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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 3 — Jan. 30, 2012
  • pp: 2335–2345
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Two-dimensional inside-out Eaton Lens: Design technique and TM-polarized wave properties

Yong Zeng and Douglas H. Werner  »View Author Affiliations


Optics Express, Vol. 20, Issue 3, pp. 2335-2345 (2012)
http://dx.doi.org/10.1364/OE.20.002335


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Abstract

In this paper we perform a theoretical and numerical study of two-dimensional inside-out Eaton lenses under transverse-magnetic-polarized excitation. We present one example design and test its performance by utilizing full-wave Maxwell solvers. With the help of the WKB approximation, we further investigate the finite-wavelength effect analytically and demonstrate one necessary condition for perfect imaging at the level of wave optics, i.e. imaging with unlimited resolution, by the lens.

© 2012 OSA

More than half a century ago, it was proposed that gradient index lenses [1

1. C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

], such as the Maxwell fish-eye lens [2

2. J. C. Maxwell, “Solutions of problems,” Camb. Dublin Math. J. 8, 188 (1854).

], the Luneburg lens [3

3. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

] and the Eaton lens [4

4. J. E. Eaton, “On spherically symmetric lenses,” Trans. IRE Antennas Propag. 4, 66–71 (1952).

], can be free of geometrical aberrations and form sharp images, at least at the geometrical-optics level (see Reference [5

5. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).

] and [6

6. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

] for more details). Taking the inside-out Eaton lens as an example, which was proposed by Miñano in 2006 [7

7. J. C. Miñano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express 14, 9627–9635 (2006). [PubMed]

], its refractive index n(r) equals (2a/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

6. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

, 7

7. J. C. Miñano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express 14, 9627–9635 (2006). [PubMed]

]. 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

8. Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Materials 8, 639–642 (2009).

17

17. J. A. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. II. Wave theory,” J. Opt. Soc. Am. A 25, 2980–2990 (2008).

], partially because of the developments in metamaterials [18

18. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [PubMed]

, 19

19. L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, 2009).

] and transformation optics [20

20. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [PubMed]

23

23. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Materials 9, 387–396 (2010).

]. Metamaterials are manmade media whose effective permittivities and permeabilities are governed by their deeply subwavelength structures as well as their constituent materials [19

19. L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, 2009).

]. For instance, properly integrating split ring resonators with metallic rods will result in a metamaterial with an effectively negative index of refraction [24

24. 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). [PubMed]

]. 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

20. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [PubMed]

, 21

21. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [PubMed]

].

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).

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

25. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).

,26

26. J. J. Sakurai, Modern Quantum Mechanics, Revised ed. (Addison-Wesley, 1994).

], 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

4. J. E. Eaton, “On spherically symmetric lenses,” Trans. IRE Antennas Propag. 4, 66–71 (1952).

]
n(r)=2ar1.
(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

6. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

] (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

13. U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).

], 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

6. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

], 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
1ɛ(r)2H+1ɛ(r)H=ω2c2H,
(2)
with ∇ = xex + yey and H as Hz, this equation can be reformulated in cylindrical coordinates as
2Hr2+(1r1ɛdɛdr)Hr+1r22Hθ2+k02ɛH=0,
(3)
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 nfn(r)einθɛ/r, where the function fn satisfies
fn+(k02ɛn2r2+ɛ23r2ɛ2+2rɛɛ+2r2ɛɛ4r2ɛ2)fn=0.
(4)
In the region r ≤ 1 where ɛ(r) = 1, this reduces to
fn+(k02n21/4r2)fn=0,
(5)
which has the general solutions
fn(r)=r[anHn(1)(k0r)+bnHn(2)(k0r)],
(6)
where Hn(1) and Hn(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
fn+s(r)k02fn=0,
(7)
with
s(r)=2rrn21/4r2k02r+1r2(2r)2k02.
(8)
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 γn.

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

25. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).

,26

26. J. J. Sakurai, Modern Quantum Mechanics, Revised ed. (Addison-Wesley, 1994).

]. More specifically, we assume that fn(r) has the form Aneik0τ(r) for positive s(r), and τ(r) can be further expanded in terms of k0,
τ(r)=τ0(r)+1k0τ1(r)+1k02τ2(r)+.
(9)
Similar arguments also hold for τ′ (r) as well as τ″ (r). By collecting the leading-order terms, it is found that
(dτ0dr)2=s(r),dτ1dr=iτ02τ0.
(10)
Consequently the first order solution can be expressed as
fn(r)An+s(r)1/4exp[ik0rnrs(r)dr]+Ans(r)1/4exp[ik0rnrs(r)dr],
(11)
with rn representing the turning point where s(r) = 0. It should be mentioned that rn depends on the mode order n: the larger the n, the smaller the turning point rn. 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 fn(r) = Bnek0τ(r), such that the resultant first-order approximation is given by
fn(r)Bn+|s(r)|1/4exp[k0rnr|s(r)|dr].
(12)
Here only the solution that is exponentially decaying in the r direction is included. Furthermore, s(r) ∼ (rnr) approaches zero linearly in the vicinity of the turning point rn. The solution therefore can be approximated as Ai(k02/3s), with Ai being the Airy function [25

25. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).

]. When k0 is large enough, we can asymptotically match Eq. (11) and (12) around the turning point, and finally achieve An+=iAn. As a direct result, in the region where s(r) is positive, we have
fn(r)Ans(r)1/4exp[ik0rnrs(r)dr]+Ans(r)1/4exp[iπ2ik0rnrs(r)dr],
(13)
which implies that an out-going wave will be totally reflected around the turning point rn, 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
k0rnrs(r)dr=k0rnrs0dr+k0rn1[s(r)s0]drk0r+γn,
(14)
where γn does not depend on r, and
k02s0=k02n21/4r2
(15)
represents the coefficient shown in Eq. (5). In the vicinity of r = 1, fn(r) can be rewritten in the form
fn(r)Ans01/4[ei(k0r+γn)+ei(k0r+γn+π/2)].
(16)
Coincidentally, a source at position r0 with r0 < 1, i.e. inside the Eaton lens, generates radiation according to
i4H0(1)(k0|rr0|)=i4Jn(k0r<)Hn(1)(k0r>)einϕ,
(17)
where r< = min{r,r0}, and r> = max{r,r0} [25

25. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).

, 27

27. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2004).

]. The total magnetic field in the region between r0 and 1 is therefore given by
i4einϕJn(k0r0)[Hn(1)(k0r)+CnHn(2)(k0r)],
(18)
with Cn 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
Jn(k0r0)[Hn(1)(k0r)+CnHn(2)(k0r)]2cos(k0r0βn)k0πrr0[ei(k0rβn)+Cnei(k0rβn)],
(19)
with βn = (2n + 1)π/4. Again, we asymptotically match the above equation with fn(r)/r from Eq. (16), and finally achieve
Cnei[(n+1)π+2γn]=(1)n+1e2iγn,An2s01/4k0πr0cos(k0r0βn).
(20)

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
Cn=(1)n+1eiϕ,
(21)
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

27. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2004).

]. Comparing Eq. (21) with Eq. (20), we obtain the following necessary condition for perfect imaging by an inside-out Eaton lens: ei2γn must be constant and independent of the mode order n. The function ei2γn can then be employed to partially evaluate the performance of a lens. For instance, we calculate the phase factor γn by using Eq. (14), and the results are shown in Fig. 2(b). Evidently, the values of γn 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

28. R. H. Good Jr., “The generalization of the WKB method to radial wave equations,” Phys. Rev. 90, 131–137 (1953).

, 29

29. B. Durand and L. Durand, “Improved WKB radial wave functions in several bases,” Phys. Rev. A 33, 2887–2898 (1986). [PubMed]

]. For example, we can take fn(r)ξr1/2gn[ξ(r)], with ξr = dξ/dr and ξ(r) is a solution of the equation [29

29. B. Durand and L. Durand, “Improved WKB radial wave functions in several bases,” Phys. Rev. A 33, 2887–2898 (1986). [PubMed]

]
κ02(ξ)ξr2κ2(r)=14r214ξ2ξr2+ξr1/2d2dr2ξr1/2,
(22)
where
κ2(r)=k02s(r)14r2=2rrk02n2r2r+1r2(2r)2,
(23)
and κ02(ξ) satisfies the following equation
d2dξ2gn(ξ)+[κ02(ξ)+14ξ2]gn(ξ)=0.
(24)
One possible example is r = eξ as well as gn(ξ) = eS(ξ), as proposed by Langer [30

30. R. E. Langer, “On the connection formulas and the solutions of the wave equation,” Phys. Rev. 51, 669–676 (1937).

]. A different approach is discussed in [29

29. B. Durand and L. Durand, “Improved WKB radial wave functions in several bases,” Phys. Rev. A 33, 2887–2898 (1986). [PubMed]

] by choosing gn(ξ) as the Riccati-Bessel functions. When the right hand side of Eq. (22) is negligible, we can obtain
fn(r)=iDnξr1/2ξeiϕn[Hn(1)(ξ)Hn(2)(ξ)e2iϕn]
(25)
for r smaller than the turning points rn. Here ξ(r) is determined by an implicit equation
rtrκ(r)dr=ξ2n2ntan1(ξ2n2)/n2,
(26)
with boundary condition ξ(r = 1) = k0, and rt is determined by ξ(rt) = n. The phase factor ϕn is given by
ϕn=rtrnκ(r)dr.
(27)
By comparing Eq. (25) with Eq. (18), we find that
Dn=14Jn(k0r0)eiϕnξr/ξ|r=1,Cn=e2iϕn.
(28)
Again, we can obtain a similar necessary condition for perfect imaging by an inside-out Eaton lens: ei(2ϕn+) 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)
ɛe=(1f)+(1+f)ɛm(1f)ɛm+(1+f),
(29)
with f being the filling fraction of the wires and ɛm being the permittivity of the ideal metal [27

27. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2004).

, 31

31. Y. Zeng, Q. Wu, and D. H. Werner, “Electrostatic theory for designing lossless negative permittivity metamaterials,” Opt. Lett. 35, 1431–1433 (2010). [PubMed]

]. Since ɛe should be equal to the permittivity of the Eaton lens, (2 − r)/r, the filling fraction is then given by
f(r)=(r1)1+ɛm1ɛm.
(30)
Notice that the first-order surface mode will be excited when ɛm = −1 [31

31. Y. Zeng, Q. Wu, and D. H. Werner, “Electrostatic theory for designing lossless negative permittivity metamaterials,” Opt. Lett. 35, 1431–1433 (2010). [PubMed]

]. 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 ap+1ap is set to be apπ/30, with ap and ap+1 being the inner and outer radius of the layer, respectively. Consequently we have
ap=a1(1+π/30)p1,
(31)
where a1 is assumed to be 0.95. Furthermore the following relation
(ap1)1+ɛm1ɛmπrp2ap2(π/30)2,
(32)
is employed to calculate the radius rp of the wire in the p-th layer. In the current design, ɛm = −0.6 + 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 ɛm = −0.6.

We employ a finite-element full-wave Maxwell solver to verify the performance of the above design [32

32. COMSOL, www.comsol.com.

], 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=|H(r0)H(r0)|2,
(33)
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 Jn(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 ɛm = −0.6 + 0.01i. The amplitude of the magnetic field |H| is plotted.
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(ɛm) = −0.6.

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, ei2γn or ei(2ϕn+) 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

6. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

]. 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
d2rdξ2=n2(r)2,
(A.1)
where n is the refractive index and the parameter ξ is given by = 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
drdt=cnkk,dkdt=ckn2n(r),
(A.2)
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×drdξ=nkr×k,
(A.3)
which leads to
dLdξ=r×d2rd2ξ=12r×n2=dn2drr×r2r=0,
(A.4)
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
d2zdξ2=z2rdn2dr=1r3z=1|z|3z,
(A.5)
by substituting the refractive index of the Eaton lens n(r)=(2r)/r. The solution, following Eq. (6.13) and (6.14) of Reference [6

6. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

], can be expressed as
z=eiα[cos(2ξ)+isinγsin(2ξ)+cosγ],dξ=2|z|dξ,
(A.6)
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 ɛm and ɛd are their respective dielectric functions. The average electric field 〈E〉 over one unit area surrounding the point x is defined as
E(x)=1AAE(x)dx=fEm(x)+(1f)Ed(x),
(B.1)
with f being the volume fraction of inclusions. A similar expression can be obtained for the average polarization
P(x)=fPm(x)+(1f)Pd(x).
(B.2)
We further assume that the following constitutive relations are valid
Pm(x)=ɛ0(ɛm1)Em(x),Pd(x)=ɛ0(ɛd1)Ed(x),
(B.3)
and the average permittivity tensor of the composite medium is defined by
P(x)=ɛ0(ɛ¯eI¯)E(x).
(B.4)
Combining the above equations we can obtain the effective permittivity ɛ̄e. Clearly the resultant ɛ̄e depends on the relationship between 〈Em(x)〉 and 〈Ed(x)〉 [33

33. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998).

].

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
(ɛ(r)ϕ)=0.
(B.5)
By matching the boundary conditions we can prove that ϕm/ϕ0 = 2ɛd/(ɛd + ɛm), where ϕm 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

34. Y. Zeng, J. Liu, and D. H. Werner, “General properties of two-dimensional conformal transformations in electrostatics,” Opt. Express 19, 20035–20047 (2011). [PubMed]

]. It is finally found that the average permittivity is scalar and can be expressed as
ɛe=ɛd(1f)ɛd+(1+f)ɛm(1f)ɛm+(1+f)ɛd,
(B.6)
which is consistent with the Maxwell-Garnett dielectric function. Equivalently we can express the filling fraction as
f(r)=(ɛeɛd)(ɛm+ɛd)(ɛe+ɛd)(ɛmɛd).
(B.7)

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.

References and links

1.

C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).

2.

J. C. Maxwell, “Solutions of problems,” Camb. Dublin Math. J. 8, 188 (1854).

3.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

4.

J. E. Eaton, “On spherically symmetric lenses,” Trans. IRE Antennas Propag. 4, 66–71 (1952).

5.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53, 69–152 (2009).

6.

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).

7.

J. C. Miñano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express 14, 9627–9635 (2006). [PubMed]

8.

Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Materials 8, 639–642 (2009).

9.

N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Materials 9, 129–132 (2010).

10.

V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Maxwell fish-eye and Eaton lenses emulated by microdroplets,” Opt. Lett. 35, 3396–3398 (2010). [PubMed]

11.

D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express 18, 21238–21251 (2010). [PubMed]

12.

T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnology 6, 151–155 (2011).

13.

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).

14.

A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express 19, 5156–5162 (2011). [PubMed]

15.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011). [PubMed]

16.

A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photonics 5, 357–359 (2011).

17.

J. A. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. II. Wave theory,” J. Opt. Soc. Am. A 25, 2980–2990 (2008).

18.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [PubMed]

19.

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, 2009).

20.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [PubMed]

21.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [PubMed]

22.

D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: An overview of the theory and its application,” IEEE Antennas Prop. Mag. 52, 24–45 (2010).

23.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Materials 9, 387–396 (2010).

24.

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). [PubMed]

25.

W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).

26.

J. J. Sakurai, Modern Quantum Mechanics, Revised ed. (Addison-Wesley, 1994).

27.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2004).

28.

R. H. Good Jr., “The generalization of the WKB method to radial wave equations,” Phys. Rev. 90, 131–137 (1953).

29.

B. Durand and L. Durand, “Improved WKB radial wave functions in several bases,” Phys. Rev. A 33, 2887–2898 (1986). [PubMed]

30.

R. E. Langer, “On the connection formulas and the solutions of the wave equation,” Phys. Rev. 51, 669–676 (1937).

31.

Y. Zeng, Q. Wu, and D. H. Werner, “Electrostatic theory for designing lossless negative permittivity metamaterials,” Opt. Lett. 35, 1431–1433 (2010). [PubMed]

32.

COMSOL, www.comsol.com.

33.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998).

34.

Y. Zeng, J. Liu, and D. H. Werner, “General properties of two-dimensional conformal transformations in electrostatics,” Opt. Express 19, 20035–20047 (2011). [PubMed]

OCIS Codes
(000.3860) General : Mathematical methods in physics
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Physical Optics

History
Original Manuscript: November 9, 2011
Revised Manuscript: December 5, 2011
Manuscript Accepted: January 3, 2012
Published: January 18, 2012

Citation
Yong Zeng and Douglas H. Werner, "Two-dimensional inside-out Eaton Lens: Design technique and TM-polarized wave properties," Opt. Express 20, 2335-2345 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2335


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References

  1. C. Gomez-Reino, M. V. Perez, and C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer, 2002).
  2. J. C. Maxwell, “Solutions of problems,” Camb. Dublin Math. J.8, 188 (1854).
  3. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).
  4. J. E. Eaton, “On spherically symmetric lenses,” Trans. IRE Antennas Propag.4, 66–71 (1952).
  5. U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt.53, 69–152 (2009).
  6. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover Publications, 2010).
  7. J. C. Miñano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express14, 9627–9635 (2006). [PubMed]
  8. Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Materials8, 639–642 (2009).
  9. N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Materials9, 129–132 (2010).
  10. V. N. Smolyaninova, I. I. Smolyaninov, A. V. Kildishev, and V. M. Shalaev, “Maxwell fish-eye and Eaton lenses emulated by microdroplets,” Opt. Lett.35, 3396–3398 (2010). [PubMed]
  11. D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express18, 21238–21251 (2010). [PubMed]
  12. T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnology6, 151–155 (2011).
  13. U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys.11, 093040 (2009).
  14. A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express19, 5156–5162 (2011). [PubMed]
  15. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science332, 1291–1294 (2011). [PubMed]
  16. A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photonics5, 357–359 (2011).
  17. J. A. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. II. Wave theory,” J. Opt. Soc. Am. A25, 2980–2990 (2008).
  18. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett.85, 3966–3969 (2000). [PubMed]
  19. L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, 2009).
  20. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006). [PubMed]
  21. U. Leonhardt, “Optical conformal mapping,” Science312, 1777–1780 (2006). [PubMed]
  22. D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: An overview of the theory and its application,” IEEE Antennas Prop. Mag.52, 24–45 (2010).
  23. H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Materials9, 387–396 (2010).
  24. 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). [PubMed]
  25. W. C. Chew, Waves and Fields in Inhomogeneous Media (IEEE Press, 1995).
  26. J. J. Sakurai, Modern Quantum Mechanics, Revised ed. (Addison-Wesley, 1994).
  27. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, 2004).
  28. R. H. Good, “The generalization of the WKB method to radial wave equations,” Phys. Rev.90, 131–137 (1953).
  29. B. Durand and L. Durand, “Improved WKB radial wave functions in several bases,” Phys. Rev. A33, 2887–2898 (1986). [PubMed]
  30. R. E. Langer, “On the connection formulas and the solutions of the wave equation,” Phys. Rev.51, 669–676 (1937).
  31. Y. Zeng, Q. Wu, and D. H. Werner, “Electrostatic theory for designing lossless negative permittivity metamaterials,” Opt. Lett.35, 1431–1433 (2010). [PubMed]
  32. COMSOL, www.comsol.com .
  33. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998).
  34. Y. Zeng, J. Liu, and D. H. Werner, “General properties of two-dimensional conformal transformations in electrostatics,” Opt. Express19, 20035–20047 (2011). [PubMed]

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