Two-dimensional inside-out Eaton Lens: Design technique and TM-polarized wave properties

Yong Zeng; Douglas H. Werner

doi:10.1364/OE.20.002335

More than half a century ago, it was proposed that gradient index lenses [1], such as the Maxwell fish-eye lens [2], the Luneburg lens [3] and the Eaton lens [4], can be free of geometrical aberrations and form sharp images, at least at the geometrical-optics level (see Reference [5] and [6] for more details). Taking the inside-out Eaton lens as an example, which was proposed by Miñano in 2006 [7], its refractive index n(r) equals $\sqrt{(2 a / r) - 1}$ for 1 ≤ r/a ≤ 2 and 1 otherwise, where r represents the distance from the center of the lens and a its inner radius (see Fig. 1). It can be analytically proven that light rays emitted from a source at position r₀, with r₀/a < 1, will be focused exactly at position −r₀ [6, 7]. Therefore, like Maxwell’s fish-eye lens, aberrationless imaging can be obtained by an inside-out Eaton lens, where both the source and the image are inside the optical device. Recently, there has been a renaissance of scientific interest in these gradient index lenses [8–17], partially because of the developments in metamaterials [18, 19] and transformation optics [20–23]. Metamaterials are manmade media whose effective permittivities and permeabilities are governed by their deeply subwavelength structures as well as their constituent materials [19]. For instance, properly integrating split ring resonators with metallic rods will result in a metamaterial with an effectively negative index of refraction [24]. Transformation optics, on the other hand, is inspired by an intriguing property of Maxwell’s equations, i.e., their form is invariant under arbitrary coordinate transformations, assuming the field quantities and the material properties are transformed accordingly [20, 21].

Fig. 1 Inside-out Eaton lens (1 ≤ r ≤ 2). The source is located at r₀ = 0.5. Light rays (blue) are described by Hamilton’s Eq. (A.2).

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Most of the current work has been focused on the Luneburg lens and Maxwell’s fish eye, while the inside-out Eaton lens has been far less studied. In this paper we propose one realistic design of a two-dimensional Eaton lens which consists of an assembly of metallic wires with variable radii. We validate the performance of our design with full-wave simulations. To further investigate the effect of the finite wavelength, we employ the WKB approach to solve the corresponding wave equations of transverse-magnetic modes [25,26], and obtain one necessary condition for perfect imaging, i.e. imaging with unlimited resolution.

The original Eaton lens has a spherical geometry and an index of refraction which is given by [4]

n (r) = \sqrt{\frac{2 a}{r} - 1} .

Throughout this paper, the inner radius a is set to be 1, whereby the variables such as r are dimensionless, and the design presented here can be applied to different excitation wavelengths from the infrared to optical regime. Since the refractive-index profile possesses spherical symmetry, the entire ray trajectory lies in a plane which is orthogonal to a conserved angular momentum L [6] (See Appendix A). Consequently, the design can be simplified and represented as a two-dimensional cylinder with a similar refractive index profile. We further assume that the cylinder lies in the xy plane, as well as the propagation plane of the light ray. Moreover, we only consider TM-polarized light where the magnetic-field vector H points in the z direction. By sacrificing the impedance matching, we can further assume that the two-dimensional cylinder is purely electrical, i.e. μ = 1, with its permittivity ɛ(r) given by n²(r). Under the same conditions, the TE-polarized mode is described by the Helmholtz equation [13], and its solutions inside the lens can be expressed in terms of the Whittaker functions analytically. A detailed treatment will be presented elsewhere.

A general ray-optics description of media with spherically-symmetric index profiles can be found in Ref. [6], which included an abbreviated extension to the inside-out Eaton lens. For convenience, a much more detailed treatment of the inside-out Eaton lens is given in Appendix A. We therefore only consider the wave interpretation here. Starting from the following wave equation

\frac{1}{ɛ (r)} \nabla^{2} H + \nabla \frac{1}{ɛ (r)} \cdot \nabla H = - \frac{ω^{2}}{c^{2}} H,

with ∇ = ∂_xe_x + ∂_ye_y and H as H_z, this equation can be reformulated in cylindrical coordinates as

\frac{\partial^{2} H}{\partial r^{2}} + (\frac{1}{r} - \frac{1}{ɛ} \frac{d ɛ}{d r}) \frac{\partial H}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} H}{\partial θ^{2}} + k_{0}^{2} ɛ H = 0,

with k₀ = ω/c being the wave number in free space. We now assume that the magnetic field H(r,θ) can be expanded as

\sum_{n} f_{n} (r) e^{i n θ} \sqrt{ɛ / r}

, where the function f_n satisfies

f_{n}^{″} + (k_{0}^{2} ɛ - \frac{n^{2}}{r^{2}} + \frac{ɛ^{2} - 3 r^{2} {ɛ^{'}}^{2} + 2 r ɛ ɛ^{'} + 2 r^{2} ɛ ɛ^{″}}{4 r^{2} ɛ^{2}}) f_{n} = 0 .

In the region r ≤ 1 where ɛ(r) = 1, this reduces to

f_{n}^{″} + (k_{0}^{2} - \frac{n^{2} - 1 / 4}{r^{2}}) f_{n} = 0,

which has the general solutions

f_{n} (r) = \sqrt{r} [a_{n} H_{n}^{(1)} (k_{0} r) + b_{n} H_{n}^{(2)} (k_{0} r)],

where

H_{n}^{(1)}

and

H_{n}^{(2)}

are the n-th order Hankel functions of the first and second kind, respectively. In the region 1 < r < 2, we can rewrite Eq. (4) as

f_{n}^{″} + s (r) k_{0}^{2} f_{n} = 0,

with

s (r) = \frac{2 - r}{r} - \frac{n^{2} - 1 / 4}{r^{2} k_{0}^{2}} - \frac{r + 1}{r^{2} {(2 - r)}^{2} k_{0}^{2}} .

A few examples are plotted in Fig. 2. For a modest value of n, s(r) generally monotonically decreases from a positive value to negative infinity. It is, however, always negative when n is large enough.

Fig. 2 (a) Dependence of the function s(r) on the mode order n. Here the wavelength is 0.3. (b) Effect of the wavelength λ on the phase factor γ_n.

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We employ the one-dimensional WKB approximation (also known as geometrical optics approximation in the optics community), developed by Wentzel, Kramers and Brillouin in 1926, to analytically solve Eq. (7) for a modest value of n [25,26]. More specifically, we assume that f_n(r) has the form A_ne^ik₀τ(r) for positive s(r), and τ(r) can be further expanded in terms of k₀,

τ (r) = τ_{0} (r) + \frac{1}{k_{0}} τ_{1} (r) + \frac{1}{k_{0}^{2}} τ_{2} (r) + \dots .

Similar arguments also hold for τ′ (r) as well as τ″ (r). By collecting the leading-order terms, it is found that

{(\frac{d τ_{0}}{d r})}^{2} = s (r), \frac{d τ_{1}}{d r} = \frac{i τ_{0}^{″}}{2 τ_{0}^{'}} .

Consequently the first order solution can be expressed as

f_{n} (r) \sim \frac{A_{n}^{+}}{s {(r)}^{1 / 4}} exp [i k_{0} \int_{r_{n}}^{r} \sqrt{s (r^{'})} d r^{'}] + \frac{A_{n}^{-}}{s {(r)}^{1 / 4}} exp [- i k_{0} \int_{r_{n}}^{r} \sqrt{s (r^{'})} d r^{'}],

with r_n representing the turning point where s(r) = 0. It should be mentioned that r_n depends on the mode order n: the larger the n, the smaller the turning point r_n. In other words, different-order modes have quite different propagation lengths inside the lens. The first term on the right hand side corresponds to an out-going wave because its phase increases with distance, while the second term corresponds to an in-coming wave. Similar procedures can be applied to a negative s(r) by assuming f_n(r) = B_ne^−k₀τ(r), such that the resultant first-order approximation is given by

f_{n} (r) \sim \frac{B_{n}^{+}}{{| s (r) |}^{1 / 4}} exp [- k_{0} \int_{r_{n}}^{r} \sqrt{| s (r^{'}) |} d r^{'}] .

Here only the solution that is exponentially decaying in the r direction is included. Furthermore, s(r) ∼ (r_n − r) approaches zero linearly in the vicinity of the turning point r_n. The solution therefore can be approximated as

Ai (- k_{0}^{2 / 3} s)

, with Ai being the Airy function [25]. When k₀ is large enough, we can asymptotically match Eq. (11) and (12) around the turning point, and finally achieve

A_{n}^{+} = i A_{n}^{-}

. As a direct result, in the region where s(r) is positive, we have

f_{n} (r) \sim \frac{A_{n}}{s {(r)}^{1 / 4}} exp [i k_{0} \int_{r_{n}}^{r} \sqrt{s (r^{'})} d r^{'}] + \frac{A_{n}}{s {(r)}^{1 / 4}} exp [- i \frac{π}{2} - i k_{0} \int_{r_{n}}^{r} \sqrt{s (r^{'})} d r^{'}],

which implies that an out-going wave will be totally reflected around the turning point r_n, accompanied by a phase variation of π/2.

We now assume Eq. (13) can be extended to the region where r is slightly smaller than 1. Additionally, its phase factor can be approximated as

k_{0} \int_{r_{n}}^{r} \sqrt{s (r^{'})} d r^{'} = k_{0} \int_{r_{n}}^{r} \sqrt{s_{0}} d r^{'} + k_{0} \int_{r_{n}}^{1} [\sqrt{s (r^{'})} - \sqrt{s_{0}}] d r^{'} \approx k_{0} r + γ_{n},

where γ_n does not depend on r, and

k_{0}^{2} s_{0} = k_{0}^{2} - \frac{n^{2} - 1 / 4}{r^{2}}

represents the coefficient shown in Eq. (5). In the vicinity of r = 1, f_n(r) can be rewritten in the form

f_{n} (r) \sim \frac{A_{n}}{s_{0}^{1 / 4}} [e^{i (k_{0} r + γ_{n})} + e^{- i (k_{0} r + γ_{n} + π / 2)}] .

Coincidentally, a source at position r₀ with r₀ < 1, i.e. inside the Eaton lens, generates radiation according to

- \frac{i}{4} H_{0}^{(1)} (k_{0} | r - r_{0} |) = - \frac{i}{4} \sum_{- \infty}^{\infty} J_{n} (k_{0} r_{<}) H_{n}^{(1)} (k_{0} r_{>}) e^{i n ϕ},

where r_< = min{r,r₀}, and r_> = max{r,r₀} [25, 27]. The total magnetic field in the region between r₀ and 1 is therefore given by

- \frac{i}{4} \sum_{- \infty}^{\infty} e^{in ϕ} J_{n} (k_{0} r_{0}) [H_{n}^{(1)} (k_{0} r) + C_{n} H_{n}^{(2)} (k_{0} r)],

with C_n representing the amplitude of the reflected n-th order wave. When k₀ is large enough, employing the large argument approximations of the Hankel functions and the Bessel function leads to

J_{n} (k_{0} r_{0}) [H_{n}^{(1)} (k_{0} r) + C_{n} H_{n}^{(2)} (k_{0} r)] \approx \frac{2 cos (k_{0} r_{0} - β_{n})}{k_{0} π \sqrt{r r_{0}}} [e^{i (k_{0} r - β_{n})} + C_{n} e^{- i (k_{0} r - β_{n})}],

with β_n = (2n + 1)π/4. Again, we asymptotically match the above equation with

f_{n} (r) / \sqrt{r}

from Eq. (16), and finally achieve

C_{n} \sim e^{- i [(n + 1) π + 2 γ_{n}]} = {(- 1)}^{n + 1} e^{- 2 i γ_{n}}, A_{n} \sim \frac{2 s_{0}^{1 / 4}}{k_{0} π \sqrt{r_{0}}} cos (k_{0} r_{0} - β_{n}) .

As mentioned, an inside-out Eaton lens will convert light rays emitted from a source at position r₀, with r₀ < 1, to the opposite location −r₀ which results in a aberrationless image. To extend this property to waves, it is required that

C_{n} = {(- 1)}^{n + 1} e^{- i ϕ},

with ϕ being the phase difference between the source and image. This relation can be easily obtained by time reversing the source radiation process [27]. Comparing Eq. (21) with Eq. (20), we obtain the following necessary condition for perfect imaging by an inside-out Eaton lens: e^i2γ_n must be constant and independent of the mode order n. The function e^i2γ_n can then be employed to partially evaluate the performance of a lens. For instance, we calculate the phase factor γ_n by using Eq. (14), and the results are shown in Fig. 2(b). Evidently, the values of γ_n are almost constant when λ is close to zero (the ray-optics region), while they vary strongly when λ is on the order of the lens size.

Note that there are many ways Eq. (7) can be solved [28, 29]. For example, we can take $f_{n} (r) \to ξ_{r}^{- 1 / 2} g_{n} [ξ (r)]$ , with ξ_r = dξ/dr and ξ(r) is a solution of the equation [29]

κ_{0}^{2} (ξ) ξ_{r}^{2} - κ^{2} (r) = \frac{1}{4 r^{2}} - \frac{1}{4 ξ^{2}} ξ_{r}^{2} + ξ_{r}^{1 / 2} \frac{d^{2}}{d r^{2}} ξ_{r}^{- 1 / 2},

where

κ^{2} (r) = k_{0}^{2} s (r) - \frac{1}{4 r^{2}} = \frac{2 - r}{r} k_{0}^{2} - \frac{n^{2}}{r^{2}} - \frac{r + 1}{r^{2} {(2 - r)}^{2}},

and

κ_{0}^{2} (ξ)

satisfies the following equation

\frac{d^{2}}{d ξ^{2}} g_{n} (ξ) + [κ_{0}^{2} (ξ) + \frac{1}{4 ξ^{2}}] g_{n} (ξ) = 0 .

One possible example is r = e^ξ as well as g_n(ξ) = e^S(ξ), as proposed by Langer [30]. A different approach is discussed in [29] by choosing g_n(ξ) as the Riccati-Bessel functions. When the right hand side of Eq. (22) is negligible, we can obtain

f_{n} (r) = i D_{n} ξ_{r}^{- 1 / 2} \sqrt{ξ} e^{- i ϕ_{n}} [H_{n}^{(1)} (ξ) - H_{n}^{(2)} (ξ) e^{2 i ϕ_{n}}]

for r smaller than the turning points r_n. Here ξ(r) is determined by an implicit equation

\int_{r_{t}}^{r} κ (r) d r = \sqrt{ξ^{2} - n^{2}} - n {tan}^{- 1} \sqrt{(ξ^{2} - n^{2}) / n^{2}},

with boundary condition ξ(r = 1) = k₀, and r_t is determined by ξ(r_t) = n. The phase factor ϕ_n is given by

ϕ_{n} = \int_{r_{t}}^{r_{n}} κ (r) d r .

By comparing Eq. (25) with Eq. (18), we find that

D_{n} = - \frac{1}{4} J_{n} (k_{0} r_{0}) e^{i ϕ_{n}} {\sqrt{ξ_{r} / ξ} |}_{r = 1}, C_{n} = - e^{2 i ϕ_{n}} .

Again, we can obtain a similar necessary condition for perfect imaging by an inside-out Eaton lens: e^i(2ϕ_n+nπ) must be constant and independent of the mode order n. It should be emphasized that for the design presented below our theory only provides a qualitative interpretation. This is because the metallic wire-based design deviates from the ideal Eaton lens in two ways; 1) due to the limitations of effective medium theory, and 2) because of material losses.

In the following, we present a conceptual design consisting of a vacuum (background) and ideal metallic wires. First, the Maxwell-Garnett formula is used to approximate the effective permittivity of the composite (See Appendix B)

ɛ_{e} = \frac{(1 - f) + (1 + f) ɛ_{m}}{(1 - f) ɛ_{m} + (1 + f)},

with f being the filling fraction of the wires and ɛ_m being the permittivity of the ideal metal [27, 31]. Since ɛ_e should be equal to the permittivity of the Eaton lens, (2 − r)/r, the filling fraction is then given by

f (r) = (r - 1) \frac{1 + ɛ_{m}}{1 - ɛ_{m}} .

Notice that the first-order surface mode will be excited when ɛ_m = −1 [31]. The designed lens, shown in Fig. 3(a), consists of 8 layers of metallic wires with different radii. The above equation is then used to determine the positions as well as the radii of these wires. The thickness of the p-th layer a_p₊₁ − a_p is set to be a_pπ/30, with a_p and a_p₊₁ being the inner and outer radius of the layer, respectively. Consequently we have

a_{p} = a_{1} {(1 + π / 30)}^{p - 1},

where a₁ is assumed to be 0.95. Furthermore the following relation

(a_{p} - 1) \frac{1 + ɛ_{m}}{1 - ɛ_{m}} \approx \frac{π r_{p}^{2}}{a_{p}^{2} {(π / 30)}^{2}},

is employed to calculate the radius r_p of the wire in the p-th layer. In the current design, ɛ_m = −0.6 + iδ with δ being very small, which leads to the wire radii shown in Fig. 3(b). Note that around 3 eV, the real part of permittivity of gold or silver is about −0.6. Evidently, the wire radius increases rapidly, with a minimum of 0.007 at the inner boundary and a maximum of 0.06 at the outer boundary.

Fig. 3 (a) A schematic of the design. (b) The radius of the metallic wire as a function of its position. Here the permittivity of the metal is ɛ_m = −0.6.

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We employ a finite-element full-wave Maxwell solver to verify the performance of the above design [32], and plot the results with δ = 0.01 in Fig. 4. In this case we set λ = 0.3, a value much smaller than the lens size while large enough to ensure the validity of Eq. (29). Notice that the layer thickness is 0.1 and 0.2 at the inner and outer boundaries, respectively. Therefore it is questionable to apply Eq. (29) at the outer layers, but because the higher-order modes do not propagate to the outer layer the inability to apply Eq. (29) in the outer regions does not degrade the performance of our design seriously. Clearly, an image is always observed at the location opposite to the source. To evaluate the quality of the image, we define

η = {| \frac{H - (r_{0})}{H (r_{0})} |}^{2},

which is a measure of the intensity of the image relative to that of the source. The larger the value of η, the better the lens performance. In Fig. 4, the source location r₀ is gradually increased from 0.2 to 0.5, with an increment of 0.1. The corresponding value of η is found to be 0.21, 0.27, 0.15 and 0.15, respectively. Along the azimuthal direction, we observe a few intensity minima, where the total number of these minima depends on the source location r₀. This phenomenon is very likely induced by the finite wavelength λ. One direct consequence is that different modes radiated from the source have quite different amplitudes, as indicated by J_n(k₀r₀) of Eq. (17). For example, the zeroth-order and third-order modes dominate the radiation when r₀ = 0.2, while the ninth-order mode is the strongest one when r₀ = 0.5. We further investigate the influence of the metallic absorption in Fig. 5, by setting r₀ = 0.2 and λ = 0.3. Two different values of δ, 0.1 and 0.5, are considered, and the corresponding intensity ratio η is found to be 0.11 and 0.06, respectively. Although the image quality is degraded with the increasing of the metallic absorption, the basic function of the Eaton lens, i.e. forming an image, is preserved. We can partially interpret this by the fact that all the higher-order modes, such as n > 25, are totally reflected around r = 1. Only the low-order modes can propagate into the lens and hence be absorbed by the metal.

Fig. 4 Full-wave simulations of the design, with the excitation source placed at different locations. r₀ equals (a) 0.2, (b) 0.3, (c) 0.4, and (d) 0.5. The wavelength is fixed at 0.3, and the permittivity of metal is ɛ_m = −0.6 + 0.01i. The amplitude of the magnetic field |H| is plotted.

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Fig. 5 The effect of metallic loss δ on the lens performance. The source is located at r₀ = 0.2. The wavelength λ is set to be 0.3 and Re(ɛ_m) = −0.6.

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It should be emphasized that a physical detector or physical drain influences the lens performance significantly. For example, Leonhardt found that a perfect image can be formed by Maxwell’s fish eye, but only when the image is detected (see Ref. [13] and the related comments and replies). In the simulations above we do not include a detector, and it is found that our design does not form a subwavelength image. Although it is likely that an appropriate detector may improve the image quality considerably, it is however beyond the scope of this paper.

To summarize, we have studied two-dimensional inside-out Eaton lenses. The corresponding full wave equation is analytically solved with the help of the WKB approximation. One necessary condition for perfect imaging is further found, i.e, e^i2γ_n or e^i(2ϕ_n+nπ) must be independent of the mode order n. Furthermore, a general design procedure for the lens, based on effective medium theory, is developed. We present one example consisting of metal wires with different radii, and further verify the design with a full-wave Maxwell solver. Its dependence on source location as well as metallic absorption is also investigated.

A. A ray-optics theory of the Eaton lens

A detailed description regarding the ray-optics theory of spherically-symmetric media can be found in Reference [6]. This theory is specifically used to treat the Eaton lens here for the reader’s convenience.

In geometric optics, there are two different but equivalent ways to describe the trajectory of a light ray. The first one is the Newtonian Euler-Lagrange equation

\frac{d^{2} r}{d ξ^{2}} = \frac{\nabla n^{2} (r)}{2},

where n is the refractive index and the parameter ξ is given by dξ = dr/n. We can interpret the above equation by using Newton’s law, ma = −∇U, for a mechanical particle with unit mass moving at ”time” ξ under the influence of potential U = −n²/2 + E, with E being an arbitrary constant. The second way is based on Hamilton’s equation

\frac{d r}{d t} = \frac{c}{n} \frac{k}{k}, \frac{d k}{d t} = \frac{c k}{n^{2}} \nabla n (r),

with k being the wave vector and c being the speed of light in free space. Notice that by treating frequency ω = ck/n as the Hamiltonian, the above equation resembles the standard form of Hamilton’s equation.

We can define an angular momentum as

L = r \times \frac{d r}{d ξ} = \frac{n}{k} r \times k,

which leads to

\frac{d L}{d ξ} = r \times \frac{d^{2} r}{d^{2} ξ} = \frac{1}{2} r \times \nabla n^{2} = \frac{d n^{2}}{d r} \frac{r \times r}{2 r} = 0,

when the refractive-index profile n(r) is spherically symmetric. The above equation suggests that the angular momentum L is conserved. Hence, a family of light rays propagating in the xy plane at the beginning will always stay in the same plane. This fact implies that a two-dimensional Eaton lens with similar refractive-index profile n(r) functions identically to the three-dimensional version.

To solve the two-dimensional Newtonian Euler-Lagrange equation, it is convenient to introduce the complex number z = x + iy, and further reformulate the equation as

\frac{d^{2} z}{d ξ^{2}} = \frac{z}{2 r} \frac{d n^{2}}{d r} = - \frac{1}{r^{3}} z = - \frac{1}{{| z |}^{3}} z,

by substituting the refractive index of the Eaton lens

n (r) = \sqrt{(2 - r) / r}

. The solution, following Eq. (6.13) and (6.14) of Reference [6], can be expressed as

z = e^{i α} [cos (2 ξ^{'}) + i sin γ sin (2 ξ^{'}) + cos γ], d ξ = 2 | z | d ξ^{'},

which describes displaced ellipses rotated by the angle α.

B. Maxwell-Garnett formula

Consider a two-component mixture composed of inclusions embedded in an otherwise homogeneous matrix, where ɛ_m and ɛ_d are their respective dielectric functions. The average electric field 〈E〉 over one unit area surrounding the point x is defined as

〈 E (x) 〉 = \frac{1}{A} \int_{A} E (x^{'}) d x^{'} = f 〈 E_{m} (x) 〉 + (1 - f) 〈 E_{d} (x) 〉,

with f being the volume fraction of inclusions. A similar expression can be obtained for the average polarization

〈 P (x) 〉 = f 〈 P_{m} (x) 〉 + (1 - f) 〈 P_{d} (x) 〉 .

We further assume that the following constitutive relations are valid

〈 P_{m} (x) 〉 = ɛ_{0} (ɛ_{m} - 1) 〈 E_{m} (x) 〉, 〈 P_{d} (x) 〉 = ɛ_{0} (ɛ_{d} - 1) 〈 E_{d} (x) 〉,

and the average permittivity tensor of the composite medium is defined by

〈 P (x) 〉 = ɛ_{0} ({\bar{ɛ}}_{e} - \bar{I}) \cdot 〈 E (x) 〉 .

Combining the above equations we can obtain the effective permittivity ɛ̄_e. Clearly the resultant ɛ̄_e depends on the relationship between 〈E_m(x)〉 and 〈E_d(x)〉 [33].

We now assume that the inclusion has the shape of a cylinder, and its radius is far smaller than the wavelength so that its optical properties can be well described by the electrostatic equation

\nabla \cdot (ɛ (r) ϕ) = 0 .

By matching the boundary conditions we can prove that ϕ_m/ϕ₀ = 2ɛ_d/(ɛ_d + ɛ_m), where ϕ_m is the total potential inside the cylinder when the external electric field −∇ϕ₀ is homogeneous. This relation is further used to obtain the electric field [34]. It is finally found that the average permittivity is scalar and can be expressed as

ɛ_{e} = ɛ_{d} \frac{(1 - f) ɛ_{d} + (1 + f) ɛ_{m}}{(1 - f) ɛ_{m} + (1 + f) ɛ_{d}},

which is consistent with the Maxwell-Garnett dielectric function. Equivalently we can express the filling fraction as

f (r) = \frac{(ɛ_{e} - ɛ_{d}) (ɛ_{m} + ɛ_{d})}{(ɛ_{e} + ɛ_{d}) (ɛ_{m} - ɛ_{d})} .

We greatly appreciate the valuable suggestions of the referees. This work was supported in part by the Penn State MRSEC under NSF grant No. DMR 0213623.

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Two-dimensional inside-out Eaton Lens: Design technique and TM-polarized wave properties

Abstract

A. A ray-optics theory of the Eaton lens

B. Maxwell-Garnett formula

References and links

Cited By

Figures (5)

Equations (46)

Optics Express