modulator and its further photodetection with the purpose of the analysis of the
correlation process of random quantities and the conditions of the phase noise
suppression.
5.5.1 Description of Mathematical Models of the Laser
and the Mach–Zehnder Modulator at OEO Analysis
as the Correlator Enclosed by the Positive FB
The laser. At analysis, we assume that the laser is the high-coherent source of
optical oscillations. In other words, the laser spectral line width is less than 100 MHz
at the average generation frequency about 200 THz. We suppose that oscillations of
the normalized strength of the electromagnetic field in the laser output are close to
sinusoidal with the noisy phase component φ Lm (t) and the normalized amplitude
noises m Lm (t): E L (t) ¼ [E 0L + m Lm (t)] cos [2πν 0L t + φ 0L + φ Lm (t)] . Here E L (t), E 0L ,
m Lm (t) are normalized dimensionless quantities of the instantaneous strength, the
amplitude of EMF strength and the EMF amplitude noises, relatively. These quantities are normalized to the maximal value of the strength E Lmax ,i.e.: E 0L ¼ E 0LL /
E Lmax , E L (t) ¼ E LL (t)/E Lmax , and m Lm (t) ¼ m Lm (t)/E Lmax , where ν 0L is the average
laser generation frequency, φ 0L is the initial constant phase shift, t is the current time.
In the OEO, at fulfillment of the self-excitation conditions, in the RF section of this
oscillator, the RF oscillations u ¼ u g (t) occur. At that, in the electrical input of the
MZ modulator, from the output of the nonlinear amplifier through the coupler C, the
radio-frequency signal passes during of the oscillation generation process, which
instantaneous voltage is u g (t) ¼ [U 10 + m em (t)] cos [2πft + ϕ 0e + φ em (t)] , where
U 01 ¼ U 01MZ ¼ U 01C is the oscillation amplitude of the first harmonic in the electric
input of the MZ modulator or in the C output, f is the radio frequency of oscillations,
ϕ 0e is the constant phase shift, φ em (t) are phase fluctuations, m em (t) are amplitude
fluctuations.
The Mach–Zehnder modulator in OEO. During the analysis in this section, we
suppose that the voltage u g (t) effect in the one of two optical MZ channels (Fig. 5.10)
upon the refraction index of the electro-optical index (the lithium niobate), from
which the optical channels of MZ modulator are made, which leads to the phase
modulation of the optical oscillation. Combining of two optical emissions on the PD
area, which passed the different optical channels of the MZ modulator, leads to
intensity modulation of the laser emission. Delayed output emissions E 1L and E 2L of
the first and second channels of the МZ modulator pass to the input of the photodetector PD. Let the fiber optical system be formed by the single lengthy optical fiber
(OF) and the delay difference is determined as ΔT M ¼ T 2M À T 1M . In Fig. 5.10, the
structure of OEO MZ with the delay in the optical path is presented. In this structure,
we see the switch Sw. If this switch is closed, at fulfillment of self-oscillation
conditions in the closed OEO MZ loop, the oscillation generation occurs. When
258
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
correlation process of random quantities and the conditions of the phase noise
suppression.
5.5.1 Description of Mathematical Models of the Laser
and the Mach–Zehnder Modulator at OEO Analysis
as the Correlator Enclosed by the Positive FB
The laser. At analysis, we assume that the laser is the high-coherent source of
optical oscillations. In other words, the laser spectral line width is less than 100 MHz
at the average generation frequency about 200 THz. We suppose that oscillations of
the normalized strength of the electromagnetic field in the laser output are close to
sinusoidal with the noisy phase component φ Lm (t) and the normalized amplitude
noises m Lm (t): E L (t) ¼ [E 0L + m Lm (t)] cos [2πν 0L t + φ 0L + φ Lm (t)] . Here E L (t), E 0L ,
m Lm (t) are normalized dimensionless quantities of the instantaneous strength, the
amplitude of EMF strength and the EMF amplitude noises, relatively. These quantities are normalized to the maximal value of the strength E Lmax ,i.e.: E 0L ¼ E 0LL /
E Lmax , E L (t) ¼ E LL (t)/E Lmax , and m Lm (t) ¼ m Lm (t)/E Lmax , where ν 0L is the average
laser generation frequency, φ 0L is the initial constant phase shift, t is the current time.
In the OEO, at fulfillment of the self-excitation conditions, in the RF section of this
oscillator, the RF oscillations u ¼ u g (t) occur. At that, in the electrical input of the
MZ modulator, from the output of the nonlinear amplifier through the coupler C, the
radio-frequency signal passes during of the oscillation generation process, which
instantaneous voltage is u g (t) ¼ [U 10 + m em (t)] cos [2πft + ϕ 0e + φ em (t)] , where
U 01 ¼ U 01MZ ¼ U 01C is the oscillation amplitude of the first harmonic in the electric
input of the MZ modulator or in the C output, f is the radio frequency of oscillations,
ϕ 0e is the constant phase shift, φ em (t) are phase fluctuations, m em (t) are amplitude
fluctuations.
The Mach–Zehnder modulator in OEO. During the analysis in this section, we
suppose that the voltage u g (t) effect in the one of two optical MZ channels (Fig. 5.10)
upon the refraction index of the electro-optical index (the lithium niobate), from
which the optical channels of MZ modulator are made, which leads to the phase
modulation of the optical oscillation. Combining of two optical emissions on the PD
area, which passed the different optical channels of the MZ modulator, leads to
intensity modulation of the laser emission. Delayed output emissions E 1L and E 2L of
the first and second channels of the МZ modulator pass to the input of the photodetector PD. Let the fiber optical system be formed by the single lengthy optical fiber
(OF) and the delay difference is determined as ΔT M ¼ T 2M À T 1M . In Fig. 5.10, the
structure of OEO MZ with the delay in the optical path is presented. In this structure,
we see the switch Sw. If this switch is closed, at fulfillment of self-oscillation
conditions in the closed OEO MZ loop, the oscillation generation occurs. When
258
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .