p
2
þ 1=T EF
ð
Þp þ 2π f 0e
ð
Þ
2
h
i
u MZ ¼ 1=T EF
ð
ÞK 0FODL pi A u PD
ð Þexp Àj2π f 0e T FOS
ð
Þ ;
ð3:52Þ
in which the instantaneous voltage on the load resistance R PD of PD is u PD ¼ R PD Á i PD ,
and the instantaneous PD current i PD is
i PD ¼ 0:5 E 0L
ð Þ
2 K 0FOS:MZ cos π U 0MZ À u MZ
ð
Þ =2U 0MZπ
½
Þ ,
ð3:53Þ
where (E 0L )
2 is the power of laser emission, U 0MZ is the DC bias voltage of the MZ
modulator, and K 0FOS.MZ is the FODL transfer function on the frequency f 0e .
Following the Evtianov approach to abbreviation of differential equations
[14, 15], in the first approximation for slowly changing amplitude U 10MZ and
phase Φ MZ of oscillations in the OEO output we have: u MZ ¼ U 10MZ Re[exp
( j2πft À Φ MZ )].
Let us perform a normalization dividing the right and left parts of Eq. (3.52)
equation by (2πf 0e )
2 , p/(2πf 0e ), and introduce the Q-factor Q EF ¼ T EF Á f 0e :
p=2π f 0e
ð
Þ
2 þ 1=Q EF
ð
ÞÁ p=2π f 0e
ð
Þþ1
h
i
u MZ
p=2π f 0e
ð
Þ
¼
K 0FODL
Q eF
i A u PD
ð Þexp Àj2π f 0e T FOS
ð
Þ ;
ð3:54Þ
According to Evtianov method, we must find the biased operator. In order to
obtain the abbreviated differential equations of the first approximation, it is necessary [14]: to represent a solution in the form of harmonic oscillation with slowly
changing amplitude and phase and to substitute it into the initial Eq. (3.47) or (3.54).
The current should be represented by the Fourier series keeping on the right the main
harmonic only. Then we use the bias theorem of the operation calculus; then we
expand the biased operators in the Taylor on p/(2πf 0e ), keeping the first (nonzero)
terms in this expansion.
We choose 2πf 0e as the reference frequency. Replacing p to j2πf 0e + p 1 , combining terms in groups according to the smallness order, keeping only the first order of
smallness terms, we obtain the expression for the abbreviated conductance in the first
approximation in the left part of Eq. (3.54) [14]:
p=2π f 0e
ð
Þ
2 þ 1=Q EF
ð
Þ p=2π f 0e
ð
Þþ1
h
i
p=2π f 0e
ð
Þ
!
j2π f 0e þp 1
1
Q EF
þ 2p 1
ð3:55Þ
where the arrow “ !
j2π f 0e þp 1 ” designates the representation of the control conductance
in the abbreviated form. At abbreviation, we expand the current in the right part I 1A
into in-phase and quadratic components:
3.4 Equations of Amplitude and Phase Balance for the Laser and for OEO
121
2
þ 1=T EF
ð
Þp þ 2π f 0e
ð
Þ
2
h
i
u MZ ¼ 1=T EF
ð
ÞK 0FODL pi A u PD
ð Þexp Àj2π f 0e T FOS
ð
Þ ;
ð3:52Þ
in which the instantaneous voltage on the load resistance R PD of PD is u PD ¼ R PD Á i PD ,
and the instantaneous PD current i PD is
i PD ¼ 0:5 E 0L
ð Þ
2 K 0FOS:MZ cos π U 0MZ À u MZ
ð
Þ =2U 0MZπ
½
Þ ,
ð3:53Þ
where (E 0L )
2 is the power of laser emission, U 0MZ is the DC bias voltage of the MZ
modulator, and K 0FOS.MZ is the FODL transfer function on the frequency f 0e .
Following the Evtianov approach to abbreviation of differential equations
[14, 15], in the first approximation for slowly changing amplitude U 10MZ and
phase Φ MZ of oscillations in the OEO output we have: u MZ ¼ U 10MZ Re[exp
( j2πft À Φ MZ )].
Let us perform a normalization dividing the right and left parts of Eq. (3.52)
equation by (2πf 0e )
2 , p/(2πf 0e ), and introduce the Q-factor Q EF ¼ T EF Á f 0e :
p=2π f 0e
ð
Þ
2 þ 1=Q EF
ð
ÞÁ p=2π f 0e
ð
Þþ1
h
i
u MZ
p=2π f 0e
ð
Þ
¼
K 0FODL
Q eF
i A u PD
ð Þexp Àj2π f 0e T FOS
ð
Þ ;
ð3:54Þ
According to Evtianov method, we must find the biased operator. In order to
obtain the abbreviated differential equations of the first approximation, it is necessary [14]: to represent a solution in the form of harmonic oscillation with slowly
changing amplitude and phase and to substitute it into the initial Eq. (3.47) or (3.54).
The current should be represented by the Fourier series keeping on the right the main
harmonic only. Then we use the bias theorem of the operation calculus; then we
expand the biased operators in the Taylor on p/(2πf 0e ), keeping the first (nonzero)
terms in this expansion.
We choose 2πf 0e as the reference frequency. Replacing p to j2πf 0e + p 1 , combining terms in groups according to the smallness order, keeping only the first order of
smallness terms, we obtain the expression for the abbreviated conductance in the first
approximation in the left part of Eq. (3.54) [14]:
p=2π f 0e
ð
Þ
2 þ 1=Q EF
ð
Þ p=2π f 0e
ð
Þþ1
h
i
p=2π f 0e
ð
Þ
!
j2π f 0e þp 1
1
Q EF
þ 2p 1
ð3:55Þ
where the arrow “ !
j2π f 0e þp 1 ” designates the representation of the control conductance
in the abbreviated form. At abbreviation, we expand the current in the right part I 1A
into in-phase and quadratic components:
3.4 Equations of Amplitude and Phase Balance for the Laser and for OEO
121