Page 312 - Fiber Optic Communications Fund

Optical Amplifiers 293

the mean noise current due to ASE–ASE noise beating is

B H (f = 0).
< I sp−sp >=< I out,sp−sp (0) >= R ASE o ̃ e (6.255)
If the electrical filter is an ideal low-pass filter given by
( )
f
H (f)= rect , (6.256)
e
2f e
Eq. (6.255) becomes
< I >= R B . (6.257)
sp−sp ASE o
Squaring Eq. (6.252) and then averaging, we find
∞ ∞
′
′′
′
′
2
′′
< I out,sp−sp (0) > = ∫ ∫ < I in,sp−sp (t )I in,sp−sp (t )H (−t )H (−t ) > dt dt ′′
e
e
−∞ −∞
∞ ∞
′′
′
′′
2
2
2
′
2
= R 2 < [n (t )+ n (t )][n (t )+ n (t )] >
∫ ∫ Fr Fi Fr Fi
−∞ −∞
′
′′
′
′′
×H (−t )H (−t )dt dt . (6.258)
e e
Using the moment theorem (see Eq. (6.238)), we obtain
′′
′
2
′
2
′′
2
′′
2
′
2
< n (t )n (t ) > = < n (t ) >< n (t ) > +2 < n (t )n (t )> , (6.259)
Fr Fr Fr Fr Fr Fr
2 ′ 2 ′′ 2 ′ 2 ′′ ′ ′′ 2
< n (t )n (t ) > = < n (t ) >< n (t ) > +2 < n (t )n (t )> , (6.260)
Fi Fi Fi Fi Fi Fi
2
′
′′
′
′′
2
′
2
2
2
′′
< n (t )n (t ) > = < n (t ) >< n (t ) > +2 < n (t )n (t )> . (6.261)
Fr Fi Fr Fi Fr Fi
′
′
2
2
< n (t ) > = < n (t ) >= B ∕2, (6.262)
Fr Fi ASE o
ASE
′′
′′
′
′
′′
′
< n (t )n (t ) > = < n (t )n (t ) >=  (t − t ), (6.263)
opt
Fr
Fi
Fi
Fr
2
′
′′
< n (t )n (t ) > = 0, (6.264)
Fr
Fi
where
2
̃
[ (t)] = |H (f)| . (6.265)
opt
opt
Using Eqs. (6.259)–(6.264) in Eq. (6.258), we find
2 ′ 2 ′ 2 ′′ 2 ′′ 2 2 2 2 ′ ′′
< [n (t )+ n (t )][n (t )+ n (t )] >= B +  (t − t ), (6.266)
Fr Fi Fr Fi ASE o ASE opt
2
2
2 2
2
< I out,sp−sp (0) >= R ASE [B  + B ], (6.267)
oe
o
e
′′
′′
′
′
′′
2
′
B 2 oe = ∫∫  (t − t )H (t )H (t )dt dt , (6.268)
e
e
opt
[ ] 2
′
 = H (t )dt ′ , (6.269)
e ∫ e
2
2
2
sp−sp =< I out,sp−sp > − < I out,sp−sp > . (6.270)