Abstract

We consider a model of an inhomogeneous dielectric slab first studied by Shvartzburg, Petite and Auby [J. Opt. Soc. Am. B 16, 966 (1999)] and several variations of that model and study the excitation of s-polarized surface electromagnetic waves on the surface of inhomogeneous dielectric media. Using the invariant imbedding theory of wave propagation in stratified media, we calculate the reflectance and the absorptance of an s wave incident obliquely on a dielectric slab in the Otto configuration, as a function of incident angle and frequency. We also calculate the spatial distribution of the electric field intensity in the inhomogeneous region. We find that in all cases we have considered, s-polarized surface waves are excited at certain incident angles and frequencies. We discuss the physical mechanism of the surface wave generation and the possibility of experimental observations of these effects.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).
  2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
    [CrossRef] [PubMed]
  3. E. Ozbay, "Plasmonics: Merging photonics and electronics at nanoscale dimensions," Science 311, 189-193 (2006).
    [CrossRef] [PubMed]
  4. A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Rep. 408, 131-314 (2005).
    [CrossRef]
  5. S. A. Maier and H. A. Atwater, "Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures," J. Appl. Phys. 98, 011101 (2005).
    [CrossRef]
  6. K. A. Willets and R. P. Van Duyne, "Localized surface plasmon resonance spectroscopy and sensing," Annu. Rev. Phys. Chem. 58, 267-297 (2007).
    [CrossRef]
  7. X. D. Hoa, A. G. Kirk, and M. Tabrizian, "Towards integrated and sensitive surface plasmon resonance biosensors: A review of recent progress," Biosens. Bioelectron. 23, 151-160 (2007).
    [CrossRef] [PubMed]
  8. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
    [CrossRef]
  9. G. I. Stegeman, J. D. Valera, C. T. Seaton, J. Sipe, and A. A. Maradudin, "Nonlinear s-polarized surface plasmon polaritons," Solid State Commun. 52, 293-297 (1984).
    [CrossRef]
  10. W. Chen and A. 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]
  11. H. W. Sch¨urmann, V. S. Serov, and Yu. V. Shestopalov, "TE-polarized waves guided by a lossless nonlinear three-layer structure," Phys. Rev. E 58, 1040-1050 (1998).
    [CrossRef]
  12. W. Youfa, W. Qi, and B. Jiashan, "Nonlinear TE surface waves on an antiferromagnetic crystal," J. Appl. Phys. 84, 6233-6238 (1998).
    [CrossRef]
  13. V. E. Kravtsov, E. I. Firsov, V. A. Yakovlev, and G. N. Zhizhin, "TE-surface polaritons at the interface of the isotropic media," Solid State Commun. 50, 741-743 (1984).
    [CrossRef]
  14. A. Shvartzburg, G. Petite, and N. Auby, "S-polarized surface electromagnetic waves in inhomogeneous media: exactly solvable models," J. Opt. Soc. Am. B 16, 966-970 (1999).
    [CrossRef]
  15. A. B. Shvartsburg, V. Kuzmiak, and G. Petite, "Optics of subwavelength gradient nanofilms," Phys. Rep. 452, 33-88 (2007).
    [CrossRef]
  16. R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (Wiley, 1976).
  17. V. I. Klyatskin, "The imbedding method in statistical boundary-value wave problems," Prog. Opt. 33, 1-127 (1994).
    [CrossRef]
  18. R. Rammal and B. Doucot, "Invariant-imbedding approach to localization. I. General framework and basic equations," J. Phys. (Paris) 48, 509-526 (1987).
    [CrossRef]
  19. K. Kim, H. Lim, and D.-H. Lee, "Invariant imbedding equations for electromagnetic waves in stratified magnetic media: Applications to one-dimensional photonic crystals," J. Korean Phys. Soc. 39, L956-L960 (2001).
  20. K. Kim, D.-H. Lee, and H. Lim, "Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media," Europhys. Lett. 69, 207-213 (2005).
    [CrossRef]
  21. K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. I. Exact calculation of the mode conversion coefficient in cold, unmagnetized plasmas," Phys. Plasmas 12, 062101 (2005).
    [CrossRef]
  22. K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. II. Mode conversion in cold, magnetized plasmas with perpendicular inhomogeneity," Phys. Plasmas 13, 042103 (2006).
    [CrossRef]
  23. K. Kim, D. K. Phung, F. Rotermund, and H. Lim, "Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses," Opt. Express 16, 1150-1164 (2008).
    [CrossRef] [PubMed]
  24. K. Kim, F. Rotermund, and H. Lim, "Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: Exact calculation based on the invariant imbedding method," Phys. Rev. B 77, 024203 (2008).
    [CrossRef]
  25. D. J. Yu and K. Kim, "Influence of bottom topography on the propagation of linear shallow water waves: an exact approach based on the invariant imbedding method," Waves Random Complex Media 18, 325-341 (2008).
    [CrossRef]
  26. I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, "The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface," Surf. Sci. 377-379, 436-439 (1997).
    [CrossRef]
  27. T. Inaoka, "Elementary excitations at doped polar semiconductor surfaces with carrier-depeltion layers," Appl. Surf. Sci. 169-170, 51-56 (2001).
    [CrossRef]
  28. J.W. L. Yim, R. E. Jones, K. M. Yu, J.W. AgerIII, W. Walukiewicz, W. J. Schaff, and J. Wu, "Effects of surface states on electrical characteristics of InN and In1-xGaxN," Phys. Rev. B 76, 041303(R) (2007).
    [CrossRef]
  29. T. M. Dunster, "Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter," SIAM J. Math. Anal. 21, 995-1018 (1990).
    [CrossRef]
  30. L. A. A. Pettersson, L. Hultman, and H. Arwin, "Porosity depth profiling of thin porous silicon layers by use of variable-range spectroscopic ellipsometry: a porosity graded-layer model," Appl. Opt. 37, 4130-4136 (1998).
    [CrossRef]
  31. X. Wang, H. Masumoto, Y. Someno, and T. Hirai, "Design and experimental approach of optical reflection filters with graded refractive index profiles," J. Vac. Sci. Technol. A 17, 206-211 (1999).
    [CrossRef]
  32. H. Ren and S.-T. Wu, "Inhomogeneous nanoscale polymer-dispersed liquid crystals with gradient refractive index," Appl. Phys. Lett. 81, 3537-3539 (2002).
    [CrossRef]

2008 (3)

K. Kim, F. Rotermund, and H. Lim, "Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: Exact calculation based on the invariant imbedding method," Phys. Rev. B 77, 024203 (2008).
[CrossRef]

D. J. Yu and K. Kim, "Influence of bottom topography on the propagation of linear shallow water waves: an exact approach based on the invariant imbedding method," Waves Random Complex Media 18, 325-341 (2008).
[CrossRef]

K. Kim, D. K. Phung, F. Rotermund, and H. Lim, "Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses," Opt. Express 16, 1150-1164 (2008).
[CrossRef] [PubMed]

2007 (3)

K. A. Willets and R. P. Van Duyne, "Localized surface plasmon resonance spectroscopy and sensing," Annu. Rev. Phys. Chem. 58, 267-297 (2007).
[CrossRef]

X. D. Hoa, A. G. Kirk, and M. Tabrizian, "Towards integrated and sensitive surface plasmon resonance biosensors: A review of recent progress," Biosens. Bioelectron. 23, 151-160 (2007).
[CrossRef] [PubMed]

A. B. Shvartsburg, V. Kuzmiak, and G. Petite, "Optics of subwavelength gradient nanofilms," Phys. Rep. 452, 33-88 (2007).
[CrossRef]

2006 (2)

E. Ozbay, "Plasmonics: Merging photonics and electronics at nanoscale dimensions," Science 311, 189-193 (2006).
[CrossRef] [PubMed]

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. II. Mode conversion in cold, magnetized plasmas with perpendicular inhomogeneity," Phys. Plasmas 13, 042103 (2006).
[CrossRef]

2005 (4)

K. Kim, D.-H. Lee, and H. Lim, "Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media," Europhys. Lett. 69, 207-213 (2005).
[CrossRef]

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. I. Exact calculation of the mode conversion coefficient in cold, unmagnetized plasmas," Phys. Plasmas 12, 062101 (2005).
[CrossRef]

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Rep. 408, 131-314 (2005).
[CrossRef]

S. A. Maier and H. A. Atwater, "Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures," J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

2002 (1)

H. Ren and S.-T. Wu, "Inhomogeneous nanoscale polymer-dispersed liquid crystals with gradient refractive index," Appl. Phys. Lett. 81, 3537-3539 (2002).
[CrossRef]

2001 (2)

T. Inaoka, "Elementary excitations at doped polar semiconductor surfaces with carrier-depeltion layers," Appl. Surf. Sci. 169-170, 51-56 (2001).
[CrossRef]

K. Kim, H. Lim, and D.-H. Lee, "Invariant imbedding equations for electromagnetic waves in stratified magnetic media: Applications to one-dimensional photonic crystals," J. Korean Phys. Soc. 39, L956-L960 (2001).

1999 (2)

A. Shvartzburg, G. Petite, and N. Auby, "S-polarized surface electromagnetic waves in inhomogeneous media: exactly solvable models," J. Opt. Soc. Am. B 16, 966-970 (1999).
[CrossRef]

X. Wang, H. Masumoto, Y. Someno, and T. Hirai, "Design and experimental approach of optical reflection filters with graded refractive index profiles," J. Vac. Sci. Technol. A 17, 206-211 (1999).
[CrossRef]

1998 (4)

L. A. A. Pettersson, L. Hultman, and H. Arwin, "Porosity depth profiling of thin porous silicon layers by use of variable-range spectroscopic ellipsometry: a porosity graded-layer model," Appl. Opt. 37, 4130-4136 (1998).
[CrossRef]

H. W. Sch¨urmann, V. S. Serov, and Yu. V. Shestopalov, "TE-polarized waves guided by a lossless nonlinear three-layer structure," Phys. Rev. E 58, 1040-1050 (1998).
[CrossRef]

W. Youfa, W. Qi, and B. Jiashan, "Nonlinear TE surface waves on an antiferromagnetic crystal," J. Appl. Phys. 84, 6233-6238 (1998).
[CrossRef]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

1997 (1)

I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, "The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface," Surf. Sci. 377-379, 436-439 (1997).
[CrossRef]

1994 (1)

V. I. Klyatskin, "The imbedding method in statistical boundary-value wave problems," Prog. Opt. 33, 1-127 (1994).
[CrossRef]

1990 (1)

T. M. Dunster, "Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter," SIAM J. Math. Anal. 21, 995-1018 (1990).
[CrossRef]

1988 (1)

1987 (1)

R. Rammal and B. Doucot, "Invariant-imbedding approach to localization. I. General framework and basic equations," J. Phys. (Paris) 48, 509-526 (1987).
[CrossRef]

1984 (2)

G. I. Stegeman, J. D. Valera, C. T. Seaton, J. Sipe, and A. A. Maradudin, "Nonlinear s-polarized surface plasmon polaritons," Solid State Commun. 52, 293-297 (1984).
[CrossRef]

V. E. Kravtsov, E. I. Firsov, V. A. Yakovlev, and G. N. Zhizhin, "TE-surface polaritons at the interface of the isotropic media," Solid State Commun. 50, 741-743 (1984).
[CrossRef]

Arwin, H.

Atwater, H. A.

S. A. Maier and H. A. Atwater, "Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures," J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

Auby, N.

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Benedek, G.

I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, "The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface," Surf. Sci. 377-379, 436-439 (1997).
[CrossRef]

Chen, W.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Doucot, B.

R. Rammal and B. Doucot, "Invariant-imbedding approach to localization. I. General framework and basic equations," J. Phys. (Paris) 48, 509-526 (1987).
[CrossRef]

Dunster, T. M.

T. M. Dunster, "Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter," SIAM J. Math. Anal. 21, 995-1018 (1990).
[CrossRef]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Firsov, E. I.

V. E. Kravtsov, E. I. Firsov, V. A. Yakovlev, and G. N. Zhizhin, "TE-surface polaritons at the interface of the isotropic media," Solid State Commun. 50, 741-743 (1984).
[CrossRef]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Hirai, T.

X. Wang, H. Masumoto, Y. Someno, and T. Hirai, "Design and experimental approach of optical reflection filters with graded refractive index profiles," J. Vac. Sci. Technol. A 17, 206-211 (1999).
[CrossRef]

Hoa, X. D.

X. D. Hoa, A. G. Kirk, and M. Tabrizian, "Towards integrated and sensitive surface plasmon resonance biosensors: A review of recent progress," Biosens. Bioelectron. 23, 151-160 (2007).
[CrossRef] [PubMed]

Hultman, L.

Inaoka, T.

T. Inaoka, "Elementary excitations at doped polar semiconductor surfaces with carrier-depeltion layers," Appl. Surf. Sci. 169-170, 51-56 (2001).
[CrossRef]

Ipatova, I. P.

I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, "The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface," Surf. Sci. 377-379, 436-439 (1997).
[CrossRef]

Jiashan, B.

W. Youfa, W. Qi, and B. Jiashan, "Nonlinear TE surface waves on an antiferromagnetic crystal," J. Appl. Phys. 84, 6233-6238 (1998).
[CrossRef]

Kim, K.

D. J. Yu and K. Kim, "Influence of bottom topography on the propagation of linear shallow water waves: an exact approach based on the invariant imbedding method," Waves Random Complex Media 18, 325-341 (2008).
[CrossRef]

K. Kim, F. Rotermund, and H. Lim, "Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: Exact calculation based on the invariant imbedding method," Phys. Rev. B 77, 024203 (2008).
[CrossRef]

K. Kim, D. K. Phung, F. Rotermund, and H. Lim, "Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses," Opt. Express 16, 1150-1164 (2008).
[CrossRef] [PubMed]

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. II. Mode conversion in cold, magnetized plasmas with perpendicular inhomogeneity," Phys. Plasmas 13, 042103 (2006).
[CrossRef]

K. Kim, D.-H. Lee, and H. Lim, "Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media," Europhys. Lett. 69, 207-213 (2005).
[CrossRef]

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. I. Exact calculation of the mode conversion coefficient in cold, unmagnetized plasmas," Phys. Plasmas 12, 062101 (2005).
[CrossRef]

K. Kim, H. Lim, and D.-H. Lee, "Invariant imbedding equations for electromagnetic waves in stratified magnetic media: Applications to one-dimensional photonic crystals," J. Korean Phys. Soc. 39, L956-L960 (2001).

Kirk, A. G.

X. D. Hoa, A. G. Kirk, and M. Tabrizian, "Towards integrated and sensitive surface plasmon resonance biosensors: A review of recent progress," Biosens. Bioelectron. 23, 151-160 (2007).
[CrossRef] [PubMed]

Klyatskin, V. I.

V. I. Klyatskin, "The imbedding method in statistical boundary-value wave problems," Prog. Opt. 33, 1-127 (1994).
[CrossRef]

Kravtsov, V. E.

V. E. Kravtsov, E. I. Firsov, V. A. Yakovlev, and G. N. Zhizhin, "TE-surface polaritons at the interface of the isotropic media," Solid State Commun. 50, 741-743 (1984).
[CrossRef]

Kuzmiak, V.

A. B. Shvartsburg, V. Kuzmiak, and G. Petite, "Optics of subwavelength gradient nanofilms," Phys. Rep. 452, 33-88 (2007).
[CrossRef]

Lee, D.-H.

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. II. Mode conversion in cold, magnetized plasmas with perpendicular inhomogeneity," Phys. Plasmas 13, 042103 (2006).
[CrossRef]

K. Kim, D.-H. Lee, and H. Lim, "Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media," Europhys. Lett. 69, 207-213 (2005).
[CrossRef]

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. I. Exact calculation of the mode conversion coefficient in cold, unmagnetized plasmas," Phys. Plasmas 12, 062101 (2005).
[CrossRef]

K. Kim, H. Lim, and D.-H. Lee, "Invariant imbedding equations for electromagnetic waves in stratified magnetic media: Applications to one-dimensional photonic crystals," J. Korean Phys. Soc. 39, L956-L960 (2001).

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Lim, H.

K. Kim, D. K. Phung, F. Rotermund, and H. Lim, "Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses," Opt. Express 16, 1150-1164 (2008).
[CrossRef] [PubMed]

K. Kim, F. Rotermund, and H. Lim, "Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: Exact calculation based on the invariant imbedding method," Phys. Rev. B 77, 024203 (2008).
[CrossRef]

K. Kim, D.-H. Lee, and H. Lim, "Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media," Europhys. Lett. 69, 207-213 (2005).
[CrossRef]

K. Kim, H. Lim, and D.-H. Lee, "Invariant imbedding equations for electromagnetic waves in stratified magnetic media: Applications to one-dimensional photonic crystals," J. Korean Phys. Soc. 39, L956-L960 (2001).

Maier, S. A.

S. A. Maier and H. A. Atwater, "Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures," J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

Maradudin, A. A.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Rep. 408, 131-314 (2005).
[CrossRef]

W. Chen and A. 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]

G. I. Stegeman, J. D. Valera, C. T. Seaton, J. Sipe, and A. A. Maradudin, "Nonlinear s-polarized surface plasmon polaritons," Solid State Commun. 52, 293-297 (1984).
[CrossRef]

Maslov, A. Yu.

I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, "The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface," Surf. Sci. 377-379, 436-439 (1997).
[CrossRef]

Masumoto, H.

X. Wang, H. Masumoto, Y. Someno, and T. Hirai, "Design and experimental approach of optical reflection filters with graded refractive index profiles," J. Vac. Sci. Technol. A 17, 206-211 (1999).
[CrossRef]

Ozbay, E.

E. Ozbay, "Plasmonics: Merging photonics and electronics at nanoscale dimensions," Science 311, 189-193 (2006).
[CrossRef] [PubMed]

Panzarini, G.

I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, "The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface," Surf. Sci. 377-379, 436-439 (1997).
[CrossRef]

Petite, G.

Pettersson, L. A. A.

Phung, D. K.

Qi, W.

W. Youfa, W. Qi, and B. Jiashan, "Nonlinear TE surface waves on an antiferromagnetic crystal," J. Appl. Phys. 84, 6233-6238 (1998).
[CrossRef]

Rammal, R.

R. Rammal and B. Doucot, "Invariant-imbedding approach to localization. I. General framework and basic equations," J. Phys. (Paris) 48, 509-526 (1987).
[CrossRef]

Ren, H.

H. Ren and S.-T. Wu, "Inhomogeneous nanoscale polymer-dispersed liquid crystals with gradient refractive index," Appl. Phys. Lett. 81, 3537-3539 (2002).
[CrossRef]

Rotermund, F.

K. Kim, D. K. Phung, F. Rotermund, and H. Lim, "Propagation of electromagnetic waves in stratified media with nonlinearity in both dielectric and magnetic responses," Opt. Express 16, 1150-1164 (2008).
[CrossRef] [PubMed]

K. Kim, F. Rotermund, and H. Lim, "Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: Exact calculation based on the invariant imbedding method," Phys. Rev. B 77, 024203 (2008).
[CrossRef]

Sch¨urmann, H. W.

H. W. Sch¨urmann, V. S. Serov, and Yu. V. Shestopalov, "TE-polarized waves guided by a lossless nonlinear three-layer structure," Phys. Rev. E 58, 1040-1050 (1998).
[CrossRef]

Seaton, C. T.

G. I. Stegeman, J. D. Valera, C. T. Seaton, J. Sipe, and A. A. Maradudin, "Nonlinear s-polarized surface plasmon polaritons," Solid State Commun. 52, 293-297 (1984).
[CrossRef]

Serov, V. S.

H. W. Sch¨urmann, V. S. Serov, and Yu. V. Shestopalov, "TE-polarized waves guided by a lossless nonlinear three-layer structure," Phys. Rev. E 58, 1040-1050 (1998).
[CrossRef]

Shestopalov, Yu. V.

H. W. Sch¨urmann, V. S. Serov, and Yu. V. Shestopalov, "TE-polarized waves guided by a lossless nonlinear three-layer structure," Phys. Rev. E 58, 1040-1050 (1998).
[CrossRef]

Shvartsburg, A. B.

A. B. Shvartsburg, V. Kuzmiak, and G. Petite, "Optics of subwavelength gradient nanofilms," Phys. Rep. 452, 33-88 (2007).
[CrossRef]

Shvartzburg, A.

Sipe, J.

G. I. Stegeman, J. D. Valera, C. T. Seaton, J. Sipe, and A. A. Maradudin, "Nonlinear s-polarized surface plasmon polaritons," Solid State Commun. 52, 293-297 (1984).
[CrossRef]

Smolyaninov, I. I.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Rep. 408, 131-314 (2005).
[CrossRef]

Someno, Y.

X. Wang, H. Masumoto, Y. Someno, and T. Hirai, "Design and experimental approach of optical reflection filters with graded refractive index profiles," J. Vac. Sci. Technol. A 17, 206-211 (1999).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman, J. D. Valera, C. T. Seaton, J. Sipe, and A. A. Maradudin, "Nonlinear s-polarized surface plasmon polaritons," Solid State Commun. 52, 293-297 (1984).
[CrossRef]

Tabrizian, M.

X. D. Hoa, A. G. Kirk, and M. Tabrizian, "Towards integrated and sensitive surface plasmon resonance biosensors: A review of recent progress," Biosens. Bioelectron. 23, 151-160 (2007).
[CrossRef] [PubMed]

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Udod, L. V.

I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, "The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface," Surf. Sci. 377-379, 436-439 (1997).
[CrossRef]

Valera, J. D.

G. I. Stegeman, J. D. Valera, C. T. Seaton, J. Sipe, and A. A. Maradudin, "Nonlinear s-polarized surface plasmon polaritons," Solid State Commun. 52, 293-297 (1984).
[CrossRef]

Van Duyne, R. P.

K. A. Willets and R. P. Van Duyne, "Localized surface plasmon resonance spectroscopy and sensing," Annu. Rev. Phys. Chem. 58, 267-297 (2007).
[CrossRef]

Wang, X.

X. Wang, H. Masumoto, Y. Someno, and T. Hirai, "Design and experimental approach of optical reflection filters with graded refractive index profiles," J. Vac. Sci. Technol. A 17, 206-211 (1999).
[CrossRef]

Willets, K. A.

K. A. Willets and R. P. Van Duyne, "Localized surface plasmon resonance spectroscopy and sensing," Annu. Rev. Phys. Chem. 58, 267-297 (2007).
[CrossRef]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Wu, S.-T.

H. Ren and S.-T. Wu, "Inhomogeneous nanoscale polymer-dispersed liquid crystals with gradient refractive index," Appl. Phys. Lett. 81, 3537-3539 (2002).
[CrossRef]

Yakovlev, V. A.

V. E. Kravtsov, E. I. Firsov, V. A. Yakovlev, and G. N. Zhizhin, "TE-surface polaritons at the interface of the isotropic media," Solid State Commun. 50, 741-743 (1984).
[CrossRef]

Youfa, W.

W. Youfa, W. Qi, and B. Jiashan, "Nonlinear TE surface waves on an antiferromagnetic crystal," J. Appl. Phys. 84, 6233-6238 (1998).
[CrossRef]

Yu, D. J.

D. J. Yu and K. Kim, "Influence of bottom topography on the propagation of linear shallow water waves: an exact approach based on the invariant imbedding method," Waves Random Complex Media 18, 325-341 (2008).
[CrossRef]

Zayats, A. V.

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Rep. 408, 131-314 (2005).
[CrossRef]

Zhizhin, G. N.

V. E. Kravtsov, E. I. Firsov, V. A. Yakovlev, and G. N. Zhizhin, "TE-surface polaritons at the interface of the isotropic media," Solid State Commun. 50, 741-743 (1984).
[CrossRef]

Annu. Rev. Phys. Chem. (1)

K. A. Willets and R. P. Van Duyne, "Localized surface plasmon resonance spectroscopy and sensing," Annu. Rev. Phys. Chem. 58, 267-297 (2007).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

H. Ren and S.-T. Wu, "Inhomogeneous nanoscale polymer-dispersed liquid crystals with gradient refractive index," Appl. Phys. Lett. 81, 3537-3539 (2002).
[CrossRef]

Appl. Surf. Sci. (1)

T. Inaoka, "Elementary excitations at doped polar semiconductor surfaces with carrier-depeltion layers," Appl. Surf. Sci. 169-170, 51-56 (2001).
[CrossRef]

Biosens. Bioelectron. (1)

X. D. Hoa, A. G. Kirk, and M. Tabrizian, "Towards integrated and sensitive surface plasmon resonance biosensors: A review of recent progress," Biosens. Bioelectron. 23, 151-160 (2007).
[CrossRef] [PubMed]

Europhys. Lett. (1)

K. Kim, D.-H. Lee, and H. Lim, "Theory of the propagation of coupled waves in arbitrarily inhomogeneous stratified media," Europhys. Lett. 69, 207-213 (2005).
[CrossRef]

J. Appl. Phys. (2)

S. A. Maier and H. A. Atwater, "Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures," J. Appl. Phys. 98, 011101 (2005).
[CrossRef]

W. Youfa, W. Qi, and B. Jiashan, "Nonlinear TE surface waves on an antiferromagnetic crystal," J. Appl. Phys. 84, 6233-6238 (1998).
[CrossRef]

J. Korean Phys. Soc. (1)

K. Kim, H. Lim, and D.-H. Lee, "Invariant imbedding equations for electromagnetic waves in stratified magnetic media: Applications to one-dimensional photonic crystals," J. Korean Phys. Soc. 39, L956-L960 (2001).

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

J. Phys. (Paris) (1)

R. Rammal and B. Doucot, "Invariant-imbedding approach to localization. I. General framework and basic equations," J. Phys. (Paris) 48, 509-526 (1987).
[CrossRef]

J. Vac. Sci. Technol. A (1)

X. Wang, H. Masumoto, Y. Someno, and T. Hirai, "Design and experimental approach of optical reflection filters with graded refractive index profiles," J. Vac. Sci. Technol. A 17, 206-211 (1999).
[CrossRef]

Nature (2)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, "Surface plasmon subwavelength optics," Nature 424, 824-830 (2003).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Plasmas (2)

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. I. Exact calculation of the mode conversion coefficient in cold, unmagnetized plasmas," Phys. Plasmas 12, 062101 (2005).
[CrossRef]

K. Kim and D.-H. Lee, "Invariant imbedding theory of mode conversion in inhomogeneous plasmas. II. Mode conversion in cold, magnetized plasmas with perpendicular inhomogeneity," Phys. Plasmas 13, 042103 (2006).
[CrossRef]

Phys. Rep. (2)

A. B. Shvartsburg, V. Kuzmiak, and G. Petite, "Optics of subwavelength gradient nanofilms," Phys. Rep. 452, 33-88 (2007).
[CrossRef]

A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, "Nano-optics of surface plasmon polaritons," Phys. Rep. 408, 131-314 (2005).
[CrossRef]

Phys. Rev. B (1)

K. Kim, F. Rotermund, and H. Lim, "Disorder-enhanced transmission of a quantum mechanical particle through a disordered tunneling barrier in one dimension: Exact calculation based on the invariant imbedding method," Phys. Rev. B 77, 024203 (2008).
[CrossRef]

Phys. Rev. E (1)

H. W. Sch¨urmann, V. S. Serov, and Yu. V. Shestopalov, "TE-polarized waves guided by a lossless nonlinear three-layer structure," Phys. Rev. E 58, 1040-1050 (1998).
[CrossRef]

Prog. Opt. (1)

V. I. Klyatskin, "The imbedding method in statistical boundary-value wave problems," Prog. Opt. 33, 1-127 (1994).
[CrossRef]

Science (1)

E. Ozbay, "Plasmonics: Merging photonics and electronics at nanoscale dimensions," Science 311, 189-193 (2006).
[CrossRef] [PubMed]

SIAM J. Math. Anal. (1)

T. M. Dunster, "Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter," SIAM J. Math. Anal. 21, 995-1018 (1990).
[CrossRef]

Solid State Commun. (2)

V. E. Kravtsov, E. I. Firsov, V. A. Yakovlev, and G. N. Zhizhin, "TE-surface polaritons at the interface of the isotropic media," Solid State Commun. 50, 741-743 (1984).
[CrossRef]

G. I. Stegeman, J. D. Valera, C. T. Seaton, J. Sipe, and A. A. Maradudin, "Nonlinear s-polarized surface plasmon polaritons," Solid State Commun. 52, 293-297 (1984).
[CrossRef]

Surf. Sci. (1)

I. P. Ipatova, A. Yu. Maslov, L. V. Udod, G. Benedek, and G. Panzarini, "The enhancement factor of hyper-Raman scattering from an inhomogeneous semiconductor surface," Surf. Sci. 377-379, 436-439 (1997).
[CrossRef]

Waves Random Complex Media (1)

D. J. Yu and K. Kim, "Influence of bottom topography on the propagation of linear shallow water waves: an exact approach based on the invariant imbedding method," Waves Random Complex Media 18, 325-341 (2008).
[CrossRef]

Other (3)

J.W. L. Yim, R. E. Jones, K. M. Yu, J.W. AgerIII, W. Walukiewicz, W. J. Schaff, and J. Wu, "Effects of surface states on electrical characteristics of InN and In1-xGaxN," Phys. Rev. B 76, 041303(R) (2007).
[CrossRef]

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

R. Bellman and G. M. Wing, An Introduction to Invariant Imbedding (Wiley, 1976).

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 (7)

Fig. 1.
Fig. 1.

Spatial variations of (1-ε), when an inhomogeneous dielectric slab located in 0≤zD is in contact with the air in the region z>D. The dielectric permittivity of the inhomogeneous slab is given by Eq. (2) with the parameter values N 0=3×1016 cm-3, εL =12, m eff=0.01me , γ=0, b=-0.53, ω=5×1013/sec and α=0.5,1,1.5.

Fig. 2.
Fig. 2.

Dispersion relations of s-polarized surface waves, q(ω), for the model with α=1 in Eq. (2). The parameters used in the calculation are N 0=3×1016 cm-3, εL =12, m eff=0.01me , a=10-4 cm, γ=0 and b=-0.4,-0.53,-0.72,-1.1,-2.1. The dashed line represents the dispersion of light in a vacuum.

Fig. 3.
Fig. 3.

Incident angle at which an s-polarized surface wave is excited plotted versus the frequency of the incident wave, when α=1, N 0=3×1016 cm-3, εL =12, m eff=0.01me , a=10-4 cm, γ=0, b=-0.53 and ε 1=2.25. The point P corresponds to ω=5.2×1013/sec and θ=64.88°.

Fig. 4.
Fig. 4.

(a), (c) Reflectance and (b), (d) absorptance of an s wave of frequency ω=5.2×1013/sec incident obliquely on an inhomogeneous dielectric slab, the permittivity of which is given by Eq. (2) with α=1, in the Otto configuration, plotted versus incident angle. In Figs. 4–7, we use the common parameter values N0 =3×1016 cm-3, εL =12, m eff=0.01me , a=10-4 cm, ε 1=2.25, εd =1, D=100a and b=-0.53. The damping parameter γ is 0.001 in (a) and (b) and 0.01 in (c) and (d). The thickness of the air gap, L-D, is equal to 15a. Note that there appear sharp reflection dips and absorption peaks at θ=64.88°, which corresponds precisely to the point P in Fig. 3.

Fig. 5.
Fig. 5.

(a), (c) Reflectance and (b), (d) absorptance of an s wave incident at θ=65° on an inhomogeneous dielectric slab, the permittivity of which is given by Eq. (2) with α=1, in the Otto configuration, plotted versus frequency. L is 115a and γ is equal to 0.001 in (a) and (b) and 0.01 in (c) and (d).

Fig. 6.
Fig. 6.

Spatial distributions of the electric field intensity inside the inhomogeneous medium, when an s wave of frequency ω=5.2×1013/sec is incident at θ=64.88° and 64° on an inhomogeneous dielectric slab, the permittivity of which is given by Eq. (2) with α=1, in the Otto configuration. L is 115a and γ is 0.001. The vertical dashed line indicates the boundary between the inhomogeneous medium located in 0≤z≤100a and the air gap located in 100az≤115a. The wave is assumed to be incident from the right side.

Fig. 7.
Fig. 7.

(a–d) Absorptance A of an s wave incident obliquely on an inhomogeneous dielectric slab, the permittivity of which is given by Eq. (2) with α=0.5, 0.75, 1.25 and 1.5, in the Otto configuration, plotted versus incident angle. L is 120a and γ is 0.001. The frequency ω is 5×1013/sec in (a) and (b) and 5.05×1013/sec in (c) and (d). (e–h) Spatial distributions of the electric field intensity, |E|2, corresponding to θ=68.82° in (a), θ=53.01° in (b), θ=48.12° in (c) and θ=47° in (d). The vertical dashed line indicates the boundary between the inhomogeneous medium and the air gap. The wave is assumed to be incident from the region where z≥120a.

Equations (11)

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

d 2 E d z 2 + [ k 0 2 ε ( z ) q 2 ] E = 0 ,
ε ( z ) = ε L { 1 ω p 2 ω ( ω + i γ ) [ 1 1 b + F 2 ( z ) b ] } ,
F ( z ) = ( 1 + D z a ) α , ω p 2 = 4 π e 2 N 0 ε L m eff ,
d 2 Q d u 2 + 1 u d Q d u + ( s 2 ρ 2 u 2 ) Q = 0 ,
Q ( u ) = E ( z ) F ( z ) , u = 1 + D z a ,
ρ 2 = ε L b ( ω p a c ) 2 + 1 4 , s 2 = ( q a ) 2 ε L ( ω a c ) 2 + ε L ( ω p a c ) 2 ( 1 1 b ) .
( q a ) 2 = ( ω a c ) 2 + [ 1 2 + 1 K ρ ( s ) K ρ ( s u ) u u = 1 ] 2 ,
E ˜ ( x , z ) = { [ e i p ( L z ) + r ( L ) e i p ( z L ) ] e iqx , z > L t ( L ) e ipz + iqx , z < 0 .
1 i ε 1 k 0 d r d l = 2 cos θ r ( l ) + ε ˜ ( l ) 1 2 cos θ [ 1 + r ( l ) ] 2 ,
1 i ε 1 k 0 d t d l = cos θ t ( l ) + ε ˜ ( l ) 1 2 cos θ [ 1 + r ( l ) ] t ( l ) ,
1 i ε 1 k 0 E l = cos θ E ( z , l ) + ε ˜ ( l ) 1 2 cos θ [ 1 + r ( l ) ] E ( z , l ) ,

Metrics