Photonic integrated circuit (PIC) comprising a waveguide comprising epsilon-near-zero (ENZ) material cladding

Abstract

A photonic integrated circuit (PIC) is disclosed. The PIC comprises: a planner substrate; and at least one waveguide, comprising: an air core having a polygonal cross-section; and a cladding comprising Epsilon-Near -Zero (ENZ) material having a refractive index lower than 1, covering the air core from all side faces, wherein the cladding is attached to the planner substrate.

Description

PHOTONIC INTEGRATED CIRCUIT (PIC) COMPRISING A WAVEGUIDE COMPRISING EPSILON-NEAR-ZERO (ENZ) MATERIAL CLADDING CROSS REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of priority of Israeli Patent Application No.319950, filed 27 March 2025. The contents of the above application are all incorporated by reference as if fully set forth herein in its entirety.FIELD OF THE INVENTION

[0002] The present invention relates generally to waveguides for high power or long wavelengths lasers. More specifically, the present invention relates to a waveguide comprising Epsilon-Near-Zero (ENZ) material cladding.BACKGROUND OF THE INVENTION

[0003] Waveguides are fundamental components in various optical and electromagnetic applications, including telecommunications, radar systems, and high-power laser systems. They guide electromagnetic waves by confining them within a specific structure, thereby minimizing energy loss and enhancing signal integrity. Traditional waveguides are typically made from materials such as metals or dielectrics, which have well-defined electromagnetic properties.

[0004] Guiding waves at optical frequencies relies on total internal reflection, which requires a medium where the wave propagates to have a higher refractive index than its surroundings. This is typically achieved in fibers, with a core that has a slightly larger index than the cladding, or in photonic integrated circuits (PICs) where the waveguide core is constructed from a higher index material than the surrounding media. Although there are many advantages to guiding light is such a material, it also possesses many downsides, such as intrinsic losses of the material, wavelength dependence, undesired nonlinear effects, and optical damage threshold limiting high power implications. One optional solution may include a waveguide having an air core (e.g., hollow core). Air core waveguides, as known in the art, are constructed from photonic crystals, which are hard to fabricate, cannot be connected to typical optical elements, and are lossy and expensive. These issues present challenges in developing efficient and practical hollow core waveguides.

[0005] High-power laser applications demand waveguides that can handle intense electromagnetic fields without significant losses or damage. Conventional materials often fall short in these high-power scenarios due to limitations in their thermal and electromagnetic properties.

[0006] In some applications, for example, but not limited to military applications, 4 pm laser sources are crucial for infrared countermeasures against heat-seeking missiles. When a quantum cascade laser (QCL) is used, the QCL needs to be protected from the environment. Therefore, the QCL is located at a distal point from where the laser is to be fired. The laser beam generated by the QCL is directed through a set of optical elements (e.g., mirrors, prisms, lenses, etc.). These elements are bulky and need to be carefully aligned for the laser beam to work properly. Guiding the light to the output with fibers and photonic integrated circuits (PICs) is extremely beneficial since it circumvents the problems that arise in free space. However, the materials that can be used for this wavelength range (3-5pm) are scarce and cannot withstand relatively high powers (typically <500mW).

[0007] In addition, in other applications requiring high powers and / or narrow bandwidth, such as, coherent beam combining (CBC) for laser defense systems, the use of lasers in PICs is essential. At increased powers (e.g., ~lGW / cm2), nonlinear effects occur, creating unwanted artifacts. Specifically, stimulated Brillouin scattering widens the bandwidth of the light beam, reducing the coherence length and preventing the CBC. In addition, the use of higher power is limited by the optical damage threshold of the fibers at ~50GW / cm2. Furthermore, in CBC all the beam channels must be in phase, hence each channel requires a phase modulator connected in a close loop control system in order to fix the phase noise gained from the environment. In current technologies, this is done by electro-optical (EO) based phase modulators and can only be implemented for low powers (< 1W @ 1064nm) and suffers from coupling losses.

[0008] Recent advancements in material science have introduced Epsilon-Near-Zero (ENZ) materials and their closely related near zero index (NZI) materials, which at certain wavelengths, the real part of their refractive index is smaller than 1. Recently, transparent conducting oxides (TCOs) have been shown to exhibit these properties across a large spectral range from the visible to mid-infrared (MIR) wavelengths. This makes them ideal candidates to use as claddings to an air waveguide. These materials offer higher refractive index contrast than photonic crystals, which allows for better confinement and results withreduced losses. In addition, they are uniform bulk materials, which eliminate the problems arising from the complicated geometry imposed by the photonic crystals.

[0009] The present invention relates to a waveguide structure that incorporates ENZ material cladding to enhance the performance of high-power laser systems. By leveraging the distinctive properties of ENZ materials, this waveguide design aims to achieve superior confinement of electromagnetic waves, reduce energy loss, and improve thermal management, thereby addressing the limitations of traditional waveguide materials.SUMMARY OF THE INVENTION

[0010] Some aspects of the invention may be directed to a photonic integrated circuit (PIC) comprising: a planner substrate; and at least one waveguide, comprising: an air core having a polygonal cross-section; and a cladding comprising Epsilon-Near-Zero (ENZ) material having a refractive index lower than 1, covering the air core from all side faces, wherein the cladding is attached to the planner substrate.

[0011] In some embodiments, the PIC may further comprise two or more waveguides combined into a single waveguide, all cladded by the ENZ material. In some embodiments, the PIC may further comprise a single waveguide split into two or more waveguides, all cladded by the ENZ material. In some embodiments, the substrate may be a material adherable to the ENZ material.

[0012] In some embodiments, the substrate may include a multilayered structure comprising at least three layers two dielectric layers attached to a metallic layer from both sides. In some embodiments, the two dielectric layers may include the same dielectric material. In some embodiments, the PIC may further include a tunable electric power source in electrical communication with the metallic layer, and at least one of: (a) a metallic layer attached to the cladding; and (b) a grounding connection electrically connected to the cladding.

[0013] In some embodiments, the PIC may further include a tunable electric power source in electrical communication with the cladding.

[0014] In some embodiments, the PIC may further include a controller configured to control the tunable electric power source to provide a voltage to the cladding. In some embodiments, the controller may further be configured to determine the voltage based on a required refractive index of the cladding. In some embodiments, the controller may further be configured to: receive an indication of a phase change of a laser beam traveling in the atleast one air core; and determine the required refractive index of the cladding based on the indication. In some embodiments, the controller may further be configured to: receive a required phase change; determine the voltage based on a phase change; and control a provision of the determined voltage.

[0015] In some embodiments, the air core may have at least one dimension of between 1 to 5 pm. In some embodiments, a thickness of the cladding may be at least 10 nm. In some embodiments, the ENZ material may be a transparent conducting oxide (TCOs).

[0016] In some embodiments, the ENZ material may be doped with a dopant selected to provide to the cladding a specific refractive index at a required wavelength or a specific ENZ wavelength. In some embodiments, selecting the dopant may include at one least of, selecting the type of dopant, and a concentration of the dopant. In some embodiments, the specific refractive index may be for ENZ wavelength of between 1 to 5 pm. In some embodiments, the cladding may be heat-treated to result in a specific refractive index at a required wavelength or a specific ENZ wavelength. In some embodiments, the specific refractive index at the required wavelength or the specific ENZ wavelength may determine at least one of, a temperature and a duration of a heat treatment. In some embodiments, the specific refractive index may be for a wavelength of between 1 to 5 pm.

[0017] In some embodiments, the polygonal cross-section may be selected from, a rectangular cross-section, a hexagonal cross-section, an octagonal cross-section, and a triangular cross-section.

[0018] Some additional aspects of the invention may be directed to a method of controlling a refractive index of a waveguide included in the PIC according to some embodiments of the invention. The method may include: receiving a required refractive index; determining a voltage level based on the required refractive index; and controlling a tunable electric power source to provide the voltage level to a cladding of the waveguide, wherein the cladding comprises Epsilon-Near-Zero (ENZ) material having a refractive index lower than 1, covering a polygonal air core from all side faces.

[0019] In some embodiments, the method may further include receiving an indication of a phase change of a laser beam traveling in the air core; and determining the required refractive index of the cladding based on the indication.

[0020] In some embodiments, the method may further include receiving a required phase change; and determining the voltage level based on a phase change.BRIEF DESCRIPTION OF THE DRAWINGS

[0021] The subject matter regarded as the invention is particularly pointed out and distinctly claimed in the concluding portion of the specification. The invention, however, both as to organization and method of operation, together with objects, features, and advantages thereof, may best be understood by reference to the following detailed description when read with the accompanying drawings in which:

[0022] Figs. 1A and IB are illustrations of waveguides according to some embodiments of the invention;

[0023] Figs. 2 A, 2B, and 2C are illustrations of some nonlimiting examples of crosssections of waveguides according to some embodiments of the invention;

[0024] Fig. 3A is an illustration of a photonic integrated circuit (PIC) according to some embodiments of the invention;

[0025] Fig. 3B includes a perspective view and two cross section views of a PIC comprising a multi-mode interference (MMI) structure according to some embodiments of the invention;

[0026] Fig. 3C includes a perspective view and two cross section views of a PIC comprising a Y-junction structure according to some embodiments of the invention;

[0027] Figs. 4A and 4B are illustrations of PICs with multilayered substrates according to some embodiments of the invention;

[0028] Fig. 5 is a block diagram depicting a computing device which may be included in a waveguide system according to some embodiments of the invention; and

[0029] Fig. 6 is a flowchart of a method of controlling a refractive index of a waveguide according to some embodiments of the invention;

[0030] Fig. 7 is a graph showing optical properties of Indium Tin Oxide (ITO) as a function of wavelength according to some embodiments of the invention;

[0031] Figs. 8 A and 8B show graphs of the refractive index and the effective index of ITO as a function of the voltage according to some embodiments of the invention;

[0032] Figs. 9A and 9B show the propagation loss and the modulation length (n phase shift) a function of the voltage in a waveguide according to some embodiments of the invention;

[0033] Fig. 10 shows simulations of TE and TM modes in a waveguide according to some embodiments of the invention; and

[0034] Figs. 11 and 12 are illustrations of two fabrication methods for making a PIC according to some embodiments of the invention.

[0035] It will be appreciated that for simplicity and clarity of illustration, elements shown in the figures have not necessarily been drawn to scale. For example, the dimensions of some of the elements may be exaggerated relative to other elements for clarity. Further, where considered appropriate, reference numerals may be repeated among the figures to indicate corresponding or analogous elements.DETAILED DESCRIPTION OF THE PRESENT INVENTION

[0036] One skilled in the art will realize the invention may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The foregoing embodiments are therefore to be considered in all respects illustrative rather than limiting of the invention described herein. Scope of the invention is thus indicated by the appended claims, rather than by the foregoing description, and all changes that come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein.

[0037] Conventional waveguides face several limitations when used in high-power laser applications. These limitations may include optical damage thresholds that restrict the maximum power levels that can be transmitted through the waveguide without causing material degradation. In some cases, conventional waveguides may experience nonlinear effects at high power levels, which can distort the transmitted optical signals and reduce system performance.

[0038] Some aspects of the invention may be directed to a waveguide having an air core cladded with ENZ material. These waveguides may have several advantages over prior art waveguides, such as, high transparency at broad wavelength regimes, easy integration with other optical devices, and high optical damage thresholds. The ENZ cladding may include TCOs which include doped semiconductors. In some embodiments, the optical properties of the ENZ materials may depend on applied voltage. Therefore, an ultrafast modulator based on such waveguide may be realized by applying a voltage across the cladding material, causing a refractive index change, which in turn changes the effective index of the guidedmode. This may result in either a phase modulation or amplitude modulation in a single homogeneous medium.

[0039] Some additional aspects of the invention disclose photonic integrated circuit that may address these limitations through the use of waveguides having an air core and ENZ material cladding with a refractive index lower than 1. In some embodiments, this configuration may enable light confinement in the air core, which can eliminate materialbased optical damage since the light propagates primarily through air rather than through solid materials. The air core design may reduce nonlinear effects that typically occur in solid materials at high power levels.

[0040] In some embodiments, the ENZ material cladding approach may enable operation across a wide wavelength range from near to mid-infrared. The air core may provide transparency across this broad spectral range without the material absorption limitations that affect conventional solid-core waveguides. In some cases, this wide wavelength operation capability may enable the photonic integrated circuit to support diverse applications spanning multiple spectral regions without requiring different waveguide materials for different wavelength ranges.

[0041] A waveguide according to embodiments of the invention may be integrated into a PIC and / or in applications that require high-power lasers at a relatively wide range of wavelengths.

[0042] As used herein, a "high-power laser" refers to a continuous-wave (CW) laser having an output power of at least 1W or a pulsed laser with a peak power of at least lOOkW. For example, a laser beam propagating inside a waveguide according to some embodiments of the invention may have a power of, 1W, 1.5W, 2W, 2.5W, 3W, 4W, 5W, 10W, 20W,50W, 100W, 250W, 500W, 750W, IkW or any value or range in between. In yet another example, a pulsed laser propagating inside a waveguide according to some embodiments of the invention may have a peak power of at least lOOkW, at least 200kW, at least 300kW, at least 500kW, at least 700kW, at least 1MW, at least 5MW, at least 10MW and or any value or range in between.As used herein, a “wide wavelength range” may be defined as a range of 0.5 to 6 pm, for example, 1 to 5 pm.

[0043] As used herein an “ENZ wavelength” refers to the specific wavelength at which a material's permittivity (real part) equals zero. This phenomenon is observed in epsilon-near-zero (ENZ) materials, which have unique optical properties that make them useful in various applications, such as nanophotonics and nonlinear optics.

[0044] In some embodiments, the laser propagating in the waveguide may be in the near to mid infrared (NIR-MIR) wavelengths having a wavelength of between 1 to 5 pm.

[0045] Reference is now made to Figs. 1A and IB which are illustrations of waveguides according to some embodiments of the invention. A waveguide 100 may include an air core 10 having a polygonal cross-section, and a cladding 20 comprising ENZ material having a refractive index lower than 1, covering air core 10 from all side faces. In some embodiments, air core 10 may be rectangular, as illustrated, or may have any other polygonal cross-section as discussed with respect to Figs. 2A, 2B and 2C herein below.

[0046] In some embodiments, the width w and / or high h of the air core may be determined based on the required wavelength and may be between 1 to 5 pm and any value or range in between. A nonlimiting example, for an air core dimensions may be 2 pm by 2 pm.

[0047] In some embodiments, cladding 20 may include an ENZ material. The ENZ material may include, for example several TCOs (e.g., Indium Tin Oxide (ITO), Fluorinedoped Tin Oxide (FTO), Cadmium Oxide (CdO), Aluminum doped Zinc Oxide (AZO), Molybdenum doped CdO (MoCdO), Gallium-doped Zinc Oxide (GZO), Niobium-doped Titanium Dioxide (NTO), etc.), which shown to have a refractive index lower than 1 in the near infrared regime. The TCOs may be doped with dopants and may optionally be heat treated (e.g., annealed) in order to alter the refractive index. The alteration may allow to optimize the refractive index for specific wavelength, or wavelength range within the NIR-MIR range.

[0048] In some embodiments, the optical properties of the TCO cladding 20 (e.g., ITO) can be calculated by characterizing the permittivity using the Drude model which describes metals and degenerately doped semiconductors. In this model, the permittivity dispersion can be expressed by equation (1):

[0049] Where, emis the high frequency permittivity, and y is the free electron damping. tUp is the plasma frequency, given by: jp= jNe2 / e0m*, with N being the charge densityand m* the effective electron mass. Assuming relatively small damping (y « to), it can be seen that when to = a)p, the real part of the permittivity reaches zero. The refractive index is related to the permittivity by: n(to) =(to), therefore when the permittivity approaches zero, so does the refractive index. Since the charge density may vary for different materials, this ENZ behavior may be found over a wide frequency range from UV to MIR. An example of the of the permittivity (real part) and the corresponding refractive index can be seen in Fig. 5 of the Examples Section.

[0050] In some embodiments, the thickness ‘d’ of the cladding is at least 10 nm, for example, 11 nm, 15 nm, 20 nm, 25 nm, 30 nm, 40nm, 50 nm, 75 nm, 100 nm, 150 nm, 200 nm, 250 nm, 500 nm 1 pm, 5 pm, 10 pm. and any value or range in between. Therefore, the total height of the waveguide T may be the Hight of air core 10 and twice the thickness, h+ 2d.Modulation with a dopant

[0051] In some embodiments, the ENZ material may be doped with a dopant selected to provide to the cladding a specific refractive index at a required wavelength or a specific ENZ wavelength. The selection of the dopant may include selecting at least one of, the type of dopant, and the concentration of the dopant. In some embodiments, the specific refractive index is for a wavelength of between 1 to 5 pm.

[0052] Some nonlimiting examples for selecting a dopant and concentration that results in a shifted ENZ wavelength and hence a refractive index n<l for a required wavelength regime are given in tables 1, 2, and 3. Tables 1 and 2 show the effect of Oxygen and electron concentration in ITO, and table 3 show the effect of Molybdenum concentration in MoCdO.

[0053] Table 1

[0054] Table 2

[0055] Table 3

[0056] As shown in tables 1, 2 and 3, there is a nonlinear dependency between the concentration of the dopant and the ENZ wavelength.Modulation with heat treatment

[0057] In some embodiments, the cladding is heat-treated to result in a specific refractive index at a required wavelength or permittivity at a required wavelength. The specific refractive index (n<l) or permittivity for a desired wavelength may determine at least one of, the temperature and the duration of a heat treatment. Tables 4, and 5 show nonlimiting examples for the dependency between heat treatment temperature (table 4) and heat treatment duration (table 5) on the ENZ wavelength, and hence on the refractive index of ITO.

[0058] Table 4

[0059] Table 5

[0060] As shown in tables 4 and 5, the higher the temperature and the longer the treatment duration the higher the ENZ wavelength is.Modulation with a voltage

[0061] In some embodiments, the refractive index and the ENZ wavelength may be adjusted or tuned by providing voltage to cladding 20. Such a system is illustrated in Fig.IB and may include at least one waveguide 100 comprising air core 10 and cladding 20, as discussed herein above. Waveguide 100 may further include a tunable electric power source 30 in electrical communication with cladding 20.

[0062] In some embodiments, tunable electric power source 30 may be any device that allow to adjust the output power (e.g., voltage) to meet specific requirements. Some nonlimiting examples for tunable electric power source 30 may include, function / arbitrary waveform generators (AWG), source measure unit (SMU), programmable DC Power Supplies, Variable Frequency Drives (VFDs), Adjustable AC Power Sources, and the like.

[0063] In some embodiments, applying voltages to cladding 20 may result in ultrafast (e.g., in sub picoseconds) modulation / changes in the refractive index or ENZ wavelength. The voltage may change the refractive index of cladding 20 which in turn alters the effective index of waveguide 100. Some nonlimiting examples for ITO are given and discussed in the Examples section with respect to Figs. 8 A and 8B.

[0064] In some embodiments, once the voltage dependent effective index is known, the propagation loss and modulation length can be calculated as well. A nonlimiting example, for ITO is given and discussed in the Examples section with respect to Figs. 9A and 9B.

[0065] In some embodiments, this may assist in adjusting the effective index of waveguide 100 in response to changes in the external (e.g., environmental) conditions acting on waveguide 100. These conditions may result in a distortion and shift of phase of the laser traveling in waveguide 100, thereby harming the CBC. Since the responsiveness of the cladding to the voltage is faster than the changes in the CBC (THz faster than the CBC), the correction is achieved in real time and the performance of the laser system may not be damaged.

[0066] In some embodiments, tuning the voltage may be controlled by a controller 40, discussed in detail with respect to Fig. 6. In some embodiments, controlling tunable electric power source 30 may be conducted in real time, for example, in order to compensate for amplitude or phase change damaging the CBC. In some embodiments, the voltage may be determined based on the required refractive index of cladding 20. For example, storage system 6 of controller 40 may include lookup tables, formulas, and / or graphs, such as, graphs 6A-7B discussed below, that correlate between the required refractive index and the applied voltage for various claddings 20.

[0067] In some embodiments, determining the required refractive index may be done in real time in response to a distortion and shift of phase of the laser traveling in waveguide 100 due to the external conditions acting on waveguide 100. Such an automatic adjustment of the refractive index is discussed with respect to the flowchart in Fig.4.

[0068] Reference is now made to Figs. 2A, 2B an 2C which show some nonlimiting examples for cross cross-section of the waveguides according to some embodiments of the invention. Fig. 2A illustrates a triangular air core 111 cladded by a triangular cladding 121. Fig. 2B illustrates a hexagonal air core 2111 cladded by a hexagonal cladding 221. Fig. 2C illustrates a trapezoid air core 311 cladded by a trapezoid cladding 321. Other optional polygonal cross-section for the air core may include an octagonal cross-section, a random polygonal cross-section, a regular polygonal cross-section, and the like. In some embodiments, the cladding (e.g., cladding 212, 221, and 321) may be substantially the same as cladding 20 discussed herein above.

[0069] Reference is now made to Figs. 3A, 3B and 3C which are illustrations of various components in photonic integrated circuits (PICs) according to some embodiments of the invention. In some embodiments, a photonic integrated circuit (PIC) 300a m 300b and 300c may include a planar substrate 200 and at least one waveguide 100.

[0070] With respect to Fig. 3A, PIC 300a may include substrate 200 supporting waveguide 100, which may include air core 10 having a polygonal cross-section and ENZ cladding 20 comprising ENZ material having a refractive index lower than 1 , covering air core 10 from all side faces. ENZ cladding 20 may be attached to planar substrate 200, providing structural support and integration capabilities for the optical system.

[0071] In some embodiments, substrate 200 may include any material with good adhesion to the ENZ material. For example, substrate 200 may include silicon-on-insulator (SOI) or silicon dioxide (SiO2) materials, which may provide suitable adhesion properties for ENZ cladding 20. In some cases, substrate 200 may include any material with an adhesion layer configured to enhance bonding between substrate 200 and ENZ cladding 20.

[0072] With continued reference to Fig. 3B, in some embodiments, PIC 300b may include two or more waveguides (e.g., strip or rib waveguide) combined into a single waveguide, all cladded by the ENZ material. This configuration may allow multiple optical channels to be merged, enabling the combination of several low power light channels to achieve a higher power output channel. The combining functionality may be implementedthrough tapered waveguide sections or the junction configurations where the separate waveguides converge into the single output waveguide. This configuration may work in reverse direction where the junction functions as splitter where a single optical path in a single waveguide is split into two or more optical paths.

[0073] In some embodiments, PIC 300b may incorporate multi-mode interference (MMI) structures for combining or splitting optical signals. The MMI structure may include a multi-mode waveguide section 10b having a wider air core than the input and output waveguides 10, with all cladded by the ENZ material. In some embodiments, the MMI structure may receive optical signals from one or more input waveguides 10 and may produce one or more output optical signals at output waveguides 10 based on self-imaging principles. The multi-mode waveguide section 10b may support multiple propagating modes that interfere constructively at specific locations along the propagation direction, creating one or more images of the input field distribution. In some embodiments, an MMI combiner may receive optical signals from two or more input waveguides positioned at one end of multi-mode section 10b and may combine these signals into a single output waveguide 10 at the opposite end. In some embodiments, an MMI splitter may receive an optical signal from a single input waveguide 10 and may split the signal into two or more output waveguides. The dimensions of the multi-mode section, including its width and length, may be selected based on the desired number of input and output ports and the operating wavelength. The MMI structures may provide advantages including compact size, broad optical bandwidth, and fabrication tolerance compared to other splitting or combining approaches.

[0074] Referring to Fig. 3C, in some embodiments, PIC 300c may include a single strip waveguide 10 split into two or more planar waveguides, or vice versa, all cladded by the ENZ material. This splitting configuration may enable optical signal distribution and routing applications. The splitting may be achieved through branching structures where the single input waveguide divides into multiple output paths, allowing for signal distribution across different optical channels.

[0075] In some embodiments, PIC 300c may incorporate optical splitting and combining elements configured to manipulate optical paths within the planar substrate. The splitting elements may include Y-junction structures where a single input waveguide 10 branches into two or more output waveguides 10, all cladded by the ENZ material. The combiningelements may include inverse Y-junction 10c or tapered waveguide structures where multiple input waveguides converge into a single output waveguide. In some embodiments, these splitting and combining elements may enable power scaling applications where several low power light channels are combined to achieve a high-power output channel. The planar architecture of the PIC may allow multiple optical channels to be precisely combined or separated within the same substrate, enabling dense multiplexing and routing in a compact footprint.

[0076] In some embodiments, PICs 300a, 300b and 300c may enable compact routing of light with bending radii in the order of hundreds of microns. This compact routing capability may be achieved due to the low refractive index properties of the ENZ material, which may allow for tighter bending without significant optical losses. The reduced bending radii may contribute to overall device miniaturization and increased integration density.

[0077] In some embodiments, entire PICs 300a, 300b and 300c may be fabricated in the millimeter scale. The compact nature of the ENZ-based waveguides may allow for complete optical systems to be implemented within millimeter-scale footprints, providing significant size advantages over conventional optical systems that may require larger form factors.

[0078] In some embodiments, PICs 300a, 300b and 300c may implement conventional photonic circuit building blocks such as splitters, modulators, and multiplexers for light manipulation and computation. These building blocks may be integrated directly onto substrate 200, providing comprehensive optical processing capabilities within a single platform. Splitters may divide optical signals into multiple paths, modulators may alter signal properties such as amplitude or phase, and multiplexers may combine multiple wavelength channels.

[0079] In some embodiments, PICs 300a, 300b and 300c may implement switches to choose between different routes with varying lengths for active optical delay lines in a coherent beam combining scheme. These switches may provide dynamic routing capabilities, allowing optical signals to be directed through paths of different optical lengths to achieve precise phase control for coherent beam combining applications.

[0080] In some embodiments, PICs 300a, 300b and 300c may implement 2x2 couplers for heterodyne measurements to achieve high signal-to-noise ratios (SNRs). The 2x2 couplers may enable interference between signal and reference beams, facilitating sensitive detection schemes that may improve measurement precision and system performance.

[0081] In some embodiments, PICs 300a, 300b and 300c may incorporate diverse functionalities such as modulators, detectors, filters, and nonlinear elements directly on a single platform. This integration may reduce reliance on external optical components and may provide a comprehensive optical processing system within the confines of substrate 200. Modulators may provide signal conditioning, detectors may enable optical-to-electrical conversion, filters may provide wavelength selectivity, and nonlinear elements may enable advanced optical processing functions such as frequency conversion or optical switching.

[0082] Referring to Figs. 4A and 4B, which are illustrations of PICs 400a and 400b with multilayered substrates according to some embodiments of the invention. PIC 400a and 400b may include planner substrate 200 and at least one waveguide 100, discussed herein above. In some embodiments, substrate 200 may comprise a multilayered structure comprising at least three layers with two dielectric layers 220a and 220b attached to a metallic layer 210 from both sides. This multilayered configuration may provide enhanced electrical control capabilities for the PIC while maintaining structural integrity and optical performance. The multilayered substrate 200 may enable precise voltage application to the ENZ cladding through the embedded metallic layer 210.

[0083] In some embodiments, the two dielectric layers 220a and 220b may comprise the same dielectric material. Using the same dielectric material for both dielectric layers 220 may provide symmetric electrical and thermal properties across the multilayered structure, which may contribute to uniform performance characteristics and simplified manufacturing processes. The dielectric layers 220a and 220b may provide electrical isolation while allowing for controlled field distribution across the structure.

[0084] In some embodiments, first dielectric layer 220a, may be the base layer on which metallic layer 210 may be applied. Using any known application method. Second dielectric layer 220b may be applied on top of metallic layer 210 and may be used as a buffer layer between the ENZ cladding and metallic layer 210. As some ENZ materials are conductive, for example, ITOs or similar oxides, in order to apply a voltage to the EMZ, a buffer dielectric layer must be placed between the metallic layer connected to a tunable electric power source and the ENZ calling.

[0085] In some embodiments, PIC 400a and 400b may further comprise a tunable electric power source 30 in electrical communication with metallic layer 210. PIC 400a may include a top metal layer 110 attached to ENZ cladding 20 and tunable electric power source 30 maybe connected between metallic layers 210 and 110. In some embodiments, top metal layer 110 may be applied above ENZ cladding 20 and may serve as an electrode for applying electrical fields to the cladding material.

[0086] PIC 400b may include a grounding connection 35 electrically connected to ENZ cladding 20. The grounding connection may provide a reference potential for the electrical circuit, enabling controlled voltage differences across ENZ cladding 20.

[0087] These electrical configurations may enable voltage application across ENZ cladding 20, allowing for dynamic control of the refractive index properties of the cladding material. The combination of tunable electric power source 30, metallic layer 210, and the electrode configuration may create a controllable electrical environment for modulating the optical properties of the ENZ material.

[0088] In some embodiments, PIC 400a and 400b may further comprise a computing device 40, illustrated in Fig. 5, configured to control tunable electric power source 30 to provide a voltage to ENZ cladding 20. Computing device 40 may include processing capabilities to determine appropriate voltage levels and timing for achieving desired optical modulation effects. The control functionality may enable real-time adjustment of the ENZ material properties in response to changing operational requirements or environmental conditions.

[0089] In some embodiments, computing device 40 may be further configured to determine the voltage based on a required refractive index of ENZ cladding 20. Computing device 40 may utilize stored calibration data, mathematical models, or lookup tables to establish the relationship between applied voltage and resulting refractive index changes in the ENZ material. This capability may allow for precise control of the optical properties of waveguide 100 by selecting appropriate voltage levels to achieve target refractive index values.

[0090] In some embodiments, computing device 40 may be further configured to receive an indication of a phase change of a laser beam traveling in air core 10 and determine the required refractive index of ENZ cladding 20 based on the indication. The phase change indication may be provided by optical sensors, interferometric measurements, or other detection systems that monitor the optical characteristics of the laser beam. Computing device 40 may analyze the phase change information to calculate the necessary refractive index adjustment needed to compensate for the detected phase variations.

[0091] In some embodiments, computing device 40 may be further configured to receive a required phase change, determine the voltage based on the phase change, and control provision of the determined voltage. This functionality may enable computing device 40 to proactively adjust the optical properties of waveguide 100 to achieve specific phase modulation targets. Computing device 40 may calculate the necessary voltage adjustments to produce the desired phase change and coordinate with tunable electric power source 30 to implement the voltage changes in a controlled manner.

[0092] In some embodiments, the voltage control system may operate in real-time to maintain optimal optical performance. Computing device 40 may continuously monitor optical parameters and adjust voltage levels to compensate for environmental variations, thermal effects, or other factors that may influence the optical characteristics of the PIC. The dynamic voltage control may enable stable operation and consistent optical performance across varying operational conditions.

[0093] Reference is now made to Fig. 5, which is a block diagram depicting a computing device or a controller 40, which may be included within an embodiment of a waveguide 100 system, according to some embodiments.

[0094] Controller 40 may include a processor 2 that may be, for example, a central processing unit (CPU) processor, a chip or any suitable computing or computational device, an operating system 3, a memory 4, executable code 5, a storage system 6, input devices 7 and output devices 8. Processor 2 (or one or more controllers or processors, possibly across multiple units or devices) may be configured to carry out methods described herein, and / or to execute or act as the various modules, units, etc. More than controllers 40 may be included in, and one or more a controller 40 may act as the components of a system according to embodiments of the invention.

[0095] Operating system 3 may be or may include any code segment (e.g., one similar to executable code 5 described herein) designed and / or configured to perform tasks involving coordination, scheduling, arbitration, supervising, controlling or otherwise managing operation of a controller 40, for example, scheduling execution of software programs or tasks or enabling software programs or other modules or units to communicate. Operating system 3 may be a commercial operating system. It will be noted that an operating system 3 may be an optional component, e.g., in some embodiments, a system may include a computing device that does not require or include an operating system 3.

[0096] Memory 4 may be or may include, for example, a Random Access Memory (RAM), a read only memory (ROM), a Dynamic RAM (DRAM), a Synchronous DRAM (SD-RAM), a double data rate (DDR) memory chip, a Flash memory, a volatile memory, a non-volatile memory, a cache memory, a buffer, a short term memory unit, a long term memory unit, or other suitable memory units or storage units. Memory 4 may be or may include a plurality of possibly different memory units. Memory 4 may be a computer or processor non-transitory readable medium, or a computer non-transitory storage medium, e.g., a RAM. In one embodiment, a non-transitory storage medium such as memory 4, a hard disk drive, another storage device, etc. may store instructions or code which when executed by a processor may cause the processor to carry out methods as described herein.

[0097] Executable code 5 may be any executable code, e.g., an application, a program, a process, task or script. Executable code 5 may be executed by processor 2 possibly under control of operating system 3. For example, executable code 5 may be an application that may control a refractive index of a waveguide, such as, waveguide 100, as further described herein. Although, for the sake of clarity, a single item of executable code 5 is shown in Fig.3, a system according to some embodiments of the invention may include a plurality of executable code segments similar to executable code 5 that may be loaded into memory 4 and cause processor 2 to carry out methods described herein.

[0098] Storage system 6 may be or may include, for example, a flash memory as known in the art, a memory that is internal to, or embedded in, a micro controller or chip as known in the art, a hard disk drive, a CD-Recordable (CD-R) drive, a Blu-ray disk (BD), a universal serial bus (USB) device or other suitable removable and / or fixed storage unit. Lookup tables, formulas, and / or graphs that correlate between the required refractive index and the applied voltage for various claddings 20 may be stored in storage system 6 and may be loaded from storage system 6 into memory 4 where it may be processed by processor 2. In some embodiments, some of the components shown in Fig. 1 may be omitted. For example, memory 4 may be a non-volatile memory having the storage capacity of storage system 6. Accordingly, although shown as a separate component, storage system 6 may be embedded or included in memory 4.

[0099] Input devices 7 may be or may include any suitable input devices, components or systems, e.g., a detachable keyboard or keypad, a mouse and the like. Output devices 8 may include one or more (possibly detachable) displays or monitors, speakers and / or any othersuitable output devices. Any applicable input / output (I / O) devices may be connected to A controller 40 as shown by blocks 7 and 8. For example, a wired or wireless network interface card (NIC), a universal serial bus (USB) device or external hard drive may be included in input devices 7 and / or output devices 8. It will be recognized that any suitable number of input devices 7 and output device 8 may be operatively connected to A controller 40 as shown by blocks 7 and 8.

[0100] A system according to some embodiments of the invention may include components such as, but not limited to, a plurality of central processing units (CPU) or any other suitable multi-purpose or specific processors or controllers (e.g., similar to element 2), a plurality of input units, a plurality of output units, a plurality of memory units, and a plurality of storage units.

[0101] Reference is now made to Fig. 6, which is a flowchart of a method of controlling a refractive index of a waveguide according to some embodiments of the invention. The method of Fig. 6 may be executed by controller 40 or by any other suitable computing device. In step 410, the controller may receive a required refractive index. For example, the controller may receive the required refractive index from a user via a user device, from a database, or automatically based on measurements form sensors.

[0102] In a nonlimiting example, the controller may receive an indication of a phase change of a laser beam traveling in air core 10. For example, in a CBC system, the indication may be received from one or more optical detectors, each being positioned and configured to measure the intensity of a respective optical interference signal with respect to a reference optical beam. These measurements may lead to generating a power output value, indicative of the overall intensity of the respective optical interference signal. Power values may be used for calculating a phase change between a specific optical beam traveling in waveguide 100 and the reference optical beam. The controller may then determine the required refractive index of the cladding based on the indication. The required change in the refractive index may be calculated by Anwhere A is the phase change required, obtained bythe interferometric measurements, A is the wavelength, and L is the length in which the refractive index is changed.

[0103] In step 420, the controller may determine a voltage level based on the required refractive index. For example, controller 40 may use lookup tables, formulas, and / or graphs,stored in storage system 6, that correlates between the required refractive index and the applied voltage for various claddings 20.

[0104] In step 430, the controller may control tunable electric power source 30 to provide the voltage level to cladding 20 of waveguide 100.

[0105] In some embodiments, the method may include receiving a required phase change and determining the voltage level based on the phase change.

[0106] The phase change may cause changes in the optical path length (OPL) of the beam traveling in waveguide 100. Therefore, adjusting the required refractive index, by adjusting the voltage, may change the OPL. Such a correlation is shown indirectly in Fig. 7A. Fig. 7A shows that there is a larger refractive index contrast when applying voltage which translates into larger confinement.Examples

[0107] Reference is now made to Fig. 7 which shows graphs of the optical properties of ITO, according to some embodiments of the invention. The refractive index (left side) and real part of the permittivity (right side) as a function of wavelength, are shown. Shaded area marks the regime where light can be guided in air (n<l). The dashed line represents the ENZ wavelength at Z=l225nm. The values used for the fit were co_p=3.O7*lOA15 [rad / s], e_oo=4.01, y=2.149*1014[rad / s],

[0108] In addition, the plasma frequency a)pwas dynamically modified by applying voltage to the ITO film. The carrier density is proportional to the applied voltage and can be written as N (F) = No+ aV, where a is the proportionality term which is dependent on the thickness and permittivity of the material. The effective electron mass is also dependent on 1 ■ 1 1 1 1 1 the carrier density, and can be expressed by: m .

[0109] The following calculation and simulations were conducted on a waveguide 100 comprising rectangular air core having 2 pm x 2 pm cross section, cladded by 25 nm thick ITO cladding.

[0110] Reference is now made to Figs. 8A and 8B, which include graphs showing the dependency of refractive index at a wavelength of 1.2 pm (Fig. 8 A) and the effective index of the waveguide (Fig. 8B) in the voltage, according to some embodiments of the invention. As shown, there is a nonlinear correlation between the applied volage and the refractiveindex. Knowing this correlation may allow to change the refractive index of the ITO cladding and the effective index of the waveguide by providing voltage level.

[0111] Once the voltage dependent effective index is known, the propagation loss and modulation length can be calculated as well. The results for a wavelength of 1.2 pm are plotted in Figs. 9 A and 9B.

[0112] Reference is now made to Figs. 9A and 9B which include graphs showing the dependency of propagation loss (Fig. 9A) and modulation length (Lpi) (e.g., OPL) in the voltage, according to some embodiments of the invention. Figs. 9A and 9B clearly demonstrate that both the loss and the modulation length decrease with increasing voltages. This can be explained by the higher confinement achieved at larger voltages, resulting in a smaller fraction of the mode propagating in the lossy ITO. A cross section of both TEO and TMO modes at 0V and 2V showing this behavior is plotted in Figs. 10A, 10B, 10C and 10B. Figs 10A and 10B show the fundamental TM mode and TE mode, respectively, while no voltage is applied, and Figs. 10C and 10D show the TM mode and TE mode, respectively, during the application of 2V.

[0113] Therefore, the proposed structure of a waveguide, according to some embodiment of the invention, comprising an air core and an ENZ cladding, was found to be suitable for high-power laser applications, where conventional core materials are limited by optical damage. Air's high linearity may also help to eliminate unwanted nonlinear effects that usually arise at higher optical powers. Additionally, air's dispersion-free and transparent nature allows this design to be effective across a wide wavelength range, including the NIR-MIR regime where transparent materials are scarce. This may also ensure that the temporal shape of the propagating pulse remains unaltered.

[0114] Reference is now made to Figs. 11 and 12 which are illustrations of steps of two fabrication methods according to some embodiments of the invention. In some embodiments, a fabrication method may be employed to produce a PIC according to embodiments described herein. Both fabrication methods may begin with evaporating an ENZ material on a substrate. The substrate may be any material with good adhesion to the ENZ material, such as silicon-on-insulator (SOI) or silicon dioxide (SiO2), or any material with an adhesion layer configured to enhance bonding between the substrate and the ENZ material. In the first embodiment of Fig. Il a relatively thin layer of ENZ material having a thickness of at least 10 nm may be deposited, for example, 11 nm, 15 nm, 20 nm, 25 nm, 30nm, 40nm, 50 nm, 75 nm, 100 nm, 150 nm, 200 nm, 250 nm, 500 nm 1 m, 5 pm, 10 pm. and any value. In the second embodiment of Fig. 12 a relatively thick layer of ENZ material may be applied, of at least 1 pm, for, example, 5 pm, 10 pm, 25 pm, 30 pm and any value or range in between.

[0115] Following the initial ENZ material deposition, a photoresist may be applied (e.g., spin-coated) onto ENZ material surface. In the first embodiment of Fig. 11, the photoresist layer may be a relatively thick layer of at least 1 pm, for, example, 5 pm, 10 pm, 25 pm, 30 pm and any value or range in between. In the second embodiment of Fig. 12 the the photoresist layer may be relatively thin having a thickness of at least 10 nm may be deposited, for example, 11 nm, 15 nm, 20 nm, 25 nm, 30 nm, 40nm, 50 nm, 75 nm, 100 nm, 150 nm, 200 nm, 250 nm, 500 nm 1 pm, 5 pm, 10 pm. and any value.

[0116] The photoresist may provide a patternable layer that enables selective material removal during subsequent processing steps. The spin-coating or other coating processes may result in a uniform photoresist layer with controlled thickness across the substrate surface.

[0117] In some embodiments, waveguides may be patterned using any type of lithography, for example, electron beam (e-beam) lithography. The e-beam lithography process may enable precise definition of waveguide geometries with sub-micron resolution capabilities. The patterning may define the locations and dimensions of air cores that will form the optical waveguides within the PIC structure. The lithography may provide the precision needed to create waveguides with polygonal cross-sections, including rectangular, triangular, hexagonal, octagonal, or other polygonal configurations.

[0118] After patterning, the substrate may be developed to remove exposed or unexposed photoresist areas, depending on the photoresist type used. The development process may reveal the patterned waveguide structures and prepare the substrate for subsequent processing steps. The developed substrate may exhibit defined openings or patterns corresponding to the intended waveguide geometries. In the embodiment of Fig. 11, the tunnel form in the photoresist may be located at the air core location. In the embodiments of Fig. 12, the element of photoresist material left after lithography may be located at the air core location.

[0119] In some embodiments, in the embodiment of Fig. 11, a sacrificial layer may be deposited following the development step. The sacrificial layer may serve as a temporaryprotective or structural element during the fabrication process. The sacrificial layer may be deposited using techniques such as physical vapor deposition, chemical vapor deposition, or other suitable deposition methods. The sacrificial layer material may be selected based on its compatibility with subsequent processing steps and its ability to be selectively removed later in the fabrication sequence.

[0120] Following sacrificial layer deposition, in both the embodiment of Fig. 11 and the embodiment pf Fig.12, the remaining photoresist may be lifted off from the substrate. The lift-off process may remove unwanted photoresist material while preserving the desired structural elements and sacrificial layer components. This step may result in a substrate with defined patterns and sacrificial layer elements positioned according to the intended waveguide design.

[0121] In both embodiments, an ENZ material may be evaporated again onto the substrate structure. This second ENZ material deposition may form the cladding layers that surround the air cores of the waveguides. The second evaporation step may deposit ENZ material over the sacrificial layer and exposed substrate areas, creating the structural framework for the waveguide cladding.

[0122] The fabrication process may conclude with removal of the sacrificial layer. The entire structure may be immersed in a suitable etchant or solvent that selectively removes the sacrificial layer material without damaging the ENZ cladding or substrate. The sacrificial layer removal may create the air cores within the ENZ cladding, resulting in completed waveguides with air cores surrounded by ENZ material cladding.

[0123] In some embodiments, this fabrication approach may enable the production of compact, millimeter-scale photonic devices with integrated functionalities. The resulting PICs may incorporate multiple optical components and functionalities within a single substrate, providing comprehensive optical processing capabilities in a compact form factor. The fabrication method may allow for the integration of various photonic circuit building blocks such as splitters, modulators, multiplexers, and other optical components within the same fabrication sequence.

[0124] In some embodiments, the fabrication process may be compatible with standardized semiconductor processes. This compatibility may enable the use of existing electronic manufacturing infrastructure for PIC production, potentially reducing manufacturing costs and improving scalability. The semiconductor process compatibilitymay facilitate integration with electronic components and may enable the use of established fabrication facilities and equipment for PIC manufacturing.

[0125] In some embodiments, the standardized semiconductor process compatibility may enable mass production capabilities and quality control measures that are well-established in the semiconductor industry. The fabrication approach may leverage existing process control methodologies, metrology techniques, and manufacturing protocols that have been developed for semiconductor device production. This compatibility may contribute to improved yield rates and consistent device performance across production batches.

[0126] Therefore, the proposed structure of a waveguide, according to some embodiment of the invention, comprising an air core and an ENZ cladding, was found to be suitable for high-power laser applications, where conventional core materials are limited by optical damage. Air's high linearity may also help to eliminate unwanted nonlinear effects that usually arise at higher optical powers. Additionally, air's dispersion-free and transparent nature allows this design to be effective across a wide wavelength range, including the NIR-MIR regime where transparent materials are scarce. This may also ensure that the temporal shape of the propagating pulse remains unaltered.

[0127] Unless explicitly stated, the method embodiments described herein are not constrained to a particular order or sequence. Furthermore, all formulas described herein are intended as examples only and other or different formulas may be used. Additionally, some of the described method embodiments or elements thereof may occur or be performed at the same point in time.

[0128] While certain features of the invention have been illustrated and described herein, many modifications, substitutions, changes, and equivalents may occur to those skilled in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the invention.

[0129] Various embodiments have been presented. Each of these embodiments may of course include features from other embodiments presented, and embodiments not specifically described may include various features described herein.

Claims

CLAIMS1. A photonic integrated circuit (PIC) comprising;a planner substrate; andat least one waveguide, comprising:an air core having a polygonal cross-section; anda cladding comprising Epsilon-Near-Zero (ENZ) material having a refractive index lower than 1 , covering the air core from all side faces, wherein the cladding is attached to the planner substrate.

2. The PIC of claim 1, further comprising:two or more waveguides combined into a single waveguide, all cladded by the ENZ material.

3. The PIC of claim 1, further comprising:a single waveguide split into two or more waveguides, all cladded by the ENZ material.

4. The PIC of any one of claims 1 to 3, wherein the substrate comprising a material adherable to the ENZ material.

5. The PIC of claim 4, wherein the substrate comprises a multilayered structure comprising at least three layers two dielectric layers attached to a metallic layer from both sides.

6. The PIC of claim 5, wherein the two dielectric layers comprises the same dielectric material.

7. The PIC of claim 5 or claim 6, further comprising:a tunable electric power source in electrical communication with the metallic layer, and at least one of: (a) a metallic layer attached to the cladding; and (b) a grounding connection electrically connected to the cladding.

8. The PIC of any one of claims 1 to 4, further comprising:a tunable electric power source in electrical communication with the cladding.

9. The PIC of claim 7 or claim 8, further comprising: a controller configured to control the tunable electric power source to provide a voltage to the cladding.

10. The PIC of claim 9, wherein the controller is further configured to determine the voltage based on a required refractive index of the cladding.

11. The PIC of claim 9, wherein the controller is further configured to:receive an indication of a phase change of a laser beam traveling in the at least one air core; anddetermine the required refractive index of the cladding based on the indication.

12. The PIC of any one of claims 9 to 11, wherein the controller is further configured to:receive a required phase change;determine the voltage based on a phase change; andcontrol a provision of the determined voltage.

13. The PIC of any one of claims 1 to 12, wherein the air core has at least one dimension of between 1 to 5 pm.

14. The PIC of any one of claims 1 to 13, wherein a thickness of the cladding is at least 10 nm.

15. The PIC of any one of claims 1 to 14, wherein the ENZ material is a transparent conducting oxide (TCOs).

16. The PIC of any one of claims 1 to 15, wherein the ENZ material is doped with a dopant selected to provide to the cladding a specific refractive index at a required wavelength or a specific ENZ wavelength.

17. The PIC of claim 16, wherein selecting the dopant comprises at one least of, selecting the type of dopant, and a concentration of the dopant.

18. The PIC of claim 16 or claim 17, wherein the specific refractive index is for ENZ wavelength of between 1 to 5 pm.

19. The PIC of any one of claims 1 to 18, wherein the cladding is heat-treated to result in a specific refractive index at a required wavelength or a specific ENZ wavelength.

20. The PIC of claim 19, wherein the specific refractive index at the required wavelength or the specific ENZ wavelength determines at least one of, a temperature and a duration of a heat treatment.

21. The PIC of claim 19 or claim 20, wherein the specific refractive index is for a wavelength of between 1 to 5 pm.

22. The PIC according to any one of claims 1 to 21, wherein the polygonal crosssection is selected from, a rectangular cross-section, a hexagonal cross-section; an octagonal cross-section, and a triangular cross-section.

23. A method of controlling a refractive index of a waveguide included in the PIC of any one of claims 1 to 22, comprising:receiving a required refractive index;determining a voltage level based on the required refractive index; andcontrolling a tunable electric power source to provide the voltage level to a cladding of the waveguide, wherein the cladding comprises Epsilon-Near-Zero (ENZ) material having a refractive index lower than 1 , covering a polygonal air core from all side faces.

24. The method of claim 23, further comprising:receiving an indication of a phase change of a laser beam traveling in the air core; and determining the required refractive index of the cladding based on the indication.

25. The method of claim 23 or claim 24, further comprising:receiving a required phase change; anddetermining the voltage level based on a phase change.

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