Abstract

Invisibility cloaking based on optical transformation involves materials singularity at the branch cut points. Many interesting optical devices, such as the Eaton lens, also require planar media index singularities in their implementation. We show a method to transmute two singularities simultaneously into harmless topological defects formed by anisotropic permittivity and permeability tensors. Numerical simulation is performed to verify the functionality of the transmuted conformal cloak consisting of two kissing Maxwell fish eyes.

© 2013 Optical Society of America

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  1. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [CrossRef]
  2. Q. N. Wu, Y. D. Xu, H. Li, and H. Y. Chen, “Cloaking and imaging at the same time,” Euro. Phys. Lett. 101, 34004 (2013).
    [CrossRef]
  3. T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New J. Phys. 10, 115038 (2008).
    [CrossRef]
  4. Y. G. Ma, C. K. Ong, T. Tyc, and U. Leonhardt, “An omnidirectional retroreflector based on the transmutation of dielectric singularities,” Nat. Mater. 8, 639–642 (2009).
    [CrossRef]
  5. J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
    [CrossRef]
  6. J. Perczel, C. Garcia-Meca, and U. Leonhardt, “Partial transmutation of singularities in optical instruments,” J. Opt. 13, 075103 (2011).
    [CrossRef]
  7. Y. G. Ma, F. Sun, Y. Zhang, Y. Jin, and C. K. Ong, “Approaches to achieve broadband optical transformation devices with transmuted singularity,” J. Opt. Soc. Am. A 29, 124–129 (2012).
    [CrossRef]
  8. T. Xu, Y. C. Liu, C. K. Ong, Y. Zhang, and Y. G. Ma, “Perfect invisibility cloaking by isotropic media,” Phys. Rev. A 86, 043827 (2012).
    [CrossRef]
  9. H. Y. Chen, U. Leonhardt, and T. Tyc, “Conformal cloak for waves,” Phys. Rev. A 83, 055801 (2011).
    [CrossRef]
  10. N. Wang, Y. G. Ma, R. F. Huang, and C. K. Ong, “Far field free-space measurement of three dimensional hole-in-Teflon invisibility cloak,” Opt. Express 21, 5941–5948 (2013).
    [CrossRef]
  11. H. F. Ma, W. X. Jiang, X. M. Yang, X. Y. Zhou, and T. J. Cui, “Compact-sized and broadband carpet cloak and free-space cloak,” Opt. Express 17, 19947–19959 (2009).
    [CrossRef]

2013

2012

Y. G. Ma, F. Sun, Y. Zhang, Y. Jin, and C. K. Ong, “Approaches to achieve broadband optical transformation devices with transmuted singularity,” J. Opt. Soc. Am. A 29, 124–129 (2012).
[CrossRef]

T. Xu, Y. C. Liu, C. K. Ong, Y. Zhang, and Y. G. Ma, “Perfect invisibility cloaking by isotropic media,” Phys. Rev. A 86, 043827 (2012).
[CrossRef]

2011

H. Y. Chen, U. Leonhardt, and T. Tyc, “Conformal cloak for waves,” Phys. Rev. A 83, 055801 (2011).
[CrossRef]

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[CrossRef]

J. Perczel, C. Garcia-Meca, and U. Leonhardt, “Partial transmutation of singularities in optical instruments,” J. Opt. 13, 075103 (2011).
[CrossRef]

2009

H. F. Ma, W. X. Jiang, X. M. Yang, X. Y. Zhou, and T. J. Cui, “Compact-sized and broadband carpet cloak and free-space cloak,” Opt. Express 17, 19947–19959 (2009).
[CrossRef]

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

2008

T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New J. Phys. 10, 115038 (2008).
[CrossRef]

2006

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

Chen, H. Y.

Q. N. Wu, Y. D. Xu, H. Li, and H. Y. Chen, “Cloaking and imaging at the same time,” Euro. Phys. Lett. 101, 34004 (2013).
[CrossRef]

H. Y. Chen, U. Leonhardt, and T. Tyc, “Conformal cloak for waves,” Phys. Rev. A 83, 055801 (2011).
[CrossRef]

Cui, T. J.

Garcia-Meca, C.

J. Perczel, C. Garcia-Meca, and U. Leonhardt, “Partial transmutation of singularities in optical instruments,” J. Opt. 13, 075103 (2011).
[CrossRef]

Huang, R. F.

Jiang, W. X.

Jin, Y.

Leonhardt, U.

J. Perczel, C. Garcia-Meca, and U. Leonhardt, “Partial transmutation of singularities in optical instruments,” J. Opt. 13, 075103 (2011).
[CrossRef]

H. Y. Chen, U. Leonhardt, and T. Tyc, “Conformal cloak for waves,” Phys. Rev. A 83, 055801 (2011).
[CrossRef]

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[CrossRef]

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

T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New J. Phys. 10, 115038 (2008).
[CrossRef]

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

Li, H.

Q. N. Wu, Y. D. Xu, H. Li, and H. Y. Chen, “Cloaking and imaging at the same time,” Euro. Phys. Lett. 101, 34004 (2013).
[CrossRef]

Liu, Y. C.

T. Xu, Y. C. Liu, C. K. Ong, Y. Zhang, and Y. G. Ma, “Perfect invisibility cloaking by isotropic media,” Phys. Rev. A 86, 043827 (2012).
[CrossRef]

Ma, H. F.

Ma, Y. G.

N. Wang, Y. G. Ma, R. F. Huang, and C. K. Ong, “Far field free-space measurement of three dimensional hole-in-Teflon invisibility cloak,” Opt. Express 21, 5941–5948 (2013).
[CrossRef]

Y. G. Ma, F. Sun, Y. Zhang, Y. Jin, and C. K. Ong, “Approaches to achieve broadband optical transformation devices with transmuted singularity,” J. Opt. Soc. Am. A 29, 124–129 (2012).
[CrossRef]

T. Xu, Y. C. Liu, C. K. Ong, Y. Zhang, and Y. G. Ma, “Perfect invisibility cloaking by isotropic media,” Phys. Rev. A 86, 043827 (2012).
[CrossRef]

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

Ong, C. K.

N. Wang, Y. G. Ma, R. F. Huang, and C. K. Ong, “Far field free-space measurement of three dimensional hole-in-Teflon invisibility cloak,” Opt. Express 21, 5941–5948 (2013).
[CrossRef]

Y. G. Ma, F. Sun, Y. Zhang, Y. Jin, and C. K. Ong, “Approaches to achieve broadband optical transformation devices with transmuted singularity,” J. Opt. Soc. Am. A 29, 124–129 (2012).
[CrossRef]

T. Xu, Y. C. Liu, C. K. Ong, Y. Zhang, and Y. G. Ma, “Perfect invisibility cloaking by isotropic media,” Phys. Rev. A 86, 043827 (2012).
[CrossRef]

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

Perczel, J.

J. Perczel, C. Garcia-Meca, and U. Leonhardt, “Partial transmutation of singularities in optical instruments,” J. Opt. 13, 075103 (2011).
[CrossRef]

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[CrossRef]

Sun, F.

Tyc, T.

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[CrossRef]

H. Y. Chen, U. Leonhardt, and T. Tyc, “Conformal cloak for waves,” Phys. Rev. A 83, 055801 (2011).
[CrossRef]

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

T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New J. Phys. 10, 115038 (2008).
[CrossRef]

Wang, N.

Wu, Q. N.

Q. N. Wu, Y. D. Xu, H. Li, and H. Y. Chen, “Cloaking and imaging at the same time,” Euro. Phys. Lett. 101, 34004 (2013).
[CrossRef]

Xu, T.

T. Xu, Y. C. Liu, C. K. Ong, Y. Zhang, and Y. G. Ma, “Perfect invisibility cloaking by isotropic media,” Phys. Rev. A 86, 043827 (2012).
[CrossRef]

Xu, Y. D.

Q. N. Wu, Y. D. Xu, H. Li, and H. Y. Chen, “Cloaking and imaging at the same time,” Euro. Phys. Lett. 101, 34004 (2013).
[CrossRef]

Yang, X. M.

Zhang, Y.

Y. G. Ma, F. Sun, Y. Zhang, Y. Jin, and C. K. Ong, “Approaches to achieve broadband optical transformation devices with transmuted singularity,” J. Opt. Soc. Am. A 29, 124–129 (2012).
[CrossRef]

T. Xu, Y. C. Liu, C. K. Ong, Y. Zhang, and Y. G. Ma, “Perfect invisibility cloaking by isotropic media,” Phys. Rev. A 86, 043827 (2012).
[CrossRef]

Zhou, X. Y.

Euro. Phys. Lett.

Q. N. Wu, Y. D. Xu, H. Li, and H. Y. Chen, “Cloaking and imaging at the same time,” Euro. Phys. Lett. 101, 34004 (2013).
[CrossRef]

J. Opt.

J. Perczel, C. Garcia-Meca, and U. Leonhardt, “Partial transmutation of singularities in optical instruments,” J. Opt. 13, 075103 (2011).
[CrossRef]

J. Opt. Soc. Am. A

Nat. Mater.

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

New J. Phys.

J. Perczel, T. Tyc, and U. Leonhardt, “Invisibility cloaking without superluminal propagation,” New J. Phys. 13, 083007 (2011).
[CrossRef]

T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New J. Phys. 10, 115038 (2008).
[CrossRef]

Opt. Express

Phys. Rev. A

T. Xu, Y. C. Liu, C. K. Ong, Y. Zhang, and Y. G. Ma, “Perfect invisibility cloaking by isotropic media,” Phys. Rev. A 86, 043827 (2012).
[CrossRef]

H. Y. Chen, U. Leonhardt, and T. Tyc, “Conformal cloak for waves,” Phys. Rev. A 83, 055801 (2011).
[CrossRef]

Science

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

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Figures (5)

Fig. 1.
Fig. 1.

Zhukovsky transformation scheme. (a) Transformed physical space and (b) Cartesian virtual space. The green line segment in (b) is the branch cut connecting the two Riemann sheets and map into a green circle in the physical space. The blue light rays travel parallel to the branch cut and will travel through the region without being disturbed. The pink light rays travel perpendicularly to the branch cut, hit the branch cut, and travel into the lower Riemann sheet space. The lower sheet consists of two contact Maxwell fish-eye lenses [red circles in (b)]. When the light rays meet the mirror (boundary of the lens), they will travel in a loop (in pink) in the lens, and then they will be reflected back to the upper sheet and continue their journey in the original direction. The contacted Maxwell fish eyes in the lower sheet are mapped into a red eye-shaped profile in (a). The region outside the lens in the lower sheet in (b) will not be reached by the light and is mapped into the inside of the eyes shaped in (a) where the object is hidden.

Fig. 2.
Fig. 2.

z-component permittivity distribution of the transmuted cloak. The big dashed red circle is the branch cut of Zhukovski transform in the z plane, and the other two small dashed circles are the transmutation region. This figure gives the contour of some permittivity values, from which we can see it ranges from 0.15 to 167. The white eye-shaped region in the middle corresponds to the outside of the Maxwell fish-eye mirror and is the hidden region for the object.

Fig. 3.
Fig. 3.

Permittivity of the z-component distribution of the transmutation region in the left small dashed circle of Fig. 2. (a) and (b) represent the distribution after transmutation; (c) and (d) are before transmutation. (a) and (c) correspond to the upper sheet; (b) and (d) correspond to the lower sheet of the Zhukovski transform. Comparing (a) and (c), we can see that after transmutation, the permittivity value is from 0.15 to 0.5, and no singularity appears. Comparing (b) and (d), we know the permittivity is from 1 to 16, while (d) is from 0 to 30. Therefore, the material singularity is removed and reasonable values are achieved.

Fig. 4.
Fig. 4.

Logarithm representation of the refractive index distribution of the original cloak before transmutation with two kissing Maxwell fish eyes in the lower sheet. The two blue dots are the indices of the singularities.

Fig. 5.
Fig. 5.

(a) Electric field distribution with the original index profile [Eq. (10)]. (b) Electric field distribution with truncated index profile: replace the refractive index smaller than 0.5 by 0.5. (c) Electric field distribution with truncated index profile: replace the refractive index smaller than unity by unity. (d) Electric field distribution with index profile after transmutation [Eq. (8)].

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

w=z+a2zorz=12(w±w24a2).
n(z)=|dwdz|n=|1a2z2|n,
ε=μ=gγΛg1ΛTdetΛγ,
R=r(r);φ=φ;andz=z,
Λ=(drdr00010001),
g=(n(z)2000n(z)2r20001),
γ=(1000r20001).
ε=μ=rn(z)2rdrdr(drdr00010001)(1n(z)20001r2n(z)20001)(drdr00010001)(1000r20001)=(rrdrdr000rrdrdr000n(z)2rrdrdr).
εz=|1a2(±a+reiφ)2|2n22rA,
n(w)={1upper sheet21+|w2a|2/(2a)2lower sheet&|w2a|<2a21+|w+2a|2/(2a)2lower sheet&|w+2a|<2alower sheet&|w2a|2a&|w+2a|2a.
εz(r)={|1a2(±a+r/Aeiφ)2|212rAupper sheet|1a2(±a+r/Aeiφ)2|212rA(21+|w2a|2/(2a)2)2lower sheet&|w2a|<2a|1a2(±a+r/Aeiφ)2|212rA(21+|w+2a|2/(2a)2)2lower sheet&|w+2a|<2a,
n(z)={|1a2z2||z|a|1a2z2|21+|w2a|2/(2a)2|z|<a&|w2a|<2a|1a2z2|21+|w+2a|2/(2a)2|z|<a&|w+2a|<2a|z|<a&|w2a|2a&|w+2a|2a.

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