From these spectra, we see that at this AC voltage amplitude U 1MZ ¼ 2U 0MZπ (the
fundamental harmonic) applied to the MZ modulator, the central optical harmonic
disappears in the spectrum (or PSD level becomes equal to zero on the optical
frequency). The spectrum of optical harmonics in the optical channel of OEO MZ
at the modulation index x ¼ U 1MZ /U 0MZπ ¼ 2.4 is presented in Fig. 3.16a, c.
The vector presentation of the EMF amplitude summation on PD is shown
without selection of the optical harmonic (Fig. 3.16d) and with summation of two
side harmonics at the carrier suppression (Fig. 3.16e) and with summation of the
carrier and the one of side harmonics (Fig. 3.16f). Figure 3.16d–f is the vector
diagrams, which clearly show the process of the central harmonic suppression with
the frequency ν 0 (or the constant optical emission with E 0L ). Here E PD is the field
strength on the photodetector area. In Fig. 3.16d we see the non-suppressed central
harmonic with the frequency ν 0 . Figure 3.16e illustrates the suppression condition of
the harmonic with the frequency ν 0 . Here E 0L , E PD are laser field strengths in the
optical input of the MZ modulator and on the PD area, relatively. The coefficient K m
introduced in Fig. 3.16d–f clearly shows that the amplitude of harmonics on the left
(ν 0 À f ) and on the right (ν 0 + f ) with respect to the central optical frequency
ν 0 represents the product K m E 0L . In the case shown in Fig. 3.16f, at nonsymmetric
suppression, for example, of the left harmonic (ν 0 À f ) suppression, the total vector
E PD is determined by both AC and DC components.
3.2.2.1 Approximation of Bessel Functions by Quadratic and Cubic
Functions
We take into consideration in calculations of the first and second harmonics only for
appropriate Bessel functions the following approximation for small x ¼ U 1MZ /
U 0MZπ , and x 2 [0;1]: J 0 x
ð Þ % 1 À
x
2
4 ; J 1 x
ð Þ %
x
2 À
x
3
16 and J 2 х
ð Þ %
x
2
8 . Assumed
approximation allows obtaining of the relatively simple expression for AC
Fig. 3.15 Functions of DC
component of the optical
emission and AC
component of fundamental
harmonic in the MZ output.
Numbers “1” and “2”
designates the values of DC
bias voltage in the
“quadrature” and “nonquadrature” modes, they are
equal relatively 0.5 and 1.0
and correspond in this figure
to phases φ 0MZ ¼
πU0MZ
U0MZπ ¼
π=2 and π, relatively
106 3 Modulation Methods of Laser Emission in Optoelectronic oscillator (OEO) and OEO. . .
fundamental harmonic) applied to the MZ modulator, the central optical harmonic
disappears in the spectrum (or PSD level becomes equal to zero on the optical
frequency). The spectrum of optical harmonics in the optical channel of OEO MZ
at the modulation index x ¼ U 1MZ /U 0MZπ ¼ 2.4 is presented in Fig. 3.16a, c.
The vector presentation of the EMF amplitude summation on PD is shown
without selection of the optical harmonic (Fig. 3.16d) and with summation of two
side harmonics at the carrier suppression (Fig. 3.16e) and with summation of the
carrier and the one of side harmonics (Fig. 3.16f). Figure 3.16d–f is the vector
diagrams, which clearly show the process of the central harmonic suppression with
the frequency ν 0 (or the constant optical emission with E 0L ). Here E PD is the field
strength on the photodetector area. In Fig. 3.16d we see the non-suppressed central
harmonic with the frequency ν 0 . Figure 3.16e illustrates the suppression condition of
the harmonic with the frequency ν 0 . Here E 0L , E PD are laser field strengths in the
optical input of the MZ modulator and on the PD area, relatively. The coefficient K m
introduced in Fig. 3.16d–f clearly shows that the amplitude of harmonics on the left
(ν 0 À f ) and on the right (ν 0 + f ) with respect to the central optical frequency
ν 0 represents the product K m E 0L . In the case shown in Fig. 3.16f, at nonsymmetric
suppression, for example, of the left harmonic (ν 0 À f ) suppression, the total vector
E PD is determined by both AC and DC components.
3.2.2.1 Approximation of Bessel Functions by Quadratic and Cubic
Functions
We take into consideration in calculations of the first and second harmonics only for
appropriate Bessel functions the following approximation for small x ¼ U 1MZ /
U 0MZπ , and x 2 [0;1]: J 0 x
ð Þ % 1 À
x
2
4 ; J 1 x
ð Þ %
x
2 À
x
3
16 and J 2 х
ð Þ %
x
2
8 . Assumed
approximation allows obtaining of the relatively simple expression for AC
Fig. 3.15 Functions of DC
component of the optical
emission and AC
component of fundamental
harmonic in the MZ output.
Numbers “1” and “2”
designates the values of DC
bias voltage in the
“quadrature” and “nonquadrature” modes, they are
equal relatively 0.5 and 1.0
and correspond in this figure
to phases φ 0MZ ¼
πU0MZ
U0MZπ ¼
π=2 and π, relatively
106 3 Modulation Methods of Laser Emission in Optoelectronic oscillator (OEO) and OEO. . .