E 12L
j
j
2 ¼
E
2
0L
2
1 À cos φ 0L1 u g
À Á À φ 0L2
Â
Ã
È
É
¼
E
2
0L
2
2 sin
2
φ 0L1 u g
À Á À φ 0L2
Â
à :
ð3:33Þ
From the last formula, it follows that
E 12L
j
j ¼ E 0L
j jÁ sin φ 0L1 u g
À Á À φ 0L2
Â
à :
ð3:34Þ
In other words, we derived the connection formulas both for the EMF amplitude
squared and for the amplitude. These formulas allows formation of expressions for |
E 12L |
2 and also for |E 12L | at utilization of expansions on the Bessel functions.
The function (E 12L )
2 of oscillation amplitude U 0M depends upon the MZ operation point choice or the DC bias voltage U 0MZ . The DC bias changes and regulates of
the phase incursion φ 0L2 . The MZ operation point can be varied by the constant bias
voltage U 0MZ . Let us designate x ¼ U 1MZ /U 0MZπ , φ 0MZ ¼
πU 0MZ
U 0MZπ
.
Now we can present Fig. 3.15, which shows the functions of DC component of
the optical emission and AC component of fundamental harmonic in the MZ output.
From Fig. 3.15 we see that the laser emission modulation can be performed in two
modes: “quadrature” and “non-quadrature,” at DC bias there is the minimum at
x ¼ U 1MZ /U 0MZπ ¼ 1.
In OEO, the optical filter is used, which effectively suppresses the optical
harmonics with carriers higher than second ν 2 ¼ ν 0 Æ 2f. Therefore, we present
below the expressions for E 12L with limited number of harmonics without the
account of higher harmonics. The expression for (E 12L )
2 can be wrote using the
expansion to even J 2k and odd J 2kÀ1 Bessel functions. Let us introduce the constant
half-wavelength bias voltage on MZ U 0MZπ , which provides the phase difference
φ 0L2 in 180
between optical oscillations of the first and second MZ channels. Now
we use the trigonometric formula for the cosine of two angle sum for carrier, the first
and second harmonics and we can write:
E L12
ð
Þ
2 ¼
E
2
0L
2
γf
1 þ γ
2
ð
Þ
2γ
Á cos 2πνt þ ϕ 0e
ð
Þ
À cos φ 0MZ
ð
ÞÁJ 0 x
ð Þ Á cos 2πνt þ ϕ 0e
ð
Þ
À sin φ 0MZ
ð
ÞÁJ 1 x
ð Þ Á cos 2πνt þ 2πft þ ϕ 0e
ð
Þ
À sin φ 0MZ
ð
ÞÁJ 1 x
ð Þ Á cos 2πνt À 2πft À ϕ 0e
ð
Þ
À cos φ 0MZ
ð
ÞÁJ 2 x
ð Þ Á cos 2πνt þ 2 Á 2πft þ 2 Á ϕ 0e
ð
Þ
À cos φ 0MZ
ð
ÞÁJ 2 x
ð Þ Á cos 2πνt À 2 Á 2πft À 2 Á ϕ 0e
ð
Þ g
:
ð3:35Þ
Figure 3.16 shows the spectrum of optical harmonics in the optical harmonics in
the optical channel of OEO with MZ at modulation index x ¼ U 1MZ /U 0MZπ ¼ 1
(Fig. 3.16a), and also the values of Bessel functions of differential orders, which are
defined the harmonic amplitudes at variation of the modulation index (Fig. 3.16b).
The optical harmonic spectrum at x ¼ U 1MZ /U 0MZπ ¼ 2.4 is presented in Fig. 3.16c.
3.2 Methods of Modulation and Heterodyning of Laser Emissions at DM and MZ. . .
105
j
j
2 ¼
E
2
0L
2
1 À cos φ 0L1 u g
À Á À φ 0L2
Â
Ã
È
É
¼
E
2
0L
2
2 sin
2
φ 0L1 u g
À Á À φ 0L2
Â
à :
ð3:33Þ
From the last formula, it follows that
E 12L
j
j ¼ E 0L
j jÁ sin φ 0L1 u g
À Á À φ 0L2
Â
à :
ð3:34Þ
In other words, we derived the connection formulas both for the EMF amplitude
squared and for the amplitude. These formulas allows formation of expressions for |
E 12L |
2 and also for |E 12L | at utilization of expansions on the Bessel functions.
The function (E 12L )
2 of oscillation amplitude U 0M depends upon the MZ operation point choice or the DC bias voltage U 0MZ . The DC bias changes and regulates of
the phase incursion φ 0L2 . The MZ operation point can be varied by the constant bias
voltage U 0MZ . Let us designate x ¼ U 1MZ /U 0MZπ , φ 0MZ ¼
πU 0MZ
U 0MZπ
.
Now we can present Fig. 3.15, which shows the functions of DC component of
the optical emission and AC component of fundamental harmonic in the MZ output.
From Fig. 3.15 we see that the laser emission modulation can be performed in two
modes: “quadrature” and “non-quadrature,” at DC bias there is the minimum at
x ¼ U 1MZ /U 0MZπ ¼ 1.
In OEO, the optical filter is used, which effectively suppresses the optical
harmonics with carriers higher than second ν 2 ¼ ν 0 Æ 2f. Therefore, we present
below the expressions for E 12L with limited number of harmonics without the
account of higher harmonics. The expression for (E 12L )
2 can be wrote using the
expansion to even J 2k and odd J 2kÀ1 Bessel functions. Let us introduce the constant
half-wavelength bias voltage on MZ U 0MZπ , which provides the phase difference
φ 0L2 in 180
between optical oscillations of the first and second MZ channels. Now
we use the trigonometric formula for the cosine of two angle sum for carrier, the first
and second harmonics and we can write:
E L12
ð
Þ
2 ¼
E
2
0L
2
γf
1 þ γ
2
ð
Þ
2γ
Á cos 2πνt þ ϕ 0e
ð
Þ
À cos φ 0MZ
ð
ÞÁJ 0 x
ð Þ Á cos 2πνt þ ϕ 0e
ð
Þ
À sin φ 0MZ
ð
ÞÁJ 1 x
ð Þ Á cos 2πνt þ 2πft þ ϕ 0e
ð
Þ
À sin φ 0MZ
ð
ÞÁJ 1 x
ð Þ Á cos 2πνt À 2πft À ϕ 0e
ð
Þ
À cos φ 0MZ
ð
ÞÁJ 2 x
ð Þ Á cos 2πνt þ 2 Á 2πft þ 2 Á ϕ 0e
ð
Þ
À cos φ 0MZ
ð
ÞÁJ 2 x
ð Þ Á cos 2πνt À 2 Á 2πft À 2 Á ϕ 0e
ð
Þ g
:
ð3:35Þ
Figure 3.16 shows the spectrum of optical harmonics in the optical harmonics in
the optical channel of OEO with MZ at modulation index x ¼ U 1MZ /U 0MZπ ¼ 1
(Fig. 3.16a), and also the values of Bessel functions of differential orders, which are
defined the harmonic amplitudes at variation of the modulation index (Fig. 3.16b).
The optical harmonic spectrum at x ¼ U 1MZ /U 0MZπ ¼ 2.4 is presented in Fig. 3.16c.
3.2 Methods of Modulation and Heterodyning of Laser Emissions at DM and MZ. . .
105