instability (or to chaotic generation). At that, the amplitude and frequency are
defined ambiguously for the same conditions.
4.6.2.7 The Cubic Parametric Function for Odd Powers
of the Characteristic Polynomial
From an analysis of plots in Fig. 4.30b we state that at the cubic parametric function
for odd powers of the characteristic polynomial, the branches of unstable and stable
1500
600
400
200
0
-200
-400
-600
400
300
200
100
0
-100
-200
-300
300
200
100
0
-100
-200
-300
200
0
-200
-400
1000
500
0
x
x
x
x
x
-500
-1000
-1500
0.5 1.0
A = 0.001
A = 0.0075
A = 0.005
a)
b)
A = 0.0081
A = 0.01
1) A = 0.07
3) A = 0.082
2) A = 0.075
4) A = 0.089
1.5
0.5 1.0 1.5
0.5 1.0 1.5
0.5 1.0 1.5
0.5 1.0 1.5
amplitude
x =
E
10n (s =
v /
v
0F ) (u.a.)
amplitude
x =
E
10n (s =
v /
v
0F ) (u.a.)
amplitude
x =
E
10n (s =
v /
v
0F ) (u.a.)
amplitude
x =
E
10n (s =
v /
v
0F ) (u.a.)
amplitude
x =
E
10n (s =
v /
v
0F ) (u.a.)
s = (v / v 0F (u.a.)
normalized optical
frequency
s = (v / v 0F ) (u.a.)
normalized optical
frequency
s = (v / v 0F ) (u.a.)
normalized optical
frequency
s = (v / v 0F ) (u.a.)
normalized optical
frequency
s = (v / v 0F )
1)
s = (v / v 0F )
2)
s = (v / v 0F )
3)
s = (v / v 0F )
4)
s = (v / v 0F )
5)
300
200
100
0
–100
–200
–300
300
200
100
0
-100
-200
-300
400
200
0
-200
-400
400
200
0
-200
-400
0.5 1.0 1.5
0.5 1.0 1.5
0.5 1.0 1.5
0.5 1.0 1.5
Fig. 4.29 The resonance characteristic of QWLD at nonlinear parametric variation of the resonator
Q-factor Q 0F ¼ 1/[0.001(1 À Ax
2
)]. The function of amplitude x ¼ E 10n versus the laser optical
frequency s ¼ j(ν/ν 0F ) at K 00 N 0 ¼ 4.9, T 1n G 00 ¼ 0.00001. For (a)[s
4 + 0.001(1 À Ax
2
)
s
3 + 2.001s
2 + 0.001(1 À Ax
2
)s + 1]x ¼ s
2 Á K 00 N 0 x/(1 + T 1n G 00 x
2
); (1/Q 0F + 1/
Q 02 ) ¼ 0.001 Á (1 À Ax
2
). For (b) [s
2 + 0.01s + 1.001] Á [s
2 + 0.0001(1 À Ax
2 )s + 1.001]
x ¼ s
2 Á K 00 N 0 x/(1 + T 1n G 00 x
2
); (1/Q 0F + 1/Q 02 ) ¼ 0.0001 Á (1 À Ax
2
)
4.6 Solutions of Nonlinear DEs of Fourth Order and Second Order for the Laser
189