The formula of FBG Fiber Bragg Grating characteristics and ...optoelec-engineering.com/tech/The...
Transcript of The formula of FBG Fiber Bragg Grating characteristics and ...optoelec-engineering.com/tech/The...
The formula of FBG(Fiber Bragg Grating) characteristics and principle of the differential FBG optical circuit
Opto-Electronic Engineering Laboratory Corporation 1. FBG characteristics The equations of the FBG wavelength deviations by the strain forces and by the temperature deviations, is shown below. (eq.1) λB ; filter wavelength of FBG, λB= 1550[nm], n ; refractive index, n=1.45, αs ; temperature coefficient, αs=0.55 [u-strain/ deg C], εs ; external stress strain [u-strain], GF ; conversion coefficient of strain, GF=0.78, δT ; temperature deviation [deg C]
Formula of FBG transfer function ; the normalized power transmittance (eq.2) Formula of FBG transfer function ; the normalized reflection , (eq.3) Here, κ; the coupling coefficient for the 1-st order for unblazed Bragg grating
δ; the detuning parameter L; the grating length
Ω; effective wave number λB; Bragg wavelengths Λ ; period of index modulation m ; mode index neff ; the effective index of guided mode(Clad) η ; the overlap integral between the forward and reverse propagating guided modes calculated over
sFsB
B GTTn
nεδα
λδλ
⋅+⋅⎟⎠⎞
⎜⎝⎛ +
∂∂
⋅=1
( )( )
( )
( )
( )
( )
( )
2
222
1
1cos
⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
m
m
m
m
m
mm L
T
κδ
κδ
κδκ
222
22
1cos
1sin
⎟⎠⎞
⎜⎝⎛−
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⎟⎠⎞
⎜⎝⎛−
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡⎟⎠⎞
⎜⎝⎛−−
=
κδ
κδκ
κδκ
L
L
R 12
>⎟⎠⎞
⎜⎝⎛κδ
Bλπηκ =
Λ−Ω=πδ
λπ effn2
=Ω
Λ= effB n2λ
Fn ⋅Δ=η
the fiber core of Bragg grating Δn ; the amplitude of induced refractive index perturbation 0.1×10^-4 to 0.5×10^-4 Δn=n1-n2 F ; the fractional modal power in the core V ; Normalized Frequency a ; Core Radius λ; Optical Wavelength Δ ; relative refractive index difference n1 ; Core Refractive Index n2 ; Clad Refractive Index d ; Core Diameter Vc ; Cut-off V-value λc ; Cut-off Wavelength ・Parameter Example ・Simulation Result
Fig-1 Simulation Result of FBG transfer function
⎥⎦⎤
⎢⎣⎡ −= 2
11V
F
Δ= 221anV
λπ
1
2121
22
21
2 nnn
nnn −
≈−
=Δ
Δ= 221anV
cc λ
π
a d n1 n2 n1-n2 Δum um %
5 10 1.45 1.4448 0.0052 0.36V Δn neff λB L m
nm mm2.49 0.00005 1.4448 1550 10 1.0000
Λ F η κ Ω δnm
536.4 0.84 4.192E-05 84.964 5.86E+06 1889.9
0
10
20
30
40
50
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70
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100
110
1549.5 1549.6 1549.7 1549.8 1549.9 1550 1550.1 1550.2 1550.3 1550.4 1550.5
Refle
ction & Transmittance[%]
WaveLength[nm]
Simulation of FBG CharacteristicsR=50%, BW=0.2nm
Reflection
Transmittance
2. Characteristics of the differential FBG optical circuit The equations of the differential FBG optical circuit by the strain forces and by the temperature deviations, is shown below. (eq.4)
Fig-2 the differential FBG optical circuit
・Simulation Result of the differential FBG optical circuit via (eq.2)&(eq.3) FBG1&2 Condition ; R=80%, BW=0.3nm
Fig-3 Transfer characteristics of the differential FBG optical circuit with pre-tension to FBG1 via simulation
( ) ( )0
0
21
0
22
20
11
10
21 11sF
B
BBs
B
B
B
B
B
BB GTTTn
nTn
nεδ
λλλαδ
λλ
λλ
λδλδλ
⋅+⋅−
⋅+⋅⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
∂∂⋅−⋅
∂∂⋅=
−
( )2
210
BBB
δλδλλ +=
FBG1(Sensing)
FBG2(Reference)
3dB CUPAR Cutting
Connector
Connector
Optical Source
Optical Monitor
Strain Sensor
Transforming W.L. to Optical Power Level
The Optical Circuits of Differential FBG Sensor
0
10
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30
40
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70
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100
110
1549.5 1549.6 1549.7 1549.8 1549.9 1550 1550.1 1550.2 1550.3 1550.4 1550.5
Refle
ction & Transmittance[%]
WaveLength[nm]
Simulation of FBG CharacteristicsR=80%, BW=0.3nm
Reflection
Transmittance
FBG1
FBG2
Fig-4 Strain sensing Signal at optical receiver in the case of Fig-3 The Strain sensing Signal is reflected by FBG1 and going through FBG2.
Fig-5 Simulation of shifted motion for FBG1 wavelength by external compressive stress
0
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100
110
1549.5 1549.6 1549.7 1549.8 1549.9 1550 1550.1 1550.2 1550.3 1550.4 1550.5
Refle
ction & Transmittance[%]
WaveLength[nm]
Simulation of FBG Characteristics
Transmittance
Rshift0
Rshift1
Rshift2
Rshift3
Rshift4
Rshift5
Rshift6
Rshift7
Rshift8
Rshift9
Rshift10
Rshift11
FBG1
FBG2
0
10
20
30
40
50
60
70
80
90
100
110
0.0E+00
1.0E‐08
2.0E‐08
3.0E‐08
4.0E‐08
5.0E‐08
6.0E‐08
7.0E‐08
8.0E‐08
9.0E‐08
1.0E‐07
1.1E‐07
1549.5 1549.6 1549.7 1549.8 1549.9 1550 1550.1 1550.2 1550.3 1550.4 1550.5
Transm
ittance[%]
Differen
tial FBG
Sen
sor Outpu
t Pow
er[W
]
WaveLength[nm]
Simulation of FBG Characteristics
Shift0
Transmittance
FBG2
Strain sensing Signal
Fig-6 Strain sensing Signal at optical receiver in the case of Fig-5 The strain sensing signal generates the wavelength deviation of FBG1, and is converted into the optical power level via FBG2. As a result, the strain sensing signal is converted into the optical power level without optical wavelength meter. FBG1 and FBG2 are mutual complementary pair, so the differential FBG optical circuit cancels the wavelength deviation of FBG as to a temperature factor. It is necessary for the operation of the differential FBG optical circuit to use the broadband optical source such as LED, SLED, ASE.
Fig-7 Simulation result of strain sensing signal via the differential FBG optical circuit
0
10
20
30
40
50
60
70
80
90
100
110
0.0E+00
1.0E‐08
2.0E‐08
3.0E‐08
4.0E‐08
5.0E‐08
6.0E‐08
7.0E‐08
8.0E‐08
9.0E‐08
1.0E‐07
1.1E‐07
1549.5 1549.6 1549.7 1549.8 1549.9 1550 1550.1 1550.2 1550.3 1550.4 1550.5
Transm
ittance[%]
Differen
tial FBG
Sen
sor Outpu
t Pow
er[W
]
WaveLength[nm]
Simulation of FBG Characteristics
Shift0
Shift1
Shift2
Shift3
Shift4
Shift5
Shift6
Shift7
Shift8
Shift9
Shift10
Shift11
Transmittance
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0 20 40 60 80 100 120 140 160 180 200
Optical Level of Differen
tiantial FBG
Sen
sor Outpu
t[uW
]
FBG Strain [u‐strain]
Strain Characteristics of Differential FBG SensorR=80%, BW=0.3nm
Fig-8 Actual experimental result of the differential FBG optical circuit 3. Strain Reduction formula of each deviation factor for the differential FBG optical circuit Basic equation of the differential FBG optical circuit is (eq.4). 1) Strain reduction term as to thermal expansion of FBG
(eq.5)
2) Strain reduction term as to refractive index deviation between FBG1 and FBG2
(eq.6) References [1] KAMINENI SRIMANNARAYANA et al, Fiber Bragg grating and long period grating sensor for simul- taneous measurement and discrimination of strain and temperature effects, Optica Applicata, Vol. XXXVIII, No. 3, 2008
0.0
0.5
1.0
1.5
2.0
2.5
‐2000 ‐1000 0 1000 2000
Normalized
Optical Loss [dB]
Distortion [u‐strain]
Long Gage SensorDistortion Characteristics
23、2nd
( ) ( )FB
BsBs
B
BB
FFBG G
TG
11
0
2211
0
21_ ×⋅
⋅−=
−×= δ
λλαλα
λδλδλ
ε αα
( )0
22
21
1
10
21_
1111
BFBB
B
nBB
FnFBG G
TTn
nTn
nG λδλλ
λδλδλ
ε⋅
×⋅⎟⎟⎠
⎞⎜⎜⎝
⎛⋅
∂∂⋅−⋅
∂∂⋅=
−×=