Abstract

We study Raman scattering in active media placed in proximity to different types of metal nanostructures, at wavelengths that display either Fabry–Perot or plasmonic resonances, or a combination of both. We use a semiclassical approach to derive equations of motion for Stokes and anti-Stokes fields that arise from quantum fluctuations. Our calculations suggest that local field enhancement yields Stokes and anti-Stokes conversion efficiencies between 5 and 7 orders of magnitudes larger compared to cases without the metal nanostructure. We also show that to first order in the linear susceptibility the local field correction induces a dynamic, intensity-dependent frequency detuning that at high intensities tends to quench Raman gain.

© 2012 Optical Society of America

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  5. I. Walmsley, “Quantum noise limit to the beam-pointing stability in stimulated Raman generation,” J. Opt. Soc. Am. B 8, 805–812 (1991).
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  6. M. Scalora, J. W. Haus, and C. M. Bowden, “Macroscopic quantum fluctuations in high-gain optical amplifiers,” Phys. Rev. Lett. 69, 3310–3313 (1992).
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  7. J. R. Ackerhalt and P. W. Milonni, “Solitons in four-wave mixing,” Phys. Rev. A 33, 3185–3198 (1986).
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  36. M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express 19, 2064–2078 (2011).
    [CrossRef]
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    [CrossRef]
  38. I. Baltog, N. Primeau, R. Reinisch, and J. L. Coutaz, “Observation of stimulated surface-enhanced Raman scattering through grating excitation of surface plasmons,” J. Opt. Soc. Am. B 13,656–660 (1996).
    [CrossRef]
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  42. P. Johansson, H. Xu, and M. Kall, “Surface-enhanced Raman scattering and fluorescence near metal nanoparticles,” Phys. Rev. B 72, 035427 (2005).
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  43. D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband non-focusing of light using kissing nanowires,” New J. Phys. 12, 093030 (2010).
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  44. A. S. Manka, M. Scalora, J. P. Dowling, and C. M. Bowden, “Pulse propagation in a Raman pumped, four-level medium that exhibits inversionless gain,” Opt. Commun. 115, 283–290 (1995).
    [CrossRef]
  45. M. Lax, “Quantum noise. IV: Quantum theory of noise sources,” Phys. Rev. 145, 110–129 (1966).
    [CrossRef]
  46. J. C. Englund and C. M. Bowden, “Spontaneous generation of Raman solitons from quantum noise,” Phys. Rev. Lett. 57, 2661–2663 (1986).
    [CrossRef]
  47. J. C. Englund and C. M. Bowden, “Spontaneous generation of phase waves and solitons in stimulated Raman scattering: quantum-mechanical model of stimulated Raman scattering,” Phys. Rev. A 42, 2870–2889 (1990).
    [CrossRef]
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    [CrossRef]
  52. M. Scalora, J. P. Dowling, M. Tocci, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Dipole emission rates in one-dimensional photonic band gap structures,” Appl. Phys. B 60, S57–S61 (1995).
  53. M. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
    [CrossRef]
  54. R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Harter, “Four-wave parametric interactions in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
    [CrossRef]
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    [CrossRef]
  56. A. G. Brolo, E. Arctander, R. Gordon, B. Leathem, and K. L. Kavanagh, “Nanohole-enhanced Raman scattering,” Nano Lett. 4, 2015–2018 (2004).
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2011 (4)

D. de Ceglia, M. A. Vincenti, M. Scalora, N. Akozbek, and M. J. Bloemer, “Plasmonic band edge effects on the transmission properties of metal gratings,” AIP Advances 1, 032151 (2011).
[CrossRef]

R. R. Frontiera, A.-I. Henry, N. L. Gruenke, and R. P. Van Duyne, “Surface-enhanced femtosecond stimulated Raman spectroscopy,” J. Phys. Chem. Lett. 2, 1199–1203 (2011).
[CrossRef]

R. R. Frontiera and R. A. Mathies, “Femtosecond stimulated Raman spectroscopy,” Laser Photon. Rev. 5, 102–113 (2011).
[CrossRef]

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express 19, 2064–2078 (2011).
[CrossRef]

2010 (2)

N. Djaker, R. Hostein, E. Devaux, T. W. Ebbesen, H. Rigneault, and J. Wenger, “Surface enhanced Raman scattering on a single nanometric aperture,” J. Phys. Chem. C 114, 16250–16256 (2010).
[CrossRef]

D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband non-focusing of light using kissing nanowires,” New J. Phys. 12, 093030 (2010).
[CrossRef]

2009 (7)

M. I. Mishchenko, “Electromagnetic scattering by nonspherical particles: a tutorial review,” J. Quant. Spectrosc. Radiat. Trans. 110, 808–832 (2009).
[CrossRef]

X. Wei, L. Zhang, J. Zhang, X. Hu, and L. Sun, “A comparison of surface enhanced Raman scattering property between silver electrodes and periodical silver nanowire arrays,” Appl. Surf. Science 255, 6612–6614 (2009).
[CrossRef]

Z. Zhuang, X. Shang, X. Wang, W. Ruan, and B. Zhao, “Density functional theory study on surface enhanced Raman scattering of 4,4’-azopyridine on silver,” Spectrochim. Acta, Part A 72, 954–958 (2009).
[CrossRef]

S. Franzen, “Intrinsic limitations on the |E|4 dependence of the enhancement factor for surface enhanced Raman scattering,” J. Phys. Chem. C 113, 5912–5919 (2009).
[CrossRef]

W. H. P. Pernice, M. Li, and H. X. Tang, “Theoretical investigation of the transverse optical force between a silicon nanowire waveguide and a substrate,” Opt. Express 17, 1806–1816 (2009).
[CrossRef]

A. Gopinath, S. V. Boriskina, B. M. Reinhard, and L. D. Negro, “Deterministic aperiodic arrays of metal nanoparticles for surface-enhanced Raman scattering (SERS),” Opt. Express 17, 3741–3753 (2009).
[CrossRef]

X. Juntao and M. Premaratne, “Analysis of the optical force dependency on beam polarization: dielectric/metallic spherical shell in a Gaussian beam,” J. Opt. Soc. Am. B 26, 973–980(2009).
[CrossRef]

2008 (6)

A. Kocabas, G. Ertas, S. Senlik, and A. Aydinli, “Plasmonic band gap structures for surface enhanced Raman scattering,” Opt. Express 16, 12469–12473 (2008).
[CrossRef]

R. Zhilong, S. Vo, and J. S. Harris, “A review of progress on nano-aperture VCSEL,” Chin. Opt. Lett. 6, 748–754 (2008).
[CrossRef]

L. K. Ausman and G. C. Schatz, “Whispering-gallery mode resonators: Surface enhanced Raman scattering without plasmons,” J. Phys. Chem. 129, 054704 (2008).
[CrossRef]

G. P. Wiederrecht, G. A. Wurtz, and A. Bouhelier, “Ultrafast hybrid plasmonics,” Chem. Phys. Lett. 461, 171–179 (2008).
[CrossRef]

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[CrossRef]

Y. J. Liu, Z. Y. Zhang, Q. Zhao, and Y. P. Zhao, “Revisiting the separation dependent surface enhance Raman scattering,” Appl. Phys. Lett. 93, 173106 (2008).
[CrossRef]

2007 (4)

P. Kukura, D. W. McCamant, and R. A. Mathies, “Femtosecond stimulated Raman spectroscopy,” Annu. Rev. Phys. Chem. 58, 461–488 (2007).
[CrossRef]

C. H. Gan, G. Gbur, and T. D. Visser, “A new role for surface plasmons,” Opt. Photon. News 18, 36 (2007).
[CrossRef]

A. S. Zelenina, R. Quidant, and M. N. Vesperinas, “Enhanced optical forces between coupled resonant metal nanoparticles,” Opt. Lett. 32, 1156–1158 (2007).
[CrossRef]

M. M. Dvoynenko and J. Wang, “Finding electromagnetic and chemical enhancement factors of surface-enhanced Raman scattering,” Opt. Lett. 32, 3552–3554 (2007).
[CrossRef]

2006 (2)

J. J. Baumberg, N. Perney, M. E. Zoorob, M. Charlton, S. Mahnkopf, and C. M. Netti, “Tuning localized plasmons in nanostructured substrates for surface-enhanced Raman scattering,” Opt. Express 14, 847–857 (2006).
[CrossRef]

M. Baia, L. Baia, S. Astilean, and J. Popp, “Surface enhanced Raman scattering efficiency of truncated tetrahedral Ag nanoparticle arrays mediated by electromagnetic couplings,” Appl. Phys. Lett. 88, 143121 (2006).
[CrossRef]

2005 (1)

P. Johansson, H. Xu, and M. Kall, “Surface-enhanced Raman scattering and fluorescence near metal nanoparticles,” Phys. Rev. B 72, 035427 (2005).
[CrossRef]

2004 (2)

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

D. F. P. Pile and D. K. Gramotnev, “Channel plasmon-polariton in a triangular groove on a metal surface,” Opt. Lett. 29, 1069–1071 (2004).
[CrossRef]

2002 (1)

S. Corni and J. Tomasi, “Surface enhanced Raman scattering from a single molecule absorbed on a metal particle aggregate: a theoretical study,” J. Chem. Phys. 116, 1156–1164, (2002).
[CrossRef]

2001 (2)

L. Gunnarsson, E. Bjerneld, H. Xu, S. Petronis, B. Kasemo, and M. Kall, “Interparticle coupling effects in nanofabricated substrates for surface enhance Raman scattering,” Appl. Phys. Lett. 78, 802–804 (2001).
[CrossRef]

C. Cappelli, S. Corni, and J. Tomasi, “Electronic and vibrational dynamic solvent effects on Raman spectra,” J. Chem. Phys. 115, 5531–5535 (2001).
[CrossRef]

1999 (1)

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

1998 (2)

T. Vo-Dinh, “Surface-enhanced Raman spectroscopy using metallic nanostructures,” TrAC Trends Anal. Chem. 17, 557–582 (1998).
[CrossRef]

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

1996 (2)

M. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
[CrossRef]

I. Baltog, N. Primeau, R. Reinisch, and J. L. Coutaz, “Observation of stimulated surface-enhanced Raman scattering through grating excitation of surface plasmons,” J. Opt. Soc. Am. B 13,656–660 (1996).
[CrossRef]

1995 (4)

M. Scalora, J. P. Dowling, M. Tocci, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Dipole emission rates in one-dimensional photonic band gap structures,” Appl. Phys. B 60, S57–S61 (1995).

A. S. Manka, M. Scalora, J. P. Dowling, and C. M. Bowden, “Pulse propagation in a Raman pumped, four-level medium that exhibits inversionless gain,” Opt. Commun. 115, 283–290 (1995).
[CrossRef]

T. Vo-Dinh, “SERS chemical sensors and biosensors: new tools for environmental and biological analysis,” Sens. Actuators B 29, 183–189 (1995).
[CrossRef]

H. Nakai and H. Nakatsuji, “Electronic mechanism of the surface enhanced Raman scattering,” J. Chem. Phys. 103, 2286–2294 (1995).
[CrossRef]

1992 (2)

M. Scalora, J. W. Haus, and C. M. Bowden, “Macroscopic quantum fluctuations in high-gain optical amplifiers,” Phys. Rev. Lett. 69, 3310–3313 (1992).
[CrossRef]

J. C. Englund and C. M. Bowden, “Spontaneous generation of phase waves and solitons in stimulated Raman scattering II: quantum statistics of Raman soliton generation,” Phys. Rev. A 46, 578–591 (1992).
[CrossRef]

1991 (2)

S. J. Kuo, D. T. Smithey, and M. G. Raymer, “Beam pointing fluctuations in gain-guided amplifiers,” Phys. Rev. Lett. 66, 2605–2608 (1991).
[CrossRef]

I. Walmsley, “Quantum noise limit to the beam-pointing stability in stimulated Raman generation,” J. Opt. Soc. Am. B 8, 805–812 (1991).
[CrossRef]

1990 (1)

J. C. Englund and C. M. Bowden, “Spontaneous generation of phase waves and solitons in stimulated Raman scattering: quantum-mechanical model of stimulated Raman scattering,” Phys. Rev. A 42, 2870–2889 (1990).
[CrossRef]

1986 (2)

J. C. Englund and C. M. Bowden, “Spontaneous generation of Raman solitons from quantum noise,” Phys. Rev. Lett. 57, 2661–2663 (1986).
[CrossRef]

J. R. Ackerhalt and P. W. Milonni, “Solitons in four-wave mixing,” Phys. Rev. A 33, 3185–3198 (1986).
[CrossRef]

1982 (2)

M. Nevière and R. Reinisch, “Electromagnetic study of the surface-plasmon-resonance contribution to surface-enhanced Raman scattering,” Phys. Rev. B 46, 5403–5408 (1982).
[CrossRef]

M. Raymer, K. RzaCewski, and J. Mostowski, “Pulse-energy statistics in stimulated Raman scattering,” Opt. Lett. 7, 71–73 (1982).
[CrossRef]

1981 (1)

R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Harter, “Four-wave parametric interactions in a strongly driven two-level system,” Phys. Rev. A 24, 411–423 (1981).
[CrossRef]

1974 (1)

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

1966 (1)

M. Lax, “Quantum noise. IV: Quantum theory of noise sources,” Phys. Rev. 145, 110–129 (1966).
[CrossRef]

1964 (1)

N. Bloembergen and Y. R. Shen, “Coupling between vibrations and light waves in Raman laser media,” Phys. Rev. Lett. 12, 504–507 (1964).
[CrossRef]

1962 (1)

G. Eckardt, R. W. Hellwarth, F. J. McClung, S. E. Scharz, D. Weiner, and E. J. Woodbury, “Stimulated Raman scattering from organic liquids,” Phys. Rev. Lett. 9, 455–457 (1962).
[CrossRef]

Ackerhalt, J. R.

J. R. Ackerhalt and P. W. Milonni, “Solitons in four-wave mixing,” Phys. Rev. A 33, 3185–3198 (1986).
[CrossRef]

Akozbek, N.

D. de Ceglia, M. A. Vincenti, M. Scalora, N. Akozbek, and M. J. Bloemer, “Plasmonic band edge effects on the transmission properties of metal gratings,” AIP Advances 1, 032151 (2011).
[CrossRef]

Anker, J. N.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. Van Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[CrossRef]

Arctander, E.

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

Astilean, S.

M. Baia, L. Baia, S. Astilean, and J. Popp, “Surface enhanced Raman scattering efficiency of truncated tetrahedral Ag nanoparticle arrays mediated by electromagnetic couplings,” Appl. Phys. Lett. 88, 143121 (2006).
[CrossRef]

Aubry, A.

D. Y. Lei, A. Aubry, S. A. Maier, and J. B. Pendry, “Broadband non-focusing of light using kissing nanowires,” New J. Phys. 12, 093030 (2010).
[CrossRef]

Ausman, L. K.

L. K. Ausman and G. C. Schatz, “Whispering-gallery mode resonators: Surface enhanced Raman scattering without plasmons,” J. Phys. Chem. 129, 054704 (2008).
[CrossRef]

Aydinli, A.

Baia, L.

M. Baia, L. Baia, S. Astilean, and J. Popp, “Surface enhanced Raman scattering efficiency of truncated tetrahedral Ag nanoparticle arrays mediated by electromagnetic couplings,” Appl. Phys. Lett. 88, 143121 (2006).
[CrossRef]

Baia, M.

M. Baia, L. Baia, S. Astilean, and J. Popp, “Surface enhanced Raman scattering efficiency of truncated tetrahedral Ag nanoparticle arrays mediated by electromagnetic couplings,” Appl. Phys. Lett. 88, 143121 (2006).
[CrossRef]

Baltog, I.

Baumberg, J. J.

Bertolotti, M.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguanno, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, “Dispersive properties of finite, one-dimensional photonic band gap structures: applications to nonlinear quadratic interactions,” Phys. Rev. E 60, 4891–4898 (1999).
[CrossRef]

Bhandari, D.

D. Bhandari, “Surface-enhanced Raman scattering: Substrate development and applications in analytical detection,” Ph.D. dissertation (University of Tennessee, 2011).

Bjerneld, E.

L. Gunnarsson, E. Bjerneld, H. Xu, S. Petronis, B. Kasemo, and M. Kall, “Interparticle coupling effects in nanofabricated substrates for surface enhance Raman scattering,” Appl. Phys. Lett. 78, 802–804 (2001).
[CrossRef]

Bloembergen, N.

N. Bloembergen and Y. R. Shen, “Coupling between vibrations and light waves in Raman laser media,” Phys. Rev. Lett. 12, 504–507 (1964).
[CrossRef]

Bloemer, M. J.

D. de Ceglia, M. A. Vincenti, M. Scalora, N. Akozbek, and M. J. Bloemer, “Plasmonic band edge effects on the transmission properties of metal gratings,” AIP Advances 1, 032151 (2011).
[CrossRef]

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M. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
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M. Scalora, J. P. Dowling, M. Tocci, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Dipole emission rates in one-dimensional photonic band gap structures,” Appl. Phys. B 60, S57–S61 (1995).

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M. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
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M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express 19, 2064–2078 (2011).
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[CrossRef]

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N. Djaker, R. Hostein, E. Devaux, T. W. Ebbesen, H. Rigneault, and J. Wenger, “Surface enhanced Raman scattering on a single nanometric aperture,” J. Phys. Chem. C 114, 16250–16256 (2010).
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M. Scalora, J. P. Dowling, M. Tocci, M. J. Bloemer, C. M. Bowden, and J. W. Haus, “Dipole emission rates in one-dimensional photonic band gap structures,” Appl. Phys. B 60, S57–S61 (1995).

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N. Djaker, R. Hostein, E. Devaux, T. W. Ebbesen, H. Rigneault, and J. Wenger, “Surface enhanced Raman scattering on a single nanometric aperture,” J. Phys. Chem. C 114, 16250–16256 (2010).
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A. S. Manka, M. Scalora, J. P. Dowling, and C. M. Bowden, “Pulse propagation in a Raman pumped, four-level medium that exhibits inversionless gain,” Opt. Commun. 115, 283–290 (1995).
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R. R. Frontiera and R. A. Mathies, “Femtosecond stimulated Raman spectroscopy,” Laser Photon. Rev. 5, 102–113 (2011).
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L. Gunnarsson, E. Bjerneld, H. Xu, S. Petronis, B. Kasemo, and M. Kall, “Interparticle coupling effects in nanofabricated substrates for surface enhance Raman scattering,” Appl. Phys. Lett. 78, 802–804 (2001).
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N. Djaker, R. Hostein, E. Devaux, T. W. Ebbesen, H. Rigneault, and J. Wenger, “Surface enhanced Raman scattering on a single nanometric aperture,” J. Phys. Chem. C 114, 16250–16256 (2010).
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D. de Ceglia, M. A. Vincenti, M. Scalora, N. Akozbek, and M. J. Bloemer, “Plasmonic band edge effects on the transmission properties of metal gratings,” AIP Advances 1, 032151 (2011).
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X. Wei, L. Zhang, J. Zhang, X. Hu, and L. Sun, “A comparison of surface enhanced Raman scattering property between silver electrodes and periodical silver nanowire arrays,” Appl. Surf. Science 255, 6612–6614 (2009).
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M. Tocci, M. Scalora, M. J. Bloemer, J. P. Dowling, and C. M. Bowden, “Measurement of spontaneous-emission enhancement near the one-dimensional photonic band edge of semiconductor heterostructures,” Phys. Rev. A 53, 2799–2803 (1996).
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R. R. Frontiera, A.-I. Henry, N. L. Gruenke, and R. P. Van Duyne, “Surface-enhanced femtosecond stimulated Raman spectroscopy,” J. Phys. Chem. Lett. 2, 1199–1203 (2011).
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Supplementary Material (2)

» Media 1: MOV (1654 KB)     
» Media 2: MOV (1251 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

(a) Metal grating that contains aperture arrays surrounded by a Raman active medium (light blue area) that also fills the slits. (b) Metal grating that contains grooves only, surrounded by Raman active medium. (c) Unit cell of a 1D periodic array composed of metal nanowires placed near a metal substrate.

Fig. 2.
Fig. 2.

Energy level diagram for the generation of Stokes (redshifted) and anti-Stokes (blueshifted) photons of frequencies ω S and ω AS , respectively, when photons of frequency ω P excite the atom. The dashed lines are virtual states located far below levels | 1 and | 3 . A pump photon excites the system to level | 2 producing a Stokes photon, while a second pump photon brings it back down to the ground state via the virtual state below state | 3 generating anti-Stokes light.

Fig. 3.
Fig. 3.

Transmission and absorption versus wavelength for the silver grating in Fig. 1(a), for a = 32 nm , w = 300 nm and p = 566 nm . A plasmonic bandgap appears when λ s p p ; i.e., λ s p = 578 nm . A broad, FP resonance is centered at 1115 nm . The solid green, dashed red, and dotted blue arrows indicate pump, Stokes, and anti-Stokes tuning, respectively, for three configurations. The model allows the fields to be tuned across a wide wavelength range that includes the FP resonance and the plasmonic bandgap. The numbered groups (AS, P , S ) stand for anti-Stokes, pump, and Stokes, respectively.

Fig. 4.
Fig. 4.

Total electric field intensities for pumping configurations P 1 (a) and P 2 (b) shown in Fig. 3. The local intensity is largest when the λ p 1115 nm , at the FP resonance (a). The resonance to the right of the plasmonic bandgap, 625 nm, (b) is reminiscent of a FP mode. The resonance to the left of the gap, 575 nm, (c) is a hybrid mode that simultaneously displays a surface wave with the characteristic shape of plasmonic excitation and a cavity mode [35]. See Media 1 to follow the dynamical development of the fields.

Fig. 5.
Fig. 5.

Incident pump spectrum for a 240 fs pulse tuned to 1115 nm, and typical transmitted (label T ) and reflected (label R ) generated Stokes (subscript S ), and anti-Stokes (subscript AS) spectra, centered at 1130 nm and 1100 nm, respectively. The amplification factor is determined with respect to a layer of active material 4 nm thick, in the absence of metal.

Fig. 6.
Fig. 6.

Transmitted ( T AS ) and reflected ( R AS ) anti-Stokes (AS) spectra obtained by tuning the AS field to (a) 575 nm; (b) 625 nm; and (c) 1105 nm. 120 fs pump pulses are tuned to 1115 nm. Inset, normalized Raman enhancement factors, according to Eq. (21) (dashed red curve), and Eq. (22) (solid blue curve). Comparing Figs. 6(a), 6(b), and 6(c) and the inset shows that Eq. (21) predicts fairly well the relative conversion efficiencies near the plasmonic bandgap. Both Eqs. (21) and (22) fail near the FP resonance, while neither gives insight regarding the proportions of forward and backward generation.

Fig. 7.
Fig. 7.

Top: Reflection and absorption spectra versus incident wavelength for the grating in Fig. 1(b): a = 300 nm , w = 45 nm , t = 200 nm , and p = 1030 nm . The solid green, dashed red and dotted blue arrows indicate pump, Stokes, and anti-Stokes tuning conditions, respectively, for three possible configurations. Reflection (absorption) minima (maxima) above 350 nm correspond to surface wave excitations. Bottom: Electric field intensity distributions at (a) 1066 nm; and (b) 545 nm, normalized with respect to incident field intensity. In Media 2 we show the fields for tuning conditions ( AS 3 , P 3 , S 3 ), for a 350 fs pump pulse. The actual position of the S 3 wave is at a wavelength longer than what is displayed in the figure.

Fig. 8.
Fig. 8.

Transmitted ( T AS ) and reflected ( R AS ) anti-Stokes (AS) spectra obtained by tuning 350 fs pump pulses at 1066 nm, and the AS field in turn to 545 nm (a), and 1065 nm (b). We note that in both cases, the transmitted signal is far smaller compared to the reflected component, but is still nearly 2 orders of magnitude more intense compared to the case when the metal grating is absent, as the generated signal tunnels through the metal. Inset, Raman enhancement factors G (dashed, red curve) and G s (solid, blue curve).

Fig. 9.
Fig. 9.

(a) Linear reflection and absorption spectra for the grating depicted in Fig. 1(c). The bandwidth of the resonance near 600 nm is 60 nm wide, and can easily accommodate an ultrashort pulse. (b) Snapshot of the total local electric and (c) magnetic field intensities excited by an incident 50 fs pulse tuned at absorption peak at 600 nm . The local electric field intensity is enhanced by 4 orders of magnitude, and manifests singular behavior when the nanowire touches the substrate.

Fig. 10.
Fig. 10.

Transmitted ( T , blue curves, right axis) and reflected ( R , red curves, left axis) Stokes ( S ) and anti-Stokes (AS) spectra obtained by tuning 50 fs pump pulses to 600 nm, and S and AS fields tuned to 590 nm and 610 nm, as indicated.

Equations (41)

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P ¨ f + γ ˜ f P ˙ f = n 0 , f e 2 m f * ( λ 0 c ) 2 E + 5 3 E F m f * c 2 ( P f ) ,
P ¨ 1 + γ ˜ 01 P ˙ 1 + ω ˜ 0 , 1 2 P 1 = n 0 , 1 e 2 m b * E ,
P ¨ 2 + γ ˜ 02 P ˙ 1 + ω ˜ 0 , 2 2 P 2 = n 0 , 2 e 2 m b * E ,
H = ω 01 | 1 1 | + ω 03 | 3 3 | + Δ | 2 2 | μ E ( | 1 0 | + | 0 1 | + | 1 2 | + | 2 1 | + | 3 0 | + | 0 3 | + | 3 2 | + | 2 3 | ) ,
E = E y j + E z k = ( l E l , y ( r , t ) e i ω l t + E l , y * ( r , t ) e i ω l t ) j + ( l E l , z ( r , t ) e i ω l t + E l , z * ( r , t ) e i ω l t ) k H = H x i = ( l H l , x ( r , t ) e i ω l t + H l , x * ( r , t ) e i ω l t ) i ,
| ψ ( t ) = C 0 ( t ) | 0 + C 1 ( t ) | 1 + C 2 ( t ) | 2 + C 3 ( t ) | 3 ,
i ρ ˙ i j = 1 [ ρ i j , H ] ,
ρ i j = ψ ( t ) | i j | ψ ( t ) .
i ρ ˙ i j = 1 ( ψ ( t ) | i j | H | ψ ( t ) ψ ( t ) | H | i j | ψ ( t ) ) .
μ = μ 0 μ ^ ( | 1 0 | + | 0 1 | + | 1 2 | + | 2 1 | + | 3 0 | + | 0 3 | + | 3 2 | + | 2 3 | ) ,
ψ | μ | ψ = μ 0 μ ^ ( ρ 10 + ρ 01 + ρ 12 + ρ 21 + ρ 30 + ρ 03 + ρ 32 + ρ 23 ) ,
P = N μ 0 μ ^ ( ρ 10 + ρ 01 + ρ 12 + ρ 21 + ρ 30 + ρ 03 + ρ 32 + ρ 23 ) ,
i ρ ˙ 11 k = μ k E k ( ρ 10 k ρ 01 k + ρ 12 k ρ 21 k ) ,
i ρ ˙ 22 k = μ k E k ( ρ 12 k + ρ 32 k ρ 21 k ρ 23 k ) ,
i ρ ˙ 33 k = μ k E k ( ρ 03 k + ρ 23 k ρ 30 k ρ 32 k ) ,
i ρ ˙ 00 k = μ k E k ( ρ 10 k ρ 01 k + ρ 30 k ρ 03 k ) ,
i ρ ˙ 01 k = ω 01 ρ 01 k + μ k E k ( ρ 11 k ρ 00 k + ρ 31 k ρ 02 k ) ,
i ρ ˙ 12 k = ( Δ ω 01 ) ρ 12 k + μ k E k ( ρ 02 k + ρ 22 k ρ 11 k ρ 13 k ) ,
i ρ ˙ 02 k = Δ ρ 02 k + μ k E k ( ρ 12 k ρ 01 k + ρ 23 k ρ 03 k ) ,
i ρ ˙ 03 k = ω 03 ρ 03 k + μ k E k ( ρ 13 k + ρ 33 k ρ 00 k ρ 02 k ) ,
i ρ ˙ 32 k = ( ω 03 Δ ) ρ 32 k + μ k E k ( ρ 31 k + ρ 33 k ρ 02 k ρ 22 k ) .
Q ˙ k = i ( Δ δ ) Q k Ω k W k e i δ t ,
W ˙ k = 2 Ω k [ Q k e i δ t + Q k * e i δ t ] .
Q ˙ k = i ( Δ δ ) Q k μ 0 2 2 2 ω 01 ( l E l , k ( r , t ) E l + 1 , k * ( r , t ) ) W k ,
W ˙ k = μ 0 2 2 ω 01 ( Q k * l E l , k ( r , t ) E l + 1 , k * ( r , t ) + c.c. ) .
Q ˙ k = ( γ ˜ i Δ ˜ ) Q k + μ 0 2 λ 0 2 2 ω 01 c ( l E l , k ( r , τ ) E l + 1 , k * ( r , τ ) ) + F k ( r , τ ) .
P k = χ L E k + i χ L ( Q k E k e i δ t Q k * E k e i δ t ) ,
P ω p , k NL = i χ L [ Q k E S , k Q k * E AS , k ] P ω S , k NL = i χ L Q k * E P , k P ω AS , k NL = i χ L Q k E P , k ,
G = | E ω p local | 2 | E ω p inc | 2 | E ω S rad | 2 | E ω p inc | 2 ,
G s = | E ω p local | 4 / | E ω p inc | 4 .
E L = E + 4 π 3 P ,
Q ˙ = Γ Q + μ 0 2 2 ω 01 E L 2 e i δ τ ,
Q ˙ = ( Γ + i μ 0 2 2 ω 01 8 π χ L 3 | E Total | 2 ) Q + μ 0 2 2 ω 01 ( E P E S * + E AS E P * ) ( 1 + 8 π χ L 3 i 16 π 2 χ L 2 9 | Q | 2 ) ,
Q i 3 8 π χ L ( E P E S * + E AS E P * ) ( | E P | 2 + | E S | 2 + | E AS | 2 ) .
P ω S NL ( E S + E AS * ) P ω AS NL ( E S * + E AS ) .
H τ = × E , E τ = × H 4 π ( P L + P NL ) τ ,
H τ = × E , E τ = × H .
H ˜ x τ = i k y E ˜ z + i k z E ˜ y E ˜ y τ = i k z H ˜ x E ˜ z τ = i k y H ˜ x .
H ˜ x ( δ τ ) = H ˜ x ( 0 ) i k y δ τ 2 ( E ˜ z ( 0 ) + E ˜ z ( δ τ ) ) + i k z δ τ 2 ( E ˜ y ( 0 ) + E ˜ y ( δ τ ) ) E ˜ y ( δ τ ) = E ˜ y ( 0 ) + i k z δ τ 2 ( H ˜ x ( 0 ) + H ˜ x ( δ τ ) ) E ˜ z ( δ τ ) = E ˜ z ( 0 ) i k y δ τ 2 ( H ˜ x ( 0 ) + H ˜ x ( δ τ ) ) .
H ˜ x ( δ τ ) = H ˜ x ( 0 ) ( 1 ( k y 2 + k z 2 ) δ τ 2 4 ) ( 1 + ( k y 2 + k z 2 ) δ τ 2 4 ) + ( i k z E ˜ y ( 0 ) i k y E ˜ z ( 0 ) ) δ τ ( 1 + ( k y 2 + k z 2 ) δ τ 2 4 ) .
H x τ = i β ( H x + E z sin θ i + E y cos θ i ) E y τ = i β ( E y + H x cos θ i ) 4 π ( J y i β P y ) E z τ = i β ( E z + H x sin θ i ) 4 π ( J z i β P z ) ,

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