# A High-Sensitivity Magnetic Field Sensor Based on PDMS Flexible Resonator
## 2 Materials and Methods
### 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 [PERSON]'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 [PERSON]'s ratio is 0.49 [35]. The [PERSON]'s modulus of the neodymium iron boron magnet (NdFeB) is 150 GPa. Figure 1: Magnetic sensing principle of sandwich structure. 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]:  
\[ \begin{array}{l}n_{x}=n_{0x}+C_{1}\sigma_{xx}+C_{2}\big{(}\sigma_{yy}+\sigma _{zz}\big{)}\\ n_{y}=n_{0y}+C_{1}\sigma_{yy}+C_{2}\big{(}\sigma_{xx}+\sigma_{zz}\big{)}\\ n_{z}=n_{0z}+C_{1}\sigma_{zz}+C_{2}\big{(}\sigma_{xx}+\sigma_{yy}\big{)}\\ \end{array}  \tag{3}\]  
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. \(\sigma_{xx}\), \(\sigma_{yy}\) and \(\sigma_{zz}\) are the principal stresses in the \(x\)-axis, \(y\)-axis and \(z\)-axis, respectively. \(C_{1}\) and \(C_{2}\) are the photo-elastic constants of the material. Since the photo-elastic constants of PDMS are \(C_{1}=C_{2}=C=-1.75\times 10^{-10}\) m\({}^{2}\)/N [38], Equation (3) can be rewritten as:  
\[n=n_{0}+C\big{(}\sigma_{xx}+\sigma_{yy}+\sigma_{zz}\big{)} \tag{4}\]  
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 \(-1.46\times 10^{-10}\) mT\({}^{-1}\). The rate of change in the refractive index due to different magnetic fields is \(-1.04\times 10^{-10}\) mT\({}^{-1}\); therefore, the magnetic field sensitivity of the sandwich structure due to refractive index change is \(1.6\times 10^{-7}\) nm/mT. The inset shows the strain change in the PDMS flexible resonator induced by the magnetic field. Figure 2: Simulation of size change in PDMS flexible resonator due to interlayer deformation. 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 \(\times\) 10\({}^{-7}\) nm/mT.