Anomalous Wavelength Scaling of Tightly Coupled Terahertz

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Anomalous Wavelength Scaling of Tightly-Coupled Terahertz Metasurfaces Jihun Kang, Seojoo Lee, Bong Joo Kang, Won Tae Kim, Fabian Rotermund, and Q-Han Park ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.8b05806 • Publication Date (Web): 29 May 2018 Downloaded from http://pubs.acs.org on May 29, 2018

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ACS Applied Materials & Interfaces

Anomalous Wavelength Scaling of Tightly-Coupled Terahertz Metasurfaces

Ji-Hun Kang1†, Seo-Joo Lee2, Bong Joo Kang3, Won Tae Kim3, Fabian Rotermund3, and Q-Han Park2*

1

Department of Physics, University of California at Berkeley, Berkeley, CA 94720, United States

2

Department of Physics, Korea University, Seoul 02841, Republic of Korea

3

Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic

of Korea

*

†

Corresponding author: [email protected] Present address: Department of Physics and Astronomy, Seoul National University, Seoul 08826,

Republic of Korea Keywords: metasurfaces, metamaterials, scaling effect, terahertz spectroscopy, resonators

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ACS Applied Materials & Interfaces 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract We theoretically and experimentally demonstrate the drastic changes in the wavelength scaling of tightly-coupled metasurfaces caused by deep sub-wavelength variations in the distance between the unit resonators but no change in the length scale of the units themselves. This coupling-dependent wavelength scaling is elucidated by our model metasurfaces of ring resonators arranged with deep sub-wavelength lattice spacing g, and we show that narrower g results in rapider changes in wavelength scaling. Also, by using terahertz time-domain spectroscopy, we experimentally observed a significant shift of the spectral response arising from very small variations in lattice spacing, confirming our theoretical predictions.


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Conceiving metamaterials and their optical responses demands an understanding of macroscopic light interaction with composite structures of sub-wavelength unit resonators. In many cases the unit resonators are designed to be loosely-coupled, and taking into account the optical properties and the length scale of the individual units is typically sufficient to attain a reasonable prediction of the optical responses as well as wavelength scaling within the operating spectral range1-5. However, now that recent research into dense metamaterials has realized ultra-high-index6-8 and reconfigurable metamaterials9-13, ascribing optical responses to single unit properties is becoming insufficient. In particular, it has been demonstrated that, when the distance between the units enters deep sub-wavelength regimes, an appreciable resonance wavelength shift can emerge from interunit coupling14-17. Important consequences of this are that the simple wavelength scaling predicted by the length scale of the individual unit is no longer valid for dense metamaterials, and that a drastic change in the wavelength scaling arising from deep subwavelength variations in the distance between the unit resonators can be expected when the units are tightly coupled. Because such strong change in the wavelength scaling from small geometric variations in metamaterials could be a key principle in designing functionalized metamaterials offering the dynamic manipulation of optical responses, a quantitative account of the effect of the interunit coupling on the wavelength scaling is important for further development of dense metamaterials. In this work, we theoretically and experimentally demonstrate how the strong coupling between the unit resonators plays an important role in drastic change in wavelength scaling. In particular, through rigorous analytical calculations of the dense metasurfaces of tightly-coupled rectangular ring resonators, we quantitatively discuss wavelength scaling by introducing a factor α, the ratio of the normalized resonance shift ∆λres/λres and the normalized lattice deformation ∆d/d, i.e.,



α≡(∆λres/λres)/(∆d/d), where λres and d are the resonance wavelength and the lattice size of the metasurface, respectively. We demonstrate that α can be anomalously large, implying that drastic resonance wavelength changes can arise from small variations in the overall lattice size. To verify our theoretical model, we performed terahertz time-domain spectroscopy measurements, and observed a large scaling rate α, which is in good agreement with our theoretical predictions. The optical response of a metasurface is governed by the interaction of light with its constituent resonator elements. For tightly-coupled resonators, the conventional effective medium approaches used with dilute metamaterials, such as the Maxwell-Garnet theory, cannot be applied18. Consider an exemplary metasurface composed of sub-wavelength rectangular ring resonators, as shown in Figure 1. The resonators are spaced periodically with the lattice-spacing parameters gx and gy in the x- and y- directions, respectively. When the spacing gx,y is comparable to the wavelength, the metasurface becomes a loosely-coupled one of which the optical properties can be approximately determined by the response of individual ring resonators. In particular, the resonant light-metasurface interaction occurs in this case when the total path length of the current on the resonator induced by the incident light is comparable to the halfwavelength. Quantitatively, when we compare the two ratios, δλres/λres and δL/L where δA denotes an infinitesimal variation in A, and λres and L are the resonance wavelength and the path length of the current, respectively, the relationship δλres/λres ≈ δL/L approximately holds. However, when the spacing gx,y enters the deep sub-wavelength scale so that the resonators are tightly-coupled, nearby resonators can interact with each other through charges induced by the incident light19. These capacitive interactions significantly change the scattering behavior of the incident light by opening a highly efficient transmission channel in between the resonators mediated by a strong funneling process20. This implies that, in tightly-coupled cases, a small


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change in the lattice spacing or rearrangement of the resonator involving deep sub-wavelength variation can yield a strong modification of the optical properties of metasurfaces. For a rigorous account of light scattering by dense metasurfaces, we adopted a coupled mode theory and performed analytic investigations (see the Supporting Information). To simplify the problem, we assumed that the rectangular rings are made of a perfect electric conductor with side lengths a and b (b ≥ a) smaller than the wavelength λ of the incident light. Figure 2a shows the zero-th order transmission spectrum through the metasurface. The analytic result shows good agreement with the more rigorous finite-difference time-domain (FDTD) result. In particular, we find that there is a resonant dip in the transmission spectrum and that resonance is highly dependent on lattice spacing. Our analytic theory predicts that the resonant dip occurs as the lattice spacing becomes the critical spacing gt that satisfies the following condition given by the geometric parameters of the metasurface (see the Supporting Information), 2 3  2π g t  λ d y hπ − 2 ln  − 3. = 4 gt  d x  4d x b

(1)

Shown in Figure 2b is the critical lattice spacing, predicted by Eq. (1), which is in very good agreement with the FDTD result. In particular, as the spacing gets smaller, a larger change in the wavelength is required to satisfy Eq. (1). This implies that, even if the size of the resonator is fixed, tight-coupling of the unit resonators enables just a small variation in the overall lattice to shift the resonance wavelength to longer wavelengths as well as to allow strong changes in the spectral response of metasurfaces. We also confirm that those trends can be more pronounced when the coupling is stronger. In order to discuss those behaviors more quantitatively, we rewrite Eq. (1) in terms of the resonance wavelength λres for a given spacing gx,y such that



λres =

2b2 dy

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 2g π  d x hπ − 2ln  x  + 3. d y gx  dx 

(2)

Assume that the metasurface is rearranged by changing the lattice spacing g. For simplicity, we consider square resonators with isotropic lattice spacing gx = gy = g. Then, one can readily find that infinitesimal variations in lattice size and the resonance wavelength have the following dependency, δλ res δ d δg d  hπ 2 g   hπ  2 gπ α= α , α ≡ −1 − = +2− − 2 ln    λ res d d d  g 2g  g  d

−1

   + 3 .  

(3)

Here, dx = dy = d is the lattice size. The ratio of the two variations is related by the scaling rate α which implicates two main behaviors. First, α is shown to be negative, which means that the resonance wavelength increases as the lattice spacing gets narrower. This trend is clearly demonstrated in Fig. 3a which shows resonance wavelength as a function of the lattice-spacing g and the thickness h. Equation (3) also suggests that a narrower lattice spacing accelerates the change in resonance wavelength. Shown in Fig. 3b is the scaling rate α against g and h. We note that, in principle, the scaling rate can be anomalously high since that demonstrated in Fig. 3b reaches about 500. However, we note that the shift of the resonance wavelength is also observable in loosely-coupled metasurfaces, even though the shift with varying g is less pronounced compared to that in the tightly-coupled case (see Fig. S2 in the Supporting Information). In order to experimentally verify the anomalous wavelength scaling, we used terahertz (THz) time-domain spectroscopy and performed THz transmission measurements using the setup illustrated in Fig. 4. The metasurfaces are fabricated on 500-µm-thick SiO2 substrates by


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standard electron-beam lithography and the lift-off process. Metal resonators are fabricated in a square ring style, as shown in Fig. 5a. To make the resonators, 85-nm-thick gold and 5-nm-thick Ti are deposited by electron-beam evaporation on the surface of the substrate. As shown in Fig. 5b and c, we prepared isotropic samples of two different spacing sizes gx = gy = either 500 or 1000 nm, respectively, with a fixed metal line width w = 500 nm and periodicity dx = dy = 30 µm. Measured THz time-domain signals, transmitted through a bare SiO2 substrate and the two metasurfaces, are shown in Fig. 5d. The corresponding transmission spectra of the reference and metasurfaces, obtained by applying the discrete fast-Fourier-transform to the measured timedomain signals, are shown in Fig. 5e. Each metasurface transmission spectrum is normalized by that of reference. Therefore, the transmission amplitudes of the metasurfaces are defined as Tmeta/TSiO2 where Tmeta and TSiO2 are the measured transmission amplitudes of the metamaterials and bare SiO2 substrate, respectively. The measured and FDTD-calculated transmission amplitudes of the two metasurfaces with different lattice-spacing are shown in Fig. 5f. For the metasurface sample with lattice spacing g = 1000 nm the resonant transmission dip occurs at the wavelength near 228 µm, whereas for the metasurface with g = 500 nm the dip is shifted to near 265 µm. Therefore, the scaling rate α in Eq. (3) can be approximately calculated as

α≈

∆λres d ≈ −9.7. λres ∆g

(4)

Also, we can see that our experimental result agrees well with the FDTD numerical result in terms of both the resonance wavelengths and the overall shapes of the transmission curves. We point out that the red-shift of the resonance wavelength with reduced lattice-spacing predicted by our analytic theory in Eq. (2) is clearly apparent in Fig. 5f. However, it should be noted that, due to the substrate effect21-22, the resonance wavelengths in Fig. 5f are strongly



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modified from the resonance condition in Eq. (2), which is defined for free-standing metasurfaces. Specifically, Eq. (2) predicts that, if the two metasurfaces used in Fig. 5 are freestanding, the resonance wavelengths for g = 500 and 1000 nm will be approximately 153 and 132 µm, respectively. A quantitative correction can be made by introducing the correction factor C, implicating the substrate effect on the resonant plasmonic structure through an averaging of the permittivities in the proximity of the structure22, sub = λres / C , λres

2 + 1) , C ≡ 2 / ( nsub

(5)

where nsub is the real part of the refractive index of the substrate and λres is the modified sub

resonance wavelength with the substrate. Within the spectral range of our interest, the nsub of SiO2 is 1.96. The modified resonance wavelengths given nsub = 1.96 for g = 500 and 1000 nm are therefore 239 and 206 µm, respectively, representing discrepancies of approximately 10% compared to the experimental and FDTD results in Fig. 5f. In conclusion, we have theoretically and experimentally demonstrated anomalous wavelength scaling in metasurfaces enabled by tight-coupling of unit resonators. Through analytic calculations of the model metasurface, we have rigorously defined the ratio of the normalized resonance shift and the normalized lattice deformation, and showed that this can be anomalously large, implying that the drastic shifts in resonance wavelength can be caused by deep subwavelength variations in the lattice spacing of the tightly-coupled unit resonators. In particular, from the experimental study based on THz time-domain spectroscopy, we have observed an appreciable shift in the resonance wavelength caused by applying deep sub-wavelength latticespacing variations to the metasurface. We also have revealed that such anomalous wavelength scaling does not require any structural complexity, as all results are demonstrated based on


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simple rectangular resonators. Our study enables a quantitative account of the drastic changes in the optical responses of metasurfaces resulting from strong interaction between unit resonators and gains the understanding which is essential for the engineering of functionalized metamaterials offering the dynamic manipulation of optical responses.

Acknowledgements This work was supported by the Center for Advanced Meta-Materials (CAMM) funded by the Ministry of Science, ICT and Future Planning as Global Frontier Project (CAMM2014M3A6B3063710). J. H. Kang was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2018R1C1B6009007). B. J. Kang, W. T. Kim and F. Rotermund acknowledge

support

from

NRF

funded

by

the

Korea

government

(MSIP)

(2017R1A4A1015426).

Associated Content Supporting Information. Detailed information about the analytic calculation of the resonance wavelength and the scaling factor, and supporting figures.

Competing Financial Interests The authors declare no competing financial interest.




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Figure 1. Schematics of a metasurface consisting of rectangular ring resonators. An x-polarized plane wave is normally incident on the metasurface.



Figure 2. Lattice-spacing-dependent transmission spectra. (a) Analytically and numerically calculated zero-th order transmission with b=2a, w=0.04a, and h=0.1a. (b) Wavelengthdependent critical lattice spacing with b=a, w=0.04a and h=0.05a.


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Figure 3. (a) Resonance wavelength and (b) scaling rate α against varied lattice spacing g and thickness h. Here, we considered an isotropic metasurface of b=a and w=0.04a.



Figure 4. A schematic of THz time-domain spectroscopy.


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Figure 5. THz transmission measurements. (a) SEM image of a metasurface sample of spacingsize gx,y = g = 500 nm. Enlarged images of samples with (b) g = 500 and (c) g = 1000 nm. The zoomed-in area is denoted by the yellow dotted-square in (a). (d) Experimentally measured timedomain signal for bare SiO2 substrate and metasurface samples. (e) Fourier-transformed spectra obtained from (d). (f) Spacing-dependent transmission spectra. In experimental results, transmission is defined by Tmeta/TSiO2 where Tmeta and TSiO2 are the measured transmission amplitudes of the metamaterials and bare SiO2 substrate, respectively.



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Anomalous Wavelength Scaling of Tightly Coupled Terahertz

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