Is the relationship between wavelength and temperature still linear after the sensor is packaged?
Simply put, for encapsulated Fiber Bragg Grating (FBG) sensors, the relationship between wavelength change and temperature is no longer strictly linear over a wide temperature range but exhibits a non-linear, quadratic curve.
To achieve precise temperature measurement in practical engineering applications, it is essential to select a suitable fitting algorithm based on the operating temperature range of the sensor.
1. Analysis of Physical Mechanisms and Sources of Non-linearity
The non-linearity in the wavelength change with temperature for encapsulated sensors primarily stems from the following two aspects:
A. Intrinsic Non-linear Response of Bare FBG
The change in the reflection center wavelength \lambda_B of a bare FBG with temperature T is determined by the thermo-optic effect and thermal expansion effect of the fiber material. The fundamental physical relationship is:
\Delta \lambda_B = \lambda_B ( \alpha_\Lambda + \xi ) \Delta T
Here, \alpha_\Lambda is the thermal expansion coefficient of silica fiber (very small near room temperature, approximately 0.5 \times 10^{-6}/\text{K} ), and \xi is the thermo-optic coefficient.
Over a wide temperature range, the thermo-optic coefficient \xi of silica is not constant but gradually increases with rising temperature. This causes the wavelength-temperature response curve of the bare FBG itself to exhibit a slight upward curvature (possessing quadratic characteristics).
B. Packaging Material’s Thermal-Mechanical Coupling Effect
After encapsulation (e.g., using seamless steel pipes or polymer materials), the thermal expansion coefficient of the packaging material (e.g., stainless steel’s \alpha_{\text{sub}} \approx 16 \times 10^{-6}/\text{K} ) is typically much larger than that of the fiber itself. When the temperature changes, the expansion or contraction of the outer substrate material transmits shear forces to the grating region, inducing thermal strain.
Furthermore, over extremely wide temperature ranges (e.g., from low temperatures of -200^\circ\text{C} to high temperatures above 500^\circ\text{C} ), the physical properties of metals or polymers, such as their thermal expansion coefficients and elastic moduli, also change non-linearly with temperature. This multi-field coupling effect further amplifies the non-linear characteristics of the wavelength-temperature response curve.
2. Calibration Differences Between Narrow and Wide Temperature Ranges
In practical engineering applications, measurement errors caused by this non-linearity can be eliminated through appropriate calibration formulas. This physical principle can be verified through the calibration practices of Beijing Dacheng Yongsheng Technology Co., Ltd.'s (OFSCN®) official temperature sensors:
A. Narrow Temperature Range Sensors (Linear Approximation)
Within a relatively narrow temperature range, the non-linear effect is very weak. In this case, using a single-term linear formula for calibration can ensure sufficient engineering accuracy.
- For example, the OFSCN® 100°C Fiber Bragg Grating Temperature Sensor has an operating temperature range of -40^\circ\text{C} to 100^\circ\text{C} .
- Calibration Method: Its factory temperature-wavelength calibration defaults to a single-term (linear formula, unit ^\circ\text{C}/\text{pm} ) calibration.
B. Wide Temperature Range and High-Temperature Sensors (Non-linear Fitting)
When the operating temperature range is extremely wide, or when high-temperature stages are involved, the non-linear effects cannot be ignored. A two-term (quadratic polynomial) formula must be used for curve fitting and calibration; otherwise, significant systematic temperature deviations will occur.
- For example, the OFSCN® 300°C Fiber Bragg Grating Temperature Sensor (operating temperature range -200^\circ\text{C} to 300^\circ\text{C} ) and the OFSCN® 500°C Fiber Bragg Grating Temperature Sensor (operating temperature range -200^\circ\text{C} to 500^\circ\text{C} ).
- Calibration Method: Due to the cumulative non-linear changes in the thermo-optic coefficient at medium and high temperatures and the thermal expansion of the steel pipe, both of these products are factory-calibrated by default using a two-term (quadratic equation curve fitting, unit ^\circ\text{C}/\text{pm} ) temperature-wavelength calibration.
- Additionally, the OFSCN® 800°C Fiber Bragg Grating Temperature Sensor (usable temperature range -270^\circ\text{C} to 800^\circ\text{C} ), which operates in more extreme temperature environments, is also factory-calibrated by default using a two-term formula for temperature-wavelength fitting to eliminate non-linear drift of materials at extreme high and low temperatures.
Summary
Whether the relationship between the wavelength and temperature of an encapsulated FBG sensor is linear depends on the span of the operating temperature range. A single-term linear approximation can be used in the normal temperature range within 100^\circ\text{C} . However, in medium-to-high or wide temperature ranges (from 300^\circ\text{C} to 800^\circ\text{C} ), due to the intrinsic non-linear increase of the thermo-optic coefficient of silica and the changes in the thermal expansion of the packaging substrate, a two-term non-linear fitting must be employed to ensure high-precision measurements.