Abstract

We study theoretically various design considerations for efficient generation of second harmonic using a nonlinear substrate patterned with nano-antennas. The analysis is focused on a gap Bowtie nano-antenna array recessed in LiNbO3 which is shown to be preferable over on surface structures due to field enhancement, field profile and linear and non-linear polarization considerations. In addition, we develop the nano-antenna counterpart of the Boyd-Klienmann model in order to analyze the impact of a Gaussian shaped fundamental beam on the generated second harmonic. Finally, we show that the dielectric properties of the substrate lead to preferable directions for the incident fundamental harmonic and the emission of the second harmonic. Our analyses lead to several design rules which can enhance second and high harmonic generation from nano-antennas arrays by several orders of magnitude.

© 2013 Optical Society of America

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2013 (1)

2012 (4)

2011 (4)

2010 (2)

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. A82(4), 043828 (2010).
[CrossRef]

E. H. Barakat, M. P. Bernal, and F. I. Baida, “Second harmonic generation enhancement by use of annular aperture arrays embedded into silver and filled by lithium niobate,” Opt. Express18(7), 6530–6536 (2010).
[CrossRef] [PubMed]

2009 (3)

2008 (3)

P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett.101, 116805 (2008).

S. Kim, J. H. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature453(7196), 757–760 (2008).
[CrossRef] [PubMed]

A. Alù and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics2(5), 307–310 (2008).
[CrossRef]

2007 (2)

M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett.98(2), 026104 (2007).
[CrossRef] [PubMed]

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics1(11), 641–648 (2007).
[CrossRef]

2006 (2)

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett.6(5), 1027–1030 (2006).
[CrossRef]

W. Fan, S. Zhang, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Second harmonic generation from patterned GaAs inside a subwavelength metallic hole array,” Opt. Express14(21), 9570–9575 (2006).
[CrossRef] [PubMed]

2005 (3)

F. J. González and G. D. Boreman, “Comparison of dipole, bowtie, spiral and log-periodic IR antennas,” Infrared Phys. Technol.46(5), 418–428 (2005).
[CrossRef]

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005).
[CrossRef] [PubMed]

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nano-antennas,” Phys. Rev. Lett.94(1), 017402 (2005).
[CrossRef] [PubMed]

2004 (1)

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nano-antennas resonant in the visible,” Nano Lett.4(5), 957–961 (2004).
[CrossRef]

2003 (2)

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett.90(1), 013903 (2003).
[CrossRef] [PubMed]

A. Bouhelier, M. R. Beversluis, and L. Novotny, “Characterization of nanoplasmonic structures by locally excited photoluminescence,” Appl. Phys. Lett.83(24), 5041–5043 (2003).
[CrossRef]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Technol.47(11), 2075–2084 (1999).
[CrossRef]

1997 (1)

R. D. Grober, R. J. Schoelkopf, and D. E. Prober, “Optical antenna: Towards a unity efficiency near-field optical probe,” Appl. Phys. Lett.70(11), 1354–1356 (1997).
[CrossRef]

1987 (1)

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, and D. B. Rutledge, “Bow-tie antennas on a dielectric half-space: theory and experiment,” IEEE Trans. Antenn. Propag.35(6), 622–631 (1987).
[CrossRef]

1985 (1)

M. Kominami, D. M. Pozar, and D. H. Schaubert, “Dipole and slot elements and arrays on semi-infinite substrates,” IEEE Trans. Antenn. Propag.33(6), 600–607 (1985).
[CrossRef]

1974 (1)

D. F. Nelson and R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys.45(8), 3688–3689 (1974).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys.39(8), 3597–3639 (1968).
[CrossRef]

1909 (1)

A. Sommerfeld, “Über die ausbreitung der wellen in der drahtlosen telegraphie,” Ann. Phys.333(4), 665–736 (1909).
[CrossRef]

Abdenour, A.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett.6(5), 1027–1030 (2006).
[CrossRef]

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. A82(4), 043828 (2010).
[CrossRef]

Alù, A.

A. Alù and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics2(5), 307–310 (2008).
[CrossRef]

Ambekar, R.

K. D. Ko, A. Kumar, K. H. Fung, R. Ambekar, G. L. Liu, N. X. Fang, and K. C. Toussaint., “Nonlinear optical response from arrays of Au bowtie nanoantennas,” Nano Lett.11(1), 61–65 (2011).
[CrossRef] [PubMed]

Bachelier, G.

Baida, F. I.

Barakat, E. H.

Bar-Lev, D.

Bernal, M. P.

Berthelot, J.

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(1), 013903 (2003).
[CrossRef] [PubMed]

Beversluis, M. R.

A. Bouhelier, M. R. Beversluis, and L. Novotny, “Characterization of nanoplasmonic structures by locally excited photoluminescence,” Appl. Phys. Lett.83(24), 5041–5043 (2003).
[CrossRef]

Bharadwaj, P.

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. A82(4), 043828 (2010).
[CrossRef]

Boag, A.

Y. Yifat, Z. Iluz, D. Bar-Lev, M. Eitan, Y. Hanein, A. Boag, and J. Scheuer, “High load sensitivity in wideband infrared dual-Vivaldi nanoantennas,” Opt. Lett.38(2), 205–207 (2013).
[CrossRef] [PubMed]

Y. Yifat, Z. Iluz, M. Eitan, I. Friedler, Y. Hanein, A. Boag, and J. Scheuer, “Quantifying the radiation efficiency of nano antennas,” Appl. Phys. Lett.100(11), 111113 (2012).
[CrossRef]

Boreman, G. D.

F. J. González and G. D. Boreman, “Comparison of dipole, bowtie, spiral and log-periodic IR antennas,” Infrared Phys. Technol.46(5), 418–428 (2005).
[CrossRef]

Bouhelier, A.

J. Berthelot, G. Bachelier, M. Song, P. Rai, G. Colas des Francs, A. Dereux, and A. Bouhelier, “Silencing and enhancement of second-harmonic generation in optical gap antennas,” Opt. Express20(10), 10498–10508 (2012).
[CrossRef] [PubMed]

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett.90(1), 013903 (2003).
[CrossRef] [PubMed]

A. Bouhelier, M. R. Beversluis, and L. Novotny, “Characterization of nanoplasmonic structures by locally excited photoluminescence,” Appl. Phys. Lett.83(24), 5041–5043 (2003).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys.39(8), 3597–3639 (1968).
[CrossRef]

Brueck, S. R. J.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett.6(5), 1027–1030 (2006).
[CrossRef]

W. Fan, S. Zhang, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Second harmonic generation from patterned GaAs inside a subwavelength metallic hole array,” Opt. Express14(21), 9570–9575 (2006).
[CrossRef] [PubMed]

Busch, K.

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. A82(4), 043828 (2010).
[CrossRef]

Cherukulappurath, S.

P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett.101, 116805 (2008).

Colas des Francs, G.

Compton, R. C.

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, and D. B. Rutledge, “Bow-tie antennas on a dielectric half-space: theory and experiment,” IEEE Trans. Antenn. Propag.35(6), 622–631 (1987).
[CrossRef]

Danckwerts, M.

M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett.98(2), 026104 (2007).
[CrossRef] [PubMed]

de Ceglia, D.

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express19(3), 2064–2078 (2011).
[CrossRef] [PubMed]

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. A82(4), 043828 (2010).
[CrossRef]

Dereux, A.

Deutsch, B.

Eisler, H. J.

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005).
[CrossRef] [PubMed]

Eitan, M.

Y. Yifat, Z. Iluz, D. Bar-Lev, M. Eitan, Y. Hanein, A. Boag, and J. Scheuer, “High load sensitivity in wideband infrared dual-Vivaldi nanoantennas,” Opt. Lett.38(2), 205–207 (2013).
[CrossRef] [PubMed]

Y. Yifat, Z. Iluz, M. Eitan, I. Friedler, Y. Hanein, A. Boag, and J. Scheuer, “Quantifying the radiation efficiency of nano antennas,” Appl. Phys. Lett.100(11), 111113 (2012).
[CrossRef]

Engheta, N.

A. Alù and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics2(5), 307–310 (2008).
[CrossRef]

Fan, W.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett.6(5), 1027–1030 (2006).
[CrossRef]

W. Fan, S. Zhang, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Second harmonic generation from patterned GaAs inside a subwavelength metallic hole array,” Opt. Express14(21), 9570–9575 (2006).
[CrossRef] [PubMed]

Fang, N. X.

K. D. Ko, A. Kumar, K. H. Fung, R. Ambekar, G. L. Liu, N. X. Fang, and K. C. Toussaint., “Nonlinear optical response from arrays of Au bowtie nanoantennas,” Nano Lett.11(1), 61–65 (2011).
[CrossRef] [PubMed]

Feth, N.

Friedler, I.

Y. Yifat, Z. Iluz, M. Eitan, I. Friedler, Y. Hanein, A. Boag, and J. Scheuer, “Quantifying the radiation efficiency of nano antennas,” Appl. Phys. Lett.100(11), 111113 (2012).
[CrossRef]

Fromm, D. P.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nano-antennas,” Phys. Rev. Lett.94(1), 017402 (2005).
[CrossRef] [PubMed]

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nano-antennas resonant in the visible,” Nano Lett.4(5), 957–961 (2004).
[CrossRef]

Fung, K. H.

K. D. Ko, A. Kumar, K. H. Fung, R. Ambekar, G. L. Liu, N. X. Fang, and K. C. Toussaint., “Nonlinear optical response from arrays of Au bowtie nanoantennas,” Nano Lett.11(1), 61–65 (2011).
[CrossRef] [PubMed]

Ghenuche, P.

P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett.101, 116805 (2008).

Gieseler, J.

González, F. J.

F. J. González and G. D. Boreman, “Comparison of dipole, bowtie, spiral and log-periodic IR antennas,” Infrared Phys. Technol.46(5), 418–428 (2005).
[CrossRef]

Grober, R. D.

R. D. Grober, R. J. Schoelkopf, and D. E. Prober, “Optical antenna: Towards a unity efficiency near-field optical probe,” Appl. Phys. Lett.70(11), 1354–1356 (1997).
[CrossRef]

Hahn, J. W.

Halas, N. J.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics1(11), 641–648 (2007).
[CrossRef]

Hanein, Y.

Y. Yifat, Z. Iluz, D. Bar-Lev, M. Eitan, Y. Hanein, A. Boag, and J. Scheuer, “High load sensitivity in wideband infrared dual-Vivaldi nanoantennas,” Opt. Lett.38(2), 205–207 (2013).
[CrossRef] [PubMed]

Y. Yifat, Z. Iluz, M. Eitan, I. Friedler, Y. Hanein, A. Boag, and J. Scheuer, “Quantifying the radiation efficiency of nano antennas,” Appl. Phys. Lett.100(11), 111113 (2012).
[CrossRef]

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(1), 013903 (2003).
[CrossRef] [PubMed]

Hecht, B.

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005).
[CrossRef] [PubMed]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Technol.47(11), 2075–2084 (1999).
[CrossRef]

Iluz, Z.

Y. Yifat, Z. Iluz, D. Bar-Lev, M. Eitan, Y. Hanein, A. Boag, and J. Scheuer, “High load sensitivity in wideband infrared dual-Vivaldi nanoantennas,” Opt. Lett.38(2), 205–207 (2013).
[CrossRef] [PubMed]

Y. Yifat, Z. Iluz, M. Eitan, I. Friedler, Y. Hanein, A. Boag, and J. Scheuer, “Quantifying the radiation efficiency of nano antennas,” Appl. Phys. Lett.100(11), 111113 (2012).
[CrossRef]

Jin, J. H.

S. Kim, J. H. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature453(7196), 757–760 (2008).
[CrossRef] [PubMed]

Kim, S.

S. Kim, J. H. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature453(7196), 757–760 (2008).
[CrossRef] [PubMed]

Kim, S. W.

S. Kim, J. H. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature453(7196), 757–760 (2008).
[CrossRef] [PubMed]

Kim, Y.

S. Kim, J. H. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature453(7196), 757–760 (2008).
[CrossRef] [PubMed]

Kim, Y. J.

S. Kim, J. H. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature453(7196), 757–760 (2008).
[CrossRef] [PubMed]

Kino, G.

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nano-antennas resonant in the visible,” Nano Lett.4(5), 957–961 (2004).
[CrossRef]

Kino, G. S.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nano-antennas,” Phys. Rev. Lett.94(1), 017402 (2005).
[CrossRef] [PubMed]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys.39(8), 3597–3639 (1968).
[CrossRef]

Ko, K. D.

K. D. Ko, A. Kumar, K. H. Fung, R. Ambekar, G. L. Liu, N. X. Fang, and K. C. Toussaint., “Nonlinear optical response from arrays of Au bowtie nanoantennas,” Nano Lett.11(1), 61–65 (2011).
[CrossRef] [PubMed]

Kominami, M.

M. Kominami, D. M. Pozar, and D. H. Schaubert, “Dipole and slot elements and arrays on semi-infinite substrates,” IEEE Trans. Antenn. Propag.33(6), 600–607 (1985).
[CrossRef]

Krishna, S.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett.6(5), 1027–1030 (2006).
[CrossRef]

Kumar, A.

K. D. Ko, A. Kumar, K. H. Fung, R. Ambekar, G. L. Liu, N. X. Fang, and K. C. Toussaint., “Nonlinear optical response from arrays of Au bowtie nanoantennas,” Nano Lett.11(1), 61–65 (2011).
[CrossRef] [PubMed]

Lal, S.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics1(11), 641–648 (2007).
[CrossRef]

Lee, J. Y.

Linden, S.

Link, S.

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics1(11), 641–648 (2007).
[CrossRef]

Liu, G. L.

K. D. Ko, A. Kumar, K. H. Fung, R. Ambekar, G. L. Liu, N. X. Fang, and K. C. Toussaint., “Nonlinear optical response from arrays of Au bowtie nanoantennas,” Nano Lett.11(1), 61–65 (2011).
[CrossRef] [PubMed]

Lovera, A.

Malloy, K. J.

W. Fan, S. Zhang, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Second harmonic generation from patterned GaAs inside a subwavelength metallic hole array,” Opt. Express14(21), 9570–9575 (2006).
[CrossRef] [PubMed]

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett.6(5), 1027–1030 (2006).
[CrossRef]

Martin, O. J. F.

McPhedran, R. C.

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, and D. B. Rutledge, “Bow-tie antennas on a dielectric half-space: theory and experiment,” IEEE Trans. Antenn. Propag.35(6), 622–631 (1987).
[CrossRef]

Mikulyak, R. M.

D. F. Nelson and R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys.45(8), 3688–3689 (1974).
[CrossRef]

Moerner, W. E.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nano-antennas,” Phys. Rev. Lett.94(1), 017402 (2005).
[CrossRef] [PubMed]

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nano-antennas resonant in the visible,” Nano Lett.4(5), 957–961 (2004).
[CrossRef]

Mühlschlegel, P.

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005).
[CrossRef] [PubMed]

Nelson, D. F.

D. F. Nelson and R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys.45(8), 3688–3689 (1974).
[CrossRef]

Niegemann, J.

Niesler, F. B. P.

Novotny, L.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics5(2), 83–90 (2011).
[CrossRef]

S. Palomba and L. Novotny, “Near-field imaging with a localized nonlinear light source,” Nano Lett.9(11), 3801–3804 (2009).
[CrossRef] [PubMed]

P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon.1(3), 438–483 (2009).
[CrossRef]

M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett.98(2), 026104 (2007).
[CrossRef] [PubMed]

A. Bouhelier, M. R. Beversluis, and L. Novotny, “Characterization of nanoplasmonic structures by locally excited photoluminescence,” Appl. Phys. Lett.83(24), 5041–5043 (2003).
[CrossRef]

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett.90(1), 013903 (2003).
[CrossRef] [PubMed]

Osgood, R. M.

W. Fan, S. Zhang, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Second harmonic generation from patterned GaAs inside a subwavelength metallic hole array,” Opt. Express14(21), 9570–9575 (2006).
[CrossRef] [PubMed]

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett.6(5), 1027–1030 (2006).
[CrossRef]

Palomba, S.

S. Palomba and L. Novotny, “Near-field imaging with a localized nonlinear light source,” Nano Lett.9(11), 3801–3804 (2009).
[CrossRef] [PubMed]

Panoiu, N. C.

Panoiu, N.-C.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett.6(5), 1027–1030 (2006).
[CrossRef]

Park, I. Y.

S. Kim, J. H. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature453(7196), 757–760 (2008).
[CrossRef] [PubMed]

Park, S.

Pendry, J. B.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Technol.47(11), 2075–2084 (1999).
[CrossRef]

Pohl, D. W.

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005).
[CrossRef] [PubMed]

Popovic, Z.

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, and D. B. Rutledge, “Bow-tie antennas on a dielectric half-space: theory and experiment,” IEEE Trans. Antenn. Propag.35(6), 622–631 (1987).
[CrossRef]

Pozar, D. M.

M. Kominami, D. M. Pozar, and D. H. Schaubert, “Dipole and slot elements and arrays on semi-infinite substrates,” IEEE Trans. Antenn. Propag.33(6), 600–607 (1985).
[CrossRef]

Prober, D. E.

R. D. Grober, R. J. Schoelkopf, and D. E. Prober, “Optical antenna: Towards a unity efficiency near-field optical probe,” Appl. Phys. Lett.70(11), 1354–1356 (1997).
[CrossRef]

Quidant, R.

P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett.101, 116805 (2008).

Rai, P.

Rebeiz, G. M.

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, and D. B. Rutledge, “Bow-tie antennas on a dielectric half-space: theory and experiment,” IEEE Trans. Antenn. Propag.35(6), 622–631 (1987).
[CrossRef]

Rivier, S.

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Technol.47(11), 2075–2084 (1999).
[CrossRef]

Roppo, V.

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express19(3), 2064–2078 (2011).
[CrossRef] [PubMed]

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. A82(4), 043828 (2010).
[CrossRef]

Rutledge, D. B.

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, and D. B. Rutledge, “Bow-tie antennas on a dielectric half-space: theory and experiment,” IEEE Trans. Antenn. Propag.35(6), 622–631 (1987).
[CrossRef]

Scalora, M.

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express19(3), 2064–2078 (2011).
[CrossRef] [PubMed]

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. A82(4), 043828 (2010).
[CrossRef]

Schaubert, D. H.

M. Kominami, D. M. Pozar, and D. H. Schaubert, “Dipole and slot elements and arrays on semi-infinite substrates,” IEEE Trans. Antenn. Propag.33(6), 600–607 (1985).
[CrossRef]

Scheuer, J.

Schoelkopf, R. J.

R. D. Grober, R. J. Schoelkopf, and D. E. Prober, “Optical antenna: Towards a unity efficiency near-field optical probe,” Appl. Phys. Lett.70(11), 1354–1356 (1997).
[CrossRef]

Schuck, P. J.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nano-antennas,” Phys. Rev. Lett.94(1), 017402 (2005).
[CrossRef] [PubMed]

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nano-antennas resonant in the visible,” Nano Lett.4(5), 957–961 (2004).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, “Über die ausbreitung der wellen in der drahtlosen telegraphie,” Ann. Phys.333(4), 665–736 (1909).
[CrossRef]

Song, M.

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Technol.47(11), 2075–2084 (1999).
[CrossRef]

Sundaramurthy, A.

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nano-antennas,” Phys. Rev. Lett.94(1), 017402 (2005).
[CrossRef] [PubMed]

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nano-antennas resonant in the visible,” Nano Lett.4(5), 957–961 (2004).
[CrossRef]

Taminiau, T. H.

P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett.101, 116805 (2008).

Thyagarajan, K.

Tong, P. P.

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, and D. B. Rutledge, “Bow-tie antennas on a dielectric half-space: theory and experiment,” IEEE Trans. Antenn. Propag.35(6), 622–631 (1987).
[CrossRef]

Toussaint, K. C.

K. D. Ko, A. Kumar, K. H. Fung, R. Ambekar, G. L. Liu, N. X. Fang, and K. C. Toussaint., “Nonlinear optical response from arrays of Au bowtie nanoantennas,” Nano Lett.11(1), 61–65 (2011).
[CrossRef] [PubMed]

van Hulst, N.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics5(2), 83–90 (2011).
[CrossRef]

van Hulst, N. F.

P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett.101, 116805 (2008).

Vincenti, M. A.

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express19(3), 2064–2078 (2011).
[CrossRef] [PubMed]

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. A82(4), 043828 (2010).
[CrossRef]

Wegener, M.

Yifat, Y.

Y. Yifat, Z. Iluz, D. Bar-Lev, M. Eitan, Y. Hanein, A. Boag, and J. Scheuer, “High load sensitivity in wideband infrared dual-Vivaldi nanoantennas,” Opt. Lett.38(2), 205–207 (2013).
[CrossRef] [PubMed]

Y. Yifat, Z. Iluz, M. Eitan, I. Friedler, Y. Hanein, A. Boag, and J. Scheuer, “Quantifying the radiation efficiency of nano antennas,” Appl. Phys. Lett.100(11), 111113 (2012).
[CrossRef]

Zhang, S.

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett.6(5), 1027–1030 (2006).
[CrossRef]

W. Fan, S. Zhang, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Second harmonic generation from patterned GaAs inside a subwavelength metallic hole array,” Opt. Express14(21), 9570–9575 (2006).
[CrossRef] [PubMed]

Adv. Opt. Photon. (1)

Ann. Phys. (1)

A. Sommerfeld, “Über die ausbreitung der wellen in der drahtlosen telegraphie,” Ann. Phys.333(4), 665–736 (1909).
[CrossRef]

Appl. Phys. Lett. (3)

R. D. Grober, R. J. Schoelkopf, and D. E. Prober, “Optical antenna: Towards a unity efficiency near-field optical probe,” Appl. Phys. Lett.70(11), 1354–1356 (1997).
[CrossRef]

A. Bouhelier, M. R. Beversluis, and L. Novotny, “Characterization of nanoplasmonic structures by locally excited photoluminescence,” Appl. Phys. Lett.83(24), 5041–5043 (2003).
[CrossRef]

Y. Yifat, Z. Iluz, M. Eitan, I. Friedler, Y. Hanein, A. Boag, and J. Scheuer, “Quantifying the radiation efficiency of nano antennas,” Appl. Phys. Lett.100(11), 111113 (2012).
[CrossRef]

IEEE Trans. Antenn. Propag. (2)

M. Kominami, D. M. Pozar, and D. H. Schaubert, “Dipole and slot elements and arrays on semi-infinite substrates,” IEEE Trans. Antenn. Propag.33(6), 600–607 (1985).
[CrossRef]

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, and D. B. Rutledge, “Bow-tie antennas on a dielectric half-space: theory and experiment,” IEEE Trans. Antenn. Propag.35(6), 622–631 (1987).
[CrossRef]

IEEE Trans. Microwave Technol. (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Technol.47(11), 2075–2084 (1999).
[CrossRef]

Infrared Phys. Technol. (1)

F. J. González and G. D. Boreman, “Comparison of dipole, bowtie, spiral and log-periodic IR antennas,” Infrared Phys. Technol.46(5), 418–428 (2005).
[CrossRef]

J. Appl. Phys. (2)

G. D. Boyd and D. A. Kleinman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys.39(8), 3597–3639 (1968).
[CrossRef]

D. F. Nelson and R. M. Mikulyak, “Refractive indices of congruently melting lithium niobate,” J. Appl. Phys.45(8), 3688–3689 (1974).
[CrossRef]

Nano Lett. (4)

W. Fan, S. Zhang, N.-C. Panoiu, A. Abdenour, S. Krishna, R. M. Osgood, K. J. Malloy, and S. R. J. Brueck, “Second harmonic generation from a nanopatterned isotropic nonlinear material,” Nano Lett.6(5), 1027–1030 (2006).
[CrossRef]

S. Palomba and L. Novotny, “Near-field imaging with a localized nonlinear light source,” Nano Lett.9(11), 3801–3804 (2009).
[CrossRef] [PubMed]

K. D. Ko, A. Kumar, K. H. Fung, R. Ambekar, G. L. Liu, N. X. Fang, and K. C. Toussaint., “Nonlinear optical response from arrays of Au bowtie nanoantennas,” Nano Lett.11(1), 61–65 (2011).
[CrossRef] [PubMed]

D. P. Fromm, A. Sundaramurthy, P. J. Schuck, G. Kino, and W. E. Moerner, “Gap-dependent optical coupling of single “bowtie” nano-antennas resonant in the visible,” Nano Lett.4(5), 957–961 (2004).
[CrossRef]

Nat. Photonics (3)

S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics1(11), 641–648 (2007).
[CrossRef]

A. Alù and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics2(5), 307–310 (2008).
[CrossRef]

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics5(2), 83–90 (2011).
[CrossRef]

Nature (1)

S. Kim, J. H. Jin, Y. J. Kim, I. Y. Park, Y. Kim, and S. W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature453(7196), 757–760 (2008).
[CrossRef] [PubMed]

Opt. Express (7)

E. H. Barakat, M. P. Bernal, and F. I. Baida, “Second harmonic generation enhancement by use of annular aperture arrays embedded into silver and filled by lithium niobate,” Opt. Express18(7), 6530–6536 (2010).
[CrossRef] [PubMed]

M. A. Vincenti, D. de Ceglia, V. Roppo, and M. Scalora, “Harmonic generation in metallic, GaAs-filled nanocavities in the enhanced transmission regime at visible and UV wavelengths,” Opt. Express19(3), 2064–2078 (2011).
[CrossRef] [PubMed]

J. Scheuer, “Ultra-high enhancement of the field concentration in Split Ring Resonators by azimuthally polarized excitation,” Opt. Express19(25), 25454–25464 (2011).
[CrossRef] [PubMed]

S. Park, J. W. Hahn, and J. Y. Lee, “Doubly resonant metallic nanostructure for high conversion efficiency of second harmonic generation,” Opt. Express20(5), 4856–4870 (2012).
[CrossRef] [PubMed]

J. Berthelot, G. Bachelier, M. Song, P. Rai, G. Colas des Francs, A. Dereux, and A. Bouhelier, “Silencing and enhancement of second-harmonic generation in optical gap antennas,” Opt. Express20(10), 10498–10508 (2012).
[CrossRef] [PubMed]

K. Thyagarajan, S. Rivier, A. Lovera, and O. J. F. Martin, “Enhanced second-harmonic generation from double resonant plasmonic antennae,” Opt. Express20(12), 12860–12865 (2012).
[CrossRef] [PubMed]

W. Fan, S. Zhang, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M. Osgood, “Second harmonic generation from patterned GaAs inside a subwavelength metallic hole array,” Opt. Express14(21), 9570–9575 (2006).
[CrossRef] [PubMed]

Opt. Lett. (2)

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. A82(4), 043828 (2010).
[CrossRef]

Phys. Rev. Lett. (4)

P. Ghenuche, S. Cherukulappurath, T. H. Taminiau, N. F. van Hulst, and R. Quidant, “Spectroscopic mode mapping of resonant plasmon nanoantennas,” Phys. Rev. Lett.101, 116805 (2008).

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett.90(1), 013903 (2003).
[CrossRef] [PubMed]

M. Danckwerts and L. Novotny, “Optical frequency mixing at coupled gold nanoparticles,” Phys. Rev. Lett.98(2), 026104 (2007).
[CrossRef] [PubMed]

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nano-antennas,” Phys. Rev. Lett.94(1), 017402 (2005).
[CrossRef] [PubMed]

Science (1)

P. Mühlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science308(5728), 1607–1609 (2005).
[CrossRef] [PubMed]

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E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008).

D. N. Nikogosyan, Nonlinear Optical Crystals: A Complete Survey (Springer, 2005).

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University, 2007)

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 3rd ed. (Prentice-Hall, 2009).

T. Volk and M. Wohlecke, Lithium Niobate—Defects, Photorefraction and Ferroelectric Switching (Springer, 2008).

C. A. Balanis, Antenna Theory Analysis and Design, 3rd ed. (Wiley, 2005).

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Figures (7)

Fig. 1
Fig. 1

(a) The wavelength dependence of the power enhancement factors of nano-antenna arrays on top and recessed in a y-cut congruent LiNbO3. The enhancement factor is calculated at the middle of the gap (0, ± 20nm,0) for an S polarized (E-plane) normal incident plane wave. (b) and (c) describes the dimensions of the array and the orientation of the ordinary (x) and extraordinary (z) axes of the crystal relative to the nano-antenns, for the on surface and recessed arrays respectively.

Fig. 2
Fig. 2

Power enhancement factor (logarithmic scale) of the FH field in the LiNbO3 substrate for Au gap Bowtie nano-antennas. (a) on top of the substrate; (b) recessed in the substrate.

Fig. 3
Fig. 3

A FH beam incident on a nano-antenna array comprising uniformly spaced and identical elements. The axes are aligned according to the LiNbO3 crystal system defined above.

Fig. 4
Fig. 4

Normalized SH AF of NxN nano-antenna array (N = 151); (a) The dependence of the AF on y/yr, for a FH beam that applies 2W0/d = N-1; (b) The FWHM of AF due to different FH waist

Fig. 5
Fig. 5

The dependence of the SH AF on the waist of the FH beam, calculated on an NxN nano-antenna array, N = 151.

Fig. 6
Fig. 6

Power enhancement cross sections of the FH for normal incident wave at 1.56μm, propagating from the air (panels (a) and (b)) or from the substrates (panels (c) and (d)). The cross sections are along the E-plane (panels (a) and (c)) and the H-plane (panels (b) and (d)), which correspond to the YZ and XY planes respectively.

Fig. 7
Fig. 7

SH irradiance at a 1m equidistance hemisphere due to a 10×10 Bowtie nano-antenna array recessed in y-cut LiNbO3. The array is positioned at (0,0,0), where the coordinate system is aligned to the crystal axes and the E-plane is YZ plane whereas the H-plane is the XY plane; (a) The SH irradiance to the air; (b) The SH irradiance into the LiNbO3; the dashed lines represent equi-polar angle circles.

Equations (12)

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P ˜ ¯ (2) = ε 0 χ (2) E ˜ ¯ E ˜ ¯
×× E ˜ ¯ + 1 ε 0 c 2 d 2 D ˜ ¯ (1) d t 2 = 1 ε 0 c 2 d 2 P ˜ ¯ (2) d t 2
( P x_SH P y_SH P z_SH )2 ε 0 ( d 11 d 12 d 13 d 14 d 15 d 16 d 21 d 22 d 23 d 24 d 25 d 26 d 31 d 32 d 33 d 34 d 35 d 36 )( E x_FH 2 E y_FH 2 E z_FH 2 2 E y E z 2 E x E z 2 E x E y )
E(r,y)= E 0 i y r q(y) e ik r 2 2q(y)
E 0 2 = 4 P FH c ε 0 π W 0 2
S ˜ ( z ) | Gap = 1 ε 0 c 2 d 2 P ˜ ¯ (2) d t 2 | Gap ( ω SH c ) 2 χ (2) E ˜ ¯ E ˜ ¯ | Gap = ( ω SH c ) 2 χ (2) G 2 E 0 2 ( i y r q(y) ) 2 e ik r 2 q(y) | Gap
E SH [m,n]=η(θ,ϕ) d x d z ( ω SH c ) 2 χ (2) G 2 E 0 2 ( i y r q(y) ) 2 e ik ( m d x M+1 2 d x ) 2 q(y) e ik ( n d z N+1 2 d z ) 2 q(y) m,n=1..M,N
E SH_farfield = E SH_Single_Element AF
E SH_farfield ( ψ x , ψ z )= m=1 M n=1 N E SH [m,n] e i( m1 ) ψ x e i( n1 ) ψ z ψ x =k d x sinθcosϕ , ψ z =k d z sinθsinϕ
E SH_farfield ( ψ x , ψ z )=C P FH 2 d x d z π W 0 2 ( i y r q(y) ) 2 m= M1 2 M1 2 e ik d x 2 m 2 q(y) e im ψ x n= N1 2 N1 2 e ik d z 2 n 2 q(y) e in ψ z C= 2η ( ω SH c ) 2 χ (2) G 2 c ε 0
AF( ψ x , ψ z )=A F x ( ψ x )A F z ( ψ z ) A F x,z ( ψ x,z )= 1 2π ( i y r q(y) ) 1i y y r l= π π e w 0 2 8 d x,z 2 ( 1i y y r ) ( ψ x,z υ2πl ) 2 sin( υN,M 2 ) sin( υ 2 ) dυ
AF( ψ x,z ) | diffraction_lobe = A F x ( ψ x ) | diffraction_lobe A F z ( ψ z ) | diffraction_lobe A F x,z ( ψ x,z ) | diffraction_lobe 1 2π 1 1i y y r π π e w 0 2 8 d x,z 2 ( 1i y y r ) υ 2 sin( υN,M 2 ) sin( υ 2 ) dυ

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