This example demonstrates a temperature sensor based on fiber Bragg gratings (FBG). The temperature-dependent change of the refractive indices of the fiber, consequently the shift of its Bragg wavelength, is used as a measure of the temperature.
Overview
Understand the simulation workflow and key results
Optical fiber Bragg grating (FBG) to be considered in this example is made of a core with alternating refractive indices and a constant periodicity. It is known that the index variation along the major axis of the fiber can induce the coupling of counter-propagating modes at the Bragg wavelength ( λ Bragg ) given by the following equation:
$$\lambda_{Bragg} = 2 \Lambda n_{eff}$$
where \(n_{eff}\) is the effective index of the fundamental mode of the fiber at the Bragg wavelength and \(\Lambda\) is the period of the grating. The uniform FBG acts as a wavelength-selective mirror at the Bragg wavelength. At each refractive index discontinuity along the fiber axis, a weak Fresnel reflection occurs. When all the reflections from the interfaces are accumulated, the gratings yield a conspicuous reflection band around the Bragg wavelength, surrounded by side lobes [1].
The above equation can be extended to include the effect of temperature (\(T\)) on the refractive index, and consequently the Bragg wavelength:
$$\Delta\lambda_{Bragg} = 2 \Lambda (\alpha n_{eff} + \eta) \Delta T$$
where \(\alpha\) and \(\eta\) stand for the thermal expansion coefficient (\(\frac{1}{\Lambda} \frac{\partial\Lambda}{\partial T}\)) and the thermo-optic coefficient (\(\frac{\partial n_{eff}}{\partial T}\)) of the grating material, respectively. The change in temperature (\(\Delta T\)) results in the change in the refractive indices of the core and the cladding by an amount determined by the value of \(\eta\) (typically, 8.3e-6 o C -1 ) [4], ultimately causing the Bragg wavelength shift. The expansion of the fiber can also contribute to the shift of the Bragg wavelength. However, we will ignore the latter effect since \(\alpha n_{eff} \) (typically, 0.55e-6*1. 4725 = 0.809e-6 o C -1 ) is an order of magnitude smaller than \(\eta\). We adopt a 2 nd order dependency of \(\eta\), as it has been proven more accurate than a linear model, especially at temperatures above 400 o C [2].
Step 1: FDE - Calculate the required period of grating and temperature-dependent \(n_{eff}\).
We first obtain the effective index of the grating at the target wavelength using the FDE solver and calculate the required period (\(\Lambda\)) of the grating. We calculate the \(n_{eff}\) at the high- and low-indexed regions and average them as a starting point of the design.
Step 2: EME - Calculate the temperature-dependent transmission/reflection response of the grating.
We analyze the transmission/reflection properties of the grating over many periods. Only a single period of the grating is included in the simulation region, but a broadband response of the long grating can be obtained by using the "Periodicity" and "Wavelength sweep" features. We then sweep the temperature and export the transmission/reflection responses as S-parameters for the ensuing circuit simulations.
Step 3: INTERCONNECT – Photonic circuit simulation.
The temperature-dependent S-parameters are imported into the INTERCONNECT using the Optical Time Variant S-parameter element for simulation of the FBG Temperature sensor. We sweep the temperature and measure the reflection spectra of the sensor at different temperatures. This circuit model can be useful when including the effect of additional PIC elements on the overall performance of the FBG.
Run and Results
Instructions for running the model and discussion of key results
Step 1: Calculate the required period of grating and temperature-dependent n_eff.
- Open the simulation file, [[fbg_temperature.lms]].
- Run the script file, [[fbg_temperature_step_1_fde.lsf]].
The fiber in consideration is composed of a core with n=1.4725/1.4728 (L/H) and R = 4.8 μm and a cladding with n=1.466 and R = 62 μm.
The script will activate the FDE solver object and run the simulations at room temperature for two different locations in the grating (high and low index regions). The average of the effective indices is taken to represent the overall index of the grating and used for the estimation of the required grating period. The field profile of the fundamental mode in consideration in this example is visualized below. As expected, the mode is very well confined to the core region of the optical fiber.
Step 2: EME - Calculate the temperature-dependent transmission/reflection response of the grating.
- Run the script [[fbg_temperature_step_2_eme.lsf]].
The period calculated from the previous step is automatically used in the “model” parameters. The script runs the EME solver and calculates the S-parameters of the Bragg grating. We have two cells in the simulation region, each representing the high- and low-index regions.
The script calculates all the S-parameters (\(S_{11}, S_{12}, S_{21}, S_{22}\)) over the given temperature range. But here we will be mainly focusing on the reflection (\({|S_{11}|}^2\)) of the grating as shown below. The peak reflections (corresponding to the Bragg wavelengths) are observed to be around 90% and show red-shift as the temperature increases from 25 o C up to 1.000 o C.
The plot of the Bragg wavelength vs. temperature shows a shift of 15.6 nm at 1.000 o C. with respect to the value at room temperature. The results show a good match with the published results [2,3,4].
The Script Prompt also displays the wavelength shift (Δλ bragg ) as well as the sensitivity of the grating for the given temperature range. The sensitivity is defined as follows:
Sensitivity = (λ Bragg at Tmax - λ Bragg at Tmin )/(T max -T min )
The sensitivity (9.4 pm/ o C) from the simulation is reasonably in good agreement with the published result (7.2 pm/ o C) [3,5], considering the lack of information about some material properties in the reference. The discrepancy might mostly come from the differences in the material parameters, which are not fully supplied in the references.
The script also extracts the temperature-dependent S-parameter and saves it into an S-parameter file format ( fbg_s_param_T.dat ) for INTERCONNECT circuit simulations in the next step.
Step 3: INTERCONNECT – Photonic circuit simulation.
- Open the [[fbg_temperature_step_3_interconnect.icp]] and run the simulation. (The FBG Temperature Sensor element has the [[fbg_s_param_T.dat]] from Step 2 already imported.)
- Run the "Gain_vs_Temperature” sweep in the Optimizations and Sweeps window.
The circuit simulation of the FBG temperature requires three elements:
- Optical Network Analyzer (ONA) that functions both as a source and a detector.
- Optical Time Variant S-parameter element that represents the FBG temperature sensor.
- DC source that is used as a temperature controller and is connected to the FBG temperature sensor element.
The left figure below shows the schematic design of the circuit simulation. Press the run button and the simulation will calculate the Reflectance spectrum of the temperature sensor at a room temperature of 25 °C. The figure on the right shows the Reflectance spectrum, which can be obtained by right-clicking on ONA and then displaying results.
Next, run the “Gain_vs_Temperature” sweep in the Optimization and Sweep tab to calculate the Reflectance spectra for a series of temperatures. Using the sweep parameter the reflection spectrum for an editable series of temperatures is generated.
The following plot shows the spectra for a temperature range from 25 °C up to 1000 °C. According to a publication [4], a Bragg wavelength shift of 4 nm was reported over a temperature range from 100 °C to 500 °C. Our simulation result shows a similar value of 4.5 nm over the same temperature range.
Important model settings
Description of important objects and settings used in this model
Model Setting
All the key design parameters such as refractive index, temperature, radius and wavelength are set in the Setup tab of the “model” using script, easing the setup of the simulation.
Thermo-optic coefficients
Both refractive indices of cladding and core materials are calculated based on the two thermo-optic coefficients, eta1 and eta2, accounting for the 2nd order behavior. In this model, we assume that both cladding and core materials share the same \(\eta\) values.
Periodicity
Periodicity is a useful feature when the simulation model includes identical periodic structures. This allows calculation of the response of gratings with many periods from a single period response, avoiding the creation and simulation of the long gratings.
Two cells were defined in the periodic group definition table in the EME setup tab of the EME solver object. Each of the cells corresponds to core-L and core-H regions of the Bragg grating. We set the periodicity of the grating to 10,000 in the "periodic group definition" table on the right side of the EME setup tab. This means that the unit cell (composed of 2 cell groups) will be propagated 10000 times, and the final length of the device will be 10000 times the length of both cells, which represent the period (\(\Lambda\)) of the grating.
The grating’s length can be easily adjusted by pressing override periodicity and increasing the periodicity number in the EME Analysis window. Thus, reflectance can be increased by increasing the length of the Bragg grating (periodicity value) or by increasing the refractive index difference (\(\Delta n\)) between the L-core and H-core.
Wavelength sweep
In Optimizations and Sweeps window, a wavelength sweep can be introduced instead of using EME Analysis window. This approach is more time consuming, as it calculates the mode profiles and s-parameters for each wavelength individually and returns more accurate results. This approach is ideal when the material is dispersive in a specific wavelength region, meaning that the effective index and the mode profile change with wavelength. Otherwise, it can be used for verification with EME Analysis sweep. It is recommended for use when the number of wavelengths is small. However, in this application example we use the EME Analysis sweep approach that scans the wavelength directly using a perturbative approach. The figures below show the reflectance spectrum, for both approaches, with 300 wavelength data points.
Taking the model further
Information and tips for users that want to further customize the model
Thermal expansion coefficient
We ignored the Bragg wavelength shift due to the thermal expansion in this example as the contribution by the thermal expansion coefficient (\(\alpha\)) is an order of magnitude smaller than the one by the thermo-optic coefficient (\(\eta\)) within the temperature range in consideration. However, both contributions might need to be included at higher temperatures since \(\alpha\) increases significantly with increasing temperature compared to \(\eta\) [6].
The inclusion of \(\alpha\) in the simulation means that the period of the grating changes at different temperatures. Just as the temperature-dependent refractive index was set up in the "model" Script tab using \(\eta\), the temperature-dependent period can be also set up in the same tab using \(\alpha\).
Expanding the circuit model
The circuit model currently includes only the basic elements. If you want to include the effect of interrogator circuit elements, you can do that by putting the required elements in the right place in the existing circuit model and connecting them appropriately.
Additional resources
Additional documentation, examples and training material
Related Publications
- Damien Kinet, Patrice Mégret, Keith W. Goossen, Liang Qiu, Dirk Heider and Christophe Caucheteur, “Fiber Bragg Grating Sensors toward Structural Health Monitoring in Composite Materials: Challenges and Solutions”,Sensors 2014, 14, 7394-7419, doi:10.3390/s140407394
- Wenyuan Wang, Yongqin Yu, Youfu Geng, and Xuejin Li "Measurements of thermo-optic coefficient of standard single mode fiber in large temperature range", Proc. SPIE 9620, 2015 International Conference on Optical Instruments and Technology: Optical Sensors and Applications, 96200Y (10 August 2015); https://doi.org/10.1117/12.2193091
- O. Hill and G. Meltz, "Fiber Bragg grating technology fundamentals and overview," in Journal of Lightwave Technology, vol. 15, no. 8, pp. 1263-1276, Aug. 1997, doi: 10.1109/50.618320.
- Hsieh TS, Chen YC, Chiang CC. “Analysis and Optimization of Thermodiffusion of an FBG Sensor in the Gas Nitriding Process.” Micromachines (Basel). 2016 Dec 12;7(12):227. doi: 10.3390/mi7120227. PMID: 30404399; PMCID: PMC6190027.
- Du Yanliang, Li Jianzhi, Liu Chenxi, “A Novel Fiber Bragg Grating Temperature Compensated Strain Sensor”, 2008 First International Conference on Intelligent Networks and Intelligent Systems, DOI 10.1109/ICINIS.2008.27
- “The Effect of Temperature and Pressure on the Refractive index of Some Oxide Glasses”, Roy M. Waxler, G.W.Cleek, Journal of Research of the National Bureau of Standards – A.Physics and Chemistry, Vol 77A, No.6, November-December 1973.
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Appendix
Additional background information and theory
INTERCONNECT: Simulating the sensor with Waveguide Bragg Grating element
Step 3 describes the temperature-dependent S-parameter element as FBG sensor, used in circuit simulation. However, INTERCONNECT provides an alternative FBG temperature sensor component. A Tunable Bragg Grating element can be used where only the effective refractive index change, with regards to temperature, should be imported. The schematic design of this alternative is depicted below:
Script [[neff_vs_T_appendix.lsf]] calculates and plots, the effective refractive index with regard to temperature. Temperature is in Kelvin units; thus, it begins from 298 Kelvin (25 o C) up to 1273 Kelvins (1000 o C). Script will export, in text format (“neff_vs_T_appendix.txt”), the data file that will be used in “INTERCONNECT: Element Library - Waveguide Bragg Grating element”.
Open [[fbg_temperature_tunable_bragg_grating.icp]] and import the data file to the measurement filename of the Temperature sensor element. Then, run the “Gain(dB)_vs_Temperature” parameter sweep. Visualization of the Reflection Gain results can be seen below. The wavelength shift, between 100 o C and 500 o C, is roughly 4.5 nm which agrees with the Lumerical application component wavelength shift, as well as with the reference [4].