dE
2
0L =dt ¼ G 0 E
2
0L N À E
2
0L =T 0F ,
dN=dt ¼ α N00 Á J 0L þ α N01 Á J 1L À
N
T 1L
À G 0 NE
2
0L ,
dφ=dt ¼ 2πν 0P N
ð Þ À 2πν 0 þ σ 0L À ρ 0L E
2
0L ,
d
2 J 1L =dt
2
þ
1
Q EF
dJ 1L =dt þ 2π f 0e
ð
Þ
2 J 1L ¼
E
2
00L K FOS K PD
Q EF
dE
2
0L J 1L , t À T FOS
ð
Þ
dt
:
8
> > > > > > > <
> > > > > > > :
ð5:35Þ
Variables included in the DE system were defined earlier. The distinguished
feature of this system of differential equations is the fact that they connect the
intensity E
2
0L of laser emission with the AC component of the current.
5.3.1 OEO Oscillation Amplitude in the Small-Signal
Single-Frequency Mode
Let the excitation condition be fulfilled in the laser and in OEO and the singlefrequency mode of the OEO RF generation in the frequency f. Considering the
modulation of the pumping current in the form J L ¼ J L00 + J 1L and
J L1 ¼ J 10L cos (2πft + φ L ), where J 1L is much lesser than J L00 , we shall find the
solution of the differential equation system (Eq. 5.35).
We investigate DE (Eq. 5.35) of the laser under conditions of the quasi-stationary
mode, i.e., with the steady-state values E
2
00L , N 00L , φ 00L . For example, in the point A
on the phase portrait in Fig. 5.2. In the small-signal mode of the OEO laser
modulation in the DE system (Eq. 5.64), we take the variable change:
(1) E
2
0L ¼ E
2
00L þ e
2
L , (2) N 0L ¼ N 00L + n l , (3) φ ¼ φ 00L + φ L , where E
2
00L , N 00L ,
φ 00L are solutions of differential equations (Eq. 5.64) in the steady state, and e
2
L , n l ,
φ L are small deviations from the steady-state values E
2
00L , N 00L , φ 00L , relatively.
We assume that in the small-signal mode E 00L ) e l , N 00L ) n l, φ 00L ) φ L . We
introduce in differential equations (Eq. 5.35) the normalized time τ ¼ t/T 1L and
suppose that G ¼ G 0 T 1L . We take into consideration that in the steady-state mode the
following equalities are fulfilled: G 0 E 00L (N 00L À 1/T 0P ) ¼ 0, A 2 ¼
ħνJ L0
eV a
À
N 00L
T 1L
À
G 0 N 00L E
2
00L ¼ J L00 .
Then, differential equations for small deviation of laser parameters e l , n l , and φ L
in the single-frequency mode, at positive feedback, without account of noises, will
be written as:
224
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
2
0L =dt ¼ G 0 E
2
0L N À E
2
0L =T 0F ,
dN=dt ¼ α N00 Á J 0L þ α N01 Á J 1L À
N
T 1L
À G 0 NE
2
0L ,
dφ=dt ¼ 2πν 0P N
ð Þ À 2πν 0 þ σ 0L À ρ 0L E
2
0L ,
d
2 J 1L =dt
2
þ
1
Q EF
dJ 1L =dt þ 2π f 0e
ð
Þ
2 J 1L ¼
E
2
00L K FOS K PD
Q EF
dE
2
0L J 1L , t À T FOS
ð
Þ
dt
:
8
> > > > > > > <
> > > > > > > :
ð5:35Þ
Variables included in the DE system were defined earlier. The distinguished
feature of this system of differential equations is the fact that they connect the
intensity E
2
0L of laser emission with the AC component of the current.
5.3.1 OEO Oscillation Amplitude in the Small-Signal
Single-Frequency Mode
Let the excitation condition be fulfilled in the laser and in OEO and the singlefrequency mode of the OEO RF generation in the frequency f. Considering the
modulation of the pumping current in the form J L ¼ J L00 + J 1L and
J L1 ¼ J 10L cos (2πft + φ L ), where J 1L is much lesser than J L00 , we shall find the
solution of the differential equation system (Eq. 5.35).
We investigate DE (Eq. 5.35) of the laser under conditions of the quasi-stationary
mode, i.e., with the steady-state values E
2
00L , N 00L , φ 00L . For example, in the point A
on the phase portrait in Fig. 5.2. In the small-signal mode of the OEO laser
modulation in the DE system (Eq. 5.64), we take the variable change:
(1) E
2
0L ¼ E
2
00L þ e
2
L , (2) N 0L ¼ N 00L + n l , (3) φ ¼ φ 00L + φ L , where E
2
00L , N 00L ,
φ 00L are solutions of differential equations (Eq. 5.64) in the steady state, and e
2
L , n l ,
φ L are small deviations from the steady-state values E
2
00L , N 00L , φ 00L , relatively.
We assume that in the small-signal mode E 00L ) e l , N 00L ) n l, φ 00L ) φ L . We
introduce in differential equations (Eq. 5.35) the normalized time τ ¼ t/T 1L and
suppose that G ¼ G 0 T 1L . We take into consideration that in the steady-state mode the
following equalities are fulfilled: G 0 E 00L (N 00L À 1/T 0P ) ¼ 0, A 2 ¼
ħνJ L0
eV a
À
N 00L
T 1L
À
G 0 N 00L E
2
00L ¼ J L00 .
Then, differential equations for small deviation of laser parameters e l , n l , and φ L
in the single-frequency mode, at positive feedback, without account of noises, will
be written as:
224
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .