Abstract

An anamorphic fractional Fourier transform (AFrFT) lens based on graded index (GRIN) materials and designed with the help of transformation optics is proposed. Cross sections of the new lens are mapped from those of a standard quadratic GRIN lens via gradually varied conformal transformations. This lens can afford complicated anamorphic patterns in the fractional Fourier domain for any fractional order, possibly leading to many new applications. Three samples are shown, which offer higher distinguishability in the fractional Fourier domain, a more precisely recognized matched filter, and stronger security of the AFrFT-based optical encryption. With metamaterials development, including three-dimensional printing technologies, GRIN media fabrication has become more convenient; thus, the proposed lens may have vast application prospects in signal processing. The design also demonstrates the ability and flexibility of the transformation optics in exploring new Fourier optics devices.

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

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References

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2017 (3)

J. Chen, J. Hu, X. B. Yang, and X. Y. Lu, “Manipulating and detecting the chirpiness of spatial chirp signals via fractional Fourier lenses designed by transformation optics,” Appl. Opt. 56(32), 9119–9125 (2017).
[Crossref] [PubMed]

B. Zhang, Y. X. Guo, H. Zirath, and Y. P. Zhang, “Investigation on 3-D-Printing Technologies for Millimeter-Wave and Terahertz Applications,” Proc. IEEE 105(4), 723–736 (2017).
[Crossref]

L. Yuan, Q. Ran, and T. Zhao, “Image authentication based on double-image encryption and partial phase decryption in nonseparable fractional Fourier domain,” Opt. Laser Technol. 88, 111–120 (2017).
[Crossref]

2016 (3)

W. X. Jiang, S. Ge, T. Han, S. Zhang, M. Q. Mehmood, C. W. Qiu, and T. J. Cui, “Shaping 3D Path of Electromagnetic Waves Using Gradient-Refractive-Index Metamaterials,” Adv. Sci. (Weinh.) 3(8), 1600022 (2016).
[Crossref] [PubMed]

E. MacDonald and R. Wicker, “Multiprocess 3D printing for increasing component functionality,” Science 353(6307), aaf2093 (2016).
[Crossref] [PubMed]

D. V. Isakov, Q. Lei, F. Castles, C. J. Stevens, C. R. M. Grovenor, and P. S. Grant, “3D printed anisotropic dielectric composite with meta-material features,” Mater. Des. 93, 423–430 (2016).
[Crossref]

2015 (3)

J. B. Pendry, Y. Luo, and R. Zhao, “Transforming the optical landscape,” Science 348(6234), 521–524 (2015).
[Crossref] [PubMed]

A. Žukauskas, I. Matulaitiene, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
[Crossref]

Q. Ran, L. Yuan, and T. Zhao, “Image encryption based on nonseparable fractional Fourier transform and chaotic map,” Opt. Commun. 348, 43–49 (2015).
[Crossref]

2013 (2)

W. X. Jiang, C. W. Qiu, T. C. Han, Q. Cheng, H. F. Ma, S. Zhang, and T. J. Cui, “Broadband all-dielectric magnifying lens for far-field high-resolution imaging,” Adv. Mater. 25(48), 6963–6968 (2013).
[Crossref] [PubMed]

J. Hu, X. Liu, and G. K. Hu, “Constraint condition on transformation relation for generalized acoustics,” Wave Motion 50(2), 170–179 (2013).
[Crossref]

2012 (4)

J. Hu and X. Y. Lu, “Determining the full transformation relations in the transformation method,” Appl. Phys., A Mater. Sci. Process. 109(4), 971–977 (2012).
[Crossref]

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun. 3(1), 870 (2012).
[Crossref] [PubMed]

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11(11), 917–924 (2012).
[Crossref] [PubMed]

C. W. Qiu, A. Akbarzadeh, T. C. Han, and A. J. Danner, “Photorealistic rendering of a graded negative-index metamaterial magnifier,” New J. Phys. 14(3), 033024 (2012).
[Crossref]

2011 (2)

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three dimensional photonic metamaterials,” Nat. Photonics 5(9), 523–530 (2011).
[Crossref]

Y. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011).
[Crossref] [PubMed]

2010 (1)

Z. Chang, J. Hu, and G. K. Hu, “Transformation method and wave control,” Lixue Xuebao 26(6), 889–898 (2010).

2009 (1)

2008 (2)

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[Crossref] [PubMed]

R. Tao, J. Lang, and Y. Wang, “Optical image encryption based on the multiple-parameter fractional Fourier transform,” Opt. Lett. 33(6), 581–583 (2008).
[Crossref] [PubMed]

2007 (1)

2006 (2)

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 (2)

B. Hennelly, D. Kelly, and J. T. Sheridan, “Wavelength-controlled variable-order optical fractional Fourier transform,” Opt. Lett. 29(5), 427–429 (2004).
[Crossref] [PubMed]

N. K. Nischal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4–6), 253–259 (2004).

2003 (3)

2001 (1)

2000 (2)

1998 (1)

1997 (1)

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).

1996 (1)

1995 (2)

1994 (2)

D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33(26), 6188–6193 (1994).
[Crossref] [PubMed]

L. B. Almeida, “The fractional Fourier transform and time–frequency representations,” IEEE Trans. Signal Process. 42(11), 3084–3091 (1994).
[Crossref]

1993 (4)

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10(10), 2181–2186 (1993).
[Crossref]

H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101(3–4), 163–169 (1993).
[Crossref]

D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation I,” J. Opt. Soc. Am. A 10(9), 1875–1881 (1993).
[Crossref]

H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101(3–4), 163–169 (1993).
[Crossref]

1991 (1)

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, and C. Puliafito, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

1990 (1)

C. Ferreira and C. Vazquez, “Anamorphic multiple matched filter for character recognition, performance with signals of equal size,” J. Mod. Opt. 37(8), 1343–1354 (1990).
[Crossref]

1989 (1)

C. Ferreira, M. J. Buades, and A. Moya, “Anamorphic correlator for character recognition. Detection of characters of different size,” J. Opt. 20(4), 181–185 (1989).
[Crossref]

1988 (1)

M. S. Millan, C. Ferreira, A. Pons, and P. Andres, “Application of anamorphic systems to directional pseudocolor encoding,” Opt. Eng. 27(2), 129–134 (1988).
[Crossref]

1986 (1)

E. Bonet, C. Ferreira, P. Andres, and A. Pons, “Nonsymmetrical Fourier correlator to increase the angular discrimination in character recognition,” Opt. Commun. 58(3), 155–160 (1986).
[Crossref]

1985 (3)

1984 (1)

Akbarzadeh, A.

C. W. Qiu, A. Akbarzadeh, T. C. Han, and A. J. Danner, “Photorealistic rendering of a graded negative-index metamaterial magnifier,” New J. Phys. 14(3), 033024 (2012).
[Crossref]

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time–frequency representations,” IEEE Trans. Signal Process. 42(11), 3084–3091 (1994).
[Crossref]

Alù, A.

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun. 3(1), 870 (2012).
[Crossref] [PubMed]

Andres, P.

M. S. Millan, C. Ferreira, A. Pons, and P. Andres, “Application of anamorphic systems to directional pseudocolor encoding,” Opt. Eng. 27(2), 129–134 (1988).
[Crossref]

E. Bonet, C. Ferreira, P. Andres, and A. Pons, “Nonsymmetrical Fourier correlator to increase the angular discrimination in character recognition,” Opt. Commun. 58(3), 155–160 (1986).
[Crossref]

P. Andres, C. Ferreira, and E. Bonet, “Fraunhofer diffraction patterns from apertures illuminated with nonparallel light in nonsymmetrical Fourier transformers,” Appl. Opt. 24(10), 1549–1552 (1985).
[Crossref] [PubMed]

Arsenault, H. H.

Belkin, M. A.

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun. 3(1), 870 (2012).
[Crossref] [PubMed]

Bitran, Y.

Bonet, E.

E. Bonet, C. Ferreira, P. Andres, and A. Pons, “Nonsymmetrical Fourier correlator to increase the angular discrimination in character recognition,” Opt. Commun. 58(3), 155–160 (1986).
[Crossref]

P. Andres, C. Ferreira, and E. Bonet, “Fraunhofer diffraction patterns from apertures illuminated with nonparallel light in nonsymmetrical Fourier transformers,” Appl. Opt. 24(10), 1549–1552 (1985).
[Crossref] [PubMed]

Buades, M. J.

C. Ferreira, M. J. Buades, and A. Moya, “Anamorphic correlator for character recognition. Detection of characters of different size,” J. Opt. 20(4), 181–185 (1989).
[Crossref]

Castles, F.

D. V. Isakov, Q. Lei, F. Castles, C. J. Stevens, C. R. M. Grovenor, and P. S. Grant, “3D printed anisotropic dielectric composite with meta-material features,” Mater. Des. 93, 423–430 (2016).
[Crossref]

Chang, W.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, and C. Puliafito, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Chang, Z.

Z. Chang, J. Hu, and G. K. Hu, “Transformation method and wave control,” Lixue Xuebao 26(6), 889–898 (2010).

Chen, J.

Cheng, Q.

W. X. Jiang, C. W. Qiu, T. C. Han, Q. Cheng, H. F. Ma, S. Zhang, and T. J. Cui, “Broadband all-dielectric magnifying lens for far-field high-resolution imaging,” Adv. Mater. 25(48), 6963–6968 (2013).
[Crossref] [PubMed]

Choma, M.

Cui, T. J.

W. X. Jiang, S. Ge, T. Han, S. Zhang, M. Q. Mehmood, C. W. Qiu, and T. J. Cui, “Shaping 3D Path of Electromagnetic Waves Using Gradient-Refractive-Index Metamaterials,” Adv. Sci. (Weinh.) 3(8), 1600022 (2016).
[Crossref] [PubMed]

W. X. Jiang, C. W. Qiu, T. C. Han, Q. Cheng, H. F. Ma, S. Zhang, and T. J. Cui, “Broadband all-dielectric magnifying lens for far-field high-resolution imaging,” Adv. Mater. 25(48), 6963–6968 (2013).
[Crossref] [PubMed]

Danner, A. J.

C. W. Qiu, A. Akbarzadeh, T. C. Han, and A. J. Danner, “Photorealistic rendering of a graded negative-index metamaterial magnifier,” New J. Phys. 14(3), 033024 (2012).
[Crossref]

Dorsch, R. G.

Erden, M. F.

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).

Ferreira, C.

D. Mendlovic, Y. Bitran, R. G. Dorsch, C. Ferreira, J. Garcia, and H. M. Ozaktaz, “Anamorphic fractional Fourier transform: optical implementation and applications,” Appl. Opt. 34(32), 7451–7456 (1995).
[Crossref] [PubMed]

C. Ferreira and C. Vazquez, “Anamorphic multiple matched filter for character recognition, performance with signals of equal size,” J. Mod. Opt. 37(8), 1343–1354 (1990).
[Crossref]

C. Ferreira, M. J. Buades, and A. Moya, “Anamorphic correlator for character recognition. Detection of characters of different size,” J. Opt. 20(4), 181–185 (1989).
[Crossref]

M. S. Millan, C. Ferreira, A. Pons, and P. Andres, “Application of anamorphic systems to directional pseudocolor encoding,” Opt. Eng. 27(2), 129–134 (1988).
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E. Bonet, C. Ferreira, P. Andres, and A. Pons, “Nonsymmetrical Fourier correlator to increase the angular discrimination in character recognition,” Opt. Commun. 58(3), 155–160 (1986).
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P. Andres, C. Ferreira, and E. Bonet, “Fraunhofer diffraction patterns from apertures illuminated with nonparallel light in nonsymmetrical Fourier transformers,” Appl. Opt. 24(10), 1549–1552 (1985).
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D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, and C. Puliafito, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
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Fu, L.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
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Gadonas, R.

A. Žukauskas, I. Matulaitiene, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
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Garcia, J.

García, J.

Ge, S.

W. X. Jiang, S. Ge, T. Han, S. Zhang, M. Q. Mehmood, C. W. Qiu, and T. J. Cui, “Shaping 3D Path of Electromagnetic Waves Using Gradient-Refractive-Index Metamaterials,” Adv. Sci. (Weinh.) 3(8), 1600022 (2016).
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Giessen, H.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
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Grant, P. S.

D. V. Isakov, Q. Lei, F. Castles, C. J. Stevens, C. R. M. Grovenor, and P. S. Grant, “3D printed anisotropic dielectric composite with meta-material features,” Mater. Des. 93, 423–430 (2016).
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Greenleaf, A.

A. Greenleaf, M. Lassas, and G. Uhlmann, “On non-uniqueness for Calderón’s inverse problem,” Math. Res. Lett. 10(5), 685–693 (2003).
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Gregory, K.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, and C. Puliafito, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
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Grovenor, C. R. M.

D. V. Isakov, Q. Lei, F. Castles, C. J. Stevens, C. R. M. Grovenor, and P. S. Grant, “3D printed anisotropic dielectric composite with meta-material features,” Mater. Des. 93, 423–430 (2016).
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Guo, H.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[Crossref] [PubMed]

Guo, Y. X.

B. Zhang, Y. X. Guo, H. Zirath, and Y. P. Zhang, “Investigation on 3-D-Printing Technologies for Millimeter-Wave and Terahertz Applications,” Proc. IEEE 105(4), 723–736 (2017).
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Han, T.

W. X. Jiang, S. Ge, T. Han, S. Zhang, M. Q. Mehmood, C. W. Qiu, and T. J. Cui, “Shaping 3D Path of Electromagnetic Waves Using Gradient-Refractive-Index Metamaterials,” Adv. Sci. (Weinh.) 3(8), 1600022 (2016).
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Han, T. C.

W. X. Jiang, C. W. Qiu, T. C. Han, Q. Cheng, H. F. Ma, S. Zhang, and T. J. Cui, “Broadband all-dielectric magnifying lens for far-field high-resolution imaging,” Adv. Mater. 25(48), 6963–6968 (2013).
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C. W. Qiu, A. Akbarzadeh, T. C. Han, and A. J. Danner, “Photorealistic rendering of a graded negative-index metamaterial magnifier,” New J. Phys. 14(3), 033024 (2012).
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Hee, M.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, and C. Puliafito, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
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Hu, G.

Hu, G. K.

J. Hu, X. Liu, and G. K. Hu, “Constraint condition on transformation relation for generalized acoustics,” Wave Motion 50(2), 170–179 (2013).
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Z. Chang, J. Hu, and G. K. Hu, “Transformation method and wave control,” Lixue Xuebao 26(6), 889–898 (2010).

Hu, J.

J. Chen, J. Hu, X. B. Yang, and X. Y. Lu, “Manipulating and detecting the chirpiness of spatial chirp signals via fractional Fourier lenses designed by transformation optics,” Appl. Opt. 56(32), 9119–9125 (2017).
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J. Hu, X. Liu, and G. K. Hu, “Constraint condition on transformation relation for generalized acoustics,” Wave Motion 50(2), 170–179 (2013).
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J. Hu and X. Y. Lu, “Determining the full transformation relations in the transformation method,” Appl. Phys., A Mater. Sci. Process. 109(4), 971–977 (2012).
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Z. Chang, J. Hu, and G. K. Hu, “Transformation method and wave control,” Lixue Xuebao 26(6), 889–898 (2010).

J. Hu, X. Zhou, and G. Hu, “Design method for electromagnetic cloak with arbitrary shapes based on Laplace’s equation,” Opt. Express 17(3), 1308–1320 (2009).
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D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, and C. Puliafito, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
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Isakov, D. V.

D. V. Isakov, Q. Lei, F. Castles, C. J. Stevens, C. R. M. Grovenor, and P. S. Grant, “3D printed anisotropic dielectric composite with meta-material features,” Mater. Des. 93, 423–430 (2016).
[Crossref]

Izatt, J.

Jiang, W. X.

W. X. Jiang, S. Ge, T. Han, S. Zhang, M. Q. Mehmood, C. W. Qiu, and T. J. Cui, “Shaping 3D Path of Electromagnetic Waves Using Gradient-Refractive-Index Metamaterials,” Adv. Sci. (Weinh.) 3(8), 1600022 (2016).
[Crossref] [PubMed]

W. X. Jiang, C. W. Qiu, T. C. Han, Q. Cheng, H. F. Ma, S. Zhang, and T. J. Cui, “Broadband all-dielectric magnifying lens for far-field high-resolution imaging,” Adv. Mater. 25(48), 6963–6968 (2013).
[Crossref] [PubMed]

Joseph, J.

Kaiser, S.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[Crossref] [PubMed]

Kelly, D.

Kivshar, Y. S.

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11(11), 917–924 (2012).
[Crossref] [PubMed]

Kosek, W.

Kutay, M. A.

Lang, J.

Lassas, M.

A. Greenleaf, M. Lassas, and G. Uhlmann, “On non-uniqueness for Calderón’s inverse problem,” Math. Res. Lett. 10(5), 685–693 (2003).
[Crossref]

Lei, Q.

D. V. Isakov, Q. Lei, F. Castles, C. J. Stevens, C. R. M. Grovenor, and P. S. Grant, “3D printed anisotropic dielectric composite with meta-material features,” Mater. Des. 93, 423–430 (2016).
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U. Leonhardt, “Optical conformal mapping,” Science 312(5781), 1777–1780 (2006).
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D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, and C. Puliafito, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Liu, N.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[Crossref] [PubMed]

Liu, S.

Liu, X.

J. Hu, X. Liu, and G. K. Hu, “Constraint condition on transformation relation for generalized acoustics,” Wave Motion 50(2), 170–179 (2013).
[Crossref]

Liu, Y.

Y. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011).
[Crossref] [PubMed]

Lohmann, A.

Lohmann, A. W.

Lu, X. Y.

J. Chen, J. Hu, X. B. Yang, and X. Y. Lu, “Manipulating and detecting the chirpiness of spatial chirp signals via fractional Fourier lenses designed by transformation optics,” Appl. Opt. 56(32), 9119–9125 (2017).
[Crossref] [PubMed]

J. Hu and X. Y. Lu, “Determining the full transformation relations in the transformation method,” Appl. Phys., A Mater. Sci. Process. 109(4), 971–977 (2012).
[Crossref]

Luo, Y.

J. B. Pendry, Y. Luo, and R. Zhao, “Transforming the optical landscape,” Science 348(6234), 521–524 (2015).
[Crossref] [PubMed]

Ma, H. F.

W. X. Jiang, C. W. Qiu, T. C. Han, Q. Cheng, H. F. Ma, S. Zhang, and T. J. Cui, “Broadband all-dielectric magnifying lens for far-field high-resolution imaging,” Adv. Mater. 25(48), 6963–6968 (2013).
[Crossref] [PubMed]

MacDonald, E.

E. MacDonald and R. Wicker, “Multiprocess 3D printing for increasing component functionality,” Science 353(6307), aaf2093 (2016).
[Crossref] [PubMed]

Malinauskas, M.

A. Žukauskas, I. Matulaitiene, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
[Crossref]

Matulaitiene, I.

A. Žukauskas, I. Matulaitiene, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
[Crossref]

Mehmood, M. Q.

W. X. Jiang, S. Ge, T. Han, S. Zhang, M. Q. Mehmood, C. W. Qiu, and T. J. Cui, “Shaping 3D Path of Electromagnetic Waves Using Gradient-Refractive-Index Metamaterials,” Adv. Sci. (Weinh.) 3(8), 1600022 (2016).
[Crossref] [PubMed]

Mendlovic, D.

Millan, M. S.

M. S. Millan, C. Ferreira, A. Pons, and P. Andres, “Application of anamorphic systems to directional pseudocolor encoding,” Opt. Eng. 27(2), 129–134 (1988).
[Crossref]

Moya, A.

C. Ferreira, M. J. Buades, and A. Moya, “Anamorphic correlator for character recognition. Detection of characters of different size,” J. Opt. 20(4), 181–185 (1989).
[Crossref]

Niaura, G.

A. Žukauskas, I. Matulaitiene, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
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N. K. Nischal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4–6), 253–259 (2004).

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A. Sahin, M. A. Kutay, and H. M. Ozaktas, “Nonseparable two-dimensional fractional fourier transform,” Appl. Opt. 37(23), 5444–5453 (1998).
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M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).

H. M. Ozaktas and D. Mendlovic, “Fractional Fourier Optics,” J. Opt. Soc. Am. A 12(4), 743–751 (1995).
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D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33(26), 6188–6193 (1994).
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D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation I,” J. Opt. Soc. Am. A 10(9), 1875–1881 (1993).
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H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101(3–4), 163–169 (1993).
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H. M. Ozaktas and D. Mendlovic, “Fourier transforms of fractional order and their optical interpretation,” Opt. Commun. 101(3–4), 163–169 (1993).
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Ozaktaz, H. M.

Paipulas, D.

A. Žukauskas, I. Matulaitiene, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
[Crossref]

Pendry, J. B.

J. B. Pendry, Y. Luo, and R. Zhao, “Transforming the optical landscape,” Science 348(6234), 521–524 (2015).
[Crossref] [PubMed]

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

Pons, A.

M. S. Millan, C. Ferreira, A. Pons, and P. Andres, “Application of anamorphic systems to directional pseudocolor encoding,” Opt. Eng. 27(2), 129–134 (1988).
[Crossref]

E. Bonet, C. Ferreira, P. Andres, and A. Pons, “Nonsymmetrical Fourier correlator to increase the angular discrimination in character recognition,” Opt. Commun. 58(3), 155–160 (1986).
[Crossref]

Puliafito, C.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, and C. Puliafito, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Qiu, C. W.

W. X. Jiang, S. Ge, T. Han, S. Zhang, M. Q. Mehmood, C. W. Qiu, and T. J. Cui, “Shaping 3D Path of Electromagnetic Waves Using Gradient-Refractive-Index Metamaterials,” Adv. Sci. (Weinh.) 3(8), 1600022 (2016).
[Crossref] [PubMed]

W. X. Jiang, C. W. Qiu, T. C. Han, Q. Cheng, H. F. Ma, S. Zhang, and T. J. Cui, “Broadband all-dielectric magnifying lens for far-field high-resolution imaging,” Adv. Mater. 25(48), 6963–6968 (2013).
[Crossref] [PubMed]

C. W. Qiu, A. Akbarzadeh, T. C. Han, and A. J. Danner, “Photorealistic rendering of a graded negative-index metamaterial magnifier,” New J. Phys. 14(3), 033024 (2012).
[Crossref]

Ran, Q.

L. Yuan, Q. Ran, and T. Zhao, “Image authentication based on double-image encryption and partial phase decryption in nonseparable fractional Fourier domain,” Opt. Laser Technol. 88, 111–120 (2017).
[Crossref]

Q. Ran, L. Yuan, and T. Zhao, “Image encryption based on nonseparable fractional Fourier transform and chaotic map,” Opt. Commun. 348, 43–49 (2015).
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B. Zhu, S. Liu, and Q. Ran, “Optical image encryption based on multifractional Fourier transforms,” Opt. Lett. 25(16), 1159–1161 (2000).
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Sahin, A.

A. Sahin, M. A. Kutay, and H. M. Ozaktas, “Nonseparable two-dimensional fractional fourier transform,” Appl. Opt. 37(23), 5444–5453 (1998).
[Crossref] [PubMed]

M. F. Erden, H. M. Ozaktas, A. Sahin, and D. Mendlovic, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).

Sarunic, M.

Schuman, J.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, and C. Puliafito, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Schurig, D.

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

Schweizer, H.

N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7(1), 31–37 (2008).
[Crossref] [PubMed]

Sheridan, J. T.

Singh, K.

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
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Soukoulis, C. M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three dimensional photonic metamaterials,” Nat. Photonics 5(9), 523–530 (2011).
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Stevens, C. J.

D. V. Isakov, Q. Lei, F. Castles, C. J. Stevens, C. R. M. Grovenor, and P. S. Grant, “3D printed anisotropic dielectric composite with meta-material features,” Mater. Des. 93, 423–430 (2016).
[Crossref]

Stinson, W.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, and C. Puliafito, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Swanson, E.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, and C. Puliafito, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Szoplik, T.

Tao, R.

Uhlmann, G.

A. Greenleaf, M. Lassas, and G. Uhlmann, “On non-uniqueness for Calderón’s inverse problem,” Math. Res. Lett. 10(5), 685–693 (2003).
[Crossref]

Unnikrishnan, G.

Vazquez, C.

C. Ferreira and C. Vazquez, “Anamorphic multiple matched filter for character recognition, performance with signals of equal size,” J. Mod. Opt. 37(8), 1343–1354 (1990).
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Wang, Y.

Wegener, M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three dimensional photonic metamaterials,” Nat. Photonics 5(9), 523–530 (2011).
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Wicker, R.

E. MacDonald and R. Wicker, “Multiprocess 3D printing for increasing component functionality,” Science 353(6307), aaf2093 (2016).
[Crossref] [PubMed]

Xin, Y.

Yang, C.

Yang, X. B.

Yuan, L.

L. Yuan, Q. Ran, and T. Zhao, “Image authentication based on double-image encryption and partial phase decryption in nonseparable fractional Fourier domain,” Opt. Laser Technol. 88, 111–120 (2017).
[Crossref]

Q. Ran, L. Yuan, and T. Zhao, “Image encryption based on nonseparable fractional Fourier transform and chaotic map,” Opt. Commun. 348, 43–49 (2015).
[Crossref]

Zalevsky, Z.

Zhang, B.

B. Zhang, Y. X. Guo, H. Zirath, and Y. P. Zhang, “Investigation on 3-D-Printing Technologies for Millimeter-Wave and Terahertz Applications,” Proc. IEEE 105(4), 723–736 (2017).
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Zhang, S.

W. X. Jiang, S. Ge, T. Han, S. Zhang, M. Q. Mehmood, C. W. Qiu, and T. J. Cui, “Shaping 3D Path of Electromagnetic Waves Using Gradient-Refractive-Index Metamaterials,” Adv. Sci. (Weinh.) 3(8), 1600022 (2016).
[Crossref] [PubMed]

W. X. Jiang, C. W. Qiu, T. C. Han, Q. Cheng, H. F. Ma, S. Zhang, and T. J. Cui, “Broadband all-dielectric magnifying lens for far-field high-resolution imaging,” Adv. Mater. 25(48), 6963–6968 (2013).
[Crossref] [PubMed]

Zhang, X.

Y. Liu and X. Zhang, “Metamaterials: a new frontier of science and technology,” Chem. Soc. Rev. 40(5), 2494–2507 (2011).
[Crossref] [PubMed]

Zhang, Y. P.

B. Zhang, Y. X. Guo, H. Zirath, and Y. P. Zhang, “Investigation on 3-D-Printing Technologies for Millimeter-Wave and Terahertz Applications,” Proc. IEEE 105(4), 723–736 (2017).
[Crossref]

Zhao, R.

J. B. Pendry, Y. Luo, and R. Zhao, “Transforming the optical landscape,” Science 348(6234), 521–524 (2015).
[Crossref] [PubMed]

Zhao, T.

L. Yuan, Q. Ran, and T. Zhao, “Image authentication based on double-image encryption and partial phase decryption in nonseparable fractional Fourier domain,” Opt. Laser Technol. 88, 111–120 (2017).
[Crossref]

Q. Ran, L. Yuan, and T. Zhao, “Image encryption based on nonseparable fractional Fourier transform and chaotic map,” Opt. Commun. 348, 43–49 (2015).
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Zhao, Y.

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun. 3(1), 870 (2012).
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Zheludev, N. I.

N. I. Zheludev and Y. S. Kivshar, “From metamaterials to metadevices,” Nat. Mater. 11(11), 917–924 (2012).
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Zhou, X.

Zhu, B.

Zirath, H.

B. Zhang, Y. X. Guo, H. Zirath, and Y. P. Zhang, “Investigation on 3-D-Printing Technologies for Millimeter-Wave and Terahertz Applications,” Proc. IEEE 105(4), 723–736 (2017).
[Crossref]

Žukauskas, A.

A. Žukauskas, I. Matulaitiene, D. Paipulas, G. Niaura, M. Malinauskas, and R. Gadonas, “Tuning the refractive index in 3D direct laser writing lithography: towards GRIN microoptics,” Laser Photonics Rev. 9(6), 706–712 (2015).
[Crossref]

Adv. Mater. (1)

W. X. Jiang, C. W. Qiu, T. C. Han, Q. Cheng, H. F. Ma, S. Zhang, and T. J. Cui, “Broadband all-dielectric magnifying lens for far-field high-resolution imaging,” Adv. Mater. 25(48), 6963–6968 (2013).
[Crossref] [PubMed]

Adv. Sci. (Weinh.) (1)

W. X. Jiang, S. Ge, T. Han, S. Zhang, M. Q. Mehmood, C. W. Qiu, and T. J. Cui, “Shaping 3D Path of Electromagnetic Waves Using Gradient-Refractive-Index Metamaterials,” Adv. Sci. (Weinh.) 3(8), 1600022 (2016).
[Crossref] [PubMed]

Appl. Opt. (10)

D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, “Graded-index fibers, Wigner-distribution functions, and the fractional Fourier transform,” Appl. Opt. 33(26), 6188–6193 (1994).
[Crossref] [PubMed]

T. Szoplik, W. Kosek, and C. Ferreira, “Nonsymmetric Fourier transforming with an anamorphic system,” Appl. Opt. 23(6), 905–909 (1984).
[Crossref] [PubMed]

T. Szoplik and H. H. Arsenault, “Rotation-variant optical data processing using the 2-D nonsymmetric Fourier transform,” Appl. Opt. 24(2), 168–172 (1985).
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A. Sahin, M. A. Kutay, and H. M. Ozaktas, “Nonseparable two-dimensional fractional fourier transform,” Appl. Opt. 37(23), 5444–5453 (1998).
[Crossref] [PubMed]

P. Andres, C. Ferreira, and E. Bonet, “Fraunhofer diffraction patterns from apertures illuminated with nonparallel light in nonsymmetrical Fourier transformers,” Appl. Opt. 24(10), 1549–1552 (1985).
[Crossref] [PubMed]

T. Szoplik and H. H. Arsenault, “Shift and scale-invariant anamorphic Fourier correlator using multiple circular harmonic filters,” Appl. Opt. 24(19), 3179–3183 (1985).
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J. García, D. Mendlovic, Z. Zalevsky, and A. Lohmann, “Space-variant simultaneous detection of several objects by the use of multiple anamorphic fractional-Fourier-transform filters,” Appl. Opt. 35(20), 3945–3952 (1996).
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G. Unnikrishnan, J. Joseph, and K. Singh, “Fractional fourier domain encrypted holographic memory by use of an anamorphic optical system,” Appl. Opt. 40(2), 299–306 (2001).
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Figures (14)

Fig. 1
Fig. 1 The relation of the FrFT order p between the detection position b in the GRIN lens.
Fig. 2
Fig. 2 The sketch of the conformal mapping in a cross section of the lens. (a) The original space; (b) the mapped space, where the center of original circular area is mapped to a given position.
Fig. 3
Fig. 3 The positions of the anamorphic centers from the input facet to the output facet in the lens are shown by the red dashed line.
Fig. 4
Fig. 4 Comparison of the ideal mapped wave fields with the outputs of the AFrFT lens. The lines in the field are the contour lines. (a) The output of a circle function in the normal GRIN lens is obtained by the wave optics module of COMSOL Multiphysics, which is the source image requiring transformation; (b) and (d) the ideal mappings of the source image obtained by directly using Eq. (4) through MATLAB, and the anamorphic centers are (0.2 mm, 0.2 mm) and (0.4 mm, 0 mm), respectively; (c) and (e) the output results of the same input as (a) in the designed AFrFT lens obtained by wave optics module of COMSOL Multiphysics, and the anamorphic centers at the output facet of the lens is designed at (0.2 mm, 0.2 mm) and (0.4 mm, 0 mm), respectively.
Fig. 5
Fig. 5 Comparison of the distinguishability of the conventional GRIN lens with that of the AFrFT lens. (a) and (e) The distribution of the wave fields in the output facets of conventional GRIN lenses with FrFT orders of 1 and 0.9, respectively. (c) and (g) The corresponding amplitude distributions along the lines in (a) and (e), respectively. (b) and (f) The distribution of the wave fields in the output facets of the AFrFT lenses with FrFT orders of 1 and 0.9, respectively. (d) and (h) The corresponding amplitude distributions along the lines in (b) and (f), respectively.
Fig. 6
Fig. 6 Sketch of the matched filter correlation recognition based on a 4f system.
Fig. 7
Fig. 7 The block diagrams of the two types of matched filter correlation recognition systems. (a) The traditional matched filter correlation recognition systems, system A; and (b) the matched filter correlation recognition systems based on the proposed AFrFT, system B.
Fig. 8
Fig. 8 Two images with nuance used as the inputs of both of the systems. (a) Image I1; and (b) image I2.
Fig. 9
Fig. 9 The result of matching correlation recognition of system A. (a)(c)€, and (g): input image is I1; (b)(d)(f), and (h): input image is I2. (a) and (b) the spectra of the input image; (c) and (d) the synthesis spectra of the input image and the temple(SS*); (e) and (f) the output wave fields; and (g) and (h) the 3D contour plots of the correlation peaks.
Fig. 10
Fig. 10 The result of matching correlation recognition of system B. (a), (c), (e) and (g), and (g): input image is I1; (b)(d)(f), and (h): input image is I2. (a) and (b) the spectra of the input image; (c) and (d) the synthesis spectra of the input images and the temples; (e) and (f) the output wave fields; and (g) and (h) the 3D contour plots of the correlation peaks.
Fig. 11
Fig. 11 The device diagram of the encryption system. (a) The traditional encryption system and (b) the proposed encryption system, where the conventional FrFT is replaced by the designed AFrFT.
Fig. 12
Fig. 12 Flow charts of the encryption and decryption process of the sample.
Fig. 13
Fig. 13 Simulation results of the encryption system. (a) Original image; (b) encrypted image; (c) reconstructed image with all correct keys; (d)-(k) reconstructed images with correct FrFT order but with just one correct coordinate of the anamorphic center, where (d)-(g) use the incorrect radial coordinate keys, and (h)-(k) use the incorrect angular coordinate keys.
Fig. 14
Fig. 14 MSE plotted as a function of errors in the decryption keys. (a) MSE with respect to the FrFT order error, (b) MSE with respect to the angular coordinate error of the anamorphic center, and (c) MSE with respect to the radial coordinate error of the anamorphic center.

Tables (2)

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Table 1 Comparison of the ideal mapping with the results obtained by the AFrFT lens

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Table 2 Comparison of the correlation peak values between systems A and B

Equations (11)

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g p (u,v)= F p [ f( x,y ) ](u,v)= f( x,y ) K p ( x,y;u,v ) dxdy,
K px ( x,u )={ A p exp( 2π s 2 ixucscα)exp[ i π s 2 ( x 2 + u 2 )cotα ],αnπ; δ(ux), α=2nπ; δ(u+x), α=(2n+1)π,
b=pL
w= z+a 1+ a ¯ z
u=f(u',v')= ( u +l)(1+l u +m v )( v +m)(m u l v ) (1+l u +m v ) 2 + (m u l v ) 2 , v=g(u',v')= ( v +m)(1+l u +m v )+( u +l)(m u l v ) (1+l u +m v ) 2 + (m u l v ) 2 ,
l=l(z)= l 0 L 2 z 2 , m=m(z)= m 0 L 2 z 2 ,
K p ( x,y;u,v )={ A p 2 exp{ix[ ζ(ul)η(vm) ζ 2 + η 2 ]cscα} ×exp[ 1 2 i( x 2 + [ ζ(ul)η(vm) ζ 2 + η 2 ] 2 )cotα ], αnπ; ×exp{iy[ ζ(vm)+η(ul) ζ 2 + η 2 ]cscα} ×exp[ 1 2 i( y 2 + [ ζ(vm)+η(ul) ζ 2 + η 2 ] 2 )cotα ] δ(ux)δ(vy), α=2nπ; δ(u+x)δ(v+y), α=(2n+1)π,
ε'= ε 0 diag[ λ 1 λ 2 λ 3 , λ 2 λ 1 λ 3 , λ 3 λ 1 λ 2 ], μ'= μ 0 diag[ λ 1 λ 2 λ 3 , λ 2 λ 1 λ 3 , λ 3 λ 1 λ 2 ],
ε'=diag[ ε 0 , ε 0 , 1 λ 2 ε 0 ], μ'=diag[ μ 0 , μ 0 , 1 λ 2 μ 0 ],
( 1 n 2 1 n c 2 )[( 1 n 2 1 n z 2 ) sin 2 θ+( 1 n 2 1 n c 2 ) cos 2 θ]=0,
MSE= 1 M×N i=1 M j=1 N | h 2 (i,j) h 1 (i,j) | 2

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