## Group theoretical description of artificial electromagnetic metamaterials

Optics Express, Vol. 15, Issue 4, pp. 1639-1646 (2007)

http://dx.doi.org/10.1364/OE.15.001639

Acrobat PDF (222 KB)

### 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

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]

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]

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 **314**977–980 (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 **444**597–600 (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]

## 2. Methodology and Symmetry

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

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]

_{2v}point group which contains the following elements [E, c

_{2}, σ(

*xy*), σ(

*xz*)] where we use the Schoenflies notation.

## 3. Calculated Electromagnetic Modes

_{2v}group we can utilize the character table for this analysis (see Table 1). The body of the table lists the characters, (trace of a matrix representation) of the group. The first column lists the irreps of the group, the top of columns 2–5 lists the symmetry operations, and the last 2 columns lists a number of linear and quadratic functions that transform as the various irreps of the group. The bottom row lists the characters of our chosen SRR basis. With the character table we can assign a basis set to the SRR and see how this basis set transforms under the C

_{2v}symmetry operations. For brevity we only consider the outer split ring, however the inner ring is also easily handled by this method. We want our basis to represent areas of electrical activity, thus we choose the basis shown in the left column of Fig. 2(b) (also shown in Fig. 1(b) for the SRR). Regions marked with arrows represent areas which can be polarized by an external electric field (i.e. currents can flow in these directions). These are similar to the P orbitals used in molecular orbital group theory (MOGT). The next step is to write out matrices which describe how this basis transforms under C

_{2v}. For example, there are five vectors which make up our basis for the SRR, and since the identity leaves the particle unchanged, this would be a 5×5 matrix with 1’s along the diagonal. The result of each symmetry operation on the basis vectors are summarized in Table 2. Rows list the symmetry element and the columns the basis vectors. The body of the table indicates where the basis vector ends up after the each symmetry operation. The characters for our chosen basis can also be determined from Table 2. We construct a table based on the following rules: If a particular basis vector gets mapped back to itself or minus itself under the symmetry operation then we assign a 1 or -1 to the body of the table respectively, else we assign a zero. We then sum each row to give the character of each symmetry for our chosen basis. The characters of our basis are shown as the bottom row of Table 1.

*h*is the order of the group (

*h*=4 in this case),

*n*is the number of symmetry operation in each class,

_{c}*χ*(

*g*) are the characters of the original representation, and

*χ*are the characters of the m

_{m}^{th}irreducible representation. Using Eq. (2) we find our basis is spanned by Γ

_{SRR}=2A

_{1}+3B

_{2}.

*χ*

_{κ}{

*g*) is the character of the k

^{th}irrep,

*g*is the symmetry operation, and ϕ

_{i}is the basis function. Equation (3) can also be written as a matrix dot product,

*gϕ*part of Eq. (3) (the body of Table 2) and the other matrix is the

_{i}*χ*(

_{k}*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,

*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

*x*̂-axis transforms as A

_{1}, since it transforms in the same manner as the function x. Thus

*y*̂-polarized light transforms as B

_{2}symmetry. The function R

_{α}represents rotation about the

*α*axis, where

*α*=

*x*̂,

*y*̂,

*z*̂. Thus a magnetic field (axial) vector polarized along the

*z*̂-direction also transforms as B

_{2}symmetry. We summarize these results in Fig. 2 for: (a) planar spirals, (b) SRRs, (c) Omega particles[26] (d) an SRR and its enantiomer[27] (e) symmetric ring resonator.

*x*̂-axis (A

_{1}) 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 E

_{x}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.

*ε*(

_{xx}*ω*). The frequency dependence of

*ε*(

_{xx}*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 **444**597–600 (2006). [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. **96**107401 (2006). [CrossRef] [PubMed]

*y*̂-polarized (B

_{2}) electric fields much more exotic behavior is predicted. Notice that for the irrep of B

_{2}both y and R

_{z}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, 24, 25] Furthermore, since E

_{y}and B

_{z}are the same basis, they will occur at the same frequency. In other words, an E field polarized along the

*y*̂-axis will result in a resonant response at a frequency

*ω*

_{0}, and a magnetic field polarized along the

*z*̂-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]

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

_{z}. 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 B

_{z}.

*ε*and

*μ*response functions take the form,

*x*̂-axis and traveling along the

*y*̂-axis.

## 5. Predictions for Ideal Magnetic Metamaterials

_{α}but with little or no occurrences of linear

*x*̂,

*y*̂,

*z*̂ 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. 3B

_{g}and 2B

_{u}. 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.

_{1g}+A

_{2g}+B

_{1g}+B

_{2g}+2E

_{u}. Thus the only linear modes determined are a magnetic mode (A

_{2g}) and an electric mode (E

_{u}). 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 E

_{u}mode implies the electric response of the particle will be identical along the

*x*̂ and

*y*̂ directions with no cross coupling terms (

*ε*,=

_{xy}*ε*=0). Thus if we want to construct a 3D

_{yx}*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: T

_{h},T

_{d},I

_{h}, and O

_{h}. The two simplest of which to visualize are I

_{h}, an icosahedron, and O

_{h}, a cube (depicted in Fig. 3(b)) or octahedron.

## 6. Conclusion

## 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. |

2. | V.G. Veselago, “The Electrodynamics of Substances with Simultaneously Negative Values of |

3. | W.E. Kock, |

4. | R.N. Bracewell, “Analogues of An Ionized Medium: Applications to the Ionosphere,” |

5. | W. Rotman, “Plasma Simulation by Artificial Dielectrics and Parallel-Plate Media,” IRE Trans. Antennas Propag. |

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. |

7. | J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. |

8. | R.A. Shelby, D.R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science |

9. | D. Schurig and D. R. Smith, “Negative Index Lens Aberrations,” Phys. Rev. E |

10. | J. B. Pendry, D. Schurig, and D.R. Smith, “Controlling electromagnetic fields,” Science |

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 |

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 |

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 |

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 |

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. |

16. | W.J. Padilla, D.R. Smith, and D.N. Basov, “Spectroscopy of Metamaterials from Infrared to Optical Frequencies,” J. Opt. Soc. Am. B |

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 |

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 |

22. | J.A. Kong, |

23. | S.F.A Kettle, |

24. | Daniel C. Harris and Michael D. Bertolucci, |

25. | Melvin Lax, |

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. |

**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

- 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]
- V. G. Veselago, "The Electrodynamics of Substances with Simultaneously Negative Values of ε and μ,"Soviet Physics USPEKI 10, 509-514 (1968). [CrossRef]
- W. E. Kock, Metallic delay lenses, Bell System Technical J. 27, 58 (1948).
- R. N. Bracewell, "Analogues of An Ionized Medium: Applications to the Ionosphere," Wireless Engineer (Iliff & Sons Ltd., London, 1954), p. 320-326.
- W. Rotman, "Plasma Simulation by Artificial Dielectrics and Parallel-Plate Media," IRE Trans. Antennas Propag. AP10, 82-95 (1962). [CrossRef]
- 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]
- J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- R. A. Shelby, D. R. Smith, S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
- D. Schurig and D. R. Smith, "Negative Index Lens Aberrations," Phys. Rev. E 70, 065601(R) (2004). [CrossRef]
- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 17801782 (2006). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- W. J. Padilla, "Group theoretical description of artificial magnetic metamaterials utilized for negative index of refraction," http://xxx.lanl.gov/abs/cond-mat/0508307
- Here we only consider point groups and do not consider other symmetries, i.e. translations (lattice groups), screw axis and glide planes (space groups).
- For a review of the conditions of effective media applicable to metamaterials see ref. [17] and the references therein.
- R. Marqués, F. Medina, R. Rafii-El-Idrissi, "Role of Bianisotropy in Negative Permeability and Left-Handed Metamaterials," Phys. Rev. B 65, 144440 (2002). [CrossRef]
- J. A. Kong, Electromagnetic Wave Theory (John Wiley & Sons, Inc., New York, 1990).
- S. F. A. Kettle, Symmetry and Structure (John Wiley & Sons, West Sussex, England, 1995).
- Daniel C. Harris and Michael D. Bertolucci, Symmetry and Spectroscopy: An Introduction to Vibrational and Electronic Spectroscopy (Dover Publications Inc., Mineola, NY, 1989).
- Melvin Lax, Symmetry Principles in Solid State and Molecular Physics (Dover Publications Inc., Mineola, NY, 2001).
- N. Engheta, M. M. I. Saadun, "Novel pseudochiral or Ω medium and its application", Proc. Progr. Electromag. Res. Syms., PIERS 1991, Cambridge, MA, July 1991.
- 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.
- 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]

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