Terahertz metasurfaces with a high refractive index enhanced by the strong nearest neighbor coupling

Siyu Tan; Fengping Yan; Leena Singh; Wei Cao; Ningning Xu; Xiang Hu; Ranjan Singh; Mingwei Wang; Weili Zhang

doi:10.1364/OE.23.029222

1. Introduction

One of the goals of developing artificially engineered metamaterials is to realize the desired effective refractive indices which are not attainable in naturally occurring materials such as negative refractive index [1–4], zero index [5,6] or extremely high index [7–13]. There have been several efforts to achieve negative refractive index since the first experimental demonstration in the microwave regime. However, metamaterials exhibiting high indices of refraction have only been sparsely studied. High refractive indices deliver a platform for terahertz metamaterials with extremely low losses while ensuring that the unit cells are effectively subwavelength in size that exclude the diffraction effects. Enhanced refractive index also opens up avenues in developing high-density integrated terahertz optoelectronic devices that would enable enhanced confinement with low losses. A large collection of different high refractive index materials would expand the feasibility of achieving potential applications such as cloaking [14–16], optical delay lines [17], enhanced imaging resolution [18] and functional devices with gradient index profile [13,19].

As one of the most fundamental parameters that describes the electromagnetic properties of a material, the refractive index can be determined by $n = \sqrt{ε_{r} μ_{r}}$ , where ε_r, μ_r are the relative permittivity and permeability, respectively. With the unique advantage of engineered electromagnetic properties, metamaterials enable the realization of exotic refractive indices by manipulating their effective permittivity and permeability simultaneously. A few naturally occurring materials such as Cesium Iodide (CsI) can also exhibit a high refractive index at terahertz frequencies [20]. By rationally designing the periodic structures and corresponding geometrical parameters, metamaterials provide extremely large tunability of both the refractive index and the resonance frequency. Furthermore, the ultra-thin characteristic of metasurfaces also throws up opportunities for developing functional terahertz devices with compact size and light weight. Previous studies have revealed that subwavelength capacitor-array structures exhibit a significant enhancement of the permittivity [7,21]. Unfortunately, in these designs, there also exists a strong diamagnetic response [22,23], which suppresses the effective permeability and further limits the enhancement of the refractive index. Shen et al. proposed a three-dimensional broadband high-refractive index metamaterial formed by a cubic array of capacitors with air slits on their surfaces [9]. The existence of such slits prevents the formation of large-area current loops, which leads to a dilute diamagnetic response. Similarly, in the terahertz regime, strongly coupled I-shaped structures were proposed by M. Choi and associates. They fabricated the structures using conventional photolithography [11] and electrohydrodynamic jet printing [12]. These I-shaped structures sustain small diamagnetic responses and enhanced permittivities simultaneously which allowed the realization of unnaturally high refractive index metamaterials. In these designs, the strong coupling between the nearest neighbors enable a large amount of charge accumulation on the surface of the subwavelength resonators and ultimately leads to the formation of a significant dipole moment in a unit cell. Microscopically, these subwavelength resonators can be equivalent to the polarized atoms driven by an external electric field. The re-radiating electric dipoles with retardation largely lower the phase velocity of the incident wave [24], which is the characteristic of a high refractive index effect in these structures.

We propose an alternative design based on a free-standing Z-shaped terahertz metamaterial with a high refractive index of 14.36 at normal incidence. The proposed structure sustains a pronounced electric response to enhance the refractive index at a lower frequency than the I-shaped structure. In the current design, we incorporate additional inductance instead of further reducing the separation between the nearest neighbors which actually offers higher design flexibility in terms of ease of fabrication. The significant enhancement in the refractive index has been verified by both numerical simulations and measurements. We also propose an equivalent circuit model to explain the observed behavior. The tuning of the effective refractive index as a function of the change in the periodicity of the metasurface structure has potential applications in transformation optics or gradient-index devices. In this work, we probe the impact of the nearest neighbor coupling on the peak value of the refractive index and its frequency tunability in the terahertz regime.

2. Proposed structure

The unit cell of the proposed free-standing high-index terahertz material is schematically illustrated in Fig. 1(a), together with the polarization of the incident wave. A thin Z-shaped metallic patch is symmetrically embedded in the dielectric substrate. The fabrication process begun with spin-coating the polyimide ( $ε_{p} = 2.96 + 0.27 i$ ) onto a silicon wafer. After heat-curing cycles, a 10-μm-thick flexible polyimide film was formed on the silicon wafer. Then, a metallic patch, made of 200 nm aluminum with geometric dimensions of a = 76 μm, c = 4.04 μm and w = 6 μm, was patterned on the polyimide layer by conventional photolithography and thermal evaporation. Repeating the polyimide coating and heat-curing cycles, a flexible, free-standing terahertz metamaterial with an overall thickness d = 20 μm was fabricated after peeling-off the polyimide layer with the metasurface from the silicon wafer on which the fabrication was initiated. The microscopic image of the fabricated sample is shown in Fig. 1(b). The entire sample consists of 125 × 125 unit cells with a periodicity of p = 80 μm. Therefore the separation between the nearest neighbors is defined as g = p – a = 4 μm.

Fig. 1 (a) A unit cell of the proposed metamaterial consists of a thin Z-shaped aluminum patch symmetrically embedded in the polyimide dielectric substrate. The relevant geometrical dimensions are: p = 80 μm, a = 76 μm, c = 4.04 μm, w = 6 μm and d = 20 μm. (b) Microscopic images of the fabricated sample. (c) Photograph of a flexibility test for the fabricated metamaterials.

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3. Experimental results and discussion

In order to experimentally verify the enhancement of the effective refractive index, the Z-shaped metamaterials were characterized at normal incidence using photoconductive-antenna-based terahertz time-domain spectroscopy (THz-TDS) from which the complex transmission coefficient can be obtained [25]. The THz-TDS system consisted of a GaAs photoconductive transmitter and a silicon-on-sapphire photoconductive receiver; each was optically excited with 26-fs ultrafast optical pulses of 10 mW average power. The THz-TDS system was configured in an 8F confocal geometry with a 3.5 mm frequency-independent beam waist. The amplitude transmission is defined as the ratio of the Fourier-transformed spectra of the metasurface sample and free space. Numerical calculations had been conducted using a commercial Finite Integration Method solver, CST microwave studio. The measured amplitude transmission and simulated S-parameters (S₁₁ and S₂₁) are shown in Figs. 2(a) and 2(b), respectively. We observed a transmission dip at 0.333 THz as indicated by the red dotted line in Fig. 2(a). Owing to an x-polarized incident electric field, this is a dipolar resonance that leads to a charge accumulation on each arm of the Z-shaped metallic patch. From the simulated S-parameters in Fig. 2(b), most of the incident wave was reflected in backwards direction rather than being scattered at non-zero angles with respect to the surface normal. According to the diffraction theory, the proposed structure can be assumed to be an infinite periodic grating. Since the lattice constant (p = 80 μm) is much smaller than the resonance wavelength (~900 μm), the first order diffraction of the structure actually does not exist at normal incidence. We also looked at the orthogonally polarized component and found that it has a negligible intensity (~0.9%) compared to the x-polarized component at the resonance frequency. Considering the deviations in the fabrication process and the material parameters used for the simulation, the overall agreement between the measured and simulated transmission is reasonably good. The relatively thin free-standing sample is unable to temporally separate the main transmission signal and the following echoes due to the Fabry-Perot multi-reflections. In order to achieve an accurate complex refractive index from this overlapping time-domain measurement, a customized code was used to calculate the Fourier transform of the measured time-domain data by solving the following equation [26]

T = \frac{4 {\tilde{n}}_{e f f} {\tilde{n}}_{a i r}}{{({\tilde{n}}_{e f f} + {\tilde{n}}_{a i r})}^{2}} \cdot \exp [- i ({\tilde{n}}_{e f f} - {\tilde{n}}_{a i r}) k_{0} d] \cdot FP,

where

FP = \frac{1}{1 - {(\frac{{\tilde{n}}_{e f f} - {\tilde{n}}_{a i r}}{{\tilde{n}}_{e f f} + {\tilde{n}}_{a i r}})}^{2} \cdot \exp [- 2 i {\tilde{n}}_{e f f} k_{0} d]}

describes the Fabry-Pérot effect of the backward and forward reflections in the sample. d is the thickness of the measured sample. Figures 2(c) and 2(d) present the comparison of the refractive index extracted from the THz-TDS measurements and the numerical results from the simulated complex transmission by the S-parameter retrieval method [27,28]. A peak refractive index of n = 14.36 was realized at 0.315 THz. Then, the extracted complex refractive index was back-substituted into the Eq. (1) to ensure the accuracy. A good agreement between the measured data and back-substitution indicates that the extraction procedure is accurate. It is worth noting that the effective homogeneity condition, which requires that the thickness along the propagation direction is smaller than the operating wavelength, remains valid throughout the frequency band of operation.

Fig. 2 (a) Measured (red solid dots) and back-substituted (blue solid line) amplitude transmission spectra; (b) Simulated S₂₁ (black solid line) and S₁₁ (magenta solid line); (c) Experimentally extracted complex refractive index; (d) Numerically retrieved values from the simulated S-parameters.

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In order to further elucidate the physical nature of the enhancement of the refractive index, the electromagnetic field components were calculated at the resonance frequency 0.333 THz, as shown in Figs. 3(a) and 3(b) respectively. The electric field is mainly localized in the gap area between the nearest neighbors, indicating a strong coupling between them. Driven by the incident electric field, a large number of charges have accumulated on the two metallic arms to form a strong dipole moment in the unit cell, leading to a significant enhancement in the effective permittivity value. Furthermore, with just a single-layer thin metallic patch (d = 200 nm) embedded in the dielectric substrate, only a very small area of the metallic patch is subtended by the induced current loop, which does not support a high magnetic moment thus leading to an extremely low diamagnetic effect [9,11]. As shown in Fig. 3(b), the magnetic field is very weak in most parts of the structure except near the central stripe. Figures 3(c) and 3(d) present the numerically extracted effective permittivity and permeability obtained through $ε_{e f f} = N_{e f f} / Z_{e f f}$ and $μ_{e f f} = N_{e f f} \cdot Z_{e f f}$ , where N_eff and Z_eff are the effective refractive index and normalized impedance retrieved from S-parameters, respectively. The permeability remains near unity in the entire frequency domain of interest with only a weak anti-resonance around the resonance frequency. This anti-resonance behavior is an intrinsic property of the metamaterial with finite spatial periodicity [29,30]. A significant electric resonance can be observed with the peak permittivity value of 205.78, which is nearly two orders of magnitude larger than the substrate permittivity ( $Re [ε_{p}] = 2.96$ ). Therefore the Z-shaped structures sustain a pure electric response, which is similar to the I-shaped resonators or the ELC resonators [31]. Owing to anisotropy of the proposed structure, the nearest neighbor coupling is extremely weak for the incident electric field being parallel to the metallic arms of the Z-shaped structure (y-polarized), which ultimately leads to a less pronounced electric resonance. Thus, in the current work, we only discuss the scenario for the x-polarized incidence.

Fig. 3 Simulated distributions of the (a) electric field (b) magnetic field on the cut plane of z = 0 (which cuts through the center of the 200 nm metallic patch) at the resonance frequency of 0.333 THz. (c) and (d) numerically obtained values of the effective permittivity and permeability from the S-parameters retrieval method.

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In order to systematically explore the enhancement of the refractive index by the strong nearest neighbor coupling between adjacent units and the corresponding resonant behavior of the Z-shaped structure, four sets of Z-shaped metamaterials with different nearest neighbor gap widths “g” were fabricated and experimentally characterized. The lattice constant p and line width w of the metallic structures were constant in all the samples. Owing to the changes in the geometric dimensions, the resonance frequencies show a pronounced blue shift in Fig. 4(a) with the increase in the gap width. Since the gap width is much smaller than the lattice constant for all the samples, the periodic Z-shaped structure can be simplified to an array of finite cascaded line capacitors. The density of accumulated charges is inversely proportional to the gap width [11]. Therefore, the coupling strength between the nearest neighbors becomes weaker as the separation increases, leading to a lower enhancement in the effective permittivity which eventually limits the enhancement in the effective refractive index of the metasurface. Figure 4(b) shows the experimentally extracted and numerically calculated refractive indices as a function of gap width and the inset shows the corresponding numerically extracted effective permittivity. The decreasing trend of the refractive indices is consistent with empirical asymptotic analysis in [11], which indicates that higher refractive indices could therefore be achieved by further reducing the gap width due to enhanced coupling between the nearest neighbors.

Fig. 4 (a) Measured amplitude transmission spectra of the Z-shaped samples with different gap widths. (b) Corresponding refractive indices from the experiment (symbols) and from the numerical calculation (dash lines). Inset: numerically obtained values of the effective permittivity versus frequency for four gap widths.

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In addition,we calculated the surface current distributions of the proposed structure and an I-shaped resonator with the same geometric parameters for comparison, as shown in Figs. 5(a) and 5(b), respectively. A dipolar current distribution of the induced current can be observed on the central microstrip of both resonators. In the Z-shaped structure, there also exist parallel currents in the same direction on the two metallic arms. Therefore, a larger effective inductance is introduced in the Z-shaped structure, which contributes to a lower resonance frequency than that of the I-shaped resonator (the average frequency shift is approximately 21.6%). According to the distribution of the electric field and the induced surface current, an equivalent circuit model of the proposed Z-shaped metamaterial is illustrated in Fig. 5(c) [31,32]. The capacitance formed by the coupling between the adjacent units (C₁) and the lateral capacitance between the two arms in a unit cell (C₂) can be calculated using the same finite line capacitance model [33], with

C_{1, 2} = α_{1, 2} ε_{0} ε_{e f f} a \frac{K (k_{01, 2})}{K (\sqrt{1 - k_{01, 2}^{2}})},

respectively, where K is the complete elliptical integral of the first kind for

k_{01} = g / (g + 2 w)

and

k_{02} = (a - 2 w) / a

;

ε_{e f f} = β ε_{p} + (1 - β)

is the effective relative permittivity calculated by taking the contribution of both free space and polyimide (

Re [ε_{p}] = 2.96

) into consideration, where β is a fitting parameter ranging from 0 to 1, which represents the contribution of the polyimide substrate [34]; α₁ and α₂ are dimensionless fitting factors representing the effective length and the width of the metallic arms that actually have direct impact on the resonance. By fitting these three factors with the experimental data sets, the best fits we obtained for the measurements are α₁ = 0.31, α₂ = 0.11, and β = 0.80. Furthermore, under the approximation of a uniform induced current flow through the two arms and the central strip of the Z-shaped structures, two different types of self-inductance and an additional mutual inductance between the two parallel arms can be calculated by Bueno’s theory [35]. By ignoring the ultrathin thickness of the metallic patch, the surface self-inductance of a microstrip with a length l and width w in Neumann model can be expressed as

L = \frac{μ_{0} l}{4 π} {2 \sin h^{- 1} (\frac{l}{w}) + 2 (\frac{l}{w}) \sin h^{- 1} (\frac{w}{l}) + \frac{2}{3} [{(\frac{l}{w})}^{2} + \frac{w}{l} - \frac{{(w^{2} + l^{2})}^{3 / 2}}{l w^{2}}]} .

The mutual inductance of the two parallel arms with the same current direction can be expressed as

M_{1} = \frac{μ_{0}}{4 π} [2 a \sin h^{- 1} (\frac{a}{h}) + 2 (h - \sqrt{h^{2} + a^{2}})],

where a is the length of the metallic arms and h = a - 2w is the separation spacing between them. After substituting the geometric parameters of the proposed Z-shaped structures into the aforementioned three equations, we can obtain the values of all the components contained in the equivalent circuit model, as shown in Table 1. Since the two lateral capacitances are parallel to each other, the resonance frequency of the equivalent circuit can then be calculated as

f_{r} = \frac{1}{2 π \sqrt{(2 L_{1} + 2 M_{1} + L_{2}) (C_{1} + C_{2} / 2)}},

where L₁ and L₂ are the self-inductances of the metallic arms and the central beam, respectively. From Table 1, the increase in the gap width contributes to a near-linear decrease to all the circuit component parameters, thus explaining the blue-shift behavior of the resonance frequency shown in Fig. 4(a). Figure 5(d) presents the comparison between the experimental results and calculated resonance frequencies by the circuit model, which indicates an excellent agreement.

Fig. 5 (a) and (b) A comparison of the surface current distribution of Z-shaped and I-shaped resonators. (c) Equivalent circuit model of the Z-shaped metamaterials with L₁: the self-inductance of two parallel metallic arms and M₁: the mutual-inductance between them; L₂: the self-inductance of the central microstrip; C₁: the capacitance between the proximity units; C₂: the lateral capacitance between two arms inside a unit cell. (d) Measured (red dots) and calculated (solid blue line) resonance frequency of the circuit model as a function of the gap width g.

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Table 1. Circuit Component Parameters in the Equivalent Circuit Model for Various Gap Widths g (p = 80 μm, w = 6 μm, α₁ = 0.31, α₂ = 0.11, β = 0.80)

View Table

4. Conclusion

In conclusion, we have experimentally demonstrated a flexible free-standing Z-shaped metasurface exhibiting nearest-neighbor coupling-dependent high refractive index in the terahertz regime. The peak value of the refractive index reaches 14.36 at 0.315 THz. The significant enhancement in the refractive index is obtained due to a negligible diamagnetic response and an enhanced electric resonance response in the strong coupling regime between the adjacent meta-atoms. The tunable behavior of the effective refractive indices as a function of the gap width has been experimentally verified by characterizing four sets of metamaterials fabricated with varying nearest neighbor coupling distances in the metasurface lattice. An equivalent circuit model is used to explain the impact of geometric parameters on the resonance behavior of the metasurface that determines the peak refractive index. The proposed flexible free-standing Z-shaped metasurfaces could further improve the design flexibility and simple fabrication of cloaking devices, delay-lines or graded index lenses in the terahertz regime. The strong coupling induced enhancement in the peak refractive index could be exploited to obtain a high index response in other frequency regimes using a similar concept.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) (Grant Nos. 61275091, 61327006, 11174158) and the U.S. National Science Foundation (Grand No. ECCS-1232081). S. Tan acknowledges the China Scholarship Council for financial support.

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g (μm)	4	6	8	10
L₁ (pH)	57.1226	55.2350	53.3582	51.4926
L₂ (pH)	82.7274	79.7412	76.7731	73.8231
M₁ (pH)	8.2490	8.0669	7.8852	7.7038
C₁ (fF)	0.9979	0.8555	0.7604	0.6886
C₂ (fF)	0.1524	0.1491	0.1463	0.1436

Terahertz metasurfaces with a high refractive index enhanced by the strong nearest neighbor coupling

Abstract

1. Introduction

2. Proposed structure

3. Experimental results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Tables (1)

Equations (6)

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