where the quadrature S LIm (ω) and in-phase S LRe (ω) PSD are equal, relatively
Eqs. (6.20) and (6.21).
Expressions (6.26) and (6.27) give a possibility to compute PSD of amplitude and
phase noises in OEO MZ.
6.5 Differential Fluctuation Equations of OEO MZ
Using expressions (6.20)–(6.27), according to the standard approach, the differential
equation is obtained for the instantaneous voltage u (in the OEO MZ output) for the
closed loop of OEO MZ, which contains RF FODL and the Mach–Zehnder optical
modulator taking into account the detected fluctuations of the optical carrier
[2, 3]. At that, the differential equation for the instantaneous voltage u(t) of OEO
MZ is written as:
d
2 u
dt 2 þ
1
T F
du
dt
þ 2π f F0
ð
Þ
2 u ¼
dS NA P 0L K 0FODL Á u t À T FOLD
ð
Þ
½
dt
þ Ψ n ,
ð6:28Þ
where Ψ n is the noise component of the PD voltage formed by the fluctuations of
amplitude and phase of the QWLD optical carrier, at that, Ψ n ¼ S NA K 0FODL μ n . The
equivalent circuit of examined here OEO with account of noises Ψ n ¼ u N (t) is
presented in Fig. 6.12. The noise component Ψ n is added with the RF voltage of
PD. The differential equation (6.28) obtained by authors is used for the analysis of
PSD of the phase noise and for formation of OEO MZ spectrum taking into account
the noises effect in the transients, and analysis results are presented below in the
section of computer modeling.
The differential equation (6.28) with fluctuations permits to transfer to abbreviated differential equations of OEO for fluctuations of amplitude m L (t) and phase
ψ m (t) of oscillations with the help of the standard operation of averaging over the fast
motions. The solution of these abbreviated differential equations allows searching of
necessary required analytical dependences for PSD of the amplitude S mOEO and
phase S ΨOEO fluctuations of the RF oscillations in OEO MZ.
Now we obtain expressions for the PSD of phase noises of RF oscillations in
OEO. For this, we examine the abbreviated differential fluctuation equations of OEO
MZ (Fig. 6.12a).
The differential equation (6.28) for the instantaneous voltage u in the OEO output
does not permit a possibility to connect the appropriate input and output currents in
the closed OEO system for each “block” of NA, F, RF FODL. This can be done at
formation of symbolic abbreviated differential equations with fluctuations for OEO
MZ and at introduction of symbolic (operator) normalized control conductance
Y con ( p). The form of equations completely coincides with the same equations for
the transistor oscillator [2, 3].
6.5 Differential Fluctuation Equations of OEO MZ
311
Eqs. (6.20) and (6.21).
Expressions (6.26) and (6.27) give a possibility to compute PSD of amplitude and
phase noises in OEO MZ.
6.5 Differential Fluctuation Equations of OEO MZ
Using expressions (6.20)–(6.27), according to the standard approach, the differential
equation is obtained for the instantaneous voltage u (in the OEO MZ output) for the
closed loop of OEO MZ, which contains RF FODL and the Mach–Zehnder optical
modulator taking into account the detected fluctuations of the optical carrier
[2, 3]. At that, the differential equation for the instantaneous voltage u(t) of OEO
MZ is written as:
d
2 u
dt 2 þ
1
T F
du
dt
þ 2π f F0
ð
Þ
2 u ¼
dS NA P 0L K 0FODL Á u t À T FOLD
ð
Þ
½
dt
þ Ψ n ,
ð6:28Þ
where Ψ n is the noise component of the PD voltage formed by the fluctuations of
amplitude and phase of the QWLD optical carrier, at that, Ψ n ¼ S NA K 0FODL μ n . The
equivalent circuit of examined here OEO with account of noises Ψ n ¼ u N (t) is
presented in Fig. 6.12. The noise component Ψ n is added with the RF voltage of
PD. The differential equation (6.28) obtained by authors is used for the analysis of
PSD of the phase noise and for formation of OEO MZ spectrum taking into account
the noises effect in the transients, and analysis results are presented below in the
section of computer modeling.
The differential equation (6.28) with fluctuations permits to transfer to abbreviated differential equations of OEO for fluctuations of amplitude m L (t) and phase
ψ m (t) of oscillations with the help of the standard operation of averaging over the fast
motions. The solution of these abbreviated differential equations allows searching of
necessary required analytical dependences for PSD of the amplitude S mOEO and
phase S ΨOEO fluctuations of the RF oscillations in OEO MZ.
Now we obtain expressions for the PSD of phase noises of RF oscillations in
OEO. For this, we examine the abbreviated differential fluctuation equations of OEO
MZ (Fig. 6.12a).
The differential equation (6.28) for the instantaneous voltage u in the OEO output
does not permit a possibility to connect the appropriate input and output currents in
the closed OEO system for each “block” of NA, F, RF FODL. This can be done at
formation of symbolic abbreviated differential equations with fluctuations for OEO
MZ and at introduction of symbolic (operator) normalized control conductance
Y con ( p). The form of equations completely coincides with the same equations for
the transistor oscillator [2, 3].
6.5 Differential Fluctuation Equations of OEO MZ
311