7.1.1 The Influence of the Coherence and the Width
of the Laser Spectral Line on the PSD of the Phase
Noise in OEO
We again address to consideration of the structure in Fig. 7.1.
The result E 12L of adding of two optical oscillations E 1L and E 2L with different
delays is determined by the sum in Fig. 7.1: E 12L ¼ E 1L þ E 2L ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
2
1L þ E
2
2L
q
exp jarg E 12L
ð
Þ
½
.
The field strengths of two oscillations on PD passed through the MZ modulator
on the first and second optical channels are equal relatively:
E 1L t
ð Þ¼ E 1L
j jRe exp j2πνÁ t ÀT FOS ÀT 1M
ð
Þ þ jφ 0L1
½
exp þjψ m1 t ÀT FOS ÀT 1M
ð
Þ
½
f
g ;
E 2L t
ð Þ¼ E 2L
j jRe exp j2πνÁ t ÀT FOS ÀT 2M
ð
Þ þ jφ 0L2
½
exp þjψ m2 t ÀT FOS ÀT 2M
ð
Þ
½
f
g :
If we assume that the delay difference in the MZ optical channels is equal to
ΔT M ¼ T 1M À T 2M for the resulting ACF E 12L on the PD area at impact on the MZ
optical input of the laser optical emission, we must distinguish the observation times
À2ΔT M < |τ| < 2ΔT M and |τ| > ΔT M at calculation and at integration over τ.
So, for |τ| > ΔT M , the ACF of the strength E 12L is determined by the expression:
R ηL τ
ð Þ ¼ exp À
2ΔT M
T c
cos 2πντ
ð
Þ. At À2ΔT M < |τ| < 2ΔT M , the ACF E 12L is
determined by the expression:
R ηL τ
ð Þ ¼ exp À
2τ
T c
cos 2πντ
ð
Þ:
ð7:3Þ
We attract our attention to the fact that at open switch Sw, the structure in
Fig. 7.1b represents the correlator structure with the difference or differential delay
ΔT M .
7.1.1.1 The Open Sw
Figure 7.2 shows the model of the single-dimension optical circuit (L is the laser, PD
is the photodetector) with the difference delay ΔT M (a), the visual representation of
integration of the laser ACF R ηL (τ) (b), the visual representation of integration and
adding of the laser ACF R ηL (τ) with delayed laser ACF R ηL (τ À ΔT M ) for in-phase
(c) and out-or-phase (d).
We emphasize that at open Sw (Fig. 7.1), the correlation function of the process in
the output of the adding device with delay lines with the difference in delays
ΔT M ¼ T 1M À T 2M can be obtained, as in the structure of the classical adder with
two delay lines [2, 3], in the form:
370
7 Optoelectronic oscillator (OEO) as the Time and Spatial Correlator of Random. . .
of the Laser Spectral Line on the PSD of the Phase
Noise in OEO
We again address to consideration of the structure in Fig. 7.1.
The result E 12L of adding of two optical oscillations E 1L and E 2L with different
delays is determined by the sum in Fig. 7.1: E 12L ¼ E 1L þ E 2L ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
E
2
1L þ E
2
2L
q
exp jarg E 12L
ð
Þ
½
.
The field strengths of two oscillations on PD passed through the MZ modulator
on the first and second optical channels are equal relatively:
E 1L t
ð Þ¼ E 1L
j jRe exp j2πνÁ t ÀT FOS ÀT 1M
ð
Þ þ jφ 0L1
½
exp þjψ m1 t ÀT FOS ÀT 1M
ð
Þ
½
f
g ;
E 2L t
ð Þ¼ E 2L
j jRe exp j2πνÁ t ÀT FOS ÀT 2M
ð
Þ þ jφ 0L2
½
exp þjψ m2 t ÀT FOS ÀT 2M
ð
Þ
½
f
g :
If we assume that the delay difference in the MZ optical channels is equal to
ΔT M ¼ T 1M À T 2M for the resulting ACF E 12L on the PD area at impact on the MZ
optical input of the laser optical emission, we must distinguish the observation times
À2ΔT M < |τ| < 2ΔT M and |τ| > ΔT M at calculation and at integration over τ.
So, for |τ| > ΔT M , the ACF of the strength E 12L is determined by the expression:
R ηL τ
ð Þ ¼ exp À
2ΔT M
T c
cos 2πντ
ð
Þ. At À2ΔT M < |τ| < 2ΔT M , the ACF E 12L is
determined by the expression:
R ηL τ
ð Þ ¼ exp À
2τ
T c
cos 2πντ
ð
Þ:
ð7:3Þ
We attract our attention to the fact that at open switch Sw, the structure in
Fig. 7.1b represents the correlator structure with the difference or differential delay
ΔT M .
7.1.1.1 The Open Sw
Figure 7.2 shows the model of the single-dimension optical circuit (L is the laser, PD
is the photodetector) with the difference delay ΔT M (a), the visual representation of
integration of the laser ACF R ηL (τ) (b), the visual representation of integration and
adding of the laser ACF R ηL (τ) with delayed laser ACF R ηL (τ À ΔT M ) for in-phase
(c) and out-or-phase (d).
We emphasize that at open Sw (Fig. 7.1), the correlation function of the process in
the output of the adding device with delay lines with the difference in delays
ΔT M ¼ T 1M À T 2M can be obtained, as in the structure of the classical adder with
two delay lines [2, 3], in the form:
370
7 Optoelectronic oscillator (OEO) as the Time and Spatial Correlator of Random. . .