Abstract

The dispersion properties of transverse electric nonlinear waves in a three-layer slab waveguide which consists of a double negative material (DNM) guiding layer sandwiched between an intensity-dependent refractive index substrate and semi-infinite linear dielectric cover are investigated. The dispersion properties for self-focusing and self-defocusing substrate nonlinearity are presented. The effects of the negative parameters of the DNM on the dispersion characteristics are investigated.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Stegeman, C. Seaton, and J. Ariyasu, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
    [CrossRef]
  2. W. Tomlinson, “Surface wave at a nonlinear interface,” Opt. Lett. 5, 323–325 (1980).
    [CrossRef]
  3. W. Chen and A. Maradudin, “S-polarized guided and surface electromagnetic waves supported by a nonlinear dielectric film,” J. Opt. Soc. Am. B 5, 529–538 (1988).
    [CrossRef]
  4. A. Aceves, P. Varatharajab, A. Newell, E. Wright, G. Stegeman, D. Heatley, J. Moloney, and H. Adachihara, “Particle aspects of collimated light channel propagation at nonlinear interfaces and in waveguides,” J. Opt. Soc. Am. B 7, 963–974 (1990).
    [CrossRef]
  5. R. Sammut, Q. Li, and C. Pask, “Variational approximations and mode stability in planar nonlinear waveguides,” J. Opt. Soc. Am. B 9, 884–890 (1992).
    [CrossRef]
  6. C. Angelis and G. Nalesso, “Spatial soliton switching modes of nonlinear optical slab waveguides,” J. Opt. Soc. Am. B 10, 55–59 (1993).
    [CrossRef]
  7. V. Veselago, “The electrodynamics of substance with simultaneously negative index values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
    [CrossRef]
  8. A. Grbic and G. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92, 5930–5935 (2002).
    [CrossRef]
  9. D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
    [CrossRef]
  10. I. Shadrivov, A. Sukhorukov, Y. Kivshar, A. Zharov, A. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
    [CrossRef]
  11. S. Taya, M. Shabat, and H. Khalil, “Enhancement of sensitivity in optical sensors using left-handed materials,” Optik 120, 504–508 (2009).
    [CrossRef]
  12. M. Abadla, S. Taya, and M. Shabat, “Four-layer slab waveguide sensors supported with left handed materials,” Sens. Lett. 9, 1823–1829 (2011).
  13. S. Taya, T. El-Agez, H. Kullab, M. Abadla, and M. Shabat, “Theoretical study of slab waveguide optical sensor with left handed material as a core layer,” Opt. Appl. 42, 193–205 (2012).
  14. H. Kullab, S. Taya, and T. El-Agez, “Metal-clad waveguide sensor using a left-handed material as a core layer,” J. Opt. Soc. Am. B 29, 959–964 (2012).
    [CrossRef]
  15. A. Hamid, “Axially slotted antenna on a circular or elliptic cylinder coated with metamaterials,” Progress Electromagn. Res. 51, 329–341 (2005).
    [CrossRef]
  16. M. Antoniades and G. Eleftheriades, “Compact linear lead/lag metamaterial phase shifters for broadband applications,” IEEE Antennas Wireless Propag. Lett. 2, 103–106 (2003).
    [CrossRef]
  17. J. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef]
  18. H. Kogelnik and V. Ramaswamy, “Scaling rules for thin film optical waveguides,” Appl. Opt. 13, 1857–1862 (1974).
    [CrossRef]
  19. S. Chelkowski and J. Chrostowski, “Scaling rules for slab waveguides with nonlinear substrate,” Appl. Opt. 26, 3681–3686 (1987).
    [CrossRef]
  20. M. Fontaine, “Scaling rules for nonlinear thin film optical waveguides,” Appl. Opt. 29, 3891–3899 (1990).
    [CrossRef]
  21. I. Qadoura, S. Taya, and K. El-Wasife, “Scaling rules for a slab waveguide structure comprising nonlinear and negative index materials,” Int. J. Microw. Opt. Technol. 7, 349–357 (2012).
  22. S. Taya and I. Qadoura, “Guided modes in slab waveguides with negative index cladding and substrate,” Optik 124, 1431–1436 (2013).
    [CrossRef]
  23. A. Boardman and P. Egan, “Novel nonlinear surface and guided TE waves in asymmetric LHM waveguides,” J. Opt. A 11, 114032 (2009).
    [CrossRef]
  24. A. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. 22, 319–324 (1986).
    [CrossRef]
  25. S. Taya, E. El-Farram, and M. Abadla, “Symmetric multilayer slab waveguide structure with a negative index material: TM case,” Optik 123, 2264–2268 (2012).
    [CrossRef]
  26. M. Fontaine, “Universal dispersion and power curves for transverse magnetic waves propagating in slab waveguides with a nonlinear self‐focusing substrate,” J. Appl. Phys. 69, 2826–2834 (1991).
    [CrossRef]

2013 (1)

S. Taya and I. Qadoura, “Guided modes in slab waveguides with negative index cladding and substrate,” Optik 124, 1431–1436 (2013).
[CrossRef]

2012 (4)

S. Taya, E. El-Farram, and M. Abadla, “Symmetric multilayer slab waveguide structure with a negative index material: TM case,” Optik 123, 2264–2268 (2012).
[CrossRef]

I. Qadoura, S. Taya, and K. El-Wasife, “Scaling rules for a slab waveguide structure comprising nonlinear and negative index materials,” Int. J. Microw. Opt. Technol. 7, 349–357 (2012).

S. Taya, T. El-Agez, H. Kullab, M. Abadla, and M. Shabat, “Theoretical study of slab waveguide optical sensor with left handed material as a core layer,” Opt. Appl. 42, 193–205 (2012).

H. Kullab, S. Taya, and T. El-Agez, “Metal-clad waveguide sensor using a left-handed material as a core layer,” J. Opt. Soc. Am. B 29, 959–964 (2012).
[CrossRef]

2011 (1)

M. Abadla, S. Taya, and M. Shabat, “Four-layer slab waveguide sensors supported with left handed materials,” Sens. Lett. 9, 1823–1829 (2011).

2009 (2)

S. Taya, M. Shabat, and H. Khalil, “Enhancement of sensitivity in optical sensors using left-handed materials,” Optik 120, 504–508 (2009).
[CrossRef]

A. Boardman and P. Egan, “Novel nonlinear surface and guided TE waves in asymmetric LHM waveguides,” J. Opt. A 11, 114032 (2009).
[CrossRef]

2005 (1)

A. Hamid, “Axially slotted antenna on a circular or elliptic cylinder coated with metamaterials,” Progress Electromagn. Res. 51, 329–341 (2005).
[CrossRef]

2004 (1)

I. Shadrivov, A. Sukhorukov, Y. Kivshar, A. Zharov, A. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

2003 (2)

D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

M. Antoniades and G. Eleftheriades, “Compact linear lead/lag metamaterial phase shifters for broadband applications,” IEEE Antennas Wireless Propag. Lett. 2, 103–106 (2003).
[CrossRef]

2002 (1)

A. Grbic and G. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92, 5930–5935 (2002).
[CrossRef]

2000 (1)

J. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

1993 (1)

1992 (1)

1991 (1)

M. Fontaine, “Universal dispersion and power curves for transverse magnetic waves propagating in slab waveguides with a nonlinear self‐focusing substrate,” J. Appl. Phys. 69, 2826–2834 (1991).
[CrossRef]

1990 (2)

1988 (1)

1987 (1)

1986 (1)

A. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. 22, 319–324 (1986).
[CrossRef]

1985 (1)

G. Stegeman, C. Seaton, and J. Ariyasu, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[CrossRef]

1980 (1)

1974 (1)

1968 (1)

V. Veselago, “The electrodynamics of substance with simultaneously negative index values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Abadla, M.

S. Taya, T. El-Agez, H. Kullab, M. Abadla, and M. Shabat, “Theoretical study of slab waveguide optical sensor with left handed material as a core layer,” Opt. Appl. 42, 193–205 (2012).

S. Taya, E. El-Farram, and M. Abadla, “Symmetric multilayer slab waveguide structure with a negative index material: TM case,” Optik 123, 2264–2268 (2012).
[CrossRef]

M. Abadla, S. Taya, and M. Shabat, “Four-layer slab waveguide sensors supported with left handed materials,” Sens. Lett. 9, 1823–1829 (2011).

Aceves, A.

Adachihara, H.

Angelis, C.

Antoniades, M.

M. Antoniades and G. Eleftheriades, “Compact linear lead/lag metamaterial phase shifters for broadband applications,” IEEE Antennas Wireless Propag. Lett. 2, 103–106 (2003).
[CrossRef]

Ariyasu, J.

G. Stegeman, C. Seaton, and J. Ariyasu, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[CrossRef]

Boardman, A.

A. Boardman and P. Egan, “Novel nonlinear surface and guided TE waves in asymmetric LHM waveguides,” J. Opt. A 11, 114032 (2009).
[CrossRef]

I. Shadrivov, A. Sukhorukov, Y. Kivshar, A. Zharov, A. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

A. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. 22, 319–324 (1986).
[CrossRef]

Chelkowski, S.

Chen, W.

Chrostowski, J.

Egan, P.

A. Boardman and P. Egan, “Novel nonlinear surface and guided TE waves in asymmetric LHM waveguides,” J. Opt. A 11, 114032 (2009).
[CrossRef]

I. Shadrivov, A. Sukhorukov, Y. Kivshar, A. Zharov, A. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

A. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. 22, 319–324 (1986).
[CrossRef]

El-Agez, T.

H. Kullab, S. Taya, and T. El-Agez, “Metal-clad waveguide sensor using a left-handed material as a core layer,” J. Opt. Soc. Am. B 29, 959–964 (2012).
[CrossRef]

S. Taya, T. El-Agez, H. Kullab, M. Abadla, and M. Shabat, “Theoretical study of slab waveguide optical sensor with left handed material as a core layer,” Opt. Appl. 42, 193–205 (2012).

Eleftheriades, G.

M. Antoniades and G. Eleftheriades, “Compact linear lead/lag metamaterial phase shifters for broadband applications,” IEEE Antennas Wireless Propag. Lett. 2, 103–106 (2003).
[CrossRef]

A. Grbic and G. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92, 5930–5935 (2002).
[CrossRef]

El-Farram, E.

S. Taya, E. El-Farram, and M. Abadla, “Symmetric multilayer slab waveguide structure with a negative index material: TM case,” Optik 123, 2264–2268 (2012).
[CrossRef]

El-Wasife, K.

I. Qadoura, S. Taya, and K. El-Wasife, “Scaling rules for a slab waveguide structure comprising nonlinear and negative index materials,” Int. J. Microw. Opt. Technol. 7, 349–357 (2012).

Fontaine, M.

M. Fontaine, “Universal dispersion and power curves for transverse magnetic waves propagating in slab waveguides with a nonlinear self‐focusing substrate,” J. Appl. Phys. 69, 2826–2834 (1991).
[CrossRef]

M. Fontaine, “Scaling rules for nonlinear thin film optical waveguides,” Appl. Opt. 29, 3891–3899 (1990).
[CrossRef]

Grbic, A.

A. Grbic and G. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92, 5930–5935 (2002).
[CrossRef]

Hamid, A.

A. Hamid, “Axially slotted antenna on a circular or elliptic cylinder coated with metamaterials,” Progress Electromagn. Res. 51, 329–341 (2005).
[CrossRef]

Heatley, D.

Khalil, H.

S. Taya, M. Shabat, and H. Khalil, “Enhancement of sensitivity in optical sensors using left-handed materials,” Optik 120, 504–508 (2009).
[CrossRef]

Kivshar, Y.

I. Shadrivov, A. Sukhorukov, Y. Kivshar, A. Zharov, A. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Kogelnik, H.

Kullab, H.

S. Taya, T. El-Agez, H. Kullab, M. Abadla, and M. Shabat, “Theoretical study of slab waveguide optical sensor with left handed material as a core layer,” Opt. Appl. 42, 193–205 (2012).

H. Kullab, S. Taya, and T. El-Agez, “Metal-clad waveguide sensor using a left-handed material as a core layer,” J. Opt. Soc. Am. B 29, 959–964 (2012).
[CrossRef]

Li, Q.

Maradudin, A.

Moloney, J.

Nalesso, G.

Newell, A.

Pask, C.

Pendry, J.

J. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

Qadoura, I.

S. Taya and I. Qadoura, “Guided modes in slab waveguides with negative index cladding and substrate,” Optik 124, 1431–1436 (2013).
[CrossRef]

I. Qadoura, S. Taya, and K. El-Wasife, “Scaling rules for a slab waveguide structure comprising nonlinear and negative index materials,” Int. J. Microw. Opt. Technol. 7, 349–357 (2012).

Ramaswamy, V.

Sammut, R.

Schurig, D.

D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

Seaton, C.

G. Stegeman, C. Seaton, and J. Ariyasu, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[CrossRef]

Shabat, M.

S. Taya, T. El-Agez, H. Kullab, M. Abadla, and M. Shabat, “Theoretical study of slab waveguide optical sensor with left handed material as a core layer,” Opt. Appl. 42, 193–205 (2012).

M. Abadla, S. Taya, and M. Shabat, “Four-layer slab waveguide sensors supported with left handed materials,” Sens. Lett. 9, 1823–1829 (2011).

S. Taya, M. Shabat, and H. Khalil, “Enhancement of sensitivity in optical sensors using left-handed materials,” Optik 120, 504–508 (2009).
[CrossRef]

Shadrivov, I.

I. Shadrivov, A. Sukhorukov, Y. Kivshar, A. Zharov, A. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Smith, D.

D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

Stegeman, G.

Sukhorukov, A.

I. Shadrivov, A. Sukhorukov, Y. Kivshar, A. Zharov, A. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Taya, S.

S. Taya and I. Qadoura, “Guided modes in slab waveguides with negative index cladding and substrate,” Optik 124, 1431–1436 (2013).
[CrossRef]

S. Taya, E. El-Farram, and M. Abadla, “Symmetric multilayer slab waveguide structure with a negative index material: TM case,” Optik 123, 2264–2268 (2012).
[CrossRef]

S. Taya, T. El-Agez, H. Kullab, M. Abadla, and M. Shabat, “Theoretical study of slab waveguide optical sensor with left handed material as a core layer,” Opt. Appl. 42, 193–205 (2012).

H. Kullab, S. Taya, and T. El-Agez, “Metal-clad waveguide sensor using a left-handed material as a core layer,” J. Opt. Soc. Am. B 29, 959–964 (2012).
[CrossRef]

I. Qadoura, S. Taya, and K. El-Wasife, “Scaling rules for a slab waveguide structure comprising nonlinear and negative index materials,” Int. J. Microw. Opt. Technol. 7, 349–357 (2012).

M. Abadla, S. Taya, and M. Shabat, “Four-layer slab waveguide sensors supported with left handed materials,” Sens. Lett. 9, 1823–1829 (2011).

S. Taya, M. Shabat, and H. Khalil, “Enhancement of sensitivity in optical sensors using left-handed materials,” Optik 120, 504–508 (2009).
[CrossRef]

Tomlinson, W.

Varatharajab, P.

Veselago, V.

V. Veselago, “The electrodynamics of substance with simultaneously negative index values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Wright, E.

Zharov, A.

I. Shadrivov, A. Sukhorukov, Y. Kivshar, A. Zharov, A. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Appl. Opt. (3)

IEEE Antennas Wireless Propag. Lett. (1)

M. Antoniades and G. Eleftheriades, “Compact linear lead/lag metamaterial phase shifters for broadband applications,” IEEE Antennas Wireless Propag. Lett. 2, 103–106 (2003).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. 22, 319–324 (1986).
[CrossRef]

Int. J. Microw. Opt. Technol. (1)

I. Qadoura, S. Taya, and K. El-Wasife, “Scaling rules for a slab waveguide structure comprising nonlinear and negative index materials,” Int. J. Microw. Opt. Technol. 7, 349–357 (2012).

J. Appl. Phys. (3)

M. Fontaine, “Universal dispersion and power curves for transverse magnetic waves propagating in slab waveguides with a nonlinear self‐focusing substrate,” J. Appl. Phys. 69, 2826–2834 (1991).
[CrossRef]

G. Stegeman, C. Seaton, and J. Ariyasu, “Nonlinear electromagnetic waves guided by a single interface,” J. Appl. Phys. 58, 2453–2459 (1985).
[CrossRef]

A. Grbic and G. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92, 5930–5935 (2002).
[CrossRef]

J. Opt. A (1)

A. Boardman and P. Egan, “Novel nonlinear surface and guided TE waves in asymmetric LHM waveguides,” J. Opt. A 11, 114032 (2009).
[CrossRef]

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

Opt. Appl. (1)

S. Taya, T. El-Agez, H. Kullab, M. Abadla, and M. Shabat, “Theoretical study of slab waveguide optical sensor with left handed material as a core layer,” Opt. Appl. 42, 193–205 (2012).

Opt. Lett. (1)

Optik (3)

S. Taya, M. Shabat, and H. Khalil, “Enhancement of sensitivity in optical sensors using left-handed materials,” Optik 120, 504–508 (2009).
[CrossRef]

S. Taya, E. El-Farram, and M. Abadla, “Symmetric multilayer slab waveguide structure with a negative index material: TM case,” Optik 123, 2264–2268 (2012).
[CrossRef]

S. Taya and I. Qadoura, “Guided modes in slab waveguides with negative index cladding and substrate,” Optik 124, 1431–1436 (2013).
[CrossRef]

Phys. Rev. E (1)

I. Shadrivov, A. Sukhorukov, Y. Kivshar, A. Zharov, A. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004).
[CrossRef]

Phys. Rev. Lett. (2)

J. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef]

Progress Electromagn. Res. (1)

A. Hamid, “Axially slotted antenna on a circular or elliptic cylinder coated with metamaterials,” Progress Electromagn. Res. 51, 329–341 (2005).
[CrossRef]

Sens. Lett. (1)

M. Abadla, S. Taya, and M. Shabat, “Four-layer slab waveguide sensors supported with left handed materials,” Sens. Lett. 9, 1823–1829 (2011).

Sov. Phys. Usp. (1)

V. Veselago, “The electrodynamics of substance with simultaneously negative index values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[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 (8)

Fig. 1.
Fig. 1.

Nonlinear slab waveguide configuration with a DNM is used as a guiding layer.

Fig. 2.
Fig. 2.

Universal dispersion curves of the slab waveguide with DNM guiding layer and self-focusing nonlinear substrate for a=0 and m=0, 1. The dotted lines describe modes having maxima in the substrate (ys<0).

Fig. 3.
Fig. 3.

Universal dispersion curves of the slab waveguide with DNM guiding layer and self-focusing nonlinear substrate for a=10 and m=0, 1. The dotted lines correspond to cases having ys<0.

Fig. 4.
Fig. 4.

Universal dispersion curves of the slab waveguide with DNM guiding layer and self-focusing nonlinear substrate for a=0 and a=10 for the fundamental mode.

Fig. 5.
Fig. 5.

Universal dispersion curves of the slab waveguide with DNM guiding layer and self-defocusing nonlinear substrate for a=0, 10 and m=0, 1.

Fig. 6.
Fig. 6.

Universal dispersion curves of the slab waveguide with DNM and PIM guiding layer and self-focusing nonlinear substrate for a=0 for the fundamental mode.

Fig. 7.
Fig. 7.

Universal dispersion curves of the slab waveguide with DNM and PIM guiding layer and self-defocusing nonlinear substrate for a=0 for the fundamental mode.

Fig. 8.
Fig. 8.

Universal dispersion curves of the slab waveguide with DNM guiding layer and nonlinear substrate (self-focusing and self-defocusing) for the fundamental mode for a=0 and different values of the DNM permeability.

Equations (15)

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

V=ktnf2ns2,
a=ns2nc2nf2ns2,
b=N2ns2nf2ns2,
bI=αEo22(nf2ns2),
(2+k2ni2)Ez(y,x)=0,
Ez(y)=Eocosφcos(qftφ)eqc(yt),
Hx(y)=qciωμoμcEocosφcos(qftφ)eqc(yt),
Ez(y)=Eocosφcos(qftφ),
Hx(y)=qfiωμoμfEocosφsin(qftφ).
Ez(y)=Aqscosh[qsk(yys)]forα>0,
Ez(y)=Eo[cosh(qsky)σsinh(qsky)]1,
σ=f(ys)(1αEo22qs2)1/2={tanh(qskys)forα0coth(qskys)forα0,
Hx(y)=qskEoiωμoμs[sinh(qsky)σcosh(qsky)][cosh(qsky)σsinh(qsky)]2.
qft=tan1(μfμcqcqf)+tan1(μfμskqsqfσ)+πm,
V1btan1(μfμca+b1b)tan1(μfμsbbI1b)πm=0.

Metrics