Abstract

We present a step-by-step analysis of four-wave mixing (FWM) in one-dimensional stacks of metallo-dielectric structures, pointing out various channels of plasmonic and Fabry–Perot enhancement. We start from the derivation of oblique incidence FWM at a single interface and then extend these expressions into a transfer-matrix-based formalism to quantitatively study films and multilayer geometries. Throughout our analysis, we consider typical examples, such as a single silver interface, a thin silver film, and Fabry–Perot multilayers. In this way, we offer an intuitive view of the surprisingly rich dynamics supported by even the simplest of nonlinear plasmonic systems.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. W. Boyd, Nonlinear Optics (Academic, 2008).
  2. M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6, 737–748 (2012).
    [CrossRef]
  3. N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
    [CrossRef]
  4. J. Rudnick and E. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
    [CrossRef]
  5. J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
    [CrossRef]
  6. G. A. Farias and A. A. Maradudin, “Second-harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30, 3002–3015 (1984).
    [CrossRef]
  7. H. B. Jiang, L. Li, W. C. Wang, J. B. Zheng, Z. M. Zhang, and Z. Chen, “Reflected second-harmonic generation at a silver surface,” Phys. Rev. B 44, 1220–1224 (1991).
    [CrossRef]
  8. A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
    [CrossRef]
  9. M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
    [CrossRef]
  10. Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
    [CrossRef]
  11. F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
    [CrossRef]
  12. M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
    [CrossRef]
  13. A. Benedetti, M. Centini, C. Sibilia, and M. Bertolotti, “Engineering the second harmonic generation pattern from coupled gold nanowires,” J. Opt. Soc. Am. B 27, 408–416 (2010).
    [CrossRef]
  14. C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Second-harmonic generation in metallic nanoparticles: clarification of the role of the surface,” Phys. Rev. B 86, 115451 (2012).
    [CrossRef]
  15. R. S. Bennink, Y. Yoon, R. W. Boyd, and J. E. Sipe, “Accessing the optical nonlinearity of metals with metal-dielectric photonic bandgap structures,” Opt. Lett. 24, 1416–1418 (1999).
    [CrossRef]
  16. N. N. Lepeahkin, A. Schweinaberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
    [CrossRef]
  17. N. A. Papadogiannics, P. A. Loukakos, and S. D. Moustaizis, “Observation of the inversion of second and third harmonic generation efficiencies on a gold surface in the femtosecond regime,” Opt. Commun. 166, 113–139 (1999).
    [CrossRef]
  18. H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, “Large third-order optical nonlinearity in Au:SiO2 composite films near the percolation threshold,” Appl. Phys. Lett. 70, 119291 (1997).
  19. F. Hache, D. Ricard, and C. Flytzanis, “Optical nonlinearities of small metal particles: surface-mediated resonance and quantum size effects,” Opt. Lett. 12, 1647–1655 (1986).
  20. M. Lippitz, M. A. van Dijk, and M. Orrit, “Third-harmonic generation from single gold nanoparticles,” Nano Lett. 5, 799–802 (2005).
    [CrossRef]
  21. M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett. 98, 026104 (2007).
    [CrossRef]
  22. F. Hache, D. Richard, C. Flytzanis, and U. Kreibig, “The optical Kerr effect in small metal particles and metal colloide: the case of gold,” Appl. Phys. A 47, 347–357 (1988).
    [CrossRef]
  23. N. N. Lepeshkin, W. Kim, V. P. Safonov, J. G. Zhu, R. L. Armstrong, C. W. White, R. A. Zuhr, and V. M. Shalaev, “Optical nonlinearities of metal-dielectric composites,” J. Nonlinear Opt. Phys. Mater. 8, 191–210 (1999).
    [CrossRef]
  24. J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing,” Phys. Rev. Lett. 103, 266802 (2009).
    [CrossRef]
  25. J. Renger, R. Quidant, and L. Novotny, “Enhanced nonlinear response from metal surfaces,” Opt. Express 19, 1777–1785 (2011).
    [CrossRef]
  26. J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface-enhanced nonlinear four wave mixing,” Phys. Rev. Lett. 104, 046803 (2010).
    [CrossRef]
  27. S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11, 34–38 (2011).
    [CrossRef]
  28. P. Genevet, J. Tetienne, E. Gatzogiannis, R. Blanchard, M. A. Kats, M. O. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. 10, 4880–4883 (2010).
    [CrossRef]
  29. E. Poutrina, C. Ciracì, D. J. Gauthier, and D. R. Smith, “Enhancing four-wave-mixing processes by nanowire arrays coupled to a gold film,” Opt. Express 20, 11005 (2012).
    [CrossRef]
  30. A. D. Boardman, Electromagnetic Surface Modes (Wiley, New York, 1982).
  31. E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for silver, gold, copper, lithium, sodium, aluminum, allium, indium, zinc, and cadmium,” J. Phys. Chem. 91, 634–643 (1987).
    [CrossRef]
  32. J. D. McMullen, “Optical parametric interactions in isotropic materials using a phase-corrected stack of nonlinear dielectric plates,” J. Appl. Phys. 46, 3076–3081 (1975).
    [CrossRef]
  33. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
    [CrossRef]
  34. X. H. Wang and B. Y. Gu, “Nonlinear frequency conversion in 2D χ(2) photonic crystals and novel nonlinear double-circle construction,” Eur. Phys. J. B 24, 323–326 (2001).
    [CrossRef]
  35. D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: analysis using optical transfer matrix techniques,” J. Opt. Soc. Am. B 6, 910–916 (1989).
    [CrossRef]
  36. D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: extension of optical transfer matrix approach to include anisotropic materials,” J. Opt. Soc. Am. B 8, 367–373 (1991).
    [CrossRef]
  37. S. Enoch and H. Akhouayri, “Second-harmonic generation in multilayered devices: theoretical tools,” J. Opt. Soc. Am. B 15, 1030–1041 (1998).
    [CrossRef]
  38. S. Lim, “Second harmonic generation of magnetic and dielectric multilayers,” J. Phys. 18, 4329–4343 (2006).
  39. J. Yuan, “Computing for second harmonic generation in one-dimensional nonlinear photonic crystals,” Opt. Commun. 282, 2628–2633 (2009).
    [CrossRef]
  40. J. Li, Z. Li, and D. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E 75, 056606 (2007).
    [CrossRef]
  41. P. Szczepański, T. Osuch, and Z. Jaroszewicz, “Modeling of amplification and light generation in one-dimensional photonic crystal using a multiwavelength transfer matrix approach,” Appl. Opt. 48, 5401–5406 (2009).
    [CrossRef]
  42. A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E 82, 036608 (2010).
    [CrossRef]
  43. N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
    [CrossRef]
  44. M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic bandgap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
    [CrossRef]
  45. B. Temelkuran and E. Ozbay, “Experimental demonstration of photonic crystal based waveguides,” Appl. Phys. Lett. 74, 486–488 (1999).
    [CrossRef]
  46. S. Larouche and D. R. Smith, “A retrieval method for nonlinear metamaterials,” Opt. Commun. 283, 1621–1627 (2010).
    [CrossRef]
  47. S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterials,” Appl. Phys. Lett. 97, 011109 (2010).
    [CrossRef]

2012 (3)

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6, 737–748 (2012).
[CrossRef]

C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Second-harmonic generation in metallic nanoparticles: clarification of the role of the surface,” Phys. Rev. B 86, 115451 (2012).
[CrossRef]

E. Poutrina, C. Ciracì, D. J. Gauthier, and D. R. Smith, “Enhancing four-wave-mixing processes by nanowire arrays coupled to a gold film,” Opt. Express 20, 11005 (2012).
[CrossRef]

2011 (2)

J. Renger, R. Quidant, and L. Novotny, “Enhanced nonlinear response from metal surfaces,” Opt. Express 19, 1777–1785 (2011).
[CrossRef]

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11, 34–38 (2011).
[CrossRef]

2010 (7)

P. Genevet, J. Tetienne, E. Gatzogiannis, R. Blanchard, M. A. Kats, M. O. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. 10, 4880–4883 (2010).
[CrossRef]

J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface-enhanced nonlinear four wave mixing,” Phys. Rev. Lett. 104, 046803 (2010).
[CrossRef]

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E 82, 036608 (2010).
[CrossRef]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

A. Benedetti, M. Centini, C. Sibilia, and M. Bertolotti, “Engineering the second harmonic generation pattern from coupled gold nanowires,” J. Opt. Soc. Am. B 27, 408–416 (2010).
[CrossRef]

S. Larouche and D. R. Smith, “A retrieval method for nonlinear metamaterials,” Opt. Commun. 283, 1621–1627 (2010).
[CrossRef]

S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterials,” Appl. Phys. Lett. 97, 011109 (2010).
[CrossRef]

2009 (5)

P. Szczepański, T. Osuch, and Z. Jaroszewicz, “Modeling of amplification and light generation in one-dimensional photonic crystal using a multiwavelength transfer matrix approach,” Appl. Opt. 48, 5401–5406 (2009).
[CrossRef]

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

J. Yuan, “Computing for second harmonic generation in one-dimensional nonlinear photonic crystals,” Opt. Commun. 282, 2628–2633 (2009).
[CrossRef]

J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing,” Phys. Rev. Lett. 103, 266802 (2009).
[CrossRef]

2007 (2)

J. Li, Z. Li, and D. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E 75, 056606 (2007).
[CrossRef]

M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett. 98, 026104 (2007).
[CrossRef]

2006 (2)

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef]

S. Lim, “Second harmonic generation of magnetic and dielectric multilayers,” J. Phys. 18, 4329–4343 (2006).

2005 (1)

M. Lippitz, M. A. van Dijk, and M. Orrit, “Third-harmonic generation from single gold nanoparticles,” Nano Lett. 5, 799–802 (2005).
[CrossRef]

2004 (1)

N. N. Lepeahkin, A. Schweinaberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
[CrossRef]

2003 (1)

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef]

2001 (1)

X. H. Wang and B. Y. Gu, “Nonlinear frequency conversion in 2D χ(2) photonic crystals and novel nonlinear double-circle construction,” Eur. Phys. J. B 24, 323–326 (2001).
[CrossRef]

1999 (4)

N. N. Lepeshkin, W. Kim, V. P. Safonov, J. G. Zhu, R. L. Armstrong, C. W. White, R. A. Zuhr, and V. M. Shalaev, “Optical nonlinearities of metal-dielectric composites,” J. Nonlinear Opt. Phys. Mater. 8, 191–210 (1999).
[CrossRef]

N. A. Papadogiannics, P. A. Loukakos, and S. D. Moustaizis, “Observation of the inversion of second and third harmonic generation efficiencies on a gold surface in the femtosecond regime,” Opt. Commun. 166, 113–139 (1999).
[CrossRef]

B. Temelkuran and E. Ozbay, “Experimental demonstration of photonic crystal based waveguides,” Appl. Phys. Lett. 74, 486–488 (1999).
[CrossRef]

R. S. Bennink, Y. Yoon, R. W. Boyd, and J. E. Sipe, “Accessing the optical nonlinearity of metals with metal-dielectric photonic bandgap structures,” Opt. Lett. 24, 1416–1418 (1999).
[CrossRef]

1998 (2)

S. Enoch and H. Akhouayri, “Second-harmonic generation in multilayered devices: theoretical tools,” J. Opt. Soc. Am. B 15, 1030–1041 (1998).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic bandgap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

1997 (1)

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, “Large third-order optical nonlinearity in Au:SiO2 composite films near the percolation threshold,” Appl. Phys. Lett. 70, 119291 (1997).

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

1991 (2)

H. B. Jiang, L. Li, W. C. Wang, J. B. Zheng, Z. M. Zhang, and Z. Chen, “Reflected second-harmonic generation at a silver surface,” Phys. Rev. B 44, 1220–1224 (1991).
[CrossRef]

D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: extension of optical transfer matrix approach to include anisotropic materials,” J. Opt. Soc. Am. B 8, 367–373 (1991).
[CrossRef]

1989 (1)

1988 (1)

F. Hache, D. Richard, C. Flytzanis, and U. Kreibig, “The optical Kerr effect in small metal particles and metal colloide: the case of gold,” Appl. Phys. A 47, 347–357 (1988).
[CrossRef]

1987 (1)

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for silver, gold, copper, lithium, sodium, aluminum, allium, indium, zinc, and cadmium,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

1986 (1)

F. Hache, D. Ricard, and C. Flytzanis, “Optical nonlinearities of small metal particles: surface-mediated resonance and quantum size effects,” Opt. Lett. 12, 1647–1655 (1986).

1984 (1)

G. A. Farias and A. A. Maradudin, “Second-harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30, 3002–3015 (1984).
[CrossRef]

1980 (1)

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
[CrossRef]

1975 (1)

J. D. McMullen, “Optical parametric interactions in isotropic materials using a phase-corrected stack of nonlinear dielectric plates,” J. Appl. Phys. 46, 3076–3081 (1975).
[CrossRef]

1971 (1)

J. Rudnick and E. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
[CrossRef]

1968 (1)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

1962 (1)

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[CrossRef]

Ahorinta, R.

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

Akhouayri, H.

Akozbek, N.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

Albers, W. M.

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

Armstrong, R. L.

N. N. Lepeshkin, W. Kim, V. P. Safonov, J. G. Zhu, R. L. Armstrong, C. W. White, R. A. Zuhr, and V. M. Shalaev, “Optical nonlinearities of metal-dielectric composites,” J. Nonlinear Opt. Phys. Mater. 8, 191–210 (1999).
[CrossRef]

Bartal, G.

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11, 34–38 (2011).
[CrossRef]

Benedetti, A.

Bennink, R. S.

N. N. Lepeahkin, A. Schweinaberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
[CrossRef]

R. S. Bennink, Y. Yoon, R. W. Boyd, and J. E. Sipe, “Accessing the optical nonlinearity of metals with metal-dielectric photonic bandgap structures,” Opt. Lett. 24, 1416–1418 (1999).
[CrossRef]

Bertolotti, M.

Bethune, D. S.

Beversluis, M.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef]

Blanchard, R.

P. Genevet, J. Tetienne, E. Gatzogiannis, R. Blanchard, M. A. Kats, M. O. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. 10, 4880–4883 (2010).
[CrossRef]

Bloembergen, N.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[CrossRef]

Bloemer, M. J.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic bandgap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

Boardman, A. D.

A. D. Boardman, Electromagnetic Surface Modes (Wiley, New York, 1982).

Bouhelier, A.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef]

Bowden, C. M.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic bandgap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

Boyd, R. W.

N. N. Lepeahkin, A. Schweinaberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
[CrossRef]

R. S. Bennink, Y. Yoon, R. W. Boyd, and J. E. Sipe, “Accessing the optical nonlinearity of metals with metal-dielectric photonic bandgap structures,” Opt. Lett. 24, 1416–1418 (1999).
[CrossRef]

R. W. Boyd, Nonlinear Optics (Academic, 2008).

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Capasso, F.

P. Genevet, J. Tetienne, E. Gatzogiannis, R. Blanchard, M. A. Kats, M. O. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. 10, 4880–4883 (2010).
[CrossRef]

Centini, M.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

A. Benedetti, M. Centini, C. Sibilia, and M. Bertolotti, “Engineering the second harmonic generation pattern from coupled gold nanowires,” J. Opt. Soc. Am. B 27, 408–416 (2010).
[CrossRef]

Chang, R. K.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

Chen, Z.

H. B. Jiang, L. Li, W. C. Wang, J. B. Zheng, Z. M. Zhang, and Z. Chen, “Reflected second-harmonic generation at a silver surface,” Phys. Rev. B 44, 1220–1224 (1991).
[CrossRef]

Ciracì, C.

C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Second-harmonic generation in metallic nanoparticles: clarification of the role of the surface,” Phys. Rev. B 86, 115451 (2012).
[CrossRef]

E. Poutrina, C. Ciracì, D. J. Gauthier, and D. R. Smith, “Enhancing four-wave-mixing processes by nanowire arrays coupled to a gold film,” Opt. Express 20, 11005 (2012).
[CrossRef]

Danckwerts, M.

M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett. 98, 026104 (2007).
[CrossRef]

de Ceglia, D.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

Dowling, J. P.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic bandgap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

Enkrich, C.

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef]

Enoch, S.

Farias, G. A.

G. A. Farias and A. A. Maradudin, “Second-harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30, 3002–3015 (1984).
[CrossRef]

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Flytzanis, C.

F. Hache, D. Richard, C. Flytzanis, and U. Kreibig, “The optical Kerr effect in small metal particles and metal colloide: the case of gold,” Appl. Phys. A 47, 347–357 (1988).
[CrossRef]

F. Hache, D. Ricard, and C. Flytzanis, “Optical nonlinearities of small metal particles: surface-mediated resonance and quantum size effects,” Opt. Lett. 12, 1647–1655 (1986).

Fu, J. S.

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, “Large third-order optical nonlinearity in Au:SiO2 composite films near the percolation threshold,” Appl. Phys. Lett. 70, 119291 (1997).

Fukui, M.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
[CrossRef]

Gatzogiannis, E.

P. Genevet, J. Tetienne, E. Gatzogiannis, R. Blanchard, M. A. Kats, M. O. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. 10, 4880–4883 (2010).
[CrossRef]

Gauthier, D. J.

Genevet, P.

P. Genevet, J. Tetienne, E. Gatzogiannis, R. Blanchard, M. A. Kats, M. O. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. 10, 4880–4883 (2010).
[CrossRef]

Gu, B. Y.

X. H. Wang and B. Y. Gu, “Nonlinear frequency conversion in 2D χ(2) photonic crystals and novel nonlinear double-circle construction,” Eur. Phys. J. B 24, 323–326 (2001).
[CrossRef]

Hache, F.

F. Hache, D. Richard, C. Flytzanis, and U. Kreibig, “The optical Kerr effect in small metal particles and metal colloide: the case of gold,” Appl. Phys. A 47, 347–357 (1988).
[CrossRef]

F. Hache, D. Ricard, and C. Flytzanis, “Optical nonlinearities of small metal particles: surface-mediated resonance and quantum size effects,” Opt. Lett. 12, 1647–1655 (1986).

Hartschuh, A.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef]

Hoyer, W.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

Huang, D.

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E 82, 036608 (2010).
[CrossRef]

S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterials,” Appl. Phys. Lett. 97, 011109 (2010).
[CrossRef]

Jaroszewicz, Z.

Jha, S. S.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

Jiang, H. B.

H. B. Jiang, L. Li, W. C. Wang, J. B. Zheng, Z. M. Zhang, and Z. Chen, “Reflected second-harmonic generation at a silver surface,” Phys. Rev. B 44, 1220–1224 (1991).
[CrossRef]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Kats, M. A.

P. Genevet, J. Tetienne, E. Gatzogiannis, R. Blanchard, M. A. Kats, M. O. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. 10, 4880–4883 (2010).
[CrossRef]

Kauranen, M.

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6, 737–748 (2012).
[CrossRef]

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

Kim, W.

N. N. Lepeshkin, W. Kim, V. P. Safonov, J. G. Zhu, R. L. Armstrong, C. W. White, R. A. Zuhr, and V. M. Shalaev, “Optical nonlinearities of metal-dielectric composites,” J. Nonlinear Opt. Phys. Mater. 8, 191–210 (1999).
[CrossRef]

Klein, M. W.

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef]

Koch, S. W.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

Kreibig, U.

F. Hache, D. Richard, C. Flytzanis, and U. Kreibig, “The optical Kerr effect in small metal particles and metal colloide: the case of gold,” Appl. Phys. A 47, 347–357 (1988).
[CrossRef]

Larouche, S.

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E 82, 036608 (2010).
[CrossRef]

S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterials,” Appl. Phys. Lett. 97, 011109 (2010).
[CrossRef]

S. Larouche and D. R. Smith, “A retrieval method for nonlinear metamaterials,” Opt. Commun. 283, 1621–1627 (2010).
[CrossRef]

Lee, C. H.

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

Lepeahkin, N. N.

N. N. Lepeahkin, A. Schweinaberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
[CrossRef]

Lepeshkin, N. N.

N. N. Lepeshkin, W. Kim, V. P. Safonov, J. G. Zhu, R. L. Armstrong, C. W. White, R. A. Zuhr, and V. M. Shalaev, “Optical nonlinearities of metal-dielectric composites,” J. Nonlinear Opt. Phys. Mater. 8, 191–210 (1999).
[CrossRef]

Li, J.

J. Li, Z. Li, and D. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E 75, 056606 (2007).
[CrossRef]

Li, L.

H. B. Jiang, L. Li, W. C. Wang, J. B. Zheng, Z. M. Zhang, and Z. Chen, “Reflected second-harmonic generation at a silver surface,” Phys. Rev. B 44, 1220–1224 (1991).
[CrossRef]

Li, Z.

J. Li, Z. Li, and D. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E 75, 056606 (2007).
[CrossRef]

Liao, H. B.

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, “Large third-order optical nonlinearity in Au:SiO2 composite films near the percolation threshold,” Appl. Phys. Lett. 70, 119291 (1997).

Lim, S.

S. Lim, “Second harmonic generation of magnetic and dielectric multilayers,” J. Phys. 18, 4329–4343 (2006).

Linden, S.

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef]

Lippitz, M.

M. Lippitz, M. A. van Dijk, and M. Orrit, “Third-harmonic generation from single gold nanoparticles,” Nano Lett. 5, 799–802 (2005).
[CrossRef]

Liu, J.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

Loukakos, P. A.

N. A. Papadogiannics, P. A. Loukakos, and S. D. Moustaizis, “Observation of the inversion of second and third harmonic generation efficiencies on a gold surface in the femtosecond regime,” Opt. Commun. 166, 113–139 (1999).
[CrossRef]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

Manka, A. S.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic bandgap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

Maradudin, A. A.

G. A. Farias and A. A. Maradudin, “Second-harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30, 3002–3015 (1984).
[CrossRef]

McMullen, J. D.

J. D. McMullen, “Optical parametric interactions in isotropic materials using a phase-corrected stack of nonlinear dielectric plates,” J. Appl. Phys. 46, 3076–3081 (1975).
[CrossRef]

Moloney, J. V.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

Moustaizis, S. D.

N. A. Papadogiannics, P. A. Loukakos, and S. D. Moustaizis, “Observation of the inversion of second and third harmonic generation efficiencies on a gold surface in the femtosecond regime,” Opt. Commun. 166, 113–139 (1999).
[CrossRef]

Novotny, L.

J. Renger, R. Quidant, and L. Novotny, “Enhanced nonlinear response from metal surfaces,” Opt. Express 19, 1777–1785 (2011).
[CrossRef]

J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface-enhanced nonlinear four wave mixing,” Phys. Rev. Lett. 104, 046803 (2010).
[CrossRef]

J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing,” Phys. Rev. Lett. 103, 266802 (2009).
[CrossRef]

M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett. 98, 026104 (2007).
[CrossRef]

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef]

Orrit, M.

M. Lippitz, M. A. van Dijk, and M. Orrit, “Third-harmonic generation from single gold nanoparticles,” Nano Lett. 5, 799–802 (2005).
[CrossRef]

Osuch, T.

Ozbay, E.

B. Temelkuran and E. Ozbay, “Experimental demonstration of photonic crystal based waveguides,” Appl. Phys. Lett. 74, 486–488 (1999).
[CrossRef]

Palomba, S.

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11, 34–38 (2011).
[CrossRef]

J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing,” Phys. Rev. Lett. 103, 266802 (2009).
[CrossRef]

Papadogiannics, N. A.

N. A. Papadogiannics, P. A. Loukakos, and S. D. Moustaizis, “Observation of the inversion of second and third harmonic generation efficiencies on a gold surface in the femtosecond regime,” Opt. Commun. 166, 113–139 (1999).
[CrossRef]

Park, Y.

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11, 34–38 (2011).
[CrossRef]

Pershan, P. S.

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[CrossRef]

Pethel, A. S.

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic bandgap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

Piredda, G.

N. N. Lepeahkin, A. Schweinaberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
[CrossRef]

Poutrina, E.

C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Second-harmonic generation in metallic nanoparticles: clarification of the role of the surface,” Phys. Rev. B 86, 115451 (2012).
[CrossRef]

E. Poutrina, C. Ciracì, D. J. Gauthier, and D. R. Smith, “Enhancing four-wave-mixing processes by nanowire arrays coupled to a gold film,” Opt. Express 20, 11005 (2012).
[CrossRef]

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E 82, 036608 (2010).
[CrossRef]

S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterials,” Appl. Phys. Lett. 97, 011109 (2010).
[CrossRef]

Quidant, R.

J. Renger, R. Quidant, and L. Novotny, “Enhanced nonlinear response from metal surfaces,” Opt. Express 19, 1777–1785 (2011).
[CrossRef]

J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface-enhanced nonlinear four wave mixing,” Phys. Rev. Lett. 104, 046803 (2010).
[CrossRef]

J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing,” Phys. Rev. Lett. 103, 266802 (2009).
[CrossRef]

Renger, J.

J. Renger, R. Quidant, and L. Novotny, “Enhanced nonlinear response from metal surfaces,” Opt. Express 19, 1777–1785 (2011).
[CrossRef]

J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface-enhanced nonlinear four wave mixing,” Phys. Rev. Lett. 104, 046803 (2010).
[CrossRef]

J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing,” Phys. Rev. Lett. 103, 266802 (2009).
[CrossRef]

Ricard, D.

F. Hache, D. Ricard, and C. Flytzanis, “Optical nonlinearities of small metal particles: surface-mediated resonance and quantum size effects,” Opt. Lett. 12, 1647–1655 (1986).

Richard, D.

F. Hache, D. Richard, C. Flytzanis, and U. Kreibig, “The optical Kerr effect in small metal particles and metal colloide: the case of gold,” Appl. Phys. A 47, 347–357 (1988).
[CrossRef]

Rodríguez, F. J.

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

Roppo, V.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

Rose, A.

S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterials,” Appl. Phys. Lett. 97, 011109 (2010).
[CrossRef]

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E 82, 036608 (2010).
[CrossRef]

Rudnick, J.

J. Rudnick and E. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
[CrossRef]

Safonov, V. P.

N. N. Lepeshkin, W. Kim, V. P. Safonov, J. G. Zhu, R. L. Armstrong, C. W. White, R. A. Zuhr, and V. M. Shalaev, “Optical nonlinearities of metal-dielectric composites,” J. Nonlinear Opt. Phys. Mater. 8, 191–210 (1999).
[CrossRef]

Scalora, M.

C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Second-harmonic generation in metallic nanoparticles: clarification of the role of the surface,” Phys. Rev. B 86, 115451 (2012).
[CrossRef]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic bandgap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

Schatz, G. C.

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for silver, gold, copper, lithium, sodium, aluminum, allium, indium, zinc, and cadmium,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

Schweinaberg, A.

N. N. Lepeahkin, A. Schweinaberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
[CrossRef]

Scully, M. O.

P. Genevet, J. Tetienne, E. Gatzogiannis, R. Blanchard, M. A. Kats, M. O. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. 10, 4880–4883 (2010).
[CrossRef]

Shalaev, V. M.

N. N. Lepeshkin, W. Kim, V. P. Safonov, J. G. Zhu, R. L. Armstrong, C. W. White, R. A. Zuhr, and V. M. Shalaev, “Optical nonlinearities of metal-dielectric composites,” J. Nonlinear Opt. Phys. Mater. 8, 191–210 (1999).
[CrossRef]

Sheng, P.

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, “Large third-order optical nonlinearity in Au:SiO2 composite films near the percolation threshold,” Appl. Phys. Lett. 70, 119291 (1997).

Sibilia, C.

Sipe, J. E.

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

R. S. Bennink, Y. Yoon, R. W. Boyd, and J. E. Sipe, “Accessing the optical nonlinearity of metals with metal-dielectric photonic bandgap structures,” Opt. Lett. 24, 1416–1418 (1999).
[CrossRef]

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
[CrossRef]

Smith, D. R.

C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Second-harmonic generation in metallic nanoparticles: clarification of the role of the surface,” Phys. Rev. B 86, 115451 (2012).
[CrossRef]

E. Poutrina, C. Ciracì, D. J. Gauthier, and D. R. Smith, “Enhancing four-wave-mixing processes by nanowire arrays coupled to a gold film,” Opt. Express 20, 11005 (2012).
[CrossRef]

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E 82, 036608 (2010).
[CrossRef]

S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterials,” Appl. Phys. Lett. 97, 011109 (2010).
[CrossRef]

S. Larouche and D. R. Smith, “A retrieval method for nonlinear metamaterials,” Opt. Commun. 283, 1621–1627 (2010).
[CrossRef]

So, V. C. Y.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
[CrossRef]

Stegeman, G. I.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
[CrossRef]

Stern, E.

J. Rudnick and E. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
[CrossRef]

Szczepanski, P.

Temelkuran, B.

B. Temelkuran and E. Ozbay, “Experimental demonstration of photonic crystal based waveguides,” Appl. Phys. Lett. 74, 486–488 (1999).
[CrossRef]

Tetienne, J.

P. Genevet, J. Tetienne, E. Gatzogiannis, R. Blanchard, M. A. Kats, M. O. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. 10, 4880–4883 (2010).
[CrossRef]

van Dijk, M. A.

M. Lippitz, M. A. van Dijk, and M. Orrit, “Third-harmonic generation from single gold nanoparticles,” Nano Lett. 5, 799–802 (2005).
[CrossRef]

van Hulst, N.

J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface-enhanced nonlinear four wave mixing,” Phys. Rev. Lett. 104, 046803 (2010).
[CrossRef]

J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing,” Phys. Rev. Lett. 103, 266802 (2009).
[CrossRef]

Vincenti, M. A.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

Wang, F. X.

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

Wang, W. C.

H. B. Jiang, L. Li, W. C. Wang, J. B. Zheng, Z. M. Zhang, and Z. Chen, “Reflected second-harmonic generation at a silver surface,” Phys. Rev. B 44, 1220–1224 (1991).
[CrossRef]

Wang, X. H.

X. H. Wang and B. Y. Gu, “Nonlinear frequency conversion in 2D χ(2) photonic crystals and novel nonlinear double-circle construction,” Eur. Phys. J. B 24, 323–326 (2001).
[CrossRef]

Wegener, M.

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef]

White, C. W.

N. N. Lepeshkin, W. Kim, V. P. Safonov, J. G. Zhu, R. L. Armstrong, C. W. White, R. A. Zuhr, and V. M. Shalaev, “Optical nonlinearities of metal-dielectric composites,” J. Nonlinear Opt. Phys. Mater. 8, 191–210 (1999).
[CrossRef]

Wong, G. K. L.

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, “Large third-order optical nonlinearity in Au:SiO2 composite films near the percolation threshold,” Appl. Phys. Lett. 70, 119291 (1997).

Xiao, R. F.

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, “Large third-order optical nonlinearity in Au:SiO2 composite films near the percolation threshold,” Appl. Phys. Lett. 70, 119291 (1997).

Yin, X.

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11, 34–38 (2011).
[CrossRef]

Yoon, Y.

Yu, P.

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, “Large third-order optical nonlinearity in Au:SiO2 composite films near the percolation threshold,” Appl. Phys. Lett. 70, 119291 (1997).

Yuan, J.

J. Yuan, “Computing for second harmonic generation in one-dimensional nonlinear photonic crystals,” Opt. Commun. 282, 2628–2633 (2009).
[CrossRef]

Zayats, A. V.

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6, 737–748 (2012).
[CrossRef]

Zeman, E. J.

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for silver, gold, copper, lithium, sodium, aluminum, allium, indium, zinc, and cadmium,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

Zeng, Y.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

Zhang, D.

J. Li, Z. Li, and D. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E 75, 056606 (2007).
[CrossRef]

Zhang, S.

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11, 34–38 (2011).
[CrossRef]

Zhang, X.

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11, 34–38 (2011).
[CrossRef]

Zhang, Z. M.

H. B. Jiang, L. Li, W. C. Wang, J. B. Zheng, Z. M. Zhang, and Z. Chen, “Reflected second-harmonic generation at a silver surface,” Phys. Rev. B 44, 1220–1224 (1991).
[CrossRef]

Zheng, J. B.

H. B. Jiang, L. Li, W. C. Wang, J. B. Zheng, Z. M. Zhang, and Z. Chen, “Reflected second-harmonic generation at a silver surface,” Phys. Rev. B 44, 1220–1224 (1991).
[CrossRef]

Zhu, J. G.

N. N. Lepeshkin, W. Kim, V. P. Safonov, J. G. Zhu, R. L. Armstrong, C. W. White, R. A. Zuhr, and V. M. Shalaev, “Optical nonlinearities of metal-dielectric composites,” J. Nonlinear Opt. Phys. Mater. 8, 191–210 (1999).
[CrossRef]

Zuhr, R. A.

N. N. Lepeshkin, W. Kim, V. P. Safonov, J. G. Zhu, R. L. Armstrong, C. W. White, R. A. Zuhr, and V. M. Shalaev, “Optical nonlinearities of metal-dielectric composites,” J. Nonlinear Opt. Phys. Mater. 8, 191–210 (1999).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. A (1)

F. Hache, D. Richard, C. Flytzanis, and U. Kreibig, “The optical Kerr effect in small metal particles and metal colloide: the case of gold,” Appl. Phys. A 47, 347–357 (1988).
[CrossRef]

Appl. Phys. Lett. (3)

H. B. Liao, R. F. Xiao, J. S. Fu, P. Yu, G. K. L. Wong, and P. Sheng, “Large third-order optical nonlinearity in Au:SiO2 composite films near the percolation threshold,” Appl. Phys. Lett. 70, 119291 (1997).

B. Temelkuran and E. Ozbay, “Experimental demonstration of photonic crystal based waveguides,” Appl. Phys. Lett. 74, 486–488 (1999).
[CrossRef]

S. Larouche, A. Rose, E. Poutrina, D. Huang, and D. R. Smith, “Experimental determination of the quadratic nonlinear magnetic susceptibility of a varactor-loaded split ring resonator metamaterials,” Appl. Phys. Lett. 97, 011109 (2010).
[CrossRef]

Eur. Phys. J. B (1)

X. H. Wang and B. Y. Gu, “Nonlinear frequency conversion in 2D χ(2) photonic crystals and novel nonlinear double-circle construction,” Eur. Phys. J. B 24, 323–326 (2001).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631–2654 (1992).
[CrossRef]

J. Appl. Phys. (2)

J. D. McMullen, “Optical parametric interactions in isotropic materials using a phase-corrected stack of nonlinear dielectric plates,” J. Appl. Phys. 46, 3076–3081 (1975).
[CrossRef]

M. Scalora, M. J. Bloemer, A. S. Pethel, J. P. Dowling, C. M. Bowden, and A. S. Manka, “Transparent, metallo-dielectric, one-dimensional, photonic bandgap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

N. N. Lepeshkin, W. Kim, V. P. Safonov, J. G. Zhu, R. L. Armstrong, C. W. White, R. A. Zuhr, and V. M. Shalaev, “Optical nonlinearities of metal-dielectric composites,” J. Nonlinear Opt. Phys. Mater. 8, 191–210 (1999).
[CrossRef]

J. Opt. Soc. Am. B (4)

J. Phys. (1)

S. Lim, “Second harmonic generation of magnetic and dielectric multilayers,” J. Phys. 18, 4329–4343 (2006).

J. Phys. Chem. (1)

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for silver, gold, copper, lithium, sodium, aluminum, allium, indium, zinc, and cadmium,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

Nano Lett. (2)

M. Lippitz, M. A. van Dijk, and M. Orrit, “Third-harmonic generation from single gold nanoparticles,” Nano Lett. 5, 799–802 (2005).
[CrossRef]

P. Genevet, J. Tetienne, E. Gatzogiannis, R. Blanchard, M. A. Kats, M. O. Scully, and F. Capasso, “Large enhancement of nonlinear optical phenomena by plasmonic nanocavity gratings,” Nano Lett. 10, 4880–4883 (2010).
[CrossRef]

Nat. Mater. (1)

S. Palomba, S. Zhang, Y. Park, G. Bartal, X. Yin, and X. Zhang, “Optical negative refraction by four-wave mixing in thin metallic nanostructures,” Nat. Mater. 11, 34–38 (2011).
[CrossRef]

Nat. Photonics (1)

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6, 737–748 (2012).
[CrossRef]

Opt. Commun. (3)

N. A. Papadogiannics, P. A. Loukakos, and S. D. Moustaizis, “Observation of the inversion of second and third harmonic generation efficiencies on a gold surface in the femtosecond regime,” Opt. Commun. 166, 113–139 (1999).
[CrossRef]

J. Yuan, “Computing for second harmonic generation in one-dimensional nonlinear photonic crystals,” Opt. Commun. 282, 2628–2633 (2009).
[CrossRef]

S. Larouche and D. R. Smith, “A retrieval method for nonlinear metamaterials,” Opt. Commun. 283, 1621–1627 (2010).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

R. S. Bennink, Y. Yoon, R. W. Boyd, and J. E. Sipe, “Accessing the optical nonlinearity of metals with metal-dielectric photonic bandgap structures,” Opt. Lett. 24, 1416–1418 (1999).
[CrossRef]

F. Hache, D. Ricard, and C. Flytzanis, “Optical nonlinearities of small metal particles: surface-mediated resonance and quantum size effects,” Opt. Lett. 12, 1647–1655 (1986).

Phys. Rev. (2)

N. Bloembergen, R. K. Chang, S. S. Jha, and C. H. Lee, “Optical second-harmonic generation in reflection from media with inversion symmetry,” Phys. Rev. 174, 813–822 (1968).
[CrossRef]

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[CrossRef]

Phys. Rev. A (1)

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

Phys. Rev. B (7)

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79, 235109 (2009).
[CrossRef]

F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80, 233402 (2009).
[CrossRef]

J. Rudnick and E. Stern, “Second-harmonic radiation from metal surfaces,” Phys. Rev. B 4, 4274–4290 (1971).
[CrossRef]

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
[CrossRef]

G. A. Farias and A. A. Maradudin, “Second-harmonic generation in reflection from a metallic grating,” Phys. Rev. B 30, 3002–3015 (1984).
[CrossRef]

H. B. Jiang, L. Li, W. C. Wang, J. B. Zheng, Z. M. Zhang, and Z. Chen, “Reflected second-harmonic generation at a silver surface,” Phys. Rev. B 44, 1220–1224 (1991).
[CrossRef]

C. Ciracì, E. Poutrina, M. Scalora, and D. R. Smith, “Second-harmonic generation in metallic nanoparticles: clarification of the role of the surface,” Phys. Rev. B 86, 115451 (2012).
[CrossRef]

Phys. Rev. E (2)

J. Li, Z. Li, and D. Zhang, “Second harmonic generation in one-dimensional nonlinear photonic crystals solved by the transfer matrix method,” Phys. Rev. E 75, 056606 (2007).
[CrossRef]

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E 82, 036608 (2010).
[CrossRef]

Phys. Rev. Lett. (5)

J. Renger, R. Quidant, N. van Hulst, and L. Novotny, “Surface-enhanced nonlinear four wave mixing,” Phys. Rev. Lett. 104, 046803 (2010).
[CrossRef]

M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett. 98, 026104 (2007).
[CrossRef]

J. Renger, R. Quidant, N. van Hulst, S. Palomba, and L. Novotny, “Free-space excitation of propagating surface plasmon polaritons by nonlinear four-wave mixing,” Phys. Rev. Lett. 103, 266802 (2009).
[CrossRef]

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[CrossRef]

N. N. Lepeahkin, A. Schweinaberg, G. Piredda, R. S. Bennink, and R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Phys. Rev. Lett. 93, 123902 (2004).
[CrossRef]

Science (1)

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science 313, 502–504 (2006).
[CrossRef]

Other (2)

R. W. Boyd, Nonlinear Optics (Academic, 2008).

A. D. Boardman, Electromagnetic Surface Modes (Wiley, New York, 1982).

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 (6)

Fig. 1.
Fig. 1.

(a) A single interface between linear dielectric medium 1 and nonlinear metallic medium 2. (b) A single metal layer. (c) Multilayer stack with n layers in total and r layers of metal. (d) Silver–dielectric–silver structure.

Fig. 2.
Fig. 2.

Norms of the FWM field generated from an air–silver interface as a function of excitation angles. Solid lines represent k1znl=0. Dashed lines denote the surface plasmon excitation condition derived from Eq. (16).

Fig. 3.
Fig. 3.

(a) and (b) Norms of the reflected (red, circle) and transmitted (blue, square) FWM fields generated from a silver film in air as a function of film thickness. (a) Propagating FWM fields. Excitation angles: θa=θc=20°, θb=30°. (b) Evanescent FWM fields. Excitation angles: θa=θc=50°, θb=40°. Lines denote the analytical results; markers represent the finite element simulations results. (c) and (d) Norms of the reflected (c) and transmitted (d) FWM fields from a 20 nm silver-film in air as a function of the incidents angles (θa,θb) of pumping fields. Black and white dots represent the two sets of chosen angles in (a) and (b), respectively.

Fig. 4.
Fig. 4.

(a), (b) Norms of the reflected (a) and transmitted (b) FWM fields generated from a silver film in the Kretschmann configuration as the function of incidents angles (θa,θb) of pumping fields. Dashed lines denote the surface plasmon excitation condition derived from Eq. (16). (c), (d) FWM field patterns at points I (c) and II (d).

Fig. 5.
Fig. 5.

(a) Norms of the reflected FWM fields generated from a silver–dielectric–silver structure in air as a function of dielectric thickness d2. Lines denote the analytical results; circles represent the finite element simulations. Excitation angles: θa=5°, θb=20°. Inset: Reflected pumping fields; red-fa; blue-fb. (b) Norms of the reflected FWM fields from a silver–dielectric–silver structure in air as a function of the incidents angles (θa,θb) of pumping fields. White dot: θa=5°, θb=20°.

Fig. 6.
Fig. 6.

Norms of the FWM fields generated from a silver–dielectric–silver structure with different sandwiched nonlinear dielectrics.

Equations (91)

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

E(t)=E0+q12E(ωq)exp(jωqt),
P(t)=P0+q12P(ωq)exp(jωqt),
Pi(3)(ωnl)=14ε0(abc)jklχijkl(3)(ωnl;ωa,ωb,ωc)Ej(ωa)Ek(ωb)El(ωc),
χxxxx(3)=χyyyy(3)=χzzzz(3)χyyzz(3)=χzzyy(3)=χzzxx(3)=χxxzz(3)=χxxyy(3)=χyyxx(3)χyzyz(3)=χzyzy(3)=χzxzx(3)=χxzxz(3)=χxyxy(3)=χyxyx(3)χyzzy(3)=χzyyz(3)=χzxxz(3)=χxzzx(3)=χxyyx(3)=χyxxy(3)χxxxx(3)=χxxyy(3)+χxyxy(3)+χxyyx(3).
Pi(3)(ωnl)=D4ε0χ(3)((Ea·Eb)Eic+(Ea·Ec)Eib+(Eb·Ec)Eia),
Hs(r,ωnl)=jωnl(×P(3)(r,ωnl))Q2+ωnl2ε(ωnl)μ(ωnl),
Es(r,ωnl)=ωnl2μ(ωnl)P(3)(r,ωnl)(·P(3)(r,ωnl)ε(ωnl))Q2+ωnl2ε(ωnl)μ(ωnl),
H1ynl=H2ynl+Hsy,
E1xnl=E2xnl+Esx.
E(r,ωq)=[A+exp(jk·r)+Aexp(jk·r)]exp(jωqt).
ωε1/k1zE1x=ωε2/k2zE2x++Hsy+,
E1x=E2x++Esx+.
E1x=k1z[Px(3)(k2)2(Px(3)Qx+Pz(3)Qz)Qx(Px(3)QzPz(3)Qx)k2z](ε2k1z+ε1k2z)(Q2k22).
E1z=k1x[Px(3)(k2)2(Px(3)Qx+Pz(3)Qz)Qx(Px(3)QzPz(3)Qx)k2z](ε2k1z+ε1k2z)(Q2k22).
R=ε1k2zε2k1zε1k2z+ε2k1z.
kx=ε1ε2ε1+ε2ωc.
kxnl=2kxakxb,
Hi(x,z,ωq)=exp(jωqt)[Hi+(x,z)+Hi(x,z)]y,
Hi(z)=(Hi+(z)Hi(z)).
Hi+1(zi,i+1)=Kii+1Hi(zi,i+1),
Kii+1=12(1+εi+1kizεiki+1,z1εi+1kizεiki+1,z1εi+1kizεiki+1,z1+εi+1kizεiki+1,z)
H(zi,i+1)=ΦiH(zi1,i),
Φi=(ϕi00ϕi1),
K=(K11K12K21K22)=K23Φ2K12.
(H3+(z23)0)=K(H1+(z12)H1(z12)).
(H2+(z12)H2(z12))=K12(H1+(z12)H1(z12)).
Ex=1jεiωHyz,
Ez=1jεiωHyx.
Mii+1=12(1+μi+1kizμiki+1,z1μi+1kizμiki+1,z1μi+1kizμiki+1,z1+μi+1kizμiki+1,z).
Ei+1(zi,i+1)=Mii+1Ei(zi,i+1).
E(zi,i+1)=ΦiE(zi1,i).
M13=M23Φ2M12,
(E3+(z23)0)=M13(E1+(z12)E1(z12)),
(E2+(z12)E2(z12))=M12(E1+(z12)E1(z12)).
P(3)(ωnl)=i{x,y,z}QPi(3,Q)(ωnl)i,
Pi(3,Q)(ωnl)=D4ε0χ(3)(Ai(Qz)(ωnl)exp(jQzz)Ai(Qz)(ωnl)exp(jQzz)),
Ai(Qz)(ωnl)={j=x,y,z[(Aja+Ajb+)Aic++(Aja+Ajc+)Aib++(Ajb+Ajc+)Aia+],Qz=+kza+kzb+kzcj=x,y,z[(AjaAjb+)Aic++(AjaAjc+)Aib++(Ajb+Ajc+)Aia],Qz=kza+kzb+kzcj=x,y,z[(Aja+Ajb)Aic++(Aja+Ajc+)Aib+(AjbAjc+)Aia+],Qz=+kzakzb+kzcj=x,y,z[(Aja+Ajb+)Aic+(Aja+Ajc)Aib++(Ajb+Ajc)Aia+],Qz=+kza+kzbkzc,
Ai(Qz)(ωnl)={j=x,y,z[(AjaAjb)Aic+(AjaAjc)Aib+(AjbAjc)Aia],Qz=+kza+kzb+kzcj=x,y,z[(Aja+Ajb)Aic+(Aja+Ajc)Aib+(AjbAjc)Aia+],Qz=kza+kzb+kzcj=x,y,z[(AjaAjb+)Aic+(AjaAjc)Aib++(Ajb+Ajc)Aia],Qz=+kzakzb+kzcj=x,y,z[(AjaAjb)Aic++(AjaAjc+)Aib+(AjbAjc+)Aia],Qz=+kza+kzbkzc.
Hs(Q)(r,ωnl)=jω(×P(3,Q)(r,ωnl))Q2+ωnl2ε(ωnl)μ(ωnl),
Es(Q)(r,ωnl)=ω2μ(ωnl)P(3,Q)(r,ωnl)(·P(3,Q)(r,ωnl)ε(ωnl))Q2+ωnl2ε(ωnl)μ(ωnl),
H1(x,z12,ωnl)=K21nlH2(x,z12,ωnl)+QKs1(Q)Hs(Q)(x,z12,ωnl)+QNs1Px(Q)(x,z12,ωnl).
H3(x,z23,ωnl)=K23nlΦ2nlH2(x,z12,ωnl)+QKs3(Q)Φs(Q)Hs(Q)(x,z12,ωnl)+QNs3Φs(Q)Px(Q)(x,z12,ωnl),
Ksi(Q)=12(1+εiQzε2kzi1εiQzε2kzi1εiQzε2kzi1+εiQzε2kzi),
Nsi=12(εiωε2kizεiωε2kizεiωε2kizεiωε2kiz),
Φs(Q)=(exp(jQzd)00exp(jQzd)).
H3(z23)=K13H1(z12)+K23W2(z12),
W2(z12)=Q(Ks2(Q)Φs(Q)Φ2Ks2(Q))Hs(Q)(z12)+Q(Ns2Φs(Q)Φ2Ns2)Px(3,Q)(z12).
(H3+0)K13(0H1)=(1(K13)120(K13)22)(H3+H1)=K23W2.
(H3+(z23)H1(z12))=(1(K13)120(K13)22)1K23W2(z12).
(E3+(z23)E1(z12))=(Fxx+Fzz)(H3+(z23)H1(z12)),
Fx=(k3zε3ω00k1zε1ω),
Fz=(kxε3ω00kxε1ω).
E3(z23)=M13E1(z12)+M23S2(z12),
S2(z12)=Q(Ms2(Q)Φs(Q)Φ2Ms2(Q))Es(Q)(z12),
Ms2(Q)=12(1+Qzk2z1Qzk2z1Qzk2z1+Qzk2z).
(E3+(z23)E1(z12))=(1(M13)120(M13)22)1M23S2(z12).
K1n=Kn1nΦn1K12,
M1n=Mn1nΦn1M12,
Hn(zn1,n)=K1nH1(z1,2)+rKrnWr(zr1,r),
Wr(zr1,r)=Qr(Ksr(Qr)Φs(Qr)ΦrKsr(Qr))Hs(Qr)(zr1,r)+Qr(NsrΦs(Qr)ΦrNsr)Px(Qr)(zr1,r).
(Hn+(zn1,n)H1(z1,2))=(1(K1n)120(K1n)22)1rKrnWr(zr1,r),
En(zn1,n)=M1nE1(z1,2)+rMrnSr(zr1,r),
Sr(zr1,r)=Qr(Msr(Qr)Φs(Qr)ΦrMsr(Qr))Es(Qr)(zr1,r).
(En+(zn1,n)E1(z1,2))=(1(M1n)120(M1n)22)1rMrnSr(zr1,r).
×E=jωB,
×H=jωD,
·D=0,
·B=0,
D=ε(ω)E+PNL,
B=μ(ω)H.
×E=jωμ(ω)H,
×H=jωε(ω)E+jωPNL.
[2ω2ε(ω)μ(ω)]H=ω2μ(×PNL),
[2ω2ε(ω)μ(ω)]E=ω2μPNL(·PNLε),
××H=(·H)2H,
·H=0,
××E=(·E)2E,
·E=·PNL
Hs(α,Q)=jω(×P(α,Q)(r,ωnl))Q2+ωnl2ε(ωnl)μ(ωnl),
Es(α,Q)=ω2μ(ω)P(α,Q)(r,ω)(·P(α,Q)(r,ω)ε(ω))Q2+ω2ε(ω)μ(ω).
Es(α,Q)=×Hs(α,Q)jωε(ω)P(α,Q)ε(ω),
Hs(α,Q)=×Es(α,Q)jωμ(ω).
Hiynl=Hjynl+Hsy(α,Q),
Eixnl=Ejxnl+Esx(α,Q).
Hiy++Hiy=Hjy++Hjy+Hsy(α,Q)++Hsy(α,Q),
Eix++Eix=Ejx++Ejx+Esx(α,Q)++Esx(α,Q).
(Hiy+Hiy)=Kji(Hjy+Hjy)+Ksi(Q)(Hsy(Q)+Hsy(Q))+Nsi(Px(Q)+Px(Q)),
Ksi(Q)=12(1+εiQzεjkzi1εiQzεjkzi1εiQzεjkzi1+εiQzεjkzi),
Nsi=12(εiωεjkziεiωεjkziεiωεjkziεiωεjkzi).
(Eiy+Eiy)=Mji(Ejy+Ejy)+Msi(Q)(Esy(Q)+Esy(Q)),
Msi=12(1+μiQzμjkzi1μiQzμjkzi1μiQzμjkzi1+μiQzμjkzi).

Metrics