and for larger time constant T F of the RF filter and for the larger delay time in FOS
T FOS .
If the following inequality is satisfied: T FOS > T eMZ þ T F þ
T 2M þT 1M
2
þ
T 2M ÀT 1M
2
1 À γ
ð
Þ
Δν
f RF
, then the influence of the quasi-static parameter variations of
the OEO MZ optical part can be estimated by the formula for the relative deviations
from the steady-state values ΔΔν and ΔΔ(1 À γ):
Δ f gen
f gen
¼
Δν
f RF
Á
T 2M ÀT 1M
2
1 À γ
ð
Þþ
Δ 1Àγ
ð
Þ
1Àγ
ð
Þ
Δν
f F
h
i
Á
T eMZ þT F
T FOS
:
We would like to note the specific case if different optical emissions transmitted
in the different MZ optical channels OC1 and OC2 after MZ are not connected. In
other words, optical oscillations with different delays propagate through different
optical fibers of the similar or different length. In such a case, the quantity
T 2M ÀT 1M
2
Á 1 À γ
ð
Þ becomes significantly important. For instance, at
T 2M ÀT 1M
2
Á
1 À γ
ð
Þ¼10
À9
À 10
À7
μs , deviations of the laser optical frequency Δν, even at
account of the stabilizing effect of the long optical fiber with the delay T FOS , for
example, 1 μs, should be taking into consideration.
6.4.4 Fluctuations of the Photodetector Photo-Current
in OEO MZ
Now we find the voltage fluctuations in the PD load, which are caused by amplitude
m L (t) and phase Δψ L (t) fluctuations of the electromagnetic field strength E 12L ,
considering that these fluctuations are small compared to its mean values and
neglecting the harmonics of the noise currents. We may prove using expressions
(6.2) and (6.3) for laser fluctuations with spectral densities of amplitude S mL (ν) and
phase S ψL (ν), relatively, that fluctuations μ n of PD i PDΣ are determined by the
detected amplitude m L (t) and the difference phase fluctuations Δψ L ¼ ψ m2 À ψ m1
and by AN-PN fluctuations of the electrical component strength of the laser electromagnetic field and it can be written in the complex form as:
μ n ¼ μ AN þ μ PN‐AN þ jμ PN ,
ð6:17Þ
where μ AN , μ PN ‐ AN , μ PM are fluctuation of PD current caused, relatively, by the
amplitude (АN) m L (t), АN-PN conversion, and phase (PN) Δψ L (t). Values of these
fluctuations of output PD current are results of heterodyning and depend upon the
oscillation amplitude U 10MZ , for them the following expressions are true:
μ AN ¼ m
2
L U 10MZ cos 2π f 0 t
ð
Þ, μ PN ¼ 2 Δψ L
ð
Þ
2 U 10MZ cos 2π f 0 t
ð
Þ:
ð6:18Þ
308
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .
T FOS .
If the following inequality is satisfied: T FOS > T eMZ þ T F þ
T 2M þT 1M
2
þ
T 2M ÀT 1M
2
1 À γ
ð
Þ
Δν
f RF
, then the influence of the quasi-static parameter variations of
the OEO MZ optical part can be estimated by the formula for the relative deviations
from the steady-state values ΔΔν and ΔΔ(1 À γ):
Δ f gen
f gen
¼
Δν
f RF
Á
T 2M ÀT 1M
2
1 À γ
ð
Þþ
Δ 1Àγ
ð
Þ
1Àγ
ð
Þ
Δν
f F
h
i
Á
T eMZ þT F
T FOS
:
We would like to note the specific case if different optical emissions transmitted
in the different MZ optical channels OC1 and OC2 after MZ are not connected. In
other words, optical oscillations with different delays propagate through different
optical fibers of the similar or different length. In such a case, the quantity
T 2M ÀT 1M
2
Á 1 À γ
ð
Þ becomes significantly important. For instance, at
T 2M ÀT 1M
2
Á
1 À γ
ð
Þ¼10
À9
À 10
À7
μs , deviations of the laser optical frequency Δν, even at
account of the stabilizing effect of the long optical fiber with the delay T FOS , for
example, 1 μs, should be taking into consideration.
6.4.4 Fluctuations of the Photodetector Photo-Current
in OEO MZ
Now we find the voltage fluctuations in the PD load, which are caused by amplitude
m L (t) and phase Δψ L (t) fluctuations of the electromagnetic field strength E 12L ,
considering that these fluctuations are small compared to its mean values and
neglecting the harmonics of the noise currents. We may prove using expressions
(6.2) and (6.3) for laser fluctuations with spectral densities of amplitude S mL (ν) and
phase S ψL (ν), relatively, that fluctuations μ n of PD i PDΣ are determined by the
detected amplitude m L (t) and the difference phase fluctuations Δψ L ¼ ψ m2 À ψ m1
and by AN-PN fluctuations of the electrical component strength of the laser electromagnetic field and it can be written in the complex form as:
μ n ¼ μ AN þ μ PN‐AN þ jμ PN ,
ð6:17Þ
where μ AN , μ PN ‐ AN , μ PM are fluctuation of PD current caused, relatively, by the
amplitude (АN) m L (t), АN-PN conversion, and phase (PN) Δψ L (t). Values of these
fluctuations of output PD current are results of heterodyning and depend upon the
oscillation amplitude U 10MZ , for them the following expressions are true:
μ AN ¼ m
2
L U 10MZ cos 2π f 0 t
ð
Þ, μ PN ¼ 2 Δψ L
ð
Þ
2 U 10MZ cos 2π f 0 t
ð
Þ:
ð6:18Þ
308
6 Operation Analysis of Optoelectronic oscillator (OEO) with External. . .