Abstract

The Hamiltonian of an optical medium is important in both the design and the description of optical devices in the geometrical optics limit. The results calculated in this article show in detail how ray tracing in anisotropic materials in arbitrary coordinate systems and curved spaces can be carried out. Writing Maxwell’s equations in the most general form, we derive a coordinate-free form for the eikonal equation and hence the Hamiltonian of a general purpose medium. The expression works for both orthogonal and non-orthogonal coordinate systems, and we show how it can be simplified for biaxial and uniaxial media in orthogonal coordinate systems. In order to show the utility of the equations in a real case, we study both the isotropic and the uniaxially transmuted birefringent Eaton lens and derive the ray trajectories in spherical coordinates for each case.

© 2010 Optical Society of America

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References

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  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef] [PubMed]
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  3. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [CrossRef] [PubMed]
  4. U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  7. J. E. Eaton, “On spherically symmetric lenses,” IRE Trans. Antennas Propag. 4, 66–71 (1952).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  16. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

2009 (3)

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).
[CrossRef]

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
[CrossRef]

N. A. Mortensen, “Prospects for poor-man’s cloaking with low-contrast all-dielectric optical elements,” J. Eur. Opt. Soc. Rapid Publ. 4, 09008 (2009).
[CrossRef]

2008 (3)

T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New J. Phys. 10, 115038 (2008).
[CrossRef]

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

M. Sluijter, D. K. G. de Boer, and J. J. M. Braat, “General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media,” J. Opt. Soc. Am. A 25, 1260–1273 (2008).
[CrossRef]

2007 (1)

Q. Cheng-Wei, Y. Hai-Ying, L. Le-Wei, S. Zouhdi, and Y. Tat-Soon, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007).
[CrossRef]

2006 (2)

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

1993 (2)

1990 (1)

1964 (1)

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

1962 (1)

1952 (1)

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

Braat, J. J. M.

Cheng-Wei, Q.

Q. Cheng-Wei, Y. Hai-Ying, L. Le-Wei, S. Zouhdi, and Y. Tat-Soon, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007).
[CrossRef]

Chipman, R. A.

Cummer, S. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

de Boer, D. K. G.

Eaton, J. E.

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

Hai-Ying, Y.

Q. Cheng-Wei, Y. Hai-Ying, L. Le-Wei, S. Zouhdi, and Y. Tat-Soon, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007).
[CrossRef]

Hillman, L. W.

Leonhardt, U.

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).
[CrossRef]

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
[CrossRef]

T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New J. Phys. 10, 115038 (2008).
[CrossRef]

Le-Wei, L.

Q. Cheng-Wei, Y. Hai-Ying, L. Le-Wei, S. Zouhdi, and Y. Tat-Soon, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

McClain, S. C.

Mortensen, N. A.

N. A. Mortensen, “Prospects for poor-man’s cloaking with low-contrast all-dielectric optical elements,” J. Eur. Opt. Soc. Rapid Publ. 4, 09008 (2009).
[CrossRef]

Pendry, J. B.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 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. Express 14, 9794–9804 (2006).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Quan-Ting, L.

Rahm, M.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

Roberts, D. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

Schurig, D.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 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. Express 14, 9794–9804 (2006).
[CrossRef] [PubMed]

Sluijter, M.

Smith, D. R.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 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. Express 14, 9794–9804 (2006).
[CrossRef] [PubMed]

Stavroudis, O. N.

Tat-Soon, Y.

Q. Cheng-Wei, Y. Hai-Ying, L. Le-Wei, S. Zouhdi, and Y. Tat-Soon, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007).
[CrossRef]

Tyc, T.

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
[CrossRef]

T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New J. Phys. 10, 115038 (2008).
[CrossRef]

Zouhdi, S.

Q. Cheng-Wei, Y. Hai-Ying, L. Le-Wei, S. Zouhdi, and Y. Tat-Soon, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007).
[CrossRef]

Appl. Opt. (1)

IRE Trans. Antennas Propag. (1)

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

J. Eur. Opt. Soc. Rapid Publ. (1)

N. A. Mortensen, “Prospects for poor-man’s cloaking with low-contrast all-dielectric optical elements,” J. Eur. Opt. Soc. Rapid Publ. 4, 09008 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

New J. Phys. (2)

T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New J. Phys. 10, 115038 (2008).
[CrossRef]

U. Leonhardt, “Perfect imaging without negative refraction,” New J. Phys. 11, 093040 (2009).
[CrossRef]

Opt. Express (1)

Photonics Nanostruct. Fundam. Appl. (1)

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell’s equations,” Photonics Nanostruct. Fundam. Appl. 6, 87–95 (2008).
[CrossRef]

Phys. Rev. B (1)

Q. Cheng-Wei, Y. Hai-Ying, L. Le-Wei, S. Zouhdi, and Y. Tat-Soon, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef] [PubMed]

Science (2)

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

U. Leonhardt and T. Tyc, “Broadband invisibility by non-Euclidean cloaking,” Science 323, 110–112 (2009).
[CrossRef]

Other (1)

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, 1964).

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Figures (3)

Fig. 1
Fig. 1

Ray trajectories inside an isotropic Eaton lens.

Fig. 2
Fig. 2

Ray trajectories inside the Eaton lens transmuted via R ( r ) for the (a) in-plane polarization and (b) out-of-plane polarization.

Fig. 3
Fig. 3

Plots of refractive indices (a) before transmutation n ( r ) and (b) after transmutation n r ( R ) and n ϕ ( R ) = n θ ( R ) .

Equations (31)

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e i j k E k , j = B i t ,
e i j k H k , j = D i t ,
D , i i = 0 ,
B , i i = 0 ,
D i = ε 0 ε i j E j ,
B i = μ 0 μ i j H j ,
E j = E j   exp ( i k 0 k m r m i ω t ) ,
H j = 1 η 0 H j   exp ( i k 0 k m r m i ω t ) ,
e i j k k j E k μ i j H j = 0 ,
e i j k k j H k + ε i j E j = 0.
M p k E k = 0 ,
H = det ( M ) = [ i j k ] M ( p ) i M ( q ) j M ( r ) k ,
H = det ( M ) = 1 det ( n ) ( k n k det ( n ) ) 2 ,
d r d τ = k H ,
d k d τ = r H ,
e i j k k j inc n k = e i j k k j ref n k = e i j k k j tran n k ,
H ( k tran ) = 0 ,
H s ( k ref ) = 0 ,
e i j k k j inc n k = e i j k k j ref n k = e i j k k j tran n k ,
H ( k ref ) = 0 ,
H s ( k tran ) = 0.
ε = ( n 1 2 0 0 0 n 2 2 0 0 0 n 3 2 ) .
M = ε + K ,
K = ( ( k 2 2 + k 3 2 ) k 1 k 2 k 1 k 3 k 1 k 2 ( k 1 2 + k 3 2 ) k 2 k 3 k 1 k 3 k 2 k 3 ( k 1 2 + k 2 2 ) ) .
H = det ( M ) = k 1 4 n 1 2 + ( k 2 2 + k 3 2 n 1 2 ) ( k 2 2 n 2 2 + ( k 3 2 n 2 2 ) n 3 2 ) + k 1 2 ( k 2 2 ( n 2 2 + n 1 2 ) n 1 2 ( n 2 2 + n 3 2 ) + k 3 2 ( n 1 2 + n 3 2 ) ) ,
H = det ( M ) = H o H e = ( k 1 2 + k 2 2 + k 3 2 n o 2 ) ( k 1 2 n e 2 + ( k 2 2 + k 3 2 n e 2 ) n o 2 ) ,
n ( r ) = 2 a r 1 ,
H = k r 2 + k ϕ 2 n ( r ) 2 .
ε ( i i ) ( R ) = { n 2 r 2 R 2 , n 2 ( d r d R ) 2 , n 2 ( d r d R ) 2 } ,
μ ( i i ) ( R ) = { r 2 R 2 ( d R d r ) 2 , 1 , 1 } ,
H = ( k r 2 + k ϕ 2 n ϕ ( R ) 2 ) ( k ϕ 2 n ϕ ( R ) 2 + ( k r 2 n ϕ ( R ) 2 ) n r ( R ) 2 ) ,

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