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US 8040586 B2 Abstract One exemplary metamaterial is formed from a plurality of individual unit cells, at least a portion of which have a different permeability than others. The plurality of individual unit cells are arranged to provide a metamaterial having a gradient index along at least one axis. Such metamaterials can be used to form lenses, for example.
Claims(54) 1. An electromagnetic apparatus, comprising:
a composite structure that includes a plurality of elements having respective adjustable individual responses, the plurality of respective adjustable individual responses providing one or more adjustable effective medium parameters that are spatially varying and that include an adjustable effective permittivity; and,
wherein at least some of the respective adjustable individual responses are adjustable in response to one or more external inputs.
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15. An electromagnetic apparatus, comprising:
a composite structure that includes a plurality of elements having respective adjustable individual responses, the plurality of respective adjustable individual responses providing one or more adjustable effective medium parameters that are spatially varying and that include an adjustable effective permeability; and,
wherein at least some of the respective adjustable individual responses are adjustable in response to one or more external inputs.
16. The electromagnetic apparatus of
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19. An electromagnetic apparatus comprising:
a composite structure that includes a plurality of elements having respective adjustable individual responses, the plurality of respective adjustable individual responses providing one or more adjustable effective medium parameters that are spatially varying and that include an adjustable effective refractive index;
wherein at least some of the respective adjustable individual responses are adjustable in response to one or more external inputs.
20. An electromagnetic apparatus, comprising:
a composite structure that includes a plurality of elements having respective adjustable individual responses, the plurality of respective adjustable individual responses providing one or more adjustable effective medium parameters that are spatially varying and that include an adjustable effective wave impedance; and,
wherein at least some of the respective adjustable individual responses are adjustable in response to one or more external inputs.
21. An electromagnetic apparatus, comprising:
a composite structure that includes a plurality of elements having a plurality of individual responses, the plurality of individual responses providing an averaged electromagnetic response at a first frequency;
wherein one or more of the elements contain one or more active devices arranged to receive one or more signal inputs, the one or more signal inputs having a high impedance at the first frequency and a low impedance at a second frequency substantially less than the first frequency.
22. The electromagnetic apparatus of
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33. An electromagnetic apparatus, comprising:
a composite structure that includes a plurality of elements having a plurality of individual responses, the plurality of individual responses being configured to define an averaged electromagnetic response that varies as a function of position within the composite structure;
where one or more of the elements contain discrete circuit components.
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47. An electromagnetic apparatus comprising:
a composite structure that includes a plurality of elements having a plurality of individual responses, the plurality of individual responses providing effective medium parameters of the composite structure, the plurality of individual responses characterized by respective resonance frequencies;
wherein at least one of the elements has an adjustable capacitance, the adjustable capacitance being adjustable in response to one or more external inputs and providing a correspondingly adjustable resonance frequency.
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Description This is a continuation of application Ser. No. 11/658,358 filed Jan. 22, 2007, now U.S. Pat. No. 7,538,946, which is a national stage entry of International Application PCT/US2005/026052, which is a non-provisional of U.S. provisional application No. 60/590,702, filed Jul. 23, 2004. This invention was made with Government assistance under DOE Grant No. DE-FG03-93ER40793 and NSF Grant Nos. DMR-9623949 and DMR-9724535. The Government has certain rights in this invention. A field of the invention is metamaterials. Another field of the invention is composite metamaterials. Another field of the invention is lenses and optics. Another field of the invention is magnetics. There have been demonstrations, including those by the present inventors and their colleagues, in which electromagnetic material response�either previously unobserved or otherwise difficult to achieve in conventional materials�has been obtained in artificially structured materials referred to herein as metamaterials. An example of unusual metamaterial response can be found in negative index metamaterials, which possess simultaneously negative permittivity (ε) and permeability (μ) over a finite frequency band. The fundamental nature of negative refraction has revealed a key role that metamaterials can play in materials physics, as negative index is a material property not available in existing materials. A generic description of material response can be found in the Drude-Lorentz model, which leads to the following frequency dispersive forms for ε and μ: Metamaterials can be designed that have either electric or magnetic resonances where there are no equivalent existing materials. Electric and magnetic resonances can be situated at any frequency in metamaterial structures. In particular, by combining electric and magnetic structures, it is possible to arrive at a material with a frequency band over which both ε and μ are simultaneously negative. The refractive index, n, for such a material, determined by taking the square root of the product εμ, is real, indicating the material is transparent to radiation. However, it has been shown that the correct choice for the sign of the square root is negative when both ε and μ are negative. Thus, materials for which ε and μ are both negative can be also characterized as negative index materials (NIMs). Prior art metamaterials include a collection of macroscopic cells that constitutes an array of split ring resonators. These examples are described in previous work by some of the present inventors and their colleagues. U.S. Patent Publication No. US-2001-0038325-A1, and its application Ser. No. 09/811,376, filed Mar. 16, 2001, entitled Left Handed Composite Media, now U.S. Pat. No. 6,791,432, are also incorporated by referenced herein. The demonstration of negative refractive index materials have confirmed various theories concerning the properties that would be possessed by negative refractive index materials. Many basic electromagnetic and optical principles need to be reconsidered as the basic physical explanations have always considered right handed magnetic materials and positive refractive indexes. A summary of some inventive features will be provided here to highlight some aspects of the invention. Additional inventive features may be found in the preferred embodiment description, attached hereto. In some embodiments of the invention, metamaterials are structured to achieve optical effects. Optics and optical effects, as used herein, includes the manipulation of electromagnetic waves, as well as visible wavelengths. In an embodiment of the invention, negative index metamaterials are optimized to produce negative index lenses. In another embodiment of the invention, metamaterials are modified to be form diffractive optical devices. In another embodiment of the invention, metamaterials are modified to form gradient index optical devices. In embodiments of the invention, optical devices have effects that may be scaled. Scaling may be used to fabricate metamaterials, including negative index lenses, diffractive optical devices, and gradient index optical devices over a wide range of frequencies, i.e., from low (RF, microwave) frequencies to high (mm, THz) frequencies. Negative index metamaterial lenses of the invention demonstrate reduced aberrations compared to positive index lenses. For an example embodiment planoconcave negative index metamaterial lens of the invention, an index value of −0.61 minimizes aberrations. As exemplary metamaterials of the invention are formed from macroscopic cells, and because physical characteristics (dimensions, type of dielectric material, relative positions, geometry, etc.) may be varied, it is possible to optimize optical effects in other negative index lenses and in the devices of the invention in general. The effects are more easily achieved at THz and lower, but the nature of the metamaterials will permit realization of the optical effects at visible wavelengths as well. Exemplary metamaterials of the invention are formed by a plurality of macroscopic cells. This presents a number of advantages in the formation of optical and other devices. In the case of diffractive optical devices of the invention, the surface profile of the metamaterial is tailored with a diffractive surface that produces broader frequency bandwidth (less chromatic aberration) and other advantages. The properties of the metamaterial may also be tailored cell-by-cell to produce a gradient index optic. Gradient index lenses are used in many optical applications. Metamaterials offer advantages in that the index profile can be specifically tailored as needed to provide focusing, beam steering, beam shaping or other optical functionality. Because metamaterials are based on macroscopic cells, active control and tuning on a cell-by-cell basis of the material may be realized. Combining this control, which can be implemented by active electronic devices or by electromechanical devices, with the gradient index concept, adaptive optical devices may be realized. Embodiments of the present invention are directed to various new metamaterials that significantly extend the range of metamaterial properties, enabling the potential for new physical and optical behavior, as well as unique electromagnetic devices. A metamaterial is an artificially structured material that includes a plurality of elements that are patterned in one or more dimensions, with each element having physical dimensions less than or on the order of an incident wavelength in the direction of wave propagation, and where each element is configured to exhibit desired electric and magnetic polarizations in response to an applied electromagnetic field. In many (but not all) metamaterials, the elements are made from conductors and are supported by a dielectric substrate. Preferred examples of elements include lattices of straight wire conductors and lattices of split ring resonators. As used herein, the term �supported� as used in the term �supported by a dielectric substrate� (or �host�) is intended to be broadly interpreted, and is not intended to be limited to being held on a surface of a substrate. A conductor may be contained or embedded within a dielectric substrate, for instance, and be �supported by� the dielectric substrate. The terms �dielectric� and �dielectric host� as used herein are intended to broadly refer to an electrically insulating material whose dielectric constant is greater than or equal to about +1, and preferably greater than +1. A dielectric host may be a gas such as air, or may be a substrate such as a dielectric polymer, glass, quartz, and the like. Exemplary metamaterials consist of a dielectric host supporting repeated unit cells of identical elements (i.e., a periodic structure) such as conductors. Other exemplary metamaterials may be formed from a heterogeneous collection of elements designed to produce gradients in one or more of the effective medium parameters of permittivity, permeability, refractive index or wave impedance. The present invention is directed to metamaterials designed to accomplish specific functions. For example, embodiments of the present invention are directed to metamaterials designed to exhibit a spatial variation in index. Other embodiments of the present invention are directed to methods for fabricating metamaterials. Embodiments of the invention may find useful applications in the fields of optics and lenses, among others. It will be appreciated that as used herein the term �optics� and �lenses� are intended to be broadly interpreted and not limited to apparatuses operable only at optical wavelengths. A �lens� may include an apparatus useful to manipulate electromagnetic waves, for example, that are not in the visible frequency. Exemplary metamaterials of the invention, including metamaterial lenses, interact with propagating free-space waves, and are not bound in any dimension. The present invention may be further discussed and described through the exemplary embodiments discussed below. A. Enhanced Diffraction from a Grating on the Surface of a Negative Index Metamaterial One aspect of the present invention is directed to negative index materials having surface characteristics optimized to tune refraction and diffraction. It has been discovered that the coupling to diffracted beams is enhanced at the interface between a positive and a negative index material. Because diffractive optics are essentially patterned gratings, embodiments of the present invention utilize a specified grating in a negative index material as a lens element. The design of the grating may be based on Eq. 1 below. Using this relationship, a negative index grating lens may be optimized using traditional optical methods. One important benefit of this embodiment of the invention is that the efficiency is so much greater using metamaterials, so that much more compact lenses than are available in the prior art are realizable. An exemplary embodiment includes a negative index material formed at whatever frequency is appropriate and fashioned into a grating structure designed to focus or otherwise manipulate light or other electromagnetic waves. Numerical simulation as well as by measurements on negative index metamaterial wedge samples shows the unavoidable stepping of the refraction interface�due to the finite unit-cell size inherent to metamaterials�can give rise to a well-defined diffracted beam in addition to the negatively refracted beam. The direction of the diffracted beam is consistent with elementary diffraction theory; however, the coupling to this higher order beam is much larger than would be the case for a positive index material. Recent demonstrations of artificial materials with negative refractive index (n) have initiated an exploration into the use of these materials to investigate new physics and to develop new applications. So many exotic and remarkable electromagnetic phenomena have been predicted to occur in negative index materials, such as reversed Cerenkov radiation and reversed Doppler shifts, that even the most basic of electromagnetic and optical phenomena must be carefully reexamined in the context of negative index media. For example, in an analysis of the imaging properties of a planar slab of n=−1, it has been predicted that resolution exceeding that of any positive index optical component may be realizable. An artificial medium composed of two interspersed lattices of conducting elements was fabricated and reported to have a negative index of refraction. A wedge sample composed of this material, as shown in In the sample used, the unit cell size of 5 mm was roughly a factor of six smaller than the free space wavelength, so that the material could be expected to be reasonably characterized by effective medium theory. The finite unit cell size, however, led to an unavoidable stepping of the surface: to achieve the 18.4 degree refraction surface, the metamaterial surface was cut in steps of three unit cells by one unit cell, as indicated in The surface stepping on the metamaterial sample constitutes a grating, which could be expected to generate a diffracted beam in addition to the zeroth order refracted beam. The condition for the generation of zeroth and higher order beams can be determined from the well known grating formula
where θm is the angle of incidence to the normal to the interface from the medium side and θ is the refraction angle. Eq. 1 accounts both for refraction (second term) as well as diffraction (first term). The elementary arguments leading to Eq. 1 do not allow the relative coupling of the incident beam to the various possible outgoing beams to be determined. A theoretical analysis of the plane wave diffraction from a grating between positive and negative index materials predicts an enhanced coupling to the diffracted orders. This enhanced coupling can be understood by the following argument. A wave incident on a periodically patterned surface will couple to all transmitted and reflected waves whose wave vectors along the interface match that of the incoming wave (kx) to within a reciprocal lattice vector (i.e., kx+mπ/d, where m is an integer). This set of modes includes both propagating components�the zeroth order refracted wave and higher diffracted orders�in addition to evanescent components for which (kx+mπ/d)>ω/c. The reflection and transmission coefficients for the evanescent waves generated at the surface between a positive and a negative index medium have far greater magnitude than those generated at the surface between two media of the same index sign. In a perturbative sense, the grating modulation leads to a coupling between the incident beam and all diffracted beams, mediated by the evanescent components. As these components can have very large magnitude between positive and negative media, the coupling between the incident beam and higher orders is also correspondingly much larger. To explore the properties of diffracted beams in a negative index sample, we simulate a wave incident on the interface between a negative index wedge and free space, as shown in A surface stepping added to a positive index wedge produces a single refracted beam identical to that of the smooth wedge; however, a surface stepping added to the negative index wedge shown in The parameters for the geometry in the simulation presented in The different curves in Below 8.5 GHz, the right hand side of Eq. 1 exceeds unity, and a diffracted beam is no longer possible. We thus expect that the coupling strength of the diffracted beam would approach zero near this frequency, which is consistent with the simulation results. The numerical study of While the simulations presented here are based on continuous, homogeneous materials, the artificially structured negative index metamaterials can also be approximated as continuous materials. It is thus believed and expected that the surface stepping in such metamaterials will lead to the same diffraction phenomenon found in the simulations described above on homogeneous, stepped wedge samples. The simulations above indicate that for the sample used in To further study and clarify experimentally the issue of higher order beams, we perform an angle-resolved mapping of the fields transmitted by each of two different metamaterial wedge samples as a function of frequency. One of the samples is that used in The apparatus used for the experiments is based on a parallel plate waveguide as is known. An incident beam having minimal transverse phase variation is generated by coupling microwaves from an X-band coax-to-waveguide adapter (HP X281A) into a channel of parallel plate wave guide. Absorber (Microsorb Technologies Inc. MTL-73) is patterned so as to gently widen along the path of the beam from the 0.9″ width of the adapter, forming an exit aperture of roughly 15 cm (6″). The channel is connected to a parallel plate semicircular central chamber, in the center of which is placed the metamaterial sample. The length of the channel (coax adapter to exit aperture) is 40 cm. A waveguide detector is positioned at the radius of the semicircular plates, a distance of 40 cm from the sample, and is capable of being swept over an angular range of nearly 180 degrees. As a control, the angular distribution of power refracted from a Teflon sample, with the same dimensions and surface step size as the sample of In contrast to positive index media, negative index media is inherently frequency dispersive. The expected frequency region of negative refraction for the sample used by in The agreement is quite good considering the simplicity of the theory, which assumes a continuous medium. These results suggest that not only does the bulk metamaterial behave as a continuous material with negative refractive index, but also the surface stepping contributes to the properties and can be modeled as steps in an otherwise continuous material. It has been discovered, and confirmed by both Eq. 1 and the simulations, that by reducing the size of the unit cell by a modest amount, effectively reducing the refraction surface grating length, the diffracted beam can be eliminated. An embodiment of the present invention includes a new metamaterial sample with stepped surface having a unit step size of 2.5 mm�half the size of that used in A map of the transmitted power as a function of frequency and angle for the 2.5 mm sample is shown in The simulations presented here, as well as the experimental data in B. Negative Index Lens Aberrations Another aspect of the present invention is related to negative index lenses. In recent years, there has been great interest in negative index artificial materials (�NIM�). One area of interest includes the perfect lens concept. A perfect lens is a flat slab of index minus one, which can focus images with resolution exceeding that possible with positive index optics. Focusing by a curved surface on negative index media is possible. Traditional spherical profile lenses composed of negative index media have several advantages over their positive index counterparts: they are more compact, they can be perfectly matched to free space. Further, it has been discovered that they can also have superior focusing performance. Exemplary metamaterial lenses of the invention are configured as composite structures whose electromagnetic properties are made to vary spatially throughout the composite. Importantly, both the permittivity and the permeability can be varied independently throughout the structure of the invention, leading to previously unrealized optical devices. An exemplary embodiment is an artificially structured composite metamaterial consisting of a dielectric substrate or host material supporting a plurality of conducting elements, each of which is designed so as to exhibit desired electric and magnetic polarizations in response to an electromagnetic field, the composite exhibiting a desired spatial variation in the permittivity and/or permeability along one or more axes, such that at least one element differs in its electric or magnetic polarization from the other elements. As used herein the term �spatial variation� is intended to be broadly interpreted as meaning a variation with spatial position. For example, a metamaterial having a permeability that exhibits spatial variation may have a permeability that varies with position along one or more of the X, Y and Z axis in the metamaterial. Importantly, the spatial variation in magnetic permeability is independent from said electric permittivity�permeability and permittivity may be �tuned� separately from one another in metamaterials of the invention. Such metamaterials have many useful and beneficial applications. For example, in some exemplary metamaterials of the invention a ratio of the magnetic permeability to the electric permittivity is maintained substantially constant and substantially equal to the same ratio for a material adjacent to or surrounding the metamaterial (with examples including free space or a second material in which the composite metamaterial is embedded), so that impedance matching is achieved. Also, the sign of the metamaterial permeability and permittivity can be controlled, with both being negative in some exemplary metamaterials to provide a negative index metamaterial. These and other advantages and benefits will be clear to those knowledgeable in the art when considering the detailed discussion of exemplary embodiments that follows. The monochromatic imaging quality of a lens can be characterized by the five Seidel aberrations: spherical, coma, astigmatism, field curvature and distortion. These well known corrections to the simple Gaussian optical formulas are calculated from a fourth order expansion of the deviation of a wave front from spherical. (A spherical wave front converges to an ideal point focus in ray optics). The coefficients in this expansion quantify the non-ideal focusing properties of an optical element for a given object and image position. We find that there is an asymmetry of several of the Seidel aberrations with respect to index about zero. Considering that an interface with a relative index of +1 is inert and one of relative index −1 is strongly refractive, this asymmetry is not surprising. However, the present discovery that the asymmetry can yield superior focusing properties for negative index lenses is a surprising and unexpected result. Negative index media are necessarily frequency dispersive, which implies increased chromatic aberration and reduced bandwidth. However, diffractive optics, which possesses a similar limitation, has found utility in narrow band applications. To confirm the analytical aberration results, custom ray tracing code has been developed that does not rely on the sign of the index to determine the path of the ray, but relies only on the permittivity, ε, the permeability, μ, Maxwell's equations and conservation of energy. Between interfaces, in homogenous media, the ray propagates in a straight line following the direction of the Poynting vector. Refraction across an interface, from a region labeled 1 into a region labeled 2, is handled as follows. Wave solutions are sought that satisfy the dispersion relation (obtained from Maxwell's equations) in region 2,
where k2 is the wave vector in region 2. The solutions must also satisfy a boundary match to the incident wave, requiring
It has been discovered that the form of the expressions for the Seidel aberrations of thin spherical lenses found in the optics literature are unchanged by the consideration of negative index media. This conclusion is confirmed by re-deriving these expressions, from first principles, using only the definition of optical path length and Fermat's Principle. We interpret the optical path length, OPL=∫Cn(s)ds, to be the phase change (in units of free space wavelength) that a wave would undergo along the path C, if C is oriented parallel to the Poynting vector. The optical path may have contributions that are negative where the Poynting vector and the wave vector are antiparallel, i.e. where the index is negative. These aberration formula are further corroborated by agreement with the results of our ray tracing. The wave aberration, ΔOPL, is the difference in optical path length of a general ray and a reference ray, where the reference ray passes through the optic axis in the aperture stop and the general ray is parameterized by its coordinate in the aperture stop, r, and its coordinate in the image plane, h as shown by To be in the Gaussian optic limit, where spherical interfaces yield perfect imaging, r and h must be near zero. A series expansion of the wave aberration in these parameters First examined is the important case of a source object at infinite distance. This is a position factor of −1. We are left with two parameters that can be used to reduce aberrations, n and q. We will set the value of q to eliminate one of the aberrations and compare the remaining aberrations as a function of index. We will restrict our attention to moderate values of index. At large absolute values of index, the aberrations approach the same value independent of sign, but dielectric lenses with high index have significant reflection coefficients due to the impedance mismatch to free space. The usual ordering of the aberrations is from highest to lowest in the order of r, the aperture coordinate. This is the ordering of most image degradation to least if one is forming images with significant lens aperture, but small to moderate image size, which is a common occurrence in applications. Thus, spherical aberration is an obvious target for elimination. However, there are no roots of C200 for values of index greater than one, which is why this aberration is referred to as spherical aberration, since it appears to be inherent to spherical lenses. The preferred practice is to eliminate coma (the next in line), and it so happens that the resulting lens has a value for the spherical aberration that is very near the minimum obtainable. Adjusting the shape factor, q, is often called lens bending. If we bend the lens for zero coma, that is find the roots of C110 with respect to q we obtain In Note that there are two values of index where q=1, which represent a plano-concave/convex lens. Setting (10) equal to one we obtain,
Several ray tracing diagrams with both meridional rays and ray spot diagrams are shown for specific values of index in It is a surprising and significant result that negative index permits an entire family of spherical aberration free spherical lenses that can focus a distant object to a real focus, as illustrated by The solution with the negative sign in the expression for qs (solid curves) has less coma for moderate negative values of index, so ray tracing diagrams are shown for that solution. We note that at n=−1, the field curvature is also zero, thus this lens has only two of the five Seidel aberrations, coma and astigmatism. For a positive index reference we use the zero coma, n=φ lens from above. Here again, negative index lenses achieve a tighter focus than a comparable positive index lens. Now we examine the case of |p|<1, which is a real object and real image both at finite position. Since p and q are both free parameters, we can conceivably eliminate two aberrations. If we eliminate spherical aberration and coma the resulting lens is called aplanatic. It is a well known, though not accurate, result that a spherical lens can only have virtual aplanatic focal pairs. The more accurate statement is that only negative index spherical lenses can have real aplanatic focal pairs. If we set C200 and C110 to zero and solve for p and q, we obtain four solutions, the two non-trivial ones are given by Ray diagrams are shown for lenses with magnifications of −1, −2 and −3. Also shown is a reference positive index lens for each. The reference lenses (which cannot be aplanatic) are of moderate index, φ, with the same magnification and f/# as the lenses they are compared to. They are bent for zero coma but also have spherical aberration near the minimum possible for the configuration. Again, the negative index lenses produce superior foci. The lens of index −1 and magnification −1 is particularly interesting. At this index value the field curvature is also zero. This remarkable lens configuration has only one of the five Seidel aberrations, astigmatism. This is confirmed by ray tracing which shows a one dimensional �spot� at the image plane. This is perfect focusing in the sagittal plane. Perfect focusing also occurs in the meridional plane, in front of sagittal focus. One may ask why this asymmetric lens, q=−1, performs so well in a symmetric configuration, p=0. This lens can be equivalently viewed as a biconcave doublet with one component. We found that all biconcave doublets with arbitrary indices of �n have identical focusing properties. The only observable difference is in the internal rays, which are always symmetric about the planer interface, but make more extreme angles at higher index magnitude. Fabrication of any of these negative index lenses can be done using periodically structured artificial materials. Artificial material designs can operate at frequencies from megahertz through terahertz, where there are numerous communication and imaging applications. For example, lens antennas could benefit both by a reduction in aberrations, which translates directly into increased gain, and by a reduction of mass, afforded by low density artificial materials. Furthermore, these lenses are even easier to implement than a perfect lens, since they lack its severe structure period per wavelength requirements and are more tolerant to losses. Negative index lenses at visible light frequencies may also be possible, by using photonic crystals, which have shown potential for negative refraction. Using the current optical system design paradigm, aberrations are minimized by combining elements with coefficients of opposite sign. However, more elements mean greater complexity and cost. Taking advantage of an expanded parameter space that includes negative index can reduce the number of required elements, with lenses having only one element believed to be possible. By way of further illustration, additional exemplary lenses of the invention are described. One exemplary metamaterial lens of the invention is comprised of homogenous isotropic refractive media with two opposing faces defined by co-axial spherical surfaces, such that the thickness of the lens is thin compared to its diameter and the distances to its intended focal points. In the geometric limit, the lens focuses parallel light (light from a very distant or collimated source) to a real image point with zero spherical aberration. To achieve this the shape factor, An additional exemplary metamaterial lens is as above but configured for the specific case that n=−1. The lens focuses parallel light to a real image point with zero spherical aberration and zero field curvature aberration. An additional exemplary metamaterial lens of the invention comprises homogenous isotropic refractive media with two opposing faces defined by co-axial spherical surfaces, such that the thickness of the lens is thin compared to its diameter and the distances to its intended focal points. In the geometric limit, the lens focuses light from a point source at a finite location to a real aplanatic point, that is a point with zero spherical aberration and zero coma aberration. To achieve this the shape factor, q, the position factor, p, (P≡1 (2f/S′), where f is the focal length and S′ is the distance from the lens to the image), and the refractive index, n, are adjusted such that, q=�2(n+1), and P=∓[n+1)/(n−1)]. We note that real aplanatic images are only possible when n<0. Such a lens an also be configured for the specific case that n=−1. In this case the field curvature aberration is also zero. An additional exemplary metamaterial lens of the invention comprises a bi-concave doublet lens composed of two plano-concave component lenses contiguously joined on their flat sides. The two component lenses are of equal thickness and have spherical surfaces of equal curvature. The refractive media comprising the component lenses are related by, n1=−n2. This doublet lens possesses all the properties of the lens above�that is it can focus a real image with zero spherical aberration, zero coma aberration and zero field curvature aberration. Implementation requires n<0, but can be realized with metamaterials. Although metamaterial lenses have been discussed and illustrated herein having negative indexes, it will be appreciated that positive index lenses made of metamaterials will likewise be useful and beneficial. C. A Gradient Index Metamaterial An additional aspect of the present invention is directed to a metamaterial having a gradient index. One embodiment of this aspect of the invention includes a structured metamaterial based on conducting split ring resonators (SRR's), which has an effective index of refraction with a constant spatial gradient. The gradient is experimentally confirmed by measuring the deflection of a microwave beam by a planar slab of the composite metamaterial over a broad range of frequencies. The gradient index metamaterial of the invention represents an alternative approach to the development of gradient index lenses and similar optics that may be advantageous, especially at higher frequencies. In particular, the gradient index material of the invention may be suited for terahertz applications, where the magnetic resonant response of SRRs has recently been demonstrated. An example of unusual metamaterial response can be found in negative index metamaterials, which possess simultaneously negative permittivity (ε) and permeability (μ) over a finite frequency band. The negative index metamaterials thus far demonstrated have been formed from periodic arrays of conducting elements, the size and spacing of which are much less than the wavelengths of interest. The shape of the repeated conducting element determines the electromagnetic response of the collective, which can be approximated as having an electric or a magnetic resonance. Application of effective medium theory to the overall periodically patterned composite allows a description in terms of bulk isotropic or anisotropic ε and μ. The split ring resonator (SRR), shown in the insets to In the prior art, metamaterials have been constructed from repeated unit cells containing identical elements, such that the resulting medium can be considered homogeneous in the sense that the averaged electromagnetic response does not vary over the structure. An embodiment of the present invention, on the other hand, includes metamaterials whose averaged electromagnetic properties vary as a function of position. Such spatially dispersive materials are of interest, for example, as they can be utilized in a variety of applications, including lensing and filtering. An embodiment of the present invention is a metamaterial based on SRR's in which a pattern of spatial dispersion is introduced by a slight change in the properties of each successive element along a direction perpendicular to the direction of propagation. The resulting metamaterial has a constant gradient index along this axis of the metamaterial, which can be confirmed by beam deflection experiments. While an SRR medium is known to have a predominantly magnetic response, this is not of direct interest here; rather, we are concerned with the =e refractive index n(ω) of the SRR medium, found from n(ω)=√{square root over (ε(ω)μ(ω))}{square root over (ε(ω)μ(ω))}, with μ(ω) given by Eq. 1 and ε(ω) approximated as constant. This form of the dispersion, ω=ck/n(ω), can be compared with that obtained from a numerical solution of Maxwell's equations for a single unit cell. To obtain the dispersion diagram numerically, we compute the eigen frequencies for a single unit cell ( The resonant frequency of an SRR, ωr, depends rather sensitively on the geometrical parameters and local dielectric environment for the SRR. Since μ(ω) depends strongly on ωr (Eq. 1), relatively small changes to the basic repeated unit cell can result in substantial changes to the permeability of the composite, especially near the resonance. The change in index n(ω)=√{square root over (ε(ω)μ(ω))}{square root over (ε(ω)μ(ω))} with change in resonant frequency can be calculated using Eq. 1. For convenience we neglect damping and set ε(ω)=1 as the primary role of the permittivity over the frequency band of interest will be to rescale the dispersion curves. At low frequencies (ω<<ωr) the index changes linearly with small changes in the resonance frequency, or Those knowledgeable in the art will appreciate that there are a variety of modifications to the SRR or its environment that can be used to introduce a variation in ωr. For example, adjustments to the size, amount, spacing or geometry of the conductor arrays or to the dielectric can be made. One exemplary method is to adjust the depth of cut of the dielectric substrate material surrounding the SRR. This method is compatible with exemplary sample fabrication, in which SRRs are patterned on copper clad circuit boards using a numerically controlled micromilling machine. The removal of dielectric material from the region near the SRR (ε�3.8 for FR4 circuit board) changes the local dielectric environment of the SRR, effecting a change in the resonance frequency. In Because the SRR exhibits a resonant frequency ωr that increases linearly as a function of the substrate cut depth, it is a convenient element from which to design a gradient index metamaterial. In particular, a metamaterial of the present invention includes one of a linear array of SRRs in which ωr varies as a function of cell number. For example, if a metamaterial having a substrate cut depth that advances linearly as a function of cell number, ωr will then also advance linearly as a function of cell number; that is, ωr becomes linearly proportional to distance. Using this relationship in Eqs. 2 and 3, we see that the gradient of the index will thus be approximately constant as a function of distance, at least for frequencies far enough away from ωr. A constant gradient metamaterial has been experimentally confirmed by observing the deflection of a beam incident on a planar metamaterial slab whose index varies linearly (in a direction perpendicular to incident radiation). To calculate this deflection, we consider two normally incident but offset rays entering a gradient index planar slab of thickness t, as shown in The rays will acquire different phase advances as they propagate through the slab. Assuming the two rays enter at locations x and x+Δx along the slab face, then the acquired phase difference of the two beams traversing the slab, An exemplary gradient index metamaterial of the invention includes a dielectric substrate, at least two conductor lattices interspersed with one another and supported by the dielectric substrate. The at least two conductor lattices and dielectric are dimensioned to provide a gradient in effective permeability along at least one axis. As used herein the term �dimensioned� is intended to be broadly interpreted and includes forming components having particular dimensions. For example, dimensioning a dielectric and conductor lattice may include setting the size of the conductors, the spacing between the conductors, the type of dielectric used, the amount of dielectric used, the capacitance of one of the conductors, and the like. It will be understood that although in an exemplary metamaterial a depth of cut of dielectric is used to dimension unit cells to result in a gradient index, other methods of dimensioning can be carried out. One exemplary constant gradient index metamaterial includes a linear array of SRR's in which the substrate depth is a linearly increasing function of cell number in the direction perpendicular to the propagation. The resulting array should then deflect an incident beam by an angle that can be predicted by the dispersion diagrams in The curves in To fabricate exemplary gradient index metamaterial samples, an LPKF micromilling machine was used to mill varying length strips of SRRs (number of cells) from copper clad (single side) FR4 circuit board substrates. Several samples were fabricated with thicknesses (in the direction of propagation) of 1, 3, or 5 unit cells. The composite metamaterials are composed of roughly forty strips spaced one unit cell apart, each strip having the substrate milled to a different depth. The resonance frequency of each SRR strip is measured in an angular resolved microwave spectrometer (ARMS). The measured resonance frequencies of each strip versus the depth of cut are plotted in The composite gradient index samples are measured in the ARMS apparatus. To confirm the gradient in the sample, a microwave beam was directed normally onto the face of the sample (as in A detailed comparison of the measured and calculated angle of deflection versus frequency is shown in Another embodiment of a gradient metamaterial of the invention includes gradient index lenses. A parabolic (as opposed to linear) distribution of index in the slab along an axis perpendicular to the direction of wave propagation results in a structure that focuses radiation rather than deflecting it. Examples of such gradient index lenses include radial gradient index rod lenses, used at optical frequencies, and Luneberg lenses, used at microwave frequencies. Gradient index rod lenses use optical glass materials that are ion doped via thermal diffusion. This process can produce only modest variations of the refractive index, (less than 0.2), and is limited to fairly small diameter rods, (less than 1 cm). Luneberg spherical or hemispherical lenses, which require the fairly wide index range of n=1 to n=2, can be implemented as stepped index devices with no particular size limitation. Both devices employ gradients in the dielectric permittivity only, and thus have limited impedance matching to the surrounding media. Gradient index metamaterials may provide a useful alternative approach to the development of optics. With the increased range of material response now identified in metamaterials, including negative refractive index, considerably more flexibility and improved performance from traditional and planar lenses formed from artificially patterned media should be possible. Gradient index metamaterials of the invention that include magnetic permeability gradients, for example, could be used to develop materials whose index varies spatially but which remains matched to free space. Moreover, gradient index metamaterials of the invention are believed to be realizable at higher frequencies including THz. Metamaterials having a gradient index may prove beneficial in numerous additional applications. A metamaterial of the invention, for example, includes lens formed from a metamaterial having a gradient index. The gradient index may be configured to provide a wide variety of focusing effects. A circular metamaterial having planar surfaces and formed from a plurality of unit cells, for example, can be configured with a first index in a center region and a gradually decreasing (or increasing) index moving outward radially. This can result in a variety of focusing effects from a substantially flat metamaterial. Other geometries, including non-planar surfaces and perimeter shapes other than circular, can likewise be used. Also, a plurality of metamaterials having gradient indexes may be arranged in a stacked configuration with one another to direct rays as desired. Rays could be directed, for example, to pass �around� an object, thereby rendering it substantially �invisible.� D. Fabrication and Characterization of a Negative-Refractive-Index Composite Metamaterial Another aspect of the present invention is directed to composite metamaterials. An exemplary negative index includes 2.7 mm thick composite panels having negative refractive index between 8.4 and 9.2 GHz. The exemplary composite metamaterial is fabricated using conventional commercial multilayer circuit-board lithography; three-dimensional physical (as opposed to electromagnetic) structure is introduced by the use of vias to form sections of the scattering elements in the direction perpendicular to the circuit board surfaces. From scattering parameter measurements, the complex permittivity, permeability, index, and impedance of the composite is unambiguously determined. The measurements enable the quantitative determination of the negative index band and associated losses. The extracted material parameters are shown to be in excellent agreement with simulation results. Prior art metamaterials have been constructed and used to demonstrate experimentally negative refraction at microwave frequencies. This material made use of an array of dual split-ring resonators (SRRs) that provided a negative interspersed with an array of wires that provided a negative ε. Because the region of negative c overlapped the region of negative μ associated with the SRR's, the composite had a frequency band of negative index. In this material, the SRR's and wires were patterned by optical lithography on either side of a circuit board substrate. The SRR/wire structures have proven difficult to fabricate. For example, the SRR elements impose a specific burden on fabrication that adds a layer of complexity to current negative index metamaterial designs. Unlike straight wires, the SRR elements typically require a significant length in the direction of wave propagation in order to provide a strong magnetic response. To meet this constraint, previous circuit-board based SRR designs have required sectioning a planar SRR circuit board into strips, or using sheets of the planar circuit boards, oriented so that the incident wave direction is in the plane and the SRR axes lie perpendicular to the propagation direction of the incident wave. One exemplary metamaterial of the present invention utilizes multilayer circuit board techniques to fabricate a negative index metamaterial structure as shown by One exemplary composite metamaterial is assembled from three laminated dielectric layers 50, 52 and 54. The top 50 and bottom 54 layers consist of Rogers 4003 circuit board laminates (ε=3.38, tan d=0.003), with a prepreg layer 52 of Gore Speed Board (ε=2.56, tan d=0.004) sandwiched therebetween. An exemplary layer 52 is a 0.0015″ thick layer of Gore Speedboard. The total thickness of the structure (layers 50, 52, and 54) is therefore about 0.065 in. Further exemplary dimensions are indicated in Table I. The layers are bound laminated together, for example using an adhesive at the interfaces between the Gore and Rogers circuit. Two sides of the nominally rectangular SRR element are patterned on the outside faces of the Rogers circuit board with thin conductor strips 58 made of copper. The remaining two perpendicular sides or legs of the conductor are formed from vias�plated through holes�that extend through laminate layers 50, 52 and 54 of the circuit board. One side 60 is formed by a through via extending continuously through all three layers of circuit board. The last side of the SRR is formed by two blind vias 62 and 64, that each terminates at respective sides of the center laminate layer 54 at circular plates 66. The dielectric layer 54 defines a gap between the plates 66. Additional dielectric layers 68 may be provided to cover the uppermost and lowermost surfaces of the laminate. Small annular rings introduce the capacitive gap. It will be appreciated that many equivalent structures will be possible within practice of the invention in addition to that illustrated by Many other particular embodiments are possible. Indeed, those knowledgeable in the art will appreciate that one of the important advantages of the component metamaterials of the invention, with an example being that of Still further, the embodiment of Further dimensioning to a unit cell such as that of Using the driven solution in (HFSS) (Ansoft), a finite element based software package that solves Maxwell's equations, the S-parameters were simulated for variations of the unit cell shown in To confirm the expected properties of the negative index composite, the magnitude and phase of the S-parameters (S11 and S21) were measured. The experiments were performed in free space. In the experiment, an Agilent 8510B vector network analyzer was used to sweep microwaves over a frequency range 7-13 GHz. Two microwave horns (Rozendal Associates Inc., Santee, CA) were used as the source and detector. Lens assemblies mounted on the horns produced a focused spot at a distance of approximately 30.5 cm (12 in.). The sample was placed at the focus. For transmission experiments, a confocal setup was used, in which both the source and detector horns were placed one focal length from the sample. Calibration was performed using a �through� measurement, in which the transmitted power was measured in the absence of any material. For the reflection measurement, the horns were moved to the same side of the sample. Because of the finite size of the horn/lens assemblies, the two horns were offset from each other such that the power was incident at an angle of 16� from the surface normal of the sample. The reflection was calibrated by measuring the reflected power from an aluminum plate�assumed a perfect reflector (with a 180� phase shift) for this measurement. The magnitude and phase of the S-parameters for a single layer of the negative index composite are presented in A method of demonstrating negative index in past work has been to first measure the transmitted power through a sample of SRRs alone, identifying the frequency range of the stop band where μ<0; then measure the power transmitted through a wire structure alone; and finally measure the power transmitted through the composite structure. This method is convenient when phase data is not available and when the frequency of negative index forms a well-defined pass band. As can be seen from The full retrieval of the material parameters for a metamaterial proceeds by a measurement of the transmitted and reflected amplitudes and phases from a slab of finite thickness. For continuous, isotropic materials, the transmission and reflection coefficients have analytic forms that can be readily inverted. For example, the inversion of the scattering equations leads to the following form that allows determination of the refractive index: The retrieved z and n are shown in A negative index frequency band occurs in the measured sample between 8.4 and 9.2 GHz. The agreement between the simulated and measured data over the negative index region is excellent, both quantitatively and qualitatively. Any existing disagreement between the four sets of curves could be further minimized by making slight alterations to the material parameters, for example, adjusting the conductivity used for the copper elements. Some of the disagreement might be caused by the non-normal incidence used in the S11 measurement. The permittivity (ε) and the permeability (μ) are simply related to n and z according to ε=n/z and μ=n z. The retrieved frequency dependent ε and μ are presented in The real part of a exhibits a zero, below which it is negative. Also, μ of the composite exhibits a characteristic resonant form, due primarily to the response of the SRR's, which has a region where the real part of μ is negative. The frequency band where both real parts of ε and μ are negative is consistent with the negative index band found in In summary, one exemplary aspect of the present invention is directed to a composite metamaterial that possesses a frequency band over which the refractive index is negative. An exemplary structure has numerous advantages and benefits, including, for example, in terms of fabrication since it can be entirely assembled using conventional multilayer circuit board technology that obviates the need for cutting and further assembly steps. The exemplary structure is amenable to numerical simulations which show excellent agreement with the measured structures. A full S-parameters retrieval provides complete information on the material parameters of an exemplary sample in a direct manner. While indirect methods, such as Snell's law measurements, can provide important complementary information, the S-parameters measurement and retrieval can form the basis of a semi-automated metamaterial characterization procedure. E. Additional Metamaterial Embodiments of the Invention Metamaterials have become of great interest, in the context of surface plasmon optics, in the context of negative index metamaterials, as well in other contexts. In negative index metamaterials, however, the negative response has associated with it inherent limitations. The frequency region near the resonant frequency displays increased absorption, where the imaginary part of either ε or μ (depending on which is resonant) becomes relatively large. This behavior is both consistent with and inherent to resonance behavior, and ultimately places key restrictions on negative materials. For passive negative materials:
An alternative way of understanding these limitations is in terms of the stored energy density. The familiar expression for the field energy density in a material is When the details of the Drude-Lorentz medium are taken into account, a more general expression for the energy density of a dispersive medium can be found as follows: Materials with either ε or μ negative are known. In naturally occurring materials, the resonances that give rise to the Drude-Lorentz forms occur within generally restricted frequency ranges. Electric resonances, for example, tend to occur in the high THz frequencies or higher, and result from phonon modes, plasma-like oscillations of the conduction electrons, or other fundamental processes. Magnetic resonances generally occur in inherently magnetic materials, associated with such processes as ferromagnetic or antiferromagnetic resonance. These resonances tend to die out in the higher GHz frequencies, and are absent in all but a few specialized systems at THz frequencies. Metamaterials can be designed that have either electric or magnetic resonances where there are no equivalent existing materials. Electric and magnetic resonances can be situated at any frequency, up to THz frequencies, in metamaterial structures. In particular, by combining electric and magnetic structures, it is possible to arrive at a material with a frequency band over which both ε and μ are simultaneously negative. The refractive index, n, for such a material, determined by taking the square root of the product εμ, is real, indicating the material is transparent to radiation. However, it has been shown that the correct choice for the sign of the square root is negative when both ε and μ are negative. Thus, materials for which ε and μ are both negative can be also characterized as negative index materials (NIMs). NIMs are of interest because, among other reasons, they allow previously unavailable solutions of Maxwell's equations. As such, NIMs represent a striking example of the utility of metamaterials. Yet, although remarkable physical phenomena have been predicted for NIMs, the fundamental limitations of negative materials must be kept in mind. For example, it has been suggested that a surface with ε=μ=−1 can be reflectionless. This statement, however, may be limited to a steady-state condition; if a wave front from free space impinges on such a surface, reflections associated with transients may occur until the steady state solution is reached. The effort over the past several years has proven that negative index metamaterials can be designed, fabricated and characterized. Negative refraction in steady state experiments has been demonstrated. The collection of initial work has placed negative refractive index on solid ground: We are now in a position to move forward and further develop the materials and methods that will make these novel materials useful. Exemplary embodiments of the present invention are directed to novel and useful metamaterials. Although several embodiments are directed to negative index metamaterials, they may likewise find utility in the form of positive index metamaterials. E(1). Dual Polarization NIM Development As yet, there has been no demonstration of a material with negative index for waves having arbitrary incident polarization. The present invention includes such a metamaterial, with an exemplary embodiment illustrated in FIG. 20. Fabrication of such an exemplary structure requires the unit cell to contain an additional magnetic loop oriented perpendicular to the first, as well as an additional wire element aligned with the second electric polarization direction. The exemplary metamaterial illustrated in The exemplary design shown in E(2) Component-Based NIM Structures To achieve the magnetic response associated with NIM's, the underlying unit cell contains a resonant circuit element that couples to the magnetic field component of the incident EM wave. While in typical prior art a double split ring resonator (DSRR) has been employed to provide the magnetic response, a single ring resonator, properly designed, can achieve an equivalent magnetic response, and is easier from a fabrication point-of-view. The present invention extends the functionality of existing NIMs in the microwave frequency range by embedding miniaturized packaged electronic components within the unit cell. This step allows greater design flexibility, by concentrating a critical element of the electrical performance within a single modular package that can be changed, upgraded or modified as needed. The basic SRR has the simplified equivalent circuit shown in For structures including (but not limited to) low frequency structures, rather than setting the capacitance in the SRR circuit by means of geometry it has been discovered that a (standard) packaged capacitor can be substituted. The resonant frequency of the SRR can thus be set by the chosen capacitor value. However, using packaged capacitors or other components pushes the limitations of the simplified circuit presented. Other factors become important: dielectric losses due the loss tangent in regions of high electric fields, such as the capacitor dielectric material, or additional parasitic inductance or resistance of the capacitor itself. These additional effects must be included in the modeling of such elements, which alter the circuit as indicated in
The resonant form of the impedance indicates that the packaged capacitor, itself, will have a resonant frequency. This self-resonance due to parasitic inductance can limit the frequency range over which packaged components can be utilized. Moreover, the reactance of the capacitance and that of the parasitic inductance cannot be separated by a single measurement of the resonant frequency; instead, the high frequency behavior as the inductive contribution dominates is needed to determine the relative contributions. Yet, the current on resonance will depend on these relative contributions, and as the losses may depend strongly on the current, this can be critical in the performance of the NIM. The choice of capacitor to use is dependant on several considerations. The dielectric loss, represented by the loss tangent δ is important. The lowest dielectric losses are exhibited by air gap dielectric, although construction of such structures is usually difficult, and the relatively low dielectric constant of free space makes higher capacitance difficult to achieve. The parasitic inductance of most conventional capacitor constructions results in low self-resonant frequencies, making these type capacitors useless for the present application. This is a problem with many embedded capacitor approaches as well, where capacitors are achieved during the layup of circuit boards using specialized processes and materials. The control of dimensions and restrictions on the dielectric materials usually results in poor performance of high speed capacitors, with losses and uncontrolled inductance contributing to the difficulties. Fortunately, recent developments of single layer capacitors (SLC) offer an attractive alternative. These are lithographically produced with a high degree of control of the dimensions, and using the latest dielectric materials design specifically for capacitor applications. The simple construction allows very low parasitic inductance, and these capacitors are now commercially available with rated self resonant frequency well above 10 GHz. Existing numerical simulation techniques will not readily be able to characterize and predict the behavior of such structures owing to the complexity and the small dimensions. Further, capacitors are normally characterized by their DC values, and not by their high frequency characteristics. The present invention includes a hybrid approach to analyze the exemplary structures, combining more traditional electrical engineering characterization of components in terms of complex impedance with the established full-wave solutions of negative index materials in terms of full free space S-parameters. By carefully modeling simple structures in HFSS or MWS, such as the SLC, and verifying the expected frequency dependence of the inductive/capacitive reactances, we establish rules which will enable efficient design of structures with active components. This hybrid modeling connects with use of active components, where the influence of parasitic elements should be considered in addition to the nonlinear component. E(3) Circuit Board Based 3-D NIM's An additional aspect of the present invention is directed to a circuit board based metamaterial. Exemplary embodiments include a modular metamaterial made of a plurality of individual circuit board pieces that may be fit together. Each circuit board portion includes a plurality of unit cells, with each unit cell comprising a substantially planar circuit boards of a dielectric substrate supporting interspersed arrays of conductors. The cells may include a central gap. The planar circuit boards are arranged in a three dimensions perpendicular to one another and once assembled form a three-dimensional array. A plurality of strip boards 120 may be fit together to form a three-dimensional grid-like array as illustrated in The arrays 106 and 126 are in conductive communication with one another through engagement of the pins 128 with the sockets 128. Further, the arrays 126 are in communication with arrays 126 on other of the strip boards 120 through cooperating engagement of the slots 130. Electrical connections between elements throughout the structure are formed by reflow soldering techniques. It will be appreciated that the exemplary modular circuit board construction of the invention, with that of E(4) Control of Losses in the NIM Unit Cell through Optimization of Current-Voltage Ratio Losses within the NIM occur due to both conductive losses associated with the conductors (usually copper) and the loss tangent of any dielectric material that experiences electric fields. In the simple picture of the ring resonator presented previously, the resonant current of the LC circuit is By varying the inductive/capacitive ratio, losses are minimized by altering the relative magnitudes of the induced currents (leading to ohmic losses) to the induced voltages (leading to dielectric losses). That is, the resonant circuit which normally oscillates between energy stored in the electric field (CV2/2) and the magnetic field (LI2/2), can alter the V/I ratio by varying the L/C ratio. There may be practical limits to this�there is a minimum induction and capacitance set by the SRR which must couple to the propagating EM mode. If either loss mechanism can be reduced, then a net lower loss benefit can be realized. The use of lower loss dielectrics, include air, is one avenue to reduce losses. Shown in E(5) Metamaterials Useful for THz Frequencies Metamaterials utilized in the prior art at microwave frequencies include dual split ring resonator (DSRR), with four DSRR's shown schematically in In Metamaterials of the invention include the structure(s) of An exemplary method of the invention includes the steps of forming a sacrificial layer on a first dielectric substrate, forming a mold in the sacrificial layer, and depositing a conductor in the mold to form a conductor ring such as a SRR, DSR, LRR, or the like. The sacrificial layer is then removed whereby the conductor ring is supported on a surface of said the dielectric substrate. A second dielectric layer is formed covering on the surface of the first dielectric substrate and covering the conductor ring. A second sacrificial layer is formed on the second dielectric layer, and a second mold formed in the second sacrificial layer. A second conductor is deposited in the second mold, and the second sacrificial layer is removed wherein the second conductor is supported on the second dielectric layer over the conductor ring. A plurality of conductor resonator rings may be formed in this manner, or these steps may be used to form a single unit cell with a plurality of the cells subsequently being combined together. The first and second dielectric layers, the conductor ring and the second conductor thus form a negative index medium having a simultaneous negative permittivity and permeability over a frequency band. Those knowledgeable in the art will appreciate that these steps may be practiced with a variety of different dielectrics, conductors, sacrificial layers, dimensions, and the like. In some embodiments, extremely small scales are used to form microchips. By way of further description, an exemplary method of the invention is outlined in Next, the topography is planaraized by applying a spin-on glass layer 256 as shown in By repeating these exemplary steps, a series of repeated SRRs and thin wires can be fabricated. Or, these steps may be applied to simultaneously form a plurality of unit cells as illustrated in E(6) Spatial Filtering with NIMs One exemplary application for passive NIMs of the invention is spatial filtering in which an incident field distribution undergoes processing by a planar material. The concept of spatial filtering is illustrated in E(7) Dynamic Tuning of Metamaterials Through Active Components An additional aspect of the present invention is directed to use of active components in metamaterials, including in NIM's. Exemplary NIM's of the invention include embedded active and non-linear devices to achieve external control of some of the response parameters of the NIMs. Active control of the effective index can be achieved, for example, by embedding gain elements in either the electric-field or magnetic-field elements of an SRR/continuous-wire based NIM. This is a more complicated construct than the linear-response NIMs which do not include any discrete active components. The active elements should be embedded in a fashion that does not overly perturb the electric and magnetic field distributions and maintains control over gain and phase shift in the active elements. Another complication is presented by the need to provide power and ground contacts for the active elements without disturbing the electromagnetic properties of the NIM at the relevant negative index band of frequencies. In exemplary NIMs of the present invention, the relevant power and ground leads are added by interspersing the power leads, designed with tuning stubs that yield high (non-transmissive) impedances in the negative-index band, yet maintain low impedance at low (power) frequencies. The nonlinear and gain elements are integrated with both the continuous wire and the SRR constructs of the NIM material: Varactor diodes may be placed in the capacitive gap region of the magnetically-active SRRs, as may PIN diodes. HEMT transistors may be used to either amplify the electric field signal by embedding them in the wire sections of the LHM, or alternatively may be used to amplify the magnetic field component by embedding them in the rings of the SRR elements. Exemplary active elements believed useful include the Agilent ATF50000 and ATF30000 series pseudomorphic HEMT transistors, which typically have gain of 15-35 dB in the 1-6 GHz band and somewhat less gain but operable up to 18 GHz. High frequency PIN diodes from the same manufacturer may allow two-terminal implementations for beam steering and wave phase shifting. Varactor diodes from MicroSemi (MTV2100 and the GC1500A) will be employed to enable tunability of the capacitive part of a SRR/wire implementation. More than one active component may be embedded, with an example including one active component embedded in a ring conductor gap and a second embedded in a straight wire conductor lattice. A schematic illustration of one exemplary active NIM operating in the X-Band microwave range is shown in E(8) Reduced Loss Metamaterials Via Active Circuits Active circuits can be applied to balance the resistive losses in resistive networks. This potentially important technique may have value across all frequency ranges, especially in the optical where pumped materials are easily obtained. However, at low frequencies, the technique can be studied and developed using off-the-shelf amplifiers and equivalent components, allowing the basic concepts of controlled losses to be explored. Negative Impedance Converters (NICs) are known in the art. A simple schematic is shown in If a net phase response is present in the circuit, then that phase response adds to that of the remainder of the material, and can be used to further control the negative index characteristics of the material. Such a circuit may be restricted to operation at frequencies where the basic components needed operate, e.g. in the low GHz range. The size of the components may also restrict their usage to the longer wavelengths. E(9) Nonlinear Metamaterials The combination of nonlinearity with metamaterial structures, which themselves possess frequency dispersion, can lead to potentially useful types of behavior. For example, because the local electric fields are considerably enhanced in the capacitive regions of the resonant metamaterial elements, a nonlinear material strategically positioned in such a structure can be strongly enhanced. It has been shown that if a material possessing a Kerr nonlinearity (i.e., an index that has the form n=n1+n2E2) is embedded into an array of SRRs, then a form of optical bistability can result. While there are a variety of nonlinear materials available toward optical frequencies (e.g., BaxSr1-xTiO3), it is advantageous in one exemplary targeted frequency range to utilize diode or similar solid-state elements to achieve the desired nonlinear response. An aspect of the present invention is directed to a nonlinear NIM that makes use of an embedded diode circuit to introduce the analogous Kerr nonlinearity into the SRR medium. The composite material will exhibit a refractive index that will be switchable from positive to negative based on the intensity of an incident field. Because the use of active components such as diodes needs to be carefully biased and balanced, the range of operation in terms of incident intensity is believed to be narrowly prescribed. While the narrow bandwidth of NIMs can be a limitation for linear frequency domain applications, an alternative approach to leveraging the unique properties of NIMs is to search for wave propagation phenomena and applications that make use of the dispersive properties of NIMs. Electromagnetic pulses, for example, contain a band of frequency components so that the shape of the pulse is in general altered by transmission through a dispersive medium. A pulse interacting with a NIM will undergo a negative phase shift for each frequency component that lies within the negative index band. While there is no particular distinction between pulses that propagate entirely within linear negative versus positive index media, there are NIM structures that can have a significant impact on wave propagation, including exemplary NIM's of the invention. Examples include structures that are formed from the combination of dispersive positive and negative materials. Another class of structures is NIMs with nonlinearities. A variety of applications have been suggested for NIMs that exhibit different types of nonlinearity. At low frequencies, as in the previous examples, nonlinear components�such as properly biased diodes�can be utilized to produce negative index metamaterials that exhibit nonlinear response to incident electromagnetic fields. E(10) Active High Frequency NIMs There are two preferred routes for embedding active control and nonlinear elements in a NIM: integration of discrete electronic components with circuit-board-based NIM structures, as discussed above, and the integration of NIM structures with custom-fabricated semiconductor materials, the latter including active elements. The former approach, using discrete components, is simpler to implement and to test design methodologies, but may be limited to frequencies below 20 GHz. The latter approach, with custom-fabricated active elements lithographically patterned on the same semiconductor substrate as the LHM structures, will ultimately allow much higher central operating frequencies for the active elements, which may approach 100-200 GHz and above, but engenders significantly higher costs and longer lead times in the development effort. Alternatively, approaches combining the two, involving the use of discrete but unpackaged components, with chip die sizes of order 100 μm, may allow significantly higher frequency operation with sufficiently sophisticated circuit-board-like technology, using perhaps semiconductor (e.g. Si and poly-Si) substrates patterned using MEMS technology, enabling the addition of vias and the corresponding interconnect capability, to the rigid, ultraflat surface afforded by a highly polished rigid semiconductor. E(11) Tunable THz NIMs The dielectric function of TiO2 can be modified at THz frequencies via an applied electric field. An exemplary device of the invention shown schematically in Changes in transmission occur in the device due to modification of the THz response of TiO2 by an applied electric field. These changes have the form of resonances occurring close to the vibrational frequencies of TiO2, where field-induced modification of ε1(ω) can exceed 50-80%. Notably, SRR arrays can be readily integrated in the bottom-contact. FET geometry thus enabling this novel approach to the device tunabilty. While specific embodiments of the present invention are shown and described herein, it should be understood that other modifications, substitutions and alternatives are apparent to one of ordinary skill in the art. Such modifications, substitutions and alternatives can be made without departing from the spirit and scope of the invention, which should be determined from the appended claims. Patent Citations
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The Jack McDonald Family Home PageUpdated September 5, 2000 |
John Henry McDonald
3952 Elmcrest Drive
Pensacola, Florida 32504-6741
850-479-3097 |
Ferguson, Giblin, Fleming,Gillespie, McDonald,Lawrence, Burns, Sibley, and Marshall---all in Essex county MA. Also Giblin, Gillespie, and Vickers in Hyde, Cheshire, England. Also Boyle, Fleming, Ferguson, and Ney in Ireland. Dates would be basically 18th and 19th century. John Henry McDonald 3952 Elmcrest Drive Pensacola, Florida 32504-6741 (850)479-3097 firstname.lastname@example.com |
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Striking dark-violet colored stems, leaf veins and pods with beautiful lilac-colored blossoms that are held out above the foliage on long straight stems. The entire plant is purple! Hyacinth Bean seed pods and beans are poisonous at certain stages!
In Asia and Africa, hyacinth beans are grown for food and can be eaten if prepared properly. We do not recommend growing for food, but rather as an ornamental flower. Can grow 10-15 feet tall if given adequate support. (Annual)
Each packet contains approximately 10 seeds.


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Each year the Secretary of Defense Environmental Security Awards recognize installation and team or individual efforts in six categories -- natural resource conservation, cultural resource management, environmental quality, pollution prevention, recycling and environmental cleanup. A total of 15 awards -- 11 installation and four individual -- were presented.
Our forces today recognize that defense environmental protection is good management, good citizenship, good stewardship and a good way to protect the health and welfare of our forces," said Secretary Cohen in remarks at the ceremony. "Your successes stand out among the countless actions we take to protect the environment as we protect the nation.
The award recipients included:
For Natural Resources Conservation: LARGE INSTALLATION Marine Corps Base Camp Pendleton, Oceanside, Calif. *Naval Air Station Miramar, San Diego, Calif. *Eielson Air Force Base, Alaska *Fort Carson and Pinon Canyon Maneuver Site, Colo.
SMALL INSTALLATION Naval Surface Warfare Center, Indian Head, Md. *Newport Chemical Depot, Newport, Ind. *Marine Corps Base Kaneohe Bay, Hawaii *Fairchild Air Force Base, Wash. INDIVIDUAL Catherine M. Zielske, Naval Amphibious Base Little Creek, Norfolk, Va. * Karl Dautermann, 410th Base Support Battalion, Bad Kreuznach, Germany * Charles D. Peterson, Marine Corps Base Camp Lejeune, N.C. * Gerald T. Johnson, Fairchild Air Force Base, Wash. For Cultural Resources Management
INSTALLATION Fort Carson and Pinon Canyon Maneuver Site, Colo. *Marine Corps Air Ground Combat Center, Twentynine Palms, Calif. *Naval Air Station Patuxent River, Md. *F.E. Warren Air Force Base, Wyo.
INDIVIDUAL Stephen A. Chomko, Fort Carson, Colorado Springs, Colo. * Stephen R. Wise, Marine Corps Recruit Depot, Parris Island, S.C. * Ronald N. Johnson, South Naval Facilities Engineering Command, Charleston, S.C. * Gene V. Augustine, Elmendorf Air Force Base, Alaska For Environmental Quality
INDUSTRIAL INSTALLATION Naval Surface Warfare Center, Indian Head, Md. *U.S. Army Kwajalein Atoll/Kwajalein Missile Range, Republic of the Marshall Islands *Marine Corps Logistics Base, Barstow, Calif. *Tinker Air Force Base, Okla.
NON-INDUSTRIAL INSTALLATION Luke Air Force Base, Ariz. *U.S. Army Transportation Center, Forts Eustis and Story, Va. *Marine Corps Base Camp Lejeune, N.C. *National Naval Medical Center, Bethesda, Md.
INDIVIDUAL Kenneth P. Stachiw, Aberdeen Proving Ground, Md. * Susan Gibson, Marine Corps Air Station Camp Pendleton, Oceanside, Calif. * Patrick F. Spahn, National Naval Medical Center, Bethesda, Md. * Daniel A. Stone, Hill Air Force Base, Utah *Cdr. William R. Arguello, USN, Defense Special Weapons Agency, Albuquerque, N.M.
For Pollution Prevention INDUSTRIAL INSTALLATION Corpus Christi Army Depot, Corpus Christi, Texas *Marine Corps Logistics Base, Barstow, Calif. *Puget Sound Naval Shipyard, Bremerton, Wash. *Hill Air Force Base, Utah
NON-INDUSTRIAL INSTALLATION Fort Lewis, Wash. *Marine Corps Base Camp Lejeune, N.C. *Naval Construction Battalion Center, Port Hueneme, Calif. *Kirtland Air Force Base, N.M.
WEAPONS SYSTEM ACQUISITION TEAM New Attack Submarine Program, Arlington, Va. *Bradley Fighting Vehicle System Acquisition Team, Warren, Mich. *Direct Reporting Program Manager Advanced Amphibious Assault Vehicle, Woodbridge, Va. *Aeronautical Systems Center Environmental Integrated Product Team, Wright-Patterson Air Force Base, Ohio
For Recycling INDUSTRIAL INSTALLATION Marine Corps Logistics Base, Barstow, Calif. *Tobyhanna Army Depot, Pa. *Naval Submarine Base Bangor, Silverdale, Wash. *Hill Air Force Base, Utah
Non-INDUSTRIAL INSTALLATION McChord Air Force Base, Wash. *Fort Hood, Texas *Marine Corps Base Camp Lejeune, N.C. *Naval Weapons Station Yorktown, Va.
INDIVIDUAL MSgt. Douglas F. Skitch, USAF, McChord Air Force Base, Wash. * Abdeslem Houmina, Tobyhanna Army Depot, Pa. * Timothy J. Harrington, Marine Corps Recruit Depot Parris Island, S.C. * Frank Space, Naval Air Station Jacksonville, Fla.
For Environmental Cleanup INSTALLATION Naval Air Station North Island, San Diego, Calif. *Fort Wainwright, Alaska *Marine Corps Base Camp Lejeune, N.C. *McChord Air Force Base, Wash. *Defense Distribution Region East, New Cumberland Site, Pa.
(* -- Indicates honorable mention in that category.)


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Not EnoughSubmitted by Loyal Democrat (not verified) on Mon, 04/04/2011 - 8:14pm.
If I had made slanderous comments to a news reporter about someone who was a public figure, my boss would have fired me. If McEwen's remarks are not reflective of the Congressman's office, then Mike McIntyre should do the honorable thing and fire Mr. McEwen, otherwise it gives the impression that he approves of and endorses them. In this case, a written apology is not enough. One those words hit the internet, they are there forever. »
Reply |


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by Earthling » Fri Dec 16, 2011 11:10 am
by jman111 » Fri Dec 16, 2011 11:21 am
by Henry Vilas » Fri Dec 16, 2011 12:00 pm
Earthling wrote: A majority of citizens see the train as a waste of taxpayer dollars...
by lukpac » Fri Dec 16, 2011 12:02 pm
Earthling wrote:(whether it's state or federal money WE are still paying for it).
by Earthling » Fri Dec 16, 2011 12:17 pm
lukpac wrote:Earthling wrote:(whether it's state or federal money WE are still paying for it).The difference is now we're paying twice. We're paying state taxes for improvements that are not being covered by federal dollars, and we're paying federal taxes for improvements in other states, which would have happened in WI. That is to say, we're paying more and getting less.
by lukpac » Fri Dec 16, 2011 12:21 pm
Earthling wrote:He is asking for it back now:http://www.huffingtonpost.com/2011/03/30/walker-rejected-12-millio_n_842481.html
by Mean Scenester » Fri Dec 16, 2011 2:08 pm
lukpac wrote:Earthling wrote:He is asking for it back now:http://www.huffingtonpost.com/2011/03/30/walker-rejected-12-millio_n_842481.html"is"? Welcome to last March.Hint - the request was denied.
by rabble » Fri Dec 16, 2011 4:17 pm
by Detritus » Fri Dec 16, 2011 5:29 pm
rabble wrote:Well, I no longer think that Earthling is a Blaska sock puppet. Even Blaska would have remembered that one.
by ilikebeans » Sun Jan 22, 2012 9:45 pm
The money was part of a larger federal package that was given to Wisconsin to implement the federal health care law passed by Congress and President Barack Obama in 2010. After receiving criticism from conservatives that he was helping implement "ObamaCare," Walker said this week that the state would return the money and stop the implementation.But that announcement didn't highlight the $11 million, which would have gone toward improving enrollment systems for existing state Medicaid health programs for the poor. It would have helped pay for some improvements to BadgerCare Plus Medicaid programs that the Walker administration had wanted to pursue anyway because state officials say they would save taxpayers money.
by Beaver » Sat Jul 07, 2012 4:09 pm
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State of the Union
I keep seeing articles about how the President will likely focus on how the economy is doing. It's likely to bu upbeat hopeful but cautious. So I won't be watching it. This cartoon not only expresses my opinion of the current state of our economy, I'm willing to bet that the President feels about the same way.


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Today the ACLU launches Free Future, a new blog dedicated to protecting civil liberties in the digital age. We hope to make it a vigorous forum for the daily exploration of issues related to technology – in particular, how we can ensure that individual freedom, control, and self-determination are enhanced rather than compromised by new advances in science and technology.
read more
This is the start of something new. Not just a New Year, but here at ACLU.org we have some new and exciting programs launching, and more coming up in the near future. Starting this week, we'll showcase the hot issues from the last s... Read Post
In Court: Uncovering Stingrays, A Troubling New Location Tracking Device [ACLU of Northern CA – Linda Lye] The ACLU and Electronic Frontier Foundation have filed an amicus brief in what will be the first case in the country to addre... Read Post
We hope you liked last week's round-up. Let us know! Is this weekly review of issues helpful or would you do it differently? Share your thoughts: email@example.com Free Speech: You Have the Right! Snapping a picture of a public buildin... Read Post


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Boxing Champions | Boxing Schedule | Boxing Video | Boxing History | Pound For Pound | Lounge | The Wire | Audio | Arcade |
Big Bad Booty Daddy Join Date: Sep 2003
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Comments Thread For: Dulorme Not Worried About Abregu's Power, Pressure
Welterweight contender Thomas Dulorme (16-0, 12KOs) is expecting a very tough fight with veteran Juan Carlos Abregu (33-1, 27KOs). The only defeated on Abregu's record was to WBO welerweight champion Timothy Bradley. The October 27th bout will be televised by HBO. It will be the first twelve round fight for Dulorme.
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"It is my intent to provide patients with an individualized treatment plan to improve their function and quality of life. Since people are living longer and are more physically active, it is my goal to help patients regain and maintain an active lifestyle, regardless of their age." — Dr. Teunis
Dr. Teunis recently relocated to Sugar Land to join her husband, Scott. Candice and Scott love to ski, golf, travel and cook together. They always try to improve their made-from-scratch pesto pizza. She is also trained in classical piano and voice.
Watch the video below for more information on Dr. Teunis.
By selecting an appointment time you will be transferred from the Memorial Hermann website to our scheduling partner's website - Healthpost.com. Please read the Healthpost Privacy Policy and Terms of Use for more information about this site.
Always verify insurance coverage with your provider prior to receiving care.
Phone: (281) 344-1715 Fax: (713) 452-4134


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Thursday, March 05, 2015 5 comments I got up to use the bathroom at 3:30am and I must have awakened the dogs. Maybe they had to go too! LOL. Anyway, after enduring a full hour of whining and howling, I got up and let them out. Then of course i couldn't get back to sleep. After... Read more
Tuesday, March 03, 2015 4 comments This past weekend, I went to Maine to attend a dear friend's memorial service. For somebody who is used to warmer temperatures, this was a big challenge. Did I mention that I dislike very cold weather? I had a great visit with my friend's fam... Read more
Thursday, February 26, 2015 7 comments I am going to Maine this weekend for my close friend's funeral. It has been many years since I have flown off to a place that I have never before visited. Plus, it is a very cold area that gets a lot of snow. Hopefully, that won't be an issue... Read more
Wednesday, February 25, 2015 2 comments I decided that I would give up all treats, candy etc. except sugar free Popsicles for Lent. It has been a nightly struggle, but so far I have been successful. I now realize that when I eat too much sugar, it makes me crave more sugar. I have ... Read more
Tuesday, February 17, 2015 4 comments Yesterday, I was on the go from 6:00am to 6:10pm. I started with making dinner in the crockpot and plugged it in. I moved on to my allergy shot at 8am, then on to 9:00am deep water class. I did 120 minutes in the water, all of that time spent ... Read more
Thursday, February 12, 2015 4 comments This morning's physical therapy session was an eye opener! My therapist asked me all kinds of questions about my first orthopedist's follow up on my avulsion fracture. There were lots of raised eyebrows when I said that the doctor checked the ... Read more
Thursday, February 12, 2015 3 comments Rough morning. I woke up with a headache and my foot was burning. Not a good start as these things go. I ate a good breakfast and now I am off to my evaluation at physical therapy. I hope it goes well!... Read more
Tuesday, February 10, 2015 4 comments Yesterday, I had a second opinion appointment for my foot. My foot issues have been dragging on since last October. It turns out that the local orthopedist, who I have seen at least 3 times, was totally wrong. What a relief! So much for goin... Read more
Saturday, February 07, 2015 7 comments I have read several blogs today by people who were disgusted with themselves. They made great progress for a long time and then lost their way. They all had a common thread. Getting back on track and staying motivated is hard. Forgiving your... Read more
Wednesday, February 04, 2015 6 comments I have had a very rough couple of months. I have some persistent health issues as well as some personal losses. With it all happening at once (when it rains it pours - thank you Morton salt!), it can be pretty overwhelming. A friend in our de... Read more
Tuesday, January 27, 2015 8 comments I had been dreading the funeral we attended this afternoon, for the most part because the young man who was killed was the 15 year old son of a friend. To further throw a wrench into our day, hubby's van broke down right before my hair appointm... Read more
Tuesday, January 27, 2015 4 comments Today started off with a bang. I had a reaction to my allergy shot and last night my arm got super red, swollen and very sore. Great! This morning I met hubby for breakfast: 3 egg whites, two pieces of whole wheat toast and a banana plus hot ... Read more
Saturday, January 24, 2015 5 comments We took our friend out for breakfast this morning. We cried with him over the loss of his beloved son. We offered him a place to go when his grief was so overwhelming at home that he just had to get away for an hour. We made him laugh at our ... Read more
Thursday, January 22, 2015 5 comments My week started off with hubby breaking a tooth and exposing a nerve. It went downhill after that. He went to the dentist on Monday and she scheduled a procedure for him on Tuesday. Since he was in such pain, I went with him on his business ca... Read more
Monday, January 19, 2015 2 comments I wrote a nice blog yesterday and proceeded to lose it to a blip on my computer. I am giving it another shot today. :) I had a migraine yesterday and hubby broke off half a tooth, so we skipped our Sunday morning breakfast group in favor of cra... Read more


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Cheap Flights to Mayotte
Did You Know?
The lengthiest nonstop flight to Mayotte lasts 10:00.
Air Austral provides 39% of the nonstop flights to Mayotte.
65% of all flights to Mayotte arrive during afternoon hours.
The longest flight into Mayotte is 4,989 miles.
The shortest flight into Mayotte is 160 miles.
Mayotte has 15 medium haul flights arriving each week.
There are 31 nonstop flights arriving in Mayotte per week.
The smallest aircraft flying into Mayotte is the ATR 42-500 with 46 seats.


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Medford, Ore. — In the middle of a drought, it's more important than ever to know where your water is going and use what water we do have wisely.
OSU Master Gardeners said you should make a priority list. They say established trees and shrubs need to be saved first because it's more costly to replace them.
They also suggest reducing lawn watering. OSU Master Gardener, Jane Moyer said you only need one to two inches of water per week to keep your lawn green.
You can learn more about watering priorities and irrigation tricks at the OSU Extension Service. The Sustainable Watering Practices class is August 11th at 7 p.m. at 569 Hanley Rd. in Central Point.
OSU Master Gardeners can be contacted at the following numbers:
Jackson County – (541) 776-7371
Josephine County – (541) 476-6613
Klamath County – (541) 883-7131
Coos County – (541) 572-5263
Douglas County – (541) 672-4461


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Pull the other one schmidt.... you had to do it otherwise your share price would have started to fall...
I hate Corporate b*******t
I agree - a very sad day - unless M$ know something we don't....
So the total number of adwords customers is no more than 300k?
Given ,if it were more, they would have used that 300k figure .
Now I can just ignore MSN search and PPC as being irrelevant.
There are 400,000 GAW accts (end of 2005). It's expected to rise by about 100K per year for the next two years.
Not likely, since Google isn't a monopoly and hasn't engaged in anticompetitive behavior.
(Contrary to popular belief, it isn't illegal to be more successful than your competitors.)


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- Mishits
Lucas at home on the range
ESPN staff
June 5, 2014
'Don't take it personally Kolo, I'm just practising for my holidays...' © Getty Images
When your club has narrowly missed out on the league title and your country doesn't pick you for the World Cup, there's only one thing left to do: wrangle some cattle. At least, that's what you do if you're Lucas Leiva.
The Liverpool midfielder has been "reminding my childhood and having fun" as he takes his mind off missing out on a place in Brazil's World Cup squad, lassoing cows on horseback back in the old country.
Hopefully he doesn't let this one slip - right, Stevie?