Abstract
This work presents a vertical hyperbolic metamaterial (vHMM) consisting of a vertically stacked metal-dielectric multilayer that operates in the visible spectrum. The vHMM is designed by exploiting the relation between negative refraction and effective permittivity along the perpendicular direction of the layers (ε ⊥). When a vHMM has a high loss tangent defined by tan δ ⊥ ≡ Im(ε ⊥)/Re(ε ⊥), even a vHMM composed of relatively thick layers can generate negative refraction. A fabricable vHMM composed of gold and copolymer resist (EL8) which exhibits negative refraction at the wavelengths between 450 and 550 nm is designed using critical layer thickness analysis. The largest negative refraction is observed at the wavelength of 500 nm, where the angle of refraction reaches −1.03°. The corresponding loss tangent and equivalent refractive index are 1.08 and −0.47, respectively. However, negative refraction is not observed at the wavelengths longer than 550 nm due to low tan δ ⊥. We uncover that the tan δ ⊥ of a vHMM is the dominant condition for generating negative refraction rather than the ratio of layer thickness to wavelength.
1 Introduction
Hyperbolic metamaterials (HMMs) have extraordinary optical properties as a consequence of the hyperbolic shape of the isofrequency surface [1], [2], [3], [4]. The hyperbolic dispersion of the HMM provides numerous unusual characteristics including negative refraction [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], sub-wavelength-resolution imaging [16], [17], [18], [19], [20], [21], [22], [23], [24], observation of the optical spin Hall effect [25, 26], anomalous scaling laws [27], thermal emission [28, 29], multidimensional images [30], extreme mechanical hardness [31], biochemical sensing [32], [33], [34], [35], [36], excitation of surface waves [37], excitation of high-k modes [38], and perfect absorption [39, 40].
The negative refraction using HMMs has been mainly studied by considering multilayered stacks of horizontal metal and dielectric layers, i.e., horizontal HMMs (hHMMs) [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. If the thickness of each layer is sufficiently small compared to the wavelength of the incident light (λ), then the hyperbolic dispersion relation of the hHMM can be explained by effective medium theory (EMT) [41, 42]. However, the working frequency ranges for the negative refraction of the hHMMs are generally narrow as the hyperbolic dispersion is a consequence of optical resonances [43]. This limits the negative refraction to occur only near the resonance frequency.
Multilayered structures composed of vertical alternating metal and dielectric layers, i.e., vertical HMMs (vHMMs), have been proposed as a solution to overcome the limitation of the hHMMs [25, 43]. However, the fabrication of vHMMs that exhibit negative refraction at visible frequencies is extremely difficult as the structures’ layers must be sufficiently thin to demonstrate the hyperbolic dispersion. Nano trench structures with a high aspect ratios (AR) up to 20 have been reported but operate in the mid-infrared range [32, 37] or only the reflectance of the structure has been reported [44]. To overcome this difficulty, vHMMs composed of metal and air have been proposed [25, 45], but the negative refraction has not been studied. Recently, a critical layer thickness of the vHMM has been proposed to be the maximum layer thickness divided by λ at which the negative refraction occurs [46]. The critical layer thickness is related to the effective permittivity along the perpendicular direction of the layer (ε ⊥) of the vHMM. The critical layer thickness tends to increase as the value of Im(ε ⊥)/Re(ε ⊥) increases. Therefore, vHMMs with relatively thick layers can exhibit hyperbolic dispersion and provide negative refraction, if the loss tangent defined by tan δ ⊥ ≡ Im(ε ⊥)/Re(ε ⊥) is sufficiently high.
In this study, a vHMM composed of gold and copolymer resist (EL8) is experimentally demonstrated to show negative refraction in visible frequencies. We use critical layer thickness analysis to find fabricable thicknesses of 150 nm and 100 nm for the gold and the dielectric layers, respectively. Despite the relatively large thickness of layers compared to λ, the vHMM exhibits hyperbolic dispersion to induce negative refraction and can be easily realized using a commercially focused ion-beam (FIB) milling method.
2 Methods
2.1 Simulation
Commercially available Comsol Multiphysics was used. The predefined free triangular mesh and extremely fine size were applied. The relative error estimate for every calculation was <10−11.
2.2 Experiment
A supercontinuum laser (YSL, SC-Pro 7) was used for a light source. The beam was linearly polarized after a polarizer (LPVISE100-A) and reached the vHMM prism device. The light that transmitted through the vHMM was captured by a CCD camera (Lumenera, Infinity 2-1R). The angle of refraction was measured and obtained from the position of the captured image compared to the one without the vHMM.
3 Fundamentals of HMM
A vertically stacked metal-dielectric multilayered structure is a typical example of a vHMM (Figure 1(a)). If the vHMM has sufficiently thin layers, its permittivity can be calculated using EMT [47] as
where ε ∥ is the effective permittivity along the parallel direction of the layers; ε d and ε m are permittivities of the dielectric and metallic materials, respectively; f is the filling ratio of the dielectric to the vHMM; and d d and d m are the dielectric and metallic layer thicknesses, respectively.
Characteristics of vHMM.
(a) Schematics of a vertically stacked hyperbolic metamaterial (vHMM). The x-direction is perpendicular to the layers. (b) Isofrequency surfaces of a type-I HMM (left) and a type-II HMM (right). The vHMM is a type-II HMM. The wave vector and the Poynting vector of the type-II HMM have different signs along the x-direction; this characteristic implies that the vHMM can provide negative refraction.
When transverse-magnetic polarized light is incident, the dispersion relation of the HMM is described as [3]
where k x , k y , and k z are the x, y, and z-directional wave vectors in the HMM, k 0 is the wave vector in a vacuum, ω is the angular frequency, and c is the speed of light in a vacuum. The direction of ε ⊥ is parallel to the x-direction. Equation (4) shows a hyperbolic isofrequency surface if the signs of Re(ε ∥) and Re(ε ⊥) are different (Figure 1(b)). The wave vector and Poynting vector therefore have opposite signs, which indicates negative refraction [13]. If Re(ε ∥) is positive, the HMM is classified as type-I; with hHMMs being a typical example, while if Re(ε ∥) is negative, the HMM is classified as type-II; with vHMMs being an example of a type-II HMM.
4 Results and discussion
4.1 Design and fabrication
The biggest challenge to the fabrication of vHMMs that exhibit negative refraction at visible frequencies is that they must have both extremely thin layers and a high AR. Some previous studies considered the vHMM structure numerically, but practical the realization of the proposed structures is extremely difficult. One proposed vHMM is composed of aluminum and silicon dioxide with a layer thickness of 30 nm and AR = 50 [43]. Another was composed of silver and magnesium fluoride with a layer thickness of 18 nm and AR = 111 [46]. However, these structures cannot be fabricated using commercial fabrication techniques.
The working frequency range of the vHMM composed of gold and EL8 for the negative refraction is broad, and the hHMM composed of the same materials cannot provide negative refraction at any λ (Figure 2(a)). Theoretically, the vHMM can provide negative refraction in the whole visible frequencies. In addition, the vHMM has a sufficiently high tan δ ⊥ at λ ≈ 500 nm and f = 0.4 (Figure 2(b)). Therefore, the vHMM composed of gold and EL8 is suitable to demonstrate negative refraction experimentally. The relative permittivity of gold varies with wavelength and both Re(ε m ) and Im(ε m ) tend to decrease as the wavelength increases, but the relative permittivity of EL8 does not (Figure 2(c)). In the vHMM that has f = 0.4, Re(ε ∥) is negative and Re(ε ⊥) is positive in the whole range 450 ≤ λ ≤ 650 nm (Figure 2(d)). Im(ε ⊥) increases in the range 450 ≤ λ ≤ 520 nm, and Im(ε ⊥) is even larger than Re(ε ⊥) at λ ≈ 500 nm. Im(ε ⊥) is sharply decreasing after λ = 520 nm where the slope of Im(ε m ) is gradual. If Im(ε d ) is negligible, tan δ ⊥ is expressed as
Properties of a vHMM composed of gold and EL8.
(a) Working frequency range of a vHMM composed of gold and EL8 for the negative refraction. The hHMM composed of the same materials cannot provide negative refraction. (b) tan δ ⊥ of the vHMM in the visible range. The vHMM that has a filling ratio f = 0.4 has a high tan δ ⊥ near λ = 500 nm. (c) Relative permittivities of gold (yellow lines) and EL8 (black lines). (d) Effective permittivity of the vHMM with the filling ratio f = 0.4.
When the absolute value of Re(ε ⊥) is sufficiently larger than Im(ε ⊥), tan δ ⊥ becomes low. tan δ ⊥ can be measured by using total internal reflection ellipsometry [48].
The vHMM is fabricated using two-step FIB milling (FEI, Helios NanoLab 650) and polymer-resin spin-coating. Firstly, a 400 nm-thick layer of gold is deposited using an electron-beam evaporator on cleaned fused silica (Figure 3(a), i and ii). Then a 150 nm-wide gold grating with a height of 400 nm is patterned on the fused silica substrate using FIB (Figure 3(a), iii) with a beam current of 80 pA for 10 min to cover the 10 µm2 gold grating pattern. EL8 is spin-coated on the gold grating at 5000 rpm for 1 min and then baked at 90 °C for 5 min (Figure 3(a), iv). Finally, a grayscale FIB milling process is conducted to fabricate prism vHMM structures (Figure 3(a), v). The fabricated vHMM (Figure 3(b)) has d m = 150 and d d = 100 nm, and f = 0.4. The height of the sample is 400 nm, so the AR is 4. The transmitted light propagates toward the refracted direction in the air, so the angle of refraction can be measured despite the short height of the vHMM (Figure 4(a)). Fabrication defects such as tilted sidewalls of fins and air gaps within the gold patterns can cause a discrepancy in measurement. The inclined sidewalls of fins are caused by the redeposition of sputtered materials during the FIB milling process [49, 50]. For low beam current during the process, the defect can be reduced [49, 50]. In addition, vacuum baking during the spin-coating process may reduce the air gap [51]. However, the tilted sidewall of the fin can create the hyperbolic shape of the isofrequency surface in a vHMM [15, 52], and the permittivity of the air gap is not much different from that of the resist to induce negative refraction.
Fabrication of a vHMM.
(a) Fabrication process of a vHMM. (i) Preparation of a cleaned fused silica substrate; (ii) deposition of 400 nm-thick gold layer; (iii) first FIB milling to pattern the gold grating structures; (iv) spin-coating EL8; (v) second FIB milling to yield the final prism structure. Scale bar: 500 nm. (b) Scanning electron microscope image of the fabricated vHMM. Scale bar: 3 μm.
Observation of negative refraction.
(a) Schematic of negative refraction in a vHMM. If negative refraction occurs, the transmitted light propagates toward the left of the dashed line and the angle of refraction θ r becomes negative. (b) Captured images (left) and the intensity profiles (right). The images are captured at the left side of the dashed line where negative refraction occurs. (c) Simulated and measured angles of refraction of the vHMM. Error bars are smaller than the point markers. (d) Simulated electric field profile at λ = 500 nm. The simulated θ r = −1.03°.
4.2 Demonstration of negative refraction
The angle of refraction θ r (Figure 4(a)) of the sample is measured at λ = 450, 500, 550, 600, and 650 nm. θ r is calculated using a charge-coupled device (CCD) camera to measure the position of the captured image pattern. The captured images are converted to intensity profiles (Figure 4(b)). If negative refraction occurs, the captured image pattern will appear to the left of the dashed line shown in Figure 4(b); as seen in images captured at 450 ≤ λ ≤ 550 nm. The measured data demonstrates a similar trend to the simulated results (Figure 4(c)). The experimental results give θ r = −0.40° at λ = 450 nm, −1.03° at 500 nm, and −0.69° at 550 nm. The angles of refraction are equivalent to those of negative-index metamaterials that have refractive indices of −0.18, −0.47, and −0.31 respectively. Because the tan δ ⊥ is sufficiently high, negative refraction is observed at 450 ≤ λ ≤ 550 nm despite large d m /λ (Table 1). The simulated electric field profiles (Figure 4(d)) give θ r = −1.4° at λ = 450 nm, −1.61° at 500 nm, and −0.41° at 550 nm when the structure is fabricated ideally. The negative phase propagation appears in the vHMM [5, 53]. Thus, light passing through the thicker part of the vHMM experiences phase advance compared with the other, and the transmitted light is refracted toward the negative direction. However, θ r is 0.31° at λ = 600 nm, 0.50° at 650 nm. Because the absolute value of Re(ε m ) is sufficiently larger than Im(ε m ) at 600 ≤ λ ≤ 650 nm, tan δ ⊥ becomes too low and negative refraction is not observed, but Re(ε ∥) and Im(ε ∥) are independent to exhibit negative refraction [46]. At λ = 650 nm, d m /λ is the smallest, but the tan δ ⊥ is too low to observe negative refraction (Table 1).
Representative parameters of the vHMM.
| Wavelength (nm) | Angle of refraction (°) | Loss tangent (tan δ ⊥ = Im(ε ⊥)/Re(ε ⊥)) | Ratio of structural geometry to wavelength (d m /λ) |
|---|---|---|---|
| 450 | −0.40 ± 0.01 | 0.706 | 0.33 |
| 500 | −1.03 ± 0.01 | 1.08 | 0.30 |
| 550 | −0.69 ± 0.01 | 0.353 | 0.27 |
| 600 | 0.31 ± 0.01 | 0.0862 | 0.25 |
| 650 | 0.50 ± 0.01 | 0.0299 | 0.23 |
4.3 Effect of diffraction
We consider the diffraction which may be in the vHMM (Figure 5). Because tangential components should be continuous at the interface, the transmitted angle θ t of the diffracted light is expressed as
where m is an integer characterizing the diffractive modes, k i and k t are incident and transmitted wave vectors, respectively, β i and β t are transverse components of k i and k t , respectively, θ i is an incident angle, n is a refractive index of the fused silica substrate, and p is a period of the vHMM. The argument of sin−1 in Equation (9) only has an absolute value smaller than unity at m = 0 because of both small p (λ/p ≥ 1.8) and θ i (θ i < 5°) in our experiment. Therefore, we can conclude that the diffraction does not occur (m = 0) in the experiment.
Schematic of diffraction in the vHMM. θ t can be varied despite constant θ i if the diffraction occurs. However, the diffraction is not valid in the vHMM because both p and θ i are sufficiently small.
5 Conclusions
In conclusion, we have experimentally demonstrated negative refraction using a vHMM composed of gold and EL8 without diffraction. The layers have thicknesses d m = 150 nm and d d = 100 nm, and the maximum height is 400 nm. AR is up to 4 and is realized using a commercial FIB milling process. The fabricated vHMM provides negative refraction at λ = 450, 500, and 550 nm, and which θ r are −0.40°, −1.03°, and −0.69°, respectively. However, negative refraction is not observed at λ = 600 or 650 nm. The vHMM has sufficiently high enough tan δ ⊥ at 450 ≤ λ ≤ 550 nm, so negative refraction occurs even though the ratio d m /λ is large. This result implies that the tan δ ⊥ of the vHMM is more dominant to exhibit hyperbolic dispersion and generate negative refraction than the ratio d m /λ.
Funding source: National Research Foundation of Korea
Award Identifier / Grant number: NRF-2017H1A2A1043322
Award Identifier / Grant number: NRF-2019R1A2C3003129
Award Identifier / Grant number: NRF-2021R1A6A3A13038935
Funding source: Chung Mong-Koo Foundation
Award Identifier / Grant number: Hyundai Motor Chung Mong-Koo fellowship
Funding source: POSCO
Award Identifier / Grant number: POSCO-POSTECH-RIST Convergence Research Center program
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Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
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Research funding: This work was financially supported by the POSCO-POSTECH-RIST Convergence Research Center program funded by POSCO, POSTECH-Samsung Semiconductor Research Center (IO201215-08187-01) funded by Samsung Electronics, and the National Research Foundation (NRF) grants (NRF-2019R1A2C3003129) funded by the Ministry of Science and ICT of the Korean government. Y.Y. acknowledges the Hyundai Motor Chung Mong-Koo fellowship, and the NRF fellowship (NRF-2021R1A6A3A13038935) funded by the Ministry of Education (MOE) of the Korean government. S.S. acknowledges the NRF Global Ph.D. fellowship (NRF-2017H1A2A1043322) funded by the MOE of the Korean government.
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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