distortions. Remarkably, that the level of nonlinear distortions depends upon the
initial conditions. The differential second-order equation solutions with the cosine
nonlinearity, which are presented in Fig. 3.22 for different K 0FODL /Q EF ¼ 5,
β 00 ¼ 2.0 and various initial conditions. These plots allow estimation how the
modulation index β 00 of the MZ modulator influences on development of oscillating
processes. In particular, we may speak the growing of the modulation index from
β 00 ¼ 2.0 to β 00 ¼ 10.5 leads not only to the growth of the second harmonic
(Fig. 3.22c), but to the frequency doubling (Fig. 3.22d).
Thus, we perform the primary analysis of the modulation methods and investigate
the main modes of OEO MZ taking into consideration the OEO block-representation
on the base of the amplitude and phase balance equations, the symbolic and
abbreviated differential equations. This analysis permits to answer the questions
about the oscillation amplitude and frequency in the steady-state modes. The derived
differential equations allow investigation of transients of OEO amplitude and
4
2
0
x
= U
1MZ / U
0MZπ
–2
–4
0
(s 2 + 0.0005 s + 1) x = 5 s cos(0.1 x)
1 2
a)
b)
c)
d)
e)
f)
1.5
x
x
2
1
1
–2
1.0
0.5
–0.5
–3 –2 –1
1
2
–2 –1
1
2
s
s
x
2
3
1
1
–2
–3 –2 –1
1 2 3
s
–1.0
–1.5
s = j ( f / f 0e )
3 4 5
2
0
1
3
1.5
1.0
0.5
–0.5
–2
1
2
s
x
–1.0
–1.5
x
= U
1MZ / U
0MZπ
x
= U
1MZ / U
0MZπ
–2
–1
–3
0
(s 2 + 0.0005 s + 1) x = 5 s cos(3 x)
(s 2 + 0.05 s 1 + 1) x = 3 s cos(5.1x)
(s 2 + 0.01 s + 1) x = –2.5 cos –10.5 x
4
(s 2 + 0.01 s + 1) x = 5 cos(10.1x)
1 2
s = j ( f / f 0e )
s = j ( f / f 0e )
3 4 5
S
(s 2 + 0.01 s 1 + 1) x = –2.5 s sin –10.5 x + 1 × 4
S
Fig. 3.21 Functions of amplitude versus frequency in OEO MZ at constant MZ bias voltage at
modulator operation in the “non-quadrature” mode (m 00 π ¼ π) for gains K 0FOLD /Q EF ¼ 5 (a),
K 0FOLD /Q EF ¼ 5 (b). In figure the solvable equations are shown at 1/Q EF ¼ 0.0005 (a), at
nonlinearity cos[β 00 x], at the modulation index β 00 ¼ 0.1 (a). For the gain K 0FOLD /Q EF ¼ 5 and
the nonlinearity cos[β 00 x], β 00 ¼ 3.0 for 1/Q EF ¼ 0.0005 (b). For the gain K 0FOLD /Q EF ¼ 2.5 and the
nonlinearity cos[φ 0 + m 00 π/2 + β 00 x] ¼ Àcos[(π/4) + (π/2) À 10.5x], for β 00 ¼ 10.5, 1/Q EF ¼ 0.01
(c). Functions of amplitude versus frequency in OEO MZ at different values of constant MZ bias
voltage at different gains. Above figures, there are solvable equations (d–f)
3.4 Equations of Amplitude and Phase Balance for the Laser and for OEO
127
initial conditions. The differential second-order equation solutions with the cosine
nonlinearity, which are presented in Fig. 3.22 for different K 0FODL /Q EF ¼ 5,
β 00 ¼ 2.0 and various initial conditions. These plots allow estimation how the
modulation index β 00 of the MZ modulator influences on development of oscillating
processes. In particular, we may speak the growing of the modulation index from
β 00 ¼ 2.0 to β 00 ¼ 10.5 leads not only to the growth of the second harmonic
(Fig. 3.22c), but to the frequency doubling (Fig. 3.22d).
Thus, we perform the primary analysis of the modulation methods and investigate
the main modes of OEO MZ taking into consideration the OEO block-representation
on the base of the amplitude and phase balance equations, the symbolic and
abbreviated differential equations. This analysis permits to answer the questions
about the oscillation amplitude and frequency in the steady-state modes. The derived
differential equations allow investigation of transients of OEO amplitude and
4
2
0
x
= U
1MZ / U
0MZπ
–2
–4
0
(s 2 + 0.0005 s + 1) x = 5 s cos(0.1 x)
1 2
a)
b)
c)
d)
e)
f)
1.5
x
x
2
1
1
–2
1.0
0.5
–0.5
–3 –2 –1
1
2
–2 –1
1
2
s
s
x
2
3
1
1
–2
–3 –2 –1
1 2 3
s
–1.0
–1.5
s = j ( f / f 0e )
3 4 5
2
0
1
3
1.5
1.0
0.5
–0.5
–2
1
2
s
x
–1.0
–1.5
x
= U
1MZ / U
0MZπ
x
= U
1MZ / U
0MZπ
–2
–1
–3
0
(s 2 + 0.0005 s + 1) x = 5 s cos(3 x)
(s 2 + 0.05 s 1 + 1) x = 3 s cos(5.1x)
(s 2 + 0.01 s + 1) x = –2.5 cos –10.5 x
4
(s 2 + 0.01 s + 1) x = 5 cos(10.1x)
1 2
s = j ( f / f 0e )
s = j ( f / f 0e )
3 4 5
S
(s 2 + 0.01 s 1 + 1) x = –2.5 s sin –10.5 x + 1 × 4
S
Fig. 3.21 Functions of amplitude versus frequency in OEO MZ at constant MZ bias voltage at
modulator operation in the “non-quadrature” mode (m 00 π ¼ π) for gains K 0FOLD /Q EF ¼ 5 (a),
K 0FOLD /Q EF ¼ 5 (b). In figure the solvable equations are shown at 1/Q EF ¼ 0.0005 (a), at
nonlinearity cos[β 00 x], at the modulation index β 00 ¼ 0.1 (a). For the gain K 0FOLD /Q EF ¼ 5 and
the nonlinearity cos[β 00 x], β 00 ¼ 3.0 for 1/Q EF ¼ 0.0005 (b). For the gain K 0FOLD /Q EF ¼ 2.5 and the
nonlinearity cos[φ 0 + m 00 π/2 + β 00 x] ¼ Àcos[(π/4) + (π/2) À 10.5x], for β 00 ¼ 10.5, 1/Q EF ¼ 0.01
(c). Functions of amplitude versus frequency in OEO MZ at different values of constant MZ bias
voltage at different gains. Above figures, there are solvable equations (d–f)
3.4 Equations of Amplitude and Phase Balance for the Laser and for OEO
127