Optoelectronic Semiconductor Devices Principals and Characteristics

3.4.2 
PHASE CONDITION
 
The relationship between gain and loss described in the previous section does not completely determines 
the lasing conditions. The emission spectra are closely related to lasing, and only spontaneous emission 
corresponding to the lasing mode makes a contribution to the lasing. The phase of the propagating light 
after a round trip is required to coincide with the initial phase. 
The following phase condition has to be satisfied (determined by a Fabry-Perot geometry):
(14) 
2 , (
1,2,3,...)
m
m
L m
λ =
=
where
m - the mode index or mode number, 
λ
m
- the wavelength of the mode in the optical cavity. 
If we define the refractive index of the active region as n
r
, formula (14) can be rewritten as:
0
2
r
m
n
λ


=




L
(15) 
where λ
0
- the wavelength in a vacuum. 

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Figure 21.: Illustration of gain profile and emission spectra (longitudinal modes). 
Lasing occurs every wavelength mode as we can see from the formula (14) and the Figure 
21.
 By using 
formula (15) and considering the next mode, (m+1) mode, we can find the mode spacing ∆λ
m
as
(
)
0
1
r
m
n
λ
λ


− ∆
+ 



2L
=
eff
(16) 
and then
(17) 
2
0
/ 2
m
n L
λ
λ
∆
=
where n
r
is replaced by the effective refractive index, n
eff
, which given by:
0
0
1
r
eff
r
r
dn
n
n
n
d
λ
λ






=
−











(18) 
When the gain reaches the threshold level, lasing occurs at the wavelength satisfying formula (14).