Abstract

Finding the wavevectors (eigenvalues) and wavefronts (eigenvectors) in nanostructured metasurfaces is cast as a problem of finding the complex roots of a non-linear equation. A new algorithm is introduced for solving this problem; example eigenvalues are obtained and compared against the results from a popular, yet much more computationally expensive method built on a matrix eigenvalue problem. In contrast to the conventional solvers, the proposed method always returns a set of ‘exact’ individual eigenvalues. First, by using the Lehmer-Schur algorithm, we isolate individual complex roots from others, then use a zero-polishing method applied at the very final stage of ultimate eigenvalue localization. Exceptional computational performance, scalability, and accuracy are demonstrated.

© 2014 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Recent advances in planar optics: from plasmonic to dielectric metasurfaces

Patrice Genevet, Federico Capasso, Francesco Aieta, Mohammadreza Khorasaninejad, and Robert Devlin
Optica 4(1) 139-152 (2017)

Loss-compensated and active hyperbolic metamaterials

Xingjie Ni, Satoshi Ishii, Mark D. Thoreson, Vladimir M. Shalaev, Seunghoon Han, Sangyoon Lee, and Alexander V. Kildishev
Opt. Express 19(25) 25242-25254 (2011)

Experimental verification of two-dimensional spatial harmonic analysis at oblique light incidence

Zhengtong Liu, Kuo-Ping Chen, Xingjie Ni, Vladimir P. Drachev, Vladimir M. Shalaev, and Alexander V. Kildishev
J. Opt. Soc. Am. B 27(12) 2465-2470 (2010)

References

  • View by:
  • |
  • |
  • |

  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,” Science 334(6054), 333–337 (2011).
    [Crossref] [PubMed]
  2. M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Natl. Acad. Sci. U.S.A. 109(31), 12364–12368 (2012).
    [Crossref]
  3. F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-Plane Reflection and Refraction of Light by Anisotropic Optical Antenna Metasurfaces with Phase Discontinuities,” Nano Lett. 12(3), 1702–1706 (2012).
    [Crossref] [PubMed]
  4. 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]
  5. S. Larouche and D. R. Smith, “Reconciliation of generalized refraction with diffraction theory,” Opt. Lett. 37(12), 2391–2393 (2012).
    [Crossref] [PubMed]
  6. X. Ni, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin, planar, Babinet-inverted plasmonic metalenses,” Light Sci. Appl. 2(4), e72 (2013).
  7. S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
    [Crossref] [PubMed]
  8. A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar Photonics with Metasurfaces,” Science 339(6125), 1232009 (2013).
    [Crossref] [PubMed]
  9. L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. 95(7), 071112 (2009).
    [Crossref]
  10. S. Ishii, A. V. Kildishev, V. M. Shalaev, K.-P. Chen, and V. P. Drachev, “Metal nanoslit lenses with polarization-selective design,” Opt. Lett. 36(4), 451–453 (2011).
    [Crossref] [PubMed]
  11. R. E. Collin, “Reflection and Transmission at a Slotted Dielectric Interface,” Can. J. Phys. 34(4), 398–411 (1956).
    [Crossref]
  12. C. B. Burckhardt, “Diffraction of a Plane Wave at a Sinusoidally Stratified Dielectric Grating,” J. Opt. Soc. Am. 56(11), 1502–1508 (1966).
    [Crossref]
  13. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
    [Crossref]
  14. F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,” J. Opt. Soc. Am. 63(1), 37–45 (1973).
    [Crossref]
  15. R. Magnusson and T. K. Gaylord, “Analysis of multiwave diffraction of thick gratings,” J. Opt. Soc. Am. 67(9), 1165–1170 (1977).
    [Crossref]
  16. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68(9), 1206–1210 (1978).
    [Crossref]
  17. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The Finitely Conducting Lamellar Diffraction Grating,” Opt. Acta (Lond.) 28(8), 1087–1102 (1981).
    [Crossref]
  18. M. G. Moharam and T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73(9), 1105–1112 (1983).
    [Crossref]
  19. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10(12), 2581–2591 (1993).
    [Crossref]
  20. E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11(9), 2494–2502 (1994).
    [Crossref]
  21. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13(5), 1019–1023 (1996).
    [Crossref]
  22. J. M. Jarem and P. P. Banerjee, “A nonlinear, transient analysis of two- and multi-wave mixing in a photorefractive material using rigorous coupled-wave diffraction theory,” Opt. Commun. 123(4-6), 825–842 (1996).
    [Crossref]
  23. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13(4), 779–784 (1996).
    [Crossref]
  24. E. Popov and M. Nevière, “Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18(11), 2886–2894 (2001).
    [Crossref] [PubMed]
  25. B. Momeni and B. Rashidian, “Improved coupled wave analysis of two-dimensional planar multiple gratings,” IEEE Trans. Antenn. Propag. 52(1), 165–171 (2004).
    [Crossref]
  26. A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73(7), 075107 (2006).
    [Crossref]
  27. X. J. Ni, Z. T. Liu, A. Boltasseva, and A. V. Kildishev, “The validation of the parallel three-dimensional solver for analysis of optical plasmonic bi-periodic multilayer nanostructures,” Appl. Phys., A Mater. Sci. Process. 100(2), 365–374 (2010).
    [Crossref]
  28. A. V. Kildishev and U. K. Chettiar, “Cascading optical negative index metamaterials,” J. Appl. Comput. Electromagnetics Soc. 22, 172–183 (2007).
  29. S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).
  30. P. Yeh, A. Yariv, and C.-S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67(4), 423–438 (1977).
    [Crossref]
  31. F. Tisseur and K. Meerbergen, “The Quadratic Eigenvalue Problem,” SIAM Rev. 43(2), 235–286 (2001).
    [Crossref]
  32. X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
    [Crossref] [PubMed]
  33. F. S. Acton, Numerical Methods that Work (Math. Assoc. Amer., Washington, DC, 1990).
  34. J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium resfponse of nanolayered metamaterials,” Appl. Phys. Lett. 90(19), 191109 (2007).
    [Crossref]
  35. J. W. Brown and R. V. Churchill, Complex Variables and Applications (McGraw Hill, 2004).
  36. M. J. D. Powell, A Hybrid Method for Nonlinear Equations (Gordon and Breach, 1970).
  37. GNU Scientific Library, (1996–2013). http://www.gnu.org/s/gsl/ .
  38. X. Ni, Z. Liu, F. Gu, M. G. Pacheco, J. Borneman, and A. V. Kildishev, “PhotonicsSHA-2D: Modeling of single-period multilayer optical gratings and metamaterials,” (2012), doi: . https://nanohub.org/resources/6977 .
    [Crossref]
  39. Z. Liu, K. P. Chen, X. Ni, V. P. Drachev, V. M. Shalaev, and A. V. Kildishev, “Experimental verification of two-dimensional spatial harmonic analysis at oblique light incidence,” J. Opt. Soc. Am. B 27(12), 2465–2470 (2010).
    [Crossref]
  40. A. V. Kildishev, J. D. Borneman, X. J. Ni, V. M. Shalaev, and V. P. Drachev, “Bianisotropic effective parameters of optical metamagnetics and negative-index materials,” Proc. IEEE 99(10), 1691–1700 (2011).
    [Crossref]
  41. M. Frigo and S. G. Johnson, “The design and implementation of FFTW 3,” Proc. IEEE 93(2), 216–231 (2005).
    [Crossref]

2013 (2)

X. Ni, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin, planar, Babinet-inverted plasmonic metalenses,” Light Sci. Appl. 2(4), e72 (2013).

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar Photonics with Metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

2012 (5)

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Natl. Acad. Sci. U.S.A. 109(31), 12364–12368 (2012).
[Crossref]

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-Plane Reflection and Refraction of Light by Anisotropic Optical Antenna Metasurfaces with Phase Discontinuities,” Nano Lett. 12(3), 1702–1706 (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]

S. Larouche and D. R. Smith, “Reconciliation of generalized refraction with diffraction theory,” Opt. Lett. 37(12), 2391–2393 (2012).
[Crossref] [PubMed]

2011 (4)

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,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

A. V. Kildishev, J. D. Borneman, X. J. Ni, V. M. Shalaev, and V. P. Drachev, “Bianisotropic effective parameters of optical metamagnetics and negative-index materials,” Proc. IEEE 99(10), 1691–1700 (2011).
[Crossref]

S. Ishii, A. V. Kildishev, V. M. Shalaev, K.-P. Chen, and V. P. Drachev, “Metal nanoslit lenses with polarization-selective design,” Opt. Lett. 36(4), 451–453 (2011).
[Crossref] [PubMed]

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
[Crossref] [PubMed]

2010 (2)

Z. Liu, K. P. Chen, X. Ni, V. P. Drachev, V. M. Shalaev, and A. V. Kildishev, “Experimental verification of two-dimensional spatial harmonic analysis at oblique light incidence,” J. Opt. Soc. Am. B 27(12), 2465–2470 (2010).
[Crossref]

X. J. Ni, Z. T. Liu, A. Boltasseva, and A. V. Kildishev, “The validation of the parallel three-dimensional solver for analysis of optical plasmonic bi-periodic multilayer nanostructures,” Appl. Phys., A Mater. Sci. Process. 100(2), 365–374 (2010).
[Crossref]

2009 (1)

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. 95(7), 071112 (2009).
[Crossref]

2007 (2)

A. V. Kildishev and U. K. Chettiar, “Cascading optical negative index metamaterials,” J. Appl. Comput. Electromagnetics Soc. 22, 172–183 (2007).

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium resfponse of nanolayered metamaterials,” Appl. Phys. Lett. 90(19), 191109 (2007).
[Crossref]

2006 (1)

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73(7), 075107 (2006).
[Crossref]

2005 (1)

M. Frigo and S. G. Johnson, “The design and implementation of FFTW 3,” Proc. IEEE 93(2), 216–231 (2005).
[Crossref]

2004 (1)

B. Momeni and B. Rashidian, “Improved coupled wave analysis of two-dimensional planar multiple gratings,” IEEE Trans. Antenn. Propag. 52(1), 165–171 (2004).
[Crossref]

2001 (2)

1996 (3)

1994 (1)

1993 (1)

1983 (1)

1981 (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The Finitely Conducting Lamellar Diffraction Grating,” Opt. Acta (Lond.) 28(8), 1087–1102 (1981).
[Crossref]

1978 (1)

1977 (2)

1973 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

1966 (1)

1956 (2)

R. E. Collin, “Reflection and Transmission at a Slotted Dielectric Interface,” Can. J. Phys. 34(4), 398–411 (1956).
[Crossref]

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The Finitely Conducting Lamellar Diffraction Grating,” Opt. Acta (Lond.) 28(8), 1087–1102 (1981).
[Crossref]

Aieta, F.

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Natl. Acad. Sci. U.S.A. 109(31), 12364–12368 (2012).
[Crossref]

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-Plane Reflection and Refraction of Light by Anisotropic Optical Antenna Metasurfaces with Phase Discontinuities,” Nano Lett. 12(3), 1702–1706 (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]

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,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The Finitely Conducting Lamellar Diffraction Grating,” Opt. Acta (Lond.) 28(8), 1087–1102 (1981).
[Crossref]

Aoust, G.

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Natl. Acad. Sci. U.S.A. 109(31), 12364–12368 (2012).
[Crossref]

Avrutsky, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium resfponse of nanolayered metamaterials,” Appl. Phys. Lett. 90(19), 191109 (2007).
[Crossref]

Banerjee, P. P.

J. M. Jarem and P. P. Banerjee, “A nonlinear, transient analysis of two- and multi-wave mixing in a photorefractive material using rigorous coupled-wave diffraction theory,” Opt. Commun. 123(4-6), 825–842 (1996).
[Crossref]

Benisty, H.

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73(7), 075107 (2006).
[Crossref]

Blanchard, R.

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Natl. Acad. Sci. U.S.A. 109(31), 12364–12368 (2012).
[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]

Boltasseva, A.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar Photonics with Metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

X. J. Ni, Z. T. Liu, A. Boltasseva, and A. V. Kildishev, “The validation of the parallel three-dimensional solver for analysis of optical plasmonic bi-periodic multilayer nanostructures,” Appl. Phys., A Mater. Sci. Process. 100(2), 365–374 (2010).
[Crossref]

Borneman, J. D.

A. V. Kildishev, J. D. Borneman, X. J. Ni, V. M. Shalaev, and V. P. Drachev, “Bianisotropic effective parameters of optical metamagnetics and negative-index materials,” Proc. IEEE 99(10), 1691–1700 (2011).
[Crossref]

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The Finitely Conducting Lamellar Diffraction Grating,” Opt. Acta (Lond.) 28(8), 1087–1102 (1981).
[Crossref]

Burckhardt, C. B.

Capasso, F.

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-Plane Reflection and Refraction of Light by Anisotropic Optical Antenna Metasurfaces with Phase Discontinuities,” Nano Lett. 12(3), 1702–1706 (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]

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Natl. Acad. Sci. U.S.A. 109(31), 12364–12368 (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,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Catrysse, P. B.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. 95(7), 071112 (2009).
[Crossref]

Chen, K. P.

Chen, K.-P.

Chettiar, U. K.

A. V. Kildishev and U. K. Chettiar, “Cascading optical negative index metamaterials,” J. Appl. Comput. Electromagnetics Soc. 22, 172–183 (2007).

Collin, R. E.

R. E. Collin, “Reflection and Transmission at a Slotted Dielectric Interface,” Can. J. Phys. 34(4), 398–411 (1956).
[Crossref]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The Finitely Conducting Lamellar Diffraction Grating,” Opt. Acta (Lond.) 28(8), 1087–1102 (1981).
[Crossref]

David, A.

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73(7), 075107 (2006).
[Crossref]

Drachev, V. P.

Elser, J.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium resfponse of nanolayered metamaterials,” Appl. Phys. Lett. 90(19), 191109 (2007).
[Crossref]

Fan, S.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. 95(7), 071112 (2009).
[Crossref]

Frigo, M.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW 3,” Proc. IEEE 93(2), 216–231 (2005).
[Crossref]

Gaburro, Z.

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Natl. Acad. Sci. U.S.A. 109(31), 12364–12368 (2012).
[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]

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-Plane Reflection and Refraction of Light by Anisotropic Optical Antenna Metasurfaces with Phase Discontinuities,” Nano Lett. 12(3), 1702–1706 (2012).
[Crossref] [PubMed]

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,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Gaylord, T. K.

Genevet, P.

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Natl. Acad. Sci. U.S.A. 109(31), 12364–12368 (2012).
[Crossref]

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-Plane Reflection and Refraction of Light by Anisotropic Optical Antenna Metasurfaces with Phase Discontinuities,” Nano Lett. 12(3), 1702–1706 (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]

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,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Granet, G.

Guizal, B.

Han, S.

He, Q.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Hong, C.-S.

Ishii, S.

Jarem, J. M.

J. M. Jarem and P. P. Banerjee, “A nonlinear, transient analysis of two- and multi-wave mixing in a photorefractive material using rigorous coupled-wave diffraction theory,” Opt. Commun. 123(4-6), 825–842 (1996).
[Crossref]

Johnson, S. G.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW 3,” Proc. IEEE 93(2), 216–231 (2005).
[Crossref]

Kaspar, F. G.

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]

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-Plane Reflection and Refraction of Light by Anisotropic Optical Antenna Metasurfaces with Phase Discontinuities,” Nano Lett. 12(3), 1702–1706 (2012).
[Crossref] [PubMed]

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Natl. Acad. Sci. U.S.A. 109(31), 12364–12368 (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,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Kildishev, A. V.

X. Ni, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin, planar, Babinet-inverted plasmonic metalenses,” Light Sci. Appl. 2(4), e72 (2013).

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar Photonics with Metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

A. V. Kildishev, J. D. Borneman, X. J. Ni, V. M. Shalaev, and V. P. Drachev, “Bianisotropic effective parameters of optical metamagnetics and negative-index materials,” Proc. IEEE 99(10), 1691–1700 (2011).
[Crossref]

S. Ishii, A. V. Kildishev, V. M. Shalaev, K.-P. Chen, and V. P. Drachev, “Metal nanoslit lenses with polarization-selective design,” Opt. Lett. 36(4), 451–453 (2011).
[Crossref] [PubMed]

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
[Crossref] [PubMed]

Z. Liu, K. P. Chen, X. Ni, V. P. Drachev, V. M. Shalaev, and A. V. Kildishev, “Experimental verification of two-dimensional spatial harmonic analysis at oblique light incidence,” J. Opt. Soc. Am. B 27(12), 2465–2470 (2010).
[Crossref]

X. J. Ni, Z. T. Liu, A. Boltasseva, and A. V. Kildishev, “The validation of the parallel three-dimensional solver for analysis of optical plasmonic bi-periodic multilayer nanostructures,” Appl. Phys., A Mater. Sci. Process. 100(2), 365–374 (2010).
[Crossref]

A. V. Kildishev and U. K. Chettiar, “Cascading optical negative index metamaterials,” J. Appl. Comput. Electromagnetics Soc. 22, 172–183 (2007).

Knop, K.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

Lalanne, P.

Larouche, S.

Lee, S.

Li, L.

Li, X.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

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, Z.

Liu, Z. T.

X. J. Ni, Z. T. Liu, A. Boltasseva, and A. V. Kildishev, “The validation of the parallel three-dimensional solver for analysis of optical plasmonic bi-periodic multilayer nanostructures,” Appl. Phys., A Mater. Sci. Process. 100(2), 365–374 (2010).
[Crossref]

Magnusson, R.

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The Finitely Conducting Lamellar Diffraction Grating,” Opt. Acta (Lond.) 28(8), 1087–1102 (1981).
[Crossref]

Meerbergen, K.

F. Tisseur and K. Meerbergen, “The Quadratic Eigenvalue Problem,” SIAM Rev. 43(2), 235–286 (2001).
[Crossref]

Moharam, M. G.

Momeni, B.

B. Momeni and B. Rashidian, “Improved coupled wave analysis of two-dimensional planar multiple gratings,” IEEE Trans. Antenn. Propag. 52(1), 165–171 (2004).
[Crossref]

Morris, G. M.

Nevière, M.

Ni, X.

Ni, X. J.

A. V. Kildishev, J. D. Borneman, X. J. Ni, V. M. Shalaev, and V. P. Drachev, “Bianisotropic effective parameters of optical metamagnetics and negative-index materials,” Proc. IEEE 99(10), 1691–1700 (2011).
[Crossref]

X. J. Ni, Z. T. Liu, A. Boltasseva, and A. V. Kildishev, “The validation of the parallel three-dimensional solver for analysis of optical plasmonic bi-periodic multilayer nanostructures,” Appl. Phys., A Mater. Sci. Process. 100(2), 365–374 (2010).
[Crossref]

Noponen, E.

Podolskiy, V. A.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium resfponse of nanolayered metamaterials,” Appl. Phys. Lett. 90(19), 191109 (2007).
[Crossref]

Popov, E.

Rashidian, B.

B. Momeni and B. Rashidian, “Improved coupled wave analysis of two-dimensional planar multiple gratings,” IEEE Trans. Antenn. Propag. 52(1), 165–171 (2004).
[Crossref]

Rytov, S. M.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Salakhutdinov, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium resfponse of nanolayered metamaterials,” Appl. Phys. Lett. 90(19), 191109 (2007).
[Crossref]

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]

Shalaev, V. M.

X. Ni, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin, planar, Babinet-inverted plasmonic metalenses,” Light Sci. Appl. 2(4), e72 (2013).

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar Photonics with Metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

A. V. Kildishev, J. D. Borneman, X. J. Ni, V. M. Shalaev, and V. P. Drachev, “Bianisotropic effective parameters of optical metamagnetics and negative-index materials,” Proc. IEEE 99(10), 1691–1700 (2011).
[Crossref]

S. Ishii, A. V. Kildishev, V. M. Shalaev, K.-P. Chen, and V. P. Drachev, “Metal nanoslit lenses with polarization-selective design,” Opt. Lett. 36(4), 451–453 (2011).
[Crossref] [PubMed]

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
[Crossref] [PubMed]

Z. Liu, K. P. Chen, X. Ni, V. P. Drachev, V. M. Shalaev, and A. V. Kildishev, “Experimental verification of two-dimensional spatial harmonic analysis at oblique light incidence,” J. Opt. Soc. Am. B 27(12), 2465–2470 (2010).
[Crossref]

Smith, D. R.

Sun, S.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

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,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Thoreson, M. D.

Tisseur, F.

F. Tisseur and K. Meerbergen, “The Quadratic Eigenvalue Problem,” SIAM Rev. 43(2), 235–286 (2001).
[Crossref]

Turunen, J.

Verslegers, L.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. 95(7), 071112 (2009).
[Crossref]

Weisbuch, C.

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73(7), 075107 (2006).
[Crossref]

Xiao, S.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Xu, Q.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Yariv, A.

Yeh, P.

Yu, N.

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-Plane Reflection and Refraction of Light by Anisotropic Optical Antenna Metasurfaces with Phase Discontinuities,” Nano Lett. 12(3), 1702–1706 (2012).
[Crossref] [PubMed]

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Natl. Acad. Sci. U.S.A. 109(31), 12364–12368 (2012).
[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]

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,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Yu, Z.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. 95(7), 071112 (2009).
[Crossref]

Zhou, L.

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Appl. Phys. Lett. (3)

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]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Planar metallic nanoscale slit lenses for angle compensation,” Appl. Phys. Lett. 95(7), 071112 (2009).
[Crossref]

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium resfponse of nanolayered metamaterials,” Appl. Phys. Lett. 90(19), 191109 (2007).
[Crossref]

Appl. Phys., A Mater. Sci. Process. (1)

X. J. Ni, Z. T. Liu, A. Boltasseva, and A. V. Kildishev, “The validation of the parallel three-dimensional solver for analysis of optical plasmonic bi-periodic multilayer nanostructures,” Appl. Phys., A Mater. Sci. Process. 100(2), 365–374 (2010).
[Crossref]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

Can. J. Phys. (1)

R. E. Collin, “Reflection and Transmission at a Slotted Dielectric Interface,” Can. J. Phys. 34(4), 398–411 (1956).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

B. Momeni and B. Rashidian, “Improved coupled wave analysis of two-dimensional planar multiple gratings,” IEEE Trans. Antenn. Propag. 52(1), 165–171 (2004).
[Crossref]

J. Appl. Comput. Electromagnetics Soc. (1)

A. V. Kildishev and U. K. Chettiar, “Cascading optical negative index metamaterials,” J. Appl. Comput. Electromagnetics Soc. 22, 172–183 (2007).

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (5)

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

Light Sci. Appl. (1)

X. Ni, S. Ishii, A. V. Kildishev, and V. M. Shalaev, “Ultra-thin, planar, Babinet-inverted plasmonic metalenses,” Light Sci. Appl. 2(4), e72 (2013).

Nano Lett. (1)

F. Aieta, P. Genevet, N. Yu, M. A. Kats, Z. Gaburro, and F. Capasso, “Out-of-Plane Reflection and Refraction of Light by Anisotropic Optical Antenna Metasurfaces with Phase Discontinuities,” Nano Lett. 12(3), 1702–1706 (2012).
[Crossref] [PubMed]

Nat. Mater. (1)

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref] [PubMed]

Opt. Acta (Lond.) (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The Finitely Conducting Lamellar Diffraction Grating,” Opt. Acta (Lond.) 28(8), 1087–1102 (1981).
[Crossref]

Opt. Commun. (1)

J. M. Jarem and P. P. Banerjee, “A nonlinear, transient analysis of two- and multi-wave mixing in a photorefractive material using rigorous coupled-wave diffraction theory,” Opt. Commun. 123(4-6), 825–842 (1996).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. B (1)

A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73(7), 075107 (2006).
[Crossref]

Proc. IEEE (2)

A. V. Kildishev, J. D. Borneman, X. J. Ni, V. M. Shalaev, and V. P. Drachev, “Bianisotropic effective parameters of optical metamagnetics and negative-index materials,” Proc. IEEE 99(10), 1691–1700 (2011).
[Crossref]

M. Frigo and S. G. Johnson, “The design and implementation of FFTW 3,” Proc. IEEE 93(2), 216–231 (2005).
[Crossref]

Proc. Natl. Acad. Sci. U.S.A. (1)

M. A. Kats, P. Genevet, G. Aoust, N. Yu, R. Blanchard, F. Aieta, Z. Gaburro, and F. Capasso, “Giant birefringence in optical antenna arrays with widely tailorable optical anisotropy,” Proc. Natl. Acad. Sci. U.S.A. 109(31), 12364–12368 (2012).
[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,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar Photonics with Metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

SIAM Rev. (1)

F. Tisseur and K. Meerbergen, “The Quadratic Eigenvalue Problem,” SIAM Rev. 43(2), 235–286 (2001).
[Crossref]

Sov. Phys. JETP (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Other (5)

F. S. Acton, Numerical Methods that Work (Math. Assoc. Amer., Washington, DC, 1990).

J. W. Brown and R. V. Churchill, Complex Variables and Applications (McGraw Hill, 2004).

M. J. D. Powell, A Hybrid Method for Nonlinear Equations (Gordon and Breach, 1970).

GNU Scientific Library, (1996–2013). http://www.gnu.org/s/gsl/ .

X. Ni, Z. Liu, F. Gu, M. G. Pacheco, J. Borneman, and A. V. Kildishev, “PhotonicsSHA-2D: Modeling of single-period multilayer optical gratings and metamaterials,” (2012), doi: . https://nanohub.org/resources/6977 .
[Crossref]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

2D cross-sections of single-periodic planar lamellar structure, (a) with s different layers and (b) with two layers (a binary medium).

Fig. 2
Fig. 2

Schematic representation of the Lehmer-Schur algorithm for Material 1. The red crosses ( × ) correspond to the values of kx detected though several consecutive splittings; the black crosses ( × ) indicate the remaining values of kx yet to be found.

Fig. 3
Fig. 3

Relative errors in real (a) and imaginary (b) parts of the propagating constant for the lowest eigenmode as a function of the incident wavelength λ obtained for test Material 2.

Fig. 4
Fig. 4

Convergence of the SHA method (non-circular markers of different color) with increase of the total number of modes to the results of the new method (blue circles) simulated for test Material 3. The results for SHA with 300 modes (brown triangles) and for proposed solver (blue circles) are indiscernible.

Fig. 5
Fig. 5

(a) Scaling of the current implementation of the algorithm. (b) Computation time of the new solver and the MEP-based SHA solver vs. number of modes (N).

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

x k 0 x , y k 0 y , k 0 δ δ ,
h i ± ( x , y ; t ) = a i ± e ± g i y e ± ι k x x e ι ω t ,
g i 2 = k x 2 ε i .
( h i + + h i ) y i = ( h i + 1 + + h i + 1 ) y i ,
g i γ i ( h i + h i ) y i = g i + 1 γ i + 1 ( h i + 1 + h i + 1 ) y i ,
h i + + h i = ( h i + s + + h i + s ) e α δ , h i + h i = ( h i + s + h i + s ) e α δ .
m i d i , i a i = m i + 1 d i + 1 , i a i + 1 ,
a i = t i a i + 1 ,
g i 0.
a s = d s , s 1 m s 1 m 1 d 1 , 0 a 1 e α δ ,
a 1 = t a 1 ,
( t i ) a 1 = 0 .
1 Tr ( t ) + | t | = 0.
| t i | = | d i , s 1 | | m i | 1 | m i + 1 | | d i + 1 , i | = ( g i γ i ) 1 g i + 1 γ i + 1 ,
| t s | = | d s , s | 1 | m s | 1 | m 1 | | i e α δ | = e 2 α δ g 1 γ g n γ n .
| t | = e 2 α δ .
Tr ( e α δ t ) = 2 e α δ cos h ( α δ ) .
K = ( i = 1 s 1 Δ i m i 1 m i + 1 ) Δ s m s 1 m 1 .
Tr ( Κ ) = 2 cos h ( α δ ) .
Κ = Δ 1 m 1 1 m 2 Δ 2 m 2 1 m 1 ,
Κ 11 = e δ ˜ 1 4 γ ˜ 1 γ ˜ 2 [ ( γ ˜ 1 + γ ˜ 2 ) 2 e δ ˜ 2 ( γ ˜ 1 γ ˜ 2 ) 2 e δ ˜ 2 ] ,
Κ 22 = e δ ˜ 1 4 γ ˜ 1 γ ˜ 2 [ ( γ ˜ 1 + γ ˜ 2 ) 2 e δ ˜ 2 ( γ ˜ 1 γ ˜ 2 ) 2 e δ ˜ 2 ] .
( γ ˜ 1 γ ˜ 2 + γ ˜ 2 γ ˜ 1 ) sin h δ ˜ 1 sin h δ ˜ 2 + 2 cos h δ ˜ 1 cos h δ ˜ 2 = 2 cos h ( α δ ) .
( γ ˜ 1 γ ˜ 2 + γ ˜ 2 γ ˜ 1 ) sin h δ ˜ 1 sin h δ ˜ 2 + 2 cos h δ ˜ 1 cos h δ ˜ 2 = 2 ,
Γ l n f ( z ) d z = ι 2 π ( Z P ) ,
1 2 π Δ Γ arg ( f ( z ) ) = Z P ,

Metrics