On to the PD optical input (or to light-sensitive area), the following delayed laser
emission intensity falls, which represents a sum of AC and DC component:
E 12L
ð
Þ
2 ¼ E 01L
ð
Þ
2 1 þ γ
2
À
Á þ 0:5 E 01L
ð
Þ
2 γ cos ½ πU 0MZ =2U 0MZπ
ð
Þ
À U 10MZ =U 0MZπ
ð
Þ cos 2πft þ ψ mMZ
ð
Þg: ð5:63Þ
The AC component of the PD photocurrent is formed by the AC component of
intensity. The amplified oscillations of the first harmonic, which passed the filter F,
act in the input of RF amplifier. The transfer function on the first harmonic K F of this
filter can be defined as the RF circuit with the transformer coupling:
d
2 U 10MZ
dt 2 þ
1
T F
dU 10MZ
dt
þ 2π f eF0
ð
Þ
2 U 10MZ ¼
dS NY U 10MZ
½
dt
:
ð5:64Þ
Therefore, for the total intensity, we must differentiate in time of the variable
(E 12L )
2
:
E 12L
ð
Þ
2 ¼ 0:5 E 01L
ð
Þ
2 γf
1 þ γ
2
ð
Þ
2γ
þ cos ½ πU 0MZ =2U 0MZπ
ð
Þ
À U 10MZ =U 0MZπ
ð
Þ cos 2πft þ ψ mMZ
ð
Þg ð5:65Þ
of the voltage of the first harmonic: U 10MZ ¼ [E 12L (t À T FOS )]
2 K DL [1 À (1 + γ
2 )/2] .
Finally, we have for OEO MZ:
d
2 E n
dt 2 þ
1
T 0F
dE n
dt
þ 2πν 0F
ð
Þ
2 E n ¼ À
2ω
2 P n
ε n
þ ξ E ;
d
2 P n
dt 2 þ
1
T 2
dP n
dt
þ 2πν 12
ð
Þ
2 P n ¼
p
2
e
h
NE n þ ξ P ;
dN
dt
¼ α N0 J 0N À
N
T 1
À
1
h
P n E n þ ξ N ;
d
2 U 10MZ
dt 2 þ
1
T n
Á
dU 10MZ
dt
þ 2π f eF0
ð
Þ
2 U 10MZ ¼
dS NY E n , U 10MZ
½
dt
þ ξ U :
8
> > > > > > > > > > > <
> > > > > > > > > > > :
ð5:66Þ
The first three equations in the differential equation system of OEO МZ are
independent with regard to the fourth, which can be introduced in the case of OEO
DM for determination only of the amplitude of laser optical oscillations and PSD of
amplitude and phase noises. In equations, the normalized laser power depends on the
laser equation solution and is determined by the transfer function of the MZ
modulator, which depends on the laser optical frequency closed to ν OF .
5.4 Fluctuation Differential Equations of OEO with the Langevinian Noise Sources
249
emission intensity falls, which represents a sum of AC and DC component:
E 12L
ð
Þ
2 ¼ E 01L
ð
Þ
2 1 þ γ
2
À
Á þ 0:5 E 01L
ð
Þ
2 γ cos ½ πU 0MZ =2U 0MZπ
ð
Þ
À U 10MZ =U 0MZπ
ð
Þ cos 2πft þ ψ mMZ
ð
Þg: ð5:63Þ
The AC component of the PD photocurrent is formed by the AC component of
intensity. The amplified oscillations of the first harmonic, which passed the filter F,
act in the input of RF amplifier. The transfer function on the first harmonic K F of this
filter can be defined as the RF circuit with the transformer coupling:
d
2 U 10MZ
dt 2 þ
1
T F
dU 10MZ
dt
þ 2π f eF0
ð
Þ
2 U 10MZ ¼
dS NY U 10MZ
½
dt
:
ð5:64Þ
Therefore, for the total intensity, we must differentiate in time of the variable
(E 12L )
2
:
E 12L
ð
Þ
2 ¼ 0:5 E 01L
ð
Þ
2 γf
1 þ γ
2
ð
Þ
2γ
þ cos ½ πU 0MZ =2U 0MZπ
ð
Þ
À U 10MZ =U 0MZπ
ð
Þ cos 2πft þ ψ mMZ
ð
Þg ð5:65Þ
of the voltage of the first harmonic: U 10MZ ¼ [E 12L (t À T FOS )]
2 K DL [1 À (1 + γ
2 )/2] .
Finally, we have for OEO MZ:
d
2 E n
dt 2 þ
1
T 0F
dE n
dt
þ 2πν 0F
ð
Þ
2 E n ¼ À
2ω
2 P n
ε n
þ ξ E ;
d
2 P n
dt 2 þ
1
T 2
dP n
dt
þ 2πν 12
ð
Þ
2 P n ¼
p
2
e
h
NE n þ ξ P ;
dN
dt
¼ α N0 J 0N À
N
T 1
À
1
h
P n E n þ ξ N ;
d
2 U 10MZ
dt 2 þ
1
T n
Á
dU 10MZ
dt
þ 2π f eF0
ð
Þ
2 U 10MZ ¼
dS NY E n , U 10MZ
½
dt
þ ξ U :
8
> > > > > > > > > > > <
> > > > > > > > > > > :
ð5:66Þ
The first three equations in the differential equation system of OEO МZ are
independent with regard to the fourth, which can be introduced in the case of OEO
DM for determination only of the amplitude of laser optical oscillations and PSD of
amplitude and phase noises. In equations, the normalized laser power depends on the
laser equation solution and is determined by the transfer function of the MZ
modulator, which depends on the laser optical frequency closed to ν OF .
5.4 Fluctuation Differential Equations of OEO with the Langevinian Noise Sources
249