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

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

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

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

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

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

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⋅

×⋅⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂⋅−⋅

∂∂⋅=

−×=