the ratio ΔT M /T c . As a rule, ΔT M /T c is small, i.e., less than 0.001, and
exp À
ΔT M
T c
% 1.
5.5.6 The Autocorrelation Function of the Strength Square
of the MZ Field
Let us determine autocorrelation functions for the random photodetection process at
the simultaneous effect on the optical and electrical inputs, relatively, of the laser
optical emission E L (t) and the electrical oscillation u g (t). Let the electrical field of the
optical carrier, which passes to the MZ optical input, be expressed as:
E L (t) ¼ E 0L cos [2πν 0 t + φ 0L + φ Lm (t)] , where E 0L is the oscillation amplitude, ν 0L
is the average laser generation frequency, φ 0L is the initial phase, φ Lm (t) is the noisy
component or the random phase. The correlation function R MZ (τ) is:
R MZE t,tÀτ
ð
Þ¼R MZ τ
ð Þ¼ E
2
12L t
ð ÞÁ E
2
12L tÀτ
ð Þ
Â
à Ã
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
2
1L þE
2
2L
q
2
exp j2arg E 12L
ð
Þ
½
Á
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
2
1L þE
2
2L
q
2
"
#
exp Àj2arg E 12L tÀτ
ð Þ
½
f
g
*
+
¼ E
2
1L t
ð ÞþE
2
2L t
ð Þ
Â
à Á E
2
1L tÀτ
ð
ÞþE
2
2L tÀτ
ð
Þ
Â
à Ã
,
ð5:88Þ
where E
2
1L , E
2
2L are the amplitude squares of the normalized strength of the laser
emission in PD, which passed through the first and the second optical channels in
MZ modulator (Figs. 5.10 and 5.11a).
We can show that ACF R MZ (τ) for the observation time |τ| > |ΔT M | is determined
by the expression
R E
2
12 t, t þ τ
ð
Þ
Â
à ¼ E
4
12L þ E
4
12L cos 2πν 0L ΔT M
ð
exp ÀΔν Á ΔT M
j
j
ð
Þ
þE
4
12L cos 2 Á 2πν 0L ΔT M
½
exp ÀΔν L Á 2 ΔT M
j
j
ð
Þ :
ð5:89Þ
The formula (Eq. 5.89) shows that ACF is determined by three terms: the first
(E
4
12L ) is the effect of DC component passing to the PD area, the second term is the
result of influence of the oscillation on the optical frequency ν 0 , while the third term
is the result of the effect of the oscillation on the optical frequency 2ν 0 . At influence
in the MZ electrical input of the voltage u g (t) ¼ U 10 cos [2πf 0 t + ϕ 0e + φ em (t)] , where
U 10 is the voltage oscillation amplitude in the MZ electrical input, f 0 is the average
laser generation frequency, ϕ 0e is the initial phase, and φ em (t) is the noisy phase
component or the random phase.
ACF of MZ is determined by a sum of three components R MZ0 , R MZ1 , R MZ2 :
R MZ (u, t, t + τ) ¼ R MZ0 + R MZ1 + R MZ2. At that, R MZ0 corresponds to the impact of
5.5 Correlator’s Mathematical Model in OEO MZ
267
exp À
ΔT M
T c
% 1.
5.5.6 The Autocorrelation Function of the Strength Square
of the MZ Field
Let us determine autocorrelation functions for the random photodetection process at
the simultaneous effect on the optical and electrical inputs, relatively, of the laser
optical emission E L (t) and the electrical oscillation u g (t). Let the electrical field of the
optical carrier, which passes to the MZ optical input, be expressed as:
E L (t) ¼ E 0L cos [2πν 0 t + φ 0L + φ Lm (t)] , where E 0L is the oscillation amplitude, ν 0L
is the average laser generation frequency, φ 0L is the initial phase, φ Lm (t) is the noisy
component or the random phase. The correlation function R MZ (τ) is:
R MZE t,tÀτ
ð
Þ¼R MZ τ
ð Þ¼ E
2
12L t
ð ÞÁ E
2
12L tÀτ
ð Þ
Â
à Ã
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
2
1L þE
2
2L
q
2
exp j2arg E 12L
ð
Þ
½
Á
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
2
1L þE
2
2L
q
2
"
#
exp Àj2arg E 12L tÀτ
ð Þ
½
f
g
*
+
¼ E
2
1L t
ð ÞþE
2
2L t
ð Þ
Â
à Á E
2
1L tÀτ
ð
ÞþE
2
2L tÀτ
ð
Þ
Â
à Ã
,
ð5:88Þ
where E
2
1L , E
2
2L are the amplitude squares of the normalized strength of the laser
emission in PD, which passed through the first and the second optical channels in
MZ modulator (Figs. 5.10 and 5.11a).
We can show that ACF R MZ (τ) for the observation time |τ| > |ΔT M | is determined
by the expression
R E
2
12 t, t þ τ
ð
Þ
Â
à ¼ E
4
12L þ E
4
12L cos 2πν 0L ΔT M
ð
exp ÀΔν Á ΔT M
j
j
ð
Þ
þE
4
12L cos 2 Á 2πν 0L ΔT M
½
exp ÀΔν L Á 2 ΔT M
j
j
ð
Þ :
ð5:89Þ
The formula (Eq. 5.89) shows that ACF is determined by three terms: the first
(E
4
12L ) is the effect of DC component passing to the PD area, the second term is the
result of influence of the oscillation on the optical frequency ν 0 , while the third term
is the result of the effect of the oscillation on the optical frequency 2ν 0 . At influence
in the MZ electrical input of the voltage u g (t) ¼ U 10 cos [2πf 0 t + ϕ 0e + φ em (t)] , where
U 10 is the voltage oscillation amplitude in the MZ electrical input, f 0 is the average
laser generation frequency, ϕ 0e is the initial phase, and φ em (t) is the noisy phase
component or the random phase.
ACF of MZ is determined by a sum of three components R MZ0 , R MZ1 , R MZ2 :
R MZ (u, t, t + τ) ¼ R MZ0 + R MZ1 + R MZ2. At that, R MZ0 corresponds to the impact of
5.5 Correlator’s Mathematical Model in OEO MZ
267