Δ f gen ¼
S ΨAG F ¼ Δ f gen
À
Á Á D Y
4T
2
0L
¼ G 22 G 12 K
2
ΓPN Á K
4
PD Á D Y
Δν
T
2
0L P 0L Δ f gen
2
ð6:82Þ
or for T
2
0L F
2
% 1:
Δ f gen ¼ G 12 G 22
ffiffi ffi
2
p
sin π=4 À FT FOS
ð
Þ
Â
à À σ U
È
É 2
1 À 1 þ σ U
ð
Þcos FT FOS
½
þσ U
f
g
2
Á K
4
PD Á D Y
Δν
P 0L
% G 12 G 22
1 À σ U
f
g
2
1 À 1 þ σ U
ð
Þcos ΔνT FOS
½
þσ U
f
g
2
Á K
4
PD Á D Y
Δν
P 0L
:
ð6:83Þ
The first limit case. From Eqs. (6.83), it follows that at given Δν L , a choice of
ΔT M and T FOS strongly affects onto the radio frequency natural spectral width of
OEO. Let us examine the first limit case for expression (6.83) for
G 12 G 22 ¼ [1 À γ k exp (ÀΔν L Á ΔT M )] % 1 (i.e., when optical phases of the coherent
laser emission at arrival to PD are completely non-correlated (independent of each
other) with the correlation coefficient, which is essentially less than 1. In this case,
the correlation coefficient K Ψ12 ! 0. This, for instance, is true at utilization in OEO
of differential RF FODL with two optical fibers, at the time delay in FOS
T FOS ¼ 10
À9 s (the geometrical FOS length is close to 1 m), but the MZ delay
difference is equal to ΔT M ¼ 10
À4 s (which corresponds to the FOS geometric
length of 20 km), the natural width of the laser spectral line is equal to Δν L % 10
kHz, i.e., the product ΔT M Á Δν L ¼ 1, we from formula (6.83) obtain that
Δ f gen ¼ 2G 22 G 12 K
2
ΓPN Á K
4
PD
Δν L
D Y
%
2G 22 G 12 K
4
PD
P 0L K FOLD
j
jD Y
Δν L :
ð6:84Þ
Taking into account the last formula for the RF generation spectral line width, we
obtain the formula:
Δ f gen ¼ Δ f 0:5 % G 12 G 22
1 À σ U
f
g
2
1 À 1 þ σ U
ð
Þcos ΔνT FOS
½
þσ U
f
g
2
Á K
4
PD Á D Y
Δν
P 0L
% G 12 G 22
Δν L
P 0L K FOLD
j
j1 þ Δν L T FOS
½
2
:
ð6:85Þ
The connection of the laser emission spectral line width Δν L with the RF
generation bandwidth Δf gen can be defined from Eqs. (6.85) by the approximated
expression (at G 12 G 22 % 1):
Δν L % Δ f 0:5 Á P 0L K FODL
j
jÁ 4,
ð6:86Þ
6.5 Differential Fluctuation Equations of OEO MZ
343
S ΨAG F ¼ Δ f gen
À
Á Á D Y
4T
2
0L
¼ G 22 G 12 K
2
ΓPN Á K
4
PD Á D Y
Δν
T
2
0L P 0L Δ f gen
2
ð6:82Þ
or for T
2
0L F
2
% 1:
Δ f gen ¼ G 12 G 22
ffiffi ffi
2
p
sin π=4 À FT FOS
ð
Þ
Â
à À σ U
È
É 2
1 À 1 þ σ U
ð
Þcos FT FOS
½
þσ U
f
g
2
Á K
4
PD Á D Y
Δν
P 0L
% G 12 G 22
1 À σ U
f
g
2
1 À 1 þ σ U
ð
Þcos ΔνT FOS
½
þσ U
f
g
2
Á K
4
PD Á D Y
Δν
P 0L
:
ð6:83Þ
The first limit case. From Eqs. (6.83), it follows that at given Δν L , a choice of
ΔT M and T FOS strongly affects onto the radio frequency natural spectral width of
OEO. Let us examine the first limit case for expression (6.83) for
G 12 G 22 ¼ [1 À γ k exp (ÀΔν L Á ΔT M )] % 1 (i.e., when optical phases of the coherent
laser emission at arrival to PD are completely non-correlated (independent of each
other) with the correlation coefficient, which is essentially less than 1. In this case,
the correlation coefficient K Ψ12 ! 0. This, for instance, is true at utilization in OEO
of differential RF FODL with two optical fibers, at the time delay in FOS
T FOS ¼ 10
À9 s (the geometrical FOS length is close to 1 m), but the MZ delay
difference is equal to ΔT M ¼ 10
À4 s (which corresponds to the FOS geometric
length of 20 km), the natural width of the laser spectral line is equal to Δν L % 10
kHz, i.e., the product ΔT M Á Δν L ¼ 1, we from formula (6.83) obtain that
Δ f gen ¼ 2G 22 G 12 K
2
ΓPN Á K
4
PD
Δν L
D Y
%
2G 22 G 12 K
4
PD
P 0L K FOLD
j
jD Y
Δν L :
ð6:84Þ
Taking into account the last formula for the RF generation spectral line width, we
obtain the formula:
Δ f gen ¼ Δ f 0:5 % G 12 G 22
1 À σ U
f
g
2
1 À 1 þ σ U
ð
Þcos ΔνT FOS
½
þσ U
f
g
2
Á K
4
PD Á D Y
Δν
P 0L
% G 12 G 22
Δν L
P 0L K FOLD
j
j1 þ Δν L T FOS
½
2
:
ð6:85Þ
The connection of the laser emission spectral line width Δν L with the RF
generation bandwidth Δf gen can be defined from Eqs. (6.85) by the approximated
expression (at G 12 G 22 % 1):
Δν L % Δ f 0:5 Á P 0L K FODL
j
jÁ 4,
ð6:86Þ
6.5 Differential Fluctuation Equations of OEO MZ
343