Page 246 - Photodetection and Measurement - Maximizing Performance in Optical Systems

Multiple Channel Detection

Multiple Channel Detection 239

sensors, there is also the propagation group delay to consider, even with rela-
tively slow modulation. This is given by: delay = distance/velocity, where we use
the group-velocity in the fiber (v g ), not the phase-velocity (v p ). Both are func-
tions of wavelength. Phase velocity is:
c
v p l () = (11.1)
n p l ()

where n p is the normal tabulated refractive index at wavelength l and c is the
speed of light. The group velocity and index are given by:

c
v g l () = (11.2)
n g l ()

n g(l) = n p(l) -ldn p(l)/dl (11.3)

For silica-based fibers in the infrared n g ª n p so we can simply assume that
the wave in the fiber is slowed by the phase index, giving the propagation veloc-
ity as:

.
¥
2 9979 10 8
8
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ª 210 ms (11.4)
/
.
147
and a “slowness” of about 5ns/m. The different lengths of monitor-arm and
sensor-arm fibers give us the possibility to adjust the two reflections to be in
quadrature at the detector. If this is done, then a synchronous detector aligned
with the signal will show zero, or at least very greatly suppressed response to
the disturbing monitor-arm reflection.
For quadrature of the two waveforms, they need a relative phase displacement
of p/2, or one quarter of a modulation period. Equating the time delay that we
need, dt = 1/(4·f mod ) to the group delay in the extra fiber of length L s , we obtain:
c
L s = (11.5)
8 nf mod
g
With a 100kHz modulation frequency this corresponds to a fiber of length
L s = 255m. With the low loss of fiber (they are the highest time-bandwidth
product delay lines available), it is often convenient to pad out the length of one
line to achieve this quadrature characteristic. Suppression of the monitor reflec-
tion response by a further 100:1 should be straightforward with this technique.
In practice it is also possible to vary the modulation frequency under computer
control. This would be very useful to accurately and actively null out the
monitor-arm response, allowing for length changes in the fiber due to substi-
tution of alternative connectorized fiber lengths. The combination of frequency
synthesizer and lock-in amplifier is a very powerful workstation for fiber sensor

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