Abstract

We present the formation of a singular (diabolical) point in k-space from a periodic metal-dielectric waveguide array. The singularity originates from the balance between alternating normal and anomalous coupling. We numerically demonstrate a strong diffraction anomaly (conical-like diffraction) near the singular point. We also show the evolution of the diffraction pattern with band deformation. The resultant peculiar propagation dynamics of surface plasmon polaritons could provide a new toolset for manipulating light on the nano-scale.

© 2010 OSA

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  1. M. Berry, “Making waves in physics. Three wave singularities from the miraculous 1830s,” Nature 403(6765), 21 (2000).
    [CrossRef] [PubMed]
  2. M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton's diabolical point at the heart of crystal optics,” in Progress in Optics 50, E. Wolf, ed., (Elsevier B. V, 2007), pp. 13–50.
  3. D. R. Yarkony, “Diabolical Conical Intersections,” Rev. Mod. Phys. 68(4), 985–1013 (1996).
    [CrossRef]
  4. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
    [CrossRef] [PubMed]
  5. O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
    [CrossRef] [PubMed]
  6. F. J. Garcia de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007).
    [CrossRef]
  7. X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006).
    [CrossRef] [PubMed]
  8. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
    [CrossRef] [PubMed]
  9. L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902 (2009).
    [CrossRef] [PubMed]
  10. G. Bartal, G. Lerosey, and X. Zhang, “Subwavelength dynamic focusing in plasmonic nanostructures using time reversal,” Phys. Rev. B 79(20), 201103 (2009).
    [CrossRef]
  11. A. A. Sukhorukov and Y. S. Kivshar, “Discrete gap solitons in modulated waveguide arrays,” Opt. Lett. 27(23), 2112–2114 (2002).
    [CrossRef]
  12. M. V. Berry and M. Wilkinson, “F. R. S, and M. Wilkinson, “Diabolic points in the spectra of triangles,” Proc. R. Soc. Lond. A Math. Phys. Sci. 392(1802), 15–43 (1984).
    [CrossRef]
  13. T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
    [CrossRef] [PubMed]
  14. A. Locatelli, M. Conforti, D. Modotto, and C. De Angelis, “Diffraction engineering in arrays of photonic crystal waveguides,” Opt. Lett. 30(21), 2894–2896 (2005).
    [CrossRef] [PubMed]
  15. R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
    [CrossRef]
  16. εAu=ε∞−ωp2/ω(ω+iγc)whereε∞=10, ωp=1.4×1016, and γc=1.1×1014.
  17. K. Nomura and A. H. MacDonald, “Quantum transport of massless Dirac fermions,” Phys. Rev. Lett. 98(7), 076602 (2007).
    [CrossRef] [PubMed]
  18. O. Bahat-Treidel, O. Peleg, and M. Segev, “Symmetry breaking in honeycomb photonic lattices,” Opt. Lett. 33(19), 2251–2253 (2008).
    [CrossRef] [PubMed]

2009 (2)

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902 (2009).
[CrossRef] [PubMed]

G. Bartal, G. Lerosey, and X. Zhang, “Subwavelength dynamic focusing in plasmonic nanostructures using time reversal,” Phys. Rev. B 79(20), 201103 (2009).
[CrossRef]

2008 (2)

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[CrossRef]

O. Bahat-Treidel, O. Peleg, and M. Segev, “Symmetry breaking in honeycomb photonic lattices,” Opt. Lett. 33(19), 2251–2253 (2008).
[CrossRef] [PubMed]

2007 (5)

K. Nomura and A. H. MacDonald, “Quantum transport of massless Dirac fermions,” Phys. Rev. Lett. 98(7), 076602 (2007).
[CrossRef] [PubMed]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[CrossRef] [PubMed]

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

F. J. Garcia de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007).
[CrossRef]

2006 (1)

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006).
[CrossRef] [PubMed]

2005 (1)

2002 (2)

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[CrossRef] [PubMed]

A. A. Sukhorukov and Y. S. Kivshar, “Discrete gap solitons in modulated waveguide arrays,” Opt. Lett. 27(23), 2112–2114 (2002).
[CrossRef]

2000 (1)

M. Berry, “Making waves in physics. Three wave singularities from the miraculous 1830s,” Nature 403(6765), 21 (2000).
[CrossRef] [PubMed]

1996 (1)

D. R. Yarkony, “Diabolical Conical Intersections,” Rev. Mod. Phys. 68(4), 985–1013 (1996).
[CrossRef]

1984 (1)

M. V. Berry and M. Wilkinson, “F. R. S, and M. Wilkinson, “Diabolic points in the spectra of triangles,” Proc. R. Soc. Lond. A Math. Phys. Sci. 392(1802), 15–43 (1984).
[CrossRef]

Bahat-Treidel, O.

Bartal, G.

G. Bartal, G. Lerosey, and X. Zhang, “Subwavelength dynamic focusing in plasmonic nanostructures using time reversal,” Phys. Rev. B 79(20), 201103 (2009).
[CrossRef]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[CrossRef] [PubMed]

Berry, M.

M. Berry, “Making waves in physics. Three wave singularities from the miraculous 1830s,” Nature 403(6765), 21 (2000).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry and M. Wilkinson, “F. R. S, and M. Wilkinson, “Diabolic points in the spectra of triangles,” Proc. R. Soc. Lond. A Math. Phys. Sci. 392(1802), 15–43 (1984).
[CrossRef]

Bräuer, A.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[CrossRef] [PubMed]

Catrysse, P. B.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902 (2009).
[CrossRef] [PubMed]

Chan, C. T.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006).
[CrossRef] [PubMed]

Christodoulides, D. N.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

Conforti, M.

De Angelis, C.

Fan, S.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902 (2009).
[CrossRef] [PubMed]

Fan, X.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006).
[CrossRef] [PubMed]

Freedman, B.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

Garcia de Abajo, F. J.

F. J. Garcia de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007).
[CrossRef]

Geim, A. K.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

Genov, D. A.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[CrossRef]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[CrossRef] [PubMed]

Kivshar, Y. S.

Lederer, F.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[CrossRef] [PubMed]

Lee, J. C. W.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006).
[CrossRef] [PubMed]

Lerosey, G.

G. Bartal, G. Lerosey, and X. Zhang, “Subwavelength dynamic focusing in plasmonic nanostructures using time reversal,” Phys. Rev. B 79(20), 201103 (2009).
[CrossRef]

Liu, Y.

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[CrossRef] [PubMed]

Locatelli, A.

MacDonald, A. H.

K. Nomura and A. H. MacDonald, “Quantum transport of massless Dirac fermions,” Phys. Rev. Lett. 98(7), 076602 (2007).
[CrossRef] [PubMed]

Manela, O.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

Modotto, D.

Nomura, K.

K. Nomura and A. H. MacDonald, “Quantum transport of massless Dirac fermions,” Phys. Rev. Lett. 98(7), 076602 (2007).
[CrossRef] [PubMed]

Novoselov, K. S.

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

Oulton, R. F.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[CrossRef]

Peleg, O.

O. Bahat-Treidel, O. Peleg, and M. Segev, “Symmetry breaking in honeycomb photonic lattices,” Opt. Lett. 33(19), 2251–2253 (2008).
[CrossRef] [PubMed]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

Pertsch, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[CrossRef] [PubMed]

Peschel, U.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[CrossRef] [PubMed]

Pile, D. F. P.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[CrossRef]

Segev, M.

O. Bahat-Treidel, O. Peleg, and M. Segev, “Symmetry breaking in honeycomb photonic lattices,” Opt. Lett. 33(19), 2251–2253 (2008).
[CrossRef] [PubMed]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

Sorger, V. J.

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[CrossRef]

Sukhorukov, A. A.

Verslegers, L.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902 (2009).
[CrossRef] [PubMed]

Wang, G. P.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006).
[CrossRef] [PubMed]

Wilkinson, M.

M. V. Berry and M. Wilkinson, “F. R. S, and M. Wilkinson, “Diabolic points in the spectra of triangles,” Proc. R. Soc. Lond. A Math. Phys. Sci. 392(1802), 15–43 (1984).
[CrossRef]

Yarkony, D. R.

D. R. Yarkony, “Diabolical Conical Intersections,” Rev. Mod. Phys. 68(4), 985–1013 (1996).
[CrossRef]

Yu, Z.

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902 (2009).
[CrossRef] [PubMed]

Zentgraf, T.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[CrossRef] [PubMed]

Zhang, X.

G. Bartal, G. Lerosey, and X. Zhang, “Subwavelength dynamic focusing in plasmonic nanostructures using time reversal,” Phys. Rev. B 79(20), 201103 (2009).
[CrossRef]

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[CrossRef]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[CrossRef] [PubMed]

Nat. Mater. (1)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[CrossRef] [PubMed]

Nat. Photonics (1)

R. F. Oulton, V. J. Sorger, D. A. Genov, D. F. P. Pile, and X. Zhang, “A hybrid plasmonic waveguide for subwavelength confinement and long-range propagation,” Nat. Photonics 2(8), 496–500 (2008).
[CrossRef]

Nature (1)

M. Berry, “Making waves in physics. Three wave singularities from the miraculous 1830s,” Nature 403(6765), 21 (2000).
[CrossRef] [PubMed]

Opt. Lett. (3)

Phys. Rev. B (1)

G. Bartal, G. Lerosey, and X. Zhang, “Subwavelength dynamic focusing in plasmonic nanostructures using time reversal,” Phys. Rev. B 79(20), 201103 (2009).
[CrossRef]

Phys. Rev. Lett. (6)

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97(7), 073901 (2006).
[CrossRef] [PubMed]

Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99(15), 153901 (2007).
[CrossRef] [PubMed]

L. Verslegers, P. B. Catrysse, Z. Yu, and S. Fan, “Deep-subwavelength focusing and steering of light in an aperiodic metallic waveguide array,” Phys. Rev. Lett. 103(3), 033902 (2009).
[CrossRef] [PubMed]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007).
[CrossRef] [PubMed]

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002).
[CrossRef] [PubMed]

K. Nomura and A. H. MacDonald, “Quantum transport of massless Dirac fermions,” Phys. Rev. Lett. 98(7), 076602 (2007).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

M. V. Berry and M. Wilkinson, “F. R. S, and M. Wilkinson, “Diabolic points in the spectra of triangles,” Proc. R. Soc. Lond. A Math. Phys. Sci. 392(1802), 15–43 (1984).
[CrossRef]

Rev. Mod. Phys. (2)

F. J. Garcia de Abajo, “Colloquium: Light scattering by particle and hole arrays,” Rev. Mod. Phys. 79(4), 1267–1290 (2007).
[CrossRef]

D. R. Yarkony, “Diabolical Conical Intersections,” Rev. Mod. Phys. 68(4), 985–1013 (1996).
[CrossRef]

Other (2)

M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton's diabolical point at the heart of crystal optics,” in Progress in Optics 50, E. Wolf, ed., (Elsevier B. V, 2007), pp. 13–50.

εAu=ε∞−ωp2/ω(ω+iγc)whereε∞=10, ωp=1.4×1016, and γc=1.1×1014.

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Figures (4)

Fig. 1
Fig. 1

(a) Schematic of a waveguide array with alternating coupling coefficients C+>0 and C<0 . (b) Two-sheeted iso-frequency surfaces for dimensionless propagation constant K(κ,η) . At η=1 and κ=0 , the band gap closes and the singular point appears. At exactly η=1 , the two bands collapse to a single band (not shown) since the array is no longer binary. (c) Band diagram for the structure (a) and corresponding diffraction coefficient D for η = 0.5, −0.5, and −0.99.

Fig. 2
Fig. 2

(a) Band diagram with a diabolical point for the metal-dielectric array with a unit cell composition: metal (Au, 8nm)/low index dielectric (n = 1.34, 50nm)/high index dielectric (n = 3.48, 200nm)/low index dielectric (n = 1.34, 50nm) at the wavelength of 1550nm. (b) Eigenmodes (Ex ) at the upper band edge (top) and at the lower band edge (bottom) at κ=0 , when 1η<0 . (c)-(d), Numerically simulated h-SPP propagation of a Gaussian beam in the metal-dielectric array having the band structure in (a). For normal incidence in (c), α22° , and for 45° incidence in (d), α'16° and β'26° . The blue arrows indicate the launching directions; electric field intensity is normalized to the intensity maximum.

Fig. 3
Fig. 3

(a) Band diagram of the metal-dielectric array with metal thickness of 10nm. Other lattice parameters are the same as in Fig. 2. (b) Simulated h-SPP diffraction pattern of a normally incident Gaussian beam in the array with band structure of (a). (c) Band diagram of the array with metal thickness of 15nm. (d) h-SPP propagation corresponding to the band in (c).

Fig. 4
Fig. 4

(a). Numerically simulated h-SPP propagation of a normally incident Gaussian beam in the metal-dielectric array in Fig. 2(c) with metal loss. (b) Top: band diagram with a diabolical point for the metal-dielectric array with a unit cell composition: metal (Au, 9.8nm)/low index dielectric (n = 1.34, 100nm)/high index dielectric (n = 2.0, 500nm)/low index dielectric (n = 1.34, 100nm) at the wavelength of 1550nm, bottom: numerically simulated h-SPP propagation of a normally incident Gaussian beam with metal loss.

Equations (4)

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

(iz+β0)E2n+cE2n1+c+E2n+1=0
(iz+β0)E2n+1+c+E2n+cE2n+2=0
H|ψ=K|ψ,
H(η,κ)=(0ηeiκ+1ηeiκ+10),|ψ=(AB).

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