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 s F s B B G T T n n ε δ α λ δλ ⋅ + ⋅ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ∂ ∂ ⋅ = 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 1 cos ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = m m m m m m m L T κ δ κ δ κ δ κ 2 2 2 2 2 1 cos 1 sin ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = κ δ κ δ κ κ δ κ L L R 1 2 > ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ κ δ B λ πη κ = Λ − Ω = π δ λ π eff n 2 = Ω Λ = eff B n 2 λ F n ⋅ Δ = η
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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
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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
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
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
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