Abstract

We investigate parametric down-conversion in a hexagonally poled nonlinear photonic crystal, pumped by a dual pump with a transverse modulation that matches the periodicity of the χ(2) nonlinear grating. A peculiar feature of this resonant configuration is that the two pumps simultaneously generate photon pairs over an entire branch of modes, via quasi-phase matching with both fundamental vectors of the reciprocal lattice of the nonlinearity. The parametric gain of these modes depends thus coherently on the sum of the two pump amplitudes and can be controlled by varying their relative intensities and phases. We find that a significant enhancement of the source conversion efficiency, comparable to that of one-dimensionally poled crystals, can be achieved by a dual symmetric pump. We also show how the four-mode coupling arising among shared modes at resonance can be tailored by changing the dual pump parameters.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Concurrent optical parametric down-conversion in χ(2) nonlinear photonic crystals

L. Chen, P. Xu, Y. F. Bai, X. W. Luo, M. L. Zhong, M. Dai, M. H. Lu, and S. N. Zhu
Opt. Express 22(11) 13164-13169 (2014)

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2018 (2)

A. Gatti, E. Brambilla, K. Gallo, and O. Jedrkiewicz, “Golden ratio entanglement in hexagonally poled nonlinear crystals,” Phys. Rev. A 98, 053827 (2018).
[Crossref]

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

2017 (3)

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

Z. Yellas, M. W. Lee, R. Kremer, K.-H. Chang, M. R. Beghoul, L.-H. Peng, and A. Boudrioua, “Multiwavelength generation from multi-nonlinear optical process in a 2D PPLT,” Opt. Express 25, 30253–30258 (2017).
[Crossref] [PubMed]

2016 (1)

2015 (1)

2014 (2)

2013 (3)

E. Megidish, A. Halevy, H. S. Eisenberg, A. Ganany-Padowicz, N. Habshoosh, and A. Arie, “Compact 2D nonlinear photonic crystal source of beamlike path entangled photons,” Opt. Express 21, 6689–6696 (2013).
[Crossref] [PubMed]

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

H. H. Lim, S. Kurimura, T. Katagai, and I. Shoji, “Temperature-dependent sellmeier equation for refractive index of 1.0 mol % Mg-doped stoichiometric lithium tantalate,” Jpn. J. Appl. Phys. 52, 032601 (2013).
[Crossref]

2012 (3)

E. Brambilla, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion,” Phys. Rev. A 85, 063834 (2012).
[Crossref]

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

M. Levenius, V. Pasiskevicius, and K. Gallo, “Angular degrees of freedom in twin-beam parametric down-conversion,” Appl. Phys. Lett. 101, 121114 (2012).
[Crossref]

2011 (1)

2008 (2)

H.-C. Liu and A. H. Kung, “Substantial gain enhancement for optical parametric amplification and oscillation in two-dimensional χ(2) nonlinear photonic crystals,” Opt. Express 16, 9714–9725 (2008).
[Crossref] [PubMed]

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, “Parametric solitons in two-dimensional lattices of purely nonlinear origin,” Phys. Rev. Lett. 100, 053901 (2008).
[Crossref] [PubMed]

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)

2003 (1)

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003).
[Crossref]

2000 (2)

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

S. Saltiel and Y. S. Kivshar, “Phase matching in nonlinear χ(2) photonic crystals,” Opt. Lett. 25, 1204–1206 (2000).
[Crossref]

1998 (1)

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
[Crossref]

1997 (1)

M. J. Werner and P. D. Drummond, “Pulsed quadrature-phase squeezing of solitary waves in χ(2) parametric waveguides,” Phys. Rev. A 56, 1508–1518 (1997).
[Crossref]

1993 (1)

Arie, A.

Assanto, G.

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, “Parametric solitons in two-dimensional lattices of purely nonlinear origin,” Phys. Rev. Lett. 100, 053901 (2008).
[Crossref] [PubMed]

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]

Bai, Y. F.

L. Chen, P. Xu, Y. F. Bai, X. W. Luo, M. L. Zhong, M. Dai, M. H. Lu, and S. N. Zhu, “Concurrent optical parametric down-conversion in χ(2) nonlinear photonic crystals,” Opt. Express 22, 13164–13169 (2014).
[Crossref] [PubMed]

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

Banfi, G. P.

Baronio, F.

Beghoul, M. R.

Berger, V.

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
[Crossref]

Björk, G.

K. Stensson, G. Björk, and K. Gallo, “Green-pumped parametric downconversion in hexagonally poled MgO:LiTaO3,” in Advanced Solid State Lasers, (Optical Society of America, 2014), p. ATu3A.5.
[Crossref]

Boudrioua, A.

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

Z. Yellas, M. W. Lee, R. Kremer, K.-H. Chang, M. R. Beghoul, L.-H. Peng, and A. Boudrioua, “Multiwavelength generation from multi-nonlinear optical process in a 2D PPLT,” Opt. Express 25, 30253–30258 (2017).
[Crossref] [PubMed]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

M. Lazoul, A. Boudrioua, L.-M. Simohamed, and L.-H. Peng, “Multi-resonant optical parametric oscillator based on 2D-PPLT nonlinear photonic crystal,” Opt. Lett. 40, 1861–1864 (2015).
[Crossref] [PubMed]

Brambilla, E.

A. Gatti, E. Brambilla, K. Gallo, and O. Jedrkiewicz, “Golden ratio entanglement in hexagonally poled nonlinear crystals,” Phys. Rev. A 98, 053827 (2018).
[Crossref]

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

E. Brambilla, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion,” Phys. Rev. A 85, 063834 (2012).
[Crossref]

A. Gatti and E. Brambilla, “Engineering multipartite entanglement in nonlinear photonic crystals,” Preprint.

Broderick, N. G. R.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

Chang, K.-H.

Chen, L.

Chikh-Touami, H.

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

Chou, M.-H.

Conforti, M.

Dai, M.

Danielius, R.

Drummond, P. D.

M. J. Werner and P. D. Drummond, “Pulsed quadrature-phase squeezing of solitary waves in χ(2) parametric waveguides,” Phys. Rev. A 56, 1508–1518 (1997).
[Crossref]

Dubietis, A.

Eisenberg, H. S.

Gallo, K.

A. Gatti, E. Brambilla, K. Gallo, and O. Jedrkiewicz, “Golden ratio entanglement in hexagonally poled nonlinear crystals,” Phys. Rev. A 98, 053827 (2018).
[Crossref]

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

M. Conforti, F. Baronio, M. Levenius, and K. Gallo, “Broadband parametric processes in χ(2) nonlinear photonic crystals,” Opt. Lett. 39, 3457–3460 (2014).
[Crossref] [PubMed]

M. Levenius, V. Pasiskevicius, and K. Gallo, “Angular degrees of freedom in twin-beam parametric down-conversion,” Appl. Phys. Lett. 101, 121114 (2012).
[Crossref]

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, “Parametric solitons in two-dimensional lattices of purely nonlinear origin,” Phys. Rev. Lett. 100, 053901 (2008).
[Crossref] [PubMed]

K. Stensson, G. Björk, and K. Gallo, “Green-pumped parametric downconversion in hexagonally poled MgO:LiTaO3,” in Advanced Solid State Lasers, (Optical Society of America, 2014), p. ATu3A.5.
[Crossref]

Ganany-Padowicz, A.

Gatti, A.

A. Gatti, E. Brambilla, K. Gallo, and O. Jedrkiewicz, “Golden ratio entanglement in hexagonally poled nonlinear crystals,” Phys. Rev. A 98, 053827 (2018).
[Crossref]

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

E. Brambilla, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion,” Phys. Rev. A 85, 063834 (2012).
[Crossref]

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003).
[Crossref]

A. Gatti and E. Brambilla, “Engineering multipartite entanglement in nonlinear photonic crystals,” Preprint.

Gong, Y. X.

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Gong, Y.-X.

Y.-X. Gong, S. Zhang, P. Xu, and S. N. Zhu, “Scheme for generating distillation-favorable continuous-variable entanglement via three concurrent parametric down-conversions in a single χ(2) nonlinear photonic crystal,” Opt. Express 24, 6402–6412 (2016).
[Crossref] [PubMed]

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

Habshoosh, N.

Halevy, A.

Hanna, D. C.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

Hloupis, G.

Jedrkiewicz, O.

A. Gatti, E. Brambilla, K. Gallo, and O. Jedrkiewicz, “Golden ratio entanglement in hexagonally poled nonlinear crystals,” Phys. Rev. A 98, 053827 (2018).
[Crossref]

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

E. Brambilla, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion,” Phys. Rev. A 85, 063834 (2012).
[Crossref]

Jin, H.

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Katagai, T.

H. H. Lim, S. Kurimura, T. Katagai, and I. Shoji, “Temperature-dependent sellmeier equation for refractive index of 1.0 mol % Mg-doped stoichiometric lithium tantalate,” Jpn. J. Appl. Phys. 52, 032601 (2013).
[Crossref]

Kivshar, Y. S.

Kremer, R.

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

Z. Yellas, M. W. Lee, R. Kremer, K.-H. Chang, M. R. Beghoul, L.-H. Peng, and A. Boudrioua, “Multiwavelength generation from multi-nonlinear optical process in a 2D PPLT,” Opt. Express 25, 30253–30258 (2017).
[Crossref] [PubMed]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

Kung, A. H.

Kurimura, S.

H. H. Lim, S. Kurimura, T. Katagai, and I. Shoji, “Temperature-dependent sellmeier equation for refractive index of 1.0 mol % Mg-doped stoichiometric lithium tantalate,” Jpn. J. Appl. Phys. 52, 032601 (2013).
[Crossref]

Lazoul, M.

Lee, H.-J.

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

Lee, M.

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

Lee, M. W.

Leng, H. Y.

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

Levenius, M.

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

M. Conforti, F. Baronio, M. Levenius, and K. Gallo, “Broadband parametric processes in χ(2) nonlinear photonic crystals,” Opt. Lett. 39, 3457–3460 (2014).
[Crossref] [PubMed]

M. Levenius, V. Pasiskevicius, and K. Gallo, “Angular degrees of freedom in twin-beam parametric down-conversion,” Appl. Phys. Lett. 101, 121114 (2012).
[Crossref]

Lim, H. H.

H. H. Lim, S. Kurimura, T. Katagai, and I. Shoji, “Temperature-dependent sellmeier equation for refractive index of 1.0 mol % Mg-doped stoichiometric lithium tantalate,” Jpn. J. Appl. Phys. 52, 032601 (2013).
[Crossref]

Liu, H.-C.

Lu, M. H.

Lugiato, L. A.

E. Brambilla, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion,” Phys. Rev. A 85, 063834 (2012).
[Crossref]

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003).
[Crossref]

Luo, X. W.

L. Chen, P. Xu, Y. F. Bai, X. W. Luo, M. L. Zhong, M. Dai, M. H. Lu, and S. N. Zhu, “Concurrent optical parametric down-conversion in χ(2) nonlinear photonic crystals,” Opt. Express 22, 13164–13169 (2014).
[Crossref] [PubMed]

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Megidish, E.

Moutzouris, K.

Offerhaus, H. L.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

Pasiskevicius, V.

M. Levenius, V. Pasiskevicius, and K. Gallo, “Angular degrees of freedom in twin-beam parametric down-conversion,” Appl. Phys. Lett. 101, 121114 (2012).
[Crossref]

M. Tiihonen and V. Pasiskevicius, “Two-dimensional quasi-phase-matched multiple-cascaded four-wave mixing in periodically poled ktiopo4,” Opt. Lett. 31, 3324–3326 (2006).
[Crossref] [PubMed]

Pasquazi, A.

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, “Parametric solitons in two-dimensional lattices of purely nonlinear origin,” Phys. Rev. Lett. 100, 053901 (2008).
[Crossref] [PubMed]

Peng, L.-H.

Z. Yellas, M. W. Lee, R. Kremer, K.-H. Chang, M. R. Beghoul, L.-H. Peng, and A. Boudrioua, “Multiwavelength generation from multi-nonlinear optical process in a 2D PPLT,” Opt. Express 25, 30253–30258 (2017).
[Crossref] [PubMed]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

M. Lazoul, A. Boudrioua, L.-M. Simohamed, and L.-H. Peng, “Multi-resonant optical parametric oscillator based on 2D-PPLT nonlinear photonic crystal,” Opt. Lett. 40, 1861–1864 (2015).
[Crossref] [PubMed]

Piskarskas, A.

Podenas, D.

Richardson, D. J.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

Ross, G. W.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

Saltiel, S.

San Miguel, M.

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003).
[Crossref]

Shoji, I.

H. H. Lim, S. Kurimura, T. Katagai, and I. Shoji, “Temperature-dependent sellmeier equation for refractive index of 1.0 mol % Mg-doped stoichiometric lithium tantalate,” Jpn. J. Appl. Phys. 52, 032601 (2013).
[Crossref]

Simohamed, L.-M.

Stavrakas, I.

Stensson, K.

K. Stensson, G. Björk, and K. Gallo, “Green-pumped parametric downconversion in hexagonally poled MgO:LiTaO3,” in Advanced Solid State Lasers, (Optical Society of America, 2014), p. ATu3A.5.
[Crossref]

Stivala, S.

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, “Parametric solitons in two-dimensional lattices of purely nonlinear origin,” Phys. Rev. Lett. 100, 053901 (2008).
[Crossref] [PubMed]

Tamosauskas, G.

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

Tiihonen, M.

Trapani, P. Di

Triantis, D.

Werner, M. J.

M. J. Werner and P. D. Drummond, “Pulsed quadrature-phase squeezing of solitary waves in χ(2) parametric waveguides,” Phys. Rev. A 56, 1508–1518 (1997).
[Crossref]

Xie, Z. D.

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

Xu, P.

Y.-X. Gong, S. Zhang, P. Xu, and S. N. Zhu, “Scheme for generating distillation-favorable continuous-variable entanglement via three concurrent parametric down-conversions in a single χ(2) nonlinear photonic crystal,” Opt. Express 24, 6402–6412 (2016).
[Crossref] [PubMed]

L. Chen, P. Xu, Y. F. Bai, X. W. Luo, M. L. Zhong, M. Dai, M. H. Lu, and S. N. Zhu, “Concurrent optical parametric down-conversion in χ(2) nonlinear photonic crystals,” Opt. Express 22, 13164–13169 (2014).
[Crossref] [PubMed]

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

Yang, J.

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

Yellas, Z.

Yu, W. J.

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Zambrini, R.

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003).
[Crossref]

Zhang, S.

Zhao, G.

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Zhong, M. L.

L. Chen, P. Xu, Y. F. Bai, X. W. Luo, M. L. Zhong, M. Dai, M. H. Lu, and S. N. Zhu, “Concurrent optical parametric down-conversion in χ(2) nonlinear photonic crystals,” Opt. Express 22, 13164–13169 (2014).
[Crossref] [PubMed]

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Zhu, S. N.

Y.-X. Gong, S. Zhang, P. Xu, and S. N. Zhu, “Scheme for generating distillation-favorable continuous-variable entanglement via three concurrent parametric down-conversions in a single χ(2) nonlinear photonic crystal,” Opt. Express 24, 6402–6412 (2016).
[Crossref] [PubMed]

L. Chen, P. Xu, Y. F. Bai, X. W. Luo, M. L. Zhong, M. Dai, M. H. Lu, and S. N. Zhu, “Concurrent optical parametric down-conversion in χ(2) nonlinear photonic crystals,” Opt. Express 22, 13164–13169 (2014).
[Crossref] [PubMed]

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

Appl. Phys. B: Lasers Opt. (1)

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Shared optical parametric generation interactions in square lattice nonlinear photonic crystals,” Appl. Phys. B: Lasers Opt. 123, 113 (2017).
[Crossref]

Appl. Phys. Lett. (1)

M. Levenius, V. Pasiskevicius, and K. Gallo, “Angular degrees of freedom in twin-beam parametric down-conversion,” Appl. Phys. Lett. 101, 121114 (2012).
[Crossref]

J. Opt. (1)

H. Chikh-Touami, R. Kremer, H.-J. Lee, M. Lee, L.-H. Peng, and A. Boudrioua, “Experimental investigation of optical parametric generation enhancement in nonlinear photonic crystal of LiTaO3,” J. Opt. 19, 065503 (2017).
[Crossref]

Jpn. J. Appl. Phys. (1)

H. H. Lim, S. Kurimura, T. Katagai, and I. Shoji, “Temperature-dependent sellmeier equation for refractive index of 1.0 mol % Mg-doped stoichiometric lithium tantalate,” Jpn. J. Appl. Phys. 52, 032601 (2013).
[Crossref]

Opt. Express (5)

Opt. Lett. (5)

Opt. Mater. Express (1)

Opt. Quant. Electron. (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]

Phys. Rev. A (5)

A. Gatti, R. Zambrini, M. San Miguel, and L. A. Lugiato, “Multiphoton multimode polarization entanglement in parametric down-conversion,” Phys. Rev. A 68, 053807 (2003).
[Crossref]

E. Brambilla, O. Jedrkiewicz, L. A. Lugiato, and A. Gatti, “Disclosing the spatiotemporal structure of parametric down-conversion entanglement through frequency up-conversion,” Phys. Rev. A 85, 063834 (2012).
[Crossref]

M. J. Werner and P. D. Drummond, “Pulsed quadrature-phase squeezing of solitary waves in χ(2) parametric waveguides,” Phys. Rev. A 56, 1508–1518 (1997).
[Crossref]

Y.-X. Gong, P. Xu, Y. F. Bai, J. Yang, H. Y. Leng, Z. D. Xie, and S. N. Zhu, “Multiphoton path-entanglement generation by concurrent parametric down-conversion in a single χ(2) nonlinear photonic crystal,” Phys. Rev. A 86, 023835 (2012).
[Crossref]

A. Gatti, E. Brambilla, K. Gallo, and O. Jedrkiewicz, “Golden ratio entanglement in hexagonally poled nonlinear crystals,” Phys. Rev. A 98, 053827 (2018).
[Crossref]

Phys. Rev. Lett. (4)

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
[Crossref]

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, “Hexagonally poled Lithium Niobate: A two-dimensional nonlinear photonic crystal,” Phys. Rev. Lett. 84, 4345–4348 (2000).
[Crossref] [PubMed]

H. Jin, P. Xu, X. W. Luo, H. Y. Leng, Y. X. Gong, W. J. Yu, M. L. Zhong, G. Zhao, and S. N. Zhu, “Compact engineering of path-entangled sources from a monolithic quadratic nonlinear photonic crystal,” Phys. Rev. Lett. 111, 023603 (2013).
[Crossref] [PubMed]

K. Gallo, A. Pasquazi, S. Stivala, and G. Assanto, “Parametric solitons in two-dimensional lattices of purely nonlinear origin,” Phys. Rev. Lett. 100, 053901 (2008).
[Crossref] [PubMed]

Sci. Rep. (1)

O. Jedrkiewicz, A. Gatti, E. Brambilla, M. Levenius, G. Tamosauskas, and K. Gallo, “Golden ratio gain enhancement in coherently coupled parametric processes,” Sci. Rep. 8, 11616 (2018).
[Crossref] [PubMed]

Other (2)

K. Stensson, G. Björk, and K. Gallo, “Green-pumped parametric downconversion in hexagonally poled MgO:LiTaO3,” in Advanced Solid State Lasers, (Optical Society of America, 2014), p. ATu3A.5.
[Crossref]

A. Gatti and E. Brambilla, “Engineering multipartite entanglement in nonlinear photonic crystals,” Preprint.

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Figures (9)

Fig. 1
Fig. 1 a) Hexagonally poled χ ( 2 ) crystal pumped by two waves, symmetrically tilted with respect to the z-axis. b) A spatial resonance is achieved by matching the pump transverse wave-vectors ± q p with the transverse components ± G x e x of the lattice vectors G 1 and G 2.
Fig. 2
Fig. 2 Quasi phase-matching in a hexagonally poled LiTaO3 crystal pumped at 532nm by a dual pump at spatial resonance with the lattice: q 0 p e x = G x e x, with G x = 2 π / ( 3 Λ ) = 0.466 μ m 1. (a) QPM surfaces in the 3D Fourier space, and (b) its section at qy = 0.
Fig. 3
Fig. 3 Photon-number distribution in the ( q x , q y )-plane (top) and in the ( λ , q x )-planes (bottom) for a single pump (a,d), two symmetric pumps (b,e) and two antisymmetric pumps (c,f), from numerical simulations of Eqs. (1), in the same NPC of Fig. 2. The pumps are plane-waves, g ¯ = 0.4 mm   1, and results are shown after 7mm of propagation ( g ¯ z = 2.8). In (d,e) the scale was truncated to 1% of the peak value. For a dual symmetric pump the Σ0 branch is significantly more intense than for a single pump of equal energy, while it is absent for antisymmetric pumping. Lines of hot spots at q x = ± G x are clearly visible in panels (a) and (b).
Fig. 4
Fig. 4 Comparison between the use of a dual symmetric pump (red triangles) and a single pump (blue squares) with the same energy. The gain enhancement factor γ is evaluated from numerical simulations of Eqs. (1), The results for plane-wave pumps in (a,b) are very close to the γ t h predicted by the parametric model [Eq. (13)]. The lower panels (c,d) are obtained for Gaussian pumps, of waists 500 μm and 200 μm along the x and y axis. Other parameters as in Fig. 3.
Fig. 5
Fig. 5 Example of four-mode coupling process among shared modes at two conjugate frequencies ω p 2 ± Ω s
Fig. 6
Fig. 6 Eigenvalues Λ + (upper surface) and Λ (lower surface) of the 4-mode propagation Eqs. (5), normalized to g ¯ = | g 1 | 2 + | g 2 | 2 as a function of the ratio | r | = | g 2 g 1 | of the amplitudes of the two pumps and of their phase difference ϕ 2 ϕ 1.
Fig. 7
Fig. 7 Evaluation of the gain enhancement factor γ in the hot-spots at q x = ± G x, from numerical simulations of Eqs. (1). Comparison between the single pump (blue square) and the dual symmetric pump (red triangles), for (a) plane-wave pumps, (b) Gaussian pumps. The crystal and pump parameters are the same as in Figs. 3 and 4.
Fig. 8
Fig. 8 (a) For = g 2 / g 1 ϵ , the 4-mode process (19) is equivalent to two independent standard parametric processes of gains Λ + and Λ mixed on a beam splitter. Panel (b) and (c) show the eigenvalues Λ ± and the ratio between the transmission and reflection coefficients of the beam-splitter as a function of r respectively.
Fig. 9
Fig. 9 Results of simulations away from resonance, for two symmetric pumps with q 0 p = 1.2 G x. (a) Intensity distribution in the ( q x , q y ) plane, showing four lines of hot spots at q x = ± q 0 p and q x = ± G x. (b) Intensity distribution in the ( λ , q x ) plane (qy = 0). Other parameters as in Fig. 3.

Equations (45)

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d ( x , z ) e i G z z [ d 01 e i G x x + d 10 e i G x x ] = 2 d 01 e i G z z cos  ( G x x )
z A ^ s ( w s , z ) = χ d 3 w p ( 2 π ) 3 2 A ^ p ( w p , z ) [ A ^ s ( w p w s G x , z ) e i D ( w s , w p w s G x ) z + A ^ s ( w p w s + G x , z ) e i D ( w s , w p w s + G x ) z ]
z A ^ p ( w p , z ) = χ 2 d 3 w s ( 2 π ) 3 2 A ^ s ( w s , z ) [ A ^ s ( w p w s G x , z ) e i D ( w s , w p w s G x ) z + A ^ s ( w p w s + G x , z ) e i D ( w s , w p w s + G x ) z ]
D ( w s , w p w s ± G x ) = [ k s z ( w s ) + k s z ( w p w s ± G x ) k p z ( w p ) + G z ] ,
A p ( q , Ω p ) = d t 2 π d x 2 π e i Ω t i q x A p ( x , t ) = ( 2 π ) 3 / 2 δ ( Ω ) δ ( q y ) [ α 1 δ ( q x q 0 p ) + α 2 δ ( q x + q 0 p ) ]
Σ 11 : D ( w s , w 0 p w s + G x ) = 0 [ q s + q i = ( q 0 p + G x ) e x ]
Σ 12 : D ( w s , w 0 p w s G x ) = 0 [ q s + q i = ( q 0 p G x ) e x ]
Σ 21 : D ( w s , w 0 p w s + G x ) = 0 [ q s + q i = ( q 0 p G x ) e x ]
Σ 22 : D ( w s , w 0 p w s G x ) = 0 [ q s + q i = ( q 0 p + G x ) e x ]
q 0 p = G x A p ( z , x , t ) = α 1 e i G x x + α 2 e i G x x
Σ 12 , Σ 21 Σ 0 : D ( w s , w s ) = 0
Σ 11 : D ( w s , w s + 2 G x ) = 0 ,
Σ 22 : D ( w s , w s 2 G x ) = 0
A ^ s z ( w s ) = ( g 1 + g 2 ) A ^ s ( w s ) e i D ( w s , w s ) z + g 1 A ^ s ( w s + 2 G x ) e i D ( w s , w s + 2 G x ) z + g 2 A ^ s ( w s 2 G x ) e i D ( w s , w s 2 G x ) z
A ^ s z ( w s ) = γ g ¯ e i ϕ 1 A ^ s ( w i ) e i D ( w s , w i ) z
A ^ s z ( w i ) = γ g ¯ e i ϕ 1 A ^ s ( w s ) e i D ( w s , w i ) z
g ¯ = | g 1 | 2 + | g 2 | 2
γ = { g 1 + g 2 g ¯ = 1 + r 1 + | r | 2 Σ 0 g 1 g ¯ = 1 1 + | r | 2 Σ 11 g 2 g ¯ = r 1 + | r | 2 Σ 22
D ( w s , w s + 2 G x ) = D ( w s , w s ) = 0 Σ 0 Σ 11
D ( w s , w s 2 G x ) = D ( w s , w s ) = 0 Σ 0 Σ 22
D ( w s , w s + 2 G x ) = D ( w s , w s 2 G x ) = 0 Σ 22 Σ 11
b ^ s : = A ^ s ( + G x , q s y , Ω s ) b ^ i : = A ^ s ( + G x , q s y , Ω s ) modes at q x = + G x
c ^ s : = A ^ s ( G x , q s y , Ω s ) c ^ i : = A ^ s ( G x , q s y , Ω s ) modes at q x = G x
d b ^ s d z = [ g 1 b ^ i + ( g 1 + g 2 ) c ^ i ] e i D ¯ z
d c ^ s d z = [ ( g 1 + g 2 ) b ^ i + g 2 c ^ i ] e i D ¯ z
d b ^ i d z = [ g 1 * b ^ s + ( g 1 + g 2 ) * c ^ s ] e i D ¯ z
d c ^ i d z = [ ( g 1 + g 2 ) * b ^ s + g 2 * c ^ s ] e i D ¯ z
Λ ± = [ 2 | g 1 + g 2 | + | g 1 | + | g 2 | 2 ± 1 2 ( | g 1 | 2 | g 2 | 2 ] 2 + 4 | g 1 + g 2 | 4 + 4 [ 2 I m ( g 1 g 2 * ) ] 2 ] 1 2 g ¯ 1 + r 2 | 1 + r 2 ± 1 2 5 ( 1 + r 2 ) + 6 r | = g ¯ × { 5 ± 1 2 r = 0 3 2 , 1 2 r = 1 1 2 , 1 2 r = 1 for r = g 2 g 1 ϵ
( δ ^ j σ ^ j ) = ( cos Θ sin Θ sin Θ cos Θ ) ( b ^ j c ^ j ) j = s , i
d σ ^ s d z = Λ + σ ^ i e i D ¯ z
d σ ^ i d z = Λ + σ ^ s e i D ¯ z
d δ ^ s d z = Λ δ ^ i e i D ¯ z
d δ ^ i d z = Λ δ ^ s e i D ¯ z
k s z ( w s ) k s ( Ω s ) q s 2 2 k s ( Ω s ) ,
D ( w s , w s + R l m ) = k s z ( w s ) + k s z ( w s + R l m ) k p z ( w 0 p ) + G z
k s ( Ω ) + k s ( Ω s ) k p z + G z | R l m | 2 2 ( k s ( Ω s ) + k s ( Ω s ) ) k s ( Ω s ) + k s ( Ω s ) 2 k s ( Ω s ) k s ( Ω ) | q k s ( Ω s ) k s ( Ω s ) + k s ( Ω s ) R l m | 2 , ( l , m = 1 , 2 )
D 0 ( Ω s ) = k s ( Ω ) + k s ( Ω s ) k p z + G z | R l m | 2 2 ( k s ( Ω s ) + k s ( Ω s ) ) 2 k s k p z + G z | R l m | 2 4 k s + k s   Ω s 2 0
q l m = k s ( Ω ) k s ( Ω ) + k s ( Ω ) R l m R l m 2
Q l m = [ 2 k s ( Ω ) k s ( Ω ) k s ( Ω ) + k s ( Ω ) D 0 ( Ω s ) ] 1 2 k s D 0 ( Ω s )
2 k s k p z + G z | 2 G x | 2 4 k s = 0
Σ 11 : | q s G x e x | k s k s   | Ω s | ,
Σ 22 : | q s + G x e x | k s k s   | Ω s | ,
Σ 0 : | q s | ( G x 2 + k s k s   Ω s 2 ) 1 / 2 ,
Σ l m Σ p q : D ( w s , w s + R l m ) = D ( w s , w s + R p q ) = 0 ,
k s z ( w s + R l m ) = k s z ( w s + R p q ) q s x = R l m + R p q 2

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