6.5.9 The Formula for PSD of the Phase Noise in OEO MZ
Let us rewrite Eq. (6.61) by dividing of the right and left parts by P OEO and then we
shall have the final formula in the form:
S ΨOEO
P OEO
¼ K
2
ΓFN
8D PN S μIm
F
2 T
2
0L
1
P 0L
:
ð6:62Þ
The formula (6.62) is the basis for engineering calculations of the ratio of PSD of
the phase noise to the power of RF oscillations of OEO MZ. Figure 6.20a, b show
suppression coefficients of the phase noise in
K
2
ΓPN
P 0L
and present the functions of the
coefficient σ U (P 0L ) versus the laser power P 0L at σ U < 0 (c) and σ U > 0 (d).
It is necessary to note that σ U can take both positive and negative values, which is
defined by the OEO stability in the steady-state point: σ U Á cos (FT FOS ) < 0. At cos
(FT BLZ ) > 0, the condition σ U < 0 should be fulfilled and vice versa: at cos
[FT FOS ] < 0, the condition σ U > 0 should be fulfilled. Substituting σ U ¼
1
3 Â
1 À
4Y 00
P 0L
1
S 01
into the formulas (6.57) and (6.58), we obtain two various formulas
for cos[FT FOS ] and σ U : 1 (1) for cos[FT FOS ] > 0, σ U < 0, σ U ¼ þ
1
3 À
1
3
Y 00
P 0L
4S 03
S 01
< 0;
and (2) for cos[FT FOS ] < 0, σ U > 0, σ U ¼ À
1
3 þ
1
3
Y 00
P 0L
4S 03
S 01
> 0 . We note that in
Fig. 6.20a, b, the presented functions correspond to values P 0L /Y 00 < 1 below of the
OEO generation threshold, and relatively to P 0L /Y 00 > 1 above this threshold. The
formula (6.62) contains the quadrature component of the laser PSD S μIm , the laser
time constant T 0L (equal approximately to laser coherence time T c ), the square of the
suppression coefficient K
2
ΓPN , which is defined by the expression (6.55) for cos
[FT FOS ] < 0, σ U > 0, σ U ¼ À
1
3 þ
1
3
Y 00
P 0L
4S 03
S 01
> 0, including the offset F, the delay
time T FOS in the optical fiber, parameters of the oscillating system of OEO MZ: Y 00 ,
S 01 , S 03 and the laser power P 0L . K
2
ΓPN insignificantly depends on the laser power.
With laser power P 0L growth, variations of K
2
ΓPN are tenth parts of unit, which is
caused with the increase of RF oscillations power P OEO ¼ P G ¼ U
2
10MZ = 2R PD
ð
Þ
(increase is proportional to the noise). The total decrease of S ΨOEO /P OEO with P 0L
growth is more than ten times, at growth of the laser power is related not to OEO
oscillating system parameters, but directly with the quadrature components of the
laser PSD S μIm .
Figure 6.21 show the plots of suppression coefficients of the phase noise in OEO
MZ K
2
ΓPN ,
K
2
ΓPN
P 0L
, the power of OEO RF oscillations P OEO depending on the laser
power in the linear (а) and logarithm (b) scales in the abscissa axis.
Formulas (6.54) and (6.55) for calculation of suppression coefficients of PSD of
the amplitude and phase noises permit to calculate K
2
ΓPN F
ð Þ not only in the steadystate generation mode in OEO MZ, but in the case, when self-excitation conditions
are not satisfied as well. We computed by the formula (6.55) the function K
2
ΓPN F
ð Þ,
which is presented at σ U ¼ 1.02 below excitation threshold of OEO MZ, which
6.5 Differential Fluctuation Equations of OEO MZ
327
Let us rewrite Eq. (6.61) by dividing of the right and left parts by P OEO and then we
shall have the final formula in the form:
S ΨOEO
P OEO
¼ K
2
ΓFN
8D PN S μIm
F
2 T
2
0L
1
P 0L
:
ð6:62Þ
The formula (6.62) is the basis for engineering calculations of the ratio of PSD of
the phase noise to the power of RF oscillations of OEO MZ. Figure 6.20a, b show
suppression coefficients of the phase noise in
K
2
ΓPN
P 0L
and present the functions of the
coefficient σ U (P 0L ) versus the laser power P 0L at σ U < 0 (c) and σ U > 0 (d).
It is necessary to note that σ U can take both positive and negative values, which is
defined by the OEO stability in the steady-state point: σ U Á cos (FT FOS ) < 0. At cos
(FT BLZ ) > 0, the condition σ U < 0 should be fulfilled and vice versa: at cos
[FT FOS ] < 0, the condition σ U > 0 should be fulfilled. Substituting σ U ¼
1
3 Â
1 À
4Y 00
P 0L
1
S 01
into the formulas (6.57) and (6.58), we obtain two various formulas
for cos[FT FOS ] and σ U : 1 (1) for cos[FT FOS ] > 0, σ U < 0, σ U ¼ þ
1
3 À
1
3
Y 00
P 0L
4S 03
S 01
< 0;
and (2) for cos[FT FOS ] < 0, σ U > 0, σ U ¼ À
1
3 þ
1
3
Y 00
P 0L
4S 03
S 01
> 0 . We note that in
Fig. 6.20a, b, the presented functions correspond to values P 0L /Y 00 < 1 below of the
OEO generation threshold, and relatively to P 0L /Y 00 > 1 above this threshold. The
formula (6.62) contains the quadrature component of the laser PSD S μIm , the laser
time constant T 0L (equal approximately to laser coherence time T c ), the square of the
suppression coefficient K
2
ΓPN , which is defined by the expression (6.55) for cos
[FT FOS ] < 0, σ U > 0, σ U ¼ À
1
3 þ
1
3
Y 00
P 0L
4S 03
S 01
> 0, including the offset F, the delay
time T FOS in the optical fiber, parameters of the oscillating system of OEO MZ: Y 00 ,
S 01 , S 03 and the laser power P 0L . K
2
ΓPN insignificantly depends on the laser power.
With laser power P 0L growth, variations of K
2
ΓPN are tenth parts of unit, which is
caused with the increase of RF oscillations power P OEO ¼ P G ¼ U
2
10MZ = 2R PD
ð
Þ
(increase is proportional to the noise). The total decrease of S ΨOEO /P OEO with P 0L
growth is more than ten times, at growth of the laser power is related not to OEO
oscillating system parameters, but directly with the quadrature components of the
laser PSD S μIm .
Figure 6.21 show the plots of suppression coefficients of the phase noise in OEO
MZ K
2
ΓPN ,
K
2
ΓPN
P 0L
, the power of OEO RF oscillations P OEO depending on the laser
power in the linear (а) and logarithm (b) scales in the abscissa axis.
Formulas (6.54) and (6.55) for calculation of suppression coefficients of PSD of
the amplitude and phase noises permit to calculate K
2
ΓPN F
ð Þ not only in the steadystate generation mode in OEO MZ, but in the case, when self-excitation conditions
are not satisfied as well. We computed by the formula (6.55) the function K
2
ΓPN F
ð Þ,
which is presented at σ U ¼ 1.02 below excitation threshold of OEO MZ, which
6.5 Differential Fluctuation Equations of OEO MZ
327