series-connected in the loop: a laser, a polarizer, the optical fiber, a photodetector,
the RF amplifier, the RF filter, and the RF directional coupler. In the structure of
OEO MZ are included: a laser, a polarizer, the Mach–Zehnder modulator, the optical
fiber, a photodetector, the RF amplifier, the RF filter and the RF coupler. (All
designations are explained in the figure caption.)
Considering that the laser is high-coherent, i.e., the spectral line width is significantly less than 100 MHz at average generation frequency of 200 THz, we assume
that oscillations of the electromagnetic field (EMF) intensity are close to harmonic
with the noisy phase component φ Lm (t)
1 at the laser output and amplitude noises
m Lm (t):
E L t
ð Þ ¼ E 0L þ m Lm t
ð Þ
½
cos 2πν 0L t þ φ 0L þ φ Lm t
ð Þ
½
,
ð3:1Þ
where E 0L is the amplitude of EMF intensity, ν 0 is the average oscillation frequency,
φ 0L is the initial constant phase shift, and t is the current time.
In OEO system, at fulfillment of excitation conditions, in the RF part of such an
oscillator, RF oscillations u ¼ u g (t) ¼ u c (t) occur. At that, the RF signal from the
nonlinear amplifier output passes to the electrical input of MZ through the coupler C
during the oscillation generation, and the instantaneous voltage of this signal is
u g t
ð Þ ¼ U 10 þ m em t
ð Þ
½
cos 2πft þ ϕ 0e þ φ em t
ð Þ
½
,
ð3:2Þ
where U 01 ¼ U 01MZ ¼ U 01C is the oscillation amplitude of the fundamental harmonic in the electrical input of the MZ modulator and at the coupler output, f is the
RF oscillation frequency, ϕ 0e is the constant phase shift, φ em (t) are the phase
fluctuations, and m em (t) are the amplitude fluctuations. These expressions for a
signal will be used later.
3.1.3 Heterodyne Transformation and Self-Heterodyning
in OEO
Let us consider the features of the heterodyne reception of the laser emission with
phase fluctuations in OEO.
As an illustration to the discussing issue, Fig. 3.3 shows the diagrams of the
heterodyne reception (Fig. 3.3a) and the self-heterodyning mode in OEO (Fig. 3.3b).
The picture on the spectrum “transfer” from the optical range into the RF range is
presented in Fig. 3.3c.
Figure 3.3a shows the laser local oscillator, which plays the role of the heterodyne
block. The signals mixing of the laser and the laser local oscillator is performed in
the photodetector, which is the nonlinear element for heterodyning.
1 Here and later, indices L show the relation to the laser in OEO.
80 3 Modulation Methods of Laser Emission in Optoelectronic oscillator (OEO) and OEO. . .
the RF amplifier, the RF filter, and the RF directional coupler. In the structure of
OEO MZ are included: a laser, a polarizer, the Mach–Zehnder modulator, the optical
fiber, a photodetector, the RF amplifier, the RF filter and the RF coupler. (All
designations are explained in the figure caption.)
Considering that the laser is high-coherent, i.e., the spectral line width is significantly less than 100 MHz at average generation frequency of 200 THz, we assume
that oscillations of the electromagnetic field (EMF) intensity are close to harmonic
with the noisy phase component φ Lm (t)
1 at the laser output and amplitude noises
m Lm (t):
E L t
ð Þ ¼ E 0L þ m Lm t
ð Þ
½
cos 2πν 0L t þ φ 0L þ φ Lm t
ð Þ
½
,
ð3:1Þ
where E 0L is the amplitude of EMF intensity, ν 0 is the average oscillation frequency,
φ 0L is the initial constant phase shift, and t is the current time.
In OEO system, at fulfillment of excitation conditions, in the RF part of such an
oscillator, RF oscillations u ¼ u g (t) ¼ u c (t) occur. At that, the RF signal from the
nonlinear amplifier output passes to the electrical input of MZ through the coupler C
during the oscillation generation, and the instantaneous voltage of this signal is
u g t
ð Þ ¼ U 10 þ m em t
ð Þ
½
cos 2πft þ ϕ 0e þ φ em t
ð Þ
½
,
ð3:2Þ
where U 01 ¼ U 01MZ ¼ U 01C is the oscillation amplitude of the fundamental harmonic in the electrical input of the MZ modulator and at the coupler output, f is the
RF oscillation frequency, ϕ 0e is the constant phase shift, φ em (t) are the phase
fluctuations, and m em (t) are the amplitude fluctuations. These expressions for a
signal will be used later.
3.1.3 Heterodyne Transformation and Self-Heterodyning
in OEO
Let us consider the features of the heterodyne reception of the laser emission with
phase fluctuations in OEO.
As an illustration to the discussing issue, Fig. 3.3 shows the diagrams of the
heterodyne reception (Fig. 3.3a) and the self-heterodyning mode in OEO (Fig. 3.3b).
The picture on the spectrum “transfer” from the optical range into the RF range is
presented in Fig. 3.3c.
Figure 3.3a shows the laser local oscillator, which plays the role of the heterodyne
block. The signals mixing of the laser and the laser local oscillator is performed in
the photodetector, which is the nonlinear element for heterodyning.
1 Here and later, indices L show the relation to the laser in OEO.
80 3 Modulation Methods of Laser Emission in Optoelectronic oscillator (OEO) and OEO. . .