Abstract

A transformation-optics wedge waveguide designed for the simultaneous collection and directional collimation of pulsed dipole radiation is described and tested with numerical simulation. Azimuthal compression of free space toward a narrow fan-shaped waveguide sector allows dipole pulse radiation in free space to be transformed into a directional non-dispersive pulse propagating within that sector. The collection and collimation ability of the proposed structure is compared with classical approaches using metallic wedge mirrors and parabolic mirrors, which inherently allow multiple internal reflections and thus generate significant pulse distortion and low light-collection efficiency. It is shown that the optical pulse generated by the dipole and propagated through the proposed transformation-optics waveguide maintains its original shape within the structure, and demonstrates enhanced optical power.

© 2013 OSA

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  17. X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon gration structure,” Appl. Phys. Lett. 95(18), 184102 (2009).
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  24. D. H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” New J. Phys. 10(11), 115023 (2008).
    [Crossref]
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2013 (1)

N. Gregersen, P. Kaer, and J. Mørk, “Modeling and design of high-efficiency single-photon sources,” IEEE Sel. Top. Quantum Electron. 19(5), 9000516 (2013).
[Crossref]

2011 (3)

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. 36(13), 2530–2532 (2011).
[Crossref] [PubMed]

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Gőtzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics 5(3), 166–169 (2011).
[Crossref]

P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Coordinate-transformation-based ultra-directive emission,” Electron. Lett. 47(10), 580–582 (2011).
[Crossref]

2010 (5)

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A 81(3), 033837 (2010).
[Crossref]

J. P. Turpin, A. T. Massoud, Z. H. Jiang, P. L. Werner, and D. H. Werner, “Conformal mappings to achieve simple material parameters for transformation optics devices,” Opt. Express 18(1), 244–252 (2010).
[Crossref] [PubMed]

B. Wang and K.-M. Huang, “Shaping the radiation pattern with mu and epsilon-near-zero metamaterials,” PIER 106, 107–119 (2010).
[Crossref]

Y. Ebenstein and L. A. Bentolila, “Single-molecule detection: Focusing on the objective,” Nat. Nanotechnol. 5(2), 99–100 (2010).
[Crossref] [PubMed]

S. McDaniel and S. Blair, “Increased OLED radiative efficiency using a directive optical antenna,” Opt. Express 18(16), 17477–17483 (2010).
[Crossref] [PubMed]

2009 (3)

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3(12), 687–695 (2009).
[Crossref]

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon gration structure,” Appl. Phys. Lett. 95(18), 184102 (2009).
[Crossref]

D. H. Kwon and D. H. Werner, “Flat focusing lens designs having minimized reflection based on coordinate transformation techniques,” Opt. Express 17(10), 7807–7817 (2009).
[Crossref] [PubMed]

2008 (6)

J. J. Zhang, Y. Luo, S. Xi, H. S. Chen, L. X. Ran, B.-I. Wu, and J. A. Kong, “Directive emission obtained by coordinate transformation,” PIER 81, 437–446 (2008).
[Crossref]

D. H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” New J. Phys. 10(11), 115023 (2008).
[Crossref]

J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[Crossref] [PubMed]

M. Rahm, D. A. Roberts, J. B. Pendry, and D. R. Smith, “Transformation-optical design of adaptive beam bends and beam expanders,” Opt. Express 16(15), 11555–11567 (2008).
[Crossref] [PubMed]

D.-H. Kwon and D. H. Werner, “Polarization splitter and polarization rotator designs based on transformation optics,” Opt. Express 16(23), 18731–18738 (2008).
[Crossref] [PubMed]

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008).
[Crossref]

2007 (2)

2006 (5)

R. Zhou, X. Chen, and W. Lu, “Strong focusing properties and far-field focus in the two-dimensional photonic-crystal-based concave lens,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(1), 016610 (2006).
[Crossref] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14(21), 9794–9804 (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]

U. Leonhardt, “Note on conformal invisibility devices,” New J. Phys. 8(7), 118 (2006).
[Crossref]

1987 (1)

E. Yablonovitch, “Inhibited Spontaneous Emission In Solid-State Physics And Electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[Crossref] [PubMed]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Bentolila, L. A.

Y. Ebenstein and L. A. Bentolila, “Single-molecule detection: Focusing on the objective,” Nat. Nanotechnol. 5(2), 99–100 (2010).
[Crossref] [PubMed]

Blair, S.

Burokur, S. N.

P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Coordinate-transformation-based ultra-directive emission,” Electron. Lett. 47(10), 580–582 (2011).
[Crossref]

Chen, H. S.

J. J. Zhang, Y. Luo, S. Xi, H. S. Chen, L. X. Ran, B.-I. Wu, and J. A. Kong, “Directive emission obtained by coordinate transformation,” PIER 81, 437–446 (2008).
[Crossref]

Chen, X.

R. Zhou, X. Chen, and W. Lu, “Strong focusing properties and far-field focus in the two-dimensional photonic-crystal-based concave lens,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(1), 016610 (2006).
[Crossref] [PubMed]

Chen, X. W.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Gőtzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics 5(3), 166–169 (2011).
[Crossref]

de Lustrac, A.

P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Coordinate-transformation-based ultra-directive emission,” Electron. Lett. 47(10), 580–582 (2011).
[Crossref]

Dong, J. W.

Ebenstein, Y.

Y. Ebenstein and L. A. Bentolila, “Single-molecule detection: Focusing on the objective,” Nat. Nanotechnol. 5(2), 99–100 (2010).
[Crossref] [PubMed]

Eghlidi, H.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Gőtzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics 5(3), 166–169 (2011).
[Crossref]

Feng, Y.

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon gration structure,” Appl. Phys. Lett. 95(18), 184102 (2009).
[Crossref]

Furusawa, A.

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3(12), 687–695 (2009).
[Crossref]

Germain, D.

P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Coordinate-transformation-based ultra-directive emission,” Electron. Lett. 47(10), 580–582 (2011).
[Crossref]

Gotzinger, S.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Gőtzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics 5(3), 166–169 (2011).
[Crossref]

Gregersen, N.

N. Gregersen, P. Kaer, and J. Mørk, “Modeling and design of high-efficiency single-photon sources,” IEEE Sel. Top. Quantum Electron. 19(5), 9000516 (2013).
[Crossref]

Hao, Y.

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon gration structure,” Appl. Phys. Lett. 95(18), 184102 (2009).
[Crossref]

Huang, K.-M.

B. Wang and K.-M. Huang, “Shaping the radiation pattern with mu and epsilon-near-zero metamaterials,” PIER 106, 107–119 (2010).
[Crossref]

Jiang, T.

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon gration structure,” Appl. Phys. Lett. 95(18), 184102 (2009).
[Crossref]

Jiang, Z. H.

Kaer, P.

N. Gregersen, P. Kaer, and J. Mørk, “Modeling and design of high-efficiency single-photon sources,” IEEE Sel. Top. Quantum Electron. 19(5), 9000516 (2013).
[Crossref]

Kidwai, O.

Kong, J. A.

J. J. Zhang, Y. Luo, S. Xi, H. S. Chen, L. X. Ran, B.-I. Wu, and J. A. Kong, “Directive emission obtained by coordinate transformation,” PIER 81, 437–446 (2008).
[Crossref]

Kukura, P.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Gőtzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics 5(3), 166–169 (2011).
[Crossref]

Kwon, D. H.

D. H. Kwon and D. H. Werner, “Flat focusing lens designs having minimized reflection based on coordinate transformation techniques,” Opt. Express 17(10), 7807–7817 (2009).
[Crossref] [PubMed]

D. H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” New J. Phys. 10(11), 115023 (2008).
[Crossref]

Kwon, D.-H.

Lederer, F.

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A 81(3), 033837 (2010).
[Crossref]

Lee, K. G.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Gőtzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics 5(3), 166–169 (2011).
[Crossref]

Leonhardt, U.

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

U. Leonhardt, “Note on conformal invisibility devices,” New J. Phys. 8(7), 118 (2006).
[Crossref]

Lettow, R.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Gőtzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics 5(3), 166–169 (2011).
[Crossref]

Li, J.

J. Li and J. B. Pendry, “Hiding under the Carpet: A New Strategy for Cloaking,” Phys. Rev. Lett. 101(20), 203901 (2008).
[Crossref] [PubMed]

Liang, W. Y.

Lu, W.

R. Zhou, X. Chen, and W. Lu, “Strong focusing properties and far-field focus in the two-dimensional photonic-crystal-based concave lens,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(1), 016610 (2006).
[Crossref] [PubMed]

Luo, Y.

J. J. Zhang, Y. Luo, S. Xi, H. S. Chen, L. X. Ran, B.-I. Wu, and J. A. Kong, “Directive emission obtained by coordinate transformation,” PIER 81, 437–446 (2008).
[Crossref]

Massoud, A. T.

McDaniel, S.

Mørk, J.

N. Gregersen, P. Kaer, and J. Mørk, “Modeling and design of high-efficiency single-photon sources,” IEEE Sel. Top. Quantum Electron. 19(5), 9000516 (2013).
[Crossref]

O’Brien, J. L.

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3(12), 687–695 (2009).
[Crossref]

Pendry, J. B.

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Rahm, M.

Ran, L. X.

J. J. Zhang, Y. Luo, S. Xi, H. S. Chen, L. X. Ran, B.-I. Wu, and J. A. Kong, “Directive emission obtained by coordinate transformation,” PIER 81, 437–446 (2008).
[Crossref]

Renn, A.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Gőtzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics 5(3), 166–169 (2011).
[Crossref]

Roberts, D. A.

Rockstuhl, C.

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A 81(3), 033837 (2010).
[Crossref]

Sandoghdar, V.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Gőtzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics 5(3), 166–169 (2011).
[Crossref]

Schmiele, M.

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A 81(3), 033837 (2010).
[Crossref]

Schurig, D.

Segerink, F. B.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008).
[Crossref]

Shields, A. J.

A. J. Shields, “Semiconductor quantum light sources,” Nat. Photonics 1(4), 215–223 (2007).
[Crossref]

Sipe, J. E.

Smith, D. R.

Stefani, F. D.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008).
[Crossref]

Taminiau, T. H.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008).
[Crossref]

Tichit, P.-H.

P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Coordinate-transformation-based ultra-directive emission,” Electron. Lett. 47(10), 580–582 (2011).
[Crossref]

Turpin, J. P.

Van Hulst, N. F.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008).
[Crossref]

Varma, V. S.

M. Schmiele, V. S. Varma, C. Rockstuhl, and F. Lederer, “Designing optical elements from isotropic materials by using transformation optics,” Phys. Rev. A 81(3), 033837 (2010).
[Crossref]

Vuckovic, J.

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3(12), 687–695 (2009).
[Crossref]

Wang, B.

B. Wang and K.-M. Huang, “Shaping the radiation pattern with mu and epsilon-near-zero metamaterials,” PIER 106, 107–119 (2010).
[Crossref]

Wang, H. Z.

Werner, D. H.

Werner, P. L.

Wu, B.-I.

J. J. Zhang, Y. Luo, S. Xi, H. S. Chen, L. X. Ran, B.-I. Wu, and J. A. Kong, “Directive emission obtained by coordinate transformation,” PIER 81, 437–446 (2008).
[Crossref]

Xi, S.

J. J. Zhang, Y. Luo, S. Xi, H. S. Chen, L. X. Ran, B.-I. Wu, and J. A. Kong, “Directive emission obtained by coordinate transformation,” PIER 81, 437–446 (2008).
[Crossref]

Xu, X.

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon gration structure,” Appl. Phys. Lett. 95(18), 184102 (2009).
[Crossref]

Yablonovitch, E.

E. Yablonovitch, “Inhibited Spontaneous Emission In Solid-State Physics And Electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[Crossref] [PubMed]

Zhang, J. J.

J. J. Zhang, Y. Luo, S. Xi, H. S. Chen, L. X. Ran, B.-I. Wu, and J. A. Kong, “Directive emission obtained by coordinate transformation,” PIER 81, 437–446 (2008).
[Crossref]

Zhao, J.

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon gration structure,” Appl. Phys. Lett. 95(18), 184102 (2009).
[Crossref]

Zhou, R.

R. Zhou, X. Chen, and W. Lu, “Strong focusing properties and far-field focus in the two-dimensional photonic-crystal-based concave lens,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(1), 016610 (2006).
[Crossref] [PubMed]

Zhukovsky, S. V.

Appl. Phys. Lett. (1)

X. Xu, Y. Feng, Y. Hao, J. Zhao, and T. Jiang, “Infrared carpet cloak designed with uniform silicon gration structure,” Appl. Phys. Lett. 95(18), 184102 (2009).
[Crossref]

Electron. Lett. (1)

P.-H. Tichit, S. N. Burokur, D. Germain, and A. de Lustrac, “Coordinate-transformation-based ultra-directive emission,” Electron. Lett. 47(10), 580–582 (2011).
[Crossref]

IEEE Sel. Top. Quantum Electron. (1)

N. Gregersen, P. Kaer, and J. Mørk, “Modeling and design of high-efficiency single-photon sources,” IEEE Sel. Top. Quantum Electron. 19(5), 9000516 (2013).
[Crossref]

Nat. Nanotechnol. (1)

Y. Ebenstein and L. A. Bentolila, “Single-molecule detection: Focusing on the objective,” Nat. Nanotechnol. 5(2), 99–100 (2010).
[Crossref] [PubMed]

Nat. Photonics (4)

J. L. O’Brien, A. Furusawa, and J. Vučković, “Photonic quantum technologies,” Nat. Photonics 3(12), 687–695 (2009).
[Crossref]

A. J. Shields, “Semiconductor quantum light sources,” Nat. Photonics 1(4), 215–223 (2007).
[Crossref]

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Gőtzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics 5(3), 166–169 (2011).
[Crossref]

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008).
[Crossref]

New J. Phys. (2)

D. H. Kwon and D. H. Werner, “Transformation optical designs for wave collimators, flat lenses and right-angle bends,” New J. Phys. 10(11), 115023 (2008).
[Crossref]

U. Leonhardt, “Note on conformal invisibility devices,” New J. Phys. 8(7), 118 (2006).
[Crossref]

Opt. Express (7)

Opt. Lett. (1)

Phys. Rev. (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).

Phys. Rev. A (1)

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[Crossref]

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

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[Crossref]

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[Crossref]

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Figures (7)

Fig. 1
Fig. 1

Classical approaches for collecting dipole radiation using (a) a parabolic mirror, and (b) a wedge mirror waveguide; compared with (c) ideal dipole collection and collimation

Fig. 2
Fig. 2

Geometries of the transformation-optics wedge waveguide for optical dipole collimation in (a) the original space, and (b) the transformed space

Fig. 3
Fig. 3

Spatial distribution of permittivity: (a), ε xx , (b) ε xy , (c) ε yy and (d) ε zz .

Fig. 4
Fig. 4

Comparison of continuous dipole radiation waves in (a) free space, and (b) in the proposed dipole collimating structure with α=179.9° . (c) Spatial time-averaged power flow distribution and (d) the spatial average of the power flow in the compressed region for various α .

Fig. 5
Fig. 5

Pulsed dipole radiation distributions in free space at (a) t = 15fs, t = 20fs, t = 25fs, and t = 30fs; and (b) pulse propagation in the space-time domain

Fig. 6
Fig. 6

Pulsed dipole radiation distribution in the classical wedge mirror at (a) t = 15fs, t = 20fs, t = 25fs, and t = 30fs and (b) pulse propagation in the space-time domain

Fig. 7
Fig. 7

Pulsed dipole radiation in the proposed dipole pulse collimator at (a) t = 0fs, (b) t = 5fs, (c) t = 10fs, (d) t = 15fs, and (e) pulse propagation in the space-time domain.

Equations (12)

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r'(r,ϕ,z)=r,
ϕ'(r,ϕ,z)={ β+( ϕ+α )( 180β )/( 180α ) for 180 ϕ α ϕβ/α for α ϕ α β+( ϕα )( 180β )/( 180α ) for α ϕ 180 ,
z'(r,ϕ,z)=z,
ε'=Jε J T /det( J ),
μ'=Jμ J T /det( J ),
J=( r'/r r'/ϕ r /z ϕ'/r ϕ /ϕ ϕ /z z /r z /ϕ z /z ).
J=( 1 0 0 0 p i 0 0 0 1 ),
p i ={ β/α for i=1 ( 180β )/( 180α ) for i=2 ,
ε'/ ε 0 =μ'/ μ 0 =( 1/ p i 0 0 0 p i 0 0 0 1/ p i ).
( A ρ A ϕ A z )=( cosϕ sinϕ 0 sinϕ cosϕ 0 0 0 1 )( A x A y A z )= T cr ( A x A y A z ).
D r = T cr 1 ε T cr E r .
ε ' r / ε 0 =μ ' r / μ 0 =( ε xx ε xy 0 ε yx ε yy 0 0 0 ε zz )=( ( 1/ p i ) cos 2 ϕ+ p i sin 2 ϕ ( 1/ p i p i )cosϕsinϕ 0 ( 1/ p i p i )cosϕsinϕ ( 1/ p i ) sin 2 ϕ+ p i cos 2 ϕ 0 0 0 1/ p i ).

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