Hyperbolic Dispersion via Symmetric and Antisymmetric Orderings of

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Hyperbolic dispersion via symmetric and antisymmetric orderings of artificial magnetic dipole array Xiangfan Chen, Biqin Dong, Chen Wang, Fan Zhou, and Cheng Sun ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b00934 • Publication Date (Web): 01 Oct 2018 Downloaded from http://pubs.acs.org on October 6, 2018

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Hyperbolic dispersion via symmetric and antisymmetric orderings of artificial magnetic dipole array Xiangfan Chen1, 2, Biqin Dong1, Chen Wang1, Fan Zhou1, and Cheng Sun1, † 1. Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA 2. The Polytechnic School, Arizona State University, Mesa, AZ 85212, USA † To whom correspondence should be addressed. E-mail: [email protected] Abstract: We report a theoretical and experimental study in achieving the hyperbolic dispersion via the symmetric and antisymmetric hybridization of two dimensional (2D) artificial magnetic dipoles using metasurfaces possessing artificial magnetic resonance at optical frequencies. Due to the mutual competition among the longitudinal and transverse coupling of the artificial magnetic dipoles, the metasurfaces exhibit distinct red/blue shifts for the incident wavevectors along the orthogonal directions. The resulting saddle-shape isofrequency surfaces possess the hyperbolic dispersion at the frequencies above and below the saddle-point. Our findings not only provide distinct k-space topology beyond the hyperboloid isofrequency surfaces being reported in the literature, but can also inspire new approaches for the design of hyperbolic materials. Keywords: Hyperbolic Dispersion; Metasurface; Artificial Magnetic Dipole Array TOC Graphic

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Hyperbolic metamaterials (HMMs) refer to a class of man-made materials possess extreme anisotropy, with the orthogonal permittivity or permeability tensor components of the opposite signs

1, 2

. As the

isofrequency surface opens into an open hyperboloid, it allows for propagating waves with infinitely large wavevectors. This unique property of propagating high-k waves has thus enabled a wide range of applications, including sub-diffraction imaging 3, negative refraction of the light emission engineering

6, 7

4, 5

, and spontaneous

. Recently, a visible-frequency hyperbolic metasurface composed of smooth,

high-quality silver/air grating nanostructures with high aspect ratios has been reported to realize the diffraction-free surface plasmon polariton propagation, as well as the dispersion-dependent plasmonic spin-Hall effect

8

.

Luminescent hyperbolic metasurface, in which distributed semiconducting quantum

wells display extreme absorption and emission polarization anisotropy, has also been demonstrated in an unconventional multilayer architecture

9

. Understanding the underlying mechanism and further

developing novel strategies in controlling the hyperbolic dispersion over a broad frequency range represents a long-standing interest in the research community. While the HMMs have been implemented in broad electromagnetic spectra, from microwave to optical frequencies, the common strategies in realizing hyperbolic dispersion were rather limited to the anisotropic arrangements (multilayer structures or nanowire structures) of metallic components under the effective medium approximation

1, 2

. The unique k-space topologies of the hyperboloid isofrequency

surfaces obtained from the multilayer structures and nanowire structures further result in the striking difference in the characteristic responses of Type I HMMs and Type II HMMs 2. Their distinct structureproperty relations in HMM design have thus motivated us to explore the new strategy in realizing hyperbolic dispersion other than the aforementioned approaches. Recognizing the fact that HMMs can be considered as polaritonic crystals where the coupled states of light and matter support the propagating high-k waves 2, 10, 11, we hypothesize that manipulating the interactions among the constituting polaritonic components can provide physics insights and potentially new strategies in manipulating the hyperbolic dispersion. In particular, we studied the hyperbolic dispersion originated from the symmetric and


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antisymmetric hybridization of two dimensional (2D) artificial magnetic dipoles. The resulting saddleshape isofrequency surfaces possess the hyperbolic dispersion at the frequencies above and below the saddle-point, while possessing the distinct k-space topology differs from the hyperboloid isofrequency surfaces being reported in the literature. Plasmonic metamaterial engineered in a deep subwavelength scale in visible optical frequencies has been reported to allow transverse electric polarized incidence to be strongly coupled to surface plasmons with efficiency close to 100% 12. Metamaterials compromising an array of split-ring resonators (SRRs) exhibiting artificial magnetic resonances

13-16

have been widely used to investigate the coupled

phenomena from ordered artificial magnetic dipoles. The transverse coupling between neighboring magnetic dipoles by the exchange of the conduction current was first studied in a one-dimensional (1D) array of SRRs 17. Simultaneous coupling of magnetic and electric dipole was further investigated in 2D array of SRRs, where symmetric ordering can be conveniently realized under the normal incident excitation 18. The competition between the in-plane coupling of electric dipole and out-of-plane coupling of magnetic dipole results in the observable blueshifts and redshifts of the magnetic resonance. Conversely, the antisymmetric ordering is more difficult to realize because of the alternating directions of artificial magnetic moments. One solution was reported that used a hybrid metal-dielectric structure consisting of a 2D array of silicon spheres and copper SRRs

19

. The antisymmetric arrangement of

artificial magnetic dipoles was demonstrated by detuning the resonances of the silicon spheres and SRRs. The antisymmetric coupling of artificial magnetic dipoles has also been demonstrated in a pair of two identical SRRs, and it was studied as a function of the separation gap and relative orientation between these two SRRs

20

. The stereo-SRR dimer comprising vertically coupled SRR pairs was further

demonstrated in realizing the anti-phase arrangement by the hybridized resonance of a pair of SRRs in the close-proximity

21, 22

. In a recent study, Decker et al. reported the retarded long-range interaction among

the 2D array of SRRs using oblique-incidence excitation 23. Inspired by this work, here, we explored the unique optical properties originated from the symmetric and antisymmetric hybridization of 2D artificial


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magnetic dipoles by using a metasurface possessing artificial magnetic resonance at optical frequencies. Specifically, we investigated how the varying spatial and orientational arrangements of the artificial magnetic dipoles can be controlled by altering the incident wavevectors. Furthermore, we experimentally demonstrated hyperbolic dispersion by exploiting the mutual competition among the longitudinal and transverse coupling of the 2D array of dipoles at optical frequencies.

Figure.1. (a) Schematic illustration of the metasurface comprising U-shaped nanowire resonators (UNWRs) fabricated by the nanotransfer printing process. (b) Scanning electron micrograph of UNWRs on a glass substrate; the inset shows an enlarged cross-sectional image of the UNWRs. (c) Physical dimension of the U-shaped resonator, in which the periodicity of the resonator p = 300 nm, the base thickness b = 40 nm, two freestanding prongs with height h = 75 nm and thickness t = 35 nm, and the gap width g = 90 nm. (d) Schematic illustration of the metasurface for manipulating the coupling between artificial magnetic dipoles by changing the incident wavevector. (e) The projection of an artificial magnetic dipole array at the resonant frequency corresponding to the situation in (d). Fig. 1a illustrates the metasurface, which comprises a large array of U-shaped nanowires resonators (UNWRs) fabricated on a glass substrate using a nanotransfer printing method (see Supplemental 4 ACS Paragon Plus Environment

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Material for details on fabrication processes). The metasurface covers an area of 25×25 mm2 with the Ushaped cross-section aligned vertically (Fig. 1b). The U-shaped nanowire structure has a periodicity of 300 nm, which is significantly smaller than the targeting wavelength range from 650 nm to 900 nm. Thus, all the diffracted modes are in the form of non-propagating evanescent waves and their contribution to the far-field measurements can be neglected. We previously reported that vertical UNWRs support continuously geometric scaling of magnetic resonance towards optical frequencies

14, 24

. When excited

under linearly polarized light with the magnetic field parallel to the length of the UNWRs, the induced magnetic resonance produces strongly amplified, coherent magnetic dipole moments along the length of the UNWRs. Here, the collective behavior of the coupled dipoles is systematically studied by tailoring the longitudinal and transverse coupling of artificial magnetic dipoles within the UNWR metasurface. As illustrated in Fig. 1d, the oblique incident planewave results in the spatially alternating phase along the metasurface, with the corresponding in-plane wavevector ∥ = , = ∙ ∙ , ∙ ∙ , where is the free-space wavevector, and are the corresponding incident angles, respectively. The induced 2D array of magnetic dipoles are spatially distributed with their phase coherent to the incident electromagnetic fields (Fig. 1e). Their spatial and orientational ordering can thus be reconfigured by controlling the angle of incidence. The transverse coupling (TC) states and the longitudinal coupling (LC) states coexist in the 2D array of elementary magnetic dipole moments, and they compete mutually to determine the hybridized modes of the UNWR metasurface 10, 25 (see Supplemental Material for details on the hybridization model of magnetic dipoles). Fig. 2a illustrates the energy level of the resulting hybridized modes. We define the symmetric order as the ground state , , which is excited by the normal incident light with the magnetic field polarized along the length of the UNWR. Its resonant frequency is determined by the competing symmetric LC (s-LC) and symmetric TC (s-TC) modes resulting from the in-phase arrangement of the 2D magnetic dipole moments. The magnetic field distribution (Hx) in the xy-plane was simulated using a commercial software (FDTD Solutions, Lumerical). As shown in Fig. 2a, the magnetic fields are localized 5 ACS Paragon Plus Environment

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within the UNWRs due to the presence of the magnetic resonance. The in-phase arrangement of the induced magnetic dipoles resembles the SM order.

Figure. 2. (a) Energy-level diagram of the hybridized modes, including , , , , , , and the corresponding configurations of magnetic dipoles at resonant frequency under incidence with varying in-plane wavevectors (kx, ky). (b) Calculated dispersion relation of interchange energy for the L-


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ASM ( , ) and T-ASM ( , ) modes. (c) The calculated saddle-shape dispersion surface of the hybridized states, and (d) the corresponding isofrequency contours for the unique hyperbolic dispersions. Changing the incident angle then results in a non-zero in-plane wavevector, thereby selectively exciting the corresponding antisymmetric order in a 2D array of magnetic dipoles. The plane waves incident in the xz-plane excite the longitudinally antisymmetric (L-ASM) order , ) at the optical frequencies. Its resonant frequency is determined by the competing symmetric TC (s-TC) and antisymmetric LC (a-LC) modes and thus, blueshifts with respective to the ground state , . The FDTD simulation of the magnetic field distribution (Hx) is shown in Fig. 2a for the cases of in-plane wavevector ∥ = 0.25 ∙ , 0 and ∥ = 0.5 ∙ , 0; the formation of the antisymmetric order along the longitudinal direction can be clearly seen. The interaction energy between the neighboring dipoles is inversely proportional to the inter-dipole spacing, which is determined by the in-plane wavevectors. Similarly, the plane waves incident in the yz-plane excite the transversely antisymmetric (T-ASM) order , ). The competing antisymmetric TC (a-TC) and symmetric LC (s-LC) modes result in the redshift of its resonant frequency respective to the ground state , . The FDTD simulation of the cases of ∥ = 0, 0.25 ∙ and ∥ = 0, 0.5 ∙ reveal the formation of the antisymmetric order along the transverse direction. Overall, the L-ASM state , and the T-ASM state , represents the upper and lower bound of the system’s energy among all the possible hybridized modes, respectively. The light incident from an arbitrary angle represents a more general case, which results in the non-zero x- and y-components of the in-plane wavevectors. The resulting biaxially antisymmetric (BASM) order is characterized by the presence of the antisymmetric order in both the longitudinal and transverse directions. All four hybridized modes (s-TC, a-TC, s-LC, and a-LC) coexist in the B-ASM orders and the resonant frequency of B-ASM state /,& is determined by the mutual competition among them. The FDTD simulation of the magnetic field distributions (Hx) in the xy-plane are shown for


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the cases of ∥ = 0.25 ∙ , 0.25 ∙ and ∥ = 0.5 ∙ , 0.5 ∙ . It is worthwhile to note that reconfiguring the symmetric and antisymmetric orders can be accomplished without needing to physically alter the structure of the UNWR metasurface. This represents a clear advantage to the previously reported methods, such as the use of stereo-metamaterials

22

and hybrid metamaterials 19, which rely on the pre-

fabricated spatial arrangements of the magnetic dipoles. Consequently, controlling the spatial and orientational order of the artificial magnetic dipoles allows for effective manipulation of the quasi-static interaction energy H (Supplemental Materials). The resulting resonant frequency of the corresponding symmetric and antisymmetric modes can be calculated using the hybridization model (Supplemental Material)

10, 25

. The calculated dispersion relation of interchange

energy for the L-ASM and T-ASM modes are plotted in Fig. 2b. As expected, the resonant frequencies of the L-ASM mode , increase monotonically with the increase of in-plane wavevector (kx). In contrast, the resonant frequencies of the T-ASM mode , decrease monotonically with the increase of in-plane wavevector (ky). As a result, the calculated dispersion surface of the hybridized states (Fig. 2c) possesses a distinct saddle-shape, which results in the unique hyperbolic dispersion relations (Fig. 2d). The open form of the isofrequency contours allows for propagating waves with infinitely large wavevectors and evanescent waves do not even exist. This unique property of propagating high-k waves gives rise to a multitude of novel applications such as engineering the broadband photonic density of states and sub-diffraction-limited imaging

8, 26-30

. Thus, the reported UNWR metasurface offers a

fundamentally new capability in controlling the extraordinary optical properties. The dispersion relation of the UNWR metasurface was further calculated numerically using the FDTD method (Supplemental Materials). The experimentally measured complex permittivity of gold was used in the simulation 31. As illustrated in Fig. 3a, the plane waves incident in the xz-plane excite the LASM ordering of the 2D magnetic dipole array. The simulated transmission spectra of the UNWR metasurface with varying in-plane wavevectors is shown in Fig. 3b, where the magnetic resonance can be characterized as the minimum of the transmission spectra. Note that strong magnetic resonance within the 8 ACS Paragon Plus Environment

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UNWRs is evident at f0 = 366.94 THz (λ = 817 nm) under normal incidence. For oblique incidence, the resonant frequencies of the L-ASM modes blueshift monotonically with the increase of in-plane wavevector, which is consistent with the analytical model. A pronounced blueshift of magnetic resonance to 382.88 THz (λ = 783 nm) is observed for ∥ = 0.5 ∙ , 0. The simulated distribution of the magnetic field in the xz-plane, shown in Fig. 3c, further validates the reconfigurable symmetric and antisymmetric orders by changing the incident wavevectors.


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Figure. 3. Schematic illustration of realizing L-ASM (a) and T-ASM (d) by sweeping incident angle ϕ in xz-plane and yz-plane, respectively. 2D contours of transmission spectra under oblique incidence with normalized in-plane wave vectors kx/k0 (b) and ky/k0 (e), the brightness of each point denotes transmitted intensity. White lines denote resonant frequencies and the red circles are chosen for calculating magnetic field distribution. Magnetic field distribution Hx in xz-plane (c) and yz-plane (f) at resonant frequency with varying in-plane wavevectors k//. (g) Saddle-shape dispersion relation for the magnetic resonance in the UNWR metasurface. (h) Isofrequency contours for hyperbolic dispersion relation in the frequency range 364 THz to 384 THz. On the other hand, Fig. 3d illustrates the case when the plane waves incident in the yz-plane were used to excite the T-ASM order. The simulated transmission spectra with varying in-plane wavevectors is shown in Fig. 3e. The resonant frequencies of the T-ASM modes redshift monotonically with the increase of in-plane wavevector, which is also consistent with the analytical model. The simulated distribution of the magnetic field in the yz-plane shown in Fig. 3f further validates the capability of the reconfiguring the symmetric and antisymmetric orders by changing the incident wavevectors. The reduced redshift to 362.51 THz (λ=827 nm) at ∥ = 0, 0.5 ∙ is likely due to the relatively low packing density of the discretized arrangement of UNWRs along the y-direction. The dispersion surface obtained via the numerical simulation also exhibits the unique saddle shape (Fig. 3g), which again leads to the hyperbolic dispersion shown in Fig. 3h.


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Figure. 4. (a) Experimental setup for angle-resolved transmission spectra measurement. (b) and (c) 2D contour of the experimentally measured transmission spectra for the L-ASM mode with sweeping kx/k0 and the T-ASM mode with sweeping ky/k0 , respectively. (d) Dispersion curves for the magnetic resonance in the UNWRs metasurface. Blue (red) curves are the simulation results of the L-ASM (T-ASM) mode, and the blue circles (red square) are the experimental results of the L-ASM (T-ASM) mode. The fabricated UNWR metasurface was experimentally characterized using angle-resolved optical spectroscopy, as illustrated in Fig. 4a (For details, see Supplemental Material) 32. The broadband emission from a halogen lamp was illuminated on the metasuface sample using a condenser lens (NA = 0.55). The transmission was then collected using an objective lens (NA = 0.8). The resulting back focal plane image was subsequently projected onto the entrance slit of a grating spectrometer; thus, the angular-dependent transmission spectrum can be measured simultaneously. The measured angular-dependent transmission of the corresponding , and , modes mapped on the wavevector-frequency plane are


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shown in Fig. 4b and 4c, respectively. Under the normal incidence, the distinct magnetic resonance is recognized as the dip in the transmission spectrum centered at f0 = 366.94 THz (λ = 817 nm). As expected, the planwaves incident in the xz-plane excite the , modes, resulting in an pronounced blueshift to 379.48 THz (λ = 790 nm) of the resonant frequency shown in Fig. 4b. In contrast, the resonant frequencies redshift to 362.95 THz (λ = 826 nm) with the planwaves incident in yz-plane exciting the , mode (Fig. 4c). As shown in Fig. 4, the experimentally measured dispersion relations are in good agreement with the numerical simulations. The observed opposite trends in the resonant frequency shifts along the orthogonal directions result in the unique saddle-shape dispersion surface. It is worthwhile to note that the saddle-surface provide a new k-space topology which possesses the hyperbolic dispersion at the frequencies above and below the saddle-point, and it’s different from the two topologies being reported previously 2. This work provides new opportunities in further manipulating the unique hyperbolic dispersions that exhibit the characteristic open-form isofrequency contours. The unique property of the supporting propagating waves with infinitely large wavevectors will enable a broad range of device applications 8, 26-28. Conventional wisdom suggests that realization of the hyperbolic dispersions requires the highly anisotropic electric permittivity or magnetic permeability tensors with the subjecting tensor components exhibiting opposite signs 8, 26-28. Here, it is worthwhile to note that our structure provides the possibility to experimentally achieve hyperbolic dispersion by exploiting the mutual competition among the longitudinal and transverse coupling of the 2D array of dipoles at optical frequencies. Furthermore, the spatially confined electromagnetic field associated with magnetic resonance offers promising opportunities in enabling the dynamic tuning of the optical properties using enhanced nonlinear lightmatter interactions

14

. In conclusion, the symmetric and antisymmetric hybridization of 2D artificial

magnetic dipoles reported here promises rich physical behaviors, e.g. topological transition process 33, and novel engineering applications, e.g. manipulation of the surface waves

8, 9

, arising from the properties of

magnetic dipoles and their interactions prompted by their spatial and orientational ordering. 12 ACS Paragon Plus Environment

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MATERIALS AND METHODS Sample fabrication. The nanotransfer printing (nTP) process flow for sample fabrication is schematically illustrated in Fig. S1. The poly[(mercaptopropyl)methylsiloxane] (PMMS, United Chemical Technologies) based soft stamp containing periodical grating with the periodicity of 300 nm was replicated from a silicon mold using the nanoimprinting process. Two consecutive angular depositions of Au at 45° incident were used to form the U-shaped resonator on the surface of the stamp, which was subsequently bonded to the glass substrate using a thermally activated or UV-cured epoxy layer. The PMMS template was then peeled off after the curing of the epoxy layer, leaving the arrayed U-shaped resonator strongly bonded to the glass substrate. Numerical simulation. Full-field electromagnetic wave calculations were performed using FDTD Solutions (Lumerical), a commercially available, finite-difference time-domain (FDTD) simulation software package. The 3D simulations were performed for the metasurface with an 8 units area of 2400 × 2400 nm2 at the xy-plane using periodic boundary conditions, and the plane wave was incident with an oblique angle ( and ) with respect to the z-direction. Magnetic field distributions at resonance were detected by 2D field profile monitors in the xy-, xz-, yz-plane. Optical characterization. The transmission spectra of the UNWR metasurface was characterized using angle-resolved optical spectroscopy combined with an inverted optical microscope (DMI 3000M, Leica) and matching grating spectrometer (SR-303i, Andor Technology). A high numerical aperture (NA) objective lens (50x, NA = 0.8) was used to collect all the light propagating through the sample, with a maximum incident angle determined by the condenser’s NA (NA = 0.55). A 100W halogen lamp was used as the illumination source, while a rotating linear polarizer (LPVISB100, Thorlabs) was used to control the polarization of the incident light. The transmission spectra were normalized with respect to the bare glass substrate.


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ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website. Detailed description of the sample fabrication, hybridization model of magnetic dipoles in two dimensions, numerical simulation of transmission spectra under oblique incidence, angle-resolved spectra measurement. AUTHOR INFORMATION Corresponding Author Cheng Sun, E-mail: [email protected] ORCID Xiangfan Chen: 0000-0002-5627-7530 Biqin Dong: 0000-0003-2283-4498 Cheng Sun: 0000-0002-2744-0896 ACKNOWLEDGEMENTS This work is supported by the National Science Foundation (NSF) under Grant number EEC-1530734 and DBI-1353952. This work utilized Northwestern University Micro/Nano Fabrication Facility (NUFAB), which is partially supported by Soft and Hybrid Nanotechnology Experimental (SHyNE) Resource (NSF ECCS-1542205), the Materials Research Science and Engineering Center (NSF DMR1720139), the State of Illinois, and Northwestern University. Competing interests The authors declare no competing financial interests.


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(23) Decker, M.; Feth, N.; Soukoulis, C. M.; Linden, S.; Wegener, M. Retarded long-range interaction in split-ring-resonator square arrays. Phys Rev B 2011, 84, 085416. (24) Zhou, F.; Wang, C.; Dong, B. Q.; Chen, X. F.; Zhang, Z.; Sun, C. Scalable nanofabrication of Ushaped nanowire resonators with tunable optical magnetism. Opt Express 2016, 24, 6367-6380. (25) Liu, N.; Giessen, H. Coupling Effects in Optical Metamaterials. Angew Chem Int Edit 2010, 49, 9838-9852. (26) Kruk, S. S.; Wong, Z. J.; Pshenay-Severin, E.; O'Brien, K.; Neshev, D. N.; Kivshar, Y. S.; Zhang, X. Magnetic hyperbolic optical metamaterials. Nat Commun 2016, 7, 11329. (27) Poddubny, A.; Iorsh, I.; Belov, P.; Kivshar, Y. Hyperbolic metamaterials. Nat Photonics 2013, 7, 948-957. (28) Lu, D.; Kan, J. J.; Fullerton, E. E.; Liu, Z. W. Enhancing spontaneous emission rates of molecules using nanopatterned multilayer hyperbolic metamaterials. Nat Nanotechnol 2014, 9, 48-53. (29) Jacob, Z.; Kim, J. Y.; Naik, G. V.; Boltasseva, A.; Narimanov, E. E.; Shalaev, V. M. Engineering photonic density of states using metamaterials. Appl Phys B-Lasers O 2010, 100, 215-218. (30) Mirmoosa, M. S.; Kosulnikov, S. Y.; Simovski, C. R. Magnetic hyperbolic metamaterial of highindex nanowires. Phys Rev B 2016, 94, 075138. (31) Johnson, P. B.; Christy, R. W. Optical Constants of Noble Metals. Phys Rev B 1972, 6, 4370-4379. (32) Wagner, R.; Heerklotz, L.; Kortenbruck, N.; Cichos, F. Back focal plane imaging spectroscopy of photonic crystals. Appl Phys Lett 2012, 101, 081904. (33) Krishnamoorthy, H. N. S.; Jacob, Z.; Narimanov, E.; Kretzschmar, I.; Menon, V. M. Topological Transitions in Metamaterials. Science 2012, 336, 205-209.


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Figure Captions Figure.1. (a) Schematic illustration of the metasurface comprising U-shaped nanowire resonators (UNWRs) fabricated by the nanotransfer printing process. (b) Scanning electron micrograph of UNWRs on a glass substrate; the inset shows an enlarged cross-sectional image of the UNWRs. (c) Physical dimension of the U-shaped resonator, in which the periodicity of the resonator p = 300 nm, the base thickness b = 40 nm, two freestanding prongs with height h = 75 nm and thickness t = 35 nm, and the gap width g = 90 nm. (d) Schematic illustration of the metasurface for manipulating the coupling between artificial magnetic dipoles by changing the incident wavevector. (e) The projection of an artificial magnetic dipole array at the resonant frequency corresponding to the situation in (d). Figure. 2. (a) Energy-level diagram of the hybridized modes, including , , , , , , and the corresponding configurations of magnetic dipoles at resonant frequency under incidence with varying in-plane wavevectors (kx, ky). (b) Calculated dispersion relation of interchange energy for the LASM ( , ) and T-ASM ( , ) modes. (c) The calculated saddle-shape dispersion surface of the hybridized states, and (d) the corresponding isofrequency contours for the unique hyperbolic dispersions. Figure. 3. Schematic illustration of realizing L-ASM (a) and T-ASM (d) by sweeping incident angle ϕ in xz-plane and yz-plane, respectively. 2D contours of transmission spectra under oblique incidence with normalized in-plane wave vectors kx/k0 (b) and ky/k0 (e), the brightness of each point denotes transmitted intensity. White lines denote resonant frequencies and the red circles are chosen for calculating magnetic field distribution. Magnetic field distribution Hx in xz-plane (c) and yz-plane (f) at resonant frequency with varying in-plane wavevectors k//. (g) Saddle-shape dispersion relation for the magnetic resonance in the UNWR metasurface. (h) Isofrequency contours for hyperbolic dispersion relation in the frequency range 364 THz to 384 THz.


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Figure. 4. (a) Experimental setup for angle-resolved transmission spectra measurement. (b) and (c) 2D contour of the experimentally measured transmission spectra for the L-ASM mode with sweeping kx/k0 and the T-ASM mode with sweeping ky/k0 , respectively. (d) Dispersion curves for the magnetic resonance in the UNWRs metasurface. Blue (red) curves are the simulation results of the L-ASM (T-ASM) mode, and the blue circles (red square) are the experimental results of the L-ASM (T-ASM) mode.


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Hyperbolic Dispersion via Symmetric and Antisymmetric Orderings of

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