Abstract

We use the theory of optical waveguides to study analytically the voltage-dependent response of a surface plasmon polariton (SPP) at the interface between a photorefractive liquid crystal cell and a semi-infinite gold layer. For sufficiently large electric fields the alignment of the liquid crystal can be calculated analytically. The resulting correction to the SPP dispersion relation is then determined in terms of the applied field and the liquid crystal surface alignment relative to the SPP propagation direction. The approximate analytic techniques developed here are shown to be accurate when compared to rigorous diffraction theory and experimental measurements. The approximate equations are a powerful tool of general application. They can be used to study SPP propagation at the interface between a metal and any nonhomogeneous or anisotropic dielectric and are also applicable to self-assembled monolayers and biosensing applications.

© 2011 Optical Society of America

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  3. I. R. Hooper and J. R. Sambles, “Differential ellipsometric surface plasmon resonance sensors with liquid crystal polarization modulators,” Appl. Phys. Lett. 85, 3017–3019(2004).
    [CrossRef]
  4. X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
    [CrossRef] [PubMed]
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  6. K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14, R597 (2002).
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    [CrossRef] [PubMed]
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    [CrossRef]
  15. K. R. Welford and J. R. Sambles, “Detection of surface director reorientation in a nematic liquid crystal,” Appl. Phys. Lett. 50, 871–873 (1987).
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  16. R. A. Innes and J. R. Sambles, “Optical nonlinearity in a nematic liquid crystal using surface plasmon-polaritons,” Opt. Commun. 64, 288–292 (1987).
    [CrossRef]
  17. R. A. Innes, S. P. Ashworth, and J. R. Sambles, “Large optical bistability in a nematic liquid crystal using surface plasmon-polaritons,” Phys. Lett. A 135, 357–362 (1989).
    [CrossRef]
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    [CrossRef]
  19. W. Dickson, G. A. Wurtz, P. R. Evans, R. J. Pollard, and A. V. Zayats, “Electronically controlled surface plasmon dispersion and optical transmission through metallic hole arrays using liquid crystal,” Nano Lett. 8, 281–286 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  31. J. Li, S. T. Wu, S. Brugioni, R. Meucci, and S. Faetti, “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97, 073501 (2005).
    [CrossRef]
  32. E. P. Raynes, C. V. Brown, and J. F. Stromer, “Method for the measurement of the K22 nematic elastic constant,” Appl. Phys. Lett. 82, 13–15 (2003).
    [CrossRef]

2008 (4)

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
[CrossRef] [PubMed]

U. Bortolozzo and S. Residori, and J. P. Huignard, “Beam coupling in photorefractive liquid crystal light valves,” J. Phys. D 41, 224007 (2008).
[CrossRef]

G. Cook, A. V. Glushchenko, V. Reshetnyak, A. T. Griffith, M. A. Saleh, and D. R. Evans, “Nanoparticle doped organic-inorganic hybrid photorefractives,” Opt. Express 16, 4015–4022(2008).
[CrossRef] [PubMed]

W. Dickson, G. A. Wurtz, P. R. Evans, R. J. Pollard, and A. V. Zayats, “Electronically controlled surface plasmon dispersion and optical transmission through metallic hole arrays using liquid crystal,” Nano Lett. 8, 281–286 (2008).
[CrossRef]

2007 (1)

2006 (1)

2005 (3)

J. Li, C. H. Wen, S. Gauza, R. Lu, and S. T. Wu, “Refractive indices of liquid crystals for display applications,” J. Display Technol. 1, 51–61 (2005).
[CrossRef]

J. Li, S. T. Wu, S. Brugioni, R. Meucci, and S. Faetti, “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97, 073501 (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]

2004 (1)

I. R. Hooper and J. R. Sambles, “Differential ellipsometric surface plasmon resonance sensors with liquid crystal polarization modulators,” Appl. Phys. Lett. 85, 3017–3019(2004).
[CrossRef]

2003 (3)

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232(2003).
[CrossRef] [PubMed]

E. P. Raynes, C. V. Brown, and J. F. Stromer, “Method for the measurement of the K22 nematic elastic constant,” Appl. Phys. Lett. 82, 13–15 (2003).
[CrossRef]

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

2002 (2)

R. H. Self, C. P. Please, and T. J. Sluckin, “Deformation of nematic liquid crystals in an electric field,” Eur. J. Appl. Math. 13, 1–23 (2002).
[CrossRef]

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14, R597 (2002).
[CrossRef]

1999 (1)

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sensors Actuators B Chem 54, 3–15(1999).
[CrossRef]

1995 (1)

Y. Wang, “Voltage-induced color-selective absorption with surface plasmons,” Appl. Phys. Lett. 67, 2759–2761 (1995).
[CrossRef]

1993 (1)

M. Malmqvist, “Biospecific interaction analysis using biosensor technology,” Nature 361, 186–187 (1993).
[CrossRef] [PubMed]

1990 (2)

1989 (1)

R. A. Innes, S. P. Ashworth, and J. R. Sambles, “Large optical bistability in a nematic liquid crystal using surface plasmon-polaritons,” Phys. Lett. A 135, 357–362 (1989).
[CrossRef]

1987 (3)

K. R. Welford, J. R. Sambles, and M. G. Clark, “Guided modes and surface plasmon-polaritons observed with a nematic liquid crystal using attenuated total reflection,” Liq. Cryst. 2, 91–105(1987).
[CrossRef]

K. R. Welford and J. R. Sambles, “Detection of surface director reorientation in a nematic liquid crystal,” Appl. Phys. Lett. 50, 871–873 (1987).
[CrossRef]

R. A. Innes and J. R. Sambles, “Optical nonlinearity in a nematic liquid crystal using surface plasmon-polaritons,” Opt. Commun. 64, 288–292 (1987).
[CrossRef]

1981 (1)

1980 (1)

W. P. Chen and J. M. Chen, “Surface plasma wave study of submonolayer Cs and Cs O covered Ag surfaces,” Surf. Sci. 91, 601–617 (1980).
[CrossRef]

1978 (1)

I. Pockrand, “Surface plasma oscillations at silver surfaces with thin transparent and absorbing coatings,” Surf. Sci. 72, 577–588 (1978).
[CrossRef]

1972 (1)

1968 (1)

E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Zeitschrift Fuer Naturforschung, Teil A 23, 2135–2136 (1968).

1957 (1)

R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. 106, 874–881 (1957).
[CrossRef]

Ashworth, S. P.

R. A. Innes, S. P. Ashworth, and J. R. Sambles, “Large optical bistability in a nematic liquid crystal using surface plasmon-polaritons,” Phys. Lett. A 135, 357–362 (1989).
[CrossRef]

Atwater, H. A.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232(2003).
[CrossRef] [PubMed]

Barnes, W. L.

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

Berreman, D. W.

Born, M.

M. Born and E. Wolf, Principles of Optics, Electromagnetic Theory of Propagation Interference and Diffraction of Light, 6th ed. (Cambridge University, 1980).
[PubMed]

Bortolozzo, U.

U. Bortolozzo and S. Residori, and J. P. Huignard, “Beam coupling in photorefractive liquid crystal light valves,” J. Phys. D 41, 224007 (2008).
[CrossRef]

Brown, C. V.

E. P. Raynes, C. V. Brown, and J. F. Stromer, “Method for the measurement of the K22 nematic elastic constant,” Appl. Phys. Lett. 82, 13–15 (2003).
[CrossRef]

Brugioni, S.

J. Li, S. T. Wu, S. Brugioni, R. Meucci, and S. Faetti, “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

Buchnev, O.

Chen, J. M.

W. P. Chen and J. M. Chen, “Use of surface plasma waves for determination of the thickness and optical constants of thin metallic films,” J. Opt. Soc. Am. 71, 189–191 (1981).
[CrossRef]

W. P. Chen and J. M. Chen, “Surface plasma wave study of submonolayer Cs and Cs O covered Ag surfaces,” Surf. Sci. 91, 601–617 (1980).
[CrossRef]

Chen, W. P.

W. P. Chen and J. M. Chen, “Use of surface plasma waves for determination of the thickness and optical constants of thin metallic films,” J. Opt. Soc. Am. 71, 189–191 (1981).
[CrossRef]

W. P. Chen and J. M. Chen, “Surface plasma wave study of submonolayer Cs and Cs O covered Ag surfaces,” Surf. Sci. 91, 601–617 (1980).
[CrossRef]

Clark, M. G.

K. R. Welford, J. R. Sambles, and M. G. Clark, “Guided modes and surface plasmon-polaritons observed with a nematic liquid crystal using attenuated total reflection,” Liq. Cryst. 2, 91–105(1987).
[CrossRef]

Cook, G.

Dasari, R. R.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14, R597 (2002).
[CrossRef]

Dereux, A.

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

Dickson, W.

W. Dickson, G. A. Wurtz, P. R. Evans, R. J. Pollard, and A. V. Zayats, “Electronically controlled surface plasmon dispersion and optical transmission through metallic hole arrays using liquid crystal,” Nano Lett. 8, 281–286 (2008).
[CrossRef]

Dyadyusha, A.

Ebbesen, T. W.

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

Elston, S. J.

S. J. Elston and J. R. Sambles, “Surface plasmon-polaritons on an anisotropic substrate,” J. Mod. Opt. 37, 1895–1902(1990).
[CrossRef]

Evans, D. R.

Evans, P. R.

W. Dickson, G. A. Wurtz, P. R. Evans, R. J. Pollard, and A. V. Zayats, “Electronically controlled surface plasmon dispersion and optical transmission through metallic hole arrays using liquid crystal,” Nano Lett. 8, 281–286 (2008).
[CrossRef]

Faetti, S.

J. Li, S. T. Wu, S. Brugioni, R. Meucci, and S. Faetti, “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

Fan, X.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
[CrossRef] [PubMed]

Feld, M. S.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14, R597 (2002).
[CrossRef]

Fontana, E.

Gauglitz, G.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sensors Actuators B Chem 54, 3–15(1999).
[CrossRef]

Gauza, S.

Gaylord, T. K.

Glushchenko, A. V.

Glytsis, E. N.

Griffith, A. T.

Harel, E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232(2003).
[CrossRef] [PubMed]

Homola, J.

J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sensors Actuators B Chem 54, 3–15(1999).
[CrossRef]

Hooper, I. R.

I. R. Hooper and J. R. Sambles, “Differential ellipsometric surface plasmon resonance sensors with liquid crystal polarization modulators,” Appl. Phys. Lett. 85, 3017–3019(2004).
[CrossRef]

Huignard, J. P.

U. Bortolozzo and S. Residori, and J. P. Huignard, “Beam coupling in photorefractive liquid crystal light valves,” J. Phys. D 41, 224007 (2008).
[CrossRef]

Innes, R. A.

R. A. Innes, S. P. Ashworth, and J. R. Sambles, “Large optical bistability in a nematic liquid crystal using surface plasmon-polaritons,” Phys. Lett. A 135, 357–362 (1989).
[CrossRef]

R. A. Innes and J. R. Sambles, “Optical nonlinearity in a nematic liquid crystal using surface plasmon-polaritons,” Opt. Commun. 64, 288–292 (1987).
[CrossRef]

Itzkan, I.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14, R597 (2002).
[CrossRef]

Kaczmarek, M.

Kik, P. G.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232(2003).
[CrossRef] [PubMed]

Kneipp, H.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14, R597 (2002).
[CrossRef]

Kneipp, K.

K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari, and M. S. Feld, Surface-enhanced Raman scattering and biophysics,” J. Phys. Condens. Matter 14, R597 (2002).
[CrossRef]

Koel, B. E.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232(2003).
[CrossRef] [PubMed]

Kretschmann, E.

E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Zeitschrift Fuer Naturforschung, Teil A 23, 2135–2136 (1968).

Li, J.

J. Li, C. H. Wen, S. Gauza, R. Lu, and S. T. Wu, “Refractive indices of liquid crystals for display applications,” J. Display Technol. 1, 51–61 (2005).
[CrossRef]

J. Li, S. T. Wu, S. Brugioni, R. Meucci, and S. Faetti, “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

Lu, R.

Maier, S. A.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232(2003).
[CrossRef] [PubMed]

Malmqvist, M.

M. Malmqvist, “Biospecific interaction analysis using biosensor technology,” Nature 361, 186–187 (1993).
[CrossRef] [PubMed]

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]

Meltzer, S.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232(2003).
[CrossRef] [PubMed]

Meucci, R.

J. Li, S. T. Wu, S. Brugioni, R. Meucci, and S. Faetti, “Infrared refractive indices of liquid crystals,” J. Appl. Phys. 97, 073501 (2005).
[CrossRef]

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2000).

Please, C. P.

R. H. Self, C. P. Please, and T. J. Sluckin, “Deformation of nematic liquid crystals in an electric field,” Eur. J. Appl. Math. 13, 1–23 (2002).
[CrossRef]

Pockrand, I.

I. Pockrand, “Surface plasma oscillations at silver surfaces with thin transparent and absorbing coatings,” Surf. Sci. 72, 577–588 (1978).
[CrossRef]

Pollard, R. J.

W. Dickson, G. A. Wurtz, P. R. Evans, R. J. Pollard, and A. V. Zayats, “Electronically controlled surface plasmon dispersion and optical transmission through metallic hole arrays using liquid crystal,” Nano Lett. 8, 281–286 (2008).
[CrossRef]

Raether, H.

E. Kretschmann and H. Raether, “Radiative decay of non radiative surface plasmons excited by light,” Zeitschrift Fuer Naturforschung, Teil A 23, 2135–2136 (1968).

Raynes, E. P.

E. P. Raynes, C. V. Brown, and J. F. Stromer, “Method for the measurement of the K22 nematic elastic constant,” Appl. Phys. Lett. 82, 13–15 (2003).
[CrossRef]

Requicha, A. A. G.

S. A. Maier, P. G. Kik, H. A. Atwater, S. Meltzer, E. Harel, B. E. Koel, and A. A. G. Requicha, “Local detection of electromagnetic energy transport below the diffraction limit in metal nanoparticle plasmon waveguides,” Nat. Mater. 2, 229–232(2003).
[CrossRef] [PubMed]

Reshetnyak, V.

Residori, S.

U. Bortolozzo and S. Residori, and J. P. Huignard, “Beam coupling in photorefractive liquid crystal light valves,” J. Phys. D 41, 224007 (2008).
[CrossRef]

Reznikov, Y.

Ritchie, R. H.

R. H. Ritchie, “Plasma losses by fast electrons in thin films,” Phys. Rev. 106, 874–881 (1957).
[CrossRef]

Saleh, M. A.

Sambles, J. R.

I. R. Hooper and J. R. Sambles, “Differential ellipsometric surface plasmon resonance sensors with liquid crystal polarization modulators,” Appl. Phys. Lett. 85, 3017–3019(2004).
[CrossRef]

S. J. Elston and J. R. Sambles, “Surface plasmon-polaritons on an anisotropic substrate,” J. Mod. Opt. 37, 1895–1902(1990).
[CrossRef]

R. A. Innes, S. P. Ashworth, and J. R. Sambles, “Large optical bistability in a nematic liquid crystal using surface plasmon-polaritons,” Phys. Lett. A 135, 357–362 (1989).
[CrossRef]

K. R. Welford, J. R. Sambles, and M. G. Clark, “Guided modes and surface plasmon-polaritons observed with a nematic liquid crystal using attenuated total reflection,” Liq. Cryst. 2, 91–105(1987).
[CrossRef]

R. A. Innes and J. R. Sambles, “Optical nonlinearity in a nematic liquid crystal using surface plasmon-polaritons,” Opt. Commun. 64, 288–292 (1987).
[CrossRef]

K. R. Welford and J. R. Sambles, “Detection of surface director reorientation in a nematic liquid crystal,” Appl. Phys. Lett. 50, 871–873 (1987).
[CrossRef]

Self, R. H.

R. H. Self, C. P. Please, and T. J. Sluckin, “Deformation of nematic liquid crystals in an electric field,” Eur. J. Appl. Math. 13, 1–23 (2002).
[CrossRef]

Shopova, S. I.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
[CrossRef] [PubMed]

Sluckin, T. J.

R. H. Self, C. P. Please, and T. J. Sluckin, “Deformation of nematic liquid crystals in an electric field,” Eur. J. Appl. Math. 13, 1–23 (2002).
[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]

Stromer, J. F.

E. P. Raynes, C. V. Brown, and J. F. Stromer, “Method for the measurement of the K22 nematic elastic constant,” Appl. Phys. Lett. 82, 13–15 (2003).
[CrossRef]

Sun, Y.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
[CrossRef] [PubMed]

Suter, J. D.

X. Fan, I. M. White, S. I. Shopova, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: a review,” Anal. Chim. Acta 620, 8–26 (2008).
[CrossRef] [PubMed]

Wang, Y.

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Figures (5)

Fig. 1
Fig. 1

(A) Hybrid photorefractive–plasmonic liquid crystal cell with hemispherical prism to allow SPP coupling in the standard Kretschmann geometry. (B) Waveguide structure under study: the system is simplified to a three-layer structure. At leading order this simplifies further to a two-layer system where both layers are homogeneous and isotropic. (C) Definition of angles used relative to coordinate axes. The SPP propagates in the x direction and the alignment of the liquid crystal is determined in terms of the Euler angles θ and ϕ.

Fig. 2
Fig. 2

Comparison between numerical reflection spectrum and analytic approximation for the SPP effective index with n o = 1.6 and n e = 1.7 . The numerical effective index is calculated directly from the minimum of the reflection spectrum, and the analytic approximation is given by Eq. (23).

Fig. 3
Fig. 3

Comparison between numerical reflection spectrum and analytic approximation for the SPP effective index with n o = 1.5100 and n e = 1.7104 . This gives ϵ c = 0.39 and ϵ lc = 0.65 . The numerical effective index is calculated directly from the minimum of the reflection spectrum, and the analytic approximation is given by Eq. (23).

Fig. 4
Fig. 4

The experimental setup to generate and analyze SPP. P-polarized light is required to observe the dip in the reflection spectrum due to coupling to an SPP. S-polarized light is used to normalize the signal.

Fig. 5
Fig. 5

Comparison between the approximate theoretical prediction and experimental measurements of the SPP effective index with n o = 1.5100 and n e = 1.7104 . Corresponding to ϵ c = 0.39 and ϵ l c = 0.65 . No fitting parameters were used to obtain these plots.

Tables (1)

Tables Icon

Table 1 Numerical Values of Constants for a Typical Hybrid Plasmonic Photorefractive LC Cell Filled with the Liquid Crystal Compound E7

Equations (28)

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ϵ ( z ) = ϵ u ( z ) I + η Δ ϵ ( z ) ,
ϵ u ( z ) = { ϵ m z < 0 ϵ d 0 < z < L ϵ d z > L , Δ ϵ ( z ) = { 0 z < 0 0 0 < z < L ϵ lc [ n ^ ( z ) n ^ ( z ) 1 3 I ] + ϵ c I z > L ,
( D 0 i i ϵ u D 0 ) ( E 0 H 0 ) = 0 ,
H 0 ( x ) = { h m e i k m · x 0 z < 0 h d e i k d · x 0 z > 0 , E 0 ( x ) = { 1 ϵ m e m e i k m · x 0 z < 0 1 ϵ d e d e i k d · x 0 z > 0 .
e ν = ( ± i β ν 0 α 0 ) , k ν = ( α 0 0 i β ν ) , h ν = ( 0 1 0 ) ,
α 0 = ϵ d ϵ m ϵ d + ϵ m , β m = ϵ m ϵ m ϵ d + ϵ m , β d = ϵ d ϵ d ϵ d + ϵ m .
( D 0 i i ϵ u D 0 ) ( E 1 H 1 ) = ( 1 × E 0 i Δ ϵ E 0 1 × H 0 ) .
H 0 * · 1 × E 0 E 0 * · 1 × H 0 i E 0 * · Δ ϵ E 0 d z 0 = 0.
n eff spp = α 0 + η 1 P 0 k 0 L E 0 * · Δ ϵ E 0 d z 0 ,
P 0 = e ^ x · [ E 0 * × H 0 + E 0 × H 0 * ] d z 0 .
F = L 0 L 1 1 2 ( z 0 θ ) 2 + 1 2 sin 2 θ ( z 0 ϕ ) 2 1 2 χ i ( z 0 ψ ) 2 1 2 χ a cos 2 θ ( z 0 ψ ) 2 d z 0 ,
d 2 θ d z 0 2 sin 2 θ ( d ϕ d z 0 ) 2 1 2 χ a E z 2 sin 2 θ = 0 ,
d d z 0 [ ( 1 cos 2 θ ) d ϕ d z 0 ] = 0 ,
d d z 0 [ ( 1 + α θ cos 2 θ ) d ψ d z 0 ] = 0 ,
2 θ z 0 2 sin 2 θ ( d ϕ d z 0 ) 2 1 δ 2 sin 2 θ = 0 ,
d ϕ i d ζ = C 0 1 cos 2 θ i ,
d 2 θ i d ζ 2 sin 2 θ i = C 0 2 sin 2 θ i ( 1 cos 2 θ i ) 2 .
1 2 ( d θ i d ζ ) 2 + 1 2 cos ( 2 θ i ) = C 0 2 2 ( 1 cos 2 θ i ) + C 1 ,
θ i = arctan { 2 C 2 e 2 ζ C 2 2 e 2 2 ζ 1 } ,
C 2 = 1 + sin ( θ p 0 ) cos ( θ p 0 ) 1 + θ p 0 + O ( θ p 0 2 ) ,
θ = arctan { 2 ( 1 + θ p 0 ) e 2 ( z 0 L 0 ) / δ e 2 2 ( z 0 L 0 ) / δ ( 1 + 2 θ p 0 ) 1 } + arctan { 2 ( 1 + θ p 1 ) e 2 ( L 1 z 0 ) / δ e 2 2 ( L 1 z 0 ) / δ ( 1 + 2 θ p 1 ) 1 } .
n ^ = 1 e 2 2 ( z 0 L 0 ) / δ + 1 ( 2 e 2 ( z 0 L 0 ) / δ cos ( ϕ p 0 ) 2 e 2 ( z 0 L 0 ) / δ sin ( ϕ p 0 ) e 2 2 ( z 0 L 0 ) / δ 1 ) .
γ = 2 ε 0 Δ ε K V k 0 β d L z .
n eff spp = α 0 [ 1 + η ( κ h + κ n h ) e 2 L β d ] ,
κ h = 1 6 { 3 ϵ m ϵ c ϵ d ( ϵ d + ϵ m ) ( ϵ d + 2 ϵ m ) ϵ m ϵ lc ϵ d ( ϵ d 2 ϵ m 2 ) } ,
κ n h = 2 ( ϵ d cos 2 ϕ p 0 + ϵ m ) ϵ m ϵ lc ϵ d ( ϵ d 2 ϵ m 2 ) 0 e ξ e γ ξ ( e γ ξ + 1 ) 2 d ξ ,
0 e ξ e γ ξ ( e γ ξ + 1 ) 2 d ξ = 1 γ ( 1 2 + n = 1 ( 1 ) n 1 + n γ ) ,
0 e ξ e γ ξ ( e γ ξ + 1 ) 2 d ξ 1 4 ( 1 1 2 γ 2 + γ 4 ) + O ( γ 6 ) .

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