
COHERENCE 317
Using the same denitions and subscripts in the time domain as we did in the frequency
domain, the autocorrelation falls to half at t
1/2
dened by
γ
(
τ
)
= e
−
(
t
1/2
/t
1/e
)
2
= 1/2 (10.35)
t
1/2
=
√
ln 2t
1/e
. (10.36)
Using Equations 10.32 and 10.34,
t
1/2
=
√
ln 2
2
√
2
ω
1/e
= ln 2
2
√
2
2πf
1/2
≈
0.31
f
1/2
=
0.62
f
FWHM
. (10.37)
If the path lengths differ by
z
1/2
= ct
1/2
, (10.38)
γ
t
1/2
= e
t
1/2
t
1/e
=
1
2
by Equation 10.34 and the fringe visibility will drop by a factor of two. To evaluate the results
numerically, it is convenient to express this distance in wavelengths.
z
1/2
λ
= ft
1/2
= ln 2 ×
2
√
2
π
f
f
FWHM
≈ 0.62
f
f
FWHM
, (10.39)
We dene z
1/2
as the coherence length of the source, and ...