Abstract

The geometric phase is a common phenomenon in a variety of physical systems, with significant applications in optics. We report the first experimental demonstration of the adiabatic geometric phase in nonlinear frequency conversion, wherein the coupling between the signal and idler frequencies constitutes the intrinsic two-level dynamics of the system. We observe a variety of effects associated with the geometric phase, including the adiabatic broadening of bandwidth, asymmetric transmission for opposite propagation directions, conjugation of the phase for orthogonal eigenstates, and the nonreciprocity associated with the pump field bias. Our work paves the way towards all-optically controlled geometric phase elements for wavefront shaping, isolation, guiding, and quantum optical applications, harnessing spectral and spatial correlations.

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

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References

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2019 (3)

2018 (8)

T. Stav, A. Faerman, E. Maguid, D. Oren, V. Kleiner, E. Hasman, and M. Segev, “Quantum entanglement of the spin and angular momentum of photons using metamaterials,” Science 361, 1101–1104 (2018).
[Crossref]

A. Karnieli and A. Arie, “Fully controllable adiabatic geometric phase in nonlinear optics,” Opt. Express 26, 4920–4932 (2018).
[Crossref]

A. Karnieli and A. Arie, “All-optical Stern–Gerlach effect,” Phys. Rev. Lett. 120, 053901 (2018).
[Crossref]

N. Shitrit, J. Kim, D. S. Barth, H. Ramezani, Y. Wang, and X. Zhang, “Asymmetric free-space light transport at nonlinear metasurfaces,” Phys. Rev. Lett. 121, 046101 (2018).
[Crossref]

T. Xu, K. Switkowski, X. Chen, S. Liu, K. Koynov, H. Yu, H. Zhang, J. Wang, Y. Sheng, and W. Krolikowski, “Three-dimensional nonlinear photonic crystal in ferroelectric barium calcium titanate,” Nat. Photonics 12, 591–595 (2018).
[Crossref]

D. Wei, C. Wang, H. Wang, X. Hu, D. Wei, X. Fang, Y. Zhang, D. Wu, Y. Hu, J. Li, S. Zhu, and M. Xiao, “Experimental demonstration of a three-dimensional lithium niobate nonlinear photonic crystal,” Nat. Photonics 12, 596–600 (2018).
[Crossref]

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

A. Karnieli and A. Arie, “Frequency domain Stern–Gerlach effect for photonic qubits and qutrits,” Optica 5, 1297–1303 (2018).
[Crossref]

2017 (5)

2016 (1)

S. Slussarenko, A. Alberucci, C. P. Jisha, B. Piccirillo, E. Santamato, G. Assanto, and L. Marrucci, “Guiding light via geometric phases,” Nat. Photonics 10, 571–575 (2016).
[Crossref]

2015 (6)

A. Shapira, L. Naor, and A. Arie, “Nonlinear optical holograms for spatial and spectral shaping of light waves, Sci. Bull. 60, 1403–1415 (2015).
[Crossref]

G. Milione, T. A. Nguyen, J. Leach, D. A. Nolan, and R. R. Alfano, “Using the nonseparability of vector beams to encode information for optical communication,” Opt. Lett. 40, 4887–4890 (2015).
[Crossref]

N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nat. Photonics 9, 180–184 (2015).
[Crossref]

M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Gradient nonlinear Pancharatnam–Berry metasurfaces,” Phys. Rev. Lett. 115, 207403 (2015).
[Crossref]

G. Li, S. Chen, N. Pholchai, B. Reineke, P. W. H. Wong, E. Y. B. Pun, K. Wai Cheah, T. Zentgraf, and S. Zhang, “Continuous control of the nonlinearity phase for harmonic generations,” Nat. Mater. 14, 607–612(2015).
[Crossref]

S. Trajtenberg-Mills, I. Juwiler, and A. Arie, “On-axis shaping of second-harmonic beams,” Laser Photon. Rev. 9, L40–L44 (2015).
[Crossref]

2014 (1)

H. Suchowski, G. Porat, and A. Arie, “Adiabatic processes in frequency conversion,” Laser Photon. Rev. 8, 333 (2014).
[Crossref]

2012 (1)

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
[Crossref]

2010 (1)

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

2006 (1)

L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam–Berry phase optical elements for wave front shaping in the visible domain: switchable helical mode generation,” Appl. Phys. Lett. 88, 221102 (2006).
[Crossref]

2005 (2)

Y. Zhang, Y.-W. Tan, H. L. Stormer, and P. Kim, “Experimental observation of the quantum Hall effect and Berry’s phase in graphene,” Nature 438, 201–204 (2005).
[Crossref]

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

2003 (1)

2001 (1)

1986 (1)

R. Y. Chiao and Y.-S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933 (1986).
[Crossref]

1984 (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).
[Crossref]

1956 (1)

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. A. 44, 247–262 (1956).
[Crossref]

Abdelsalam, K.

T. Li, K. Abdelsalam, S. Fathpour, and J. B. Khurgin, “Wide bandwidth, nonmagnetic linear optical isolators based on frequency conversion,” in Conference on Lasers and Electro-Optics (OSA, 2019), paper FW3B.7.

Achouri, K.

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

Alberucci, A.

S. Slussarenko, A. Alberucci, C. P. Jisha, B. Piccirillo, E. Santamato, G. Assanto, and L. Marrucci, “Guiding light via geometric phases,” Nat. Photonics 10, 571–575 (2016).
[Crossref]

Alemán-Castaneda, L. A.

Alfano, R. R.

Alonso, M. A.

Alù, A.

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Gradient nonlinear Pancharatnam–Berry metasurfaces,” Phys. Rev. Lett. 115, 207403 (2015).
[Crossref]

Ambrosio, A.

Arie, A.

A. Karnieli and A. Arie, “All-optical Stern–Gerlach effect,” Phys. Rev. Lett. 120, 053901 (2018).
[Crossref]

A. Karnieli and A. Arie, “Fully controllable adiabatic geometric phase in nonlinear optics,” Opt. Express 26, 4920–4932 (2018).
[Crossref]

A. Karnieli and A. Arie, “Frequency domain Stern–Gerlach effect for photonic qubits and qutrits,” Optica 5, 1297–1303 (2018).
[Crossref]

S. Trajtenberg-Mills and A. Arie, “Shaping light beams in nonlinear processes using structured light and patterned crystals,” Opt. Mater. Express 7, 2928–2942 (2017).
[Crossref]

A. Shapira, L. Naor, and A. Arie, “Nonlinear optical holograms for spatial and spectral shaping of light waves, Sci. Bull. 60, 1403–1415 (2015).
[Crossref]

S. Trajtenberg-Mills, I. Juwiler, and A. Arie, “On-axis shaping of second-harmonic beams,” Laser Photon. Rev. 9, L40–L44 (2015).
[Crossref]

H. Suchowski, G. Porat, and A. Arie, “Adiabatic processes in frequency conversion,” Laser Photon. Rev. 8, 333 (2014).
[Crossref]

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

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

Assanto, G.

S. Slussarenko, A. Alberucci, C. P. Jisha, B. Piccirillo, E. Santamato, G. Assanto, and L. Marrucci, “Guiding light via geometric phases,” Nat. Photonics 10, 571–575 (2016).
[Crossref]

Bahabad, A.

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

Barth, D. S.

N. Shitrit, J. Kim, D. S. Barth, H. Ramezani, Y. Wang, and X. Zhang, “Asymmetric free-space light transport at nonlinear metasurfaces,” Phys. Rev. Lett. 121, 046101 (2018).
[Crossref]

Belkin, M. A.

M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Gradient nonlinear Pancharatnam–Berry metasurfaces,” Phys. Rev. Lett. 115, 207403 (2015).
[Crossref]

Berry, M. V.

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. London Ser. A 392, 45–57 (1984).
[Crossref]

Bomzon, Z.

Bouchard, F.

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1, 437 (2019).
[Crossref]

Boyd, R. W.

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

Caloz, C.

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

Capasso, F.

Chen, R.

Chen, S.

G. Li, S. Chen, N. Pholchai, B. Reineke, P. W. H. Wong, E. Y. B. Pun, K. Wai Cheah, T. Zentgraf, and S. Zhang, “Continuous control of the nonlinearity phase for harmonic generations,” Nat. Mater. 14, 607–612(2015).
[Crossref]

Chen, X.

T. Xu, K. Switkowski, X. Chen, S. Liu, K. Koynov, H. Yu, H. Zhang, J. Wang, Y. Sheng, and W. Krolikowski, “Three-dimensional nonlinear photonic crystal in ferroelectric barium calcium titanate,” Nat. Photonics 12, 591–595 (2018).
[Crossref]

Chiao, R. Y.

R. Y. Chiao and Y.-S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57, 933 (1986).
[Crossref]

Cohen, E.

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1, 437 (2019).
[Crossref]

Deck-Léger, Z.-L.

C. Caloz, A. Alù, S. Tretyakov, D. Sounas, K. Achouri, and Z.-L. Deck-Léger, “Electromagnetic nonreciprocity,” Phys. Rev. Appl. 10, 047001 (2018).
[Crossref]

Devlin, R. C.

Ellenbogen, T.

N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nat. Photonics 9, 180–184 (2015).
[Crossref]

Faerman, A.

T. Stav, A. Faerman, E. Maguid, D. Oren, V. Kleiner, E. Hasman, and M. Segev, “Quantum entanglement of the spin and angular momentum of photons using metamaterials,” Science 361, 1101–1104 (2018).
[Crossref]

Fan, S.

K. Wang, Y. Shi, A. S. Solntsev, S. Fan, A. A. Sukhorukov, and D. N. Neshev, “Non-reciprocal geometric phase in nonlinear frequency conversion,” Opt. Lett. 42, 1990–1993 (2017).
[Crossref]

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. Rev. Lett. 109, 033901 (2012).
[Crossref]

Fang, X.

D. Wei, C. Wang, H. Wang, X. Hu, D. Wei, X. Fang, Y. Zhang, D. Wu, Y. Hu, J. Li, S. Zhu, and M. Xiao, “Experimental demonstration of a three-dimensional lithium niobate nonlinear photonic crystal,” Nat. Photonics 12, 596–600 (2018).
[Crossref]

Fathpour, S.

T. Li, K. Abdelsalam, S. Fathpour, and J. B. Khurgin, “Wide bandwidth, nonmagnetic linear optical isolators based on frequency conversion,” in Conference on Lasers and Electro-Optics (OSA, 2019), paper FW3B.7.

Flemens, N.

P. Krogen, H. Suchowski, H. Liang, N. Flemens, K.-H. Hong, F. X. Kärtner, and J. Moses, “Generation and multi-octave shaping of mid-infrared intense single-cycle pulses,” Nat. Photonics 11, 222–226(2017).
[Crossref]

Gefen, Y.

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1, 437 (2019).
[Crossref]

Gomez-Diaz, J. S.

M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Gradient nonlinear Pancharatnam–Berry metasurfaces,” Phys. Rev. Lett. 115, 207403 (2015).
[Crossref]

Hasman, E.

T. Stav, A. Faerman, E. Maguid, D. Oren, V. Kleiner, E. Hasman, and M. Segev, “Quantum entanglement of the spin and angular momentum of photons using metamaterials,” Science 361, 1101–1104 (2018).
[Crossref]

Z. Bomzon, V. Kleiner, and E. Hasman, “Pancharatnam–Berry phase in space-variant polarization-state manipulations with subwavelength gratings,” Opt. Lett. 26, 1424–1426 (2001).
[Crossref]

Hendler, N.

N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nat. Photonics 9, 180–184 (2015).
[Crossref]

Holzlöhner, R.

Hong, K.-H.

P. Krogen, H. Suchowski, H. Liang, N. Flemens, K.-H. Hong, F. X. Kärtner, and J. Moses, “Generation and multi-octave shaping of mid-infrared intense single-cycle pulses,” Nat. Photonics 11, 222–226(2017).
[Crossref]

Hu, X.

D. Wei, C. Wang, H. Wang, X. Hu, D. Wei, X. Fang, Y. Zhang, D. Wu, Y. Hu, J. Li, S. Zhu, and M. Xiao, “Experimental demonstration of a three-dimensional lithium niobate nonlinear photonic crystal,” Nat. Photonics 12, 596–600 (2018).
[Crossref]

Hu, Y.

D. Wei, C. Wang, H. Wang, X. Hu, D. Wei, X. Fang, Y. Zhang, D. Wu, Y. Hu, J. Li, S. Zhu, and M. Xiao, “Experimental demonstration of a three-dimensional lithium niobate nonlinear photonic crystal,” Nat. Photonics 12, 596–600 (2018).
[Crossref]

Jisha, C. P.

S. Slussarenko, A. Alberucci, C. P. Jisha, B. Piccirillo, E. Santamato, G. Assanto, and L. Marrucci, “Guiding light via geometric phases,” Nat. Photonics 10, 571–575 (2016).
[Crossref]

Juwiler, I.

S. Trajtenberg-Mills, I. Juwiler, and A. Arie, “On-axis shaping of second-harmonic beams,” Laser Photon. Rev. 9, L40–L44 (2015).
[Crossref]

Karimi, E.

E. Cohen, H. Larocque, F. Bouchard, F. Nejadsattari, Y. Gefen, and E. Karimi, “Geometric phase from Aharonov–Bohm to Pancharatnam–Berry and beyond,” Nat. Rev. Phys. 1, 437 (2019).
[Crossref]

Karnieli, A.

Kärtner, F. X.

P. Krogen, H. Suchowski, H. Liang, N. Flemens, K.-H. Hong, F. X. Kärtner, and J. Moses, “Generation and multi-octave shaping of mid-infrared intense single-cycle pulses,” Nat. Photonics 11, 222–226(2017).
[Crossref]

Keren-Zur, S.

N. Segal, S. Keren-Zur, N. Hendler, and T. Ellenbogen, “Controlling light with metamaterial-based nonlinear photonic crystals,” Nat. Photonics 9, 180–184 (2015).
[Crossref]

Khorasaninejad, M.

Khurgin, J. B.

T. Li, K. Abdelsalam, S. Fathpour, and J. B. Khurgin, “Wide bandwidth, nonmagnetic linear optical isolators based on frequency conversion,” in Conference on Lasers and Electro-Optics (OSA, 2019), paper FW3B.7.

Kim, J.

N. Shitrit, J. Kim, D. S. Barth, H. Ramezani, Y. Wang, and X. Zhang, “Asymmetric free-space light transport at nonlinear metasurfaces,” Phys. Rev. Lett. 121, 046101 (2018).
[Crossref]

Kim, P.

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Nat. Photonics (5)

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Opt. Lett. (3)

Opt. Mater. Express (1)

Optica (2)

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Supplementary Material (1)

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Figures (3)

Fig. 1.
Fig. 1. Wedged rotation scheme. (a) The parameter-space surface is elongated along the z direction. The Hamiltonian follows the black line from the north pole (the idler frequency in the corresponding Bloch sphere) to the south pole (signal frequency). (b) Illustration of the experiment. The idler and pump beams enter the crystal; the signal emerging is shaped as HG01. Inset is a microscopic image of the fabricated crystal surface after selective etching that reveals the poled surface. The poling in the two sections has the opposite phase. (c) Far-field image of the measured signal field at 631 nm. (d) Measured (blue dots) and simulated (red) photon conversion efficiency dependence on idler wavelength. Black dashed curve shows the efficiency for a regular 10 mm long periodically poled crystal with a period of 11.78 μm.
Fig. 2.
Fig. 2. Geometric phase lens. (a), (b) show the experimental setups. In (a) the idler (1550 nm, CW) and pump (1064.5 nm, pulsed) enter the crystal, and the signal is measured using a regular CCD camera. In (b) the signal (633 nm, CW) and pump fields enter the geometric phase crystal, and the idler is measured with a cooled InGaAs CCD camera. F1 was used to filter the pump and signal/idler wavelength, L1 was used for focusing, and L2 was used for imaging consecutive planes from the first/second crystal facet (inside/outside the crystal) until 10 mm after. (c), (d) and (e), (f) show the idler and signal propagation measurements, at different crystal orientations, shown in black and white at the bottom. Top insets are transverse pictures of the beam at planes depicted by the white arrows. The yellow scale bar is 0.2 mm.
Fig. 3.
Fig. 3. Circular rotation scheme. (a) The crystal design has two paths, each following the same circular trajectory: one clockwise and the second counterclockwise. (b) B follows the black line around the unit vector n^, which forms an angle Θ0 with the z axis. (c) The experiment. The pulsed 1064.5 nm pump and 1550 nm CW idler are combined using a dichroic mirror (DM) in an adiabatic KTP crystal for conversion to the 631 nm signal. The idler is filtered using the filter (F1) and the synchronized pump and signal pulses are focused using the lens (L1) into our geometric phase crystal. The pump is filtered by F2, and the output is imaged to a CCD camera using L2. We imaged consecutive planes from the crystal exit facet. (d1), (d2) The measured propagation after the crystal, when the pump is on (d2) or off—i.e., filtered before the crystal (d1). Inset is the transverse image of the beam at the exit facet of the crystal.

Equations (3)

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B=κ(cosφx^+sinφy^)+Δk2z^,
HSFG(z)=σ·B(z),
γ=12Ω,