expressed through the AC physical quantities of the laser: the population, the dipole
moment, the lifetime on the upper operation level, the time constant of the laser
optical filter, the Langevinian noise sources of the optical emission, population, and
polarization.
The transfer from Eqs. (5.43) and (5.45) to the abbreviated equations and their
linearization permits not only to determine in the steady-state mode the intensity of
the laser emission, but to note down the abbreviated equations with fluctuations,
from which we obtain the values for the power spectral density of phase and
amplitude noises.
At analysis of Eqs. (5.43) and (5.45), we may make one more important conclusion: three last terms of the Langevinian sources in the right part of Eq. (5.45)
determine the total noises of the ξ SN impact accordion to the formula:
ξ SN ¼ ξ N E n
T 1 exp Àj2πν 0 T 1
ð
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ 2πν 0 T 1
ð
Þ
2
q
þ
ξ P
À 2πν 0
ð
Þ
2 þ j 1=T 2
ð
Þ 2πν 0
ð
Þþ 2πν 12
ð
Þ
2
þ ξ E :
ð5:46Þ
We can neglect by the second and third terms included in the last expression
(Eq. 5.46) at large pumping excess over the threshold value. Therefore, we can
consider that the main fluctuation impact in the laser is the noise of the inversed
population ξ N . These noises depend upon the strength value E n and the carrier
lifetime T 1 . The typical feature of fluctuations is the presence of inertia property,
which is determined by the exponential function argument Àj2πν 0 T 1 .
We introduce the designation for short notation:
Q p
ð Þ ¼ p
2
þ
1
T 0F
p þ 2πν 0F
ð
Þ
2
!
p
2
þ
1
T 2
p þ 2πν 12
ð
Þ
2
!
:
ð5:47Þ
If to take into account the delay in the circular optical channel of the laser
resonator by the T L time, the expression for the symbolic equation (Eq. 5.45) takes
the form of the symbolic equation with the inertial active element:
Q p
ð ÞE n ¼
S NE E n
ð Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ 2πν 0 T 1
ð
Þ
2
q
exp Àj2πν 0 T 1 þ T L
ð
Þ
½
þ ξ SN ,
ð5:48Þ
where S NE (E n ) ¼ [α 001 E n À β 001 E n |E n |
2 ], and new coefficients α 001 and β 001 are:
α 001 ¼ N 0
2p
2
e
ε 0 ħ T 1, β 001 ¼
2p
2
e
ε 0 ħ N 0 G 0n T
2
1 . We note that in Eq. (5.48), the inertia property
depends not only upon the delay time T L of oscillations in the optical channel of the
laser resonator, but it determines the carrier lifetime T 1 on the upper operation level.
In the analog model presented in Fig. 5.6, we see that the noise impact sources are
included in the circular laser circuit with the help of adders. The obvious representation of the laser with Langevinian noise sources simplifies understanding of
5.4 Fluctuation Differential Equations of OEO with the Langevinian Noise Sources
239
moment, the lifetime on the upper operation level, the time constant of the laser
optical filter, the Langevinian noise sources of the optical emission, population, and
polarization.
The transfer from Eqs. (5.43) and (5.45) to the abbreviated equations and their
linearization permits not only to determine in the steady-state mode the intensity of
the laser emission, but to note down the abbreviated equations with fluctuations,
from which we obtain the values for the power spectral density of phase and
amplitude noises.
At analysis of Eqs. (5.43) and (5.45), we may make one more important conclusion: three last terms of the Langevinian sources in the right part of Eq. (5.45)
determine the total noises of the ξ SN impact accordion to the formula:
ξ SN ¼ ξ N E n
T 1 exp Àj2πν 0 T 1
ð
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ 2πν 0 T 1
ð
Þ
2
q
þ
ξ P
À 2πν 0
ð
Þ
2 þ j 1=T 2
ð
Þ 2πν 0
ð
Þþ 2πν 12
ð
Þ
2
þ ξ E :
ð5:46Þ
We can neglect by the second and third terms included in the last expression
(Eq. 5.46) at large pumping excess over the threshold value. Therefore, we can
consider that the main fluctuation impact in the laser is the noise of the inversed
population ξ N . These noises depend upon the strength value E n and the carrier
lifetime T 1 . The typical feature of fluctuations is the presence of inertia property,
which is determined by the exponential function argument Àj2πν 0 T 1 .
We introduce the designation for short notation:
Q p
ð Þ ¼ p
2
þ
1
T 0F
p þ 2πν 0F
ð
Þ
2
!
p
2
þ
1
T 2
p þ 2πν 12
ð
Þ
2
!
:
ð5:47Þ
If to take into account the delay in the circular optical channel of the laser
resonator by the T L time, the expression for the symbolic equation (Eq. 5.45) takes
the form of the symbolic equation with the inertial active element:
Q p
ð ÞE n ¼
S NE E n
ð Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ 2πν 0 T 1
ð
Þ
2
q
exp Àj2πν 0 T 1 þ T L
ð
Þ
½
þ ξ SN ,
ð5:48Þ
where S NE (E n ) ¼ [α 001 E n À β 001 E n |E n |
2 ], and new coefficients α 001 and β 001 are:
α 001 ¼ N 0
2p
2
e
ε 0 ħ T 1, β 001 ¼
2p
2
e
ε 0 ħ N 0 G 0n T
2
1 . We note that in Eq. (5.48), the inertia property
depends not only upon the delay time T L of oscillations in the optical channel of the
laser resonator, but it determines the carrier lifetime T 1 on the upper operation level.
In the analog model presented in Fig. 5.6, we see that the noise impact sources are
included in the circular laser circuit with the help of adders. The obvious representation of the laser with Langevinian noise sources simplifies understanding of
5.4 Fluctuation Differential Equations of OEO with the Langevinian Noise Sources
239