A High-Sensitivity Magnetic Field Sensor Based on PDMS Flexible Resonator

Abstract

High-sensitivity resonator magnetic sensing requires a significant magnetostrictive response, while the narrow linewidth mode of a high-Q resonator can provide a high-precision frequency resolution. Therefore, a polydimethylsiloxane (PDMS) flexible resonator with both a low Young’s modulus and high optical transmittance is an ideal platform for realizing high-sensitivity magnetic sensing. Based on the sandwich structure of the PDMS flexible resonator, the mechanism of the magnetic field sensitivity of the PDMS flexible resonator sandwich structure is studied, and the impact of changes in the refractive index and radius on the sensor device is analyzed. In order to optimize the sensitivity of the sensor, when an external magnetic field acts on the sandwich structure, the impacts of three aspects on the sensitivity of the sensor are simulated and analyzed: different coupling positions of PDMS flexible resonator, different radii, and PDMS mixing ratios. The trend of sensitivity change is obtained, and the physical explanation of the sensitivity trend is analyzed. By optimizing these three aspects, the magnetic field sensitivity is eventually calculated as 19.02 nm/mT. Based on the existing experimental conditions and the preparation technology of the PDMS flexible resonator, the measured magnetic field sensitivity is 4.23 nm/mT.

1. Introduction

A whispering gallery mode (WGM) optical resonator has high quality factor and compact-mode volume and has become an important component in current optics which is fully utilized in practical applications [1,2,3,4,5,6,7,8,9] such as basic science and engineering physics. With the continuous development of micro-nano processing technology, multiple choices of optical resonator shapes and materials exist, such as optical crystal materials with small absorption coefficients, such as calcium fluoride [10,11] (CaF₂), magnesium fluoride [12,13] (MgF₂), etc. A crystal material micro-resonator is insensitive to the surrounding environment humidity and has the advantages of a high nonlinear coefficient, a low absorption coefficient, etc., showing important application value in nonlinear optics. Silica and silicon nitride materials [14,15] are prepared via lithography and other processing technologies to form micro-ring resonators. These resonators are easy to integrate on-chip and are widely used in the field of optical communication. Polymer materials with small Young’s moduli and good plasticity, such as polymethyl methacrylate [16,17] (PMMA) and polydimethylsiloxane [18,19] (PDMS), can be applied to micro-lasers and sensors. Due to the very narrow resonance mode linewidth of WGM optical resonators, it has highly competitive high sensitivity and has been applied in various types of sensing research.

In recent years, magnetic field sensor devices have been used in various industrial fields such as geological exploration [20], aviation and navigation [21], and medical instruments [22]. Compared with traditional magnetic field sensors, such as sensor devices prepared based on the Hall effect [23], they are limited by the measurement distance and external magnetic field. Optical magnetic field sensors based on WGM resonators are well suited to make up for the shortcomings of traditional magnetic field sensors, with the advantages of having a fast magnetic field response, strong anti-interference capability, good reliability, and easy integration, which have been gradually developed in recent years. Various types of WGM-resonator-based magnetic sensors have been reported. For example, Zhang [24] et al. reported a highly sensitive bidirectional magnetic field sensor by packaging an optofluidic micro-resonator filled with a magnetic fluid which achieved a maximum sensitivity of 304.80 pm/mT and a detection limit of 0.0656 mT. Guo [25] et al., from Harbin Institute of Technology reported a hollow microbubble resonator based on fixed Terfenol-D, which achieved a tunable Q factor of 2.07 × 10⁵ by tuning the WGMs with magnetic fields and a magnetic field sensitivity of 0.081 pm/mT. Yu [26] et al., reported a microcapillary resonator combined with magnetostrictive materials, achieving a DC magnetic field sensitivity of 0.1703 dB/Oe through AC magnetic field modulation. Rakhi et al. reported the deflection vibration response and thermal moment of a resonator via a magnetic field [27] and illustrated the thermoelastic vibrations of a rotating resonator caused by a laser pulse heat source and sinusoidal heating in the context of three-phase thermo-elasticity [28].

Therefore, most magnetic sensors based on WGM resonators are two main types. One type is the magneto-optical sensor. Magnetic fluid is widely used as a key sensitive material for magneto-optical sensing schemes because of their tunable refractive index (RI) under different external magnetic fields. A microcapillary or a hollow microfiber is filled with a magnetic fluid. As the magnetic field is changed, the RI around the WGM changes, leading to a shift in the transmission spectrum [24,29]. Although the magnetic fluid can provide a significant response to the applied magnetic field, the magnetic fluid usually affects the optical quality of the resonator, making the detection limit of the sensor lower. Another type is a resonator combined with a magnetic material, usually consisting of a WGM resonator and a magnetostrictive material [25,30,31], where an external magnetic field induces mechanical forces to modify the transmission spectrum while being able to maintain a high Q-factor. The magnetic-field-induced strain of most magnetostrictive materials is very small, typically less than a few ppm/mT [32], and the bandwidth and sensitivity are limited by the degree of combination of the magnetostrictive material and the WGM resonator.

As research on magnetic field sensing based on WGM resonators progresses, the Young’s modulus is becoming one of the factors limiting the sensitivity of such sensing because WGM resonators exhibit excellent sensitivity to mechanical deformation. Therefore, compared to crystalline materials, we choose the PDMS flexible resonator with a relatively small Young’s modulus as the magnetic field sensing platform. We have solved the problem of PDMS flexible resonators not having the characteristic of sensing magnetic fields by using a magnet sandwich. However, the mechanism of magnetic field detection based on the PDMS flexible resonator sandwich structure is still unclear. In this paper, we model the sandwich structure and investigate the sensitivity mechanism of magnetic field induction in the sandwich structure by analyzing three factors: the coupling position, different sizes and different mixing ratios of PDMS. The calculated results show that when all three factors influencing the structure are simultaneously controlled, a magnetic field sensitivity of 19.02 nm/mT is achieved. By optimizing the structural parameters, it is relatively easy to achieve magnetic field sensitivity at the nm/mT level. Therefore, this study provides theoretical support for magnetic field sensing based on PDMS flexible resonators. Relying on the existing experimental conditions and PDMS flexible resonator preparation technology, the sandwich structure is successfully fabricated, and the measured magnetic field sensitivity is 4.23 nm/mT.

2. Materials and Methods

2.1. PDMS Flexible Resonator Magnetic Sensing Structure

Resonant wavelength can be derived from the resonance condition of WGM:

(1)$\lambda \text{=}\frac{2\pi r{n}_r}{m}$

where λ is the resonant wavelength in vacuum, r is the radius of the resonator, m is the mode number and n_r is the effective refractive index. Since m is a constant for a specific mode, the magnetic field applied to the sandwich structure will cause changes in the radius and refractive index of the PDMS flexible resonator, resulting in a shift in the optical resonant wavelength:

(2)$\frac{\Delta \lambda }{\lambda }=\frac{\Delta R}{R}+\frac{\Delta n}{n}$

The principle of the magnetic field sensor based on the PDMS flexible resonator is shown in Figure 1. The PDMS flexible resonator can be optically injected and output through a tapered optical fiber. The electromagnetic coil generates a magnetic field that acts on the sandwich structure, which in turn causes the magnetic iron layer to act on the PDMS flexible resonator, resulting in the drift in the resonant wavelength of the resonance spectrum. The change in resonant wavelength under different magnetic fields is obtained, which is the sensitivity of the magnetic field sensor.

In order to verify the response of the designed magnetic sensing structure, a numerical simulation based on COMSOL 6.0 simulation software is used. The simulation calculation of the sensitivity of this magnetic sensor device is achieved by using a two-dimensional axisymmetric model with a magnetic field and a solid module.

2.2. Study on Magnetic Sensing Mechanism

PDMS is a highly transparent and hydrophobic elastomeric organosilicon compound. The PDMS-based flexible resonator optical magnetic sensor utilizes a PDMS flexible resonator as the core, with a magnetic sandwich structure for the magnetic field sensing function. By applying a magnetic field, the sandwich structure causes changes in the radius and refractive index of the PDMS flexible resonator. Finite element simulation is used to establish a model of the PDMS flexible resonator with the sandwich structure. The lower magnetic iron is fixed, and the magnetic field is provided by an electromagnetic coil to observe the changes in the radius of the PDMS flexible resonator. The experimental device is set with the following parameters: the radius (R) of the PDMS flexible resonator is 1.5 mm, the height (h) is 2 mm and the diameter of the magnetic iron in the sandwich structure is 4 mm. The Young’s modulus of the PDMS polymer and curing agent mixture ratio of 10:1 is 1.47 MPa [33], the refractive index is 1.41 [34] and Poisson’s ratio is 0.49 [35]. The Young’s modulus of the neodymium iron boron magnet (NdFeB) is 150 GPa.

We ignore the magnetic field sensitivity due to refractive index change. By applying different magnetic fields and linearly fitting the change in radius of the PDMS flexible resonator, the linear relationship between the magnetic field and PDMS flexible resonator radius is obtained, as shown in Figure 2. The inset illustrates the surface displacement of the PDMS flexible resonator caused by a magnetic field of 0.06 mT. The radius variation is obtained by fitting at different magnetic fields, and the magnetic field sensitivity is 2.59 nm/mT at this time.

The next discussion is about the influence of the refractive index on sensing. The refractive index change caused by stress is called the photo-elastic effect [36,37]. The Neumann–Maxwell equation provides the relationship between stress and the refractive index as follows [38]:

(3)$\begin{array}{l}{n}_x=n_{0x}+C_1{\sigma }_{xx}+C_2\left({\sigma }_{yy}+{\sigma }_{zz}\right)\\ n_y=n_{0y}+C_1{\sigma }_{yy}+C_2\left({\sigma }_{xx}+{\sigma }_{zz}\right)\\ n_z=n_{0z}+C_1{\sigma }_{zz}+C_2\left({\sigma }_{xx}+{\sigma }_{yy}\right)\end{array}$

Here, n_x, n_y and n_z are the refractive indices along the three principal stress directions, and n_0x, n_0y and n_0z are the refractive indices of the unstressed material. σ_xx, σ_yy and σ_zz are the principal stresses in the x-axis, y-axis and z-axis, respectively. C₁ and C₂ are the photo-elastic constants of the material. Since the photo-elastic constants of PDMS are C₁ = C₂ = C = −1.75 × 10⁻¹⁰ m²/N [38], Equation (3) can be rewritten as:

(4)$n=n_0+C\left({\sigma }_{xx}+{\sigma }_{yy}+{\sigma }_{zz}\right)$

The relationship between magnetic field and refractive index of the PDMS flexible resonator is obtained by applying different magnetic fields, and from Figure 3, there is a linear relationship between the magnetic field and refractive index, with a slope of −1.46 × 10⁻¹⁰ mT⁻¹. The rate of change in the refractive index due to different magnetic fields is −1.04 × 10⁻¹⁰ mT⁻¹; therefore, the magnetic field sensitivity of the sandwich structure due to refractive index change is 1.6 × 10⁻⁷ nm/mT. The inset shows the strain change in the PDMS flexible resonator induced by the magnetic field.

The effect of PDMS flexible resonator radius variation and refractive index variation on the magnetic field sensitivity can be obtained from the above simulations. The sensitivity of radius change is 2.59 nm/mT, and the sensitivity of refractive index change is 1.6 × 10⁻⁷ nm/mT. Therefore, the effect of refractive index change on the magnetic field sensitivity is negligible. A basis is provided for the following discussion of the factors affecting the magnetic field sensitivity of the sandwich structure.

The experiments are performed by mixing PDMS prepolymer and curing agent in a mixing ratio of 10:1 in a Petri dish, stirring for 10 min and then vacuuming the PDMS from the Petri dish using a syringe. The mixed PDMS is placed on a slide coated with superhydrophobic material, and then a PDMS flexible resonator is formed on the slide. To maintain the stability of the sandwich structure in the experiment, we use three PDMS flexible resonators to achieve the sandwich structure. Therefore, the sensitivity in the experiment is smaller than the theoretical analysis sensitivity. The process of PDMS flexible resonator preparation is shown in Figure 4.

The experimental measurement setup for the PDMS flexible resonator sandwich structure is shown in Figure 5. The incident light from the center wavelength 1550 nm tunable laser is passed through an isolator so that the incident light cannot be reflected back into the laser, thus protecting the protection laser. The transmission spectrum of the sandwich structure is obtained by coupling a tapered fiber with a diameter of about 2.1 μm into and out of the resonator. A polarization controller is used to optimize the coupling strength. The light coming out of the tapered fiber is connected to a photodetector and transmitted to an oscilloscope to obtain the transmission spectrum, and a power meter (FMH-87107) is used for data acquisition and analysis. By attaching the tapered fiber to the resonator surface, the purpose is to ensure a stable coupling state during the whole experiment and to reduce the effect of ambient fluctuation noise. An electromagnetic coil is placed on the top side of the sandwich structure to generate an external magnetic field and calibrate it with a Gauss meter.

3. Results and Discussion

To optimize the sandwich structure to obtain the maximum sensitivity, we mainly analyze the structure: from the location of the fiber-coupled resonator, the radius size of the resonator and different mixing ratios of PDMS. The following simulations are carried out in three ways:

3.1. Position of PDMS Flexible Resonator Coupling

We analyze the position of PDMS flexible resonator coupling, and due to the compression of the magnetic field on the resonator, there are differences in stress and strain at different coupling positions. We simulate and calculate the magnetic sensing sensitivity corresponding to different coupling positions. Figure 6a shows the magnetic field sensitivity at coupling positions of 0.3 h, 0.35 h, 0.55 h, 0.6 h, 0.65 h and 0.7 h, where h is the height of the resonator. It can be seen that the magnetic field sensitivity gradually increases with the upward shift of the coupling position. Figure 6b quantitatively describes the differences in sensitivity at different positions, and the sensitivity at the 0.7 h coupling position is 113% higher than that at the 0.3 h coupling position. The closer the coupling position is to the location where the force is applied, the greater the deformation, and the greater the magnetic field sensitivity as the coupling position increases. The reason for choosing 0.7 h as the upper limit of the coupling position is that in the actual measurement process, a too-high coupling position makes it difficult to obtain a good resonant spectrum.

3.2. Radius of the PDMS Flexible Resonator

Based on the model parameters mentioned above, by keeping the coupling position at 0.3 h unchanged, the change in the radius under different magnetic fields is obtained by changing the size of the PDMS flexible resonator radius, and the magnetic field sensitivity of the PDMS flexible resonator with different radius is fitted, as shown in Figure 7a.

The magnetic field sensitivities for different PDMS flexible resonator radii are shown in Figure 7a, The PDMS flexible resonator radii are 1.5 mm, 1.2 mm, 1 mm, 0.9 mm, 0.75 mm and 0.65 mm,. The magnetic field sensitivity is obtained by linearly fitting the relationship between the magnetic field and the amount of radiation. The magnetic field sensitivity is minimized at a radius of 1.5 mm. By varying the radius of the PDMS flexible resonator, the magnetic field sensitivity of the sandwich structure is also varied. The sensitivity is calculated to be 3.41 nm/mT when the radius is 1.2 mm, 4.04 nm/mT when the radius is 1 mm and 4.11 nm/mT when the radius is 0.9 mm. Figure 7b depicts the trend of the magnetic field sensitivity with the change in the radius of the PDMS flexible resonator. With a decreasing radius, the magnetic field sensitivity gradually increases. When the radius of the PDMS flexible resonator is reduced to 0.65 mm, the magnetic field sensitivity of the sandwich structure is 5.14 nm/mT. The gradual increase between the radius of 1 mm and 0.9 mm shows that the different radii of the PDMS flexible resonator have a greater effect on the magnetic field sensitivity of the sandwich structure. The sensor is made by an external magnetic field acting on the magnet squeezing the PDMS resonator, and the PDMS resonator undergoes a change in radius due to the force. As the radius of the PDMS resonator gets larger, the larger it gets, the harder it is for the resonator to change. Therefore, the sensitivity decreases as the resonator radius increases. It provides a dimensional reference for the subsequent PDMS flexible resonator preparation process.

3.3. PDMS with Different Mixing Ratios

The Young’s modulus of the PDMS cured with different mixing ratios is obtained by referring to the literature, as shown in Table 1.

In Table 1, we can obtain the Young’s modulus of the PDMS with different mixing ratios. As the ratio of the PDMS pre-polymer and curing agent gradually becomes smaller, it leads to the Young’s modulus of PDMS also gradually decreasing. The smaller the Young’s modulus, the softer the PDMS flexible resonator, which makes the magnet sense the external magnetic field under the external magnetic field and thus translates into pressure on the PDMS, which directly affects the degree of the deformation of the PDMS flexible resonator and leads to the increased sensitivity of the sandwich structure. The Young’s modulus parameters in the table are brought into the COMSOL parameters to simulate and calculate the sensitivity.

Based on the unchanged parameters of the PDMS flexible resonator with a radius of 1.5 mm and a coupling position at 0.3 h, the Young’s modulus of the PDMS flexible resonator is adjusted by changing the mixing ratio of PDMS pre-polymer and curing agent. By introducing the Young’s modulus of PDMS with different mixing ratios into the model, the magnetic field sensitivity of the sandwich structure under different magnetic fields is obtained, as shown in Figure 8a. The mixing ratio of the PDMS flexible resonator is 6:1, and its magnetic field sensitivity is almost doubled compared with the mixing ratio of the PDMS flexible resonator of 10:1, which shows the influence of Young’s modulus on the magnetic field sensor.

From the results in Figure 8a, it is concluded that the decrease in Young’s modulus also affects the magnetic field sensitivity of the sandwich structure, which is 3.2 nm/mT, 3.63 nm/mT and 4.58 nm/mT for a PDMS polymer to curing agent ratio of 8:1, 7:1 and 6:1, respectively, while Figure 8b quantitatively depicts the difference in sensitivity between different mixing ratios of PDMS. As the PDMS prepolymer ratio decreases, the Young’s modulus gradually decreases, leading to an increase in the deformation of the PDMS flexible resonator, which leads to an increase in the magnetic field sensitivity of the sandwich structure. The simulation results lay the foundation for the subsequent structure optimization and illustrate the feasibility of this structure.

From the above three results, it can be seen that in the PDMS-flexible-resonator-based sandwich structure, all three adjustments affect the sandwich sensor device to a greater extent: as the size of the PDMS flexible resonator decreases, the magnetic field sensitivity of the sandwich structure becomes larger, and when we design the radius adjustment to 0.65 mm, the magnetic field sensitivity reaches 5.14 nm/mT. When we change the coupling position of the PDMS flexible resonator, the magnetic field sensitivity increases as the coupling position approaches the top of the PDMS flexible resonator, and the magnetic field sensitivity is 5.51 nm/mT at the limit coupling position of 0.7 h which we can approach experimentally; the magnetic field sensitivity of the sandwich structure is affected by adjusting the Young’s modulus using the PDMS proportional ratio, and the magnetic field sensitivity reaches 4.58 nm/mT at the Young’s modulus of 0.93 MPa. Therefore, by changing the radius of the PDMS flexible resonator, adjusting the coupling position and the PDMS ratio can directly affect the sensitivity of the sandwich structure, which provides a theoretical basis for a magnetic field sensor with a PDMS flexible resonator.

Through the above analysis of the optimized sandwich structure, it is clear that all three variables affect the PDMS flexible resonator in the sandwich structure. We optimize the three variables of the PDMS flexible resonator via finite element simulation to calculate the magnetic field sensitivity of the optimized sandwich structure. We reduce the radius of the PDMS flexible resonator to 0.65 mm, using a mixing ratio of PDMS polymer and curing agent of 6:1; conduct sensing tests at the 0.6 h coupling position of the PDMS flexible resonator; and perform simulations by applying an external magnetic field to affect the sandwich structure to obtain the relationship between the external magnetic field and the amount of change in the radius of the PDMS flexible resonator, and the simulation results are shown in Figure 9.

Figure 9 shows that the magnetic field sensitivity of the sandwich structure is significantly improved when we optimize the PDMS flexible resonator radius size, coupling position and mixing ratio simultaneously. The magnetic field sensitivity is 19.02 nm/mT in the range of 0.01 mT to 0.11 mT. For the actual preparation process, the preparation difficulty increases with the decrease in the radius of the PDMS flexible resonator. Therefore, this optimization is an important guideline for the actual preparation process, so the sensitivity of the PDMS flexible resonator magnetic sensor can be effectively improved by upgrading the PDMS flexible resonator preparation process. Through the simulation analysis of the PDMS-based flexible resonator sandwich structure model, a theoretical foundation is laid for the feasibility of the scheme, and PDMS-based flexible resonator magnetic field sensing becomes a possibility for WGM-based high-sensitivity magnetic field sensing and provides a reference.

Due to the limitation of the preparation process, the radius of the PDMS flexible resonator is stable between 2.8 mm and 3 mm. We prepare the sandwich structure using NdFeB magnets with 8 mm diameters, and the PDMS mixing ratio is 6:1. We measure the shift in the resonance spectrum under different magnetic fields using a sandwich structure prepared with a PDMS flexible resonator with a radius of 2.8 mm, as shown in the shaded part of Figure 10a. At different magnetic fields, we read the shift of one resonance peak, as shown in Figure 10b, and measure the slope of 1.41 nm/mT via linear fitting; R² = 0.9862. In fact, it is difficult to prepare the sandwich structure with one PDMS flexible resonator, and we improve the stability of the sandwich using three PDMS flexible resonators. Compared to the sandwich structure with one PDMS flexible resonator, the structure in our experiment actually divides the magnetic field sensitivity equally, and the final magnetic field sensitivity should be 4.23 nm/mT, which is three times that of one PDMS flexible resonator. This is almost equal to the magnetic field sensitivity obtained from our simulation. Magnetic field sensing structures prepared using a PDMS flexible resonator provide an idea for miniaturization and, more importantly, such structures can be freely tuned to provide magnetic field sensitivity in the order of nm/mT even with the limitations of the preparation process. During the experiment, only the laser affects the resonator, and the overall temperature of the sensor does not exceed 1 degree Celsius, and the resulting variation is negligible. This approach opens a window for ultra-high-sensitivity magnetic field sensing applications.

4. Conclusions

PDMS is an elastic polymeric silicon compound with high light transmittance and a small Young’s modulus. Traditional sensing methods detect a single resonant wavelength shift under external stimuli, while the small Young’s modulus of PDMS becomes a natural material property advantage of this sensing method, which undoubtedly makes PDMS flexible resonators become high-quality platforms for sensor devices. In this paper, based on the sandwich structure of PDMS flexible resonators, we investigate the mechanism of magnetic field sensitivity of the PDMS flexible resonator sandwich structure and analyze the effect of the change in the refractive index and radius on the sensor device. In order to optimize the sensor sensitivity, the impacts of three aspects on the sensor sensitivity are simulated and analyzed when the external magnetic field is applied to the sandwich structure: different coupling positions of PDMS flexible resonators and different radii and PDMS mixing ratios. The sensitivity trends are obtained, and the physical interpretation of the sensitivity trends is analyzed. By optimizing these three aspects, the magnetic field sensitivity is finally calculated as 19.02 nm/mT. Based on the existing experimental conditions and the PDMS flexible resonator preparation technique, the measured magnetic field sensitivity is 4.23 nm/mT. To maintain the long-term stability of the sensor, we keep the humidity of the experimental environment stable. This sensor paves the way for the development of polymer-resonator-based magnetic field sensors.

Author Contributions

J.R. conceived the idea, revised the paper and wrote the paper; W.X. designed the model, E.X., W.X. and J.R. analyzed the data. J.T. provided scientific support and conceptual advice. All authors discussed the results and commented on the manuscript. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Table

Figure 1. Magnetic sensing principle of sandwich structure.

Figure 2. Simulation of size change in PDMS flexible resonator due to interlayer deformation.

Figure 3. Simulation of effective refractive index change due to photo-elastic effect.

Figure 4. Preparation of PDMS flexible resonator.

Figure 5. Schematic diagram of the experimental setup.

Figure 6. (a) Magnetic field sensitivity at different coupling positions. (b) Trend of magnetic field sensitivity with coupling position.

Figure 7. (a) Magnetic field sensitivity for optimized PDMS flexible resonator radius. (b) Trend of magnetic field sensitivity with radius.

Figure 8. (a) Magnetic field sensitivity for different mixing ratios. (b) Trend of magnetic field sensitivity with mixing ratio.

Figure 9. Sensitivity obtained by optimizing the sandwich structure.

Figure 10. The spectral shift (a) and the fitting curve (b) of sandwich structures in the magnetic field range of 0.01 mT to 0.08 mT. Grey part: spectral shift due to external magnetic field action.

Table 1

Young’s modulus of PDMS with different mixing ratios [33].

Mixing Ratio	Young’s Modulus
10:1	1.47 MPa
8:1	1.34 MPa
7:1	1.2 MPa
6:1	0.93 MPa

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