## 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]artificial magnetism,[6] reduced lens aberrations,[9] invisibility cloaking,[10, 11], and metamaterial electronics.[12] 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] or higher to the infrared regime[14] 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] or THz time domain spectroscopy[15] may prove useful. Further since a magnetic and electric resonant response enter similarly into the Fresnel equations,[16] 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 *ω _{p}* 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.

## 2. Methodology and Symmetry

We present a group theoretical method capable of determining electromagnetic properties for artificial magnetic and electric EM metamaterials.[18] The theory is based simply upon the symmetry operations of the constituent particles about a point in space, i.e. point group theory.[19] 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] 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] thus let us review the constitutive relations for these materials, which can be written as:[22]

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 C_{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

We now turn to determination of the EM modes of the particles. Since we have determined SRRs to belong to the C_{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.

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]

where *h* is the order of the group (*h*=4 in this case), *n _{c}* is the number of symmetry operation in each class,

*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}.

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,

where ${\varphi}_{i}^{\prime}$ is the SALC, *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,

where the right most matrix is the *gϕ _{i}* part of Eq. (3) (the body of Table 2) and the other matrix is the

*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,

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 *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.

An electric field polarized along the *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, 15]

For *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] as well as by simulation.[28]

## 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 B_{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}.

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,

Thus we see that the spiral element causes polarization rotation for a wave polarized along the *x*̂-axis and traveling along the *y*̂-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 R_{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.

Group theoretical analysis is carried out for the particle depicted in Fig. 3(a), and we find the following modes Γ=A_{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

We have demonstrated a new method capable of determining various properties of the electromagnetic response for artificial magnetic metamaterials. A specific example has been worked out for the most common element utilized for negative magnetic response and we find excellent agreement with all previous work thus verifying the validity of the theory. This analysis has also been carried out and detailed for other various metamaterials. We have predicted 2 ideal magnetic particles, including one for 3D isotropic magnetic response. Whether one wants to take advantage of the exotic electromagnetic properties that emerge from the bianisotropic nature of artificial metamaterials, or construct artificial materials free from this complication, these novel methods are valuable for determining the expected response.

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.

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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).

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

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**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.

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