Let us transfer from Eq. (5.41) to symbolic fluctuation equations. For this, we use
the Laplace transform: d
2 /dt
2
¼ p
2 and d/dt ¼ p. We note that at transfer from the
operator record to the frequency record, we can replace the operator p by
jω ¼ ( j2πν À j2πν 0n ), where ν 0n ¼ ν 0 , ν is the QWLD generation frequency.
Besides, we introduce the constant coefficients A NE1 ¼ 2/ε n and B NE1 ¼ p
2
e = 2πh
ð
Þ.
The system of differential equations (Eq. 5.41) in the operator form represents the
following system of symbolic fluctuation equations for the laser:
p
2
þ
1
T 0F
p þ 2πv 0n
ð
Þ
2
!
E n ¼ À
1
ε n
p
2 P n þ ξ E ,
p
2
þ
1
T 2
p þ 2πv 12
ð
Þ
2 P n ¼
p
2
e
2πħ
NE n þ ξ P , pN ¼ α N0 À
1
T 1
N À
1
T 1ind
NE n E
Ã
n þ ξ n :
8
> > > <
> > > :
ð5:42Þ
We consider the case when the resonator Q-factor of the laser essentially exceeds
the Q-factor of the spectral emission line of the active element, i.e., the conditions
T 2 ( T 1 , T 2 ( T 0F are fulfilled. We express P n from the second equation of
Eq. (5.42) P n ¼
p
2
e
2πħ
1
p 2 þ 1=T 2
ð
Þpþ 2πv 12
ð
Þ
2 NE n . We take into consideration that N(t) is
the slowly changing function compared to E n (t) and P n (t). Pumping α N0 in Eq. (5.42)
can be expressed as α N0 ¼ α N00 Á J 0L + α N01 i 1L ¼ α N00 Á J 0L + α N01 Á J 1L , where α N00
and α N01 are constants, I 0L ¼ J 0L is the DC component of the pumping current,
i 1L ¼ J 1L is the AC component of the pumping current. Now we have from Eq. (5.42)
the system from two symbolic equations with fluctuations for QWLD:
p
2
þ
2
T 0F
p þ þ 2πv 0n
ð
Þ
2
!
p
2
þ
2
T 2
p þ 2πv 12
ð
Þ
2
!
E n ¼
2p
2
e
2πħε n
ω
2 p
2 NE n þ ξ EP ,
pN ¼ α N00 Á J 0L þ α N01 Á J 1L À
1
T 1
N À
1
T 1
NE
2
n þ ξ NP :
8
> > <
> > :
ð5:43Þ
This system of fluctuation Eq. (5.43) is similar in mathematical notation to the
equation system for the double-circuit RF oscillator with inertial auto-bias [1, 2]
(or pumping) with noise sources ξ EP and ξ NP . The distinguish feature of Eq. (5.43) is
the nonlinearity presence in the form of multiplication operation N Á E n or the
multiplicative laser nonlinearity. The second fluctuation equation in Eq. (5.43)
takes into account the carriers’ noises, which are created by the laser spontaneous
emission due to the spontaneous carrier transfer from the upper energy level to the
lower in the double-level laser model in the dipole approximation.
5.4 Fluctuation Differential Equations of OEO with the Langevinian Noise Sources
237
the Laplace transform: d
2 /dt
2
¼ p
2 and d/dt ¼ p. We note that at transfer from the
operator record to the frequency record, we can replace the operator p by
jω ¼ ( j2πν À j2πν 0n ), where ν 0n ¼ ν 0 , ν is the QWLD generation frequency.
Besides, we introduce the constant coefficients A NE1 ¼ 2/ε n and B NE1 ¼ p
2
e = 2πh
ð
Þ.
The system of differential equations (Eq. 5.41) in the operator form represents the
following system of symbolic fluctuation equations for the laser:
p
2
þ
1
T 0F
p þ 2πv 0n
ð
Þ
2
!
E n ¼ À
1
ε n
p
2 P n þ ξ E ,
p
2
þ
1
T 2
p þ 2πv 12
ð
Þ
2 P n ¼
p
2
e
2πħ
NE n þ ξ P , pN ¼ α N0 À
1
T 1
N À
1
T 1ind
NE n E
Ã
n þ ξ n :
8
> > > <
> > > :
ð5:42Þ
We consider the case when the resonator Q-factor of the laser essentially exceeds
the Q-factor of the spectral emission line of the active element, i.e., the conditions
T 2 ( T 1 , T 2 ( T 0F are fulfilled. We express P n from the second equation of
Eq. (5.42) P n ¼
p
2
e
2πħ
1
p 2 þ 1=T 2
ð
Þpþ 2πv 12
ð
Þ
2 NE n . We take into consideration that N(t) is
the slowly changing function compared to E n (t) and P n (t). Pumping α N0 in Eq. (5.42)
can be expressed as α N0 ¼ α N00 Á J 0L + α N01 i 1L ¼ α N00 Á J 0L + α N01 Á J 1L , where α N00
and α N01 are constants, I 0L ¼ J 0L is the DC component of the pumping current,
i 1L ¼ J 1L is the AC component of the pumping current. Now we have from Eq. (5.42)
the system from two symbolic equations with fluctuations for QWLD:
p
2
þ
2
T 0F
p þ þ 2πv 0n
ð
Þ
2
!
p
2
þ
2
T 2
p þ 2πv 12
ð
Þ
2
!
E n ¼
2p
2
e
2πħε n
ω
2 p
2 NE n þ ξ EP ,
pN ¼ α N00 Á J 0L þ α N01 Á J 1L À
1
T 1
N À
1
T 1
NE
2
n þ ξ NP :
8
> > <
> > :
ð5:43Þ
This system of fluctuation Eq. (5.43) is similar in mathematical notation to the
equation system for the double-circuit RF oscillator with inertial auto-bias [1, 2]
(or pumping) with noise sources ξ EP and ξ NP . The distinguish feature of Eq. (5.43) is
the nonlinearity presence in the form of multiplication operation N Á E n or the
multiplicative laser nonlinearity. The second fluctuation equation in Eq. (5.43)
takes into account the carriers’ noises, which are created by the laser spontaneous
emission due to the spontaneous carrier transfer from the upper energy level to the
lower in the double-level laser model in the dipole approximation.
5.4 Fluctuation Differential Equations of OEO with the Langevinian Noise Sources
237