6. Fluctuations of the laser intensities are negligible small because the laser
pumping level is over the threshold.
7. The phase fluctuation model is described by the model of the Wiener random
process, which is the limit case of the process of random wandering at aspiration
of the number of steps for infinity.
8. The probability distribution density of the random quantity (the ψ m phase)
depends on time, has the normal Gaussian density with zero mean: P ψ m
ð Þ ¼
1
ffiffiffi
tα
p
ffiffiffiffi
2π
p
exp À
ψ
2
m
2αt
, where t is time, α is the physical parameter, for example, for
the laser the spectral line width (on the level 0.5).
9. The random process increments (i.e., increments of the phase difference during
the time interval (t,t + τ) are stationary in time) and the probability distribution
density P(Δψ m ) of the random quantity (the phase difference) does not depend
on time, has the normal Gaussian density with zero mean Δψ m ¼ ψ m1 (t) À
ψ m1 (t + τ): PðΔψ m Þ ¼
1
ffiffiffiffi
2π
p σ Δψ m
expðÀ
Δψ
2
m
2σ 2
Δψ m
Þ, where σ
2
Δψ m
is the dispersion of the
random phase difference Δψ m ¼ ψ m1 (t) À ψ m1 (t + τ), which depends only upon
the difference of the time moments τ.
10. For the dispersion of the random phase difference Δψ m ¼ ψ m1 (t) À ψ m1 (t + τ),
the true expression for the dispersion σ
2
Δψ m
is: σ
2
Δψ m
¼ αt þ α t þ τ
ð
ÞÀ2αt ¼
α τ
j j, in which α is the constant width of the laser spectral line α ¼ Δν L ¼ 1/T c ,
and T c is the laser time of coherence (or the time constant).
Figure 5.2b, c shows the representation of the laser electromagnetic field strength
in the complex space, i.e., in the phase space formed by the vector components with
the module and the phase (the argument). From the point of view of the statistical
analysis, this vector has fluctuations of the module (Fig. 5.2b) and the phase
(Fig. 5.2c). Taking into consideration the quantum description of the field, the vector
end can lay in any point of the phase space region, which has the minimal area πħ/2.
This region of ambiguity can be a circle, which leads to symmetrical distribution of
fluctuations.
5.4.2 Symbolic Constitutive Differential Equations
of the Laser with Fluctuations
In this section, we analyze the symbolic fluctuation equations of the laser for the
single-frequency generation mode with account of the differential equation for the
population.
As in Chaps. 2 and 3, now we consider the single-frequency QWLD, which
resonator allows performing of the traveling-wave mode (Fig. 2.2a). We note that
formation and analysis of fluctuation equations of the laser with the resonator Fabry–
Perot (the resonator of plane-parallel mirrors), in which the standing-wave mode in
the cavity occurs.
234
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .
pumping level is over the threshold.
7. The phase fluctuation model is described by the model of the Wiener random
process, which is the limit case of the process of random wandering at aspiration
of the number of steps for infinity.
8. The probability distribution density of the random quantity (the ψ m phase)
depends on time, has the normal Gaussian density with zero mean: P ψ m
ð Þ ¼
1
ffiffiffi
tα
p
ffiffiffiffi
2π
p
exp À
ψ
2
m
2αt
, where t is time, α is the physical parameter, for example, for
the laser the spectral line width (on the level 0.5).
9. The random process increments (i.e., increments of the phase difference during
the time interval (t,t + τ) are stationary in time) and the probability distribution
density P(Δψ m ) of the random quantity (the phase difference) does not depend
on time, has the normal Gaussian density with zero mean Δψ m ¼ ψ m1 (t) À
ψ m1 (t + τ): PðΔψ m Þ ¼
1
ffiffiffiffi
2π
p σ Δψ m
expðÀ
Δψ
2
m
2σ 2
Δψ m
Þ, where σ
2
Δψ m
is the dispersion of the
random phase difference Δψ m ¼ ψ m1 (t) À ψ m1 (t + τ), which depends only upon
the difference of the time moments τ.
10. For the dispersion of the random phase difference Δψ m ¼ ψ m1 (t) À ψ m1 (t + τ),
the true expression for the dispersion σ
2
Δψ m
is: σ
2
Δψ m
¼ αt þ α t þ τ
ð
ÞÀ2αt ¼
α τ
j j, in which α is the constant width of the laser spectral line α ¼ Δν L ¼ 1/T c ,
and T c is the laser time of coherence (or the time constant).
Figure 5.2b, c shows the representation of the laser electromagnetic field strength
in the complex space, i.e., in the phase space formed by the vector components with
the module and the phase (the argument). From the point of view of the statistical
analysis, this vector has fluctuations of the module (Fig. 5.2b) and the phase
(Fig. 5.2c). Taking into consideration the quantum description of the field, the vector
end can lay in any point of the phase space region, which has the minimal area πħ/2.
This region of ambiguity can be a circle, which leads to symmetrical distribution of
fluctuations.
5.4.2 Symbolic Constitutive Differential Equations
of the Laser with Fluctuations
In this section, we analyze the symbolic fluctuation equations of the laser for the
single-frequency generation mode with account of the differential equation for the
population.
As in Chaps. 2 and 3, now we consider the single-frequency QWLD, which
resonator allows performing of the traveling-wave mode (Fig. 2.2a). We note that
formation and analysis of fluctuation equations of the laser with the resonator Fabry–
Perot (the resonator of plane-parallel mirrors), in which the standing-wave mode in
the cavity occurs.
234
5 Optoelectronic oscillator (OEO) Differential Equations as the Laser System with. . .