(resonators), delay lines and filters on the base of optical fibers and disks, the
photodetector (or detectors).
In the further chapters, owing to this theoretical approach, we write the DE
system of the laser taking into account the noisy Langevinian fluctuation sources
[6]. The noise of spontaneous (radiation) emission having the quantum nature plays
the role of the main one among such sources. Then we develop the method based on
the OEO representation as the unified oscillating system, in which the motion
processes develop in optical and radio-frequency ranges taking fluctuations into
account.
The analog fluctuation model of the laser formed on the base of constitutional
equation with utilization of Langevinian noise sources allows determination of phase
and amplitude noise in OEO MZ. In this interpretation, OEO is the quantum
generator spanned by the positive feedback or opto-electronic network containing
the optical fiber, the photodetector, the RF filter, and the RF amplifier
4.1.1 OEO as the Laser Spanned by the Feedback Loop
We remind once more that at analysis of OEO DМ and OEO MZ, we consider them
as systems, in which the laser is spanned by the positive feedback (FB) loop, which
consists of series-connected the optical fiber, the photodetector, the RF amplifier, the
RF amplifier, and the RF filter (Fig. 4.1a, b). In OEO DМ, the loop of positive
feedback is closed (connected) through the coupler C to the laser, while in OEO MZ
the FB loop connects through the coupler C to the Mach–Zehnder modulator.
4.1.1.1 The Positive Feedback Loop in OEO DM Spanned the Laser
The closed loop of positive FB contains (Fig. 4.1a) series-connected the
“polarizer” P, the optical fiber OF, the photodetector PD, the amplifier A, the
filter F, and the coupler C.
For simplicity, we neglect by actions of the optical polarizer (P), by dispersion of
the optical fiber, and we consider that PD is wideband, and the RF amplifier has no
inertial properties. The transfer function of FB loop K BZ we define for OEO DM and
OEO MZ as: K FB ¼
E
2
1L
i 1L
or K FB ¼
E
2
1L
u 1MZ
, where E
2
1L is the normalized strength square
(laser power), i 1L is AC component of the laser pumping current, u 1MZ is AC voltage
component of the MZ modulator. In this case, the operator controlling transfer
function
of
this
FB
loop
has
a
form:
K FB ¼ K BZ ¼
E
2
1L
i 1L
¼
1=T EF
ð
Þ p
p 2 þ 1=T EF
ð
Þ pþ 2πf 0e
ð
Þ
2 K OF exp ÀpT OF
ð
ÞK PD S NY , where p ¼ d/dt is the Laplace operator,
f 0e is the fundamental frequency of the RF filter, T EF is the time constant of the RF
filter, T OF is time delay in the optical fiber, K OF is the transfer function of OF, K PD is
136
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .
photodetector (or detectors).
In the further chapters, owing to this theoretical approach, we write the DE
system of the laser taking into account the noisy Langevinian fluctuation sources
[6]. The noise of spontaneous (radiation) emission having the quantum nature plays
the role of the main one among such sources. Then we develop the method based on
the OEO representation as the unified oscillating system, in which the motion
processes develop in optical and radio-frequency ranges taking fluctuations into
account.
The analog fluctuation model of the laser formed on the base of constitutional
equation with utilization of Langevinian noise sources allows determination of phase
and amplitude noise in OEO MZ. In this interpretation, OEO is the quantum
generator spanned by the positive feedback or opto-electronic network containing
the optical fiber, the photodetector, the RF filter, and the RF amplifier
4.1.1 OEO as the Laser Spanned by the Feedback Loop
We remind once more that at analysis of OEO DМ and OEO MZ, we consider them
as systems, in which the laser is spanned by the positive feedback (FB) loop, which
consists of series-connected the optical fiber, the photodetector, the RF amplifier, the
RF amplifier, and the RF filter (Fig. 4.1a, b). In OEO DМ, the loop of positive
feedback is closed (connected) through the coupler C to the laser, while in OEO MZ
the FB loop connects through the coupler C to the Mach–Zehnder modulator.
4.1.1.1 The Positive Feedback Loop in OEO DM Spanned the Laser
The closed loop of positive FB contains (Fig. 4.1a) series-connected the
“polarizer” P, the optical fiber OF, the photodetector PD, the amplifier A, the
filter F, and the coupler C.
For simplicity, we neglect by actions of the optical polarizer (P), by dispersion of
the optical fiber, and we consider that PD is wideband, and the RF amplifier has no
inertial properties. The transfer function of FB loop K BZ we define for OEO DM and
OEO MZ as: K FB ¼
E
2
1L
i 1L
or K FB ¼
E
2
1L
u 1MZ
, where E
2
1L is the normalized strength square
(laser power), i 1L is AC component of the laser pumping current, u 1MZ is AC voltage
component of the MZ modulator. In this case, the operator controlling transfer
function
of
this
FB
loop
has
a
form:
K FB ¼ K BZ ¼
E
2
1L
i 1L
¼
1=T EF
ð
Þ p
p 2 þ 1=T EF
ð
Þ pþ 2πf 0e
ð
Þ
2 K OF exp ÀpT OF
ð
ÞK PD S NY , where p ¼ d/dt is the Laplace operator,
f 0e is the fundamental frequency of the RF filter, T EF is the time constant of the RF
filter, T OF is time delay in the optical fiber, K OF is the transfer function of OF, K PD is
136
4 Semiclassical Theory and Laser Differential Equations for Optoelectronic. . .