is determined as: Δφ 0M2 ¼ 2πν 0 B M U 0M ¼ 2πν 0 n
3
M r 33 L M U 0M = cd M
ð
Þ, where the
effective length of the modulator electrodes L M ¼ 0.02 m, the distance between
modulator electrodes is d M ¼ 10 μm, the coefficient r 33 ¼ 30.8 Á 10
À12 m/V, the
effective refraction index of the material n M ¼ 2.3 and is several degrees at input
voltages U 0MZ not more than 0.01 V. At that, the phase modulation index determining as B UM ¼ 100 % Á Δφ 0M2 /2π ¼ 100 % Á ν 0 U 0MZ B M is several percents. We note
that at large modulation signals, the expression for an argument argE 12L should be
presented by the nonlinear dependence through the Bessel functions.
The modulator effectiveness can be estimated by the coefficient η, which is
defined as the ratio: η ¼
Δφ 0M2
U 0MZ L ¼
2πΔn eff
λU 0M
. To determine the MZ effectiveness, we
introduce the coefficient α ch defining as a ratio of the phase shift increment and the
optical losses Δα pt L M in MZ (or the ratio of the real part of the MZ refraction index
to its imaginary part): α ch ¼
Δn ef Re
Δn efIm
¼
2Δφ 0M2
Δα pt L M
.
At small input modulation signal in MZ, we may use the linearization of the
argument argE 12L and to present the expression for the strength on the PD area E L12
as: E 12L ¼ E 1L + E 2L ¼ |E 12L | exp [j arg (E 12L )], in which the module is |E 12L | ¼ |
M z E 0L |, where M z is the module of the transfer function of the MZ modulator.
Using the fact that E
2
0L is the normalized intensity in the MZ input, the coefficient
γ ¼ k 01 /k 02 and k 01 % k 02 % 0.5, and the intensity module |E 12L |in the MZ output is in
Eqs. (3.30)À(3.34). In the small mode, at γ % 1, the following expression is obtained
from Eq. (3.31) for the small argument deviations φ 1 and φ 2 from its average values
of Δ Δφ 0M2
ð
Þ¼ arctan
sin Δφ 1 þγ sin Δφ 2
cos Δφ 1 þγ cos Δφ 2
n
o
. And for the small argument deviations
Δ Δφ 0M2
ð
Þ%
Δφ 1 þ γΔφ 2
2 À Δφ 1
2 À γΔφ 2
2
%
γΔφ 2
2 À γ Δφ 2
ð
Þ
2
:
ð6:6Þ
From Eqs. (3.30) to (3.34), (6.6), the important conclusion follows that small
variations of the irregularity coefficient γ of channel OC1 and OC2 excitation lead to
variations of the difference phase shift Δφ 0M2 . This means that for well operation, in
OEO MZ should be satisfied the requirements to excitation uniformity of MZ optical
channels. From this requirement, the demand to the spatial laser coherence follows
and the demand to polarization of laser emission.
For the sake of simplicity, we may consider that in the small-signal mode
argE 12L ¼ [2πν 0 (T 2M À T 1M )]. In the small-signal mode, the coefficient α ch is
defined as α ch ¼ 2E
2
0L
dΔφ 02M
dE
2
0L
. This means that at the output power maximum of the
laser emission E
2
0L , there is the maximum of the α ch parameter, while the power
variations are minimal.
⁄
ä
Fig. 6.11 (continued) (curve 1—1 dB, curve 2—5 dB, curve 3—10 dB). (b) The calculated
functions of the MZ transfer function module |M eZ | in the small-signal mode Eqs. (6.8) and (6.9).
The laser optical wavelength λ ¼ 1550 nm(ν 0 ¼ 128 THz). For curve (А) α 1m ¼ 0.27 dB/cm,
n M ¼ 2.4, curve (B) α 1m ¼ 0.41 dB/cm, n M ¼ 3.18, (C) α 1m ¼ 0.68 dB/cm, n M ¼ 4.24
6.4 Characteristics and the Transfer Function of the MZ Modulator in OEO
303
3
M r 33 L M U 0M = cd M
ð
Þ, where the
effective length of the modulator electrodes L M ¼ 0.02 m, the distance between
modulator electrodes is d M ¼ 10 μm, the coefficient r 33 ¼ 30.8 Á 10
À12 m/V, the
effective refraction index of the material n M ¼ 2.3 and is several degrees at input
voltages U 0MZ not more than 0.01 V. At that, the phase modulation index determining as B UM ¼ 100 % Á Δφ 0M2 /2π ¼ 100 % Á ν 0 U 0MZ B M is several percents. We note
that at large modulation signals, the expression for an argument argE 12L should be
presented by the nonlinear dependence through the Bessel functions.
The modulator effectiveness can be estimated by the coefficient η, which is
defined as the ratio: η ¼
Δφ 0M2
U 0MZ L ¼
2πΔn eff
λU 0M
. To determine the MZ effectiveness, we
introduce the coefficient α ch defining as a ratio of the phase shift increment and the
optical losses Δα pt L M in MZ (or the ratio of the real part of the MZ refraction index
to its imaginary part): α ch ¼
Δn ef Re
Δn efIm
¼
2Δφ 0M2
Δα pt L M
.
At small input modulation signal in MZ, we may use the linearization of the
argument argE 12L and to present the expression for the strength on the PD area E L12
as: E 12L ¼ E 1L + E 2L ¼ |E 12L | exp [j arg (E 12L )], in which the module is |E 12L | ¼ |
M z E 0L |, where M z is the module of the transfer function of the MZ modulator.
Using the fact that E
2
0L is the normalized intensity in the MZ input, the coefficient
γ ¼ k 01 /k 02 and k 01 % k 02 % 0.5, and the intensity module |E 12L |in the MZ output is in
Eqs. (3.30)À(3.34). In the small mode, at γ % 1, the following expression is obtained
from Eq. (3.31) for the small argument deviations φ 1 and φ 2 from its average values
of Δ Δφ 0M2
ð
Þ¼ arctan
sin Δφ 1 þγ sin Δφ 2
cos Δφ 1 þγ cos Δφ 2
n
o
. And for the small argument deviations
Δ Δφ 0M2
ð
Þ%
Δφ 1 þ γΔφ 2
2 À Δφ 1
2 À γΔφ 2
2
%
γΔφ 2
2 À γ Δφ 2
ð
Þ
2
:
ð6:6Þ
From Eqs. (3.30) to (3.34), (6.6), the important conclusion follows that small
variations of the irregularity coefficient γ of channel OC1 and OC2 excitation lead to
variations of the difference phase shift Δφ 0M2 . This means that for well operation, in
OEO MZ should be satisfied the requirements to excitation uniformity of MZ optical
channels. From this requirement, the demand to the spatial laser coherence follows
and the demand to polarization of laser emission.
For the sake of simplicity, we may consider that in the small-signal mode
argE 12L ¼ [2πν 0 (T 2M À T 1M )]. In the small-signal mode, the coefficient α ch is
defined as α ch ¼ 2E
2
0L
dΔφ 02M
dE
2
0L
. This means that at the output power maximum of the
laser emission E
2
0L , there is the maximum of the α ch parameter, while the power
variations are minimal.
⁄
ä
Fig. 6.11 (continued) (curve 1—1 dB, curve 2—5 dB, curve 3—10 dB). (b) The calculated
functions of the MZ transfer function module |M eZ | in the small-signal mode Eqs. (6.8) and (6.9).
The laser optical wavelength λ ¼ 1550 nm(ν 0 ¼ 128 THz). For curve (А) α 1m ¼ 0.27 dB/cm,
n M ¼ 2.4, curve (B) α 1m ¼ 0.41 dB/cm, n M ¼ 3.18, (C) α 1m ¼ 0.68 dB/cm, n M ¼ 4.24
6.4 Characteristics and the Transfer Function of the MZ Modulator in OEO
303