Abstract

Dielectric nanocavities are emerging as a versatile and powerful tool for the linear and nonlinear manipulation of light at the nanoscale. In this work, we exploit the effective coupling of electric and toroidal modes in AlGaAs nanodimers to locally enhance both electric and magnetic fields while minimizing the optical scattering, thereby optimizing their second-harmonic generation efficiency with respect to the case of a single isolated nanodisk. We also demonstrate that proper near-field coupling can provide further degrees of freedom to control the polarization state and the radiation diagram of the second-harmonic field.

© 2018 Chinese Laser Press

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  17. L. Ghirardini, L. Carletti, V. Gili, G. Pellegrini, L. Duò, M. M. Finazzi, D. Rocco, A. Locatelli, C. De Angelis, I. Favero, M. Ravaro, G. Leo, A. A. Lemaitre, and M. Celebrano, “Polarization properties of second-harmonic generation in AlGaAs optical nanoantennas,” Opt. Lett. 42, 559–562 (2017).
    [Crossref]
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    [Crossref]
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    [Crossref]
  20. P. Albella, M. A. Poyli, M. K. Schmidt, S. A. Maier, F. Moreno, J. J. Sáenz, and J. Aizpurua, “Low-loss electric and magnetic field-enhanced spectroscopy with subwavelength silicon dimers,” J. Phys. Chem. C 117, 13573–13584 (2013).
    [Crossref]
  21. U. Zywietz, M. K. Schmidt, A. B. Evlyukhin, C. Reinhardt, J. Aizpurua, and B. N. Chichkov, “Electromagnetic resonances of silicon nanoparticle dimers in the visible,” ACS Photon. 2, 913–920 (2015).
    [Crossref]
  22. R. M. Bakker, D. Permyakov, Y. F. Yu, D. Markovich, R. Paniagua-Domínguez, L. Gonzaga, A. Samusev, Y. Kivshar, B. Luk’yanchuk, and A. I. Kuznetsov, “Magnetic and electric hotspots with silicon nanodimers,” Nano Lett. 15, 2137–2142 (2015).
    [Crossref]
  23. S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, “The refractive index of AlxGa1−xAs below the band gap: accurate determination and empirical modeling,” J. Appl. Phys. 87, 7825–7837 (2000).
    [Crossref]
  24. E. E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys. Rev. E 65, 046609 (2002).
    [Crossref]
  25. A. E. Miroshnichenko, A. B. Evlyukhin, Y. Feng Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Commun. 6, 8069 (2015).
    [Crossref]
  26. G. Grinblat, Y. Li, M. P. Nielsen, R. F. Oulton, and S. A. Maier, “Enhanced third harmonic generation in single germanium nanodisks excited at the anapole mode,” Nano Lett. 16, 4635–4640 (2016).
    [Crossref]
  27. D. Bouchet, M. Mivelle, J. Proust, B. Gallas, I. Ozerov, M. F. Garcia-Parajo, A. Gulinatti, I. Rech, Y. De Wilde, N. Bonod, V. Krachmalnicoff, and S. Bidault, “Enhancement and inhibition of spontaneous photon emission by resonant silicon nanoantennas,” Phys. Rev. Appl. 6, 064016 (2016).
    [Crossref]
  28. D. Rocco, L. Carletti, A. Locatelli, and C. De Angelis, “Controlling the directivity of all-dielectric nanoantennas excited by integrated quantum emitters,” J. Opt. Soc. Am. B 34, 1918–1922 (2017).
    [Crossref]
  29. D. de Ceglia, M. A. Vincenti, C. De Angelis, A. Locatelli, J. W. Haus, and M. Scalora, “Role of antenna modes and field enhancement in second harmonic generation from dipole nanoantennas,” Opt. Express 23, 1715–1729 (2015).
    [Crossref]

2018 (1)

A. Krasnok, M. Tymchenko, and A. Alù, “Nonlinear metasurfaces: a paradigm shift in nonlinear optics,” Mater. Today 21, 8–21 (2018).
[Crossref]

2017 (6)

L. Wang, S. Kruk, L. Xu, M. Rahmani, D. Smirnova, A. Solntsev, I. Kravchenko, D. Neshev, and Y. Kivshar, “Shaping the third-harmonic radiation from silicon nanodimers,” Nanoscale 9, 2201–2206 (2017).
[Crossref]

L. Ghirardini, L. Carletti, V. Gili, G. Pellegrini, L. Duò, M. M. Finazzi, D. Rocco, A. Locatelli, C. De Angelis, I. Favero, M. Ravaro, G. Leo, A. A. Lemaitre, and M. Celebrano, “Polarization properties of second-harmonic generation in AlGaAs optical nanoantennas,” Opt. Lett. 42, 559–562 (2017).
[Crossref]

D. Rocco, L. Carletti, A. Locatelli, and C. De Angelis, “Controlling the directivity of all-dielectric nanoantennas excited by integrated quantum emitters,” J. Opt. Soc. Am. B 34, 1918–1922 (2017).
[Crossref]

I. Staude and J. Schilling, “Metamaterial-inspired silicon nanophotonics,” Nat. Photonics 11, 274–284 (2017).
[Crossref]

S. S. Kruk, R. Camacho-Morales, L. Xu, M. Rahmani, D. A. Smirnova, L. Wang, H. H. Tan, C. Jagadish, D. N. Neshev, and Y. S. Kivshar, “Nonlinear optical magnetism revealed by second-harmonic generation in nanoantennas,” Nano Lett. 17, 3914–3918 (2017).
[Crossref]

M. Guasoni, L. Carletti, D. N. Neshev, and C. De Angelis, “Theoretical model for pattern engineering of harmonic generation in all-dielectric nanoantennas,” IEEE J. Quantum Electron. 53, 6100205 (2017).
[Crossref]

2016 (9)

G. Grinblat, Y. Li, M. P. Nielsen, R. F. Oulton, and S. A. Maier, “Enhanced third harmonic generation in single germanium nanodisks excited at the anapole mode,” Nano Lett. 16, 4635–4640 (2016).
[Crossref]

D. Bouchet, M. Mivelle, J. Proust, B. Gallas, I. Ozerov, M. F. Garcia-Parajo, A. Gulinatti, I. Rech, Y. De Wilde, N. Bonod, V. Krachmalnicoff, and S. Bidault, “Enhancement and inhibition of spontaneous photon emission by resonant silicon nanoantennas,” Phys. Rev. Appl. 6, 064016 (2016).
[Crossref]

A. I. Kuznetsov, A. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354, aag2472 (2016).
[Crossref]

A. S. Shorokhov, E. V. Melik-Gaykazyan, D. A. Smirnova, B. Hopkins, K. E. Chong, D.-Y. Choi, M. R. Shcherbakov, A. E. Miroshnichenko, D. N. Neshev, A. A. Fedyanin, and Y. S. Kivshar, “Multifold enhancement of third-harmonic generation in dielectric nanoparticles driven by magnetic Fano resonances,” Nano Lett. 16, 4857–4861 (2016).
[Crossref]

S. Liu, M. B. Sinclair, S. Saravi, G. A. Keler, Y. Yang, J. Reno, F. Setzpfandt, I. Stause, T. Pertsch, and I. Brener, “Resonantly enhanced second-harmonic generation using III-V semiconductor all-dielectric metasurfaces,” Nano Lett. 16, 5426–5432 (2016).
[Crossref]

R. Camacho-Morales, M. Rahmani, S. Kruk, L. Wang, L. Xu, D. A. Smirnova, A. S. Solntsev, A. Miroshnichenko, H. H. Tan, F. Karouta, S. Naureen, K. Vora, L. Carletti, C. De Angelis, C. Jagadish, Y. S. Kivshar, and D. N. Neshev, “Nonlinear generation of vector beams from AlGaAs nanoantennas,” Nano Lett. 16, 7191–7197 (2016).
[Crossref]

H. Linnenbank, Y. Grynko, J. Förstner, and S. Linden, “Second harmonic generation spectroscopy on hybrid plasmonic/dielectric nanoantennas,” Light Sci. Appl. 5, e16013 (2016).
[Crossref]

L. Carletti, A. Locatelli, D. N. Neshev, and C. De Angelis, “Shaping the radiation pattern of second-harmonic generation from AlGaAs dielectric nanoantennas,” ACS Photon. 3, 1500–1507 (2016).
[Crossref]

V. F. Gili, L. Carletti, A. Locatelli, D. Rocco, M. Finazzi, L. Ghirardini, I. Favero, C. Gomez, A. Lemaître, M. Celebrano, C. De Angelis, and G. Leo, “Monolithic AlGaAs second-harmonic nanoantennas,” Opt. Express 24, 15965–15971 (2016).
[Crossref]

2015 (6)

D. de Ceglia, M. A. Vincenti, C. De Angelis, A. Locatelli, J. W. Haus, and M. Scalora, “Role of antenna modes and field enhancement in second harmonic generation from dipole nanoantennas,” Opt. Express 23, 1715–1729 (2015).
[Crossref]

L. Carletti, A. Locatelli, O. Stepanenko, G. Leo, and C. De Angelis, “Enhanced second-harmonic generation from magnetic resonance in AlGaAs nanoantennas,” Opt. Express 23, 26544–26550 (2015).
[Crossref]

M. R. Shcherbakov, A. S. Shorokhov, D. N. Neshev, B. Hopkins, I. Staude, E. V. Melik-Gaykazyan, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Nonlinear interference and tailorable third-harmonic generation from dielectric oligomers,” ACS Photon. 2, 578–582 (2015).
[Crossref]

A. E. Miroshnichenko, A. B. Evlyukhin, Y. Feng Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Commun. 6, 8069 (2015).
[Crossref]

U. Zywietz, M. K. Schmidt, A. B. Evlyukhin, C. Reinhardt, J. Aizpurua, and B. N. Chichkov, “Electromagnetic resonances of silicon nanoparticle dimers in the visible,” ACS Photon. 2, 913–920 (2015).
[Crossref]

R. M. Bakker, D. Permyakov, Y. F. Yu, D. Markovich, R. Paniagua-Domínguez, L. Gonzaga, A. Samusev, Y. Kivshar, B. Luk’yanchuk, and A. I. Kuznetsov, “Magnetic and electric hotspots with silicon nanodimers,” Nano Lett. 15, 2137–2142 (2015).
[Crossref]

2014 (2)

P. Albella, R. Alcaraz de la Osa, F. Moreno, and S. A. Maier, “Electric and magnetic field enhancement with ultralow heat radiation dielectric nanoantennas: considerations for surface-enhanced spectroscopies,” ACS Photon. 1, 524–529 (2014).
[Crossref]

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced third-harmonic generation in silicon nanoparticles driven by magnetic response,” Nano Lett. 14, 6488–6492 (2014).
[Crossref]

2013 (2)

P. Albella, M. A. Poyli, M. K. Schmidt, S. A. Maier, F. Moreno, J. J. Sáenz, and J. Aizpurua, “Low-loss electric and magnetic field-enhanced spectroscopy with subwavelength silicon dimers,” J. Phys. Chem. C 117, 13573–13584 (2013).
[Crossref]

L. Zou, W. Withayachumnankul, C. M. Shah, A. Mitchell, M. Bhaskaran, S. Sriram, and C. Fumeaux, “Dielectric resonator nanoantennas at visible frequencies,” Opt. Express 21, 1344–1352 (2013).
[Crossref]

2012 (1)

2002 (1)

E. E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys. Rev. E 65, 046609 (2002).
[Crossref]

2000 (1)

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, “The refractive index of AlxGa1−xAs below the band gap: accurate determination and empirical modeling,” J. Appl. Phys. 87, 7825–7837 (2000).
[Crossref]

Aizpurua, J.

U. Zywietz, M. K. Schmidt, A. B. Evlyukhin, C. Reinhardt, J. Aizpurua, and B. N. Chichkov, “Electromagnetic resonances of silicon nanoparticle dimers in the visible,” ACS Photon. 2, 913–920 (2015).
[Crossref]

P. Albella, M. A. Poyli, M. K. Schmidt, S. A. Maier, F. Moreno, J. J. Sáenz, and J. Aizpurua, “Low-loss electric and magnetic field-enhanced spectroscopy with subwavelength silicon dimers,” J. Phys. Chem. C 117, 13573–13584 (2013).
[Crossref]

Albella, P.

P. Albella, R. Alcaraz de la Osa, F. Moreno, and S. A. Maier, “Electric and magnetic field enhancement with ultralow heat radiation dielectric nanoantennas: considerations for surface-enhanced spectroscopies,” ACS Photon. 1, 524–529 (2014).
[Crossref]

P. Albella, M. A. Poyli, M. K. Schmidt, S. A. Maier, F. Moreno, J. J. Sáenz, and J. Aizpurua, “Low-loss electric and magnetic field-enhanced spectroscopy with subwavelength silicon dimers,” J. Phys. Chem. C 117, 13573–13584 (2013).
[Crossref]

Alcaraz de la Osa, R.

P. Albella, R. Alcaraz de la Osa, F. Moreno, and S. A. Maier, “Electric and magnetic field enhancement with ultralow heat radiation dielectric nanoantennas: considerations for surface-enhanced spectroscopies,” ACS Photon. 1, 524–529 (2014).
[Crossref]

Alù, A.

A. Krasnok, M. Tymchenko, and A. Alù, “Nonlinear metasurfaces: a paradigm shift in nonlinear optics,” Mater. Today 21, 8–21 (2018).
[Crossref]

Bakker, R. M.

R. M. Bakker, D. Permyakov, Y. F. Yu, D. Markovich, R. Paniagua-Domínguez, L. Gonzaga, A. Samusev, Y. Kivshar, B. Luk’yanchuk, and A. I. Kuznetsov, “Magnetic and electric hotspots with silicon nanodimers,” Nano Lett. 15, 2137–2142 (2015).
[Crossref]

A. E. Miroshnichenko, A. B. Evlyukhin, Y. Feng Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Commun. 6, 8069 (2015).
[Crossref]

Belov, P. A.

Bhaskaran, M.

Bidault, S.

D. Bouchet, M. Mivelle, J. Proust, B. Gallas, I. Ozerov, M. F. Garcia-Parajo, A. Gulinatti, I. Rech, Y. De Wilde, N. Bonod, V. Krachmalnicoff, and S. Bidault, “Enhancement and inhibition of spontaneous photon emission by resonant silicon nanoantennas,” Phys. Rev. Appl. 6, 064016 (2016).
[Crossref]

Bonod, N.

D. Bouchet, M. Mivelle, J. Proust, B. Gallas, I. Ozerov, M. F. Garcia-Parajo, A. Gulinatti, I. Rech, Y. De Wilde, N. Bonod, V. Krachmalnicoff, and S. Bidault, “Enhancement and inhibition of spontaneous photon emission by resonant silicon nanoantennas,” Phys. Rev. Appl. 6, 064016 (2016).
[Crossref]

Bouchet, D.

D. Bouchet, M. Mivelle, J. Proust, B. Gallas, I. Ozerov, M. F. Garcia-Parajo, A. Gulinatti, I. Rech, Y. De Wilde, N. Bonod, V. Krachmalnicoff, and S. Bidault, “Enhancement and inhibition of spontaneous photon emission by resonant silicon nanoantennas,” Phys. Rev. Appl. 6, 064016 (2016).
[Crossref]

Brener, I.

S. Liu, M. B. Sinclair, S. Saravi, G. A. Keler, Y. Yang, J. Reno, F. Setzpfandt, I. Stause, T. Pertsch, and I. Brener, “Resonantly enhanced second-harmonic generation using III-V semiconductor all-dielectric metasurfaces,” Nano Lett. 16, 5426–5432 (2016).
[Crossref]

M. R. Shcherbakov, A. S. Shorokhov, D. N. Neshev, B. Hopkins, I. Staude, E. V. Melik-Gaykazyan, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Nonlinear interference and tailorable third-harmonic generation from dielectric oligomers,” ACS Photon. 2, 578–582 (2015).
[Crossref]

M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced third-harmonic generation in silicon nanoparticles driven by magnetic response,” Nano Lett. 14, 6488–6492 (2014).
[Crossref]

Brongersma, M. L.

A. I. Kuznetsov, A. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354, aag2472 (2016).
[Crossref]

Camacho-Morales, R.

S. S. Kruk, R. Camacho-Morales, L. Xu, M. Rahmani, D. A. Smirnova, L. Wang, H. H. Tan, C. Jagadish, D. N. Neshev, and Y. S. Kivshar, “Nonlinear optical magnetism revealed by second-harmonic generation in nanoantennas,” Nano Lett. 17, 3914–3918 (2017).
[Crossref]

R. Camacho-Morales, M. Rahmani, S. Kruk, L. Wang, L. Xu, D. A. Smirnova, A. S. Solntsev, A. Miroshnichenko, H. H. Tan, F. Karouta, S. Naureen, K. Vora, L. Carletti, C. De Angelis, C. Jagadish, Y. S. Kivshar, and D. N. Neshev, “Nonlinear generation of vector beams from AlGaAs nanoantennas,” Nano Lett. 16, 7191–7197 (2016).
[Crossref]

Carletti, L.

M. Guasoni, L. Carletti, D. N. Neshev, and C. De Angelis, “Theoretical model for pattern engineering of harmonic generation in all-dielectric nanoantennas,” IEEE J. Quantum Electron. 53, 6100205 (2017).
[Crossref]

L. Ghirardini, L. Carletti, V. Gili, G. Pellegrini, L. Duò, M. M. Finazzi, D. Rocco, A. Locatelli, C. De Angelis, I. Favero, M. Ravaro, G. Leo, A. A. Lemaitre, and M. Celebrano, “Polarization properties of second-harmonic generation in AlGaAs optical nanoantennas,” Opt. Lett. 42, 559–562 (2017).
[Crossref]

D. Rocco, L. Carletti, A. Locatelli, and C. De Angelis, “Controlling the directivity of all-dielectric nanoantennas excited by integrated quantum emitters,” J. Opt. Soc. Am. B 34, 1918–1922 (2017).
[Crossref]

V. F. Gili, L. Carletti, A. Locatelli, D. Rocco, M. Finazzi, L. Ghirardini, I. Favero, C. Gomez, A. Lemaître, M. Celebrano, C. De Angelis, and G. Leo, “Monolithic AlGaAs second-harmonic nanoantennas,” Opt. Express 24, 15965–15971 (2016).
[Crossref]

L. Carletti, A. Locatelli, D. N. Neshev, and C. De Angelis, “Shaping the radiation pattern of second-harmonic generation from AlGaAs dielectric nanoantennas,” ACS Photon. 3, 1500–1507 (2016).
[Crossref]

R. Camacho-Morales, M. Rahmani, S. Kruk, L. Wang, L. Xu, D. A. Smirnova, A. S. Solntsev, A. Miroshnichenko, H. H. Tan, F. Karouta, S. Naureen, K. Vora, L. Carletti, C. De Angelis, C. Jagadish, Y. S. Kivshar, and D. N. Neshev, “Nonlinear generation of vector beams from AlGaAs nanoantennas,” Nano Lett. 16, 7191–7197 (2016).
[Crossref]

L. Carletti, A. Locatelli, O. Stepanenko, G. Leo, and C. De Angelis, “Enhanced second-harmonic generation from magnetic resonance in AlGaAs nanoantennas,” Opt. Express 23, 26544–26550 (2015).
[Crossref]

Celebrano, M.

Chichkov, B. N.

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M. R. Shcherbakov, D. N. Neshev, B. Hopkins, A. S. Shorokhov, I. Staude, E. V. Melik-Gaykazyan, M. Decker, A. A. Ezhov, A. E. Miroshnichenko, I. Brener, A. A. Fedyanin, and Y. S. Kivshar, “Enhanced third-harmonic generation in silicon nanoparticles driven by magnetic response,” Nano Lett. 14, 6488–6492 (2014).
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Figures (5)

Fig. 1.
Fig. 1. (a) Extinction efficiency of the isolated cylinder versus the radius of its section: the continuous (dashed) line refers to modeling (experimental data); (b) extinction efficiency decomposed in electric dipole p, magnetic dipole m, toroidal dipole t, electric quadrupole Qe, and magnetic quadrupole Qm contributions. (c) Electric field enhancement for a nanocylinder with radius 320 nm. The white arrows indicate the electric field vector (in-plane components). (d) Magnitude of the volume integral of the normalized nonlinear current density ρSHG inside the cylinder as a function of the radius.
Fig. 2.
Fig. 2. AlGaAs-on-AlOx nanodimers. (a) SEM image of a part of the array. The principal axis of the dimer is oriented parallel to the [100] direction of the AlGaAs crystal. (b) Pictorial view of the proposed structure. Calculated Cartesian decomposition of the dimer photonic modes in the case of light linearly polarized either along (c) the x axis or (d) the y axis for different dimer radii at λ=1550  nm. (e) Phase difference between p and t multipoles in the case of incident light polarized along the x (blue curve) or the y axis (red curve). A black dashed line corresponding to π is plotted as a reference.
Fig. 3.
Fig. 3. (a) Electric–magnetic field distribution in the xy plane at the fundamental wavelength in the dimer structure for the two incident (inc) polarizations, as indicated by the white arrows: for a dimer with (a), (d) a radius of 335 nm and (b), (e) with a radius of 320 nm; in (a) and (b) the height is 200 nm; in (d) and (e) it is 300 nm. (c) Schematic description of the coupling mechanism in the dimer for the two orthogonal incident polarizations: p indicates the electric dipole, t is the toroidal dipole, and M is the magnetic field loop. (f) Electric and magnetic field enhancement as a function of the dimer radius for the x (blue curve) and y (orange curve) incident polarizations.
Fig. 4.
Fig. 4. (a) Numerical calculations of the SH efficiency for x-polarized pump beam (blue line) and y-polarized pump beam (red line) as a function of cylinder radius with respect to the SHG of the single cylinder (black line). (b) Measured SHG for the dimer structure as a function of radius for two pump polarizations: along the x axis (blue curve) and along the y axis (red curve). The black line represents the measured SHG for the single cylinder structure. (c) Calculated volume integral of the nonlinear polarization in the isolated cylinder (black line) and in the dimer for x- (blue line) and y- (red line) polarized pump beams.
Fig. 5.
Fig. 5. Simulated (top) and experimentally obtained (bottom) SH emission patterns decomposed in co-polarized and cross-polarized (with respect to the pump beam) contributions for the dimer structure with r=335  nm [(a) x-polarized pump] and with r=320  nm [(b) y-polarized pump].

Equations (1)

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|ρSHG|=|2ωϵ0i,j,kχijk(2)EjωEkωdV|,

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