the input of the correlator, the low-pass filter (LPF) is connected to the output of the
multiplier, the random process in the LPF output is η(t). LPF performs the integration
operation of the multiplication results of two random quantities.
At closing of the switch Sw, owing to oscillation excitation in the closed OEO,
the process of the laser phase fluctuations occurs, which is caused by the carrier noise
or by the spontaneous emission. In OEO, operations of multiplication and integration of LPF signals are performed by the photodetector.
5.5.3 The Mach–Zehnder Correlator-Interferometer
Let us examine the structure in Fig. 5.11 for modeling of the statistical process in
OEO with utilization of the traditional correlator at opened switch Sw and at absence
of electrical oscillations in the electrical input of the Mach–Zehnder modulation. At
that, for determination of the laser emission spectrum, it is necessary to calculate of
the autocorrelation function R(t, t + τ) of the laser random field [4]. ACF is determined by the coherent time T c , which is inverse proportional to the laser spectral line
width Δν L :
R L t, t þ τ
ð
Þ¼
Z 1
À1
E L t
ð ÞE
Ã
L t À τ
ð
Þdt ¼ E
2
L0 exp À
τ
T c
exp j2πν 0L τ
ð
Þ , ð5:82Þ
where ν L ¼ ν 0L ¼ ν 0 is the current optical frequency of the laser, and
E L (t) ¼ E L0 exp [j2πν 0L t + ψ m (t)], while the conjugated and delayed by the time τ
the quantity is E
Ã
L t À τ
ð
Þ¼E L0 exp Àj2πν 0L t À j2πν 0L τ À ψ m t À τ
ð
Þ
ð
Þ .
The autocorrelation function R 00 (t, t À τ) of the complex envelope of the laser
random field strength is determined as R L00 t, t þ τ
ð
Þ¼ E L0 t
ð Þ Á E
Ã
L0 t À τ
ð
Þ
¼
E
2
L0 exp ÀΔν L Á τ
ð
Þ .
In order to calculate the required laser spectral density, we can use the Wiener–
Khinchin formula:
S L ¼ 4
Z 1
0
R L τ
ð Þ cos 2πf τ
ð
Þdτ
¼ 4
Z 1
0
E
2
L exp ÀΔν L τ þ j2πντ
ð
Þ cos 2πf τ
ð
Þdτ:
ð5:83Þ
The laser emission spectrum is Lorentzian and can be written as:
S L ¼
1= Δν L
ð
Þ
1 þ ν À ν 0L
ð
Þ=Δν L
½
2
¼
Δν L
Δν 2
L þ ν À ν 0L
ð
Þ
½
2
:
ð5:84Þ
5.5 Correlator’s Mathematical Model in OEO MZ
261
multiplier, the random process in the LPF output is η(t). LPF performs the integration
operation of the multiplication results of two random quantities.
At closing of the switch Sw, owing to oscillation excitation in the closed OEO,
the process of the laser phase fluctuations occurs, which is caused by the carrier noise
or by the spontaneous emission. In OEO, operations of multiplication and integration of LPF signals are performed by the photodetector.
5.5.3 The Mach–Zehnder Correlator-Interferometer
Let us examine the structure in Fig. 5.11 for modeling of the statistical process in
OEO with utilization of the traditional correlator at opened switch Sw and at absence
of electrical oscillations in the electrical input of the Mach–Zehnder modulation. At
that, for determination of the laser emission spectrum, it is necessary to calculate of
the autocorrelation function R(t, t + τ) of the laser random field [4]. ACF is determined by the coherent time T c , which is inverse proportional to the laser spectral line
width Δν L :
R L t, t þ τ
ð
Þ¼
Z 1
À1
E L t
ð ÞE
Ã
L t À τ
ð
Þdt ¼ E
2
L0 exp À
τ
T c
exp j2πν 0L τ
ð
Þ , ð5:82Þ
where ν L ¼ ν 0L ¼ ν 0 is the current optical frequency of the laser, and
E L (t) ¼ E L0 exp [j2πν 0L t + ψ m (t)], while the conjugated and delayed by the time τ
the quantity is E
Ã
L t À τ
ð
Þ¼E L0 exp Àj2πν 0L t À j2πν 0L τ À ψ m t À τ
ð
Þ
ð
Þ .
The autocorrelation function R 00 (t, t À τ) of the complex envelope of the laser
random field strength is determined as R L00 t, t þ τ
ð
Þ¼ E L0 t
ð Þ Á E
Ã
L0 t À τ
ð
Þ
¼
E
2
L0 exp ÀΔν L Á τ
ð
Þ .
In order to calculate the required laser spectral density, we can use the Wiener–
Khinchin formula:
S L ¼ 4
Z 1
0
R L τ
ð Þ cos 2πf τ
ð
Þdτ
¼ 4
Z 1
0
E
2
L exp ÀΔν L τ þ j2πντ
ð
Þ cos 2πf τ
ð
Þdτ:
ð5:83Þ
The laser emission spectrum is Lorentzian and can be written as:
S L ¼
1= Δν L
ð
Þ
1 þ ν À ν 0L
ð
Þ=Δν L
½
2
¼
Δν L
Δν 2
L þ ν À ν 0L
ð
Þ
½
2
:
ð5:84Þ
5.5 Correlator’s Mathematical Model in OEO MZ
261