In this research paper, multiple trapped modes within a bi-periodic silicon particle array were studied numerically both through finite difference time domain method and finite element method. Anti-phased electric dipole, magnetic dipole, and magnetic quadrupole collective resonances in the two arrays result in three distinct trapped modes, which interact with the corresponding “bright modes” generating three sharp asymmetric Fano profiles in reflection. The lineshapes can be easily tailored by particle radius to achieve very narrow linewidth, while still remains 100% modulation depth. As a three-channel refractive sensor, the sensitivities for three working frequencies are 155nm/RIU, 725nm/RIU, and 190nm/RIU.
© 2014 Optical Society of America
Metamaterial has attracted much attention in the past decade for its great ability to manipulate light by designing effective electromagnetic parameters as needed and found various applications, such as negative refraction, cloaking, perfect absorber, and filter [1–4], which are dominated by the resonant nature of their response. One of the interests in this field concentrates on the trapped mode resulting from the anti-phased electric dipole resonance in metamaterial and asymmetric plasmonic structures with strong magnetic response [5–11]. Due to its weak coupling to incident light, the scattering loss of such trapped mode is strongly eliminated accompanying with sharp asymmetric Fano profile with a high Q factor. The trapped mode is highly sensitive to local media making it suitable for applications such as biochemical sensor and modulator, in which a slight perturbation can result in a considerable change in reflection or transmission [6–11]. However, the intrinsic Ohmic loss and saturation effects in the metallic metamaterials preclude a number of applications such as lasing and switching where high modulation depth and Q factor are essential for the performance.
Recently, metamaterial based on dielectric with high refractive index has also been proposed as an alternative way towards future novel devices [12–22]. Due to the low loss nature of dielectric, the dissipation and power consumption can be greatly reduced opening more versatile route for constructing metamaterial with lower loss and higher Q factor. In near-infrared and visible range, strong magnetic resonance has been observed in silicon particles which tend to be an excellent choice for the building block of all-dielectric metamaterial for its outstanding property and mature manufacturing process [23–26]. Perfect absorber , reflector , and magnetic field-enhanced spectroscopy  based on silicon were studied extensively. More recently, it has been shown theoretically and experimentally that asymmetric silicon bars can also support trapped mode with Q factor exceeding 200, which is far superior to its counterpart of metal [30, 31]. However, for sensing applications, sensors with higher Q factor and multi-working frequencies are urgently needed, but up to now have not been demonstrated yet.
In this research paper, we proposed a novel scheme of bi-periodic silicon particle array (BSPA) containing two sets of silicon particle arrays (SPAs) interlacing with each other. Anti-phased electric dipole, magnetic dipole, and magnetic quadrupole collective resonances (CR) result in three distinct trapped modes contrasting to the previous studies, in which trapped mode simply originates from anti-phased electric dipole resonance. The interaction of such trapped modes and corresponding “bright modes” gives rise to three sharp asymmetric Fano profiles which can be further tailored effectively by tuning the radii of particles in the two arrays. Due to the lossless nature of silicon in near-infrared range, the Q factor of the electric dipole trapped mode can exceed 5400 while the modulation depth still remains 100%. The sensitivities of such BSPA as a three-channel refractive sensor can reach as high as 155nm/RIU, 725nm/RIU, and 190nm/RIU (RIU represents the refractive index units) for these three working frequencies which are on the same order of most plasmonic sensors that are intrinsically sensitive to local media [32, 33]. The structure also has potentials for multi-window electromagnetically induced transparency, filter, and modulator.
2. Formation of multi-trapped modes in BSPA
Figure 1(a) shows the scheme of proposed BSPA, two SPAs interlace with each other in X-Y plane and have a lateral displacement of half period in both directions. The periods of SPAs are (1000nm) in both X and Y directions and the radii are denoted by and . The BSPA can also be seen as a superlattice with unit cell indicated by red dash line in Fig. 1(a). The refractive index of silicon is 3.5 in near-infrared range  while the surrounding media is assumed to be air (). In the simulations, the incident light is normal to the array plane with polarization along the X direction. Periodic condition is used in each boundary of the unit cell to reduce the amount of computation. Reflection spectra are carried out using commercial finite difference time domain software package (FDTD solutions) and field distribution was plotted by Comsol Multiphysics using finite element method.
The scattering cross section of a single silicon particle can be calculated using Mie theory. Specifically, the scattered electromagnetic field is represented by an infinite series of vector spherical harmonics and , each weighted by amplitude coefficients and . In general, the field is a superposition of normal modes of fields which can be interpreted as the fields generated by electric and magnetic dipoles and multipoles with details in . In particular, for silicon particle with radius of 230nm in near-infrared range, the scattering property dominates by electric dipole, magnetic dipole, and magnetic quadrupole resonances with obvious separations in spectra . When silicon particles are arranged in ordered array, the lateral scattering is strongly depressed and the particles oscillate collectively to form lattice CR, accounting for the lattice collective excitation of the constituent particle members in the array . Figure 1(b) shows the reflection spectra of two SPAs, SPA with radius of 230nm indicated by green dash line has three distinct peaks. The peak at 1265nm originates from electric dipole CR while the peaks at 1620nm and 1163nm correspond to magnetic dipole and quadrupole CR, respectively. The resonant peaks have a blue shift as particle radius reduces to 225nm for these three resonances. In order to interpret the resonances clearly, we plot the electric field distribution of a unit cell of SPA at different resonant frequencies. The left side of Fig. 2 shows the X component electric field distribution at the plane of Z = 0, while the right side lustrates the electric field vector on the particle surface. At 1163nm, the SPA is on the magnetic quadrupole CR with electric field forming closed circles with different circulating directions at the top and bottom, which can be equivalent to two magnetic dipoles oscillating with opposed phase along Y direction. Due to the sharp resonance, magnetic quadrupole CR provides very high electric field enhancement of 50. At 1265nm, the electric fields with strong enhancement near the particle surface point from one pole to another pole mimicking the radiation pattern of electric dipole in the far-field and indicating the excitation of electric dipole CR. At 1620nm, magnetic dipole CR dominates the radiation with most of the energy confined within the particle due to the cavity-type mode nature. This resonant mode with strong circulating displacement is essential for the construction of magnetic response in all-dielectric metamaterial.
Furthermore, when the central wavelength of the CR approaches geometric resonant wavelength corresponding to the diffractive Raleigh anomalies, the coupling effects lead to an anomaly diffractive CR accompanying with significant narrowing of transmittance and reflectance spectra of the array, which has been studied both theoretically and experimentally [36–38]. Geometric resonant wavelength can be calculated by , where is the refraction index of the surrounding media and is the period of the particle array. At the wavelength of geometric resonance, there is a transition from a diffracted order propagating in the plane of the array to an evanescent diffracted order. Within the framework of Fano resonances, diffractive CR can be described as the interference between a broad CR due to radiation damping and a narrow resonance given by the Rayleigh anomaly. This interference gives rise to the asymmetric line shapes characteristic for Fano resonances. Due to the excitation of dipole and quadrupole CRs in silicon nanoparticle array, the coupling between geometric resonance and CRs are abundant . In this research paper, we concentrated on the coupling effects between different modes (dipole and quadrupole) in a bi-periodic silicon nanoparticle array with different radii. The geometric parameters are chosen to make sure the wavelengths of CRs are far from the geometric resonant wavelength (1000nm), so the coupling between geometric resonance and following considered trapped modes can be safely ignored.
In the case of BSPA, as shown in Fig. 1(b), three trapped modes appear at 1148nm, 1327nm, and 1648nm accompanying with three distinct Fano profiles. The origin of the trapped mode at 1648nm can be described as follows: Each SPA supports a magnetic dipole CR mode with resonant frequency determined by the particle radius. When two sets of SPAs are combined to form a BSPA, the CR in each SPA hybridize to generate anti-phased magnetic dipole CR mode and in-phased magnetic dipole CR mode through lattice coupling effects. For the anti-phased magnetic dipole CR mode, one SPA oscillates in synchronization with the incident light while the other in anti-synchronization to generate destructive re-radiated fields in the far-field and elimination of scattering loss. This resonant mode has a long lifetime due to its weak coupling to free space radiation and therefore appears to be “trapped” in the vicinity of the BSPA surface. For the in-phased magnetic dipole CR mode, both SPAs oscillate in synchronization or anti-synchronization with the incident light, thus, the re-radiated fields of two sets of SPAs interfere constructively and can be seen as a “bright mode” due to its effective coupling to incident light and enhanced radiation. The superposition of such “bright mode” and “trapped mode” also gives rise to the sharp asymmetric Fano profile in reflection.
Figure 3 shows the magnetic field amplitude distribution and electric field vector on the particle surface of a unit cell at different wavelengths near the magnetic dipole trapped mode. The silicon particles can be treated as magnetic dipoles and oscillate collectively with the incident light. The coupling between the particles is relatively week and the fields mainly concentrate inside the particle forming closed circle. When the incident wavelength is 1600nm (Fig. 3(a)), two SPAs oscillate in phase and the electric field in the SPA rolls towards the same direction generating constructive re-radiated fields in the far-field. SPA with radius of 225nm dominates the radiative property due to its shorter resonant CR wavelength. At 1648nm (Fig. 3(b)), two SPAs oscillate with phase difference of and possess inversed electrical field pattern. Due to the same strength of resonant amplitude, the re-radiated fields of two SPAs completely cancel each other in the far-field and the reflection is totally eliminated. One can also notice the enhanced magnetic field in the center of each particle resulting from strong circulating displacement current. At 1700nm (Fig. 3(c)), two SPAs also oscillate in phase. Different from that at 1600nm, in this case, SPA with radius of 230nm plays the leading role for the radiation.
The response of BSPA can be interpreted by a system of harmonic oscillators. The oscillating SPAs are represented by two large harmonic oscillators with different resonant frequencies of and as shown in Fig. 4(a). Two oscillators are connected to solid walls, while also couple to each other through a third small oscillator which accounts for the far-field interference of the two SPAs. The incident electromagnetic wave can be treated as the external force exerted on the two large oscillators, while friction force exerted on the small oscillator stands for the scattering losses in the BSPA. Normally, the small oscillator will be forced by the heavy oscillator to oscillate for all driving frequencies, leading to energy dissipation through the friction of small oscillator, in other words scattering loss. However, due to the difference in resonant frequencies, an anti-symmetric mode can be established at a specific frequency. In such mode two heavy oscillators oscillate with same amplitudes and opposite phases resulting in net zero force on the small oscillator. Thus, the small oscillator remains still and the scattering losses in the system are strongly minimized; all the energy will be stored in the oscillations of the large oscillators.
Two combined SPAs also hybridize to form anti-phased magnetic quadrupole CR and electric dipole CR at 1148nm and 1327nm with total elimination of reflection. Figure 4(b) shows the electric field vector on the particle surface of BSPA at 1148nm. Two SPAs are on the magnetic quadrupole CRs with electric field mainly confined inside the particles. The electric field directions of the central particle and outer particle oppose to each other resulting from the phase difference of the two resonant SPAs. At 1327nm shown in Fig. 4(c), the central particle and outer particle possess the same electric field pattern of electric dipole indicating the excitation of electric dipole CRs in both SPAs. Due to the inversed phase in central and outer particles, the radiative fields from the two SPAs strongly cancel each other in the far-field and the energy pumped on the BSPA accumulates in the resonance of particles leading to strong field enhancement.
3. BSPA based three-channel refractive sensor
The lineshapes of the Fano resonance can be easily tailored by tuning the radius of the silicon particle. Figure 5(a) shows the evolution of reflection spectra for electric dipole trapped mode as the radius increasing from 220nm to 230nm while is fixed at 230nm. The modulation depth defined as remains 100% for and , while the linewidth becomes sharper and Q factor increases as shown in Fig. 5(b). The Q factor is calculated by , where is the central wavelength of the resonance and is the full width at half maximum. It is noted that the modulation depth of the three trapped modes always remains 100% for any radius which is very different from that of metal where 100% modulation depth and narrow linewidth can hardly be achieved at the same time. For the case of , there is no difference between the resonant frequencies of two SPAs and the “trapped mode” cannot be directly excited by normally incident field, thus, the Fano resonance also disappears. For plasmonic structures made of metal, due to the limitation of scattering loss and Ohmic loss, the Q factor is usually on the order of 10, precluding a number of important applications, such as lasing, biochemical sensing, SERS, and enhanced nonlinear effects. Because of the strong elimination of scattering loss and lossless nature of silicon in infrared range, the BSPA structure possesses a very high Q factor up to 5400 for , which can be further improved by reducing the difference of radii between two SPAs.
The BSPA can work as a three-channel refractive sensor. Figure 5(c) shows the reflection spectra under surrounding media of different indexes for electric dipole trapped mode. The spectrum has a red shift as the index increases, while the lineshape remains unchanged. For magnetic dipole and quadrupole trapped modes, the same trend of spectrum shift also appears and not shown here. Figure 5(d) shows the resonance wavelength shifts under different refractive index changes for three trapped modes. The sensitivity of the BSPA structure describing the ratio between resonance wavelength change and refractive index change can be obtained from Fig. 5(d) by calculating the slop of the lines corresponding to the resonance shift and index change. The sensitivity can also be qualitatively represented by:32, 33]. However, for magnetic dipole and quadrupole trapped modes, the electric fields are mainly confined inside the particle rather than the surrounding media due to the cavity-type mode nature of these modes. So that the sensitivities of these two modes are relatively low at 190nm/RIU for magnetic dipole trapped mode and 155nm/RIU for magnetic quadrupole trapped mode.
In general, the refractive sensor detects the relative intensity change at a fixed wavelength induced by a small index change , the performance greatly depends on the linewidth and modulation depth of the resonance. The incomplete destructive interference and broad linewidth strongly decreases the FOM value which is introduced by Becker et al to describe the performance of plasmonic sensors to measure the index change in terms of intensity change . The FOM is defined as , where represents the reflection intensity, is the intensity change caused by a small index change . The distinct advantage of BSPA is its ability to achieve 100% modulation depth and narrow linewidth simultaneously, thus, to improve FOM greatly. The FOM of the three working frequencies are 23, 2900, and 93 at corresponding original resonance dips. The values are much larger than its counterparts of metal usually less than 10.
For experimental realization, it is hard to fabricate ordered silicon sphere array under present technology. An alternative way is to replace the silicon spheres by silicon disks which also support Mie-like resonances. The silicon disks can be readily fabricated by electron beam lithography followed by ion beam etching process. Further experiment related to bi-periodic silicon disk array is currently in preparation.
In this research paper, we studied, for the first time to our knowledge, multi-trapped modes originating from far-field interference of lattice collective resonance in all-dielectric particle arrays. Very different from that of metamaterial made of metal, in such two sets of particle arrays made of silicon the trapped modes originate not only from the anti-phased electric dipole CR, but also from the anti-phased magnetic dipole and quadrupole CR. The lineshape can be tailored simply by tuning particle radius; in particular, the Q factor of trapped mode caused by anti-phased electric dipole can exceed 5400. The structure can be used as a three-channel refractive sensor with sensitivities up to 155nm/RIU, 725nm/RIU, and 190nm/RIU for three working frequencies. Also, the FOMs are orders higher than most plasmonic sensors.
The authors thank financial support from the National Key Basic Research program of China under Grant No. 2013CB328702 and the National Natural Science Foundation of China under Grant Nos. 50836002 and 51176041.
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