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

Optics Express, Vol. 20, Issue 3, pp. 2335-2345 (2012)

http://dx.doi.org/10.1364/OE.20.002335

Acrobat PDF (1223 KB)

### 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

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

*n*(

*r*) equals

*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

**r**, with

_{0}*r*

_{0}/

*a*< 1, will be focused exactly at position −

**r**[6, 7

_{0}7. J. C. Miñano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express **14**, 9627–9635 (2006). [PubMed]

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

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

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]

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

*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

*n*

^{2}(

*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.

*∂*

_{x}**e**

*+*

_{x}*∂*

_{y}**e**

*and*

_{y}*H*as

*H*, this equation can be reformulated in cylindrical coordinates as with

_{z}*k*

_{0}=

*ω*/

*c*being the wave number in free space. We now assume that the magnetic field

*H*(

*r*,

*θ*) can be expanded as

*f*satisfies In the region

_{n}*r*≤ 1 where

*ɛ*(

*r*) = 1, this reduces to which has the general solutions where

*n*-th order Hankel functions of the first and second kind, respectively. In the region 1 <

*r*< 2, we can rewrite Eq. (4) as with 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.

*n*[25,26]. More specifically, we assume that

*f*(

_{n}*r*) has the form

*A*

_{n}e^{ik0τ(r)}for positive

*s*(

*r*), and

*τ*(

*r*) can be further expanded in terms of

*k*

_{0}, Similar arguments also hold for

*τ*′ (

*r*) as well as

*τ*″ (

*r*). By collecting the leading-order terms, it is found that Consequently the first order solution can be expressed as with

*r*representing the turning point where

_{n}*s*(

*r*) = 0. It should be mentioned that

*r*depends on the mode order

_{n}*n*: the larger the

*n*, the smaller the turning point

*r*. 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

_{n}*s*(

*r*) by assuming

*f*(

_{n}*r*) =

*B*

_{n}e^{−k0τ(r)}, such that the resultant first-order approximation is given by 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*. The solution therefore can be approximated as

_{n}*k*

_{0}is large enough, we can asymptotically match Eq. (11) and (12) around the turning point, and finally achieve

*s*(

*r*) is positive, we have which implies that an out-going wave will be totally reflected around the turning point

*r*, accompanied by a phase variation of

_{n}*π*/2.

*r*is slightly smaller than 1. Additionally, its phase factor can be approximated as where

*γ*does not depend on

_{n}*r*, and represents the coefficient shown in Eq. (5). In the vicinity of

*r*= 1,

*f*(

_{n}*r*) can be rewritten in the form Coincidentally, a source at position

**r**

_{0}with

*r*

_{0}< 1,

*i.e.*inside the Eaton lens, generates radiation according to where

*r*

_{<}= min{

*r,r*

_{0}}, and

*r*

_{>}= max{

*r,r*

_{0}} [25, 27]. The total magnetic field in the region between

*r*

_{0}and 1 is therefore given by with

*C*representing the amplitude of the reflected

_{n}*n*-th order wave. When

*k*

_{0}is large enough, employing the large argument approximations of the Hankel functions and the Bessel function leads to with

*β*= (2

_{n}*n*+ 1)

*π*/4. Again, we asymptotically match the above equation with

**r**, with

_{0}*r*

_{0}< 1, to the opposite location −

**r**which results in a aberrationless image. To extend this property to waves, it is required that with

_{0}*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*

^{i2γn}

*must be constant and independent of the mode order n*. The function

*e*

^{i2γn}can then be employed to partially evaluate the performance of a lens. For instance, we calculate the phase factor

*γ*by using Eq. (14), and the results are shown in Fig. 2(b). Evidently, the values of

_{n}*γ*are almost constant when

_{n}*λ*is close to zero (the ray-optics region), while they vary strongly when

*λ*is on the order of the lens size.

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

*ξ*=

_{r}*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]

*r*=

*e*as well as

^{ξ}*g*(

_{n}*ξ*) =

*e*

^{S(ξ)}, as proposed by Langer [30]. 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]

*g*(

_{n}*ξ*) as the Riccati-Bessel functions. When the right hand side of Eq. (22) is negligible, we can obtain for

*r*smaller than the turning points

*r*. Here

_{n}*ξ*(

*r*) is determined by an implicit equation with boundary condition

*ξ*(

*r*= 1) =

*k*

_{0}, and

*r*is determined by

_{t}*ξ*(

*r*) =

_{t}*n*. The phase factor

*ϕ*is given by By comparing Eq. (25) with Eq. (18), we find that Again, we can obtain a similar necessary condition for perfect imaging by an inside-out Eaton lens:

_{n}*e*

^{i(2ϕn+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.

*f*being the filling fraction of the wires and

*ɛ*being the permittivity of the ideal metal [27, 31

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

*ɛ*should be equal to the permittivity of the Eaton lens, (2 −

_{e}*r*)/

*r*, the filling fraction is then given by Notice that the first-order surface mode will be excited when

*ɛ*= −1 [31

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

*p*-th layer

*a*

_{p}_{+1}−

*a*is set to be

_{p}*a*/30, with

_{p}π*a*and

_{p}*a*

_{p}_{+1}being the inner and outer radius of the layer, respectively. Consequently we have where

*a*

_{1}is assumed to be 0.95. Furthermore the following relation is employed to calculate the radius

*r*of the wire in the

_{p}*p*-th layer. In the current design,

*ɛ*= −0.6 +

_{m}*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.

32. COMSOL, www.comsol.com.

*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 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

*r*

_{0}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

*r*

_{0}. 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}*k*

_{0}

*r*

_{0}) of Eq. (17). For example, the zeroth-order and third-order modes dominate the radiation when

*r*

_{0}= 0.2, while the ninth-order mode is the strongest one when

*r*

_{0}= 0.5. We further investigate the influence of the metallic absorption in Fig. 5, by setting

*r*

_{0}= 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.

*i.e*,

*e*

^{i2γn}or

*e*

^{i(2ϕn+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

*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,

*m*

**a**= −∇

*U*, for a mechanical particle with unit mass moving at ”time”

*ξ*under the influence of potential

*U*= −

*n*

^{2}/2 +

*E*, with

*E*being an arbitrary constant. The second way is based on Hamilton’s equation 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.

*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.

*z*=

*x*+

*iy*, and further reformulate the equation as by substituting the refractive index of the Eaton lens

*α*.

## B. Maxwell-Garnett formula

*ɛ*and

_{m}*ɛ*are their respective dielectric functions. The average electric field 〈

_{d}**E**〉 over one unit area surrounding the point

**x**is defined as with

*f*being the volume fraction of inclusions. A similar expression can be obtained for the average polarization We further assume that the following constitutive relations are valid and the average permittivity tensor of the composite medium is defined by Combining the above equations we can obtain the effective permittivity

*ɛ̄*. Clearly the resultant

_{e}*ɛ̄*depends on the relationship between 〈

_{e}**E**

*(*

_{m}**x**)〉 and 〈

**E**

*(*

_{d}**x**)〉 [33].

*ϕ*/

_{m}*ϕ*

_{0}= 2

*ɛ*/(

_{d}*ɛ*+

_{d}*ɛ*), where

_{m}*ϕ*is the total potential inside the cylinder when the external electric field −∇

_{m}*ϕ*

_{0}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]

## References and links

1. | C. Gomez-Reino, M. V. Perez, and C. Bao, |

2. | J. C. Maxwell, “Solutions of problems,” Camb. Dublin Math. J. |

3. | R. K. Luneburg, |

4. | J. E. Eaton, “On spherically symmetric lenses,” Trans. IRE Antennas Propag. |

5. | U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. |

6. | U. Leonhardt and T. G. Philbin, |

7. | J. C. Miñano, “Perfect imaging in a homogeneous three-dimensional region,” Opt. Express |

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

9. | N. Kundtz and D. R. Smith, “Extreme-angle broadband metamaterial lens,” Nat. Materials |

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. |

11. | D. R. Smith, Y. Urzhumov, N. B. Kundtz, and N. I. Landy, “Enhancing imaging systems using transformation optics,” Opt. Express |

12. | T. Zentgraf, Y. Liu, M. H. Mikkelsen, J. Valentine, and X. Zhang, “Plasmonic Luneburg and Eaton lenses,” Nat. Nanotechnology |

13. | U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. |

14. | A. Di Falco, S. C. Kehr, and U. Leonhardt, “Luneburg lens in silicon photonics,” Opt. Express |

15. | A. Vakil and N. Engheta, “Transformation optics using graphene,” Science |

16. | A. J. Danner, T. Tyc, and U. Leonhardt, “Controlling birefringence in dielectrics,” Nat. Photonics |

17. | J. A. Lock, “Scattering of an electromagnetic plane wave by a Luneburg lens. II. Wave theory,” J. Opt. Soc. Am. A |

18. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

19. | L. Solymar and E. Shamonina, |

20. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

21. | U. Leonhardt, “Optical conformal mapping,” Science |

22. | D.-H. Kwon and D. H. Werner, “Transformation electromagnetics: An overview of the theory and its application,” IEEE Antennas Prop. Mag. |

23. | H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Materials |

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. |

25. | W. C. Chew, |

26. | J. J. Sakurai, |

27. | J. D. Jackson, |

28. | R. H. Good Jr., “The generalization of the WKB method to radial wave equations,” Phys. Rev. |

29. | B. Durand and L. Durand, “Improved WKB radial wave functions in several bases,” Phys. Rev. A |

30. | R. E. Langer, “On the connection formulas and the solutions of the wave equation,” Phys. Rev. |

31. | Y. Zeng, Q. Wu, and D. H. Werner, “Electrostatic theory for designing lossless negative permittivity metamaterials,” Opt. Lett. |

32. | COMSOL, www.comsol.com. |

33. | C. F. Bohren and D. R. Huffman, |

34. | Y. Zeng, J. Liu, and D. H. Werner, “General properties of two-dimensional conformal transformations in electrostatics,” Opt. Express |

**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

Sort: Year | Journal | Reset

### References

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

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.