1. Introduction

Metamaterials built by artificial subwavelength structures have attracted enormous interest in recent years due to their unique electromagnetic properties, which are not available in natural materials [1,2]. Unit cells such as metallic split ring resonators [3], continuous wires [4], fishnets [5] and cut-wire pairs [6] have been proposed as building blocks for metamaterials. Functional devices based on metamaterials, such as superlenses [7], invisibility cloaks [8], perfect absorbers [9], polarizers [10], and filters [11–16] have been reported in previous works. With regard to metamaterial-based filters, some integrated transmission line filters, including band-pass and band-stop filters, have been designed in the microwave range [11–14]. Recently some works have been focused on THz band-pass filters, which filtrate THz electromagnetic signals in free space [15,16]. However, to the best of our knowledge, there is no related work being carried out on space transmission band-stop filters for the suppression of undesired responses or for the elimination of interfering signals. Actually, metamaterials with either electric or magnetic responses are inherent transmission band-stop filters owing to their transmission dips. However, the bandwidth is always narrow due to the resonant nature of the response, which greatly limits certain applications in which wider forbidden bands are required. In addition, the coupling effect between the unit cells of metamaterials is an important topic that triggers great interest [17–19]. In this paper, we demonstrate that a band-stop filter with a broad bandwidth can be achieved by making use of the plasmon hybridization of nanostructures with periodic coupled cut-wire pairs in the optical frequency range. The bandwidth of the filter is tunable by coupling the adjacent unit cells, which can be analyzed qualitatively by a simple RLC coupling circuit model. The tunable broadband optical filter of the proposed structure can readily be scaled up to the structures that are working in terahertz and microwave frequency ranges as well.

2. Structures and Design

For its simplicity of design for modulation of permeability in the optical range with the participation of surface plasmons [6,20,21], a cut-wire pair structure is used in this paper as the basic nanostructure of an optical band-stop filter, as shown in Fig. 1 . The unit cell has the dimensions of $Px=300$nm and $Py=120$nm in the x and y directions. The single metal cut-wire has the dimension of $a=160$nm and $b=40$nm. We introduce a curvature of 20 nm at the corners of the wire to approach the actual fabrication [22,23]. The symmetrical metal wires and the central dielectric layer have thicknesses of $t_m=20$nm and $t_d=150$nm, respectively. The structure with the aforementioned dimensions can be fabricated using focus-ion-beam milling or electron-beam lithography [24]. The simulation is performed by using commercial software (CST Microwave Studio), where a plane wave polarized in the x direction illuminates normally to the structure from the top, as depicted in Fig. 1. Corresponding periodic boundary conditions are considered. A Drude model is used to describe the realistic characteristics of Ag at optical frequencies where the high-frequency bulk permittivity is${\epsilon }_{\infty }=4.2$, the corresponding plasma frequency is ${\omega }_{\mathit{\text{p}}}=1.346\times {10}^4$THz, and the collision frequency is $\gamma =96.17$THz [25]. Quartz with refraction index $n=1.46$is adopted as the central dielectric layer material.

 

Fig. 1 Schematic of the coupled cut-wire pair structure. (a) Top view and (b) side view located at the dashed-dotted line marked in (a).

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3. Simulation Results and Discussion

Figure 2(a) shows a simulated transmission of the coupled cut-wire pairs by a black solid line with the aforementioned dimensional parameters. For reference, the transmission of the nanostructure with the unit cell of a single cut-wire layer is also presented (red, short dashed line). Compared to the case of a single cut-wire layer nanostructure, the transmission stop band of the coupled cut-wire pair nanostructure is expanded due to plasmon hybridization, which originates from longitudinal coupling (parallel to the transmission direction). The two resonances of the anti-symmetrical plasmon mode (corresponding to the lower transmission minimum at $f_h=365.7$THz) and the symmetrical plasmon mode (corresponding to the higher transmission minimum at $f_e=409.4$ THz) are located at each side of the bare plasmon mode resonance at $f_0=387.1$THz [18,26]. As depicted in Fig. 2(a), the two transmission stop bands resulting from plasmon hybridization are combined to be a continuously broader stop band. The retrieved frequency-dependent material parameters are given in Fig. 2(b). Magnetic resonance and dielectric resonance take place at f_h and f_e, respectively, which is consistent with the corresponding hybrid anti-symmetrical and symmetrical plasmon resonance positions. Thus a negative μ band and negative ε band are realized in the vicinity of the corresponding resonant frequencies. The closed two single negative bands bring a wider-range impedance z mismatch between the structure and free space, as shown in inset of Fig. 2(b), which finally shows the broader transmission stop band. Different from the previous works [6,20] that show the overlap of the negative μ band and negative ε band to achieve a negative refractive index of the metamaterials, our design brings the two single negative bands close in order to expand the stop band. Meanwhile, the dielectric thickness t_d, plays a key role in obtaining a broad and continuous stop band, because it primarily determines the hybridized electric and magnetic resonant frequencies of the cut-wire pair [18,26]. It should be noted that both of the hybridized resonances at f_h and f_e are much stronger than the bare plasmon mode resonance at f₀, thus leading to a steep transformation between pass and stop bands, as shown in Fig. 2(a). The transmission out of the resonant band is higher than 0.9 with the consideration of the dielectric loss in silver.

 

Fig. 2 (a) Transmission spectra for the coupled cut-wire pairs and single cut-wire layer nanostructures. (b) Retrieved frequency dependent Real(μ) and Real(ε) for the coupled cut-wire pairs. The insert shows the retrieved frequency dependent Real(z).

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The bandwidth of the stop band can be tuned by the transverse coupling (vertical to transmission direction) between the unit cells without a distinct change in the shape of the transmission spectra. Recently, novel properties have been reported by introducing coupling between the unit cells. It is pointed out that both electric and magnetic dipole couplings contribute to the resonant properties of single-layered split-ring metamaterial arrays, and the effects on bandwidth of transverse couplings are also discussed by analyzing the effective extinction cross section in [19]. We demonstrate that the bandwidth of our band-stop filter can be modulated by both magnetic coupling in the y direction and electric coupling in the x direction.

Figures 3(a) and 3(b) reveal that the stop band is expanded with a decrease of space Px and Py between the unit cells. The 3 dB bandwidth, which is defined as the range between f_high (the higher-frequency point at which 50% energy transmits through the structure) and f_low (the lower-frequency point at which 50% energy transmits through the structure) in this paper is increased from 56.6 THz to 147.7 THz with a decrease of Py from 360 nm to 90 nm, as shown in Fig. 3(c). Similarly, it is increased from 108.2 THz to 182.2 THz with a decrease of P_x from 380 nm to 180 nm, as shown in Fig. 3(d). The shift of the central frequency, defined as$f_{cen}=(f_{high}+f_{low})/2$, is opposite in the case of a decrease of P_y and P_x, namely, it decreases with P_y, while it increases with P_x, as shown in Figs. 3(c) and 3(d). The shift tends to saturate at a larger space for both cases because of the negligible coupling.

 

Fig. 3 Transmission spectra for (a) different Py values at a fixed$Px=300$nm, and (b) different Px values at a fixed$Py=120$nm, where other parameters are fixed as$a=160$nm, $b=40$nm, $t_m=20$nm, and$t_d=150$nm. Resonant frequency and 3 dB bandwidth of the coupled cut-wire pairs nanostructure as functions of (c) Py and (d) Px.

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It can be concluded unambiguously that both stop band expansion and central frequency shift resulted from the coupling of the unit cells. The resonant frequency and bandwidth of the single cut-wire layer nanostructure as functions of Py and Px are also plotted in Figs. 4(a) and 4(b), respectively, where we can find that the variations of resonance frequency and bandwidth with Py and Px for the single cut-wire layer nanostructure are similar to coupled cut-wire pair nanostructures. The resonant frequency of the single cut-wire layer nanostructure is overlapped approximately with the central frequency of the coupled cut-wire pairs nanostructure for the same Py and Px, as depicted in the insets in Figs. 4(a) and 4(b). It indicates two unique properties: one is that the transverse coupling between unit cells has little influence on the symmetrical plasmon hybridization between the cut-wire pairs; the other is that the stop band shift of the cut-wire pairs with transverse coupling originates from the shift of bare plasmonic resonant frequency for single cut-wire layer nanostructures. For conveniently comprehending the origin of the stop band shift and expansion properties of coupled cut-wire pairs, an equivalent RLC circuit model is introduced to analyze the resonant characteristics of the single cut-wire layer as shown in Fig. 4(c). The resonant frequency and the bandwidth can be calculated as follows [27]:

 

Fig. 4 Resonant frequency and 3dB bandwidth of the single cut-wire layer nanostructure as functions of (a) Py and (b) Px. (c) and (d) The equivalent RLC circuits for magnetic and electric couplings. Insets in (a) and (b) are the resonant frequencies (RF) of the single cut-wire layer nanostructure as a function of central frequencies (CF) of the coupled cut-wire pair nanostructures at different Py and Px values.

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Stop band expansion and resonant frequency blue shift with the decrease of Py are caused by the magnetic coupling in the y direction. The incident light excites electric current I in the metal wires, which stimulates a directionally opposite magnetic field in the intermediate areas between the adjacent cut wires, as shown in Fig. 4(c). Therefore, an extra equivalent negative mutual inductance (-Lc) is introduced in the equivalent RLC circuit because of the cancellation of the magnetic field. Meanwhile, the absolute value of mutual inductance increases with the enhancement of coupling, which reduces the total inductance in the circuit. As a result, both the resonant frequency and the bandwidth increase with Py decreasing, based on Eqs. (1) and (2). On the other hand, the electric coupling between the adjacent unit cells in the x direction accounts for the stop band expansion and the red shift of the resonant frequency. An equivalent parallel capacitance C_c is introduced in the equivalent RLC circuit because of E-field coupling. The increasing capacitance with the enhancement of coupling decreases the resonant frequency, as shown in Fig. 4(b). It should be noticed that the silver in the optical frequency range induces energy loss. The enhanced E-field around the silver by the strong electric coupling greatly increases the loss [22,23]. The extra coupling resistance R_c increases the total resistance R_total in the RLC circuit, as sketched in Fig. 4(d). The increase of R_total therefore expands the bandwidth, according to Eq. (2), which agrees with the results shown in Fig. 4(b). From analysis of the single cut-wire layer nanostructure, we can deduce that the stop band shift of the coupled cut-wire pair nanostructure originates from a decrease of equivalent inductance by magnetic coupling in the y direction and the increase of equivalent capacitance by electric coupling in the x direction, respectively. The stop band expansion of the coupled cut-wire pair nanostructure originates from a decrease of equivalent inductance by magnetic coupling in the y direction and an increase of equivalent resistance by electric coupling in the x direction, respectively. However, the enhanced electric resistance also decreases the transmission in the transmission band. As plotted in Fig. 3(b), when Px equals to 180 nm, the transmission at the transmission band decreases to about 0.8.

4. Conclusions

In summary, we demonstrate a band-stop filter in the optical frequency range by a coupled cut-wire pair nanostructure. Wider bandwidth and steeper transformations between pass and stop bands are realized by making use of plasmon hybridization as compared to bare plasmonic resonance. The transmission out of the resonant band is higher than 0.9. The bandwidth of the filter is tunable by the magnetic and electric couplings between adjacent unit cells. This outstanding property provides good guidance for metamaterial design that works in broadband frequencies. The origin of the coupling effect is studied in simple equivalent RLC circuit models. The band-stop filter with tunable bandwidth is a great supplementation for application in metamaterials.