In this example, a bend sensor based on fiber Bragg grating (FBG) is demonstrated. The change of both physical length and strain-dependent refractive index of the fiber, are calculated by altering the bend radius of the sensor. The detection of the bend radius is determined by the shift of the Bragg wavelength from the reflection/transmission spectrum. The sensitivity of the sensor is also estimated and compared with reference.
Overview
Understand the simulation workflow and key results
Step 1: Analytical method - Lumerical.
Simulation model consists of two structures, a core and a cladding. The core is separated into two subsections named coreL, and coreH, with n=1.474/1.4743 (L/H) and R = 5 μm. Their length sum equals a Bragg period. The cladding has n=1.466 and R = 62 μm. The structure group contains a script that constructs the spatially varying refractive index based on the formulas and constants for glass. More information can be found at Spatially-varying-refractive-index-due-to-strain .
In the structure group we included an option (“bending”) allowing the user to enable or disable the bending and set the radius of curvature. Set “bending” at 0 means that bend is disabled, and the structure is at neutral condition. This is portrayed at the top image in the figure below. On the other hand, the bottom image depicts the sensor under bend. Set “bending” to 1 corresponds to negative bending, while 2 to positive bending. A positive bending is considered when the sensor is compressed, while under a negative bending the sensor is stretched. The dashed line at the bottom image shows the center of the fiber when its length stays unchanged under bend.
Step 2: Bending simulation with Ansys Mechanical.
With ANSYS Mechanical we simulated the strain distribution of a single period FBG grating under mechanical bend. A single period is adequate for our analysis due to uniform strain distribution along bent fiber axis (z-axis). More information can be found at Verification of B ragg grating strain uniformity . The model consists of two structures, a double core fiber (green/brown color - inset picture) and a base (beige color). The top surface of the base is defined by a bend radius that varied throughout this analysis. We applied mechanical displacement (-Y direction – blue arrow) on the edges of the fiber, subjecting the fiber to be pushed towards the base to form the bending.
Next, we calculate the strain distribution of grating’s cross section. Figure below portrays the XY strain distribution map of the cladding. The inset picture depicts the strain map of the stretched core that is exported separately due to the different mesh size from the cladding. For core regions, a high mesh (200 nm) was used as it is needed for high resolution strain map analysis.
The left picture below depicts the “Analysis group” in Lumerical model where variables, such as refractive indices, radius, wavelength etc. are defined. Using directional deformation as input, we obtain the change of period’s length under mechanical force.
Step 3: (n,k) import material – Interpolate Bragg period XY refractive index map.
In the analytical method, only the variation of refractive index due to bend is considered. Interpolated strain data are imported into a structure group, where a script calculates the refractive index from stress-strain Hook’s law equations. The change of Bragg period length due to bending is not included in this analysis. Due to that, the positive bend will cause an increase at the refractive index, and consequently a red shift at the reflected spectrum due to compression of glass material. The opposite effect is expected for negative bend, where Bragg period stretches, and refractive index tends to decrease.
In Ansys Mechanical simulation method, the calculation of refractive index takes place in python script. Then the refractive index map is directly imported in structure groups of coreH, coreL and cladding. Importnk2 script command is used to import the spatially varying refractive index structure.
Run and Results
Instructions for running the model and discussion of key results
In this example, we use FBG’s sensitivity to strain to create a bend sensor application example. At this type of sensor, the measurand causes a shift in the Bragg wavelength, due to applied strain (Δε). Thus, the Bragg wavelength shift becomes :
$$\Delta\lambda_{Bragg} = \lambda_{Bragg} * (1 - p_{e} )* \Delta \epsilon$$
where p e (typically, 0.21) is the effective photo elastic coefficient of the fiber and Δλ Bragg is the Bragg wavelength.
The wavelength shift varies linearly with strain, and consequently with bending curvature. Sensor’s performance is dominated by the expansion/compression of grating’s period (ΔΛ) as it is proved to have larger impact than the effective refractive index change (Δn eff ) [1,2]. For more information regarding Bragg condition please refer to Appendix .
Step 1: Calculate the required period of grating.
- Open the simulation file “fbg_bend_analytical_method.lms.”
- Run script “grating_period.lsf.”
First, calculate the grating period length for the aimed Bragg wavelength. The script will activate the FDE solver object and run the simulations for two separate locations in the grating (high and low index regions). The average of the effective indices represents the overall index of the grating and used for the estimation of the required grating period.
3. Run script “fbg_bend_analytical_method_FDE.lsf”.
The grating period from the previous step is now used in the model to calculate the loss and effective refractive index versus bend diameter (mm). Maximum bend diameter is set at 4 mm, according to reference [3]. As expected, the mode loss starts increasing below 6 mm diameter due to weakly confined modes in the fiber for such steep bends.
The following left picture validates that bend diameters larger than 20 mm have negligible effect on the sensor’s performance. For such large bend diameters, the sensor can be considered flat due to the miniature dimension of the fiber. The picture on the right portrays the change of effective refractive index of the fundamental mode for both positive and negative bend mode. As expected, the mode is still well confined to the core region of the optical fiber at the minimum bend, and the compression causes an increase of refractive index value, while the stretch causes a decrease.
4. Run script “fbg_bend_analytical_method_EME.lms”.
The script runs the EME solver and calculates the S-parameters of the Bragg grating, concentrating at reflection . It calculates the Bragg wavelength shift with regards to bend diameter, and bend curvature. Both figures below portray these results for a set of bend diameters. The left figure portrays the 30 mm bend diameter with negligible to no effect at sensor’s response, while for a bend diameter of 7 mm the wavelength shift increases in nonlinear behavior. The right figure portrays the wavelength behavior regarding curvature. Its response exhibits a linear behavior, which agrees with references [3]. For positive bending (legend “Diff Braggc” ) the wavelength increases while for the negative bending shift (legend “Diff Braggx” ) exhibits the opposite effect.
Additionally, the script will calculate and print in the Script Prompt the Bragg wavelength shift (Δλ bragg ) for both negative and positive bends. The output is depicted below:
Finally, the script will plot the reflectance response for various bend diameters. The figure below exhibits the reflectance spectrum for bending diameters varying from 30 mm down to 7 mm. Positive bending decreases λ bragg due to consideration of only strain analysis while negative bend returns a decreasing wavelength shift.
Step 2: Calculate strain distribution and physical length – Python refractive index map interpolation.
- Open a workbench project and import the simulation file “associate_file_Mechanical_Bend_Sensor.wbpz”.
- Press “Solve” to run the simulation.
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Export (a) - “Deformation” and (b) - “Equivalent Elastic Strain” data.
- Open folder “physical_length_change” and Run python file “length_deformation.py”.
- Open folder “RI_strain_change” and Run python file “run_interpolation.py”.
Steps of calculation and export for both (a) strain distribution, and (b) period length change, are depicted in the figure below. The simulation is completed with the results ready for export, as indicated by the green tick at Solution(A6) .
We manually export the length deformation in the files named “compressed_Zdeform.txt” and “stretched_Zdeform.txt” respectively. Both files are used in a python script ( “ length_deformation.py" ) to average the deformation to a single value. More information can be found in Model setting - Mechanical simulation method .
Both negative and positive bend period lengths are used as input in Lumerical model. The figure below shows the change of length (for a single Bragg period) with regards to wavelength, for radius ranging from 3.5 mm to 30 mm. The change of period length increases for short bend radius as expected. Any minor length difference between the negative and positive bending can be addressed to the difference between the tensile and compressive strength of fused silica according to references [7].
Next, we export the raw strain data for both negative and positive bends and import them within a series of python script files for refractive index calculation and interpolation on a rectilinear grid. The files “cladding_map.txt,” “compressed_core_map.txt” and “stretched_core_map.txt” contain the raw strain data exported from ANSYS Mechanical. A Python script ( “run_interpolation.py ”) is used to run a series of python files that export the interpolated refractive index maps of cladding, coreH, and coreL. The refractive index maps are calculated based on strain data from Hook’s law equations.
The figure below depicts the interpolated refractive index XY map of fiber’s cross section with bend radius of 3.5 mm. The negative bend (left figure) shows the refractive index change of the core which is higher than the positive bend (right figure). That can be also addressed to the properties of fused silica. Also, it can be clearly seen the varying refractive index distribution of the cladding due to bending.
Step 3: Calculate reflection response and Bragg wavelength shift.
- Open the simulation file “fbg_bend_mechanical_import.lms”.
- Open and Run script “fbg_bend_mechanical_method_EME.lsf”.
Length deformation is used in “model properties”, while the refractive index distribution is imported with importnk2 script command. Script file “fbg_bend_mechanical_method_EME.lsf” calculates with EME solver the reflection spectrum \({|S_{11}|}^2\) for negative, positive, and neutral bending. It also plots the reflectance spectrum according to the selected bend radius. The figure below depicts the reflectance spectra for all tested bend radii.
The light blue reflectance peak set at 1550 nm represents the reflected signal of a neutral sensor. Negative bending causes λ bragg to shift towards longer wavelengths (red-shift), while positive bending to shorter wavelengths (blue-shift). The absolute value of wavelength shifts for negative and positive bending are not equal due to the different compressive and tensile strength properties of fused silica. Also, the reflectance intensity decreases for short bend radii. This is due to the bending losses of the waveguided mode in the optical fiber, as expected [7].
The short video below animates the sensor’s performance where both cores, stretched and compressed, can be seen when force is applied on the edges causing the sensor to bend. The inset image animates the reflectance spectrum where Bragg wavelength shift increases/decreases under negative/positive bending conditions.
The figure below depicts the linear dependency of the Bragg wavelength shift (Δλ bragg ) due to bend curvature increase. The maximum shift corresponds to 5 nm, and it is obtained for a maximum bend curvature of 290 m -1 . A small bend curvature corresponds to a large bend radius where it can be considered as a flat surface, with regard to the miniature size of the fiber. This means that sensor’s wavelength shift will be negligible for small curvatures, while a large curvature will increase the Bragg wavelength of the sensor to higher values.
From this plot we can calculate the sensitivity of the sensor which is 20 pm/m -1 for both negative and positive bending. References indicate a variety of bend sensor sensitivities up to 100 pm/m -1 , however absolute values can deviate due to lack of specific design information such as design distances, material etc. from the references [8].
Next figure depicts sensor’s Bragg wavelength (λ bragg ) with regards to bend radius. There is an exponential increase/decrease depending on the bend type that exhibits very well correlation behavior with a similar bend sensor from references [6]. The sensor’s response starts to increase rapidly when the bend diameter drops below 40 mm.
Important model settings
Description of important objects and settings used in this model
Model setting - Mechanical simulation method.
The following video animates how to import the “associate_file_Mechanical_Bend_Sensor.wbpz” model in workbench and visualize the four designs for various bend radius.
The length deformation for a single Bragg period is exported from mechanical simulation. The explanatory video below shows how to run the python script “ length_deformation.py ” that prints the deformed lengths.
Both deformed lengths are used as input in the “Analysis group” window of the “fbg_bend_mechanical_import.lms”. For the negative bend we use parameter “dz_str” while for the positive bend “dz_compr”. The image below depicts the tabs where we should use as input.
The strain calculation follows the same process where the following video explains how the refractive index maps for coreH, coreL and cladding are obtained. The interpolated refractive index maps are exported in “.txt” format files.
Finally, we open the “fbg_bend_mechanical_import.lms” and set the correct path for all structure groups included in the model. After inserting the path where the interpolated strain data is placed, it is recommended to go to the Script tab and press the test button. The script tab is portrayed in the image below by a red box. If a wrong path is inserted, then an error code will be printed in the Script output.
Verification of Bragg grating strain uniformity
A single period of an FBG was used to simplify our model and avoid simulating time consuming and high-capacity simulations for thousands of periods that define a 5 mm length Bragg grating. Therefore, before moving to a single period simulation, we established that the strain distribution along the fiber will be uniform for an applied bend.
Our analysis included a design of multiple grating periods at various locations along the fiber. The left figure below depicts the design with the positions of gratings. The right figure portrays the strain measurement path (vertical orange line). The maximum strain can be seen at the top surface of the grating (stretched) with red color, while the bottom (compressed) is depicted with blue color.
The location of the gratings along the fiber varied from -100 μm (center) up to 400 μm (edge). The following plot portrays the strain values with regards to fiber core’s diameter, for all gratings at various locations. This plot proves that regardless of grating’s location on the bend fiber, the XY cross-section strain distribution along the sensor is kept uniform under bend. That proof provides a simulation advantage to analyze a single grating period instead of a lengthy and simulation time-consuming grating.
Taking the model further
Information and tips for users that want to further customize the model
Protective layers around fiber sensor
In this analysis, the design of the optical fiber consists of a core and cladding only. Normally, optical fiber is equipped with a protective coating and plastic jacket. The picture below shows a typical formation of an optical fiber.
Mechanical simulations in this application example did not include any protective layers due to simplicity purposes. Users can extend this model by adding extra layers, creating a more detailed version of optical fiber. The rest of the workflow, such as force application and data extraction should be kept the same. It is expected a decrease in wavelength shift results, as including extra layers around the fiber will decrease the sensitivity of the sensor.
Optimization and Sweep
The figure below depicts the Optimizations and Sweeps window. A bend radius sweep can be introduced to calculate the loss or n eff for FDE solver or the s-parameters form EME solver, instead of using EME Analysis window. This specific approach is more time consuming, as it calculates the mode profiles and s-parameters for each wavelength individually and returns more accurate results.
This example includes a large set of parameters. For simplicity reasons we chose specific values for core radius, cladding radius, refractive indices, bend radius etc. according to references found in literature. The design of the double core fiber used in this application is depicted in the figure below. The distance between the top and bottom core was set at 40 μm. An analysis of this distance variation and its effect on the sensitivity of the sensor could be extensively conducted by as ANSYS OptiSLang optimization tool.
Interconnect circuit.
A complete sensor circuit can be built using INTERCONNECT. The user can follow the workflow of Step 3: INTERCONNECT – Photonic circuit simulation to connect the sensor component with other optical elements such as a detector and a source. A basic model should include an Optical Network Analyzer (ONA) used as a source and detector, an Optical Time Variant S-parameter that represents the actual bend sensor and finally a DC source representing a radius controller.
Additional resources
Additional documentation, examples, and training material
Related publications
- V. V. S. Ch. Swamy, Saidi Reddy Parne, Sanjjev Afzulpurkar, Shivanand Prabhudesai, “Design and development of pressure sensor based on Fiber Bragg Grating (FBG) for ocean applications”. DOI: 10.1051/epjap/2020200036
- Jun Huang, Zude Zhou, Dongsheng Zhang, and Qin Wei, «A Fiber Bragg Grating Pressure Sensor and Its Application to Pipeline Leakage Detection», http://dx.doi.org/10.1155/2013/590451
- Alexander C. Thompson, William G.A. Brown, Paul R. Stoddart, “Bend effects on fibre Bragg gratings in standard and low bend loss optical fibres”, Conference Paper · January 2011, DOI: 10.1109/ACOFT.2010.5929958.
- Peter J. Cadusch, Alexander C. Thompson, Paul R. Stoddart, Scott A. Wad, “Modeling of Bend Effects on Fiber Bragg Gratings” Faculty of Engineering and Industrial Sciences, Swinburne University of Technology. DOI: 10.1117/12.915939
- Pauli Kiiveri, Joona Koponen, Juha Harra “Stress-Induced Refractive Index Changes in Laser Fibers and Preforms”, DOI: 10.1109/JPHOT.2019.2943208
- F M Araújo, L A Ferreira, J L Santos and F Farahi, “Temperature and strain insensitive bending measurements with D-type fibre Bragg gratings”, DOI: 10.1088/0957-0233/12/7/314
- Mechanical-properties-of-fused-quartz
- Xianfeng Chen et al. “Highly Sensitive Bend Sensor Based on Bragg Grating in Eccentric Core Polymer Fiber”, DOI: 10.1109/LPT.2010.2046482.
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