Abstract

The second-order nonlinear frequency conversion in three-dimensional nonlinear photonic crystals is theoretically studied using coupled wave equations. A universal theoretical model is obtained, with a unified expression combining birefringence phase match, quasi-phase match, nonlinear Raman-Nath diffraction, nonlinear Čerenkov radiation and nonlinear Bragg diffraction. They are demonstrated in the numerical simulation. With the phase-matching conditions in lower dimensions extended to three dimensions, more various phenomena can be seen and corresponding mechanisms can be explained. This research enables the control of second-harmonic generation more efficiently and has potential applications in more complicated nonlinear photonic crystals.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref] [PubMed]

2017 (1)

2016 (1)

2014 (2)

2012 (3)

Y. Sheng, Q. Kong, V. Roppo, K. Kalinowski, Q. Wang, C. Cojocaru, and W. Krolikowski, “Theoretical study of Čerenkov-type second-harmonic generation in periodically poled ferroelectric crystals,” J. Opt. Soc. Am. B 29, 312–318 (2012).
[Crossref]

Y. Sheng, Q. Kong, W. Wang, K. Kalinowski, and W. Krolikowski, “Theoretical investigations of nonlinear Raman–Nath diffraction in the frequency doubling process,” J. Phys. B: At. Mol. Opt. Phys. 45, 055401 (2012).
[Crossref]

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. 108, 223901 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (1)

A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photonic Rev. 4, 355–373 (2010).
[Crossref]

2009 (1)

2008 (2)

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008).
[Crossref] [PubMed]

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[Crossref]

2007 (1)

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quant. Electron. 39, 361–375 (2007).
[Crossref]

2006 (1)

M. Lauritano, A. Parini, G. Bellanca, S. Trillo, M. Conforti, A. Locatelli, and C. D. Angelis, “Bistability, limiting, and self-pulsing in backward second-harmonic generation: a time-domain approach,” J. Opt. A: Pure Appl. Opt. 8, S494–S501 (2006).
[Crossref]

2005 (2)

S. Inoue and Y. Aoyagi, “Design and fabrication of two-dimensional photonic crystals with predetermined nonlinear optical properties,” Phys. Rev. Lett. 94, 103904 (2005).
[Crossref] [PubMed]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[Crossref] [PubMed]

2002 (1)

2000 (1)

N. Broderick, G. Ross, H. Offerhaus, D. Richardson, and D. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345 (2000).
[Crossref] [PubMed]

1998 (2)

Y. J. Ding, J. U. Kang, and J. B. Khurgin, “Theory of backward second-harmonic and third-harmonic generation using laser pulses in quasi-phase-matched second-order nonlinear medium,” IEEE J. Quant. Electron. 34, 966–974 (1998).
[Crossref]

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136 (1998).
[Crossref]

1997 (1)

G. Pan, R. Kesavamoorthy, and S. A. Asher, “Optically nonlinear Bragg diffracting nanosecond optical switches,” Phys. Rev. Lett. 78, 3860 (1997).
[Crossref]

1995 (1)

1994 (1)

M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47, 25–33 (1994).
[Crossref]

1992 (1)

M. M. Fejer, G. 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]

1988 (1)

J. Wen and M. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[Crossref]

1971 (1)

W. Gandrud, “On the possibility of phase-matched infrared mixing using induced birefringence,” IEEE J. Quantum Electron. 7, 132–133 (1971).
[Crossref]

1970 (1)

P. Tien, R. Ulrich, and R. Martin, “Optical second harmonic generation in form of coherent Cerenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. 17, 447–450 (1970).
[Crossref]

Akhmatkhanov, A.

An, N.

X. Zhao, Y. Zheng, H. Ren, N. An, X. Deng, and X. Chen, “Nonlinear Cherenkov radiation at the interface of two different nonlinear media,” Opt. Express 24, 12825–12830 (2016).
[Crossref] [PubMed]

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. 108, 223901 (2012).
[Crossref] [PubMed]

Angelis, C. D.

M. Lauritano, A. Parini, G. Bellanca, S. Trillo, M. Conforti, A. Locatelli, and C. D. Angelis, “Bistability, limiting, and self-pulsing in backward second-harmonic generation: a time-domain approach,” J. Opt. A: Pure Appl. Opt. 8, S494–S501 (2006).
[Crossref]

Aoyagi, Y.

S. Inoue and Y. Aoyagi, “Design and fabrication of two-dimensional photonic crystals with predetermined nonlinear optical properties,” Phys. Rev. Lett. 94, 103904 (2005).
[Crossref] [PubMed]

Arie, A.

A. Arie and N. Voloch, “Periodic, quasi-periodic, and random quadratic nonlinear photonic crystals,” Laser Photonic Rev. 4, 355–373 (2010).
[Crossref]

S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling processes,” Opt. Lett. 34, 848–850 (2009).
[Crossref] [PubMed]

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[Crossref]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008).
[Crossref] [PubMed]

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quant. Electron. 39, 361–375 (2007).
[Crossref]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[Crossref] [PubMed]

Asher, S. A.

G. Pan, R. Kesavamoorthy, and S. A. Asher, “Optically nonlinear Bragg diffracting nanosecond optical switches,” Phys. Rev. Lett. 78, 3860 (1997).
[Crossref]

Assanto, G.

Bahabad, A.

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quant. Electron. 39, 361–375 (2007).
[Crossref]

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[Crossref] [PubMed]

Bang, O.

Baturin, I.

Bellanca, G.

M. Lauritano, A. Parini, G. Bellanca, S. Trillo, M. Conforti, A. Locatelli, and C. D. Angelis, “Bistability, limiting, and self-pulsing in backward second-harmonic generation: a time-domain approach,” J. Opt. A: Pure Appl. Opt. 8, S494–S501 (2006).
[Crossref]

Berger, V.

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136 (1998).
[Crossref]

Bosenberg, W.

Bowden, C. M.

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 2003).

Breazeale, M.

J. Wen and M. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[Crossref]

Broderick, N.

N. Broderick, G. Ross, H. Offerhaus, D. Richardson, and D. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345 (2000).
[Crossref] [PubMed]

Busacca, A. C.

Byer, R.

Byer, R. L.

M. M. Fejer, G. 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]

Chen, J.

Chen, X.

Cojocaru, C.

Conforti, M.

M. Lauritano, A. Parini, G. Bellanca, S. Trillo, M. Conforti, A. Locatelli, and C. D. Angelis, “Bistability, limiting, and self-pulsing in backward second-harmonic generation: a time-domain approach,” J. Opt. A: Pure Appl. Opt. 8, S494–S501 (2006).
[Crossref]

Curcio, L.

Deng, X.

X. Zhao, Y. Zheng, H. Ren, N. An, X. Deng, and X. Chen, “Nonlinear Cherenkov radiation at the interface of two different nonlinear media,” Opt. Express 24, 12825–12830 (2016).
[Crossref] [PubMed]

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. 108, 223901 (2012).
[Crossref] [PubMed]

Ding, Y. J.

Y. J. Ding, J. U. Kang, and J. B. Khurgin, “Theory of backward second-harmonic and third-harmonic generation using laser pulses in quasi-phase-matched second-order nonlinear medium,” IEEE J. Quant. Electron. 34, 966–974 (1998).
[Crossref]

Eckardt, R.

Fejer, M.

Fejer, M. M.

M. M. Fejer, “Nonlinear optical frequency conversion,” Phys. Today 47, 25–33 (1994).
[Crossref]

M. M. Fejer, G. 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]

Ferraro, P.

P. Ferraro, S. Grilli, and P. D. Natale, Ferroelectric Crystals for Photonic Applications: Including Nanoscale Fabrication and Characterization Techniques, vol. 91 (Springer Science & Business Media, 2013).

Fischer, R.

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008).
[Crossref] [PubMed]

Galun, E.

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[Crossref]

Gandrud, W.

W. Gandrud, “On the possibility of phase-matched infrared mixing using induced birefringence,” IEEE J. Quantum Electron. 7, 132–133 (1971).
[Crossref]

Gayer, O.

O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[Crossref]

Grilli, S.

P. Ferraro, S. Grilli, and P. D. Natale, Ferroelectric Crystals for Photonic Applications: Including Nanoscale Fabrication and Characterization Techniques, vol. 91 (Springer Science & Business Media, 2013).

Habshoosh, N.

A. Arie, N. Habshoosh, and A. Bahabad, “Quasi phase matching in two-dimensional nonlinear photonic crystals,” Opt. Quant. Electron. 39, 361–375 (2007).
[Crossref]

Hanna, D.

N. Broderick, G. Ross, H. Offerhaus, D. Richardson, and D. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345 (2000).
[Crossref] [PubMed]

Inoue, S.

S. Inoue and Y. Aoyagi, “Design and fabrication of two-dimensional photonic crystals with predetermined nonlinear optical properties,” Phys. Rev. Lett. 94, 103904 (2005).
[Crossref] [PubMed]

Jundt, D. H.

M. M. Fejer, G. 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]

Kalinowski, K.

Y. Sheng, Q. Kong, W. Wang, K. Kalinowski, and W. Krolikowski, “Theoretical investigations of nonlinear Raman–Nath diffraction in the frequency doubling process,” J. Phys. B: At. Mol. Opt. Phys. 45, 055401 (2012).
[Crossref]

Y. Sheng, Q. Kong, V. Roppo, K. Kalinowski, Q. Wang, C. Cojocaru, and W. Krolikowski, “Theoretical study of Čerenkov-type second-harmonic generation in periodically poled ferroelectric crystals,” J. Opt. Soc. Am. B 29, 312–318 (2012).
[Crossref]

Kang, J. U.

Y. J. Ding, J. U. Kang, and J. B. Khurgin, “Theory of backward second-harmonic and third-harmonic generation using laser pulses in quasi-phase-matched second-order nonlinear medium,” IEEE J. Quant. Electron. 34, 966–974 (1998).
[Crossref]

Kesavamoorthy, R.

G. Pan, R. Kesavamoorthy, and S. A. Asher, “Optically nonlinear Bragg diffracting nanosecond optical switches,” Phys. Rev. Lett. 78, 3860 (1997).
[Crossref]

Khurgin, J. B.

Y. J. Ding, J. U. Kang, and J. B. Khurgin, “Theory of backward second-harmonic and third-harmonic generation using laser pulses in quasi-phase-matched second-order nonlinear medium,” IEEE J. Quant. Electron. 34, 966–974 (1998).
[Crossref]

Kivshar, Y. S.

S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling processes,” Opt. Lett. 34, 848–850 (2009).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008).
[Crossref] [PubMed]

Kong, Q.

Y. Sheng, Q. Kong, W. Wang, K. Kalinowski, and W. Krolikowski, “Theoretical investigations of nonlinear Raman–Nath diffraction in the frequency doubling process,” J. Phys. B: At. Mol. Opt. Phys. 45, 055401 (2012).
[Crossref]

Y. Sheng, Q. Kong, V. Roppo, K. Kalinowski, Q. Wang, C. Cojocaru, and W. Krolikowski, “Theoretical study of Čerenkov-type second-harmonic generation in periodically poled ferroelectric crystals,” J. Opt. Soc. Am. B 29, 312–318 (2012).
[Crossref]

Krolikowski, W.

Y. Sheng, Q. Kong, V. Roppo, K. Kalinowski, Q. Wang, C. Cojocaru, and W. Krolikowski, “Theoretical study of Čerenkov-type second-harmonic generation in periodically poled ferroelectric crystals,” J. Opt. Soc. Am. B 29, 312–318 (2012).
[Crossref]

Y. Sheng, Q. Kong, W. Wang, K. Kalinowski, and W. Krolikowski, “Theoretical investigations of nonlinear Raman–Nath diffraction in the frequency doubling process,” J. Phys. B: At. Mol. Opt. Phys. 45, 055401 (2012).
[Crossref]

S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling processes,” Opt. Lett. 34, 848–850 (2009).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008).
[Crossref] [PubMed]

Lauritano, M.

M. Lauritano, A. Parini, G. Bellanca, S. Trillo, M. Conforti, A. Locatelli, and C. D. Angelis, “Bistability, limiting, and self-pulsing in backward second-harmonic generation: a time-domain approach,” J. Opt. A: Pure Appl. Opt. 8, S494–S501 (2006).
[Crossref]

Lifshitz, R.

R. Lifshitz, A. Arie, and A. Bahabad, “Photonic quasicrystals for nonlinear optical frequency conversion,” Phys. Rev. Lett. 95, 133901 (2005).
[Crossref] [PubMed]

Locatelli, A.

M. Lauritano, A. Parini, G. Bellanca, S. Trillo, M. Conforti, A. Locatelli, and C. D. Angelis, “Bistability, limiting, and self-pulsing in backward second-harmonic generation: a time-domain approach,” J. Opt. A: Pure Appl. Opt. 8, S494–S501 (2006).
[Crossref]

Magel, G.

M. M. Fejer, G. 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]

Martin, R.

P. Tien, R. Ulrich, and R. Martin, “Optical second harmonic generation in form of coherent Cerenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. 17, 447–450 (1970).
[Crossref]

Myers, L. E.

Natale, P. D.

P. Ferraro, S. Grilli, and P. D. Natale, Ferroelectric Crystals for Photonic Applications: Including Nanoscale Fabrication and Characterization Techniques, vol. 91 (Springer Science & Business Media, 2013).

Neshev, D. N.

S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling processes,” Opt. Lett. 34, 848–850 (2009).
[Crossref] [PubMed]

S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008).
[Crossref] [PubMed]

Offerhaus, H.

N. Broderick, G. Ross, H. Offerhaus, D. Richardson, and D. Hanna, “Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345 (2000).
[Crossref] [PubMed]

Pan, G.

G. Pan, R. Kesavamoorthy, and S. A. Asher, “Optically nonlinear Bragg diffracting nanosecond optical switches,” Phys. Rev. Lett. 78, 3860 (1997).
[Crossref]

Parini, A.

M. Lauritano, A. Parini, G. Bellanca, S. Trillo, M. Conforti, A. Locatelli, and C. D. Angelis, “Bistability, limiting, and self-pulsing in backward second-harmonic generation: a time-domain approach,” J. Opt. A: Pure Appl. Opt. 8, S494–S501 (2006).
[Crossref]

Pierce, J.

Ren, H.

X. Zhao, Y. Zheng, H. Ren, N. An, X. Deng, and X. Chen, “Nonlinear Cherenkov radiation at the interface of two different nonlinear media,” Opt. Express 24, 12825–12830 (2016).
[Crossref] [PubMed]

H. Ren, X. Deng, Y. Zheng, N. An, and X. Chen, “Nonlinear Cherenkov radiation in an anomalous dispersive medium,” Phys. Rev. Lett. 108, 223901 (2012).
[Crossref] [PubMed]

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S. M. Saltiel, D. N. Neshev, W. Krolikowski, A. Arie, O. Bang, and Y. S. Kivshar, “Multiorder nonlinear diffraction in frequency doubling processes,” Opt. Lett. 34, 848–850 (2009).
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S. M. Saltiel, D. N. Neshev, R. Fischer, W. Krolikowski, A. Arie, and Y. S. Kivshar, “Generation of second-harmonic conical waves via nonlinear Bragg diffraction,” Phys. Rev. Lett. 100, 103902 (2008).
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Sheng, Y.

Y. Sheng, Q. Kong, W. Wang, K. Kalinowski, and W. Krolikowski, “Theoretical investigations of nonlinear Raman–Nath diffraction in the frequency doubling process,” J. Phys. B: At. Mol. Opt. Phys. 45, 055401 (2012).
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Appl. Opt. (1)

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O. Gayer, Z. Sacks, E. Galun, and A. Arie, “Temperature and wavelength dependent refractive index equations for MgO-doped congruent and stoichiometric LiNbO3,” Appl. Phys. B 91, 343–348 (2008).
[Crossref]

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P. Tien, R. Ulrich, and R. Martin, “Optical second harmonic generation in form of coherent Cerenkov radiation from a thin-film waveguide,” Appl. Phys. Lett. 17, 447–450 (1970).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 All the types of phase-matching condition during χ(2) processes in 1D and 2D NPCs, n is the refractive index of NPCs, k is the wave vector, G is the reciprocal vector in NPCs, subscripts 1 and 2 denote FW and SH, respectively. (a) BPM processes in birefringent crystals; (b) QPM processes; (c) NBD generated by the compensation of transverse reciprocal vectors; (d) Spontaneously transverse phase-matching compensated by reciprocal vectors generates NRND; (e) Spontaneously longitudinal phase-matching generates NCR without reciprocal vectors compensation.
Fig. 2
Fig. 2 (a) Schematic of phase match between wave vectors and reciprocal vectors in 3D NPCs; (b) xy section plane of (a); (c) Spontaneous transverse phase match compensated by reciprocal vectors generates multiple-order NRNDs in 3D NPCs; (d) Spontaneous longitudinal phase match compensated by reciprocal vectors generates multiple-order NCRs in 3D NPCs.
Fig. 3
Fig. 3 3D NPC structures and the corresponding simulated SH intensities. Coordinate θ x = arcsin k x k 2 and coordinate θ y = arcsin k y k 2 . (a) Cylindrical 3D NPC; (c) Cubical 3D NPC; (b) and (d) are the corresponding simulated SH intensities, respectively.
Fig. 4
Fig. 4 (a) NBD is generated at the position of 0-order NCR ring; (b)NBD is generated at the position of 1st-order NCR ring; (c) NBD is generated at the position of 3rd-order NCR ring.
Fig. 5
Fig. 5 Dimension reduction from 3D NPCs to 1D NPCs. (a) Standard cubical 3D NPC; (b) Reduced 1D (0, 0, l) type NPC; (c) and (d) Simulation results of generated SH in 1D (0, 0, l) type NP; (c) Phase-mismatching condition with Λ z = 15 μm; (d) Phase-matching condition with Λ z = 5.28 μm; The maximum intensity of (d) is 104 times to that of (c). (e) Reduced 1D (n, 0, 0) type NPC and corresponding phase-matching vectors; (f) Multiple-order NRNDs generated in x-direction with Λ x = 10 μm in 1D (n, 0, 0) type NPC.
Fig. 6
Fig. 6 Dimension reduction from 3D NPCs to 2D NPCs. (a) Reduced 2D (n, m, 0) type NPC and corresponding phase-matching vectors; (b) Simulation results of generated SH when Λ x = 14.14 μm, discrete 0-order NCR ring is generated; (c) Other types of 2D NPCs.

Equations (7)

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{ A 1 ( r ) z = 0 , A 2 ( r ) z i 2 k 2 2 A 2 ( r ) = i ω 2 k 2 c d eff g ( x , y , z ) [ A 1 ( r ) e x 2 + y 2 w 2 ] 2 e i ( k 2 2 k 1 ) z ,
I 2 ( z , k x , k y ) = β 2 2 I 1 2 | 0 z g ( x , y , z ) e i ( k 2 2 k 1 k x 2 + k y 2 2 k 2 ) z e 2 ( x 2 + y 2 ) w 2 + i ( k x x + k y y ) d x d y d z | 2 ,
I 2 = w 4 π 2 4 β 2 2 I 1 2 z 2 | l m n C n m l sinc [ ( k z 2 k 1 + 2 l π Λ z ) z 2 ] e w 2 ( k x Λ x 2 n π ) 2 8 Λ x 2 e w 2 ( k y Λ y 2 m π ) 2 8 Λ y 2 | 2 ,
I 2 = w 4 π 2 4 β 2 2 I 1 2 z 2 sinc 2 [ ( k 2 2 k 1 ) z 2 ] .
I 2 = w 4 π 2 4 β 2 2 I 1 2 z 2 | l C 00 l sinc [ ( k z 2 k 1 + 2 l π Λ z ) z 2 ] | 2 e w 2 4 ( k x 2 + k y 2 ) ,
I 2 = w 4 π 2 4 β 2 2 I 1 2 z 2 sinc 2 [ ( k z 2 k 1 ) z 2 ] | n C n 00 e w 2 ( k x Λ x 2 n π ) 2 8 Λ x 2 | 2 e w 2 4 k y 2 .
I 2 = w 4 π 2 4 β 2 2 I 1 2 z 2 sinc 2 [ ( k z 2 k 1 ) z 2 ] | m n C n m 0 e w 2 ( k x Λ x 2 n π ) 2 8 Λ x 2 e w 2 ( k y Λ y 2 m π ) 2 8 Λ y 2 | 2 .

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