S 1L F
ð Þ %
S 1ΨOEO
P G
¼ K
2
ΓPN
G 12 Á K
2
PD S 1βPN
P 0L F
2 T
2
0L
þ
P 0L
P G
Á
2eS 0PD Á R PD
1
þ
D Y kT
P 0L
!
&
'
,
ð6:71Þ
S 2L F
ð Þ %
S 2ΨOEO
P G
¼ K
2
ΓPN
G 12 K
2
PD S 2βPN
P 0L F
2 T
2
0L
þ
P 0L
P G
Á
2eS 0PD Á R PD
1
þ
D Y kT
P 0L
!
&
'
:
ð6:72Þ
We note that in result of PSD adding S 1L (F) and S 2L (F) in the photo-current of
PD2, we obtain:
S PD2 F
ð Þ ¼ S 1L F
ð Þ þ S 2L F
ð Þ À 2 k 01 =k 02
ð
ÞK 21 S 1L F
ð Þ Ã S 2L F
ð Þ,
ð6:73Þ
where “Ô is the convolution operation, K 21 is the correlation function
K 21 ¼ exp (À2Δν L Á ΔT M À 2Δν L Á ΔT FOS ) of the random process, Δν L is the
natural width of the spectral line of the laser emission, ΔT M is the difference of
delays in MZ and FOS channels, γ k is the irregularity coefficient of optical channel
excitation γ k ¼ k 01 /k 02 , k 01 and k 02 is the excitation coefficients of optical channel
OC1 and OC2 in the MZ modulator: k 01 ¼ P 1L /P 0L and k 02 ¼ P 2L /P 0L ; P 1L and P 2L
is the optical power in the optical channel OC1 and OC2 modulator MZ.
From expression (6.73), the important conclusion follows that at equalization in
the optical power in OC1 and OC2, the following relation is fulfilled:
S 1L (F) % S 2L (F). At small Δν L Á ΔT M ! 0, ΔT M ( ΔT FOS and γ k ¼ k 01 /k 02 close
to 1, we may use the expression:
S PD2 F
ð Þ ¼ S 1L F
ð Þ 1 À k 01 =k 02
ð
ÞÁexp À2Δν L Á ΔT M À 2Δν L Á T FOS
ð
Þ
½
: ð6:74Þ
Then this expression can be written as:
S PD2 F
ð Þ ¼ G 22 K
2
ΓPN
Á
G 12 K
2
PD S 2βPN
P 0L F
2 T
2
0L
þ
P 0L
P G
Á
2eS 0PD Á R PD
1
þ
D Y kT
P 0L
!
&
'
K
2
PD ,
ð6:75Þ
where G 22 ¼ [1 À (k 01 /k 02 ) Á exp (À2Δν L Á ΔT M À 2Δν L Á T FOS )], G 12 ¼ [1 À (A 1 /
A 2 ) Á exp (À2Δν L Á ΔT M À 2Δν L Á T FOS )]. Thus, during simplification:
K
2
PD S 2βPN
P 0L F
2 T
2
0L
)
P 0L
P G
Á
2eS 0PD ÁR PD
1
þ
D Y kT
P 0L
h
i
, we obtain the following expression:
6.5 Differential Fluctuation Equations of OEO MZ
337
ð Þ %
S 1ΨOEO
P G
¼ K
2
ΓPN
G 12 Á K
2
PD S 1βPN
P 0L F
2 T
2
0L
þ
P 0L
P G
Á
2eS 0PD Á R PD
1
þ
D Y kT
P 0L
!
&
'
,
ð6:71Þ
S 2L F
ð Þ %
S 2ΨOEO
P G
¼ K
2
ΓPN
G 12 K
2
PD S 2βPN
P 0L F
2 T
2
0L
þ
P 0L
P G
Á
2eS 0PD Á R PD
1
þ
D Y kT
P 0L
!
&
'
:
ð6:72Þ
We note that in result of PSD adding S 1L (F) and S 2L (F) in the photo-current of
PD2, we obtain:
S PD2 F
ð Þ ¼ S 1L F
ð Þ þ S 2L F
ð Þ À 2 k 01 =k 02
ð
ÞK 21 S 1L F
ð Þ Ã S 2L F
ð Þ,
ð6:73Þ
where “Ô is the convolution operation, K 21 is the correlation function
K 21 ¼ exp (À2Δν L Á ΔT M À 2Δν L Á ΔT FOS ) of the random process, Δν L is the
natural width of the spectral line of the laser emission, ΔT M is the difference of
delays in MZ and FOS channels, γ k is the irregularity coefficient of optical channel
excitation γ k ¼ k 01 /k 02 , k 01 and k 02 is the excitation coefficients of optical channel
OC1 and OC2 in the MZ modulator: k 01 ¼ P 1L /P 0L and k 02 ¼ P 2L /P 0L ; P 1L and P 2L
is the optical power in the optical channel OC1 and OC2 modulator MZ.
From expression (6.73), the important conclusion follows that at equalization in
the optical power in OC1 and OC2, the following relation is fulfilled:
S 1L (F) % S 2L (F). At small Δν L Á ΔT M ! 0, ΔT M ( ΔT FOS and γ k ¼ k 01 /k 02 close
to 1, we may use the expression:
S PD2 F
ð Þ ¼ S 1L F
ð Þ 1 À k 01 =k 02
ð
ÞÁexp À2Δν L Á ΔT M À 2Δν L Á T FOS
ð
Þ
½
: ð6:74Þ
Then this expression can be written as:
S PD2 F
ð Þ ¼ G 22 K
2
ΓPN
Á
G 12 K
2
PD S 2βPN
P 0L F
2 T
2
0L
þ
P 0L
P G
Á
2eS 0PD Á R PD
1
þ
D Y kT
P 0L
!
&
'
K
2
PD ,
ð6:75Þ
where G 22 ¼ [1 À (k 01 /k 02 ) Á exp (À2Δν L Á ΔT M À 2Δν L Á T FOS )], G 12 ¼ [1 À (A 1 /
A 2 ) Á exp (À2Δν L Á ΔT M À 2Δν L Á T FOS )]. Thus, during simplification:
K
2
PD S 2βPN
P 0L F
2 T
2
0L
)
P 0L
P G
Á
2eS 0PD ÁR PD
1
þ
D Y kT
P 0L
h
i
, we obtain the following expression:
6.5 Differential Fluctuation Equations of OEO MZ
337