R ηPD τ
ð Þ ¼ 2R ξL τ
ð Þ þ R ξL τ À ΔT M
ð
ÞþR ξL τ þ ΔT M
ð
Þ ,
ð7:4Þ
where R ηPD (τ) is the correlation function of the stationary random process in the PD
input (or in the MZ output) R ξL τ
ð Þ ¼ exp À
2τ
T c
cos 2πντ
ð
Þ, where T c is the laser
coherence time. Integration of the adding (or subtracting) the result of mutually
delayed ACFs (Eq. 7.4) determines the energy spectrum or PSD:
S ηPD f
ð Þ ¼
ð
1
À1
R ηPD τ
ð Þ cos 2πf τ
ð
Þdτ:
ð7:5Þ
From Fig. 7.2, we see that at adding in out-of-phase (d), the PSD level in the PD
input will be minimal and is equal to zero. Hence, noises introduced by the DC
component will be minimal in the PD load. At in-phase adding (Fig. 7.2c), the
1.0
R hL (t)
R hL (t) = exp(- )cos(2pv t)
2t
T c
0.5
-0.5
2
4
6
8
1 0
t
-1.0
L
PD
a)
b)
c)
d)
1.0
R hL (t)
R hL (t) = exp(- )cos(2pv t)
2t
T c
0.5
-0.5 0.5
2
4
6
8
1 0
t
-1.0
1.0
R hL (t)
0.5
-0.5
2
4
6
8
1 0
t
-1.0
1.0
R hL (t)
R hL (t) = exp()cos(2pv (t-ΔT M ))
T c
0.5
-0.5
2
4
6
8
1 0
t
-1.0
R hL (t) = exp(- )cos(2pv t)
2t
T c
ΔT M
ΔT M
1.0
R hL (t)
R hL (t) = exp(-2
)cos(2pv (t-ΔT M ))
T c
0.5
-0.5
2
4
6
8
1 0
-1.0
ΔT M
ΔT M
t-ΔT M
ΔT M
t-ΔT M
c
c
Fig. 7.2 The model of the single-dimension optical structure (L is the laser, PD is the photodetector) with the difference delay ΔT M (a), the visual representation of the laser ACF integration R ηL (τ)
(b), the visual representation of the integration and adding of the laser ACF R ηL (τ) with the delayed
Laser ACF R ηL (τ À ΔT M ) in-phase (c) and out-of-phase (d)
7.1 OEO as the Time and Spatial Correlator of Random Variables
371