Fiber Optic Sensors
Calculate the strain and temperature response of FBG and Mach-Zehnder fiber sensors, the fringes of an air-gap Fabry-Pérot cavity, and the evanescent penetration depth.
Total Shift
219.1 pm
Strain Sensitivity
1.236 pm/µε
1 − p_e = 0.7973
Temperature Sensitivity
9.56 pm/K
K_T = 6.165×10⁻⁶/K (SMF-28)
Shifted λ_B
1550.219 nm
Formulas
Δλ_B/λ_B = (1 − p_e)·ε + K_T·ΔT (Kersey 1997)
p_e = (n²/2)[p₁₂ − ν(p₁₁ + p₁₂)], p₁₁ 0.113, p₁₂ 0.252, ν 0.16
K_T = (1/n)dn/dT + α = 6.165×10⁻⁶/K (SMF-28, 20 °C)
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
FBG, MZI: linear (1 − p_e)ε + K_T·ΔT, free fibre. EFPI: lossless plane-wave Airy cavity, no gap diffraction loss (d ≤ 50 µm), gap carries ε. Evanescent: planar decay; water dn/dT −1e-4/K near 25 °C.
- Kersey A. D. et al. (1997). Fiber grating sensors. J. Lightwave Technol. 15, 1442. Strain and temperature response. doi:10.1109/50.618377
- Bertholds A., Dändliker R. (1988). Determination of the individual strain-optic coefficients in single-mode optical fibres. J. Lightwave Technol. 6, 17. p₁₁ = 0.113, p₁₂ = 0.252. doi:10.1109/50.3956
- Cosme M. et al. (2026). Accurate determination of the temperature sensitivity of UV-induced fiber Bragg gratings. Sensors 26, 435. SMF-28: K_T = 6.165e-6 /K at 20 °C. doi:10.3390/s26020435
- Born M., Wolf E. (1999). Principles of Optics, 7th ed., §7.6.1 (Airy formulae, coefficient of finesse). Cambridge University Press. doi:10.1017/CBO9781139644181
- Harrick N. J. (1967). Internal Reflection Spectroscopy, ch. 2 (evanescent penetration depth). Wiley.
- Hale G. M., Querry M. R. (1973). Optical constants of water in the 200-nm to 200-μm wavelength region. Appl. Opt. 12, 555. n ≈ 1.318 at 1550 nm. doi:10.1364/AO.12.000555
- Thormählen I., Straub J., Grigull U. (1985). Refractive index of water and its dependence on wavelength, temperature, and density. J. Phys. Chem. Ref. Data 14, 933. doi:10.1063/1.555743