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Review article

Optics and Lasers in EngineeringJournal2024, Optics and Lasers in Engineering

Wenlong Huang, ... Xufeng Jing

1 Introduction

Metamaterial is a kind of synthetic structural material [1–5] with strange electromagnetic, acoustic or mechanical properties. The first physical metamaterial was developed by John Pendry, and David R.Smith was the first experimentally demonstrated a material with a negative refractive index [6]. This innovation has promoted the development of materials science and has a wide range of application prospects in optics, electronics and other fields. The term "metamaterial" was coined by Walser in 2001 [7]. In the last two decades, metamaterials have achieved some unique physical phenomena and functions, such as electromagnetic cloaks [8–10], negative refraction [11,12], perfect absorption [13–18], as well as negative permeability [6] and negative dielectric constant [19] as shown in Fig. 1. Although metamaterials can produce some unique physical phenomena and functions, there are many disadvantages of metamaterials, such as: high loss, complex manufacturing process, high manufacturing cost, and large weight. In addition, the narrow operating spectrum is also one of the disadvantages of metamaterials, and their efficiency is relatively low, which may hinder the promotion of metamaterials in practical applications.

Fig. 1. The history and future of terahertz dynamic metasurfaces [7,20].

The researchers overcame this shortcoming by evolving the 3D metamaterial into a 2D form that compresses the direction of the wave to a near-zero thickness, resulting in what is known as a metasurface [21,22]. In recent years, the concept of active tunable metasurface has been proposed in order to realize dynamic function and frequency modulation. Compared with passive metasurface, active dynamic metasurface usually has the advantages of wide frequency band, wide adjustable range and low loss, which brings great vitality to the development of metasurface. The core idea of the active tunable metasurface is to load the active device to each unit to realize the functions of polarization conversion [23–26], beam steering [27–32], wave absorption [33,34], etc., while keeping the physical structure of the unit unchanged. At present, the adjustment methods of tunable metasurface mainly include electrical control [23,35–39], temperature control [40–43], optical control [44–47] and so on. The commonly used electronic devices in electrical control are: PIN diode and varactor diode. Electrical control has lower system complexity, more flexible adjustment form and stronger beam adjustment ability. Temperature control can be achieved by doping phase change materials such as alum dioxide in the metasurface, and light control can be achieved by using photoresist and photodiodes to sense light changes. With the continuous development of new materials in the field of electronic information [48–51], researchers further compound some tunable sensitive materials (such as liquid crystal [52–56], graphene [57–61] and lithium niobate [62]) with the metasurface to achieve dynamic metasurface.

Metasurfaces have been rapidly developed over the past two decades and are widely used in many new devices [63–67] in the microwave [31,68], terahertz [69–74], and visible light regions [75]. The refraction and reflection of light at the interface of two homogeneous isotropic media follow Snell's law. In 2011, the generalized Snell's law was proposed [76], and space-gradient metasurfaces were used to manipulate electromagnetic waves by introducing phase transitions at two media interfaces [77]. In order to explore the possible connection between metasurfaces and digital information, Cui et al. proposed a new theory of digitally encoded programmable metasurfaces in 2014 [35], opening a new chapter in the study of metasurfaces. At present, metasurface have been widely used in beam control [69,78–81], scattering reduction [82–84], holographic imaging [85–90], orbital angular momentum [91–97],cloaking [98–101], optical encryption [102], metalens [103–107] and wireless communication [108–112]. More importantly, digital coded metasurface builds a bridge between the physical world and the digital world, which enables researchers to explore metasurface from the perspective of information science and forms a research system of information metasurface. However, the active control of the programmable metasurface still requires manual intervention to change the control instructions or programs to achieve the switching of different electromagnetic characteristics, such as switching different phase coding states, different polarization coding states, etc. Therefore, intelligent metasurface [113,114] will be an important direction of future metasurface development.

The dynamic control technology of metasurface electromagnetic wave beam can be applied to the fields of communication [108,111,112,115], radar [82,83,116–118], solar energy [119], biomedicine [120–123], etc., and provide technical support for the further development of these fields. The dynamic regulation of electromagnetic wave can be realized, and the transmission efficiency of electromagnetic wave can be improved by adjusting the reflection and transmission characteristics of electromagnetic wave beam on the metasurface. The dynamic control technology of electromagnetic wave beam on the metasurface can reduce electromagnetic interference to a certain extent and improve the performance of communication and radar systems. The traditional communication and radar systems need to transmit and receive signals through multiple antennas and many power amplifiers, but the metasurface electromagnetic wave beam dynamic regulation technology can realize signal transmission and reception by adjusting the structure on the metasurface, thus reducing power consumption and energy consumption. Through the metasurface electromagnetic wave beam dynamic regulation technology can realize the signal directional transmission and reception, to reduce the risk of signal interference and loss, improve the reliability of communication and radar system.

This paper introduces liquid crystal tuning, graphene tuning, GST tuning, vanadium dioxide tuning, optical tuning method, metasurface-tuning based on varactor diode and PIN diode. The spatio-temporal coding metasurface and the reconfigurable metasurface which can realize their functions in frequency domain and space domain are also introduced. Finally, the beam control and prospect of intelligent metasurface are introduced.

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Review article

Two-photon lithography for three-dimensional fabrication in micro/nanoscale regime: A comprehensive review

Optics & Laser TechnologyJournal2021, Optics & Laser Technology

V. Harinarayana, Y.C. Shin

6 Metamaterials

Materials with excellent mechanical properties, optical properties, multiple functionalities with the least weight possible are always in high demand. The design of materials with such characteristics is a continual goal for researchers and industries. Metamaterials is a rapidly emerging branch of science, which deals with the development of material properties out of the ordinary. Metamaterials are artificial materials or composites whose properties usually do not exist in nature, do not depend on the material composition, or do not depend on individual atoms, but largely depend on the fabricated structure topology. Metamaterials are derived from the Greek word “µɛα” meaning beyond materials. TPL is an ideal technology for the fabrication of three-dimensional metamaterials as most of the elements do not require post-processing. Historically, the pursuit of metamaterials and phononic crystals was the primary motivation for the development of TPL technology.

Electromagnetic metamaterials are of great attraction to researchers, which usually involve cells with subwavelength geometries. Among the many types of electromagnetic metamaterials, plasmonic metamaterials stand out because of the unprecedented optical functionalities like a negative index of refraction, negative permeability, negative permittivity that can be obtained at will and cannot be realized in materials available in nature. These materials make good use of surface plasmons, which are collective oscillations of free electrons at metal-dielectric interfaces. Surface plasmon resonance (SPR) occurs when a light wave of a certain wavelength reaches the interface, and most of the irradiant energy is transferred to a surface plasmon of a shorter wavelength. The creation of plasmons on the surface of nanoparticles is called localized surface plasmon resonance (LSPR) and is guided by the shape, spacing and orientation of the nanoparticles. Manipulation of these parameters can result in metamaterials with unique mechanical and optical functionalities.

In 1968, Veselago [91] suggested the phenomenon of negative refractive index by materials whose permittivity ( and permeability (µ) were negative, which indicates the velocity of light within the material would also be negative. Although there are materials in nature with negative permittivity, there is none with both parameters being negative. Metamaterials were first experimentally proposed by Pendry et al. in 1996 [92]. Thin metallic wires with radius in the order of ~ 1 µm were manufactured and assembled to form a simple cubic lattice structure illustrated in Fig. 14. The effective plasma frequency of the artificial material was depressed by up to 6 orders of magnitude, which subsequently changed the dielectric function of the material to a negative value. The effective permittivity is derived from the Drude-Lorentz model and is expressed as:

Fig. 14. Infinite wires arranged in a simple cubic lattice, joined at the corners of the structure [92].

(10)

where is the dielectric function, is the frequency of oscillation, is the plasma frequency of the material and is the damping term representing dissipation of plasmon’s energy into the system (~0.1. Negative gives rise to a variety of electromagnetic structures decorating the surface of metals with a complexity controlled by the geometry of the surface [92].

Metamaterials have been studied for several unique properties and applications viz. negative index of refraction [93–96], invisibility cloaking [97–99], and superlensing [100]. TPL is a promising method for the fabrication of true 3D plasmonic metamaterials. Although TPL is limited in options for direct fabrication of 3D metallic structures, polymer-based resins can be embedded with metallic components by incorporating various deposition techniques. Split ring resonators (SRR) are a common application of electromagnetic metamaterials used to produce artificial magnetism. Smith and coworkers [93], as represented in Fig. 15, realized 3D structures in the millimeter scale, operated in the microwave regime with a two-dimensional array of repeated unit cells of copper strips and SRRs on interlocking strips that exhibit a frequency band where the effective index of refraction is negative. However, to achieve magnetic resonance at optical frequencies, the SRR structure must be less than 100 nm in structure dimension with a gap less than 10 nm. Moreover, the scaling principle also starts to break down at higher frequencies as the metal significantly deviates from an ideal conductor [101].

Fig. 15. left-handed metamaterial (LHM) with square copper SRRs and copper wire strips on fiberglass circuit board material [93].

However, Dolling et al. [102] overcame this problem by employing nanoscopic plate pairs or cut-wire pairs to produce negative permeability and negative permittivity directly, without employing SRRs as shown in Fig. 16 (a). In their design, the split in a normal SRR is opened, which consequently decreases the capacitance, C, and in turn increases the LC resonance frequency given by . Additionally, the bottom arm in the U-shaped structure is removed to produce cut-wire pair with length, l, thickness, t, width w, and the spacing between the wires, d. This furthermore decreases the net capacitance and increases the magnetic resonance frequency. Although it is straight-forward to achieve this design, the ratio of wavelength to the lattice constant, must be very high for true metamaterials. In this case, the ratio is typically low (~2). However, when l = w, the cut-wire pairs behave like nanoscopic plate pairs with more pronounced optical resonance relative to cut-wire pairs. In contrast to the theoretical studies, a negative refractive index, however, was not obtained although negative permittivity and permeability were achieved.

Fig. 16. (a) Schematic of adiabatic transition from SRRs to cut-wire pairs. (bottom left) one cut wire placed on top of another with a certain dielectric spacer layer in between. (b) Measure transmittance (red) and reflectance (blue) for cut-wire pairs. Insets correspond to electron micrographs. (c) measure transmittance (red) and reflectance (blue) for arrays of plate pairs. From [102].

In 2007, Dolling and colleagues [103] claimed to be the first ones to fabricate one, two and three-functional layer metamaterials. The process started with a single functional layer producing negative magnetic permeability above the magnetic resonance frequency and negative electric permittivity below the effective plasma frequency successively generating a negative index of refraction. To expand the structure to three dimensions, multiple layers of metal-dielectric-metal were stacked as represented in Fig. 17 (i). They used 31 nm thick silver and 21 nm thick MgF2 as the metal and dielectric layers respectively resulting in a high ratio of wavelength to lattice constant (~27), although they could experimentally fabricate only up to 3 functional layers (7 real layers) through electron beam lithography (EBL). Fabrication of thicker structures is much more difficult through EBL because of the dependence of total thickness on the thickness of the patterned electron beam resist. The total thickness is limited to, at maximum, 80% of the thickness of the resist, which is close to ~ 100 nm for EBL. The issue of a stepped wall also arises if the thickness is high.

Fig. 17. (i)(a) Top view of once unit cell of the functional metamaterial. (b) Side view. (c) magnetic field (linear scale) for N = 3 and a wavelength of 1430 nm. (d) Electric field (linear scale) for same plane and wavelength. (ii) Measured (solid) and calculated (dashed) normal incidence transmittance (red) and reflectance (blue) for N = 1,2,3 functional layers respectively. Insets correspond to electron micrographs with a 400 nm scale. (iii) Effective refractive index (top) and permeability (bottom). Real parts, solid, imaginary parts, dashed. From [103].

Though stacking up 2D layers via EBL or focused ion beam (FIB) forms 3D metamaterials in principle, inherently they are 2.5D, i.e., 2D structures with multiple functional layers. Furthermore, the process is slow, expensive and requires a lot of skill to achieve proper alignment. In order to obtain true 3D metamaterial structures, TPL can be employed. It is best suited, and the process can be expedited by integrating optics like MLAs, DMDs as discussed earlier.

Gansel et al. [104] fabricated a true 3D helical free-standing metamaterial structure via TPL as depicted in Fig. 18. The process started with the usual split-ring resonators and adiabatically pulling one end of this planar SRR out of the fabricating plane, thereby resulting in a circular helix of gold structures with electromagnetic modes close to that of an SRR. They used a positive-tone photoresist to fabricate the template and infilled it with gold by electrochemical deposition because of its excellent optical properties at mid-infrared wavelengths. Subsequently, they removed the polymer through plasma etching. Initially, for the purpose of electrochemical deposition, the glass substrate on which fabrication was performed was deposited with a thin film (~25 nm) of indium tin oxide (ITO) acting as the cathode. However, the electroplating process is not applicable to many complex designs like structures with interlockings, suspended features and chirality. The parameters like electroplating time, the direction of electroplating, bath temperature, current density, solution concentration define the thickness of the coating. Manipulation of these parameters simultaneously to obtain the desired coating is cumbersome and time-consuming. Similarly, in 2017, Frenzel et al. [105] fabricated the elastic counterpart of optical activity in 3D chiral structures. i.e., 3D chiral mechanical metamaterials with a twisting degree of freedom extending beyond the Cauchy elasticity.

Fig. 18. Flow of fabrication of gold helix plasmonic metamaterials. (top left) A positive-tone photoresist is deposited onto a glass substrate with a ~ 25 nm thin layer of ITO (green). (top right) An array of helices fabricated in the photoresist. (Bottom right) After electrochemical deposition of gold onto the template. (bottom left) self-standing helical structures obtained after removing the photoresist via plasma etching. From [104].

Mu and coworkers [106] fabricated metallic pyramids plasmonic metamaterial via TPL followed by electron beam evaporation. Initially, keeping a copper grid as the substrate, a negative photoresist was used to fabricate the pyramid template via TPL. This was followed by electron beam evaporation of silver onto the hollow pyramid surfaces. These metallic pyramids can be used as surface-enhanced Raman spectroscopy (SERS) substrates for their electromagnetic properties. However, the stepped wall effect became prominent as the height of the structure increases as shown in Fig. 19.

Fig. 19. SEM images of hollow silver-coated metallic pyramid structures with 50.60.70.80.90 µm in height, respectively. From [106].

Electroless plating is an alternative metal deposition technique to overcome the limitations in electroplating. It is a wet-chemical metallization technique on a catalytic surface based on an autocatalytic redox reaction without any external current dependency. A reducing agent added to the solution reduces the metal particles from their ionic state before deposition. The quality of deposition is determined by the adhesion between the metal particles and the surface. Radke and coworkers [107] used this technique along with TPL to fabricate three-dimensional bichiral plasmonic crystals as shown in Fig. 20. Initially, they fabricated a 3D bichiral crystal template via TPL in a negative-tone photoresist. Eventually, this template was coated with silver via electroless silver plating. However, during the process of electroless plating, the glass substrate was coated with silver as well. In order to overcome this effect, the crystal had to be detached from the coated substrate and placed onto a neat glass substrate. However, detaching the crystal from the bottom created holes in the silver film at the point of contact. Therefore, they set up vertical posts at the corners as shown in Fig. 20 (b). These posts served as spacers between the crystal and the substrate making it easier to remove after the plating process. Although, when compared to electroplating, the process is faster and accurate, it is cumbersome and requires expertise to carefully remove the structure from the substrate and place it onto a cleaner glass substrate. Similarly, Chen et al. [108] fabricated 3D silver-coated polymeric microstructures. In their method, the 3D microstructure template after TPA was functionalized with alkylamines by treating with NH2(CH2)3NHLi. This strong nucleophile cleaves exposed acrylate esters, creating surface-bound amine termination. Gold particles were then bound to the amines by immersing the template structure in aqueous AuCl4- and reducing with NaBH4. Silver metal was then deposited onto the structure via electroless plating. The deposition is effective only on the surfaces treated with both aminolysis and AuCl4-/ NaBH4 reagents. In this way the extra step of transferring the microstructure to a clean substrate was eliminated. However, the main disadvantage of such techniques is that the density of the metal-binding sites onto the template cannot be controlled. The quality of metallization depends on the adhesion between the metal and the surface, which cannot be manipulated easily. Therefore, site-selective electroless plating is a viable alternative.

Fig. 20. (a) Flow of fabrication. A glass cover slip serves as the substrate on which a negative-tone photoresist is deposited via spin-coating. TPL is used to fabricate the template structure. Post-baking and developing generate a free-standing template of the 3D bichiral crystal structure. All surfaces of the template including the substrate are coated with a conformal silver via electroless plating. To facilitate transmission spectroscopy, the crystal is detached from the template and deposited onto a clean glass substrate with a thin glass capillary. (b) oblique view of the bichiral structure after electroless plating. From [107].

Kawata et al. [109] demonstrated the fabrication of 3D metal/polymer microstructures via TPL and site-selective electroless silver plating. In their experiment, activated and non-activated resins were utilized for the fabrication of complex metal/polymer 3D microstructures. Initially, a laser beam was focused onto the non-activated monomer and after the exposure, the uncured resin was washed away with acetone. Eventually, a small amount of activated resin was dripped onto the polymerized structure and the same procedure was followed to form pairs of polymer layers on a glass substrate. Later, the fabricated polymer samples were immersed in an aqueous solution of AgNO3 for a period of close to ~ 6 hrs. After the treatment, the activated resin parts appeared slightly darker due to the deposition of Ag nanoparticles onto the surface. Finally, when silver was coated onto the structures via electroless plating, the activated resin parts appeared opaque while the other parts appeared transparent in the transmission image as seen in Fig. 21, thus confirming the deposition of silver only in the desired sites. The main disadvantages of this technique are: there are limited metal-binding materials that can be utilized and the required resolution and structural integrity for optical metamaterials are not met in most cases. However, in 2012, Vasilantonakis and colleagues [110,111] demonstrated the possibility of fabrication of optical nanophotonic devices via TPL and selective electroless plating. 3D metallic woodpile structures with features below 100 nm were fabricated using an organic–inorganic, zirconium-silicon hybrid material doped with a metal-binding monomer followed by metallization through electroless plating. The metalized structures exhibited ohmic conductivity, comparable to pure silver.

Fig. 21. Microscope images of pairs of polymer sheets made via TPL with activated and non-activated resin. (Top) after soaking in AgNO

aqueous solution for 6 h. (bottom) After electroless plating. From [109].

Similarly, Formanek and coworkers [112] demonstrated the 3D fabrication of metallic micro/nano structures based on TPL combined with electroless plating. MLAs were employed to produce numerous structures of the same pattern spread out over a large area. They also claimed their experiment to be one of the firsts to demonstrate the fabrication of true 3D metallic complex structures obtained from TPL. In their fabrication process, TPL was conducted inside a chemically modified resin through MLAs producing multiple structures on a previously deactivated glass substrate. Initially, to obtain a deactivated substrate, glass slides were cleaned, dried and soaked in a 5% solution of dimethyldichlorosilane in toluene for 1 min and later washed with methanol producing a layer of hydrophobic coating. Subsequently, after the microfabrication process, the surface of the structures was pretreated with SnCl2 to improve metal adhesion with the polymerized resin. Finally, silver was deposited via electroless plating producing thickness-controlled, uniformly coated self-standing metallic 3D structures as shown in Fig. 22 (b). This metallization technique can produce numerous highly conducting structures by making the substrate hydrophobic or can also produce hundreds of isolated insulators on a metallic coated substrate.

Fig. 22. (a) SEM images of 2D polymer structures selectively coated with small silver grains. (b) SEM images of a silver-coated polymer structure consisting of a cube supporting a helical structure. From [112].

Rill et al. [113] demonstrated the fabrication of photonic metamaterials via TPL combined with silver chemical vapor deposition. Their experiment started with a glass substrate covered with a 2 µm thick, fully polymerized SU8 film. Subsequently, another film was spin-coated on the substrate and the template was fabricated via TPL. The template was later coated with SiO2 via ALD with SiCl4 as the precursor, which was performed to provide mechanical stability, thermal resistance, and chemical protection to the template during CVD of silver at 160 °C. The activation of the coated surface was performed by exposing it to O2 plasma for 15 min. Each static cycle during CVD deposited approximately ~ 5 nm of silver. The structures fabricated by them are results of 10 CVD static cycles, i.e., deposition thickness close to ~ 50 nm. The crucial aspects that can be inferred from their method are that the coating is uniform even in 3D, silver exhibits good dc conductivity, magnetic permeability and exhibits magnetic resonance at near-infrared frequencies with Re(µ) less than 0 and the process is quick.

Line of sight deposition techniques such as thermal evaporation and sputter coating can also be integrated with TPL to realize true three-dimensional metamaterial structures rapidly. Sputtering techniques are mostly used for the deposition of metal and oxide films by controlling the crystalline structure and the surface roughness. For effective sputtering, the bombarding ions and the atoms being bombarded must be of the same atomic weight to maximize the momentum transfer. Several groups [114,115] have utilized this technique along with TPL to deposit metal nanoparticles in order to obtain desired magnetic properties. Sadeqi and colleagues [116] took it further by fabricating several kinds of metamaterials via TPL. Metallization was performed by two techniques namely, stamping (manually dipping the template into a metal paste) and sputtering. They compared the two fabricated structures and proved experimentally that the sputtering technique produced relatively uniform thickness coating. However, in one of their novel metamaterials embedded geometrical optics (MEGO) designs called an omni-directional hemispherical moth-eye absorber which resembles a moth-eye as shown in Fig. 23 (a), the coating was performed via stamping. This was because sputtering and wet etching are not suitable for curved substrates. They also claim that this metamaterial is the first ever realization of an angle-insensitive narrow-band metamaterial absorber fabricated on a curved substrate. They state that such elements can be incorporated in future cloaking devices for enhanced optical properties. Integration of an optical parabolic reflector with a frequency selective metamaterial-based transmissive filter was also performed to realize a unique parabolic MEGO reflector device as shown in Fig. 24. It was designed in such a way that the MEGO reflector reflects the beam at a single focal point for selective frequencies where a detector can be placed.

Fig. 23. a) CAD model of the moth-eye MEGO absorber. b) fabricated and silver-coated structure. c) schematic of the device in different propagation angles as a function of . d) transmission spectrum of the absorber as a function of . From [116].

Fig. 24. Fabrication flow of MEGO parabolic reflector. From [116].

Fabrication of complex structures incorporating hybrid polymer templates with multiple photoresists deposited on top of another and polymerized sequentially has also been demonstrated [117]. Integration of various techniques to write on the different resists can be performed to obtain enhanced material properties with superior resolution. Generally, two different photoresists are deposited on the same substrate over one another and the top layer is used as the sacrificial stencil mask which is later removed via appropriate post-treatment. This technology can be utilized to a good advantage for the deposition of metal nanoparticles uniformly onto the polymer templates for various photonic and metamaterial applications. However, while TPL allows for the fabrication of arbitrary complex structures, the feature sizes obtained are too large for optical frequency applications. Although STED and PInSR are considered to produce enhanced structures with superior resolution, the metallization of the template selectively becomes an issue.

Staude and team [118] overcame this problem by exploiting the technology of multiple photoresists and integrating several technologies like TPL and EBL in combination with metal-evaporation and standard lift-off procedure to produce high resolution 3D metallic structures coated selectively and evenly as shown in Fig. 25. In their technique, initially, the glass substrate was spin-coated with IP-L negative tone photoresist and TPL was performed and developed to produce permanent 3D polymeric structures. Eventually, the developed photoresist was sputter-coated with ITO to prevent charge accumulation at the surface during EBL. Later, a sufficiently thick layer of PMMA was spun onto the same substrate over the polymerized structure. EBL was then employed to write high resolution pattern on the PMMA. After the development of PMMA, a 50 nm thick gold film was deposited onto the sample pattern by electron beam evaporation. Finally, a standard lift-off procedure was used to remove the PMMA and these gold patterns were embedded onto the two-photon polymerized structures uniformly at desired locations as shown in Fig. 26. However, the standard lift-off procedure does have some major drawbacks such as retention and redeposition.

Fig. 25. Schematic of the process of hybrid 3D nanofabrication via TPL and EBL. From [118].

Fig. 26. SEM images of several structures realized from hybrid nanofabrication technology. (a) selective metallization. (b) Short gold nanowires aligned with respect to the TPL photoresist lines to form Ω-shaped structures. (c)-(d) 3D nanoantennae inspired by various designs. (e) A 2D array of upright standing SRRs. From [118].

Although TPL has been considered a promising technology for the development of photonic crystal (PhC) design largely due to its 3D processing capability, many researchers have failed to fabricate structures demonstrating the photonic bandgap (PBG) effect. Considering a log pile structure, it is certain that the rod diameter is not uniform due to the acceleration and deceleration effects of the piezo stage moving with a uniform scan speed. This directly hinders the flow of non-polymerized resin resulting in incomplete resin-air lattices. Similarly, the height and width of the structure are defined by the laser exposure dose and numerical aperture of the objective. Since they cannot be manipulated independently, the lattice parameters including in-plane filling ratio and layer interval are difficult to optimize separately. These factors have been identified as the major reasons for no PBG effect in PhC fabricated via TPL. Sun et al. [35] proposed and demonstrated a technique to overcome this problem. They quantified the entire 3D space of fabrication into uniformly spaced voxels instead of defining a rod by single-line scanning. During fabrication, the laser raster scans the entire 3D space but stops only at certain prescribed points for a given duration and irradiates the resin to define the structure. Additionally, structure deformation in the transition layer was also taken care of by using a pre-compensation strategy considering that the shrinkage rate from liquid to solid is material dependent and experimentally measurable.

In another approach by Yinan and coworkers [119], a self-developed sol–gel material prepared using a sol–gel organic–inorganic hybrid technology called the SZ2080 was utilized for the fabrication of photonic crystals via TPL. This material possessed a unique property of very negligible shrinkage eventuating before polymerization. This material was employed to fabricate 3D woodpile structures with periodicity in every four layers. Although the ratio of the refractive indices of SZ2080 and air was around 1.5 (less than2), band gaps were observed along the stacking direction.

Besides magnetic resonance imaging, a perfect lens with resolution beyond the governing diffraction limit is one of the most sought out subjects. Resolving two points with spacing less than , where n is the index of refraction, is impossible with conventional optical systems. This is because the features of the object are carried by evanescent waves which exponentially decay before reaching the image plane. In order to overcome this, Pendry [120] proposed a negative index of refraction (NIR) metamaterial which uses the properties of both evanescent and propagating waves forming a perfect lens for imaging beyond the diffraction limit. Fig. 27 depicts the working of a NIR metamaterial for imaging.

Fig. 27. A NIR metamaterial initially bending light to a negative angle with the normal and eventually converging to a point on the back of the lens forming an image. From [120].

Transformational optics is a rapidly growing branch of optics that deals with precise control of light path by spatially tailoring the material property governed by Fermat’s principle, the law of refraction, and Snell’s law. Invisibility cloaking and superlensing are some of the applications of transformational optics. As depicted in Fig. 28 (a), a superlens consists of a thin silver slab separated from the object (“NANO”) by a spacer layer and coated with an imaging material on the opposite side carefully designed in a such a way that the surface plasmons match the evanescent waves from the object. The key aspect for perfect lensing is the enhancement of the evanescent waves by the surface plasmons. Although experimentally many groups failed to achieve resonance with accurate thickness, Fang et al. [100], as shown in Fig. 28 (a), experimentally demonstrated this technique using a 35 nm silver film evaporated over a PMMA spacer layer and a subsequent 120 nm thick coating of a negative photoresist and the object being placed at a distance of 40 nm from the film. Atomic force microscopy (AFM) was used to develop the image obtained from the superlens, which resulted in a 60 nm half-pitch resolution. The captured image replicated the fine features of the object mask in all directions consistently as seen in Fig. 28 (b).

Fig. 28. (a) The object to be captured is inscribed onto a 50 nm thick chrome (Cr); at left is an array of 60 nm wide slots of 120 nm pitch. At right is the inscription of the word “NANO”. All separated from the 35 nm silver film by a 40 nm PMMA spacer layer. The image is recorded by the photoresist on the opposite side. (b) FIB image of “NANO” (top); AFM of the captured image (center) on the photoresist; AFM of the captured image (bottom) without the 35 nm silver film. The average line width obtained with the silver film was found to be ~ 89 nm and without the film was found to be ~ 321 nm. From [100].

Mechanical metamaterials are also of great interest to researchers for their myriad of applications in mechanics. Among the various types of mechanical metamaterials, lightweight lattice structures are found to be fascinating. Most of the rigid materials with high strength-to-weight and stiffness-to-weight ratios such as diamond, metallic glass, or ceramics possess excellent strength and are light. However, their sub-par elastic nature and very low toughness hinder their suitability to many applications in lightweight mechanics. Composites are a good alternative, but again the increase in weight makes them less viable to lightweight applications. Many naturally occurring materials such as sea sponge euplectella aspergillium [121] and diatom shells [122], which are coincidently stiff, tough, and light, are believed to arise mainly from the design of the structural components. Meza et al. [123] fabricated a strong, stiff, energy absorbing, hollow-tube nanolattice with an octet truss geometry that exclusively consists of brittle ceramic and alumina as shown in Fig. 29 (D-E). This structure exhibited almost complete recoverability after compressions in excess of 50% strain. It was also found that Young’s modulus varied with the relative density as , and the ultimate strength varied with a density as which differed from the analytical values because of the hollow tubes and nodes. The fabrication was performed starting with the two-photon polymerization of the 3D scaffold. Subsequently, thin layers of alumina were then deposited onto the scaffold via the atomic layer deposition (ALD) technique. The outermost walls of the structure after coating were removed via focused ion beam milling and the polymer scaffold was removed via O2 plasma etching resulting in a 3D free standing nanolattice. Because ALD is a layer-by-layer approach, the thickness of the resulting layers of structures is entirely controlled by the number of deposition cycles.

Fig. 29. (A) CAD image of the octet truss design. (B) A single unit cell. (C) Elliptical cross-section of the hollow octet truss tube. (D) SEM image of the alumina octet-truss nanolattice. (E) Zoomed-in image. Inset showing the hollow tube. (F) TEM dark-field image with diffraction grating o the alumina nanolattice tube wall. From [123].

In 2016, Bauer and group [124] took it further by fabricating ultra-strong, lightweight, nano-architectured glassy carbon lattices via TPL and pyrolysis of the polymeric structures which could sustain compression stresses of up to ~ 310 MPa at a density of 0.35 gcm−3. The polymeric nanolattices and nano-honeycomb structures were constructed by direct laser writing and subsequent pyrolysis in vacuum at a temperature of 900 °C. During pyrolysis, these structures shrank isotropically by 80% in volume. Undistorted structures were obtained by placing the structures on pedestals and coiled springs for easy removal from the substrate as shown in Fig. 30. Interestingly, these significantly reduced feature sizes displayed unprecedented strength corresponding to the theoretical strength of bulk glassy carbon. Furthermore, these nanolattices represent the smallest lattice structures produced so far in published literature. It is evident that such structures possess great potential for mechanical and optical metamaterial applications.

Fig. 30. (a) Polymeric 3D structure before pyrolysis. (b) Magnified image of a single unit cell. (c) shrunk nanolattice on a pedestal for easy removal from the substrate. (d) magnified image of the shrunk nanolattice. From [124].

In order to characterize mechanical metamaterials appropriately, the height of the overall structure must be high. TPL allows fabrication of structures typically with height in the range of tens of micrometers only. However, Buckmann et al. [125] proposed a novel “Dip-in” 3D DLW technique as shown in Fig. 31 (i). This method makes use of the photoresist itself as the immersion oil between the objective and the substrate extending the fabrication to millimeters in height with sub-micrometer feature sizes. They fabricated true 3D crystalline metamaterials in the micrometer range exhibiting adjustable Poisson’s ratios including negative values, i.e., compressing the material in the axial direction leads to a contraction in at least one of the lateral directions in uniaxial structures as well as huge mechanical nonlinearities starting from zero Poisson’s ratio. The structural design of their metamaterial was inspired by the bow-tie functional element represented in Fig. 31 (ii) (a). Fabrication of the 3D structures was performed by grouping these functional elements in different orientations as shown in Fig. 32. These structures were later subjected to compression loading along the z-direction to determine the Poisson’s ratio.

Fig. 31. (i) a) Schematic of regular DLW. b) Schematic of novel “Dip-in” 3D DLW. (ii) a) Bow-tie functional element. b) Triclinic crystal structure with four-fold rotational axis (left) and six-fold rotational axis (right) respectively. (c) top view of the respective images in (ii)b). From [125].

Fig. 32. (a)-(c) triclinic structures with the four-fold axis of rotation. (d) triclinic structure with a six-fold axis of rotation. Lower row images are magnified images of the respective images on the top row. From [125].

Similarly, in 2012, the same group [126] demonstrated the fabrication of pentamode three-dimensional mechanical metamaterials applying their novel “dip-in” 3D DLW with double cone lengths in the ten-micrometer range and with figures of merit (FOM) exceeding 103.

Pentamodes, also sometimes known as metafluids, were first proposed by Milton and Cherkaev in 1995 [127]. They are unique because of their property that avoids coupling of compression and shear waves by increasing the bulk modulus, B, to an extremely large value ( relative to the shear modulus, G. However, for the stability they replaced the fine point-like tips suggested by Milton and Cherkaev with thicker overlap values (d ~ 0.55 µm) leading to the figure of merit values in the range of ~ 103. Figs. 33 and 34 represent the design and TPL fabrication of the pentamode metamaterial, respectively. Furthermore, they believe that a reduction of the critical dimension to 0.1 µm can increase the FOM to 104.

Fig. 33. (a) Depiction of pentamode metamaterial suggested by Milton and Cherkaev. (b) Illustration of the approximated pentamode ideal with a finite diameter, d in the connecting regions of the touching cones. (Right) Dependence of FOM with the diameter, d. From [126].

Fig. 34. (a) Electron micrograph of a polymer pentamode mechanical metamaterial fabricated by “dip-in” 3D DLW with h = 16.15 µm, D = 3 µm and d = 0.55 µm. (b) another pentamode metamaterial with h = 16.15 µm, D = 3 µm and d = 1 µm. From [126].

Cloaking has been demonstrated in electromagnetism at various frequencies by many groups [91–93,128,129]. Although the implementation of cloaking in 3D structures is a cumbersome task, TPL has been a promising technology to achieve 3D cloaking. Fischer and his team [77] were among the first groups to demonstrate three-dimensional invisibility cloaking in the microwave frequencies (~1.5 µm). They employed STED-inspired-TPL to fabricate the 3D woodpile-like polymeric structures. In 2011, the same group [130] demonstrated 3D invisibility cloaking in the optical frequencies by miniaturizing all the features in their previous experiment by a factor of 2. The lattice spacing was scaled down to 350 nm from 800 nm and according to Maxwell relations, the operating wavelength scaled down from 1.5 µm to 0.7 µm (visible red). The modifications in this experiment from the previous experiment included the employment of a different photoinitiator and a suitable phase mask for enhanced lateral and axial resolutions. In order to investigate the action of cloaking, they fabricated two structures: 1) reference structure and 2) cloaking structure with a cos2-function like indentation on the top surface of both as shown in Fig. 35 (a). Both these structures were sputter-coated with 100 nm of gold after TPL for enhanced visibility prior to the indentation. Cloaking of the indentation was effectively seen when the surface was exposed to light with a wavelength in the range of 500–900 nm as illustrated in Fig. 35 (c).

Fig. 35. a) Electron micrograph of (top) polymer reference and (bottom) cloaking structures. (b) Corresponding FIB cuts of the structures. (Right) Optical micrographs of illumination wavelengths ranging from 500 to 900 nm. From [130]. The length scale is 10 µm.

In 2013, Buckmann and his group [131] ventured into mechanical cloaking utilizing pentamode metamaterials via their novel “dip-in” 3D DLW technique. Experimental demonstration of a core–shell-based elasto-mechanical cloak was achieved. As shown in Fig. 36, the structure consisted of three crucial components. First, the rigid core–shell wall protects any object placed inside it. Second and third being the homogeneous, isotropic surroundings with high FOM to make the core–shell geometry appear elastically as its surroundings. Intuitively, the system can be compared to a spring-mass mechanical system with three different spring constants: one is very stiff and the other two are made softer to compensate for the effect exhibited by the stiff spring and finally, the effective spring constant is the same as that of system with three identical springs. The structure had a macroscopic volume of 2 mm3 with 1024 face-centered cubic unit cells and a lattice constant of 125 µm. Three structures were fabricated: 1) without the core–shell rigid wall acting as the reference, 2) Core-shell wall with a homogeneous isotropic surrounding of B/G ratio ~ 120, and 3) cloaking structure consisting of the core–shell, homogeneous and isotropic surroundings with two regions of B/G ~ 908 near the core–shell wall and B/G ~ 120 away from the wall. Finally, these structures were subjected to compression loading from the top using a hard silicon stamp and the results obtained successfully displayed elastic “unfeelability” as shown in the plots in Fig. 37. Detailed reviews on optical, mechanical, and acoustic metamaterials fabricated using several techniques can be found in ref. [132–135].

Fig. 36. a) Pentamode mechanical metamaterial with a rigid core–shell element surrounded by a flexible homogeneous, isotropic polymer material. b) magnified view of the near surroundings of the core–shell element. From [131].

Fig. 37. (left) a) Reference structure without the core–shell element. b) Reference structure with the rigid hollow semi-cylinder (obstacle). c) Cloaking structure with core–shell element and varying surroundings as specified. (d-f) magnified view of (c) with dimensions farther and near surroundings respectively. (Right) Optical photograph of the cloaking structure when subjected to compression loading from the top. As shown the strain is similar for both the reference and cloaking structures. From [131].

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Research progress on terahertz achromatic broadband polarization wave plates

Optics & Laser TechnologyJournal2023, Optics & Laser Technology

Yandong Gong, ... Kai Pang

3.3 Based on metamaterial

In past few years, the discovery of metamaterials has provided new prospects for the engineering of THz WP [20]. Metamaterial is a non-natural artificial microstructure material. It possesses a subwavelength periodic structure and, via flexible design, may accomplish specific functions that ordinary materials cannot. Metasurface is a simpler two-dimensional version of metamaterial. It may alter the amplitude and phase of the light source at any point on the wave plate by adjusting the resonance strength and the abrupt phase induced by resonance, to regulate the propagation, polarization, and form of the light beam.

3.3.1 Metal-Insulator-Metal (MIM) structures

A thin insulating layer is sandwiched between two metal layers to create this sort of WP. The insulating layer is designed to have a specific dielectric constant, and metal structures are often composed of elements with electric or magnetic dipole resonance, such as metal nanorods or line pairs can be equivalent to electric or magnetic dipoles, and the scattered light excited by electric or magnetic dipole resonance has a phase difference with the incident wave, driven by strong birefringence [21]. MIM wave plates provide great transmission efficiency and minimal insertion loss. Grady et al. [22] reported a HWP consisting of an array of metal cutting lines, where the metal film and the array of cutting lines form an F-P-like cavity to excite dipole oscillations, emulating an ultra-thin and efficient linear polarization converter. A dual-wavelength achromatic deflector aiming to circumvent the strong wavelength dependence has been constructed and verified experimentally by Ding et al. [23], which can deflect the normal incident wave in one anomalous direction at is 240 or 750 . A sufficiently thick metal film can reflect all the incident light, so MIM structures are often used in the design of reflective WP. Xia et al. [24] developed a THz HWP in reflection mode based on a metasurface with anisotropic structure, comprising of two pairs of patches in a cell, working in the broadband frequency range of 0.67 to 1.66 THz, and the polarization conversion rate (PCR) exceeds 88% (in Fig. 3(a)). These may, however, be hard to produce, and the metal layers can introduce significant optical losses and absorption.

Fig. 3. THz WP based on metamaterial. (a) A schematic illustration of a broadband HWP, reflectance, and PCR of incident x-polarized and y-polarized waves [24]. (b) A conceptual description of the metasurface based on two identical dielectrics and its simulation of electric field distributions, related diffraction angles, measured cross-polarized transmission spectra, and quality analysis of the output polarization state component [26]. (c) The polarization performance and schematic concept of a broadband QWP [28]. (d) Structure and schematic diagram of chiral metasurface and numerical simulation of cross-polarization transmission coefficients, rotation azimuth angle, ellipticity, and circular dichroism [29].

3.3.2 Dielectric metasurface

This sort is made by patterning a thin dielectric layer in a way that allows it to function as a WP. Benefits of a dielectric metasurface include superior transmission efficiency, low insertion loss, and strong temperature stability. Using elliptical air holes, Zi et al. [25] mentioned an all-dielectric metasurface THz QWP. Given that the period of subwavelength air holes is considerably shorter than the wavelength range of THz waves, and the elliptical air holes exhibit spatial asymmetry, the effective refractive index along with direction is different, resulting in birefringence, but its bandwidth is insufficient. The geometric phase is introduced to span the whole phase space to maximize the operating bandwidth. Yang et al. [26] provided a dielectric material based on Mie resonance instead of metal components, added a phase gradient to separate the orthogonal polarization states, and obtained an output wave with a PCR of 67.5%, resulting in a wide-band THz HWP seen in Fig. 3(b). They, though, on the reverse hand, may have a restricted bandwidth and be difficult to generate with great accuracy.

3.3.3 Hybrid Metal-Dielectric structures

Metal and dielectric materials are combined in a single metasurface structure to create this kind of wave plate. The ability to accomplish broadband operation, excellent transmission efficiency, and reduced insertion loss are all advantages of hybrid wave plates. Torres et al [27] discussed achieving form by modifying the aperture array with anomalous transmission to trigger complementary capacitive and inductive responses, yielding a single-layer QWP working in two separate THz bands. The transmission light dispersion, yet another side effect, is rather considerable, leading to a restricted working band of 16.8% and 2.9% in the experiment at 1 and 2.2 THz. Liu et al. [28] stated a wavy resonator-based single-layer THz QWP illustrated in Fig. 3(c). The combined arrangement of two-line resonators, with a thickness of roughly 37 , has a broadband impact on the metasurface due to the amplitude and phase characteristics in the non-resonant zone. The polarization state and conversion efficiency can also be effectively adjusted by controlling the incidence angle. The size of the WP described above is on the order of , which is more in line with existing miniaturization needs and, to some extent, the complexity of THz functional devices is simplified.

3.3.4 Chiral metamaterials

The WP of this kind are made by constructing a metamaterial with a chiral response and have a broadband function and remarkable transmission efficiency. Lv et al. [29] conveyed a bilayer chiral metasurface formed of distorted S-shaped metal patterns with broken symmetry in the direction of light transmission indicated in Fig. 3(d), causing high coupling of orthogonal linear polarization states between the two layers. The asymmetrical transmission (AT) effect is remains stable within an incidence angle of 60°, making it appropriate for manipulating various linearly polarized waves. The anisotropy and chirality of the metamaterial give rise to cross-polarization conversion with a ratio over 0.8 and a full width half maximum bandwidth greater than 0.50 THz. Whereas, they can be difficult to manufacture and may have poor polarization selectivity.

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Review article

The state-of-the-art and emerging design approaches of double-tuned RF coils for X-nuclei, brain MR imaging and spectroscopy: A review

Magnetic Resonance ImagingJournal2020, Magnetic Resonance Imaging

Chang-Hoon Choi, ... N. Jon Shah

2.2.4 Metamaterial-inspired design

More recently, interest in using metamaterials for MRI has increased. Metamaterials are artificial composite periodic structured materials, which are characterised by the values of effective permittivity and permeability, and offer a unique platform for controlling the propagation of electromagnetic fields [146,147]. The use of metamaterials has proven to be a promising approach in MRI to boost SNR, particularly at ultra-high field where the RF penetration and B1 homogeneity are deprived [148–152]. Metamaterials are capable of shaping the main B1 field distribution of the deeper level efficiently and can be used to enhance SNR and homogeneity in the target regions of interest.

The metamaterial contains one or more conductor paths/patterns tuned to the appropriate frequencies which can be used to achieve single- or double-tuning, or more. Most of the double-resonant applications using metamaterials are for close frequencies, i.e.1H/19F [153–156]. The double-tuned coil introduced by Hurshkainen et al operates at hybridised eigenmodes in two mutually orthogonal periodic structures [14]. In this study, depending on the opted eigenmodes, the field distribution was independently controlled at two close frequencies. One notable advantage of this metamaterial-inspired feeding loop is that it does not need any physical lumped capacitors or inductors for tuning and matching. Furthermore, using a similar technique, Ivanov et al demonstrated that the metamaterial-inspired coil could be tuned for a broad range X-nuclei [157]. Vergara Gomez et al used two parallel coupled-wire structure to resonate it at the 1H and 19F frequencies. They demonstrated that the SNR of the coil was as high as that of the surface coil and that the coverage was not worse than that of a large birdcage volume coil [155]. Moreover, Yang et al utilised an interdigitated capacitive metasurface to show the capability of double-tuning. They positioned this material between the feeding coil and the object and evaluated the B1 in comparison to that of the single-tuned conventional coil. They observed that the metamaterial inspired double-tuned coil provided a comparable performance for the X-nucleus signal and significant enhancement for the 1H signal [156]. Schmidt and Webb proposed a novel dual-band metamaterial pad, as shown in Fig. 9a. This design combines an electric dipole mode for the X-nuclei (here 31P) frequency and a magnetic dipole mode for the 1H frequency [158]. They constructed multiple patterns in one plane and acquired data from both 1H images and 31P spectra with substantial improvement in SNR - a maximum of 1.8 times higher for 31P and 2.1 times for 1H, as shown in Fig. 9b and c. Despite these clear advantages, using such a metamaterial is a relatively new approach and there are a number of different structures and arrangements to be investigated in terms of designing a double- or multi-tuned coil.

Fig. 9. A schematic diagram of a double-tuned,

P metamaterial pad and its characteristic view (a) which combines two different patterned metamaterial structures. Fig. 9b shows the B

maps with (w) and without (w/o) the support of the metamaterial pad. The substantial enhancement of efficiencies at both frequencies can be clearly seen with the metamaterial. The Fig. 9 a) is reused and b) is redrawn from the reference (https://pubs.acs.org/doi/10.1021/acsami.7b06949) [158] with the copyright permission granted from ASC.

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Review article

Ultrafast laser nano-structuring of photopolymers: a decade of advances

Physics ReportsJournal2013, Physics Reports

Mangirdas Malinauskas, ... Saulius Juodkazis

5.2 Metamaterials

Metamaterials are artificial materials with properties that do not exist in nature; these properties are due to structure and not material composition. Their name derives from the greek word ‘meta’, which means beyond, because these materials have properties that extend beyond materials found naturally. In Metamaterials, an assembly of structures can replace the role that atoms and molecules have in conventional materials, resulting in a composite structure with electromagnetic properties beyond anything that can be found naturally, or chemically synthesized. An excellent tutorial on Metamaterials can be found in the website of Prof. David R. Smith (Duke University) [199].

Photonic Metamaterials consist of nanostructured metallo-dielectric subwavelength building components, and allow the realization of many new and unusual optical properties, such as negative refractive index, magnetism at optical frequencies, perfect absorption, and enhanced optical nonlinearities. Several applications of Metamaterials have been proposed, including ultrahigh-resolution imaging systems, compact polarization optics and cloaking devices ([200] and references herein). The realization of these applications requires the fabrication of large-scale metallo-dielectric structures, a very challenging task. There has been some limited research into the direct fabrication of metallic 3D structures using multiphoton reduction of metal ions. The quality of the structures, however, has been compromised by the reduced transparency of the metal ion solutions at the laser wavelengths used (500–800 nm) [201,202]. Metallic woodpile structures have also been realized experimentally at micron wavelengths using traditional lithographic techniques [203–205]. However, lithographic techniques can accommodate only a very limited number of layers, and aligning each layer with the previous one is difficult.

DLW is the only inherently 3D fabrication technique, with the potential to fabricate 3D structures, but the majority of the materials structurable by DLW are dielectrics. A popular approach is to use DLW to make dielectric structures, and subsequently metallize them. The most successful approaches and their advantages and disadvantages are listed in below:

1.DLW of positive photoresists and filling with gold using electroplating [117,206]. Here, voids are created in a positive tone photoresist using DLW; these are subsequently filled with gold using classic electroplating (Fig. 24). The main advantage of this technique is that there is no need to remove the photoresist as the refractive index contrast between the gold and the dielectric material is very high. The main disadvantage is that the number of designs that can be structured is limited, as the right apertures of the material removal and gold filling have to be allowed; therefore, this is fundamentally a 2.5D structuring technique.2.DLW of dielectric structures and non-selective metallization with electroless plating. Here, a standard photolithographic material is used, such as SU-8, for the fabrication of the structures, and subsequently their surface is covered with silver using classic electroless plating [144,207–209]. Additional processing, to enable the metal adhesion on the surface, is required and the quality, structural integrity and resolution of the structures depends on the building material and the surface-processing step. The advantage of this technique is that any photopolymer can be used; the disadvantages are that, as the density of the metal binding sites on the structure cannot be controlled, the metallization quality can vary. In addition, along with the surface of the structures, the substrate is also activated; the metallization is therefore not selective, often requiring an extra step to remove the structures from the metallized substrate [207].3.DLW of dielectric structures and selective metallization with electroless plating. Here, a composite doped with the metal binding sites is employed for the structure fabrication [129,168,169]. The main advantages in this case are that the metallization is selective, and the density and distribution of the metal binding sites can be controlled. The main disadvantages are that specific metal-binding materials need to be used, and in most cases these were not able to provide the required resolution and structural integrity required for optical Metamaterials. Only very recently it was shown that it is possible to use this method for the fabrication of optical nanophotonic devices [169].4.DLW of dielectric structures and metallization with Chemical Vapor Deposition (CVD). Here DLW is used to fabricate 3D structures by any material, typically SU-8 [143]. The surface of the structures is subsequently activated using plasma etch. Finally, they are covered with silver using CVD. The main advantage of this technique is that any photopolymer can be used, and the resolution and final quality of the structures will depend on that. The main disadvantages are firstly, CVD can only penetrate a small number of structure layers, allowing only a small number of unit cells, and secondly, there is no selectivity, as with the plasma etch, the substrate is activated as well as the structure.

It should also be noted that there has been recently some work recently structuring by DLW transparent conducting materials, such as ionogels [210,211]. However, neither the conductivity nor the resolution of these materials are sufficient for applications in optical Metamaterials.

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Review article

Dielectric nanoresonators for light manipulation

Physics ReportsJournal2017, Physics Reports

Zhong-Jian Yang, ... Hai-Qing Lin

7.3.1 Metamaterials

Metamaterials are artificial electromagnetic media consisting of periodic subwavelength metal/dielectric structures at a scale smaller than their operation wavelengths. They exhibit unusual properties that are not found in nature. Metal/dielectric structures as the building blocks are usually resonantly coupled to incident electromagnetic fields. Over the last decade, great progresses have been made on metamaterials that exhibit unique properties, such as negative refraction [249,250], perfect lensing [251,252] and cloaking [253,254]. Most of reported metamaterials are constructed with resonant metallic structures. However, their high intrinsic material losses limit the efficiencies and functionalities of the resultant metamaterials. The losses can be reduced by decreasing the light propagation length in the metamaterials made of metallic structures. The reduction in the propagation length can be realized by designing two-dimensional metamaterials, which are also often called metasurfaces. Another way to reduce the loss is to use low-loss DNRs as the building blocks [7,9,46]. Earlier dielectric metamaterials (DMMs) were demonstrated in frequencies mainly from the mid-infrared to microwave region [7,71,86,255,256], where the DNRs are at the micrometer scale or have larger sizes. Recently, using DNRs to construct DMMs working at optical frequencies has become an attractive and important research direction. Up to date, most of the realized DMMs based on DNRs are dielectric metasurfaces (DMSs). DMSs will be discussed in the next section. In the following part of this section, we will focus on three-dimensional DMMs. The research on DNR-based three-dimensional DMMs is still in its infancy probably owing to the limitations in current fabrication technologies. However, there have already been works demonstrating that three-dimensional DMMs can exhibit unique fantastic properties.

Light inside zero-index metamaterials (ZIMs) experiences no spatial phase changes and possesses an extremely large phase velocity. ZIMs can therefore be utilized for realizing many photonic devices [257,258]. Impedance-matched ZIMs require both the permittivity and permeability to be zero. ZIMs have been proposed to be achieved at the Dirac point in the band structure of DMMs [14]. Fig. 25 (a)–(f) show an experimental realization of an impedance-matched dielectric ZIM at optical frequencies [15]. The fabricated ZIM consists of 200-m-long Si rods that support electric and magnetic resonance modes and are separated by low-index SiO rods. Two features that characterize the ZIM properties have been experimentally investigated to confirm the realization of an impedance-matched ZIM at optical frequencies. One feature of ZIMs is that light incident from free space can only be transmitted over a narrow range of the incidence angle. This effect is caused by the phase-matching at the interface, which requires that the wavevector along the interface be conserved. As a result, the wavevector is restricted to extremely small values, causing light incident at high angles to be reflected while nearly normally incident light is transmitted. The Fourier-plane images of the transmitted light confirm the angular selectivity of transmission in the fabricated ZIM (Fig. 25 (b) and (c)). The other feature is that incoherent isotropic emitters placed within a ZIM tend to radiate in the direction normal to the air–ZIM interface. In the experiments, PbS semiconductor quantum dots were placed within the fabricated ZIM to act as the emitters (Fig. 25 (d)). The quantum dots have a PL peak centered at 1420 nm. The Fourier-plane images (Fig. 25 (e) and (f)) show good angular confinement in the y-direction from the ZIM in comparison with the unstructured sample. These experiments confirm the successful realization of an impedance-matched ZIM.

Fig. 25 (g) and (h) demonstrate a multilayered DMM with high broadband reflectivity [259]. As revealed by the false-color SEM image (Fig. 25 (g)), the multilayer sample is composed of 300-nm-thick GaAs DNRs separated by 300-nm-thick AlGaO native oxide layers. The fabrication process allows for the formation of a low-refractive-index native oxide layer between the resonators and the semiconductor substrate. The measured reflectivity spectrum of the structure agrees well with the simulation results (Fig. 25 (h)). The multilayer structure exhibits a higher reflectivity than the gold mirror over a broad (200 nm) spectral range.

Fig. 25. DNR-based metamaterials. (a) False-color focused ion beam image of a ZIM before spin-coating PMMA. The inset shows the cross-section of the structure after PMMA filling. The fabricated sample has a total of 11 Si and SiO layers that are alternate with each other. The widths of the Si rods are 270, 280, 310, 320 and 380 nm, from top to bottom, respectively. (b) Fourier-plane image of a beam passing through the fabricated ZIM within the low-index band at 1450 nm. (c) Fourier-plane image of the illumination beam showing the uniform intensity over the measured angular range. The color bars for (b) and (c) are the same. (d) Schematic of the directional emission from the quantum dots embedded within the ZIM. (e) Fourier-plane image of the emission from the quantum dots deposited on a substrate. The intensity has been scaled by a factor of two. (f) Fourier-plane image of the emission from the quantum dots embedded within the ZIM, showing the enhanced rate and directivity of the spontaneous emission. The color bars for (e) and (f) are the same. Reprinted with permission from Ref. [15]. Copyright 2013 Nature Publishing Group. (g) False-color SEM image of a sample consisting of three layers of GaAs nanoresonators separated by AlGaO native oxide. The green, brown, and yellow colors represent the AlGaO layers, the GaAs resonators, and the GaAs substrate, respectively. The tapered nanoposts have a diameter of 350 nm at the top and a diameter of 370 nm at the bottom. (h) Experimental reflectivity spectra of the sample shown in (g). The horizontal black dashed line represents 97% reflectivity from a gold mirror. Reprinted with permission from Ref. [259]. Copyright 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (i) Measured CD spectra. The CD signal, expressed in degrees, is induced by the stacking of chiral structures along the direction normal to the substrate. The inset SEM images show the helix types with the box colors corresponding to the colors of the measured CD spectra. Reprinted with permission from Ref. [260]. Copyright 2016 American Physical Society.

Helix-shaped DMMs for the chiral effect have also been studied recently [260]. Chiral metamaterials are capable of inducing chiral responses with circularly polarized light because of the three-dimensional variation of the building blocks along the light propagation direction. Circular dichroism (CD) is usually used to characterize the chiral response of a chiral structure. The CD signal can be expressed as:

(21)

where is the ellipticity, and are the transmissions of left- and right-circularly polarized light at normal incidence, respectively. In the reported work, high CD signals have been observed from the helix metamaterials. The helix nanowire (Fig. 25 (i), inset) consists of a thick amorphous shell of carbon encapsulating a tiny core of small precipitates of amorphous Ga. It was fabricated on the basis of focused ion beam-induced deposition. Fig. 25 (i) shows that the difference on the light transmission curve between the two circular polarization states gives two CD bands, with sign inversion occurring at 560 nm. The CD signal is induced by the matching of the effective light wavelength with the vertical pitch of the helical structure. The transmission bands do not shift with the number of the stacked elements, where an element corresponds to a half-pitch helix, while the CD peak value evolves from 1 degree for the unit cell up to 6°for a 1.5-turn helix. The results agree well with the numerical electrodynamic calculations. These remarkably high values of CD in the visible range are noted to be obtained by matching the structural features of the helix nanowires with the probing light.

DMMs can also be used to effectively manipulate evanescent waves to realize subdiffraction light confinement in waveguides [261,262] and super-resolution lenses [263,264]. Strong confinement of light is important as the diffraction limit of light is a fundamental barrier to the integration of nanoscale electronic devices with conventional photonic devices. The light confinement below the diffraction limit by use of DMMs is demonstrated in Fig. 26 (a)–(d) [261]. The conventional light confinement relies on total internal reflection (TIR) (Fig. 26 (a)). The requirement of TIR for light propagating from medium 1 to medium 2 is that >, where and are the refractive indexes of the two media. According to the study [261], this requirement can be relaxed to:

(22)

where is defined as the dielectric constant of medium 2 in the direction perpendicular to the interface. The z-axis is parallel to the interface and the x-axis is normal to it. This condition is called relaxed-TIR. On the basis of the momentum conservation of light parallel to the interface, the evanescent wave decay constant for TM-polarized waves in medium 2 is given by:

(23)

The penetration depth, or the skin depth, of the evanescent field penetrating into medium 2 is therefore governed by the square root of the ratio of the permittivity components . With relatively small and when , strong confinement of light can be realized (Fig. 26 (b)). This strong anisotropy cannot be satisfied by natural dielectric media. However, it can be realized by artificially structured media using available lossless DMMs. In one demonstrated approach, a multilayer structure composed of two alternating materials with a high index contrast and layer thicknesses far below the wavelength of light forms an anisotropic cladding. The anisotropic dielectric constant can be predicted by use of the multilayer effective medium theory. The simulated electric energy density in the waveguide with an all-dielectric anisotropic cladding ( and ) is shown in Fig. 26 (d). Numerical calculation reveals that about 36% of the total power can be confined inside a lower-index glass, while only 1% of the total power lies inside the core without the cladding (Fig. 26 (c)).

A super-resolution lens at visible frequencies has been realized by a three-dimensional all-dielectric metamaterial composed of assembled 15-nm NPs (Fig. 26 (e)–(i)) [263]. The formed metamaterial-based solid immersion lens (mSIL) can effectively image subwavelength structures. Fig. 26 (f) shows a gold-coated line pattern with 45-nm features. In the images created by the hemispherical mSIL, the dots can be distinguished from the lines under an illumination of white, green ( 540 nm), or blue light ( 470 nm) (Fig. 26 (g)–(i)). These subwavelength features are far beyond the resolution of traditional optical microscopes. Full-wave three-dimensional electromagnetic calculations show that the excitation of the evanescent waves in the coupled high-refractive-index and deep-subwavelength particles is responsible for the super-resolution of the mSIL. The evanescent waves are subdiffraction-confined at the nanoscale comparable to the size of the NPs. As a result, they carry subwavelength information about the object. Furthermore, owing to the lack of energy dissipation in the all-dielectric medium, the evanescent waves can be effectively guided and coupled into propagating waves, which produces far-field super-resolution images.

Fig. 26. DNR-based deep subwavelength waveguides and metalenses. (a) Conventional waveguide based on TIR. As the core size is decreased, most of the power lies outside and decays slowly in the cladding. (b) Waveguide based on relaxed-TIR. Relaxed-TIR preserves the conventional waveguiding mechanism. Furthermore, the light decays fast in the cladding as the optical momentum in the cladding is transformed using anisotropy. (c) and (d) Simulated distributions of the electric energy densities for the waveguide without cladding and the waveguide with all-dielectric nonmagnetic cladding ( and ), respectively. The color bar shown in (d) also applies for (c). When the anisotropic cladding is added, the fraction of the power inside the core to the total power is increased from less than 1% to 36%. Reprinted with permission from Ref. [261]. Copyright 2014 Optical Society of America. (e) Schematic illustration of an all-dielectric mSIL, which is constructed by densely packed NPs of 15 nm in size. (f) SEM image of the wafer pattern with a 45-nm gold-coated pitch. (g)-(i) Optical microscopy images of the mSIL focusing on the surface of the wafer pattern in (f) with a magnification factor of 3.1. The mSIL in (g)–(i) is illuminated under white, green ( nm), and blue light ( nm), respectively. Reprinted with permission from Ref. [263]. Copyright 2016 American Association for the Advancement of Science.

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Review article

A review of additive manufacturing of metamaterials and developing trends

Materials TodayJournal2021, Materials Today

Junxiang Fan, ... Yusheng Shi

Abstract

The concept of metamaterials originates from the proposal of left-hand materials with negative refractive index, followed by which, varieties of metamaterials with kinds of fantastic properties that cannot be found in natural materials, such as zero/negative Poisson’s ratio, electromagnetic/acoustic/thermal cloaking effect, etc., were come up with. According to their application fields, the metamaterials are roughly classified into four categories, electromagnetic metamaterials, acoustic metamaterials, thermal metamaterials, and mechanical metamaterials. By designing structures and arranging the distribution of materials with different physical parameters, the function of metamaterials can be realized in theory. Additive manufacturing (AM) technology provides a more direct and efficient way to achieve a sample of metamaterial and experiment verification due to the great advantages in fabricating complex structures. In this review, we introduce the typical metamaterials in different application situations and their design methods. In particular, we are focused on the fabrication of metamaterials and the application status of AM technology in them. Furthermore, we discuss the limits of present metamaterials in the aspect of design method and the disadvantages of existing AM technology, as well as the development tendency of metamaterials.

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Review article

A review of additive manufacturing of metamaterials and developing trends

Materials TodayJournal2021, Materials Today

Junxiang Fan, ... Yusheng Shi

Introduction

Metamaterials (MMs) refer to artificial structures or composite materials with extraordinary physical properties that can not be found in natural materials, such as electromagnetic/acoustic cloak [1–5], zero/negative Poisson's ratio [6–10], negative refractive index [11–15], etc. To realize these magic phenomena, some carefully designed structures with single material or multi-material were put forward, consisting of identical or gradually changing cell arrays. Similar to atoms or molecules in natural materials, these cells are basic units that determine the properties of MMs and are therefore called “artificial atoms”. According to their functionalities, currently developed MMs can be roughly classified into four categories: electromagnetic metamaterials (EMMs), acoustic metamaterials (AMMs), thermal metamaterials (TMMs), and mechanical metamaterials (MMMs), as shown in Fig. 1. All Acronyms and their corresponding meanings appeared in this review are summarized in Table 1 to make it more convenient for readers.

Figure 1. The classification of MMs based on their functionalities.

Table 1. Acronyms.

Acronym	Meaning
AAMs	Acoustic absorption metamaterials
ACCs	Acoustic cloaking carpets
ACMs	Acoustic cloaking metamaterials
AFMs	Acoustic focusing metamaterials
ALs	Acoustic lenses
AM	Additive manufacturing
AMMs	Acoustic metamaterials
BCC	Body-centered cubic
CLIP	Continuous liquid interface production
CPS	Compliant porous structure
DLW	Direct laser writing
DMRs	Decorated membrane resonators
EM	Electromagnetic
EMMs	Electromagnetic metamaterials
FCC	Face-centered cubic
FDM	Fused deposition modeling
FEM	Finite element method
FIB	Focused Ion Beam
HMMs	Hyperbolic metamaterials
IR-SPRs	Infrared surface polariton resonances
MMMs	Mechanical metamaterials
MMs	Metamaterials
NPR	Negative Poisson’s ratio
PCB	Printed circuit board
PCs	Photonic crystals
PDMS	Polydimethylsiloxane
PMMs	Pentamode metamaterials
SC	Simple-cubic
SLA	Stereolithography apparatus
SLM	Selective laser melting
SLS	Selective laser sintering
SRRs	Split-ring resonators
TMMs	Thermal metamaterials
TPMS	Triple periodic minimal surface
TPP	Two-photon polymerization

The research of MMs derived from the proposal of left-hand materials by Soviet physicist Veselago in 1968 [11], such materials would lead to many very interesting physical phenomena, such as negative refraction, electromagnetic stealth or absorption, etc. Another representative research achievement of EMMs is the proposal of the photonic crystal [16,17], in which electromagnetic waves of a certain frequency range could not propagate because there was a photonic bandgap in their band structure, just like electronic bandgap in semiconductors. That made it promising in the fields of lasers and high-quality microwave antennas. However, before the structure of EMMs with negative permittivity and permeability was designed and manufactured [12,18], the research in this field only remained a theoretical hypothesis. Subsequently, the word “metamaterials” [19] was used to refer to those who achieved exotic performance beyond the limits of natural materials, and the concept was widely accepted. These exciting works attracted more and more researchers to devote themselves to the field of EMMs. Varieties of EMMs with exotic properties were proposed, such as Electromagnetic cloak [1–3], Electromagnetic wave absorbers [20–22], Terahertz electromagnetic metamaterials [23–25], etc.

Inspired by photonic crystals, Liu et al.[26] put forward locally resonant phononic crystals, which opened the prelude to the study of AMMs. With further research, AMMs with both negative equivalent modulus and equivalent density, the two most important parameters for acoustic materials, were realized by dispersing soft rubber in water [13]. That double negative AMMs could achieve a negative refractive index. It was widely accepted that acoustic MMs all relied on locally resonant units to realize their extraordinary properties [27] until Norris [28] proposed pentamode metamaterials (PMMs) with effective density and bulk modulus could be regulated in a large range without resonant phenomenon. The mentioned PMMs had fluid properties and could decouple effective bulk modulus and density, meaning that the most important two parameters could be designed separately without affecting each other. Afterward, varieties of acoustic MMs, such as acoustic cloaking metamaterials (ACMs) [5,29–31], acoustic absorption metamaterials (AAMs) [32–35], acoustic focusing metamaterials (AFMs) [36–38], were proposed.

With further research of MMs, great progress has been made in the study of mechanical and thermal metamaterials. Auxetics referred to materials with negative Poisson’s ratio (NPR) [6,39] and was first introduced by Evans in 1991 [40], playing an important role in MMMs [41]. This phenomenon is mainly attributed to their unique microstructure and composites designs [40,42] and can improve mechanical properties such as enhanced shear moduli, indentation resistance, fracture toughness, and impact energy absorption, etc. [41]. The research of TMMs was mainly targeted at controlling the heat flux by carefully arranging the distribution of different materials and designing microstructures. Furthermore, the TMMs could be combined with other kinds of MMs to achieve multi-function.

The fantastic properties of MMs are mainly realized by designing the structure and combining various materials, the traditional processing methods, such as casting, welding, molding, etc., were time-and-labor-costing in fabricating them, and some sophisticated lattice structures even can not be manufactured. Thanks to the appearance of AM technology, the fabrication and experiment verification of MMs could be realized more conveniently and efficiently. By discretizing a 3D model of an object into several thin layers and accumulating them layer by layer, the AM technology is capable of fabricating any complicated structures in theory. Besides, AM has been greatly enriched after nearly 40 years of development since it was patented in 1979, more than 20 AM technologies have been recognized [43]. Moreover, new technologies are still emerging [44–50]. For example, Daniel Oran et al. [47] proposed a nano-scale AM technology by volumetric deposition and controlled shrinkage, which could use more than one functional material at the same time. Moreover, materials with different properties, including metals, semiconductors, and biomolecules could be utilized.

On the whole, according to the forming material state, AM technologies could be divided into wire-based, liquid-based, powder-based, and mixed liquid–powder-based types. The applicable materials contain metals, polymers and ceramics [43], and the manufacturing dimension ranges from nano-scale [45,47,49,51–54] to meter-scale, which can greatly meet the ultra-high requirements of most MMs. Moreover, AM technologies could work by automatic control software and equipment, which greatly saves laborers. The examples of the applications of AM technologies in MMs were listed in Table 2. However, it should be noted that different AM technologies have different characteristics, such as the forming materials, size, resolution, and surface quality are all of significant difference. In the aspect of MMs manufacturing, it is necessary to select the appropriate technology according to the structure and the characteristics of the required material. Fig. 2 illustrated the manufacturing characteristics (mainly referred to manufacturing size and resolution) and applied materials of some typical AM technologies, which would reveal the limitations of AM technologies in fabricating MMs with various dimensions. Though the figure is mapped with the reference to acoustic and electromagnetic waves, it has general applicability in balancing the manufacturing resolution and size of MMs. It needs to be clear that although AM technology develops rapidly, there are still some limits at the aspect of manufacturing some types of MMs, such as ultra-fine nano-scale complex structures, multi-material systems, ultra-large structures, and so on. These problems will be discussed in detail in the text.

Table 2. The examples of the applications of AM technologies in MMs.

Types of MMs	Functionality	Materials	Dimension of the sample/mm	Types of AM technologies	Reference
EMMs	Electromagnetic cloak	Photoresist	0.09 × 0.09 × 0.01	DLW	[55]
		Photo-curable resin	246 × 100 × 6	SLA	[56]
	Millimeter-Wave waveguide	Cu-15Sn powder	3.10 × 3.10 × 1.5	SLM	[57]
	Electromagnetic absorption	Conductive ABS	300 × 300 × 6.5	FDM	[58]
AMMs	Acoustic absorption	ABS	49 × 25 × 80	FDM	[35]
		Polymer	2.3 × 2.3 × 10.6		[34]
	Acoustic cloak	Plastics	–	SLA	[59]
TMMs	Ultralow thermal conductivity	Epoxy resin	–	SLA	[60]
	Improving thermal conductivity	Hexagonal boron nitride (hBN)/TPU	80 × 80 × 1	FDM	[61]
	Thermal camouflages	AlSi10Mg	6.4 × 16.8	SLM	[62]
MMMs	Elasto-mechanical unfeelability cloak	Polymer	1 × 1 × 2	DLW	[63]
	negative Poisson’s ratio	ABS + maerials	50 × 50 × 5	Multi-material 3D printing	[64]
	Vanishing shear modulus	Ti6Al4V	34.64 × 60 × 10	SLM	[65]

Figure 2. The manufacturing characteristics and applied materials of some typical AM technologies. The triangle, circle and square represent the maximum part size, minimum part size, and manufacturing resolution, respectively. The position of the symbol represents the dimension corresponded to the wavelength. The distance between two symbols represents the achievable amount of unit cells per AM technology.

In this review, we will introduce the fundamental principles of different MMs and the typical types of MMs in various fields, as well as the applications of AM technologies in them. Furthermore, the limitations of the existed MMs and AM technologies will be concluded. We emphasize that AM technology is a suitable method for MMs fabrication and conversely the research of MMs could promote the development of AM technologies.

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Review article

Superconducting Metamaterials

Physics ReportsJournal2018, Physics Reports

N. Lazarides, G.P. Tsironis

1 Introduction

1.1 Metamaterials & synthetic media: Concepts and perspectives

Metamaterials represent a new class of materials generated by the arrangement of artificial structural elements, designed to achieve advantageous and/or unusual properties that do not occur in natural materials. In particular, naturally occurring materials show a limited range of electrical and magnetic properties, thus restricting our ability to manipulate light and other forms of electromagnetic waves. The functionality of metamaterials, on the other hand, relies on the fact that their constitutive elements can be engineered so that they may achieve access to a widely expanded range of electromagnetic properties. Although metamaterials are often associated with negative refraction, this is only one manifestation of their possible fascinating behaviors; they also demonstrate negative permittivity or permeability, cloaking capabilities [1], perfect lensing [2], high frequency magnetism [3], classical electromagnetically induced transparency [4–7], as well as dynamic modulation of Terahertz (THz) radiation [8], among other properties. High-frequency magnetism, in particular, exhibited by magnetic metamaterials, is considered one of the “forbidden fruits” in the Tree of Knowledge that has been brought forth by metamaterial research [9]. Their unique properties allow them to form a material base for other functional devices with tuning and switching capabilities [9–11]. The scientific activity on metamaterials which has exploded since their first experimental demonstration [12,13], has led to the emergence of a new, rapidly growing interdisciplinary field of science. This field has currently progressed to the point where physicist, material scientists and engineers are now pursuing applications, in a frequency region that spans several orders of magnitude, from zero [14–18] to THz [19–25] and optical [3,26–28]. Historically, the metamaterial concept goes back to 1967 [29], when V. Veselago investigated hypothetical materials with simultaneously negative permeability and permittivity with respect to their electromagnetic properties. He showed that simultaneously negative permeability and permittivity result in a negative refractive index for such a medium, which would bend the light the “wrong” way. The realization of materials with simultaneously negative permeability and permittivity, required for negative refractive index, had however to wait until the turn of the century, when D. Smith and his collaborators demonstrated for the first time a structure exhibiting negative refraction in the microwaves [12]. The first metamaterial was fabricated by two interpenetrating subsystems, one them providing negative permittivity while the other negative permeability within the same narrow frequency band. Specifically, an array of thin metallic wires and an array of metallic rings with a slit (split-ring resonators), which were fabricated following the “recipies” in the seminal works of J. B. Pendry, provided the negative permeability [30] and the negative permittivity [31], respectively. The wires and the split-rings act as electrically small resonant “particles”, undertaking the role of atoms in natural materials; however, they are themselves made of conventional materials (highly conducting metals). Accordingly, a metamaterial represents a higher level of structural organization of matter, which moreover is man-made.

The key element for the construction of metamaterials has customarily been the split-ring resonator (SRR), which is a subwavelength “particle”; in its simplest version it is just a highly conducting metallic ring with a slit. The SRR and all its subsequent versions, i.e., U particles, H particles, or like particles, double and/or multislit SRR molecules, are resonant particles which effectively act as artificial “magnetic atoms” [32]. The SRRs can be regarded as inductive–resistive–capacitive () oscillators, featuring a self-inductance , a capacitance , and a resistance , in an electromagnetic field whose wavelength much larger than their characteristic dimension. As long as a metamaterial comprising SRRs is concerned, the wavelength of the electromagnetic field has to be much larger than its unit cell size; then the field really “sees” the structure as a homogeneous medium at a macroscopic scale and the macroscopic concepts of permittivity and permeability become meaningful. The (effective) homogeneity is fundamental to the metamaterial concept, as it is the ability to structure a material on a scale less than the wavelength of the electromagnetic field of interest. Although in microwaves this is not a problem, downsizing the scale of metamaterial elements to access the optical frequency range may be a non-trivial issue. The advent of metamaterials has led to structures with many different designs of elemental units and geometries, that may extend to one [33,34], two [13,17], or three dimensions [35]. One of the most investigated metamaterial designs which does not contain SRRs is the fishnet structure and its versions in two [36], quasi-two [37], and three dimensions [38,39]. However, all these metamaterials have in common that they owe their extraordinary electromagnetic properties more to their carefully designed and constructed internal structure rather than, e.g., chemical composition of their elements. Metamaterials comprising of split-rings or some other variant of resonant elements, are inherently discrete; discreteness effects do not however manifest themselves as long as the metamaterial responds linearly (low-field intensities) and the homogeneous medium approximation holds. The coupling effects, however, in relatively dense SRR metamaterials are of paramount importance for a thorough understanding of certain aspects of their behavior, since they introduce spectral splitting and/or resonant frequency shifts [40–46]. The SRRs are coupled to each other through non-local magnetic and/or electric dipole–dipole interaction, with relative strength depending on the relative orientation of the SRRs in an array. However, due to the nature of the interaction, the coupling energy between neighboring SRRs is already much less than the characteristic energy of the metamaterial; thus in most cases next-nearest and more distant neighbor interactions can be safely neglected. SRR-based metamaterials support a new kind of propagating waves, referred to as magnetoinductive waves, for metamaterials where the magnetic interaction between its units is dominant. They exhibit phonon-like dispersion curves and they can transfer energy [33,47], and they have been experimentally investigated both in linear and nonlinear SRR-based metamaterials [48–50]. It is thus possible to fabricate contact-free data and power transfer devices which make use of the unique properties of the metamaterial structure, and may function as a frequency-selective communication channel for devices via their magneto-inductive wave modes [51].

Unfortunately, metamaterials structures comprising of resonant metallic elements revealed unfavorable characteristics that render them unsuitable for most practical applications. The SRRs, in particular, suffer from high Ohmic losses at frequencies close to their resonance, where metamaterials acquire their extraordinary properties. Moreover, those properties may only appear within a very narrow band, that is related to the weak coupling between elements. High losses thus hamper any substantial progress towards the practical use of these metamaterials in novel devices. Many applications are also hampered by the lack of tuning capabilities and relatively bulky size. However, another breakthrough came with the discovery ofnon-resonant, transmission line negative refractive index metamaterials [52,53], which very quickly led to several applications, at least in the microwaves [54]. Transmission line metamaterials rely on the appropriate combination of inductive–capacitive () lumped elements into large networks. The tremendous amount of activity in the field of metamaterials since has been summarized in various reviews [3,26,28,55–61] and books [11,62–72].

1.2 Nonlinear, superconducting, and active metamaterials

Dynamic tunability is a property that is required for applications [73]; in principle, one should be able to vary the effective (macroscopic) parameters of a metamaterial in real time, simply by varying an applied field. Tunability provides the means for fabricating meta-devices with switching capabilities [9,10], among others, and it can be achieved by the introduction of nonlinearity. Nonlinearity adds new degrees of freedom for metamaterial design that allows for both tunability and multistability — another desired property, that may offer altogether new functionalities and electromagnetic characteristics [74], as well as wide-band negative permeability [75]. It was very soon after the first demonstration of metamaterials, named at that time as negative refractive index materials, when it became clear that the SRR structure has considerable potential to enhance nonlinear effects due to the intense electric fields which can be generated in their slits [76]. Following these ideas, several research groups have demonstrated nonlinear metamaterial units, by filling the SRR slits with appropriate materials, e.g., with a strongly nonlinear dielectric [77], or with a photo-sensitive semiconductor. Other approaches have made use of semiconducting materials, e.g., as substrates, on which the actual metamaterial is fabricated, that enables modulation of THz transmission by [78]. However, the most convenient method for introducing nonlinearity in SRR-based metamaterials was proved to be the insertion of nonlinear electronic components into the SRR slits, e.g., a variable capacitance diode (varactor) [79,80]. The dynamic tunability of a two-dimensional metamaterial comprising varactor-loaded SRRs by the power of an applied field has been demonstrated experimentally [81]. Both ways of introducing nonlinearity affect the capacitance of the SRRs which becomes field-dependent; in the equivalent electrical circuit picture, in which the SRRs can be regarded as lumped element electrical oscillators, the capacitance acquires a voltage dependence and in turn a field-dependent magnetic permeability. Nonlinear transmission line metamaterials are reviewed in Ref. [82].

Nonlinearity does not however help in the reduction of losses; in nonlinear metamaterials the losses continue to be a serious problem. The quest for loss compensation in metamaterials is currently following two different pathways: a “passive” one, where the metallic elements are replaced by superconducting ones [58,83], and an “active” one, where appropriate constituents are added to metallic metamaterials that provide gain through external energy sources. In order to fabricate both nonlinear and active metamaterials, gain-providing electronic components such as tunnel (Esaki) diodes [84] or particular combinations of other gain-providing devices have to be utilized. The Esaki diode, in particular, features a negative resistance part in its current–voltage characteristics, and therefore can provide both gain and nonlinearity in a conventional (i.e., metallic) metamaterial. Tunnel diodes which are biased so that they operate at the negative resistance region of their characteristics may also be employed for the construction of symmetric metamaterials, that rely on balanced gain and loss [85]. symmetric systems correspond to a new paradigm in the realm of artificial or “synthetic” materials that do not obey separately the parity () and time () symmetries; instead, they do exhibit a combined symmetry [86,87]. The notions of symmetric systems originate for non-Hermitian quantum mechanics [88,89], but they have been recently extended to optical lattices [90,91]. The use of active components which are incorporated in metamaterial unit elements has been actually proposed several years ago [92], and it is currently recognized as a very promising technique of compensating losses [93]. Low-loss and active negative index metamaterials by incorporating gain material in areas with high local field have been demonstrated in the optical [94]. Recently, transmission lines with periodically loaded tunnel diodes which have the negative differential resistance property have been realized and tested as low-loss metamaterials, in which intrinsic losses are compensated by gain [95]. Moreover, a combination of transistors and a split-ring has been shown to act as a loss-compensated metamaterial element [96]. In the latter experiment, the quality factor for the combined system exhibits huge enhancement compared with that measured for the split-ring alone.

The “passive” approach to loss reduction employs superconducting materials, i.e, materials exhibiting absence of dc resistance below a particular temperature, known as the critical temperature, . A rough classification of the superconducting materials is made on the basis of their critical temperature; according to that, there are low- and high- superconducting materials. The former include primarily elemental and binary compounds, like Niobium (Nb), Niobium di-Selenide (NbSe) and more recently Niobium Nitride (NbN), while the most known representatives of the latter are the superconducting perovskites such as Yttrium–Barium–Copper-Oxide (YBCO). The latter is the most commonly used perovskite superconductor which typically has a critical temperature , well above the boiling point of liquid Nitrogen. The last few years, there has been an increasing interest in superconducting metamaterials that exploit the zero resistance property of superconductivity, targeting at severe reduction of losses and the emergence of intrinsic nonlinearities due to the extreme sensitivity of the superconducting state to external stimuli [10,58]. The direct approach towards fabrication of superconducting metamaterials relies on the replacement of the metallic split-rings of the conventional SRR-based metamaterials by superconducting ones. More sophisticated realizations of superconducting metamaterials result from the replacement of the metallic SRRs by rf SQUIDs (Superconducting QUantum Interference Devices) [97]; those SQUID metamaterials are discussed below.

Superconducting metamaterials are not however limited to the above mentioned realizations, but they also include other types of artificial metamaterials; thin superconducting plates have been used in a particular geometrical arrangement to “beat the static” [98] and make possible a zero frequency metamaterial (dc metamaterial) [14–17,99]. Other types of superconducting metamaterials in the form of heterostructures, where superconducting layers alternate with ferromagnetic or non-magnetic metallic layers have been shown to exhibit electromagnetically induced transparency [5,6,100], switching capabilities [101], magnetic cloaking, and concentration [102]. Recently, tunable electromagnetically induced transparency has been demonstrated in a Niobium Nitride (NbN) terahertz (THz) superconducting metamaterial. An intense THz pulse is used to induce nonlinearities in the NbN thin film and thereby tune the electromagnetically induced transparency-like behavior [7]. Furthermore, the dynamic process of parity-time () symmetry breaking was experimentally demonstrated in a hybridized metasurface which consists of two coupled resonators made from metal and NbN [103]. Negative refraction index metamaterials in the visible spectrum, based on MgB/SiC composites, have been also realized [104], following prior theoretical investigations [105]. Moreover, there is substantial evidence for negative refraction index in layered superconductors above the plasma frequency of the Josephson plasma waves [106], that was theoretically investigated by several authors [107,108]. Other types of superconducting metamaterials include those made of magnetically active planar spirals [109], as well as those with rather special (“woodcut”) geometries [110], two-dimensional arrays of Josephson junctions [111], as well as superconducting “left-handed” transmission lines [112,113]. Recently, in a novel one-dimensional Josephson metamaterial composed of a chain of asymmetric SQUIDs, strong Kerr nonlinearity was demonstrated [114]. Moreover, the Kerr constant was tunable over a wide range, from positive to negative values, by a magnetic flux threading the SQUIDs.

1.3 Superconducting metamaterials from zero to Terahertz frequencies

There are several demonstrations of superconducting metamaterial elements which exhibit tunability of their properties by varying the temperature or the applied magnetic field [22,24,115–119]. We should also mention the theoretical investigations (nonlinear circuit modeling) on a multi-resonant superconducting split-ring resonator [120], and on a “meta-atom” composed of a direct current (dc) SQUID and a superconducting rod attached to it, which exhibits both electric and magnetic resonant response [121]. Superconducting split-rings combined into two-dimensional planar arrays form superconducting metamaterials exhibiting tunability and switching capabilities at microwave and THz frequencies [22–25,116,122–127]. Up to the time of writing, metamaterials comprising superconducting SRRs employ one of the following geometries:

(i) square SRRs with rectangular cross-section in the double, narrow-side coupled SRR geometry [115,116,128];

(ii) circular, asymmetrically split-rings [117,129,130];

(iii) square SRRs with rectangular cross-section in the single SRR geometry [22];

(iv) electric inductive–capacitive SRRs of two different types [131].

Also, novel metamaterial designs including a “woodcut” type superconducting metamaterial, and niobium-connected asymmetrically split-ring metamaterials were demonstrated [130]. All these metamaterials were fabricated in the planar geometry, using either conventional, low superconductors such as niobium (Nb) and niobium nitride films, or the most widely used member of the high superconductor family, i.e., the yttrium–barium–copper-oxide (). The experiments were performed in microwaves and in the (sub-)THz range ().

All these superconducting metamaterials share a common feature: they all comprise resonant sub-wavelength superconducting elements, that exhibit a strong response at one particular frequency, i.e., the resonance frequency, . That resonance frequency is tunable under external fields, such as temperature, constant (dc) and time-periodic (ac) magnetic fields, and applied current, due to the extreme sensitivity of the superconducting state to external stimuli. (Note however that for some geometries there can be more than one strong resonances.) The experimental investigation of the resonances and their ability for being shifted either to higher or lower frequencies relies on measurements of the complex transmission spectra, with dips signifying the existence of resonances. However, not only the frequency of a resonance but also its quality is of great interest in prospective applications. That quality is indicated by the depth of the dip of the complex transmission magnitude in the corresponding transmission spectrum, as well as its width, and quantified by the corresponding quality factor . In general, the quality factor increases considerably as the temperature decreases below the critical one at . Other factors, related to the geometry and material issues of the superconducting SRRs that comprise the metamaterial, also affect the resonance frequency . Thus, the resonance properties of a metamaterial can be engineered to achieve the desired values, just like in conventional metamaterials. However, for superconducting metamaterials, the thickness of the superconducting film seems to be an important parameter, because of the peculiar magnetic properties of superconductors. Using proper design, it is possible to switch on and off the resonance in superconducting metamaterials in very short time-scales, providing thus the means of manufacturing devices with fast switching capabilities.

1.4 Summary of earlier work in superconducting metamaterials

In this Subsection, a brief account is given on the progress in the development and applications of superconducting (both classical and quantum) metamaterials, i.e., metamaterials utilizing either superconducting materials or devices. A more detailed and extended account is given in two review articles on the subject [58,83], as well as in Chapter 5.5 of a recently published book [11]. The status of the current research on SQUID metamaterials is discussed separately in the next Subsection (Section 1.5). In the older of the two review articles [58], the properties of superconductors which are relevant to superconducting metamaterials, and the advantages of superconducting metamaterials over their normal metal counterparts are discussed. The author reviews the status of superconductor–ferromagnet composites, dc superconducting metamaterials, radio frequency (rf) superconducting metamaterials, and superconducting photonic crystals (although the latter fall outside the domain of what are usually called metamaterials). There is also a brief discussion on SQUID metamaterials, with reference to the theoretical works in which it was proposed to use an array of rf SQUIDs as a metamaterial [132,133]. In the second review article [83], a more detailed account on the advantages of superconducting metamaterials over their normal metal counterparts was given, along with an update on the status of superconducting metamaterials. Moreover, analogue electromagnetically-induced transparency superconducting metamaterials and superconducting SRR-based metamaterials are also reviewed. In this review article, there is also a discussion on the first experiments on SQUID metamaterials [34,134,135] which have confirmed earlier theoretical predictions. However, a lot of experimental and theoretical work on SQUID metamaterials has been performed after the time of writing of the second review article in this field. The present review article aims to give an up-to-date and extended account of the theoretical and experimental work on SQUID metamaterials and reveal their extraordinary nonlinear dynamic properties. SQUID metamaterials provide a unique testbed for exploring complex spatio-temporal dynamics. In the quantum regime, a prototype model for a “basic” superconducting quantum metamaterial (SCQMM) which has been investigated by several authors is reviewed, which exhibits novel physical properties. Some of these properties are discussed in detail.

Superconductivity is a macroscopic quantum state of matter which arises from the interaction between electrons and lattice vibrations; as a result, the electrons form pairs (Cooper pairs) which condense into a single macroscopic ground state. The latter is the equilibrium thermodynamic state below a transition (critical) temperature . The ground state is separated by a temperature-dependent energy gap from the excited states with quasi-particles (quasi-electrons). As mentioned earlier, the superconductors are roughly classified into high and low critical temperature ones (high and low, respectively). In some circumstances, the Cooper pairs can be described in terms of a single macroscopic quantum wavefunction , whose squared magnitude is interpreted as the local density of superconducting electrons (), and whose phase is coherent over macroscopic dimensions. Superconductivity exhibits several extraordinary properties, such as zero dc resistance and the Meissner effect. Importantly, it also exhibits macroscopic quantum phenomena including fluxoid quantization and the Josephson effects at tunnel (insulating) barriers and weak links. When two superconductors and are brought close together and separated by a thin insulating barrier, there can be tunneling of Cooper pairs from one superconductor to the other. This tunneling produces a supercurrent (Josephson current) between and , , where is the critical current of the Josephson junction and is the gauge-invariant Josephson phase, with and being the phases of the macroscopic quantum wavefunctions of and , respectively, the electromagnetic vector potential in the region between and , and the flux quantum ( is Planck’s constant and the electron’s charge). Depending on whether is time-dependent or not, the appearance of the supercurrent is referred to as the ac or the dc Josephson effect.

Superconductors bring three unique advantages to the development of metamaterials in the microwave and sub-THz frequencies which have been analyzed in Refs. [58,83]. Namely, (i) low losses (one of the key limitations of conventional metamaterials), (ii) the possibility for higher compactification of superconducting metamaterials compared to other realizations (superconducting SRRs can be substantially miniaturized while still maintaining their low-loss properties), and (iii) strong nonlinearities inherent to the superconducting state, which allow for tunability and provide switching capabilities. The limitations of superconducting metamaterials arise from the need to create and maintain a cryogenic environment, and to bring signals to and from the surrounding room temperature environment. Quite fortunately, closed-cycle cryocoolers have become remarkably small, efficient and inexpensive since the discovery of high superconductors, so that they are now able to operate for several years unattended, and moreover they can accommodate the heat load associated with microwave input and output transmission lines to room temperature. Superconductors can also be very sensitive to variations in temperature, stray magnetic field, or strong rf power which can alter their properties and change the behavior of the metamaterial. Thus, careful temperature control and high quality magnetic shielding are often required for reliable performance of superconducting metamaterials.

Superconducting metamaterials exhibit intrinsic nonlinearity because they are typically made up of very compact elementary units, resulting in strong currents and fields within them. Nonlinearity provides tunability through the variation of external fields. For example, a connected array of asymmetrically-split Nb resonators shows transmission tunable by current at sub-THz frequencies due to localized heating and the entrance of magnetic vortices [129]. The change in superfluid density by a change in temperature was demonstrated for a superconducting thin film SRR [128]. Later, it was demonstrated that the resonant frequency of a Nb SRR changes significantly with the entry of magnetic vortices [116]. Similar results showing complex tuning with magnetic field were later demonstrated at microwaves using high superconducting SRRs [119] and at sub-THz frequencies with similarly designed Nb SRRs [130]. The nonlinearity associated with the resistive transition of the superconductor was exploited to demonstrate a bolometric detector at sub-THz frequencies using the collective properties of an asymmetrically split-ring array made of Nb [130]. Thermal tuning has been accomplished at THz frequencies by varying the temperature in high (YBCO) metamaterial [24] and NbN electric inductive–capacitive thin film resonators [124]. Enhancement of thermal tunability was accomplished by decreasing the thickness of the high superconducting films which make up square SRRs [136]. Fast nonlinear response can be obtained in superconducting films in THz time-domain experiments. In such an experiment it was found that intense THz pulses on a NbN metamaterial could break significant number of Cooper pairs to produce a large quasi-particle density which increases the effective surface resistance of the film and modulates the depth of the SRR resonance [25,126]. The tuning of high (YBCO) SRR metamaterial with variable THz beam intensity has demonstrated that the resonance strength decreases and the resonance frequency shifts as the intensity is increased [127].

A natural opportunity to create a negative real part of the effective magnetic permeability is offered by a gyromagnetic material for frequencies above the ferromagnetic resonance [137]. However, the imaginary part of is quite large near the resonance and limits the utility of such a metamaterial. A hybrid metamaterial, resulting from the combination of the gyromagnetic material mentioned earlier with a superconductor can help to reduce losses [138]. A superlattice consisting of high superconducting and manganite ferromagnetic layers (YBCO/LSMO) was created and it was shown to produce a negative index band in the vicinity of the ferromagnetic resonance () at magnetic fields between 2.9 and [107]. More recently, a metamaterial composed of permanent magnetic ferrite rods and metallic wires was fabricated. This metamaterial exhibits not only negative refraction but also near-zero refraction, without external magnetic field [139].

The concept of a dc metamaterial operating at very low frequencies that could make up a dc magnetic cloak has been proposed and investigated in Ref. [14]. The first realization of such a metamaterial, which is based on non-resonant elements, consists of an array of superconducting plates [15]. The superconducting elements exclude a static magnetic field, and provide the foundation for the diamagnetic effect, when that field is applied normal to the plates. The strength of the response depends on the ratio between the dimension of the plates and the lattice spacing. An experimental demonstration of a dc metamaterial cloak was made using an arrangement of Pb thin film plates [15]. Subsequent theoretical work has refined the dc magnetic cloak design and suggested that it can be implemented with high superconducting thin films [16]. It was later demonstrated experimentally that a specially designed cylindrical superconductor–ferromagnet bilayer can exactly cloak uniform static magnetic fields [99].

Superconducting rf metamaterials consisting of two-dimensional Nb spirals developed on quartz substrates show strong tunability as the transition temperature is approached [109,122]. Rf metamaterials have great potential in applications such as magnetic resonance imaging devices for non-invasive and high resolution medical imaging [140]. The superconducting rf metamaterials have many advantages over their normal metal counterparts, such as reducing considerably the Ohmic losses, compact structure, and sensitive tuning of resonances with temperature or rf magnetic field, which makes them promising for rf applications. Similar, high rf metamaterials (in which the spirals are made by YBCO) were also fabricated, which enable higher operating temperatures and greater tunability [141].

The elementary units (i.e., the meta-atoms) in a metamaterial can be combined into meta-molecules so that the interactions between the meta-atoms can give rise to qualitatively new effects, such as the classical analogue of the electromagnetically induced transparency (EIT). This effect has been observed in asymmetrically-split ring metamaterials in which Fano resonances have been measured as peaks in the transmission spectrum, corresponding to metamaterial induced transparency [117]. Metamaterials consisting of normal metal–superconductor hybrid meta-molecules can create strong classical EIT effects [5]. The meta-molecule consists of a gold (Au) strip with end caps and two superconducting (Nb) SRRs (the “bright” and the “dark” element, respectively). A tunable transparency window which could even be switched off completely by increasing the intensity of the signal propagating through the meta-molecule was demonstrated [5,101]. EIT effects were also observed in the THz domain utilizing NbN bright and dark resonators to create a transparency window [100]. Further experiments on all-superconducting (NbN) metamaterials utilizing strongly coupled SRR-superconducting ring elements showed enhanced slow-light features [6].

A quantum metamaterial is meant to be an artificial optical medium that (a) comprise quantum coherent unit elements whose parameters can be tailored, (b) the quantum states of (at least some of) these elements can be directly controlled, and (c) maintain global coherence for sufficiently long time. These properties make a quantum metamaterial a qualitatively different system [142,143]. Superconducting quantum metamaterials offer nowadays a wide range of prospects from detecting single microwave photons to quantum birefringence and superradiant phase transitions [83]. They may also play a role in quantum computing and quantum memories. The last few years, novel superconducting devices, which can be coupled strongly to external electromagnetic field, can serve as the quantum coherent unit elements of superconducting quantum metamaterials (SCQMMs). For example, at ultra-low temperatures, superconducting loops containing Josephson junctions exhibit a discrete energy level spectrum and thus behave in many aspects as quantum meta-atoms. It is very common to approximate such devices as two-level quantum systems, referred to as superconducting qubits, whose energy level splitting corresponds to a frequency of the order of a few GHz. The interaction between light and a SCQMM is described by photons coupling to the artificial two-level systems, i.e., the superconducting qubits. The condition of keeping the energy of thermal fluctuations , where is Boltzmann’s constant and the temperature, below the energy level splitting of the qubit, where is Planck’s constant and the transition frequency, requires temperatures well below . In the past few years, research on superconducting qubits has made enormous progress that paves the way towards superconducting qubit-based quantum metamaterials.

There are several theoretical investigations on the physics of one-dimensional arrays of superconducting qubits coupled to transmission-line resonators [144–152]. Moreover, two-dimensional [153] and three-dimensional [35] SCQMMs based on Josephson junction networks were proposed. A more extended discussion of the theoretical works on SCQMMs is given in Section 5.1. Still there is little progress in the experimental realization of such systems. The first SCQMM was implemented in 2014 [154], and comprises flux qubits arranged in a double chain geometry. In that prototype system, the dispersive shift of the resonator frequency imposed by the SCQMM was observed. Moreover, the collective resonant coupling of groups of qubits with the quantized mode of a photon field was identified, despite of the relatively large spread of the qubit parameters. Recently, an experiment on an SCQMM comprising an array of twin flux qubits, was demonstrated [155]. The qubit array is embedded directly into the central electrode of an Al coplanar waveguide; each qubit contains Josephson junctions, and it is strongly coupled to the electromagnetic waves propagating through the system. It was observed that in a broad frequency range, the transmission coefficient through that SCQMM depends periodically on the external magnetic field. Moreover, the excitation of the qubits in the array leads to a large resonant enhancement of the transmission. We undoubtedly expect to see more experiments with arrays of superconducting qubits placed in transmission lines or waveguides in the near future.

1.5 SQUID metamaterials

The rf SQUIDs, mentioned above, are highly nonlinear superconducting devices which are long known in the Josephson community and encompass the Josephson effect [156]. The simplest version of a SQUID is made by a superconducting ring which is interrupted by a Josephson junction (JJ); the latter is typically formed by two superconductors separated by a thin insulating (dielectric) layer. The current through the insulating layer and the voltage across the junction are then determined by the celebrated Josephson relations and crucially affect the electromagnetic behavior of the rf SQUID. SQUIDs have found numerous technological applications in modern science [157–160]; they are most commonly used as magnetic field sensors, since they can detect even tiny magnetic fields and measure their intensity with unprecedented precision. SQUID metamaterials constitute the direct superconducting analogue of conventional (metallic) nonlinear (i.e., varactor loaded) SRR-based magnetic metamaterials, which result from the replacement of the nonlinear SRRs by rf SQUIDs. The latter possess inherent nonlinearity due to the Josephson element. Similarly to the conventional (metallic), SRR-based magnetic metamaterials, the SQUIDs are coupled magnetically to each other through magnetic dipole–dipole interactions. Several years ago, theoretical investigations have suggested that rf SQUID arrays in one and two dimensions can operate as magnetic metamaterials both in the classical [133] and in the quantum regime [132], and they may exhibit negative and/or oscillating effective magnetic permeability in a particular frequency band which encloses the resonance frequency of individual SQUIDs. Recent experiments on single rf SQUIDs in a waveguide demonstrated directly the feasibility of constructing SQUID-based thin-film metasurfaces [118]. Subsequent experiments on one-dimensional, quasi-two-dimensional, and truly two-dimensional SQUID metamaterials have revealed a number of several extraordinary properties such as negative diamagnetic permeability [34,118], broad-band tunability [34,134], self-induced broad-band transparency [161], dynamic multistability and switching [135], as well as coherent oscillations [162]. Moreover, nonlinear localization [163] and nonlinear band-opening (nonlinear transmission) [164], as well as the emergence of dynamic states referred to as chimera states in current literature [165,166], have been demonstrated numerically in SQUID metamaterial models. Those counter-intuitive dynamic states, which have been discovered numerically in rings of identical phase oscillators [167], are reviewed in Refs. [168,169]. Moreover, numerical investigations on SQUID metamaterials on Lieb lattices which possess a flat band in their frequency spectrum, reveal the existence of flat-band localized states in the linear regime and the more well-known nonlinearly localized states in the nonlinear regime [170]. The interaction of an electromagnetic wave with a diluted concentration of a chain of SQUIDs in a thin film suggests a mechanism for the excitation of magnetization waves along the chain by a normally incident field [171]. In the linear limit, a two-dimensional array of rf SQUIDs acts as a metasurface that controls the polarization of an electromagnetic wave [172].

SQUID arrays have been also integrated in larger devices in order to take advantage of their extraordinary properties; notably, amplification and squeezing of quantum noise has been recently achieved with a tunable SQUID-based metamaterial [173]. Other important developments demonstrate clearly that SQUID-based metamaterials enable feedback control of superconducting cubits [174], observation of Casimir effects [175], measurements of nanomechanical motion below the standard quantum limit [176], and three-wave mixing [177]. At sufficiently low (sub-Kelvin) temperatures, SQUID metamaterials provide access to the quantum regime, where rf SQUIDs can be manipulated as flux and phase qubits [178,179]. Recently, the technological advances that led to nano-SQUIDs make possible the fabrication of SQUID metamaterials at the nanoscale [180].

From the above discussion it should be clear that the field of superconducting metamaterials, in which superconductivity plays a substantial role in determining their properties, has expanded substantially. In this review, we focus on the SQUID metamaterials, that represent an area of the field of superconducting metamaterials, which however has already reached a level of maturity. We also focus on SCQMMs, and in particular on a prototype model for a chain of charge qubits in a transmission-line resonator [144]. The SCQMMs are related to the (classical, i.e., not truly quantum) SQUID metamaterials in that they also encompass the Josephson effect. In Section 2, we describe the SQUID metamaterial models used for simulating real systems in current research, we provide the corresponding dispersion of flux waves which can propagate in SQUID metamaterials, and we present numerical results (along with selected experimental ones), which reveal novel properties such as wide-band tunability, energy transmission, and multistability. In Section 3, we present and discuss results on nonlinear localization in SQUID metamaterials, which leads to the generation of states referred to as discrete breathers. In that Section, we also emphasize the possibility for the emergence of chimera states in SQUID metamaterial models with either nonlocal or local (nearest-neighbor) coupling between their elements (i.e., the SQUIDs). In Section 4, the dynamical model for SQUID metamaterials on a Lieb lattice is presented, along with its full frequency spectrum. The latter contains a flat band, which allows for the formation of flat-band localized states in the linear regime. The case of nonlinearly localized states, which can be formed in the nonlinear regime, as well as the transition between the two regimes, is investigated. In Section 5, we describe a model SCQMMs (a chain of charge qubits in a superconducting transmission-line resonator) and discuss the possibility for having propagating self-induced transparent or superradiant pulses in that medium. Most importantly, those pulses induce quantum coherence effects in the medium itself, by exciting population inversion pulses in the qubit subsystem. Moreover, the speed of the propagating pulses can be controlled by proper engineering of the parameters of the qubits. The most important points made in this review are summarized in Section 6.

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Review article

Superconducting Metamaterials

Physics ReportsJournal2018, Physics Reports

N. Lazarides, G.P. Tsironis

1.1 Metamaterials & synthetic media: Concepts and perspectives

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