The self-excitation condition (Eq. 5.31) has the clear physical sense: the equivalent gain in the feedback loop of OEO α N0 G 0 E
4
0L K 0DL multiplied by the average
slope of the RF amplifier S NY00 should exceed losses in the resonator of the RF
amplifier
1
Q 0EF
and the reduced losses in the optical resonator of the laser
1
Q 0L
. The
slope of the nonlinear dependence should be more than zero: S NY00 e
2
L ¼ 0
À
Á > 0,
and the detuning of the natural frequencies β 0L ¼ ω
2
0L À ω
2
0e
À
Á =ω
2
0e introduces in the
OEO self-excitation conditions of small correction: the increase of β 0L leads to the
growth of the excitation threshold of OEO.
We should note that in the case, if the right part of DE (Eq. 5.28), i.e., the operator
controlling transfer function, contains the operator in the first order p 00 or the third
order p
3
00 , the self-excitation conditions take the form, which is traditional for RF
oscillators: α N0 G 0 E
4
0L K 0DL S NY00 >
1
Q 0EF
þ
1
Q 0L , α N0 G 0 E
4
0L K 0DL S NY00 >
1þβ 0L
Q 0EF
þ
1
Q 0L .
These conditions take a form: Q 0EF > Q 0L :Q 0L α N0 G 0 E
4
0L K 0DL S NY00 > 1.
The last inequality can be treated by analogy with the well-known (from the
oscillation theory) self-excitation condition of RF oscillator as: S Á R cont > 1.
In S Á R cont > 1, the slope S of the nonlinear function of the nonlinear element is
equal to the derivative of S ¼ S NY00 , and the controlling resistance R cont is:
R cont ¼ Q 0L α N0 G 0 E
4
0L K 0DL .
We note that R cont is the product of terms reflecting the transfer function on the
closed feedback loop of OEO: the Q-factors Q 0L , pumping α N0 , the saturation
coefficient G 0 , the square of intensity E
4
0L , the transfer function on the feedback
loop K 0FODL. Remarkably that the function of the normalized field strength of the
laser has the fourth order.
Oscillation stability conditions of OEO DM in the steady-state point e
2
L ¼ e
2
L0
for the appropriate stationary values of E
2
00L , N 00 , and J 0L are found in the similar
manner and have the form of inequality:
α N0 G 0 E
4
0L K 0DL S NY00 < 2 þ β 0L þ
1
Q 0EF
1
Q 0L
À
1
Q 0EF
þ
1
Q 0L
1 þ β 0L
ð
Þ
2 : ð5:32Þ
For inequality (Eq. 5.31) fulfillment, the necessary condition is the condition
S NY00 e
2
L0
À Á < 0 fulfillment in the steady-state point e
2
L ¼ e
2
L0 , and the slope module
in the initial state must be more than the slope module of the nonlinear function in the
steady-state point.
We note that in the case when the right part of Eq. (5.28), i.e., the controlling
operator transfer function, contains the operator in the first order p 00 or in the third
order p
3
00 , and the oscillation stability conditions in the steady-state point take the
form, which is traditional for the RF oscillators at S NY00 e
2
L0
À Á < 0 :
α N0 G 0 E
4
0L K 0DL S NY00 <
1
Q 0EF
þ
1
Q 0L , and α N0 G 0 E
4
0L K 0DL S NY00 <
1þβ 0L
Q 0EF
þ
1
Q 0L .
5.2 Stability Conditions: Self-Excitation and Oscillation Existence Conditions in. . .
221
4
0L K 0DL multiplied by the average
slope of the RF amplifier S NY00 should exceed losses in the resonator of the RF
amplifier
1
Q 0EF
and the reduced losses in the optical resonator of the laser
1
Q 0L
. The
slope of the nonlinear dependence should be more than zero: S NY00 e
2
L ¼ 0
À
Á > 0,
and the detuning of the natural frequencies β 0L ¼ ω
2
0L À ω
2
0e
À
Á =ω
2
0e introduces in the
OEO self-excitation conditions of small correction: the increase of β 0L leads to the
growth of the excitation threshold of OEO.
We should note that in the case, if the right part of DE (Eq. 5.28), i.e., the operator
controlling transfer function, contains the operator in the first order p 00 or the third
order p
3
00 , the self-excitation conditions take the form, which is traditional for RF
oscillators: α N0 G 0 E
4
0L K 0DL S NY00 >
1
Q 0EF
þ
1
Q 0L , α N0 G 0 E
4
0L K 0DL S NY00 >
1þβ 0L
Q 0EF
þ
1
Q 0L .
These conditions take a form: Q 0EF > Q 0L :Q 0L α N0 G 0 E
4
0L K 0DL S NY00 > 1.
The last inequality can be treated by analogy with the well-known (from the
oscillation theory) self-excitation condition of RF oscillator as: S Á R cont > 1.
In S Á R cont > 1, the slope S of the nonlinear function of the nonlinear element is
equal to the derivative of S ¼ S NY00 , and the controlling resistance R cont is:
R cont ¼ Q 0L α N0 G 0 E
4
0L K 0DL .
We note that R cont is the product of terms reflecting the transfer function on the
closed feedback loop of OEO: the Q-factors Q 0L , pumping α N0 , the saturation
coefficient G 0 , the square of intensity E
4
0L , the transfer function on the feedback
loop K 0FODL. Remarkably that the function of the normalized field strength of the
laser has the fourth order.
Oscillation stability conditions of OEO DM in the steady-state point e
2
L ¼ e
2
L0
for the appropriate stationary values of E
2
00L , N 00 , and J 0L are found in the similar
manner and have the form of inequality:
α N0 G 0 E
4
0L K 0DL S NY00 < 2 þ β 0L þ
1
Q 0EF
1
Q 0L
À
1
Q 0EF
þ
1
Q 0L
1 þ β 0L
ð
Þ
2 : ð5:32Þ
For inequality (Eq. 5.31) fulfillment, the necessary condition is the condition
S NY00 e
2
L0
À Á < 0 fulfillment in the steady-state point e
2
L ¼ e
2
L0 , and the slope module
in the initial state must be more than the slope module of the nonlinear function in the
steady-state point.
We note that in the case when the right part of Eq. (5.28), i.e., the controlling
operator transfer function, contains the operator in the first order p 00 or in the third
order p
3
00 , and the oscillation stability conditions in the steady-state point take the
form, which is traditional for the RF oscillators at S NY00 e
2
L0
À Á < 0 :
α N0 G 0 E
4
0L K 0DL S NY00 <
1
Q 0EF
þ
1
Q 0L , and α N0 G 0 E
4
0L K 0DL S NY00 <
1þβ 0L
Q 0EF
þ
1
Q 0L .
5.2 Stability Conditions: Self-Excitation and Oscillation Existence Conditions in. . .
221