dE
2
10L =dt ¼ G 0 E
2
10L N 0L À E
2
10L =T 0F ,
dN 0L =dt ¼ α 00 Á J 0L þ α N01 Á J 1L À
N 0L
T 1L
À G 0 N 0L E
2
10L ,
dΦ 10L =dt ¼ 2 Á π Á ν 0P N 0L
ð
ÞÀ2πν 0 þ σ 0L À ρ 0L Á E
2
10L ,
dJ 10L
dt
¼ 2π f 0e
E
2
10L K DL
2R L Q EF
I 1A U PD , E
2
10L
À
Á
cos½2π f 0e T FOS À 2π f 0e Á
J 10L
2Q EF
,
dΦ 10J
dt
¼ 2π f 0e
E
2
10L K DL
2Q EF
I 1A U PD , E
2
10L
À
Á
sin½2π f 0e T FOS Þ,
8
> > > > > > > > > > > > <
> > > > > > > > > > > > :
ð5:15Þ
where the coefficient K DL is K DL ¼ cos [Δϕ OF ](1/T eF )K OF K PD .
The transfer function of the feedback network K FBN ¼ K DL for OEO DM can be
defined as K DL ¼ K FBN ¼
i L
E n
j j
2 ¼
i L
E L
j j
2 , where E L is the normalized strength in the
QWLD output, which is equal in magnitude to the value in the FOS input, i L ¼ i m is
the AC current modulation component in the QWLD in the OEO DM.
The presented system of abbreviated equations (Eq. 5.15) from five equations
gives a possibility to obtain the steady-state quantities of E
2
10L , N 0L , Φ 10L , J 10L , Φ 10J ,
as well as the frequency deviations of the laser optical emission and the OEO radio
frequency from their reference values. The system (Eq. 5.15) gives a possibility to
investigate the dynamics of the transients in OEO DM. The second equation for the
population N 0L in the system (Eq. 5.26) complicates the analysis but takes into
account the generation development more accurately.
The analog model of QWLD, which is obtained on the base of abbreviated
equations deduced in this section, is presented in Fig. 2.2. The pumping block is
modeled by the circuit, as shown in Fig. 2.2.
The description of the main components of the analog model (Fig. 2.2) is
presented in Chap. 3. We remind that the laser model consists of two closed loops,
in one of which the oscillations of the strength E n and the polarization P n are
circulated, while in the other loop, the population is circulated. The analog laser
model reflects the functional connection of the laser model parameters. The model
gives understanding how the oscillation amplitude establishes and how the time
constants T 0F , T 2 , T 1 differences between frequencies ν 0n and ν 12 , and also the Qfactor of the RF filter Q EF and its natural frequency f F0 ¼ f 0e affect laser dynamics.
Now we transfer to the stability analysis of OEO oscillating system and investigate the self-excitation conditions and existence conditions in ОЕО DM.
5.2 Stability Conditions: Self-Excitation and Oscillation
Existence Conditions in ОЕО DM
For stability analysis of differential equations system, we consider oscillations of E n
and i L with average frequencies ν 0 and f 0 . As initial, we take the system of
differential equations of OEO DM for the strength amplitude square E
2
n of the
214
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
2
10L =dt ¼ G 0 E
2
10L N 0L À E
2
10L =T 0F ,
dN 0L =dt ¼ α 00 Á J 0L þ α N01 Á J 1L À
N 0L
T 1L
À G 0 N 0L E
2
10L ,
dΦ 10L =dt ¼ 2 Á π Á ν 0P N 0L
ð
ÞÀ2πν 0 þ σ 0L À ρ 0L Á E
2
10L ,
dJ 10L
dt
¼ 2π f 0e
E
2
10L K DL
2R L Q EF
I 1A U PD , E
2
10L
À
Á
cos½2π f 0e T FOS À 2π f 0e Á
J 10L
2Q EF
,
dΦ 10J
dt
¼ 2π f 0e
E
2
10L K DL
2Q EF
I 1A U PD , E
2
10L
À
Á
sin½2π f 0e T FOS Þ,
8
> > > > > > > > > > > > <
> > > > > > > > > > > > :
ð5:15Þ
where the coefficient K DL is K DL ¼ cos [Δϕ OF ](1/T eF )K OF K PD .
The transfer function of the feedback network K FBN ¼ K DL for OEO DM can be
defined as K DL ¼ K FBN ¼
i L
E n
j j
2 ¼
i L
E L
j j
2 , where E L is the normalized strength in the
QWLD output, which is equal in magnitude to the value in the FOS input, i L ¼ i m is
the AC current modulation component in the QWLD in the OEO DM.
The presented system of abbreviated equations (Eq. 5.15) from five equations
gives a possibility to obtain the steady-state quantities of E
2
10L , N 0L , Φ 10L , J 10L , Φ 10J ,
as well as the frequency deviations of the laser optical emission and the OEO radio
frequency from their reference values. The system (Eq. 5.15) gives a possibility to
investigate the dynamics of the transients in OEO DM. The second equation for the
population N 0L in the system (Eq. 5.26) complicates the analysis but takes into
account the generation development more accurately.
The analog model of QWLD, which is obtained on the base of abbreviated
equations deduced in this section, is presented in Fig. 2.2. The pumping block is
modeled by the circuit, as shown in Fig. 2.2.
The description of the main components of the analog model (Fig. 2.2) is
presented in Chap. 3. We remind that the laser model consists of two closed loops,
in one of which the oscillations of the strength E n and the polarization P n are
circulated, while in the other loop, the population is circulated. The analog laser
model reflects the functional connection of the laser model parameters. The model
gives understanding how the oscillation amplitude establishes and how the time
constants T 0F , T 2 , T 1 differences between frequencies ν 0n and ν 12 , and also the Qfactor of the RF filter Q EF and its natural frequency f F0 ¼ f 0e affect laser dynamics.
Now we transfer to the stability analysis of OEO oscillating system and investigate the self-excitation conditions and existence conditions in ОЕО DM.
5.2 Stability Conditions: Self-Excitation and Oscillation
Existence Conditions in ОЕО DM
For stability analysis of differential equations system, we consider oscillations of E n
and i L with average frequencies ν 0 and f 0 . As initial, we take the system of
differential equations of OEO DM for the strength amplitude square E
2
n of the
214
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .