We take into consideration that in OEO DM, two oscillations α N0 ¼
α N00 Á J 0L þ α N01 K DL E n E
Ã
n ¼ α N00 þ α N01 K DL E n
j j
2 are added on the PD area.
Although E n is, in the general case, E n ¼ E 0n + E 1n , here in the right part, the
second term α N01 K DL E 1n |E 1n |
2 represents only the AC component of E 1n , since the
DC component is filtered out by the “PD-A-F” circuit. The DC component of
pumping independently affects the QWLD active substance, i.e., QWLD is supplied
by the pumping current through the independent external circuit. Cutting down of
algebraic transformations, under condition of the laser operation in the single-state
single-frequency quasi-stationary mode (i.e., at average population level N 02 above
the threshold value), from the equation systems (Eqs. 5.42 and 5.43), we obtain for
the “Laser” and for OEO DM the following system of symbolic equations with
fluctuating noise sources at modulation by the AC pumping current i 1L :
Fluctuation differential equations for OEO DM:
p
2 þ
1
T 0F
p þ ð2πν 0F Þ
2
!
p
2 þ
1
T 2
p þ ð2πν 12 Þ
2
!
E n ¼ α 00 E n þ α N01 J 1L E n À β 00 E n jE n j
2 þ ξ ENP ,
p
2 þ ð1=T eF Þp þ ð2π f 0e Þ
2
h
i
J 1L ¼ cos½Δϕ OF Á jE n j
2 Á
1
T eF
Á K OF K PD S NY pexpðÀpT FOS ÞJ 1L ,
8
> > > <
> > > :
ð5:62Þ
where α 00 ¼ N 02
2p
2
e 2πν
ð
Þ
2
ε 0 h
À
1
T 0F
; β 00 ¼ p
4
e N 02
T 1 exp Àj2πν 0 T 1
ð
Þ
2πν 12
ð
Þ
2 h
2 1þ 2πν 0 T 1
ð
Þ
2
½
1=2 . The noise component ξ ENP is ξ ENP ¼ ξ N E n η 00 þ
ξ P
2πν 12
ð
Þ
2 Q P
þ ξ E , where Q P ¼ (2πν 12 )
2 + (1/T 2 )
(2πν 12 ) + (2πν 12 )
2 and η 00 ¼
T 1 exp Àj2πν 0 T 1
ð
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1þ 2πν 0 T 1
ð
Þ
2
p
.
Equation (5.62) permit to calculate PSD of the amplitude and phase noises for
OEO DM, the instantaneous values of E L , the amplitude, frequency and phase of
oscillations as well as to take into account contributions of different physical
quantities including the noisy impact ξ ENP and the spontaneous emission of the
carrier noise in the output emission of the laser as well as the current oscillations of
the first harmonic of OEO DM.
The analog model of OEO DM is presented in Fig. 5.6a and is constructed
according to the differential equations system (Eq. 5.62) as QWLD enclosed by
the positive feedback loop “Feedback Chain.”
The specific feature of equation (Eq. 5.62) is that they are similar in the form to
well-studied equations in radio electronics for the double-circuit autonomous oscillator with linear-cubic function of the nonlinear inertial element of the AC voltage
versus the current. But, coefficients included in Eq. (5.62) are expressed through the
laser variable physical quantities: the population, the dipole moment, the lifetime on
the upper operation level, the time constant of the laser optical filter, Langevinian
noise sources ξ ENP of the optical emission, the population, and polarization.
Equation (5.62) reflects the laser properties as the oscillating system in operation
mode above the threshold at modulation of the AC pumping current i 1L . In the left
part of Eq. (5.62) the operator “conductance” is presented. It is defined by the laser
physical quantities: the transition frequency between the first and the second energy
levels ν 12 , the natural frequency of the optical filter included in the laser ν 0F , the time
5.4 Fluctuation Differential Equations of OEO with the Langevinian Noise Sources
245
α N00 Á J 0L þ α N01 K DL E n E
Ã
n ¼ α N00 þ α N01 K DL E n
j j
2 are added on the PD area.
Although E n is, in the general case, E n ¼ E 0n + E 1n , here in the right part, the
second term α N01 K DL E 1n |E 1n |
2 represents only the AC component of E 1n , since the
DC component is filtered out by the “PD-A-F” circuit. The DC component of
pumping independently affects the QWLD active substance, i.e., QWLD is supplied
by the pumping current through the independent external circuit. Cutting down of
algebraic transformations, under condition of the laser operation in the single-state
single-frequency quasi-stationary mode (i.e., at average population level N 02 above
the threshold value), from the equation systems (Eqs. 5.42 and 5.43), we obtain for
the “Laser” and for OEO DM the following system of symbolic equations with
fluctuating noise sources at modulation by the AC pumping current i 1L :
Fluctuation differential equations for OEO DM:
p
2 þ
1
T 0F
p þ ð2πν 0F Þ
2
!
p
2 þ
1
T 2
p þ ð2πν 12 Þ
2
!
E n ¼ α 00 E n þ α N01 J 1L E n À β 00 E n jE n j
2 þ ξ ENP ,
p
2 þ ð1=T eF Þp þ ð2π f 0e Þ
2
h
i
J 1L ¼ cos½Δϕ OF Á jE n j
2 Á
1
T eF
Á K OF K PD S NY pexpðÀpT FOS ÞJ 1L ,
8
> > > <
> > > :
ð5:62Þ
where α 00 ¼ N 02
2p
2
e 2πν
ð
Þ
2
ε 0 h
À
1
T 0F
; β 00 ¼ p
4
e N 02
T 1 exp Àj2πν 0 T 1
ð
Þ
2πν 12
ð
Þ
2 h
2 1þ 2πν 0 T 1
ð
Þ
2
½
1=2 . The noise component ξ ENP is ξ ENP ¼ ξ N E n η 00 þ
ξ P
2πν 12
ð
Þ
2 Q P
þ ξ E , where Q P ¼ (2πν 12 )
2 + (1/T 2 )
(2πν 12 ) + (2πν 12 )
2 and η 00 ¼
T 1 exp Àj2πν 0 T 1
ð
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1þ 2πν 0 T 1
ð
Þ
2
p
.
Equation (5.62) permit to calculate PSD of the amplitude and phase noises for
OEO DM, the instantaneous values of E L , the amplitude, frequency and phase of
oscillations as well as to take into account contributions of different physical
quantities including the noisy impact ξ ENP and the spontaneous emission of the
carrier noise in the output emission of the laser as well as the current oscillations of
the first harmonic of OEO DM.
The analog model of OEO DM is presented in Fig. 5.6a and is constructed
according to the differential equations system (Eq. 5.62) as QWLD enclosed by
the positive feedback loop “Feedback Chain.”
The specific feature of equation (Eq. 5.62) is that they are similar in the form to
well-studied equations in radio electronics for the double-circuit autonomous oscillator with linear-cubic function of the nonlinear inertial element of the AC voltage
versus the current. But, coefficients included in Eq. (5.62) are expressed through the
laser variable physical quantities: the population, the dipole moment, the lifetime on
the upper operation level, the time constant of the laser optical filter, Langevinian
noise sources ξ ENP of the optical emission, the population, and polarization.
Equation (5.62) reflects the laser properties as the oscillating system in operation
mode above the threshold at modulation of the AC pumping current i 1L . In the left
part of Eq. (5.62) the operator “conductance” is presented. It is defined by the laser
physical quantities: the transition frequency between the first and the second energy
levels ν 12 , the natural frequency of the optical filter included in the laser ν 0F , the time
5.4 Fluctuation Differential Equations of OEO with the Langevinian Noise Sources
245