RF FODL is nonuniform and nonideal. It contains the non-axis-symmetric planar
optical waveguides, Y- and Х-couplers of the MZ modulator, and the axissymmetric fiber-optical light guide of FOS. We can take into consideration the
dependence of the laser emission phase fluctuations in the transverse section, if we
introduce the spatial parameter R.
We can show that the laser spectrum S ψL (ν) ¼ S ψL (ν, R) is the function of the
spatial parameter R and the expression is true: S ψL (ν, R) % S ψL (ν) exp (γ Lψ R), where
γ Lψ is the spatial constant of the phase fluctuations of the laser output emission.
Now we explain the physical sense of the expression S ψL (ν,
R) % S ψL (ν) exp (γ Lψ R). In particular, we define the spatial parameter R as the radius
(or the distance in the transverse section from the optical axis to the analysis point
with the radius R). At that, the phase fluctuations of the laser emission in the
transverse section increase at receding from the optical axis.
The plot presented in Fig. 5.12 shows that at the delay difference in MZ optical
channels τ D ¼ ΔT M and less than 10
À8
μs, the value is close to 1 with the accuracy to
the third sign at the laser spectral width Δν L ¼ 1 kHz. This means that statistical
random quantities of phase fluctuations of the laser emission are dependent in
outputs of channels OC1 and OC2.
The linearly polarized optical emission E L from the laser output passes to the
modulator input. Then, the time-function of instantaneous field strength E L on the
central frequency ν 0 of QWLD generation taking into account of amplitude and
phase laser fluctuations is determined by expression: E L ¼ (E 0L + m lL ) exp [j
(2πν 0 t À φ 0L À ψ m )], where m L ¼ m L (t, R), ψ m ¼ ψ m (t, R) are amplitude and
phase field fluctuations E L of the laser, which are defined by spectral densities,
relatively, R is the index accounting the spatial dependence of the laser optical
emission (for instance, the polarization dependence), E 0L ¼ E 0L (R) is the partial
amplitude of the laser emission, φ 0L ¼ φ 0L (R) is the partial phase incursion of the
laser emission.
The total output laser emission is defined as the result of integration on the PD
area over all values of the spatial index R of the laser emission. The spatial index R,
which in the very simplest case is the distance or the radius from the optical axis to
the specific point in the perpendicular section to the optical axis. Laser emission has
the spatial distribution E L (t, R) over the some section, which is in distant from
QWLD output for some distance.
The physical sense of the integration over the R index is quite understandable: we
perform integration over values of the field strength amplitudes taking into account
the spatial distribution of the field amplitude.
298
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .
optical waveguides, Y- and Х-couplers of the MZ modulator, and the axissymmetric fiber-optical light guide of FOS. We can take into consideration the
dependence of the laser emission phase fluctuations in the transverse section, if we
introduce the spatial parameter R.
We can show that the laser spectrum S ψL (ν) ¼ S ψL (ν, R) is the function of the
spatial parameter R and the expression is true: S ψL (ν, R) % S ψL (ν) exp (γ Lψ R), where
γ Lψ is the spatial constant of the phase fluctuations of the laser output emission.
Now we explain the physical sense of the expression S ψL (ν,
R) % S ψL (ν) exp (γ Lψ R). In particular, we define the spatial parameter R as the radius
(or the distance in the transverse section from the optical axis to the analysis point
with the radius R). At that, the phase fluctuations of the laser emission in the
transverse section increase at receding from the optical axis.
The plot presented in Fig. 5.12 shows that at the delay difference in MZ optical
channels τ D ¼ ΔT M and less than 10
À8
μs, the value is close to 1 with the accuracy to
the third sign at the laser spectral width Δν L ¼ 1 kHz. This means that statistical
random quantities of phase fluctuations of the laser emission are dependent in
outputs of channels OC1 and OC2.
The linearly polarized optical emission E L from the laser output passes to the
modulator input. Then, the time-function of instantaneous field strength E L on the
central frequency ν 0 of QWLD generation taking into account of amplitude and
phase laser fluctuations is determined by expression: E L ¼ (E 0L + m lL ) exp [j
(2πν 0 t À φ 0L À ψ m )], where m L ¼ m L (t, R), ψ m ¼ ψ m (t, R) are amplitude and
phase field fluctuations E L of the laser, which are defined by spectral densities,
relatively, R is the index accounting the spatial dependence of the laser optical
emission (for instance, the polarization dependence), E 0L ¼ E 0L (R) is the partial
amplitude of the laser emission, φ 0L ¼ φ 0L (R) is the partial phase incursion of the
laser emission.
The total output laser emission is defined as the result of integration on the PD
area over all values of the spatial index R of the laser emission. The spatial index R,
which in the very simplest case is the distance or the radius from the optical axis to
the specific point in the perpendicular section to the optical axis. Laser emission has
the spatial distribution E L (t, R) over the some section, which is in distant from
QWLD output for some distance.
The physical sense of the integration over the R index is quite understandable: we
perform integration over values of the field strength amplitudes taking into account
the spatial distribution of the field amplitude.
298
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .