Abstract

We study nonlinear propagation of surface plasmon polaritons along an interface between metal and nonlinear Kerr dielectric. We demonstrate numerically self-focusing of a plasmon beam at large powers and the formation of slowly decaying spatial soliton in the presence of losses. We develop an analytical model for describing the evolution of spatial plasmon-solitons and observe a good agreement with numerical results.

© 2009 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Self-focusing of femtosecond surface plasmon polaritons

Andreas Pusch, Ilya V. Shadrivov, Ortwin Hess, and Yuri S. Kivshar
Opt. Express 21(1) 1121-1127 (2013)

Spatial modulation instability of coupled surface plasmon polaritons in a dielectric–metal–dielectric structure

Monisha Kumar, K. Porsezian, P. Tchofo-Dinda, Ph. Grelu, T. Mithun, and T. Uthayakumar
J. Opt. Soc. Am. B 34(1) 198-206 (2017)

Spatial optical solitons in planar glass waveguides

J. S. Aitchison, Y. Silberberg, A. M. Weiner, D. E. Leaird, M. K. Oliver, J. L. Jackel, E. M. Vogel, and P. W. E. Smith
J. Opt. Soc. Am. B 8(6) 1290-1297 (1991)

More Recommended Articles

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  1. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer-Verlag, Berlin2007).
  2. Plasmonic Nanoguides and Circuits, Ed. S.I. Bozhevolnyi (Pan Stanford Publ, Singapore2009).
  3. V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” Sov. Phys. JETP Lett.32, 512–515 ( 1980) [In Russian: Pisma JETF 32, 532–535 ( 1980)].
  4. G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 ( 1985).
    [Crossref]
  5. A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright“Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 ( 1987).
    [Crossref] [PubMed]
  6. J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 ( 1985).
    [Crossref]
  7. A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear plasmonic slot waveguides,” Opt. Express 16, 21209–21214 ( 2008).
    [Crossref] [PubMed]
  8. A. R. Davoyan, I. V. Shadrivov, and Yu. S. Kivshar, “Nonlinear plasmonic slot waveguides: erratum,” Opt. Express 17, 4833–4833 ( 2009).
    [Crossref]
  9. A. D. Boardman, G. S. Cooper, A. A. Maradudin, and T. P. Shen, “Surface-polariton solitons,” Phys. Rev. B 34, 8273–8278 ( 1986).
    [Crossref]
  10. E. Feigenbaum and M. Orenstein, “Plasmon-soliton,” Opt. Lett. 32, 674–676 ( 2007).
    [Crossref] [PubMed]
  11. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge2002).
    [PubMed]
  12. B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B 44, 13556–13572 ( 1991).
    [Crossref]
  13. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons. From Fibers to Photonic Crystals (Elsevier Science, USA, 2003).
  14. M. Asobe, T. Kanamori, and K. Kubodera, “Applications of highly nonlinear chalcogenite glass fibers in ultrafast all optical switches,” IEEE J. Quantum Electron. 29, 2325–2333 ( 1993).
    [Crossref]
  15. P. B. Johnson and R. W. Christy, “Optical constants of nobel metals,” Phys. Rev. B 6, 4370–4379 ( 1972).
    [Crossref]
  16. Yo. Lui, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 ( 2007).
    [Crossref]
  17. L. Brzozowski, E. H. Sargent, A. S. Thorpe, and M. Extavour, “Direct measurements of large near-band edge nonlinear index change from 1.48 to 1.55 µ m in InGaAs/InAlGaAs multiquantum wells,” Appl. Phys. Lett. 82, 4429–4431 ( 2003).
    [Crossref]
  18. S. O. Demokritov, B. Hillebrands, and A. N. Slavin, “Brillouin light scattering studies of confined spin waves: linear and nonlinear confinment,” Phys. Rep. 348, 441–489 ( 2001).
    [Crossref]
  19. K. OkamotoFundamentals and Applications of Optical Waveguides (Academic Press, 2000).

2009 (1)

2008 (1)

2007 (2)

E. Feigenbaum and M. Orenstein, “Plasmon-soliton,” Opt. Lett. 32, 674–676 ( 2007).
[Crossref] [PubMed]

Yo. Lui, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 ( 2007).
[Crossref]

2003 (1)

L. Brzozowski, E. H. Sargent, A. S. Thorpe, and M. Extavour, “Direct measurements of large near-band edge nonlinear index change from 1.48 to 1.55 µ m in InGaAs/InAlGaAs multiquantum wells,” Appl. Phys. Lett. 82, 4429–4431 ( 2003).
[Crossref]

2001 (1)

S. O. Demokritov, B. Hillebrands, and A. N. Slavin, “Brillouin light scattering studies of confined spin waves: linear and nonlinear confinment,” Phys. Rep. 348, 441–489 ( 2001).
[Crossref]

1993 (1)

M. Asobe, T. Kanamori, and K. Kubodera, “Applications of highly nonlinear chalcogenite glass fibers in ultrafast all optical switches,” IEEE J. Quantum Electron. 29, 2325–2333 ( 1993).
[Crossref]

1991 (1)

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B 44, 13556–13572 ( 1991).
[Crossref]

1987 (1)

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright“Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 ( 1987).
[Crossref] [PubMed]

1986 (1)

A. D. Boardman, G. S. Cooper, A. A. Maradudin, and T. P. Shen, “Surface-polariton solitons,” Phys. Rev. B 34, 8273–8278 ( 1986).
[Crossref]

1985 (2)

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 ( 1985).
[Crossref]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 ( 1985).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of nobel metals,” Phys. Rev. B 6, 4370–4379 ( 1972).
[Crossref]

Agranovich, V. M.

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” Sov. Phys. JETP Lett.32, 512–515 ( 1980) [In Russian: Pisma JETF 32, 532–535 ( 1980)].

Agrawal, G. P.

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons. From Fibers to Photonic Crystals (Elsevier Science, USA, 2003).

Ariyasu, J.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 ( 1985).
[Crossref]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 ( 1985).
[Crossref]

Asobe, M.

M. Asobe, T. Kanamori, and K. Kubodera, “Applications of highly nonlinear chalcogenite glass fibers in ultrafast all optical switches,” IEEE J. Quantum Electron. 29, 2325–2333 ( 1993).
[Crossref]

Babichenko, V. S.

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” Sov. Phys. JETP Lett.32, 512–515 ( 1980) [In Russian: Pisma JETF 32, 532–535 ( 1980)].

Bartal, G.

Yo. Lui, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 ( 2007).
[Crossref]

Boardman, A. D.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright“Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 ( 1987).
[Crossref] [PubMed]

A. D. Boardman, G. S. Cooper, A. A. Maradudin, and T. P. Shen, “Surface-polariton solitons,” Phys. Rev. B 34, 8273–8278 ( 1986).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge2002).
[PubMed]

Brzozowski, L.

L. Brzozowski, E. H. Sargent, A. S. Thorpe, and M. Extavour, “Direct measurements of large near-band edge nonlinear index change from 1.48 to 1.55 µ m in InGaAs/InAlGaAs multiquantum wells,” Appl. Phys. Lett. 82, 4429–4431 ( 2003).
[Crossref]

Chernyak, V. Ya.

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” Sov. Phys. JETP Lett.32, 512–515 ( 1980) [In Russian: Pisma JETF 32, 532–535 ( 1980)].

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of nobel metals,” Phys. Rev. B 6, 4370–4379 ( 1972).
[Crossref]

Cooper, G. S.

A. D. Boardman, G. S. Cooper, A. A. Maradudin, and T. P. Shen, “Surface-polariton solitons,” Phys. Rev. B 34, 8273–8278 ( 1986).
[Crossref]

Davoyan, A. R.

Demokritov, S. O.

S. O. Demokritov, B. Hillebrands, and A. N. Slavin, “Brillouin light scattering studies of confined spin waves: linear and nonlinear confinment,” Phys. Rep. 348, 441–489 ( 2001).
[Crossref]

Extavour, M.

L. Brzozowski, E. H. Sargent, A. S. Thorpe, and M. Extavour, “Direct measurements of large near-band edge nonlinear index change from 1.48 to 1.55 µ m in InGaAs/InAlGaAs multiquantum wells,” Appl. Phys. Lett. 82, 4429–4431 ( 2003).
[Crossref]

Feigenbaum, E.

Genov, D. A.

Yo. Lui, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 ( 2007).
[Crossref]

Hillebrands, B.

S. O. Demokritov, B. Hillebrands, and A. N. Slavin, “Brillouin light scattering studies of confined spin waves: linear and nonlinear confinment,” Phys. Rep. 348, 441–489 ( 2001).
[Crossref]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of nobel metals,” Phys. Rev. B 6, 4370–4379 ( 1972).
[Crossref]

Kanamori, T.

M. Asobe, T. Kanamori, and K. Kubodera, “Applications of highly nonlinear chalcogenite glass fibers in ultrafast all optical switches,” IEEE J. Quantum Electron. 29, 2325–2333 ( 1993).
[Crossref]

Kivshar, Yu. S.

Kubodera, K.

M. Asobe, T. Kanamori, and K. Kubodera, “Applications of highly nonlinear chalcogenite glass fibers in ultrafast all optical switches,” IEEE J. Quantum Electron. 29, 2325–2333 ( 1993).
[Crossref]

Lui, Yo.

Yo. Lui, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 ( 2007).
[Crossref]

Maier, S. A.

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer-Verlag, Berlin2007).

Maradudin, A. A.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright“Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 ( 1987).
[Crossref] [PubMed]

A. D. Boardman, G. S. Cooper, A. A. Maradudin, and T. P. Shen, “Surface-polariton solitons,” Phys. Rev. B 34, 8273–8278 ( 1986).
[Crossref]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 ( 1985).
[Crossref]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 ( 1985).
[Crossref]

Mysyrowicz, A.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B 44, 13556–13572 ( 1991).
[Crossref]

Okamoto, K.

K. OkamotoFundamentals and Applications of Optical Waveguides (Academic Press, 2000).

Orenstein, M.

Prade, B.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B 44, 13556–13572 ( 1991).
[Crossref]

Sargent, E. H.

L. Brzozowski, E. H. Sargent, A. S. Thorpe, and M. Extavour, “Direct measurements of large near-band edge nonlinear index change from 1.48 to 1.55 µ m in InGaAs/InAlGaAs multiquantum wells,” Appl. Phys. Lett. 82, 4429–4431 ( 2003).
[Crossref]

Seaton, C. T.

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 ( 1985).
[Crossref]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 ( 1985).
[Crossref]

Shadrivov, I. V.

Shen, T. P.

A. D. Boardman, G. S. Cooper, A. A. Maradudin, and T. P. Shen, “Surface-polariton solitons,” Phys. Rev. B 34, 8273–8278 ( 1986).
[Crossref]

Slavin, A. N.

S. O. Demokritov, B. Hillebrands, and A. N. Slavin, “Brillouin light scattering studies of confined spin waves: linear and nonlinear confinment,” Phys. Rep. 348, 441–489 ( 2001).
[Crossref]

Stegeman, G. I.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright“Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 ( 1987).
[Crossref] [PubMed]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 ( 1985).
[Crossref]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 ( 1985).
[Crossref]

Thorpe, A. S.

L. Brzozowski, E. H. Sargent, A. S. Thorpe, and M. Extavour, “Direct measurements of large near-band edge nonlinear index change from 1.48 to 1.55 µ m in InGaAs/InAlGaAs multiquantum wells,” Appl. Phys. Lett. 82, 4429–4431 ( 2003).
[Crossref]

Twardowski, T.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright“Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 ( 1987).
[Crossref] [PubMed]

Vinet, J. Y.

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B 44, 13556–13572 ( 1991).
[Crossref]

Wallis, R. F.

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 ( 1985).
[Crossref]

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 ( 1985).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge2002).
[PubMed]

Wright, E. M.

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright“Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 ( 1987).
[Crossref] [PubMed]

Zhang, X.

Yo. Lui, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 ( 2007).
[Crossref]

Appl. Phys. Lett. (1)

L. Brzozowski, E. H. Sargent, A. S. Thorpe, and M. Extavour, “Direct measurements of large near-band edge nonlinear index change from 1.48 to 1.55 µ m in InGaAs/InAlGaAs multiquantum wells,” Appl. Phys. Lett. 82, 4429–4431 ( 2003).
[Crossref]

IEEE J. Quantum Electron. (1)

M. Asobe, T. Kanamori, and K. Kubodera, “Applications of highly nonlinear chalcogenite glass fibers in ultrafast all optical switches,” IEEE J. Quantum Electron. 29, 2325–2333 ( 1993).
[Crossref]

J. Appl. Phys. (2)

G. I. Stegeman, C. T. Seaton, J. Ariyasu, R. F. Wallis, and A. A. Maradudin, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 ( 1985).
[Crossref]

J. Ariyasu, C. T. Seaton, G. I. Stegeman, A. A. Maradudin, and R. F. Wallis, “Nonlinear surface polaritons guided by metal films,” J. Appl. Phys. 58, 2460–2466 ( 1985).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rep. (1)

S. O. Demokritov, B. Hillebrands, and A. N. Slavin, “Brillouin light scattering studies of confined spin waves: linear and nonlinear confinment,” Phys. Rep. 348, 441–489 ( 2001).
[Crossref]

Phys. Rev. A (1)

A. D. Boardman, A. A. Maradudin, G. I. Stegeman, T. Twardowski, and E. M. Wright“Exact theory of nonlinear p-polarized optical waves,” Phys. Rev. A 35, 1159–1164 ( 1987).
[Crossref] [PubMed]

Phys. Rev. B (3)

A. D. Boardman, G. S. Cooper, A. A. Maradudin, and T. P. Shen, “Surface-polariton solitons,” Phys. Rev. B 34, 8273–8278 ( 1986).
[Crossref]

P. B. Johnson and R. W. Christy, “Optical constants of nobel metals,” Phys. Rev. B 6, 4370–4379 ( 1972).
[Crossref]

B. Prade, J. Y. Vinet, and A. Mysyrowicz, “Guided optical waves in planar heterostructures with negative dielectric constant,” Phys. Rev. B 44, 13556–13572 ( 1991).
[Crossref]

Phys. Rev. Lett. (1)

Yo. Lui, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength discrete solitons in nonlinear metamaterials,” Phys. Rev. Lett. 99, 153901 ( 2007).
[Crossref]

Other (6)

K. OkamotoFundamentals and Applications of Optical Waveguides (Academic Press, 2000).

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons. From Fibers to Photonic Crystals (Elsevier Science, USA, 2003).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University Press, Cambridge2002).
[PubMed]

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer-Verlag, Berlin2007).

Plasmonic Nanoguides and Circuits, Ed. S.I. Bozhevolnyi (Pan Stanford Publ, Singapore2009).

V. M. Agranovich, V. S. Babichenko, and V. Ya. Chernyak, “Nonlinear surface polaritons,” Sov. Phys. JETP Lett.32, 512–515 ( 1980) [In Russian: Pisma JETF 32, 532–535 ( 1980)].

Supplementary Material (2)

» Media 1: MOV (499 KB)     
» Media 2: MOV (486 KB)     

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 (3)

Fig. 1.
Fig. 1.

(a) Schematic of a metal-dielectric interface shown with a transverse structure of the surface plasmon. (b) Geometry used in the numerical FDTD calculations.

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2.

Magnetic field distribution at the metal-dielectric interface for: (a) Linear diffraction (low power), R 0=0.35µm. (b) Self-focusing and soliton generation for |E|2⇍3×106 V 2/µm 2 (Media 1). (c) Same as in (b) but for low losses (Media 2). Animations corresponding to (b) and (c) show that soliton formation with gradual increase of the incident power.

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3.

(a) Evolution of the beam width for: 1 - linear regime, 2 - nonlinear regime with realistic losses, 3 - soliton formation in nonlinear regime with low losses. Dashed lines correspond to the results obtained from Eq. (4). (b) Soliton profiles: analytical (solid) and FDTD simulations (dots). (c) Soliton intensity vs. the effective index: solid - analytical results; crosses - FDTD data.

Download Full Size | PPT Slide | PDF

Equations (5)

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

{ x < 0 : E z 0 = e κ 1 x , E x 0 = − ( β ⁄ κ 1 ) e κ 1 x , κ 1 2 = β 2 − ε metal ; x > 0 : E z 0 = e − κ 2 x , E x 0 = ( β ⁄ κ 2 ) e − κ 2 x , κ 2 2 = β 2 − ε lin
x : ∂ 2 E x ∂ y 2 + ∂ 2 E x ∂ z 2 − ∂ 2 E z ∂ x ∂ z + ε E x = 0
z : ∂ 2 E x ∂ x 2 + ∂ 2 E z ∂ y 2 − ∂ 2 E x ∂ z ∂ x + ε E z = 0
E = ( x A E x 0 + i z B E z 0 ) e − i β z ,
− 2 i β D ∂ A ∂ z + ∂ 2 A ∂ y 2 + 2 I ∣ A ∣ 2 A + i Γ A = 0 ,

Metrics