Figure 1 is the schematic structure of the proposed sensor, which is a PC silicon slab with a defect waveguide and two cascaded cavities. The holes in the slab form a square lattice. The period of the lattice, the radius of the holes and the thickness of the slab are denoted by a, r, and h, respectively. The defect waveguide is formed by removing a row of holes (W1 waveguide, namely, width of waveguide is 2a). Two additional holes at each end of W1 waveguide are also

removed to enhance the coupling efficiency between PC waveguide and stripe waveguide. One cavity is formed by filling eight holes neighboring the waveguide with one kind of MF (i.e. MF1). The filled four holes form a symmetrical triangular shape. Another cavity is formed by filling eighteen holes neighboring the waveguide with another kind of MF (i.e. MF2). The filled nine holes form a 3 × 3 square symmetrically. Two dielectric stripe waveguides are connected to the incident and outgoing ends of the PC waveguide with width of 2a and thickness of h. The lattice period was selected as 450 nm so that the device can work at about 1550 nm wavelength. This type of PC device can be fabricated using electron beam lithography and electron-cyclotron-resonance ion-stream plasma etching using SOI substrates. The underlying SiO2 layer can be removed by selective wet etching using hydrofluoric acid to create an air-bridge PC membrane [22] [23].

As shown in Figure 1, light is launched from the left dielectric stripe waveguide into the PC waveguide, and the outgoing light is monitored in the right dielectric stripe waveguide. When the frequency of incident light coincides with the resonance frequency of either of the cavities, light in the PC waveguide will be coupled into the cavity. Dips will appear in the transmission spectrum at the resonant frequencies. According to perturbation theory, when there is a tiny perturbation of cavity’s geometry or RI, the dip wavelength will shift. MF has the property of magnetically tunable RI [2] [3]. It is supposed that the magnetic field direction is in the plane of the PC slab and along the PC waveguide. The magnetic particles can be considered as magnetic dipoles. When external magnetic field is applied, the direction of magnetic dipoles will tend to be the same as external magnetic field direction. The magnetic particles will be lined up end to end, then form magnetic chains along the magnetic field direction [2] [3]. When magnetic field increases beyond a critical value, the RI of MF increases. When the temperature of MF increases, the thermal motion of magnetic particles becomes violent, which impedes the formation of magnetic chains. Then, the RI of MF decreases with temperature. When the ambient temperature increases, the RI of silicon slab increases (see Table 2 later).

The effect of RI of single MF cavity on the resonant wavelength is shown in Figure 2. λ₁ is the resonant wavelength of MF cavity 1. λ₂ and λ₃ are the resonant wavelengths of MF cavity 2. In Figure 2(a), only the RI of MF cavity 1 is changed. When the RI of MF cavity 1 increases, the spectral line of MF cavity 1 shifts to long wavelength, while the two spectral lines of cavity 2 keep unchanged. In Figure 2(b), only the RI of MF cavity 2 is changed. When the RI of MF cavity 2 increases, the two spectral lines of MF cavity 2 shift to long wavelength, while the spectral line of cavity 1 keeps unchanged. It can be seen that the spectral lines of the two cavities are far away enough that they work independently without affecting each other.

Figure 3 shows the distribution of electric field at three resonant wavelengths. The light in the waveguide is coupled into the MF micro-cavity, which forms a strong electric field distribution within and around the cavity. It is obvious that the strong electric field is mostly located within the MF holes. The field nodes occur at silicon material between the MF holes, where light field is weak. Therefore, it is reasonable to consider that the light heats the MF in the holes. As a preliminary exploration, we have assumed that the two MF cavities have the same temperature [19].

The RI variations of the two MF cavities and Si slab can be described as [2] [3].

[ Δ n m f 1 Δ n m f 2 Δ n S i ] = [ K H 1 0 K T m f 1 K H 2 0 K T m f 2 0 K S i T c 0 ] [ Δ H Δ T c Δ T m f ] (1)

where Δ n m f 1 , Δ n m f 2 , and Δ n S i are the RI variations of the two MF cavities and Si slab, respectively. Δ H , Δ T c , and Δ T m f are the change of magnetic field, Si slab temperature and MF temperature, respectively. K H 1 and K H 2 are the magneto-optical coefficients of the two kinds of MF, respectively. K T m f 1 , K T m f 2 , and K S i T c are the thermo-optical coefficients of the two kinds of MF and Si slab, respectively. When the ambient temperature changes, the RI of silicon slab and air holes changes (see Table 2 later). Considering the very small thermo-optical coefficient of air, the influence of ambient temperature on the RI of air is ignored and only the influence of ambient temperature on the RI of silicon slab is considered. Then, the shift of resonant wavelength can be described as [2] [3]

[ Δ λ 1 Δ λ 2 Δ λ 3 ] = [ K 11 0 K 1 3 0 K 22 K 23 0 K 32 K 33 ] [ Δ n m f 1 Δ n m f 2 Δ n S i ] = [ K 11 0 K 1 3 0 K 22 K 23 0 K 32 K 33 ] [ K H 1 0 K T m f 1 K H 2 0 K T m f 2 0 K S i T c 0 ] [ Δ H Δ T c Δ T m f ] = [ K 11 K H 1 K 13 K S i T c K 11 K T m f 1 K 22 K H 2 K 23 K S i T c K 22 K T m f 2 K 32 K H 2 K 33 K S i T c K 32 K T m f 2 ] [ Δ H Δ T c Δ T m f ] (2)

where Δ λ 1 is the wavelength shift of cavity 1 (filled with MF1). Δ λ 2 and Δ λ 3 are the wavelength shifts of the two modes of cavity 2 (filled with MF2), respectively. K 11 , K 1 3 , K 22 , K 23 , K 32 , and K 33 are wavelength sensitivities of three dips responding to RI variations.

Some reported works use the constant effective RI of silicon slab for two-dimensional (2D) calculation [19] [24]. In fact, the effective RI of silicon slab varies with wavelength. The variational finite-difference time-domain (varFDTD) method can be used to calculate the effective RI of silicon slab with wavelength [25]. It is faster than 3D-FDTD method. The varFDTD method is based on collapsing a 3D geometry into a 2D set of effective RIs that can be solved with 2D-FDTD. The lattice constant a = 450 nm, the radius of normal air holes is r = 0.38a. The thickness of the PC slab is h = 0.7a. The simulation area is truncated with a perfectly matched layer to prevent leakage of light reflection. The PC waveguide port is connected with rectangular silicon dielectric waveguide. Simulation was implemented using MODE SOLUTIONS of Lumerical [26].

The RI of the silicon slab is 3.47809 at temperature of 295 K and wavelength of 1550 nm [27]. The thermo-optical coefficient of silicon is almost constant at around 300 K, i.e. 1.8 × 10⁻⁴/K [28]. The RIs of two types of MFs at different magnetic fields and temperatures can be found in [19]. The material data used in the simulations are shown in Table 1 and Table 2. The effects of error of RI from [19] were tested, and it was found the shifts of dips were so little that software can’t distinguish them, so error of RI was ignored in simulations.

Equation (2) indicates that the sensitivity of the sensor depends on the thermo-optical coefficients of MF and silicon slab, magneto-optical coefficient of MF, sensitivity of spectral line to the RI change. The sensitivity of spectral lines to RI change depends on the structure of the cavity, the coupling between waveguide and cavity, and the structure of the silicon slab. In this work, the radius of holes and thickness of silicon slab are changed to investigate their influence on the three spectral lines. When the radius of holes [29] [30] or the thickness

[29] - [35] of Si slab changes, the effective refractive index of Si slab will change, then the guided mode and cavity modes will change accordingly. Qualitatively, the coupling strength between cavity modes and guided mode, and then the deepness of dips and Q factors will change accordingly. The depth of spectral line and Q factor are taken as the evaluation index. First, the radius of the hole increases from 0.36a to 0.39a with a step 0.005a. The corresponding changes of the three spectral lines are shown in Figure 4. Figure 4(a) shows that the spectral line blue-shifts with the radius increase, which can be explained by the variational principle [6]. When radius increases, the effective RI of silicon slab decreases and then the resonant frequency increases. Moreover, three spectral lines are gradually deepening, especially at λ₁ and λ₃, which is much clear as shown in Figure 4(b). Finally, the Q value is calculated by using Lorentz fitting of the spectral lines, which is shown in Figure 4(c). Q = λ / Δ λ , where Δ λ is full width at half-maximum (FWHM) and λ is resonant wavelength. At λ₁, Q value is relatively small and gets maximum value at 0.37a. Q value is almost constant at λ₂. Q value decreases as the radius increases at λ₃.

The thickness of PC slab is critical to its properties, such as band gap [30] - [35], guided mode [31] [33] and Q factor [29] [35]. For optimizing the silicon slab thickness, the hole radius r is kept at 0.38a, and the slab thickness increases from 0.4a to a. The change of three spectral lines is shown in Figure 5(a). Figure 5(a) displays that the spectral line redshifts with the slab thickness, which can be explained by the variational principle [6]. As the slab thickness increases, the effective RI of silicon slab increases and then the resonant frequency decreases. Furthermore, the three spectral lines gradually deepen. The explicit result is shown in Figure 5(b). Finally, the Q value is calculated by using Lorentz fitting and the results are shown in Figure 5(c). Q value decreases as slab thickness increases at λ₁. At λ₂, the Q value reaches the maximum when the thickness is 0.5a. At λ₃, the Q value reaches the maximum when the thickness is 0.6a.