## Abstract

Negative refraction is currently achieved by driving the magnetic permeability and electrical permittivity simultaneously negative, thus requiring two separate resonances in the refracting material. The introduction of a single chiral resonance leads to negative refraction of one polarization, resulting in improved and simplified designs of negatively refracting materials and opening previously unknown avenues of investigation in this fast-growing subject.

Negative refraction is an intriguing and counter-intuitive phenomenon that has attracted much attention. Not only does light bend the “wrong” way at a normal/negative interface, but there are even more surprising properties, such as the ability to construct a “perfect” lens for which the resolution is limited not by the wavelength but by the quality of manufacture (*1*, *2*). Negative refraction never occurs in nature, and we rely on artificial materials, metamaterials, to realize the effect as discussed in (*3*). In this paper, I discuss the consequences of chirality and show that it offers an alternative to the present routes to negative refraction. I produce a practical design that is chiral, has many advantages, and exhibits novel properties.

In the original description of negative refraction (*4*), it was stated that when the electrical permittivity and magnetic permeability are both negative light bends the wrong way at an interface. It was only much later, with the ability to construct artificial metamaterials, that the properties could be realized (*5*–*7*): The original prescription for a sub-wavelength array of thin metallic wires combined with resonant metallic rings has been extensively investigated, and negative refraction at microwave frequencies has been confirmed by several investigators (*8*–*13*). Although referred to as “left-handed” materials, I stress that the sense in which this term was used has nothing to do with chirality. Therefore I prefer to use the expression “negatively refracting” to avoid confusion.

The nonchiral designs suffer some limitations. They use two sets of resonant structures, one for the electric and the other for the magnetic response, and these structures have to be very carefully designed to resonate in the same frequency range. Figure 1 shows a typical schematic band structure where it was assumed that
(1)
where μ is the magnetic permeability, ω is the frequency, and ϵ is the effective electric permittivity. The negatively dispersing band between ω_{2} and ω_{3} is responsible for a negative refractive index. The structure of the metamaterial is required to be as fine as possible so that the fields experience an effectively homogeneous material. The present generation of designs go some way to achieving this but rarely do better than a wavelength-to-structure ratio of 10:1. Also, producing a magnetic resonance at optical frequencies will be particularly difficult if low loss is required.

Chiral materials exhibit a different refractive index for each polarization. The dispersion of wave vector, *k*, with ω is shown (Fig. 2A). Formally speaking we introduce a tensor, χ,
(2)
which defines the response of the medium to an electromagnetic field:
(3)
where **D** is the electric displacement vector; **E**, the electric field intensity; **B**, the magnetic induction field; and **H**, the magnetic field intensity. The Supporting Online Material (SOM) Text relates χ to ω(*k*).

Next, consider another medium filled with resonant electric dipoles so that (4)

The response is shown in Fig. 2B. The resonance induces a band gap: a range of frequencies in which the electrical permittivity is negative and where there are no allowed states. The medium is not chiral, so the two transverse polarizations are degenerate. In addition to the transverse modes there is a single longitudinal mode, which is degenerate with the longitudinal modes at *k* = 0 and ω = ω_{0}.

Now consider what happens if dipole resonators are inserted into the chiral medium. The combined response is given by
(5)
and the schematic dispersion is plotted (Fig. 2C). Chirality splits the degenerate transverse modes and in doing so creates a range of frequencies just below ω_{0}′ where the group velocity,
(6)
has the opposite sign to the phase velocity,
(7)
but only for one polarization. This is the signature of negative refraction. One can define a refractive index for each polarization,
(8)
where *c*_{0} is the velocity of light in free space, and one of them is negative. An impedance can also be defined (SOM Text).

The bands shown in Fig. 2C are remarkable in several ways. In contrast to the nonchiral situation where there are two resonances and therefore two gaps, here there is only one gap. Hence, the transition to negative refraction is smooth and continuous at ω_{0}′ with no gap. Only a chiral material can achieve this because it requires that the intersecting bands at *k* = 0 do not hybridize as they normally would do. Also, the bands at this point have finite group velocity but infinite phase velocity. For the chiral material, the minimum frequency, ω_{0}″, occurs as a finite wave vector, and hence the density of states diverges at this frequency.

In general when light in a vacuum is incident upon a slab of the new resonant chiral material, two refracted beams will be observed because a refracting surface can mix polarizations. However suppose the parameters are chosen such that
(9)
and
(10)
where *Z*_{0} is the ratio of electric to magnetic fields for waves in the vacuum, that is, the impedance of the vacuum, and *Z*_{+} is the ratio of electric to magnetic fields in the medium for the polarization that shows negative refraction (SOM Text). Then detailed calculations show that the slab is perfectly transparent to the + polarization. All the properties predicted for an isotropic nonchiral medium with
(11)
will be reproduced in this new medium but only for one of the polarizations. This includes the ability to focus the near fields and hence reproduce an image with resolution unlimited by wavelength.

This recipe for chiral negative refraction is a general one. Only the ingredients of a resonant system producing a band gap and chirality are needed. A moment's study of Fig. 2C will show that any minimum in ω(*k*) at *k* = 0 will produce negative refraction when split in this way.

Next I suggest a practical realization of a resonant chiral structure. I do this by winding a continuous insulated metal tape onto a cylinder so that it forms an overlapping helix (Fig. 3). More details of the performance of this structure are available (figs. S8 to S13). This is a chiral variant of the so-called Swiss roll structure used to produce negative permeability in the MHz range of frequencies (*14*). The structure is resonant because of inductance in the coiled helix and capacitance between the inner and outer layers of the helix. When current flows along the helix, not only does it produce a magnetic polarization along the axis, but it also produces an electric polarization because some of the current flows parallel to the axis. Typical values for the parameters are
(12)
where *a* is the lattice constant of the log-pile structure and *r, d*, θ, and *N* are defined in Fig. 3. These parameters give negative refraction at around 100 MHz. The design can be tuned over a wide range of frequencies (SOM Text). Achieving strong chirality in the optical region of the spectrum is more difficult, but some promising design studies have been made (*15*).

The class of negatively refracting materials introduced here with the prescribed properties should open previously unknown avenues of investigation. Specific designs are greatly simplified with very compact internal structure, typically on a scale less than 1/100th of the free space wavelength at the resonant frequency. The structures offer further opportunities to extend the negative refraction concept.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/306/5700/1353/DC1

SOM Text

Figs. S1 to S13