1. Introduction

Negative refraction, first introduced in a left-handed material (LHM) by Veselago in the 1960s [1], exhibiting a negative index of refraction due to simultaneously negative permeability and permittivity, has recently attracted much attention due to the enormous implication of “perfect” or super lens (beating the diffraction limit) [2]. In such a material, the electric field, the magnetic field and the wave vector of an electromagnetic wave obey the left-hand rule (instead of the right-hand rule that conventional materials obey), and the phase velocity and the Poynting vector (group velocity) are in opposite directions (thus called a backward wave). LHMs in microwave region have been fabricated experimentally [3]. The impact would be much larger if we can realize negative refraction at optical frequencies (beating the diffraction limit can give revolutionary breakthroughs to e.g. the optical storage industries). In the present paper we study a chiral way [4] of beating the diffraction limit at optical frequencies.

A chiral medium, which exhibits electromagnetic handedness, has the following constitutive relations [5]

where κ is the chirality parameter (assumed to be positive in this paper), ε and µ are the permittivity and permeability of the chiral medium, respectively (ε ₀ and µ ₀ are the vacuum permittivity and permeability). In a chiral material, an electric or magnetic excitation will produce simultaneously both the electric and magnetic polarizations. Some natural and optically chiral media can be considered as a homogeneous medium (sugar solution in water is probably the cheapest optical chiral medium). Until last year, all scientists working on chiral media believed that the chirality parameter should satisfy $\kappa <\sqrt{\mu \epsilon }⁄\sqrt{{\mu }_0{\epsilon }_0}$ (see e.g. [5]). Inspired by the recent emerging left-handed materials, we now realized that we did a mistake in the past by setting such an unnecessary restriction [6]. $\kappa >\sqrt{\mu \epsilon }⁄\sqrt{{\mu }_0{\epsilon }_0}$ can occur at least at or near the resonant frequency of the permittivity of a chiral medium (called chiral nihility [7]), and then backward wave will occur at one of the two circularly polarized eigenwaves [6].

Since a chiral medium with appropriate parameters can support backward waves, a natural question is whether a slab of such a chiral medium can be used as a perfect or super lens. In this paper, we study the focusing properties of a chiral slab, whose impedance may in general be mismatched to that of air (then the two circularly polarized eigenwaves will be coupled to each other and consequently one can not consider the focusing of the backward polarized eigenwave separately). For simplicity, we consider in this paper only the two-dimensional case (the point source is a line current, the electric and magnetic field vectors are z-independent but may have all the three components).

2. Theory for negative refraction and focusing

First consider the propagation of a time harmonic plane wave exp[i($\stackrel{\rightharpoonup }{\text{k}}$ ·$\stackrel{\rightharpoonup }{\text{r}}$ -ωt)] in a homogeneous chiral medium, where $\stackrel{\rightharpoonup }{\text{k}}$ =k_x $\stackrel{\rightharpoonup }{\text{e}}$ _x +k_y $\stackrel{\rightharpoonup }{\text{e}}$ _y . From Maxwell’s equations, we have

From Eqs. (1)-(2), we can derive the following dispersion relation for the chiral medium

which has four solutions k±=±k _±, where $k_{+}=\omega \left(\kappa \sqrt{{\mu }_0{\epsilon }_0}+\sqrt{\mu \epsilon }\right)$ and $k_{-}=\omega \left(\kappa \sqrt{{\mu }_0{\epsilon }_0}-\sqrt{\mu \epsilon }\right)$. For a given wave vector $\stackrel{\rightharpoonup }{\text{k}}$ , there are two (left or right) circularly polarized plane waves, for which the Cartesian components of the electromagnetic fields satisfy the following relations

By setting κ=0, the above 2 sets of relations will be reduced to the conventional relations for a left or right circularly polarized plane wave in an isotropic dielectric medium [8].

When $\kappa \sqrt{{\mu }_0{\epsilon }_0}>\sqrt{\mu \epsilon }$, the set of relations (Eq. (4)) associated with k_[corresponding to a right circularly polarized plane wave (the rotation of the electric field is clockwise when the observer is facing into the oncoming energy stream)] will give $S_x=-(\sqrt{\epsilon }{k}_x⁄\sqrt{\mu }{k}_{-}){\mid E_z\mid }^2$, $S_y=-(\sqrt{\epsilon }{k}_y⁄\sqrt{\mu }{k}_{-}){\mid E_z\mid }^2$ and S_z =0. This indicates that the wave vector and the time averaged Poynting vector are in opposite directions, and thus this right circularly polarized plane wave is a backward wave. The continuation of the tangential components of the electromagnetic fields will then give rise to the phenomenon of negative refraction when a propagating plane wave obliquely incident on a chiral medium from air. The other circularly polarized plane wave in the chiral medium is a conventional forward wave and positive refraction occurs at an air-chiral interface. The refractive indexes for plane waves associated with k ₊ and k_ are $n_{+}=\sqrt{\mu \epsilon ⁄{\mu }_0{\epsilon }_0}+\kappa$ and $n_{-}=\sqrt{\mu \epsilon ⁄{\mu }_0{\epsilon }_0}-\kappa$, respectively. n ₊ is always positive, but n_ is negative when $\kappa \sqrt{{\mu }_0{\epsilon }_0}>\sqrt{\mu \epsilon }$. In the rest of the present paper we will consider only the case of $\kappa \sqrt{{\mu }_0{\epsilon }_0}>\sqrt{\mu \epsilon }$.

 

Fig. 1. Configuration for the focusing of a chiral slab for an object in air. Superscripts “+” and “-” of the electric field spectra indicate the left-and right-going wave streams, respectively.

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The configuration for the focusing of a chiral slab for an object in air is shown in Fig. 1. The distance between the object and the left surface of the slab is d ₀ and the thickness of the chiral slab is d ₁. As a general situation, the field from the object consists of left and right circularly polarized plane waves. The electric field emanating from the object between the object and the left surface of the slab can be decomposed as

where ${\stackrel{\mathit{\rightharpoonup }}{E}}_{0+}^{+}$(k _0,y) and ${\stackrel{\mathit{\rightharpoonup }}{E}}_{0-}^{+}$(k _0,y) are the (right-going) spatial spectra of the electric fields of the left and right circularly polarized plane waves, respectively, and $k_{0,x}^2$+$k_{0,y}^2$=$k_0^2$ (here k ₀=2π/λ ₀ and λ ₀ is the wavelength in the air). Similarly, the electric field on the right side of the slab can be decomposed as

where ${\stackrel{\mathit{\rightharpoonup }}{E}}_{2+}^{+}$(k _0,y) and ${\stackrel{\mathit{\rightharpoonup }}{E}}_{2-}^{+}$(k _0,y) are the (right-going) spatial spectra of the electric fields of the left and right circularly polarized plane waves transmitting through the chiral slab, respectively.

In Fig.1, inside the chiral slab E${\rightharpoonup }_{1+}^{+}$, E${\rightharpoonup }_{1+}^{-}$ represent the left circularly polarized waves and E${\rightharpoonup }_{1-}^{+}$, E${\rightharpoonup }_{1-}^{-}$represent the right circularly polarized waves (the argument k _0,y is omitted in the field spectrum notations). The continuation conditions of the tangential electric and magnetic fields at the two surfaces of the chiral slab lead to

where a=exp(ik _1+,x d ₁), b=exp (-ik _1-x d ₁), c=exp(-ik _1+x d ₁), d=exp(ik _1-,x d ₁), e=exp(ik _0,x d ₁), $k_{1+,x}=\sqrt{k_{+}^2-k_{0,y}^2}$ and $k_{1-,x}=\sqrt{k_{-}^2-k_{0,y}^2}$. The Cartesian components for each wave in Eq. (7) satisfy Eq. (4) (all in terms of the z components of the electric field spectra). Therefore, from Eqs. (4) and (7), we can obtain a system of equations for the z components of the electric field spectra in three different regions. After solving this system of equations, we can obtain the other components for the electric and magnetic field spectra from Eq. (4). This is actually how the numerical results for focusing by a chiral slab are obtained in the next section. Note that in general the left and right circularly polarized plane waves are coupled to each other due to the cross-polarization reflection (see e.g. [5, 9]).

As a special case of the above formulas we can prove below theoretically that such a chiral slab under certain matching conditions can give a perfect image of an object source whose field can be presented by an angular spectrum of right circularly polarized plane waves. If µ/ε=µ ₀/ε ₀ (i.e., matched impedance), the system of Eq. (7) can be split into two decoupled sub-systems. The sub-system (after using Eq. (4)) associated with backward waves is

In this special case, an incident right circularly polarized plane wave in the air only excites the right circularly polarized plane waves associated with $k_{-}=\omega \left(\kappa \sqrt{{\mu }_0{\epsilon }_0}-\sqrt{\mu \epsilon }\right)$ inside the chiral slab, and the reflected and transmitted plane waves are also right circularly polarized. If at the same time we have $\kappa \sqrt{{\mu }_0{\epsilon }_0}-\sqrt{\mu \epsilon }=\sqrt{{\mu }_0{\epsilon }_0}$ (i.e., refractive index match), we can find easily from Eq. (8) that the transmitted field spectrum $E_{2-,z}^{+}$=exp(-2ik _0,x d ₁)$E_{0-,z}^{+}$ and there is no reflected wave. Therefore, an evanescent wave (i.e., k _0,x=i|k _0,x|) can be amplified (by a factor (|k _0,x|d ₁) after transmitting through the chiral slab. Then from Eq. (6), we have

which indicates that the image at a distance (d ₁-d ₀) (with d ₁>d ₀) from the right surface of the chiral slab is identical to the object at a distance d ₀ away from the left surface of the chiral slab (see Fig. 1).

3. Numerical results

It is interesting and necessary to study numerically the focusing by a slab of mismatched chiral medium (which still supports backward waves). First we consider the case when the object is a line source composed of only right circularly polarized plane wave components (i.e., the incident field is obtained by removing the left circularly polarized plane wave components from the incident field generated by a unit current line source; nevertheless, some left circularly polarized plane wave components could be generated and coupled to the right circularly polarized plane wave components in the reflected, transmitted or internal field due to the cross-polarization reflection). Thus, the z component of the incident electric field can be written as

where ${\omega }_0=2\pi \sqrt{{\mu }_0{\epsilon }_0}⁄{\lambda }_0$. The other components of the incident electromagnetic fields can then be obtained from Eq. (4). As the first numerical example, we choose material parameters ε=ε ₀, µ=µ ₀ and κ=1.975 (thus the refractive index for right circularly polarized plane waves is n_=0.975 and has a small mismatch from that of the air). The thickness d ₁ of the chiral slab is λ ₀ and d ₀=0.5λ ₀. Solving system (7) of equations and using Eq. (6), we obtain the normalized electric field intensity distribution on the right side of the chiral slab (as shown in Fig. 2(a)). From Fig. 2(a) one sees that a subwavelength focusing is achieved. Fig. 2(b) shows the focusing for another numerical example with ε=0.1ε ₀, µ=µ ₀, $\kappa =2\sqrt{\mu \epsilon ⁄{\mu }_0{\epsilon }_0}$, d ₀=0.5λ ₀, and d ₁=0.3λ ₀ (both the impedance and refractive index are mismatched quite much from those of the air).

Finally, we calculate the corresponding intensity distribution of the transmitted field for an incident field generated by a unit current line source (without removing the left circularly polarized plane wave components) and the results are shown in Figs. 2(c) and 2(d). By comparing Figs. 2(c) and 2(d) with Figs. 2(a) and 2(b), respectively, one sees that the left circularly polarized plane wave components in the incident field degrades a bit the focusing performance (in the first example some strong surface plasmon effects occur and we have to illustrate the focusing more clearly in a smaller area with the field intensity renormalized for this reduced area; see the inset of Fig. 2(c)).

 

Fig. 2. The normalized electric field intensity distribution on the right side of a chiral slab. (a) and (c): A case with a small mismatch in the refractive index (the parameters are chosen as ε=ε ₀, µ=µ ₀, κ=1.975, d ₀=0.5λ ₀ and d ₁=λ ₀); (b) and (d): The case with a large mismatch in both the refractive index and the impedance (the parameters are chosen as ε=0.1ε ₀, $\mu ={\mu }_0,\kappa =2\sqrt{\mu \epsilon ⁄{\mu }_0{\epsilon }_0},$ d ₀=0.5λ ₀ and d ₁=0.3λ ₀). The left circularly polarized plane wave components in the incident field (generated by a unit current line source) are removed for (a) and (b), but kept for (c) and (d). The electric field intensity distribution in the white box in (c) is renormalized for the reduced area and shown in the inset.

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4. Conclusion

Backward waves can propagate in a chiral medium when the chiral parameter $\kappa \sqrt{{\mu }_0{\epsilon }_0}>\sqrt{\mu \epsilon }$ (with κ>0). A slab of such a chiral medium with both the refractive index and impedance matched to those of the air can give a perfectly sharp focusing for right circularly polarized plane wave components of a point source. Other than this special case, the left and right circularly polarized plane waves are in general coupled to each other due to the cross-polarization reflection. Some numerical examples have been given to show the focusing performance when the chiral medium has some mismatch in the refractive index or impedance. Similar negative refraction results are expected when κ<0.