Abstract

It is theoretically shown that multi-component discrete vector surface waves can exist in arrays of coupled waveguides. These mutually trapped surface states primarily reside in the first waveguide of a semi-infinite array. The existence and stability of such surface waves are systematically investigated.

© 2005 Optical Society of America

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  1. W.L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003).
    [CrossRef] [PubMed]
  2. P. Yeh, A. Yariv, and A.Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104 (1978).
    [CrossRef]
  3. D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94, 013901 (2005).
    [CrossRef] [PubMed]
  4. A.D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic phenomena, V.M. Agranovich, A.A. Maradudin, H.-E. Ponath, and G.I. Stegeman, eds. (Elsevier Science Publishers B.V., New York, 1991).
  5. D. Mihalache, M. Bertootti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Progress in Optics 27, 229 (1989).
    [CrossRef]
  6. M. Miyagi and S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 24, 53 (1972).
  7. W.J. Tomlinson, “Surface wave at a nonlinear interface,” Opt. Lett. 5323 (1980).
    [CrossRef] [PubMed]
  8. N. N. Akhmediev, “A novel class of non-linear surface waves-asymmetric modes in symmetric layered structures,” Zh. Eksp. Teor. Fiz. 83, 545 (1982).
  9. V. K. Fedyanin and D. Mihalache, “P-polarized nonlinear surface polaritons in layered structures,” Z. Phys. B 47, 167 (1982).
    [CrossRef]
  10. N.N. Akhmediev, V.I. Korneev, and Y.V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams,” Sov. Phys. JETP 61, 62 (1985).
  11. U. Langbein, F. Lederer, and H.E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167, (1983).
    [CrossRef]
  12. C.T. Seaton, J.D. Valera, R.L. Shoemaker, G.I. Stegeman, J.T. Chilwell, and S.D. Smith, “Calculations of nonlinear TE waves guided by thin dielectic films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 782 (1985).
    [CrossRef]
  13. G.S. Garcia Quirino, J.J. Sanchez-Mondragon, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571 (1995).
    [CrossRef] [PubMed]
  14. M. Cronin-Golomb, “Photorefractive surface waves,” Opt. Lett. 20, 2075 (1995).
    [CrossRef] [PubMed]
  15. D.N Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817 (2003).
    [CrossRef] [PubMed]
  16. K.G. Makris, S. Suntsov, D.N. Christodoulides, G.I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett. 30, 2466 (2005).
    [CrossRef] [PubMed]
  17. E.A. Ostrovskaya, N.N. Akhmediev, G.I. Stegeman, J.U. Kang, and J.S. Aitchison, “Mixed-mode spatial solitons in semiconductor waveguides,” J. Opt. Soc. Am. B 14, 880 (1997).
    [CrossRef]
  18. R.R. Malendevich, L. Friedrich, G.I. Stegeman, J.M. Soto-Crespo, N.N. Akhmediev, and J.S. Aitchison, “Radiation-related polarization instability of Kerr spatial vector solitons” J. Opt. Soc. Am. B 19, 695 (2002).
    [CrossRef]
  19. J. Meier, J. Hudock, D. Christodoulides, G.I. Stegeman, Y. Silberberg, R. Morandotti, and J.S. Aitchison, “DiscreteVector Solitons in Kerr NonlinearWaveguide Arrays,” Phys. Rev. Lett. 91, 143907 (2003).
    [CrossRef] [PubMed]

2005 (2)

D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94, 013901 (2005).
[CrossRef] [PubMed]

K.G. Makris, S. Suntsov, D.N. Christodoulides, G.I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett. 30, 2466 (2005).
[CrossRef] [PubMed]

2003 (3)

D.N Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817 (2003).
[CrossRef] [PubMed]

W.L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003).
[CrossRef] [PubMed]

J. Meier, J. Hudock, D. Christodoulides, G.I. Stegeman, Y. Silberberg, R. Morandotti, and J.S. Aitchison, “DiscreteVector Solitons in Kerr NonlinearWaveguide Arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

2002 (1)

1997 (1)

1995 (2)

G.S. Garcia Quirino, J.J. Sanchez-Mondragon, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571 (1995).
[CrossRef] [PubMed]

M. Cronin-Golomb, “Photorefractive surface waves,” Opt. Lett. 20, 2075 (1995).
[CrossRef] [PubMed]

1989 (1)

D. Mihalache, M. Bertootti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Progress in Optics 27, 229 (1989).
[CrossRef]

1985 (2)

N.N. Akhmediev, V.I. Korneev, and Y.V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams,” Sov. Phys. JETP 61, 62 (1985).

C.T. Seaton, J.D. Valera, R.L. Shoemaker, G.I. Stegeman, J.T. Chilwell, and S.D. Smith, “Calculations of nonlinear TE waves guided by thin dielectic films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 782 (1985).
[CrossRef]

1983 (1)

U. Langbein, F. Lederer, and H.E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167, (1983).
[CrossRef]

1982 (2)

N. N. Akhmediev, “A novel class of non-linear surface waves-asymmetric modes in symmetric layered structures,” Zh. Eksp. Teor. Fiz. 83, 545 (1982).

V. K. Fedyanin and D. Mihalache, “P-polarized nonlinear surface polaritons in layered structures,” Z. Phys. B 47, 167 (1982).
[CrossRef]

1980 (1)

1978 (1)

P. Yeh, A. Yariv, and A.Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104 (1978).
[CrossRef]

1972 (1)

M. Miyagi and S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 24, 53 (1972).

Aitchison, J.S.

Akhmediev, N. N.

N. N. Akhmediev, “A novel class of non-linear surface waves-asymmetric modes in symmetric layered structures,” Zh. Eksp. Teor. Fiz. 83, 545 (1982).

Akhmediev, N.N.

Artigas, D.

D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94, 013901 (2005).
[CrossRef] [PubMed]

Barnes, W.L.

W.L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003).
[CrossRef] [PubMed]

Bertootti, M.

D. Mihalache, M. Bertootti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Progress in Optics 27, 229 (1989).
[CrossRef]

Boardman, A.D.

A.D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic phenomena, V.M. Agranovich, A.A. Maradudin, H.-E. Ponath, and G.I. Stegeman, eds. (Elsevier Science Publishers B.V., New York, 1991).

Chilwell, J.T.

C.T. Seaton, J.D. Valera, R.L. Shoemaker, G.I. Stegeman, J.T. Chilwell, and S.D. Smith, “Calculations of nonlinear TE waves guided by thin dielectic films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 782 (1985).
[CrossRef]

Cho, A.Y.

P. Yeh, A. Yariv, and A.Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104 (1978).
[CrossRef]

Christodoulides, D.

J. Meier, J. Hudock, D. Christodoulides, G.I. Stegeman, Y. Silberberg, R. Morandotti, and J.S. Aitchison, “DiscreteVector Solitons in Kerr NonlinearWaveguide Arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

Christodoulides, D.N

D.N Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817 (2003).
[CrossRef] [PubMed]

Christodoulides, D.N.

Cronin-Golomb, M.

Dereux, A.

W.L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003).
[CrossRef] [PubMed]

Ebbesen, T.W.

W.L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003).
[CrossRef] [PubMed]

Egan, P.

A.D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic phenomena, V.M. Agranovich, A.A. Maradudin, H.-E. Ponath, and G.I. Stegeman, eds. (Elsevier Science Publishers B.V., New York, 1991).

Fedyanin, V. K.

V. K. Fedyanin and D. Mihalache, “P-polarized nonlinear surface polaritons in layered structures,” Z. Phys. B 47, 167 (1982).
[CrossRef]

Friedrich, L.

Garcia Quirino, G.S.

G.S. Garcia Quirino, J.J. Sanchez-Mondragon, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571 (1995).
[CrossRef] [PubMed]

Hache, A.

Hudock, J.

J. Meier, J. Hudock, D. Christodoulides, G.I. Stegeman, Y. Silberberg, R. Morandotti, and J.S. Aitchison, “DiscreteVector Solitons in Kerr NonlinearWaveguide Arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

Kang, J.U.

Korneev, V.I.

N.N. Akhmediev, V.I. Korneev, and Y.V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams,” Sov. Phys. JETP 61, 62 (1985).

Kuz’menko, Y.V.

N.N. Akhmediev, V.I. Korneev, and Y.V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams,” Sov. Phys. JETP 61, 62 (1985).

Langbein, U.

U. Langbein, F. Lederer, and H.E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167, (1983).
[CrossRef]

A.D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic phenomena, V.M. Agranovich, A.A. Maradudin, H.-E. Ponath, and G.I. Stegeman, eds. (Elsevier Science Publishers B.V., New York, 1991).

Lederer, F.

D.N Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817 (2003).
[CrossRef] [PubMed]

U. Langbein, F. Lederer, and H.E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167, (1983).
[CrossRef]

A.D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic phenomena, V.M. Agranovich, A.A. Maradudin, H.-E. Ponath, and G.I. Stegeman, eds. (Elsevier Science Publishers B.V., New York, 1991).

Makris, K.G.

Malendevich, R.R.

Meier, J.

J. Meier, J. Hudock, D. Christodoulides, G.I. Stegeman, Y. Silberberg, R. Morandotti, and J.S. Aitchison, “DiscreteVector Solitons in Kerr NonlinearWaveguide Arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

Mihalache, D.

D. Mihalache, M. Bertootti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Progress in Optics 27, 229 (1989).
[CrossRef]

V. K. Fedyanin and D. Mihalache, “P-polarized nonlinear surface polaritons in layered structures,” Z. Phys. B 47, 167 (1982).
[CrossRef]

A.D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic phenomena, V.M. Agranovich, A.A. Maradudin, H.-E. Ponath, and G.I. Stegeman, eds. (Elsevier Science Publishers B.V., New York, 1991).

Miyagi, M.

M. Miyagi and S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 24, 53 (1972).

Morandotti, R.

J. Meier, J. Hudock, D. Christodoulides, G.I. Stegeman, Y. Silberberg, R. Morandotti, and J.S. Aitchison, “DiscreteVector Solitons in Kerr NonlinearWaveguide Arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

Nishida, S.

M. Miyagi and S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 24, 53 (1972).

Ostrovskaya, E.A.

Ponath, H.E.

U. Langbein, F. Lederer, and H.E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167, (1983).
[CrossRef]

Sanchez-Mondragon, J.J.

G.S. Garcia Quirino, J.J. Sanchez-Mondragon, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571 (1995).
[CrossRef] [PubMed]

Seaton, C.T.

C.T. Seaton, J.D. Valera, R.L. Shoemaker, G.I. Stegeman, J.T. Chilwell, and S.D. Smith, “Calculations of nonlinear TE waves guided by thin dielectic films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 782 (1985).
[CrossRef]

Shoemaker, R.L.

C.T. Seaton, J.D. Valera, R.L. Shoemaker, G.I. Stegeman, J.T. Chilwell, and S.D. Smith, “Calculations of nonlinear TE waves guided by thin dielectic films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 782 (1985).
[CrossRef]

Sibilia, C.

D. Mihalache, M. Bertootti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Progress in Optics 27, 229 (1989).
[CrossRef]

Silberberg, Y.

D.N Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817 (2003).
[CrossRef] [PubMed]

J. Meier, J. Hudock, D. Christodoulides, G.I. Stegeman, Y. Silberberg, R. Morandotti, and J.S. Aitchison, “DiscreteVector Solitons in Kerr NonlinearWaveguide Arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

Smith, S.D.

C.T. Seaton, J.D. Valera, R.L. Shoemaker, G.I. Stegeman, J.T. Chilwell, and S.D. Smith, “Calculations of nonlinear TE waves guided by thin dielectic films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 782 (1985).
[CrossRef]

Soto-Crespo, J.M.

Stegeman, G.I.

K.G. Makris, S. Suntsov, D.N. Christodoulides, G.I. Stegeman, and A. Hache, “Discrete surface solitons,” Opt. Lett. 30, 2466 (2005).
[CrossRef] [PubMed]

J. Meier, J. Hudock, D. Christodoulides, G.I. Stegeman, Y. Silberberg, R. Morandotti, and J.S. Aitchison, “DiscreteVector Solitons in Kerr NonlinearWaveguide Arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

R.R. Malendevich, L. Friedrich, G.I. Stegeman, J.M. Soto-Crespo, N.N. Akhmediev, and J.S. Aitchison, “Radiation-related polarization instability of Kerr spatial vector solitons” J. Opt. Soc. Am. B 19, 695 (2002).
[CrossRef]

E.A. Ostrovskaya, N.N. Akhmediev, G.I. Stegeman, J.U. Kang, and J.S. Aitchison, “Mixed-mode spatial solitons in semiconductor waveguides,” J. Opt. Soc. Am. B 14, 880 (1997).
[CrossRef]

C.T. Seaton, J.D. Valera, R.L. Shoemaker, G.I. Stegeman, J.T. Chilwell, and S.D. Smith, “Calculations of nonlinear TE waves guided by thin dielectic films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 782 (1985).
[CrossRef]

Stepanov, S.

G.S. Garcia Quirino, J.J. Sanchez-Mondragon, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571 (1995).
[CrossRef] [PubMed]

Suntsov, S.

Tomlinson, W.J.

Torner, L.

D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94, 013901 (2005).
[CrossRef] [PubMed]

Valera, J.D.

C.T. Seaton, J.D. Valera, R.L. Shoemaker, G.I. Stegeman, J.T. Chilwell, and S.D. Smith, “Calculations of nonlinear TE waves guided by thin dielectic films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 782 (1985).
[CrossRef]

Yariv, A.

P. Yeh, A. Yariv, and A.Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104 (1978).
[CrossRef]

Yeh, P.

P. Yeh, A. Yariv, and A.Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104 (1978).
[CrossRef]

Appl. Phys. Lett. (1)

P. Yeh, A. Yariv, and A.Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104 (1978).
[CrossRef]

IEEE J. Quantum Electron. (1)

C.T. Seaton, J.D. Valera, R.L. Shoemaker, G.I. Stegeman, J.T. Chilwell, and S.D. Smith, “Calculations of nonlinear TE waves guided by thin dielectic films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 782 (1985).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nature (2)

D.N Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817 (2003).
[CrossRef] [PubMed]

W.L. Barnes, A. Dereux, and T.W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

U. Langbein, F. Lederer, and H.E. Ponath, “A new type of nonlinear slab-guided waves,” Opt. Commun. 46, 167, (1983).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. A (1)

G.S. Garcia Quirino, J.J. Sanchez-Mondragon, and S. Stepanov, “Nonlinear surface optical waves in photorefractive crystals with a diffusion mechanism of nonlinearity,” Phys. Rev. A 51, 1571 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

J. Meier, J. Hudock, D. Christodoulides, G.I. Stegeman, Y. Silberberg, R. Morandotti, and J.S. Aitchison, “DiscreteVector Solitons in Kerr NonlinearWaveguide Arrays,” Phys. Rev. Lett. 91, 143907 (2003).
[CrossRef] [PubMed]

D. Artigas and L. Torner, “Dyakonov surface waves in photonic metamaterials,” Phys. Rev. Lett. 94, 013901 (2005).
[CrossRef] [PubMed]

Progress in Optics (1)

D. Mihalache, M. Bertootti, and C. Sibilia, “Nonlinear wave propagation in planar structures,” Progress in Optics 27, 229 (1989).
[CrossRef]

Sci. Rep. Res. Inst. Tohoku Univ. Ser. B (1)

M. Miyagi and S. Nishida, Sci. Rep. Res. Inst. Tohoku Univ. Ser. B 24, 53 (1972).

Sov. Phys. JETP (1)

N.N. Akhmediev, V.I. Korneev, and Y.V. Kuz’menko, “Excitation of nonlinear surface waves by Gaussian light beams,” Sov. Phys. JETP 61, 62 (1985).

Z. Phys. B (1)

V. K. Fedyanin and D. Mihalache, “P-polarized nonlinear surface polaritons in layered structures,” Z. Phys. B 47, 167 (1982).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

N. N. Akhmediev, “A novel class of non-linear surface waves-asymmetric modes in symmetric layered structures,” Zh. Eksp. Teor. Fiz. 83, 545 (1982).

Other (1)

A.D. Boardman, P. Egan, F. Lederer, U. Langbein, and D. Mihalache, “Third-order nonlinear electromagnetic TE and TM guided waves,” in Nonlinear Surface Electromagnetic phenomena, V.M. Agranovich, A.A. Maradudin, H.-E. Ponath, and G.I. Stegeman, eds. (Elsevier Science Publishers B.V., New York, 1991).

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Figures (4)

Fig. 1.
Fig. 1.

P - Δ diagrams for the linearly polarized (solid line) and elliptically polarized (dash line) discrete vector surface waves as well as for the scalar TE (dotted line) and TM (dotted-dash line) surface waves.

Fig. 2.
Fig. 2.

P-Δ for the TE (solid line) and TM (dashed line) components of the stable linearly polarized family of vector discrete surface waves

Fig. 3.
Fig. 3.

Stable propagation of the (a) TE component and (b) TM component of a linearly polarized discrete vector surface wave. (c) The eigenvalues of the perturbed problem.

Fig. 4.
Fig. 4.

Unstable propagation of the (a) TE component and (b) TM component of an elliptically polarized discrete vector surface wave. (c) The eigenvalues of the perturbed problem.

Equations (4)

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

i d a n + a n + γ ( a n + 1 + a n 1 ) + [ a n 2 + A b n 2 ] a n + B b n 2 a n * = 0 ,
i d b n b n + γ ( b n + 1 + b n 1 ) + [ b n 2 + A a n 2 ] b n + B a n 2 b n * = 0 .
A 0 2 = γ [ ( A ± B ) exp ( μ ) exp ( ν ) ] ( A ± B ) 2 1 ,
B 0 2 = γ [ ( A ± B ) exp ( ν ) exp ( μ ) ] ( A ± B ) 2 1 ,

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