What is the contribution of heating time to the wavelength shift of fiber lengthening?
For bare Fiber Bragg Gratings (Bare FBG), the contribution of fiber elongation (i.e., thermal expansion effect) to the Bragg center wavelength shift during heating is very small, accounting for only about 5% to 8% of the total temperature sensitivity.
In contrast, the vast majority (approximately 92% to 95%) of the wavelength drift is contributed by the change in the refractive index of the fiber material with temperature (i.e., the thermo-optic effect).
I. Physical Formulas and Theoretical Derivation
The Bragg reflection center wavelength of a Fiber Bragg Grating (FBG) is determined by the following formula:
\lambda_B = 2 n_{eff} \Lambda
Where:
- \lambda_B is the Bragg reflection wavelength;
- n_{eff} is the effective refractive index of the fiber core;
- \Lambda is the grating period.
When the ambient temperature changes by \Delta T , differentiating the above equation gives the relative change formula for wavelength drift:
\frac{\Delta \lambda_B}{\lambda_B} = \left( \frac{1}{n_{eff}} \frac{\partial n_{eff}}{\partial T} + \frac{1}{\Lambda} \frac{\partial \Lambda}{\partial T} \right) \Delta T
We define the two physical quantities in parentheses as:
- Thermo-optic coefficient: \xi = \frac{1}{n_{eff}} \frac{\partial n_{eff}}{\partial T} , which reflects the change in refractive index with temperature;
- Thermal expansion coefficient: \alpha = \frac{1}{\Lambda} \frac{\partial \Lambda}{\partial T} , which reflects the change in fiber elongation (increase in grating period) due to heating.
Therefore, the formula can be simplified to:
\frac{\Delta \lambda_B}{\lambda_B} = ( \xi + \alpha ) \Delta T
II. Calculation of Typical Parameter Values
For standard silica (quartz) fiber, the physical parameters near room temperature are as follows:
- Thermo-optic coefficient: \xi \approx 7.0 \times 10^{-6}\ \text{K}^{-1}
- Thermal expansion coefficient: \alpha \approx 0.55 \times 10^{-6}\ \text{K}^{-1} (due to the extremely low thermal expansion coefficient of silica glass)
In the \lambda_B = 1550\ \text{nm} band:
- Contribution of thermal expansion effect (fiber elongation):\Delta \lambda_{B, \alpha} = \lambda_B \cdot \alpha \approx 1550\ \text{nm} \times 0.55 \times 10^{-6}\ \text{K}^{-1} \approx 0.85\ \text{pm/K}
- Contribution of thermo-optic effect (refractive index change):\Delta \lambda_{B, \xi} = \lambda_B \cdot \xi \approx 1550\ \text{nm} \times 7.0 \times 10^{-6}\ \text{K}^{-1} \approx 10.85\ \text{pm/K}
- Theoretical total temperature sensitivity:
$$ \Delta \lambda_B \approx 10.85 + 0.85 = 11.7\ \text{pm/K} $$ (In actual tests, the sensitivity of bare gratings is usually around 10\ \text{pm/K} to 11\ \text{pm/K} )
Calculation of Contribution Ratio:
- Thermal expansion contribution ratio: \frac{\alpha}{\xi + \alpha} \approx \frac{0.55}{7.0 + 0.55} \approx 7.28\%
- Thermo-optic contribution ratio: \frac{\xi}{\xi + \alpha} \approx \frac{7.0}{7.0 + 0.55} \approx 92.72\%
This demonstrates that in the unconstrained bare grating state, the contribution of fiber elongation to temperature drift is negligible.
III. Industrial Implementation: How to Amplify Sensitivity Using “Thermal Expansion”?
Although the thermal expansion coefficient of bare fiber itself is extremely small, in the development of high-sensitivity FBG temperature sensors, encapsulation thermal expansion sensitization technology can be introduced.
By fixing both ends or the entire bare grating to a substrate with a larger coefficient of linear expansion (such as metals like copper or stainless steel), when the ambient temperature rises, the tensile strain generated by the metal substrate’s high thermal expansion is transmitted to the grating. At this point, due to the thermal expansion stretching of the substrate, the FBG period \Lambda is forced to increase, thereby increasing the temperature sensitivity by several times or even tens of times.
In OFSCN®'s (大成永盛) high-precision fiber grating product line, whether it’s bare fiber gratings or packaged sensors that utilize seamless steel pipes for precise temperature drift control, this physical mechanism is strictly considered:
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High-strength Bare Fiber Gratings (suitable for scenarios without additional substrate constraints or requiring self-packaging):
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Seamless Steel Tube Encapsulated Temperature Sensors (ensuring excellent thermal conductivity and calibration consistency through precise steel tube structure and specific material matching):
Through refined structural design, the temperature calibration formula (typically a first-order or second-order polynomial, in ^\circ\text{C/pm} ) for encapsulated sensors can perfectly calibrate and compensate for the wavelength drift caused by these physical effects.