Abstract

The design rules of transformation optics generally lead to spatially inhomogeneous and anisotropic impedance-matched magneto-dielectric material distributions for, e.g., free-space invisibility cloaks. Recently, simplified anisotropic non-magnetic free-space cloaks made of a locally uniaxial dielectric material (calcite) have been realized experimentally. In a two-dimensional setting and for in-plane polarized light propagating in this plane, the cloaking performance can still be perfect for light rays. However, for general views in three dimensions, various imperfections are expected. In this paper, we study two different purely dielectric uniaxial cylindrical free-space cloaks. For one, the optic axis is along the radial direction, for the other one it is along the azimuthal direction. The azimuthal uniaxial cloak has not been suggested previously to the best of our knowledge. We visualize the cloaking performance of both by calculating photorealistic images rendered by ray tracing. Following and complementing our previous ray-tracing work, we use an equation of motion directly derived from Fermat’s principle. The rendered images generally exhibit significant imperfections. This includes the obvious fact that cloaking does not work at all for horizontal or for ordinary linear polarization of light. Moreover, more subtle effects occur such as viewing-angle-dependent aberrations. However, we still find amazingly good cloaking performance for the purely dielectric azimuthal uniaxial cloak.

© 2012 OSA

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References

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  1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
    [CrossRef] [PubMed]
  2. U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
    [CrossRef] [PubMed]
  3. U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, Mineola, 2010).
  4. C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).
  5. H. Hashemi, B. Zhang, J. D. Joannopoulos, and S. G. Johnson, “Delay-Bandwidth and Delay-Loss Limitations for Cloaking of Large Objects,” Phys. Rev. Lett.104(25), 253903 (2010).
    [CrossRef] [PubMed]
  6. H. Chen and B. Zheng, “Broadband polygonal invisibility cloak for visible light,” Sci. Rep.2, 255 (2012).
    [CrossRef] [PubMed]
  7. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express14(21), 9794–9804 (2006).
    [CrossRef] [PubMed]
  8. A. Akbarzadeh and A. J. Danner, “Generalization of ray tracing in a linear inhomogeneous anisotropic medium: a coordinate-free approach,” J. Opt. Soc. Am. A27(12), 2558–2562 (2010).
    [CrossRef] [PubMed]
  9. 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,” Science314(5801), 977–980 (2006).
    [CrossRef] [PubMed]
  10. J. C. Halimeh and M. Wegener, “Time-of-flight imaging of invisibility cloaks,” Opt. Express20(1), 63–74 (2012).
    [CrossRef] [PubMed]
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  12. R. Schmied, J. C. Halimeh, and M. Wegener, “Conformal carpet and grating cloaks,” Opt. Express18(23), 24361–24367 (2010).
    [CrossRef] [PubMed]
  13. J. C. Halimeh, R. Schmied, and M. Wegener, “Newtonian photorealistic ray tracing of grating cloaks and correlation-function-based cloaking-quality assessment,” Opt. Express19(7), 6078–6092 (2011).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  17. A. Weidlich and A. Wilkie, “Realistic rendering of birefringency in uniaxial crystals,” ACM Trans. Graph.27(10), 1–12 (2008).
  18. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
    [CrossRef]
  19. T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700 nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett.107(17), 173901 (2011).
    [CrossRef] [PubMed]
  20. S. Hrabar, I. Krois, and A. Kiricenko, “Towards active dispersionless ENZ metamaterial for cloaking applications,” Metamaterials (Amst.)4(2-3), 89–97 (2010).
    [CrossRef]
  21. S. Hrabar, I. Krois, I. Bonic, and A. Kiricenko, “Negative capacitor paves the way to ultra-broadband metamaterials,” Appl. Phys. Lett.99(25), 254103 (2011).
    [CrossRef]

2012 (2)

H. Chen and B. Zheng, “Broadband polygonal invisibility cloak for visible light,” Sci. Rep.2, 255 (2012).
[CrossRef] [PubMed]

J. C. Halimeh and M. Wegener, “Time-of-flight imaging of invisibility cloaks,” Opt. Express20(1), 63–74 (2012).
[CrossRef] [PubMed]

2011 (4)

T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700 nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett.107(17), 173901 (2011).
[CrossRef] [PubMed]

J. C. Halimeh, R. Schmied, and M. Wegener, “Newtonian photorealistic ray tracing of grating cloaks and correlation-function-based cloaking-quality assessment,” Opt. Express19(7), 6078–6092 (2011).
[CrossRef] [PubMed]

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).

S. Hrabar, I. Krois, I. Bonic, and A. Kiricenko, “Negative capacitor paves the way to ultra-broadband metamaterials,” Appl. Phys. Lett.99(25), 254103 (2011).
[CrossRef]

2010 (4)

H. Hashemi, B. Zhang, J. D. Joannopoulos, and S. G. Johnson, “Delay-Bandwidth and Delay-Loss Limitations for Cloaking of Large Objects,” Phys. Rev. Lett.104(25), 253903 (2010).
[CrossRef] [PubMed]

A. Akbarzadeh and A. J. Danner, “Generalization of ray tracing in a linear inhomogeneous anisotropic medium: a coordinate-free approach,” J. Opt. Soc. Am. A27(12), 2558–2562 (2010).
[CrossRef] [PubMed]

S. Hrabar, I. Krois, and A. Kiricenko, “Towards active dispersionless ENZ metamaterial for cloaking applications,” Metamaterials (Amst.)4(2-3), 89–97 (2010).
[CrossRef]

R. Schmied, J. C. Halimeh, and M. Wegener, “Conformal carpet and grating cloaks,” Opt. Express18(23), 24361–24367 (2010).
[CrossRef] [PubMed]

2008 (2)

2007 (1)

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

2006 (4)

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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

Akbarzadeh, A.

Bonic, I.

S. Hrabar, I. Krois, I. Bonic, and A. Kiricenko, “Negative capacitor paves the way to ultra-broadband metamaterials,” Appl. Phys. Lett.99(25), 254103 (2011).
[CrossRef]

Braat, J. J. M.

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

Chen, H.

H. Chen and B. Zheng, “Broadband polygonal invisibility cloak for visible light,” Sci. Rep.2, 255 (2012).
[CrossRef] [PubMed]

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Danner, A. J.

de Boer, D. K. G.

Ergin, T.

T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700 nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett.107(17), 173901 (2011).
[CrossRef] [PubMed]

Fischer, J.

T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700 nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett.107(17), 173901 (2011).
[CrossRef] [PubMed]

Halimeh, J. C.

Hashemi, H.

H. Hashemi, B. Zhang, J. D. Joannopoulos, and S. G. Johnson, “Delay-Bandwidth and Delay-Loss Limitations for Cloaking of Large Objects,” Phys. Rev. Lett.104(25), 253903 (2010).
[CrossRef] [PubMed]

Hrabar, S.

S. Hrabar, I. Krois, I. Bonic, and A. Kiricenko, “Negative capacitor paves the way to ultra-broadband metamaterials,” Appl. Phys. Lett.99(25), 254103 (2011).
[CrossRef]

S. Hrabar, I. Krois, and A. Kiricenko, “Towards active dispersionless ENZ metamaterial for cloaking applications,” Metamaterials (Amst.)4(2-3), 89–97 (2010).
[CrossRef]

Joannopoulos, J. D.

H. Hashemi, B. Zhang, J. D. Joannopoulos, and S. G. Johnson, “Delay-Bandwidth and Delay-Loss Limitations for Cloaking of Large Objects,” Phys. Rev. Lett.104(25), 253903 (2010).
[CrossRef] [PubMed]

Johnson, S. G.

H. Hashemi, B. Zhang, J. D. Joannopoulos, and S. G. Johnson, “Delay-Bandwidth and Delay-Loss Limitations for Cloaking of Large Objects,” Phys. Rev. Lett.104(25), 253903 (2010).
[CrossRef] [PubMed]

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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Kildishev, A. V.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

Kiricenko, A.

S. Hrabar, I. Krois, I. Bonic, and A. Kiricenko, “Negative capacitor paves the way to ultra-broadband metamaterials,” Appl. Phys. Lett.99(25), 254103 (2011).
[CrossRef]

S. Hrabar, I. Krois, and A. Kiricenko, “Towards active dispersionless ENZ metamaterial for cloaking applications,” Metamaterials (Amst.)4(2-3), 89–97 (2010).
[CrossRef]

Krois, I.

S. Hrabar, I. Krois, I. Bonic, and A. Kiricenko, “Negative capacitor paves the way to ultra-broadband metamaterials,” Appl. Phys. Lett.99(25), 254103 (2011).
[CrossRef]

S. Hrabar, I. Krois, and A. Kiricenko, “Towards active dispersionless ENZ metamaterial for cloaking applications,” Metamaterials (Amst.)4(2-3), 89–97 (2010).
[CrossRef]

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science312(5781), 1777–1780 (2006).
[CrossRef] [PubMed]

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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Pendry, J. B.

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Schmied, R.

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express14(21), 9794–9804 (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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

Sluijter, M.

Smith, D. R.

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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express14(21), 9794–9804 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

Soukoulis, C. M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).

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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Wegener, M.

J. C. Halimeh and M. Wegener, “Time-of-flight imaging of invisibility cloaks,” Opt. Express20(1), 63–74 (2012).
[CrossRef] [PubMed]

J. C. Halimeh, R. Schmied, and M. Wegener, “Newtonian photorealistic ray tracing of grating cloaks and correlation-function-based cloaking-quality assessment,” Opt. Express19(7), 6078–6092 (2011).
[CrossRef] [PubMed]

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).

T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700 nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett.107(17), 173901 (2011).
[CrossRef] [PubMed]

R. Schmied, J. C. Halimeh, and M. Wegener, “Conformal carpet and grating cloaks,” Opt. Express18(23), 24361–24367 (2010).
[CrossRef] [PubMed]

Weidlich, A.

A. Weidlich and A. Wilkie, “Realistic rendering of birefringency in uniaxial crystals,” ACM Trans. Graph.27(10), 1–12 (2008).

Wilkie, A.

A. Weidlich and A. Wilkie, “Realistic rendering of birefringency in uniaxial crystals,” ACM Trans. Graph.27(10), 1–12 (2008).

Zhang, B.

H. Hashemi, B. Zhang, J. D. Joannopoulos, and S. G. Johnson, “Delay-Bandwidth and Delay-Loss Limitations for Cloaking of Large Objects,” Phys. Rev. Lett.104(25), 253903 (2010).
[CrossRef] [PubMed]

Zheng, B.

H. Chen and B. Zheng, “Broadband polygonal invisibility cloak for visible light,” Sci. Rep.2, 255 (2012).
[CrossRef] [PubMed]

ACM Trans. Graph. (1)

A. Weidlich and A. Wilkie, “Realistic rendering of birefringency in uniaxial crystals,” ACM Trans. Graph.27(10), 1–12 (2008).

Appl. Phys. Lett. (1)

S. Hrabar, I. Krois, I. Bonic, and A. Kiricenko, “Negative capacitor paves the way to ultra-broadband metamaterials,” Appl. Phys. Lett.99(25), 254103 (2011).
[CrossRef]

J. Opt. Soc. Am. A (2)

Metamaterials (Amst.) (1)

S. Hrabar, I. Krois, and A. Kiricenko, “Towards active dispersionless ENZ metamaterial for cloaking applications,” Metamaterials (Amst.)4(2-3), 89–97 (2010).
[CrossRef]

Nat. Photonics (2)

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics5, 523–530 (2011).

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics1(4), 224–227 (2007).
[CrossRef]

Opt. Express (4)

Phys. Rev. Lett. (2)

H. Hashemi, B. Zhang, J. D. Joannopoulos, and S. G. Johnson, “Delay-Bandwidth and Delay-Loss Limitations for Cloaking of Large Objects,” Phys. Rev. Lett.104(25), 253903 (2010).
[CrossRef] [PubMed]

T. Ergin, J. Fischer, and M. Wegener, “Optical phase cloaking of 700 nm light waves in the far field by a three-dimensional carpet cloak,” Phys. Rev. Lett.107(17), 173901 (2011).
[CrossRef] [PubMed]

Sci. Rep. (1)

H. Chen and B. Zheng, “Broadband polygonal invisibility cloak for visible light,” Sci. Rep.2, 255 (2012).
[CrossRef] [PubMed]

Science (3)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312(5781), 1780–1782 (2006).
[CrossRef] [PubMed]

U. Leonhardt, “Optical conformal mapping,” Science312(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,” Science314(5801), 977–980 (2006).
[CrossRef] [PubMed]

Other (4)

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, Vol. 8 (Butterworth-Heinemann, Oxford, 1984).

U. Leonhardt and T. G. Philbin, Geometry and Light: The Science of Invisibility (Dover, Mineola, 2010).

M. Born and E. Wolf, Principles of Optics, 7. Ed. (University Press, Cambridge, 1999).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., 1999).

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Figures (3)

Fig. 1
Fig. 1

Photorealistic images for the non-magnetic radial uniaxial dielectric cloak according to Eq. (14) rendered by ray tracing using Eq. (2) and Eq. (8). The scenery and the input raw images are defined in Fig. 3 in Ref. 10. (a) Reference image without metal cylinder, (b) with metal cylinder in front of the model’s head but without cloak, (c) with metal cylinder and with radial uniaxial dielectric cloak and for unpolarized light, (d) as (c) but for vertical orientation of the linear polarizer in front of the virtual camera, and (e) as (c) but for horizontal polarization.

Fig. 2
Fig. 2

As Fig. 1, but for the non-magnetic azimuthal uniaxial dielectric cloak according to Eq. (15) rather than the radial uniaxial cloak according to Eq. (14). Comparison of panels (d) of Fig. 1 and Fig. 2 shows a far superior performance of the azimuthal uniaxial dielectric cloak compared to the radial uniaxial dielectric cloak. In panels (d), the linear polarizer in front of the virtual camera is vertically orientated. This different cloaking performance is explained in Sect. 5 and illustrated in Fig. 3.

Fig. 3
Fig. 3

Illustration explaining the different cloaking behavior found in Fig. 1 and Fig. 2. (a) Radial uniaxial cylindrical cloak and (b) azimuthal uniaxial cylindrical cloak. Extraordinary light in the vertical plane carries an extraordinary polarization vector that is parallel to the vertical plane in both cases, thus providing perfect cloaking as expected. However, for extraordinary light in the horizontal plane, the extraordinary polarization vector lies in the horizontal plane in the case of the radial uniaxial cloak, leading to bad cloaking quality. In contrast, for the case of the azimuthal uniaxial cloak, the extraordinary polarization is still parallel to the vertical plane, which generally leads to good cloaking behavior.

Equations (15)

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ϵ =( ϵ o 0 0 0 ϵ o 0 0 0 ϵ e ).
d v o dt = | v o | 2   n o 2( n o v o )  v o n o  .
ϵ  1 B e = ϵ o 1   B e = 1 n o 2 B e .
v e = E e × H e w e =  1 μ 0 E e × B e w e = n o 2 ϵ 0 μ 0 (   ϵ  1   D e ) × (   ϵ  1   B e ) w e ,
( A   a )×( A   b )=| A   |  A 1 ( a × b ) ,
v e = 1 ϵ 0 μ 0  ω   M 1   k e  ,
M = 1 n o 2  | ϵ   |  ϵ  1 = n o 2 n e  2   ϵ  1 .
d v e dt = M 1 2  [ v e (   M ) v e 2( (   M ). v e ) v e ] .
n( ϕ e )= n o n e n o 2 cos 2 ( ϕ e )+ n e 2 sin 2 ( ϕ e ) ,
ϕ e =arcsin( ( c ^ . t ^ )sin( φ e )( c ^ . n ^ )cos( φ e ) ),
n i sin( φ i )= n o sin( φ o )=n( φ e ) sin( φ e ),
T o =| n ^  . ( E o  ×  H o ) n ^  . ( E i  ×  H i ) |   and    T e =| n ^  . ( E e  ×  H e ) n ^  . ( E i  ×  H i ) |,
ϵ r = μ r = ra r  ;  ϵ Θ = μ Θ = r ra  ;  ϵ z = μ z = ( b ba ) 2 ra r .
ϵ r ' = ( ra r ) 2 ( b ba ) 2  ;  ϵ Θ ' = ( b ba ) 2  ;  ϵ z ' = ( b ba ) 2 .
ϵ r ' = ( ra r ) 2 ( b ba ) 2  ;  ϵ Θ ' = ( b ba ) 2  ;  ϵ z ' = ( ra r ) 2 ( b ba ) 2 .

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