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### Transcript of The formula of FBG Fiber Bragg Grating characteristics and ...optoelec- formula of FBG_Fiber Bragg...

• 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

sFs B

B GT T n

n εδα

λ δλ

⋅+⋅⎟ ⎠ ⎞

⎜ ⎝ ⎛ +

∂ ∂

⋅= 1

( ) ( )

( )

( )

( )

( )

( )

2

22 2

1

1cos

⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ +

⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ +

⎥ ⎥

⎢ ⎢

⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ +

=

m

m

m

m

m

m m L

T

κ δ

κ δ

κ δκ

22 2

2 2

1cos

1sin

⎟ ⎠ ⎞

⎜ ⎝ ⎛−

⎥ ⎥

⎢ ⎢

⎡ ⎟ ⎠ ⎞

⎜ ⎝ ⎛−

⎥ ⎥

⎢ ⎢

⎡ ⎟ ⎠ ⎞

⎜ ⎝ ⎛−−

=

κ δ

κ δκ

κ δκ

L

L

R 1 2

>⎟ ⎠ ⎞

⎜ ⎝ ⎛ κ δ

Bλ πηκ =

Λ −Ω= πδ

λ π 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

11 V

F

Δ= 22 1anV λ π

1

21 2 1

2 2

2 1

2 n nn

n nn −

≈ −

Δ= 22 1anV c

c λ π

a d n1 n2 n1-n2 Δ um um %

5 10 1.45 1.4448 0.0052 0.36 V Δn neff λB L m

nm mm 2.49 0.00005 1.4448 1550 10 1.0000

Λ F η κ Ω δ nm

536.4 0.84 4.192E-05 84.964 5.86E+06 1889.9

10

20

30

40

50

60

70

80

90

100

110

1549.5 1549.6 1549.7 1549.8 1549.9 1550 1550.1 1550.2 1550.3 1550.4 1550.5

Re fle

ct io n  &  T ra ns m it ta nc e[ % ]

WaveLength[nm]

Simulation of FBG Characteristics R=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 11 sF

B

BB s

B

B

B

B

B

BB GTT T n

nT n

n εδ

λ λλαδ

λ λ

λ λ

λ δλδλ

⋅+⋅ −

⋅+⋅⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ ⋅

∂ ∂ ⋅−⋅

∂ ∂ ⋅=

( ) 2

21 0

BB B

δλδλλ +=

FBG1(Sensing)

FBG2(Reference)

3dB CUP AR Cutting

Connector

Connector

Optical Source

Optical  Monitor

Strain Sensor

Transforming W.L. to Optical Power Level

The Optical Circuits of Differential FBG Sensor

10

20

30

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50

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70

80

90

100

110

1549.5 1549.6 1549.7 1549.8 1549.9 1550 1550.1 1550.2 1550.3 1550.4 1550.5

Re fle

ct io n  &  T ra ns m it ta nc e[ % ]

WaveLength[nm]

Simulation of FBG Characteristics R=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

10

20

30

40

50

60

70

80

90

100

110

1549.5 1549.6 1549.7 1549.8 1549.9 1550 1550.1 1550.2 1550.3 1550.4 1550.5

Re fle

ct io n  &  T ra ns m it ta nc e[ % ]

WaveLength[nm]

Simulation of FBG Characteristics

Transmittance

Rshift0

Rshift1

Rshift2

Rshift3

Rshift4

Rshift5

Rshift6

Rshift7

Rshift8

Rshift9

Rshift10

Rshift11

FBG1

FBG2

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

Tr an sm

it ta nc e[ % ]

D iff er en

ti al  F BG

S en

so r  O ut pu

t P ow

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

10

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40

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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

Tr an sm

it ta nc e[ % ]

D iff er en

ti al  F BG

S en

so r  O ut pu

t P ow

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

O pt ic al  L ev el  o f  D iff er en

ti an ti al  F BG

S en

so r  O ut pu

t[ uW

]

FBG Strain [u‐strain]

Strain Characteristics of Differential FBG Sensor R=80%, BW=0.3nm

• Fig-8 Actual experimental result of the differential FBG optical circuit 3. Strain Reduction