Abstract

We perform a systematic study of spontaneous Raman scattering in resonant planar structures. We present a semiclassical approach that allows the description of spontaneous Raman scattering in an arbitrary multilayer, providing analytical expressions of the Raman cross sections in terms of the Fresnel coefficients of the structure and taking into account beam size effects. Large enhancements of the Raman cross section are predicted in fully dielectric structures. In particular, given our results, truncated periodic multilayers supporting Bloch surface waves might be of interest for the realization of integrated Raman sensor devices.

© 2012 Optical Society of America

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  33. M. Shinn, W. M. Robertson, “Surface plasmon-like sensor based on surface electromagnetic waves in a photonic band-gap material,” Sens. Actuators B 105, 360–364 (2005).
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  34. F. Giorgis, E. Descrovi, C. Summonte, L. Dominici, F. Michelotti, “Experimental determination of the sensitivity of Bloch surface waves based sensors,” Opt. Express 18, 8087–8093 (2010).
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    [CrossRef]
  46. J. E. Sipe, J. Becher, “Surface energy transfer enhanced by optical cavity excitation: a pole analysis,” J. Opt. Soc. Am. 72, 288–295 (1982).
    [CrossRef]
  47. T. Sfez, E. Descrovi, L. Yu, D. Brunazzo, M. Quaglio, L. Dominici, W. Nakagawa, F. Michelotti, F. Giorgis, O. J. F. Martin, H. P. Herzig, “Bloch surface waves in ultrathin waveguides: near-field investigation of mode polarization and propagation,” J. Opt. Soc. Am. B 27, 1617–1625 (2010).
    [CrossRef]
  48. M. Liscidini, D. Gerace, D. Sanvitto, D. Bajoni, “Guided Bloch surface wave polaritons,” Appl. Phys. Lett. 98, 121118 (2011).
    [CrossRef]

2012

Y. F. Chan, H. J. Xu, L. Cao, Y. Tang, D. Y. Li, X. M. Sun, “ZnO/Si arrays decorated by Au nanoparticles for surface enhanced Raman scattering study,” J. Appl. Phys. 111, 033104 (2012).
[CrossRef]

2011

V. Paeder, V. Musi, L. Hvozdara, S. Herminjard, H. P. Herzig, “Detection of protein aggregation with a Bloch surface wave based sensor,” Sens. Actuators B 157, 260–264 (2011).
[CrossRef]

M. Liscidini, D. Gerace, D. Sanvitto, D. Bajoni, “Guided Bloch surface wave polaritons,” Appl. Phys. Lett. 98, 121118 (2011).
[CrossRef]

2010

H. Qiao, B. Guan, J. J. Gooding, P. J. Reece, “Protease detection using a porous silicon based Bloch surface wave optical biosensor,” Opt. Express 18, 15174–15182 (2010).
[CrossRef]

F. Giorgis, E. Descrovi, C. Summonte, L. Dominici, F. Michelotti, “Experimental determination of the sensitivity of Bloch surface waves based sensors,” Opt. Express 18, 8087–8093 (2010).
[CrossRef]

T. Sfez, E. Descrovi, L. Yu, D. Brunazzo, M. Quaglio, L. Dominici, W. Nakagawa, F. Michelotti, F. Giorgis, O. J. F. Martin, H. P. Herzig, “Bloch surface waves in ultrathin waveguides: near-field investigation of mode polarization and propagation,” J. Opt. Soc. Am. B 27, 1617–1625 (2010).
[CrossRef]

A. Y. Panarin, S. N. Terekhov, K. I. Kholostov, V. P. Bondarenko, “SERS-active substrates based on n-type porous silicon,” Appl. Surf. Sci. 256, 6969–6976 (2010).
[CrossRef]

X. Yang, C. Shi, D. Wheeler, R. Newhouse, B. Chen, J. Z. Zhang, C. Gu, “High-sensitivity molecular sensing using hollow-core photonic crystal fiber and surface-enhanced Raman scattering,” J. Opt. Soc. Am. A 27, 977–984 (2010).
[CrossRef]

2009

M. Liscidini, J. E. Sipe, “Analysis of Bloch surface waves assisted diffraction-based biosensors,” J. Opt. Soc. Am. B 26, 279–289 (2009).
[CrossRef]

M. Liscidini, M. Galli, M. Shi, G. Dacarro, M. Patrini, D. Bajoni, J. E. Sipe, “Strong modification of light emission from a dye monolayer via Bloch surface waves,” Opt. Lett. 34, 2318–2320 (2009).
[CrossRef]

2008

F. Giorgis, E. Descrovi, A. Chiodoni, E. Froner, M. Scarpa, A. Venturello, F. Geobaldo, “Porous silicon as efficient surface enhanced Raman scattering (SERS) substrate,” Appl. Surf. Sci. 254, 7494–7497 (2008).
[CrossRef]

2007

S. Lal, S. Link, N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photon. 1, 641–648 (2007).
[CrossRef]

2006

A. Pope, A. Schulte, Y. Guo, L. K. Ono, B. R. Cuenya, C. Lopez, K. Richardson, K. Kitanovski, T. Winningham, “Chalcogenide waveguide structures as substrates and guiding layers for evanescent wave Raman spectroscopy of bacteriorhodopsin,” Vibr. Spectrosc. 42, 249–253 (2006).
[CrossRef]

E. Guillermain, V. Lysenko, T. Benyattou, “Surface wave photonic device based on porous silicon multilayers,” J. Lumin. 121, 319–321 (2006).
[CrossRef]

2005

M. Shinn, W. M. Robertson, “Surface plasmon-like sensor based on surface electromagnetic waves in a photonic band-gap material,” Sens. Actuators B 105, 360–364 (2005).
[CrossRef]

2004

H. Lin, J. Mock, D. Smith, T. Gao, M. J. Sailor, “Surface-enhanced Raman scattering from silver-plated porous silicon,” J. Phys. Chem. B 108, 1165–1167 (2004).
[CrossRef]

A. G. Brolo, E. Arctander, R. Gordon, B. Leathem, K. L. Kavanagh, “Nanohole-enhanced Raman scattering,” Nano Lett. 4, 2015–2018 (2004).
[CrossRef]

2003

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

N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095 (2003).
[CrossRef]

J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. 377, 528–539 (2003).
[CrossRef]

1998

G. Stanev, N. Goutev, Zh. S. Nickolov, “Coupled waveguides for Raman studies of thin liquid films,” J. Phys. D 31, 1782–1786 (1998).
[CrossRef]

M. Kahl, E. Voges, S. Kostrewa, C. Viets, W. Hill, “Periodically structured metallic substrates for SERS,” Sens. Actuators B 51, 285–291 (1998).
[CrossRef]

A. Campion, P. Kambhampati, “Surface-enhanced Raman scattering,” Chem. Soc. Rev. 27, 241–250 (1998).
[CrossRef]

1997

S. Nie, S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[CrossRef]

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

1996

J. S. Kanger, C. Otto, M. Slotboom, J. Greve, “Waveguide Raman spectroscopy of thin polymer layers and monolayers of biomolecules using high refractive index waveguides,” J. Phys. Chem. 100, 3288–3292 (1996).
[CrossRef]

1990

L. Kang, R. E. Dessy, “Slab waveguide in chemistry,” Crit. Rev. Anal. Chem. 21, 377–388 (1990).
[CrossRef]

1988

Raman scattering in the Kretschmann configuration employing surface plasmon structures was discussed by J. Giergiel, E. Reed, J. C. Hemminger, S. Ushioda, “Surface plasmon polariton enhancement of Raman scattering in Kretschmann geometry,” J. Phys. Chem. 92, 5357–5365 (1988).
[CrossRef]

1987

W. M. Robertson, A. L. Moretti, R. Bray, “Surface-plasmon-enhanced Brillouin scattering on silver films: double-resonance effect,” Phys. Rev. B 35, 8919–8928 (1987).
[CrossRef]

1985

M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
[CrossRef]

1982

J. E. Sipe, J. Becher, “Surface energy transfer enhanced by optical cavity excitation: a pole analysis,” J. Opt. Soc. Am. 72, 288–295 (1982).
[CrossRef]

1981

J. E. Sipe, “The dipole antenna problem in surface physics: a new approach,” Surf. Sci. 105, 489–504 (1981).
[CrossRef]

1980

J. F. Rabolt, R. Santo, J. D. Swalen, “Raman measurements on thin polymer films and organic monolayers,” Appl. Spectrosc. 34, 517–521 (1980).
[CrossRef]

1978

P. Yeh, A. Yariv, A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105(1978).
[CrossRef]

1974

Y. Levy, C. Imbert, S. Cipriani, S. Racine, R. Dupeyrat, “Raman scattering of thin films as a waveguide,” Opt. Commun. 11, 66–69 (1974).
[CrossRef]

M. Fleischmann, P. J. Hendra, A. J. McQuillan, “Raman spectra of pyridine adsorbed at a silver electrode,” Chem. Phys. Lett. 26, 163–166 (1974).
[CrossRef]

1972

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

1968

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
[CrossRef]

E. Kretschmann, H. Raether, “Radiative decay of nonradiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135–2136 (1968).

Arctander, E.

A. G. Brolo, E. Arctander, R. Gordon, B. Leathem, K. L. Kavanagh, “Nanohole-enhanced Raman scattering,” Nano Lett. 4, 2015–2018 (2004).
[CrossRef]

Aubard, J.

N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095 (2003).
[CrossRef]

Aussenegg, F. R.

N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095 (2003).
[CrossRef]

Bajoni, D.

M. Liscidini, D. Gerace, D. Sanvitto, D. Bajoni, “Guided Bloch surface wave polaritons,” Appl. Phys. Lett. 98, 121118 (2011).
[CrossRef]

M. Liscidini, M. Galli, M. Shi, G. Dacarro, M. Patrini, D. Bajoni, J. E. Sipe, “Strong modification of light emission from a dye monolayer via Bloch surface waves,” Opt. Lett. 34, 2318–2320 (2009).
[CrossRef]

Barnes, W. L.

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

Becher, J.

J. E. Sipe, J. Becher, “Surface energy transfer enhanced by optical cavity excitation: a pole analysis,” J. Opt. Soc. Am. 72, 288–295 (1982).
[CrossRef]

Benyattou, T.

E. Guillermain, V. Lysenko, T. Benyattou, “Surface wave photonic device based on porous silicon multilayers,” J. Lumin. 121, 319–321 (2006).
[CrossRef]

Bondarenko, V. P.

A. Y. Panarin, S. N. Terekhov, K. I. Kholostov, V. P. Bondarenko, “SERS-active substrates based on n-type porous silicon,” Appl. Surf. Sci. 256, 6969–6976 (2010).
[CrossRef]

Bray, R.

W. M. Robertson, A. L. Moretti, R. Bray, “Surface-plasmon-enhanced Brillouin scattering on silver films: double-resonance effect,” Phys. Rev. B 35, 8919–8928 (1987).
[CrossRef]

Brolo, A. G.

A. G. Brolo, E. Arctander, R. Gordon, B. Leathem, K. L. Kavanagh, “Nanohole-enhanced Raman scattering,” Nano Lett. 4, 2015–2018 (2004).
[CrossRef]

Brunazzo, D.

T. Sfez, E. Descrovi, L. Yu, D. Brunazzo, M. Quaglio, L. Dominici, W. Nakagawa, F. Michelotti, F. Giorgis, O. J. F. Martin, H. P. Herzig, “Bloch surface waves in ultrathin waveguides: near-field investigation of mode polarization and propagation,” J. Opt. Soc. Am. B 27, 1617–1625 (2010).
[CrossRef]

Campion, A.

A. Campion, P. Kambhampati, “Surface-enhanced Raman scattering,” Chem. Soc. Rev. 27, 241–250 (1998).
[CrossRef]

Cao, L.

Y. F. Chan, H. J. Xu, L. Cao, Y. Tang, D. Y. Li, X. M. Sun, “ZnO/Si arrays decorated by Au nanoparticles for surface enhanced Raman scattering study,” J. Appl. Phys. 111, 033104 (2012).
[CrossRef]

Chan, Y. F.

Y. F. Chan, H. J. Xu, L. Cao, Y. Tang, D. Y. Li, X. M. Sun, “ZnO/Si arrays decorated by Au nanoparticles for surface enhanced Raman scattering study,” J. Appl. Phys. 111, 033104 (2012).
[CrossRef]

Chen, B.

X. Yang, C. Shi, D. Wheeler, R. Newhouse, B. Chen, J. Z. Zhang, C. Gu, “High-sensitivity molecular sensing using hollow-core photonic crystal fiber and surface-enhanced Raman scattering,” J. Opt. Soc. Am. A 27, 977–984 (2010).
[CrossRef]

Chiodoni, A.

F. Giorgis, E. Descrovi, A. Chiodoni, E. Froner, M. Scarpa, A. Venturello, F. Geobaldo, “Porous silicon as efficient surface enhanced Raman scattering (SERS) substrate,” Appl. Surf. Sci. 254, 7494–7497 (2008).
[CrossRef]

Cho, A. Y.

P. Yeh, A. Yariv, A. Y. Cho, “Optical surface waves in periodic layered media,” Appl. Phys. Lett. 32, 104–105(1978).
[CrossRef]

Christy, R. W.

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

Cipriani, S.

Y. Levy, C. Imbert, S. Cipriani, S. Racine, R. Dupeyrat, “Raman scattering of thin films as a waveguide,” Opt. Commun. 11, 66–69 (1974).
[CrossRef]

Cuenya, B. R.

A. Pope, A. Schulte, Y. Guo, L. K. Ono, B. R. Cuenya, C. Lopez, K. Richardson, K. Kitanovski, T. Winningham, “Chalcogenide waveguide structures as substrates and guiding layers for evanescent wave Raman spectroscopy of bacteriorhodopsin,” Vibr. Spectrosc. 42, 249–253 (2006).
[CrossRef]

Dacarro, G.

M. Liscidini, M. Galli, M. Shi, G. Dacarro, M. Patrini, D. Bajoni, J. E. Sipe, “Strong modification of light emission from a dye monolayer via Bloch surface waves,” Opt. Lett. 34, 2318–2320 (2009).
[CrossRef]

Dasari, R. R.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

Dereux, A.

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

Descrovi, E.

F. Giorgis, E. Descrovi, C. Summonte, L. Dominici, F. Michelotti, “Experimental determination of the sensitivity of Bloch surface waves based sensors,” Opt. Express 18, 8087–8093 (2010).
[CrossRef]

T. Sfez, E. Descrovi, L. Yu, D. Brunazzo, M. Quaglio, L. Dominici, W. Nakagawa, F. Michelotti, F. Giorgis, O. J. F. Martin, H. P. Herzig, “Bloch surface waves in ultrathin waveguides: near-field investigation of mode polarization and propagation,” J. Opt. Soc. Am. B 27, 1617–1625 (2010).
[CrossRef]

F. Giorgis, E. Descrovi, A. Chiodoni, E. Froner, M. Scarpa, A. Venturello, F. Geobaldo, “Porous silicon as efficient surface enhanced Raman scattering (SERS) substrate,” Appl. Surf. Sci. 254, 7494–7497 (2008).
[CrossRef]

Dessy, R. E.

L. Kang, R. E. Dessy, “Slab waveguide in chemistry,” Crit. Rev. Anal. Chem. 21, 377–388 (1990).
[CrossRef]

Dominici, L.

F. Giorgis, E. Descrovi, C. Summonte, L. Dominici, F. Michelotti, “Experimental determination of the sensitivity of Bloch surface waves based sensors,” Opt. Express 18, 8087–8093 (2010).
[CrossRef]

T. Sfez, E. Descrovi, L. Yu, D. Brunazzo, M. Quaglio, L. Dominici, W. Nakagawa, F. Michelotti, F. Giorgis, O. J. F. Martin, H. P. Herzig, “Bloch surface waves in ultrathin waveguides: near-field investigation of mode polarization and propagation,” J. Opt. Soc. Am. B 27, 1617–1625 (2010).
[CrossRef]

Dupeyrat, R.

Y. Levy, C. Imbert, S. Cipriani, S. Racine, R. Dupeyrat, “Raman scattering of thin films as a waveguide,” Opt. Commun. 11, 66–69 (1974).
[CrossRef]

Ebbesen, T. W.

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

Emory, S. R.

S. Nie, S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[CrossRef]

Etchegoin, P.

E. L. Ru, P. Etchegoin, Principles of Surface-Enhanced Raman Spectroscopy: and Related Plasmonic Effects (Elsevier Science2008).

Feld, M. S.

K. Kneipp, Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

Felidj, N.

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F. Giorgis, E. Descrovi, C. Summonte, L. Dominici, F. Michelotti, “Experimental determination of the sensitivity of Bloch surface waves based sensors,” Opt. Express 18, 8087–8093 (2010).
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V. Paeder, V. Musi, L. Hvozdara, S. Herminjard, H. P. Herzig, “Detection of protein aggregation with a Bloch surface wave based sensor,” Sens. Actuators B 157, 260–264 (2011).
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M. Liscidini, M. Galli, M. Shi, G. Dacarro, M. Patrini, D. Bajoni, J. E. Sipe, “Strong modification of light emission from a dye monolayer via Bloch surface waves,” Opt. Lett. 34, 2318–2320 (2009).
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H. Qiao, B. Guan, J. J. Gooding, P. J. Reece, “Protease detection using a porous silicon based Bloch surface wave optical biosensor,” Opt. Express 18, 15174–15182 (2010).
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M. Liscidini, D. Gerace, D. Sanvitto, D. Bajoni, “Guided Bloch surface wave polaritons,” Appl. Phys. Lett. 98, 121118 (2011).
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F. Giorgis, E. Descrovi, A. Chiodoni, E. Froner, M. Scarpa, A. Venturello, F. Geobaldo, “Porous silicon as efficient surface enhanced Raman scattering (SERS) substrate,” Appl. Surf. Sci. 254, 7494–7497 (2008).
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N. Felidj, J. Aubard, G. Levi, J. R. Krenn, A. Hohenau, G. Schider, A. Leitner, F. R. Aussenegg, “Optimized surface-enhanced Raman scattering on gold nanoparticle arrays,” Appl. Phys. Lett. 82, 3095 (2003).
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A. Pope, A. Schulte, Y. Guo, L. K. Ono, B. R. Cuenya, C. Lopez, K. Richardson, K. Kitanovski, T. Winningham, “Chalcogenide waveguide structures as substrates and guiding layers for evanescent wave Raman spectroscopy of bacteriorhodopsin,” Vibr. Spectrosc. 42, 249–253 (2006).
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X. Yang, C. Shi, D. Wheeler, R. Newhouse, B. Chen, J. Z. Zhang, C. Gu, “High-sensitivity molecular sensing using hollow-core photonic crystal fiber and surface-enhanced Raman scattering,” J. Opt. Soc. Am. A 27, 977–984 (2010).
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M. Liscidini, M. Galli, M. Shi, G. Dacarro, M. Patrini, D. Bajoni, J. E. Sipe, “Strong modification of light emission from a dye monolayer via Bloch surface waves,” Opt. Lett. 34, 2318–2320 (2009).
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M. Liscidini, M. Galli, M. Shi, G. Dacarro, M. Patrini, D. Bajoni, J. E. Sipe, “Strong modification of light emission from a dye monolayer via Bloch surface waves,” Opt. Lett. 34, 2318–2320 (2009).
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[CrossRef]

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H. Lin, J. Mock, D. Smith, T. Gao, M. J. Sailor, “Surface-enhanced Raman scattering from silver-plated porous silicon,” J. Phys. Chem. B 108, 1165–1167 (2004).
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

Sketch of the pump and detection configuration under consideration in the case of spontaneous Raman scattering, θ P is the incident angle of the pump in the substrate.

Fig. 2.
Fig. 2.

(a) BSW dispersion curve and cladding light line (LL) . (b) WG dispersion curve, cladding light line, and substrate light line. (c) SP dispersion curve and cladding light line.

Fig. 3.
Fig. 3.

Structures: (a) multilayer structure with BSWs; (b) WG structure; (c) metallic structure with SP modes; (d) bare prism of the reference.

Fig. 4.
Fig. 4.

Pump enhancement factor for multilayer, WG, SP, and bare prism at (a) 532 and (b) 1064 nm. In (a), the very small rise in | E L / E | 2 as θ θ 0 for the SP and bare prism structures cannot be seen.

Fig. 5.
Fig. 5.

Differential cross sections normalized to the total cross section in free space ( n 1 = 1 ) for the scattered field in the substrate: (a)  λ = 532 nm ; (b)  λ = 1064 nm for multilayer, WG, SP, and bare prism structures.

Fig. 6.
Fig. 6.

Integrated Raman cross section versus the distance of the molecule from the surface of the structure: (a)  λ = 532 nm ; (b)  λ = 1064 nm for multilayer, WG, SP, and bare prism structures. The plots of SP and bare prism structures have been multiplied by 20.

Fig. 7.
Fig. 7.

Finite beam of light incident on the multilayer structure, and its reflected beam.

Fig. 8.
Fig. 8.

Normalized intensity distribution of incident beams at 532 nm with (a) 2 mm and (c) 50 μm FWHM, and (b) and (d) the corresponding intensity in the cladding for the BSW structure.

Fig. 9.
Fig. 9.

Normalized Raman scattering power per unit areal density of molecules, for the multilayer, WG, SP, and bare prism structures at (a) 532 and (b) 1064 nm. The plots of the SP and bare prism structures have been multiplied by 40.

Equations (77)

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

α ( t ) = α 0 + ξ α 1 ξ q ξ ( t ) ,
q ( t ) = ( 2 m ω 0 ) 1 / 2 ( a e i ω 0 t + a e i ω 0 t ) ,
E inc ( t ) = { E e i ω P t + c . c T / 2 < t < T / 2 , 0 otherwise ,
f ± ( t ) = d ω 2 π f ± ( ω ) e i ω t ,
f ( ω ) = f ( t ) e i ω t d t .
μ + ( ω ) = ( 2 m ω 0 ) 1 / 2 α 1 ( a E inc + ( ω ω 0 ) + a E inc + ( ω + ω 0 ) ) ,
E + ( r , ω ) = ω ˜ 2 4 π ϵ 0 e i ω ˜ n 1 r r ( r ^ × μ + ( ω ) ) × r ^ , H + ( r , ω ) = ω ˜ 2 n 1 c 4 π e i ω ˜ n 1 r r ( r ^ × μ + ( ω ) ) ,
: S ( r , t ) : E ( r , t ) × H + ( r , t ) H ( r , t ) × E + ( r , t ) ,
: S ( r , t ) : d t = 0 S ( r , ω ) d ω ,
S ( r , ω ) = 1 2 π ( ( E + ( r , ω ) ) × H + ( r , ω ) ( H + ( r , ω ) ) × E + ( r , ω ) ) ,
S ( r , ω ) = ω ˜ 4 n 1 c 16 π 3 ϵ 0 r ^ r 2 Γ free ( r ^ ) : ( μ + ( ω ) ) μ + ( ω ) ,
S ( r , ω ) = ω ˜ 4 n 1 c 16 π 3 ϵ 0 r ^ r 2 Γ free ( r ^ ) : ( μ + ( ω ) ) μ + ( ω ) .
μ + ( ω ) ) μ + ( ω ) T = π ( α 1 A ) 2 2 E * E δ ( ω ω P ω 0 ) + π ( α 1 S ) 2 2 E * E δ ( ω ω P + ω 0 ) ,
S = ( 2 m ω 0 ) ( n ¯ + 1 ) , A = ( 2 m ω 0 ) n ¯ ,
1 T 0 S ( r , ω ) d ω = ω ˜ S 4 n 1 ( ω S ) c 32 π 2 ϵ 0 ( α 1 S ) 2 r ^ r 2 Γ free ( r ^ , ω S ) : ( E * E ) + ω ˜ A 4 n 1 ( ω A ) c 32 π 2 ϵ 0 ( α 1 A ) 2 r ^ r 2 Γ free ( r ^ , ω A ) : ( E * E ) ,
P S = S S ( r ) · r ^ r 2 d Ω = ω ˜ S 4 n 1 ( ω S ) c 12 π ϵ 0 ( α 1 S ) 2 E * · E ,
σ S o ( r ^ ) = S S ( r ) · r ^ r 2 S P ,
S P = 2 n 1 ( ω P ) c μ 0 ( E * · E ) ,
σ S o ( r ^ ) ω ˜ S 4 n 1 ( ω S ) 4 n 1 ( ω P ) ( α 1 S 4 π ϵ 0 ) 2 Γ free ( r ^ , ω S ) : ( e ^ inc * e ^ inc ) ,
σ S o = σ S o ( r ^ ) d Ω = 2 π ω ˜ S 4 n 1 ( ω S ) 3 n 1 ( ω P ) ( α 1 S 4 π ϵ 0 ) 2 .
E L ( t ) = { E L e i ω P t + c . c T / 2 < t < T / 2 , 0 otherwise .
μ + ( ω ) = ( 2 m ω 0 ) 1 / 2 α 1 ( a E L + ( ω ω 0 ) + a E L + ( ω + ω 0 ) ) .
E + ( r , ω ) = i d κ 2 π w 1 e 1 ( κ , ω ) e i ν 1 + . r ,
ν i ± κ ± w i z ^ ,
κ = κ x x ^ + κ y y ^ ,
w i = ( n i ω ˜ ) 2 κ 2 ,
e 1 ( κ , ω ) = ω ˜ 2 4 π ϵ 0 ( s ^ γ s 1 ( κ , ω ) + p ^ 1 + γ p 1 ( κ , ω ) ) · μ + ( ω ) ,
γ s 1 ( κ , ω ) = ( e i w 1 d + e i w 1 d R 1 N s ) s ^ , γ p 1 ( κ , ω ) = ( e i w 1 d p ^ 1 + + e i w 1 d R 1 N p p ^ 1 ) ,
s ^ = κ ^ × z ^ , p ^ i ± = κ z ^ w i κ ^ ν i ,
E + ( r , ω ) ω ˜ 2 4 π ϵ 0 ( s ^ ¯ γ s 1 ( κ ¯ , ω ) + p ^ ¯ 1 + γ p 1 ( κ ¯ , ω ) ) · μ + ( ω ) e i ω ˜ n 1 r r ,
H + ( r , ω ) n 1 c ω ˜ 2 4 π ( s ^ ¯ γ p 1 ( κ ¯ , ω ) p ^ ¯ 1 + γ s 1 ( κ ¯ , ω ) ) · μ + ( ω ) e i ω ˜ n 1 r r ,
S S ( r ) = ω ˜ S 4 n 1 ( ω S ) c 32 π 2 ϵ 0 ( α 1 S ) 2 r ^ r 2 Γ clad ( r ^ , ω S ) : ( E L * E L ) ,
Γ clad ( r ^ , ω S ) = γ s 1 * ( κ ¯ , ω S ) γ s 1 ( κ ¯ , ω S ) + γ p 1 * ( κ ¯ , ω S ) γ p 1 ( κ ¯ , ω S ) .
E inc ( r , t ) = E e i κ P · R e i w N P z e i ω P t + c . c ,
E L = e L ( κ P , ω P ) E e i κ P · R 0 ,
e L ( κ P , ω P ) = ( T N 1 s ( κ P , ω P ) F s s ^ P + T N 1 p ( κ P , ω P ) F p p ^ 1 + P ) e i w 1 P d ,
S S ( r ) = ω ˜ S 4 n 1 ( ω S ) c | E | 2 32 π 2 ϵ 0 ( α 1 S ) 2 r ^ r 2 Γ clad ( r ^ , ω S ) : ( e L * e L ) .
σ S ( r ^ ) = S S ( r ) · r ^ r 2 S P ,
S P = 2 n N ( ω P ) c μ 0 ( E * · E ) .
σ S ( r ^ ) = ω ˜ S 4 n 1 ( ω S ) 4 n N ( ω P ) ( α 1 S 4 π ϵ 0 ) 2 Γ clad ( r ^ , ω S ) : ( e L * e L ) .
σ ¯ S ( r ^ ) = σ S ( r ^ ) σ S o = 3 n 1 ( ω P ) 8 π n N ( ω P ) Γ clad ( r ^ , ω S ) : ( e L * e L ) .
σ ¯ S ( r ^ ) = 3 8 π n 1 ( ω P ) n N ( ω P ) ( | γ s 1 ( κ ¯ , ω S ) . e L | 2 + | γ p 1 ( κ ¯ , ω S ) . e L | 2 ) .
s ^ ¯ = s ^ P cos ϕ + κ ^ P sin ϕ , κ ^ ¯ = s ^ P sin ϕ + κ ^ P cos ϕ .
σ ¯ clad ( θ ) = 0 2 π σ ¯ S ( r ^ ) d ϕ ,
σ ¯ clad ( θ ) = 3 8 n 1 P n N P ( | ( 1 + e 2 i w 1 d R 1 N s ) T N 1 s P F s e i ( w 1 P w 1 ) d | 2 + | ( 1 + e 2 i w 1 d R 1 N s ) w 1 P ω ˜ P n 1 P T N 1 p P F p e i ( w 1 P w 1 ) d | 2 + 2 | κ ω ˜ n 1 ( 1 + e 2 i w 1 d R 1 N p ) κ P ω ˜ P n 1 P T N 1 p P F p e i ( w 1 P w 1 ) d | 2 + | w 1 ω ˜ n 1 ( 1 e 2 i w 1 d R 1 N p ) w 1 P ω ˜ P n 1 P T N 1 p P F p e i ( w 1 P w 1 ) d | 2 + | w 1 ω ˜ n 1 ( 1 e 2 i w 1 d R 1 N p ) T N 1 s P F s e i ( w 1 P w 1 ) d | 2 ) .
σ ¯ clad tot = 0 π / 2 σ ¯ clad ( θ ) sin ( θ ) d θ .
E + ( r , ω ) ω ˜ 2 4 π ϵ 0 ( s ^ ¯ γ s N ( κ ¯ , ω ) + p ^ ¯ N γ p N ( κ ¯ , ω ) ) · μ + ( ω ) e i ω ˜ n N r r ,
H + ( r , ω ) n N c ω ˜ 2 4 π ( s ^ ¯ γ p N ( κ ¯ , ω ) p ^ ¯ N γ s N ( κ ¯ , ω ) ) · μ + ( ω ) e i ω ˜ n N r r ,
γ s N ( κ ¯ , ω ) = ( w N T 1 N s e i w 1 d w 1 ) s ^ , γ p N ( κ ¯ , ω ) = ( w N T 1 N p e i w 1 d w 1 ) p ^ 1 .
S S ( r ) = ω ˜ S 4 n N ( ω S ) c 32 π 2 ϵ 0 ( α 1 S ) 2 r ^ r 2 Γ sub ( r ^ , ω S ) : ( e L * e L ) ,
Γ sub ( r ^ , ω S ) = γ s N * ( κ ¯ , ω S ) γ s N ( κ ¯ , ω S ) + γ p N * ( κ ¯ , ω S ) γ p N ( κ ¯ , ω S ) .
σ ¯ S ( r ^ ) = 3 n 1 ( ω P ) n N ( ω S ) 8 π n 1 ( ω S ) n N ( ω P ) Γ sub ( r ^ , ω S ) : ( e L * e L ) .
σ ¯ sub ( θ ) = 3 8 n 1 P n N n 1 n N P ( | w N T 1 N s w 1 T N 1 s P F s e i ( w 1 P + w 1 ) d | 2 + | w N T 1 N s w 1 w 1 P ω ˜ P n 1 P T N 1 p P F p e i ( w 1 P + w 1 ) d | 2 + 2 | w N κ T 1 N p w 1 ω ˜ n 1 κ P ω ˜ P n 1 P T N 1 p P F p e i ( w 1 P + w 1 ) d | 2 + | w N T 1 N p ω ˜ n 1 w 1 P ω ˜ P n 1 P T N 1 p P F p e i ( w 1 P + w 1 ) d | 2 + | w N T 1 N p ω ˜ n 1 T N 1 s P F s e i ( w 1 P + w ) d | 2 ) ,
σ ¯ sub tot = π / 2 π σ ¯ sub ( θ ) sin ( θ ) d θ ,
T 1 N s τ 1 N S κ κ res S ,
σ ¯ sub ( θ ) 3 8 n 1 P n N n 1 n N P | w N w 1 T N 1 s P e i ( w 1 P + w 1 ) d | 2 × | τ 1 N S | 2 ( κ κ R S ) 2 + ( κ I S ) 2 ,
σ ¯ sub pole | tan θ R | 2 n N λ κ I S σ ¯ max pole ,
{ x = z sin θ P + x cos θ P , y = y , z = z cos θ P + x sin θ P ,
E inc ( x , y , z ) = E e i ν N P z
E inc ( x , y , z ) = E e i ν N P z e ( x 2 + y 2 ) 2 Δ 2 ,
| E inc ( x , y , z ) | 2 d x d y = | E | 2 A ,
E inc ( x , y , 0 ) = e i ν N P x sin θ P E f inc ( x , y )
f inc ( x , y ) = e x 2 cos 2 θ P 2 Δ 2 e y 2 2 Δ 2 .
T N 1 P τ N 1 P κ κ res P ,
E clad ( x , y , D ) = e i κ R P x E ¯ clad ( x , y , D ) ,
x E ¯ clad ( x , y , D ) + κ I P E ¯ clad ( x , y , D ) = i τ N 1 P E f inc ( x , y )
E ¯ clad ( x , y , D ) = τ N 1 P i κ I P E f ( x , y ) ,
f ( x , y ) = κ I P x e κ I P ( x x ) f inc ( x , y ) d x .
f ( x , y ) f inc ( x , y ) ( κ I P ) 1 f inc ( x , y ) x + f inc ( x ( κ I P ) 1 , y ) .
f ( x , y ) 2 π κ I P Δ cos θ P e y 2 2 Δ 2 e κ I P x .
S S ( r ) = ω ˜ S 4 n N ( ω S ) c 32 π 2 ϵ 0 ( α 1 S ) 2 r ^ r 2 Γ sub ( r ^ , ω S ) : ( e L * ( x 0 , y 0 ) e L ( x 0 , y 0 ) ) ,
e L ( x 0 , y 0 ) = f ( x 0 , y 0 ) e L .
S S all ( r ) = ρ ω ˜ S 4 n N ( ω S ) c 32 π 2 ϵ 0 ( α 1 S ) 2 r ^ r 2 Γ sub ( r ^ , ω S ) : ( e L * e L ) × | f ( x 0 , y 0 ) | 2 d x 0 d y 0 .
P S = S S all ( r ) · r ^ r 2 d Ω ,
1 σ S o P S P pump = ρ σ ¯ sub tot | f ( x , y ) | 2 d x d y A ,
1 σ S o P S P pump ρ σ ¯ sub tot cos θ P ,
1 σ S o P S P pump π ρ σ ¯ sub tot cos 2 θ P κ P I Δ ,

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