Localization of Distributed Fiber Sensor Employing a Sub-Ring

is
B
LB
0 B+LB 1 B
LB
is
LB
B
0 B
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JOURNAL OF ELECTRONIC SCIENCE AND TECHNOLOGY OF CHINA, VOL. 6, NO. 4, DECEMBER 2008
n=0
Using DC filter, the AC element of output is
CW
Coupler CCW
ZB
0 B=0
1 ⎛
2 ⎞
Pn( τ ) = I0
cos (
n n) (
n
)
n ⎜θτ + δ −θτ −δn
− π⎟. (3)
9× 2 ⎝ 3 ⎠
CW
CCW
CW
CCW
Fig. 2. Equivalent light path.
L
Vibration point
n=1
Vibration point
2LB
3. Localization Theory
The power of SLD is expressed as I, as the −3P PdB 3×3
coupler’s effects, both clockwise and counter clockwise
beams’ intensities are 1/3IB0B, IB0B=α I (α is the turndown
ratio of 3×3 coupler). The vibration position is at either side
of the middle point zB0B=0, and causes a phase difference θBtB
to both the clockwise (CW) and counterclockwise (CCW)
light beams. Because those two lights pass the vibration
point at different time and the 3×3 coupler introduces a
2/3 π phase difference, the interference light is a time
variant. Firstly, the CW light passes through 2×2 coupler
and will pass it n times, so the intensity transmits in sensing
1
fiber is I
1 0
; then the CCW light passes through 2×2
9 2 n +
×
coupler after sensing fiber, the outputting light intensity is
1
I
1 0
too, which travels n times in delay fiber. When
9 2 n +
×
the lights are with the same n interfere, the power range of
1
sensor output is
0
9 2 I and changes with the phase
×
n
difference. It can be expressed as follows:
1 2
Pn( τ ) I0
1 cos ⎛
⎞
= + θτ (
n
δn) θτ (
n
δ )
n ⎜ + − −
n
− π⎟
9× 2 ⎝ 3 ⎠ . (1)
In (1),
⎧ L0 + nL1
τ
n
= t −
⎪ 2c
⎨
L1
(2)
⎪ z
z0
+ n
n
δ
2
⎪ n
= =
⎩ c c
where LB0 Bis the fiber ring length, LB1B the delay fiber length,
and ZBnB the distance from vibration point to the middle of
the sensing loop.
n=2
Vibration point
LB0 B+2LB
Z
ZB
1 B=ZB 0 B+LB 1/
B ZB2B=ZB0B+LB1 ZB
1 B=0
ZB
2 B=0
[2]
It is supposed that θ() t = Acos(2 π ft)
, the
amplitude A and vibration frequency f affect the power
output totally. As the 2/3 π phase difference and the
amplitude of θ () t are small enough, the AC power has
good linearity, and it can be predigested as
where
( )
P() τ ∝ Kcos θτ ( + δ) −θτ ( − δ ) (4)
n n n n n
K
1
.
= I0
9 × 2 n
n
n
As well known, triangle function can be written equally as
p ( τ ) ∝ − 2Ksin(2 π fτ )sin(2 π fδ
) . (5)
n
In this expression, sensor output is changed with time
variant τ
n
, and its scope is related to both vibration
frequency and position. We handle with the scope
coefficients of P ( τ ) 0 , P () τ 1 , and P ( ) 2
τ , so it is equal to
⎧ ⎛ z0
⎞
⎪P0( τ) ∝− 2Ksin(2 π fδ0) =−2Ksin⎜2π
f
c
⎟
⎪
⎝ ⎠
⎪ 1 1 ⎛ z0 + L1/2⎞
⎨P1( τ) ∝− Ksin(2 π fδ1) =− Ksin 2 f
2 2
⎜ π
c
⎟
⎪
⎝
⎠
⎪
⎪ 1 1 ⎛ z0 + L1
⎞
P2() τ ∝− Ksin(2 π fδ2) =− Ksin 2 π f .
⎪ 8 8
⎜
c
⎟
⎩
⎝ ⎠
(6)
Then it is supposed that
z0
L1
A = 2π
f , B = 2π
f , (7)
c
2c
so the scope of output will be written as
⎧
⎪P0
∝−2Ksin
A
⎪ 1
⎨P1
∝− Ksin( A+
B)
(8)
⎪ 2
⎪ 1
P2
∝− Ksin( A+
2 B).
⎪⎩ 8
Let us do division as follows:
D = 4P P = cos B+ ctan Asin
B (9)
1 1 0
D = 16P P = cos 2B+ ctan Asin 2B
(10)
2 2 0
⎛ L1 ⎞ D2
+ 1
cos B = cos⎜π
f
c
⎟=
⎝ ⎠ 2D
2
⎛ z
0 ⎞ 2D1 −D2
−1
ctan A= ctan ⎜2π
f
c
⎟=
⎝ ⎠ 4 D − ( D + 1)
1
2 2
1 2
(11)
(12)