Broadband negative-refractive index terahertz metamaterial with optically tunable equivalent-energy level

Fang Ling; Zheqiang Zhong; Yuan Zhang; Renshuai Huang; Bin Zhang

doi:10.1364/OE.26.030085

1. Introduction

Metamaterials (MMs) composing synthetic elements whose scale is smaller than the operating wavelength have attracted numerous attention due to their novel electromagnetic properties that is unattainable in nature, examples of which are negative refraction [1], backward wave propagation [2] and coordinating transformation materials [3,4]. Therefore, MMs have wide applications ranging from transformation optics [5], nanoelectronics [6], absorbers [7], telecommunications [8], sensors and diagnostics [9,10] and automotive [11] to invisibility cloaks [12]. Since the negative-refractive index metamaterials (NIMs) at microwave band were experimentally demonstrated by Shelby, Smith and Schultz [13], the corresponding investigation booms due to its unusual properties. For example, NIMs can be used for restoring evanescent wave to realize imaging beyond the diffraction limit and allow a flat lens. According to the basic mechanism of NIMs condition ε₁μ₂ + ε₂μ₁ < 0 (the permittivity ε = ε₁ + iε₂, the permeability μ = μ₁ + iμ₂) [14], the double-negative refraction (DNR) NIMs with simultaneous negative ε₁ and μ₁ and single-negative refraction (SNR) NIMs with single negative ε₁ or μ₁ can be designed. Therefore, a large amount of structures have been developed and characterized to realize negative refractive index (NRI) in terahertz (THz)region, such as asymmetrical metal patches on both sides of dielectric substrate [15,16], symmetric layered fishnet [17], symmetrically aligned paired cut metal wires [18], reverse bi-layer S-shaped strings [19], etc. In general, a planar resonator provides a simple and efficient approach to manipulate the electromagnetic wave, but mainly generates a relative strong electric response which limits its magnetic response. The most common strategy of NIMs design is thus the combination of the electric and magnetic resonators with the overlapping electric and magnetic responses. However, the coupling between electric and magnetic responses always exhibits destructive interference, which makes the realization of broadband, tunability, polarization insensitivity, wide incident angle and highpass transmission somewhat difficult. Interestingly, Shuang et al. [20] experimentally demonstrated that the magnetic response could be excited by a pair of metal strips separated by a dielectric layer with parallel incident magnetic field, though the coupling theory of multilayer metasurface was still underdeveloped. Later, T. Li et al [21] successfully analyzed the physical mechanism of the stacking NIMs by the use of LC-circuit model, which demonstrated that the magnetic polaritons are induced by the strong coupling effect between the layered unit cells. In addition, Ebbesen et al. [22] experimentally demonstrated that a periodic array of subwavelength metallic holes at resonant frequency displayed highly unusual zero-order transmission spectrum without the presence of diffraction, which was several orders of magnitude higher than the theoretical value [23] at optical region. Afterwards, a quantity of NIMs with easy structures, highpass transmission and low loss based on staking metallic holes has been proposed [24–27].

In this paper, an optically tunable THz NIM has been designed and characterized. The NIMs are composed of metallic rings and ring-shaped silicon (Si) apertures, which are arrayed on the both sides of the Teflon substrate. Thus, the symmetry structure in the incident plane implies that the NIMs are wide incident angle and insensitivity to polarization. In order to study the tunability of NIMs, a theoretical model is established to calculate the conductivity of Si with different photoexcitation time and bulk Si thicknesses. Besides, the LC-circuit model is adopted to study the tunable physical mechanism of NIMs, which indicates that the energy level of NIMs can be dynamically tuned from two equivalent-energy levels to four energy levels with the increasing of Si aperture conductivity. Simultaneously, the transmission of NIMs can be tuned from lowpass into highpass, and the abrupt phase change at lower frequency is continually increasing. As a result of that, the NRI is dynamically controlled by the pump power, and the classical wedge sample of NIMs verifies the tunable negative refraction. Moreover, the phase flow indicates that the phase velocity of SNR is negative.

2. Theoretical model and discussion

Assuming there is a homogeneous bulk Si with length L and width W, and the incident pump laser is a Gaussian pulse expressed by:

I (t) = \frac{I_{0}}{\sqrt{2 π} σ} e^{- {(\frac{t - t_{0}}{\sqrt{2} σ})}^{2}}

where I₀ and σ are the intensity and the pulse width of the incident laser, respectively. The continuity equation of the carrier density of Si is given by [28]:

\frac{\partial Δ N}{\partial t} = D \frac{\partial^{2} Δ N}{\partial z^{2}} + G (z, t) - \frac{Δ N}{τ_{f}}

where D is the carrier diffusivity; τ_f is the lifetime of photo-excited carrier. The depth-dependent generation rate of free carriers is written by [29]

G (z, t) = I (t) \frac{(1 - R) α β}{L W h f} e^{- α z}

where α is the absorption efficient. β denotes the optical-to-electrical energy conversion efficiency. R represents the reflectivity of the front surface of bulk Si. h and f are Plank constant and frequency of incident wave, respectively. For the high injection density of indirect p-type Si, the Auger recombination dominates the recombination mechanism [30]: τ_Auger = C_p/(p₀² + 2p₀ΔN + ΔN²), where p₀ represents the equilibrium hole density, C_p is the Auger recombination coefficient of Si. Then, the combination rate is described as:

\frac{Δ N}{τ_{f}} = \frac{B_{r} Δ N}{τ_{A u g e r}}

where B_r denotes recombination. The boundary condition is expressed by:

{\begin{matrix} Δ N (0, t) = \frac{D}{S} \frac{\partial Δ N}{\partial z} |_{z = 0} \\ Δ N (\infty, t) = 0 \end{matrix}

where S is surface recombination velocity. The initial condition satisfies:

Δ N (z, 0) = 0

Thus, the conductivity of Si can be derived by:

σ = q μ_{p} Δ N

where μ_p represents the hole mobility of p-type Si. When the bulk Si is excited by a femtosecond pulse with center wavelength of 800 nm, power of 85 μJ, repetition rate of 1 kHz and pulse duration of 35 fs [31], and the parameters are adopted by: L = 1 cm, W = 1 cm, D = 34.6 cm²/s, C_p = 1.1 × 10⁻³⁰ and S = 1 × 10³ cm/s, the photoexcited carrier density and conductivity with different excited time are calculated according to Eqs. (1)-(7).

Figure 1 illustrates that the photoexcited carrier density of Si reduces with the increasing of penetrating depth due to the absorption of Si. Although the skin depth of Si is about 12 μm for the wavelength of 800 nm [32], the photoexcited carrier of Si concentrates on the shallow surface within 5 μm. Correspondingly, the inset of Fig. 1(b) shows the conductivity of Si with different photoexcitation powers, which indicates that the conductivity of Si is dynamically controlled by laser. Hence, the bilayer ring-shaped Si apertures with a thickness of 3.5 μm can be effectively excited and controlled by laser, as shown in Fig. 2.

Fig. 1 The three-dimensional distributions of (a) photoexcited carrier density and (b) conductivity of Si with different exciting time and penetrating depth. The inset is conductivity of Si with different photoexcitation powers.

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Fig. 2 (a) Schematic illustration of a unit cell structure of the tunable NIMs, (b) the principle of the photoexcitation.

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Figure 2(a) shows that the NIM is composed of two metallic rings with width of 4 μm, a substrate and bilayer ring-shaped aperture Si slots with a lattice constant of p = 100 μm. The geometric parameters are: d₁ = 70 μm, d₂ = 72 μm, d₃ = 96 μm, t_sub = 43 μm and the thickness of metallic ring is t_m = 0.5 μm. In order to simultaneously excite the photosensitive semiconductor Si, there are two laser beams simultaneous illuminating the both sides of the MMs, as shown in Fig. 2(b). According to previous studies [33–35], the two metallic rings can be first patterned on Teflon by conventional photolithography using positive photoresist followed by metallization and lift-off process, and then utilizing the ring pattern as a photo-mask to create an identical structure on the other side of Teflon with a negative photoresist, and followed again by lift-off process. After that process, the double-layered aluminum (Al) ring structure can be fabricated. It is worth pointing out that the accuracy of alignment has reached to nanometer for stacking four-layer structure by gold alignment marks [36], and the ring-shaped aperture Si with width of 12 μm can also be fabricated [37,38]. Therefore, the Si can be stacked on the both sides of double-layered Al ring structure.

Hereafter we adopted the commercial software CST Microwave Studio 2016 with the frequency domain solver to investigate the performance of the NIMs. The NIMs with unit cell boundary conditions in x- and y-directions and floquet ports in z-direction are applied for calculating the transmission parameters. For the floquet ports, the propagation wave vector (k) of the linearly polarized THz wave is perpendicular to the NIMs, whereas the electric field (E) and magnetic field (H) are parallel to the incident plane. The tetrahedral meshes with the adaptive meshing method are employed. In order to accurately simulate the THz MMs, the metallic rings and the dielectric substrate are the lossy Al with a frequency-independent conductivity σ = 3.56 × 10⁷ S/m and the low-loss Teflon with a relative permittivity ε_r = 2.1 and loss tanδ = 0.0002, respectively. In order to investigate the resonant mechanism of the NIMs, the single-layered MMs composed of ring resonator and ring-shaped Si aperture (S₁), single-layered MMs composed of ring-shaped Al aperture (S₂), and their corresponding bilayer MMs (B₁ and B₂) were firstly investigated, whose structural scales were the same as the proposed NIMs. Figure 3 gives the transmission and phase spectra of S₁, S₂, B₁ and B₂.

Fig. 3 (a) The normalized transmissions and (b) phase spectra of monolayer and bilayer MMs. The brown-dashed and pink-dashed curves represent the analytic transmissions of S₁ and S₂, respectively.

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There is a significant transmission dip of S₁ without photoexcitation at 1.05 THz (marked by mode I), while the transmission of S₂ exhibits highpass with a transmission peak at 0.94 THz (marked by mode II), as shown in Fig. 3(a). If the monolayer MMs becomes bi-layered MMs, the single resonant mode is split into two sub-modes. The transmission dips of B₁ locate at 0.97 THz and 1.12 THz (indexed as I-M mode and I-E mode), which forms a broadband lowpass transmission. Similarly, the transmission of B₂ exhibits broadband and highpass, whose transmission peaks locate at 0.93 THz and 1.20 THz (indexed by II-M mode and II-E mode), respectively. Figure 3(b) indicates that there are two abrupt phase differences for both S₁ and B₁, while the abrupt phase difference disappears for both S₂ and B₂. The two abrupt phase differences exhibit obvious shift, and the phase shifts of Δϕ₁^, and Δϕ₂^, of double-layer MMs are larger than Δϕ₁ and Δϕ₂ of monolayer MMs, as shown in Fig. 3(b). Therefore, the increase of resonator broadens the transmission and enhances the phase difference, and the continually tunable transmission and phase MMs can be designed by controlling the resonant mode.

Actually, the unit cell can be treated as a meta-atom [39] or meta-molecule [40], and thus, the MMs can be equivalent to a nucleus surrounded by electrons oscillating in the metal layer [41,42]. Therefore, interaction between equivalent electron and incident wave is considered as simple harmonic oscillation where the motion of the equivalent substrate nucleus is ignored. According to the methods of multi-layer [43], transmission line theory [44], effective medium theory [45] and scattering [46], we propose the transmission analytic model of S₁ and S₂. For the ring-shaped ring structure, the external and internal parts are also equivalent two electrons with simple harmonic oscillations. According to the equivalent LC circuit [47], we can calculate the resonant frequency f_r of resonator by

f_{r} = \frac{1}{2 π \sqrt{(l_{m} + Ω) C}}

where Ω is the effective mutual inductance induced by coupling. The effective inductance l_m = l / (ω_p²ε₀S), the capacitance C = ε_subS / t_sub, where S and l represent the cross-section area and length of resonator, respectively. ε₀ and t_sub denote the vacuum permittivity and thickness of substrate, respectively. According to the Lorentz oscillation, the equivalent electron resonant polarization is described as

P_{r} = N P = - N e x = \frac{N e^{2}}{m_{0}} \frac{1}{(ω_{r}^{2} - ω^{2} - i γ ω)} E

where N labels the number of atoms per unit volume, γ is the damping rate, e is the magnitude of the electric charge of the electron, ω_r is the resonant angular frequency. Therefore, the electric displacement D = ε₀E + P, thus, the relative dielectric constant is defined as D = ε₀ε_rE when assuming the metamaterial is isotropic. Hence, the complex relative dielectric constant ε_r is defined as

ε_{r} (ω) = 1 + \frac{N e^{2}}{m_{0} ε_{0}} \frac{1}{(ω_{r}^{2} - ω^{2} - i γ ω)}

On the basis of Fabry-Perot interference transmission [48], the frequency dependent transmissions of S₁ and the complementary S₂ are

\lim_{d \to 0} t (f) = r e a l | \frac{c (1 + n_{s u b})}{c (1 + n_{s u b}) - i f χ_{e}} |

\lim_{d \to 0} t (f) = 1 - r e a l | \frac{c (1 + n_{s u b})}{c (1 + n_{s u b}) - i f χ_{e}} |

where χ_e = a · real (ε_r) / t_sub, a is a coupling factor. According to the Eqs. (8)-(12), the transmissions of S₁ and S₂ are calculated in Fig. 3(a). It is clear that the analytic resonant frequencies are good agreement with the simulated resonant frequencies, as brown- and pink-dashed curves shown in Fig. 3(a). In the analytic model, the loss induced by surface wave [49], electric current and magnetic current [50] is neglected because it is very small. We have not established the effectively analytic model of induced effective mutual M induced by coupling, thus, the M is obtained by the fitting method in Fig. 3(a).

In order to clearly understand the physical mechanism of double-layered MMs, the equivalent LC-circuit model is given in Fig. 4, whose effective inductance L and capacitance C are determined by the coupling currents of metal layer and the thickness of dielectric layer. Besides, there is an additional effective mutual inductance M induced by the coupling between resonators. In order to intuitively illustrate the physical mechanism, the surface currents, equivalent energy diagrams and magnetic fields of MMs are given in Fig. 4.

Fig. 4 (a) The surface current distributions of S₁ and S₂. (b) Schematics of equivalent-energy level of S₁, B₁, S₂ and B₂. (c) The surface current distributions of B₁ and B₂. (d) LC-circuit models and (e) magnetic field distributions of B₁ and B₂. The solid and dashed single arrows represent current direction of the top and bottom resonators, respectively. The blue single arrow line indicates the current direction of ring, while the green and red single arrow lines represent the current directions of inner and outer parts of ring aperture, respectively. The black double arrow represents the coupling between different parts of MMs.

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Figure 4(a) clearly demonstrates that the transmission dip mode I is fundamental dipole resonance, while the highpass mode II is excited by surface plasmons polaritons (SPPs) and localized surface plasmons (LSPs) [51]. The SPPs resonance is determined by the lattice parameter of MMs, whereas the LSPs resonance is controlled by the aperture scale [52]. It is worth noting that the resonant mode I is totally taken over by the mode II when the ring resonator is totally short-circuited by the inner part of Al aperture. The directions of excited surface current of top and bottom resonant rings are reverse for I-M mode but are same for I-E mode. Their equivalent-energy level and LC-circuit models of B₁ are given in Figs. 4(b) and 4(c), where M and M^᾿ represent the effective mutual inductance of mode I-M and mode I-E induced by the coupling of stacked ring resonators, respectively. As the magnetic field distribution shown in Fig. 4(e), the constructive magnetic response is induced by the anti-phase currents, whereas the destructive magnetic response is excited by the in-phase currents. Therefore, the effective inductance of the MMs can be described as:

L_{I - M} = L_{e f f} + M

L_{I - E} = L_{e f f} - M^{'}

On the basis of f_r ∝ (LC)^-1/2, the original resonant mode I is split into two resonances of mode I-M and mode I-E, and I-M represents redshift, while I-E shows blueshift. Figure 4(a) indicates that the surface current of the external and inner Al apertures is antiphase for mode II, which forms a parallel-connected sub-resonant mode and greatly increases the transmission of incident THz wave. Similarly, there are two parallel-connected sub-resonant modes for B₂. According to the equivalent-energy levels, LC-circuit models and magnetic field distributions in Figs. 4(b), 4(d), and 4(e), the effective inductance of mode II-M and II-E between the two parts can be written as:

L_{1 e f f} = \frac{1}{\frac{1}{L_{e x t e r}} + \frac{1}{L_{i n t e r}}} + M_{1} + M_{2} = \frac{L_{e x t e r} L_{i n t e r}}{L_{e x t e r} + L_{i n t e r}} + M_{1} + M_{2}

L_{2 e f f} = \frac{1}{\frac{1}{L_{e x t e r}} + \frac{1}{L_{i n t e r}}} + M_{1}^{'} + M_{2}^{'} = \frac{L_{e x t e r} L_{i n t e r}}{L_{e x t e r} + L_{i n t e r}} + M_{1}^{'} + M_{2}^{'}

respectively, where the L_exter and L_inter represent the effective conductances of external and internal parts of Al aperture, respectively. The M₁ and M₁^′ are the mutual inductances of mode II-M and mode II-E, respectively. The current distributions of external and internal parts are in-phase for II-M mode, whereas that is anti-phase surface current for II-E mode. According to the magnetic field distribution, it is clear that the in-phase currents enhance the magnetic response but decrease the magnetic response for anti-phase, which is different from the double-layered MMs composed of square holes [21]. Therefore, the effective inductances of MMs with double-layered Al apertures can be described as:

L_{I I - M} = L_{1 e f f} + L_{2 e f f} + M_{3} + M_{4}

L_{I I - E} = L_{1 e f f}^{'} + L_{2 e f f}^{'} - M_{3}^{'} - M_{4}^{'}

where M₃ and M₃^᾿ are the mutual inductances between external parts of aperture for mode II-M and mode II-E, respectively. M₄ and M₄^᾿ are the mutual inductances between internal parts of aperture for mode II-M and mode II-E, respectively. That is the reason why the resonant frequency of mode II-M shifts to low frequency, while the resonant frequency of II-E mode exhibits redshift. It is noteworthy that the magnetic response is enhanced when the two equivalent-energy levels are changed to four equivalent-energy levels. Accordingly, the tunable NIMs are dynamically controlled by optically tuning the equivalent-energy level, as shown in Fig. 5.

Fig. 5 (a) The normalized transmission and (b) the transmission phase with different σ of Si. (c) The electric field and (d) the magnetic field distributions of NIMs with different states at 1.1 THz.

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The normalized transmission spectra of the THz NIMs with different excitation power are plotted in Fig. 5(a). For the NIMs without laser excitation, the transmission exhibits low-pass and broadband, and the slope efficiencies of sharp trailing and rising edges are 527% and 418% (based on ∂T/∂f [53]), respectively. With the increasing of the pump power, the low-pass broadband transmission spectrum gradually becomes a high-pass broadband because of the enhancement of the conductivity of photosensitive Si. However, the transmission of the tunable NIMs is lower than that of bilayer ring-shaped Al aperture MMs because the conductivity of Si is 3 orders smaller than Al. Correspondingly, the phase of the THz NIMs can be manipulated by the pump laser, as depicted in Fig. 5(b). The abrupt phase change of NIMs at lower frequency increases with the increasing conductivity of Si, and the phase difference Δϕ is −0.83π rad. However, it is different from the abrupt phase at lower frequency that the abrupt phase at higher frequency decreases firstly and then increases when the conductivity of Si increases. Furthermore, the magnetic response of NIMs is enhanced obviously by the strengthening of the metallic characteristic of the ring-shaped aperture, and the electric response of ring resonators becomes weak but increases in the apertures, as shown in Figs. 5(c) and 5(d). The phenomena are further confirmed that the two equivalent-energy levels are tuned to four equivalent-energy levels if the ring resonator is totally short-circuited by Si.

In practical applications of NIMs like some THz waveguides and communication systems, the polarization and the incident angle are also key factors. Hereafter we analyze the influence of the polarization and the incident angle on NIMs’ performance, as shown in Fig. 6. Due to the symmetrical characteristic of the NIMs, the normalized transmission of the NIMs is independent of polarization angles under the normally incident THz radiation whether the NIMs are excited by laser or not, as shown in Figs. 6(a) and 6(b). It can see from Fig. 6(c) that the NIMs without photoexcitation perform well over a wide range of the incident angle from 0° to 89° because the structure of the NIMs is quit isotropic in x-y plane [55]. When the conductivity of Si increases to 60000 S/m, the NIMs perform stable highpass transmission for the incident angle between 0° and 60°, while its highpass performance gradually disappears if the incident angle is larger than 60°, as depicted in Fig. 6(d). Compared with the planar reverse double split resonators [54], the double-layer closed rings and ring-shaped apertures are polarization independent and efficient with wide incident angle. Therefore, the double-layer symmetric structure performs nearly omnidirectional and polarization-independent. In order to study the characteristics of the NIMs, Fig. 7 shows the retrieved constitutive parameters of the NIMs at different states.

Fig. 6 The transmission contour of the NIMs with different polarization angles (a) without photoexcitation and (b) with σ = 6 × 10⁴ S/m. The transmission contour of the NIMs with different incident angles (c) without photoexcitation and (d) with σ = 6 × 10⁴ S/m.

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Fig. 7 The constitutive parameter of NIMs (a) without photoexcitation, (b) with conductivity σ = 1 × 10³ S/m, (c) with conductivity σ = 1 × 10⁴ S/m and (b) with conductivity σ = 6 × 10⁴ S/m. The gray and pink regions represent the DNR band and SNR band, respectively.

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If the scale of the unit cell of MMs is smaller than the incident wavelength and the structure is symmetric along the THz propagation, i.e. S₁₁ = S₂₂, the standard retrieval procedure can be applied to extract the effective constructive parameter. Figure 7 shows the transmission, the figure of merit (FOM, FOM = │Re(n) / Im(n)│) and retrieval parameters of NIMs under normal incidence as a function of frequency for different photoexcitation. In order to directly show the NRI tunability, the DNR and SNR bands are highlighted by gray and pink regions, respectively. The first DNR band lies between 0.70 THz and 0.86 THz, and the second DNR band locates between 1.24 THz and 1.56 THz, as illustrated in Fig. 7(a). Besides, there are three SNR bands, and the lowpass band locates at SNR band. For the case of σ = 1 × 10³ S/m shown in Fig. 7(b), the two DNR bands become a broadband DNR, which lies between 0.63 THz and 1.45 THz, and the SNR band with negative permittivity shifts to higher frequency. When the pump power continually increases to stimulating the conductivity σ = 1 × 10⁴ S/m, the NIMs only exhibit a broadband DNR band forming a highpass band, as shown in Fig. 7(c), and the negative permeability is relative steady. If the conductivity of Si is raised to σ = 6 × 10⁴ S/m, the highpass band is observed obviously with sharp rising and trailing edges at the DNR band, and the NIMs have a SNR band with negative permeability at higher frequency, as demonstrated in Fig. 7(d). Accordingly, the NRI property of THz NIMs is dynamically manipulated by the pump laser, which is further simulated by the classical method of wedge model structure [13,56,57].

The wedge samples with one-unit-cell stairs in z-directions are constructed, and the adjacent unit cells in x-direction are contacted but separated by vacuum with 2 μm in z-direction to avoid the interference between the adjacent unit cells in propagation direction, as given in Figs. 8(a) and 8(c). Especially, the thickness of vacuum is 60 μm for the σ = 6 × 10⁴ S/m to evidently observe the refraction effect, as shown in Fig. 8(d). The scale of the wedge along the x-directions is a_x = 6 × 100 μm, and that in z-directions are a_z₁ = 6 × 52 μm and a_z₂ = 6 × 110 μm, respectively. Hence, the correspondingly incident angles for the two cases locate at θ_i₁ = arctan (a_z₁ / a_x) ≈27.47° and θ_i₂ = arctan (a_z₂ / a_x) ≈47.73°. Figures 8(a) and 8(b) depict that the refracted angles θ_r are approximately −29.0° and −15.0° for the NIMs without photoexcitation and with σ = 1 × 10³ S/m at 1.35 THz, respectively. According to the Snell’s law n = sin θ_r / sin θ_i, the real parts of the refractive index of the NIMs at 1.35 THz are −1.05 and −0.56, which agrees well with the retrieved indexes of −1.07 and −0.53. Besides, the refracted angles θ_r are approximately −10.0° and 11.0° for the NIMs with σ = 1 × 10³ S/m and σ = 6 × 10⁴ S/m at 1.40 THz, respectively, and the corresponding refractive indices are −0.38 and 0.26, which agrees well with the retrieval indices of −0.36 and 0.25. Consequently, the refractive index of NIMs can be dynamically controlled by pump power, which can enable applications more flexible. In order to investigate the phase velocity of SNR, Fig. 9 records the phase flow of 5-layer NIMs slab with σ = 6 × 10⁴ S/m under normal incidence, respectively.

Fig. 8 Negative refraction of NIMs in the wedge sample.

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Fig. 9 (a) The transmission spectra of NIMs with different layers. Phase flows with free space and five-layered NIMs with σ = 6 × 10⁴ S/m under normal incidence (b) at 1.25 THz with a constant phase step of 60 degrees.

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Figure 9(a) illustrates that the transmission of the NIMs decreases with the increasing of stacking NIMs number, and the inset indicates that the stacked NIMs is separated by vacuum with 60 μm. In order to obviously observe the phase velocity direction with SNR, the phase flow is calculated with 5-layered NIMs slab sample by giving the time evolution of the electric field under normal incidence [57,58], as plotted in Fig. 9(b). The black arrows represent the incident wave propagation direction in free space, whereas the red arrows indicate the THz wave propagation in 5-layer NIMs slab. According to the retrieval refraction of NIMs in Fig. 7(d), the refractive indexes at 1.25 THz is SNR. As shown in Fig. 9(b), the phase velocity of SNR in the five-layered NIMs slab with a constant step of phase change 60-degree at 1.00 THz advances toward –z direction as the red arrows shown, whereas that in free space toward + z direction as the black arrows indicated. Therefore, the phase velocity of NIMs slab is also negative for SNR.

According to the characteristics of NIMs, it is clear that the NIMs have broadband NRI band, thus, it can be applied to the THz antenna aiming to increase the radiant power and enhance the gain [59]. The certain transmission windows of THz wave with minimum attenuation are located at 0.3 THz, 0.35 THz, 0.41 THz, 0.65 THz and 0.85 THz, where the atmosphere attenuation of THz electromagnetic can be below 100 dB/km [60]. For our proposed NIMs without photoexcitation, the refractive index is −0.87 with ε₁ = −1.80 and μ₁ = −0.39 at 0.85 THz. Therefore, it can be applied to THz antenna for enhancing the gain in THz wireless communication for short distance. Besides, the humidity of atmosphere can be neglected for THz band in space, therefore, the tunable NIMs THz wireless communication can also be applied to interstellar communication. It is worth pointing out that if the NIR band of NIMs is narrow, which will limit the band of antenna when it enhances the gain at the same time. Fortunately, the tunable NIMs have broadband NIR, thus, the bandwidth will not decrease obviously when the NIMs enhance the gain of antenna.

3. Conclusion

A tunable THz NIM composed of coaxial double-layer ring resonators and ring-shaped Si apertures coated on the both sides of Teflon substrate was proposed. By the use of the equivalent-energy level and the LC-circuit model, the mechanism of the NIMs has been analyzed. Theoretical results indicate that, the NIMs perform well THz incident angle- and polarization-independence, and show good tunability in controlling the transmission of THz radiation. Without laser excitation, the NIMs are excited by the resonant mode of two equivalent-energy levels whose transmission is broadband and lowpass with slope efficiencies of sharp trailing and rising edges are 527% and 418%, respectively. When the NIMs are excited by pump laser, the transmission becomes highpass because the resonant mode is dominated by four equivalent-energy levels. The constitutive retrieval parameters results reveal that the NIMs exhibit two DNR bands for the case without photoexcitation, when the conductivity of Si aperture increases to 1000 S/m, a maximum broadband DNR is achieved. When the conductivity of Si aperture further increases to 10000 S/m, the NIMs only have a broadband DNR band leading to a highpass transmission, and the negative permeability is relatively stable at the DNR band. If the conductivity continually rises to 60000 S/m, the broadband of DNR reduces and a SNR band with negative permeability appears, resulting in the highpass band with sharp rising and trailing edges. Therefore, the tunable NRI is realized by controlling the resonant mode of NIMs. Correspondingly, the tunable refraction is further investigated by the theoretical wedge sample, and the phase flow of NIMs with SNR indicates that the phase velocity is also negative for single negative refractive index. It is worth pointing out that the equivalent-energy level is valid not only for the multi-layer MMs with dielectric substrate, but also for the multi-planar arranged MMs with dielectric substrate. However, the method cannot be applied to analyze the all dielectric MMs, all metal MMs, chirality MMs and other special-shaped MMs. Additionally, we will aim at building a quantitative equivalent-energy level model.

Funding

China Innovative Talent Promotion Plans for Innovation Team in priority Fields (Grant No.2014RA4051); Fund of Tianjin Key Laboratory of Optical Thin Film (Grant No. kjwx170620).

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Broadband negative-refractive index terahertz metamaterial with optically tunable equivalent-energy level

Abstract

1. Introduction

2. Theoretical model and discussion

3. Conclusion

Funding

References

Cited By

Figures (9)

Equations (18)

Optics Express