## Abstract

Miniaturizing optical devices with desired functionality is a key prerequisite for nanoscale photonic circuits. Based on Fano resonance, an on-chip high-sensitivity sensor, composed of two waveguides coupling with a symmetry breaking ring resonator, is theoretically and numerically investigated. The established theoretical model agrees well with the finite-difference time-domain simulations, which reveals the physics of Fano resonance. Differing with the coupled cavities, the Fano resonance originates from the interference between symmetry-mode and asymmetry-mode in a single symmetry-broken cavity. The spectral responses and sensing performances of the plasmonic structure rely on the degree of asymmetry of cavity. In particular, the plasmonic sensor can detect the refractive index changes as small as 10^{−5}, and the figure of merit (FOM) of symmetry-breaking cavity structure is 17 times larger than that of symmetrical cavity system. Additionally, the sensitivity to temperature of ethanol analyte achieves 0.701 nm/^{○}C. Compared with the coupled cavities, the on-chip high-sensitivity sensor using a single cavity is more compact, which paves the way toward highly integrated photonic devices.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Surface plasmon is a kind of surface evanescent wave produced by collective oscillation of free electrons on metal surface when matching the frequency of photons in incident light waves. The surface plasmon polaritons (SPPs) [1,2] bound to the metal-dielectric interface is a class of surface plasmon, which can overcome the classical diffraction limit [3] using deep sub-wavelength metal structures [4]. The rapid developments in nanotechnology and plasmonics facilitate the manipulation of light at the nanoscale, and plasmonic devices have been widely used in sensing [5–8], optical switching [9–11], filtering [12–16] and other fields [17–19].

As one of the most prevalent devices in optical circuits, the surface plasmon resonance (SPR) optical sensor based on metal-dielectric-metal (MDM) waveguide has attracted enormous attention due to small size, high sensitivity and easy integration [20,21]. Recent trends in plasmonic sensor research have advanced towards waveguide coupled resonator structures that enable some novel nonlinear optical effects, such as Fano resonance [22–27] and electromagnetic induced transparency [28–30]. For the Fano resonance arising from the interference of the radiant mode and subradiant mode in metallic nanostructures, it exhibits tremendous field enhancements [26]. The tremendous field enhancements are quite sensitive to the structural and electromagnetic parameters, empowering that a small variation can shift the Fano resonances significantly. Compared to the symmetric Lorentzian-like profile in the conventional resonance, Fano resonance exhibits obviously asymmetric spectral line shape [31] and large spectral contrasts.

Due to its tremendous field enhancements, sharp asymmetric spectral line shape, and large spectral contrasts, Fano resonance based on optical cavities has significant applications in sensing technology. In 2012, *Lu et al.* established a nanosensor in MDM waveguide-cavity system and obtained a figure of merit (FOM) of ∼500 [32]. *Chen et al.* demonstrated a novel and compact refractive index sensor based on Fano resonance which comprises with a stub and groove resonator coupled with a MDM waveguide [33]. In addition, *Wen et al.* proposed a plasmonic MDM waveguide structure by placing a slot cavity below or above a groove [34]. To obtain the asymmetric Fano resonance line shape, various artificial nanostructures have been proposed and designed recently. However, studies on Fano resonance emerged by a single symmetry breaking cavity are relatively rare.

Here, we numerically and analytically demonstrate high-sensitivity and tunable sensing performances based on Fano resonances in a plasmonic symmetry breaking cavity nanostructure. The proposed structure composed of MDM waveguide and a symmetry breaking ring cavity. The on-chip symmetry broken design provides a solution for ultra-compact plasmonic sensor. The proposed analytical theory is in good agreement with the finite-difference time-domain (FDTD) simulations, and the magnetic field distributions imply that Fano resonance line shapes arising from the interference between symmetry-mode and asymmetry-mode in symmetry-broken cavity. Moreover, the Fano peak intensity can be tuned by the degree of asymmetry of ring cavity, which lays the groundwork for improving the performances of plasmonic sensing.

## 2. Plasmonic structure and theoretical model

The nanoscale plasmonic sensor structure is schematically shown in Fig. 1(a). The plasmonic structure is comprised of two MDM waveguides coupled with a ring resonator, and a metal bar is positioned inside the ring cavity to break the symmetry of ring cavity. In Fig. 1(a), the width of waveguide is *d*, and the coupling distances between the two waveguides and ring cavity are both *w*. *r* denotes the radius of ring cavity, and *r _{m}* presents the radius of metal bar. The degree of asymmetry is denoted by the intersection angle

*θ*between the metal bar and

*y*-axis, and

*θ*increases clockwise. For metal and dielectric materials in the structure, we select silver and air, respectively. In general, the Drude model is used to approximately describe the relative dielectric constant of silver by the equation:

*ɛ*(

_{m}*ω*)

*=ɛ*-

_{∞}*ω*/ (

_{p}*ω*+

^{2}*iωγ*).

_{p}*ɛ*= 3.7 represents the dielectric constant of silver when the frequency is infinite,

_{∞ }*ω*1.38×10

_{p }=^{16}rad/s and

*γ*= 2.37×10

_{p }^{13}rad/s represent the oscillation frequency and damping frequency of silver, respectively. The spectral responses are numerically studied by the two-dimensional FDTD method. The perfect matched layer (PML) boundary conditions are used in the FDTD simulations, and the source is Gaussian light, which is incident from the left waveguide. When the incident light is TM-polarized wave, the SPPs will be excited and confined in waveguide.

To explain the physics of spectral responses in plasmonic structure, a theoretical model is established based on the coupled mode theory (CMT), as illustrated in Fig. 1(b). Then, the characteristic equations for the evolution of cavity modes can be expressed as follows

*S*and

_{±in}*S*represent the amplitudes of incoming and outgoing waves in waveguides. $|a \rangle$ and $|{\textrm K} \rangle$, respectively, stand for the field amplitude of resonant mode and coupling coefficient between resonator and waveguides. $\left\langle {\textrm K} \right|$ is dependent on $|{\textrm K} \rangle$. They can be written as

_{±out}*κ*stands for the coupling between waveguides and cavity.

_{m}In Eq. (1), Ω, Γ* _{i}*, Γ

*and Μ matrices are expressed as*

_{e}*, Γ*

_{i}*and Μ matrices denote resonance frequencies, intrinsic loss rate, external loss rate of cavity, and coupling coefficients between resonant modes, respectively. In Ω, Γ*

_{e}*, Γ*

_{i}*and Μ matrices, if*

_{e}*p*≠

*q*,

*ω*0,

_{pq }=*γ*0,

_{ipq }=*γ*0, and

_{epq }=*μ*/ (2

_{pq }= ω_{qq}*Q*); if

_{pq}*p = q, ω*,

_{pq }= ω_{p}*γ*/ (2

_{ipq }= ω_{pq}*Q*),

_{iq}*γ*/ (2

_{epq }= ω_{pq}*Q*), and

_{eq}*μ*0. The cavity quality factors

_{pq }=*Q*,

_{iq}*Q*are related to intrinsic loss, waveguide coupling loss, respectively.

_{eq}*Q*denotes the coupling between the

_{pq}*p*th and

*q*th modes. Based on the condition

*S*0 and Eqs. (1)–(5), we can get the transmission efficiency of the system

_{-in }=*I*= |

*S*/

_{+out}*S*|

_{+in}^{2}.

When *m* = 2, the power transmission is given as

*D*

_{1}

_{ }= -jω + jω_{1}

*+*1

*/τ*

_{i}_{1}

*+*1

*/τ*

_{ω}_{1},

*D*

_{2}

*=*

_{ }*-jω + jω*

_{2}

*+*1/

*τ*

_{i}_{2}

*+*1/

*τ*

_{ω}_{2}. In the proposed symmetry breaking ring cavity, supporting multiple modes, resonance modes are orthogonal to each other. Thus, the M matrix in Eq. (1) is equal to zero matrix in the following contents, and waveguide induces the indirect coupling between different resonant modes [35,36].

## 3. Results and discussion

Figures 2(a)–2(d) display the transmission spectrum and field distributions of the structure. The geometrical parameters of the plasmonic system are set as *d *= 50 nm, *w* = 10 nm, *r *= 200 nm, *r _{m }*= 15 nm, and

*θ =*0°. Figure 2(a) illustrates a Fano lineshape and a Lorentzian lineshape in transmission spectra. The Fano peak and Lorentzian peak occur at

*λ*= 919 nm and

*λ*= 1900nm, corresponding to the first (P1) and second (P2) resonant modes of ring resonator, respectively. The analytical result (red line) is in good agreement with the FDTD simulations (black line), which proves the correctness of the established theoretical model. In order to better understand the transmission characteristics of the structure, Figures 2(b)–2(d) show the contour profiles of the field |

*H*| at different wavelengths. Figures 2(b) and 2(c) present the magnetic field |

_{z}*H*| for the wavelengths of resonant peaks at P1 and P2, respectively. As illustrated, the magnetic field distributions exhibit two resonant modes in ring cavity, and the SPPs can pass through the right port. Figure 2(d) presents the magnetic field distribution at wavelength

_{z}*λ*= 2200 nm. It is apparent that there is a very weak field distribution |

*H*| in the ring cavity, and the transmittance approaches zero.

_{z}In Fig. 3(a), we present the transmission spectra with metal bar placed at *θ = *0°, *θ = *15°, *θ = *45° and *θ = *75°, respectively. Apparently, a sharp dip appears in the transmission peak with changing *θ*, and it can be seen that the Fano lineshape exhibits in the transmission spectra. To real the spectral responses in the plasmonic system, we study the spectral features by referring to the Fano formula [26]

*q*denotes the so-called Fano parameter, which is related to the incident waves and geometric and material parameters of system. The spectral profile is dependent on

*q*.

*F*

_{0}denotes the amplitude of the Fano resonance. Γ is the resonance width.

*λ*and

*c*denote the wavelength and light velocity in vacuum, respectively.

*λ*

_{0}is the resonant wavelength.

*A*(

*λ*) denotes the background term. Basing on Eq. (7), we can obtain the Fano parameter

*q*for the spectral lineshape, which indicates the spectral features in the plasmonic system. For the Lorentzian lineshape, Fano parameter

*q*→ ±∞. Figure 3(b) presents the evolution of transmittance as function of

*θ*and response wavelength, where

*θ*varies from 0° to 360°. The Fano dip around P1 shifts from left to right of Fano peak, which leads to reciprocal transformation between Lorentzian and Fano lineshapes. The Fano dip around P2 is always on the left of Fano peak. Moreover, it is apparent that the transmission spectra vary periodically versus

*θ*, and the periods around P1 and P2 are 90° and 180°, respectively.

To get more insight into the spectral responses, Figures 3(c)–3(e) show the magnetic field distributions, corresponding to *θ* = 15°, at *λ = *907 nm (left peak), *λ = *921 nm (dip), *λ = *939 nm (right peak). With respect to the *x* axis, the field distributions in Figs. 3(c) and 3(e) are symmetrical and asymmetrical, respectively. The field distribution in Fig. 3(d) is in conformity with the Fano dip in transmission spectrum. According to the distributions of magnetic field, the resonance modes in the proposed structure are classified as symmetric mode and asymmetric mode. Combining with the numerical simulations and theoretical analyses, we can conclude that both the Fano lineshape and Lorentzian lineshape result from the interference between two different pathways related to the symmetric and asymmetric modes. The Fano spectral responses originate from the symmetric and asymmetric modes with different resonant frequencies, while the Lorentzian spectral features are attributed to the symmetric and asymmetric modes with identical resonant frequencies. Thus, the symmetry breaking ring cavity can be used to drive new resonant modes as well as provide a promising approach to generate Fano resonance.

Compared with the Lorentzian spectral features, Fano resonance with sharp slope and asymmetrical line shape holds potential applications in nanoscale ultra-sensitive sensors. To illustrate the sensing characteristics of the plasmonic system, the transmission spectra versus refractive index *n* of surrounding medium are given in Fig. 4(a), where the refractive index *n* increases from 1.00 to 1.10 with step of 0.02. It can be observed from the Fig. 4 that the transmission spectra are related to the refractive index of surrounding medium, and the two Fano resonances (FR1 and FR2) exhibits monotonically red shift with increasing the refractive index. The FR2 shifts bigger, and the transmittances of Fano peaks decrease owning to the increase of transmission losses. According to Fig. 4(a), we display the linear relationship between the Fano peak wavelength and refractive index in Fig. 4(b). As the refractive index *n* varies from 1.00 to 1.10, the wavelength shift of FR1 (red dotted line) is 92 nm, while the wavelength shift of FR2 (blue dotted line) is 181 nm. Generally, the sensitivity of the plasmonic sensor is defined as *dλ* / *dn*. Then the sensitivity around FR1 and FR2 is, respectively, 920 nm/RIU and 1810nm/RIU. It is obvious that the sensitivity is proportional to the Fano resonance wavelength, which provides a way to improve the performance of optical sensors.

In order to better characterize the performance of the plasmonic sensor, the limit of detection (LOD) and figure of merit (FOM) are key parameters. For a specific wavelength, the limit of detection is defined as [37]

which reflects the detection capability of a sensor. Basing on Eq. (8), we obtain that the detection limit around FR1 equals 2.17×10^{−5}RIU, and the detection limit around FR2 equals 1.1×10

^{−5}RIU.

In sensing system, the higher FOM indicates the better sensing performance, and the FOM could be expressed as FOM = |d*I*(*n*) / d*n*| at fixed wavelength [38]. d*I*(*n*) denotes the transmission intensity variation, and d*n* stands for the change of refractive index of the surrounding medium. As refractive index varies from 1.00 to 1.02, we plot the FOM, in Fig. 5(a), for the symmetry-broken ring cavity system with *θ* = 15°. It can be found that the maximum FOM approaches 34.4 at wavelength *λ* = 939.5nm. Figure 5(b) demonstrates the maximum value of FOM (MAX of FOM) around FR1 versus the position of metal bar *θ*, and the MAX of FOM reaches the maximum value at *θ *= 15°, which indicates that the position of metal bar provides an efficient way to modulate the sensing performance.

Figure 5(c) displays the FOM as a function of the response wavelength for different refractive index *n*, where *n* varies from 1.00 to 1.10. In this case, the change of refractive index d*n* is 0.02, and d*I*(*n*) = *I*(1.02) - *I*(1.00), *I*(1.04) - *I*(1.02), *I*(1.06) - *I*(1.04), *I*(1.08) - *I*(1.06), *I*(1.10) - *I*(1.08), respectively. Figure 5(d) is an enlarged view around FR1 in Fig. 5(c). It illustrates that the maximum of FOM approaches 34.4 in range of 1.00-1.02. The FOM around FR1 is more than three times larger than that in [38]. Thus, the range of refractive index also has an impact on the maximum of FOM. To study the sensing performance of the plasmonic sensor, another figure of merit (FOM*) is defined as [39] FOM* = d*I* / *I*d*n*. As the refractive index increases from 1.00 to 1.02, Figure 5(e) shows the FOM* of symmetrical cavity and symmetry-breaking cavity systems. For symmetrical cavity without metal bar, the maximum of FOM* is about 103 at wavelength *λ* = 1900nm; for symmetry-breaking cavity with metal bar at *θ* = 75°, the maximum of FOM* could reach 1760 at wavelength *λ* = 919nm. The maximum of FOM* for the symmetry-breaking ring cavity structure is 17 times larger than that of the symmetrical ring cavity system. Thus, the symmetry breaking can improve the figure of merit for refractive index sensing. These results could lay a foundation for the ultra-compact and high-performance plasmonic sensors.

To our knowledge, the performance of the proposed refractive index sensor is better than previously reported plasmonic sensors [16,32,40–43]. Here, to characterize the sensing performance versus temperature, the ethanol is chosen as the filling medium in waveguides and ring cavity. The temperature coefficient of ethanol is 3.94×10^{−4}, and the functional relation between refractive index and temperature can be defined as [44]

*T*

_{0}denotes the room temperature and its value is 20

^{○}C, and

*T*is the measured ambient temperature. The sensitivity of temperature sensor is defined as

*dλ*/

*dT*.

Figure 6(a) displays the transmission spectra versus different temperatures of ethanol, such as *T = *80^{○}C, *T = *60^{○}C, *T = *20^{○}C, *T = *-20^{○}C, *T = *-60^{○}C, and *T = *-100^{○}C. It is clear that the Fano resonant wavelength decreases with the increase of temperature, and the overall spectral shape remains unchanged. Equation (9) indicates that the refractive index is linearly related to temperature, and Figure 4(b) shows that the Fano peak wavelength has a linear relationship with the refractive index. Thus, in Fig. 6(b), there is a linear variation of Fano peak wavelength with the measured temperature. When *T* rises from -100^{○}C to 80^{○}C, with the metal bar placed at *θ = *75^{○}, the variations of wavelength of FR1 and FR2 are 63.13nm, 126.25nm, respectively. According to the definition of the sensitivity for the temperature sensor, the sensitivity around FR1 and FR2 approach 0.35nm/^{○}C and 0.701nm/^{○}C, respectively. As mentioned above, the symmetry breaking ring cavity structure filled with ethanol holds potential applications for the manufacture of on-chip temperature sensors.

## 4. Conclusion

In summary, we have reported a joint numerical and theoretical study on ultra-compact and high-sensitivity plasmonic sensor based on Fano resonance, which is composed of two waveguides coupling with a symmetry breaking ring resonator. The established theoretical model is validated by numerical simulations, and sets up a platform to understand the physics of Fano resonance. Differing with the coupled cavities, the Fano resonance originates from the interference between symmetry-mode and asymmetry-mode, and the Fano parameter *q* and sensing performance rely on the degree of asymmetry of the ring cavity. The sensitivity of the plasmonic sensor could approach 1810nm/RIU. In particular, the refractive index changes as small as 10^{−5} can be detected, and the FOM of symmetry-breaking cavity structure was 17 times larger than that of symmetry cavity system. Moreover, the plasmonic sensor can achieve high sensitivity to external temperature changes. This small footprint together with the tunable spectral responses can actualize active devices for fundamental study and applications in on-chip high-sensitivity sensors.

## Funding

National Natural Science Foundation of China (11564014, 61865006); Natural Science Foundation of Hunan Province (2017JJ2097, 2019JJ50481); Education Department of Hunan Province (16A067, 18B324); Xiangxi Autonomous Prefecture Science and Technology Program (2018SF5024); Natural Science Foundation of Guangdong Province (2018A030313684); Major Projects of Guangdong Education Department for Foundation Research and Applied Research (2017KTSCX134).

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