Abstract

Based on conformal mapping method, a two dimensional, multi-functional lens structure is proposed and designed in this work. The lens is an infinitely-long, gradient-index dielectric cylinder with a semi-elliptic cross-section. The lens can first be considered like a flattened Luneburg lens, which produces highly-directive electromagnetic waves by adjusting the feed position along the line connecting the two foci. It also functions like an Eaton lens. When an incoming beam impinges on the same line but outside the two foci, it will be guided through the lens structure and take a U-turn. Besides, if properly shaped, the structure can also be used as a waveguide bend. The lens can be realized using non-resonant metamaterials with inhomogeneous hole arrays. Simulation results demonstrate excellent performance of the lens and agree well with theoretical prediction. The designed lens can be used in the electromagnetic control. And it is especially useful in the real optical lens system.

© 2015 Optical Society of America

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References

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2014 (1)

F. Yang, Z. L. Mei, and T. J. Cui, “Design and experiment of perfect relay lens based on the Schwarz-Christoffel mapping,” Appl. Phys. Lett. 104, 073510 (2014).
[Crossref]

2013 (2)

H. Y. Chen, Y. D. Xu, H. Li, and T. Tyc, “Playing the tricks of numbers of light sources,” New J. Phys. 15, 093034 (2013).
[Crossref]

K. Hosseini and Z. Atlasbaf, “Analysis and synthesis of singly-curved microstrip structures utilizing modified Schwarz-Christoffel transformation,” IEEE Trans. Antennas Propag. 61(12), 5940–5947 (2013).
[Crossref]

2012 (3)

2011 (6)

K. Yao and X. Jiang, “Designing feasible optical devices via conformal mapping,” J. Opt. Soc. Am. B 28(5), 1037–1042 (2011).
[Crossref]

L. Tang, J. Yin, G. Yuan, J. Du, H. Gao, X. Dong, Y. Lu, and C. Du, “General conformal transformation method based on Schwarz-Christoffel approach,” Opt. Express 19(16), 15119–15126 (2011).
[Crossref] [PubMed]

C. García-Meca, A. Martínez, and U. Leonhardt, “Engineering antenna radiation patterns via quasi-conformal mappings,” Opt. Express 19(24), 23743–23750 (2011).
[Crossref] [PubMed]

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Moulding the flow of surface plasmons using conformal and quasiconformal mapping,” New J. Phys. 13, 033011 (2011).
[Crossref]

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
[Crossref]

H. Y. Chen, U. Leonhardt, and T. Tyc, “Conformal cloak for waves,” Phys. Rev. A 83(5), 055801 (2011).
[Crossref]

2010 (5)

2009 (1)

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53(08), 69–152 (2009).
[Crossref]

2008 (1)

2006 (3)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(23), 1780–1782 (2006).
[Crossref] [PubMed]

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

U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8, 118 (2006).
[Crossref]

2002 (2)

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflections and transmission coefficient,” Phys. Rev. B 65, 195104 (2002).
[Crossref]

W. Wen, L. Zhou, J. Li, W. Ge, C. T. Chan, and P. Sheng, “Subwavelength photonic band gaps from planar fractals,” Phys. Rev. Lett. 89, 223901 (2002).
[Crossref] [PubMed]

1996 (1)

M. L. Wu, P. L. Fan, J. M. Hsu, and C. T. Lee, “Design of ideal structures for lossless bends in optical waveguides by conformal mapping,” IEEE J. Lightw. Technol. 14(11), 2604–2614 (1996).
[Crossref]

1952 (1)

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

Atlasbaf, Z.

K. Hosseini and Z. Atlasbaf, “Analysis and synthesis of singly-curved microstrip structures utilizing modified Schwarz-Christoffel transformation,” IEEE Trans. Antennas Propag. 61(12), 5940–5947 (2013).
[Crossref]

Aubry, A.

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82(20), 205109 (2010).
[Crossref]

Bowen, P.

Chan, C. T.

W. Wen, L. Zhou, J. Li, W. Ge, C. T. Chan, and P. Sheng, “Subwavelength photonic band gaps from planar fractals,” Phys. Rev. Lett. 89, 223901 (2002).
[Crossref] [PubMed]

Chen, H.

X. Jiang, K. Yao, Q. Wu, Y. Xu, and H. Chen, “Conformal transformations to achieve unidirectional behavior of light,” New J. Phys. 14, 053023 (2012).
[Crossref]

Chen, H. Y.

H. Y. Chen, Y. D. Xu, H. Li, and T. Tyc, “Playing the tricks of numbers of light sources,” New J. Phys. 15, 093034 (2013).
[Crossref]

H. Y. Chen, U. Leonhardt, and T. Tyc, “Conformal cloak for waves,” Phys. Rev. A 83(5), 055801 (2011).
[Crossref]

Cui, J.

Cui, T. J.

F. Yang, Z. L. Mei, and T. J. Cui, “Design and experiment of perfect relay lens based on the Schwarz-Christoffel mapping,” Appl. Phys. Lett. 104, 073510 (2014).
[Crossref]

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optical lens,” Nat. Commun. 1, 124 (2010).
[Crossref]

Dhar, S.

Dong, X.

Du, C.

Du, J.

Eaton, J. E.

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

Fan, P. L.

M. L. Wu, P. L. Fan, J. M. Hsu, and C. T. Lee, “Design of ideal structures for lossless bends in optical waveguides by conformal mapping,” IEEE J. Lightw. Technol. 14(11), 2604–2614 (1996).
[Crossref]

Gan, Z.

Gao, H.

García-Meca, C.

García-Vidal, F. J.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Moulding the flow of surface plasmons using conformal and quasiconformal mapping,” New J. Phys. 13, 033011 (2011).
[Crossref]

Ge, W.

W. Wen, L. Zhou, J. Li, W. Ge, C. T. Chan, and P. Sheng, “Subwavelength photonic band gaps from planar fractals,” Phys. Rev. Lett. 89, 223901 (2002).
[Crossref] [PubMed]

Halimeh, J. C.

Hosseini, K.

K. Hosseini and Z. Atlasbaf, “Analysis and synthesis of singly-curved microstrip structures utilizing modified Schwarz-Christoffel transformation,” IEEE Trans. Antennas Propag. 61(12), 5940–5947 (2013).
[Crossref]

Hsu, J. M.

M. L. Wu, P. L. Fan, J. M. Hsu, and C. T. Lee, “Design of ideal structures for lossless bends in optical waveguides by conformal mapping,” IEEE J. Lightw. Technol. 14(11), 2604–2614 (1996).
[Crossref]

Hu, R.

Huidobro, P. A.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Moulding the flow of surface plasmons using conformal and quasiconformal mapping,” New J. Phys. 13, 033011 (2011).
[Crossref]

Hunt, J.

Jiang, X.

X. Jiang, K. Yao, Q. Wu, Y. Xu, and H. Chen, “Conformal transformations to achieve unidirectional behavior of light,” New J. Phys. 14, 053023 (2012).
[Crossref]

K. Yao and X. Jiang, “Designing feasible optical devices via conformal mapping,” J. Opt. Soc. Am. B 28(5), 1037–1042 (2011).
[Crossref]

Jiang, Z. H.

Jokerst, N. M.

Kundtz, N. B.

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
[Crossref]

Larouche, S.

Lee, C. T.

M. L. Wu, P. L. Fan, J. M. Hsu, and C. T. Lee, “Design of ideal structures for lossless bends in optical waveguides by conformal mapping,” IEEE J. Lightw. Technol. 14(11), 2604–2614 (1996).
[Crossref]

Lei, D. Y.

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82(20), 205109 (2010).
[Crossref]

Leonhardt, U.

H. Y. Chen, U. Leonhardt, and T. Tyc, “Conformal cloak for waves,” Phys. Rev. A 83(5), 055801 (2011).
[Crossref]

C. García-Meca, A. Martínez, and U. Leonhardt, “Engineering antenna radiation patterns via quasi-conformal mappings,” Opt. Express 19(24), 23743–23750 (2011).
[Crossref] [PubMed]

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53(08), 69–152 (2009).
[Crossref]

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

U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8, 118 (2006).
[Crossref]

Li, H.

H. Y. Chen, Y. D. Xu, H. Li, and T. Tyc, “Playing the tricks of numbers of light sources,” New J. Phys. 15, 093034 (2013).
[Crossref]

Li, J.

W. Wen, L. Zhou, J. Li, W. Ge, C. T. Chan, and P. Sheng, “Subwavelength photonic band gaps from planar fractals,” Phys. Rev. Lett. 89, 223901 (2002).
[Crossref] [PubMed]

Lin, L.

Liu, S.

Lu, Y.

Luo, X.

Ma, H. F.

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optical lens,” Nat. Commun. 1, 124 (2010).
[Crossref]

Ma, Y. G.

Maier, S. A.

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82(20), 205109 (2010).
[Crossref]

Markos, P.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflections and transmission coefficient,” Phys. Rev. B 65, 195104 (2002).
[Crossref]

Martínez, A.

Martín-Moreno, L.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Moulding the flow of surface plasmons using conformal and quasiconformal mapping,” New J. Phys. 13, 033011 (2011).
[Crossref]

Massoud, A. T.

Mei, Z. L.

F. Yang, Z. L. Mei, and T. J. Cui, “Design and experiment of perfect relay lens based on the Schwarz-Christoffel mapping,” Appl. Phys. Lett. 104, 073510 (2014).
[Crossref]

Nesterov, M. L.

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Moulding the flow of surface plasmons using conformal and quasiconformal mapping,” New J. Phys. 13, 033011 (2011).
[Crossref]

Ong, C. K.

Pendry, J. B.

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
[Crossref]

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82(20), 205109 (2010).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(23), 1780–1782 (2006).
[Crossref] [PubMed]

Philbin, T. G.

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53(08), 69–152 (2009).
[Crossref]

Qin, Z.

Schmied, R.

Schultz, S.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflections and transmission coefficient,” Phys. Rev. B 65, 195104 (2002).
[Crossref]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(23), 1780–1782 (2006).
[Crossref] [PubMed]

Sheng, P.

W. Wen, L. Zhou, J. Li, W. Ge, C. T. Chan, and P. Sheng, “Subwavelength photonic band gaps from planar fractals,” Phys. Rev. Lett. 89, 223901 (2002).
[Crossref] [PubMed]

Smith, D. R.

J. Hunt, T. Tyler, S. Dhar, Y. J. Tsai, P. Bowen, S. Larouche, N. M. Jokerst, and D. R. Smith, “Planar, flattened Luneburg lens at infrared wavelengths,” Opt. Express 20(2), 1706–1713 (2012).
[Crossref] [PubMed]

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(23), 1780–1782 (2006).
[Crossref] [PubMed]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflections and transmission coefficient,” Phys. Rev. B 65, 195104 (2002).
[Crossref]

Soukoulis, C. M.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflections and transmission coefficient,” Phys. Rev. B 65, 195104 (2002).
[Crossref]

Tang, L.

Tsai, Y. J.

Turpin, J. P.

Tyc, T.

H. Y. Chen, Y. D. Xu, H. Li, and T. Tyc, “Playing the tricks of numbers of light sources,” New J. Phys. 15, 093034 (2013).
[Crossref]

H. Y. Chen, U. Leonhardt, and T. Tyc, “Conformal cloak for waves,” Phys. Rev. A 83(5), 055801 (2011).
[Crossref]

Tyler, T.

Urzhumov, Y. A.

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
[Crossref]

Wang, N.

Wang, W.

Wegener, M.

Wen, W.

W. Wen, L. Zhou, J. Li, W. Ge, C. T. Chan, and P. Sheng, “Subwavelength photonic band gaps from planar fractals,” Phys. Rev. Lett. 89, 223901 (2002).
[Crossref] [PubMed]

Werner, D. H.

Werner, P. L.

Wu, B.

Wu, M. L.

M. L. Wu, P. L. Fan, J. M. Hsu, and C. T. Lee, “Design of ideal structures for lossless bends in optical waveguides by conformal mapping,” IEEE J. Lightw. Technol. 14(11), 2604–2614 (1996).
[Crossref]

Wu, Q.

X. Jiang, K. Yao, Q. Wu, Y. Xu, and H. Chen, “Conformal transformations to achieve unidirectional behavior of light,” New J. Phys. 14, 053023 (2012).
[Crossref]

Xu, Y.

X. Jiang, K. Yao, Q. Wu, Y. Xu, and H. Chen, “Conformal transformations to achieve unidirectional behavior of light,” New J. Phys. 14, 053023 (2012).
[Crossref]

Xu, Y. D.

H. Y. Chen, Y. D. Xu, H. Li, and T. Tyc, “Playing the tricks of numbers of light sources,” New J. Phys. 15, 093034 (2013).
[Crossref]

Yang, F.

F. Yang, Z. L. Mei, and T. J. Cui, “Design and experiment of perfect relay lens based on the Schwarz-Christoffel mapping,” Appl. Phys. Lett. 104, 073510 (2014).
[Crossref]

Yang, X.

Yao, K.

X. Jiang, K. Yao, Q. Wu, Y. Xu, and H. Chen, “Conformal transformations to achieve unidirectional behavior of light,” New J. Phys. 14, 053023 (2012).
[Crossref]

K. Yao and X. Jiang, “Designing feasible optical devices via conformal mapping,” J. Opt. Soc. Am. B 28(5), 1037–1042 (2011).
[Crossref]

Yin, J.

Yuan, G.

Zheng, H.

Zhou, L.

W. Wen, L. Zhou, J. Li, W. Ge, C. T. Chan, and P. Sheng, “Subwavelength photonic band gaps from planar fractals,” Phys. Rev. Lett. 89, 223901 (2002).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

F. Yang, Z. L. Mei, and T. J. Cui, “Design and experiment of perfect relay lens based on the Schwarz-Christoffel mapping,” Appl. Phys. Lett. 104, 073510 (2014).
[Crossref]

IEEE IRE Trans. Antennas Propag. (1)

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

IEEE J. Lightw. Technol. (1)

M. L. Wu, P. L. Fan, J. M. Hsu, and C. T. Lee, “Design of ideal structures for lossless bends in optical waveguides by conformal mapping,” IEEE J. Lightw. Technol. 14(11), 2604–2614 (1996).
[Crossref]

IEEE Trans. Antennas Propag. (1)

K. Hosseini and Z. Atlasbaf, “Analysis and synthesis of singly-curved microstrip structures utilizing modified Schwarz-Christoffel transformation,” IEEE Trans. Antennas Propag. 61(12), 5940–5947 (2013).
[Crossref]

J. Opt. (1)

Y. A. Urzhumov, N. B. Kundtz, D. R. Smith, and J. B. Pendry, “Cross-section comparisons of cloaks designed by transformation optical and optical conformal mapping approaches,” J. Opt. 13, 024002 (2011).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Nat. Commun. (1)

H. F. Ma and T. J. Cui, “Three-dimensional broadband and broad-angle transformation-optical lens,” Nat. Commun. 1, 124 (2010).
[Crossref]

New J. Phys. (4)

X. Jiang, K. Yao, Q. Wu, Y. Xu, and H. Chen, “Conformal transformations to achieve unidirectional behavior of light,” New J. Phys. 14, 053023 (2012).
[Crossref]

H. Y. Chen, Y. D. Xu, H. Li, and T. Tyc, “Playing the tricks of numbers of light sources,” New J. Phys. 15, 093034 (2013).
[Crossref]

U. Leonhardt, “Notes on conformal invisibility devices,” New J. Phys. 8, 118 (2006).
[Crossref]

P. A. Huidobro, M. L. Nesterov, L. Martín-Moreno, and F. J. García-Vidal, “Moulding the flow of surface plasmons using conformal and quasiconformal mapping,” New J. Phys. 13, 033011 (2011).
[Crossref]

Opt. Express (7)

Phys. Rev. A (1)

H. Y. Chen, U. Leonhardt, and T. Tyc, “Conformal cloak for waves,” Phys. Rev. A 83(5), 055801 (2011).
[Crossref]

Phys. Rev. B (2)

A. Aubry, D. Y. Lei, S. A. Maier, and J. B. Pendry, “Conformal transformation applied to plasmonics beyond the quasistatic limit,” Phys. Rev. B 82(20), 205109 (2010).
[Crossref]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflections and transmission coefficient,” Phys. Rev. B 65, 195104 (2002).
[Crossref]

Phys. Rev. Lett. (1)

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[Crossref] [PubMed]

Prog. Opt. (1)

U. Leonhardt and T. G. Philbin, “Transformation optics and the geometry of light,” Prog. Opt. 53(08), 69–152 (2009).
[Crossref]

Science (2)

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[Crossref] [PubMed]

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Figures (9)

Fig. 1
Fig. 1 The principle diagram of the conformal mapping between rectangular and elliptic grids. (a) the virtual space in the rectangular system; (b) the physical space in the elliptic coordinate system.
Fig. 2
Fig. 2 The refractive index distribution in xoy plane and the ray paths in the lens structure. (a) the refractive index in the lens; (b) and (c), ray traces inside the lens.
Fig. 3
Fig. 3 The full wave simulated z components of electric fields at 18GHz in xoy plane, where the feeding source is located at (a) x = 0, y = 0; (b) x = 20mm, y = 0; (c) x = 29mm, y = 0; and (d) x = 51.5mm, y = 0.
Fig. 4
Fig. 4 Far field patterns for the transmitted waves. (a)–(d) corresponds to the radiation pattern of each plot in Fig. 3, respectively.
Fig. 5
Fig. 5 The wave guide bend using a half lens structure. (a) the simulated z component of electric field in the wave guide bend; (b) the far field pattern corresponding to (a).
Fig. 6
Fig. 6 Relations between refraction index and the hole’s radius. (a) The refractive index analysis of the unit cell with 2 mm thickness when the dielectric constant is 2.2, 4. 4 and 12.9 respectively; (b) The refractive index analysis of the unit cell with 1mm thickness when the dielectric constant is 2.2.
Fig. 7
Fig. 7 the whole structure simulation diagram. (a–c) The distribution of electric field at the 12, 15 and 20GHz respectively when the feeding source is located at x = 20mm, y = 0; (d–f) The distribution of electric field at the 12, 15 and 20GHz respectively when the feeding source is located at x = 29mm, y = 0.
Fig. 8
Fig. 8 The parallel connection capacitor model.
Fig. 9
Fig. 9 The series connection capacitor. (a) The unit we adopt, (b) The equivalent series connection capacitor model.

Equations (7)

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w = c sin z = c sin x cosh y + i c cos x sinh y ,
{ u 2 c 2 cosh 2 y + v 2 c 2 sinh 2 y = 1 u 2 c 2 sin 2 x v 2 c 2 cos 2 x = 1 .
n = n 0 / | d w d z | .
n = 1 [ ( 1 ( u c ) 2 + ( v c ) 2 ) 2 + 4 ( u c ) 2 ( v c ) 2 ] 1 / 4 .
z = ± ( 1 + r ) 2 t 2 ( 1 r ) 2 t 2
Im ( n ) = ± Im ( cos 1 ( 1 2 t [ 1 ( r 2 t 2 ) ] ) k d )
Re ( n ) = ± Re ( cos 1 ( 1 2 t ) [ 1 ( r 2 t 2 ) ] k d ) + 2 π m k d

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