantum efficiency determined from the plot is
91% and the internal loss is 1.4 cm −1
where the transparency current density is j tr = e d n tr /τ . Thus, the plot of j thr versus 1/L (or the
optical loss) should be linear and can be extrapolated towards the transparency current density (cf.
Fig. 23.45a).
Any additional increase of the current j leads to stimulated emission with the rate
r st = d v g g thr N ph ,
(23.26)
where v g is the group velocity (mostly c 0 /n r ) and N ph is the photon density (per length) in the cavity.
The photon density increases linearly beyond the threshold
N ph =
1
e d v g g thr
( j − j thr ) .
(23.27)
The photon lifetime
1
τ ph
= v g (α i + α m ) = v g g thr
(23.28)
is introduced that describes the loss rate of photons. v g α m describes the escape rate of photons from
the cavity into the laser beam(s). Therefore,
N ph =
τ ph
e d
( j − j thr ) .
(23.29)
Since the threshold depends on the carrier density, it is advantageous to reduce the active volume
further and further. In this way, the same amount of injected carriers creates a larger carrier density.
Figure 23.46 shows the historic development of laser threshold due to design improvements.
23.4.7 Spontaneous Emission Factor
The spontaneous emission factor β is the fraction of spontaneous emission (emitted into all angles)
that is emitted into laser modes. For Fabry–Perot lasers, β is typically in the order of 10
−4 –10
−5 . The
design of a microcavity can increase β drastically by several orders of magnitude to ≈ 0.1 [1989] or
above and thus reduce the threshold current. The photon number as a function of the pump current can
671
(a)
(b)
Fig. 23.45 a Threshold current density for (three-fold InGaAs/GaAs QD stack) laser (λ = 1150 nm) at 10 ◦ C with
different cavity length vs. the optical loss (∝ 1/L). The extrapolated transparency current density is 21.5 ±0.9 A/cm 2 .
b Inverse external quantum efficiency versus cavity length. The internal quantum efficiency determined from the plot is
91% and the internal loss is 1.4 cm −1
where the transparency current density is j tr = e d n tr /τ . Thus, the plot of j thr versus 1/L (or the
optical loss) should be linear and can be extrapolated towards the transparency current density (cf.
Fig. 23.45a).
Any additional increase of the current j leads to stimulated emission with the rate
r st = d v g g thr N ph ,
(23.26)
where v g is the group velocity (mostly c 0 /n r ) and N ph is the photon density (per length) in the cavity.
The photon density increases linearly beyond the threshold
N ph =
1
e d v g g thr
( j − j thr ) .
(23.27)
The photon lifetime
1
τ ph
= v g (α i + α m ) = v g g thr
(23.28)
is introduced that describes the loss rate of photons. v g α m describes the escape rate of photons from
the cavity into the laser beam(s). Therefore,
N ph =
τ ph
e d
( j − j thr ) .
(23.29)
Since the threshold depends on the carrier density, it is advantageous to reduce the active volume
further and further. In this way, the same amount of injected carriers creates a larger carrier density.
Figure 23.46 shows the historic development of laser threshold due to design improvements.
23.4.7 Spontaneous Emission Factor
The spontaneous emission factor β is the fraction of spontaneous emission (emitted into all angles)
that is emitted into laser modes. For Fabry–Perot lasers, β is typically in the order of 10
−4 –10
−5 . The
design of a microcavity can increase β drastically by several orders of magnitude to ≈ 0.1 [1989] or
above and thus reduce the threshold current. The photon number as a function of the pump current can