photodetection and obtain formulas for oscillation spectrum in the photodetector
output taking into account phase fluctuations.
3.1.3.1 The Autocorrelation Function of the Random EMF of Laser
Emission
The autocorrelation function R(t, t + τ) of the random EMF of the laser emission
E L (t) depends on the laser coherence time T coh and is defined as R L (t, t + τ) ¼ R L (τ):
R L τ
ð Þ ¼ E L t
ð Þ Á E
Ã
L t À τ
ð
Þ
¼
Z 1
À1
E L t
ð Þ Á E
Ã
L t À τ
ð
Þdt:
ð3:4Þ
Substituting in the expression for R L (t, t + τ) oscillations of the normalized
intensity E L (t) ¼ E L0 exp[j2π Á νt + ψ m (t)] and oscillations of conjugated quantity
and delayed quantity by τ. E
Ã
L t À τ
ð
Þ¼E L0 exp Àj2π Á νt À j2π Á ντ À ψ m t À τ
ð
Þ
½
Š ,
we can write the expression for R(t, t + τ) in the form:
R t,t þτ
ð
Þ¼
Z 1
À1
E L0 exp j2π Áνt þψ m t
ð Þ
½
Š ÁE L0 exp Àj2π Áνt þj2π Áντ Àψ m t Àτ
ð
Þ
½
Š dt
¼ E
2
L0
Z 1
À1
exp j2π Áντ þψ m t
ð ÞÀψ m t Àτ
ð
Þ
½
Š dt:
ð3:5Þ
We should note that in the last formula, the phase fluctuation difference
ψ m (t) À ψ m (t À τ) is defined by the laser coherence time T coh After integration,
we have the expression for R(t, t + τ) as R t, t þ τ
ð
Þ¼E
2
L0 exp À
τ
T coh
Á
exp j2π Á ντ
ð
Þ .
To calculate the required power spectral density of the laser S L , we must use the
Wiener–Khinchin formula
S L ¼ 4
Z 1
0
R L τ
ð Þ cos 2πf τ
ð
Þdτ
¼ 4
Z 1
0
E
2
L exp À
τ
T coh
þ j2πντ
cos 2πf τ
ð
Þdτ
ð3:6Þ
3.1 Direct and External Laser Modulation in OEO
83