K
2
ΓPN ¼
cos FT FOS
ð
ÞÀ1 þ 1 À σ U
½
2
1 À 1 þ σ U
ð
Þcos FT FOS
½
þσ U
f
g
2
¼
4 sin
4 FT FOS =2
ð
ÞÀ2 sin
2 FT FOS =2
ð
Þ1 À σ U
ð
Þ
1 À 1 þ σ U
ð
Þcos FT FOS
½
þσ U
f
g
2
þ
1 À σ U
½
2
1 À 1 þ σ U
ð
Þcos FT FOS
½
þσ U
f
g
2
:
ð6:78Þ
In the last expression, we extract the second term and introduce the designation
K
2
2ΓPN ¼
1Àσ U
½
2
1À 1þσ U
ð
Þcos FT FOS
½
þ σ U
f
g
2 .
The approximate coefficient K
2
2ΓPN can be used only for rough engineering
estimates of PSD of the OEO phase noise. It correctly reflects the oscillating
character of PSD, but it does not give the exact limit values. In engineering
calculations, the suppression coefficient K
2
ΓPN can be calculated at σ U ¼ 1.0 with
utilization of the simplified formula:
K
2
2ΓPN ¼
1 À σ U
σ U
2
1
σ U
2 À 2
cos FT FOS
½
σ U
þ 1
n
o
2
4
3
5
2
:
ð6:79Þ
If the term
y M 1þFT F
½
P 0L K LZ
j
j % 1, for the coefficient K PN the following expression is true:
K
2
2ΓPN ¼
0:1
0:81 2:23À2:22 cos FT FOS
½
f
g
h
i 2 %
0:1
1:81Á 1À0:995Á cos FT FOS
½
f
g
h
i 2
. Figure 6.26b shows
functions of the suppression coefficient square K
2
ΓFM FT FOS
ð
Þof the phase noise in
OEO MZ, and Fig. 6.26b shows the suppression coefficient K ΓFM (FT FOS ) of the
phase noise in OEO MZ calculated on the expression (6.79).
From analysis of curves presented in Fig. 6.26b, we may make a conclusion that
using the approximated formula for K
2
2ΓFM FT FOS
ð
Þ (6.79) for calculation of the
suppression coefficient of the phase noise K
2
ΓFM FT FOS
ð
Þ, we must take into consideration that minimal values of K
2
2ΓFM FT FOS
ð
Þ differ by the one order than at
utilization of more exact formula for K
2
ΓFM FT FOS
ð
Þ(6.77).
The MZ modulator made on the base of Fig. 6.24 provides the effective suppression of the laser, PD and NA phase noises owing to the optical phase modulation in
both channels OC1 and OC2 and the effect caused by the self-heterodyning, which is
described in Chap. 3. We also note that effective suppression of the phase noise in
OEO will occur in the case if the noises are absent caused by the DC component of
optical emission.
6.5 Differential Fluctuation Equations of OEO MZ
341
2
ΓPN ¼
cos FT FOS
ð
ÞÀ1 þ 1 À σ U
½
2
1 À 1 þ σ U
ð
Þcos FT FOS
½
þσ U
f
g
2
¼
4 sin
4 FT FOS =2
ð
ÞÀ2 sin
2 FT FOS =2
ð
Þ1 À σ U
ð
Þ
1 À 1 þ σ U
ð
Þcos FT FOS
½
þσ U
f
g
2
þ
1 À σ U
½
2
1 À 1 þ σ U
ð
Þcos FT FOS
½
þσ U
f
g
2
:
ð6:78Þ
In the last expression, we extract the second term and introduce the designation
K
2
2ΓPN ¼
1Àσ U
½
2
1À 1þσ U
ð
Þcos FT FOS
½
þ σ U
f
g
2 .
The approximate coefficient K
2
2ΓPN can be used only for rough engineering
estimates of PSD of the OEO phase noise. It correctly reflects the oscillating
character of PSD, but it does not give the exact limit values. In engineering
calculations, the suppression coefficient K
2
ΓPN can be calculated at σ U ¼ 1.0 with
utilization of the simplified formula:
K
2
2ΓPN ¼
1 À σ U
σ U
2
1
σ U
2 À 2
cos FT FOS
½
σ U
þ 1
n
o
2
4
3
5
2
:
ð6:79Þ
If the term
y M 1þFT F
½
P 0L K LZ
j
j % 1, for the coefficient K PN the following expression is true:
K
2
2ΓPN ¼
0:1
0:81 2:23À2:22 cos FT FOS
½
f
g
h
i 2 %
0:1
1:81Á 1À0:995Á cos FT FOS
½
f
g
h
i 2
. Figure 6.26b shows
functions of the suppression coefficient square K
2
ΓFM FT FOS
ð
Þof the phase noise in
OEO MZ, and Fig. 6.26b shows the suppression coefficient K ΓFM (FT FOS ) of the
phase noise in OEO MZ calculated on the expression (6.79).
From analysis of curves presented in Fig. 6.26b, we may make a conclusion that
using the approximated formula for K
2
2ΓFM FT FOS
ð
Þ (6.79) for calculation of the
suppression coefficient of the phase noise K
2
ΓFM FT FOS
ð
Þ, we must take into consideration that minimal values of K
2
2ΓFM FT FOS
ð
Þ differ by the one order than at
utilization of more exact formula for K
2
ΓFM FT FOS
ð
Þ(6.77).
The MZ modulator made on the base of Fig. 6.24 provides the effective suppression of the laser, PD and NA phase noises owing to the optical phase modulation in
both channels OC1 and OC2 and the effect caused by the self-heterodyning, which is
described in Chap. 3. We also note that effective suppression of the phase noise in
OEO will occur in the case if the noises are absent caused by the DC component of
optical emission.
6.5 Differential Fluctuation Equations of OEO MZ
341