Dynamically tunable transmissive metasurfaces with high quality factors

Abstract

Systems and methods for dynamically tunable metasurfaces with quality factor of at least (10) are described. The metasurfaces operate in transmission mode. The metasurfaces can be used for wavefront shaping and beam stearing.

Description

DYNA CALLY TUNABLE TRANSMISSIVE METASURFACES WITH HIGH QUALITYFACTORSGOVERNMENT SPONSORED RESEARCH

[0001] This invention was made with government support under Grant No(s). FA9550- 21-1 -0312 and FA9550-18-1 -0354 awarded by FA9550-18-1 -0354. The government has certain rights in the invention.FIELD OF THE INVENTION

[0002] The present invention generally relates to systems and methods for high quality factor metasurfaces for two-dimensional wavefront manipulation.BACKGROUND OF THE INVENTION

[0003] The regulation of electromagnetic waves with traditional optical components, such as lenses and prisms, is realized through the accumulation of phase delay in the process of light propagation, which limits the reduction and integration of optical devices. Control of phase and amplitude plays an important part in wavefront modulation. Traditional optical elements, as well as diffractive elements such as gratings and holograms, can be bulky for optical set-up. Metasurface can modify the amplitude and impart an abrupt phase shift to the incident wave within the sub-wavelength scale through the light-matter interaction, and thus realize the wavefront modulation more efficiently.

[0004] In optical metasurfaces, a subwavelength-spaced array of localized resonators can be used to abruptly manipulate the phase, amplitude, polarization, and spectrum of light at an interface. Attaining strong light matter interaction and hence high quality factors in metasurfaces can be desirable. However, the required subwavelength scale wavefront control, imposes a limit on the resonator size, leading to significant radiative loss. As a result, most metasurfaces are broadband and rely on dielectric structures with limited light confinement and hence low quality factor (Q-factors less than about 15). Low quality factor means photon residence times are very short, and hence local electromagnetic fields tend to be small.

[0005] Active metasurfaces can dynamically control the wavefront of the scattered light at a subwavelength scale. Most active metasurfaces that enable dynamic wavefront shaping operate in reflection. Active metasurfaces operating in transmission are of considerable interest as they can be integrated with chip-scale light sources, yielding compact wavefront shaping devices. It is challenging to achieve dynamically tunable metasurfaces with high quality factors.BRIEF SUMMARY OF THE INVENTION

[0006] Many embodiments are directed to systems of low-loss active metasurfaces that can dynamically manipulate the transmitted light wavefront. In several embodiments, the dynamically tunable metasurfaces can be made with dielectric materials with quality factor of at least 100. In many embodiments, the metasurfaces manipulate the light in transmission mode. The dynamically tunable metasurfaces can manipulate light in various wavelength ranges from ultraviolet to visible to near infrared to infrared wavelengths.

[0007] Some embodiments include an apparatus comprising: an electromagnetic metasurface comprising a plurality of repeating unit cells with a periodicity conformally disposed on a substrate; wherein the periodicity is less than a wavelength in free space of an operating light; wherein 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 a phase of the operating light in transmission mode with a quality factor of at least 10; wherein the apparatus transforms a transmission dip to a transmission peak in quality factor resonance; and wherein a change in at least one parameter selected from the group consisting of: a nanostructure length, a nanostructure width, a nanostructure height, and the periodicity tunes the quality factor.

[0008] In some embodiments, the wavelength is selected from the group consisting of: an ultraviolet wavelength from 100 nm to 400 nm, a visible wavelength from 380 nm to 800 nm, a near infrared wavelength from 800 nm to 2500 nm, and an infrared wavelength from 780 nm to 1000 pm.

[0009] In some embodiments, the plurality of repeating unit cells is arranged in an array.

[0010] In some embodiments, the nanostructure has a shape selected from the group consisting of: a cuboid, a cube, a pillar, a cylinder, an elliptical cylinder, a trapezoid, a triangular prism, a polygonal prism, a pyramid, and a combination thereof.

[0011] In some embodiments, the nanostructure and the second substrate each comprises a lossless dielectric material with an imaginary refractive index less than or equal to 0.5 at the wavelength of operation.

[0012] In some embodiments, the first substrate comprises a material with a real part of a refractive index less than the real part of the refractive index at the wavelength of operation of the nanostructure.

[0013] In some embodiments, 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 a combination thereof.

[0014] In some embodiments, 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, magnesium fluoride, tantalum pentoxide, zirconium oxide, vanadium oxide, a germanium-antimony- tellurium alloy, titanium nitride, hafnium oxide, hafnium nitride, molybdenum diselenide, hexagonal boron nitride, black phosphorous, tungsten diselenide, tungsten disulfide, and a combination thereof.

[0015] In some embodiments, 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 a combination thereof.

[0016] In some embodiments, the wavelength is a near infrared wavelength from 800 nm to 2500 nm, the height of the nanostructure is less than or equal to 860 nm, and the quality factor is from 100 to 9800; wherein the apparatus is configured to be a portion of a wavefront shaping system.

[0017] In some embodiments, a change in the first substrate thickness tunes a spectral shape of transmittance.

[0018] In some embodiments, a thermo-optic modulation of a refractive index of the nanostructure from 0.001 to 0.01 shapes a transmitted light wavefront.

[0019] In some embodiments, the apparatus is configured to be a part of a dynamic beam steering system.

[0020] In some embodiments, the dynamic beam steering system operates in a transverse electric (TE) polarization or a transverse magnetic (TM) polarization.

[0021] In some embodiments, a plurality of nanostructures in a row is connected to form an interconnected structure; wherein at least two surfaces of the interconnected structure are electrically conductive; wherein the interconnected structure is separated from the first substrate via a plurality of pillars.

[0022] In some embodiments, the interconnected structure is heated via a voltage applied to the electrically conductive surfaces such that the refractive index of the nanostructure is thermo-optically modulated; wherein the plurality of pillars prevents thermal crosstalk.

[0023] In some embodiments, the electrically conductive surfaces comprise a material selected from the group consisting of: a doped semiconductor, a doped compound semiconductor, a metal, a metal alloy, doped gallium arsenide, doped gallium phosphide, and doped amorphous silicon.

[0024] In some embodiments, the plurality of pillars 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, magnesium fluoride, tantalum pentoxide, zirconium oxide, vanadium oxide, a germanium- antimony-tellurium alloy, titanium nitride, hafnium oxide, hafnium nitride, molybdenum diselenide, hexagonal boron nitride, black phosphorous, tungsten diselenide, tungsten disulfide, and a combination thereof.

[0025] Some embodiments further comprise a light source disposed on an opposite side of the second substrate from the first substrate.

[0026] In some embodiments, the light source is a chip scale laser.

[0027] Additional embodiments and features are set forth in part in the description that follows, and in part will become apparent to those skilled in the art upon examination of the specification or may be learned by the practice of the disclosure. A further understanding of the nature and advantages of the present disclosure may be realized by reference to the remaining portions of the specification and the drawings, which forms a part of this disclosure.BRIEF DESCRIPTION OF THE DRAWINGS

[0028] The description will be more fully understood with reference to the following figures, which are presented as exemplary embodiments of the invention and should not be construed as a complete recitation of the scope of the invention. It should be noted that the patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.

[0029] Figures 1A through 1 E illustrate all-dielectric high-Q factor metasurfaces in transmission mode in accordance with an embodiment.

[0030] Figures 2A through 2D illustrate dependence of the properties of the high-Q mode on the height of the a-Si square pillar in accordance with an embodiment.

[0031] Figures 3A through 3F illustrate metasurface structures with a transmittance peak in accordance with an embodiment.

[0032] Figures 4A through 4H illustrate beam steering using the lower-Q factor mode in accordance with an embodiment.

[0033] Figures 5A through 5F illustrate thermo-optic beam switching using the high-Q factor mode with an interconnect metasurface in accordance with an embodiment.

[0034] Figures 6A through 6C illustrate electrical and thermal analysis of thermo-optic beam switching in accordance with an embodiment.

[0035] Figures 7A and 7B illustrate thermo-optic three-level phase grating with a realistic interconnect architecture and optimized geometry in accordance with an embodiment.

[0036] Figures 8A and 8B illustrate spatial distribution of the electric field amplitude inside the metasurface unit cell in accordance with an embodiment.

[0037] Figures 9A through 9D illustrate spatial distribution of the electric field amplitude inside the metasurface unit cell in accordance with an embodiment.

[0038] Figures 10A and 10B illustrate optical response of a-Si pillar array on an SiO2 substrate in accordance with an embodiment.

[0039] Figure 11 illustrates spatial electric field profiles of a lower Q modes supported by the metasurface in the x-z plane in accordance with an embodiment.

[0040] Figure 12 illustrates Spatial electric field profiles of a lower Q modes supported by the metasurface in the y-z plane in accordance with an embodiment.

[0041] Figures 13A and 13B illustrate the optical response of a-Si pillar array suspended in air in accordance with an embodiment.

[0042] Figures 14A and 14B illustrate the transmittance and phase spectrum for an array of a-Si pillars in air in accordance with an embodiment.

[0043] Figures 15A through 15D illustrate the optical response of an a-Si pillar array on an SiO2 substrate when the incoming electric field is x-polarized in accordance with an embodiment.

[0044] Figures 16A through 16D illustrate the optical response of an a-Si pillar array on an SiO2 substrate when the incoming electric field is x-polarized in accordance with an embodiment.

[0045] Figures 17A and 17B illustrate scattering cross section of a single a-Si pillar in air as a function of wavelength and pillar height in accordance with an embodiment.

[0046] Figures 18A and 18B illustrate the scattering cross section of an isolated pillar on an SiO2 substrate as a function of wavelength and pillar height in accordance with an embodiment.

[0047] Figure 19 illustrates scattering cross section of a single a-Si pillar on an SiO2 substrate as a function of wavelength and pillar height in accordance with an embodiment.

[0048] Figures 20A through 20F illustrate spatial distribution of the x-component and y-component of the electric field E in the x-z plane in accordance with an embodiment.

[0049] Figures 21 A through 21 D illustrate spatial distribution of the x-component of the electric field E in the x-z plane in accordance with an embodiment.

[0050] Figure 22 illustrates scattering cross section of a single a-Si pillar on an SiO? substrate as a function of wavelength and pillar height in accordance with an embodiment.

[0051] Figures 23A and 23B illustrate the transmittance and the phase of the transmitted light respectively as a function of the wavelength and thickness of the spacer in accordance with an embodiment.

[0052] Figures 23C and 23D illustrate transmittance and phase spectra for different SiO2 spacer thickness in accordance with an embodiment.

[0053] Figure 24 illustrates quality factor and Fano phase of the high-Q resonance as a function of the SiCh thickness d in accordance with an embodiment.

[0054] Figures 25A through 25D illustrate spatial distribution of the electric field amplitude in the metasurface unit cell in accordance with an embodiment.

[0055] Figure 26 illustrates transmittance and phase shift as a function of the a-Si index change achieved by using a high-Q resonance in accordance with an embodiment.

[0056] Figures 27A through 27F illustrate dynamic beam switching with realistic interconnect architectures in accordance with an embodiment.

[0057] Figures 28A through 28C illustrate dependence of the transmittance and phase on the height of the pillar and bars in accordance with an embodiment.

[0058] Figures 29A through 29D illustrate transmittance and phase shift as function of the a-Si index change for the metasurface with electrodes in accordance with an embodiment.

[0059] Figures 30A through 30D illustrate transmittance and phase shift as function of the a-Si index change for the metasurface with electrodes in accordance with an embodiment.

[0060] Figures 31 A through 31 D illustrate thermo-optic beam switching using the lower-Q mode in accordance with an embodiment.

[0061] Figures 32A through 32D illustrate analytical array factor calculations for a two- level phase grating in accordance with an embodiment.

[0062] Figures 33A to 33B illustrate intensity of the electric filed in the far field as a function of the steering angle in the cases of a three-level phase grating in accordance with an embodiment.

[0063] Figure 34 illustrates intensity of the electric filed in the far field as a function of the steering angle in accordance with an embodiment.

[0064] Figures 35A and 35B illustrate transmittance and phase shift as function of the a-Si index change for an optimized metasurface with electrodes in accordance with an embodiment.

[0065] Figure 36 illustrates thermo-optic three-level phase grating with a realistic interconnect architecture and optimized geometry in accordance with an embodiment.

[0066] Figures 37A through 37C illustrate quality factors of metasurfaces with a finite number of elements in accordance with an embodiment.

[0067] Figures 38A through 38F illustrate the influence of tilted sidewalls on the metasurface performance in accordance with an embodiment.

[0068] Figures 39A through 39D illustrate the influence of the rounded comers on the metasurface performance in accordance with an embodiment.

[0069] Figures 40A through 40F illustrate the influence of the material loss on the metasurface performance in accordance with an embodiment.

[0070] Figures 41 A and 41 B illustrate high-efficiency transmissive metasurfaces in accordance with an embodiment.DETAILED DESCRIPTION OF THE INVENTION

[0071] Chip-scale ultracompact optical components that can be dynamically programmed to scatter or emit light with an arbitrarily shaped wavefront can be applied in various applications, such as light detection and ranging (LiDAR), free space optical communications, additive manufacturing, and directed energy. Active metasurfaces can be used in the chip-scale spatial light modulators. A prototypical active metasurface can include an array of geometrically identical subwavelength resonant metasurface elements which can dynamically control the phase and amplitude of light scattered by each metasurface element. (See, e.g., P. Thureja, et al., Nanophotonics 2022, 11 , 3745; thedisclosure of which is incorporated by reference.) Programmable metasurface chips with electric field effect control of the phase of light scattered by each metasurface element have been shown. (See, e.g., J. Park, et al., Nature Nanotechnology 2021 , 16, 69; the disclosure of which is incorporated by reference.) Dynamic control of the phase at a subwavelength scale may enable the experimental realization of beam steering and reconfigurable focusing using the same metasurface structure. (G. K. Shirmanesh, et al., ACS Nano 2020, 14, 6912; the disclosure of which is incorporated by reference.) However, these active metasurfaces operate in reflection. Operation in reflection mode dictates that the illuminating light source is located off-chip, which can increase the form factor of the resulting optical system. Moreover, a reflection configuration for the illuminating light source may limit versatility of the system, since part of the metasurface aperture will be blocked by the light source. In comparison, transmissive metasurfaces can yield more compact monolithic optical systems, since they allow for integration with chip-based light sources such as vertical cavity surface emitting lasers (VCSELs) or photonic cavity surface emitting lasers (PCSELs).

[0072] Dynamic amplitude-tunable transmissive metasurfaces have been experimentally demonstrated using ITO field effect modulation, conductive polymer electrochemical transitions, ionic transport, Pockels effect in organic molecules, and thermo-optic effect in silicon (Si). (See, e.g., 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; l.-C. Benea- Chelmus, et al., Nature Communications 2021 , 12, 5928; K. Zangeneh Kamali, et al., Light: Science & Applications 2023, 12, 40; the disclosures of which are incorporated by reference.) Transmittance modulation has been used to achieve diffractive beam switching with modest efficiencies. (A. She, et al., Science Advances 2018, 4, eaap9957; E. Arbabi, et al., Nature Communications 2018, 9, 812; the disclosures of which 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 reorientation of liquid crystals in which a degree of phase control may contribute to the observed dynamic beam switching. (S.-Q.Li, et al., Science 2019, 364, 1087; the disclosure of which is incorporated by reference.) Prior work has used dielectric elastomer actuators and microelectromechanical systems for adaptive metalenses or demonstrated thermo-optically tunable Si nonlocal metasurfaces operating at near-infrared wavelengths. (See, e.g., S. C. Malek, et al., Nanophotonics 2021 , 10, 655; the disclosure of which is incorporated by reference.) These works, however, do not allow for arbitrary wavefront reconfiguration of the transmitted light.

[0073] Several designs for dynamically tunable transmissive metasurfaces may use field effect control of the transmission phase to dynamically shape the transmitted light wavefront but with low optical efficiencies (<0.07%), limiting their use in practical applications. (See, e.g., A. Forouzmand, et al., Nanophotonics 2019, 8, 415; J. Park, et al., Appl. Opt. 2018, 57, 6027; the disclosures of which are incorporated by reference.) Theoretical work has demonstrated a dielectric transmissive metasurface which uses carrier injection in Si to realize subwavelength phase control in transmission as well as dynamic wavefront shaping. The optical efficiency of the reported metasurface is about 65%. The maximal phase shift reported for one-dimensional phase gradients is 215°, for an assumed refractive index of silicon An about 0.01. (See, e.g., A. Forouzmand, et al., Laser & Photonics Reviews 2020, 14, 1900353; the disclosure of which is incorporated by reference.) Inverse design may be used for high-directivity beam steering using alldielectric transmissive metasurfaces, such as those based on reorientation of liquid crystal molecules. (See, e.g., H. Chung, et al., ACS Photonics 2020, 7, 2236; the disclosure of which is incorporated by reference.) It was 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 an optical efficiency of about 100%. (See, e.g., Z. Yang, et al., Advanced Optical Materials 2022, 10, 2101893; the disclosure of which is incorporated by reference.) The concept of congener dipoles can be used to develop a dynamic phase-change metasurface design that enables transmissive phase modulation covering over 240° while maintaining transmittance exceeding 80%. (See, e.g., S. Zhuo, et al., Laser & Photonics Reviews 2023, 17, 2200403; the disclosure of which is incorporated by reference.) These recent theoreticalstudies, however, do not assess wavefront shaping capabilities of the designed high- efficiency metasurfaces. Prior work has reported a thermo-optically reconfigurable metalens, which enables a continuous tunability of its focal length from 165 pm to 135 pm when the metalens temperature is increased from 20 °C to 260 °C. (See, e.g., A. Archetti, et al., Nanophotonics 2022, 11 , 3969; the disclosure of which is incorporated by reference.)

[0074] Dielectric passive metasurfaces exhibiting high quality factors have been explored, for example the potential of non-local metasurfaces. (See, e.g., 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; the disclosures of which are incorporated by reference.) For example, all-dielectric metasurfaces supporting delocalized photonic bound states in the continuum (BICs) have shown narrow-bandwidth resonances where a large electric field enhancement is observed. (See, e.g., A. Kodigala, et al., Nature 2017, 541 , 196; the disclosure of which is incorporated by reference.) In addition to structures that support delocalized modes, individual subwavelength dielectric resonators can support quasi-BIC modes, also referred to as supercavity modes, which exhibit moderately high quality factors and are weakly coupled to the radiative continuum. (See, e.g., K. Koshelev, et al., Science 2020, 367, 288; E. Melik-Gaykazyan, et al., Nano Letters 2021 , 21 , 1765; the disclosures of which are incorporated by reference.) The high quality factor of the supercavity modes originates from interference of multiple localized modes supported by a resonator. (See, e.g., M. V. Rybin, et al., Physical Review Letters 2017, 119, 243901 ; the disclosure of which is incorporated by reference.) These features have enabled theuse of quasi-BIC metasurfaces for applications such as sensing and harmonic generation. (See, e.g., A. Tittl, et al., Science 2018, 360, 1105; G. Zograf, et al., ACS Photonics 2022, 9, 567; the disclosures of which are incorporated by reference.) Quasi- BIC mode subwavelength nanolasers have also been realized. (See, e.g., V. Mylnikov, et al., ACS Nano 2020, 14, 7338; the disclosure of which is incorporated by reference.) Quasi-BIC modes supported by an individual cylinder, however, cannot be efficiently excited by a normally incident linearly polarized light, and azimuthally polarized excitationmay be required. For transmissive metasurfaces, normal incidence illumination with linearly polarized light is important for metasurface integration with chip-scale light sources. Notably, addition of an appropriately spaced back reflector to a cylinder array enables excitation of array quasi-BIC modes with normally incident light, but a back reflector precludes use for transmissive metasurfaces. (See, e.g., G. Yang, et al., Nano Letters 2022, 22, 2001 ; the disclosure of which is incorporated by reference.) Excitation of quasi-BIC (or, supercavity) modes by a normal incident linearly polarized plane wave using a single high-index rectangular parallelepiped has also been reported. (See, e.g., L. Huang, et al., Advanced Photonics 2021 , 3, 016004; the disclosure of which is incorporated by reference.)

[0075] Many embodiments provide dielectric high quality factor (Q or Q factor) metasurface structures in transmission mode. Several embodiments use high Q subwavelength resonators as metasurface building blocks such that the metasurface can achieve dynamically tunable optical response upon modulation of the external stimulus. The metasurfaces can be used for (but not limited to) dynamic beam switching or beam steering. Many embodiments implement physically realizable interconnect architectures to enable dynamic beam steering via thermo-optic modulation.

[0076] In many embodiments, the transmissive active metasurfaces operate at near infrared wavelengths (from about 800 nm to about 2500 nm). The transmissive active metasurfaces in accordance with some embodiments can be excited by a normally incident linearly polarized light, exploiting from lower-Q modes to high-Q modes. In various embodiments, the metasurfaces can achieve quality factor of at least about 10; or at least about 20; or at least about 30; or at least about 40; or at least about 50; or at least about 60; or at least about 70; or at least about 80; or at least about 90; or at least about 100; or at least about 200; or at least about 300; or at least about 400; or at least about 500; or at least about 600; or at least about 700; or at least about 800; or at least about 900; or at least about 1000; or at least about 2000; or at least about 3000; or at least about 4000; or at least about 5000; or at least about 6000; or at least about 7000; or at least about 8000; or at least about 9000; or less than or equal to about 10000; or from about 100 to about 999; or from about 1000 to about 10000; or from about 1000 toabout 9999; or from about 3000 to about 9800. In various embodiments, lower-Q or low- Q refers to quality factor from about 100 to about 999. In certain embodiments, higher-Q or high-Q refers to quality factor from about 1000 to about 10000.

[0077] The metasurfaces in accordance with many embodiments can manipulate the wavefront of light of various wavelengths with high quality factors. The light can have wavelengths including (but not limited to) ultraviolet wavelengths from about 100 nm to about 400 nm; visible wavelengths from about 380 nm to about 800 nm; near infrared wavelengths from about 800 nm to about 2500 nm; infrared wavelengths from about 780 nm to about 1000 pm. The light being manipulated by the metasurfaces can have a single wavelength or a range of wavelengths such as broadband illumination. In order to manipulate different wavelengths of incoming light, the metasurfaces can be made of different dimensions and / or be made of different materials. In certain embodiments, the desired dimensions and / or materials of the nanostructures on the substrates can be selected for the light wavelength(s). The metasurfaces can be designed to exhibit multiple high quality optical resonances that appear at different wavelengths, and show selective wavefront manipulation capabilities at different wavelengths.

[0078] The nanostructures on the substrates can be made of various structures and / or dimensions. The nanostructures on the substrates can be arranged in an array; or in parallel lines; or in straight lines; or in curved lines; or in an aperiodic manner. A repeating unit of the nanostructures can be referred to as a unit cell. A unit cell can include at least one nanostructure; or at least two nanostructures; or at least three nanostructures; or at least four nanostructures; or at least five nanostructures. The repeating unit cells can have a periodicity P. The nanostructure has dimensions including a length L, a width W, and a height H. In several embodiments, the periodicity P of the metasurfaces is less than the wavelength of the light. In various embodiments, the periodicity P of the metasurfaces can be greater than or equal to the wavelength of the light. In some embodiments, the length L, the width W, and the height H of the nanostructure are less than the periodicity P. The length L, the width W, and the height H of the nanostructure can be the same or can be different. In various embodiments, the length L, the width W, and the height H scale linearly with the operating wavelength of the light. In several embodiments, thenanostructure can have a symmetrical shape. In some embodiments, the nanostructure can have a non-symmetrical shape to induce a polarization selective response or a chiral response. The nanostructure can have various shapes such as (but not limited to) cuboids, cubes, pillars, cylinders, elliptical cylinders, trapezoids, triangular prisms, polygonal prisms, pyramids, and any combinations thereof. As can readily be appreciated, any of a variety of shapes of the nanostructures can be utilized as appropriate to the requirements of specific applications in accordance with various embodiments of the invention. The geometrical dimensions of the nanostructures, such as length of the L and width W, or height H, can vary arbitrarily in a non-uniform manner over the aperture of the metasurface device according to the optical function of the metasurface.

[0079] In many embodiments, the metasurface structures can include a plurality of unit cells. The unit cells can have various structures and sizes. In some embodiments, a unit cell can include a nanostructure on a substrate. The nanostructures can be deposited conformally onto the substrate(s). The nanostructure can have various shapes such as (but not limited to) cuboids, cubes, pillars, cylinders, elliptical cylinders, trapezoids, triangular prisms, polygonal prisms, pyramids, and any combinations thereof. The nanostructure and the substrate can be made with high refractive index materials with a large nonlinear optical susceptibility to enhance nonlinear optical parametric conversion processes and / or lossless dielectric materials. The lossless dielectric materials can have an imaginary refractive index (also known as extinction coefficient) of less than or equal to about 0.5; or less than or equal to about 0.1 ; or less than or equal to about 0.05 at the wavelength of operation. 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 parallelpipeds. As can readily be appreciated, any of a variety of materials can be utilized as appropriate to the requirements of specific applications in accordance with various embodiments of the invention.

[0080] In several embodiments, the spectral line shape of the high-Q mode at resonance can be ‘inverted’ from exhibiting a dip in transmission to exhibiting atransmission peak by adding a second substrate. The second substrate can have a range of thicknesses. The nanostructure and the second substrate can comprise a material with a higher refractive index than the substrate. The substrate can be made of materials with a lower refractive index. Suitable materials for the nanostructures and the second substrate include materials with a large nonlinear optical susceptibility to enhance nonlinear optical parametric conversion processes and / or lossless dielectric materials. At the wavelength of operation, the real part of refractive index of the substrate material should be less than the real part of the refractive index of the nanostructure material (or high refractive index materials) in accordance with several embodiments. Examples of low refractive index materials for the substrate 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 fluoride, tantalum pentoxide, zirconium oxide, vanadium oxide, germanium-antimony-tellurium alloys, titanium nitride, hafnium oxide, hafnium nitride, molybdenum diselenide, hexagonal boron nitride, black phosphorous, tungsten diselenide, tungsten disulfide and any combinations thereof. In certain embodiments, the low refractive index substrate can be made with elastic materials including (but not limited to) elastic polymers, silicone, polydimethylsiloxane (PDMS), poly(methyl methacrylate) (PMMA), and any combinations thereof. As can readily be appreciated, any of a variety of substrate materials can be utilized as appropriate to the requirements of specific applications in accordance with various embodiments of the invention.

[0081] In several embodiments, the refractive index of the high refractive index materials can be modulated dynamically in real time using a mechanism such as (but not limited to): a thermo-optic effect, an electro-optic effect, a magneto-optic effect, or a nonlinear Kerr effect, and / or by electrical or optical injection of free charges into the unit cell materials. Some embodiments achieve refractive index modulation by thermo-optic modulation of high refractive index materials. In some embodiments, amorphous Si (a-Si) with assumed index modulation ranging between about 0.0026 to about can be achieved via thermo-optic modulation. In certain embodiments, the high refractive index materials comprise multiple quantum wells and the refractive index can be modulated via quantumconfined Stark effect. The modulation of refractive index can sculp the transmitted light wavefront both in the near and far field in accordance with several embodiments.

[0082] Many embodiments implement interconnected metasurface to realize the thermo-optic effect. In various embodiments, the nanostructures can be separated from the substrates in order to avoid thermal crosstalk during thermo-optic modulation. The nanostructures can be separated from the substrates (including the low refractive index substrate and the high refractive index second substrate) via various mechanisms such as (but not limited to) using a pedestal; or a beam; or a pillar; such that the thermal crosstalk between the nanostructures is reduced. In some embodiments, the nanostructures can include electrically conductive materials on at least two surfaces to serve as electrodes for resistive heating during thermo-optic modulation. Examples of electrically conductive materials include (but are not limited to) doped semiconductors, doped compound semiconductors, metals, metal alloys, doped gallium arsenide, doped gallium phosphide, doped amorphous silicon, indium tin oxide (ITO), cadmium oxide (CdO), aluminum doped zinc oxide (AZO). The electrically conductive materials can be in a thin film; or a plurality of thin films. Several embodiments dope the entire conductive layers. Some embodiments dope a portion of the conductive layers. The high refractive index materials can be heated via the applied voltage and / or current. The applied heat can change the refractive index of the materials.

[0083] In some embodiments, the dimensions of the substrates can have various sizes ranging from microns to millimeters or larger. Examples of one dimension of the substrate include (but are not limited to) greater than or equal to about 1 pm; greater than or equal to about 5 pm; greater than or equal to about 10 pm; greater than or equal to about 50 pm; greater than or equal to about 100 pm; greater than or equal to about 150 pm; greater than or equal to about 200 pm; greater than or equal to about 300 pm; greater than or equal to about 400 pm; greater than or equal to about 500 pm; greater than or equal to about 1 mm; greater than or equal to about 2 mm; greater than or equal to about 3 mm; greater than or equal to about 4 mm; greater than or equal to about 5 mm; greater than or equal to about 10 mm. The layer thickness of the substrate (i.e. in the z direction) can have a thickness ranging from 0.1 nm to several millimeters or larger.Dielectric and Dynamically Tunable Transmissive Metasurfaces

[0084] Many embodiments provide dielectric dynamically tunable transmissive metasurfaces with high quality factors. The metasurfaces are achievable using high-Q resonances with Q-factors less than or equal to about 10,000. Several embodiments provide metasurface structures that exhibit a transmittance peak (rather than a transmittance dip at the resonant wavelength for a variety of quality factors. In certain embodiments, the metasurface structures exhibit a transmittance peak at the resonant wavelength with a quality factor range from about 3000 to about 4000. These high quality factors enable achieving beam steering at variations of the a-Si refractive index from about 1x10-3to about 1x1 O'2Lower-Q modes (Q from about 100 to about 999) can also be used for amplitude and phase modulation, but the required a-Si refractive index modulation is higher as compared with the case of high-Q mode. Thermo-optically active elements can be addressed using interconnects that do not perturb the resonator cavity modes, and both high-Q and lower-Q modes can realize thermo-optic beam switching, with switching times less than or equal to about 10 ps. For both high-Q and lower-Q modes, dynamic switching of both TE- and TM-polarized light is possible, although at different wavelengths. Using a three-level phase grating approach, the interconnected metasurfaces are capable of dynamic beam steering, with diffraction efficiencies of about 54% and 69% for lower Q and high-Q modes, respectively. In some embodiments, the optical efficiency of the metasurface structures can be enhanced by introducing a metallic back reflector and / or a SiO2 spacer. The designed reflective metasurfaces may exhibit optical efficiencies greater than about 90%.

[0085] Thermo-optic effect is one of the possible pathways for actively modulating the designed metasurface. Several embodiments use a thin transparent conductive oxide (TCO) layer such as ITO or cadmium oxide underneath the a-Si pillars and using field effect modulation to actively control the optical response of metasurface via gating. Epsilon-near-zero transition in TCOs can be a useful mechanism for achieving phase modulation. The second TCO layer can be added as a counter-electrode. Certain embodiments combine lossless electro-optic materials such as lithium niobate (LNO) orJRD1 molecules embedded in a polymer, with high-Q metasurfaces. In this case, Pockels effect in electro-optic materials may be used for achieving electrically tunable optical response.

[0086] Many embodiments provide dielectric high-Q metasurface structures utilizing refractive index modulation by thermo-optic modulation of amorphous Si (a-Si) with index modulation An ranging between about 0.0026 to about 0.01. The metasurfaces operate in transmission mode. The transmissive active metasurfaces operate at near infrared wavelengths and can be excited by a normally incident linearly polarized light, exploiting either a high-Q mode or lower-Q modes supported by an individual a-Si rectangular parallelepiped (also referred to as a square pillar). The spectral line shape of the high-Q mode at resonance can be ‘inverted’ from exhibiting a dip in transmission to exhibiting a transmission peak by including a Si substrate separated from the square pillar by a silica (SiO2) layer of appropriately chosen thickness. In some embodiments, the resonances exhibit large (at least about 300°) transmitted light phase spectral variation near resonance, which is a prerequisite for dynamically tunable phase shifts. Some embodiments provide that refractive index modulation of modes of individual a-Si pillars can sculp the transmitted light wavefront both in the near and far field. In several embodiments, the metasurface structures can be physically interconnected to enable dynamic beam steering via thermo-optic modulation.

[0087] In many embodiments, a metasurface includes an array of unit cells. Each unit cell can include a nanostructure (such as an a-Si rectangular pillar) on a substrate (such as a silica (SiO?) substrate). The metasurface can be illuminated by a linearly polarized (x-polarized) plane wave from within the SiO2 substrate. The phase and amplitude characteristics of the transmitted light can be studied. As the metasurfaces operate in transmission mode, light sources such as (but not limited to) lasers, or chip scale lasers can be integrated with the metasurfaces. In certain embodiments, light sources can be positioned on an opposite side of the substrate from the nanostructure (such as, a-Si pillar).

[0088] Figures 1A through 1 E illustrate all-dielectric high-Q metasurfaces in transmission mode in accordance with an embodiment. Figure 1 A shows a schematic ofa transmissive metasurface. The metasurface includes an array of amorphous a-Si rectangular pillars on an SiO2 substrate. Figure 1 B shows a schematic of a unit cell of the transmissive metasurface. The pillar can have a height h of about 860 nm, and the pillar length / and width w can be about 963 nm. The metasurface period is Pxand Pycan be about 1425 nm. The a-Si resonators are illuminated by an x-polarized light from within the substrate. The refractive indices of a-Si and SiO2 are 3.734 and 1.44, respectively.

[0089] Figure 1 C shows transmittance and phase spectra of the metasurface shown in Figure 1A. The Q-factor of the supported mode is Q about 9800, as determined by fitting the transmittance spectrum with Fano form lineshape. The observed transmittance dip is accompanied by a broad spectral feature in the transmitted light phase, indicating that this unit cell motif can permit design of metasurface phase gradients either via geometric tuning or an external active control.

[0090] The spatial mode profile can be used to gain further insight about the high-Q mode. Figures 1 D and 1 E show spatial distribution of the electric field amplitude in the metasurface unit cell in x-z and y-z planes, respectively. Eo denotes the amplitude of the impinging electric field. In Figures 1 D and 1 E, the considered cross-section in which the electric field is calculated traverses the center of the a-Si resonator. Figure 1 D indicates that in the x-direction, the electric field is tightly confined within the resonator and is enhanced by a factor of almost 80. While a large electric field enhancement is inside the a-Si resonator, a modest enhancement of the electric field below and above the a-Si pillar can be observed as well. In the y-z plane, which is perpendicular to the polarization of the incoming light, a more complex electric field distribution is observed where the only nonzero component is Ex as shown in Figure 1 E. In the y direction, a non-negligible electric field enhancement is observed between neighboring metasurface unit cells, which could be an indication of inter-resonator near-field coupling in the direction perpendicular to the incoming electric field. This inter-resonator coupling can have broad implications for comprehensive transmitted light wavefront control.

[0091] Several embodiments provide that the nanostructure geometries can affect transmittance characteristics. In some embodiments, the transmittance characteristics may vary with the height of the square a-Si pillars while the width is kept fixed. Figures2A through 2D illustrate dependence of the properties of the high-Q mode on the height of the a-Si square pillar in accordance with an embodiment. Figure 2A shows transmittance as a function of wavelength and the a-Si pillar height h. Figure 2B shows quality factor and Fano phase of the designed high-Q mode a function of the a-Si pillar height h. Figures 2C and 2D show transmittance and phase as a function of wavelength for the a-Si pillar heights of h of about 850 nm and h of about 870 nm, respectively. Both the resonance lineshape and linewidth change upon increasing the pillar height as shown in Figure 2B and 2C. At certain pillar heights, asymmetric resonance lineshapes indicative of Fano resonances, namely a high-Q mode coupled to a broader mode, or a “continuum” of modes can be seen. By fitting transmittance spectra to Fano resonance lineshapes, the dependence of the resonance quality factor and the Fano phase on the pillar height can be seen. The Fano phase characterizes the phase difference between the narrow and broad resonant modes. The quality factor of the resonance increases with the pillar height, and peaks at h of about 860 nm as shown in Figure 2B, with a value of approximately 9800. At a pillar height corresponding to the maximal Q-factor, the Fano phase passes through 90° (thus, the Fano asymmetry parameter is zero), indicating a symmetric quasi-Lorentzian lineshape. If the substrate is removed and an array of a-Si pillars is considered suspended in free space, Q-factor can be as high as 221 ,000. It may be possible to achieve even higher quality factors in free space by more systematic variation of the pillar height. When changing the metasurface period, for periods exceeding 1300 nm, the spectral position of the resonance does not change significantly when increasing the metasurface period.

[0092] The spectral behavior of the transmitted light phase can be important for metasurface applications. Several embodiments examine the spectral behavior of the transmitted light phase. When the pillar height is below 860 nm (h < 860 nm), the phase of the transmitted light spans almost 360° when varying the wavelength of the transmitted light. When h is greater than about 860 nm, an abrupt change in the spectral characteristics of the transmitted light can be seen. For example, at a pillar height of h about 870 nm, the phase spans about 40° (as a function of wavelength). For h greaterthan about 860 nm, the phase variation is limited despite a high quality factor. Thus, the pillar height should be less than about 860 nm for wavefront shaping applications.

[0093] The optical mode supported by a pillar in an array can be compared to the mode 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 1 E). For an isolated a-Si pillar in free space, the Q-factor of this mode is greater than about 1000, whereas, for an a-Si resonator on a substrate, the Q-factor is reduced due to mode leakage into the substrate. Scattering cross-section calculations indicate that, besides the high-Q mode, the isolated Si pillar also supports two lower-Q modes. When comparing isolated pillar mode profiles with mode profiles of a pillar in an array, nonlocality for one of the lower-Q modes is observed since its field profile is not observed in the isolated resonator.

[0094] Several embodiments provide the dependence of the spectral position of the resonance on the metasurface period. Changing the metasurface period may not significantly affect the spectral position of the high-Q resonance. In some embodiments, for metasurfaces with a finite number of metasurface elements in either the x or y direction and for 20 metasurface periods, the quality factor of the mode converges with the quality factor of the infinite metasurface array. In the case of delocalized resonances, such as guided mode resonances, the effect of the finite grating size becomes negligible when the number of the grating periods approaches the quality factor of the resonance. This would correspond to thousands of grating periods. These observations lead to the conclusion that the high-Q mode may be quasi-local.

[0095] One shortcoming of the metasurface motif (pillar on the substrate) can be that large phase variation is accompanied by low transmittance, and operating at the transmittance dip wavelength can result in low metasurface optical efficiency. Several embodiments implement metasurface structures that enable high Q modes to be supported with large phase variation at a transmittance peak rather than a transmittance dip. Figures 3A through 3F illustrate metasurface structures with a transmittance peak in accordance with an embodiment. Figure 3A shows a schematic of a transmissive metasurface. The top panel shows the schematic of the metasurface unit cell. The bottompanel shows the metasurface comprising an array of the unit cells. The unit cell includes a high refractive index second substrate (such as a crystalline silicon (c-Si) substrate), a low refractive index substrate (such as an SiO2 spacer), and a high refractive index nanostructure (such as an a-Si rectangular pillar). A unit cell without the second substrate (or the c-Si substrate) may exhibit a transmittance dip. The second substrate (or the c-Si substrate) can enable the high Q modes with large phase variation at a transmittance peak.

[0096] Figure 3B shows transmittance and phase spectra of the metasurface with a 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 h about 845 nm, and w and / about 963 nm. The height of the a-Si resonator is slightly reduced to about 845 nm considering taller pillars may compromise variation of phase as a function of wavelength. The period values are Px about 1520 nm and Pyabout 1425 nm. Increasing the metasurface period in the x- direction to Px= 1520 nm reduces the near-field coupling between the neighboring metasurface unit cells. In Figure 3B, a SiO2 spacer thickness d of about 1450 nm can yield a large phase variation with wavelength accompanied by a transmittance peak of about 37%. The electric field profiles for a modified unit cell (Figure 3A) exhibit mode leakage into the SiO? spacer. Both Exand Ezcomponents of the electric field have nonzero values in the SiO2 spacer layer, and the electric field enhancement in a-Si pillars can be as high as 120.

[0097] In various embodiments, the spectral shape of the transmittance can be controlled by changing the thickness of the SiO2 spacer. Figure 3C shows the resonance quality factor and Fano phase as a function of the SiC>2 thickness d. By appropriately choosing the SiO2 thickness d, the Fano phase can be tuned over the range 0° to 140° (Fano phase is uniquely defined between 0° to 180°). This large variation of the Fano phase is indicative of comprehensive control of the resonant lineshape, exhibiting transmission dips, transmission peaks and asymmetric spectral shapes. Thus, by tailoring resonator mode leakage into the spacer layer, the relative phase between the high-Q mode and the continuum can be modified, resulting in a change of the transmittance spectral lineshape.

[0098] Several embodiments assess metasurface beam steering. Assume all pillar refractive indices change by an equal amount An (change in refractive index) and calculate the phase shift and the transmittance as a function of the index change An at a given wavelength. The phase shift can be defined as a difference between the phase of the transmitted light at An 0 and at An = 0. Figure 3D shows transmittance and phase shift as a function of the a-Si index change at a wavelength of about 1534.8 nm. The calculations show that a refractive index change An of about 3 x 10’3can result in a phase shift of about 250°, which is sufficient to enable beam steering.

[0099] Some embodiments implement one-dimensional beam steering where the phases of all metasurface elements along the y direction are identical while the phases of neighboring elements in the x direction differ. Calculations for periodic arrays of identical elements suggest that an index change that would yield a 180° phase shift between neighboring elements in the x direction can create a two-level phase grating and strongly suppress the zeroth order beam. However, full wave simulations indicate that the index change value derived via this method may not be optimal, as an unexpectedly large zeroth order beam is obtained. The observed low diffraction efficiency may be due to non- negligible near-field coupling between neighboring metasurface elements. The calculations indicate that an index change An of about 0.0026 between neighboring rows of a-Si resonators at fixed operating wavelength 4 of about 1535.44 nm can fully extinguish the zeroth order diffraction beam. Figure 3E shows intensity of the electric field in the far field for a two-level phase grating with 10 grating periods when the index change between neighboring a-Si pillars is An = 0.0026. Interestingly, for these parameters ( / \ = 1535.44 nm, An = 0.0026), the predicted relative phase shift between light scattered by adjacent rows of pillars is about 107°, according to the simulations of periodic arrays of identical resonators. In principle, this phase should result in a non-negligible zeroth order beam. These results indicate that such a simple ansatz for optimal beam steering may not apply for these high-Q metasurfaces.

[0100] Figure 3F shows overall transmittance of the two-level grating. In Figure 3F, the dot marks the transmittance at an operating wavelength A of about 1535.44 nm. The inset of Figure 3F shows the spatial distribution of the electric field in the x-z cross-sectionof the unit cell. The considered x-z cross-section goes through the center of the a-Si pillars.

[0101] Metasurface optical efficiency is an important design parameter. The overall transmittance of the two-level phase grating T is about 11 % (Figure 3E and 3F) which is lower than the peak transmittance of about 35% derived from the identical resonator array simulations. The observed reduction in transmittance is due to the spectral shape of the resonance (Figure 3B). As seen in Figure 3B, the transmittance falls off abruptly when moving away from the resonant wavelength. When realizing a two-level grating, the resonant wavelength of every other metasurface element is shifted, resulting in the reduction of the overall transmittance at a given operating wavelength. The transmittance spectrum of the two-level phase grating exhibits two distinct peaks (Figure 3F). The nearfield distribution of the electric field (see the inset of Figure 3F) indicates that at a given operating wavelength, the relative electric field enhancement in neighboring a-Si rectangular pillars differs quite significantly. The optimal operating wavelength, which is marked by a dot in Figure 3F, is located between the two resonant peak wavelengths resulting in a lower overall transmittance. To ensure wavefront shaping with higher optical efficiency, the transmittance should stay above a certain minimal value when spectrally moving away from the resonance.

[0102] The design of high-Q dynamic beam steering metasurfaces has two challenges: i) One cannot use intuitive phase profile to steer the beam with a high steering efficiency, and the optimal beam steering conditions have to be identified via full wave simulations; ii) diffractive switching occurs only in one direction due to significant interemitter near-field coupling in the direction perpendicular to the polarization of the incoming electric field (y-direction). Hence, it would be desirable to identify optical modes which enable two-dimensional beam steering.

[0103] Figures 4A through 4H illustrate beam steering using the lower-Q mode in accordance with an embodiment. Besides the high-Q mode, several lower-Q modes are supported with quality factors of about 200. Figure 4A shows transmittance and phase spectra of the metasurface with a unit cell shown in Figure 1 B. In Figure 4A, the metasurface period values are Px of about 1500 nm and Pyof about 1500 nm. Thegeometrical parameters are as follows: the pillar height h of about 845 nm, and the pillar length / and width w are of about 963 nm. The metasurface with a-Si rectangular pillar exhibits 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 about 1583 nm is accompanied by a large spectral variation of the phase. Figure 4B shows transmittance and phase shift as a function of the a-Si index change at a wavelength 4 of about 1583.85 nm. When changing the refractive index of a-Si pillars An by about 0.01 everywhere in the metasurface, a dynamically tunable phase shift of about 285° is observed, which is large enough to enable dynamical beam steering.

[0104] Some embodiments investigate whether the lower-Q mode can be used for dynamic beam steering, in two configurations: i) when the beam is steered in the direction perpendicular to the polarization of the incoming electric field (transverse electric or TE case); ii) when the steering direction is aligned with the polarization of the incoming light (transverse magnetic or TM case). Figure 4C shows intensity of the electric field in the far field as a function of the polar (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 intensity of the electric field in the far field for a two-level phase grating when the incoming electric field is parallel to the steering direction (TM polarization). Figure 4E shows near-filed electric field distribution in the two-level phase grating depicted in the inset of Figure 4C, which corresponds to the case of the TE polarization. Figure 4F shows near-filed electric field distribution in the two-level phase grating depicted in the inset of Figure 4D, which corresponds to the case of the TM polarization. In Figures 4E and 4F, Eo denotes the amplitude of the impinging electric field.

[0105] Figures 4G and 4H plot the intensity of the electric field in the far field as a function of the steering angle in the cases of three-level and four-level phase gratings, respectively. Figures 4G and 4H correspond to the case of the TE polarization. The insets of Figures 4G and 4H show the near-field distribution of the electric field in each case. The overall transmittance values are specified in the insets of Figures 4C, 4D, 4G, and 4H. The operating wavelength A is about 1583.85 nm.

[0106] Some embodiments use a two-level phase grating and assume that the a-Si index change between neighboring element rows An is about 0.0048 (for schematic of a grating period see insets of Figure 4C and 4D). At An of about 0.0048, neighboring metasurface rows may exhibit a phase difference of 180° at a wavelength of about 1583.85 nm (Figure 4B). Thus, a near complete suppression of the specularly transmitted beam may be expected. However, the full wave simulations show significant transmission at normal incidence, for two reasons: i) near-field coupling between neighboring metasurface elements, ii) the difference in scattered light amplitude from neighboring elements.

[0107] To fully suppress the normally transmitted beam at an operating wavelength of about 1583.85 nm, an optimization procedure for the pillar refractive indices may be used to maximize the diffraction efficiency of a desired order, resulting in full suppression of the zeroth diffraction order at an operating wavelength A of about 1583.85 nm for both TE and TM polarizations of the incoming light (Figure 4C and 4D). In the case of the TE polarization, the optimized refractive index of the one of the a-Si pillars within the grating period can retain its original value (m = 3.734), and the refractive index difference An between neighboring a-Si pillars An is about 0.0055, which yields a phase shift of about 200° (see Figure 4B), which is close to the originally chosen phase shift of about 180°. In the case of the TM polarization, the optimization procedure yields a-Si refractive indices: m of about 3.7353 and m of about 3.7440. The corresponding phase difference for adjacent a-Si pillars is about 270°, which significantly deviates from the original value of 180°. For TM polarization, full suppression of the zeroth diffraction order occurs for a phase profile which significantly differs from the optimal phase shift uncoupled resonator array, indicating that inter-element near-field coupling is more significant for TM than TE polarization.

[0108] Although diffractive switching of a two-level phase grating is observed, this analysis does not clarify whether intermediate steering angles are possible using a blazed grating-type design approach. In the case of TM polarization, using three- and four-level blazed grating phase profiles, a highly directional beam owing to near-field coupling between neighboring metasurface elements may not be obtained. However, for TEpolarization, beam steering to angles of about 20.6° and about 15.2° with reasonable diffraction efficiencies can be achieved. To steer a beam to a polar angle of about 20.6°, Figure 4B can be used to construct a three-level phase profile with relative phase between elements of about 0°, 120°, 240°, respectively, resulting in beam steering with a diffraction efficiency of about 60% at an operating wavelength A of about 1583.85 nm (Figure 4G). To steer the beam to a polar angle of about 15.2°, a phase profile of about 0°, 90°, 180°, 270° is constructed, resulting in a diffraction efficiency of 38% at an operating wavelength 4 of about 1583.85 nm and 46% at A of about 1584.16 nm. To improve the diffraction efficiency at 4 of about 1583.85 nm, a full wave optimization procedure if performed, yielding a diffraction efficiency of about 52% (Figure 4H), with a-Si refractive indices: m = 3.7340, r?2 = 3.7358, m = 3.7386 = 3.7417. Thus, for TE-polarized incident light, the metasurface can steer the beam to intermediate angles between 0° and 20.6°.

[0109] The overall transmittance of the four-level phase grating is higher as compared with the overall transmittance of the three-level phase grating (Figures 4H and 4G). This is because the transmission amplitude values of the four-level phase grating are on average higher as compared with the transmission amplitude values of the three-level phase grating (Figure 4B).

[0110] In several embodiments, both high- and lower-Q modes can be used for transmitted light wavefront manipulation. Some embodiments provide metasurface structures that enable refractive index variation and dynamic beam switching, by refractive index modulation of a-Si using the thermo-optic effect. To selectively heat rows of a-Si pillars, many embodiments implement a-Si electrodes and connect the pillars in series. In some embodiments, the high refractive index nanostructures (such as a-Si pillars) can be placed on low refractive index structures (such as SiO2 pedestals) to limit thermal crosstalk between neighboring pillars. In certain embodiments, fabrication entails first pattering a-Si on an SiO2 spacer followed by hydrofluoric acid wet etching of the SiO2 spacer into pillars on a c-Si substrate.

[0111] Figures 5A through 5F illustrate thermo-optic beam switching using the high-Q mode with an interconnect metasurface in accordance with an embodiment. Figure 5A shows a schematic of the interconnected metasurface. Figure 5B shows a schematic ofa unit cell of the interconnected metasurface. In Figures 5A and 5B, the 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 upon an SiCh pedestal to enhance thermal insulation between neighboring metasurface nanostructures. The top and bottom 50 nm-thick a-Si layers are doped, and a voltage can be applied to the top and bottom layer resulting in current flow between the electrodes through the lightly doped a-Si layer. The induced current raises the temperature T of a-Si pillars by Joule heating. The raised temperature can modify the refractive index of a-Si via the thermos-optic effect. For the unit cell, the width w and length / of the pillar are about 963 nm, the height of the pillar h is about 850 nm. The width of the bar 5 is about 50 nm, and the height of the bars is about 850 nm. The SiO? pedestal has a shape of a rectangular pillar with a length a and width b of about 200 nm, and a height of about 382 nm. The thickness of the planar SiO2 spacer d is about 380 nm. The structure is built on a high refractive index substrate (such as a c-Si substrate).

[0112] In Figures 5A and 5B, the top and bottom layers of the high refractive index nanostructures and bars are doped to form electrically conductive electrodes. As can be readily appreciated, different surfaces of the high refractive index nanostructures and bars can be doped to form electrodes as long as desired heating patterns can be achieved. In some embodiments, a pair of electrodes can be formed for each row of the interconnected metasurface (Figure 5A). The front surface and the rear surface 501 of each row of the interconnected metasurface can be doped to form electrodes such that a voltage can be applied to heat up the row. The left side and the right side 502 of each row of the interconnected metasurface can be doped to form electrodes such that a voltage can be applied to heat up the row. In various embodiments, the metasurfaces comprise a plurality of connecting electrodes. The electrodes can be doped in whole; or partially doped. The electrodes can be conductive layers made of doped semiconductors (such as doped silicon, doped GaAs, ITO, CdO, AZO) or metals.

[0113] The induced current raises the temperature T of a-Si pillars by Joule heating, with modified a-Si refractive index n(T) = nsi + An(AT). For silicon, a temperature change AT of about 10 K changes the refractive index A by about 0.00239. Dynamic beamswitching using a higher-Q mode is possible with refractive index difference An at least about 0.0026, corresponding to a relative temperature difference of at least about 11 K. For beam switching with lower-Q modes, a refractive index difference An at least about 0.006 may be needed, corresponding to a relative temperature difference AT of at least about 25 K. The interconnected metasurfaces can enable beam steering with modest temperature differences.

[0114] To construct a desired transmitted light phase profile, several embodiments heat individual rows of pillars to different temperatures. For the high-Q mode, diffractive switching with a two-level phase grating can be realized by changing the temperature of every other metasurface pixel (i.e. , electrically connected rows of metasurface elements) by at least 11 K (An at least 0.0026). Heating to create a two-level phase grating, deflects the light to angles 0 of about ±30° (Figures 5C and 5E). For an incident x-polarized electric field, this modest refractive index difference enables diffractive beam switching with a complete suppression of the zerothdiffraction order and an overall transmittance T of about 13% (Figure 5C). Diffractive beam switching can be observed for y-polarized incidence (Figure 5D) with a lower overall transmittance and at a different operating wavelength. For a-Si on an SiCh substrate, diffractive switching for the high-Q mode is observed when the steering direction aligns with the incoming plane wave (TM) polarization. The a-Si electrodes and the SiO2 pedestal serve to reduce near-field coupling of neighboring metasurface elements. Lower-Q modes are also capable of diffractive switching for both TE and TM polarized incidence, at different wavelengths.

[0115] Figure 5C shows the intensity of the electric field in the far field for a two-level phase grating when the incoming plane wave is x polarized. In Figure 5C, the operating wavelength is about 1537.15 nm. Figure 5D shows the overall transmittance spectrum corresponding the two-level phase grating studied in Figure 5C. In Figure 5E, the intensity of the electric field in the far field for a two-level phase grating when the incoming plane wave is y polarized. In Figure 5E, the operating wavelength A is about 1534.25 nm. Figure 5F shows the overall transmittance spectrum corresponding to the two-level phase grating studied in Figure 5E. Insets of Figures 5D and 5F show spatial distribution of the electric field in the x-z cross-section of the grating period at operating wavelengths in caseof x- and y-polarized incoming light, respectively. The refractive index difference between neighboring metasurface elements An is about 0.0026.

[0116] Several embodiments perform a coupled electrical and thermal analysis. Figures 6A through 6C illustrate electrical and thermal analysis of thermo-optic beam switching in accordance with an embodiment. To induce a diffraction grating analogous to the one in Figure 5A, a voltage Vi of about 1 .05 V can be applied between the top and bottom doped a-Si layers of one metasurface pixel while grounding each neighboring metasurface pixel ( / 2 = 0 V) in a periodic array. Figure 6A shows steady state spatial current density distribution in a thermo-optically controlled metasurface with Vi = 1.05 V and I / 2 - 0 V. Figure 6A illustrates the calculated current density in the silicon bars in one grating period. The largest current density and most significant heat generation occurs in the top and bottom doped a-Si layers of the connector bars. Figure 6B shows steady state spatial temperature distribution in a thermo-optically controlled metasurface with 1 / 1 = 1 .05 V and I / 2 = 0 V. Figure 6B illustrates the steady state temperature distribution for one grating period, such that the temperatures of two adjacent metasurface pixels are 7i = 312.8 K and T2 = 301.4 K, respectively, yielding refractive index difference An of about 0.0026, assuming a thermo-optic coefficient of 2.3 x W4K'1. The power density required to maintain this temperature difference in steady state is about 1.62 pW / pm2The temperature distribution in the pillars and connectors is nearly uniform due to the high thermal conductivity of silicon, even though heating occurs predominantly in the connector bars. The largest temperature gradient occurs in the silicon oxide pedestal due to its large thermal resistance, which greatly reduces the thermal crosstalk between adjacent rows. To assess metasurface dynamic performance, several embodiments provide transient electrical and thermal simulations of dynamic thermal switching between the two pixels comprising one grating period. Figure 6C shows transient temperature difference between the two rows of a thermo-optically controlled metasurface when switching the voltage with a square wave of amplitude V of about 1.05 V and a frequency of about 100 kHz. The geometrical parameters are identical to the ones of the structure in Figure 5B. A response time of about 7.3 ps is obtained for a square wave voltageprofile, indicating that thermo-optic switching is possible at frequencies up to about 140 kHz.

[0117] Several embodiments provide that metasurface beam steering performance with interconnected metasurfaces is possible when a three-level phase profile (0°, 120°, 240°) is applied to the metasurface, for both lower-Q and high-Q modes under TM- polarized plane wave illumination. This phase profile yields a maximum diffraction efficiency of about 35% for the lower-Q mode and about 28% for the high-Q mode. Optimization of the a-Si pillar refractive indices results in a modest improvement of diffraction efficiencies. To further increase diffraction efficiency, some embodiments cooptimize the geometrical parameters of the structure and refractive indices of a-Si pillars enabling enhanced diffraction efficiencies of 54% and 69% for the lower-Q and the high- Q mode, respectively. Figures 7A and 7B illustrate thermo-optic three-level phase grating with a realistic interconnect architecture and optimized geometry in accordance with an embodiment. The incoming plane wave is TM-polarized. Figure 7A shows intensity of the electric field in the far field for the case of the lower-Q mode. In Figure 7A, the operating wavelength 4 is about 1567.1 nm. Figure 7B shows intensity of the electric field in the far field for the case of the high-Q mode. In Figure 7Bm the operating wavelength A is about 1531.3 nm. The insets of Figures 7A and 7B show the spatial distribution of the electric field in the x-z cross-section of the grating period at the operating wavelengths.EXEMPLARY EMBODIMENTS

[0118] Although specific embodiments of systems and apparatuses are discussed in the following sections, it will be understood that these embodiments are provided as exemplary and are not intended to be limiting.Example 1 : Methods

[0119] The Optical simulations were performed using the finite difference time domain method (FDTD Lumerical). In our optical simulations, a normally incident linearly polarized plane wave illuminates the metasurface from within the substrate. When simulating an array of a-Si pillars, periodic boundary conditions are used in the x and ydirections, and perfectly matched layers (PML) boundary condition is used in the z direction. When considering the behavior of an isolated pillar, PML boundary conditions are used at all simulation boundaries, and the simulation volume is 3 x 3 x 3 pm3. In the simulations the materials are assumed non-dispersive. The refractive indices of a-Si, SiO2, and c-Si are taken as 3.734, 1.44, 3.43, respectively. The effect of potential fabrication non-idealities on the metasurface performance is summarized below. When considering interconnected metasurface structures to perform thermo-optic beam switching and thermo-optic beam steering, several embodiments assume the complex refractive index of the doped a-Si layers n is about 3.734+0.0013 i, which corresponds to the carrier density of 6 x 1018cm-3in the case of the n-doped Si layer and the carrier density of 1019crrr3in case of the p-doped Si layer. Complex refractive index of the lightly doped Si core n is about 3.734+0.000025 i, which corresponds to the carrier density of 3.2 x io17cm-3of the n-doped Si. In simulations used to generate Figures 1 A through 2D, the assumed mesh in the z-direction is 5 nm while the mesh in the x- and / -directions is set to 20 nm. In the beam switching and beam steering simulations, the mesh in the x- y-, and z-directions is set to 20 nm. When performing the near to far field projection in Figures 3A through 3F, the number of metasurface elements in the x-direction is 20, while for the near to far field projection in Figures 4A through 7B, the number of assumed metasurface elements in the x-direction is 100.

[0120] To optimize diffraction efficiencies, MATLAB is used to drive a multi-variable nonlinear optimization in FDTD via Lumerical's Automation Application Programming Interface (API). When performing the optimization, the sequential quadratic programming (SQP) method and the inverse of the maximal diffraction efficiency as a figure of merit are used. The phase profile is assumed periodic, and the geometry and the refractive indices of the structure within a period are varied. When optimizing diffraction efficiencies in Figure 4, only refractive indices of a-Si pillars are varied so that the maximal assumed index change is A / ? = 0.01. When optimizing diffraction efficiencies of the structure with interconnects (Figure 7) a series of optimization runs is done. First, refractive indices of the metasurface elements are optimized. In the next step, the refractive index of the metasurface element m is fixed and the structure height and the refractive indices of thetwo remaining metasurface elements are co-optimized. Next, the Pyperiod (or Pxperiod) of the structure and refractive indices of the second and third metasurface element are co-optimized while keeping the refractive index of the first metasurface element fixed. As a final optimization step, the refractive indices of all three metasurface elements are reoptimized. In the case of the lower-Q mode, the optimization yields the following values for the refractive indices of the Si pillars and the geometrical parameters of the structure: m = 3.73416, n2= 3.738299, n3= 3.73778, Px= 1440 nm, Py= 1440 nm, and h = 840.75 nm, and the observed diffraction efficiency Deff=54% (Figure 7A). In the case of the high- Q mode, the obtained parameter values are as follows: ni = 3.73416, n2= 3.7383, n3= 3.744, Px- 1520 nm, Py- 1495.92 nm, and h - 841.207 nm, and the resulting diffraction efficiency Deff= 69% (Figure 7B). When optimizing the beam steering performance of the high-Q mode, co-optimizing the pillar height and the two of the refractive indices has enabled increasing the diffraction efficiency from 36% to 53%. As a result of the optimization, the a-Si pillar height is reduced from 850 nm to 841 nm. Next, co-optimizing the period Pyand the two of the refractive indices has enabled increasing the diffraction efficiency to 69%. As a result of the optimization, the period Pyis increased from 1425 nm to 1440 nm.

[0121] Three-dimensional electrical and thermal simulations are performed using finite element method (COMSOL Multiphysics). In the electrical simulations, in the metasurface unit cell, the top and bottom 50 nm of a-Si are n-doped with assumed carrier density of 6 x 1018cm’3The carrier density of the lightly doped core is taken to be 3.2 x io17cm'3First, electrical solver is used to obtain the volumetric heat source distribution due to Joule heating. As a next step, a thermal solver is used to obtain the temperature distribution. The thermal simulations account for heat conduction and convection. The top of the metasurface is cooled via natural convection with a heat transfer coefficient of h = 5 W / m2K. The temperature at the bottom of the substrate, 50 mm from the metasurface, is fixed to 298 K with an external heat sink. The assumed ambient temperature is also 298 K. When performing thermal simulations, periodic boundary conditions in the x and y directions are used.Example 2: Extraction of the Quality Factor and Fano Phase

[0122] We extracted the quality factor and Fano phase values by fitting the transmittance spectra to the Fano formula:Here, Tbgis a constant offset plus a linear background (Tbg= B + C( ) - ay0' ), A is the resonance amplitude, a> denotes the frequency of light while a>Qis the resonant frequency, y0is the damping constant. A denotes the Fano phase, and the Fano asymmetry parameter q is related to the Fano phase A as q = -cot(A). The quality factors are calculated as Q = — .YoExample 3: Field Profile of the Hiqh-Q Mode in the Pillar Array

[0123] Figures 8A and 8B illustrate spatial distribution of the electric field amplitude inside the metasurface unit cell in accordance with an embodiment. The unit cell is shown in Figures 1A and 1 B, which shows the case of an array of a-Si pillars on an SiO2 substrate. Namely, the length and width of the pillar are taken to be I = w = 963 nm. The metasurface period is Px= Py= 1425 nm. The electric field is plotted in the x-z plane, which goes through the center of the pillar (the same as in Figure 1 D). Figure 8A shows x-component of the electric field, and Figure 8B shows z-component of the electric field. Field profiles are plotted at a resonant wavelength of the resonance dip shown in Figure 1 C. The Eycomponent is identically equal to zero (Ey= 0).

[0124] Figures 9A through 9D illustrate spatial distribution of the electric field amplitude inside the metasurface unit cell in accordance with an embodiment. The unit cell is shown in Figures 1 A and 1 B, which shows the case of an array of a-Si pillars on an SiO2 substrate. The length and width of the pillar are taken to be I = w = 963 nm. The metasurface period is Px= Py= 1425 nm. The electric field is plotted in the x-y plane, which goes through the center of the pillar. Figure 9A shows an absolute value of the electric field, Figure 9B shows x-component of the electric field, Figure 9C shows y- component of the electric field, and Figure 9D shows z-component of the electric field.Example 4: Optical Response of a Pillar Array: Effect of the Pillar Height on the ArrayPerformance

[0125] Figures 10A and 10B illustrate optical response of a-Si pillar array on an SiO2 substrate in accordance with an embodiment. The metasurface structure is shown in Figures 1 A through 2D. Namely, the length and width of the pillar are taken to be I = w = 963 nm. The metasurface period is Px= Py = 1425 nm. Figures 10A and 10B shows transmittance and the phase of a transmitted light respectively as a function of wavelength and the a-Si pillar height h. Modes 2 and 3 coalesce at a pillar height h of about 850 nm, and the highest-Q high-Q mode is observed at a pillar height h of about 860 nm.

[0126] Figure 11 illustrates spatial electric field profiles of a lower Q modes supported by the metasurface in the x-z plane in accordance with an embodiment. The x-z plane passes through the center of the pillar. In the displayed electric field profiles, the z coordinate ranges from z = -200 nm to z = 1000 nm while the top of the SiCh substrate corresponds 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 the one shown in Figure 1. x- and y-components of the electric field of mode 3 have very similar features when varying the pillar height. On the other hand, mode 2 ‘disappears’ after coalescing with mode 3. The mode profile of mode 1 is identical to the one shown in Figure 1 .

[0127] Figure 12 illustrates Spatial electric field profiles of a lower Q modes supported by the metasurface in the y-z plane in accordance with an embodiment. The y-z plane passes through the center of the pillar. In the displayed electric field profiles, the z coordinate ranges from z = -200 nm to z = 1000 nm while the top of the SiO2 substrate corresponds to the plane z = 0. The x coordinate ranges from x = -712.5 nm to x = 712.5 nm. The spatial field profile of the high-Q mode is identical to the one shown in Figure 1 . x- and y-components of the electric field of mode 3 have very similar features when varying the pillar height. On the other hand, mode 2 ‘disappears’ after coalescing with mode 3. The mode profile of mode 1 is identical to the one shown in Figure 1 . In this Figure, electric field amplitude |E| is shown. On the considered y-z plane, the only nonzero component is Ex.

[0128] Figures 13A and 13B illustrate the optical response of a-Si pillar array suspended in air in accordance with an embodiment. The assumed geometrical parameters are the same as in Figure 1 . Namely, the length and width of the pillar are taken to be I = w = 963 nm. The metasurface period is Px= Py= 1425 nm. In Figure 13A, the transmittance is shown as a function of wavelength and the a-Si pillar height h. In Figure 13B, the phase of the transmitted light is shown as a function of wavelength and the a-Si pillar height h. Compared with the case of pillars on an SiO2 substrate, the abundance of high-Q modes is observed, which are marked by a circle in Figure 13A. At the pillar height h of about 870 nm, the Q-factor of the observed high-Q mode is about 48,000.

[0129] When considering an a-Si pillar array, at a pillar height of 870 nm, coupling to the resonant mode is still possible, and the resonant spectral feature is still visible from the false color plots of the transmittance and phase spectra (Figures 13A and 13B). The extracted Q-factor of the resonance is about 48,000. In the simulation, assuming a 5 nm mesh in z direction. Running a series of simulations with finer mesh could potentially enable to identify the parameter values at which the mode with even higher Q-factor can be observed. When take the metasurface period as Px= 1520 nm and Py= 1425 nm, the Q-factor of the supported mode is about 221 ,000. Thus, increasing the metasurface period from Px= 1425 nm to Px= 1520 nm can strongly affect the quality factor of the metasurface.

[0130] Figures 14A and 14B illustrate the transmittance and phase spectrum for an array of a-Si pillars in air in accordance with an embodiment. The assumed width, length and height of the pillars are w = I = 963 nm, h = 870 nm (see the schematic in Figure 1 ). In Figure 14A, the values for the period are Px= 1425 nm and Py= 1425 nm. The extracted Q-factor of the resonance is about 48,000. The mesh is 20 nm in the x- and / -directions, and 5 nm in the z-direction. In Figure 14B, the values for the period are as compared with Figure 14A. In Figure 14B, Px= 1520 nm and Py= 1425 nm. To reduce simulation time, a mesh of 20 nm is in all three directions. The extracted Q-factor of the resonance is about 211 ,000.Example 5: Optical Response of a Pillar array: Effect of the Pillar Period on the Array Performance

[0131] Several embodiments study how the period of the metasurface affects the modes supported by the metasurface consisting of an array of a-Si pillars on an SiO2 substrate (Figure 1 ). For metasurface periods exceeding 1200 nm, three distinct modes in the transmittance and phase false color plots, which correspond to the high-Q mode at wavelength around 1540 nm and two lower-Q modes at wavelengths around 1580 nm and 1590 nm, which correspond to mode 2 and mode 3 from Figure 10. When studying the dependence of the mode position on the x-period Px, the quality of the high-Q mode gradually increases with period. For periods of Px> 1300 nm, the position of the high-Q mode does not change significantly with period. Mode 2 shifts stronger with the x-period Pxas compared with mode 3. When the y-period Pychanges, the high-Q mode shifts stronger as compared with the case when the y-period is changed. This result is also consistent with the specifics of the spatial mode profiles of the high-Q mode in the x-z and y-z planes (see Figure 1 ). When the y-period Pyincreases from 1300 nm to 1520 nm, the high-Q resonance position shifts by about 5 nm.

[0132] Figures 15A through 15D illustrate the optical response of an a-Si pillar array on an SiO2 substrate when the incoming electric field is x-polarized in accordance with an embodiment. The metasurface has the same structure as in Figure 1 and 2. Namely, the length and width of the pillar are taken to be I = w = 963 nm, the height of the pillar is h - 860 nm. The metasurface period in the y-direction is Py- 1425 nm. Figures 15A and 15B show the transmittance and the phase of a transmitted light respectively as a function of wavelength and the period in the x-direction Px. Figures 15C and 15D show the behavior of the high-Q mode by limiting the range of the x-period to [1300 nm, 1520 nm] in Figures 15A and 15B, respectively. In Figures 15C and 15D, the wavelength range is limited to [1542.7 nm, 1543.7 nm],

[0133] Figures 16A through 16D illustrate the optical response of an a-Si pillar array on an SiO2 substrate when the incoming electric field is x-polarized in accordance with an embodiment. The metasurface has the same structure as in Figure 1 and 2. Namely, the length and width of the pillar are taken to be I = w = 963 nm, the height of the pillar ish = 860 nm. The metasurface period in the x-direction is Px= 1425 nm. Figures 16A and 16B show the transmittance and the phase of a transmitted light respectively as a function of wavelength and the period in the y-direction Py. Figures 16C and 16D show the behavior of the high-Q mode by limiting the range of the y-period 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

[0134] In the optical simulations, several embodiments impose periodic boundaries in x- and y- directions implying that a periodic array of square a-Si pillars is investigated. Some embodiments investigate whether the identified photonic mode is also supported by an isolated a-Si pillar. Some embodiments simulate scattering cross section of an isolated pillar. In the revised simulation, the perfectly matched layer (PML) boundary conditions at all simulation boundaries. First, we consider the case of a single a-Si pillar in air and calculate its scattering cross section as a function of wavelength and pillar height. The length and width of the pillar are I = w = 963 nm. Figure 17A illustrates scattering cross section of a single a-Si pillar in air as a function of wavelength and pillar height in accordance with an embodiment. Figure 17B illustrates scattering cross section of a single a-Si pillar in the air where the pillar height is h = 834 nm. The highest-Q mode is observed at a wavelength A is about 1519.8 nm. As seen in Figures 17A and 17B, in the vicinity of the highest-Q mode, crossing of two other modes is observed. For a pillar height h of about 834 nm, the quality factor of the highest-Q mode is about 1000.

[0135] Several embodiments provide the case of an isolated a-Si pillar on an SiO2 substrate. Figures 18A and 18B illustrate the scattering cross section of an isolated pillar on an SiO2 substrate as a function of wavelength and pillar height in accordance with an embodiment. Figure 18A shows scattering cross section of a single a-Si pillar on an S / O2 substrate as a function of wavelength and pillar height. The length and width of the pillar is I = w = 963 nm. Figure 18B shows scattering cross section of a single a-Si pillar where the pillar height h is about 830 nm. The highest-Q mode is observed at a wavelength A of about 1519.2 nm. An overall broadening of the spectral features is observed as comparedwith the case of a single a-Si pillar in the air. For a pillar height of 830 nm, the Q-factor of the supported mode is about 676.

[0136] Figure 19 illustrates scattering cross section of a single a-Si pillar on an SiO? substrate as a function of wavelength and pillar height in accordance with an embodiment. The spatial distribution of the electric field amplitude \E\ is plotted in the a-Si pillar for the pillar heights of h = 720 nm, h = 790 nm, h = 830 nm, and h = 850 nm. The electric field is plotted in the x-z plane, which passes through the center of the pillar. The solid black lines point towards the scattering peaks at which the field profiles have been simulated.

[0137] Figures 20A through 20F illustrate 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 x-z plane in accordance with an embodiment. The x-z plane passes through the center of an isolated a-Si pillar on an SiO2 substrate. The x- and z- components of the electric field are plotted along the modal line 3 in Figure 18. Geometrical parameters of the pillar are the same as in Figure 18. The a-Si pillar height and the operating wavelength are marked at the top of each column. The x- and z- components of the electric field gradually change as the pillar height increases.

[0138] Figures 21 A through 21 D illustrate spatial distribution of the x-component of the electric field E in the x-z plane in accordance with an embodiment. The x-z plane passes through the center of an isolated a-Si pillar on an SiO2 substrate. The x-component of the electric field is plotted along the modal line 2 in Figure 18. Geometrical parameters of the pillar are the same as in Figure 18. The a-Si pillar height and the operating wavelength are marked at the top of each column. The x-components of the electric field gradually change as the pillar height increases.

[0139] To understand the nature of the considered high-Q resonance and its relation to the high-Q resonance 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 illustrates scattering cross section of a single a-Si pillar on an SiO2 substrate as a function of wavelength and pillar height in accordance with an embodiment. The spatial distribution is plotted of the electric field amplitude |E| in the a-Si pillar for the pillar heights of h = 720 nm, h = 790 nm, h = 830 nm, and h = 850 nm. The electric field is plotted in the y-z plane,which passes through the center of the pillar. The solid black lines point towards the scattering peaks at which the field profiles have been simulated. As seen in Figure 22, the spatial distribution of the electric field amplitude of the high-Q mode in the y-z plane (for example, at h = 830 nm) is identical to the spatial distribution of the electric field observed in the case of the pillar array (see Figure 1 E).Example 7: Controlling the Spectral Shape of the Resonance

[0140] Figures 23A and 23B illustrate the transmittance and the phase of the transmitted light respectively as a function of the wavelength and thickness of the SiO2 spacer d in accordance with an embodiment. The metasurface structure is illustrated in Figure 3A. The assumed geometrical parameters are the same as in Figure 3. Figures 23C and 23D show transmittance and phase spectra for the SiO2 spacer thickness for d = 510 nm and d = 1870 nm respectively in accordance with an embodiment.

[0141] Figure 24 illustrates quality factor and Fano phase of the high-Q resonance as a function of the SiO2 thickness d in accordance with an embodiment. The metasurface structure is illustrated in Figure 3A. The considered structure and geometrical parameters are identical to the one if Figure 3. Figure 3C shows that around the SiO2 spacer thickness d = 1550 nm, the Fano phase varies abruptly as a function of the SiO2 spacer thickness. Figure 24 shows that varying the SiO2 spacer thickness more finely to accurately capture the details of the variation of the Fano phase.

[0142] Figures 25A through 25D illustrate spatial distribution of the electric field amplitude in the metasurface unit cell in accordance with an embodiment. The metasurface structure is illustrated in Figure 3A. The geometrical parameters of the metasurface unit cell are as follows: the length and width of the pillar is I = w = 963 nm, the height of the pillar is h = 845 nm. The assumed period values are Px= 1520 nm and Py= 1425 nm. The thickness of the SiO2 spacer is d = 1450 nm. The spatial distribution of the electric field is plotted in the x-y plane. Figure 25A plots the absolute value of the electric field |E / Eo|. Figures 25B and 25C plot the spatial distributions of x- and z- components of the electric field, respectively. Figure 25D plots the real part of the z-component of the electric field. We observe that both x- and z-components of the electric field adopt non-zero values in the SiO2 spacer.

[0143] Figure 26 illustrates transmittance and phase shift as a function of the a-Si index change achieved by using a high-Q resonance in accordance with an embodiment. The metasurface structure is illustrated in Figure 3A. The wavelength is fixed at an operating wavelength A = 1535.44 nm. At an operating wavelength and for an index change of An = 0.0026, the phase shift extracted from a periodic array calculation is 107°.Example 8: Realistic Electrical Addressing Architectures

[0144] Several embodiments provide how the proposed metasurface design can be modified to enable thermo-optic control of the wavefront of the transmitted light. The first variant of the proposed metasurface unit cell features a-Si pillars, which are connected via an a-Si bar. Figures 27A through 27F illustrate dynamic beam switching with realistic interconnect architectures in accordance with an embodiment. Figure 27A shows a schematic of a metasurface unit cell. The square a-Si pillars connected via a-Si bars. In Figure 27A, the whole a-Si layer is lightly doped and is placed on an SiO2 substrate so that the complex refractive index of a-Si now reads as n = 3.734+0.0013 i. Several embodiments dope the whole silicon metasurface or a fraction of it. In some embodiments, a conductive layer made of a different material above or below high-index rectangular pillars can be used as the electrodes. Examples of the materials include (but are not limited to) semiconductors, doped GaAs, ITO, CdO, AZO, thin metal layers. The temperature of the metasurface pixel can be actively controlled by biasing the metasurface unit cell at the edge and flowing current in the y direction. The geometrical parameters of the metasurface unit cell are as follows: 5 = 82 nm, w = / 1 =963 nm, h = 850 nm, Px= Py= 1500 nm. Several embodiments include one or more connecting electrodes in the metasurfaces.

[0145] Figures 27B and 27C plot the electric field intensity in the far field as function of the steering angle for a two-level phase grating, which uses the unit cell shown in Figure 27A. The a-Si refractive index difference between neighboring metasurface elements is A = 0.006. In Figure 26B, the incoming plane wave is x-polarized while in Figure 27Cthe incoming electric field is y-polarized. The inset of Figure 27B shows a schematic of one period of a two-level phase grating, which is utilized to theoretically demonstrate switchable diffraction. Within a grating period, the assumed index change between neighboring a-Si pillars is An=0.006 (Figures 27A and 27B). The switchable diffraction is observed both in the case when the polarization of the electric field is perpendicular and parallel to the a-Si bars (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 overall transmittance at the operating wavelength is 7=0.3%. Note that to observe the diffractive switching, the lower-Q mode described in Figure 3 is utilized. When using the configuration shown in Figure 26A, diffractive beam switching using the higher-Q mode is not available since the spectral characteristics of the high-Q mode are strongly affected by introduced optical losses. Namely, in the lossy structure (Figure 27A), large spectral variation of the phase of the transmitted light is not observed in the case of no applied current.

[0146] To enable thermo-optic switching using also higher-Q mode, the metasurface unit cell design can be modified. Figure 27C shows a schematic of a metasurface unit cell in which the top and bottom 50 nm of a-Si are doped. The thickness of the SiO? layer is 1440 nm. In the modified unit cell design, only top and bottom 50 nm layers of the a-Si are doped while the core section of a-Si layer is practically undoped (or very lightly doped). In this implementation, the top and bottom electrodes are biased with respect to each other, and current flows in the vertical direction. The geometrical parameters of the metasurface unit cell are also modified: Px= 1520 nm and Py= 1425 nm. The height of the a-Si and the a-Si bar is h = 845 nm, and the thickness of the SiO2 spacer is 1440 nm. The width and length of the a-Si pillar are taken to be w = h = 963 nm.

[0147] Figures 27E and 27F plot the electric field intensity in the far field as function of the steering angle for a two-level phase grating, which uses the unit cell shown in Figure 27D. In Figure 27E, the a-Si refractive index difference between neighboring metasurface elements is An = 0.005, and the width of the a-Si bar is 5 = 40 nm. In Figure 27F, the a- Si refractive index difference between neighboring metasurface elements is An = 0.006,and the width of the a-Si bar is 5 = 82 nm. The insets of Figures 27E and 27F show schematics of one period of a two-level phase grating, which is utilized to theoretically demonstrate switchable diffraction. Utilizing the higher-Q mode, switchable diffraction can be shown assuming that the refractive index difference between neighboring a-Si pillars in a grating period is An = 0.005, and the width of the a-Si bar of 5 = 40 nm. In Figure 27E, the polarization direction of the electric field of the incoming light is perpendicular to the a-Si bars. In Figure 27E, the operating wavelength is 1536.15 nm, while the overall transmittance at the operating wavelength is T = 6.8 %.

[0148] Interestingly, using the two-level phase grating, highly asymmetric diffraction pattern originating from geometrical asymmetry of the metasurface unit cell can be seen, as shown in Figure 27F. In Figure 27F, the refractive index difference between neighboring a-Si pillars in a grating period An is 0.006. The width of the a-Si bar 5 is about 82 nm. The operating wavelength of the utilized lower-Q mode is 1548.4 nm, and the overall transmittance at the operating wavelength is T = 1 %.

[0149] Several embodiments provide the case when the metasurface unit cells are placed on an SiO2 pedestal and the a-Si pillars are connected in series vis a-Si bars. In this case, for the pillar height of h = 880 nm, the spectral and phase signatures of the high-Q mode are not observed. Figures 28A through 28C illustrate dependence of the transmittance and phase on the height of the pillar and bars in accordance with an embodiment. Figure 28A shows a schematic of the metasurface unit cell. In the schematic, Px= 1520 nm, Py=1 425 nm, 5 = 82 nm. The height of the SiO2 pedestal is 382 nm. The length of the pedestal is 200 nm, the width of the pedestal is also 200 nm. Note that the metasurface considered here does not include a Si substrate. Figure 28B plots transmittance as a function of wavelength and the a-Si pillar height. Figure 28C plots the phase of the transmitted light as a function of wavelength and the a-Si pillar height. When changing the height of the a-Si pillar, we simultaneously change the height of the a-Si bars. As seen in Figures 28B and 28C, at the pillar height of h = 880 nm, the spectral and phase signature of the resonance disappears.

[0150] Figures 29A through 29D illustrate transmittance and phase shift as function of the a-Si index change for the metasurface with electrodes in accordance with anembodiment. The metasurface structure is as shown in Figure 5A and 5B. The incoming light is x-polarized. Figures 29A and 29B correspond to the high-Q high-Q mode while Figures 29C and 29D correspond to the lower-Q mode. The maximal phase shift enabled by the high-Q high-Q mode is 277° while the lower-Q mode enables a phase shift of 230° when the refractive index of a-Si is changed by An = 0.01 .

[0151] Figures 30A through 30D illustrate transmittance and phase shift as function of the a-Si index change for the metasurface with electrodes in accordance with an embodiment. The metasurface structure is as shown in Figure 5A and 5B. The incoming light is y-polarized. Figures 30A and 30B correspond to the high-Q high-Q mode while Figures 30C and 30D correspond to the lower-Q mode. The maximal phase shift enabled by the high-Q high-Q mode is 320° while the lower-Q mode enables a phase shift of 275° when the refractive index of a-Si is changed by An = 0.01 .

[0152] Figures 31 A through 31 D illustrate thermo-optic beam switching using the lower-Q mode in accordance with an embodiment. The metasurface structure is as shown in Figure 5A and 5B. The optical performance of a two-level phase grating is analyzed. The unit cell of the grating is shown in the inset of Figure 31 A. The index change between neighboring a-Si pillars is An = 0.006. Figure 31 A shows the intensity of the electric field in the far field when the incoming plane wave is x polarized, the operating wavelength is A = 1575.5 nm, and the overall transmittance is T = 7.6 %. In Figure 31 B, the overall transmittance spectrum of the two-level grating is studied. In Figure 31 C, the intensity of the electric field in the far field when the incoming plane wave is y polarized. In Figure 31 C, the operating wavelength is A = 1583.2 nm, and the overall transmittance is T = 0.6 %. Figure 31 D shows the overall transmittance spectrum of the two-level grating studied in Figure 31 C.

[0153] Figures 32A through 32D illustrate analytical array factor calculations for a two- level phase grating in accordance with an embodiment. The plots calculate intensity of the electric field in the far field as a function of the steering angle in the case of a two- level phase grating. Figures 32A and 32B correspond to the case of the high-Q mode. Figures 32A and 32B assume that the refractive index difference between neighboring metasurface pixels is about 0.0026, which is the value in Figure 5. In Figures 32A and32B, the values of the electric field amplitudes and relative phases in a grating period are constructed based on Figures 29A through 2D) correspond to the case of the lower-Q mode. Figures 32C and 32D assume that the refractive index difference between neighboring metasurface pixels is about 0.006, which is the value chosen in Figure 31 . In Figures 32C and 32D, the values of the electric field amplitudes and relative phases in a grating period are constructed based on Figures 29C and 30D. In the case of the high-Q mode, the array level analytical calculations yield different results as compared with the case of full wave simulations providing evidence of near-field coupling between neighboring metasurface pixels. In the case of the lower-Q mode, the array factor calculation results and full wave simulations yield similar results.

[0154] Figures 33A to 33B illustrate intensity of the electric filed in the far field as a function of the steering angle in the cases of a three-level phase grating in accordance with an embodiment. The incoming light is y-polarized. The three-level phase grating is generated using the simulated phase shift data for an operating wavelength of A = 1583.2 n, and the phase shift values in the grating period are given as (0°, 120°, 240°). Within a grating period, the values of the real part of the refractive index are taken as (3.734, 3.734+0.003333, 3.734+0.01 ). a) plots the intensity of the electric field in the far field at an operating wavelength of A = 1583.2 nm. Figure 33B keeps the same spatial distribution of the real part of the refractive index as in Figure 33A, but the operating wavelength is now taken as A = 1583.8 nm. In Figure 33A, the target steered beam at a steering angle of 20.3°. A number of spurious diffraction orders can be seen. By changing the operating wavelength to A = 1583.8 nm, the zeroth diffraction order can be suppressed, but the other spurious diffraction orders are still observed.

[0155] Figure 34 illustrates intensity of the electric filed in the far field as a function of the steering angle in the cases of a three-level phase grating at an operating wavelength of A = 1535.81 nm, which is a resonant wavelength for a high-Q mode. The incoming light is y-polarized. Within a grating period, the values of the real part of the refractive index are taken as (3.734, 3.734+0.003333, 3.734+0.01 ). By appropriately choosing the operating wavelength, we have been able to suppress the zeroth diffraction order.

[0156] Figures 35A and 35B illustrate transmittance and phase shift as function of the a-Si index change for an optimized metasurface with electrodes in accordance with an embodiment. The incoming light is x-polarized. Figure 35A corresponds to the lower-Q mode and Figure 35B corresponds to the high-Q mode. The insets indicate the operating wavelengths for which the diffraction patterns shown in Figure 7 are observed. At an operating wavelength of A = 1567.1 nm, the maximal phase shift enabled by the lower-Q mode is 236.2°. In the case of the high-Q mode, the maximal phase shift at an operating wavelength of A = 1531 .3 nm is 265°.

[0157] Figure 36 illustrates thermo-optic three-level phase grating with a realistic interconnect architecture and optimized geometry in accordance with an embodiment. The incoming plane wave is TE-polarized. The figure plots the intensity of the electric field in the far field for the case of the lower-Q mode at an operating wavelength of A = 1581.5 nm. The employed optimization procedure has yielded the following geometrical parameters for the structure shown in Figure 5a-b: Px= 1520 nm, Py= 1100 nm, and h = 851.327 nm, and the resulting diffraction efficiency Deff = 53%. The real part of the complex refractive index for each metasurface element within the grating period is given as follows: ni = 3.734, ri2 = 3.74144,= 3.74318.Example 9: Metasurfaces with a Finite Number of Metasurface Elements

[0158] Consider an m x m array of metasurface elements, several embodiments study how the quality factor of the mode supported by the finite array increases with m. To calculate the quality factor, several embodiments calculate the scattering cross-section of the finite array and fit it to the Fano formula. When studying a finite metasurface array, the assumed geometrical parameters are the same as in Figure 1 : 1 = w = 963 nm and Px= Py= 1425 nm. To reduce the simulation time, the assumed mesh in the z-direction is 10 nm while the mesh in the x- and y-directions is set to 20 nm. When assuming periodic boundary conditions in the x- and y-directions, the quality factor of the supported mode is 8500. Note that this quality factor value (8500) is lower as compared with the one reported in Figure 1 and 2 (9800). This difference in the observed quality factors is due to the factthe in the simulations shown in Figure 1 , the mesh in the z-direction is taken to be 5 nm while the mesh in the x- and y-directions is still set to 20 nm.

[0159] Figures 37A through 37C illustrate quality factors of metasurfaces with a finite number of elements in accordance with an embodiment. Figure 37A shows quality factor of an m x m metasurface array as a function of m. Figure 37B shows quality factor of the metasurface, which is finite in the direction parallel to the incoming electric field as a function of the number of the metasurface elements in the mentioned direction nx. In Figure 37B, in the direction perpendicular to the electric field, the periodic boundary condition is assumed. Figure 37C shows quality factor of the metasurface, which is finite in the direction perpendicular to the incoming electric field as a function of the number of the metasurface elements in the mentioned direction nx. In Figure 37C, in the direction parallel to the electric field, the periodic boundary condition is assumed.

[0160] As seen in Figure 37, the quality factor of the supported mode monotonously increases when increasing the number of metasurface elements in the array. For a 12x12 array, the quality factor is 6000.

[0161] Some embodiments assume that the number of the metasurface elements is finite in the direction parallel to the incoming electric field while in the direction perpendicular to the electric field, the periodic boundary condition is assumed. The described simulation setup enables reducing the simulation time while accessing the quality factor of the metasurface for a larger number of metasurface elements nx. As seen in Figure 37B, when the number of the metasurface elements in the direction parallel to the electric field nx is nx= 10, the quality factor of the metasurfaces is about 7000. While when nx= 20, the quality factor of the metasurface is slightly below 8000.

[0162] Figure 37C assumes that the number of the metasurface elements is finite in the direction perpendicular to the incoming electric field while in the direction parallel to the electric field, the periodic boundary condition is assumed. In this case, when nx= 10, the quality fact of the metasurface is about 6000. Note though that in both cases where the metasurface is finite in the direction perpendicular or parallel to the electric field, for nx= 20, the quality factor of the metasurface is about 8000.Example 10: Effect of Potential Fabrication Imperfections

[0163] Several embodiments investigate the effect of fabrication imperfections on the metasurface performance. Some embodiments consider three different imperfections: tilted sidewalls; rounded corners; and lossy Si.

[0164] The simulations show that the metasurface is quite robust to possible fabrication imperfections such as tilted sidewalls or rounded corners. It may be necessary to slightly change the height of the structure depending on the exact tilt of the sidewalls. The challenging aspect for achieving quality factors of about 10,000 would be depositing a-Si with a very low extinction coefficient (k ~ 10'5). A detailed discussion of the effect of each of the listed non-idealities on the metasurface performance is discussed in what follows.

[0165] Tilted sidewalls. For simplicity, some embodiments implement the case of an a-Si resonator on an SiO2 substrate (see Figure 1 and 2). Assume the period of Px= Py = 1425 nm. Because of the tilted sidewalls, the a-Si resonator is not a rectangular parallelepiped anymore but is a truncated square pyramid (see the inset of Figure 38A). To assess how the extent of tilt affects the supported resonances, the top base of the truncated a-Si pyramid is fixed as 963 nm and the length of the bottom base changes. Figures 38A and 38B plot the transmittance and the phase of the transmitted light as a function of wavelength and the lower base length of the pyramid. In Figures 38A and 38B, the height of the pyramid is h = 845 nm. A broadening of the high-Q resonance with the length of the lower base of the pyramid is observed. The phase of the transmitted light exhibits a large variation in the spectral domain only for the base lengths below 1030 nm. Note that a large range of the phase variation in the spectral domain is important for exhibiting a dynamically tunable phase shift. Thus, when the pyramid height is h = 845 nm, the tilt angle should not exceed 2.27°.

[0166] Assume an a-Si truncated pyramid with a height of h = 885 nm on an SiO2 substrate (Figures 38C and 38D). The top base of the pyramid is 963 nm. When the lower and upper bases of the pyramid are equal, both high-Q and a lower-Q mode can be seen in the transmittance spectrum, but no large phase variation in the spectral domain. The quality factor of the higher-Q resonance increases with the lower base length andachieves a peak value of about 4000 when the lower base length is 1030 nm (Figure 38C). Notably, when the lower base length is increased to 1030 nm, a large phase variation in the spectral domain can be seen (Figure 38D). Thus, when the pyramid's height is 885 nm, a large variation of phase in the spectral domain even for the larger base lengths of 1100 nm that corresponds to the tilt angle of 4.43°.

[0167] Some embodiments investigate how the optical mode supported by an a-Si truncated pyramid on an SiO2 substrate changes with the pyramid height (Figures 38E and 38F). In Figures 38E and 38F, the upper base of the pyramid is 963 nm while the lower base of the pyramid is 1030 nm. Both the high-Q and the lower-Q resonances red shift when increasing the pyramid height (Figures 38E and 38F). The quality factor of the high-Q resonance increases when the pyramid height is increased from 840 nm peaking at a value of 890 nm. For the given values of the pyramid bases (963 nm and 1030 nm), a large phase variation in the spectral domain is observed for the pyramid height ranging between 850 nm and 885 nm. Thus, the metasurface is robust to a possible non-ideality originating from not having perfectly vertical sidewalls. Even when the sidewalls are not perfectly vertical, one may slightly increase the height of the resonator to access the mode with a high quality factor.

[0168] Figures 38A through 38F illustrate the influence of tilted sidewalls on the metasurface performance. Figures 38A and 38B illustrate the transmittance and the phase of the transmitted light respectively as a function of wavelength and the lower base length of the pyramid when the pyramid height is h - 845 nm. Figures 38C and 38D illustrate the transmittance and the phase of the transmitted light respectively as a function of wavelength and the lower base length of the pyramid when the pyramid height is h = 885 nm. Figures 38E and 38F illustrate the transmittance and the phase of the transmitted light respectively as a function of wavelength and the pyramid height. In Figures 38D and 38F, the upper base of the pyramid is 963 nm while the lower base of the pyramid is 1030 nm, which corresponds to the tilt angle of 2.27°.

[0169] Rounded corners. Several embodiments investigate if rounded rather than right corners affects the performance of the metasurface. Consider an a-Si parallelepiped with rounded corners and bends on a SiO? substrate. The dimensions of the structure are thesame as in Figure 1 and 2. Figures 39A through 39D illustrate the influence of the rounded corners on the metasurface performance in accordance with an embodiment. Figures 39A and 39B show the transmittance and the phase of the transmitted light respectively as a function of wavelength and the radius of curvature when the pillar height is h = 845 nm. Figures 39C and 39D show the transmittance and the phase of the transmitted light respectively as a function of wavelength and the radius of the curvature when the pillar height is h = 860 nm. Figure 39 investigates how changing the radius of the curvature of the corners and bends affects the transmittance and phase of the 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. A blue shift of both high-Q and lower-Q resonances can be seen when increasing the curvature radius. The blue shift is more prominent for the lower-Q mode. For the resonator height h = 845 nm, the high-Q resonance is very robust to non-ideality associated with rounded corners and bends. When the a-Si resonator height is h = 845 nm, the essential characteristics of the resonance are preserved even for a curvature radius of 120 nm (Figures 39A and 39B). When the a-Si resonator height is h = 860 nm, a large variation of phase in the spectral domain is seen for the radiuses of curvature below 100 nm (Figure 39C). Thus, if the expected radius of curvature is above 100 nm, the resonator height needs to be below 860 nm.

[0170] Optical loss in Si. When performing experiments, non-ideal properties of materials can affect the optical response of the metasurface. Some embodiments investigate how non-zero optical loss in a-Si affects the transmittance spectrum and phase spectrum of the transmitted. Consider a metasurface comprised of a-Si pillars on an SiO2 substrate, which has been discussed 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 spectrum for three different values of the extinction coefficient of a-Si: k = 0, k = 0.0001 , and k = 0.0005 (Figure 40). The lower-Q resonances observed at longer wavelengths are practically unaffected by the introduced optical loss. On the other hand, the high-Q resonance observed at shorter wavelengths can be strongly affected by optical loss. For example, when the pillar height is h = 845 nm, for the high-Q mode, the phase of the transmitted light exhibits limited variability in the spectral domain when theextinction coefficient is k = 0.0005 (Figure 40D). When the pillar height is h = 860 nm, introducing an extinction coefficient k - 0.0001 is sufficient for changing the spectral shape of the transmitted phase (Figure 40F). The simulations also confirm (not shown here) that when the pillar height is h = 860 nm, introducing an extinction coefficient of k = 0.00001 , does not affect the optical response of the metasurface.

[0171] Figures 40A through 40F illustrate the influence of the material loss on the metasurface performance in accordance with an embodiment. Figures 40A and 40B illustrate the transmittance and the phase of the transmitted light respectively as a function of wavelength for different values of the extinction coefficient k when the pillar height is h - 830 nm. Figures 40A and 40B illustrate the transmittance and the phase of the transmitted light respectively as a function of wavelength for different values of the extinction coefficient k when the pillar height is h = 845 nm. Figures 40A and 40B illustrate the transmittance and the phase of the transmitted light respectively as a function of wavelength for different values of the extinction coefficient when the pillar height is h = 860 nm.Example 11 : High-Efficiency Reflective High Quality Factor Metasurfaces

[0172] By adding a gold back reflector to the transmissive metasurface, several embodiments can attain high-efficiency high-Q reflective metasurfaces. The unit cell of the metasurface design is shown in Figure 41 A. The metasurface unit cell includes a gold back reflector, followed by an SiO2 spacer. An amorphous Si pillar is deposited on top of the spacer. By appropriately choosing the thickness of the SiO? spacer, several embodiments obtain high-efficiency high-Q reflective metasurfaces.

[0173] Figures 41 A and 41 B illustrate high-efficiency transmissive metasurfaces in accordance with an embodiment. Figure 41 A shows a schematic of the unit cell of the high efficiency transmissive metasurface. In Figure 41 A, the length and width of the a-Si pillar are taken to be I = w = 963 nm. The metasurface period is Px= Py= 1425 nm. The thickness of the SiO2 spacer is 516 nm while Au is optically thick. Figure 41 B shows the phase and reflectance spectra for the metasurface.EXAMPLES

[0174] Example 1 : An apparatus comprising: an electromagnetic metasurface comprising a plurality of repeating unit cells with a periodicity conformally disposed on a substrate; wherein the periodicity is less than a wavelength in free space of an operating light; wherein 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 a phase of the operating light in transmission mode with a quality factor of at least 10; wherein the apparatus transforms a transmission dip to a transmission peak in quality factor resonance; and wherein a change in at least one parameter selected from the group consisting of: a nanostructure length, a nanostructure width, a nanostructure height, and the periodicity tunes the quality factor.

[0175] Example 2: The example apparatus of example 1 , wherein the wavelength is selected from the group consisting of: an ultraviolet wavelength from 100 nm to 400 nm, a visible wavelength from 380 nm to 800 nm, a near infrared wavelength from 800 nm to 2500 nm, and an infrared wavelength from 780 nm to 1000 pm.

[0176] Example 3: The example apparatus of example 1 or 2, wherein the plurality of repeating unit cells is arranged in an array.

[0177] Example 4: The example apparatus of example 1 , 2, or 3, wherein the nanostructure has a shape selected from the group consisting of: a cuboid, a cube, a pillar, a cylinder, an elliptical cylinder, a trapezoid, a triangular prism, a polygonal prism, a pyramid, and a combination thereof.

[0178] Example 5: The example apparatus of any one of examples 1 to 4, wherein the nanostructure and the second substrate each comprises a lossless dielectric material with an imaginary refractive index less than or equal to 0.5 at the wavelength of operation.

[0179] Example 6: The example apparatus of any one of examples 1 to 5, wherein the first substrate comprises a material with a real part of a refractive index less than the real part of the refractive index at the wavelength of operation of the nanostructure.

[0180] Example 7: The example apparatus of 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, bariumtitanate, lithium niobate, tantalum pentoxide, silicon oxide, amorphous silicon, silicon, and a combination thereof.

[0181] Example 8: The example apparatus of any one of examples 1 to 7, 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, magnesium fluoride, tantalum pentoxide, zirconium oxide, vanadium oxide, a germanium-antimony-tellurium alloy, titanium nitride, hafnium oxide, hafnium nitride, molybdenum diselenide, hexagonal boron nitride, black phosphorous, tungsten diselenide, tungsten disulfide, and a combination thereof.

[0182] Example 9: The example apparatus of 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 a combination thereof.

[0183] Example 10: The example apparatus of any one of examples 1 to 9, wherein the wavelength is a near infrared wavelength from 800 nm to 2500 nm, the height of the nanostructure is less than or equal to 860 nm, and the quality factor is from 100 to 9800; wherein the apparatus is configured to be a portion of a wavefront shaping system.

[0184] Example 11 : The example apparatus of any one of examples 1 to 10, wherein a change in the first substrate thickness tunes a spectral shape of transmittance.

[0185] Example 12: The example apparatus of any one of examples 1 to 11 , wherein a thermo-optic modulation of a refractive index of the nanostructure from 0.001 to 0.01 shapes a transmitted light wavefront.

[0186] Example 13: The example apparatus of any one of examples 1 to 12, wherein the apparatus is configured to be a part of a dynamic beam steering system.

[0187] Example 14: The example apparatus of any one of examples 1 to 13, wherein the dynamic beam steering system operates in a transverse electric (TE) polarization or a transverse magnetic (TM) polarization.

[0188] Example 15: The example apparatus of any one of examples 1 to 14, wherein a plurality of nanostructures in a row is connected to form an interconnected structure;wherein at least two surfaces of the interconnected structure are electrically conductive; wherein the interconnected structure is separated from the first substrate via a plurality of pillars.

[0189] Example 16: The example apparatus of any one of examples 1 to 15, wherein the interconnected structure is heated via a voltage applied to the electrically conductive surfaces such that the refractive index of the nanostructure is thermo-optically modulated; wherein the plurality of pillars prevents thermal crosstalk.

[0190] Example 17: The example apparatus of any one of examples 1 to 16, wherein the electrically conductive surfaces comprise a material selected from the group consisting of: a doped semiconductor, a doped compound semiconductor, a metal, a metal alloy, doped gallium arsenide, doped gallium phosphide, and doped amorphous silicon.

[0191] Example 18: The example apparatus of any one of examples 1 to 17, wherein the plurality of pillars 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, magnesium fluoride, tantalum pentoxide, zirconium oxide, vanadium oxide, a germanium-antimony-tellurium alloy, titanium nitride, hafnium oxide, hafnium nitride, molybdenum diselenide, hexagonal boron nitride, black phosphorous, tungsten diselenide, tungsten disulfide, and a combination thereof.

[0192] Example 19: The example apparatus of any one of examples 1 to 18, further comprising a light source disposed on an opposite side of the second substrate from the first substrate.

[0193] Example 20: The example apparatus of any one of examples 1 to 19, wherein the light source is a chip scale laser.DOCTRINE OF EQUIVALENTS

[0194] As can be inferred from the above discussion, the above-mentioned concepts can be implemented in a variety of arrangements in accordance with embodiments of the invention. Accordingly, although the present invention has been described in certain specific aspects, many additional modifications and variations would be apparent to thoseskilled in the art. It is therefore to be understood that the present invention may be practiced otherwise than specifically described. Thus, embodiments of the present invention should be considered in all respects as illustrative and not restrictive.

[0195] As used herein, the singular terms "a," "an," and "the" may include plural referents unless the context clearly dictates otherwise. Reference to an object in the singular is not intended to mean "one and only one" unless explicitly so stated, but rather "one or more."

[0196] As used herein, the terms “approximately,” and "about" are used to describe and account for small variations. When used in conjunction with an event or circumstance, the terms can refer to instances in which the event or circumstance occurs precisely as well as instances in which the event or circumstance occurs to a close approximation. When used in conjunction with a numerical value, the terms can refer to a range of variation of less than or equal to ± 10% of that numerical value, such as less than or equal to ±5%, less than or equal to ±4%, less than or equal to ±3%, less than or equal to ±2%, less than or equal to ±1 %, less than or equal to ±0.5%, less than or equal to ±0.1 %, or less than or equal to ±0.05%.

[0197] Additionally, amounts, ratios, and other numerical values may sometimes be presented herein in a range format. It is to be understood that such range format is used for convenience and brevity and should be understood flexibly to include numerical values explicitly specified as limits of a range, but also to include all individual numerical values or sub-ranges encompassed within that range as if each numerical value and sub-range is explicitly specified. For example, a ratio in the range of about 1 to about 200 should be understood to include the explicitly recited limits of about 1 and about 200, but also to include individual ratios such as about 2, about 3, and about 4, and sub-ranges such as about 10 to about 50, about 20 to about 100, and so forth.

Claims

WHAT IS CLAIMED IS:1 . An apparatus comprising: an electromagnetic metasurface comprising a plurality of repeating unit cells with a periodicity conformally disposed on a substrate; wherein the periodicity is less than a wavelength in free space of an operating light; wherein 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 a phase of the operating light in transmission mode with a quality factor of at least 10; wherein the apparatus transforms a transmission dip to a transmission peak in quality factor resonance; and wherein a change in at least one parameter selected from the group consisting of: a nanostructure length, a nanostructure width, a nanostructure height, and the periodicity tunes the quality factor.

2. The apparatus of claim 1 , wherein the wavelength is selected from the group consisting of: an ultraviolet wavelength from 100 nm to 400 nm, a visible wavelength from 380 nm to 800 nm, a near infrared wavelength from 800 nm to 2500 nm, and an infrared wavelength from 780 nm to 1000 pm.

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

4. The apparatus of claim 1 , wherein the nanostructure has a shape selected from the group consisting of: a cuboid, a cube, a pillar, a cylinder, an elliptical cylinder, a trapezoid, a triangular prism, a polygonal prism, a pyramid, and a combination thereof.

5. The apparatus of claim 1 , wherein the nanostructure and the second substrate each comprises a lossless dielectric material with an imaginary refractive index less than or equal to 0.5 at the wavelength of operation.

6. The apparatus of claim 1 , wherein the first substrate comprises a material with a real part of a refractive index less than the real part of the refractive index at the wavelength of operation of the nanostructure.

7. The apparatus of 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 a combination thereof.

8. The apparatus of 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, magnesium fluoride, tantalum pentoxide, zirconium oxide, vanadium oxide, a germanium-antimony-tellurium alloy, titanium nitride, hafnium oxide, hafnium nitride, molybdenum diselenide, hexagonal boron nitride, black phosphorous, tungsten diselenide, tungsten disulfide, and a combination thereof.

9. The apparatus of 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 a combination thereof.

10. The apparatus of claim 1 , wherein the wavelength is a near infrared wavelength from 800 nm to 2500 nm, the height of the nanostructure is less than or equal to 860 nm, and the quality factor is from 100 to 9800; wherein the apparatus is configured to be a portion of a wavefront shaping system.11 . The apparatus of claim 1 , wherein a change in the first substrate thickness tunes a spectral shape of transmittance.

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

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

14. The apparatus of claim 13, wherein the dynamic beam steering system operates in a transverse electric (TE) polarization or a transverse magnetic (TM) polarization.

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

16. The apparatus of claim 15, wherein the interconnected structure is heated via a voltage applied to the electrically conductive surfaces such that the refractive index of the nanostructure is thermo-optically modulated; wherein the plurality of pillars prevents thermal crosstalk.

17. The apparatus of claim 16, wherein the electrically conductive surfaces comprise a material selected from the group consisting of: a doped semiconductor, a doped compound semiconductor, a metal, a metal alloy, doped gallium arsenide, doped gallium phosphide, and doped amorphous silicon.

18. The apparatus of claim 16, wherein the plurality of pillars 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, magnesium fluoride, tantalum pentoxide, zirconium oxide, vanadium oxide, a germanium-antimony-tellurium alloy, titanium nitride, hafnium oxide, hafnium nitride, molybdenum diselenide, hexagonal boron nitride, black phosphorous, tungsten diselenide, tungsten disulfide, and a combination thereof.

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

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

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