Abstract

Using the effective-index approximation along with the explicit expression for the propagation constant of gap surface plasmons established recently [Opt. Express 16, 2676-2684 (2008)], we develop an analytic description of channel plasmon polariton (CPP) modes guided along V-grooves and gaps between two metal cylindrical surfaces. The results obtained for V-grooves are compared with those reported previously. Dispersion of the main CPP characteristics is calculated for air-gold configurations, allowing one to design graded-gap waveguides for the single-mode operation supporting well-confined fundamental CPP modes in a broad wavelength range.

© 2009 OSA

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References

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  1. S. I. Bozhevolnyi, ed., Plasmonic Nanoguides and Circuits, (Pan Stanford Publishing, Singapore, 2008).
  2. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969).
    [CrossRef]
  3. S. I. Bozhevolnyi, “Effective-index modeling of channel plasmon polaritons,” Opt. Express 14(20), 9467–9476 (2006).
    [CrossRef] [PubMed]
  4. K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82(8), 1158–1160 (2003).
    [CrossRef]
  5. K. Tanaka, M. Tanaka, and T. Sugiyama, “Simulation of practical nanometric optical circuits based on surface plasmon polariton gap waveguides,” Opt. Express 13(1), 256–266 (2005).
    [CrossRef] [PubMed]
  6. K. Tanaka, T. T. Minh, and M. Tanaka, “Analysis of propagation characteristics in the surface plasmon polariton gap waveguides by method of lines,” Opt. Express 17(2), 1078–1092 (2009).
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  8. I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66(3), 035403 (2002).
    [CrossRef]
  9. D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85(26), 6323–6325 (2004).
    [CrossRef]
  10. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005).
    [CrossRef] [PubMed]
  11. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006).
    [CrossRef] [PubMed]
  12. V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Wavelength selective nanophotonic components utilizing channel plasmon polaritons,” Nano Lett. 7(4), 880–884 (2007).
    [CrossRef] [PubMed]
  13. V. S. Volkov, S. I. Bozhevolnyi, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, and T. W. Ebbsen, “Nanofocusing with channel plasmon polaritons,” Nano Lett. 9(3), 1278–1282 (2009).
    [CrossRef] [PubMed]
  14. R. B. Nielsen, I. Fernandez-Cuesta, A. Boltasseva, V. S. Volkov, S. I. Bozhevolnyi, A. Klukowska, and A. Kristensen, “Channel plasmon polariton propagation in nanoimprinted V-groove waveguides,” Opt. Lett. 33(23), 2800–2802 (2008).
    [CrossRef] [PubMed]
  15. E. Moreno, F. J. García-Vidal, S. G. Rodrigo, L. Martín-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006).
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    [CrossRef]
  20. R. Zia, A. Chandran, and M. L. Brongersma, “Dielectric waveguide model for guided surface polaritons,” Opt. Lett. 30(12), 1473–1475 (2005).
    [CrossRef] [PubMed]
  21. H. Kogelnik and V. Ramaswamy, “Scaling rules for thin-film optical waveguides,” Appl. Opt. 13, 1857- (1974).
  22. S. I. Bozhevolnyi and J. Jung, “Scaling for gap plasmon based waveguides,” Opt. Express 16(4), 2676–2684 (2008).
    [CrossRef] [PubMed]
  23. E. Merzbacher, Quantum Mechanics (Wiley, 1998).
  24. E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).
  25. A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005).
    [CrossRef]
  26. L. D. Landau, and E. M. Lifshitz, Quantum Mechanics – Non-relativistic Theory (Pergamon Press, 1960).

2009 (3)

2008 (2)

2007 (3)

2006 (3)

2005 (5)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005).
[CrossRef] [PubMed]

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005).
[CrossRef]

K. Tanaka, M. Tanaka, and T. Sugiyama, “Simulation of practical nanometric optical circuits based on surface plasmon polariton gap waveguides,” Opt. Express 13(1), 256–266 (2005).
[CrossRef] [PubMed]

A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23(1), 413–422 (2005).
[CrossRef]

R. Zia, A. Chandran, and M. L. Brongersma, “Dielectric waveguide model for guided surface polaritons,” Opt. Lett. 30(12), 1473–1475 (2005).
[CrossRef] [PubMed]

2004 (1)

D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85(26), 6323–6325 (2004).
[CrossRef]

2003 (1)

K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82(8), 1158–1160 (2003).
[CrossRef]

2002 (1)

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66(3), 035403 (2002).
[CrossRef]

1977 (1)

1969 (1)

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969).
[CrossRef]

Barchiesi, D.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005).
[CrossRef]

Boltasseva, A.

Bozhevolnyi, S. I.

V. S. Volkov, S. I. Bozhevolnyi, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, and T. W. Ebbsen, “Nanofocusing with channel plasmon polaritons,” Nano Lett. 9(3), 1278–1282 (2009).
[CrossRef] [PubMed]

R. B. Nielsen, I. Fernandez-Cuesta, A. Boltasseva, V. S. Volkov, S. I. Bozhevolnyi, A. Klukowska, and A. Kristensen, “Channel plasmon polariton propagation in nanoimprinted V-groove waveguides,” Opt. Lett. 33(23), 2800–2802 (2008).
[CrossRef] [PubMed]

S. I. Bozhevolnyi and J. Jung, “Scaling for gap plasmon based waveguides,” Opt. Express 16(4), 2676–2684 (2008).
[CrossRef] [PubMed]

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Wavelength selective nanophotonic components utilizing channel plasmon polaritons,” Nano Lett. 7(4), 880–884 (2007).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, “Effective-index modeling of channel plasmon polaritons,” Opt. Express 14(20), 9467–9476 (2006).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006).
[CrossRef] [PubMed]

E. Moreno, F. J. García-Vidal, S. G. Rodrigo, L. Martín-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006).
[CrossRef] [PubMed]

A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23(1), 413–422 (2005).
[CrossRef]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005).
[CrossRef] [PubMed]

Brongersma, M. L.

Burns, W. K.

Chandran, A.

Chang, S. H.

Chiu, T. C.

de la Chapelle, M. L.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005).
[CrossRef]

Devaux, E.

V. S. Volkov, S. I. Bozhevolnyi, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, and T. W. Ebbsen, “Nanofocusing with channel plasmon polaritons,” Nano Lett. 9(3), 1278–1282 (2009).
[CrossRef] [PubMed]

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Wavelength selective nanophotonic components utilizing channel plasmon polaritons,” Nano Lett. 7(4), 880–884 (2007).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005).
[CrossRef] [PubMed]

Dintinger, J.

Ebbesen, T. W.

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Wavelength selective nanophotonic components utilizing channel plasmon polaritons,” Nano Lett. 7(4), 880–884 (2007).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005).
[CrossRef] [PubMed]

Ebbsen, T. W.

V. S. Volkov, S. I. Bozhevolnyi, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, and T. W. Ebbsen, “Nanofocusing with channel plasmon polaritons,” Nano Lett. 9(3), 1278–1282 (2009).
[CrossRef] [PubMed]

Economou, E. N.

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969).
[CrossRef]

Fernandez-Cuesta, I.

García-Vidal, F. J.

V. S. Volkov, S. I. Bozhevolnyi, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, and T. W. Ebbsen, “Nanofocusing with channel plasmon polaritons,” Nano Lett. 9(3), 1278–1282 (2009).
[CrossRef] [PubMed]

E. Moreno, F. J. García-Vidal, S. G. Rodrigo, L. Martín-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006).
[CrossRef] [PubMed]

Gramotnev, D. K.

D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85(26), 6323–6325 (2004).
[CrossRef]

Grimault, A.-S.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005).
[CrossRef]

Hocker, G. B.

Jung, J.

Kjaer, K.

Klukowska, A.

Kristensen, A.

Laluet, J.-Y.

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Wavelength selective nanophotonic components utilizing channel plasmon polaritons,” Nano Lett. 7(4), 880–884 (2007).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006).
[CrossRef] [PubMed]

Larsen, M. S.

Leosson, K.

Macías, D.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005).
[CrossRef]

Maradudin, A. A.

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66(3), 035403 (2002).
[CrossRef]

Martin, O. J. F.

Martín-Moreno, L.

V. S. Volkov, S. I. Bozhevolnyi, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, and T. W. Ebbsen, “Nanofocusing with channel plasmon polaritons,” Nano Lett. 9(3), 1278–1282 (2009).
[CrossRef] [PubMed]

E. Moreno, F. J. García-Vidal, S. G. Rodrigo, L. Martín-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006).
[CrossRef] [PubMed]

Minh, T. T.

Moreno, E.

Nielsen, R. B.

Nikolajsen, T.

Novikov, I. V.

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66(3), 035403 (2002).
[CrossRef]

Pile, D. F. P.

D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85(26), 6323–6325 (2004).
[CrossRef]

Qiu, M.

Rodrigo, S. G.

V. S. Volkov, S. I. Bozhevolnyi, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, and T. W. Ebbsen, “Nanofocusing with channel plasmon polaritons,” Nano Lett. 9(3), 1278–1282 (2009).
[CrossRef] [PubMed]

E. Moreno, F. J. García-Vidal, S. G. Rodrigo, L. Martín-Moreno, and S. I. Bozhevolnyi, “Channel plasmon-polaritons: modal shape, dispersion, and losses,” Opt. Lett. 31(23), 3447–3449 (2006).
[CrossRef] [PubMed]

Sugiyama, T.

Tai, C.-Y.

Tanaka, K.

Tanaka, M.

Vial, A.

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005).
[CrossRef]

Volkov, V. S.

V. S. Volkov, S. I. Bozhevolnyi, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, and T. W. Ebbsen, “Nanofocusing with channel plasmon polaritons,” Nano Lett. 9(3), 1278–1282 (2009).
[CrossRef] [PubMed]

R. B. Nielsen, I. Fernandez-Cuesta, A. Boltasseva, V. S. Volkov, S. I. Bozhevolnyi, A. Klukowska, and A. Kristensen, “Channel plasmon polariton propagation in nanoimprinted V-groove waveguides,” Opt. Lett. 33(23), 2800–2802 (2008).
[CrossRef] [PubMed]

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Wavelength selective nanophotonic components utilizing channel plasmon polaritons,” Nano Lett. 7(4), 880–884 (2007).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005).
[CrossRef] [PubMed]

Yan, M.

Zia, R.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

D. K. Gramotnev and D. F. P. Pile, “Single-mode subwavelength waveguide with channel plasmon-polaritons in triangular grooves on a metal surface,” Appl. Phys. Lett. 85(26), 6323–6325 (2004).
[CrossRef]

K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82(8), 1158–1160 (2003).
[CrossRef]

J. Lightwave Technol. (1)

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

Nano Lett. (2)

V. S. Volkov, S. I. Bozhevolnyi, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Wavelength selective nanophotonic components utilizing channel plasmon polaritons,” Nano Lett. 7(4), 880–884 (2007).
[CrossRef] [PubMed]

V. S. Volkov, S. I. Bozhevolnyi, S. G. Rodrigo, L. Martín-Moreno, F. J. García-Vidal, E. Devaux, and T. W. Ebbsen, “Nanofocusing with channel plasmon polaritons,” Nano Lett. 9(3), 1278–1282 (2009).
[CrossRef] [PubMed]

Nature (1)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006).
[CrossRef] [PubMed]

Opt. Express (6)

Opt. Lett. (3)

Phys. Rev. (1)

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969).
[CrossRef]

Phys. Rev. B (2)

I. V. Novikov and A. A. Maradudin, “Channel polaritons,” Phys. Rev. B 66(3), 035403 (2002).
[CrossRef]

A. Vial, A.-S. Grimault, D. Macías, D. Barchiesi, and M. L. de la Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71(8), 085416 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005).
[CrossRef] [PubMed]

Other (5)

L. D. Landau, and E. M. Lifshitz, Quantum Mechanics – Non-relativistic Theory (Pergamon Press, 1960).

H. Kogelnik and V. Ramaswamy, “Scaling rules for thin-film optical waveguides,” Appl. Opt. 13, 1857- (1974).

S. I. Bozhevolnyi, ed., Plasmonic Nanoguides and Circuits, (Pan Stanford Publishing, Singapore, 2008).

E. Merzbacher, Quantum Mechanics (Wiley, 1998).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, New York, 1985).

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Figures (5)

Fig. 1
Fig. 1

(a). Geometry of the considered V-groove configuration. (b) Fundamental and (c) second CPP mode E z-field distributions (the size of panels: 6 × 10 µm2) calculated for an infinitely deep air-filled groove in gold with the angle θ = 25°at the wavelength λ = 1.55 µm.

Fig. 2
Fig. 2

The fundamental CPP mode characteristics, i.e. (a) the effective index and propagation length along with (b) cutoff groove depth and normalized mode width, as a function of light wavelength for the groove angles θ = 20° and 25°. Filled circles represent the results of EIM simulations obtained previously [3] for the groove angle θ = 25°.

Fig. 3
Fig. 3

(a) Geometry of the graded-gap configuration formed by two cylindrical surfaces. (b) CyPP field profiles calculated for the first three modes [Eq. (12)].

Fig. 4
Fig. 4

(a,b,c) Fundamental and (d) second CyPP mode E z-field magnitude distributions (the size of panels: 2 × 1 µm2) calculated for different gaps between 2-µm-radius gold cylinders with the smallest separations: w0 = (a,c,d) 100 and (b) 200 nm and wavelengths: λ = (a,b) 1.55 and (c,d) 1.033 µm.

Fig. 5
Fig. 5

The fundamental CyPP mode characteristics, i.e. (a) the effective index and propagation length along with (b) cutoff cylinder radius and normalized mode size, as a function of light wavelength for the cylinder gap w 0 = 100 nm and the effective cylinder radius R = 2µm.

Equations (15)

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Ez(x,y,z)   =   Y(y)Z(y,z)exp(iβx),
2Zz2     +     [εd(m)k02q2(y)]   Z   =   0       ,                                                        (2a)d2Ydy2     +     [q2(y)β2]   Y   =   0                   ,                ​ ​                                             (2b)
q2(y)k2+2krw(y);k=k0εd,r=εd(εdεm)εm ,
Z[w(y),z]   =   A   {γcosh(κdz),   z   (0.5w,0.5w)   ;exp(κm|z|),   z   (0.5w,0.5w);            with    γ   =   εmexp[0.5κmw(y)]εdcosh[0.5κdw(y)]     ,
d2Ydy2   +   [2krw(y)+k2β2]Y   =   0 .
                                                                         βn   =   k1   +   [rθ(n+1)]2     ,   n   =   0,   1,   2,   ;   
Y0(ξ=k|r|y/θ)   =   α0ξexp(ξ)   ,           Y1(ξ)   =   α1ξ(10.5ξ)exp(0.5ξ)   ,         ...   
Vcpp=4πdεd|εdεm|λθ|εm|   >   22       .
w(y)      w0   +   0.5y2(R11+R21)   =   w0   +   y2R   ,                        where      R=2R1R2R1+R2 ,
d2Yd2y   +   (k2   +   2  ​krw0      2​  kry2w02R      β2)Y   =   0 .
βn   =   k1   +   2rkw0(1      2n+12krR)   ,n   =   0,1,2, ;
Yn(ξ   =   2k|r|w02R4y)   =   αn(1)nexp(ξ22)dn{exp(ξ2)}dξn ,
Re(βn)   >   k            Rn   >   (2n+1)22k|r|         ,           n   =   0,1,2,... .
wy(ξ=22)   =   22w02Rkr4          wz   =   w0(1+2rkR) .
Vcypp   =   RminRminq2(y)k2dy      2krRln(4Rmin2w0R) .

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