EuROPEAN MICROWAVE ASSOCIATION
Multiplicative measure of planar chirality for 2D meta-materials Serge P. Boruhovich1, Sergey L. Prosvirnin1, Alexander S. Schwanecke2, Nikolay I. Zheludev2 Abstract – We describe a new efficient continuous geometrical measure of two-dimensional chirality for photonic and microwave planar meta-materials. Its properties include integrability, scale-independence, boundedness to a well defined interval and flexibility as well as applicability towards an extensive number of chiral geometries. The chirality measures of various arrangements of standard chiral and achiral template structures are evaluated and a method for the characterization of (infinite) periodic structures is suggested. Index terms – Artificially structured surface, polarization, planar chirality measure, optical activity.
I. Introduction In recent years progress of both microwave technology and photonics has found itself closely linked to the topic of electromagnetic meta-materials, structures with unique characteristics – amongst them the branch of planar chiral materials. The authors of Ref. [1] found that the effectiveness of optical activity by planar chiral surfaces is several orders of magnitude larger than that of bulk chiral structures when operating in the proximity of resonances. Consequently, these planar chiral materials have attracted a lot of attention due to a number of new polarization phenomena associated with them [2-5]. Analogously to chirality in three dimensions, a twodimensional structure is chiral if it cannot be superimposed with its mirror image without having to lift latter out of the plane. It is now understood that chirality measures can be introduced in a number of ways [6-8]. Here, we suggest a new algorithm, which is applicable to single planar chiral elements and their arrays returning a dimensionless valued chirality measure. It is independent of scale and, moreover, limited to the interval [-1,1]. In contrast to the additive nature of the measures proposed in Refs. [6, 8], we explore a multiplicative approach. A method to facilitate the characterization of (infinite) periodic structures is suggested.
Fig. 1. Three points, vertices of a scalene triangle ABC, are the simplest planar chiral object; a)-c) depict types of triangles distinguished by the algorithm depending on the relative position of their altitudes BB’ in respect to the baseline vertices AC.
any number of points in a plane, e.g. a curvilinear segment, can be achieved by summing over all possible three-point permutations [6]. Such a scalene triangle ABC, as seen on Figure 1(a), with base line AC and altitude BB’ can be subdivided into the triangles BCB’ and ABB’ enclosing the areas S1 and S2 respectively. Introducing S˜i idenoting ‘signed’ areas, according to whether they are located to the left of BB’ as seen from vertex B (positive) or to its right (negative), allows to define Q(1) = S˜1 + S˜2 . This results into Q(1) = S1 - S2 for the case of a triangle shaped like that of Figure 1(a), Q(1) = S1 + S2 for Figure 1(b), and Q(1) = - S1 - S1 for Figure 1(c). Repetition of these steps choosing BC as base and the altitude defined in respect to vertex A results analogously into a value for Q(2), while AB is chosen as base line to define Q(3). Thus, we receive values Q(i) with respect to all three altitudes of a triangle ABC. For the chirality measure of the simplest planar chiral or three-point object (triangle ABC) we consider the dimensionless parameter
Received: November 7, 2005. Revised: December 22, 2005.
II. Definition of measure The simplest chiral object in a plane is a set of three points with different distances between them, i.e. three vertices of a scalene triangle. Upon construction of a chirality measure for this set of points, an extension to
1 Institute of Radio Astronomy, Krasnoznamennaya st. 4, Kharkov, 61002, Ukraine. 2 EPSRC NanoPhotonics Portfolio Centre, School of Physics and Astronomy, University of Southampton, SO17 1BJ, UK - Tel. +44 23 80592699 - Fax 3910 - E-mail:
[email protected], www.nanophotonics.org.uk
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(1)
KABC = Q(1)Q(2)Q(3)/d6,
where d denotes the length of the largest side of the triangle. Evaluating the product of the values for Q(i) guarantees the measure to be equal to zero for isosceles and, hence, likewise for equilateral three-point objects, which are of course mirror symmetric, i.e. achiral. This multiplicative merger of the influence over the combined chirality of a triangle as seen from its vertices is in contrast to the additive approaches suggested by Refs. [6, 8]. As a set of three points composes the simplest chiral object in a plane, investigating the possibility of a most asymmetric triangle and quantifying its chirality are imperative. On the abscissa axis of a rectangular coordinate system in the plane of xOy the two points A (-1,0) and B (1,0) were chosen, while a third point C is assumed to be located within the first quadrant. Figure 2 depicts the dependence of the parameter KABC on the coordinates xC and yC for point C. Accordingly, three maxima can be found for the function KABC (xc,yc), all of which display equal values of Km = maxx ,y KABC = 0.0055 and define triangles of scaled, but congruent shape. These three different positions for C in which Km is reached correspond to triangles with internal angles of 27°, 56° and 97°. As the definition of this chirality measure differs from the approaches considered in Refs. [9] and [10], the forms of the maximum asymmetric triangles differ as well. This fact is obvious from a conclusion in Ref. [11] that any asymmetric triangle is most asymmetric in some chirality measure. c
c
Fig. 3. Examples of planar chiral structures: (a) a grid of asymmetrically located straight crosses individually tilted by ψ in respect to the main axes of the underlying grid; (b) a matrix of right-handed gammadions each comprised of four interconnected arcs spanning an angle of φ.
The shape of objects more complex than three-point elements can be represented at any given accuracy by a sufficiently large number of points located uniformly within the object. This enables to determine an overall value for the asymmetry of an object by summing over all chirality measures for any of its three-point sets. In order to provide comparability of results when using different amounts of points, we introduce normalization by the number of three-point elements used in an individual calculation. This number is obviously equal to M = N (N - 1) (N - 2)/6, where N is the total number of points. Thus we determine the chirality measure of a set of points as (2)
K=
1 N N N (1) ( 2) ( 3) 6 ∑ ∑ ∑ Qijk Qijk Qijk dijk , M i =1 j =i +1 k= j +1
where the three Q(l)ijk and dijk are the values described above for a certain selected triangle defined by the points i, j, k. The measure defined by (2) can be understood as the average chirality of all three-point elements available in the set of points describing the object. Here, this average chirality of any complex object consisting of large numbers of points is generally substantially less than the chirality measure of the maximum asymmetric triangle described above. Therefore, it is instrumental to introduce the chirality index (3)
Fig. 2. Investigating the generic template of a planar chiral structure, three points ABC, the dependence of the chirality measure K for a standard set of their positions is depicted; here, vertices A (-1,0) and B (1,0) are fixed, while C (xc, yc) is being varied within the first quadrant of the xy-plane.
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I = K/Km
as the proportionality coefficient associating the chirality measures of a planar object with the most asymmetric triangle. Thus, the chirality index I will always assume values in the interval [-1,1]. Regularly ordered planar structures, either composed of a finite number of elements or infinite periodic structures, can first of all be chiral because of the chirality of individual elements and, secondly, due to positional
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S.P. BORUHOVICH, ET AL.
asymmetry in the grid. The chirality index for any system containing finite numbers of elements can be found employing this algorithm, although it may require a considerable amount of computational time. If such a system is ordered, for example a fragment of a periodic structure, the chirality index will tend to zero as the number of elements increases, because the symmetry of the overall structure will start to dominate the chirality of individual elements. At this point the chirality index of a single periodic cell gains significance. Its calculation can be achieved by adding the border of a single unit cell to the investigated element.
III. Numerical results The algorithm specified above has been employed to calculate chirality indices of a number of elements and structures. For optics and microwave applications it is advisable to use elements with fourth order rotational symmetry, i.e., elements which pass into themselves following any planar rotation of 90° around their centre of symmetry. Straight crosses as well as left and right (handed) gammadions [2], see Figure 3, are available options, since chiral structured surfaces based on them cause polarization transformation. An individual straight cross, of course, is an achiral object. However, a grid of crosses can be chiral resulting from the asymmetric orientation of elements in their respective unit cells. In contrast a gammadion is chiral on its own. It shall be noted that in general the chirality indices of two enantiomorphous objects (for example left and right gammadions) differ by sign. Figure 4 depicts the dependence of the chirality index of a right-handed gammadion on its angle of twist. Here, results corresponding to varying amounts of
Fig. 4. Chirality index of a single right-handed gammadion depending on angle φ for a - N = 80, b - N = 160, c - N = 200.
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Fig. 5. This planar structure composed of left- and right-handed gammadions results into an overall chirality index of zero.
points N representing the object are compared. Stability and convergence of the algorithm can be observed. A complimentary test involves the calculation of the chirality index for the structure shown on Figure 5 which is composed of elements in enantiomorphous forms. Such system consisting of two left and two right-handed gammadions is symmetric, i.e. achiral, and indeed the calculated index proves to be zero for all values of the bending angle ϕ. The dependence of the chirality index on this angle ϕ for right-handed gammadion structures is presented on Figure 6. The considered structures are finite grids of 2 × 2, see
Fig. 6. Chirality index for simple structures composed of righthanded gammadions depending on angle φ: (a) - single gammadion, (b) - 2 × 2 grid, (c) - 3 × 3 grid, (d) - individual periodic cell.
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structure on its polarization characteristics is available. However, two important future perspectives can be isolated: first of all, a detailed and systematic study of the correlation between geometrical features of asymmetric planar structures, characterized both by the chirality index and diffraction polarization effects; and secondly, an investigation of elementary shapes suitable for applications based on structured surfaces.
Acknowledgements
Fig. 7. Simple structure built of straight crosses with dependence of its chirality index on angle ψ: (a) - 2 × 2 grid, (b) - individual periodic cell.
The authors would like to acknowledge the Engineering and Physical Sciences Research Council, UK for the support of joint scientific work.
References the Figure 3(b), and 3 × 3 elements. The chirality measure of such structures decreases rapidly for increasing numbers of elements. In addition the dependence for a gammadion in a single periodic cell (square cell as indicated by a dashed line on Figure 3(b)) is illustrated. Equivalent data for 2 × 2 structures built from straight crosses and a single cross in a periodic cell is shown on Figure 7. The largest absolute value for the index of an individual cell is reached for a tilt angle of ψ ≈ 22.5°. It obviously corresponds to maximum asymmetry of a cell.
IV. Conclusion An algorithm for the calculation of a chirality measure, which is both simple in realization and converging stable and quickly, has been demonstrated. Comparison of this chirality index with known data about polarization transformation effects for light diffraction on periodic grids composed of crosses and gammadions is to be an immediate aim of future research. Already polarization transformation for first order diffracted beams of normally incident waves can be considered using data from Ref. [12]. The strongest polarization effect due to chirality of a gammadion grid was observed for angles ϕ of approximately 70° and 120°. Latter corresponds to the case of the most chiral individual gammadion and former to the most asymmetrical element location in the grid. For the case of a grid built from straight crosses chirality and polarization effects can be related to the positioning of an element in the cell. Currently no more detailed information on the influence of the geometrical shape of a diffraction
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Hecht, L.; Barron, L.D.: Rayleigh and Raman optical activity from chiral surfaces. Chem. Phys. Lett. 225 (1994), 525-530. [2] Papakostas, A.; Potts, A.; Bagnall, D.M.; Prosvirnin, S.L.; Coles, H.J.; Zheludev, N.I.: Optical manifestations of planar chirality. Phys. Rev. Lett. 90 (2003), 107404. [3] Schwanecke, A.S.; Krasavin, A.; Bagnall, D.M.; Potts, A.; Zayats, A.V.; Zheludev, N.I.: Broken time reversal of light interaction with planar chiral nanostructures. Phys. Rev. Lett. 91 (2003), 247404. [4] Krasavin, A.V.; Schwanecke, A.S.; Reichelt, M.; Stroucken, T.; Koch, S.W.; Wright, E.M.; Zheludev, N.I.: Polarization conversion and focusing of light propagating through a small chiral hole in a metallic screen. Appl. Phys. Lett. 86 (1998), 201205. [5] Zhang, W.; Potts, A.; Papakostas, A.; Bagnall, D.M.: Intensity modulation and polarization rotation of visible light by dielectric planar chiral metamaterials. Appl. Phys. Lett. 86 (2005), 231905. [6] Osipov, M.A.; Pickup, B.T.; Fehervari, M.; Dunmur, D.A.: Chirality measure and chiral order parameter for a twodimensional system. Mol. Phys. 94 (1998), 283-287. [7] Petitjean, M.: Chirality and symmetry measures: A transdisciplinary review. Entropy 5 (2003), 271-312. [8] Potts, A.; Bagnall, D.M.; Zheludev, N.I.: A new model of geometric chirality for two-dimensional continuous media and planar meta-materials. J. Opt. A 6 (2004), 193-203. [9] Petitjean, M.: On the root mean square quantitative chirality and quantitative symmetry measures. J. Math. Phys. 40 (1999), 4587-4595. [10] Weinberg, N.; Mislow, K.: Distance functions as generators of chirality measures. J. Math. Chem. 11 (1993), 427450. [11] Rassat, A.; Fowler, P.W.: Any scalene triangle is the most chiral triangle. Helvetica Chimica Acta 86 (1993), 17281740. [12] Prosvirnin, S.L.; Zheludev, N.I.: Polarization effects in the diffraction of light by a planar chiral structure. Phys. Rev. E 71 (2005), 037603.
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Serge Boruhovich graduated from Mechanical and Mathematical Department of the Kharkov National University in 2001. He has been Post-Graduate Student of the Theoretical Radio Physics Department of the Institute of Radio Astronomy, Kharkov, Ukraine. His scientific interest is complex media electromagnetics. Sergey Prosvirnin is head of the Theoretical Radio Physics Department of the Institute of Radio Astronomy, Kharkov, Ukraine and professor at the Radio Physics School of the Kharkov National University. His main scientific interests are analytical-numerical methods in the theory of scattering and radiation of electromagnetic waves, theory of antennas and antenna arrays, and complex media electromagnetics.
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Alexander Schwanecke began his studies of physics at the University of Hamburg in 2000 holding a scholarship of the German National Academic Foundation. Shortly after receiving his Vordiplom he continued his efforts in Southampton graduating with an M.Phil. in 2004. Currently, he stays as a research student with the EPSRC NanoPhotonics Portfolio Centre. His main scientific interest is photonics of planar meta-materials. Nikolay Zheludev is a Professor in Nanophotonics and Coordinator of the EPSRC Portfolio Centre in NanoPhotonics at the University of Southampton.
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