abbreviated equations, which are obtained from the earlier-considered differential
equations for the laser:
dE
2
0L =dt ¼ G 0 E
2
0L N À E
2
0L =T 0F þ ξ E1 ,
dN=dt ¼ α N00 Á J 0L þ α N01 Á J L1 À
N 0L
T 1
À G 0 NE
2
0L þ ξ N1 ,
dφ 10n =dt ¼ 2πν 0P À 2πν 0 þ σ 0L þ ρ 0L E
2
0L þ ξ φ1 :
8
> > <
> > :
ð5:60Þ
In Eq. (5.60), the first two equations are related with each other through parameters E
2
0L and N. The third equation for the slowly changing phase is dependent upon
parameters E
2
0L and N, which are determined by the first two equations of Eq. (5.60)
system. There are constant coefficients (σ n and ρ n ) in the third equation, and the
natural frequency of the laser optical resonator ν 0P (N ) depends upon the population
N ¼ N 0L . The second system of abbreviated differential equations presented below is
formed for the single-state single-frequency quasi-stationary mode of the laser and
can be simplified by introduction of α n and β n parameters (for the nonlinear laser
element) excluding the population difference N:
dE 10n
dt
¼ α N00 Á J 0L þ α N01 Á J L1 À
1
T 0F
E 10n À β n E 10n
ð
Þ
2 þ ξ EE ,
dφ 10n
dt
¼ 2πν n À 2πν 0n
ð
Þ þ σ n þ ρ n E 10n
ð
Þ
2 þ ξ φφ :
8
> <
> :
ð5:61Þ
The system (Eq. 5.61) consists only of the two equations and represents the
simplified notation of the Eq. (5.60) system. The first equation for the slowly
changing amplitude E n10 consists of α n and β n parameters, which determine the
nonlinear function of the laser nonlinear element. The second equation of Eq. (5.61)
for the slowly changing phase of the laser oscillation field. In this system, we exclude
the abbreviated equation for the population N, but, nevertheless, its influence is taken
into account by coefficients α n and β n .
5.4.6 Symbolic Fluctuation Equations of OEO
To form the system of fluctuation differential equations for OEO DM (Fig. 5.6), we
consider the oscillation of the laser strength E n (Eq. 5.28) and the AC electrical
component i L (Eq. 5.29) with the amplitude of the first harmonic of the laser
pumping current J 1L : E n ¼ E 10L Re [exp(2πν 0 t)] , i L ¼ J 10L Re [exp(2πf 0 t)] , where
ν 0 and f 0 are the average oscillation optical and RF frequencies.
We take as initial the system of differential equations (Eq. 5.43) for the square
strength amplitude E n of the electrical component of EMF with the amplitude of the
first harmonic of E 10L .
244
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
equations for the laser:
dE
2
0L =dt ¼ G 0 E
2
0L N À E
2
0L =T 0F þ ξ E1 ,
dN=dt ¼ α N00 Á J 0L þ α N01 Á J L1 À
N 0L
T 1
À G 0 NE
2
0L þ ξ N1 ,
dφ 10n =dt ¼ 2πν 0P À 2πν 0 þ σ 0L þ ρ 0L E
2
0L þ ξ φ1 :
8
> > <
> > :
ð5:60Þ
In Eq. (5.60), the first two equations are related with each other through parameters E
2
0L and N. The third equation for the slowly changing phase is dependent upon
parameters E
2
0L and N, which are determined by the first two equations of Eq. (5.60)
system. There are constant coefficients (σ n and ρ n ) in the third equation, and the
natural frequency of the laser optical resonator ν 0P (N ) depends upon the population
N ¼ N 0L . The second system of abbreviated differential equations presented below is
formed for the single-state single-frequency quasi-stationary mode of the laser and
can be simplified by introduction of α n and β n parameters (for the nonlinear laser
element) excluding the population difference N:
dE 10n
dt
¼ α N00 Á J 0L þ α N01 Á J L1 À
1
T 0F
E 10n À β n E 10n
ð
Þ
2 þ ξ EE ,
dφ 10n
dt
¼ 2πν n À 2πν 0n
ð
Þ þ σ n þ ρ n E 10n
ð
Þ
2 þ ξ φφ :
8
> <
> :
ð5:61Þ
The system (Eq. 5.61) consists only of the two equations and represents the
simplified notation of the Eq. (5.60) system. The first equation for the slowly
changing amplitude E n10 consists of α n and β n parameters, which determine the
nonlinear function of the laser nonlinear element. The second equation of Eq. (5.61)
for the slowly changing phase of the laser oscillation field. In this system, we exclude
the abbreviated equation for the population N, but, nevertheless, its influence is taken
into account by coefficients α n and β n .
5.4.6 Symbolic Fluctuation Equations of OEO
To form the system of fluctuation differential equations for OEO DM (Fig. 5.6), we
consider the oscillation of the laser strength E n (Eq. 5.28) and the AC electrical
component i L (Eq. 5.29) with the amplitude of the first harmonic of the laser
pumping current J 1L : E n ¼ E 10L Re [exp(2πν 0 t)] , i L ¼ J 10L Re [exp(2πf 0 t)] , where
ν 0 and f 0 are the average oscillation optical and RF frequencies.
We take as initial the system of differential equations (Eq. 5.43) for the square
strength amplitude E n of the electrical component of EMF with the amplitude of the
first harmonic of E 10L .
244
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .