Biomedical Engineering Reference
In-Depth Information
∞
∫
1
2
T
e
f
ω
t
=
S
(
ω
)
d
ω
φ
−∞
where
S
ϕ
(ω) is the two-sided power spectral density of the source phase fluc-
tuations ϕ
n
(
t
). Once
S
ϕ
(ω) is known, an allowable system limit on τ, the dif-
ferential time delay between the two signal paths, is set.
Even if the source is relatively noisy, its effects can be minimized by mini-
mizing the absolute phase differences between the two fiber paths. The detec-
tor and loop amplifier noise are often the dominant system noise sources. In
addition to the random zero-mean fluctuations of instantaneous phase in the
laser, the system will have two major bias components to contend with: tem-
perature differentials between the two signal fibers and frequency drifts in
the laser. Both biases can cause phase biases that the loop attempts to elimi-
nate via its tracking mechanism. As bias signals become large, the piezoelec-
tric phase shifter is not able to track them since it cannot cause phase shifts of
many radians. (Note that in a classical PLL, the typical VCO can shift many
hundreds of megaradians when required.) When the phase shifter is satu-
rated, further bias input causes the loop to experience a transient of 2π radi-
ans in phase state, with the frequency state being a variable outside loop. At
this point, the loop will be nearly saturated with opposite polarity if its total
dynamic range is 2π radians; thus when the loop resets its steady state auto-
matically, it will return to the middle of the signal range. A loop dynamic
range of 4π radians minimizes the amount of time the loop is in saturation
without creation of undue performance requirements on the phase shifter.
The differential phase shift Δϕ due to frequency shift Δ
f
is Δϕ(2πΔ1)/ν
0
Δ
f
,
where ν
0
is the velocity of propagation in the fibers, and Δ
l
is the physical
path length differential. The differential phase shift due to temperature
change from steady state of the two fibers is
=
⎛
2
π
λ
⎞
⎟
Δ
φ
(TC
l T
Δ
−
TC
l T
)
Δ
(4.3)
⎜
1 1
1
2 2
2
where
TC
1
and TC
2
are the temperature coefficients of expansion in the two signal
paths
l
1
and
l
2
are the physical lengths of the two signal fibers
T
1
and
T
2
are the differences in the two signal paths' ambient temperatures
from a reference temperature
A temperature-related phase bias is introduced into the loop unless both sig-
nal paths have identical temperature coefficients of expansion and physical
lengths and are at the same ambient temperature.
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