Abstract

A frequency-domain method based surface integral equation of the tangential Poggio–Miller–Chang–Harrington–Wu–Tsai formulation is presented as a full wave analysis to evaluate surface second-harmonic generation from noble metal nanoparticles of virtually arbitrary shape. According to the similar solution of fields and boundary conditions at fundamental and second-harmonic frequency, we get the derivation of surface integral equation formulations with only half unknowns compared with the conventional surface integral equation method. Simultaneously, the condition number of impedance matrix has been sharply declined. Numerical examples of gold nanospheres of different radius are presented to demonstrate the accuracy and efficiency of the proposed method. To further research the distribution of surface nonlinear polarization and properties of the second-harmonic radiation, we apply our method to a noncentrosymmetric L-shaped gold nanoparticle studied experimentally. This method provides an efficient and promising approach for evaluation of nonlinear optical radiation generated from metal nanoparticles array and optimization design of nonlinear nanoantennas.

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Mode analysis of second-harmonic generation in plasmonic nanostructures

Gabriel D. Bernasconi, Jérémy Butet, and Olivier J. F. Martin
J. Opt. Soc. Am. B 33(4) 768-779 (2016)

Enforcing symmetries in boundary element formulation of plasmonic and second-harmonic scattering problems

Jouni Mäkitalo, Saku Suuriniemi, and Martti Kauranen
J. Opt. Soc. Am. A 31(12) 2821-2832 (2014)

Surface integral method for second harmonic generation in metal nanoparticles including both local-surface and nonlocal-bulk sources

Carlo Forestiere, Antonio Capretti, and Giovanni Miano
J. Opt. Soc. Am. B 30(9) 2355-2364 (2013)

References

  • View by:
  • |
  • |
  • |

  1. Y. R. Shen, The principle of Nonlinear Optics (John Wiley & Sons Inc.,1984).
  2. M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6, 737–748 (2012).
  3. J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B Condens. Matter 35(17), 9091–9094 (1987).
    [PubMed]
  4. P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B Condens. Matter 33(12), 8254–8263 (1986).
    [PubMed]
  5. F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 308–310 (2009).
  6. C. M. Dissanayake, M. Premaratne, I. D. Rukhlenko, and G. P. Agrawal, “FDTD modeling of anisotropic nonlinear optical phenomena in silicon waveguides,” Opt. Express 18(20), 21427–21448 (2010).
    [PubMed]
  7. M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photonics Technol. Lett. 21, 310–312 (2009).
  8. J. Mäkitalo, S. Suuriniemi, and M. Kauranen, “Boundary element method for surface nonlinear optics of nanoparticles,” Opt. Express 19(23), 23386–23399 (2011).
    [PubMed]
  9. J. Butet, B. Gallinet, K. Thyagarajan, and O. J. F. Martin, “Second harmonic generation from periodic arrays of arbitrary shape plasmonic nanostructures: a surface integral approach,” J. Opt. Soc. Am. B 30, 2970–2979 (2013).
  10. 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).
  11. A. Benedetti, M. Centini, M. Bertolotti, and C. Sibilia, “Second harmonic generation from 3D nanoantennas: on the surface and bulk contributions by far-field pattern analysis,” Opt. Express 19(27), 26752–26767 (2011).
    [PubMed]
  12. C. Forestiere, A. Capretti, and G. Miano, “Surface integral method for second harmonic generation in metal nanoparticles including both local-surface and nonlocal-bulk sources,” J. Opt. Soc. Am. B 30(9), 2355–2364 (2013).
  13. M. Luo and Q. H. Liu, “Enhancement of second-harmonic generation in an air-bridge photonic crystal slab: simulation by spectral element method,” J. Opt. Soc. Am. B 28, 2879–2887 (2011).
  14. X. Y. Z. Xiong, L. J. Jiang, W. E. I. Sha, Y. H. Lo, and W. C. Chew, “Compact Nonlinear Yagi-Uda Nanoantennas,” Sci. Rep. 6, 18872 (2016).
    [PubMed]
  15. J. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., 1999).
  16. R. W. Boyd, Nonlinear Optics, Third Edition (Academic, 2008).
  17. J. Jin, Theory and Computation of Electromagnetic Fields (Wiley, 2010).
  18. T. F. Heinz, Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman, eds. (Elsevier, 1991) Chapter 5, p. 397–405.
  19. 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, 4389–4403 (1980).
  20. P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface Integral Equation Method for General Composite Metallic and Dielectric Structures with Junctions,” Prog. Electromagnetics Res. 52(3), 81–108 (2005).
  21. P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40(6), 5391–5392 (2005).
  22. Z. G. Qian and C. C. Weng, “Fast Full-Wave Surface Integral Equation Solver for Multiscale Structure Modeling,” IEEE Trans. Antenn. Propag. 57(11), 3594–3601 (2009).
  23. P. Yla-Oijala and M. Taskinen, “Application of combined field Integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antenn. Propag. 53(3), 1168–1173 (2005).
  24. R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green’s function or its gradient on a plane triangle,” IEEE Trans. Antenn. Propag. 41(10), 1448–1455 (1993).
  25. P. Yla-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n×RWG functions,” IEEE Trans. Antenn. Propag. 51(8), 1837–1846 (2003).
  26. D. A. Dunavant, “High degree efficient symmetrical Gaussian quadrature rules for the triangle,” Int. J. Numer. Methods Eng. 21(6), 1129–1148 (1985).
  27. P. Johnson and R. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
  28. M. S. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).
  29. Y. Pavlyukh and W. Hubner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70, 245–434 (2004).
  30. S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
    [PubMed]
  31. H. Husu, R. Siikanen, J. Mäkitalo, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Metamaterials with tailored nonlinear optical response,” Nano Lett. 12(2), 673–677 (2012).
    [PubMed]
  32. B. Canfield, S. Kujala, K. Jefimovs, J. Turunen, and M. Kauranen, “Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,” Opt. Express 12(22), 5418–5423 (2004).
    [PubMed]
  33. B. Lambrecht, A. Leitner, and F. R. Aussenegg, “Femtosecond decay-time measurement of electron-plasma oscillation in nanolithographically designed silver particles,” Appl. Phys. B 64(2), 269–272 (1997).
  34. A. Benedetti, M. Centini, M. Bertolotti, and C. Sibilia, “Second harmonic generation from 3D nanoantennas: on the surface and bulk contributions by far-field pattern analysis,” Opt. Express 19(27), 26752–26767 (2011).
    [PubMed]

2016 (1)

X. Y. Z. Xiong, L. J. Jiang, W. E. I. Sha, Y. H. Lo, and W. C. Chew, “Compact Nonlinear Yagi-Uda Nanoantennas,” Sci. Rep. 6, 18872 (2016).
[PubMed]

2013 (2)

2012 (2)

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

H. Husu, R. Siikanen, J. Mäkitalo, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Metamaterials with tailored nonlinear optical response,” Nano Lett. 12(2), 673–677 (2012).
[PubMed]

2011 (4)

2010 (2)

2009 (3)

M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photonics Technol. Lett. 21, 310–312 (2009).

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 308–310 (2009).

Z. G. Qian and C. C. Weng, “Fast Full-Wave Surface Integral Equation Solver for Multiscale Structure Modeling,” IEEE Trans. Antenn. Propag. 57(11), 3594–3601 (2009).

2007 (1)

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[PubMed]

2005 (3)

P. Yla-Oijala and M. Taskinen, “Application of combined field Integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antenn. Propag. 53(3), 1168–1173 (2005).

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface Integral Equation Method for General Composite Metallic and Dielectric Structures with Junctions,” Prog. Electromagnetics Res. 52(3), 81–108 (2005).

P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40(6), 5391–5392 (2005).

2004 (2)

2003 (1)

P. Yla-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n×RWG functions,” IEEE Trans. Antenn. Propag. 51(8), 1837–1846 (2003).

1997 (1)

B. Lambrecht, A. Leitner, and F. R. Aussenegg, “Femtosecond decay-time measurement of electron-plasma oscillation in nanolithographically designed silver particles,” Appl. Phys. B 64(2), 269–272 (1997).

1993 (1)

R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green’s function or its gradient on a plane triangle,” IEEE Trans. Antenn. Propag. 41(10), 1448–1455 (1993).

1987 (1)

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B Condens. Matter 35(17), 9091–9094 (1987).
[PubMed]

1986 (1)

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B Condens. Matter 33(12), 8254–8263 (1986).
[PubMed]

1985 (1)

D. A. Dunavant, “High degree efficient symmetrical Gaussian quadrature rules for the triangle,” Int. J. Numer. Methods Eng. 21(6), 1129–1148 (1985).

1982 (1)

M. S. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).

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, 4389–4403 (1980).

1972 (1)

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

Agrawal, G. P.

Ahorinta, R.

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 308–310 (2009).

Albers, W.

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 308–310 (2009).

Al-Mudhaffar, H. M.

M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photonics Technol. Lett. 21, 310–312 (2009).

Alsunaidi, M. A.

M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photonics Technol. Lett. 21, 310–312 (2009).

Aussenegg, F. R.

B. Lambrecht, A. Leitner, and F. R. Aussenegg, “Femtosecond decay-time measurement of electron-plasma oscillation in nanolithographically designed silver particles,” Appl. Phys. B 64(2), 269–272 (1997).

Benedetti, A.

Bertolotti, M.

Butet, J.

Canfield, B.

Canfield, B. K.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[PubMed]

Capretti, A.

Centini, M.

Chen, W.

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B Condens. Matter 33(12), 8254–8263 (1986).
[PubMed]

Chew, W. C.

X. Y. Z. Xiong, L. J. Jiang, W. E. I. Sha, Y. H. Lo, and W. C. Chew, “Compact Nonlinear Yagi-Uda Nanoantennas,” Sci. Rep. 6, 18872 (2016).
[PubMed]

Christy, R.

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

Dissanayake, C. M.

Dunavant, D. A.

D. A. Dunavant, “High degree efficient symmetrical Gaussian quadrature rules for the triangle,” Int. J. Numer. Methods Eng. 21(6), 1129–1148 (1985).

Forestiere, C.

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, 4389–4403 (1980).

Gallinet, B.

Glisson, A. W.

M. S. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).

Graglia, R. D.

R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green’s function or its gradient on a plane triangle,” IEEE Trans. Antenn. Propag. 41(10), 1448–1455 (1993).

Guyot-Sionnest, P.

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B Condens. Matter 33(12), 8254–8263 (1986).
[PubMed]

Hubner, W.

Y. Pavlyukh and W. Hubner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70, 245–434 (2004).

Husu, H.

H. Husu, R. Siikanen, J. Mäkitalo, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Metamaterials with tailored nonlinear optical response,” Nano Lett. 12(2), 673–677 (2012).
[PubMed]

Järvenpää, S.

P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40(6), 5391–5392 (2005).

Jefimovs, K.

Jiang, L. J.

X. Y. Z. Xiong, L. J. Jiang, W. E. I. Sha, Y. H. Lo, and W. C. Chew, “Compact Nonlinear Yagi-Uda Nanoantennas,” Sci. Rep. 6, 18872 (2016).
[PubMed]

Johnson, P.

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

Kauranen, M.

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

H. Husu, R. Siikanen, J. Mäkitalo, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Metamaterials with tailored nonlinear optical response,” Nano Lett. 12(2), 673–677 (2012).
[PubMed]

J. Mäkitalo, S. Suuriniemi, and M. Kauranen, “Boundary element method for surface nonlinear optics of nanoparticles,” Opt. Express 19(23), 23386–23399 (2011).
[PubMed]

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 308–310 (2009).

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[PubMed]

B. Canfield, S. Kujala, K. Jefimovs, J. Turunen, and M. Kauranen, “Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,” Opt. Express 12(22), 5418–5423 (2004).
[PubMed]

Kuittinen, M.

H. Husu, R. Siikanen, J. Mäkitalo, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Metamaterials with tailored nonlinear optical response,” Nano Lett. 12(2), 673–677 (2012).
[PubMed]

Kujala, S.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[PubMed]

B. Canfield, S. Kujala, K. Jefimovs, J. Turunen, and M. Kauranen, “Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,” Opt. Express 12(22), 5418–5423 (2004).
[PubMed]

Lambrecht, B.

B. Lambrecht, A. Leitner, and F. R. Aussenegg, “Femtosecond decay-time measurement of electron-plasma oscillation in nanolithographically designed silver particles,” Appl. Phys. B 64(2), 269–272 (1997).

Laukkanen, J.

H. Husu, R. Siikanen, J. Mäkitalo, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Metamaterials with tailored nonlinear optical response,” Nano Lett. 12(2), 673–677 (2012).
[PubMed]

Lehtolahti, J.

H. Husu, R. Siikanen, J. Mäkitalo, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Metamaterials with tailored nonlinear optical response,” Nano Lett. 12(2), 673–677 (2012).
[PubMed]

Leitner, A.

B. Lambrecht, A. Leitner, and F. R. Aussenegg, “Femtosecond decay-time measurement of electron-plasma oscillation in nanolithographically designed silver particles,” Appl. Phys. B 64(2), 269–272 (1997).

Liu, Q. H.

Lo, Y. H.

X. Y. Z. Xiong, L. J. Jiang, W. E. I. Sha, Y. H. Lo, and W. C. Chew, “Compact Nonlinear Yagi-Uda Nanoantennas,” Sci. Rep. 6, 18872 (2016).
[PubMed]

Luo, M.

Mäkitalo, J.

H. Husu, R. Siikanen, J. Mäkitalo, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Metamaterials with tailored nonlinear optical response,” Nano Lett. 12(2), 673–677 (2012).
[PubMed]

J. Mäkitalo, S. Suuriniemi, and M. Kauranen, “Boundary element method for surface nonlinear optics of nanoparticles,” Opt. Express 19(23), 23386–23399 (2011).
[PubMed]

Martin, O. J. F.

Masoudi, H. M.

M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photonics Technol. Lett. 21, 310–312 (2009).

Miano, G.

Mizrahi, V.

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B Condens. Matter 35(17), 9091–9094 (1987).
[PubMed]

Pavlyukh, Y.

Y. Pavlyukh and W. Hubner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70, 245–434 (2004).

Premaratne, M.

Qian, Z. G.

Z. G. Qian and C. C. Weng, “Fast Full-Wave Surface Integral Equation Solver for Multiscale Structure Modeling,” IEEE Trans. Antenn. Propag. 57(11), 3594–3601 (2009).

Rao, M. S.

M. S. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).

Rodríguez, F.

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 308–310 (2009).

Rukhlenko, I. D.

Sarvas, J.

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface Integral Equation Method for General Composite Metallic and Dielectric Structures with Junctions,” Prog. Electromagnetics Res. 52(3), 81–108 (2005).

Sha, W. E. I.

X. Y. Z. Xiong, L. J. Jiang, W. E. I. Sha, Y. H. Lo, and W. C. Chew, “Compact Nonlinear Yagi-Uda Nanoantennas,” Sci. Rep. 6, 18872 (2016).
[PubMed]

Shen, Y. R.

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B Condens. Matter 33(12), 8254–8263 (1986).
[PubMed]

Sibilia, C.

Siikanen, R.

H. Husu, R. Siikanen, J. Mäkitalo, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Metamaterials with tailored nonlinear optical response,” Nano Lett. 12(2), 673–677 (2012).
[PubMed]

Sipe, J.

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 308–310 (2009).

Sipe, J. E.

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B Condens. Matter 35(17), 9091–9094 (1987).
[PubMed]

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, 4389–4403 (1980).

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, 4389–4403 (1980).

Stegeman, G. I.

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B Condens. Matter 35(17), 9091–9094 (1987).
[PubMed]

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, 4389–4403 (1980).

Suuriniemi, S.

Svirko, Y.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[PubMed]

Taskinen, M.

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface Integral Equation Method for General Composite Metallic and Dielectric Structures with Junctions,” Prog. Electromagnetics Res. 52(3), 81–108 (2005).

P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40(6), 5391–5392 (2005).

P. Yla-Oijala and M. Taskinen, “Application of combined field Integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antenn. Propag. 53(3), 1168–1173 (2005).

P. Yla-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n×RWG functions,” IEEE Trans. Antenn. Propag. 51(8), 1837–1846 (2003).

Thyagarajan, K.

Turunen, J.

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[PubMed]

B. Canfield, S. Kujala, K. Jefimovs, J. Turunen, and M. Kauranen, “Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,” Opt. Express 12(22), 5418–5423 (2004).
[PubMed]

Wang, F.

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 308–310 (2009).

Weng, C. C.

Z. G. Qian and C. C. Weng, “Fast Full-Wave Surface Integral Equation Solver for Multiscale Structure Modeling,” IEEE Trans. Antenn. Propag. 57(11), 3594–3601 (2009).

Wilton, D. R.

M. S. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).

Xiong, X. Y. Z.

X. Y. Z. Xiong, L. J. Jiang, W. E. I. Sha, Y. H. Lo, and W. C. Chew, “Compact Nonlinear Yagi-Uda Nanoantennas,” Sci. Rep. 6, 18872 (2016).
[PubMed]

Yla-Oijala, P.

P. Yla-Oijala and M. Taskinen, “Application of combined field Integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antenn. Propag. 53(3), 1168–1173 (2005).

P. Yla-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n×RWG functions,” IEEE Trans. Antenn. Propag. 51(8), 1837–1846 (2003).

Ylä-Oijala, P.

P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40(6), 5391–5392 (2005).

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface Integral Equation Method for General Composite Metallic and Dielectric Structures with Junctions,” Prog. Electromagnetics Res. 52(3), 81–108 (2005).

Zayats, A. V.

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

Appl. Phys. B (1)

B. Lambrecht, A. Leitner, and F. R. Aussenegg, “Femtosecond decay-time measurement of electron-plasma oscillation in nanolithographically designed silver particles,” Appl. Phys. B 64(2), 269–272 (1997).

IEEE Photonics Technol. Lett. (1)

M. A. Alsunaidi, H. M. Al-Mudhaffar, and H. M. Masoudi, “Vectorial FDTD technique for the analysis of optical second-harmonic generation,” IEEE Photonics Technol. Lett. 21, 310–312 (2009).

IEEE Trans. Antenn. Propag. (5)

Z. G. Qian and C. C. Weng, “Fast Full-Wave Surface Integral Equation Solver for Multiscale Structure Modeling,” IEEE Trans. Antenn. Propag. 57(11), 3594–3601 (2009).

P. Yla-Oijala and M. Taskinen, “Application of combined field Integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antenn. Propag. 53(3), 1168–1173 (2005).

R. D. Graglia, “On the numerical integration of the linear shape functions times the 3-D Green’s function or its gradient on a plane triangle,” IEEE Trans. Antenn. Propag. 41(10), 1448–1455 (1993).

P. Yla-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix elements with RWG and n×RWG functions,” IEEE Trans. Antenn. Propag. 51(8), 1837–1846 (2003).

M. S. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antenn. Propag. 30(3), 409–418 (1982).

Int. J. Numer. Methods Eng. (1)

D. A. Dunavant, “High degree efficient symmetrical Gaussian quadrature rules for the triangle,” Int. J. Numer. Methods Eng. 21(6), 1129–1148 (1985).

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

Nano Lett. (1)

H. Husu, R. Siikanen, J. Mäkitalo, J. Lehtolahti, J. Laukkanen, M. Kuittinen, and M. Kauranen, “Metamaterials with tailored nonlinear optical response,” Nano Lett. 12(2), 673–677 (2012).
[PubMed]

Nat. Photonics (1)

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

Opt. Express (5)

Phys. Rev. B (4)

Y. Pavlyukh and W. Hubner, “Nonlinear Mie scattering from spherical particles,” Phys. Rev. B 70, 245–434 (2004).

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, 4389–4403 (1980).

F. Wang, F. Rodríguez, W. Albers, R. Ahorinta, J. Sipe, and M. Kauranen, “Surface and bulk contributions to the second-order nonlinear optical response of a gold film,” Phys. Rev. B 80(23), 308–310 (2009).

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

Phys. Rev. B Condens. Matter (2)

J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of second-harmonic generation as a strictly surface probe,” Phys. Rev. B Condens. Matter 35(17), 9091–9094 (1987).
[PubMed]

P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General considerations on optical second-harmonic generation from surfaces and interfaces,” Phys. Rev. B Condens. Matter 33(12), 8254–8263 (1986).
[PubMed]

Phys. Rev. Lett. (1)

S. Kujala, B. K. Canfield, M. Kauranen, Y. Svirko, and J. Turunen, “Multipole interference in the second-harmonic optical radiation from gold nanoparticles,” Phys. Rev. Lett. 98(16), 167403 (2007).
[PubMed]

Prog. Electromagnetics Res. (1)

P. Ylä-Oijala, M. Taskinen, and J. Sarvas, “Surface Integral Equation Method for General Composite Metallic and Dielectric Structures with Junctions,” Prog. Electromagnetics Res. 52(3), 81–108 (2005).

Radio Sci. (1)

P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods,” Radio Sci. 40(6), 5391–5392 (2005).

Sci. Rep. (1)

X. Y. Z. Xiong, L. J. Jiang, W. E. I. Sha, Y. H. Lo, and W. C. Chew, “Compact Nonlinear Yagi-Uda Nanoantennas,” Sci. Rep. 6, 18872 (2016).
[PubMed]

Other (5)

J. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., 1999).

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

J. Jin, Theory and Computation of Electromagnetic Fields (Wiley, 2010).

T. F. Heinz, Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath and G. I. Stegeman, eds. (Elsevier, 1991) Chapter 5, p. 397–405.

Y. R. Shen, The principle of Nonlinear Optics (John Wiley & Sons Inc.,1984).

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 (4)

Fig. 1
Fig. 1 The results of spherical gold particle of radius r = 50nm. (a) SCS at fundament frequency. SCS at SH frequency with (b) ( χ , χ |||| )=(1,0), (c) ( χ , χ |||| )=(0,1). (d) Comparison of convergence property.
Fig. 2
Fig. 2 The results of spherical gold particle with radius (a)-(c) 10nm, (d)-(f) 20nm, (g)-(i) 100nm. SCS at SH frequency with (b)(d)(g) ( χ , χ |||| )=(1,0), (c)(e)(h) ( χ , χ |||| )=(0,1). (d)(f)(i) Comparison of convergence property.
Fig. 3
Fig. 3 The rounded-edge L-shaped particle.
Fig. 4
Fig. 4 The nonlinear surface polarization (a)(b) and SH radiation pattern (c)(d) from an L-shaped gold nanoparticles, with (a)(c) corresponds to incident wave with x-polarization and (b)(d) corresponds to incident wave with y-polarization.

Tables (2)

Tables Icon

Table 1 Comparison of Two Methods for SHG of Spherical Nanoparticles r = 50nm

Tables Icon

Table 2 Comparison of Two Methods for SHG of Spherical Nanoparticles r = 10nm, 20nm and 100nm

Equations (25)

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

{ E e (ω,sca) = E e (ω) E 0 (ω,inc) H e (ω,sca) = H e (ω) H 0 (ω,inc) in V e { E i (ω,sca) = E i (ω) H i (ω,sca) = H i (ω) in V i
{ × E l (ω,sca) =jω μ l H l (ω,sca) × H l (ω,sca) =jω ε l (ω) E l (ω,sca) in V l ,withl=i,e, { n×( E e (ω,sca) E i (ω,sca) )=n× E 0 (ω,inc) n×( H e (ω,sca) H i (ω,sca) )=n× H 0 (ω,inc) onS
P S = ε 0 χ (2) : E i (ω) E i (ω)
P S = ε 0 [ χ (2) nnn+ χ |||| (2) (n t 1 t 1 +n t 2 t 2 )+ χ |||| (2) ( t 1 n t 1 + t 2 n t 2 ) ]: E i (ω) E i (ω)
{ × E l (2ω) =j2ω μ l H l (2ω) × H l (2ω) =j2ω ε l (2ω) E l (2ω) in V l ,withl=i,e, { n×( E e (2ω) E i (2ω) )=j2ω P t S n×( H e (2ω) H i (2ω) )= 1 ε n× S P n S onS
C 2 (ν) x 2 (ν) = y 2 (ν)
C 2 (ν) =( L e (ν) K e (ν) L i (ν) K i (ν) K e (ν) η e 2 L e (ν) K i (ν) η i 2 L i (ν) I 0 I 0 0 I 0 I ), x 2 (ν) =( j e (ν,e) j e (ν,m) j i (ν,e) j i (ν,m) ), y 2 (ν) =( 1 2 n× π S (ν,m) 1 2 n× π S (ν,e) π S (ν,e) π S (ν,m) )
{ π S (ω,e) =n× H 0 (ω,inc) π S (ω,m) =n× E 0 (ω,inc) ,{ π S (2ω,m) =j2ω P t S π S (2ω,e) = 1 ε n× S P n S
L l (ν) { X(r) }=n×n×iν μ l S G l (ν) (r r )X( r )d S n×n× 1 iν ε l S G l (ν) (r r ) S X( r )d S
K l (ν) { X(r) }=n×n× S X( r )× G l (ν) (r r ) d S
{ × E l (ω) =jω μ l H l (ω) × H l (ω) =jω ε l (ω) E l (ω) in V l ,withl=i,e, { n×( E e (ω) E i (ω) )=n×( E e (ω,sca) + E 0 (ω,inc) E i (ω,sca) )=0 n×( H e (ω) H i (ω) )=n×( H e (ω,sca) + H 0 (ω,inc) H i (ω,sca) )=0 onS
n×( E i (ω) E e (ω) )=0,n×( H i (ω) H e (ω) )=0
{ J e (ω) = n 1 × H e (ω) M e (ω) = n 1 × E e (ω) ,{ J i (ω) = n 2 × H i (ω) M i (ω) = n 2 × E i (ω)
J e (ω) = J i (ω) , M e (ω) = M i (ω)
{ j e (ω,e) =n× H e (ω,sca) j e (ω,m) =n× E e (ω,sca) j i (ω,e) =n× H i (ω,sca) j i (ω,m) =n× E i (ω,sca) ,{ J e (ω) =n× H e (ω) M e (ω) =n× E e (ω) J i (ω) =n× H i (ω) M i (ω) =n× E i (ω)
{ J e (ω) = j e (ω,e) +n× H 0 (ω,inc) ( a ) M e (ω) = j e (ω,m) n× E 0 (ω,inc) ( b ) J i (ω) = j i (ω,e) =n× H i (ω,sca) ( c ) M i (ω) = j i (ω,m) =n× E i (ω,sca) ( d )
J e (ω,inc) = J i (ω,inc) , M e (ω,inc) = M i (ω,inc)
{ E ie (ω,sca) = L e (ω) ( J i (ω,inc) )+ K e (ω) ( M i (ω,inc) )=0 H ie (ω,sca) = K e (ω) ( J i (ω,inc) )+ η e 2 L e (ω) ( M i (ω,inc) )=0
{ E ee (ω,sca) = L e (ω) ( J e (ω,inc) )+ K e (ω) ( M e (ω,inc) )=0 H ee (ω,sca) = K e (ω) ( J e (ω,inc) )+ η e 2 L e (ω) ( M e (ω,inc) )=0
{ × E l (2ω) =jω μ l H l (2ω) × H l (2ω) =jω ε l (2ω) E l (2ω) in V l ,withl=i,e, { n×( E e (2ω) E i (2ω) )=n×( E e (2ω,sca) + E r (2ω,inc) E i (2ω,sca) )=0 n×( H e (2ω) H i (2ω) )=n×( H e (2ω,sca) + H r (2ω,inc) H i (2ω,sca) )=0 onS
{ E r (2ω,inc) =( L i (ω) + L e (ω) ) π S (2ω,e) +( K i (ω) + K e (ω) ) π S (2ω,m) H r (2ω,inc) =( K i (ω) K e (ω) ) π S (2ω,e) +( η i 2 L i (ω) + η e 2 L e (ω) ) π S (2ω,m)
C 1 (ν) x 1 (ν) = y 1 (ν)
C 1 (ν) =( L e (ν) + L i (ν) K e (ν) + K i (ν) K e (ν) K i (ν) η e 2 L e (ν) + η i 2 L i (ν) ), x 1 (ν) =( J (ν) M (ν) ),
y 1 (ω) =( n×n× E 0 (ω,inc) n×n× H 0 (ω,inc) ), y 1 (2ω) =( n×n× E r (2ω,inc) n×n× H r (2ω,inc) )
σ= lim r 1 4π r 2 | E s | 2 | E i | 2

Metrics