S RFMZ has the Lorentzian view and is defined by the laser phase noise, and the
maximal value of S RFMZ is defined by the ratio ΔT MZ /T coh . As a rule, ΔT MZ /T coh is
small, i.e., less than 0.001 and exp À
ΔT MZ
T coh
% 1.
3.1.3.3 Oscillation Spectrum at the Closed Feedback Loop
At closed Sw switch in Fig. 3.4a, at fulfillment of OEO excitation conditions, the
amplified electrical oscillation passes to the electrical input of the MZ modulator
from the PD output. We note that this oscillation is delayed by the time of fiber
optical system T FOS relative to the laser oscillation arriving to PD.
At that, on the PD area, there is the process of statistical averaging in time of the
interference result of two optical harmonics ν 0L and ν 0L + f, which are applied to the
PD area:
E 10L t
ð Þ ¼ A 1 E 0L cos 2πν 0L t þ φ 0L þ φ 10Lm t
ð Þ
½
ð 3:9Þ
and
E 20L t
ð Þ ¼ A 2 E 0L cos 2π ν 0L þ f 0
ð
Þ t þ φ 0L þ φ 20Lm t
ð Þ
½
,
ð3:10Þ
with the amplitudes A 1 E 0L and A 2 E 0L .
As shown in Chaps. 5 and 6 of this book, in this case, the RF spectrum at the PD
output S RFL (F) is defined by the formula:
S RFL F
ð Þ ¼ K
2
2ΓPN F
ð Þ Á
1= Δν L
ð
Þ
1 þ F À ν L0
ð
Þ =Δν L
½
2
Á 1 À
A 2
A 1
exp À
T FOS
T coh
&
'
, ð3:11Þ
in which K
2
2ΓPN F
ð Þ is the coefficient of phase noise suppression, which depends on
T FOS , on the laser optical power P 0L , σ U is the nonlinearity coefficient, and T F is the
time constant of the RF filter. The analytical formula for the RF FODL transfer
function K
2
2ΓPN F
ð Þ is defined and investigated in Chap. 6 of this book.
Thus, at the closed feedback loop in OEO MZ (Fig. 3.4a), the spectrum in PD
output has the Lopentzian shape, is defined by the laser phase noises, and its
Fig. 3.5 The structure of
the correlator on the base of
PD at modeling of the
statistical process in OEO
86 3 Modulation Methods of Laser Emission in Optoelectronic oscillator (OEO) and OEO. . .
maximal value of S RFMZ is defined by the ratio ΔT MZ /T coh . As a rule, ΔT MZ /T coh is
small, i.e., less than 0.001 and exp À
ΔT MZ
T coh
% 1.
3.1.3.3 Oscillation Spectrum at the Closed Feedback Loop
At closed Sw switch in Fig. 3.4a, at fulfillment of OEO excitation conditions, the
amplified electrical oscillation passes to the electrical input of the MZ modulator
from the PD output. We note that this oscillation is delayed by the time of fiber
optical system T FOS relative to the laser oscillation arriving to PD.
At that, on the PD area, there is the process of statistical averaging in time of the
interference result of two optical harmonics ν 0L and ν 0L + f, which are applied to the
PD area:
E 10L t
ð Þ ¼ A 1 E 0L cos 2πν 0L t þ φ 0L þ φ 10Lm t
ð Þ
½
ð 3:9Þ
and
E 20L t
ð Þ ¼ A 2 E 0L cos 2π ν 0L þ f 0
ð
Þ t þ φ 0L þ φ 20Lm t
ð Þ
½
,
ð3:10Þ
with the amplitudes A 1 E 0L and A 2 E 0L .
As shown in Chaps. 5 and 6 of this book, in this case, the RF spectrum at the PD
output S RFL (F) is defined by the formula:
S RFL F
ð Þ ¼ K
2
2ΓPN F
ð Þ Á
1= Δν L
ð
Þ
1 þ F À ν L0
ð
Þ =Δν L
½
2
Á 1 À
A 2
A 1
exp À
T FOS
T coh
&
'
, ð3:11Þ
in which K
2
2ΓPN F
ð Þ is the coefficient of phase noise suppression, which depends on
T FOS , on the laser optical power P 0L , σ U is the nonlinearity coefficient, and T F is the
time constant of the RF filter. The analytical formula for the RF FODL transfer
function K
2
2ΓPN F
ð Þ is defined and investigated in Chap. 6 of this book.
Thus, at the closed feedback loop in OEO MZ (Fig. 3.4a), the spectrum in PD
output has the Lopentzian shape, is defined by the laser phase noises, and its
Fig. 3.5 The structure of
the correlator on the base of
PD at modeling of the
statistical process in OEO
86 3 Modulation Methods of Laser Emission in Optoelectronic oscillator (OEO) and OEO. . .