670
23 Electricity-to-Light Conversion
where α 0 is the loss coefficient in the active medium and α g is the loss coefficient outside the active
medium.
The mirror loss is due to the incomplete reflection of the optical wave at the laser facets. This
condition is necessary, however, to observe a laser beam outside the cavity. R 1 and R 2 are the values
of reflectance of the two facets, respectively. An as-cleaved facet has a reflectance of about 30% (cf.
(23.8)). Using dielectric layers on the facets, the reflectance can be increased (high reflection, HRcoating) or decreased (antireflection, AR-coating). One round-trip through the cavity of length L has
the length 2L. The intensity loss due to reflection at the facets is expressed via exp(−2α m L) = R 1 R 2
α m =
1
2L
ln
1
R 1 R 2
.
(23.19)
If both mirrors have the same reflectance R, we have α m = −L
−1 ln R. For R = 0.3 a 1-mm cavity
has a loss of 12 cm
−1 . For the internal loss a typical value is 10 cm
−1 , very good waveguides go down
to 1–2 cm
−1 .
Lasing is only possible if the gain overcomes all losses (at least for one wavelength), i.e.
g mod = g mat ≥ α tot .
(23.20)
23.4.6 Threshold
When the laser reaches threshold, the (material) gain is pinned at the threshold value
g thr =
α i + α m
.
(23.21)
Since g ∝ n, the carrier density is also pinned at its threshold value and does not increase further
with increasing injection current. Instead, additional carriers are quickly converted into photons by
stimulated emission. The threshold carrier density is (using the linear gain model, cf. (23.13))
n thr = n tr +
α i + α m
ˆ
α α
.
(23.22)
For an active layer of thickness d, the threshold current density is
j thr ∼ =
e d n thr
τ (n thr )
,
(23.23)
where τ (n thr ) is the (minority) carrier lifetime at the threshold carrier density (cmp. Sect. 10.10) from
τ (n) =
1
A + B n + C n 2 .
(23.24)
Using (23.22), we can write (for R = R 1 = R 2 )
j thr = j tr +
e d
τ ˆ
α α
α i −
1
L
ln R
,
(23.25)
23 Electricity-to-Light Conversion
where α 0 is the loss coefficient in the active medium and α g is the loss coefficient outside the active
medium.
The mirror loss is due to the incomplete reflection of the optical wave at the laser facets. This
condition is necessary, however, to observe a laser beam outside the cavity. R 1 and R 2 are the values
of reflectance of the two facets, respectively. An as-cleaved facet has a reflectance of about 30% (cf.
(23.8)). Using dielectric layers on the facets, the reflectance can be increased (high reflection, HRcoating) or decreased (antireflection, AR-coating). One round-trip through the cavity of length L has
the length 2L. The intensity loss due to reflection at the facets is expressed via exp(−2α m L) = R 1 R 2
α m =
1
2L
ln
1
R 1 R 2
.
(23.19)
If both mirrors have the same reflectance R, we have α m = −L
−1 ln R. For R = 0.3 a 1-mm cavity
has a loss of 12 cm
−1 . For the internal loss a typical value is 10 cm
−1 , very good waveguides go down
to 1–2 cm
−1 .
Lasing is only possible if the gain overcomes all losses (at least for one wavelength), i.e.
g mod = g mat ≥ α tot .
(23.20)
23.4.6 Threshold
When the laser reaches threshold, the (material) gain is pinned at the threshold value
g thr =
α i + α m
.
(23.21)
Since g ∝ n, the carrier density is also pinned at its threshold value and does not increase further
with increasing injection current. Instead, additional carriers are quickly converted into photons by
stimulated emission. The threshold carrier density is (using the linear gain model, cf. (23.13))
n thr = n tr +
α i + α m
ˆ
α α
.
(23.22)
For an active layer of thickness d, the threshold current density is
j thr ∼ =
e d n thr
τ (n thr )
,
(23.23)
where τ (n thr ) is the (minority) carrier lifetime at the threshold carrier density (cmp. Sect. 10.10) from
τ (n) =
1
A + B n + C n 2 .
(23.24)
Using (23.22), we can write (for R = R 1 = R 2 )
j thr = j tr +
e d
τ ˆ
α α
α i −
1
L
ln R
,
(23.25)