Dynamically tunable, high-quality transmission metasurface

Abstract

A system and method for a dynamically tunable metasurface having at least 10 quality factors are described. The metasurface operates in transmit mode. The metasurface can be used for wavefront shaping and beam steering.

Description

Government-funded research

[0001] This invention was made with government support under Grant No(s): FA9550-21-1-312 and FA9550-18-1-354, granted by FA9550-18-1-0354. The U.S. Government has certain rights to this invention. [Technical Field]

[0002] The present invention relates, in general, to a system and method for a high-quality coefficient metasurface for dimensional wavefront manipulation. [Background technology]

[0003] The regulation of electromagnetic waves by traditional optical components such as lenses and prisms is achieved through the accumulation of phase delays during the process of light propagation, limiting the decompression and integration of optical devices. In wavefront modulation, control of phase and amplitude plays a crucial role. Traditional optical elements, like diffractometers such as gratings and holograms, can be bulky due to their optical configuration. Metasurfaces can modify amplitude and impart abrupt phase shifts to incident waves within subwavelength scales through light-matter interactions, thus enabling more efficient wavefront modulation.

[0004] In optical metasurfaces, the phase, amplitude, polarization, and spectrum of light at the interface can be abruptly manipulated using a subwavelength-spacing array of localized resonators. In some cases, strong photomatter interactions, and therefore high quality factors, are desirable in metasurfaces. However, the required subwavelength-scale wavefront control imposes limitations on resonator size and results in significant radiative losses. Consequently, most metasurfaces are broadband and rely on dielectric structures with limited optical confinement and therefore low quality factors (Q-factor less than approximately 15). A low quality factor means that photon residence times are very short, and therefore, local electromagnetic fields tend to be small.

[0005] Active metasurfaces can dynamically control the wavefront of scattered light on a subwavelength scale. Most active metasurfaces that enable dynamic wavefront shaping operate on reflection. Active metasurfaces that operate on transmission are of considerable interest because they can be integrated with chip-scale light sources, resulting in compact wavefront shaping devices. Achieving dynamically tunable metasurfaces with high quality factors is challenging. [Overview of the project]

[0006] Many embodiments relate to systems of low-loss active metasurfaces that can dynamically manipulate the transmitted light wavefront. In some embodiments, dynamically tunable metasurfaces can be fabricated from dielectric materials having a quality factor of at least 100. In many embodiments, the metasurface manipulates light in transmission mode. Dynamically tunable metasurfaces can manipulate light across a wide range of wavelengths, from ultraviolet to visible to near-infrared to infrared wavelengths.

[0007] Some embodiments include an electromagnetic metasurface comprising a plurality of periodic repeating unit cells conformally arranged on a substrate, wherein the periodicity is smaller than the wavelength of the working light in free space, and each of the plurality of repeating unit cells comprises a first substrate on a second substrate and a nanostructure on the first substrate, wherein the apparatus controls the phase of the working light in a transmission mode having at least 10 quality factors, the apparatus converts a transmission dip into a transmission peak in a quality factor resonant sound, and the length of the nanostructure, the width of the nanostructure, the height of the nanostructure, and the periodicity adjust the quality factors by changing at least one parameter selected from the group.

[0008] In some embodiments, the wavelength is selected from the group consisting of ultraviolet wavelengths from 100 nm to 400 nm, visible wavelengths from 380 nm to 800 nm, near-infrared wavelengths from 800 nm to 2500 nm, and infrared wavelengths from 780 nm to 1000 μm. In some embodiments, multiple repeating unit cells are arranged in an array.

[0009] In some embodiments, the shape of the nanostructure is selected from the group consisting of cuboids, rectangular prisms, prisms, pillars, elliptical pillars, trapezoids, triangular prisms, polygonal prisms, pyramidal pyramids, and combinations thereof.

[0010] In some embodiments, the nanostructure and the second substrate each include a lossless dielectric material having an imaginary refractive index of 0.5 or less at the operating wavelength.

[0011] In some embodiments, the first substrate includes a material having a real refractive index smaller than the real refractive index at the operating wavelength of the nanostructure.

[0012] In some embodiments, the nanostructures include materials selected from the group consisting of gallium arsenide, gallium phosphide, silicon carbide, titanium oxide, silicon nitride, barium titanate, lithium niobate, tantalum pentoxide, silicon oxide, amorphous silicon, silicon, and combinations thereof.

[0013] In some embodiments, the first substrate includes a material selected from the group consisting of glass, silicon oxide, silicon nitride, gold, silver, aluminum, copper, titanium, platinum, indium tin oxide, aluminum tin oxide, aluminum zinc oxide, aluminum zinc oxide, magnesium fluoride, tantalum pentoxide, zirconium oxide, vanadium oxide, germanium-antimony-tellurium alloy, titanium nitride, hafnium oxide, hafnium nitride, molybdenum diselenide, hexagonal boron nitride, black phosphorus, tungsten diselenide, tungsten disulfide, and combinations thereof.

[0014] In some embodiments, the second substrate includes a material selected from the group consisting of gold, gallium arsenide, gallium phosphide, silicon carbide, titanium oxide, silicon nitride, barium titanate, lithium niobate, tantalum pentoxide, silicon oxide, amorphous silicon, crystalline silicon, silicon, and combinations thereof.

[0015] In some embodiments, the wavelength is a near-infrared wavelength from 800 nm to 2500 nm, the height of the nanostructure is 860 nm or less, the quality factor is 100 to 9800, and the device is configured to be part of a wavefront shaping system.

[0016] In some embodiments, a change in the first substrate thickness adjusts the spectral shape of the transmittance.

[0017] In some embodiments, thermo-optic modulation of the refractive index of the nanostructure by 0.001 to 0.01 forms the transmitted light wavefront.

[0018] In some embodiments, the device is configured to be part of a dynamic beam steering system.

[0019] In some embodiments, the dynamic beam steering system operates with transverse electric (TE) polarization or transverse magnetic (TM) polarization.

[0020] In some embodiments, a row of multiple nanostructures is connected to form an interconnected structure; at least two surfaces of the interconnected structure are conductive; and the interconnected structure is separated from a first substrate via multiple pillars.

[0021] In some embodiments, the interconnected structures are heated via a voltage applied to a conductive surface so that the refractive index of the nanostructure is thermo-optically modulated, and multiple pillars prevent thermal crosstalk.

[0022] In some embodiments, the conductive surface includes a material selected from the group consisting of doped semiconductors, doped compound semiconductors, metals, metal alloys, doped gallium arsenide, doped gallium prinide, and doped amorphous silicon.

[0023] In some embodiments, the multiple pillars include materials selected from the group consisting of glass, silicon oxide, silicon nitride, gold, silver, aluminum, copper, titanium, platinum, indium tin oxide, aluminum tin oxide, aluminum zinc oxide, aluminum zinc oxide, magnesium fluoride, tantalum pentoxide, zirconium oxide, vanadium oxide, germanium-antimony-tellurium alloy, titanium nitride, hafnium oxide, hafnium nitride, molybdenum diselenide, hexagonal boron nitride, black phosphorus, tungsten diselenide, tungsten disulfide, and combinations thereof.

[0024] Some embodiments further include a light source located on the opposite side of the second substrate from the first substrate.

[0025] In some embodiments, the light source is a chip-scale laser. Additional embodiments and features are described in part below, some of which will become apparent to those skilled in the art by examining this specification, or may be known by practicing the disclosure. A further understanding of the nature and merits of the disclosure may be achieved by referring to the remainder of this specification and the drawings that form part of the disclosure. [Brief explanation of the drawing]

[0026] This description is presented as an exemplary embodiment of the invention and should not be construed as a complete enumeration of the scope of the invention; it will be better understood with reference to the following drawings. Note that the patent or application file shall include at least one drawing drawn in color. A copy of this patent or patent application publication accompanied by a color drawing will be provided by the Patent and Trademark Office upon request and payment of the required fees.

[0027] [Figure 1] Figures 1A to 1E show a total dielectric high-Q metasurface in transmission mode according to one embodiment. [Figure 2] Figures 2A to 2D show the dependence of the high-Q mode characteristics on the height of an a-Si prism according to one embodiment. [Figure 3] Figures 3A to 3F show a metasurface structure having a transmittance peak according to one embodiment. [Figure 4] Figures 4A to 4H show beam steering using a lower Q-factor mode according to one embodiment. [Figure 5] Figures 5A to 5F show thermo-optic beam switching using a high-Q mode with an interconnected metasurface according to one embodiment. [Figure 6] Figures 6A to 6C show the electrical and thermal analysis of thermo-optical beam switching according to one embodiment. [Figure 7] Figures 7A and 7B show a thermo-optic three-level phase grating with a realistic interconnection architecture and optimized geometry according to one embodiment. [Figure 8] Figures 8A and 8B show the spatial distribution of electric field amplitude inside a metasurface unit cell according to one embodiment. [Figure 9] Figures 9A to 9D show the spatial distribution of electric field amplitude inside a metasurface unit cell according to one embodiment. [Figure 10] Figures 10A and 10B show the optical reaction of an a-Si pillar array on an SiO2 substrate according to one embodiment. [Figure 11] Figure 11 illustrates a spatial electric field profile for lower Q modes supported by a metasurface in the xz plane, according to one embodiment. [Figure 12] Figure 12 illustrates a spatial electric field profile of a lower Q mode supported by a metasurface in the yz plane, according to one embodiment. [Figure 13] Figures 13A and 13B show the optical response of an a-Si pillar array suspended in air according to one embodiment. [Figure 14] Figures 14A and 14B show the transmittance and phase spectra of an array of a-Si pillars in air according to one embodiment. [Figure 15]Figures 15A to 15D show the optical reaction of an a-Si pillar array on an SiO2 substrate when the incident electric field is x-polarized, according to one embodiment. [Figure 16] Figures 16A to 16D show the optical reaction of an a-Si pillar array on an SiO2 substrate when the incident electric field is x-polarized, according to one embodiment. [Figure 17] Figures 17A and 17B show the scattering cross-section of a single a-Si pillar in air according to one embodiment, with wavelength and pillar height as functions. [Figure 18] Figures 18A and 18B show the scattering cross-sections of isolated pillars on an SiO2 substrate according to one embodiment, with respect to wavelength and pillar height. [Figure 19] Figure 19 shows the scattering cross-section of a single a-Si pillar on an SiO2 substrate according to one embodiment, with respect to wavelength and pillar height. [Figure 20] Figures 20A to 20F show the spatial distribution of the x and y components of the electric field E in the xz plane according to one embodiment. [Figure 21] Figures 21A to 21D show the spatial distribution of the x-component of the electric field E in the xz-plane according to one embodiment. [Figure 22] Figure 22 shows the scattering cross-section of a single a-Si pillar on an SiO2 substrate according to one embodiment, with respect to wavelength and pillar height. [Figure 23] Figures 23A and 23B show the transmittance and phase of transmitted light according to one embodiment, depending on the wavelength and thickness of the spacer. Figures 23C and 23D show the transmittance and phase spectra for various SiO2 spacer thicknesses according to one embodiment. [Figure 24] Figure 24 shows the quality coefficient and Fano phase of high-Q resonant tones according to the SiO2 thickness d in one embodiment. [Figure 25] Figures 25A to 25D show the spatial distribution of electric field amplitude in a metasurface unit cell according to one embodiment. [Figure 26]Figure 26 illustrates the functions of the a-Si exponential change achieved by using a high-Q resonant sound according to one embodiment, specifically the transmittance and phase shift. [Figure 27] Figures 27A to 27F show dynamic beam switching using a realistic interconnection architecture according to one embodiment. [Figure 28] Figures 28A to 28C show the dependence of transmittance and phase with respect to the height of a column and a rod according to one embodiment. [Figure 29] Figures 29A to 29D show transmittance and phase shift as functions of the a-Si index change for the metaelectrode according to one embodiment. [Figure 30] Figures 30A to 30D show the transmittance and phase shift as functions of the a-Si index change for the meta electrode according to one embodiment. [Figure 31] Figures 31A to 31D show thermo-optic beam switching using a lower Q mode according to one embodiment. [Figure 32] Figures 32A to 32D show analytical array factor calculations for a two-level phase grid according to one embodiment. [Figure 33] Figures 33A and 33B show the electric field strength in the far field as a function of the rudder angle in the case of a three-level phase grid according to one embodiment. [Figure 34] Figure 34 shows the electric field strength in the far field as a function of steering angle according to one embodiment. [Figure 35] Figures 35A and 35B illustrate transmittance and phase shift as functions of the a-Si index change on an optimized metasurface by an electrode, according to one embodiment. [Figure 36] Figure 36 shows a thermo-optic three-level phase grating with a realistic interconnection architecture and optimized geometry according to one embodiment. [Figure 37] Figures 37A to 37C show the quality coefficients of a metasurface having a finite number of elements according to one embodiment. [Figure 38]Figures 38 to 38F show the effect of inclined sidewalls on metasurface performance according to one embodiment. [Figure 39] Figures 39A to 39D show the effect of rounded corners on metasurface performance according to one embodiment. [Figure 40] Figures 40A to 40F show the effect of material loss on metasurface performance according to one embodiment. [Figure 41] Figures 41A and 41B show a highly efficient permeable metasurface according to one embodiment.

[0028] Chip-scale microoptical components that can be dynamically programmed to scatter or emit light with wavefronts of arbitrary shapes can be applied to a variety of applications such as light detection and ranging (LiDAR), free-space optical communications, additive manufacturing, and directed energy. Active metasurfaces can be used in chip-scale spatial optical modulators. A prototype active metasurface may include an array of geometrically identical subwavelength resonant metasurface elements whose phase and amplitude of the light scattered by each metasurface element can be dynamically controlled. (See, for example, P. Thureja et al., Nanophotonics 2022, 11, 3745; this disclosure is incorporated by reference.) A programmable metasurface chip with field-effect control of the phase of the light scattered by each metasurface element has been shown. (See, for example, J. Park, et al., Nature Nanotechnology 2021, 16, 69; this disclosure is incorporated by reference.) Dynamic control of phase at the subwavelength scale may enable experimental implementations of beam steering and reconfigurable focusing using the same metasurface structure. (GK Shirmanesh, et al., ACS Nano 2020, 14, 6912; its disclosure is incorporated herein by reference) However, these active metasurfaces operate in reflection. Operation in reflection mode dictates that the illumination source be located off-chip, which can increase the shape factor of the resulting optical system. Furthermore, a reflection configuration relative to the illumination source can limit the versatility of the system, as a portion of the metasurface aperture will be blocked by the light source. In comparison, transmissive metasurfaces can result in more compact monolithic optical systems because they allow integration with chip-based light sources such as vertical cavity surface-emitting lasers (VCSELs) or photonic cavity surface-emitting lasers (PCSELs).

[0029] We experimentally demonstrated dynamically amplitude-tunable wavelength-transmitting metasurfaces using Ito field-effect modulation, electrochemical transitions of conductive polymers, ion transport, the Pockels effect in organic molecules, and thermo-optic effects in silicon (Si). (For example, Y. Lee, et al., Advanced Optical Materials 2020, 8, 2001256; A. Howes, et al., Optica 2018, 5, 787; J. Karst, et al., Science 2021, 374, 612; K. Thyagarajan et al., Advanced Materials 2017, 29, 1701044; I.-C. Benea-Chelmus, et al., Nature Communications 2021, 12, 5928; K. Zangeneh Kamali, et al., Light: Science & Applications 2023, 12, 40; their disclosures are incorporated by reference.) Transmittance modulation is used to achieve diffraction beam switching with reasonable efficiency. (A. She, et al., Science Advances 2018, 4, eaap9957; E. Arbabi, et al., Nature Communications 2018, 9, 812; these disclosures are incorporated by reference.) However, dynamically tunable phase control is also a prerequisite for versatile wavefront manipulation. Experiments have demonstrated dynamic beam switching in transmission using liquid crystal reorientation, in which some degree of phase control can contribute to dynamic beam switching. (S.-Q. Li, et al., Science 2019, 364, 1087; its disclosures are incorporated by reference.) Previous studies have demonstrated dielectric elastomer actuators and micro-electromechanical systems for adaptive metalenses, or thermo-optically tunable Si nonlocal metasurfaces operating at near-infrared wavelengths. (See, for example, SC Malek, et al., Nanophotonics 2021, 10, 655; its disclosure is incorporated by reference), however, these studies do not enable arbitrary wavefront reconstruction of transmitted light.

[0030] Several designs for dynamically tunable transmission metasurfaces may use field-effect control of the transmission phase to dynamically shape the transmitted light wavefront, but with low optical efficiency (<0.07%), limiting their use in practical applications. (e.g., A. Forouzmand, et al., Nanophotonics 2019, 8, 415; J. Park, et al., Appl.Opt.2018, 57, 6027; these disclosures are incorporated by reference) Theoretical studies have demonstrated dielectric transmission metasurfaces that achieve subwavelength phase control in transmission and dynamic wavefront shape using carrier injection in Si. The reported optical efficiency of the metasurface is approximately 65%. The maximum phase shift reported for a one-dimensional phase gradient is 215° when the assumed refractive index Δn of silicon is approximately 0.01. (See, for example, A. Forouzmand, et al., Laser & Photonics Reviews 2020, 14, 1900353; its disclosure is incorporated by reference.) Inverse design may be used for highly directional beam steering using all-dielectric transmitting metasurfaces, such as those based on the reorientation of liquid crystal molecules. (See, for example, H. Chung, et al., ACS Photonics 2020, 7, 2236; its disclosure is incorporated by reference.) It has been shown that the optical anisotropy of the active liquid crystal medium surrounding the metasurface can be used to achieve large phase modulation in transmission while maintaining approximately 100% optical efficiency. (See, for example, Z. Yang, et al., Advanced Optical Materials 2022, 10, 2101893; its disclosure is incorporated by reference.) The concept of homologous dipoles may be used to develop dynamic phase-shifting metasurface designs that enable transmission phase modulation of over 240° while maintaining transmittance of over 80%. (See, for example, S. Zhuo, et al., Laser & Photonics Reviews 2023, 17, 2200403; its disclosure is incorporated by reference.) These recent theoretical studies, however, do not evaluate the wavefront shaping capability of designed high-efficiency metasurfaces.Previous studies have reported thermo-optically reconfigurable metalens that allow for continuous adjustment of focal lengths from 165 μm to 135 μm as the metalens temperature increases from 20°C to 260°C. (See, for example, A. Archetti, et al., Nanophotonics 2022, 11, 3969; its disclosure is incorporated by reference).

[0031] We explored the potential of dielectric passive metasurfaces exhibiting high-quality factors, such as nonlocal metasurfaces. (See, for example, S. Joseph, et al., Nanophotonics 2021, 10, 4175; A. Overvig, et al., Laser & Photonics Reviews 2022, 16, 2100633; Y. Zhou, et al., Nano Letters 2023, 23, 6768; K. Shastri, et al., Nature Photonics 2023, 17, 36. These disclosures suggest, for example, that all-dielectric metasurfaces supporting delocalized photonic coupling states in a continuum (BIC) can provide large electric field enhancements (see, for example, A. Kodigala, et al., Nature 2017, 541). See 196; its disclosure is incorporated by reference) In addition to structures that support delocalized modes, individual subwavelength dielectric resonators can support quasi-BIC modes, also known as supercavity modes, which exhibit moderately high quality factors and are weakly coupled to the radiating continuum. (e.g., K. Koshelev, et al., Science 2020, 367, 288; E. Melik-Gaykazyan, et al., Nano Letters 2021, 21, 1765; these disclosures are incorporated by reference) The high quality factor of the supercavity modes arises from the interference of multiple localized modes supported by the resonator. (e.g., MV Rybin, et al., Physical Review Letters 2017, 119, 243901; its disclosure is incorporated by reference) These features have enabled the use of quasi-BIC metasurfaces for applications such as sensing and harmonic generation. (e.g., A. Tittl, et al., Science Quasi-BIC mode subwavelength nanolasers have also been realized (see 2018, 360, 1105; G. Zograf, et al., ACS Photonics 2022, 9, 567; their disclosure is incorporated herein by reference).(See, for example, V. Mylnikov, et al., ACS Nano 2020, 14, 7338; its disclosure is incorporated by reference.) Quasi-BIC modes supported by individual cylinders, however, cannot be efficiently excited by perpendicularly incident linearly polarized light, and azimuthal polarization excitation may be required. For transmitted metasurfaces, perpendicularly incident illumination with linearly polarized light is important for metasurface integration with chip-scale light sources. Notably, adding appropriately spaced back reflectors to a cylindrical array allows excitation of array quasi-BIC modes with normally incident light, but the back reflectors do not allow for use for transmitted metasurfaces. (See, for example, G. Yang, et al., Nano Letters 2022, 22, 2001; its disclosure is incorporated by reference.) Excitation of quasi-BIC (or hypercavity) modes with perpendicularly incident linearly polarized plane waves using a single high refractive index cuboid has also been reported. (See, for example, L. Huang, et al., Advanced Photonics 2021, 3, 016004; its disclosure is incorporated by reference).

[0032] Many embodiments provide dielectric quality coefficient (Q or Q factor) metasurface structures in transmission mode. Some embodiments use high-Q subwavelength resonators as metasurface components so that the metasurface can achieve a dynamically tuneable optical response when modulated by external stimuli. Metasurfaces can be used (but are not limited to) for dynamic beam switching or beam steering. Many embodiments implement physically feasible interconnection architectures to enable dynamic beam steering via thermo-optic modulation.

[0033] In many embodiments, the transmission-activated metasurface operates at near-infrared wavelengths (approximately 800 nm to approximately 2500 nm). In some embodiments, the transmission-activated metasurface can be excited by perpendicularly incident linearly polarized light, utilizing low-Q to high-Q modes. In various embodiments, the metasurface can achieve the following quality factors: at least about 10, at least about 30, at least about 50, at least about 60, at least about 70, at least about 80, at least about 90, at least about 200, at least about 300, at least about 400, at least about 500, at least about 600, at least about 700, at least about 800, at least about 900, or at least about 900, or about 109, or about 1000 to about 1000, 1000 to about 9999, or about 3000 to about 9800. In various embodiments, low Q or low Q refers to a quality factor of about 100 to about 999. In certain embodiments, higher Q or high Q refers to a quality factor of approximately 1000 to approximately 10000.

[0034] Metasurfaces in many embodiments can manipulate the wavefronts of light at various wavelengths with high-quality coefficients. The light includes, but is not limited to, ultraviolet wavelengths from approximately 100 nm to 400 nm; visible light wavelengths from approximately 380 nm to 800 nm; near-infrared wavelengths from approximately 800 nm to 2500 nm; and infrared wavelengths from approximately 780 nm to 1000 μm. The light manipulated by the metasurface can have a single wavelength or a range of wavelengths, such as broadband illumination. To manipulate different wavelengths of incident light, metasurfaces can be fabricated from different dimensions and / or different materials. In certain embodiments, the desired dimensions and / or material of the nanostructure on the substrate can be selected with respect to the light wavelength. Metasurfaces can be designed to exhibit multiple high-quality optical resonances appearing at different wavelengths and to demonstrate selective wavefront manipulation capabilities at different wavelengths.

[0035] Nanostructures on a substrate can be made in a variety of structures and / or dimensions. Nanostructures on a substrate can be arranged in arrays, parallel lines, linearly, curvilinearly, or aperiodically. Repeating units of nanostructures can be called unit cells. A unit cell can contain at least one nanostructure, or at least two nanostructures, or at least three nanostructures, or at least four nanostructures, or at least five nanostructures. Repeating unit cells can have periodicity P. Nanostructures have dimensions including length L, width W, and height H. In some embodiments, the periodicity P of a metasurface is less than the wavelength of light. In various embodiments, the periodicity P of a metasurface can be greater than or equal to the wavelength of light. In some embodiments, the length L, width W, and height H of a nanostructure are less than the periodicity P. The length L, width W, and height H of a nanostructure may be the same or different. In various embodiments, the length L, width W, and height H scale linearly with respect to the operating wavelength of light. In some embodiments, nanostructures can have a symmetrical shape. In some embodiments, the nanostructure may have an asymmetric shape that induces a polarization-selective or chiral response. The nanostructure may have a variety of shapes, including (but not limited to) cuboids, pillars, cylinders, elliptical cylinders, trapezoids, triangular prisms, polygonal prisms, pyramidal pyramids, and any combination thereof. As can be easily understood, any of the various shapes of nanostructures can be utilized as appropriate to the requirements of a particular application according to various embodiments of the present invention. The geometric dimensions of the nanostructure, such as the lengths L and width W, or height H, can vary non-uniformly and arbitrarily across the openings of the metasurface device according to the optical function of the metasurface.

[0036] In many embodiments, the metasurface structure can include multiple unit cells. The unit cell can have a variety of structures and sizes. In some embodiments, the unit cell can include nanostructures on the substrate. The nanostructures can be conformally deposited on the substrate. The nanostructures can have a variety of shapes, including (but are not limited to) cuboids, cuboids, pillars, cylinders, elliptical cylinders, trapezoids, triangular prisms, polygonal prisms, pyramidal pyramids, and any combination thereof. The nanostructures and substrates can be fabricated from high refractive index materials with high nonlinear optical sensitivity to enhance nonlinear optical parametric conversion processes and / or lossless dielectric materials. The lossless dielectric material can have a virtual refractive index (also known as the absorption coefficient) of about 0.5 or less, or about 0.1 or less, or about 0.05 or less at the operating wavelength. Examples of high refractive index materials include (but are not limited to) gallium arsenide, gallium phosphide, silicon carbide, titanium oxide, silicon nitride, barium titanate, lithium niobate, tantalum pentoxide, silicon oxide, amorphous silicon, crystalline silicon, and silicon parallel tubes. As can be easily understood, any of the various materials can be used as appropriate to the requirements of specific applications according to various embodiments of the present invention.

[0037] In some embodiments, the spectral lineshape of the high-Q mode at resonance can be "inverted" from showing a transmission dip to showing a transmission peak by adding a second substrate. The second substrate can have a range of thicknesses. The nanostructure and the second substrate may include materials having a higher refractive index than the substrate. The substrate can be fabricated from materials having a lower refractive index. Suitable materials for the nanostructure and the second substrate include materials with high nonlinear optical sensitivity to enhance nonlinear optical parametric conversion processes and / or lossless dielectric materials. At the operating wavelength, the real part of the refractive index of the substrate material must be smaller than the real part of the refractive index of the nanostructure material (or high refractive index material), according to some embodiments. Examples of low refractive index materials for substrates include (but are not limited to) glass, silicon oxide, silicon nitride, gold, silver, aluminum, copper, titanium, platinum, indium tin oxide, aluminum tin oxide, aluminum zinc oxide, magnesium oxide, magnesium fluoride, tantalum pentoxide, zirconium oxide, vanadium oxide, germanium-antimony-tellurium alloy, titanium nitride, hafnium oxide, hafnium nitride, molybdenum diselenium, hexagonal boron nitride, black phosphorus, tungsten diselenium, tungsten disulfide, and any combination thereof. In certain embodiments, low refractive index substrates can be made from elastic materials including (but are not limited to) elastic polymers, silicones, polydimethylsiloxane (PDMS), poly(methyl methacrylate) (PMMA), and any combination thereof. As can be easily understood, any of the various substrate materials can be used as appropriate to the requirements of the specific application according to the various embodiments of the present invention.

[0038] In some embodiments, the refractive index of high refractive index materials can be dynamically modulated in real time using mechanisms such as thermo-optic effects, electro-optic effects, magneto-optic effects, or nonlinear Kerr effects, and / or by electrical or optical injection of free charges into the unit cell material (but not limited to these). In some embodiments, refractive index modulation is achieved by thermo-optic modulation of high refractive index materials. In some embodiments, amorphous Si (a-Si) with refractive index modulation in the range of about 0.0026 to about can be achieved by thermo-optic modulation. In certain embodiments, high refractive index materials contain multiple quantum wells, and the refractive index can be modulated via the quantum confinement Stark effect. Refractive index modulation can sculpt transmitted light wavefronts in both near-field and far-field views, according to some embodiments.

[0039] Many embodiments implement interconnected metasurfaces to achieve thermo-optic effects. In various embodiments, nanostructures can be isolated from the substrate to avoid thermal crosstalk during thermo-optic modulation. Nanostructures can be isolated from the substrate (including low-refractive-index substrates and high-refractive-index secondary substrates) via various mechanisms such as using pedestals; or beams; or pillars (not limited to these), resulting in reduced thermal crosstalk between nanostructures. In some embodiments, the nanostructure may include conductive material on at least two surfaces to act as electrodes for resistive heating during thermo-optic modulation. Examples of conductive materials include, but are not limited to, doped semiconductors, doped compound semiconductors, metals, metal alloys, doped gallium arsenide, doped gallium pruriticide, doped amorphous silicon, indium tin oxide (ITO), cadmium oxide (CdO), and aluminum-doped zinc oxide (AZO). The conductive material may be a thin film or multiple thin films. In some embodiments, the entire conductive layer is doped. In some embodiments, only a portion of the conductive layer is doped. High refractive index materials can be heated via applied voltage and / or current. The applied heat can change the refractive index of the material.

[0040] In some embodiments, the dimensions of the substrate can vary in size from microns to millimeters or larger. Examples of one substrate dimension include (but are not limited to) about 1 μm or larger, about 5 μm or larger, about 10 μm or larger, about 50 μm or larger, about 100 μm or larger, about 150 μm or larger, about 200 μm or larger, about 300 μm or larger, about 400 μm or larger, about 500 μm or larger, about 1 mm or larger, about 2 mm or larger, about 3 mm or larger, about 4 mm or larger, about 5 mm or larger, and about 10 mm or larger. The layer thickness of the substrate (i.e., in the z direction) can be in the range of 0.1 nm to several millimeters or larger.

[0041] Dielectric and dynamically tunable transmittance measurement surface Many embodiments provide dielectric dynamic tuning wavelength-tunable transmissive metasurfaces with high quality factors. The metasurfaces can be achieved using high-Q resonant tones with a Q factor of approximately 10,000 or less. Some embodiments provide metasurface structures that exhibit a transmittance peak (rather than a transmittance dip at the resonant wavelength for various quality factors). In some embodiments, the metasurface structures exhibit a transmittance peak at the resonant wavelength with a quality factor range of approximately 3000 to approximately 4000. These quality factors are approximately 1 × 10⁻⁶. -3 ~Approx. 1×10 -2This makes it possible to achieve beam steering in variations of the a-Si refractive index. Low-Q modes (Q of about 100 to about 999) can also be used for amplitude and phase modulation, but the required a-Si refractive index modulation is higher compared to the case of high-Q modes. Thermo-optically active elements can be addressed using interconnects that do not perturb the resonator cavity modes, and thermo-optic beam switching can be achieved for both high-Q and low-Q modes with switching times of about 10 μs or less. Dynamic switching of both TE-polarized and TM-polarized beams is possible for both high-Q and low-Q modes, although at different wavelengths. Using a three-level phase grating approach, interconnected metasurfaces are capable of dynamic beam steering and have diffraction efficiencies of about 54% and 69% for low-Q and high-Q modes, respectively. In some embodiments, the optical performance of the metasurface structure can be improved by introducing a metal back reflector and / or SiO2 spacer. Designed reflective metasurfaces can exhibit optical efficiencies of over 90%.

[0042] Thermo-optic effects are one possible pathway for actively modulating designed metasurfaces. Several embodiments utilize a thin transparent conductive oxide (TCO) layer, such as Ito or cadmium oxide, beneath a-Si pillars, and use field-effect modulation to actively control the optical response of the metasurface via a gating action. The epsilon-near-zero transition in the TCO can be a useful mechanism for achieving phase modulation. A second TCO layer can be added as a counter electrode. Certain embodiments combine a high-Q metasurface with a lossless electro-optic material, such as lithium niobate (LNO) or JRD1 molecules embedded in a polymer. In this case, the Pockels effect in the electro-optic material may be used to achieve an electrically tunable optical response.

[0043] Many embodiments provide dielectric high-Q metasurface structures that utilize refractive index modulation by thermo-optic modulation of amorphous Si (a-Si) having a refractive index modulation Δn in the range of about 0.0026 to about 0.01. The metasurface operates in transmit mode. The transmitted active metasurface operates at near-infrared wavelengths and can be excited by perpendicularly incident linearly polarized light, utilizing either high-Q or low-Q modes supported by individual a-Si cuboids (also called prisms). The shape of the spectral line of the high-Q mode at resonance can be "inverted" from showing a dip in transmission to showing a peak in transmission by including a Si substrate separated from the prisms by a appropriately selected thickness of silica (SiO2). In some embodiments, the resonant tone exhibits a large (at least about 300°) transmitted light phase spectral shift near the resonant tone. This is a prerequisite for dynamically tunable phase shifts. Some embodiments provide that the refractive index modulation of the modes of individual a-Si pillars can sculpt transmitted light wavefronts in both near-field and far-field views. In some embodiments, the metasurface structures can be physically interconnected to enable dynamic beam steering via thermo-optic modulation.

[0044] In many embodiments, the metasurface includes an array of unit cells. Each unit cell may include nanostructures (such as a-Si rectangular pillars) on a substrate (such as a silica (SiO2) substrate). The metasurface can be illuminated by linearly polarized (x-polarized) plane waves from within the SiO2 substrate. The phase and amplitude characteristics of the transmitted light can be studied. Since the metasurface operates in transmission mode, a light source such as a laser or chip-scale laser (but not limited to) can be integrated with the metasurface. In certain embodiments, the light source can be positioned on the opposite side of the substrate from the nanostructure (such as a-Si pillars).

[0045] Figures 1A-1E show a total dielectric high-Q metasurface in a transmission mode according to one embodiment. Figure 1A shows a schematic diagram of the transmission metasurface. The metasurface includes an array of amorphous a-Si rectangular pillars on an SiO2 substrate. Figure 1B shows a schematic diagram of the unit cell of the transmission metasurface. The pillars can have a height h of approximately 860 nm, and the pillar length l and width w can be approximately 963 nm. The metasurface period is P x P y The wavelength is approximately 1425 nm. The a-Si resonator is irradiated by x-polarized light from within the substrate. The refractive indices of a-Si and SiO2 are 3.734 and 1.44, respectively.

[0046] Figure 1C shows the transmittance and phase spectra of the metasurface shown in Figure 1A. The Q value of the supported mode is approximately 9800, as determined by fitting the transmittance spectrum to the Fano formation linearity. The observed transmittance dip is accompanied by broad spectral features in the transmitted light phase, indicating that this unit cell motif may allow for the design of a mid-surface phase gradient, either through shape adjustment or external active control.

[0047] Spatial mode profiles can be used to gain further insight into high-Q modes. Figures 1D and 1E show the spatial distribution of electric field amplitudes within the metasurface unit cell in the xz and yz planes, respectively. E0 represents the magnitude of the collision electric field. In Figures 1D and 1E, the cross section considered for calculation of the electric field traverses the center of the a-Si resonator. Figure 1D shows that in the x direction, the electric field is tightly confined within the resonator and amplified by approximately 80 times. Although there is a large electric field enhancement inside the a-Si resonator, moderate enhancements of the electric field below and above the a-Si pillars could be observed. In the yz plane perpendicular to the polarization of the incident ray, only non-zero components are E, as shown in Figure 1E. xIn this case, a more complex electric field distribution is observed. In the y-direction, a non-negligible electric field enhancement is observed between adjacent metasurface unit cells, which can serve as an indicator of near-field coupling between resonators in the direction perpendicular to the incoming electric field. This inter-resonator coupling can have broad implications for comprehensive transmission wavefront control.

[0048] Some embodiments demonstrate that the geometric shape of the nanostructure can influence the transmittance properties. In some embodiments, the transmittance properties may vary with the height of a square a-Si pillar while keeping the width fixed. Figures 2A to 2D show the dependence of the properties of high-Q modes on the height of an a-Si prism according to one embodiment. Figure 2A shows transmittance as a function of wavelength and a-Si pillar height h. Figure 2B shows the quality factor and Fano phase of the designed high-Q mode as a function of a-Si pillar height h. Figures 2C and 2D show transmittance and phase as a function of wavelength and on a-Si pillar heights of approximately 850 nm and 870 nm, respectively. As shown in Figures 2B and 2C, increasing the pillar height changes both the resonance linewidth and linewidth. At a certain pillar height, an asymmetric resonance linearity exhibiting Fano resonance, i.e., a high-Q mode coupled to a wider mode, or a "continuum" of modes, can be observed. By fitting the transmittance spectrum to a Fano resonance linearity, the dependence of the resonance quality factor and Fano phase on pillar height can be observed. The Fano phase characterizes the phase difference between narrow and wide resonance modes. The resonance quality factor increases with pillar height, peaking at approximately 860 nm h as shown in Figure 2B, with a value of approximately 9800. At pillar heights corresponding to the maximum Q value, the Fano phase passes through 90° (hence the Fano asymmetry parameter is zero), exhibiting a symmetric quasi-Lorentz linearity. If the substrate is removed and the a-Si pillar array is considered to be suspended in free space, the Q value can be as high as 221,000. With more systematic variations in pillar height, it may be possible to achieve even higher quality factors in free space. When varying the metasurface period, beyond 1300 nm, increasing the metasurface period does not significantly change the spectral position of the resonance sound.

[0049] The spectral behavior of the transmitted light phase can be important for metasurface applications. Several embodiments investigate the spectral behavior of the transmitted light phase. When the pillar height is 860 nm or less (h ≤ 860 nm), the phase of the transmitted light spans approximately 360° when the wavelength of the transmitted light is changed. When h is greater than approximately 860 nm, abrupt changes in the spectral characteristics of the transmitted light are observed. For example, at a pillar height h of approximately 870 nm, the phase spans approximately 40° (as a function of wavelength). For h greater than approximately 860 nm, the phase variation is limited despite a high quality factor. Therefore, the pillar height should be less than approximately 860 nm for wavefront forming applications.

[0050] The optical modes supported by pillars in an array can be compared to those of a single, isolated pillar. An isolated subwavelength pillar on an SiO2 substrate supports a high-Q mode with a Q factor of 676, and the high-Q mode profile is identical to that observed in the array configuration (Figure 1E). In an isolated a-Si pillar in free space, the Q factor of this mode is approximately 1000 or higher, but in an a-Si resonator on a substrate, the Q factor decreases due to mode leakage into the substrate. Scattering cross-section calculations showed that, in addition to the high-Q mode, the isolated Si pillar also supports two low-Q modes. When comparing the mode profiles of the pillars in the array with those of the isolated pillar, non-locality was observed for one of the low-Q modes because its field profile was not observed in the isolated resonator.

[0051] Some embodiments provide a dependency of the spectral position of resonant sounds on the metasurface period. Changing the metasurface period may have little effect on the spectral position of high-Q resonant sounds. In some embodiments, for a metasurface with a finite number of metasurface elements in either the x or y direction, and for 20 metasurface periods, the quality coefficient of the modes converges to the quality coefficient of an infinite metasurface array. For delocalized resonances such as waveguide mode resonances, as the number of lattice periods approaches the quality coefficient of the resonance, the effect of the finite lattice size becomes negligible. This corresponds to thousands of lattice periods. These observations lead to the conclusion that high-Q modes can be quasi-local.

[0052] One drawback of metasurface motifs (pillars on a substrate) is that large phase fluctuations are accompanied by low transmittance, and operating at the transmittance dip wavelength can result in low metasurface optical efficiency. Several embodiments implement metasurface structures that allow high-Q modes to be supported by large phase fluctuations at the transmittance peak rather than the transmittance dip. Figures 3A–3F show a metasurface structure with a transmittance peak according to one embodiment. Figure 3A shows a schematic diagram of a transmittance metasurface. The upper panel shows a schematic of a metasurface unit cell. The lower panel shows a metasurface including an array of unit cells. The unit cell includes a second substrate with a high refractive index (e.g., crystalline silicon (c-Si) substrate), a substrate with a low refractive index (e.g., SiO2 spacer), and a nanostructure with a high refractive index (e.g., a-Si rectangular pillar). A unit cell without a second substrate (or c-Si substrate) may exhibit a transmittance dip. The second substrate (or c-Si substrate) can enable high-Q modes with large phase fluctuations at the transmittance peak.

[0053] Figure 3B shows the transmittance and phase spectra of the metasurface having the unit cell shown in Figure 3A. The thickness of the SiO2 spacer d is about 1450 nm. The height, width, and length of the a-Si pillar are such that h is about 845 nm and w is about 963 nm. The height of the a-Si resonator is slightly reduced to about 845 nm considering the higher pillar, which may impair the phase variation as a function of wavelength. The period is P x about 1520 nm, P y about 1425 nm. Increasing the metasurface period in the x direction to P x = 1520 nm reduces the near-field coupling between adjacent metasurface unit cells. In Figure 3B, a SiO2 spacer thickness d of about 1450 nm can result in large phase variations with wavelengths accompanied by a transmittance peak of about 37%. The electric field profile of the modified unit cell (Figure 3A) shows mode leakage into the SiO2 spacer. The E x and E z components both have non-zero values in the SiO2 spacer layer, and the electric field enhancement of the a-Si pillar was up to 120.

[0054] In various embodiments, the spectral shape of the transmittance can be controlled by varying the thickness of the SiO2 spacer. Figure 3C shows the resonance quality factor and Fano phase as a function of the thickness d of SiO2. By appropriately selecting the SiO2 thickness d, the Fano phase can be adjusted between 0° and 140° (the Fano phase is uniquely defined between 0° and 180°). This large variation in the Fano phase indicates comprehensive control of the resonance line shape, showing transmission dips, transmission peaks, and asymmetric spectral shapes. Thus, by engineering the leakage of the resonator mode into the spacer layer, the relative phase between the high-Q mode and the continuum can be modified, resulting in a change in the transmittance spectral line shape.

[0055] Several embodiments evaluate metasurface beam steering. Assuming that all pillar refractive indices change by an equal amount Δn (refractive index change), the phase shift and transmittance are calculated as functions of the refractive index change Δn at a given wavelength. The phase shift can be defined as the difference between the phases of the transmitted light at Δn ≠ 0 and Δn = 0. Figure 3D shows the transmittance and phase shift as functions of the a-Si refractive index change at a wavelength λ of approximately 1534.8 nm. As a result, 3 × 10⁻⁶ -3 A change in refractive index of a certain magnitude Δn results in a phase shift of approximately 250°, enabling beam steering.

[0056] Some embodiments implement dimensional beam steering where the phases of all metasurface elements along the y-direction are identical, while the phases of adjacent elements in the x-direction are different. Calculations for a periodic array of identical elements suggested that an exponential change resulting in a 180° phase shift between adjacent elements in the x-direction would create a two-level phase grating, strongly suppressing the zero-order beam. However, full-wave simulations indicate that the exponential change values ​​derived through this method may not be optimal, as an unexpectedly large zero-order beam is obtained. The observed low diffraction efficiency may be due to non-negligible near-field coupling between adjacent metasurface elements. Calculations showed that an exponential change Δn of approximately 0.0026 between adjacent columns of a-Si resonators at a fixed operating wavelength λ of approximately 1535.44 nm can completely extinguish the zero-order diffracted beam. Figure 3E shows the far-field electric field intensity for a two-level phase grating with 10 grating periods when the exponential change between adjacent a-Si pillars is Δn = 0.0026. Interestingly, for these parameters (λ = 1535.44 nm, Δn = 0.0026), simulations of the same resonator's periodic array show that the predicted relative phase shift between light scattered by adjacent columns of pillars is approximately 107°. In principle, this phase should result in a non-negligible zero-order beam. These results suggest that such a simple hypothesis for optimal beam steering may not apply to these high-Q metasurfaces.

[0057] Figure 3F shows the overall transmittance of the two-level grating. In Figure 3F, the dots mark the transmittance at the operating wavelength λ of approximately 1535.44 nm. The inset in Figure 3F shows the spatial distribution of the electric field in the x-z cross-section of the unit cell. The x-z cross-section considered passes through the center of the a-Si pillar.

[0058] The optical efficiency of the measurement surface is an important design parameter. The overall transmittance of the two-level phase grating T is approximately 11% (Figures 3E and 3F), which is lower than the peak transmittance of approximately 35% derived from the same resonator array simulation. The observed decrease in transmittance is due to the spectral shape of the resonant sound (Figure 3B). As seen in Figure 3B, the transmittance attenuates sharply as it moves away from the resonant wavelength. When a two-level grating is implemented, the resonant wavelengths of every other meta-element shift, resulting in a decrease in overall transmittance at a given operating wavelength. The transmittance spectrum of the two-level phase grating shows two distinct peaks (Figure 3F). The near-field distribution of the electric field (see inset in Figure 3F) shows that at a given operating wavelength, the relative improvement of the electric field in adjacent a-Si rectangular pillars is considerably different. The optimal operating wavelength, indicated by the dots in Figure 3F, lies between the two resonant peak wavelengths, resulting in a lower overall transmittance. To ensure a favorable wavefront shape with higher optical efficiency, the transmittance should remain above a certain minimum value as it moves spectrally away from the resonance.

[0059] Designing high-Q dynamic beam steering metasurfaces presents two challenges: i) one is that the beam cannot be steered with high steering efficiency using intuitive phase profiles, and the optimal beam steering conditions must be identified through full-wave simulation; and ii) significant near-field coupling between emitters in the direction perpendicular to the polarization of the incoming electric field (y-direction) results in diffraction switching in only one direction. Therefore, it is desirable to identify the optical modes that enable dimensional beam steering.

[0060] Figures 4A to 4H show beam steering using low-Q modes according to one embodiment. In addition to high-Q modes, several low-Q modes are supported with a quality factor of approximately 200. Figure 4A shows the transmittance and phase spectrum of a metasurface having the unit cell shown in Figure 1B. In Figure 4A, the metasurface period value is approximately 1500 nm P x And, P at approximately 1500 nm y The geometric parameters are pillar height h of approximately 845 nm, and pillar length l and width w of approximately 963 nm. The metasurface with a-Si rectangular pillars exhibited a relatively low Q photonic mode at a wavelength of 1583 nm. As seen in Figure 4A, the resonant dip observed at a wavelength of approximately 1583 nm is accompanied by a large spectral variation in phase. Figure 4B shows the transmittance and phase shift as functions of the a-Si refractive index change at a wavelength λ of approximately 1583.85 nm. Varying the refractive index Δn of the a-Si pillars by approximately 0.01 throughout the metasurface resulted in a dynamically tuneable phase shift of approximately 285°, which is large enough to enable dynamic beam steering.

[0061] Several embodiments investigate whether low-Q modes can be used for dynamic beam steering in two configurations: i) when the beam is steered perpendicular to the polarization of the incident electric field (in the case of a transverse electric field or TE), and ii) when the steering direction is matched to the polarization of the incident light (in the case of a transverse magnetic field or TM). Figure 4C shows the far-field electric field intensity as a function of the polarity (steering) angle for a two-level phase grating when the incoming electric field is polarized perpendicular to the steering direction (TE polarization). Figure 4D shows the far-field electric field intensity for a two-level phase grating when the incoming electric field is parallel to the steering direction (TM polarization). Figure 4E shows the near-field electric field distribution for a two-level phase grating shown in the inset of Figure 4C, and the application corresponds to the case of TE polarization. Figure 4F shows the near-field electric field distribution for a two-level phase grating shown in the inset of Figure 4D, and the application corresponds to the case of TM polarization. In Figures 4E and 4F, E0 represents the magnitude of the collision electric field.

[0062] Figures 4G and 4H plot the electric field intensity in the far field as a function of the rudder angle for the 3-level and 4-level phase grids, respectively. Figures 4G and 4H correspond to the TE polarization case. The insets in Figures 4G and 4H show the near-field distribution of the electric field for each case. The overall transmittance values ​​are indicated in the insets in Figures 4C, 4D, 4G, and 4H. The operating wavelength λ is approximately 1583.85 nm.

[0063] Some embodiments use a two-level phase grating and assume that the change in a-Si exponent between adjacent element rows Δn is approximately 0.0048 (see insets in Figures 4C and 4D for schematic diagrams of the grating period). At Δn of approximately 0.0048, adjacent metasurface rows may exhibit a phase difference of 180° at a wavelength λ of approximately 1583.85 nm (Figure 4B). Thus, nearly complete suppression of specular transmission beams can be expected. However, full-wave simulations show significant transmission at normal incidence for two reasons: i) near-field coupling between adjacent metasurface elements, and ii) differences in scattered light amplitude from adjacent elements.

[0064] To completely suppress the normal transmitted beam at the operating wavelength λ of approximately 1583.85 nm, a pillar refractive index optimization procedure may be used to maximize the diffraction efficiency of the desired order, thereby completely suppressing the zeroth diffraction order at the operating wavelength λ of approximately 1583.85 nm for both TE and TM polarizations of the incident light (Figures 4C and 4D). In the case of TE polarization, the optimized refractive index of one a-Si pillar within the lattice period can retain its original value (n1 = 3.734), and the refractive index difference Δn between adjacent a-Si pillars is approximately 0.0055, resulting in a phase shift of approximately 200° (see Figure 4B), which is close to the originally selected phase shift of approximately 180°. For TM polarization, the optimization procedure yielded a-Si aliasing indices of approximately 3.7353 for n1 and approximately 3.7440 for n2. The corresponding phase difference between adjacent a-Si pillars is approximately 270°, which is a significant deviation from the original value of 180°. For TM polarization, complete suppression of the zeroth diffraction order occurs with a phase profile significantly different from that of an optimal phase-shifted uncoupled resonator array, indicating that near-field coupling between elements is more pronounced for TM than for TE polarization.

[0065] While diffraction switching is observed in the two-level phase grating, this analysis does not clarify whether intermediate steering angles are possible using the Blaze grating design approach. For TM polarization, using three- and four-level Blaze grating phase profiles may not yield a highly directional beam due to near-field coupling between adjacent metasurface elements. However, with TE polarization, beam steering to angles of approximately 20.6° and 15.2° can be achieved with reasonable diffraction efficiency. To steer the beam to a polar angle of approximately 20.6°, Figure 4B can be used to construct a three-level phase profile with relative phases between elements of approximately 0°, 120°, and 240°, resulting in beam steering with approximately 60% diffraction efficiency at an operating wavelength λ of approximately 1583.85 nm (Figure 4G). To steer the beam to an extreme angle of approximately 15.2°, phase profiles of approximately 0°, 90°, 180°, and 270° were constructed, resulting in diffraction efficiencies of 38% at an operating wavelength λ of approximately 1583.85 nm and 46% at λ of approximately 1584.16 nm. When full-wave optimization was performed to improve the diffraction efficiency at λ = approximately 1583.85 nm, the diffraction efficiency was approximately 52% (Figure 4H), with a-Si refractive indices of n1 = 3.7340, n2 = 3.7358, n3 = 3.7386, and n4 = 3.7417. Therefore, for TE-polarized incident light, the metasurface can steer the beam to intermediate angles between 0° and 20.6°.

[0066] The overall transmittance of a 4-level phase grating is higher than that of a 3-level phase grating (Figures 4H and 4G). This is because the transmission amplitude of the 4-level phase grating is, on average, higher than that of the 3-level phase grating (Figure 4B).

[0067] In some embodiments, both high-Q and low-Q modes can be used for transmitted light wavefront manipulation. Some embodiments provide metasurface structures that enable refractive index fluctuations and dynamic beam switching by refractive index modulation of a-Si using thermo-optic effects. To selectively heat a row of a-Si pillars, many embodiments implement a-Si electrodes and connect the pillars in series. In some embodiments, high-refractive-index nanostructures (e.g., a-Si pillars) can be placed on low-refractive-index structures (e.g., SiO2 pedestals) to limit thermal crosstalk between adjacent pillars. In certain embodiments, the fabrication involves first patterning a-Si on an SiO2 spacer, followed by hydrofluoric acid wet etching of the SiO2 spacer within the pillars on a c-Si substrate.

[0068] Figures 5A to 5F illustrate thermo-optic beam switching using a high-Q mode with interconnected metasurfaces according to one embodiment. Figure 5A shows a schematic diagram of the interconnected metasurfaces. Figure 5B shows a schematic diagram of the unit cell of the interconnected metasurfaces. In Figures 5A and 5B, square high-refractive-index nanostructures (such as a-Si pillars) are connected via high-refractive-index materials (such as a-Si bars). Each a-Si pillar is placed on an SiO2 pedestal to enhance thermal insulation between adjacent metasurface nanostructures. The top and bottom 50 nm thick a-Si layers are doped, and a voltage is applied to the top and bottom layers to allow current to flow between electrodes through the low-concentration doped a-Si layer. The induced current increases the temperature T of the a-Si pillars due to Joule heating. Increasing the temperature allows the refractive index of a-Si to be changed by the thermo-optic effect. In the unit cell, the pillar width w and length l are approximately 963 nm, and the pillar height h is approximately 850 nm. The bar width δ is approximately 50 nm, and the bar height is approximately 850 nm. The SiO2 base is rectangular in shape with a length b of approximately 200 nm and a height of approximately 382 nm. The thickness of the planar SiO2 spacer d is approximately 380 nm. The structure is built on a high refractive index substrate (such as a c-Si substrate).

[0069] In Figures 5A and 5B, the top and bottom layers of the high refractive index nanostructures and bars are doped to form conductive electrodes. As is readily apparent, different surfaces of the high refractive index nanostructures and bars can be doped to form electrodes, as long as the desired heating pattern can be achieved. In some embodiments, a pair of electrodes can be formed for each row of interconnected metasurfaces (Figure 5A). The front and back surfaces 501 of each row of interconnected metasurfaces can be doped to form electrodes so that a voltage can be applied to heat the row. The left and right sides 502 of each row of interconnected metasurfaces can be doped to form electrodes so that a voltage can be applied to heat the row. In various embodiments, the metasurface includes multiple interconnected electrodes. The electrodes may be entirely doped or partially doped. The electrodes may be conductive layers made from doped semiconductors (e.g., doped silicon, doped GaAs, ITO, CdO, AZO) or metals.

[0070] The induced current is given by the modified a-Si refractive index n(T) = n Si A temperature change of +Δn(ΔT) increases the temperature T of the a-Si pillar due to Joule heating. In silicon, a temperature change of approximately 10 K ΔT changes the refractive index Δn by approximately 0.00239. Dynamic beam switching using higher Q modes is possible with a refractive index difference Δn of at least approximately 0.0026, corresponding to a relative temperature difference of at least approximately 11 K. For beam switching with lower Q modes, a refractive index difference Δn of at least approximately 0.006 may be required, corresponding to a relative temperature difference ΔT of at least approximately 25 K. Interconnected metasurfaces can enable beam steering with moderate temperature differences.

[0071] To construct the desired transmitted light phase profile, some embodiments heat individual rows of the pillars to different temperatures. For high-Q modes, diffraction switching by a two-level phase grating can be achieved by varying the temperature of every other metasurface pixel (i.e., electrically connected rows of metasurface elements) by at least 11 K (Δn is at least 0.0026). Heating creates a two-level phase grating, deflecting the light at an angle θ of approximately ±30° (Figures 5C and 5E). For an incident x-polarized electric field, this small difference in refractive index is 0 th This completely suppresses the diffraction order, enabling the switching of the diffracted beam to an overall transmittance T of approximately 13% (Figure 5C). For y-polarized incidence (Figure 5D), diffraction beam switching can be observed at different operating wavelengths with low overall transmittance. On a-Si on an SiO2 substrate, high-Q mode diffraction switching is observed when the steering direction aligns with the incident plane wave (TM) polarization. The a-Si electrode and SiO2 pedestal reduce near-field coupling of adjacent metasurface elements. Furthermore, the low-Q mode allows for diffraction switching at different wavelengths for both TE and TM polarized incidence.

[0072] Figure 5C shows the far-field electric field intensity for a two-level phase grating when the incoming plane wave is x-polarized. In Figure 5C, the operating wavelength λ is approximately 1537.15 nm. Figure 5D shows the overall transmittance spectrum corresponding to the two-level phase grating examined in Figure 5C. Figure 5E shows the far-field electric field intensity for a two-level phase grating when the incident plane wave is y-polarized. In Figure 5E, the operating wavelength λ is approximately 1534.25 nm. Figure 5F shows the overall transmittance spectrum corresponding to the two-level phase grating examined in Figure 5E. The insets in Figures 5D and 5F show the spatial distribution of the electric field in the xz cross-section of the grating period at the operating wavelength for x-polarized and y-polarized incident light, respectively. The refractive index difference between adjacent metasurface elements Δn is approximately 0.0026.

[0073] Several embodiments perform coupled electrical and thermal analysis. Figures 6A to 6C show the electrical and thermal analysis of thermo-optic beam switching according to one embodiment. To induce a diffraction grating similar to that of Figure 5A, a voltage V1 of 1.05 V can be applied between the upper doped a-Si layer and the lower doped a-Si layer of one metasurface pixel, while grounding each adjacent metasurface pixel (V2 = 0 V) ​​in a periodic arrangement. Figure 6A shows the steady-state spatial current density distribution in a thermo-optically controlled metasurface with V1 = 1.05 V and V2 = 0 V. Figure 6A shows the calculated current density in the silicon bar for one grating period. The maximum current density and most significant heat generation occur in the upper and lower doped a-Si layers of the connector bar. Figure 6(b) shows the steady-state spatial temperature distribution in a thermo-optically controlled metasurface with V1 = 1.05 V and V2 = 0 V. Figure 6B shows that the temperatures of two adjacent metasurface pixels are T1 = 312.8K and T2 = 301.4K, respectively, and the result is 2.3 × 10⁻⁶. -4 K -1 Assuming a thermo-optic coefficient of , it exhibits a steady-state temperature distribution for one lattice period that produces a refractive index difference Δn of approximately 0.0026. The power required to maintain this steady state is approximately 1.62 μW / μm 2 The temperature distribution within the pillars and connectors is nearly uniform due to the high thermal conductivity of silicon, even though heating primarily occurs within the connector bars. The largest temperature gradient occurs in the silicon oxide pedestal due to its high thermal resistance, significantly reducing thermal crosstalk between adjacent rows. To evaluate the dynamic performance of the metasurface, several embodiments provide transient electrical and thermal simulations of dynamic thermal switching between two pixels with one grid period. Figure 6C shows the transient temperature difference between two rows of a thermo-optically controlled metasurface when switching voltage with a square wave of amplitude V at approximately 1.05 V and frequency at approximately 100 kHz. The geometric parameters are identical to those of the structure in Figure 5B. A response time of approximately 7.3 μs is obtained for the square wave voltage profile, indicating that thermo-optic switching is possible at frequencies up to approximately 140 kHz.

[0074] Several embodiments provide that interconnected metasurface beam steering performance is possible when a three-level phase profile (0°, 120°, 240°) is applied to the metasurface for both the lower Q mode and the high Q mode under TM-polarized plane wave illumination. This phase profile yielded a maximum diffraction efficiency of approximately 35% in the low Q mode and approximately 28% in the high Q mode. Optimization of the refractive index of the a-Si pillar results in a moderate improvement in diffraction efficiency. To further increase diffraction efficiency, several embodiments co-optimize the geometric parameters of the a-Si pillar structure and refractive index, enabling improvements in diffraction efficiency of 54% and 69% for the low Q mode and the high Q mode, respectively. Figures 7A and 7B show a thermo-optic three-level phase grating with a realistic interconnected architecture and optimized geometry according to one embodiment. The incident plane wave is TM-polarized. Figure 7A shows the electric field strength in the far field for the low Q mode. In Figure 7A, the operating wavelength λ is approximately 1567.1 nm. Figure 7B shows the electric field strength in the far field for the high-Q mode. In Figure 7Bm, the operating wavelength λ is approximately 1531.3 nm. The inset of Figures 7A and 7B shows the spatial distribution of the electric field in the xz cross-section of the lattice period at the operating wavelength. [Modes for carrying out the invention]

[0075] Specific embodiments of the system and apparatus are discussed in the following sections, but it should be understood that these embodiments are provided as examples and are not intended to be limiting.

[0076] Example 1: Method Optical simulations were performed using the finite difference time-domain method (FDTD Lumerical). In our optical simulations, a normally incident linearly polarized plane wave irradiates the metasurface from within the substrate. When simulating an array of a-Si pillars, periodic boundary conditions were used in the x and y directions, and a perfectly matched layer (PML) boundary condition was used in the z direction. Considering the behavior of isolated pillars, PML boundary conditions were used for all simulation boundaries, and the simulation volume was 3 × 3 × 3 μm. 3In the simulation, the material was assumed to be nondispersive. The refractive indices of a-Si, SiO2, and c-Si were assumed to be 3.734, 1.44, and 3.43, respectively. The effect of potential-processing non-idealism on metasurface performance is summarized below. When considering metasurface structures interconnected to perform thermo-optic beam switching and thermo-optic beam steering, some embodiments assume that the complex refractive index of the doped a-Si layer n is approximately 3.734 + 0.0013 i, which is 6 × 10 for the n-doped Si layer. 18 cm -3 Corresponding to the carrier density, in the case of a p-doped Si layer, it is 10 19 cm -3 This corresponds to the carrier density. The complex reflectivity of a thinly doped Si core n is approximately 3.734 + 0.000025 i, which is 3.2 × 10⁻¹⁵ i for n-doped Si. 17 cm -3 This corresponds to the carrier density. In the simulation used to generate 2D from Figure 1A, the assumed mesh in the z direction is 5 nm, while the mesh in the x and y directions is set to 20 nm. In the beam switching and beam steering simulations, the mesh in the x, y, and z directions was set to 20 nm. When performing projection from near to far distance in Figures 3A to 3F, the number of metasurface elements in the x direction is 20, and for projection from near to far distance in Figures 4A to 7B, the assumed number of metasurface elements in the x direction is 100.

[0077] To optimize diffraction efficiency, we used MATLAB to drive multivariable nonlinear optimization in FDTD via Lumerical's Automation Application Programming Interface (API). For optimization, we used successive quadratic programming (SQP) and the reciprocal of the maximum diffraction efficiency as the index of merit. We assumed the phase profile was temporal, and varied the geometric shape and refractive index of the structure over a certain period. In Figure 4, when optimizing diffraction efficiency, only the refractive index of the a-Si pillar is varied such that the assumed maximum refractive index change is Δn = 0.01. When optimizing the diffraction efficiency of a structure with interconnects (Figure 7), a series of optimization runs are performed. First, the refractive index of the metasurface element is optimized. In the next step, the refractive index of metasurface element n1 is fixed, and the structural height and refractive index of the two remaining metasurface elements are co-optimized. Next, while keeping the refractive index of the first metasurface element constant, the structural and refractive index of the second and third metasurface elements are p-optimized. y Period (or P x The period is co-optimized. As the final optimization step, the refractive index of all three metasurface elements is re-optimized. For the low Q mode, the optimization is performed for the reflectance index of the Si pillar and the geometric parameters of the structure, with n1 = 3.73416, n2 = 3.738299, n3 = 3.73778, P x = 1440nm, P y = 1440 nm, and h = 840.75 nm, and the observed diffraction efficiency D eff A value of 54% was obtained (Figure 7A). In the high-Q mode, n1 = 3.73416, n2 = 3.7383, n3 = 3.744, P x = 1520nm, P y = 1495.92 nm, h = 841.207 nm, and diffraction efficiency D eff =69% (Figure 7B). By co-optimizing both pillar height and refractive index when optimizing the beam steering performance in High-Q mode, it was possible to increase the diffraction efficiency from 36% to 53%. As a result of the optimization, the a-Si pillar height was reduced from 850 nm to 841 nm. Next, period P yBy co-optimizing both the refractive index and the diffraction efficiency, we were able to increase the diffraction efficiency to 69%. Through optimization, during period P y The wavelength increased from 1425nm to 1440nm.

[0078] Three-dimensional electromagnetic and thermal simulations were performed using the finite element method (COMSOL Multiphysics). In the electrical simulation, the top and bottom 50 nm of the a-Si in the metasurface unit cell were 6 × 10⁻¹⁶ 18 centimeter -3 . The carrier density of a light-doped core doped with n at the assumed carrier density is 3.2 × 10⁻⁶. 17 cm -3 This is considered to be the case. First, the volumetric heat source distribution due to Joule heating is determined using the electromagnetic solution. In the next step, the thermal solution is used to obtain the temperature distribution. The thermal simulation explains heat conduction and convection. The upper part of the metasurface is h=5 W / m 2 Cooling is performed by natural convection in K. The temperature at the bottom of the substrate 50 mm from the metasurface is fixed at 298 K using an external heat sink. The assumed ambient temperature is also 298 K. Periodic boundary conditions in the x and y directions are used when performing the thermal simulation.

[0079] Example 2: Extraction of Quality Factor and Fano Phase By applying the transmittance spectrum to the Fano formula, the quality factor and Fano phase value were extracted: [Mathematics 1] Here, a constant offset plus linear background () represents the resonance amplitude and the frequency of light, while the resonance frequency, damping constant, and Fano phase are also represented, and the Fano asymmetry parameter q is related to the Fano phase Δ as q - cot(Δ). The quality factor is calculated as follows: [Math 2] Example 3: Field profile of a pillar array in High-Q mode

[0080] Figures 8A and 8B show the spatial distribution of electric field amplitude inside a metasurface unit cell according to one embodiment. The unit cell is shown in Figures 1A and 1B and represents the case of an a-Si pillar arrangement on an SiO2 substrate. That is, the pillar length and width are l = w = 963 nm. The metasurface period is P x = P y = 1425 nm. The electric field is plotted in the xz plane and passes through the center of the column (same as Figure 1D). Figure 8A shows the x component of the electric field, and Figure 8B shows the z component of the electric field. The field profile is plotted at the resonant wavelength of the resonant dip shown in Figure 1C. E y The component is the same as 0 (E y = 0).

[0081] Figures 9A to 9D show the spatial distribution of electric field amplitude inside a metasurface unit cell according to one embodiment. The unit cell is shown in Figures 1A and 1B and represents the case of an a-Si pillar arrangement on an SiO2 substrate. The pillar length and width are l = w = 963 nm. The metasurface period is P x = P y = 1425 nm. The electric field is plotted on the xy-plane passing through the center of the pillar. Figure 9A shows the absolute value of the electric field, Figure 9B shows the x component of the electric field, Figure 9C shows the y component of the electric field, and Figure 9D shows the z component of the electric field.

[0082] Example 4: Optical response of pillar array: Effect of pillar height on array performance Figures 10A and 10B show the optical reaction of an a-Si pillar array on an SiO2 substrate according to one embodiment. The metasurface structure is shown in Figures 1A to 2D. Specifically, the length and width of the pillars are l = w = 963 nm. The metasurface period is P x = P y = 1425 nm. Figures 10A and 10B show the transmittance and phase of the transmitted light, respectively, as functions of wavelength and a-Si pillar height h. Modes 2 and 3 merge at a pillar height h of approximately 850 nm, and the highest Q mode, Q-mode, is observed at a pillar height h of approximately 860 nm.

[0083] Figure 11 illustrates the spatial electric field profile of a lower Q mode supported by a metasurface in the xz plane, according to one embodiment. The xz plane passes through the center of the pillar. In the displayed electric field profile, the z coordinate ranges from z = -200 nm to z = 1000 nm, with the top of the SiO2 substrate corresponding to the plane z = 0. The x coordinate ranges from x = -712.5 to x = 712.5 nm. The spatial field profile of the high-Q mode is identical to that shown in Figure 1. The x and y components of the electric field for mode 3 have very similar characteristics when the pillar height is varied. On the other hand, mode 2 "disappears" after merging with mode 3. The mode profile for mode 1 is the same as that shown in Figure 1.

[0084] Figure 12 illustrates the spatial electric field profile of lower Q modes supported by a metasurface in the yz plane according to one embodiment. The yz plane passes through the center of the pillar. In the shown electric field profile, the z coordinate ranges from z = -200 nm to z = 1000 nm, with the top of the SiO2 substrate corresponding to the plane z = 0. The x coordinate ranges from x = -712.5 nm to x = 712.5 nm. The spatial field profile for high-Q modes is identical to that shown in Figure 1. The x and y components of the electric field for mode 3 have very similar characteristics when the pillar height is varied. On the other hand, mode 2 "disappears" after merging with mode 3. The mode profile for mode 1 is identical to that shown in Figure 1. In this figure, the electric field amplitude |E| is shown in the figure. In the considered yz plane, non-zero components are E x That is all.

[0085] Figures 13A and 13B show the optical response of an a-Si pillar array suspended in air according to one embodiment. The assumed geometric parameters are the same as in Figure 1; that is, the pillar length and width are l = w = 963 nm. The metasurface period is P x = P y= 1425 nm. In Figure 13A, transmittance is shown as a function of wavelength and a-Si pillar height h. In Figure 13B, the phase of transmitted light is shown as a function of wavelength and a-Si pillar height h. Compared to pillars on an SiO2 substrate, an abundance of high-Q modes was observed, which is shown by a circle in Figure 13A. At a pillar height h of approximately 870 nm, the Q value of the observed high-Q modes is approximately 48,000.

[0086] When considering the a-Si pillar array, coupling to resonant modes is still possible at a pillar height of 870 nm, and resonant spectral features are still visible from the pseudocolor plots of transmittance and phase spectra (Figures 13A and 13B). The Q value of the extracted resonant sound is approximately 48,000. The simulation assumed a 5 nm mesh in the z direction. By performing a series of simulations with finer meshes, it may be possible to identify parameter values ​​in which modes with even higher Q values ​​can be observed. The metasurface period is set to P x = 1520nm, P y Assuming = 1425nm, the Q factor of the supported modes is approximately 221,000. Therefore, the metasurface period is P x = From 1425nm to P x Increasing the wavelength to 1520nm can strongly affect the quality factor of the metasurface.

[0087] Figures 14A and 14B show the transmittance and phase spectra of an array of a-Si pillars in air according to one embodiment. The width, length, and height of the pillars are assumed to be w = l = 963 nm and h = 870 nm (see schematic diagram in Figure 1). In Figure 14A, P x = 1425nm, P y = 1425nm. The Q value of the extracted resonant sound is approximately 48,000. The mesh is 20nm in the x and y directions and 5nm in the z direction. In Figure 14(B), the period values ​​are compared with those in Figure 14(A). In Figure 14B, P x = 1520nm and P y= 1425nm. To reduce simulation time, a 20nm mesh is present in all three directions. The Q value of the extracted resonant sound is approximately 211,000.

[0088] Example 5: Optical response of a pillar array: Effect of pillar duration on array performance Several embodiments study how the periodicity of the metasurface affects the modes supported by the metasurface, which consists of an array of a-Si pillars on an SiO2 substrate (Figure 1). During the metasurface period above 1200 nm, there are three distinct modes of false color in transmittance and phase, corresponding to high-Q modes at wavelengths around 1540 nm, and two low-Q modes at wavelengths around 1580 nm and 1590 nm, corresponding to modes 2 and 3 in Figure 10. The x-period P of the mode position. x When examining the dependence on P, the quality of the high-Q mode gradually improves over time. x > During the 1300nm period, the position of the High-Q mode does not change significantly with the period. Mode 2 is different from Mode 3 in terms of period P x The shift becomes stronger during period P. y When the y-period P changes, the High-Q modes shift more strongly compared to when the Y-period is changed. This result is consistent with the details of the spatial mode profiles of the High-Q modes in the xz and yz planes (see Figure 1). y When the wavelength increases from 1300 nm to 1520 nm, the high-Q resonance position shifts by approximately 5 nm.

[0089] Figures 15A to 15D show the optical reaction of an a-Si pillar array on an SiO2 substrate when the incident electric field is x-polarized, according to one embodiment. The metasurface has the same structure as in Figures 1 and 2. That is, the length and width of the pillars are l = w = 963 nm, and the height of the pillars is h = 860 nm. The metasurface period in the y direction is P y = 1425 nm. Figures 15A and 15B show the x-direction P xThe transmittance and phase of the transmitted light are shown according to wavelength and period, respectively. Figures 15C and 15D show the behavior of the high-Q mode when the x-period range is limited to [1300 nm and 1520 nm] in Figures 15A and 15B, respectively. In Figures 15C and 15D, the wavelength range is limited to [1542.7 nm and 1543.7 nm].

[0090] Figures 16A to 16D show the optical reaction of an a-Si pillar array on an SiO2 substrate when the incident electric field is x-polarized, according to one embodiment. The metasurface has the same structure as in Figures 1 and 2. That is, the length and width of the pillars are l = w = 963 nm, and the height of the pillars is h = 860 nm. The metasurface period in the x-direction is P x = 1425 nm. Figures 16A and 16B show the y-direction P y The transmittance and phase of the transmitted light are shown according to wavelength and period, respectively. Figures 16C and 16D show the behavior of the high-Q mode when the y-period range is limited to [1300 nm, 1520 nm] in Figures 16A and 16B, respectively. In Figures 16C and 16D, the wavelength range is limited to [1542 nm, 1548 nm]. Example 6: Modes supported by a single isolated pillar

[0091] In the optical simulations, some embodiments impose periodic boundaries in the x and y directions, implying that a periodic array of square a-Si pillars is being investigated. Some embodiments investigate whether identified photonic modes are also supported by isolated a-Si pillars. Some embodiments simulate the scattering cross-section of isolated pillars. In the revised simulations, fully matched layer (PML) boundary conditions were used for all simulation boundaries. First, the case of a single a-Si pillar in air was considered, and its scattering cross-section was calculated as a function of wavelength and pillar height. The pillar length and width are l = w = 963 nm. Figure 17A illustrates the scattering cross-section of a single a-Si pillar in air as a function of wavelength and pillar height, according to one embodiment. Figure 17B shows the scattering cross-section of a single a-Si pillar in air with a pillar height of h = 834 nm. The highest Q mode is observed at a wavelength λ of approximately 1519.8 nm. As seen in Figures 17A and 17B, crossovers of two other modes are observed in the vicinity of the highest-Q mode. For a pillar height h of approximately 834 nm, the quality factor of the highest-Q mode is approximately 1000.

[0092] Several embodiments provide cases of isolated a-Si pillars on an SiO2 substrate. Figures 18A and 18B show scattering cross-sections of isolated pillars on an SiO2 substrate according to one embodiment, depending on wavelength and pillar height. Figure 18A shows a scattering cross-section of a single a-Si pillar on an SiO2 substrate, depending on wavelength and pillar height. The pillar length and width are l = w = 963 nm. Figure 18B shows a scattering cross-section of a single a-Si pillar with a pillar height h of approximately 830 nm. The highest Q mode is observed at a wavelength λ of approximately 1519.2 nm. An overall broader spectrum of spectral features was observed compared to the case of a single a-Si pillar in air. At a pillar height of 830 nm, the Q value of the support mode is approximately 676.

[0093] Figure 19 shows a scattering cross-section of a single a-Si pillar on an SiO2 substrate according to one embodiment, depending on wavelength and pillar height. The spatial distribution of electric field amplitude |E| is plotted within the a-Si pillar for pillar heights of h = 720 nm, h = 790 nm, h = 830 nm, and h = 850 nm. The electric field is plotted in the xz plane, which passes through the center of the pillar. The solid black line indicates the direction of the field profile toward the simulated scattering peak.

[0094] Figures 20A–20F show the spatial distribution of the x-component (Figures 20A, 20C, and 20E) and y-component (Figures 20B, 20D, and 20F) of the electric field E in the xz plane according to one embodiment. The xz plane passes through the centers of isolated a-Si pillars on an SiO2 substrate. The x and z components of the electric field are plotted along mode line 3 in Figure 18. The geometric parameters of the pillars are the same as in Figure 18. The height and operating wavelength of the a-Si pillars are marked at the top of each column. The x and z components of the electric field gradually change as the height of the pillars increases.

[0095] Figures 21A to 21D show the spatial distribution of the x-component of the electric field E in the xz plane according to one embodiment. The xz plane passes through the centers of isolated a-Si pillars on an SiO2 substrate. The x-component of the electric field is plotted along mode line 2 in Figure 18. The geometric parameters of the pillars are the same as in Figure 18. The height and operating wavelength of the a-Si pillars are marked at the top of each column. The x-component of the electric field gradually changes as the pillar height increases.

[0096] To understand the relationship between the properties of the high-Q resonant sounds considered and the high-Q resonant sounds observed in the case of an array, the spatial distribution of the electric field amplitude in the y-z cross-section of the resonator is shown. Figure 22 shows the scattering cross-section of a single a-Si pillar on an SiO2 substrate according to one embodiment, depending on the wavelength and pillar height. The spatial distribution of the electric field amplitude |E| is plotted within the a-Si pillar for pillar heights of h = 720 nm, h = 790 nm, h = 830 nm, and h = 850 nm. The electric field is plotted in the yz plane passing through the center of the pillar. The solid black line indicates the direction of the field profile toward the simulated scattering peak. As seen in Figure 22, the spatial distribution of the electric field amplitude of the high-Q mode in the yz plane (e.g., at h = 830 nm) is identical to the spatial distribution of the electric field observed in the case of a pillar array (see Figure 1E).

[0097] Example 7: Controlling the spectral shape of resonance Figures 23A and 23B show the transmittance and phase of transmitted light, respectively, depending on the wavelength and thickness of the SiO2 spacer d according to the embodiment. The metasurface structure is shown in Figure 3A. The assumed geometric parameters are the same as in Figure 3, and Figures 23C and 23D show the transmittance and phase spectra for SiO2 spacer thicknesses of d = 510 nm and d = 1870 nm, respectively, according to one embodiment.

[0098] Figure 24 shows the quality coefficient and Fano phase of high-Q resonant tones according to the SiO2 thickness d in one embodiment. The meta-surface structure is shown in Figure 3A. The structure and geometric parameters considered are the same when Figure 3(C) shows that the Fano phase changes abruptly with respect to the SiO2 spacer thickness around 1550 nm. Figure 24 shows that the SiO2 spacer thickness is varied more finely to accurately capture the details of the Fano phase change.

[0099] Figures 25A to 25D show the spatial distribution of electric field amplitude in a metasurface unit cell according to one embodiment. The metasurface structure is shown in Figure 3A. The geometric parameters of the metasurface unit cell are as follows: pillar length and width are l = w = 963 nm, pillar height is h = 845 nm. The assumed period is P x = 1520nm, P y The wavelength is 1425 nm. The thickness of the SiO2 spacer is d = 1450 nm. The spatial distribution of the electric field was plotted in the x-y plane. Figure 25A plots the absolute value of the electric field |E / E0|. Figures 25B and 25C plot the spatial distribution of the x and z components of the electric field, respectively. Figure 25D plots the real part of the z component of the electric field. It was observed that both the x and z components of the electric field are non-zero at the SiO2 spacer.

[0100] Figure 26 illustrates the transmittance and phase shift as functions of a-Si exponential change achieved by using high-Q resonant tones according to one embodiment. The metasurface structure is shown in Figure 3A. The wavelength is fixed at the operating wavelength λ = 1535.44 nm. At the operating wavelength, and for an exponential change of Δn = 0.0026, the phase shift extracted from the periodic array calculation is 107°.

[0101] Example 8: A realistic electromagnetic addressing architecture Several embodiments provide how the proposed metasurface design can be modified to enable thermo-optical control of the wavefront of transmitted light. A first modification of the proposed metasurface unit cell features a-Si pillars connected via a-Si bars. Figures 27A to 27F show dynamic beam switching with a realistic interconnection architecture according to one embodiment. Figure 27A is a schematic diagram of the metasurface unit cell. Square a-Si pillars connected via a-Si bars. In Figure 27A, the entire a-Si is doped to a low concentration and placed on an SiO2 substrate such that the complex refractive index of a-Si is n = 3.734 + 0.0013 i. Several embodiments dope the entire silicon metasurface or a portion thereof. In some embodiments, conductive layers of different materials can be used as electrodes above or below the high refractive index rectangular pillars. Examples of materials include (but are not limited to) semiconductors, doped GaAs, ITO, CdO, AZO, and thin metal layers. The temperature of the metasurface pixel can be actively controlled by biasing the metasurface unit cell at the edge and passing a current in the y direction. The geometric parameters of the metasurface unit cell are as follows: ML = 82 nm, W = L1 = 963 nm, h = 850 nm, P x = P y = 1500 nm. Some embodiments include one or more connecting electrodes on the metasurface.

[0102] Figures 27(B) and 27(C) plot the far-field electric field strength as a function of the rudder angle for a two-level phase lattice using the unit cell shown in Figure 27(A). The a-Si refractive index difference between adjacent metasurface elements is Δn = 0.006. In Figure 26B, the incoming plane wave is x-polarized, while in Figure 27C, the incoming electric field is y-polarized. The inset in Figure 27B shows a schematic diagram of one period of the two-level phase lattice used to theoretically demonstrate switching diffraction. Within the lattice period, the assumed refractive index change between adjacent a-Si pillars is Δn = 0.006 (Figures 27A and 27B). Switching diffraction is observed both when the polarization of the electric field is perpendicular and parallel to the a-Si bar (Figures 27B and 27C, respectively). In Figure 27B, the operating wavelength is 1586.5 nm, and the overall transmittance at the operating wavelength is T = 1.2%. In Figure 27C, the operating wavelength is 1583.2 nm, and the total transmittance at the operating wavelength is T=0.3%. Note that the low-Q mode described in Figure 3 is used to observe diffraction switching. When using the configuration shown in Figure 26A, the spectral characteristics of the high-Q mode are strongly affected by the introduced optical loss, and therefore diffraction beam switching using the high-Q mode cannot be used. That is, in the loss structure (Figure 27A), no large spectral fluctuations in the phase of the transmitted light are observed when no current is applied.

[0103] Furthermore, the metasurface unit cell design can be modified to enable thermo-optic switching using higher-Q modes. Figure 27(C) shows a schematic diagram of a metasurface unit cell with 50 nm above and below the a-Si layer doped. The thickness of the SiO2 layer is 1440 nm. In the improved unit cell design, only the 50 nm above and below the a-Si layer is doped, while the core portion of the a-Si layer is practically undoped (or very lightly doped). In this implementation, the upper and lower electrodes are biased toward each other, and current flows in the vertical direction. The geometric parameters of the metasurface unit cell were also modified: P x = 1520nm and P y= 1425nm. The height of the a-Si and a-Si rods is h = 845nm, and the thickness of the SiO2 spacer is 1440nm. The width and length of the a-Si pillar were set to w = l1 = 963nm.

[0104] Figures 27(E) and 27(F) plot the far-field electric field strength as a function of the rudder angle for a two-level phase lattice using the unit cell shown in Figure 27(D). In Figure 27E, the a-Si refractive index difference between adjacent metasurface elements is Δn = 0.005, and the width of the a-Si bar is δ = 40 nm. In Figure 27F, the a-Si refractive index difference between adjacent metasurface elements is Δn = 0.006, and the width of the a-Si bar is δ = 82 nm. The inset of Figures 27E and 27F shows a schematic diagram of one period of the two-level phase lattice used to theoretically demonstrate switching diffraction. By utilizing higher Q modes, switching diffraction can be demonstrated assuming a refractive index difference between adjacent a-Si pillars of Δn = 0.005 and an a-Si bar width of δ = 40 nm in the lattice period. In Figure 27E, the polarization direction of the electric field of incoming light is perpendicular to the a-Si bar. In Figure 27E, the operating wavelength is 1536.15 nm, and the overall transmittance at the operating wavelength is T = 6.8%.

[0105] Interestingly, using a two-level phase grating allows us to observe highly asymmetric diffraction patterns stemming from the geometric asymmetry of the metasurface unit cell, as shown in Figure 27F. In Figure 27F, the refractive index difference between adjacent a-Si pillars at lattice period Δn is 0.006. The width δ of the a-Si bar is approximately 82 nm. The operating wavelength of the low-Q mode used is 1548.4 nm, and the overall transmittance at the operating wavelength is T=1%.

[0106] Some embodiments provide a case where the metasurface unit cell is placed on an SiO2 pedestal and a-Si pillars are connected in series with a-Si bars. In this case, no high-Q mode spectrum and phase signature are observed for a pillar height of h = 880 nm. Figures 28A to 28C show the dependence of transmittance and phase with respect to pillar and bar height according to one embodiment. Figure 28A is a schematic diagram of the metasurface unit cell. In the schematic diagram, P x = 1520nm, P y =1 425nm, π=82nm. The height of the SiO2 pedestal is 382nm. The length of the pillar is 200nm, and the width of the pillar is also 200nm. Note that the metasurface considered here does not contain a Si substrate. Figure 28B plots the transmittance as a function of wavelength and a-Si pillar height. Figure 28C plots the phase of transmitted light as a function of wavelength and a-Si pillar height. When the height of the a-Si pillar is changed, the height of the a-Si rod is changed simultaneously. As seen in Figures 28B and 28C, at a pillar height of h = 880nm, the sine of the resonant sound spectrum and phase disappears.

[0107] Figures 29A to 29D show the transmittance and phase shift as functions of a-Si exponential change on the metaelectrode according to one embodiment. The meta surface structure is as shown in Figures 5A and 5B. The incident light is x-polarized. Figures 29A and 29B correspond to the high-Q mode, and Figures 29C and 29D correspond to the low-Q mode. The maximum phase shift enabled by the high-Q mode is 277°, while the low-Q mode enabled a phase shift of 230° when the refractive index of a-Si changed by Δn = 0.01.

[0108] Figures 30A to 30D show the transmittance and phase shift as functions of a-Si exponential change on the metaelectrode according to one embodiment. The meta surface structure is as shown in Figures 5A and 5B. The incident light is y-polarized. Figures 30A and 30B correspond to the high-Q mode, and Figures 30C and 30D correspond to the low-Q mode. The maximum phase shift enabled by the high-Q mode is 320°, while the low-Q mode enabled a phase shift of 275° when the refractive index of a-Si changed by Δn = 0.01.

[0109] Figures 31A to 31D show thermo-optical beam switching using a lower Q mode according to one embodiment. The metasurface structure is as shown in Figures 5A and 5B. The optical performance of the two-level phase grating was analyzed. The unit grating of the grating is shown in the inset of Figure 31A. The refractive index change between adjacent a-Si pillars is Δn = 0.006. Figure 31A shows the far-field electric field intensity when the incoming plane wave is x-polarized, the operating wavelength λ = 1575.5 nm, and the overall transmittance T = 7.6%. In Figure 31B, the overall transmittance spectrum of the two-level grating is studied. In Figure 31C, the far-field electric field intensity is shown when the incident plane wave is y-polarized. In Figure 31C, the operating wavelength λ = 1583.2 nm, and the overall transmittance T = 0.6%. Figure 31D shows the overall transmittance spectrum of the two-level grating examined in Figure 31C.

[0110] Figures 32A to 32D show analytical array factor calculations for a two-level phased grid according to one embodiment. The plots calculate the electric field intensity in the far field as a function of the rudder angle in the case of a two-level phased grid. Figures 32A and 32B correspond to the high-Q mode case. Figures 32A and 32B assume that the refractive index difference between adjacent metasurface pixels is approximately 0.0026, which is the value in Figure 5. Figures 32A and 32B correspond to the low-Q mode case (the values ​​of electric field amplitude and relative phase at the grid period are constructed based on Figures 29A to 2D). Figures 32C and 32D assume that the refractive index difference between adjacent metasurface pixels is approximately 0.006, which is the value selected in Figure 31. Figures 32C and 32D assume that the values ​​of electric field amplitude and relative phase at the grid period are constructed based on Figures 29C and 30D. In high-Q modes, array-level analysis calculations yielded different results compared to full-wave simulations, which provided evidence of near-field coupling between adjacent metasurface pixels. In low-Q modes, array factor calculations and full-wave simulations yielded similar results.

[0111] Figures 33A and 33B show the far-field electric field intensity as a function of the steering angle in the case of a three-level phased grating according to one embodiment. The incident light is y-polarized. The three-level phased grating is generated using simulated phase shift data for the operating wavelength λ = 1583.2 nm, and the phase shift values ​​at the grating period are given as (0°, 120°, 240°). Within the grating period, the real part of the refractive index is (3.734, 3.734 + 0.003333, 3.734 + 0.01), and a) plots the far-field electric field intensity at the operating wavelength λ = 1583.2 nm. Figure 33B maintains the spatial distribution of the real part of the refractive index, similar to Figure 33A, but here the operating wavelength is λ = 1583.8 nm. In Figure 33A, the steered beam of the target has a steering angle of 20.3°. Many spurious diffraction orders are observed. By changing the operating wavelength to λ=1583.8nm, the zeroth diffraction order can be suppressed, but other spurious diffraction orders are still observed.

[0112] Figure 34 shows the electric field strength in the far field as a function of the rudder angle for a 3-level phase grating at an operating wavelength of λ = 1535.81 nm, which is the resonance wavelength for high-Q modes. The incident light is y-polarized. Within the grating period, the real part of the refractive index is set to (3.734, 3.734 + 0.003333, 3.734 + 0.01). By appropriately selecting the operating wavelength, the zeroth diffraction order could be suppressed.

[0113] Figures 35A and 35B illustrate transmittance and phase shift as functions of a-Si index change on an optimized metasurface by an electrode according to one embodiment. The incident light is x-polarized. Figure 35A corresponds to the low-Q mode, and Figure 35B corresponds to the high-Q mode. The inset shows the operating wavelength at which the diffraction pattern shown in Figure 7 is observed. At an operating wavelength of λ = 1567.1 nm, the maximum phase shift possible with the low-Q mode is 236.2°. For the high-Q mode, the maximum phase shift at an operating wavelength of λ = 1531.3 nm is 265°.

[0114] Figure 36 shows a thermo-optic three-level phase grating with a realistic interconnection architecture and optimized geometry according to one embodiment. The incident plane wave is TE polarized. The figure plots the far-field electric field intensity for lower Q modes at the operating wavelength λ = 1581.5 nm. The optimization method employed is as follows for the structure shown in Figures 5a-b: P x = 1520nm, P y Geometric parameters of = 1100 nm and h = 851.327 nm were obtained, and the resulting diffraction efficiency D eff = 53% was obtained. The real part of the complex refractive index for each metasurface element during the lattice period is given as n1 = 3.734, n2 = 3.74144, n3 = 3.74318, and so on.

[0115] Example 9: Metasurface having a finite number of metasurface elements Consider an m × m array of metasurface elements. Several embodiments examine how the quality factor of the modes supported by the finite array increases with m. To calculate the quality factor, some embodiments calculate the scattering cross-section of the finite array and fit it to Fano's formula. When studying finite metasurface arrays, the assumed geometric parameters are the same as in Figure 1: l = w = 963 nm and P x = P y = 1425 nm. To reduce simulation time, the assumed mesh in the z direction was set to 10 nm, while the mesh in the x and y directions was set to 20 nm. Assuming periodic boundary conditions in the x and y directions, the quality factor for supported modes is 8500. Note that this quality factor value (8500) is lower than that reported in Figures 1 and 2 (9800). This difference in the observed quality factor is due to the fact that in the simulation shown in Figure 1, the mesh in the z direction was set to 5 nm while the mesh in the x and y directions was still set to 20 nm.

[0116] Figures 37A to 37C show the quality coefficients of a metasurface having a finite number of elements according to one embodiment. Figure 37A shows the quality coefficient of an m × m metasurface array as function m. Figure 37B shows the quality coefficient of the metasurface, which is in the direction parallel to the incident electric field, as described above in direction n x It is finite depending on the number of metasurface elements in the region. In Figure 37B, periodic boundary conditions are assumed in the direction perpendicular to the electric field. Figure 37C shows the quality coefficient of the metasurface, which is in the direction perpendicular to the incident electric field, as described above in the direction n x It is finite depending on the number of metasurface elements in the field. In Figure 37(C), periodic boundary conditions are assumed in the direction parallel to the electric field.

[0117] As shown in Figure 37, the quality factor for supported modes increases monotonically with increasing the number of metasurface elements in the array. For a 12x12 array, the quality factor is 6000.

[0118] In some embodiments, the number of metasurface elements is finite in the direction parallel to the incoming electric field, while periodic boundary conditions are assumed in the direction perpendicular to the electric field. The simulation settings described here allow for shortening the simulation time while accessing the quality factor of the metasurface for a larger number of metasurface elements n x For example, as shown in Fig. 37(b), when the number of metasurface elements parallel to the electric field nx is n x = 10, the quality factor of the metasurface is about 7000. When n x = 20, the quality factor of the metasurface is slightly below 8000.

[0119] Fig. 37C assumes that the number of metasurface elements is finite in the direction perpendicular to the incoming electric field, while periodic boundary conditions are assumed in the direction parallel to the electric field. In this case, when n x = 10, the quality factor of the metasurface is about 6000. However, when the metasurface is finite in the perpendicular or parallel direction to the electric field, when n x = 20, the quality factor of the metasurface is about 8000. Example 10: Impact of potential manufacturing defects

[0120] Some embodiments are investigating the impact of manufacturing imperfections on metasurface performance. Some embodiments consider three different imperfections, namely, tilted sidewalls, rounded corners, and lossy Si. The simulation showed that the metasurface is quite robust to possible manufacturing imperfections such as tilted sidewalls or rounded corners. Depending on the exact tilt of the sidewalls, it may be necessary to slightly change the height of the structure. The challenging aspect for achieving a quality factor of about 10,000 is depositing a-Si with a very low absorption coefficient (k ~ 10 -5 ). A detailed discussion of the effect of each of the listed non-idealities on metasurface performance is discussed below.

[0121] To simplify the inclined sidewalls, some embodiments mount the a-Si resonator case on an SiO2 substrate (see Figures 1 and 2). x = P y The period is set to 1425 nm. Due to the inclined sidewalls, the a-Si resonator is not a rectangular prism but a truncated square pyramid (see inset in Figure 38A). To evaluate how the degree of inclination affects the supported resonant sound, the upper base of the truncated a-Si pyramid is fixed at 963 nm, and the length of the lower base is varied. Figures 38A and 38B plot the transmittance and the phase of the transmitted light as functions of wavelength and the length of the pyramid's lower base. In Figures 38A and 38B, the height of the pyramid is h = 845 nm. A broadening of the high-Q resonant sound was observed with respect to the length of the pyramid's lower base. The phase of the transmitted light showed large variations in the spectral domain only for base lengths below 1030 nm. Note that the large range of phase variation in the spectral domain is important for indicating a dynamically tunable phase shift. Therefore, when the height of the pyramid is h = 845 nm, the inclination is 2.27 o It must not exceed that limit.

[0122] Assume an a-Si truncated pyramid with height h = 885 nm on an SiO2 substrate (Figures 38C and 38D). The upper base of the pyramid is 963 nm. When the lower and upper bases of the pyramid are equal, both high-Q and low-Q modes were observed in the transmittance spectrum, but there was no large phase shift in the spectral domain. The quality coefficient of higher-Q resonant tones increases with decreasing base length, reaching a peak value of approximately 4000 when the base length is 1030 nm (Figure 38C). Notably, when the lower base length increases to 1030 nm, a large phase shift in the spectral domain is observed (Figure 38D). Therefore, when the height of the pyramid is 885 nm, 4.43 o Even for the larger base length of 1100 nm, which corresponds to the tilt angle, there is a large phase variation in the spectral region.

[0123] Several embodiments investigate how optical modes supported by a-Si truncated pyramids on an SiO2 substrate change with pyramidal height (Figures 38E and 38F). In Figures 38E and 38F, the upper base of the pyramid is 963 nm and the lower base is 1030 nm. Increasing the pyramidal height causes redshift in both high-Q and low-Q resonants (Figures 38E and 38F). The quality factor of the high-Q resonants increases when the pyramidal height peaks at values ​​between 840 nm and 890 nm. For given values ​​of the pyramidal base (963 nm and 1030 nm), large phase shifts in the spectral region are observed for pyramidal heights in the range of 850 nm to 885 nm. Thus, the metasurface is robust to possible non-idealisms resulting from not having perfectly vertical sidewalls. Even if the sidewalls are not perfectly vertical, the resonator height can be slightly increased to access modes with a quality factor.

[0124] Figures 38A-38F show the effect of inclined sidewalls on metasurface performance. Figures 38A and 38B show the transmittance and phase of transmitted light as functions of the wavelength and base length of the pyramid, respectively, when the pyramid height is h = 845 nm. Figures 38C and 38D show the transmittance and phase of transmitted light as functions of the wavelength and base length of the pyramid, respectively, when the pyramid height is h = 885 nm. Figures 38E and 38F show the transmittance and phase of transmitted light as functions of the wavelength and pyramid height, respectively. In Figures 38D and 38F, the upper base of the pyramid is 963 nm and the lower base of the pyramid is 1030 nm, which corresponds to 2.27 o It corresponds to the angle of inclination.

[0125] Several embodiments investigate whether rounded corners, rather than the upper right corner, affect the performance of the metasurface. Consider an a-Si parallelepiped with rounded corners and bends on an SiO2 substrate. The dimensions of the structure are the same as in Figures 1 and 2. Figures 39A to 39D show the effect of rounded corners on metasurface performance according to one embodiment. Figures 39A and 39B show the transmittance and phase of transmitted light as functions of wavelength and radius of curvature, respectively, when the pillar height is h = 845 nm. Figures 39C and 39D show the transmittance and phase of transmitted light as functions of wavelength and radius of curvature, respectively, when the pillar height is h = 860 nm. Figure 39 investigates how changing the radius of curvature of the corners and bends affects the transmittance and phase of transmitted light. In Figures 39A and 39B, the resonator height is h = 845 nm, while in Figures 39C and 39D, the resonator height is h = 860 nm. Increasing the radius of curvature resulted in blueshifts for both high-Q and low-Q resonants. The blueshift was more pronounced in the low-Q mode. For a resonator height of h = 845 nm, the high-Q resonance is very robust to non-idealisms associated with rounded corners and bending. For an a-Si resonator height of h = 845 nm, the essential characteristics of the resonant sound are preserved even for a radius of curvature of 120 nm (Figures 39A and 39B). For an a-Si resonator height of h = 860 nm, large phase variations in the spectral region are observed for radii of curvature less than 100 nm (Figure 39C). Therefore, if the expected radius of curvature exceeds 100 nm, the resonator height must be less than 860 nm.

[0126] Si light loss When conducting experiments, non-ideal properties of the material can affect the optical response of the metasurface. Several embodiments investigate how non-zero light loss in a-Si affects the transmitted transmittance and phase spectra. Consider a metasurface composed of a-Si pillars on an SiO2 substrate, as described in Figures 1 and 2. Consider three different pillar heights: h = 830 nm, h = 845 nm, and h = 860 nm. For each pillar height, plot the transmittance spectra for three different values ​​of the a-Si extinction coefficient: k = 0, k = 0.0001, and k = 0.0005 (Figure 40). Lower Q resonances observed at longer wavelengths are substantially unaffected by the introduced light loss. On the other hand, higher Q resonances observed at shorter wavelengths can be strongly affected by the light loss. For example, when the pillar height is h = 845 nm, in the high-Q mode, the phase of the transmitted light exhibits limited variability in the spectral region when the absorption coefficient is k = 0.0005 (Figure 40D). When the pillar height is h = 860 nm, introducing an absorption coefficient k = 0.0001 is sufficient to change the spectral shape of the transmitted phase (Figure 40F). Furthermore, simulations confirm that when the pillar height is h = 860 nm, introducing an absorption coefficient of k = 0.00001 does not affect the optical response of the metasurface (not shown here).

[0127] Figures 40A to 40F show the effect of material loss on metasurface performance according to one embodiment. Figures 40A and 40B show the transmittance and phase of transmitted light as a function of wavelength for different values ​​of the absorption coefficient k when the pillar height is h = 830 nm. Figures 40A and 40B show the transmittance and phase of transmitted light as a function of wavelength for different values ​​of the absorption coefficient k when the pillar height is h = 845 nm. Figures 40A and 40B show the transmittance and phase of transmitted light as a function of wavelength for different values ​​of the absorption coefficient k when the pillar height is h = 860 nm.

[0128] Example 11: High-efficiency reflective, high-quality factor metasurface By adding a gold back reflector to the meta - surface, some embodiments can achieve a high - efficiency high - Q reflective meta - surface. The unit cell of the meta - surface design is shown in FIG. 41A. The meta - surface unit cell includes a gold back reflector followed by a SiO2 spacer. Amorphous Si pillars are deposited on top of the spacer. By appropriately selecting the thickness of the SiO2 spacer, some embodiments obtain a high - efficiency high - Q reflective meta - surface.

[0129] FIGS. 41A and 41B show a high - efficiency transmissive meta - surface according to one embodiment. FIG. 41A shows a schematic of the unit cell of the high - efficiency transmissive meta - surface. In FIG. 41A, the length and width of the a - Si pillars are l = w = 963 nm. The meta - surface period is P x = P y = 1425 nm. The thickness of the SiO2 spacer is 516 nm and the gold is optically thick. FIG. 41B shows the phase and reflectivity spectra of the meta - surface.

Example

[0130] Example 1: An apparatus comprising an electromagnetic meta - surface including a plurality of repeating unit cells conformally arranged periodically on a substrate, wherein the periodicity is smaller than the wavelength of the operating light in free space, and each of the plurality of repeating unit cells includes a first substrate on a second substrate and nanostructures on the first substrate, the apparatus being capable of controlling the phase of the operating light in the transmission mode with a quality factor of at least 10, the apparatus being capable of converting a transmission dip to a transmission peak at the quality - factor resonance, and a change in at least one parameter selected from the group consisting of nanostructure length, nanostructure width, nanostructure height, and the periodicity for adjusting the quality factor.

[0131] Example 2: The exemplary apparatus according to Example 1, wherein the wavelength is selected from the group consisting of ultraviolet wavelengths from 100 nm to 400 nm, visible wavelengths from 380 nm to 800 nm, near - infrared wavelengths from 800 nm to 2500 nm, and infrared wavelengths from 780 nm to 1000 μm.

[0132] Example 3: An embodiment of the apparatus according to Example 1 or 2, wherein the plurality of repeating unit cells are arranged in an array. Example 4: An apparatus of Example 1, 2, or 3, wherein the nanostructure has a shape selected from the group consisting of rectangular parallelepipeds, rectangular parallelepipeds, prisms, pillars, elliptical pillars, trapezoids, triangular prisms, polygonal prisms, pyramidal pyramids, and combinations thereof. Example 5: An apparatus according to any one of Examples 1 to 4, wherein each of the nanostructure and the second substrate comprises a lossless dielectric material having a virtual refractive index of 0.5 or less at the operating wavelength. Example 6: An exemplary apparatus according to any one of Examples 1 to 5, wherein the first substrate comprises a material having a real part of refractive index smaller than the real part of refractive index at the operating wavelength of the nanostructure.

[0133] Example 7: An exemplary apparatus according to any one of Examples 1 to 6, wherein the nanostructure comprises a material selected from the group consisting of gallium arsenide, gallium phosphide, silicon carbide, titanium oxide, silicon nitride, barium titanate, lithium niobate, tantalum pentoxide, silicon oxide, amorphous silicon, silicon, and combinations thereof.

[0134] Example 8: Apparatus of any one example from Examples 1 to 7, wherein the first substrate is made of a material selected from the group consisting of glass, silicon oxide, silicon nitride, gold, silver, aluminum, copper, titanium, platinum, indium tin oxide, aluminum tin oxide, aluminum zinc oxide, magnesium fluoride, tantalum pentoxide, zirconium oxide, vanadium oxide, germanium-antimony-tellurium alloy, titanium nitride, hafnium oxide, hafnium nitride, molybdenum diselenium, hexagonal boron nitride, black phosphorus, tungsten diselenium, tungsten disulfide, and combinations thereof.

[0135] Example 9: An exemplary apparatus according to any one of Examples 1 to 8, wherein the second substrate comprises a material selected from the group consisting of gold, gallium arsenide, gallium phosphide, silicon carbide, titanium oxide, silicon nitride, barium titanate, lithium niobate, tantalum pentoxide, silicon oxide, amorphous silicon, crystalline silicon, silicon, and combinations thereof.

[0136] Example 10: The apparatus according to any one of Examples 1 to 9, wherein the wavelength is a near-infrared wavelength of 800 nm to 2500 nm, the height of the nanostructure is 860 nm or less, the quality factor is 100 to 9800, and the apparatus is configured to be part of a wavefront shaping system.

[0137] Example 11: An apparatus according to any one of Examples 1 to 10, wherein a change in the thickness of the first substrate adjusts the spectral shape of the transmittance.

[0138] Example 12: An apparatus according to any one of Examples 1 to 11, wherein thermo-optical modulation of the refractive index of nanostructures between 0.001 and 0.01 shapes the transmitted light wavefront.

[0139] Example 13: An exemplary apparatus according to any one of Examples 1 to 12, wherein the apparatus is configured to be part of a dynamic beam steering system.

[0140] Example 14: An apparatus according to any one of Examples 1 to 13, wherein the dynamic beam steering system operates with transverse electric (TE) polarization or transverse magnetic (TM) polarization.

[0141] Example 15: An example of the apparatus according to any one of Examples 1 to 14, wherein a plurality of nanostructures in a single row are connected to form an interconnected structure; at least two surfaces of the interconnected structure are conductive; and the interconnected structure is separated from the first substrate via a plurality of pillars.

[0142] Example 16: An apparatus according to any one of Examples 1 to 15, wherein the interconnected structures are heated via a voltage applied to the conductive surface such that the refractive index of the nanostructure is thermo-optically modulated, and the plurality of pillars prevent thermal crosstalk.

[0143] Example 17: The apparatus according to any one of Examples 1 to 16, wherein the conductive surface comprises a material selected from the group consisting of doped semiconductors, doped compound semiconductors, metals, metal alloys, doped gallium arsenide, doped gallium prionide, and doped amorphous silicon.

[0144] Example 18: An apparatus of any one example from Examples 1 to 17, wherein the multiple pillars include materials selected from the group consisting of glass, silicon oxide, silicon nitride, gold, silver, aluminum, copper, titanium, platinum, indium tin oxide, aluminum tin oxide, aluminum zinc oxide, magnesium fluoride, tantalum pentoxide, zirconium oxide, vanadium oxide, germanium-antimony-tellurium alloy, titanium nitride, hafnium oxide, hafnium nitride, molybdenum diselenium, hexagonal boron nitride, black phosphorus, tungsten diselenium, tungsten disulfide, and combinations thereof.

[0145] Example 19: An exemplary apparatus according to any one of Examples 1 to 18, further comprising a light source located on the opposite side of the second substrate from the first substrate. Example 20: An apparatus according to any one of Examples 1 to 19, wherein the light source is a chip-scale laser.

[0146] Principle of Equivalence As can be inferred from the above discussion, the concepts described above can be implemented in various configurations according to embodiments of the present invention. Therefore, although the present invention has been described in certain aspects, many additional modifications and variations will be apparent to those skilled in the art. Thus, it should be understood that the present invention can be implemented in ways other than those specifically described. Accordingly, embodiments of the present invention should be considered in all respects as illustrative rather than restrictive.

[0147] As used herein, the singular terms “a,” “an,” and “the” may refer to multiple objects unless the context clearly indicates otherwise. References to singular objects are not intended to mean “one and only one,” but rather “one or more.”

[0148] As used herein, the terms “approximately” and “about” are used to describe and explain small variations. When used in conjunction with an event or situation, the terms may refer to instances in which the event or situation occurs exactly, as well as instances in which the event or situation occurs very close to it. When used with a number, the terms may refer to a range of variation of that number of ±10% or less, for example, ±5%, ±4%, ±3%, ±2%, ±1%, ±0.5%, ±0.1%, or ±0.05%.

[0149] Furthermore, quantities, ratios, and other numerical values ​​may be presented in range form as used herein. Such range forms are used for convenience and brevity and include numerical values ​​explicitly designated as limits to the range, but should be understood flexibly to also include all individual numerical values ​​or subranges contained within that range, as if each numerical value and subrange were explicitly designated. For example, a ratio in the range of about 1 to about 200 should be understood to include the explicitly listed limits of about 1 and about 200, but should also be understood to include individual ratios such as about 2, about 3, and about 4, as well as subranges such as about 10 to about 50, about 20 to about 100.

Claims

1. including the device An electromagnetic metasurface comprising a plurality of periodic repeating unit cells conformally arranged on a substrate, wherein the periodicity is less than the wavelength of the working light in free space. Here, each of the multiple repeating unit cells includes a first substrate on a second substrate and a nanostructure on the first substrate. Here, the device controls the phase of the working light in transmission mode with a quality factor of at least 10. The device converts transmission dips into transmission peaks in the quality coefficient resonant tone. Here, a change in at least one parameter selected from the group consisting of the length of the nanostructure, the width of the nanostructure, the height of the nanostructure, and the periodicity adjusts the quality factor.

2. The apparatus according to claim 1, wherein the wavelength is selected from the group consisting of ultraviolet wavelengths from 100 nm to 400 nm, visible light wavelengths from 380 nm to 800 nm, near-infrared wavelengths from 800 nm to 2500 nm, and infrared wavelengths from 780 nm to 1000 μm.

3. The apparatus according to claim 1, wherein the plurality of repeating unit cells are arranged in an array.

4. The apparatus according to claim 1, wherein the nanostructure has a shape selected from the group consisting of a rectangular parallelepiped, a rectangular parallelepiped, a prism, a pillar, an elliptical pillar, a trapezoid, a triangular prism, a polygonal prism, a pyramidal pyramid, and combinations thereof.

5. The apparatus according to claim 1, wherein the nanostructure and the second substrate each include a lossless dielectric material having a virtual refractive index of 0.5 or less at the operating wavelength.

6. The apparatus according to claim 1, wherein the first substrate comprises a material having a real portion of the refractive index smaller than the real portion of the refractive index at the operating wavelength of the nanostructure.

7. The apparatus according to claim 1, wherein the nanostructure comprises a material selected from the group consisting of gallium arsenide, gallium phosphide, silicon carbide, titanium oxide, silicon nitride, barium titanate, lithium niobate, tantalum pentoxide, silicon oxide, amorphous silicon, silicon, and combinations thereof.

8. The apparatus according to claim 1, wherein the first substrate comprises a material selected from the group consisting of glass, silicon oxide, silicon nitride, gold, silver, aluminum, copper, titanium, platinum, indium tin oxide, aluminum tin oxide, aluminum zinc oxide, aluminum zinc oxide, magnesium fluoride, tantalum pentoxide, zirconium oxide, vanadium oxide, germanium-antimony-tellurium alloy, titanium nitride, hafnium oxide, hafnium nitride, molybdenum diselenium, hexagonal boron nitride, black phosphorus, tungsten diselenium, tungsten disulfide, and combinations thereof.

9. The apparatus according to claim 1, wherein the second substrate comprises a material selected from the group consisting of gold, gallium arsenide, gallium phosphide, silicon carbide, titanium oxide, silicon nitride, barium titanate, lithium niobate, tantalum pentoxide, silicon oxide, amorphous silicon, crystalline silicon, silicon, and combinations thereof.

10. The apparatus according to claim 1, wherein the wavelength is a near-infrared wavelength of 800 nm to 2500 nm, the height of the nanostructure is 860 nm or less, the quality factor is 100 to 9800, and the apparatus is configured to be part of a wavefront shaping system.

11. The apparatus according to claim 1, wherein a change in the thickness of the first substrate adjusts the spectral shape of the transmittance.

12. The apparatus according to claim 1, wherein thermo-optical modulation of the refractive index of the nanostructure, ranging from 0.001 to 0.01, shapes the transmitted light wavefront.

13. The apparatus according to claim 12, wherein the apparatus is configured to be part of a dynamic beam steering system.

14. The apparatus according to claim 13, wherein the dynamic beam steering system operates with transverse electric (TE) polarization or transverse magnetic (TM) polarization.

15. Multiple nanostructures in a row are connected to form an interconnected structure; and at least two surfaces of the interconnected structure are conductive. The apparatus according to claim 12, wherein the interconnection structure is separated from the first substrate via a plurality of pillars.

16. The apparatus according to claim 15, wherein the interconnected structures are heated via a voltage applied to the conductive surface such that the refractive index of the nanostructures is thermo-optically modulated, and the plurality of pillars prevent thermal crosstalk.

17. The apparatus according to claim 16, wherein the conductive surface comprises a material selected from the group consisting of doped semiconductors, doped compound semiconductors, metals, metal alloys, doped gallium arsenide, doped gallium prinide, and doped amorphous silicon.

18. The apparatus according to claim 16, wherein the plurality of pillars are made of a material selected from the group consisting of glass, silicon oxide, silicon nitride, gold, silver, aluminum, copper, titanium, platinum, indium tin oxide, aluminum tin oxide, aluminum zinc oxide, aluminum zinc oxide, magnesium fluoride, tantalum pentoxide, zirconium oxide, vanadium oxide, germanium-antimony-tellurium alloy, titanium nitride, hafnium oxide, hafnium nitride, molybdenum diselenium, hexagonal boron nitride, black phosphorus, tungsten diselenium, tungsten disulfide, and combinations thereof.

19. The apparatus according to claim 1, further comprising a light source disposed on the opposite side of the second substrate from the first substrate.

20. The apparatus according to claim 19, wherein the light source is a chip-scale laser.

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