5.2.2 Plots of Oscillation Stability Regions
Now we construct the plots of oscillation stability regions of OEO DM. For this, we
introduce coefficients A ¼ a 2 and B ¼ a 3 and use designations: A ¼ a 2 ¼
2 þ β 0L þ
1
Q 0EF
1
Q 0L
À α N0 G 0 E
4
0L K 0DL S NY00
, and B ¼ a 3 ¼
1þβ 0L
Q 0EF
þ
1
Q 0L
. Let us
write the inequality system, which is deduced from the stability conditions. The
system of two inequality describing the stability region (i.e., the zone of the steadystate point), which oscillations will not be excited, will take the following for:
Δ 2 > 0, Δ 3 > 0. Now we have the inequalities:
1
Q 0EF
þ
1
Q 0L
A À B > 0,
1
Q 0EF
þ
1
Q 0L
AB À
1
Q 0EF
þ
1
Q 0L
1 þ β 0L
ð
Þ
!
À B
2
> 0:
8
> > > <
> > > :
ð5:33Þ
The variety of points, which are above the straight line A ¼
1
Q 0EF
þ
1
Q 0L
À1 Á B,
satisfies the inequality A >
1
Q 0EF
þ
1
Q 0L
À1
B. As a result, the solution of the whole
system is the region of values A and B only, which satisfies the inequality (Eq. 5.33)
having the form:
A >
1
Q 0EF
þ
1
Q 0L
1 þ β 0L
ð
ÞÁ
1
B
þ
1
Q 0EF
þ
1
Q 0L
À1
B:
ð5:34Þ
Figure 5.2 shows plots of the oscillation stability regions of OEO DM without
account of the inertial nonlinearity. The stability region of values A and B is located
in the right upper quadrant (at A > 0 and B > 0) and shown in blue in Fig. 5.2. This
region in the plane (B,A) represents the region of the asymptotic stability of the initial
equation.
If we take into consideration the nonlinear inertia property in Eq. (5.28), we must
expand the function S NY00 e
2
L
À Á
exp ÀjT FOS Á f 0e
ð
Þon the in-phase and quadrature
components and to use the locus method or the Nyquist criterion as we performed in
Chap. 3. It is shown in Chap. 3 that the self-excitation conditions and the oscillation
stability conditions for the similar symbolic equation (Eq. 5.28) in the steady-state
point must be added by conditions of amplitude and phase stability.
Now we can make a conclusion: for the soft OEO excitation mode and for the
oscillation stability in the steady-state point, the slope of the nonlinear function in the
initial state e
2
L ¼ 0 must be positive, and in the steady state e
2
L ¼ e
2
L0 , this slope must
be negative, and the slope module in the initial state must be more than the slope
module in the steady-state point.
222
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
Now we construct the plots of oscillation stability regions of OEO DM. For this, we
introduce coefficients A ¼ a 2 and B ¼ a 3 and use designations: A ¼ a 2 ¼
2 þ β 0L þ
1
Q 0EF
1
Q 0L
À α N0 G 0 E
4
0L K 0DL S NY00
, and B ¼ a 3 ¼
1þβ 0L
Q 0EF
þ
1
Q 0L
. Let us
write the inequality system, which is deduced from the stability conditions. The
system of two inequality describing the stability region (i.e., the zone of the steadystate point), which oscillations will not be excited, will take the following for:
Δ 2 > 0, Δ 3 > 0. Now we have the inequalities:
1
Q 0EF
þ
1
Q 0L
A À B > 0,
1
Q 0EF
þ
1
Q 0L
AB À
1
Q 0EF
þ
1
Q 0L
1 þ β 0L
ð
Þ
!
À B
2
> 0:
8
> > > <
> > > :
ð5:33Þ
The variety of points, which are above the straight line A ¼
1
Q 0EF
þ
1
Q 0L
À1 Á B,
satisfies the inequality A >
1
Q 0EF
þ
1
Q 0L
À1
B. As a result, the solution of the whole
system is the region of values A and B only, which satisfies the inequality (Eq. 5.33)
having the form:
A >
1
Q 0EF
þ
1
Q 0L
1 þ β 0L
ð
ÞÁ
1
B
þ
1
Q 0EF
þ
1
Q 0L
À1
B:
ð5:34Þ
Figure 5.2 shows plots of the oscillation stability regions of OEO DM without
account of the inertial nonlinearity. The stability region of values A and B is located
in the right upper quadrant (at A > 0 and B > 0) and shown in blue in Fig. 5.2. This
region in the plane (B,A) represents the region of the asymptotic stability of the initial
equation.
If we take into consideration the nonlinear inertia property in Eq. (5.28), we must
expand the function S NY00 e
2
L
À Á
exp ÀjT FOS Á f 0e
ð
Þon the in-phase and quadrature
components and to use the locus method or the Nyquist criterion as we performed in
Chap. 3. It is shown in Chap. 3 that the self-excitation conditions and the oscillation
stability conditions for the similar symbolic equation (Eq. 5.28) in the steady-state
point must be added by conditions of amplitude and phase stability.
Now we can make a conclusion: for the soft OEO excitation mode and for the
oscillation stability in the steady-state point, the slope of the nonlinear function in the
initial state e
2
L ¼ 0 must be positive, and in the steady state e
2
L ¼ e
2
L0 , this slope must
be negative, and the slope module in the initial state must be more than the slope
module in the steady-state point.
222
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .