Appendix 5 Spontaneous Emission Term and Factors

Appendix 5
Spontaneous Emission Term
and Factors
Consider the spontaneous emission term describing the contribution of the
spontaneous emission to the laser oscillation, which appears as the final
term on the right-hand side of the rate equation for the photon density
(Eq. (6.21)). Since this term represents the component of the spontaneous
emission belonging to the same mode as the laser oscillation (angular
frequency !
m
), it is closely related to the gain for the oscillation mode.
Consider a laser of index-guiding type, and let E ¼ E(r) ¼ E(x, y)E(z) be the
complex electric field of the oscillation mode. The field is normalized in
the similar manner as Eq. (2.6), so that the optical energy in the resonator
corresponds to the energy of a photon:
Z
c
"
0
n
r
n
g
2
jEðrÞj
2
dV ¼ hh!
m
ðA5:1Þ
Then, from Eqs (2.50a) and (2.57), the transition probability relevant to the
photons of this mode can be written as

w
abs
n
¼
w
stm
n
¼ w
spt
¼
p
2hh
EðrÞ E h
1
jerj
2
i




2
ðE
1
þ hh!
m
À E
2
ÞðA5:2Þ
where n is the number of the photons in the resonator. Using the direct-

transition model, the net number of the stimulated emission transition
per unit volume of the active region per unit time can be calculated by
integrating w
stm
multiplied by (1/2p
3
)( f
2
À f
1
)dk. The average value of
jE(r)j
2
in the active region is given by
hjEj
2
i
a
¼
Gð2hh!
m
="
0
n
r
n
g
Þ
V
a

ðA5:3Þ
Copyright © 2004 Marcel Dekker, Inc.
G ¼
R
a
jEðrÞj
2
dV
R
c
jEðrÞj
2
dV
ðA5:4Þ
where V
a
is the volume of the active region, À is the confinement factor, and
use has been made of Eq. (A5.1). Replacing E in Eq. (A5.2) by the average
given by Eq. (A5.3), and calculating the relative time variation in the mode
photon number in the resonator of volume V
a
, we obtain an expression for
the mode gain:
GGð!
m
Þ¼G
pe
2
n
r

n
g
"
0
m
2
!
m
jMj
2
ðf
2
À f
1
Þ
r
ðhh!
m
ÞðA5:5Þ
In the derivation of the above expression, use has been made of Eqs (3.14)–
(3.17). This result is consistent with that obtained by rewriting the material
gain g given by Eq. (3.16) in G ¼ v
g
g, and then in the mode gain ÀG.
Let R
sp
(!
m
) be the number of photons spontaneously emitted per unit
time in the active region of a volume V

a
. Then R
sp
can be calculated by
integrating w
stm
given by Eq. (A5.2) multiplied by V
a
(1/2p
3
) f
2
(1 À f
1
)dk
in a similar manner as above, to yield
R
sp
ð!
m
Þ¼G
pe
2
n
r
n
g
"
0
m

2
!
m
jMj
2
f
2
ð1 À f
1
Þ
r
ðhh!
m
ÞðA5:6Þ
Accordingly, from Eqs (A5.5) and (A5.6), the spontaneous emission term
and the mode gain are correlated by
R
sp
ð!
m
Þ¼n
sp
GGð!
m
ÞðA5:7Þ
n
sp
¼
f
2

ð1 À f
1
Þ
ð f
2
À f
1
Þ
¼ 1 À exp
hh!
m
À ÁF
k
B
T
 !
À1
ðA5:8Þ
where ÁF ¼ F
c
À F
v
is the difference between the quasi-Fermi levels. The
parameter n
sp
in Eq. (A5.8) is referred to as the population inversion factor.
The optical waves in an ordinary laser structure include many radiation
modes with the propagation vector not parallel to the waveguide axis. The
majority of the spontaneous emissions belong to such radiation modes, and
therefore the guided mode component is negligibly small. Therefore, the

expression given by Eq. (3.20) for a homogeneous semiconductor can also be
used to describe approximately the spontaneous emission spectrum in a laser
structure. By integrating it, the total spontaneous emission R
sp
can be
calculated. Approximating the spontaneous emission spectrum by a
Lorentzian distribution with a half-width Á! at half-maximum and using
300 Appendix 5
Copyright © 2004 Marcel Dekker, Inc.
an approximation that spontaneous emission peak frequency % oscillation
frequency, we obtain an expression for R
sp
:
R
sp
¼
n
r
e
2
!
pm
2
c
3
"
0
jMj
2
f

2
ð1 À f
1
Þ
r
ðhh!
m
Þ
Â
Z
ðÁ!=2Þ
2
ð! À !
m
Þ
2
þðÁ!=2Þ
2
d!
¼
n
r
e
2
! Á!
2m
2
c
3
"

0
jMj
2
f
2
ð1 À f
1
Þ
r
ðhh!
m
ÞðA5:9Þ
The spontaneous emission term C
s
N/
s
at the end of the right-hand side of
the rate equation given by Eq. (6.21) was introduced, by representing the
total spontaneous emission per unit volume in the active region
approximately as R
sp
¼ N/
s
, and by representing the component belonging
to the oscillation mode per unit volume in the active region R
sp
(!
m
)/V
a

as
C
s
N/
s
¼ C
s
R
sp
. Therefore, we see from Eqs (A5.6) and (A5.9) that the
spontaneous emission factor C
s
is given by
C
s
¼ G
2pc
3
n
2
r
n
g
V
a
!
2
Á!
¼
G

4
4p
2
n
2
r
n
g
V
a
Á
ðA5:10Þ
It should be noted that the above result does not apply for lasers of gain-
guiding type, where the guided mode cannot be described independently to
the carrier injection. Since the guided mode in gain-guiding lasers has curved
wavefronts, the coupling of the spontaneous emission to the guided mode is
stronger, and the value of C
s
is several times the value given by Eq. (A5.10).
Spontaneous Emission Term and Factors 301
Copyright © 2004 Marcel Dekker, Inc.