Engineering & Science
Fiber Bragg Grating (FBG) Wavelength, Strain & Temperature Shift Formula
Calculate FBG center Bragg reflection wavelength λ_B = 2 * n_eff * Λ and coupled strain-temperature shift Δλ_B = λ_B * ((1 - p_e) * ε + (α_Λ + ξ) * ΔT) for telecom C-band structural health sensors.
How a 535.6 nm UV grating inside glass fiber measures microstrain on bridges and aircraft wings!
A Fiber Bragg Grating (FBG) inscribes a periodic refractive-index modulation of pitch Λ along a single-mode silica core (effective index n_eff = 1.447), reflecting a narrow C-band peak at λ_B = 2 * n_eff * Λ = 2 * 1.447 * 535.6 nm = 1,550.026 nm! Stretching or heating the fiber shifts BOTH pitch Λ and refractive index n_eff: Δλ_B = λ_B * ((1 - p_e) * ε_μstrain * 10^-6 + (α_Λ + ξ) * 10^-6 * ΔT), where silica photoelastic coefficient p_e ≈ 0.22 and combined thermal-expansion + thermo-optic sensitivity (α_Λ + ξ) ≈ 8.85 ppm/K!
At 1,550 nm in standard SMF-28 silica fiber, pure axial tension of 1,000 με (0.1% strain, ΔT = 0 K) shifts λ_B by +1.2090 nm (~1.21 pm/με), while adding a +25 K temperature rise adds +0.3429 nm (~13.7 pm/K) for a combined +1.5520 nm Bragg shift—showing why bridge and wind-turbine FBG arrays always pair strained sensors with an unstrained thermal-reference FBG!
| SMF-28 FBG Load Case (neff=1.447, Λ=535.6 nm, pe=0.22) | Base Bragg Peak λ_B | Wavelength Shift Δλ_B (nm) | Interrogator Sensitivity |
|---|---|---|---|
| Unstrained Isothermal Reference (0 με, 0 K) | 1,550.026 nm | 0.0000 nm | 1550 nm C-Band Channel |
| Pure Tension (1,000 με, ΔT = 0 K) | 1,550.026 nm | +1.2090 nm | 1.209 pm / με |
| Combined Load (1,000 με, ΔT = +25 K) | 1,550.026 nm | +1.5520 nm | +0.3430 nm Thermal Cross-Term |
Coupled Thermo-Mechanical Bragg Wavelength Shift Δλ_B (nm)
Computes Δλ_B (nm) from effective index B2, grating pitch B3 (nm), photoelastic constant B4, axial strain B5 (microstrain με), combined thermal sensitivity B6 (ppm/K), and temperature change B7 (K).
=IF(OR(B2 <= 1, B3 <= 0), "Invalid Grating Geometry", ROUND((2 * B2 * B3) * (((1 - B4) * B5 * 0.000001) + (B6 * B7 * 0.000001)), 4))Unperturbed Reference Bragg Wavelength λ_B0 = 2 * n_eff * Λ (nm)
Returns the baseline first-order Bragg reflection wavelength in nanometers.
=IF(OR(B2 <= 1, B3 <= 0), "Invalid Grating Geometry", ROUND(2 * B2 * B3, 3))Try the example
Use the original values or change them to test your understanding.