Abstract

A novel homogeneous illusion device with arbitrary polygonal cross section that acts as either invisible cloak or shifting medium has been proposed and designed based on coordinate transformation method. The material parameters of the device are derived and the effectiveness is verified by full-wave simulation. Results show that whether the illusion device acts as invisible cloak or shifting medium depends on a value of shifting distance which is about 2a (ais circum-radius of the outer polygon).When the shifting distance is larger than 2a, the illusion device acts as an invisible cloak, and otherwise it acts as shifting medium. The stealth effect of all kinds of illusion devices are investigated, including identical-size mapped polygonal devices or non-identical size mapped polygonal devices. The results show that the device is a novel interactive cloak and is different from pre-proposed cloaks. The shifting properties of the device are validated by two examples, including moving the target object virtually and generating illusionary image of a line source. The material parameters of the device are homogeneous, which makes it more practicable in reality. It is hoped that our works may open an avenue for designing novel invisible cloaks, and are helpful for speeding up the potential applications of the illusion devices, such as aircraft or military equipment stealth, target objects camouflage or protection.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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  5. F. Sun, B. Zheng, H. Chen, W. Jiang, S. Guo, Y. Liu, and S. He, “Transformation Optics: From Classic Theory and Applications to its New Branches,” Laser Photonics Rev. 11(6), 1700034 (2017).
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  9. T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008).
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    [Crossref] [PubMed]
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  16. F. Sun, S. Li, and S. He, “Translational illusion of acoustic sources by transformation acoustics,” J. Acoust. Soc. Am. 142(3), 1213–1218 (2017).
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    [Crossref]
  27. H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. 10(11), 113016 (2008).
    [Crossref]
  28. Z. Yao, J. Luo, and Y. Lai, “Illusion optics via one-dimensional ultratransparent photonic crystals with shifted spatial dispersions,” Opt. Express 25(25), 30931–30938 (2017).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  30. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
    [Crossref] [PubMed]
  31. J. Li and C. T. Chan, “Double-negative acoustic metamaterial,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 055602 (2004).
    [Crossref] [PubMed]
  32. Y. Huang, Y. Feng, and T. Jiang, “Electromagnetic cloaking by layered structure of homogeneous isotropic materials,” Opt. Express 15(18), 11133–11141 (2007).
    [Crossref] [PubMed]
  33. Y. Cheng, F. Yang, J. Y. Xu, and X. J. Liu, “A multilayer structured acoustic cloak with homogeneous isotropic materials,” Appl. Phys. Lett. 92(15), 151913 (2008).
    [Crossref]
  34. J. Zhang, L. Liu, Y. Luo, S. Zhang, and N. A. Mortensen, “Homogeneous optical cloak constructed with uniform layered structures,” Opt. Express 19(9), 8625–8631 (2011).
    [Crossref] [PubMed]

2018 (1)

X. Zang, P. Huang, and Y. Zhu, “Optical illusions induced by rotating medium,” Opt. Commun. 410, 977–982 (2018).
[Crossref]

2017 (5)

A. Rajput and K. V. Srivastava, “Arbitrary shaped reciprocal external cloak with nonsingular and homogeneous material parameters using expanding coordinate transformation,” Plasmonics 12(3), 771–781 (2017).
[Crossref]

Z. Yao, J. Luo, and Y. Lai, “Illusion optics via one-dimensional ultratransparent photonic crystals with shifted spatial dispersions,” Opt. Express 25(25), 30931–30938 (2017).
[Crossref] [PubMed]

F. Sun, B. Zheng, H. Chen, W. Jiang, S. Guo, Y. Liu, and S. He, “Transformation Optics: From Classic Theory and Applications to its New Branches,” Laser Photonics Rev. 11(6), 1700034 (2017).
[Crossref]

Y. Liu, Z. Liang, F. Liu, O. Diba, A. Lamb, and J. Li, “Source illusion devices for flexural Lamb waves using elastic metasurfaces,” Phys. Rev. Lett. 119(3), 034301 (2017).
[Crossref] [PubMed]

F. Sun, S. Li, and S. He, “Translational illusion of acoustic sources by transformation acoustics,” J. Acoust. Soc. Am. 142(3), 1213–1218 (2017).
[Crossref] [PubMed]

2016 (1)

J. S. Mei, Q. Wu, K. Zhang, X. J. He, and Y. Wang, “Homogeneous illusion device exhibiting transformed and shifted scattering effect,” Opt. Commun. 368, 113–118 (2016).
[Crossref]

2015 (4)

Y. Du, X. Zang, C. Shi, X. Ji, and Y. Zhu, “Shifting media induced super-resolution imaging,” J. Opt. 17(2), 025606 (2015).
[Crossref]

J. Yi, P. H. Tichit, S. N. Burokur, and A. de Lustrac, “Illusion optics: Optically transforming the nature and the location of electromagnetic emissions,” J. Appl. Phys. 117(8), 084903 (2015).
[Crossref]

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2015).
[Crossref]

W. Tang, Z. Mei, and T. Cui, “Theory, experiment and applications of metamaterials,” Sci. China Phys. Mech. 58(12), 127001 (2015).
[Crossref]

2014 (1)

X. He and L. Z. Wu, “Illusion thermodynamics: A camouflage technique changing an object into another one with arbitrary cross section,” Appl. Phys. Lett. 105(22), 221904 (2014).
[Crossref]

2013 (1)

T. Li, M. Huang, J. Yang, W. Zhu, and J. Zeng, “A novel method for designing electromagnetic shrinking device with homogeneous material parameters,” IEEE T. Magn. 49(10), 5280–5286 (2013).
[Crossref]

2012 (2)

M. Liu, Z. L. Mei, X. Ma, and T. J. Cui, “Dc illusion and its experimental verification,” Appl. Phys. Lett. 101(5), 051905 (2012).
[Crossref]

Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012).
[Crossref] [PubMed]

2011 (5)

W. X. Jiang, T. J. Cui, X. M. Yang, H. F. Ma, and Q. Cheng, “Shrinking an arbitrary object as one desires using metamaterials,” Appl. Phys. Lett. 98(20), 204101 (2011).
[Crossref]

Y. Xu, S. Du, L. Gao, and H. Chen, “Overlapped illusion optics: a perfect lens brings a brighter feature,” New J. Phys. 13(2), 023010 (2011).
[Crossref]

C. Yang, J. Yang, M. Huang, Z. Xiao, and J. Peng, “An external cloak with arbitrary cross section based on complementary medium and coordinate transformation,” Opt. Express 19(2), 1147–1157 (2011).
[Crossref] [PubMed]

J. J. Yang, M. Huang, C. F. Yang, and J. Yu, “Reciprocal invisibility cloak based on complementary media,” Eur. Phys. J. D 61(3), 731–736 (2011).
[Crossref]

J. Zhang, L. Liu, Y. Luo, S. Zhang, and N. A. Mortensen, “Homogeneous optical cloak constructed with uniform layered structures,” Opt. Express 19(9), 8625–8631 (2011).
[Crossref] [PubMed]

2010 (1)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

2009 (4)

H. Chen, “Transformation optics in orthogonal coordinates,” Laser Photonics Rev. 11(6), 1700034 (2009).

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[Crossref] [PubMed]

J. Ng, H. Chen, and C. T. Chan, “Metamaterial frequency-selective superabsorber,” Opt. Lett. 34(5), 644–646 (2009).
[Crossref] [PubMed]

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102(9), 093901 (2009).
[Crossref] [PubMed]

2008 (3)

H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. 10(11), 113016 (2008).
[Crossref]

Y. Cheng, F. Yang, J. Y. Xu, and X. J. Liu, “A multilayer structured acoustic cloak with homogeneous isotropic materials,” Appl. Phys. Lett. 92(15), 151913 (2008).
[Crossref]

T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008).
[Crossref] [PubMed]

2007 (1)

2006 (3)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

2004 (1)

J. Li and C. T. Chan, “Double-negative acoustic metamaterial,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 055602 (2004).
[Crossref] [PubMed]

2000 (1)

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(18), 4184–4187 (2000).
[Crossref] [PubMed]

Burokur, S. N.

J. Yi, P. H. Tichit, S. N. Burokur, and A. de Lustrac, “Illusion optics: Optically transforming the nature and the location of electromagnetic emissions,” J. Appl. Phys. 117(8), 084903 (2015).
[Crossref]

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102(9), 093901 (2009).
[Crossref] [PubMed]

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[Crossref] [PubMed]

J. Ng, H. Chen, and C. T. Chan, “Metamaterial frequency-selective superabsorber,” Opt. Lett. 34(5), 644–646 (2009).
[Crossref] [PubMed]

H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. 10(11), 113016 (2008).
[Crossref]

J. Li and C. T. Chan, “Double-negative acoustic metamaterial,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 055602 (2004).
[Crossref] [PubMed]

Chen, H.

F. Sun, B. Zheng, H. Chen, W. Jiang, S. Guo, Y. Liu, and S. He, “Transformation Optics: From Classic Theory and Applications to its New Branches,” Laser Photonics Rev. 11(6), 1700034 (2017).
[Crossref]

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2015).
[Crossref]

Y. Xu, S. Du, L. Gao, and H. Chen, “Overlapped illusion optics: a perfect lens brings a brighter feature,” New J. Phys. 13(2), 023010 (2011).
[Crossref]

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102(9), 093901 (2009).
[Crossref] [PubMed]

H. Chen, “Transformation optics in orthogonal coordinates,” Laser Photonics Rev. 11(6), 1700034 (2009).

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[Crossref] [PubMed]

J. Ng, H. Chen, and C. T. Chan, “Metamaterial frequency-selective superabsorber,” Opt. Lett. 34(5), 644–646 (2009).
[Crossref] [PubMed]

T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008).
[Crossref] [PubMed]

H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. 10(11), 113016 (2008).
[Crossref]

Cheng, Q.

W. X. Jiang, T. J. Cui, X. M. Yang, H. F. Ma, and Q. Cheng, “Shrinking an arbitrary object as one desires using metamaterials,” Appl. Phys. Lett. 98(20), 204101 (2011).
[Crossref]

Cheng, Y.

Y. Cheng, F. Yang, J. Y. Xu, and X. J. Liu, “A multilayer structured acoustic cloak with homogeneous isotropic materials,” Appl. Phys. Lett. 92(15), 151913 (2008).
[Crossref]

Cui, T.

W. Tang, Z. Mei, and T. Cui, “Theory, experiment and applications of metamaterials,” Sci. China Phys. Mech. 58(12), 127001 (2015).
[Crossref]

Cui, T. J.

M. Liu, Z. L. Mei, X. Ma, and T. J. Cui, “Dc illusion and its experimental verification,” Appl. Phys. Lett. 101(5), 051905 (2012).
[Crossref]

W. X. Jiang, T. J. Cui, X. M. Yang, H. F. Ma, and Q. Cheng, “Shrinking an arbitrary object as one desires using metamaterials,” Appl. Phys. Lett. 98(20), 204101 (2011).
[Crossref]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

de Lustrac, A.

J. Yi, P. H. Tichit, S. N. Burokur, and A. de Lustrac, “Illusion optics: Optically transforming the nature and the location of electromagnetic emissions,” J. Appl. Phys. 117(8), 084903 (2015).
[Crossref]

Diba, O.

Y. Liu, Z. Liang, F. Liu, O. Diba, A. Lamb, and J. Li, “Source illusion devices for flexural Lamb waves using elastic metasurfaces,” Phys. Rev. Lett. 119(3), 034301 (2017).
[Crossref] [PubMed]

Du, S.

Y. Xu, S. Du, L. Gao, and H. Chen, “Overlapped illusion optics: a perfect lens brings a brighter feature,” New J. Phys. 13(2), 023010 (2011).
[Crossref]

Du, Y.

Y. Du, X. Zang, C. Shi, X. Ji, and Y. Zhu, “Shifting media induced super-resolution imaging,” J. Opt. 17(2), 025606 (2015).
[Crossref]

Feng, Y.

Gao, L.

Y. Xu, S. Du, L. Gao, and H. Chen, “Overlapped illusion optics: a perfect lens brings a brighter feature,” New J. Phys. 13(2), 023010 (2011).
[Crossref]

Guo, S.

F. Sun, B. Zheng, H. Chen, W. Jiang, S. Guo, Y. Liu, and S. He, “Transformation Optics: From Classic Theory and Applications to its New Branches,” Laser Photonics Rev. 11(6), 1700034 (2017).
[Crossref]

Han, D.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[Crossref] [PubMed]

He, S.

F. Sun, B. Zheng, H. Chen, W. Jiang, S. Guo, Y. Liu, and S. He, “Transformation Optics: From Classic Theory and Applications to its New Branches,” Laser Photonics Rev. 11(6), 1700034 (2017).
[Crossref]

F. Sun, S. Li, and S. He, “Translational illusion of acoustic sources by transformation acoustics,” J. Acoust. Soc. Am. 142(3), 1213–1218 (2017).
[Crossref] [PubMed]

He, X.

X. He and L. Z. Wu, “Illusion thermodynamics: A camouflage technique changing an object into another one with arbitrary cross section,” Appl. Phys. Lett. 105(22), 221904 (2014).
[Crossref]

He, X. J.

J. S. Mei, Q. Wu, K. Zhang, X. J. He, and Y. Wang, “Homogeneous illusion device exhibiting transformed and shifted scattering effect,” Opt. Commun. 368, 113–118 (2016).
[Crossref]

Huang, M.

T. Li, M. Huang, J. Yang, W. Zhu, and J. Zeng, “A novel method for designing electromagnetic shrinking device with homogeneous material parameters,” IEEE T. Magn. 49(10), 5280–5286 (2013).
[Crossref]

J. J. Yang, M. Huang, C. F. Yang, and J. Yu, “Reciprocal invisibility cloak based on complementary media,” Eur. Phys. J. D 61(3), 731–736 (2011).
[Crossref]

C. Yang, J. Yang, M. Huang, Z. Xiao, and J. Peng, “An external cloak with arbitrary cross section based on complementary medium and coordinate transformation,” Opt. Express 19(2), 1147–1157 (2011).
[Crossref] [PubMed]

Huang, P.

X. Zang, P. Huang, and Y. Zhu, “Optical illusions induced by rotating medium,” Opt. Commun. 410, 977–982 (2018).
[Crossref]

Huang, Y.

Ji, X.

Y. Du, X. Zang, C. Shi, X. Ji, and Y. Zhu, “Shifting media induced super-resolution imaging,” J. Opt. 17(2), 025606 (2015).
[Crossref]

Jiang, T.

Jiang, W.

F. Sun, B. Zheng, H. Chen, W. Jiang, S. Guo, Y. Liu, and S. He, “Transformation Optics: From Classic Theory and Applications to its New Branches,” Laser Photonics Rev. 11(6), 1700034 (2017).
[Crossref]

Jiang, W. X.

W. X. Jiang, T. J. Cui, X. M. Yang, H. F. Ma, and Q. Cheng, “Shrinking an arbitrary object as one desires using metamaterials,” Appl. Phys. Lett. 98(20), 204101 (2011).
[Crossref]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Lai, Y.

Z. Yao, J. Luo, and Y. Lai, “Illusion optics via one-dimensional ultratransparent photonic crystals with shifted spatial dispersions,” Opt. Express 25(25), 30931–30938 (2017).
[Crossref] [PubMed]

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[Crossref] [PubMed]

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102(9), 093901 (2009).
[Crossref] [PubMed]

Lamb, A.

Y. Liu, Z. Liang, F. Liu, O. Diba, A. Lamb, and J. Li, “Source illusion devices for flexural Lamb waves using elastic metasurfaces,” Phys. Rev. Lett. 119(3), 034301 (2017).
[Crossref] [PubMed]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

Li, J.

Y. Liu, Z. Liang, F. Liu, O. Diba, A. Lamb, and J. Li, “Source illusion devices for flexural Lamb waves using elastic metasurfaces,” Phys. Rev. Lett. 119(3), 034301 (2017).
[Crossref] [PubMed]

J. Li and C. T. Chan, “Double-negative acoustic metamaterial,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 055602 (2004).
[Crossref] [PubMed]

Li, S.

F. Sun, S. Li, and S. He, “Translational illusion of acoustic sources by transformation acoustics,” J. Acoust. Soc. Am. 142(3), 1213–1218 (2017).
[Crossref] [PubMed]

Li, T.

T. Li, M. Huang, J. Yang, W. Zhu, and J. Zeng, “A novel method for designing electromagnetic shrinking device with homogeneous material parameters,” IEEE T. Magn. 49(10), 5280–5286 (2013).
[Crossref]

Liang, Z.

Y. Liu, Z. Liang, F. Liu, O. Diba, A. Lamb, and J. Li, “Source illusion devices for flexural Lamb waves using elastic metasurfaces,” Phys. Rev. Lett. 119(3), 034301 (2017).
[Crossref] [PubMed]

Liu, F.

Y. Liu, Z. Liang, F. Liu, O. Diba, A. Lamb, and J. Li, “Source illusion devices for flexural Lamb waves using elastic metasurfaces,” Phys. Rev. Lett. 119(3), 034301 (2017).
[Crossref] [PubMed]

Liu, L.

Liu, M.

M. Liu, Z. L. Mei, X. Ma, and T. J. Cui, “Dc illusion and its experimental verification,” Appl. Phys. Lett. 101(5), 051905 (2012).
[Crossref]

Liu, X. J.

Y. Cheng, F. Yang, J. Y. Xu, and X. J. Liu, “A multilayer structured acoustic cloak with homogeneous isotropic materials,” Appl. Phys. Lett. 92(15), 151913 (2008).
[Crossref]

Liu, Y.

Y. Liu, Z. Liang, F. Liu, O. Diba, A. Lamb, and J. Li, “Source illusion devices for flexural Lamb waves using elastic metasurfaces,” Phys. Rev. Lett. 119(3), 034301 (2017).
[Crossref] [PubMed]

F. Sun, B. Zheng, H. Chen, W. Jiang, S. Guo, Y. Liu, and S. He, “Transformation Optics: From Classic Theory and Applications to its New Branches,” Laser Photonics Rev. 11(6), 1700034 (2017).
[Crossref]

Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012).
[Crossref] [PubMed]

Luo, J.

Luo, X.

T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008).
[Crossref] [PubMed]

H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. 10(11), 113016 (2008).
[Crossref]

Luo, Y.

Ma, H.

T. Yang, H. Chen, X. Luo, and H. Ma, “Superscatterer: enhancement of scattering with complementary media,” Opt. Express 16(22), 18545–18550 (2008).
[Crossref] [PubMed]

H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. 10(11), 113016 (2008).
[Crossref]

Ma, H. F.

W. X. Jiang, T. J. Cui, X. M. Yang, H. F. Ma, and Q. Cheng, “Shrinking an arbitrary object as one desires using metamaterials,” Appl. Phys. Lett. 98(20), 204101 (2011).
[Crossref]

Ma, X.

M. Liu, Z. L. Mei, X. Ma, and T. J. Cui, “Dc illusion and its experimental verification,” Appl. Phys. Lett. 101(5), 051905 (2012).
[Crossref]

Mei, J. S.

J. S. Mei, Q. Wu, K. Zhang, X. J. He, and Y. Wang, “Homogeneous illusion device exhibiting transformed and shifted scattering effect,” Opt. Commun. 368, 113–118 (2016).
[Crossref]

Mei, Z.

W. Tang, Z. Mei, and T. Cui, “Theory, experiment and applications of metamaterials,” Sci. China Phys. Mech. 58(12), 127001 (2015).
[Crossref]

Mei, Z. L.

M. Liu, Z. L. Mei, X. Ma, and T. J. Cui, “Dc illusion and its experimental verification,” Appl. Phys. Lett. 101(5), 051905 (2012).
[Crossref]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Mortensen, N. A.

Nemat-Nasser, S. C.

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(18), 4184–4187 (2000).
[Crossref] [PubMed]

Ng, J.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[Crossref] [PubMed]

J. Ng, H. Chen, and C. T. Chan, “Metamaterial frequency-selective superabsorber,” Opt. Lett. 34(5), 644–646 (2009).
[Crossref] [PubMed]

Padilla, W. J.

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(18), 4184–4187 (2000).
[Crossref] [PubMed]

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Peng, J.

Rajput, A.

A. Rajput and K. V. Srivastava, “Arbitrary shaped reciprocal external cloak with nonsingular and homogeneous material parameters using expanding coordinate transformation,” Plasmonics 12(3), 771–781 (2017).
[Crossref]

Schultz, S.

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(18), 4184–4187 (2000).
[Crossref] [PubMed]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

Shi, C.

Y. Du, X. Zang, C. Shi, X. Ji, and Y. Zhu, “Shifting media induced super-resolution imaging,” J. Opt. 17(2), 025606 (2015).
[Crossref]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

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(18), 4184–4187 (2000).
[Crossref] [PubMed]

Srivastava, K. V.

A. Rajput and K. V. Srivastava, “Arbitrary shaped reciprocal external cloak with nonsingular and homogeneous material parameters using expanding coordinate transformation,” Plasmonics 12(3), 771–781 (2017).
[Crossref]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Sun, F.

F. Sun, B. Zheng, H. Chen, W. Jiang, S. Guo, Y. Liu, and S. He, “Transformation Optics: From Classic Theory and Applications to its New Branches,” Laser Photonics Rev. 11(6), 1700034 (2017).
[Crossref]

F. Sun, S. Li, and S. He, “Translational illusion of acoustic sources by transformation acoustics,” J. Acoust. Soc. Am. 142(3), 1213–1218 (2017).
[Crossref] [PubMed]

Tang, W.

W. Tang, Z. Mei, and T. Cui, “Theory, experiment and applications of metamaterials,” Sci. China Phys. Mech. 58(12), 127001 (2015).
[Crossref]

Tichit, P. H.

J. Yi, P. H. Tichit, S. N. Burokur, and A. de Lustrac, “Illusion optics: Optically transforming the nature and the location of electromagnetic emissions,” J. Appl. Phys. 117(8), 084903 (2015).
[Crossref]

Vier, D. C.

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(18), 4184–4187 (2000).
[Crossref] [PubMed]

Wang, Y.

J. S. Mei, Q. Wu, K. Zhang, X. J. He, and Y. Wang, “Homogeneous illusion device exhibiting transformed and shifted scattering effect,” Opt. Commun. 368, 113–118 (2016).
[Crossref]

Wu, L. Z.

X. He and L. Z. Wu, “Illusion thermodynamics: A camouflage technique changing an object into another one with arbitrary cross section,” Appl. Phys. Lett. 105(22), 221904 (2014).
[Crossref]

Wu, Q.

J. S. Mei, Q. Wu, K. Zhang, X. J. He, and Y. Wang, “Homogeneous illusion device exhibiting transformed and shifted scattering effect,” Opt. Commun. 368, 113–118 (2016).
[Crossref]

Xiao, J.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[Crossref] [PubMed]

Xiao, Z.

Xu, J. Y.

Y. Cheng, F. Yang, J. Y. Xu, and X. J. Liu, “A multilayer structured acoustic cloak with homogeneous isotropic materials,” Appl. Phys. Lett. 92(15), 151913 (2008).
[Crossref]

Xu, L.

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2015).
[Crossref]

Xu, Y.

Y. Xu, S. Du, L. Gao, and H. Chen, “Overlapped illusion optics: a perfect lens brings a brighter feature,” New J. Phys. 13(2), 023010 (2011).
[Crossref]

Yang, C.

Yang, C. F.

J. J. Yang, M. Huang, C. F. Yang, and J. Yu, “Reciprocal invisibility cloak based on complementary media,” Eur. Phys. J. D 61(3), 731–736 (2011).
[Crossref]

Yang, F.

Y. Cheng, F. Yang, J. Y. Xu, and X. J. Liu, “A multilayer structured acoustic cloak with homogeneous isotropic materials,” Appl. Phys. Lett. 92(15), 151913 (2008).
[Crossref]

Yang, J.

T. Li, M. Huang, J. Yang, W. Zhu, and J. Zeng, “A novel method for designing electromagnetic shrinking device with homogeneous material parameters,” IEEE T. Magn. 49(10), 5280–5286 (2013).
[Crossref]

C. Yang, J. Yang, M. Huang, Z. Xiao, and J. Peng, “An external cloak with arbitrary cross section based on complementary medium and coordinate transformation,” Opt. Express 19(2), 1147–1157 (2011).
[Crossref] [PubMed]

Yang, J. J.

J. J. Yang, M. Huang, C. F. Yang, and J. Yu, “Reciprocal invisibility cloak based on complementary media,” Eur. Phys. J. D 61(3), 731–736 (2011).
[Crossref]

Yang, T.

Yang, X. M.

W. X. Jiang, T. J. Cui, X. M. Yang, H. F. Ma, and Q. Cheng, “Shrinking an arbitrary object as one desires using metamaterials,” Appl. Phys. Lett. 98(20), 204101 (2011).
[Crossref]

Yao, Z.

Yi, J.

J. Yi, P. H. Tichit, S. N. Burokur, and A. de Lustrac, “Illusion optics: Optically transforming the nature and the location of electromagnetic emissions,” J. Appl. Phys. 117(8), 084903 (2015).
[Crossref]

Yu, J.

J. J. Yang, M. Huang, C. F. Yang, and J. Yu, “Reciprocal invisibility cloak based on complementary media,” Eur. Phys. J. D 61(3), 731–736 (2011).
[Crossref]

Zang, X.

X. Zang, P. Huang, and Y. Zhu, “Optical illusions induced by rotating medium,” Opt. Commun. 410, 977–982 (2018).
[Crossref]

Y. Du, X. Zang, C. Shi, X. Ji, and Y. Zhu, “Shifting media induced super-resolution imaging,” J. Opt. 17(2), 025606 (2015).
[Crossref]

Zeng, J.

T. Li, M. Huang, J. Yang, W. Zhu, and J. Zeng, “A novel method for designing electromagnetic shrinking device with homogeneous material parameters,” IEEE T. Magn. 49(10), 5280–5286 (2013).
[Crossref]

Zhang, J.

Zhang, K.

J. S. Mei, Q. Wu, K. Zhang, X. J. He, and Y. Wang, “Homogeneous illusion device exhibiting transformed and shifted scattering effect,” Opt. Commun. 368, 113–118 (2016).
[Crossref]

Zhang, S.

Zhang, X.

Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012).
[Crossref] [PubMed]

H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. 10(11), 113016 (2008).
[Crossref]

Zhang, Z. Q.

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[Crossref] [PubMed]

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102(9), 093901 (2009).
[Crossref] [PubMed]

Zheng, B.

F. Sun, B. Zheng, H. Chen, W. Jiang, S. Guo, Y. Liu, and S. He, “Transformation Optics: From Classic Theory and Applications to its New Branches,” Laser Photonics Rev. 11(6), 1700034 (2017).
[Crossref]

Zhu, W.

T. Li, M. Huang, J. Yang, W. Zhu, and J. Zeng, “A novel method for designing electromagnetic shrinking device with homogeneous material parameters,” IEEE T. Magn. 49(10), 5280–5286 (2013).
[Crossref]

Zhu, Y.

X. Zang, P. Huang, and Y. Zhu, “Optical illusions induced by rotating medium,” Opt. Commun. 410, 977–982 (2018).
[Crossref]

Y. Du, X. Zang, C. Shi, X. Ji, and Y. Zhu, “Shifting media induced super-resolution imaging,” J. Opt. 17(2), 025606 (2015).
[Crossref]

Appl. Phys. Lett. (4)

W. X. Jiang, T. J. Cui, X. M. Yang, H. F. Ma, and Q. Cheng, “Shrinking an arbitrary object as one desires using metamaterials,” Appl. Phys. Lett. 98(20), 204101 (2011).
[Crossref]

M. Liu, Z. L. Mei, X. Ma, and T. J. Cui, “Dc illusion and its experimental verification,” Appl. Phys. Lett. 101(5), 051905 (2012).
[Crossref]

X. He and L. Z. Wu, “Illusion thermodynamics: A camouflage technique changing an object into another one with arbitrary cross section,” Appl. Phys. Lett. 105(22), 221904 (2014).
[Crossref]

Y. Cheng, F. Yang, J. Y. Xu, and X. J. Liu, “A multilayer structured acoustic cloak with homogeneous isotropic materials,” Appl. Phys. Lett. 92(15), 151913 (2008).
[Crossref]

Eur. Phys. J. D (1)

J. J. Yang, M. Huang, C. F. Yang, and J. Yu, “Reciprocal invisibility cloak based on complementary media,” Eur. Phys. J. D 61(3), 731–736 (2011).
[Crossref]

IEEE T. Magn. (1)

T. Li, M. Huang, J. Yang, W. Zhu, and J. Zeng, “A novel method for designing electromagnetic shrinking device with homogeneous material parameters,” IEEE T. Magn. 49(10), 5280–5286 (2013).
[Crossref]

J. Acoust. Soc. Am. (1)

F. Sun, S. Li, and S. He, “Translational illusion of acoustic sources by transformation acoustics,” J. Acoust. Soc. Am. 142(3), 1213–1218 (2017).
[Crossref] [PubMed]

J. Appl. Phys. (1)

J. Yi, P. H. Tichit, S. N. Burokur, and A. de Lustrac, “Illusion optics: Optically transforming the nature and the location of electromagnetic emissions,” J. Appl. Phys. 117(8), 084903 (2015).
[Crossref]

J. Opt. (1)

Y. Du, X. Zang, C. Shi, X. Ji, and Y. Zhu, “Shifting media induced super-resolution imaging,” J. Opt. 17(2), 025606 (2015).
[Crossref]

Laser Photonics Rev. (2)

F. Sun, B. Zheng, H. Chen, W. Jiang, S. Guo, Y. Liu, and S. He, “Transformation Optics: From Classic Theory and Applications to its New Branches,” Laser Photonics Rev. 11(6), 1700034 (2017).
[Crossref]

H. Chen, “Transformation optics in orthogonal coordinates,” Laser Photonics Rev. 11(6), 1700034 (2009).

Nanoscale (1)

Y. Liu and X. Zhang, “Recent advances in transformation optics,” Nanoscale 4(17), 5277–5292 (2012).
[Crossref] [PubMed]

Nat. Mater. (1)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

Nat. Photonics (1)

L. Xu and H. Chen, “Conformal transformation optics,” Nat. Photonics 9(1), 15–23 (2015).
[Crossref]

New J. Phys. (2)

Y. Xu, S. Du, L. Gao, and H. Chen, “Overlapped illusion optics: a perfect lens brings a brighter feature,” New J. Phys. 13(2), 023010 (2011).
[Crossref]

H. Chen, X. Zhang, X. Luo, H. Ma, and C. T. Chan, “Reshaping the perfect electrical conductor cylinder arbitrarily,” New J. Phys. 10(11), 113016 (2008).
[Crossref]

Opt. Commun. (2)

J. S. Mei, Q. Wu, K. Zhang, X. J. He, and Y. Wang, “Homogeneous illusion device exhibiting transformed and shifted scattering effect,” Opt. Commun. 368, 113–118 (2016).
[Crossref]

X. Zang, P. Huang, and Y. Zhu, “Optical illusions induced by rotating medium,” Opt. Commun. 410, 977–982 (2018).
[Crossref]

Opt. Express (5)

Opt. Lett. (1)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

J. Li and C. T. Chan, “Double-negative acoustic metamaterial,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(5), 055602 (2004).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

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(18), 4184–4187 (2000).
[Crossref] [PubMed]

Y. Lai, H. Chen, Z. Q. Zhang, and C. T. Chan, “Complementary media invisibility cloak that cloaks objects at a distance outside the cloaking shell,” Phys. Rev. Lett. 102(9), 093901 (2009).
[Crossref] [PubMed]

Y. Lai, J. Ng, H. Chen, D. Han, J. Xiao, Z. Q. Zhang, and C. T. Chan, “Illusion optics: the optical transformation of an object into another object,” Phys. Rev. Lett. 102(25), 253902 (2009).
[Crossref] [PubMed]

Y. Liu, Z. Liang, F. Liu, O. Diba, A. Lamb, and J. Li, “Source illusion devices for flexural Lamb waves using elastic metasurfaces,” Phys. Rev. Lett. 119(3), 034301 (2017).
[Crossref] [PubMed]

Plasmonics (1)

A. Rajput and K. V. Srivastava, “Arbitrary shaped reciprocal external cloak with nonsingular and homogeneous material parameters using expanding coordinate transformation,” Plasmonics 12(3), 771–781 (2017).
[Crossref]

Sci. China Phys. Mech. (1)

W. Tang, Z. Mei, and T. Cui, “Theory, experiment and applications of metamaterials,” Sci. China Phys. Mech. 58(12), 127001 (2015).
[Crossref]

Science (3)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
[Crossref] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

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Figures (11)

Fig. 1
Fig. 1 Schematic of the illusion device. (a) The gray region bordered by polygon C is moved to the dashed region bordered by polygon B;(b)the quadrilateral a i a i+1 b i+1 b i in the virtual space is transformed into the quadrilateral a i a i+1 c i+1 c i in the physical space.
Fig. 2
Fig. 2 Electric field distribution in the vicinity of illusion device when without hidden object. The TE plane wave is incident from (a) left to right; (b) bottom to top; (c) oblique direction with an angle of π/4.
Fig. 3
Fig. 3 Electric field ( E z ) distributions in the vicinity of an aircraft shaped object covered by the illusion device [(a), (b) and(c)], and the corresponding object without the illusion device [(d), (e) and (f)] under different irradiation conditions.(a)- (d) TE plane wave propagates from left to right (horizontal direction); (b)-(e) TE plane wave propagates from bottom to top (vertical direction); (c)-(f) TE plane wave propagates from an oblique incident direction with an angle of π/4.
Fig. 4
Fig. 4 The normalized far field of the aircraft shaped object with and without the illusion device.
Fig. 5
Fig. 5 The electric field distribution in the vicinity of (a) 5-sided polygonal illusion device, (b) 6-sided polygonal illusion device, (c)20-sided polygonal illusion device, and (d) the normalized far field of the illusion devices.
Fig. 6
Fig. 6 Electric field distribution in the vicinity of the illusion device when the shifting distance is chosen as (a) d=0.2m, (b) d=0.24m.
Fig. 7
Fig. 7 Electric field distributions of the illusion devices. (a) Near electric field distribution of the device with geometric parameters of a=0.08m, b=0.055m, c=0.04m. (b) Near electric field distribution of the device with geometric parameters of a=0.08m, b=0.025m, c=0.04m. (c) Normalized far field of the devices. Black colored line is the scattering far field of the bare aircraft shaped object without illusion device, while red colored line and blue colored line indicate the far field of the illusion device with geometric parameters of b=0.055m, and b=0.025m, respectively. The white dashed quadrangle in (a) and (b) indicates the original coordinate of polygon B.
Fig. 8
Fig. 8 Material parameter distributions for the quadrilateral illusion devices with identical size of polygon B and C [(a) ~(d)], non-identical size of polygon B and C [(e) - (h)]. (a), (e) μ xx ;(b), (f) μ xy ;(c), (g) μ yy ; (d), (h) ε zz .
Fig. 9
Fig. 9 The electric field distribution in the vicinity of illusion device [(a), (b)] and aircraft shaped object in the original space [(c), (d)]. The center of the geometric is at (0,0),(0,0),(−0.1m,0) and(−0.06m,0) for (a), (b), (c) and (d) respectively.
Fig. 10
Fig. 10 Electric field ( E z ) distributions in the vicinity of the line source covered by the illusion device [(a), (b)], and the corresponding virtual line source inside continues boundary circle [(c), (d)]. (a), (b) a line source is located at (0, 0) ;(c) a line source is located at (−0.06m, 0);(d) a line source is located at (−0.1m, 0).
Fig. 11
Fig. 11 The material parameter distribution of (a)–(d) the outer triangles, (e)-(h) the inner triangles with the variation of the shifting distanced. In the simulation, the circum-radius of polygons A and C are fixed at a=0.08m, c=0.04m, respectively.

Equations (12)

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

ε'=Λε Λ T /detΛ, μ =Λμ Λ T /detΛ.
x ai =acos[(i1)2π/N], y ai =asin[(i1)2π/N], x bi =bcos[(i1)2π/N]d, y bi =bsin[(i1)2π/N], x ci =ccos[(i1)2π/N], y ci =csin[(i1)2π/N].
x = m 1 x+ m 2 y+ m 3 , y = n 1 x+ n 2 y+ n 3 , z =z.
x c i = m 1 x b i + m 2 y b i + m 3 , y c i = n 1 x b i + n 2 y b i + n 3 , x a i+1 = m 1 x a i+1 + m 2 y a i+1 + m 3 , y a i+1 = n 1 x a i+1 + n 2 y a i+1 + n 3 , x a i = m 1 x a i + m 2 y a i + m 3 , y a i = n 1 x a i + n 2 y a i + n 3 .
[ x c i y c i x a i+1 y a i+1 x a i y a i ]=[ x b i y b i 1 x a i+1 y a i+1 1 x a i y a i 1 ][ m 1 n 1 m 2 n 2 m 3 n 3 ].
[ m 1 n 1 m 2 n 2 m 3 n 3 ]= Α 1 [ x c i y c i x a i+1 y a i+1 x a i y a i ]
Λ=[ m 1 m 2 0 n 1 n 2 0 0 0 1 ].
μ ' outer =μ[ ( m 1 2 + m 2 2 )/( m 1 n 2 m 2 n 1 ) ( m 1 n 1 + m 2 n 2 )/( m 1 n 2 m 2 n 1 ) ( m 1 n 1 + m 2 n 2 )/( m 1 n 2 m 2 n 1 ) ( n 1 2 + n 2 2 )/( m 1 n 2 m 2 n 1 ) ], ε ' outer =ε/( m 1 n 2 m 2 n 1 ).
x'= p 1 x+ p 2 y+ p 3 , y'= q 1 x+ q 2 y+ q 3 , z'=z.
[ p 1 q 1 p 2 q 2 p 3 q 3 ]= Β 1 [ x c i y c i x c i+1 y c i+1 x a i+1 y a i+1 ]
Λ=[ p 1 p 2 0 q 1 q 2 0 0 0 1 ],detΛ= p 1 q 2 p 2 q 1 .
μ ' inner =μ[ ( p 1 2 + p 2 2 )/( p 1 q 2 p 2 q 1 ) ( p 1 q 1 + p 2 q 2 )/( p 1 q 2 p 2 q 1 ) ( p 1 q 1 + p 2 q 2 )/( p 1 q 2 p 2 q 1 ) ( q 1 2 + q 2 2 )/( p 1 q 2 p 2 q 1 ) ]. ε ' inner =ε/( p 1 q 2 p 2 q 1 ).

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