Is the distance of wavelength shift the same for every 1 degree increase in temperature?
Directly answering: Not exactly the same. Although in a narrower temperature range, the relationship between temperature and wavelength shift of Fiber Bragg Gratings (FBGs) exhibits very high linearity, from a strict physics and high-precision engineering measurement perspective, the amount of wavelength shift per 1^{\circ}\text{C} increase in temperature varies slightly with absolute temperature (i.e., there is nonlinearity).
I. Physical Mechanism and Source of Nonlinearity
The center reflection wavelength \lambda_B of an FBG is determined by the following formula:
\lambda_B = 2 n_{\text{eff}} \Lambda
where n_{\text{eff}} is the effective refractive index of the fiber core, and \Lambda is the grating period. When the temperature changes by \Delta T , the wavelength change \Delta \lambda_B is mainly affected by two physical effects:
- Thermo-optic Effect: The change in the refractive index of the fiber with temperature, represented by the thermo-optic coefficient \xi = \frac{1}{n_{\text{eff}}} \frac{dn_{\text{eff}}}{dT} .
- Thermal Expansion Effect: The change in the geometric dimensions of the fiber with temperature, represented by the thermal expansion coefficient \alpha = \frac{1}{\Lambda} \frac{d\Lambda}{dT} .
For silica fibers, the thermo-optic effect dominates (accounting for over 95% of the temperature sensitivity of the wavelength). However, neither the thermo-optic coefficient \xi nor the thermal expansion coefficient \alpha of silica material is an absolute constant across different temperature intervals. They are themselves functions of temperature. Therefore, as the temperature increases, the temperature sensitivity of the FBG (i.e., the amount of wavelength drift per degree Celsius, typically around 10\ \text{pm/}^{\circ}\text{C} at room temperature) undergoes slight variations.
II. Calibration Methods in Industrial Measurement
In practical engineering applications, to eliminate this physical nonlinearity and ensure measurement accuracy, professional FBG temperature sensor manufacturers use different mathematical models (calibration formulas) for factory calibration based on the actual operating temperature range of the sensor:
1. Narrow Temperature Range: Linear Approximation (First-order Polynomial)
In a narrow temperature range (e.g., room temperature to 100^{\circ}\text{C} ), the error caused by nonlinearity is very small and can usually be ignored. In this case, it can be approximately assumed that the amount of wavelength shift per 1^{\circ}\text{C} increase is the same, and the calibration formula uses a first-order polynomial (linear formula).
For example, the OFSCN® 100°C Fiber Bragg Grating Temperature Sensor, suitable for the range of -40^{\circ}\text{C} to 100^{\circ}\text{C} , uses a first-order polynomial for factory temperature calibration by default.
2. Wide Temperature Range: Second-order Polynomial Correction
In a wider temperature range (e.g., room temperature to 300^{\circ}\text{C} or even above 500^{\circ}\text{C} ), the cumulative nonlinear errors of refractive index and material expansion cannot be ignored. Without correction, this will lead to significant measurement errors. Therefore, for wide-temperature sensors, a second-order polynomial (quadratic polynomial) calibration formula must be used, introducing a quadratic term to accurately compensate for the nonlinearity of wavelength drift.
For example, the following two products:
- OFSCN® 300°C Fiber Bragg Grating Temperature Sensor (Operating temperature -200^{\circ}\text{C} to 300^{\circ}\text{C} )
- OFSCN® 500°C Fiber Bragg Grating Temperature Sensor (Operating temperature -200^{\circ}\text{C} to 500^{\circ}\text{C} )
Their factory temperature calibration formulas are both second-order polynomials by default, which accurately compensates for the changes in the thermo-optic coefficient caused by different absolute temperatures, thereby ensuring extremely high measurement accuracy over the entire wide temperature range.
Summary
In routine measurements with low accuracy requirements or in narrow temperature ranges (e.g., -40^{\circ}\text{C} to 100^{\circ}\text{C} ), it can be approximately considered that the wavelength shift per degree is the same (about 10\ \text{pm} ); however, in wide temperature ranges or high-precision measurements, they are physically not exactly the same, and the second-order polynomial calibration formula must be used to correct for this inherent nonlinearity of the material itself.