Möbius Metamaterial Strips: Opportunity, Trends, Challenges and Future
The MMS (Metamaterial Möbius Strip) is an artificial composite structure with a negative index of refraction , where n is the refractive index, εis the electrical permittivity and µ is the magnetic permeability of the medium. It has emerged as a cutting edge of science relating physics, chemistry, biology, material science, optics, acoustics and electronics. For most naturally existing materials, µ is close to 1; hence, magnetic susceptibility of natural materials is small as compared to the electric/dielectric susceptibility. This phenomenon limits the interaction of atoms to the electric component of the electromagnetic (EM) wave, leaving the magnetic component mostly unexploited. Magnetism is primarily weak at optical frequencies as well, because the relaxation times of paramagnetic and ferromagnetic processes are considerably longer than an optical period, electron movement in atoms is the only mechanism for creating the magnetic response. This is why the magnetic field component is usually not involved in light-matter interactions.
The reason for weak magnetism is mainly due to limitations of the material properties imposed by chemical composition and constituent components (atoms and molecules). On the contrary, MMS resonant nanostructures, in principle, can exhibit a broad range of magnetic permeability values.1-75 A number of stimulating phenomena and applications associated with MMS structures are discussed in part 1 (MWJ May 2016) and part 2 (MWJ June 2016) of this series. This issue addresses the prospects, challenges and future directions of MMS inspired components for various applications including the Gravitational Casimir Effect.
Recent research in the field of metamaterials69-75 has not only established interesting physical phenomena but also lead to opportunities for utilizing negative index components and devices for next generation energy-efficient electronic circuits and systems. Figure 1 compares the properties of natural and artificially engineered composite materials.1 Unlike conventional materials that interact with EM waves based on their chemical compositions, the properties of metamaterials are derived from their topologies and geometric structures.
The typical metamaterial consists of periodically or arbitrarily disseminated structured cells with dimensions and spacings much smaller than a wavelength of the incident EM waves.1-3 As a consequence, the microscopic detail of each unit cell structure cannot be sensed by EM waves. What is important to understand is the average result of the collective response of the entire assemblage, comprised of inhomogeneous matter. In other words, such a collection of inhomogeneous matter can be characterized by an equivalent homogenous material with effective constitutive relative permittivity (εr,eff) and permeability (µr,eff) at the macroscopic level. The key aspect of an MMS inspired structure is that effective permittivity (εr,eff) and permeability (µr,eff) can be controlled and tuned by a suitably designed disseminated element for broadband operation.
In classical EM theory, the characteristics of matter illustrated in Figure 1 can be described by the Drude–Lorentz model6 as
where ωp is the plasma frequency, ωo is the resonant frequency, subscripts ‘e’ and ‘m’ represent electric and magnetic response, and γ is the damping factor associated with material losses.
Figure 2 shows a typical metamaterial structure,4-5 realized by the combination of split ring resonators (SRR) and thin metallic wires. The effective relative permittivity (εr,eff) and effective relative permeability (µr,eff) obey the Drude–Lorentz6-9 model as
where σ is the conductivity of metal wire, d is the lattice constant, ω0 represents the resonant frequency, ωp,eff is the effective plasma frequency, F is the filling ratio of the SRR, and Γ is the damping term. From (3), the effective plasma frequency is ωp,eff = 7.52 × 1010 rad/sec; assuming that the metal wire conductivity σ =107 Ω-1 m-1, the metal wire radius r = 1 × 10-6 meters and the lattice constant d = 3.5 × 10-3 meters. From (4), the resonant frequency f0 = 8.324 GHz; assuming that for the SRR, d = 4 × 10-3 meters, r =1 × 10-3 meters, and s = 1 × 10-4 meters. This corresponds to a free space wavelength of 3.6 × 10-2 meters which is about 10 times larger than d.
From (3) and (4), εr,eff of metallic wire and µr,eff of the SRR exhibit the typical Drude-Lorentz characteristics,8 plotted in Figures 2b and d. Equations (3) and (4) provide approximate analytical solutions for the effective constitutive parameters (permittivity, permeability) with reasonable accuracy, valid for the simple structure shown in Figure 2. For a complex structure, however, and especially for an MMS inspired negative index metasurface, this is not true.
The alternative approach is to retrieve the effective parameters from numerical simulations9-12 described in detail.70 The first step of the retrieval procedure is to calculate the transmission and reflection of the composite MMS based on numerical algorithms, such as finite-difference time-domain (FDTD) and finite element method (FEM). Some commercial software, including ADS, Ansys HFSS, CST Microwave Studio, COMSOL Multiphysics and SONNET are widely used but none of these provide error free solutions. The refractive index and impedance are related to the transmission coefficient (t) and reflection coefficient (r) by equations9
where k0 is the wave vector in vacuum, L is the thickness of the metamaterial, and m is an integer.
From Equations (5) and (6), effective impedance and refractive index can be determined provided that the metamaterial structures act like a passive medium; this implies that the real part of zeff and the imaginary part of neff are positive.9 Note that the effective parameter retrieval process is a challenging task, especially when metamaterial structures fall into the category of anisotropic or bi-anisotropic, and EM wave is obliquely incident.12 Parameter retrieval becomes even more complex for nonlinear metamaterial composite structures.
The metamaterial structure shown in Figure 3a is a combination of metallic wires and SRRs; however, these artificially engineered structures pose fabrication problems. A simplified structure reported in literature is the Fishnet structure14-16 that consists of two layers of metal meshes separated by a dielectric layer (see Figures 3b and 3c). Paired stripes oriented parallel to the electric field provide negative εr,eff (ω), while the other pairs of stripes parallel to the magnetic field support negative µr,eff (ω). Since the dielectric thickness of the fishnet structure is easy to control, the fabrication burden is significantly eased as compared to the conventional approach of using SRRs and metallic wires.8 Moreover, in order to produce a negative refractive index, EM waves are incident normal to the fishnet surface, whereas the structure fabricated by SRRs and wires requires oblique incidence to excite SRRs with out-of-plane magnetic fields for strong magnetic resonances.
METAMATERIAL OPPORTUNITIES AND EMERGING TRENDS
The aerospace, defense and biomedical electronics sectors are viewed as the most vibrant market areas for metamaterial products applications. The rise of drones and related weight considerations, the need for improved military communications, and the burgeoning demand for new and more sophisticated biosensors are all areas where metamaterial technology can help propel things forward.
Metamaterial structures have been used in magnetic imaging, microwave circuit components, antennas, and perfect lenses with imaging resolutions beyond the diffraction limit.1-21 In conventional optical systems, it is not possible to determine two points separated less than l/2n, where n is the refractive index of the ambient medium. This fundamental limitation exists because the information of the object’s fine features and textures are carried by evanescent waves, which exponentially decay in space. All the information relevant to the sub-wavelength details of the object is lost, before reaching the imaging plane. It is interesting to note that a metamaterial slab acts as a perfect lens to recover all the lost information.20 This extraordinary property of perfect lens arises from the fact that the initially decayed evanescent waves are now amplified through the slab. Meanwhile, the propagating waves are focused due to the negative refraction and reversed phase front. As a result, a metamaterial slab, incredibly, brings both propagating and evanescent waves to a perfect focus (see Figures 4a and 4b), without suffering the traditional constraint imposed by the diffraction limit.
This shows promise in the realization of metamaterial super lenses, which are lenses that are almost free of aberrations and that can focus images below the diffraction limit. The recorded image ‘‘NANO’’ (see Figure 5b) reproduces the fine features from the object mask (see Figure 5a) in all directions with good fidelity, while the image in the control experiment without the super lens (see Figure 5c) shows a much wider line width.22 In the seminal paper, Pendry et al.,5 predicted the enhanced nonlinear optical properties by inserting nonlinear elements into the gap of SRRs, arising from the giant local-field amplification.
Figure 6 shows the typical measurement setup for an MRI measurement at 8.5 MHz for prostate cancer detection. Metamaterial inspired (µ = -1) split ring resonators loaded with capacitors and inductors enabled a 20 times increase in the magnetic field. As shown in Figure 6b, the lens resolves two magnetic sources indistinguishable without the lens.76
Metamaterial technology is an enabler to build lighter and more compact antenna systems. Figure 7 shows the performance of the typical light weight inspired monopole antenna. The data shows that the bandwidth increased to over an octave while preserving the radiation characteristics of a simple monopole.
Metamaterial technology offers unparalleled opportunities for light manipulation. Recent developments in the field have fueled new opportunities for light propagation, establishing a new paradigm for spin and quantum related phenomena in optical physics. Nonlinear metamaterials, with properties depending on the intensity of EM waves, is an emerging research topic with novel phenomena such as hysteretic transition,23 unusual wave mixing24 and solitary wave propagation.25-26
The other interesting phenomenon is the reversed Manley–Rowe relation and backward phase matching condition for second-harmonic generation (SHG) or optical parametric amplification (OPA).27 Suppose a metamaterial has a negative refractive index at the fundamental frequency ω1 and a positive refractive index at the second-harmonic frequency ω2. At ω1 the energy flow (Poynting vector) points from left to right, for example; then the wave vector must point from right to left arising from negative index sample at ω1. The phase matching condition i.e. requires that the wave vector at the second harmonic frequency ω2 also travels from the right to the left. Since the metamaterial possesses a positive refractive index at ω2, the energy flow is at the same direction as the wave vector. As a result, the second harmonic signal is maximal at the incident interface rather than at the exit interface of the metamaterial slab, in sharp contrast to SHG in normal dielectric materials (see Figure 8). Moreover, artificial magnetic metamaterials could provide additional ways to boost the nonlinear process.28 In terms of applications, tunable metamaterials29-30 and memory devices31 have been experimentally demonstrated based on nonlinear metamaterial composites.
Metamaterials may manifest fascinating phenomena in the quantum world. In principle, the metamaterial concept could be applied to any wave at any scale, including the matter wave which is the wave description of particles, such as electrons and neutrons, in quantum mechanics. Indeed, researchers have made theoretical efforts in this direction. Cheianov et al.,32 theoretically demonstrated that negative refraction and focusing of electrons can be achieved in graphene (see Figure 9), a monolayer of graphite.
Figure 10 illustrates the technology and transformation optics approach that enables unprecedented design flexibility and novel device applications.
The metamaterial technology and transformation optics shown in Figure 10 promise unparalleled opportunities; but at the same time, metamaterial composites fabrication is challenging. The real challenge is to predict the topology and geometry of negative index microstructures even though they tend to have simple shapes. The topology optimization method allows selection of geometric and topological configuration of multi-physical functional materials while taking into account the MMS material composition. Commercialization is primarily a manufacturing problem due to the lack of effective tools to economically pattern large volumes of material.
The immediate step is to improve the homogenization methodology for the design of multi-function nonlinear metamaterial devices, improving the bandwidth, and providing smaller/more compact structures. More investigation of pulse propagation in optical fiber and speed control by means of nonlinear refractive index for the space-time cloak, solitons and their variants is needed in negative refractive index composites.
The emerging future is likely to be in the area of the Gravitational Casimir effect and signal processing where the space-time cloak acts as a means of prioritizing data channels, rather than theoretically attempting to combine space-time and spatial cloaks. In addition to this, Möbius transformations that exploit hyperbolic characteristics could be interesting for a variety of Minkowski-based relativistic scenarios including spinning cosmic strings.
Gravitational Casimir Effect
Figure 11 shows the practical evidence of the Casimir force ‘F’ on parallel plates kept in vacuum. The effective force F ∝ A/d4, where A is the area of plate and d is the distance between the plates. The Casimir force67 (see Figure 12) arises from the interaction of the surfaces with the surrounding electromagnetic spectrum, and includes a complex dependence on the full dielectric function of both surfaces and the region between. On the more theoretical side, the MMI structure can produce a powerful Casimir effect (force from nothing), which will allow the transport of matter; this implies the ability to attract or push away physical matter.
As shown in Figure (12),56 the polaritonic contribution is responsible for the change in sign of the Casimir force between a metallic and a metamaterial mirror. For L ≥ λr/5 the binding TM polariton, which dominates at short distance, is overwhelmed by the joint repulsion due to the anti-binding TM and TE polaritons. This shows that, for mixed configurations as well, surface plasmons are crucial in determining both the strength and the sign of the Casimir interaction.
One of the exciting properties of MMI structures is that they can bend light in a way that is mathematically equivalent to the way space-time bends light, enabling topological exploration for the realization of low cost gravitational wave detector. Figure 13 shows the gravitational Casimir effect, with a two plate setup. The change in the refractive index of the plates causes the gravitational wave to refract, where k represents the wave vector of the incident, transmitted, and reflected gravitational waves, and γ is the corresponding angle with respect to the surface normal.68 The Casimir effect has also been investigated in weak gravitational fields to see the effect the slightly curved space time background would have on the Casimir energy.68
Gravitational Wave Reflector and the Gravitational Characteristic Impedance of Free Space
Thin Metamaterial superconducting films are predicted to be highly reflective mirrors for gravitational waves at microwave frequencies. The quantum mechanical non-localizability of the negatively charged Cooper pairs, which are protected from the localizing effect of decoherence by an energy gap, causes the pairs to undergo non-picturable, non-geodesic motion in the presence of a gravitational wave57. This non-geodesic motion, which is accelerated motion through space, leads to the existence of mass and charge super currents inside the Metamaterial superconducting film. On the other hand, the decoherence-induced localizability of the positively charged ions in the lattice causes them to undergo picturable, geodesic motion as they are carried along with space in the presence of the same gravitational wave. The resulting separation of charges leads to a virtual plasma excitation within the film that enormously enhances its interaction with the wave, relative to that of a neutral super fluid or any normal matter. The existence of strong mass super currents within a superconducting film in the presence of a gravitational wave, dubbed the H-C (Heisenberg-Coulomb) effect, implies the specular reflection of a gravitational microwave from a film whose thickness is much less than the penetration depth of the material, in close analogy with the electromagnetic case. The argument is developed by allowing classical gravitational fields, which obey Maxwell-like equations, to interact with quantum matter, which is described using the BCS and Ginzburg-Landau theories of superconductivity, as well as a collisionless plasma model.
Experiments at the frontiers of quantum mechanics and gravitation are rare. Minter57et al argue for a claim that may lead to several new types of experiments, namely, that a superconducting film whose thickness is less than the London penetration depth of the material can specularly reflect not only electromagnetic (EM) microwaves, as has been experimentally demonstrated58-59 but gravitational (GR) microwaves as well. The basic motivation lies in the well known fact that Einstein’s field equations lead, in the limits of weak GR fields and non-relativistic matter, to gravitational Maxwell-like equations,60 which in turn lead to boundary conditions for gravitational fields at the surfaces of the superconducting films homologous to those of electromagnetism. All radiation fields, whether electromagnetic or gravitational, will be treated classically, whereas the superconductors with which they interact will be treated quantum mechanically.
The first claim is that a GR microwave will generate quantum probability super currents, and thus mass and electrical super currents, inside a superconductor, due to the quantum mechanical non-localizability of the Cooper pairs within the material. The non-localizability of Cooper pairs, which is ultimately due to the Uncertainty Principle (UP), causes them to undergo non-picturable, non-geodesic motion in the presence of a GR wave. This non-geodesic motion, which is accelerated motion through space, leads to the existence of mass and charge super-currents inside a superconductor. By contrast, the localizability of the ions within the superconductor’s lattice causes them to undergo picturable, geodesic motion, i.e., free fall, in the presence of the same wave. The resulting relative motion between the Cooper pairs and the ionic lattice causes the electrical polarization of the superconductor in the presence of a GR wave, since its Cooper pairs and ions carry not only mass but oppositely signed charge as well.
Furthermore, the non-localizability of the Cooper pairs is “protected” from the normal process of localization, i.e., from decoherence, by the characteristic energy gap of the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity. The decoherence of entangled quantum systems such as Cooper pairs (which are in the spin-singlet state) is the fundamental cause of the localizability of all normal matter.61
Minter57et claimed that the mass super-currents induced by a GR wave are much stronger than what one would expect in the case of a neutral super-fluid or any normal matter, due to the electrical polarization of the superconductor caused by the wave. This is what referred to as the “Heisenberg-Coulomb (H-C) effect.” The magnitude of the enhancement due to the H-C effect (is given by the ratio of the electrical force to the gravitational force between two electrons,
The Maxwell-like representation of Einstein’s equations of general relativity describe the coupling of weak gravitational fields to slowly moving matter. In the asymptotically flat space-time coordinate system of a distant inertial observer, the four equations in SI units are:
where q is the electron charge, me is the electron mass, ε0 is the permittivity of free space, and G is Newton’s constant. The vastness of equation (9) implies the possibility of an enormous back-action of a superconductor upon an incident GR wave, leading to its reflection.
There are four conventionally accepted fundamental force of nature (i) gravitational, (ii) electromagnetic, (iii) strong nuclear, and (iv) weak nuclear. Each one is understood as the dynamics of a field. The gravitational force is modeled as a continuous classical field. Interestingly, of the four fundamental forces of nature, only gravity and electricity have long range, inverse square laws.
The pure number obtained in (9) by taking the ratio of these two inverse-square laws is therefore just as fundamental as the fine structure constant. Because this number is so large, the gravitational force is typically ignored in treatments of the relevant quantum physics. But for a semi-classical treatment of the interaction of a superconductor with a GR wave must account for both the electrodynamics and the gravito-electrodynamics of the superconductor, since both play an important role in its overall response to a GR wave.
Gravitational Characteristic Impedance of Free Space
The Maxwell like representation of the Einstein equations of general relativity can be described the coupling of weak GR fields to slowly moving matter. In the asymptotically flat space-time coordinate system of a distant inertial observer, the four equations in SI units are
where the gravitational analog of the electric permittivity εG and magnetic permeability μG of free space is given by
The value of εG is fixed by demanding that Newton’s law of gravitation be recovered from the Gauss-like law (13), whereas the value of µG is fixed by the linearization procedure from Einstein’s field equations. These two constants express the strengths of the coupling between sources (i.e., of masses and mass currents, respectively) and gravitational fields, and are analogous to the two constants ε0 (the permittivity of free space) and µG (the permeability of free space), which express the strengths of coupling between sources (charges and charge currents, respectively) and electromagnetic fields in Maxwell’s theory.
In the above set of equations, the field is the gravito-electric field, which is to be identified with the local acceleration g of a test particle produced by the mass density ρG, in the Newtonian limit of general relativity. The field is the gravito-magnetic field produced by the mass current density jG and by the gravitational analog of the Maxwell displacement current density
of the Ampere-like law (11). The resulting magnetic-like field can be regarded as a generalization of the Lense-Thirring field of general relativity. Because these equations are linear, all fields will obey the superposition principle not only outside the source (i.e., in the vacuum), but also within the matter inside the source, provided the field strengths are sufficiently weak and the matter is sufficiently slowly moving. Note that the fields and in the above Maxwell-like equations will be treated as classical fields, just like the fields and in the classical Maxwell’s equations.
As noted earlier, Cooper pairs cannot freely fall along with the ionic lattice in response to an incident GR wave because the UP forbids such pairs from having classical trajectories, i.e., from traveling along geodesics. An incident field E will therefore cause the Cooper pairs to undergo non-geodesic motion, in contrast to the geodesic motion of the ions inside the lattice. This entails the existence of mass currents (as well as charge currents) from the perspective of a local, freely falling observer who is located near the surface of the superconducting film anywhere other than at its center of mass. These mass currents will be describable by a gravitational version of Ohm’s law
where JG (w) is the mass-current density at frequency w, ðS,G = ð1S,G(w) + ið2S,G (w) is the complex mass-current conductivity of the film at the frequency w in its linear response to the fields of the incident GR wave, and -INSIDE(w) is the driving gravito-electric field inside the film at frequency w.
The existence of these mass currents can also be inferred from DeWitt’s minimal coupling rule for superconductors 62. The real part of the mass conductivity, ð1S,G(w), describes the superconductor’s dissipative response to the incident gravito-electric field, while the imaginary part, ð2S,G (w) describes its non-dissipative response to the same field. The basic assumption behind equation (16) is that the mass-current density in any superconductor responds linearly to a weak GR wave at the driving frequency. One should view ðS,G (w) as a phenomenological quantity, which, like the electrical conductivity ðS, must be experimentally determined. In any case, the resulting optics for weak GR waves will be linear, just like the linear optics for weak EM waves.
An important physical property follows from the above Maxwell-like equations, namely, the characteristic gravitational impedance of free space ZG 63-65.
This quantity is a characteristic of the vacuum, i.e., it is a property of space-time itself, and it is independent of any of the properties of matter per se. As with
ohms in the EM case,
will play a central role in all GR radiation coupling problems.
In practice, the impedance of a material object must be much smaller than this extremely small quantity before any significant portion of the incident GR-wave power can be reflected. In other words, conditions must be highly unfavorable for dissipation into heat. Because all classical material objects have extremely high levels of dissipation compared to ZG, even at very low temperatures, they are inevitably very poor GW reflectors.65-66 The question of GW reflection from macroscopically coherent quantum systems such as superconductors requires a separate analysis due to the effectively zero resistance associated with superconductors, i.e., the lack of dissipation exhibited by matter in this unique state, at temperatures near absolute zero.
Peters78 reported on the gravitational refractive index nG, which was much larger than that generated by just considering induced quadruple moments, suggesting that his model encapsulates the dominant GW interaction with matter, given as
where ρ is the density of the medium.
Minter57 et al., give the reflection coefficient of a superconducting film from an incident GW as
where δ is the EM skin depth of the superconducting film and d is the film thickness.
From (19), the gravitonic contribution to the Casimir pressure for superconducting lead (Pb) of thickness d = 2 nm at zero temperature is plotted in Figure 14.68 The EM skin depth of Pb is δ= 37 nm. This result is compared with the photonic contribution to the Casimir pressure of superconducting lead. The EM reflection coefficient is 79
where λ = 83 nm is the coherence length. The photonic contribution to the Casimir pressure is calculated by using Equation (19).
James 68 claims that if measurements of the Casimir pressure plots shown in Figure (14) solid line, then one should conclude that the H-C effect is invalid, if we are to hold on to the idea of the graviton. However, if experiments show the Casimir pressure to be an order of magnitude larger than that predicted from the photonic contribution alone, this would be the first experimental evidence for the validity of the H-C theory and the existence of gravitons. This would open a new field in the way of graviton detection.
This series of articles discussed the opportunities, emerging trends, challenges and future direction promoted by the scientists, experimentalists and technologists whose focus is in translating metamaterials into practical systems and devices. Their unique electromagnetic properties have attracted considerable attention from researchers across multiple disciplines. With the complete degree of freedom to control over material properties, what is possible is limited only by our imagination. Magneto electric couplings can be a source of new behavior in Casimir systems, metamaterial Casimir repulsive effects can lead to anti-gravity and low cost solution for levitation. As a final comment, the authors acknowledge that MMS metasurfaces can provide nearly infinite group delay, which is very helpful in understanding Einstein precession, geodetic effects and provides new evidence for refining our understanding of the relativistic corrections to Newtonian celestial mechanics. For example, multi-knots Möbius strips can be considered as tiny strings that can vibrate in multiple dimensions, and depending on how they vibrate, they might be seen in 3-dimensional space as matter, light or gravity. The vibration of the string which determines whether it appears to be matter or energy, and every form of matter or energy is the result of the vibration of the string. The authors consciously set out to describe a model of the universe as a “Möbius Universe,” unbounded in the form of Möbius loop along the plane of space-time fabrics.
The authors would like to thank Dr. Ignaz Eisele (EMFT Fraunhofer Institute, Munich, Germany) for developing prototypes and custom structures, Dr. Karl-Heinz Hoffmann (Bavarian Academy of Science, Germany), Dr. Shiban Koul (IIT-Delhi, India), Dr. Tatsuo Itoh (UCLA, USA), Dr. Matthias Rudolph (BTU Cottbus, Germany), Dr. Afshin Daryoush (Drexel University, Philadelphia, USA) and Dr. Arye Rosen (Rowan University, NJ, USA) for their valuable inputs, technical discussions and developing prototype for the theoretical proof and experimental validation. The authors would also like to thank the Indian Government in 1990s for supporting the negative index Möbius strips based RF&MW components for space applications. A special heartfelt thank you goes to our close friend Prof. D. Sundararajan who passed away on 22nd September 2014. Prof. Sundararajan and Dr. Poddar had been working on this mysterious Metamaterial Mobius strips since 1990 and were the first ones to demonstrate Möbius strips sensors for the applications in underground mine detections, memory devices, RFID for the detection of chemical explosives and weapons made of plastics, and composite materials which were often missed and undetected by RADAR.
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