3.3. SEMICONDUCTOR LASERS 95
Figure 3.10: Structure of a semiconductor laser and the Fabry–Perot cavity associated with it.
The cleaved facets act as partially reflecting mirrors.
only when the laser is pumped above a threshold level. The current needed to reach the
threshold is called thethreshold current.
A simple way to obtain the threshold condition is to study how the amplitude of
a plane wave changes during one round trip. Consider a plane wave of amplitude
E 0 , frequencyω, and wave numberk=nω/c. During one round trip, its amplitude
increases by exp[(g/ 2 )( 2 L)]because of gain (gis the power gain) and its phase changes
by 2kL, whereLis the length of the laser cavity. At the same time, its amplitude
changes by
√
R 1 R 2 exp(−αintL)because of reflection at the laser facets and because of
an internal lossαintthat includes free-carrier absorption, scattering, and other possible
mechanisms. HereR 1 andR 2 are the reflectivities of the laser facets. Even though
R 1 =R 2 in most cases, the two reflectivities can be different if laser facets are coated
to change their natural reflectivity. In the steady state, the plane wave should remain
unchanged after one round trip, i.e.,
E 0 exp(gL)√
R 1 R 2 exp(−αintL)exp( 2 ikL)=E 0. (3.3.3)By equating the amplitude and the phase on two sides, we obtain
g=αint+1
2 L
ln(
1
R 1 R 2
)
=αint+αmir=αcav, (3.3.4)2 kL= 2 mπ or ν=νm=mc/ 2 nL, (3.3.5)wherek= 2 πnν/candmis an integer. Equation (3.3.4) shows that the gaingequals
total cavity lossαcavat threshold and beyond. It is important to note thatgis not the
same as the material gaingmshown in Fig. 3.9. As discussed in Section 3.3.3, the