Metamaterial interferometer system and method

Abstract

The optical system includes a first optical element comprising a first substrate, a partial reflection coating disposed on a first surface of the first substrate, and a first metamaterial layer disposed on or adjacent to the first surface of the first substrate and having a structure that defines a continuous phase gradient along a first direction parallel to the first surface of the first substrate; and a second optical element comprising a second substrate and a second metamaterial layer disposed on or adjacent to the first surface of the second substrate and having a structure that defines a continuous phase gradient along a second direction parallel to the first surface of the first substrate, wherein at least one surface of the second optical element is curved along the second direction.

Description

[Technical Field]

[0001] (Cross-reference of related applications) This application claims priority to U.S. Provisional Application No. 63 / 470,162, filed on 31 May 2023, the entire contents of which are incorporated herein by reference.

[0002] (Technical field) This disclosure relates to the manipulation, spatial dispersion, and detection of light of different wavelengths. [Background technology]

[0003] Optical waveforms, which include frequency bands, are widely used in applications related to signal transmission, measurement, communication, and spectroscopy. In the frequency domain, different frequency components of an optical waveform can be spatially separated using dispersive optical elements such as prisms and gratings, allowing for the measurement and manipulation of individual components of the waveform. Perturbations between different frequency components can be measured using techniques such as interferometry, and interference between different frequency components of an optical waveform can be generated using these dispersive optical elements. [Overview of the Initiative]

[0004] This disclosure features optical elements, systems, and methods for interferometry and other applications relating to the detection and manipulation of frequency components of optical waveforms. The optical elements include one or more metamaterial layers defining a phase gradient along at least one direction of the layer, thereby acting as dispersion elements that spatially separate different frequency components of an optical waveform. The optical elements may be implemented in the form of a wedge or prism and may be incorporated into an optical system. For example, in some embodiments, the optical elements function as (or are part of a system that functions as) a Fizeau interferometer. The optical elements and systems may include multiple metamaterial layers to increase the spatial dispersion of frequency components of an optical waveform, and the input optical waveform may pass through one or more metamaterial layers multiple times to enhance the spatial dispersion of frequency components and the measurement sensitivity.

[0005] Implementing a Fizeau interferometer with metamaterial-based optical elements offers several significant advantages. Conventional Fizeau interferometers use precisely calibrated glass wedges or prisms to disperse frequency components, and the fidelity of the dispersion (and therefore the sensitivity of the interferometer measurement) depends in part on the orientation accuracy of the wedge's inclined surface. In the case of imperfect wedges, the measurement signal can depend on the position at which the light waveform is incident on the inclined wedge surface. A Fizeau interferometer, in which the dispersive element is an optical element having one or more metamaterial layers, can be implemented without using wedges or prisms, and is therefore not affected by measurement uncertainties that may arise from the relative position of the incident light waveform.

[0006] Furthermore, by using one or more metamaterial layers as an alternative to intersecting planes at wedge angles for dispersion, the free spectral range of the interferometer can be maintained over a relatively large area. The metamaterial layers can be fabricated as flat layers, enabling a very large free spectral range. Moreover, since metamaterials can be fabricated on a nanometer scale, the dispersion properties of the metamaterial layers can be tuned for specific applications, including particular wavelength bands.

[0007] In one embodiment, the Disclosure includes a first optical element comprising a first substrate, a partial reflective coating disposed on a first surface of the first substrate, a first metamaterial layer located on or adjacent to the first surface of the first substrate and including a structure that defines a continuous phase gradient along a first direction parallel to the first surface of the first substrate, wherein the change in phase magnitude along the first direction is at least 2π in wavelength λ, and a partial reflective layer located on or adjacent to the first metamaterial layer and on the opposite side of the metamaterial layer from the first substrate; and a second optical element comprising a second substrate and a second metamaterial layer located on or adjacent to the first surface of the second substrate and including a structure that defines a continuous phase gradient along a second direction parallel to the first surface of the first substrate, wherein the first and second optical elements are at a non-zero angle.

[0008] Embodiments of the optical system may include one or more of the following features:

[0009] The change in phase magnitude along the first direction can be at least 2π at a wavelength λ from 0.8 μm to 1.8 μm. The first substrate can be formed from at least one material selected from the group consisting of SiO2, Al2O3, and ZnS. The partial reflection coating may contain Si. The partial reflection coating may have a reflectance of 20% to 50% at wavelength λ (e.g., 25% to 35% at wavelength λ).

[0010] A continuous phase gradient along the first direction may extend over a distance of at least 1 mm (e.g., at least 2 mm) along the first direction. The continuous phase gradient along the first direction may be a linear phase gradient. A portion of the continuous phase gradient along the first direction may be a linear phase gradient.

[0011] The partial reflective layer may be a coating disposed on the surface of the first metamaterial layer. The partial reflective layer may include a third metamaterial layer distinct from the first metamaterial layer. The third metamaterial layer may be in contact with the first metamaterial layer. The third metamaterial layer may be disposed on a third substrate distinct from the first substrate. The third substrate may be positioned relative to the first substrate such that the first and third metamaterial layers are in contact with it.

[0012] A third substrate may be positioned relative to the first substrate such that a gap exists between the first and third metamaterial layers. The gap may be at least partially filled with air. At least one additional layer may be located within the gap. At least one additional layer may include a solid material. At least one additional layer includes an index matching layer having a refractive index between the refractive index of the first metamaterial layer at wavelength λ and the refractive index of the third metamaterial layer at wavelength λ.

[0013] The optical system may include an anti-reflective coating disposed on a second surface of the first substrate opposite to the first surface. The anti-reflective coating may have a reflectance of 5% or less at a wavelength λ.

[0014] The partial reflective coating, the first metamaterial layer, and the partial reflective layer may define an optical cavity, the finesse of which may be at least 2 (e.g., at least 3, at least 10). The transmission efficiency of the optical system may be at least 70% (e.g., at least 80%).

[0015] The first metamaterial layer may be located on or adjacent to a first region of the first surface of the first substrate, and an opening where the first metamaterial layer is absent may be located on or adjacent to a second region of the first surface of the first substrate, and the first and second regions of the first surface of the first substrate do not overlap. The partial reflective coating may not be located in the second region of the first surface that forms the opening. The partial reflective layer may not be located on or adjacent to the second region of the first surface that forms the opening.

[0016] At least one of the group consisting of a partial reflection coating, a first metamaterial layer, and a partial reflection layer may include a reference marker.

[0017] The first metamaterial layer may include a plurality of repeating structures formed from a first material and embedded in a second material. The first material may include Si. The first material may include TiO2. The plurality of repeating structures may include cylindrical structures. The plurality of repeating structures may include cuboidal prism structures.

[0018] The average height of the repeating structure in the first metamaterial layer may be 0.2 μm to 1.5 mm (e.g., 0.5 μm to 1.0 mm) when measured in a direction perpendicular to the first surface of the first substrate. The average maximum cross-sectional dimension of the repeating structure in the first metamaterial layer may be 50 nm to 1 mm (e.g., 200 nm to 600 nm) when measured in a direction parallel to the first surface of the first substrate. The refractive index of the first material at wavelength λ may be 3.0 to 4.0. The refractive index of the second material at wavelength λ may be 1.0 to 2.0. The difference in refractive index between the first and second materials at wavelength λ may be 1.5 to 2.5.

[0019] The second material may include at least one material selected from the group consisting of glass, fused silica, quartz, and sapphire. The second material may include at least one polymer material. The at least one polymer material may be selected from the group consisting of polymethyl methacrylate (PMMA), polycarbonate (PC), polystyrene (PS), polyester (PE), and cyclic olefin polymer (COP).

[0020] The multiple repeating structures may be a first set of multiple repeating structures, and the first metamaterial layer may further include a second set of multiple repeating structures formed from a first material and embedded in a second material, the second set of multiple repeating structures may differ from the first set of multiple repeating structures. The second set of multiple repeating structures may have a different cross-sectional shape from the first set of multiple repeating structures. The average height of the second set of multiple repeating structures may differ from the average height of the first set of multiple repeating structures measured in a direction perpendicular to the first surface of the first substrate, measured in a direction perpendicular to the first direction of the first set of multiple repeating structures. The average maximum dimension of the second set of multiple repeating structures may differ from the average maximum dimension of the first set of multiple repeating structures measured in a direction parallel to the first surface of the first substrate, measured in a direction parallel to the first surface of the first substrate.

[0021] The first metamaterial layer may include a plurality of repeating structures formed from a third material and embedded in the second material, the third material may be different from the first material. The third material may include at least one material selected from the group consisting of Si and TiO2. The plurality of repeating structures formed from the first material and the plurality of repeating structures formed from the third material may have a common cross-sectional shape. Alternatively, the plurality of repeating structures formed from the first material and the plurality of repeating structures formed from the third material may have different cross-sectional shapes.

[0022] The average height of a plurality of repeating structures formed from the first material may be the same as the average height of a plurality of repeating structures formed from the third material, measured in a direction perpendicular to the first surface of the first substrate, when measured in a direction perpendicular to the first surface of the first substrate. A plurality of repeating structures formed from the first material may have an average height measured in a direction perpendicular to the first surface of the first substrate that is different from the average height of a plurality of repeating structures formed from the third material, when measured in a direction perpendicular to the first surface of the first substrate.

[0023] The average maximum dimension of a plurality of repeating structures formed from the first material may be the same as the average maximum dimension of a plurality of repeating structures formed from the third material, measured in a direction parallel to the first surface of the first substrate, when measured in a direction parallel to the first surface of the first substrate. A plurality of repeating structures formed from the first material may have a maximum dimension measured in a direction parallel to the first surface of the first substrate that is different from the average maximum dimension of a plurality of repeating structures formed from the third material, measured in a direction parallel to the first surface of the first substrate.

[0024] The second optical element can be a prism. The second optical element can be a spectral dispersion window. The second optical element can include a second surface that is non-parallel to the first surface of the second optical element. The second optical element can include a second surface that is parallel to the first surface of the second optical element. The second optical element can be a transmissive element. The second optical element can be a reflective element. The second optical element can include a second surface on the side opposite to the first surface, and the first surface and the second surface of the second optical element can be flat.

[0025] The change in the magnitude of the phase along the second direction can be at least 2π at a wavelength λ from 0.8 μm to 1.8 μm. The second substrate can be formed from at least one material selected from the group consisting of SiO2, Al2O3, and ZnS.

[0026] The continuous phase gradient along the second direction can extend over a distance of at least 1 mm (e.g., at least 2 mm along the second direction). The continuous phase gradient along the second direction can be a linear phase gradient. A part of the continuous phase gradient along the second direction can be a linear phase gradient.

[0027] The second metamaterial layer can include a plurality of repeating structures formed from a first material and embedded in a second material. The first material can include Si. The first material can include TiO2.

[0028] The plurality of repeating structures can include a cylindrical structure. The plurality of repeating structures can include a rectangular prism structure. The average height of the repeating structures in the second metamaterial layer can be from 0.2 μm to 1.5 mm (e.g., from 0.5 μm to 1.0 mm) measured in a direction orthogonal to the first surface of the second substrate. The average maximum cross-sectional dimension of the repeating structures in the second metamaterial layer can be from 50 nm to 1 mm (e.g., from 200 nm to 600 nm) measured in a direction parallel to the first surface of the second substrate.

[0029] The refractive index of the first material at wavelength λ may range from 3.0 to 4.0. The refractive index of the second material at wavelength λ may range from 1.0 to 2.0. The difference in refractive index between the first and second materials at wavelength λ may range from 1.5 to 2.5.

[0030] The second material may include at least one material selected from the group consisting of glass, fused silica, quartz, and sapphire. The second material may include at least one polymer material. The at least one polymer material may be selected from the group consisting of polymethyl methacrylate (PMMA), polycarbonate (PC), polystyrene (PS), polyester (PE), and cyclic olefin polymer (COP).

[0031] The multiple repeating structures may be a first set of multiple repeating structures, and the second metamaterial layer may further include a second set of multiple repeating structures formed from the first material and embedded in the second material, the second set of multiple repeating structures may differ from the first set of multiple repeating structures. The second set of multiple repeating structures may have a different cross-sectional shape from the first set of multiple repeating structures. The average height of the second set of multiple repeating structures may differ from the average height of the first set of multiple repeating structures measured in a direction perpendicular to the first surface of the second substrate, measured in a direction perpendicular to the first direction of the first set of multiple repeating structures. The average maximum dimension of the second set of multiple repeating structures may differ from the average maximum dimension of the first set of multiple repeating structures measured in a direction parallel to the first surface of the second substrate, measured in a direction parallel to the first surface of the second substrate.

[0032] The second metamaterial layer may further include a plurality of repeating structures formed from a third material and embedded within the second material, the third material being different from the first material. The third material may include at least one material selected from the group consisting of Si and TiO2.

[0033] Multiple repeating structures formed from the first material and multiple repeating structures formed from the third material may have a common cross-sectional shape. Alternatively, or in addition, multiple repeating structures formed from the first material and multiple repeating structures formed from the third material may have different cross-sectional shapes.

[0034] The average height of a plurality of repeating structures formed from the first material, measured in a direction perpendicular to the first surface of the second substrate, may be the same as the average height of a plurality of repeating structures formed from the third material, measured in a direction perpendicular to the first surface of the second substrate. The plurality of repeating structures formed from the first material may have an average height measured in a direction perpendicular to the first surface of the second substrate that is different from the average height of the plurality of repeating structures formed from the third material, measured in a direction perpendicular to the first surface of the second substrate.

[0035] The maximum dimensions of a plurality of repeating structures formed from the first material, measured in a direction parallel to the first surface of the second substrate, may be the same as the maximum dimensions of a plurality of repeating structures formed from the third material, measured in a direction parallel to the first surface of the second substrate. The plurality of repeating structures formed from the first material may have a maximum dimension measured in a direction parallel to the first surface of the second substrate that differs from the average maximum dimension of a plurality of repeating structures formed from the third material, measured in a direction parallel to the first surface of the second substrate.

[0036] Embodiments of the apparatus may include any other features described herein, and may also include any combination of features described separately in relation to different embodiments, unless expressly otherwise stated.

[0037] In another embodiment, the present disclosure features an apparatus comprising any optical system described herein, the optical system comprising a plurality of first optical elements, the first surface of the first substrate of each first optical element located in a common plane, and as a result, the partial reflective coating, first metamaterial layer and partial reflective layer of each first optical element define a continuous optical cavity extending in a direction parallel to the common plane, the phase magnitude change along the continuous optical cavity exceeds 2π in wavelength λ.

[0038] Embodiments of the apparatus may include one or more of the following features:

[0039] The change in phase magnitude can exceed 8π at wavelength λ. Each of the multiple first optical elements may be identical. One or more of the multiple first optical elements may differ from one or more other first optical elements.

[0040] One or more first metamaterial layers among the plurality of first optical elements may differ from the first metamaterial layers of one or more other plurality of first optical elements. The change in phase magnitude along a first direction of one or more of the plurality of first optical elements may differ from the change in phase magnitude along a first direction of one or more other plurality of first optical elements. The plurality of first optical elements may include at least four first optical elements.

[0041] Embodiments of the apparatus may include any other features described herein, and may also include any combination of features described separately in relation to different embodiments, unless otherwise expressly stated.

[0042] In yet another aspect, the present disclosure features an apparatus comprising any optical system described herein, the optical system comprising a plurality of first optical elements, wherein the first surface of one or more first substrates of the plurality of first optical elements is offset from the first surface of the first substrate of the other one or more plurality of first optical elements in a direction perpendicular to the first surface, as a result the partial reflective coatings, metamaterial layers and partial reflective layers of the plurality of first optical elements define a plurality of optical cavities offset from each other in an orthogonal direction, and the overall change in the magnitude of the phase between the optical cavities exceeds 2π at wavelength λ.

[0043] Embodiments of the apparatus may include one or more of the following features:

[0044] The first surfaces of at least some of the first substrates of a plurality of first optical elements may be located in a common plane, thereby defining a continuous optical cavity in which the partial reflective coatings, metamaterial layers, and partial reflective layers of at least some of the first optical elements as a whole extend in a direction parallel to the common plane. The apparatus may include a plurality of continuous optical cavities, each extending in a different plane, and each continuous optical cavity may be offset from other continuous optical cavities in the apparatus. The overall change in the magnitude of the phase between the optical cavities may exceed 8π at wavelength λ.

[0045] Each first optical element of a plurality of optical elements may be identical. One or more of the plurality of first optical elements may differ from one or more other first optical elements. One or more metamaterial layers of the plurality of first optical elements may differ from the metamaterial layers of one or more other first optical elements. The magnitude of the phase change along a first direction of one or more of the plurality of first optical elements may differ from the magnitude of the phase change along a first direction of one or more other first optical elements. Each continuous optical cavity may contain a magnitude of the phase change along the continuous optical cavity, and the magnitude of the phase change of one or more of the plurality of continuous optical cavities may differ from the magnitude of the phase change of one or more other continuous optical cavities.

[0046] The plurality of first optical elements comprises at least four first optical elements.

[0047] Embodiments of the apparatus may include any combination of other features described herein, i.e., features described separately in relation to different embodiments, unless otherwise expressly stated.

[0048] In another aspect, the Disclosure features a method for generating an output optical waveform, comprising the steps of preparing any optical system described herein, generating an input optical waveform including a plurality of wavelength components, and transmitting the input optical waveform through the optical system to generate an output optical waveform, wherein at least some of the wavelength components of the input optical waveform pass through at least one metamaterial layer of the optical system multiple times as they pass through the optical system, and the output optical waveform includes spatially separated wavelength components.

[0049] Embodiments of the method may include any combination of other features described herein, i.e., features described separately in relation to different embodiments, unless otherwise expressly stated.

[0050] In yet another aspect, the Disclosure features a method for generating an output optical waveform, comprising the steps of preparing any apparatus described herein, generating an input optical waveform including a plurality of wavelength components, and transmitting the input optical waveform through the apparatus to generate an output optical waveform, wherein at least some of the wavelength components of the input optical waveform pass through at least one metamaterial layer of the apparatus multiple times as they pass through the apparatus, and the output optical waveform includes spatially separated wavelength components.

[0051] Embodiments of the method may include any combination of other features described herein, i.e., features described separately in relation to different embodiments, unless otherwise expressly stated.

[0052] In this specification, “metamaterial layer” refers to a planar or non-planar layer comprising multiple structures designed to interact with and manipulate the properties of electromagnetic waves in a manner not found in nature when the individual materials constituting the layer interact with such waves. In some embodiments, the metamaterial layer may have properties not found in nature (such as a negative refractive index). The metamaterial layer is generally formed by introducing structural elements into or on a substrate. In certain embodiments, the introduced structural elements have spatial periodicity in at least part of the layer, and the structural elements and the substrate form a composite material. Spatial periodicity may exist in one, two, and / or three dimensions of the layer, and may exist on a dimensional scale smaller than the interacting wavelength, but is not always present. Furthermore, in some embodiments, the structural elements may have regular geometric shapes, such as cylinders, cuboidal prisms, triangular prisms, cones, helices, and extruded bodies derived from two-dimensional cross-sectional shapes such as crosses, polygons, and other regular geometric shapes. In certain embodiments, the structural element may include openings, channels, and / or other features that partially or completely penetrate the element.

[0053] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as those commonly understood by those skilled in the art. Methods and materials similar to or equivalent to those described herein may be used for the implementation or testing of the subject matter described herein, but suitable methods and materials are listed below. All publications, patent applications, patents, and other documents mentioned herein are incorporated in their entirety by reference. In the event of any conflict, this specification (including definitions) shall prevail. Furthermore, materials, methods, and examples are illustrative and not intended to be limiting.

[0054] Details of one or more embodiments are described in the accompanying drawings and the description below. Other features and advantages will become apparent from the description, drawings and claims. [Brief explanation of the drawing]

[0055] [Figure 1] This is a schematic diagram of a Fizeau interferometer. [Figure 2] This is a schematic diagram of an example of a metamaterial-based interferometer. [Figure 3] This graph shows the transmitted light intensity in a conventional Fabry-Perot etalone. [Figure 4] This is a schematic diagram of an example of another metamaterial-based interferometer. [Figure 5] Figures 5A to 5C are graphs showing the spatial fringe profiles calculated for metamaterial-based interferometers with different spacings between the metamaterial layer and the partial reflection layer. [Figure 6] Figure 2 shows a graph of the fringe peak position as a function of the spacing between the metamaterial layer and the partial reflection layer in the interferometer, and the wavelength λ. [Figure 7] Figures 7A to 7C and 5A to 5C show graphs of interference fringes calculated when the reflectance of the metamaterial layer and the partial reflection layer are different. [Figure 8] Figure 8A shows a top and side view of an example of a metamaterial-based interferometer. Figure 8B also shows a top and side view of an example of a metamaterial-based interferometer. Figure 8C is a graph showing the transmittance as a function of wavelength and position, calculated using the plane wave approximation for the interferometers in Figures 8A and 8B. Figure 8D is a graph showing the transmittance as a function of wavelength and position, calculated using the total electric field simulation for the interferometers in Figures 8A and 8B. [Figure 9] This graph shows the transmission amplitude and phase of the light waveform incident on a Si column array. [Figure 10] Figures 10A to 10D are graphs showing design parameters and error estimates for an example of a metamaterial-based interferometer. [Figure 11] Figures 11A to 11D are graphs showing design parameters and error estimates for an example of another metamaterial-based interferometer. [Figure 12A] This is a schematic diagram of an example of a metamaterial-based interferometer. [Figure 12B] Figure 12A shows the optical microscope image from the interferometer. [Figure 12C] Figure 12A shows a scanning electron microscope image from the interferometer. [Figure 12D] Figure 12A shows a scanning electron microscope image from the interferometer. [Figure 12E] Figure 12A shows a scanning electron microscope image from the interferometer. [Figure 13A] These images show single-wavelength and multi-wavelength fringing for an example of an interferometer with different effective wedge angles and reflectances. [Figure 13B] These images show single-wavelength and multi-wavelength fringing for an example of an interferometer with different effective wedge angles and reflectances. [Figure 13C] These images show single-wavelength and multi-wavelength fringing for an example of an interferometer with different effective wedge angles and reflectances. [Figure 13D] These images show single-wavelength and multi-wavelength fringing for an example of an interferometer with different effective wedge angles and reflectances. [Figure 14A] Figure 13C is a graph showing the fringe pattern of the interferometer when optical waveforms are incident at different angles along the phase gradient direction of the metamaterial layer. [Figure 14B] Figure 13C is a graph showing the fringe pattern of the interferometer when optical waveforms are incident at different angles along the phase gradient direction of the metamaterial layer. [Figure 14C] Figure 13C is a graph showing the fringe pattern of the interferometer when optical waveforms are incident at different angles along the phase gradient direction of the metamaterial layer. [Figure 14D] Figure 13C is a graph showing the fringe pattern of the interferometer when optical waveforms are incident at different angles along the phase gradient direction of the metamaterial layer. [Figure 14E] Figure 13C is a graph showing the fringe pattern of the interferometer when optical waveforms are incident at different angles along the phase gradient direction of the metamaterial layer. [Figure 14F]Figure 13C is a graph showing the fringe pattern of the interferometer when optical waveforms are incident at different angles along the phase gradient direction of the metamaterial layer. [Figure 14G] Figure 13C is a graph showing the fringe pattern of the interferometer when optical waveforms are incident at different angles along the phase gradient direction of the metamaterial layer. [Figure 15A] Figure 13C is a graph showing the fringe pattern of the interferometer when optical waveforms are incident at different angles along a direction perpendicular to the phase gradient direction of the metamaterial layer. [Figure 15B] Figure 13C is a graph showing the fringe pattern of the interferometer when optical waveforms are incident at different angles along a direction perpendicular to the phase gradient direction of the metamaterial layer. [Figure 15C] Figure 13C is a graph showing the fringe pattern of the interferometer when optical waveforms are incident at different angles along a direction perpendicular to the phase gradient direction of the metamaterial layer. [Figure 16] These are images showing the measurement fringes at single and multiple wavelengths in a corrected interferometer. [Figure 17] Figure 17A is a schematic diagram of an optical system including a metamaterial-based interferometer and a metamaterial-based lens. Figure 17B is a graph showing the spatial dispersion and focusing of single-wavelength light by the metamaterial-based lens. Figure 17C is a graph showing the spatial dispersion and focusing of multi-wavelength light by the metamaterial-based lens. Figure 17D is a graph showing how a small wavelength difference is separated by the spatial dispersion and focusing of nearly monochromatic light using a metamaterial-based lens and interferometer. Figure 17E is a graph showing how a wavelength difference is separated in two dimensions by the spatial dispersion and focusing of multi-color light using a metamaterial-based lens and interferometer. Figure 17F is a graph showing the transmittance map of the apparatus in Figure 17A. [Figure 18A] This is a schematic diagram of an optical device including a metamaterial-based interferometer and a metamaterial-based prism. [Figure 18B]Figure 18A shows an example transmission spectrum graph for the interferometer of the system. [Figure 18C] These are schematic diagrams of other optical devices, including other metamaterial-based interferometers and metamaterial-based prisms. Similar symbols in the various diagrams indicate similar elements. [Modes for carrying out the invention]

[0056] (I. Introduction) In a wide variety of spectroscopic, measurement, and communication applications, optical waveforms containing light of different wavelengths are generated, detected, and measured. In many such applications, the frequency components of the optical waveform are spatially dispersed, allowing light of different wavelengths to be detected and / or manipulated. Optical elements that achieve spatial dispersion of different frequency components in an optical waveform include prisms, wedges, diffraction gratings, phase gratings, and other elements that impart frequency-dependent phases to the optical waveform that interact with the element.

[0057] An example of a device that spatially disperses the frequency components of an incident light waveform is a Fizeau interferometer, which can function as a spectrometer or, more generally, as a wavelength-dispersing element in a system performing spectroscopic analysis or measurement operations (e.g., surface geometry measurement). Figure 1 is a schematic diagram of an example of a Fizeau interferometer 100. The interferometer includes a first window 102 and a second window 112 positioned at a small angle α relative to each other. The first window 102 includes an anti-reflective coating 104, a substrate 106, and a partial reflection mirror layer 108. Similarly, the second window 112 includes an anti-reflective coating 114, a substrate 116, and a partial reflection mirror layer 118.

[0058] When the light waveform 130 is incident on the first window 102, the light waveform passes through the (nominal) anti-reflective coating 104, the substrate 106, and the mirror layer 108, then through the space between the first window 102 and the second window 112, and is incident on the mirror layer 118. Because the mirror layer 118 is partially reflective, a portion of the waveform 130 is reflected by the mirror layer 118 (as waveform 132), and a portion of the waveform 130 passes through the mirror layer 118, the substrate 116, and the anti-reflective coating 114, and exits from the second window 112 as waveform 134.

[0059] The reflected waveform 132 is reflected by the mirror layer 108, and the reflected component of waveform 132 (waveform 136) is incident on the mirror layer 118. Part of waveform 136 passes through the second window 112 (as waveform 140), and part is reflected by the mirror layer 118 (as waveform 138). Part of waveform 138 is reflected from the mirror layer 108 as waveform 142, and then incident on the mirror layer 118. Part of waveform 142 passes through the second window 112 as waveform 146, and part is reflected from the mirror layer 118 as waveform 144. This process is repeated, generating additional waveforms that pass through the second window 112.

[0060] The transverse spacing between transmitted waveforms (e.g., 134, 140, and 146) in the x-direction depends on the angle of incidence of optical waveform 130 into the first window 102, the wavelength (or frequency) of optical waveform 130, and the angle α between windows 102 and 112. The transmitted waveforms 134, 140, and 146 interfere beyond the second window 112, generating an interference fringe pattern with peaks of maximum intensity at specific locations along the x-direction.

[0061] If the optical waveform 130 contains only monochromatic light of a single wavelength, a single fringe pattern is observed at a location along the x-direction corresponding to the wavelength of the monochromatic light. However, if the optical waveform 130 contains light of multiple wavelengths, multiple fringe patterns corresponding to the wavelengths are observed, and the fringe pattern corresponding to each wavelength is located at different positions along the x-direction. In effect, windows 102 and 112 function as multiple Fabry-Perot etalons, and for an incident optical waveform 130 containing light of multiple wavelengths (or, in other words, multiple frequency components), each transmitted wavelength of light undergoes constructive interference at different locations along the x-direction, and is therefore spatially dispersed along the x-direction, which can be individually detected or otherwise manipulated by placing additional optical elements (e.g., detectors, modulators) at specific locations along the x-direction.

[0062] However, as described above, the operation of the interferometer 100 is highly dependent on various geometric variables. These include the angle of incidence of the optical waveform 130 to the first window 102, the angle α between windows 102 and 112, the refractive index of the medium in the gap between windows 102 and 112, and the geometric characteristics (and any defects) of windows 102 and 112. Therefore, before use, the interferometer 110 is usually carefully calibrated so that the mapping between wavelength and position along the x-direction is determined with high fidelity. However, it can be difficult to compensate for defects in windows 102 and 112 (e.g., surface anomalies, non-constant wedge angle α due to the curvature of either or both windows 102 and / or 112, deformation due to temperature changes and / or mechanical perturbations). Furthermore, the wedge angle α is usually not adjustable, and it can be difficult to maintain a constant angular relationship between windows 102 and 112 over long periods of time.

[0063] This disclosure features optical elements, systems, and methods for performing spectral dispersion and wavelength resolution based on Fizeau interference without using optical wedges. More specifically, the optical element includes one or more metamaterial layers that impart phase offsets to different wavelengths of a multi-wavelength optical waveform, so that each wavelength of the optical waveform is dispersed in space perpendicular to the incident direction of the multi-wavelength optical waveform.

[0064] (II. Dispersive Optical Elements) Figure 2A is a schematic diagram of an example of a metamaterial-based Fizeau interferometer 200. The interferometer includes a substrate 202, a metamaterial layer 204, a partial reflective layer 206, an optional anti-reflective layer 208, and other optional anti-reflective layers 210. It should be understood that the interferometer 200 in Figure 2A is just one example, and other configurations are possible. For example, other functional layers may be present in the interferometer 200, and certain layers (e.g., layers 208 and / or 210) may not be present in the interferometer 200.

[0065] The substrate 202 is generally optically transparent and formed from one or more materials capable of transmitting light of a specific wavelength (e.g., wavelengths used in spectroscopic measurements, measurements, and other applications) without significant absorption. In this disclosure, “significant absorption” means that the material used for the substrate 202 has a thickness of 0.1 cm -1 This means that it has the following linear absorption coefficient.

[0066] To manufacture the substrate 202, various materials may be used depending on the specific application in which the interferometer 200 is used. For example, the substrate 202 may be formed from materials such as SiO2, Al2O3, and ZnS. In some embodiments, the substrate 202 may be formed from a single material. Alternatively, in certain embodiments, the substrate 202 may be formed from multiple different materials, mixed as a homogeneous or heterogeneous mixture, or arranged as multiple layers, domains, or other structures within the substrate 202.

[0067] The partially reflective layer 206 is capable of reflecting a portion of the waveform incident on the layer and transmitting a portion of it. The layer 206 can be formed from a variety of materials, depending on the properties of the other layers in the interferometer 200 and the wavelength of light used in applications including the interferometer. For example, in some embodiments, the layer 206 is formed from Si. In certain embodiments, the layer 206 is formed from one or more metals such as chromium, gold, and / or silver. In some embodiments, the layer 206 is formed from one or more dielectric materials. The layer 206 can be formed from a single material or from a combination of multiple materials.

[0068] Although layer 206 is shown in Figure 2 as a homogeneous layer containing one or more materials, in certain embodiments, layer 206 may be implemented as a heterogeneous layer. For example, layer 206 may contain multiple layers of different materials (e.g., a layer stack in which layers of different materials are adjacent). In some embodiments, layer 206 may be implemented by forming domains of one or more materials within layers of other materials, resulting in the domains appearing substantially as connected or disconnected regions within the foundation of one or more other materials.

[0069] Layer 206 can be deposited on the substrate 202 in various ways. For example, in some embodiments, layer 206 may be applied by chemical vapor deposition (CVD) or physical vapor deposition (PVD), which allows for the introduction of one or more precursor materials into the gas phase and their deposition on the substrate 202 to form layer 206. In some embodiments, the thickness of layer 206 is controlled (e.g., by controlling the deposition time) to adjust the reflectivity of layer 206. In certain embodiments, layer 206 may be formed by sputtering one or more materials into the gas phase and depositing the sputtered atoms or ions onto the substrate 202. In some embodiments, layer 206 may be formed by solid-state methods. For example, the materials constituting layer 206 may be deposited directly onto the substrate 202 by processes such as solution-based deposition, spin coating, and / or in situ chemical processes (e.g., polymerization).

[0070] The anti-reflective layers 208 and 210 may have the same or different compositions, thicknesses, and anti-reflective properties. Generally, layer 208 is formed from one or more materials that reduce reflections caused by refractive index mismatch at the interface between these layers and other layers of the interferometer 200. A variety of materials may be used to form layers 208 and 210, including but not limited to MgF2, CaF2, quartz, SiO2, polymer materials, and other crystalline and amorphous materials. Layers 208 and 210 may each be formed independently from a single material or multiple materials. When multiple materials are used, they may form a homogeneous composite material, a heterogeneous composite as domains in other materials as described above in relation to layer 206, or a laminated layer of different materials. The reflectivity of each layer 208 and 210 may generally be 0.10 or less (e.g., 0.08 or less, 0.05 or less, 0.03 or less, 0.02 or less, 0.01 or less, or less).

[0071] In general, the anti-reflective layers 208 and 210 can be formed by processes such as those described above in relation to layer 206, i.e., (but not limited to) CVD, PVD, sputtering, solid-state physical and chemical deposition methods.

[0072] As described above, in a conventional Fizeau interferometer, the wedge formed by two surfaces effectively functions as a Fabry-Perot etalon with continuous and adiabatic spatial variation. In a conventional Fabry-Perot etalon, partial reflection of light by two adjacent surfaces causes light to travel back and forth between the surfaces multiple times before exiting through the gap between them. If the accumulation of round-trip phase is an integer multiple of 2π, then constructive interference occurs at the etalon output, and the light is transmitted through the etalon. Assuming no scattering or other losses, the light is transmitted with unit efficiency. When the phase condition is not met, some or all of the light is reflected by interference. The round-trip phase depends primarily on the phase imparted in the region between the etalon surfaces. Because the phase is dispersive with respect to wavelength, the etalon transmits only light of specific wavelengths.

[0073] Figure 3 is a graph showing the transmitted light intensity in a conventional Fabry-Perot etalon where the spacing between etalon surfaces is 5 micrometers and the reflectivity of each etalon surface is 0.8. As is clear from Figure 3, only light of a specific wavelength λ that satisfies the phase condition described above is transmitted through the etalon.

[0074] In conventional Fizeau interferometers, the two reflective surfaces of the wedge are positioned at an angle α, so the phase condition described above is satisfied for different wavelengths λ at different lateral positions in the x-coordinate direction. Therefore, conventional wedge-type Fizeau interferometers function as dispersion elements, with light of different wavelengths in the incident light waveform passing through the interferometer at different positions along the x-coordinate direction.

[0075] Figure 4 is a schematic diagram showing specific features of the interferometer 200, which includes a substrate 202, a metamaterial layer 204, and a partial reflection layer 206. In the interferometer 200, the metamaterial layer 204 produces an effect similar to the wedge in a conventional Fizeau interferometer. Layer 204 introduces a phase gradient that causes anomalous reflection. As shown in Figure 4, the optical waveform 130 incident on the interferometer passes through layer 204 and substrate 202. Each time the waveform reaches layer 206, a portion of the waveform passes through layer 206, and another portion is reflected through substrate 202. Each time the waveform passes through substrate 202, an additional phase is accumulated, depending on the thickness d of substrate 202 and its refractive index.

[0076] Furthermore, as shown in Figure 4, each time the waveform is reflected by the metamaterial layer 204, the layer, i.e., the phase gradient layer, generates momentum kick k G The equation = 2π / P is assigned, where P is the superperiod of the metamaterial layer 204. The output from the interferometer 200 can be modeled by passing through the partial reflection layer 206 and adding plane waves (weighted with appropriate coefficients) corresponding to each round trip of the optical waveform between layers 204 and 206. As shown in Figure 4, each weighted plane wave includes a phase component due to interactions with the substrate 202 and the metamaterial layer 204.

[0077] Furthermore, as shown in Figure 4, the phase accumulated by the waveform transmitted through layer 206 changes spatially. Therefore, the interferometer 200 in Figure 4 functions as a Fabry-Perot etalon, where each wavelength has a transmission peak (e.g., transmitted through layer 206) at a specific position within the superperiod P of the metamaterial layer 204. Since the metamaterial layer 204 imparts all phase delays from 0 to 2π along the period P, at a certain position along the x-coordinate direction, the above interference condition is satisfied for all wavelengths λ.

[0078] As described above, the round-trip phase accumulated in the optical waveform depends on the thickness d of the substrate 202 and the refractive index n2 of the substrate. Generally, the larger the thickness d, the larger the phase accumulated in each round trip of the optical waveform between layers 204 and 206. Furthermore, the magnitude of the phase accumulated in each round trip between layers 204 and 206 affects the spatial dispersion of the wavelength in the x-coordinate direction emitted from the interferometer 200.

[0079] Figures 5A to 5C are graphs showing the spatial fringe profiles calculated under the condition n² = 1.5 for different substrate thicknesses d (0.05 μm, 0.2 μm, 1 μm) and three different wavelengths (0.9 μm, 1.0 μm, 1.1 μm) of the substrate 202. As shown in Figures 5A to 5C, the interferometer 200 behaves in an etalon-like manner, with the periodic fringe position at each wavelength shifting laterally (i.e., in the x-coordinate direction) as a function of wavelength. In particular, the lateral shift increases with increasing d. In Figures 5A to 5C, the fringe pattern is periodic with period P, which is because it is calculated from the sum of weighted plane waves, as explained above in relation to Figure 4. In the graphs of Figures 5A to 5C, the period P = 1 mm.

[0080] Figure 6 is a graph showing the peak position of the fringe (remainder at period P) as a function of wavelength λ and thickness d. This graph follows the following analytical formula. x peak =mod (-2n2dP / λ,P) [1] x peaky is the peak intensity, and the function mod(y,P) is defined as the remainder when y is divided by P.

[0081] The graph in Figure 6 was calculated assuming that layers 204 and 206 have a reflectance of 0.5. The fringe becomes sharper as the reflectance of either or both layers increases. Figures 7A–7C show graphs of interference fringe calculated for a substrate thickness d=0.2 μm, at the same wavelengths as in Figures 5A–5C, when the reflectance of layers 204 and 206 increases from 0.3 (Figure 7A) to 0.5 (Figure 7B) and further to 0.8 (Figure 7C). As is clear from these figures, the fringe becomes significantly sharper as the reflectance of layers 204 and 206 increases. A similar phenomenon occurs in Fabry-Perot etalons, but in etalons, the sharpening is due solely to spectral interference, whereas in the examples of Figures 7A–7C, the increase in sharpening is substantially due to an increase in the "sensitivity" to the phase accumulated in each round trip of the optical waveform between layers 204 and 206. As a result, constructive interference between round trips occurs over a narrower phase range, and the spatial distribution of each fringe becomes narrower (i.e., the peaks are "sharpened").

[0082] To further understand the effect of the metamaterial layer 204, we consider an example of an interferometer 200 shown in Figures 8A and 8B. Figure 8A is a schematic top view of the interferometer, and Figure 8B is a schematic side view of the interferometer. The interferometer includes multiple titania (TiO2) pillars (refractive index 2.5), each pillar having the shape of a rectangular prism and embedded in a glass substrate. The pillars have a square cross-sectional shape (i.e., in the xy plane) and a length in the z-coordinate direction of 500 nm. In the configuration shown in Figures 8A and 8B, any phase value from 0 to 2π can be applied to the incident light waveform.

[0083] FIG. 8C is a graph showing the calculation of the transmittance at each position along the x - coordinate direction as a function of wavelength using the model of FIG. 4, and FIG. 8D is a graph showing the full - wave simulation performed on the interferometers of FIGS. 8A and 8B. The two simulations show a relatively good agreement, and most of the deviation between them is due to the wavelength dependence of the phase profile of the metamaterial layer 204.

[0084] Generally, the parameters of the metamaterial layer 204 can be adjustable based on the desired operating wavelength. As an example, the metamaterial layer 204 is designed for use in the near - infrared spectral region. For a periodic array of cylindrical silicon pillars embedded in a low - refractive - index (n = 1.48) base PMMA material, the lattice constant A of the unit cell is 600 nm in both the x - coordinate and y - coordinate directions, the height of the silicon pillars in the z - coordinate direction is 0.8 μm, the diameter D ranges from 50 nm to 400 nm, and the transmittance coefficient of the unit cell when the refractive index of Si at the central wavelength of 1.2 μm is n = 3.52 is shown in the graph of FIG. 9. In FIG. 9, the straight line indicates the transmission amplitude, while the decaying line indicates the transmission phase when passing through the silicon pillar array.

[0085] Wedge angle θ w To mimic the function of a Fizeau interferometer having a wedge angle θ, the period length P of the metamaterial layer 204 can be calculated as P = λ0 / nθ w where λ0 is the design wavelength (e.g., 1.2 μm), n is the refractive index of the substrate material (e.g., 1.48 for PMMA), and θ w is assumed to be relatively small. FIGS. 10A - 10D are graphs showing the design parameters and error estimations for a device with θ w = 0.2 degrees, and FIGS. 11A - 11D are graphs showing the design parameters and error estimations for a device with θ w = 0.1 degrees. The phase varies linearly from 0 to 2π over the entire period, and for both devices, the phase error (less than 1%) and amplitude error (less than 2%) are very small over the entire period.

[0086] Referring again to Figures 2 and 4, the interferometer 200 operates in transmission mode, i.e., the wavelength component of the optical waveform is emitted from the interferometer 200 in a direction approximately corresponding to the incident direction of the optical waveform. The reflective layer 210 ensures that the wavelength component is effectively reflected from the metamaterial layer 204, thereby achieving this output directionality. However, in some embodiments, the interferometer 200 may operate in reflection mode, and the wavelength component is emitted from the interferometer 200 in a direction similar to that which would occur if the incident optical waveform were reflected by the interferometer 200. For example, layer 210 may be implemented as a partial reflection layer (e.g., similar to layer 206 described above), and layer 206 may be implemented as a reflection layer (e.g., similar to layer 210 described above). In this configuration, the wavelength component passing through the metamaterial layer 204 acquires a phase shift in a similar manner to the interaction between the optical waveform and the metamaterial layer 204 described above.

[0087] In certain embodiments, the phase imparted to the optical waveform by the metamaterial layer 204 is polarization-insensitive. In other words, an optical waveform polarized in either S or P orientation (or a superposition thereof) acquires the same phase shift when interacting with the metamaterial layer 204. However, in some embodiments, the phase imparted to the optical waveform by the metamaterial layer 204 is polarization-dependent. For example, the metamaterial layer 204 may be manufactured to implement a first phase pattern or gradient for optical waveforms polarized in a first linear direction (e.g., a P-polarized optical waveform) and a second phase pattern or gradient for optical waveforms polarized in a second linear direction relative to the first linear direction (e.g., orthogonal to the first linear direction) (e.g., an S-polarized optical waveform). Thus, an optical waveform linearly polarized in either the first or second direction is imparted with a phase corresponding to the first or second phase pattern / gradient, respectively. A light waveform linearly polarized in a direction intermediate between the first and second directions, or a light waveform having circular or ellipsoidal polarization, is given a more complex overall phase that depends on the interaction between the polarization component of the light waveform and the first and second phase patterns / gradients.

[0088] In general, the metamaterial layer 204 can be manufactured using a variety of techniques. For example, in some embodiments, the metamaterial layer 204 is manufactured using standard electron beam lithography (EBL) techniques. Other methods that may be used to manufacture the metamaterial layers described herein include, but are not limited to, nanoimprint lithography and photolithography.

[0089] An example of the metamaterial layer 204 is shown in Figure 12A. To form the metamaterial layer 204 (consisting of multiple Si pillars embedded in a PMMA matrix), an initial fabricated substrate was prepared consisting of an SiO2 layer with an epitaxial layer of Au (as a partial reflective layer) and SiO2. Next, Si pillars were deposited on the fabricated substrate using EBL to selectively deposit Si in the unmasked areas by masking the fabricated substrate. Inductively coupled plasma (ICP) etching was used to ensure that the pillars were sufficiently separated and had smooth surfaces. Subsequently, a thin epitaxial layer of Al2O3 was formed on top of each pillar, and the mask was removed from the fabricated substrate.

[0090] Next, a PMMA layer was formed using standard polymer deposition techniques to encapsulate the Si pillars on the fabricated substrate, and a thin Au epitaxial layer was formed on top of the PMMA layer to function as a partial reflective layer.

[0091] Figure 12B is an interferometer microscope image after the mask was removed from the manufactured substrate, and Figures 12C-12E are scanning electron microscope images of the Si pillars after ICP etching. As shown in these images, the Si pillars were formed uniformly and precisely using the above process.

[0092] The interferometer manufactured using the above technology was tested by positioning the interferometer to receive light waveforms from a collimated laser light source in the near-infrared spectral region. Using a near-infrared objective lens, the light emitted from the interferometer was focused into a near-infrared camera system, and the light from the interferometer was imaged.

[0093] Figures 13A to 13D are a series of images showing the spectral responses of four different interferometers irradiated with forward incidence. In each of Figures 13A to 13D, the upper image shows the image acquired at a wavelength of 1.2 μm, and the lower image shows the spectral fringe measured at wavelengths from 1.1 μm to 1.3 μm. The interferometers in Figures 13A and 13C had an effective wedge angle of 0.1 degrees, while the interferometers in Figures 13B and 13D had an effective wedge angle of 0.2 degrees. Notably, the fringe pattern becomes more compact as the effective wedge angle increases, while the free spectral range remains almost the same. The reflectance of the partial reflection layer is lower in Figures 13A and 13C than in Figures 13B and 13D. As mentioned above, the sharpness of the interference fringe generally increases with increasing reflectance.

[0094] Figures 14A to 14G show the angular response of the interferometer in Figure 13C along the x-coordinate direction (i.e., the direction in which the phase changes) (i.e., the response as a function of the incident angle of the optical waveform), while Figures 15A to 15C show the angular response along the y-coordinate direction (i.e., the direction in which nominal phase gradient does not exist). As shown in Figures 14A to 14G, when the incident optical waveform is no longer positively incident (90 degrees), one of the fringes blurs, but the other fringes remain almost unchanged. A similar phenomenon is observed in the angular response in the y-coordinate direction, demonstrating that the manufactured interferometer exhibits similar performance even when the incident waveform deviates from positive incidence.

[0095] The fringes shown in Figures 14A–14G generally appear as curves rather than straight lines, suggesting a nonlinear phase shift across the entire metamaterial layer 204. This nonlinear droplet phase shift may be due to wavelength-dependent phase contributions and can be corrected by adjusting the diameter of the pillars along the x-direction of the metamaterial layer 204 as needed. Figure 16 shows a series of images illustrating the fringes measured from the corrected interferometer. As described above, the upper images were acquired at a wavelength of 1.2 μm, and the lower images show interference fringes measured between 1.1 μm and 1.3 μm. The central fringe in the lower image is linear within this wavelength range.

[0096] As described above, the phase shift imparted by the metamaterial layer 204 at wavelength λ is generally at least 2π, making it possible to achieve structural interference conditions. In some embodiments, the phase shift can exceed 2π (e.g., 3π or more, 4π or more, 5π or more, 6π or more, 8π or more, 10π or more, or more). In certain embodiments, a phase shift exceeding 2π can be achieved by positioning two or more interferometers in a common plane, such that reflected light exits one interferometer laterally (i.e., in the x-coordinate direction) and enters the adjacent interferometer from the side. In this configuration, the two interferometers function substantially as a single extended interferometer, providing a phase gradient greater than that of each individual interferometer.

[0097] In some embodiments, a phase shift of more than 2π can be achieved by positioning two or more interferometers adjacent to each other and offset from each other in the z-coordinate direction, resulting in light emitted from one interferometer entering the adjacent interferometer. As described above, by positioning two or more interferometers in this manner, the overall phase shift can exceed 2π (e.g., 3π or more, 4π or more, 5π or more, 6π or more, 8π or more, 10π or more, or more). Whether positioned in a common plane or offset in the z-coordinate direction, a phase shift of more than 2π can be achieved by using multiple interferometers (e.g., two or more, three or more, four or more, five or more, six or more, eight or more, ten or more, or more). When multiple interferometers are used in an apparatus, it should be noted that the interferometers may have the same configuration or may have some or all different configurations. For example, some or all of the metamaterial layers of multiple interferometers may differ in any of the properties of these layers described herein.

[0098] While the above example was designed assuming operation at a central wavelength of 1.2 μm, it should be noted that the interferometers described herein can generally function (and be designed to operate) over a wide range of wavelengths in the ultraviolet, visible, infrared, and other spectral regions. For example, in the infrared region, an interferometer can provide a phase shift of 2π or more at wavelengths from 0.8 μm to 1.8 μm (e.g., 0.9 μm to 1.7 μm, 1.0 μm to 1.6 μm, and 1.1 μm to 1.5 μm).

[0099] The reflectance of the partial reflective layer (e.g., layer 206) in the interferometer can generally be selected as desired to balance the transmitted light intensity and the sharpness of the interference fringes, as described above. In some embodiments, at the operating wavelength λ, the partial reflective layer has a reflectance of 0.2 to 0.5 (e.g., 0.25 to 0.35, 0.25 to 0.45, 0.3 to 0.4). The interferometer described herein (in particular, the metamaterial layer 204 and the partial reflective layer 206) functions substantially as an optical cavity. The finesse of the optical cavity can be controlled by adjusting the reflectance of the partial reflective layer 206. In some embodiments, the finesse of the optical cavity in the interferometer is at least 2 (e.g., at least 3, at least 4, at least 5, at least 7, at least 10, at least 15, at least 20, or more).

[0100] The lateral extent of the interferometer 200 is substantially determined by the lateral extent of the metamaterial layer 204 (e.g., in the x-direction), and more specifically by the distance over which the metamaterial layer 204 imparts a phase gradient. In some embodiments, the metamaterial layer 204 imparts a continuous phase gradient over a distance of at least 1 mm (e.g., at least 1.2 mm, at least 1.4 mm, at least 1.6 mm, at least 1.8 mm, at least 2.0 mm, at least 2.2 mm, at least 2.4 mm, at least 2.6 mm, at least 2.8 mm, at least 3.0 mm, or more).

[0101] In some embodiments, the phase gradient introduced by the metamaterial layer 204 is a continuous phase gradient. In certain embodiments, the phase gradient may not be entirely continuous. When the metamaterial layer 204 imparts a continuous phase gradient, the continuous phase gradient may be a linear phase gradient. Alternatively, in some embodiments, only a portion of the imparted phase gradient may be linear, with the rest being nonlinear. In certain embodiments, the imparted phase gradient is entirely nonlinear across the entire metamaterial layer 204.

[0102] In some embodiments, the partial reflection layer 206 may include a second metamaterial layer. Typically, but not always, the second metamaterial layer may have different properties from the metamaterial layer 204 and may be in contact with the metamaterial layer 204 or spatially separated from it. The second metamaterial layer may be placed on a second substrate (e.g., a substrate different from substrate 202), and the second substrate may be positioned such that the two metamaterial layers are either in contact in the z-coordinate direction or spaced apart from each other. If a gap exists between the metamaterial layers, the gap may be at least partially filled with air. Alternatively, or in addition, the gap may be partially or completely filled with one or more additional layers of material. In some embodiments, the additional layers of material function as index-matching layers, having a refractive index intermediate between the refractive indices of the two metamaterial layers to reduce unwanted reflections within the interferometer 200.

[0103] In some embodiments, the partial reflective layer 206 does not extend across the entire surface of the substrate 202, as shown in Figure 2A. Instead, the partial reflective layer 206 may extend only to a portion of the substrate 202 in the x-coordinate direction, forming an aperture that substantially allows the optical waveform 130 to be introduced into the interferometer 200. In certain embodiments, the metamaterial layer 204 may also not extend below the aperture to allow the introduction of the optical waveform 130.

[0104] In certain embodiments, the interferometer 200 may include one or more reference markers to facilitate the alignment of the interferometer within the optical system. The reference markers can generally take various forms, such as crosses, triangles, circles, or other geometric shapes. The reference markers may be applied by etching, lithography, or any other technique compatible with the manufacturing process described herein. The reference markers may be applied to one or more of the substrate 202 and layers 204, 206, 208, and 210.

[0105] As described above, the metamaterial layer 204 is typically fabricated as a series of repeating structures on the substrate 202. These structures can be formed from a variety of materials, not limited to Si and TiO2. Generally, any optical material with appropriate refractive index properties can be used for the repeating structures.

[0106] The repeating structure can be implemented in various shapes. In some embodiments, the structure is cylindrical. In certain embodiments, the structure is prismatic (e.g., square prism, cuboid prism, or other geometric prismatic shapes such as pentagonal, hexagonal, or octagonal prisms).

[0107] In some embodiments, the structures are located in a regular arrangement within the metamaterial layer 204, and the spacing between adjacent structures is uniform in both the x and y directions. In certain embodiments, the spacing in the x and y directions is uniform, but the uniform spacing in the x direction and the uniform spacing in the y direction are different. In some embodiments, the spacing between some structures differs in either the x direction and / or the y direction or both. As described above, the spacing in the x direction between at least some structures may be varied to compensate for wavelength-dependent phase contributions.

[0108] Generally, the structures may have a uniform height in the z-direction, or some structures may have different heights than others. In some embodiments, the average height of the structures in the z-direction is between 0.2 μm and 1.5 mm (for example, any range of 0.5 μm to 1.3 mm, 0.5 μm to 1.0 mm, 0.7 μm to 1.1 mm, or 0.2 μm to 1.5 mm).

[0109] The cross-sectional dimensions of the structures can generally be adjusted to control the reflectivity of the metamaterial layer 204. The cross-sectional dimensions and shapes of the structures may all be the same in some embodiments, or in certain embodiments, at least some structures may have different cross-sectional shapes and / or dimensions from others. For example, in some embodiments, the average maximum cross-sectional dimensions of the structures are 50 nm to 1 mm (e.g., any range of 100 nm to 800 μm, 200 nm to 500 μm, 200 nm to 100 μm, 200 nm to 1 μm, 200 nm to 900 nm, 200 nm to 800 nm, 200 nm to 700 nm, 200 nm to 600 nm, 200 nm to 500 nm, or 50 nm to 1 mm).

[0110] The structure can generally be formed from a material having a desired refractive index, but typically the refractive index of the structure is higher than that of the substrate 202. For example, in some embodiments, the refractive index of the material forming the structure is 3.0 to 4.0.

[0111] In certain embodiments, the structure of the metamaterial layer 204 is embedded in a second material, which may have a lower refractive index. For example, in some of the examples described above, the structure was embedded in PMMA. More generally, a variety of materials can be used as the embedding material, including PMMS, PDMS, polycarbonate, polyacrylic, polystyrene, polyester, fluoropolymers such as cyclic olefin polymers (COP) and CYTOP, and nonpolymer materials such as glass, fused silica, quartz, and sapphire.

[0112] When the structure is embedded in a second material within the metamaterial layer 204, in some embodiments the second material may have a refractive index of 1.0 to 2.0 at the operating wavelength λ. In some embodiments the refractive index difference between the structure and the second material is 1.5 to 2.5 (e.g., any range of 1.7 to 2.3, 1.8 to 2.2, or 1.5 to 2.5).

[0113] In some embodiments, the metamaterial layer 204 may include a plurality of repeating structures, each of which may have any of the properties described herein, and each of which repeating structures differs from the others in at least one property. Examples of properties that differ among the plurality of repeating structures include shape, cross-sectional dimensions, height, the material from which the repeating structures are manufactured, and the spacing between adjacent structures within each group of structures.

[0114] (III. Metamaterial Optical Devices and Systems) The optical devices and systems comprising interferometers described herein may include other metamaterial-based optical elements to provide additional functionality not available in conventional optical systems.

[0115] In conventional optical spectrometers, the wavelength components of an optical waveform are spatially separated by a dispersion element such as a diffraction grating or prism, and a detector is placed at the position where that component is to be measured. In the case of pixel-based detectors such as CCDs, the wavelength is substantially mapped to the pixel position, making it possible to determine the spectral intensity at a specific wavelength. However, in conventional spectrometers, the wavelength components are dispersed along a single spatial dimension, and therefore, the number of wavelengths that can be resolved is limited to the number of available detection elements along that dimension.

[0116] The number of wavelengths that can be resolved can be increased without changing the dimensions of the detector by resolving wavelengths along both spatial dimensions of the two-dimensional detector. Figure 17A shows a schematic diagram of the optical device 1700, which includes the interferometer 200 and metamaterial lens 500 described herein. These two elements have the function of improving the wavelength resolution of the pixel-based detector.

[0117] Lens 500 is manufactured in a manner generally described herein in relation to interferometers, except that at least one of the outer surfaces of lens 500 is curved. In Figure 17A, lens 500 is mounted as a cylindrical lens and is curved along the x-direction, which is the same direction in which the lens imparts a phase gradient through an internal metamaterial layer. An optical waveform 130 is incident on lens 500 and spatially dispersed along the x-direction, and each wavelength component in the waveform 130 is focused by lens 500 into a line extending in the y-direction.

[0118] The dispersed and focused wavelength components are then incident on the interferometer 200, which is positioned such that a phase change occurs in the y-coordinate direction. In this way, each linearly focused wavelength component is further spatially dispersed in the y-direction, and additional wavelength resolution is obtained within each spectral component.

[0119] The combined effect of the interferometer 200 and lens 500 can significantly increase the number of wavelengths that can be independently measured by a pixel-based detector positioned to receive spatially separated wavelength components from the interferometer 200.

[0120] Figures 17B and 17C are graphs showing the effect of lens 500 alone. As shown in Figure 17B, for an optical waveform consisting of a single wavelength λ0, lens 500 focuses the light into a single line extending in the y direction. As shown in Figure 17C, for an optical waveform consisting of multiple wavelength components, lens 500 disperses the components along the x direction and focuses each dispersed component into a line extending along the y direction.

[0121] Figures 17D and 17E are graphs showing the combined effect of lens 500 and interferometer 200. In Figure 17D, the spectral contribution at wavelength λ0-fλ (fλ≪λ0) is separated from the spectral contribution at λ0 by a phase gradient applied along the y-direction by interferometer 200. As shown in Figure 17E, for optical waveform 130 containing multiple wavelength components, each component focused on a line extending along the y-direction is further spatially dispersed by interferometer 200, resulting in wavelength resolution along both the x and y directions of the pixel-based detector.

[0122] Figure 17F is a graph showing the transmittance map of device 1700. This map shows d=10μm and r 2 The calculation was performed assuming =0.85. The fringe length along the y-direction is approximately 50 μm. Assuming a pixel-based detector with a length of 3 mm along the y-direction, the number of resolved wavelengths increases by 3 mm / 50 μm = 60 times compared to a device without interferometer 200.

[0123] Another example of a metamaterial-based optical system is shown in Figure 18A. System 1800 includes the interferometer 200 described herein and a metamaterial-based prism 600. The prism 600 consists of a substrate and a metamaterial layer 204 formed on its surface. The metamaterial layer of the prism 600 may generally have any of the properties of the metamaterial layers described herein and may be manufactured in a manner similar to that described herein.

[0124] Generally, the metamaterial layer of prism 600 imparts a nominally linear phase gradient to the light incident on the prism. When prism 600 is combined with interferometer 200, system 1800 separates the incident light in a way that neither element alone could achieve. As shown in Figure 18A, the light waveform 130 is incident on interferometer 200, and the phase gradient imparted by the interferometer is positioned along the y-direction, so the wavelength components of the light waveform 130 are dispersed in the y-direction. Figure 18B is a graph showing the transmission spectrum of an example of interferometer 200, designed to produce three peaks within the operating band of the free spectral range (indicated by the arrows in Figure 18B). Returning to Figure 18A, these three peaks are separated in the x-direction by the phase imparted by prism 600. This extends the operating range of interferometer 200 beyond a single free spectral range.

[0125] Another example of optical system 1850 is shown in Figure 18C. System 1850 includes both an interferometer 200 and a prism 600, similar to Figure 18A. However, in Figure 18C, the magnitude and direction of the phase imparted by the prism 600 vary in the x-direction. As a result, the separated wavelength components generated by the interferometer 200 from the incident light waveform 130 are deflected at different angles and positions in the x-direction. Thus, system 1850 makes it possible to measure and control individual wavelength components within a single free spectral range of the interferometer 200. For example, by selecting an appropriate variable phase for the metamaterial layer of the prism 600, wavelength components incident on the prism 600 can be spatially separated significantly in the x-direction at the far field of the prism 600, thereby effectively amplifying the output of the interferometer 200.

[0126] (Other embodiments) While this disclosure describes specific embodiments, these should not be construed as limiting the scope of this disclosure, but rather as descriptions of features in those specific embodiments. Features described in the context of individual embodiments can generally be implemented in combination in a single embodiment. Conversely, various features described in the context of a single embodiment can be implemented individually or in any suitable subcombination in multiple embodiments. Furthermore, even if features are described above as existing in a particular combination and are initially claimed as such, it is generally possible to exclude one or more features from the claimed combination, and the claimed combination can be directed towards a subcombination or a variation of a subcombination.

[0127] It should be understood that, in addition to the embodiments expressly disclosed herein, various modifications to the embodiments described can be made without departing from the spirit and scope of this disclosure. Accordingly, other embodiments are also included in the claims below.

Claims

1. An optical system, It comprises a first optical element, The first optical element is, The first substrate and A partial reflective coating disposed on the first surface of the first substrate, A first metamaterial layer is disposed on or adjacent to the first surface of the first substrate and has a structure that defines a continuous phase gradient along a first direction parallel to the first surface of the first substrate, wherein the change in the magnitude of the phase along the first direction is at least 2 at a wavelength λ, A partial reflective layer disposed on or adjacent to the first metamaterial layer and on the side of the metamaterial layer opposite to the first substrate, Equipped with, It comprises a second optical element, The second optical element is, The second circuit board, A second metamaterial layer is disposed on or adjacent to the first surface of the second substrate and has a structure that defines a continuous phase gradient along a second direction parallel to the first surface of the first substrate, Equipped with, The first and second optical elements are oriented such that the first and second directions are at a non-zero angle. Optical system.

2. The optical system according to claim 1, wherein the change in the magnitude of the phase along the first direction is at least 2 at a wavelength λ from 0.8 μm to 1.8 μm.

3. The first substrate is made of SiO ２ Al₂O ３ The optical system according to claim 1, which is formed from at least one material selected from the group consisting of , and ZnS.

4. The optical system according to claim 1, wherein the partial reflection coating comprises Si.

5. The optical system according to claim 1, wherein the partial reflection coating has a reflectance of 20% to 50% at a wavelength λ.

6. The optical system according to claim 5, wherein the partial reflection coating has a reflectance of 25% to 35% at a wavelength λ.

7. The optical system according to claim 1, wherein the continuous phase gradient along the first direction extends over a distance of at least 1 mm along the first direction.

8. The optical system according to claim 7, wherein the continuous phase gradient along the first direction extends over a distance of at least 2 mm along the first direction.

9. The optical system according to claim 1, wherein the continuous phase gradient along the first direction is a linear phase gradient.

10. The optical system according to claim 1, wherein a portion of the continuous phase gradient along the first direction is a linear phase gradient.

11. The optical system according to claim 1, wherein the partial reflective layer is a coating disposed on the surface of the first metamaterial layer.

12. The optical system according to claim 1, wherein the partial reflective layer comprises a third metamaterial layer different from the first metamaterial layer.

13. The optical system according to claim 12, wherein the third metamaterial layer is in contact with the first metamaterial layer.

14. The optical system according to claim 12, wherein the third metamaterial layer is disposed on a third substrate different from the first substrate.

15. The optical system according to claim 14, wherein the third substrate is positioned relative to the first substrate such that the first metamaterial layer and the third metamaterial layer are in contact with each other.

16. The optical system according to claim 14, wherein the third substrate is positioned relative to the first substrate such that a gap exists between the first metamaterial layer and the third metamaterial layer.

17. The optical system according to claim 16, wherein the gap is at least partially filled with air.

18. The optical system according to claim 16, wherein at least one additional layer is located within the gap.

19. The optical system according to claim 18, wherein the at least one additional layer comprises a solid material.

20. The optical system according to claim 18, wherein the at least one additional layer comprises an index matching layer whose refractive index at wavelength λ is between the refractive index of the first metamaterial layer at wavelength λ and the refractive index of the third metamaterial layer at wavelength λ.

21. The optical system according to claim 1, further comprising an anti-reflective coating disposed on a second surface of the first substrate opposite to the first surface.

22. The optical system according to claim 21, wherein the anti-reflective coating has a reflectance of 5% or less at a wavelength λ.

23. The optical system according to claim 1, wherein the partial reflective coating, the first metamaterial layer, and the partial reflective layer define an optical cavity, and the finesse of the optical cavity is at least 2.

24. The optical system according to claim 23, wherein the finesse is at least 3.

25. The optical system according to claim 24, wherein the finesse is at least 10.

26. The optical system according to claim 1, wherein the transmission efficiency of the optical system is at least 70%.

27. The optical system according to claim 26, wherein the transmission efficiency is at least 80%.

28. The optical system according to claim 1, wherein the first metamaterial layer is located on or adjacent to a first region of the first surface of the first substrate, and the apertures where the first metamaterial layer is absent are located on or adjacent to a second region of the first surface of the first substrate, and the first and second regions of the first surface of the first substrate do not overlap.

29. The optical system according to claim 28, wherein the partial reflective coating is not located in the second region of the first surface that forms the aperture.

30. The optical system according to claim 28, wherein the partial reflective layer is not located on or adjacent to the second region of the first surface forming the aperture.

31. The optical system according to claim 1, wherein at least one of the group consisting of the partial reflective coating, the first metamaterial layer, and the partial reflective layer comprises a reference marker.

32. The optical system according to claim 1, wherein the first metamaterial layer comprises a plurality of repeating structures formed from a first material and embedded in a second material.

33. The optical system according to claim 32, wherein the first material comprises Si.

34. The first material is TiO ２ The optical system according to claim 32, comprising:

35. The optical system according to claim 32, wherein the plurality of repeating structures include cylindrical structures.

36. The optical system according to claim 32, wherein the plurality of repeating structures comprises a rectangular parallelepiped prism structure.

37. The optical system according to claim 32, wherein the average height of the repeating structure in the first metamaterial layer is 0.2 μm to 1.5 mm, measured in a direction perpendicular to the first surface of the first substrate.

38. The optical system according to claim 37, wherein the average height of the repeating structure in the first metamaterial layer is 0.5 μm to 1.0 mm, measured in a direction perpendicular to the first surface of the first substrate.

39. The optical system according to claim 32, wherein the average maximum cross-sectional dimension of the repeating structure in the first metamaterial layer is 50 nm to 1 mm, measured in a direction parallel to the first surface of the first substrate.

40. The optical system according to claim 39, wherein the average maximum cross-sectional dimension of the repeating structure in the first metamaterial layer is 200 nm to 600 nm, measured in a direction parallel to the first surface of the first substrate.

41. The optical system according to claim 32, wherein the refractive index of the first material at a wavelength λ is 3.0 to 4.

0.

42. The optical system according to claim 32, wherein the refractive index of the second material at wavelength λ is 1.0 to 2.

0.

43. The optical system according to claim 32, wherein the difference in refractive index between the first material and the second material at wavelength λ is 1.5 to 2.

5.

44. The optical system according to claim 32, wherein the second material comprises at least one material selected from the group consisting of glass, fused silica, quartz, and sapphire.

45. The optical system according to claim 32, wherein the second material comprises at least one polymer material.

46. The optical system according to claim 45, wherein the at least one polymer material is selected from the group consisting of polymethyl methacrylate (PMMA), polycarbonate (PC), polystyrene (PS), polyester (PE), and cyclic olefin polymer (COP).

47. The optical system according to claim 32, wherein the plurality of repeating structures are a first plurality of repeating structures, and the first metamaterial layer further comprises a second plurality of repeating structures formed from the first material and embedded in the second material, wherein the second plurality of repeating structures are different from the first plurality of repeating structures.

48. The optical system according to claim 47, wherein the second plurality of repeating structures have a different cross-sectional shape from the first plurality of repeating structures.

49. The optical system according to claim 47, wherein the average height of the second plurality of repeating structures is measured in a direction perpendicular to the first surface of the first substrate and differs from the average height measured in a direction perpendicular to the first direction of the first plurality of repeating structures.

50. The optical system according to claim 47, wherein the average maximum dimension of the second plurality of repeating structures is measured in a direction parallel to the first surface of the first substrate and differs from the average maximum dimension of the first plurality of repeating structures measured in a direction parallel to the first surface.

51. The optical system according to claim 32, wherein the first metamaterial layer further comprises a plurality of repeating structures formed from a third material and embedded in the second material, the third material being different from the first material.

52. The third material is Si and TiO ２ The optical system according to claim 51, comprising at least one material selected from the group consisting of the following.

53. The optical system according to claim 51, wherein the plurality of repeating structures formed from the first material and the plurality of repeating structures formed from the third material have a common cross-sectional shape.

54. The optical system according to claim 51, wherein the plurality of repeating structures formed from the first material and the plurality of repeating structures formed from the third material have different cross-sectional shapes.

55. The optical system according to claim 51, wherein the average height of the plurality of repeating structures formed from the first material is measured in a direction perpendicular to the first surface of the first substrate and is the same as the average height of the plurality of repeating structures formed from the third material measured in a direction perpendicular to the first surface.

56. The optical system according to claim 51, wherein the average height of the plurality of repeating structures formed from the first material is measured in a direction perpendicular to the first surface of the first substrate and differs from the average height of the plurality of repeating structures formed from the third material measured in a direction perpendicular to the first surface.

57. The optical system according to claim 51, wherein the average maximum dimension of the plurality of repeating structures formed from the first material is measured in a direction parallel to the first surface of the first substrate and is the same as the average maximum dimension of the plurality of repeating structures formed from the third material measured in a direction parallel to the first surface.

58. The optical system according to claim 51, wherein the maximum dimension of the plurality of repeating structures formed from the first material is measured in a direction parallel to the first surface of the first substrate and differs from the average maximum dimension of the plurality of repeating structures formed from the third material measured in a direction parallel to the first surface.

59. The optical system according to claim 1, wherein the second optical element is a prism.

60. The optical system according to claim 1, wherein the second optical element is a spectral dispersion window.

61. The optical system according to claim 1, wherein the second optical element comprises a second surface that is not parallel to the first surface of the second optical element.

62. The optical system according to claim 1, wherein the second optical element comprises a second surface parallel to the first surface of the second optical element.

63. The optical system according to claim 1, wherein the second optical element is a transmissive element.

64. The optical system according to claim 1, wherein the second optical element is a reflective element.

65. The optical system according to claim 1, wherein the second optical element has a second surface opposite to the first surface, and the first and second surfaces of the second optical element are planar.

66. The optical system according to claim 1, wherein the change in the magnitude of the phase along the second direction is at least 2 at a wavelength λ from 0.8 μm to 1.8 μm.

67. The second substrate is SiO ２ Al₂O ３ The optical system according to claim 1, which is formed of at least one material selected from the group consisting of , and ZnS.

68. The optical system according to claim 1, wherein the continuous phase gradient along the second direction extends over a distance of at least 1 mm along the second direction.

69. The optical system according to claim 68, wherein the continuous phase gradient along the second direction extends over a distance of at least 2 mm along the second direction.

70. The optical system according to claim 1, wherein the continuous phase gradient along the second direction is a linear phase gradient.

71. The optical system according to claim 1, wherein a portion of the continuous phase gradient along the second direction is a linear phase gradient.

72. The optical system according to claim 1, wherein the second metamaterial layer comprises a plurality of repeating structures formed from the first material and embedded in the second material.

73. The optical system according to claim 72, wherein the first material comprises Si.

74. The first material is TiO ２ The optical system according to claim 72, comprising:

75. The optical system according to claim 72, wherein the plurality of repeating structures include cylindrical structures.

76. The optical system according to claim 72, wherein the plurality of repeating structures comprises a rectangular prism structure.

77. The optical system according to claim 72, wherein the average height of the repeating structure in the second metamaterial layer is 0.2 μm to 1.5 mm, measured in a direction perpendicular to the first surface of the second substrate.

78. The optical system according to claim 77, wherein the average height of the repeating structure in the second metamaterial layer is 0.5 μm to 1.0 mm, measured in a direction perpendicular to the first surface of the second substrate.

79. The optical system according to claim 72, wherein the average maximum cross-sectional dimension of the repeating structure in the second metamaterial layer is 50 nm to 1 mm, measured in a direction parallel to the first surface of the second substrate.

80. The optical system wherein the average maximum cross-sectional dimension of the repeating structure in the second metamaterial layer is 200 nm to 600 nm as described in claim 79, measured in a direction parallel to the first surface of the second substrate.

81. The optical system according to claim 72, wherein the refractive index of the first material at wavelength λ is 3.0 to 4.

0.

82. The optical system according to claim 72, wherein the refractive index of the second material at wavelength λ is 1.0 to 2.

0.

83. The optical system according to claim 72, wherein the difference in refractive index at wavelength λ between the first material and the second material is 1.5 to 2.

5.

84. The optical system according to claim 72, wherein the second material comprises at least one material selected from the group consisting of glass, fused silica, quartz, and sapphire.

85. The optical system according to claim 72, wherein the second material comprises at least one polymer material.

86. The optical system according to claim 85, wherein the at least one polymer material is selected from the group consisting of polymethyl methacrylate (PMMA), polycarbonate (PC), polystyrene (PS), polyester (PE), and cyclic olefin polymer (COP).

87. The optical system according to claim 72, wherein the plurality of repeating structures are a first plurality of repeating structures, and the second metamaterial layer further comprises a second plurality of repeating structures formed from the first material and embedded in the second material, wherein the second plurality of repeating structures are different from the first plurality of repeating structures.

88. The optical system according to claim 87, wherein the second plurality of repeating structures have a different cross-sectional shape from the first plurality of repeating structures.

89. The optical system according to claim 87, wherein the average height of the second plurality of repeating structures is measured in a direction perpendicular to the first surface of the second substrate and differs from the average height measured in a direction perpendicular to the first direction of the first plurality of repeating structures.

90. The optical system according to claim 87, wherein the average maximum dimension of the second plurality of repeating structures is measured in a direction parallel to the first surface of the second substrate and differs from the average maximum dimension of the first plurality of repeating structures measured in a direction parallel to the first surface.

91. The optical system according to claim 72, wherein the second metamaterial layer further comprises a plurality of repeating structures formed from a third material and embedded in the second material, the third material being different from the first material.

92. The third material is Si and TiO ２ The optical system according to claim 91, comprising at least one material selected from the group consisting of the following.

93. The optical system according to claim 91, wherein the plurality of repeating structures formed from the first material and the plurality of repeating structures formed from the third material have a common cross-sectional shape.

94. The optical system according to claim 91, wherein the plurality of repeating structures formed from the first material and the plurality of repeating structures formed from the third material have different cross-sectional shapes.

95. The optical system according to claim 91, wherein the average height of the plurality of repeating structures formed from the first material is measured in a direction perpendicular to the first surface of the second substrate and is the same as the average height of the plurality of repeating structures formed from the third material measured in a direction perpendicular to the first surface.

96. The optical system according to claim 91, wherein the average height of the plurality of repeating structures formed from the first material is measured in a direction perpendicular to the first surface of the second substrate and differs from the average height of the plurality of repeating structures formed from the third material measured in a direction perpendicular to the first surface.

97. The optical system according to claim 91, wherein the average maximum dimension of the plurality of repeating structures formed from the first material is measured in a direction parallel to the first surface of the second substrate and is the same as the average maximum dimension of the plurality of repeating structures formed from the third material measured in a direction parallel to the first surface.

98. The optical system according to claim 91, wherein the maximum dimensions of the plurality of repeating structures formed from the first material are measured in a direction parallel to the first surface of the second substrate and differ from the average maximum dimensions of the plurality of repeating structures formed from the third material measured in a direction parallel to the first surface.

99. The optical system comprises the one described in any one of claims 1 to 98, The optical system comprises a plurality of first optical elements, The first surface of the first substrate of each first optical element is located in a common plane, and the partial reflection coating, first metamaterial layer, and partial reflection layer of each first optical element define a continuous optical cavity extending in a direction parallel to the common plane. The device wherein the change in the magnitude of the phase along the continuous optical cavity is greater than 2 at wavelength λ.

100. The apparatus according to claim 99, wherein the change in the magnitude of the phase is greater than 8 at wavelength λ.

101. The apparatus according to claim 99, wherein each of the plurality of first optical elements is identical.

102. The apparatus according to claim 99, wherein one or more of the plurality of first optical elements are different from one or more other first optical elements.

103. The apparatus according to claim 102, wherein one or more first metamaterial layers among the plurality of first optical elements are different from the first metamaterial layers of one or more other first optical elements.

104. The apparatus according to claim 103, wherein the change in the magnitude of the phase along a first direction of one or more of the plurality of first optical elements is different from the change in the magnitude of the phase along a first direction of one or more other plurality of first optical elements.

105. The apparatus according to claim 99, wherein the plurality of first optical elements comprises at least four first optical elements.

106. The optical system comprises the one described in any one of claims 1 to 98, The optical system comprises a plurality of first optical elements, The first surface of one or more first substrates among the plurality of first optical elements is offset from the first surface of the first substrate of one or more other first optical elements in a direction perpendicular to the first surface of the first substrate, and as a result, the partial reflection coatings, metamaterial layers, and partial reflection layers of the plurality of first optical elements define a plurality of optical cavities that are offset from each other in an orthogonal direction. The overall change in the magnitude of the phase between the optical cavities is greater than 2 at wavelength λ in the apparatus.

107. The apparatus according to claim 106, wherein at least some of the first surfaces of the first substrates of the plurality of first optical elements are located in a common plane, and the partial reflection coatings, metamaterial layers, and partial reflection layers of at least some of the first optical elements define a continuous optical cavity as a whole that extends in a direction parallel to the common plane.

108. The apparatus according to claim 107, comprising a plurality of continuous optical cavities each extending in different planes, wherein each continuous optical cavity is offset from other continuous optical cavities in the apparatus.

109. The apparatus according to claim 106, wherein the overall change in the magnitude of the phase between the optical cavities is greater than 8 at wavelength λ.

110. The apparatus according to claim 106, wherein each of the plurality of first optical elements is identical.

111. The apparatus according to claim 106, wherein one or more of the plurality of first optical elements are different from one or more other first optical elements.

112. The apparatus according to claim 111, wherein one or more metamaterial layers among the plurality of first optical elements are different from the metamaterial layers of one or more other first optical elements.

113. The apparatus according to claim 112, wherein the change in the magnitude of the phase along a first direction of one or more of the plurality of first optical elements is different from the change in the magnitude of the phase along a first direction of one or more other plurality of first optical elements.

114. The apparatus according to claim 108, wherein each continuous optical cavity has a phase magnitude change along the continuous optical cavity, and the phase magnitude change of one or more of the multiple continuous optical cavities is different from the phase magnitude change of the other one or more of the multiple continuous optical cavities.

115. The apparatus according to claim 106, wherein the plurality of first optical elements comprises at least four first optical elements.

116. A method for generating an output optical waveform, A step of preparing the optical system according to any one of claims 1 to 98, A step of generating an input optical waveform containing multiple wavelength components, The step of transmitting the input optical waveform through the optical system to generate an output optical waveform is included, At least some of the wavelength components of the input light waveform pass through the optical system multiple times, passing through at least one metamaterial layer of the optical system. The output optical waveform comprises spatially separated wavelength components. A method for generating output light waveforms.

117. A method for generating an output optical waveform, A step of preparing the apparatus according to any one of claims 99 to 115, A step of generating an input optical waveform containing multiple wavelength components, The step of transmitting the input optical waveform through the device to generate an output optical waveform is included, At least some of the wavelength components of the input optical waveform pass through the apparatus multiple times, passing through at least one metamaterial layer of the apparatus. The output optical waveform comprises spatially separated wavelength components. A method for generating output light waveforms.

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