QUEEN’S UNIVERSITY AT KINGSTON
Department of Electrical and Computer Engineering
ELEC–486 Fiber-Optic Communications
Laser Rate Equations
The modulation dynamics of the laser are modelled by coupled rate equations which
describe the relation between the carrier density N (t), photon density S(t) and optical
phase chirp (t)
I(t)
dN (t)
=
dt
qV
N (t)
chirp (t)
dt
N0 )
n
dS(t)
= g0 (N (t)
dt
d
g0 (N (t)
N0 )
=
1
S(t)
(1 + "S(t) )
1
S(t)
(1 + "S(t) )
1
2
g0 (N (t)
N0 )
S(t)
+
p
1
N (t)
(1)
(2)
n
:
(3)
p
is the mode con…nement factor, N0 is the carrier density at transparency for which the
net gain is zero, p is the photon lifetime, is the fraction of spontaneous emission coupled
into the lasing mode, n is the electron lifetime, I(t) is the injected current, q is the electron charge, V is the active layer volume, g0 is the gain slope constant, is the linewidth
enhancement factor and " is the gain compression factor. The gain slope constant is given
by g0 = vg a0 where vg is the group velocity and a0 is the active layer gain coe¢ cient. The
output optical power is given by
p(t) =
S(t)V
2
0h
(4)
p
where 0 is the di¤erential quantum e¢ ciency. The deviation of the optical frequency about
the carrier frequency (or chirp) is related to the time derivative of the optical phase by
v(t) =
1 d
2
chirp (t)
dt
:
(5)
The small-signal frequency response for the intensity modulation is obtained by linearizing the coupled rate equations. The injection current is given by
I(t) = Idc +
I exp (j!t)
1
(6)
where Idc is the laser bias current, I is the amplitude of the modulating current, and ! is
the radian modulation frequency (! = 2 f ). The response to this harmonically varying
current is:
(7)
(8)
S(t) = Sdc + S exp (j!t)
N (t) = Ndc + N exp (j!t) :
The small-signal IM transfer function is obtained by substituting (6)-(8) into (1) and (2).
Since the time dependence has been explicity speci…ed as exp (j!t), the derivatives can be
taken and the di¤erential equations are converted to algebraic equations. The linearization
is obtained by neglecting products of small terms (e.g., S N 0).
The small-signal IM transfer function HIM (j2 f ) is
S(j2 f )
I(j2 f )
HIM (j2 f ) =
(9)
where S(j2 f ) is the small-signal amplitude of the photon density and I(j2 f ) is the
small-signal amplitude of the current. Another form of the small-signal IM transfer function
HIM (j2 f ), expressed in units of W/A, is de…ned as
p(j2 f )
I(j2 f )
HIM (j2 f ) =
(10)
where p(j2 f ) is the small-signal amplitude of the output power. Equations (9) and (10)
are the same to within the multiplicative constant V 0 h =2 p .
The analytical expression for the normalized small-signal IM response of a laser is given
by
Z
HIM (j2 f )
=
(11)
2
HIM (0)
(j2 f ) + j2 f Y + Z
where
Y = g0
and
Z = g0
Sdc
1
+
(1 + "Sdc )
n
Sdc
1
+(
(1 + "Sdc ) p
g0 (Ndc
1)
g0
N0 )
(Ndc
1
1
+
2
(1 + "Sdc )
p
N0 )
n
1
+
(1 + "Sdc )2
(12)
1
:
(13)
n p
The zero frequency (dc) IM response is given by
HIM (0) =
Sdc
g0 (1+"S
+
dc )
qV
n
!
1
:
Z
(14)
In order to use the above results, the values of Sdc and Ndc that correspond to Idc need to
be determined. For (1) and (2), the steady-state solution can be found analytically, which
is quite convenient. The results are:
2
Sdc =
(g0
n
qV
+ ")Idc
(g0 N0
p (1
) + 1) +
p
"
(g0
n
qV
+ ")Idc
2
(g0 N0
p (1
) + 1)
+4
qV
p
2
qV
(" + g0
p
1
(" + g0
n)
Idc
#1=2 9
=
;
(15)
n)
p
Ndc = Sdc
g0 N0
1
+
1 + "Sdc
p
3
g0 Sdc
+
1 + "Sdc
1
(16)
n