Abstract

We propose a novel, miniaturized astigmatic optical element based on a single biaxial hyperbolic metamaterial that enables the conversion of Hermite-Gaussian beams into vortex beams carrying an orbital angular momentum and vice versa. As an example, we design a biaxial anisotropic metamaterial that introduces a π/2 phase shift between two orthogonal components of a Hermite–Gaussian beam due to the optical path difference and at the same time astigmatically focuses these orthogonal components such that they recombine in a symmetric Laguerre-Gaussian beam. We design the proposed device using an array of silver nanowires in an MgF2 matrix. The advantages of the proposed approach over the existing bulk optics based techniques include compactness and therefore, compatibility with ultra-compact opto-electronic circuits, potential re-configurability and an increased tolerance to misalignment.

© 2013 OSA

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2013 (1)

2012 (9)

M. N. O’Sullivan, M. Mirhosseini, M. Malik, and R. W. Boyd, “Near-perfect sorting of orbital angular momentum and angular position states of light,” Opt. Express20(22), 24444–24449 (2012).
[CrossRef] [PubMed]

B. Rodenburg, M. P. J. Lavery, M. Malik, M. N. O’Sullivan, M. Mirhosseini, D. J. Robertson, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on states of light carrying orbital angular momentum,” Opt. Lett.37(17), 3735–3737 (2012).
[CrossRef] [PubMed]

J. Leach, E. Bolduc, D. J. Gauthier, and R. W. Boyd, “Secure information capacity of photons entangled in many dimensions,” Phys. Rev. A85(6), 060304 (2012).
[CrossRef]

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys.14(3), 033001 (2012).
[CrossRef]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6(7), 488–496 (2012).
[CrossRef]

M. Malik, M. N. O’Sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express20(12), 13195–13200 (2012).
[CrossRef] [PubMed]

N. Bozinovic, S. Golowich, P. Kristensen, and S. Ramachandran, “Control of orbital angular momentum of light with optical fibers,” Opt. Lett.37(13), 2451–2453 (2012).
[CrossRef] [PubMed]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science338(6105), 363–366 (2012).
[CrossRef] [PubMed]

2011 (1)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

2009 (3)

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett.103(10), 103903 (2009).
[CrossRef] [PubMed]

T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE7400, 740020 (2009).
[CrossRef]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94(23), 231124 (2009).
[CrossRef]

2008 (1)

2006 (3)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett.96(16), 163905 (2006).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation,” Appl. Phys. Lett.88(22), 221102 (2006).
[CrossRef]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: far-field imaging beyond the diffraction limit,” Opt. Express14(18), 8247–8256 (2006).
[CrossRef] [PubMed]

2004 (2)

2003 (1)

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90(7), 077405–077409 (2003).
[CrossRef] [PubMed]

2002 (1)

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett.88(5), 053601 (2002).
[CrossRef] [PubMed]

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412(6844), 313–316 (2001).
[CrossRef] [PubMed]

1999 (1)

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev.6(4), 339–344 (1999).
[CrossRef]

1994 (2)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun.112(5-6), 321–327 (1994).
[CrossRef]

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template synthesized nanoscopic gold particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem.98(11), 2963–2971 (1994).
[CrossRef]

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun.96(1-3), 123–132 (1993).
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

1990 (1)

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Ahmed, N.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6(7), 488–496 (2012).
[CrossRef]

Aieta, F.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Alekseyev, L. V.

Allen, L.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett.88(5), 053601 (2002).
[CrossRef] [PubMed]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun.96(1-3), 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Asavei, T.

T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE7400, 740020 (2009).
[CrossRef]

Barnett, S. M.

Bartal, G.

Bazhenov, V. Y.

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun.112(5-6), 321–327 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun.96(1-3), 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Bianchini, A.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys.14(3), 033001 (2012).
[CrossRef]

Blanchard, R.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

Bolduc, E.

J. Leach, E. Bolduc, D. J. Gauthier, and R. W. Boyd, “Secure information capacity of photons entangled in many dimensions,” Phys. Rev. A85(6), 060304 (2012).
[CrossRef]

Boyd, R. W.

Bozinovic, N.

Brasselet, E.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett.103(10), 103903 (2009).
[CrossRef] [PubMed]

Cai, X.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science338(6105), 363–366 (2012).
[CrossRef] [PubMed]

Capasso, F.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun.112(5-6), 321–327 (1994).
[CrossRef]

Courtial, J.

Dolinar, S.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6(7), 488–496 (2012).
[CrossRef]

Fang, H.

Fazal, I. M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6(7), 488–496 (2012).
[CrossRef]

Foss, C. A.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template synthesized nanoscopic gold particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem.98(11), 2963–2971 (1994).
[CrossRef]

Franke-Arnold, S.

Gaburro, Z.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Gauthier, D. J.

J. Leach, E. Bolduc, D. J. Gauthier, and R. W. Boyd, “Secure information capacity of photons entangled in many dimensions,” Phys. Rev. A85(6), 060304 (2012).
[CrossRef]

Genevet, P.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Gibson, G.

Golowich, S.

Hara, T.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev.6(4), 339–344 (1999).
[CrossRef]

Heckenberg, N. R.

T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE7400, 740020 (2009).
[CrossRef]

Hornyak, G. L.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template synthesized nanoscopic gold particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem.98(11), 2963–2971 (1994).
[CrossRef]

Huang, H.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6(7), 488–496 (2012).
[CrossRef]

Igasaki, Y.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev.6(4), 339–344 (1999).
[CrossRef]

Inoue, T.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev.6(4), 339–344 (1999).
[CrossRef]

Jacob, Z.

Jia, P.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Johnson-Morris, B.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science338(6105), 363–366 (2012).
[CrossRef] [PubMed]

Juodkazis, S.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett.103(10), 103903 (2009).
[CrossRef] [PubMed]

Karimi, E.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94(23), 231124 (2009).
[CrossRef]

Kats, M. A.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Kobayashi, Y.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev.6(4), 339–344 (1999).
[CrossRef]

Kolinko, P.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing by a slab of indefinite media,” Appl. Phys. Lett.84(13), 2244–2246 (2004).
[CrossRef]

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun.112(5-6), 321–327 (1994).
[CrossRef]

Kristensen, P.

Lavery, M. P. J.

Leach, J.

Li, F.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev.6(4), 339–344 (1999).
[CrossRef]

Lin, J.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

Liu, Y.

Loke, V. L. Y.

T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE7400, 740020 (2009).
[CrossRef]

MacVicar, I.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett.88(5), 053601 (2002).
[CrossRef] [PubMed]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412(6844), 313–316 (2001).
[CrossRef] [PubMed]

Malik, M.

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett.96(16), 163905 (2006).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation,” Appl. Phys. Lett.88(22), 221102 (2006).
[CrossRef]

Mari, E.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys.14(3), 033001 (2012).
[CrossRef]

Marrucci, L.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94(23), 231124 (2009).
[CrossRef]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett.96(16), 163905 (2006).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation,” Appl. Phys. Lett.88(22), 221102 (2006).
[CrossRef]

Martin, C. R.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template synthesized nanoscopic gold particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem.98(11), 2963–2971 (1994).
[CrossRef]

Min, C. J.

Mirhosseini, M.

Misawa, H.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett.103(10), 103903 (2009).
[CrossRef] [PubMed]

Mock, J. J.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing by a slab of indefinite media,” Appl. Phys. Lett.84(13), 2244–2246 (2004).
[CrossRef]

Mukohzaka, N.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev.6(4), 339–344 (1999).
[CrossRef]

Murazawa, N.

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett.103(10), 103903 (2009).
[CrossRef] [PubMed]

Nagali, E.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94(23), 231124 (2009).
[CrossRef]

Narimanov, E.

Nieminen, T. A.

T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE7400, 740020 (2009).
[CrossRef]

O’Brien, J. L.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science338(6105), 363–366 (2012).
[CrossRef] [PubMed]

O’Neil, A. T.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett.88(5), 053601 (2002).
[CrossRef] [PubMed]

O’Sullivan, M. N.

Padgett, M. J.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation,” Appl. Phys. Lett.88(22), 221102 (2006).
[CrossRef]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett.96(16), 163905 (2006).
[CrossRef] [PubMed]

Pas’ko, V.

Piccirillo, B.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94(23), 231124 (2009).
[CrossRef]

Ramachandran, S.

Ren, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6(7), 488–496 (2012).
[CrossRef]

Robertson, D. J.

Rodenburg, B.

Romanato, F.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys.14(3), 033001 (2012).
[CrossRef]

Rubinsztein-Dunlop, H.

T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE7400, 740020 (2009).
[CrossRef]

Rye, P.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing by a slab of indefinite media,” Appl. Phys. Lett.84(13), 2244–2246 (2004).
[CrossRef]

Santamato, E.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94(23), 231124 (2009).
[CrossRef]

Schurig, D.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing by a slab of indefinite media,” Appl. Phys. Lett.84(13), 2244–2246 (2004).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90(7), 077405–077409 (2003).
[CrossRef] [PubMed]

Scully, M. O.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

Smith, D. R.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing by a slab of indefinite media,” Appl. Phys. Lett.84(13), 2244–2246 (2004).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90(7), 077405–077409 (2003).
[CrossRef] [PubMed]

Sorel, M.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science338(6105), 363–366 (2012).
[CrossRef] [PubMed]

Soskin, M. S.

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

Sponselli, A.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys.14(3), 033001 (2012).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Stockert, J. A.

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template synthesized nanoscopic gold particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem.98(11), 2963–2971 (1994).
[CrossRef]

Strain, M. J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science338(6105), 363–366 (2012).
[CrossRef] [PubMed]

Tamburini, F.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys.14(3), 033001 (2012).
[CrossRef]

Tetienne, J.-P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Thidé, B.

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys.14(3), 033001 (2012).
[CrossRef]

Thompson, M. G.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science338(6105), 363–366 (2012).
[CrossRef] [PubMed]

Toyoda, H.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev.6(4), 339–344 (1999).
[CrossRef]

Tur, M.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6(7), 488–496 (2012).
[CrossRef]

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun.96(1-3), 123–132 (1993).
[CrossRef]

Vasnetsov, M.

Vasnetsov, M. V.

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412(6844), 313–316 (2001).
[CrossRef] [PubMed]

Wang, J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science338(6105), 363–366 (2012).
[CrossRef] [PubMed]

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6(7), 488–496 (2012).
[CrossRef]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412(6844), 313–316 (2001).
[CrossRef] [PubMed]

Willner, A. E.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6(7), 488–496 (2012).
[CrossRef]

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun.112(5-6), 321–327 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun.96(1-3), 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

Yan, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6(7), 488–496 (2012).
[CrossRef]

Yang, J.-Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6(7), 488–496 (2012).
[CrossRef]

Yang, Y.

Yoshida, N.

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev.6(4), 339–344 (1999).
[CrossRef]

Yu, N.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

Yu, S.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science338(6105), 363–366 (2012).
[CrossRef] [PubMed]

Yuan, X.-C.

Yue, Y.

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6(7), 488–496 (2012).
[CrossRef]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412(6844), 313–316 (2001).
[CrossRef] [PubMed]

Zhang, X.

Zhu, J.

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science338(6105), 363–366 (2012).
[CrossRef] [PubMed]

Appl. Phys. Lett. (4)

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing by a slab of indefinite media,” Appl. Phys. Lett.84(13), 2244–2246 (2004).
[CrossRef]

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation,” Appl. Phys. Lett.88(22), 221102 (2006).
[CrossRef]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett.94(23), 231124 (2009).
[CrossRef]

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett.100(1), 013101 (2012).
[CrossRef]

J. Phys. Chem. (1)

C. A. Foss, G. L. Hornyak, J. A. Stockert, and C. R. Martin, “Template synthesized nanoscopic gold particles: optical spectra and the effects of particle size and shape,” J. Phys. Chem.98(11), 2963–2971 (1994).
[CrossRef]

Laser-beams with screw dislocations in their wavefronts (1)

V. Y. Bazhenov, M. V. Vasnetsov, M. S. Soskin, V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431429–431 (1990), [Pis'ma Zh. Eksp. Teor. Fiz. 52, 1037(1990)].

Nat. Photonics (1)

J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur, and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics6(7), 488–496 (2012).
[CrossRef]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature412(6844), 313–316 (2001).
[CrossRef] [PubMed]

New J. Phys. (1)

F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys.14(3), 033001 (2012).
[CrossRef]

Opt. Commun. (2)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun.96(1-3), 123–132 (1993).
[CrossRef]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun.112(5-6), 321–327 (1994).
[CrossRef]

Opt. Express (5)

Opt. Lett. (3)

Opt. Rev. (1)

Y. Igasaki, F. Li, N. Yoshida, H. Toyoda, T. Inoue, N. Mukohzaka, Y. Kobayashi, and T. Hara, “High efficiency electrically-addressable phase-only spatial light modulator,” Opt. Rev.6(4), 339–344 (1999).
[CrossRef]

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A45(11), 8185–8189 (1992).
[CrossRef] [PubMed]

J. Leach, E. Bolduc, D. J. Gauthier, and R. W. Boyd, “Secure information capacity of photons entangled in many dimensions,” Phys. Rev. A85(6), 060304 (2012).
[CrossRef]

Phys. Rev. B (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972).
[CrossRef]

Phys. Rev. Lett. (4)

E. Brasselet, N. Murazawa, H. Misawa, and S. Juodkazis, “Optical vortices from liquid crystal droplets,” Phys. Rev. Lett.103(10), 103903 (2009).
[CrossRef] [PubMed]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett.88(5), 053601 (2002).
[CrossRef] [PubMed]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90(7), 077405–077409 (2003).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett.96(16), 163905 (2006).
[CrossRef] [PubMed]

Proc. SPIE (1)

T. Asavei, V. L. Y. Loke, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical paddle-wheel,” Proc. SPIE7400, 740020 (2009).
[CrossRef]

Science (2)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction,” Science334(6054), 333–337 (2011).
[CrossRef] [PubMed]

X. Cai, J. Wang, M. J. Strain, B. Johnson-Morris, J. Zhu, M. Sorel, J. L. O’Brien, M. G. Thompson, and S. Yu, “Integrated Compact Optical Vortex Beam Emitters,” Science338(6105), 363–366 (2012).
[CrossRef] [PubMed]

Other (2)

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular momentum (Institute of Physics Publishing, 2003), Chap. 3.

A. Sihvola, Electromagnetic Mixing Formulas and Applications, Institution of Electrical Engineers (1999).

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Figures (4)

Fig. 1
Fig. 1

Schematic of negative refraction in HMM (a) and the equi-frequency contours of the HMM in xz-plane (b) and yz-plane (c). The black circles are the EFCs of the air and the blue hyperbola are the EFCs of the hyperbolic material. ki and Si are the incident wave vector and Poynting vector, respectively. kr and Sr are the refractive wave vector and Poynting vector, respectively. Due to the indefinite property between xy-plane and z orientation, both the HG1,0 mode and the HG0,1 mode experience a negative refraction. The astigmatic beam is focused inside the indefinite beam.

Fig. 2
Fig. 2

Structure of the biaxial HMM.

Fig. 3
Fig. 3

The schematic of the device and the results including energy density and the phase distributions of the output beam around the focal length.

Fig. 4
Fig. 4

The time averaged energy density and phase distributions in the cross section of the beam along the propagation direction.

Equations (15)

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E(r,z)= E 0 w 0 w(z) exp( r 2 w 2 (z) ikzik r 2 2R(z) +iς(z) ) H 1 ( 2 w 0 (x+y) ),
θ i =arctan( w(z) z ),
ε =( ε xx 0 0 0 ε yy 0 0 0 ε zz ),
k x 2 ε zz + k z 2 ε xx = ω 2 c 2 , k y 2 ε zz + k z 2 ε yy = ω 2 c 2 .
k i, x = k i, y =sin( θ i )= tan 2 ( θ i ) 1+ tan 2 ( θ i ) , k i, z =cos( θ i )= 1 1+ tan 2 ( θ i ) ,
k i, x = k i, y = k r, x = k r, y ,
k r,xz = ε xx ( 1 k r,x 2 ε zz ) , k r,yz = ε yy ( 1 k r,y 2 ε zz ) .
n kr, xz = k r,xz 2 + k r,x 2 , n kr, yz = k r,yz 2 + k r,y 2 ;
θ kr,zx =arctan( k r, x k r, xz ) , θ kr,yz =arctan( k r, y k r,yz ),
θ Sr,xz =arctan( k r, x ε xx k r, z ε zz ) , θ Sr,yz =arctan( k r, y ε yy k r, z ε zz ).
Δ l o = dcos( θ Sr,zx + θ kr,zx ) cos( θ Sr,zx ) n kr, zx dcos( θ Sr,yz + θ kr,yz ) cos( θ Sr,yz ) n kr, yz ,
Δ l o =d( K k r,x 2 ε zz K )Δn
ε m ( ω )= ε ω p 2 ω( ω+i γ c ) .
ε || =p ε m +( 1p ) ε d ,
ε = ε d + p ε d ( ε m ε d ) ε d +( 1p )( ε m ε d ) q eff .

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