S out F
ð Þ ¼ L AC F
ð ÞK
2
2ΓPN S L F
ð Þ þ L FC F
ð ÞK
2
2ΓPN S conRF F
ð Þ þ S PD F
ð Þ
½
:
ð7:77Þ
In another variant, the laser can be the reference source, and the RF generator is
the controlling element. At that, the expression Eq. (7.76) is written as:
S out F
ð Þ ¼ L AC F
ð ÞK
2
2ΓPN S conRF F
ð Þ þ S PD F
ð Þ
½
þ K
2
2ΓPN L FC F
ð ÞS L F
ð Þ:
ð7:78Þ
For demonstration of PLL system operation in OEO, below we give descriptions
of calculated functions presented in Fig. 7.40a and obtained according Eq. (7.77) and
in Fig. 7.40b obtained according to for K
2
2ΓPN (Eqs. 6.80, 6.92, 6.93 and 7.78).
For demonstration of the function of PSD of the phase noise of OEO S(Δν) with
the PLL system versus the width of the laser spectral line Δv in Fig. 7.40a, we
present the calculated curves 2–4 for different values of T RC /Ω 0 : the ratio of the time
constant of the integration filter T RC and the Ω 0 parameter of the PLL system: the
curve corresponds to T RC /Ω 0 ¼ 0.0002, the curve 3—T RC /Ω 0 ¼ 0.02, the curve 4—
T RC /Ω 0 ¼ 2.0. At that, the curve 1 corresponds to the function of PSD of the phase
noise without PLL.
For calculation of theoretical dependences of PSD of the phase noise in OEO
(Fig. 7.40), the estimation of the width of the laser spectral line Δν ¼ Δν L is
performed according to the appropriation expression for PSD of the laser phase
noise S L ¼ Δν L
D Y
2P L F
2 . At values S L ¼ 10
À12 dB/Hz,
D Y
P L
¼ 10
À10 , at the offset from
the carrier F ¼ 1 kHz, the width of the laser spectral line is Δν L ¼ 20 Â 10
3 Hz.
Presented functions S(Δν) (Fig. 7.40a) are quadratic functions of the width of the
laser spectral line Δν ¼ Δν L , and we see that at the ratio T RC /Ω 0 ¼ 0.0002, the
decrease of PSD of the phase noise is more than 20 dB/Hz for OEO with the PLL
system compared to OEO without PLL.
In Fig. 7.40b, the presented curve 5 corresponds to the computed PSD of the
phase noise of OEO S(F) with the POLL system at the ration T RC /Ω 0 ¼ 0.0002 for
the same laser and OEO elements parameters, as the curves 1–4 (which are in
Fig. 7.40b). The benefit in reduction of PSD of the phase noise of OEO with PLL
is more than 25 dB/Hz at the offset F ¼ 1 kHz, which confirms the effectiveness of
utilization of PLL system in OEO.
So, the PLL system in OEO can be successfully used for the decrease of the OEO
phase noise. At that, we show that this system can decrease the phase noises by
7–15 dB/Hz in the range offsets from the carrier 1–10 kHz. The presence of PLL
system in OEO gives a possibility to decrease the geometrical length of the optical
fiber by 1.5–3 times at the same values of PSD of the phase noise.
7.8 Conclusions
We mark out several main mechanisms for the phase noise suppression in OEO:
7.8 Conclusions
459
ð Þ ¼ L AC F
ð ÞK
2
2ΓPN S L F
ð Þ þ L FC F
ð ÞK
2
2ΓPN S conRF F
ð Þ þ S PD F
ð Þ
½
:
ð7:77Þ
In another variant, the laser can be the reference source, and the RF generator is
the controlling element. At that, the expression Eq. (7.76) is written as:
S out F
ð Þ ¼ L AC F
ð ÞK
2
2ΓPN S conRF F
ð Þ þ S PD F
ð Þ
½
þ K
2
2ΓPN L FC F
ð ÞS L F
ð Þ:
ð7:78Þ
For demonstration of PLL system operation in OEO, below we give descriptions
of calculated functions presented in Fig. 7.40a and obtained according Eq. (7.77) and
in Fig. 7.40b obtained according to for K
2
2ΓPN (Eqs. 6.80, 6.92, 6.93 and 7.78).
For demonstration of the function of PSD of the phase noise of OEO S(Δν) with
the PLL system versus the width of the laser spectral line Δv in Fig. 7.40a, we
present the calculated curves 2–4 for different values of T RC /Ω 0 : the ratio of the time
constant of the integration filter T RC and the Ω 0 parameter of the PLL system: the
curve corresponds to T RC /Ω 0 ¼ 0.0002, the curve 3—T RC /Ω 0 ¼ 0.02, the curve 4—
T RC /Ω 0 ¼ 2.0. At that, the curve 1 corresponds to the function of PSD of the phase
noise without PLL.
For calculation of theoretical dependences of PSD of the phase noise in OEO
(Fig. 7.40), the estimation of the width of the laser spectral line Δν ¼ Δν L is
performed according to the appropriation expression for PSD of the laser phase
noise S L ¼ Δν L
D Y
2P L F
2 . At values S L ¼ 10
À12 dB/Hz,
D Y
P L
¼ 10
À10 , at the offset from
the carrier F ¼ 1 kHz, the width of the laser spectral line is Δν L ¼ 20 Â 10
3 Hz.
Presented functions S(Δν) (Fig. 7.40a) are quadratic functions of the width of the
laser spectral line Δν ¼ Δν L , and we see that at the ratio T RC /Ω 0 ¼ 0.0002, the
decrease of PSD of the phase noise is more than 20 dB/Hz for OEO with the PLL
system compared to OEO without PLL.
In Fig. 7.40b, the presented curve 5 corresponds to the computed PSD of the
phase noise of OEO S(F) with the POLL system at the ration T RC /Ω 0 ¼ 0.0002 for
the same laser and OEO elements parameters, as the curves 1–4 (which are in
Fig. 7.40b). The benefit in reduction of PSD of the phase noise of OEO with PLL
is more than 25 dB/Hz at the offset F ¼ 1 kHz, which confirms the effectiveness of
utilization of PLL system in OEO.
So, the PLL system in OEO can be successfully used for the decrease of the OEO
phase noise. At that, we show that this system can decrease the phase noises by
7–15 dB/Hz in the range offsets from the carrier 1–10 kHz. The presence of PLL
system in OEO gives a possibility to decrease the geometrical length of the optical
fiber by 1.5–3 times at the same values of PSD of the phase noise.
7.8 Conclusions
We mark out several main mechanisms for the phase noise suppression in OEO:
7.8 Conclusions
459