electrical component of EMF with the amplitude of the first harmonic E 10L of
E n ¼ E 10L Re [exp(2πν 0 t)] and the AC electrical component i L ¼ J 10L Re [exp
(2πf 0 t)] with the amplitude of the first harmonic of the laser pumping current
J 1L ¼ J 10L .
We write the laser abbreviated equation taking into account the population
equation for slowly changing functions E 0n , φ, the population difference N(t), and
we present the types of abbreviated equations, which are deduced from the considered differential equation for the laser:
dE
2
0L =dt ¼ G 0 E
2
0L N 0L À E
2
0L =T 0F ,
dN 0L =dt ¼ α N00 Á J 0L þ α N01 Á J 1L À
N 0L
T 1
À G 0 N 0L E
2
0L ,
dφ=dt ¼ 2πν 0P ðN 0L Þ À 2πν 0 þ σ 0L þ ρ 0L E
2
0L :
8
> > <
> > :
ð5:16Þ
We must add this system by differential equations for the circuit of the positive
feedback.
5.2.1 The Circuit of the Positive Feedback Spanned
the Laser: Differential Equations of OEO
The circuit of the positive feedback covered of QWLD consists of series-connected
the fiber-optical system (FOS), which includes optical filters, optical amplifiers, and
optical modulators; the photodetector (FD), the nonlinear RF amplifier, the RF filter
with the natural frequency, and the coupler. The transfer function K FB of the
“Feedback Chain” circuit (Fig. 5.1a) can be defined as for OEO DM: K DL ¼
K FB ¼
i 1L
E
2
n
¼
J 1L
E
2
n
, where E n ¼ E L is the normalized strength in the QWLD output,
which is equal to the value of the FOS input, i 1L ¼ i m is the AC current component in
the QWLD input in the structure of OEO DM and u m is the AC component of the
voltage in MZ input in the structure of OEO MZ. Now we have the following
symbolic equation:
J 1L ¼
cos Δϕ OF
ð
Þ
½
E n
j j
2 1=T EF
ð
ÞK OF K PD p exp ÀpT DL
ð
ÞS NY J 1L
ð Þ
p 2 þ 1=T EF
ð
Þp þ 2π f 0e
ð
Þ
2
h
i
:
ð5:17Þ
Taking into consideration the circuit of positive FB, we transfer to equation
system in the time domain for OEO DM:
5.2 Stability Conditions: Self-Excitation and Oscillation Existence Conditions in. . .
215
E n ¼ E 10L Re [exp(2πν 0 t)] and the AC electrical component i L ¼ J 10L Re [exp
(2πf 0 t)] with the amplitude of the first harmonic of the laser pumping current
J 1L ¼ J 10L .
We write the laser abbreviated equation taking into account the population
equation for slowly changing functions E 0n , φ, the population difference N(t), and
we present the types of abbreviated equations, which are deduced from the considered differential equation for the laser:
dE
2
0L =dt ¼ G 0 E
2
0L N 0L À E
2
0L =T 0F ,
dN 0L =dt ¼ α N00 Á J 0L þ α N01 Á J 1L À
N 0L
T 1
À G 0 N 0L E
2
0L ,
dφ=dt ¼ 2πν 0P ðN 0L Þ À 2πν 0 þ σ 0L þ ρ 0L E
2
0L :
8
> > <
> > :
ð5:16Þ
We must add this system by differential equations for the circuit of the positive
feedback.
5.2.1 The Circuit of the Positive Feedback Spanned
the Laser: Differential Equations of OEO
The circuit of the positive feedback covered of QWLD consists of series-connected
the fiber-optical system (FOS), which includes optical filters, optical amplifiers, and
optical modulators; the photodetector (FD), the nonlinear RF amplifier, the RF filter
with the natural frequency, and the coupler. The transfer function K FB of the
“Feedback Chain” circuit (Fig. 5.1a) can be defined as for OEO DM: K DL ¼
K FB ¼
i 1L
E
2
n
¼
J 1L
E
2
n
, where E n ¼ E L is the normalized strength in the QWLD output,
which is equal to the value of the FOS input, i 1L ¼ i m is the AC current component in
the QWLD input in the structure of OEO DM and u m is the AC component of the
voltage in MZ input in the structure of OEO MZ. Now we have the following
symbolic equation:
J 1L ¼
cos Δϕ OF
ð
Þ
½
E n
j j
2 1=T EF
ð
ÞK OF K PD p exp ÀpT DL
ð
ÞS NY J 1L
ð Þ
p 2 þ 1=T EF
ð
Þp þ 2π f 0e
ð
Þ
2
h
i
:
ð5:17Þ
Taking into consideration the circuit of positive FB, we transfer to equation
system in the time domain for OEO DM:
5.2 Stability Conditions: Self-Excitation and Oscillation Existence Conditions in. . .
215