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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 4 — Feb. 19, 2007
  • pp: 1639–1646
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Group theoretical description of artificial electromagnetic metamaterials

Willie J. Padilla  »View Author Affiliations


Optics Express, Vol. 15, Issue 4, pp. 1639-1646 (2007)
http://dx.doi.org/10.1364/OE.15.001639


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Abstract

Point group theoretical methods are used to determine the electromagnetic properties of metamaterials, based solely upon the symmetries of the underlying constituent particles. From the transformation properties of an electromagnetic (EM) basis under symmetries of the particles, it is possible to determine, (i) the EM modes of the particles, (ii) the form of constitutive relations (iii) magneto-optical response of a metamaterial or lack thereof. Based upon these methods, we predict an ideal planar artificial magnetic metamaterial, and determine the subset of point groups of which particles must belong to in order to yield an isotropic 3D magnetic response, and we show an example.

© 2007 Optical Society of America

1. Introduction

Metamaterials (MMs) are artificially structured materials which exhibit superior properties not inherent in the individual constituent components. A subclass of metamaterials involves novel electromagnetic (EM) response not attainable or extremely difficult to achieve with naturally occurring mater. Although EM-MMs are relatively new, there have already been several demonstrations of exotic electromagnetic behavior such as, negative index of refraction (NI),[1

1. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 84,4184–4187 (2000). [CrossRef] [PubMed]

]artificial magnetism,[6

6. J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Steward, “Magnetism from Conductors and Enhanced Nonlinear Phenomena,” IEEE Trans. Microwave Theory Tech. 47,2075–2084 (1999). [CrossRef]

] reduced lens aberrations,[9

9. D. Schurig and D. R. Smith, “Negative Index Lens Aberrations,” Phys. Rev. E 70,065601(R) (2004). [CrossRef]

] invisibility cloaking,[10

10. J. B. Pendry, D. Schurig, and D.R. Smith, “Controlling electromagnetic fields,” Science 312,17801782 (2006). [CrossRef]

, 11

11. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314977–980 (2006). [CrossRef] [PubMed]

], and metamaterial electronics.[12

12. Hou-Tong Chen, Willie J. Padilla, Joshua M. O. Zide, Arthur C. Gossard, Antoinette J. Taylor, and Richard D. Averitt, “Active Terahertz Metamaterial Devices,” Nature 444597–600 (2006). [CrossRef] [PubMed]

] Perhaps the most distinguishing property of EM metamaterials is their ability to scale with wavelength over many decades of the electromagnetic spectrum. Thus for example, EM metamaterial behavior demonstrated at microwave frequencies can be scaled to lower RF frequencies,[13

13. M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, “Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,” Science 291,849–851 (2001). [CrossRef] [PubMed]

] or higher to the infrared regime[14

14. T.J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang “Terahertz Magnetic Response from Artificial Materials,” Science 303,1494–1496 (2004). [CrossRef] [PubMed]

] by simply scaling the size of the elements. Great interest for the further extension of these exotic materials to optical frequencies (utilizing nano-sized elements) is fueling massive research efforts.

However structures utilized as EM metamaterials typically exhibit a complicated response. Given the potential of NI materials to span the electromagnetic spectrum, it is important to understand their full complex electromagnetic behavior. In particular EM metamaterial structures may exhibit bianisotropy. At microwave frequencies where complex reflection and transmission measurements (S-parameter) are common, it is difficult to characterize bianisotropic materials. This difficulty stems from the fact that these materials are necessarily described by the most general form of the constitutive relations. There may be 36 complex quantities to determine, and thus standard complex reflection and transmission measurements yield incomplete EM information.

At THz and higher frequencies where phase sensitive measurements are not common it is even more difficult, although methods such as ellipsometry[14

14. T.J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang “Terahertz Magnetic Response from Artificial Materials,” Science 303,1494–1496 (2004). [CrossRef] [PubMed]

] or THz time domain spectroscopy[15

15. W.J. Padilla, A.J. Taylor, C. Highstrete, Mark Lee, and R.D. Averitt, “Dynamical Electric and Magnetic Metamaterial Response at Terahertz Frequencies,” Phys. Rev. Lett. 96107401 (2006). [CrossRef] [PubMed]

] may prove useful. Further since a magnetic and electric resonant response enter similarly into the Fresnel equations,[16

16. W.J. Padilla, D.R. Smith, and D.N. Basov, “Spectroscopy of Metamaterials from Infrared to Optical Frequencies,” J. Opt. Soc. Am. B 23404–414 (2006) [CrossRef]

] it is difficult to determine the origin of the response. Typically, artificial magnetic metamaterials are constructed from conducting elements, and thus a full electromagnetic characterization is a necessity, since an electric response is unavoidable. Analytical theories have yielded equations able to predict the resonant frequency ω 0 and plasma frequency ωp of metamaterials. However there is a lack of suitable analytical methods capable of determining the many varying and complicated EM properties. Simulation is still heavily relied upon and has yet to establish itself as a standard routine to characterize bianisotropic response.

Fig. 1. Point group symmetries of the SRR particle. Panel (a) shows the coordinate system convention. Panel (b) shows the symmetry axis of the SRR and the C2 symmetry (rotation about the axis by 2π/n, n=2), and the electromagnetic basis (red arrows). Panels (c) and (d) demonstrate the mirror plane symmetries.

2. Methodology and Symmetry

We present a group theoretical method capable of determining electromagnetic properties for artificial magnetic and electric EM metamaterials.[18

18. W.J. Padilla, “Group theoretical description of artificial magnetic metamaterials utilized for negative index of refraction,” http://xxx.lanl.gov/abs/cond-mat/0508307

] The theory is based simply upon the symmetry operations of the constituent particles about a point in space, i.e. point group theory.[19

19. Here we only consider point groups and do not consider other symmetries, i.e. translations (lattice groups), screw axis and glide planes (space groups).

] The derivation is based upon the assumption of slowly varying external electromagnetic waves and is therefore applicable to metamaterials whose constituent components, and spacings between them, are small compared to a wavelength.[20

20. For a review of the conditions of effective media applicable to metamaterials see ref. [17] and the references therein.

] An EM basis is assigned to the particle under investigation, and transformations of this basis under the symmetries of the group yield the electromagnetic modes. This new method allows one to calculate: the EM modes, the form of the constitutive relations, and the determine whether a particle will exhibit magneto-optical response. This approach is demonstrated by way of an example.

Materials utilized for negative magnetic response have been shown to be bianisotropic,[21

21. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of Bianisotropy in Negative Permeability and Left-Handed Metamaterials,” Phys. Rev. B 65,144440 (2002). [CrossRef]

] thus let us review the constitutive relations for these materials, which can be written as:[22

22. J.A. Kong, Electromagnetic Wave Theory (John Wiley & Sons, Inc., New York, 1990).

]

[D¯B¯]=[ε̿ξ̿ζ̿μ̿][E¯H¯]
(1)
Fig. 2. Basis utilized (red arrows in left column) for magnetic metamaterials used to calculate the EM modes (red arrows in the remaining columns). The remaining columns for each row show the SALCs and modes of the SRR particle, as determined by point group symmetries. For each column, the irrep is shown above and the corresponding component of the an external electromagnetic wave, or function is shown.

In Eq. (1) the permittivities within the 2×2 matrix are tensors of second rank, also called dyadics. The terms ξ̿ and ζ̿ are called the magneto-optical permittivities, and they describe coupling of the magnetic to electric response and electric to magnetic response respectively.

As an example, let us start by examining the point group symmetries of a typical element utilized for magnetic response. In Fig. 1 (b–d) the symmetry operations of a split ring resonator (SRR) are shown. A symmetry operation brings the element into self coincidence. Thus for the SRR it can be seen that there are three symmetries which meet this criterion. In addition, all groups also contain the identity operation E. We then find that the SRR particle belongs to the C2v point group which contains the following elements [E, c2, σ(xy), σ(xz)] where we use the Schoenflies notation.

3. Calculated Electromagnetic Modes

Table 1. Character table for the C2v point group

table-icon
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Table 2. Results summarizing the effect of each symmetry operation on the basis vectors

table-icon
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Once we obtain matrix representations for each element of the group, we can use a result of the Orthogonality Theorem to determine how many times each Irreducible Representation (irrep) occurs. Note, we end up using the characters of the matrices rather than the matrices themselves. For our chosen basis we use the following equation,[23

23. S.F.A Kettle, Symmetry and Structure (John Wiley & Sons, West Sussex, England, 1995).

]

am=1hcncχ(g)χm(g)
(2)

where h is the order of the group (h=4 in this case), nc is the number of symmetry operation in each class, χ(g) are the characters of the original representation, and χm are the characters of the mth irreducible representation. Using Eq. (2) we find our basis is spanned by ΓSRR=2A1+3B2.

Having determined the irreps spanned by an arbitrary basis set, we can work out the appropriate linear combinations of basis functions that transform the matrix representations of our original representation into block diagonal form. These are called Symmetry Adapted Linear Combinations (SALCs). We use a projection operator to determine the symmetry adapted linear combinations that transforms as an irreducible representation. This is given by,

ϕi=χkg(g)i
(3)

where ϕi is the SALC, χ κ{g) is the character of the kth irrep, g is the symmetry operation, and ϕi is the basis function. Equation (3) can also be written as a matrix dot product,

ϕ(A1)ϕ(A2)ϕ(B1)ϕ(B2)=1111111111111111.e1e2e3e4e5e2e1e4e3e5e2e1e4e3e5e1e2e3e4e5
(4)

where the right most matrix is the i part of Eq. (3) (the body of Table 2) and the other matrix is the χk(g) term which is also simply the body of the character table shown in Table 1. The SALCs for A 1 and B 2 are the only products that need to be determined since our basis is spanned by these. Explicitly this is,

ϕ(A1)ϕ(A2)ϕ(B1)ϕ(B2)=2·e1+e2e1+e2e3e4e3e400000000000e1e2e2e1e3+e4e3+e42e5
(5)

where each row of the matrix on the right hand side lists the symmetry adapted linear combinations for each irreducible representation. Thus we see there are two independent solutions for A 1 and three for B 2, in accord with Eq. (2). We can normalize the SALCs determined by Eq. (5), but this is not necessary since a constant factor doesn’t affect the symmetry of the calculated modes. The response of the SRRs can now be determined by considering incident external electromagnetic fields. For example by examination of the character table for the C 2v point group we see that an electric field vector polarized along the x̂-axis transforms as A1, since it transforms in the same manner as the function x. Thus ŷ-polarized light transforms as B2 symmetry. The function Rα represents rotation about the α axis, where α=x̂,ŷ,ẑ. Thus a magnetic field (axial) vector polarized along the ẑ-direction also transforms as B2 symmetry. We summarize these results in Fig. 2 for: (a) planar spirals, (b) SRRs, (c) Omega particles[26

26. N. Engheta and M.M.I. Saadun, “Novel pseudochiral or Ω medium and its application”, Proc. Progr. Electromag. Res. Syms., PIERS 1991, Cambridge, MA, July 1991.

] (d) an SRR and its enantiomer[27

27. Enantiomer in this sense is defined as “the exact opposite” meaning that the polarization mixing which results from one orientation of the split gap can be corrected by another SRR with the split gap oriented oppositely.

] (e) symmetric ring resonator.

An electric field polarized along the x̂-axis (A1) of the SRR drives currents as shown in row (b) of Fig. 2. This would give a frequency response determined by the dimensions (length) of the SRR segments along which Ex lies. Since currents in this case are driven by the electric field, the proper quantity which characterizes its response is the dielectric function, i.e. εxx(ω). The frequency dependence of εxx(ω) is resonant and takes the approximate form of a Lorentzian.[12

12. Hou-Tong Chen, Willie J. Padilla, Joshua M. O. Zide, Arthur C. Gossard, Antoinette J. Taylor, and Richard D. Averitt, “Active Terahertz Metamaterial Devices,” Nature 444597–600 (2006). [CrossRef] [PubMed]

, 15

15. W.J. Padilla, A.J. Taylor, C. Highstrete, Mark Lee, and R.D. Averitt, “Dynamical Electric and Magnetic Metamaterial Response at Terahertz Frequencies,” Phys. Rev. Lett. 96107401 (2006). [CrossRef] [PubMed]

]

For ŷ-polarized (B2) electric fields much more exotic behavior is predicted. Notice that for the irrep of B2 both y and Rz form a suitable basis. Thus we can use a linear combination of these two functions for the basis. This predicts that the SRR should exhibit a magneto-optical response, as in accord with MOGT[23

23. S.F.A Kettle, Symmetry and Structure (John Wiley & Sons, West Sussex, England, 1995).

, 24

24. Daniel C. Harris and Michael D. Bertolucci, Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy (Dover Publications Inc., Mineola, NY, 1989).

, 25

25. Melvin Lax, Symmetry Principles in Solid State and Molecular Physics (Dover Publications Inc., Mineola, NY, 2001).

] Furthermore, since Ey and Bz are the same basis, they will occur at the same frequency. In other words, an E field polarized along the ŷ-axis will result in a resonant response at a frequency ω 0, and a magnetic field polarized along the ẑ-axis will result in a resonant response at the same frequency ω 0. These theoretical predictions are consistent with results obtained from a simple analytical model[21

21. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of Bianisotropy in Negative Permeability and Left-Handed Metamaterials,” Phys. Rev. B 65,144440 (2002). [CrossRef]

] as well as by simulation.[28

28. N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric Coupling to the Magnetic Resonance of Split Ring Resonators,” Appl. Phys. Lett. 84,2943–2945 (2004). [CrossRef]

]

4. Results for Selected Metamaterials

In Fig. 2 we show predictions for other various metamaterials. Particles are listed from lowest symmetry (top) to highest symmetry (bottom). The artificial magnetic metamaterials listed in Fig. 2(a–c) are determined to be bianisotropic, and thus we list the form of the constitutive equations governing the predicted magneto-optical response on the right side of each row. In row (d) we show a particular way of symmetrizing the SRR particle, which we predict will eliminate the magneto-optical response. The material is a bipartite lattice with each of the two sub-lattices consisting of SRRs each with the gap oriented oppositely. Thus it is predicted that any polarization rotation or mixing resulting from one unit cell is corrected by the other unit cell, resulting in no net polarization rotation. Indeed it can be seen that the theory predicts no magneto-optical activity, while still exhibiting a magnetic response for Bz. Another predicted way to eliminate ξ and ζ is to add a second gap in the SRR opposite to the first, as shown in Fig. 2(e). Again the theory predicts no magneto-optical response and thus we have eliminated bianisotropy by symmetrizing the SRR. This structure is simpler than that depicted in Fig. 2(d) and we have a predicted magnetic response for Bz.

Notice in row (a) of Fig. 2 that there are three linear functions which transform as the irrep 5A’. This predicts not only magneto-optical response shown in Fig. 2 (a), but additionally off diagonal terms in the dielectric response. For the spiral of Fig. 2 the ε and μ response functions take the form,

ε=[εxx0εxz010εzx0εzz];μ=[1000μyy0001]
(6)

Thus we see that the spiral element causes polarization rotation for a wave polarized along the x̂-axis and traveling along the ŷ-axis.

5. Predictions for Ideal Magnetic Metamaterials

Lastly let us turn to the prediction of an ideal magnetic metamaterial. With our new understanding of the irreps of a point group and their relation to magneto-optical behavior we can leverage point group tables for help. An ideal magnetic particle is one in which no magneto optical behavior is predicted. In the parlance of group theory this implies we should look for a point group which has linear basis functions with rotational functions Rα but with little or no occurrences of linear x̂,ŷ,ẑ functions. This not only ensures the elimination of the ξ and ζ terms, but also off-diagonal terms in the μ and ε response functions, like those which occur for the particle listed in Fig. 2(d), i.e. 3Bg and 2Bu. In Table 3 we show a candidate point group which should have good magnetic response with no magneto-optical activity, and no frequency dependent ε occurring near the magnetic resonance. A particle which has the symmetry of this group is shown in Fig. 3.

Group theoretical analysis is carried out for the particle depicted in Fig. 3(a), and we find the following modes Γ=A1g+A2g+B1g+B2g+2Eu. Thus the only linear modes determined are a magnetic mode (A2g) and an electric mode (Eu). The fact that the electric mode does not occur in the same irrep as the magnetic mode ensures it will not occur at the same frequency. Further the two dimensional Eu mode implies the electric response of the particle will be identical along the x̂ and ŷ directions with no cross coupling terms (εxy,=εyx=0). Thus if we want to construct a 3D isotropic magnetic metamaterial free from magneto-optical activity, then the geometry of the constituent particles are required to be one of the following point groups: Th,Td,Ih, and Oh. The two simplest of which to visualize are Ih, an icosahedron, and Oh, a cube (depicted in Fig. 3(b)) or octahedron.

Table 3. Character table for the D4h point group.

table-icon
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Fig. 3. Predicted ideal planar and ideal 3D magnetic particles. In panel (a) we show a planar magnetic particle with D4h symmetry. The currents under the A2g magnetic mode are shown. Panel (b) shows a 3D isotropic magnetic particle with Oh symmetry. The electric response of this particles is also isotropic, but importantly does not occur at the same frequency as the magnetic resonance.

6. Conclusion

We would like to acknowledge support from the Los Alamos National Laboratory LDRD Director’s Fellowship, and thank David Schurig, Rick Averitt, and David Smith for valuable feedback.

References and links

1.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite Medium with Simultaneously Negative Permeability and Permittivity,” Phys. Rev. Lett. 84,4184–4187 (2000). [CrossRef] [PubMed]

2.

V.G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of ε and μ,”Soviet Physics USPEKI 10,509–514 (1968). [CrossRef]

3.

W.E. Kock, Metallic delay lenses, Bell System Technical J. 27,58 (1948).

4.

R.N. Bracewell, “Analogues of An Ionized Medium: Applications to the Ionosphere,” Wireless Engineer (Iliff & Sons Ltd., London, 1954), p.320–326.

5.

W. Rotman, “Plasma Simulation by Artificial Dielectrics and Parallel-Plate Media,” IRE Trans. Antennas Propag. AP10,82–95 (1962). [CrossRef]

6.

J.B. Pendry, A.J. Holden, D.J. Robbins, and W.J. Steward, “Magnetism from Conductors and Enhanced Nonlinear Phenomena,” IEEE Trans. Microwave Theory Tech. 47,2075–2084 (1999). [CrossRef]

7.

J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85,3966–3969 (2000). [CrossRef] [PubMed]

8.

R.A. Shelby, D.R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292,77–79 (2001). [CrossRef] [PubMed]

9.

D. Schurig and D. R. Smith, “Negative Index Lens Aberrations,” Phys. Rev. E 70,065601(R) (2004). [CrossRef]

10.

J. B. Pendry, D. Schurig, and D.R. Smith, “Controlling electromagnetic fields,” Science 312,17801782 (2006). [CrossRef]

11.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314977–980 (2006). [CrossRef] [PubMed]

12.

Hou-Tong Chen, Willie J. Padilla, Joshua M. O. Zide, Arthur C. Gossard, Antoinette J. Taylor, and Richard D. Averitt, “Active Terahertz Metamaterial Devices,” Nature 444597–600 (2006). [CrossRef] [PubMed]

13.

M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, “Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging,” Science 291,849–851 (2001). [CrossRef] [PubMed]

14.

T.J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang “Terahertz Magnetic Response from Artificial Materials,” Science 303,1494–1496 (2004). [CrossRef] [PubMed]

15.

W.J. Padilla, A.J. Taylor, C. Highstrete, Mark Lee, and R.D. Averitt, “Dynamical Electric and Magnetic Metamaterial Response at Terahertz Frequencies,” Phys. Rev. Lett. 96107401 (2006). [CrossRef] [PubMed]

16.

W.J. Padilla, D.R. Smith, and D.N. Basov, “Spectroscopy of Metamaterials from Infrared to Optical Frequencies,” J. Opt. Soc. Am. B 23404–414 (2006) [CrossRef]

17.

Th. Koschny, P. Markos, E. N. Economou, D. R. Smith, D. C. Vier, and C. M. Soukoulis, “Impact of Inherent Periodic Structure on Effective Medium Description of Left-Handed and Related Metamaterials,” Phys. Rev. B 71,245105 (2005). [CrossRef]

18.

W.J. Padilla, “Group theoretical description of artificial magnetic metamaterials utilized for negative index of refraction,” http://xxx.lanl.gov/abs/cond-mat/0508307

19.

Here we only consider point groups and do not consider other symmetries, i.e. translations (lattice groups), screw axis and glide planes (space groups).

20.

For a review of the conditions of effective media applicable to metamaterials see ref. [17] and the references therein.

21.

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of Bianisotropy in Negative Permeability and Left-Handed Metamaterials,” Phys. Rev. B 65,144440 (2002). [CrossRef]

22.

J.A. Kong, Electromagnetic Wave Theory (John Wiley & Sons, Inc., New York, 1990).

23.

S.F.A Kettle, Symmetry and Structure (John Wiley & Sons, West Sussex, England, 1995).

24.

Daniel C. Harris and Michael D. Bertolucci, Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy (Dover Publications Inc., Mineola, NY, 1989).

25.

Melvin Lax, Symmetry Principles in Solid State and Molecular Physics (Dover Publications Inc., Mineola, NY, 2001).

26.

N. Engheta and M.M.I. Saadun, “Novel pseudochiral or Ω medium and its application”, Proc. Progr. Electromag. Res. Syms., PIERS 1991, Cambridge, MA, July 1991.

27.

Enantiomer in this sense is defined as “the exact opposite” meaning that the polarization mixing which results from one orientation of the split gap can be corrected by another SRR with the split gap oriented oppositely.

28.

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric Coupling to the Magnetic Resonance of Split Ring Resonators,” Appl. Phys. Lett. 84,2943–2945 (2004). [CrossRef]

OCIS Codes
(160.2100) Materials : Electro-optical materials
(160.3820) Materials : Magneto-optical materials
(160.4760) Materials : Optical properties

ToC Category:
Metamaterials

History
Original Manuscript: December 4, 2006
Revised Manuscript: February 1, 2007
Manuscript Accepted: February 6, 2007
Published: February 19, 2007

Citation
Willie J. Padilla, "Group theoretical description of artificial electromagnetic metamaterials," Opt. Express 15, 1639-1646 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-4-1639


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References

  1. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, S. Schultz, "Composite Medium with Simultaneously Negative Permeability and Permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000). [CrossRef] [PubMed]
  2. V. G. Veselago, "The Electrodynamics of Substances with Simultaneously Negative Values of ε and μ,"Soviet Physics USPEKI 10, 509-514 (1968). [CrossRef]
  3. W. E. Kock, Metallic delay lenses, Bell System Technical J. 27, 58 (1948).
  4. R. N. Bracewell, "Analogues of An Ionized Medium: Applications to the Ionosphere," Wireless Engineer (Iliff & Sons Ltd., London, 1954), p. 320-326.
  5. W. Rotman, "Plasma Simulation by Artificial Dielectrics and Parallel-Plate Media," IRE Trans. Antennas Propag. AP10, 82-95 (1962). [CrossRef]
  6. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Steward, "Magnetism from Conductors and Enhanced Nonlinear Phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999). [CrossRef]
  7. J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
  8. R. A. Shelby, D. R. Smith, S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
  9. D. Schurig and D. R. Smith, "Negative Index Lens Aberrations," Phys. Rev. E 70, 065601(R) (2004). [CrossRef]
  10. J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 17801782 (2006). [CrossRef]
  11. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial Electromagnetic Cloak at Microwave Frequencies," Science 314977-980 (2006). [CrossRef] [PubMed]
  12. H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor and R. D. Averitt, "Active Terahertz Metamaterial Devices," Nature 444597-600 (2006). [CrossRef] [PubMed]
  13. M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, "Microstructured Magnetic Materials for RF Flux Guides in Magnetic Resonance Imaging," Science 291, 849-851 (2001). [CrossRef] [PubMed]
  14. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang "Terahertz Magnetic Response from Artificial Materials," Science 303, 1494-1496 (2004). [CrossRef] [PubMed]
  15. W. J. Padilla, A. J. Taylor, C. Highstrete, Mark Lee, and R. D. Averitt, "Dynamical Electric and Magnetic Metamaterial Response at Terahertz Frequencies," Phys. Rev. Lett. 96107401 (2006). [CrossRef] [PubMed]
  16. W. J. Padilla, D. R. Smith, and D. N. Basov, "Spectroscopy of Metamaterials from Infrared to Optical Frequencies," J. Opt. Soc. Am. B 23404-414 (2006) [CrossRef]
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  19. Here we only consider point groups and do not consider other symmetries, i.e. translations (lattice groups), screw axis and glide planes (space groups).
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