i A u PD
ð Þexp Àj2π f 0e T FOS
ð
Þ !
j2π f 0e þp 1
 I 1A cos 2π f 0e T FOS
ð
ÞþjI 1A sin 2π f 0e T FOS
ð
Þ
½
Š
ð3:56Þ
Now we write the abbreviated equation of the first approximation
1
Q EF
þ 2p 1
Á exp jΦ 1
½
ŠÁU 10MZ ¼
K 0FODL
Q EF
 I 1A cos 2π f 0e T FOS
½
ŠþjI 1A sin 2π f 0e T FOS
½
Š
f
g
Á exp jΦ 1
½
Š
ð3:57Þ
Performing differentiation and separation of real and imaginary parts, we have
system of two real equations:
1
Q EF
þ 2p 1
Á U 10MZ ¼
K 0FODL
Q EF
Á I 1A cos 2π f 0e T FOS
½
Š ,
2U 10MZ p 1 Φ 1 ¼
K 0FOLD
Q EF
Á I 1A sin 2π f 0e T FOS
½
Š Þ
8
> > <
> > :
ð3:58Þ
Let us write equations in the time domain replacing p 1 by d/dt:
dU 10MZ
dt
¼ 2π f 0e
K 0FODL
2Q EF
I 1A cos 2π f 0e T FOS
½
ŠÀ2π f 0e Á
U 10MZ
2Q EF
dΦ 10MZ
dt
¼ 2π f 0e
K 0FODL
2Q EF
I 1A
U 10MZ
sin 2π f 0e T FOS
½
Š Þ
8
> > <
> > :
ð3:59Þ
where I 1A ¼ (S 1A /2U 0MZπ )sin[π(U 0MZ À U 10MZ )/(2U 0MZπ )].
The first Eq. (3.59) defines the oscillation amplitude transient, while the second
Eq. (3.59) defines the frequency amendment in the first approximation. The system
of Eq. (3.59) allows determination of the amplitude U 10MZ upon the generated
frequency.
3.4.3.4 Amplitude and Frequency in OEO MZ in the Steady State
In the steady state,
dU 10MZ
dt
¼ 0 and
dΦ 10MZ
dt
¼ 0 and we have the system of equations
two algebraic equations:
122 3 Modulation Methods of Laser Emission in Optoelectronic oscillator (OEO) and OEO. . .