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Abstract
A broadband negative-refractive index metamaterial, composited of two square-shaped aluminum apertures with different title angles etched on both sides of polyimide substrate, has been proposed. The calculated results indicate that the coupling between the titled and untitled square-shaped apertures is more conducive to the high-pass and broadband transmission than the combination of two untitled structure. The high-pass band of metamaterial increases from 0.43 to 0.76 THz with the increasing aperture width from 30.0 to 35.0 μm, meanwhile, the maximum bandwidths of double-negative refraction and negative-refractive index can reach up to 0.58 and 0.78 THz, respectively. The highpass transmission mainly locates at the double-negative refraction region. Furthermore, the negative refraction of the NIMs has been verified by the classical method of the wedge, and the results show many unique properties of the symmetric slab waveguide based on the proposed NIM.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Terahertz (THz), defined by the frequency range of 0.1 to 10 THz [1], has many unique electromagnetic properties. In recent years, THz technology has attracted great attention in various fields, such as communication [2], imaging [3], clocking [4], chemical sensing [5] and biomedicine [6,7] etc. However, there are rare nature materials exhibiting electromagnetic response to THz wave, limiting the development of THz applications. Fortunately, metamaterials (MMs) can provide an effective routine to solve the problem, and THz MMs have many unique properties, examples of which are cloaking [8], sensing [9] and negative refraction [10]. The negative index MMs (NIMs) have many special behaviors like negative refraction and backward-wave media, which can be applied to some important fields, for instance, slow light device [11], perfect lens [12] and superlenses [13] etc. The term MM referred to negative refractive index (NRI) can be divided into double-negative refraction (DNR) with simultaneous negative ε1 and μ1 and single-negative refraction (SNR) with negative ε1 or μ1 based on the condition ε1μ2 + ε2μ1 < 0 (the permittivity ε = ε1 + iε2, the permeability μ = μ1 + iμ2) [14].
The design of THz NIMs is often based on cut metal wire [15], fishnet structure [16–19] and three-dimensional (3D) standing structure [20–22] etc. However, for the NIMs composed of cut metal wire or fishnet structure, the transmissions are relatively low, and the bandwidth of DNR is narrow due to the weak magnetic response. Although the standing structure can induce strong magnetic response with THz wave, the bandwidth of DNR is always restricted owing to strong bianisotropy arising from the asymmetrical structure. Therefore, the practical application of these NIMs is limited.
In this paper, a highpass transmission THz NIM with broadband DNR has been designed. The NIMs consist of square shaped perforations in two aluminum films separated by a polyimide layer, where the inside square-shaped aperture is tilted 45 degrees and the outside one is not tilted. The influence of the structural parameters on the properties of the NIMs including transmission, permeability, permittivity and refractive index has been investigated, and the corresponding physical mechanism has also been analyzed in detail. The results indicate that the broadband highpass transmission mainly corresponds to the broadband DNR region. Finally, the negative refraction of the NIMs has been confirmed by the classical method of the wedge, and the application of symmetric parallel waveguide has further been discussed as well.
2. Design of the THz NIMs and its mechanism
The designed THz NIMs are composited of square shaped aluminum perforations etched on both sides of polyimide substrate, as shown in Fig. 1. In order to induce a broadband high-pass band, the inside square-shaped aperture is tilted 45 degrees while the outside one is not tilted. In order to ensure that the broadband DNR region mainly corresponds to the broadband highpass transmission, the optimized geometric parameters of the NIMs are as follows: p = 50.0 μm, tsub = 9.2 μm, tm = 0.2 μm, a = 35.0 μm, b = 24.0 μm, c = 17.0 μm, d = 8.0 μm. The NIMs can be fabricated by the method of focused ion-beam milling or standard electron-beam lithography because these methods enable the production of microstructures with feature sizes in smaller than 300 nm [23,24]. Firstly, a resist film is spin coated on top of the polyimide substrate, and then the top layer is patterned with electron beam lithography on the resist. After development of the resist, the resist residues in exposed areas are removed. Then, an aluminum layer with thickness of 0.2 μm is sequentially deposited, followed by the removal of the resist. Finally, the fabrication of the another side structure follows the same procedure to that of the top layer [25] until two aluminum layers are patterned to achieve the structure shown in Fig. 1.
Fig. 1 The Schematic illustration of the NIMs, E, H and k represent the electric field, magnetic field and propagation direction of the incident THz wave.
The electromagnetic response of the double-layered structure is numerically simulated by CST Microwave Studio based on the finite difference-time domain (FDTD). For all simulations, the magnetic and electric fields are along the x- and y-axes, respectively, and open space boundary is used in the z-axes (that is, the propagation of incident plane wave is along the z direction). The loss aluminum films with conductivity of 3.56×107 S/m and the loss free polyimide substrate with effective permittivity of 3.5 are adapted in simulations. Then, the retrieved constitutive parameters of the NIMs can be calculated by the standard retrieve method [26]. The effective refractive index (n) and impedance (Z) of the structure can be written as [26]:
where k, t, S21 and S11 are wavevector, sample thickness, transmission coefficient and reflection coefficient. Then, the effective permittivity (ε) and permeability (μ) can be calculated by [26–28]:3. Results and discussions
In order to investigate the physical mechanism of the THz NIMs, the structure is split into the untitled outside square-shaped aperture (A1), the titled inside square-shaped aperture (A2) and the untitled inside square-shaped aperture (A3). The MMs with different double-layered square-shaped aluminum apertures and polyimide substrate are calculated, as shown in Fig. 2.
Fig. 2 (a) The normalized transmission spectra of the double-layered MMs with different structures. (b) The electric field distributions of NIMs.
Figure 2(a) shows that the MMs composed of A1 have a sharp transmission at 4.70 THz, and similarly, the sharp transmission excited by A3 locates at 4.42 THz. However, there is a broadband highpass transmission between 4.33 and 4.39 THz for the MMs composed of A2. For the MMs composited of A1 and A3 (marked by A13), there is a transmission dip. If the inner structure of the MMs is changed to titled 45-degree square-shaped apertures (that is, composed of A1 and A2 indexed as A12), a broad and high-pass transmission spectrum is obtained arisen from coupling of A1 and A2 [29]. The slope efficiencies of the rising edge and the trailing edge of the broadband high-pass transmission are 135%, 284% (based on ∂T/∂f [30]), which can be manipulated as a bandpass filter due to its high-performance filtering property. In Fig. 2(b), the electric field distributions associated with f1, f2-1, and f3-1 are represented to study the resonance mechanism of NIMs. It is obvious that the electric field is mainly concentrated on the inside square-shaped aperture and the outside square-shaped aperture at f1 and f3-1, respectively. At f2-1, there is strong coupling between the inside and outside square-shaped apertures, as the electric field distributions illustrated. Thus, the high-pass band of NIMs is caused by the coupling of the corresponding electromagnetic response of the two structures. In order to study how the structural parameter a affects the performance of NIMs, the normalized transmission spectra, the electric and magnetic field distributions of the THz NIMs with different lengths a are shown in Fig. 3.
Fig. 3 (a) The normalized transmission spectra of MMs with different lengths a, the solid and dotted curves with the same color represent the MMs with the same value of a. (b) The magnetic field distributions and (c) the electric field distributions with a = 30.0 μm, a = 32.0 μm, a = 34.0 μm, a = 35.0 μm.
Figure 3(a) indicates that the high-pass bandwidth of NIMs can be tuned by the configuration of aperture parameter a. When the parameter a is smaller than 30.0 μm or greater than 35.0 μm, the transmission decreases. Thus, the range of structural parameter a is limited between 30.0 and 35.0 μm, correspondingly, the high-pass band of NIMs can be tuned between 0.43 and 0.76 THz. It is clear that the trailing edge of the high-pass bandwidth exhibits blue-shift with the increasing a, but its rising edge remains almost unchanged. Meanwhile, the resonance and transmission of the MMs composed of A1 are increased. Therefore, the higher frequency of the NIMs is determined by the outside untitled aperture rather than the inner titled aperture. To investigate how the parameter a affects resonance, the magnetic and the electric field distributions of NIMs are shown in Figs. 3(b-c), which demonstrates that the magnetic field distributions of the untitled aperture are enhanced with the increasing of a, whereas the coupling between the two apertures decreases simultaneously. Consequently, the magnetic response excited by the untitled aperture is enhanced by increasing the structural parameter a, resulting in the broadening of high-pass band of the NIMs. However, the coupling between the two apertures decreases and further makes the performance worse. Furthermore, Fig. 4 gives the retrieved constitutive parameters of the NIMs with different lengths a.
Fig. 4 The retrieved parameters of the NIMs with (a) a = 30.0 μm (b) a = 32.0 μm (c) a = 34.0 μm (d) a = 35.0 μm, respectively. The blue and pink regions represent the ranges of SNR with ε1 < 0 and μ1 < 0, respectively. The overlap area of two regions reveals the ranges of DNR with ε1 < 0 and μ1 < 0.
The DNR band is highlighted by the overlap area, where the blue and pink regions in Fig. 4 represent the bands of SNR. A broadband NRI located between 3.90 and 4.55 THz is achieved when the length a is 30 μm, which is composed of two SNR bands and a DNR band. It is worth pointing out that the transmissions are high at the DNR region, while the SNR corresponds to the scope of low transmission due to the considerably high loss [14], as shown in Fig. 4(a). The bandwidth of the DNR increases from 0.33 to 0.58 THz with the increasing of a, as the pink region shown in Fig. 4. The phenomena are mainly ascribed to the enhancement of the magnetic response, resulting in the broadening of the negative permeability in high frequency [31]. Consequently, the maximum bandwidths of NRI and DNR are 0.78 and 0.58 THz for a = 35.0 μm, respectively. Figure 5 gives the calculated transmission contour maps of the NIMs at f1, f2 and f3 for a = 30.0 μm, where the refraction of f2 is DNR but the refraction of f1 and f3 are SNR with μ1 < 0 and ε1 < 0, respectively.
Fig. 5 The contour maps of electric field Ey in x-z plane at (a) f1 = 4.00 THz, (b) f2 = 4.28 THz and (c) f3 = 4.50 THz at a = 30.0 μm.
Figure 5 implies that the propagation properties of electric fields at 4.00 and 4.50 THz are similar, and their transmission is lower than that of NIMs at 4.28 THz. Therefore, the NIMs with DNR have high transmission and low loss. Further, the classical method of wedge sample is employed to demonstrate the negative refraction [32], and the prism sample is modeled in Fig. 6. The unit cell of NIMs wedge sample in x direction is connected, while it is separated by vacuum with 2 μm in z direction to avoid the interference between the adjacent unit cells in propagation direction. It is noteworthy that the NIMs wedge only has one unit cell in y direction because the structure of NIMs is symmetric along the x and y directions.
Fig. 6 The electric field distributions of the refracted THz radiation at (a) 4.39 THz and (b) 4.41 THz for the NIMs with a = 35.0 μm.
It is clear that the refracted THz wave is bent to the same side of the THz incident wave in the normal surface when THz wave crosses the oblique interface, and the refracted angles θr of the wedge sample are −21.5° at 4.39 THz and −19.2° at 4.41 THz, respectively, as the electric field distributions shown in Fig. 6. The incident angle is 12.6° according to the relation θi = arctan (pz / px) [33]. According to the Snell’s law, i.e., n = sin θr / sin θi, the real parts of the refractive indexes Re(n) of the NIMs at 4.39 and 4.41 THz are −1.68 and −1.51, respectively. The retrieval Re(n) of the NIMs at 4.39 and 4.41 THz are −1.68 and −1.54, which are consistent with that estimated by the Snell’s law. Thus, the negative refractive characteristic of the proposed NIMs is verified, and has some potential applications in THz slow light device, perfect lens and waveguide etc. Hereafter we take the application of this NIMs in THz waveguide for example. The k1x and k2x are the transverse wave numbers along the x direction in free space and NIMs, respectively, which can be written as [34,35]:
where β and ω are the propagation constant and angular frequency of THz wave, respectively. c is the speed of light in vacuum. Therefore, the transverse electric (TE) guidance condition of even m mode can be written as:For the odd m mode,Accordingly, the TE guided modes of the symmetric slab waveguide with different thicknesses d are calculated at 4.39 and 4.41 THz in Fig. 7, respectively.Fig. 7 The TE guided modes of the symmetric slab waveguide with different thicknesses at (a) 4.39 THz and (b) 4.41 THz for the NIMs with a = 35.0 μm.
The inset of Fig. 7 shows the schematics of symmetric NIMs slab waveguide with different thicknesses d in free space, where the gray and violet areas are the free space and NIMs, respectively. The solid and dashed blue curves represent the even and odd modes with different m, respectively. It is clear that the TE1 mode of negative index waveguides has cutoff frequency. The red-, green- and brown-dashed curves are ρ = (k1xd)2 + (k2xd)2 with different values of d. The guided mode solution is expressed by the intersection point of the curve. Thus, there is no TE0 mode no matter what the value of d is, because the value of the transvers wave number k1xd is negative, while the ρ is always positive. As interaction shown in Fig. 7, the two modes for the TE2 mode are supported at 4.39 THz, whereas the NIMs waveguide do not support any TE wave at 4.41 THz when the thickness d is 24 μm. For d = 33 μm, the NIMs slab waveguide at 4.39 THz supports TE2 and TE3 modes simultaneously, and the NIMs slab waveguide at 4.41 THz only support TE2 mode. If d increases to 42 μm, the TE2, TE3, TE4 and TE5 modes are supported at 4.39 THz, whereas the TE2, TE3 and TE4 modes are supported at 4.41 THz. Compared with the conventional waveguide, the NIMs slab waveguide don’t support TE0 mode, and TE1 mode has cutoff frequency. In addition, the two co-existing modes for a mode can be supported for the NIMs waveguide. For the absence of the fundamental modes, the conventionally fast mode disappears and the result is the fact that the NIMs slab waveguide does not support fast mode. Therefore, the electromagnetic wave can be bounded in the waveguide during the propagation [35,36]. Because the TE1 mode has cutoff frequency, the NIMs slab waveguide can break the limitation of the conventional single-mode waveguide [36]. Double degeneracy of the mode is a selection criterion for the filter with high Q-value [37,38]. It overcomes the restriction that the cavity size should be increased when high Q-value is desired. Consequently, the NIMs with simple structure have a broad and high-pass DNR band, which has many advantages in symmetric slab waveguide.
4. Summary
A novel broadband NIM based on square-shaped perforations separated by polyimide layer substrate has been proposed. The high-pass band of NIMs is excited by the combination of double-layered titled and untitled square-shaped apertures, whose physical mechanism is investigated by calculating and further analyzing the electric field distribution. When the width of the outside square-shaped aperture increases from 30.0 μm to 35.0 μm, the maximum bandwidths of high-pass transmission can be tuned from 0.43 to 0.76 THz. The magnetic and electric field distributions of NIMs demonstrates that the broadening of high-pass band is caused by the blueshift of resonance frequency excited by the outside square-shaped aperture. Meanwhile, the maximum bandwidths of DNR and NRI are 0.58 and 0.78 THz when the width of outside aperture is 35.0 μm, and the negative refraction of the NIMs is verified by the classical method of wedge. Simulation results indicate that the NIMs slab waveguide has many properties different from the conventional slab waveguide, for example, the absence of the fundamental modes, double degeneracy of the modes, and TE1 mode with cutoff frequency. Due to these properties, the NIMs slab waveguide has many advantages in practical applications.
Funding
China Innovative Talent Promotion Plans for Innovation Team in priority Fields (Grant No.2014RA4051) and the Fund of Tianjin Key Laboratory of Optical Thin Film (Grant No. kjwx170620).
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