Dynamic Regulation of Multiple Fano Resonances Based on Liquid Crystal

A coupled structure, consisting of two perpendicular cavities coupled with a waveguide, has been proposed and researched numerically and theoretically. It is found that the lateral displacement S plays a key role in transmission spectrum. Whether the first- or the second-order mode can exist in the horizontal cavity has been well explained. Owing to the interaction among two discrete and a continuous states, four Fano resonance can be achieved to set appropriate S. When the coupled structure is filled with liquid crystal (LC), the refractive index can be regulated dynamically by external voltage. The coupled structure can be act as a filter which filters arbitrary wavelength within the scope of certain wavelength. At the same time, in a certain wavelength range, an optical switch can be achieved with different voltage at any wavelength. This structure has the advantages of simple, easy fabrication and tunability, which has potential applications in integrated optical devices.

Fano resonance, arising from the interaction between the continuous and discrete states, has been researched in detail [8], [9], [10]. Fano resonance with the characteristics of the sharp asymmetric line shape and strong-field enhancement, has attract researches' attentions. Owing to advantages of confining light and long propagation length, Fano resonance based Manuscript  on Metal-Insulator-Metal (MIM) structure have been intensive studied, and the applications are very broad, such as hemoglobin detection [11], cancer biomarker [12], sensing [13], [14], [15], filter [16] and so on. Moreover, as is well known, when light inject into MIM waveguide, the MIM structure can support the SPs which can be excited in the surface of metal and insulator. Ensure that only the transverse magnetic mode propagates in the waveguide, the width of waveguide is relatively small [17]. Booth. et al. developed a new technology to efficiently fabricate arbitrary micro-and nano-waveguides with varying three-dimensional cross sections, where the technology is named spherical phase-induced multicore waveguide (SPIM-WG) and the waveguide has very low transmission loss [18]. Li J. et al. construct a coupling system with a single layer of MoS 2 , silver nano cube and gold film to realize the strong interaction between light and matter, and the longitudinal spatial resolution of coupling strength can reach 1 nm [19]. Li B. et al. proposed a U-shape structure to achieve dual transparency window, and the sensitivity and figure of merit are up to 1225 nn/RIU and 62.5, respectively [20]. However, the limitation such as non-adjustable optical properties hinders its applications on some occasions. In this paper, a coupled structure which is composed of a horizontal and a vertical cavities coupled with waveguide has been studied numerically and theoretically. The results show that the lateral displacement S has a great impact on the transmission. At the same time, the existence of the first-order or the secondorder mode in the horizontal cavity has been well explained theoretically. The Coupled Method Theory (CMT) is introduced to analysis the coupled structure, which has been well demonstrated by the Finite-Different Time-Domain (FDTD). Four Fano resonance can be achieved by choosing appropriate value S. With the introduction of liquid crystal (LC), the refractive index can be regulated dynamically by external voltage. The coupled structure can be act as a filter which filters arbitrary wavelength within the scope of certain wavelength. At the same time, in a certain wavelength range, an optical switch can be achieved with different voltage at any wavelength. The coupled structure has the advantages of simple, easy fabrication and adjustable optical characteristics, which has potential applications in integrated optical devices.

II. MODEL AND METHOD
A plasmonic coupled periodic structure, consisting of a horizontal and a vertical cavities coupled with bus-waveguide, has been proposed, which is illustrated in Fig. 1  silver, which has been deposited on a SiO 2 substrate, is etched into two perpendicular cavities. And the silver can be expressed by the Drude-model as [21]: ε(ω) = ε Ý -ω p /(ω 2 +iωγ p ), where ω stands for the angle frequency, ε Ý = 3.7, ω p = 9.1ev represents the bulk plasmon frequency, and γ p = 0.018ev denotes the damping rate. When the silver layer is thick enough, three-dimensional structure can be replaced by a 2-D structure to reduce the computation. Meanwhile, the perfectly matched layer (PML) absorbing boundary conditions together with periodic boundary along y-direction are introduced to simplify calculation. The mesh and the temporal step are set as Δx = Δy = 3 nm and Δt = Δx/2c, respectively. According to Fig. 1(b), the width of bus-waveguide, vertical and horizontal cavities are w 0 , w 1 and w 2 respectively. L 1 and L 2 , respectively, represent the length of vertical-and horizontal-cavities. The distance between the two cavities, vertical cavity and bus-waveguide are g. The buswaveguide is truncated by a silver block, whose width is d. And the lateral displacement between the two cavities is represented as S. The values of the parameters are shown in Table I. In this manuscript, the external temperature is set to be 20°C.
Based on the CMT, the temporal normalized mode amplitudes a 1 of vertical cavity can be expressed as [10], [14], [22]: Here, ω is the angle frequency of the light, ω 1 denotes the resonant frequency of the vertical cavity, k 1 and k 2 stand for the attenuation rate for the internal loss of the two cavities. k 0 is the coupling coefficient between the two cavities. S i+ and S i-(i = 1, 2) are the amplitude of input and output wave. We define ϕ 1 and ϕ 2 to be the phase delay in the left and right facets of the horizontal cavity, ϕ 1 = 4πn eff (0.5L 2 -S)/λ and ϕ 2 = 4πn eff (0.5L 2 +S)/λ, where n eff is effective refractive index. In this manuscript, since the light injects from the left port, S 2+ = 0. According to the (1)-(4), the transmittance of the coupled structure can be expressed as (5) shown at the bottom of next page

III. RESULTS AND DISCUSSION
It is well known that the mechanism of Fano resonance can be attributed to the interaction between continuous and discrete states. In order to search the underlying physical mechanism, the transmission spectrum of three structures have been illustrated in Fig. 2(a). Obviously, the transmission spectrum of the block coupled with bus-waveguide is very flat, while two valleys arise in the transmission of vertical cavity coupled with bus-waveguide. When these two structures put together, due to the interaction between the continuous and discrete states, two Fano resonances which are named as LFR (left Fano resonance) and RFR (right Fano resonance) at the wavelength of 775 nm and 1550 nm appear. And the continuous and discrete states are, respectively, caused by the block and vertical cavity. The insets in Fig. 2(a) show the magnetic field distributions of 783.5 nm and 1559.5 nm in the vertial cavity coupled with bus-waveguide, where correspond to the first-and the second-order modes, respectively. Fig. 2(b) shows the wavelength shift with different L 1 . It's pretty obvious that there is a linear relationship between the Fano resonance and the length of L 1 , which has been illustrated clearly in the inset in Fig. 2(b).
In this section, a horizontal cavity with width w 2 and length L 2 is introduced, which can be seen in Fig. 1(b). Fig. 3 show the transmittance with different S. It is found that the lateral displacement S has a great impact on the transmittance. When S = 0, it is clear from Fig. 3(a) that three Fano resonances appear. And the Fano resonance I and II derive from the splitting of LFR. Moreover, when S = 145 nm, three Fano resonances emerge and the RFR splits into two Fano resonance V and VI. It can be said that different value of S has different effects on the transmission spectrum.
To search the mechanism behind the above phenomenon, the magnetic field distributions with six wavelengths in Fig. 3(a) and (b) have been drawn in Fig. 4. According to Fig. 4, we found that most of the power concentrate in the horizontal cavity in I, while most of the power concentrate in the vertical cavity in II. Moreover, the power in horizontal and vertical cavities are the second-order mode in I and II. There is basically no power in the horizontal cavity and the power in vertical cavity is the first-order mode in III. To IV, V, VI in Fig. 4, it is obvious that no power exist in the horizontal cavity and the power in vertical cavity corresponds to the second-order mode in IV. Most of the power concentrate in the horizontal cavity in V, while the power concentrate in the vertical cavity in VI. In addition, the power in horizontal and vertical cavities are the   first-order mode in V and VI. The results show that the firstand the second-order modes in the horizontal cavity depend on S. Under the assumption of neglecting the reflection phase shift of the horizontal cavity, the stable standing waves can be exited when the resonance condition is satisfied: Φ m ≈β m 2L 2 ≈2mπ (m = 1,2) [23], [24], where Φ m , β m and L 2 stand for the phase delay in the horizontal cavity, propagation constant of SPs, the length of the horizontal cavity, respectively. Based on the superposition principle of optics and the standing wave theory [25], [26], when the light input into the horizontal cavity, the input field P in will divide into two parts, P left and P right , and the two parts have the relation of P left = P right = P in /2 = P 0 . The horizontal cavity is symmetric regarding to the central line x = 0. The field P m (x,t) in the horizontal cavity with an arbitrary input position S can be expressed as where δ represents the loss coefficient of light propagation in the horizontal cavity. For the first-order mode (m = 1), we can obtain that Φ 1 ≈β 1 2L 2 ≈2π. According to Euler's formula, it is concluded the field P 1 (x,t) in horizontal cavity is expressed as Analogously, for the second order mode (m = 2), Φ 2 ≈β 2 2L 2 ≈4π, the (6) can be abbreviated as When S = 0, it is concluded that sin (β 1 S) = 0, cos(β 2 S) = 1, and P 1 (x,t) = 0 and P 2 (x,t)ࣔ0, which means that the field of the first-order mode can not exist while the second-order mode can exist in the horizontal cavity. For the case of S = 145nm, it can get that β 1 S≈πS/L≈0.25π and β 2 S≈2πS/L≈0.5π, which indicate sin(β 1 S) = 0.707 and cos(β 2 S) = 0. At this moment, it is clear that P 1 (x,t)ࣔ0 and P 2 (x,t) = 0, which means the first-order mode can exist in the horizontal cavity while the second one can not. What analysis above well explain that the value of lateral displacement S determines the existence of the first-and the second-order modes. As shown in Fig. 3(c), there are four Fano resonances in the transmission spectrum when S = 110nm. Based on the analysis above, when S = 110nm, P 1 (x,t)ࣔ0 and P 2 (x,t)ࣔ0, which signifies that the firstand the second-order modes can exist in the horizontal cavity simultaneously. Moreover, what needs to be pointed out is that the CMT results are good consistent with the FDTD ones in Fig. 3.
As is well known, Liquid Crystal (LC) is anisotropic uniaxial, whose refractive index with extraordinary (n e ) and ordinary (n o ) has a great relationship with wavelength and temperature [26]. When the coupled structure is filled with LC (E7), the refractive index can be adjusted by external electric field, which enables dynamical control the Fano resonance. The extraordinary n e and ordinary n o can be shown as [27] n e = A e + B e /λ 2 + C e /λ 4 (9) where A e , B e , C e , A o , B o and C o are Cauchy coefficients and equal to 1.6993, 0.0085, 0.0027, 1.4998, 0.0067 and 0.0004, respectively, which can be obtained from reference [28]. Fig. 5(a) shows the relation between n e , n o and wavelength. It is apparent that the refractive index drops rapidly in the visible range. As the wavelength increase further, n e and n o go flat and keep at values of 1.7187 and 1.5076, respectively. An external electric field is applied along the z-direction to adjust the refractive index of LC, where the refractive index n can be described as follows [29], [30] where θ represents the tilt angle between the long axis of the LC molecules and x axis, which has been illustrated in Fig. 6. In this manuscript, we assume that all the LC molecules are along the x-and y-direction in the horizontal cavity and vertical cavity, respectively. To ensure the refractive index in the horizontal and vertical cavities equal to n o , a lens cleaning tissue can be used to brush the PVA repeatedly along the x-and y-direction, respectively [31], [32]. In order to dynamically control the optical properties, an electrical field along the z-direction is introduced. When the voltage is greater than the threshold, the tilt angle θ will increase and the LC molecules will align with the z-direction finally, which means that the refractive index of LC in the horizontal and vertical cavities is n e , and the angle θ can be written as [33] θ= Where U is the applied voltage, V th is the threshold voltage which can be obtained from the equation [34] V th = π k 11 Δεε 0 (13)   11 is the splay elastic constant of LC, ε 0 = 8.85pF/m is the electric constant. Δε = ε -ε represents the anisotropy dielectric permittivity. And k 11 , ε , ε can be achieved in reference [35]. It is concluded that the threshold voltage V th = 0.952 V (20°C). Fig. 5(b) illustrates the dependence of refractive index on voltage. It is found that the refractive index of LC increases from n o to n e , as the voltage goes from 0 to 90 V. At the same time, as can be seen from Fig. 5(b), it is worth noting that there is a linear relationship between refractive index and voltage in the range of 10.5 V to 36.9 (the purple line), which corresponds to the refractive index of 1.5267 to 1.6492. And the linear relationship can be expressed as n = 0.00464U+1.478, which has a very important guide for us to adjust the specific refractive index. When the coupled structure is filled with LC and a external electric field is loaded to regulate the refractive index of LC, which is shown in Fig. 6. Two plate electrodes sandwich the coupled structure and the external electric field along the z-direction is applied. To simplify the calculation, we ignore the effect of electric field on LC orientation and threshold voltages in metal cavity, and we assume that the electric fields in the liquid crystal cavity are uniform. In order to seal the LC, a thin films of polyvinyl alcohol (PVA) have been spin coated on the bottom and top of the Ag layer.
As described above, the property that the refractive index of LC can be dynamically regulated by voltage can act as a adjustable optical switch. Fig. 7 illustrates the transmission spectrum with different refractive index. And Table II   of the transmission spectrum can be observed. At last, when the voltage increases to 90V, the refractive index change to 1.7187. In addition, it is worth noting that the refractive index changes very little and remains at 1.7187 when the voltage exceeds 90 V.
With the introduction of horizontal cavity with different S, several Fano resonances can be observed. Moreover, when the coupled structure is filled with LC, the refractive index of LC can be regulated dynamically through external voltage. Arbitrary wavelength can be filtered by appropriate applied voltage within the scope of certain wavelength. The proposed coupled structure can be act as a filter. Moreover, to a given wavelength, the change of the refractive index will lead to the shift of the transmission spectrum, which will result in the high and low transmittance, corresponding to the on-and off-states. For example, when the voltage is 90 V, the wavelength of Fano resonance III is 2720.6 nm and the transmittance is 0.31. As the voltage decreases to 36.9 V, the Fano resonance III shifts to 2610 nm with a transmittance of 0.32. To the wavelength of 2720.6 nm, the transmittance is about close to 0, which realizes a switch function. Analogously, the switch function can also be achieved to the other wavelength. What should be particularly pointed out is that we can achieve optical switch at any wavelength according to the equation n = 0.00464U+1.478. Fig. 8 illustrates the process of the two functions above. Compare with the literature mentioned above, the proposed coupled structure has several advantages of simple, easy fabrication and adjustable optical characteristics, which may provide guidance for the design and fabrication of integrated optical devices.

IV. CONCLUSION
In conclusion, a coupled structure, consisting of two perpendicular cavities coupled with a waveguide, has been proposed and investigated theoretically and numerically. The results indicate that the lateral displacement S has a great impacts on the transmission spectrum. When S takes a appropriate value, the first-and the second-order modes can exist simultaneously, where four Fano resonances can be observed in the transmission spectrum. As liquid crystal (LC) is filled into the coupled structure, the refractive index can be regulated dynamically through the external voltage, a filter can be realized by appropriate applied voltage within the scope of certain wavelength. Moreover, due to the shift of the transmission spectrum with different voltage, in a certain wavelength range, a device capable of switching on and off states can achieve at any wavelength. The coupled structure has the advantages of simple, easy fabrication and adjustable optical characteristics, which has potential applications in integrated optical devices.