Fiber Bragg Grating Sensor

Calculate FBG wavelength shift for strain and temperature sensing applications.

Model: Textbook approximation

Grating Period Λ (nm)Effective Index n_eff (phase)Grating Length (mm)

AC effective-index modulation δn_eff

≈ 0.8 × the core index change for SMF

Strain (µε)Temperature Change ΔT from 20 °C (K)Thermal Coefficient K_T (10⁻⁶/K)

K_T slope dK_T/dT (10⁻⁸/K²)

SMF-28 1.431 (Cosme 2026); 0 for a constant K_T

Bragg Wavelength

1550.03 nm

Peak Reflectance

93.3 %

FWHM / First-Zero Width

149.9 pm / 197.6 pm

Shifted λ_B

1550.840 nm

Strain Shift

617.9 pm

1.236 pm/µε (1 − p_e = 0.7973)

Temperature Shift

195.6 pm

9.56 pm/K at 20 °C

Total Shift

813.5 pm

Reflection Spectrum

Wavelength Shift vs Strain

Wavelength Shift vs Temperature

Cross-Sensitivity: Δλ for Strain and Temperature Together

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Key Formulas

λ_B = 2·n_eff·Λ; R_max = tanh²(πΔn·L/λ_B) (Erdogan 1997)

Δλ_B/λ_B = (1 − p_e)·ε + K_T·ΔT + ½K_T′·ΔT² (Kersey 1997; K_T′ from Cosme 2026 Table 3)

p_e = (n²/2)[p₁₂ − ν(p₁₁ + p₁₂)] = 0.2027 with p₁₁ 0.113, p₁₂ 0.252, ν 0.16

K_T = (1/n)dn/dT + α = 6.165×10⁻⁶/K for SMF-28 at 20 °C, K_T′ = 1.431×10⁻⁸/K² over −20…100 °C (Cosme 2026)

A single grating cannot separate strain from temperature: use a strain-free reference grating.

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

Uniform-grating spectrum (Erdogan). Shift Δλ/λ = (1 − p_e)ε + K_T·ΔT + ½K_T′ΔT²: p_e from strain-optic coefficients; K_T at 20 °C and its slope K_T′ measured on SMF-28 (Cosme Table 3, K_T linear in T over −20…100 °C; without K_T′ the shift is ≈ 6 % low at ΔT = 50 K). Strain and ΔT not separable.

Kersey A. D. et al. (1997). Fiber grating sensors. J. Lightwave Technol. 15, 1442. Strain and temperature response. doi:10.1109/50.618377
Erdogan T. (1997). Fiber grating spectra. J. Lightwave Technol. 15, 1277. Coupled-mode reflectance (eq. 20), bandwidth, transfer matrices. doi:10.1109/50.618322
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

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