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
!
E n
¼ p
2
α N01 Á J 1L Á E n þ α 00 E n À β 00 E n E n
j j
2
h
i
,
p
2
þ 1=T eF
ð
ÞÁp þ 2π f 0e
ð
Þ
2
h
i
i L
¼ p Á cos Δϕ OF
½
Á E n
j j
2 1=T eF
ð
ÞK OF Á K PD Á exp ÀpT FOS
ð
ÞÁS NY u PD
ð Þ,
8
> > > > > > > > <
> > > > > > > > :
ð5:6Þ
where coefficients α 00 and β 00 are (in first approximation): α 00 ¼ N 0
2p
2
e
ε 0 h η 00 ; β 00 ¼
2p
2
e
ε 0 h N 0 G 0n T 1 η
2
00 ; where η 00 ¼
T 1 exp Àj2πν 0 T 1
ð
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1þ 2πν 0 T 1
ð
Þ
2
p
. Coefficients α 00 ¼ α 0L and β 00 ¼ β 0L in
the second approximation (at consideration of nonlinearity approximation with
account of the fifth-order polynomial and at calculation of the second partial
derivative) are: α 0L ¼
6N 0 T 1 G 00
2π νÀν 0
ð
Þ T 1
ð
Þ
2 þ1
½
2
ffiffi ffi
5
p
exp ÀjArctan 2 Á 2π ν À ν 0
ð
ÞT 1
ð
½
½
and
β 0L ¼
20N 0 T 1 G 00
ð
Þ
2
2π νÀν 0
ð
Þ T 1
ð
Þ
2 þ1
½
3
ffiffiffiffiffi
10
p
exp ÀjArctan 3 2π ν À ν 0
ð
ÞT 1
ð
½
½
.
The system of Eq. (5.6) allows calculation in the specified quasi-stationary mode
of the instantaneous values of E L, the amplitude, the frequency, and the phase of
laser oscillations, and of OEO DM, and to take into consideration the influence of
different parameters in Eq. (5.6) upon the formation of current oscillations of the first
harmonic of OEO DM. The specific feature of the equation system (Eq. 5.6) is the
fact that these equations are similar in their form to well-studied radio electronics
equations for the double-circuit autonomous oscillator (the first equation of the
system) with the inertial auto-bias circuit. Nevertheless, the coefficient included in
Eq. (5.6) are expressed through the laser variable parameters: the population, the
dipole moment, the lifetime on the upper operating level, and the time constant of the
laser optical filter.
The equation system (Eq. 5.6) reflects the laser properties as the oscillating
system in the operation mode, which is upper than a threshold at modulation of
the variable pumping current i L ¼ i 1L . In the left part of Eq. (5.6), there is the
operator “conductance.” It is defined by laser physical quantities: the transition
frequency between the first and second energy levels ν 12 , the natural frequency of
the optical filter included in the laser and this frequency ν 0F is much more than the
time constant of the active medium polarization T 2 , the time constant of the optical
filter T 0F . Oscillations of E n in Eq. (5.6) occur only at the definite threshold pumping
value α N00 . In the right part of Eq. (5.6), the following expression is located:
S L ¼ α 00 E n À β 00 E n |E n |
2 , which defines the laser nonlinearity in the quasistationary mode.
We must note that the equation system of OEO DM (Eq. 5.6) may take into
consideration the heterodyning. As we discussed in Chap. 3, the heterodyning in
OEO is performed by multiplication oscillations with different delays. For this, the
multiplier cos[Δϕ OF ] is added to the second equation in Eq. (5.6) into its right part.
At that, Δϕ OF is defined by the difference of optical phases of optical calculations,
which pass to PD after passing of the optical filter (Fig. 3.1). The contribution into
5.1 Symbolic and Abbreviated Equations of OEO
209
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
!
E n
¼ p
2
α N01 Á J 1L Á E n þ α 00 E n À β 00 E n E n
j j
2
h
i
,
p
2
þ 1=T eF
ð
ÞÁp þ 2π f 0e
ð
Þ
2
h
i
i L
¼ p Á cos Δϕ OF
½
Á E n
j j
2 1=T eF
ð
ÞK OF Á K PD Á exp ÀpT FOS
ð
ÞÁS NY u PD
ð Þ,
8
> > > > > > > > <
> > > > > > > > :
ð5:6Þ
where coefficients α 00 and β 00 are (in first approximation): α 00 ¼ N 0
2p
2
e
ε 0 h η 00 ; β 00 ¼
2p
2
e
ε 0 h N 0 G 0n T 1 η
2
00 ; where η 00 ¼
T 1 exp Àj2πν 0 T 1
ð
Þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1þ 2πν 0 T 1
ð
Þ
2
p
. Coefficients α 00 ¼ α 0L and β 00 ¼ β 0L in
the second approximation (at consideration of nonlinearity approximation with
account of the fifth-order polynomial and at calculation of the second partial
derivative) are: α 0L ¼
6N 0 T 1 G 00
2π νÀν 0
ð
Þ T 1
ð
Þ
2 þ1
½
2
ffiffi ffi
5
p
exp ÀjArctan 2 Á 2π ν À ν 0
ð
ÞT 1
ð
½
½
and
β 0L ¼
20N 0 T 1 G 00
ð
Þ
2
2π νÀν 0
ð
Þ T 1
ð
Þ
2 þ1
½
3
ffiffiffiffiffi
10
p
exp ÀjArctan 3 2π ν À ν 0
ð
ÞT 1
ð
½
½
.
The system of Eq. (5.6) allows calculation in the specified quasi-stationary mode
of the instantaneous values of E L, the amplitude, the frequency, and the phase of
laser oscillations, and of OEO DM, and to take into consideration the influence of
different parameters in Eq. (5.6) upon the formation of current oscillations of the first
harmonic of OEO DM. The specific feature of the equation system (Eq. 5.6) is the
fact that these equations are similar in their form to well-studied radio electronics
equations for the double-circuit autonomous oscillator (the first equation of the
system) with the inertial auto-bias circuit. Nevertheless, the coefficient included in
Eq. (5.6) are expressed through the laser variable parameters: the population, the
dipole moment, the lifetime on the upper operating level, and the time constant of the
laser optical filter.
The equation system (Eq. 5.6) reflects the laser properties as the oscillating
system in the operation mode, which is upper than a threshold at modulation of
the variable pumping current i L ¼ i 1L . In the left part of Eq. (5.6), there is the
operator “conductance.” It is defined by laser physical quantities: the transition
frequency between the first and second energy levels ν 12 , the natural frequency of
the optical filter included in the laser and this frequency ν 0F is much more than the
time constant of the active medium polarization T 2 , the time constant of the optical
filter T 0F . Oscillations of E n in Eq. (5.6) occur only at the definite threshold pumping
value α N00 . In the right part of Eq. (5.6), the following expression is located:
S L ¼ α 00 E n À β 00 E n |E n |
2 , which defines the laser nonlinearity in the quasistationary mode.
We must note that the equation system of OEO DM (Eq. 5.6) may take into
consideration the heterodyning. As we discussed in Chap. 3, the heterodyning in
OEO is performed by multiplication oscillations with different delays. For this, the
multiplier cos[Δϕ OF ] is added to the second equation in Eq. (5.6) into its right part.
At that, Δϕ OF is defined by the difference of optical phases of optical calculations,
which pass to PD after passing of the optical filter (Fig. 3.1). The contribution into
5.1 Symbolic and Abbreviated Equations of OEO
209