Abstract

We propose and analyze a new way for obtaining an adiabatic geometric phase for light, via the sum-frequency-generation nonlinear process. The state of light is represented by the complex amplitudes at two different optical frequencies, coupled by the second order nonlinearity of the medium. The dynamics of this system is then shown to be equivalent to that of a spin-1/2 particle in a magnetic field, which in turn can be rotated adiabatically on the Bloch sphere. When the input wave itself is an eigenstate of the magnetic field equivalent, the geometric phase is manifested as a pure phase factor. Two adiabatic rotation schemes, based on specific modulations of the quasi-phase-matching poling parameters, are discussed. In the first, the geometric phase is shown to be sensitive to the pump intensity variations, as a result of the Bloch sphere deformation. The second can be utilized for the realization of nonlinear-optics-based geometric phase plates. Moreover, non-closed adiabatic trajectories are investigated, which are expected to provide a robust and broadband geometric wavefront shaping in the sum frequency.

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

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References

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  1. S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Ac. Sci. A. 44(5), 247–262 (1956).
  2. M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A,  392(1802), 45–57 (1984).
    [Crossref]
  3. J. J. Sakurai and J. Napolitano, Modern quantum mechanics, 2 edition (Addison-Wesley, 2011).
  4. M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34(11), 1401–1407 (1987).
    [Crossref]
  5. J. C. Gutiérrez-Vega, “Pancharatnam-Berry phase of optical systems,” Opt. Lett.,  36(7), 1143–1145 (2011).
    [Crossref] [PubMed]
  6. S. Slussarenko, A. Alberucci, C. P. Jisha, S. Fan, B. Piccirillo, E. Santamato, G. Assanto, and L. Marrucci, “Guiding light via geometric phases,” Nat. Phot.,  10, 571–575 (2016).
    [Crossref]
  7. Z. Bomzon, V. Kleiner, and E. Hasman, “Pancharatnam-Berry phase in space-variant polarization-state manipulations with subwavelength gratings,” Opt. Lett. 26(18), 1424–1426 (2001).
    [Crossref]
  8. L. Marrucci, C. Manzo, and D. Paparo, “Pancharatnam-Berry phase optical elements for wave front shaping in the visible domain: switchable helical mode generation,” App. Phys. Lett 88, 221102 (2006).
    [Crossref]
  9. M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science,  352(6290), 1190–1194 (2016).
    [Crossref] [PubMed]
  10. R. Y. Chiao and Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57(8), 933–936 (1986).
    [Crossref] [PubMed]
  11. A. Tomita and R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57(8), 937–940 (1986).
    [Crossref] [PubMed]
  12. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A. 78, 063821 (2008).
    [Crossref]
  13. H. Suchowski, G. Porat, A. Arie, and Y. Silberberg, “Adiabatic processes in frequency conversion,” Las. Phot. Rev. 8(3), 333–367 (2014).
    [Crossref]
  14. H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17(15), 12731–12740 (2009).
    [Crossref] [PubMed]
  15. P. Krogen, H. Suchowski, H. Liang, H. Flemens, K. H. Hong, F. X. Kartner, and J. Moses, “Generation and multi-octave shaping of mid-infrared intense single-cycle pulses,” Nat. Phot. 11, 222–226 (2017).
    [Crossref]
  16. C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasiphase-matched gratings,” Opt. Lett.,  35(18), 3093–3095 (2010).
    [Crossref] [PubMed]
  17. O. Yaakobi, L. Caspani, M. Clerici, F. Vidal, and R. Morandotti, “Complete energy conversion by autoresonant three-wave mixing in nonuniform media,” Opt. Express 21(2), 1623–1632 (2013).
    [Crossref] [PubMed]
  18. P. Mandel, P. Galatola, L. A. Lugiato, and W. Kaige, “Berry phase analogies in nonlinear optics,” Opt. Comm. 80(3), 262–266 (1991).
    [Crossref]
  19. 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] [PubMed]
  20. M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Advanced control of nonlinear beams with Pancharatnam-Berry metasurfaces,” Phys. Rev. B. 94, 214303 (2016).
    [Crossref]
  21. 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(10), 1990–1993 (2017).
    [Crossref] [PubMed]
  22. R. W. Boyd, Nonlinear optics, 3 edition (Academic Press, 2008).
  23. A. Shapira, L. Naor, and A. Arie, “Nonlinear optical holograms for spatial and spectral shaping of light waves,” Sci. Bull. 60(16), 1403–1415 (2015).
    [Crossref]
  24. S. Trajtenberg-Mills, I. Juwiler, and A. Arie, “On-axis shaping of second harmonic beams,” Las. Phot. Rev. 9(6), L40–L44 (2015).
    [Crossref]
  25. J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, “Nonlinear physical optics with transversely patterned quasi-phase-matching gratings,” J. Sel. Top. Quant. Elect. 8(3), 660–664 (2002).
    [Crossref]
  26. N. Voloch Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
    [Crossref]
  27. A. Bahabad and A. Arie, “Generation of optical vortex beams by nonlinear wave mixing”, Opt. Express 15(26), 17619–17624 (2007).
    [Crossref] [PubMed]
  28. M. J. Missey, V. Dominic, L. E. Myers, and R. C. Eckardt, “Diffusion-bonded stacks of periodically poled lithium niobate,” Opt. Lett. 9(23), 664–666 (1998).
    [Crossref]
  29. A. Karnieli and A. Arie, “All-optical Stern-Gerlach effect,” Phys. Rev. Lett. 120, 053901 (2018).
  30. S. Trajtenberg-Mills and A. Arie, “Shaping light beams in nonlinear processes using structured light and patterned crystals,” Opt. Mat. Express 7(8), 2928–2942 (2017).
    [Crossref]
  31. The specific value of the azimuthal angle of n^ does not play a role in our analysis.

2018 (1)

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

2017 (3)

S. Trajtenberg-Mills and A. Arie, “Shaping light beams in nonlinear processes using structured light and patterned crystals,” Opt. Mat. Express 7(8), 2928–2942 (2017).
[Crossref]

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(10), 1990–1993 (2017).
[Crossref] [PubMed]

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

2016 (3)

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

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science,  352(6290), 1190–1194 (2016).
[Crossref] [PubMed]

M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Advanced control of nonlinear beams with Pancharatnam-Berry metasurfaces,” Phys. Rev. B. 94, 214303 (2016).
[Crossref]

2015 (3)

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] [PubMed]

A. Shapira, L. Naor, and A. Arie, “Nonlinear optical holograms for spatial and spectral shaping of light waves,” Sci. Bull. 60(16), 1403–1415 (2015).
[Crossref]

S. Trajtenberg-Mills, I. Juwiler, and A. Arie, “On-axis shaping of second harmonic beams,” Las. Phot. Rev. 9(6), L40–L44 (2015).
[Crossref]

2014 (1)

H. Suchowski, G. Porat, A. Arie, and Y. Silberberg, “Adiabatic processes in frequency conversion,” Las. Phot. Rev. 8(3), 333–367 (2014).
[Crossref]

2013 (1)

2012 (1)

N. Voloch Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
[Crossref]

2011 (1)

2010 (1)

2009 (1)

2008 (1)

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A. 78, 063821 (2008).
[Crossref]

2007 (1)

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,” App. Phys. Lett 88, 221102 (2006).
[Crossref]

2002 (1)

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, “Nonlinear physical optics with transversely patterned quasi-phase-matching gratings,” J. Sel. Top. Quant. Elect. 8(3), 660–664 (2002).
[Crossref]

2001 (1)

1998 (1)

M. J. Missey, V. Dominic, L. E. Myers, and R. C. Eckardt, “Diffusion-bonded stacks of periodically poled lithium niobate,” Opt. Lett. 9(23), 664–666 (1998).
[Crossref]

1991 (1)

P. Mandel, P. Galatola, L. A. Lugiato, and W. Kaige, “Berry phase analogies in nonlinear optics,” Opt. Comm. 80(3), 262–266 (1991).
[Crossref]

1987 (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34(11), 1401–1407 (1987).
[Crossref]

1986 (2)

R. Y. Chiao and Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57(8), 933–936 (1986).
[Crossref] [PubMed]

A. Tomita and R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57(8), 937–940 (1986).
[Crossref] [PubMed]

1984 (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A,  392(1802), 45–57 (1984).
[Crossref]

1956 (1)

S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Ac. Sci. A. 44(5), 247–262 (1956).

Alberucci, A.

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

Alù, A.

M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Advanced control of nonlinear beams with Pancharatnam-Berry metasurfaces,” Phys. Rev. B. 94, 214303 (2016).
[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] [PubMed]

Arie, A.

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

S. Trajtenberg-Mills and A. Arie, “Shaping light beams in nonlinear processes using structured light and patterned crystals,” Opt. Mat. Express 7(8), 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(16), 1403–1415 (2015).
[Crossref]

S. Trajtenberg-Mills, I. Juwiler, and A. Arie, “On-axis shaping of second harmonic beams,” Las. Phot. Rev. 9(6), L40–L44 (2015).
[Crossref]

H. Suchowski, G. Porat, A. Arie, and Y. Silberberg, “Adiabatic processes in frequency conversion,” Las. Phot. Rev. 8(3), 333–367 (2014).
[Crossref]

N. Voloch Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
[Crossref]

H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17(15), 12731–12740 (2009).
[Crossref] [PubMed]

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A. 78, 063821 (2008).
[Crossref]

A. Bahabad and A. Arie, “Generation of optical vortex beams by nonlinear wave mixing”, Opt. Express 15(26), 17619–17624 (2007).
[Crossref] [PubMed]

Assanto, G.

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

Bahabad, A.

Belkin, M. A.

M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Advanced control of nonlinear beams with Pancharatnam-Berry metasurfaces,” Phys. Rev. B. 94, 214303 (2016).
[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] [PubMed]

Berry, M. V.

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34(11), 1401–1407 (1987).
[Crossref]

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A,  392(1802), 45–57 (1984).
[Crossref]

Bomzon, Z.

Boyd, R. W.

R. W. Boyd, Nonlinear optics, 3 edition (Academic Press, 2008).

Capasso, F.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science,  352(6290), 1190–1194 (2016).
[Crossref] [PubMed]

Caspani, L.

Chen, W. T.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science,  352(6290), 1190–1194 (2016).
[Crossref] [PubMed]

Chiao, R. Y.

R. Y. Chiao and Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57(8), 933–936 (1986).
[Crossref] [PubMed]

A. Tomita and R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57(8), 937–940 (1986).
[Crossref] [PubMed]

Clerici, M.

Devlin, R. C.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science,  352(6290), 1190–1194 (2016).
[Crossref] [PubMed]

Dominic, V.

M. J. Missey, V. Dominic, L. E. Myers, and R. C. Eckardt, “Diffusion-bonded stacks of periodically poled lithium niobate,” Opt. Lett. 9(23), 664–666 (1998).
[Crossref]

Eckardt, R. C.

M. J. Missey, V. Dominic, L. E. Myers, and R. C. Eckardt, “Diffusion-bonded stacks of periodically poled lithium niobate,” Opt. Lett. 9(23), 664–666 (1998).
[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(10), 1990–1993 (2017).
[Crossref] [PubMed]

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

Fejer, M. M.

C. R. Phillips and M. M. Fejer, “Efficiency and phase of optical parametric amplification in chirped quasiphase-matched gratings,” Opt. Lett.,  35(18), 3093–3095 (2010).
[Crossref] [PubMed]

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, “Nonlinear physical optics with transversely patterned quasi-phase-matching gratings,” J. Sel. Top. Quant. Elect. 8(3), 660–664 (2002).
[Crossref]

Flemens, H.

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

Galatola, P.

P. Mandel, P. Galatola, L. A. Lugiato, and W. Kaige, “Berry phase analogies in nonlinear optics,” Opt. Comm. 80(3), 262–266 (1991).
[Crossref]

Gomez-Diaz, J. S.

M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Advanced control of nonlinear beams with Pancharatnam-Berry metasurfaces,” Phys. Rev. B. 94, 214303 (2016).
[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] [PubMed]

Gutiérrez-Vega, J. C.

Hasman, E.

Hong, K. H.

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

Hum, D. S.

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, “Nonlinear physical optics with transversely patterned quasi-phase-matching gratings,” J. Sel. Top. Quant. Elect. 8(3), 660–664 (2002).
[Crossref]

Jisha, C. P.

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

Juwiler, I.

S. Trajtenberg-Mills, I. Juwiler, and A. Arie, “On-axis shaping of second harmonic beams,” Las. Phot. Rev. 9(6), L40–L44 (2015).
[Crossref]

N. Voloch Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
[Crossref]

Kaige, W.

P. Mandel, P. Galatola, L. A. Lugiato, and W. Kaige, “Berry phase analogies in nonlinear optics,” Opt. Comm. 80(3), 262–266 (1991).
[Crossref]

Karnieli, A.

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

Kartner, F. X.

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

Khorasaninejad, M.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science,  352(6290), 1190–1194 (2016).
[Crossref] [PubMed]

Kleiner, V.

Krogen, P.

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

Kurz, J. R.

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, “Nonlinear physical optics with transversely patterned quasi-phase-matching gratings,” J. Sel. Top. Quant. Elect. 8(3), 660–664 (2002).
[Crossref]

Lee, J.

M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Advanced control of nonlinear beams with Pancharatnam-Berry metasurfaces,” Phys. Rev. B. 94, 214303 (2016).
[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] [PubMed]

Liang, H.

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

Lugiato, L. A.

P. Mandel, P. Galatola, L. A. Lugiato, and W. Kaige, “Berry phase analogies in nonlinear optics,” Opt. Comm. 80(3), 262–266 (1991).
[Crossref]

Mandel, P.

P. Mandel, P. Galatola, L. A. Lugiato, and W. Kaige, “Berry phase analogies in nonlinear optics,” Opt. Comm. 80(3), 262–266 (1991).
[Crossref]

Manzo, C.

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

Marrucci, L.

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

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

Missey, M. J.

M. J. Missey, V. Dominic, L. E. Myers, and R. C. Eckardt, “Diffusion-bonded stacks of periodically poled lithium niobate,” Opt. Lett. 9(23), 664–666 (1998).
[Crossref]

Morandotti, R.

Moses, J.

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

Myers, L. E.

M. J. Missey, V. Dominic, L. E. Myers, and R. C. Eckardt, “Diffusion-bonded stacks of periodically poled lithium niobate,” Opt. Lett. 9(23), 664–666 (1998).
[Crossref]

Naor, L.

A. Shapira, L. Naor, and A. Arie, “Nonlinear optical holograms for spatial and spectral shaping of light waves,” Sci. Bull. 60(16), 1403–1415 (2015).
[Crossref]

Napolitano, J.

J. J. Sakurai and J. Napolitano, Modern quantum mechanics, 2 edition (Addison-Wesley, 2011).

Neshev, D. N.

Nookala, N.

M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Advanced control of nonlinear beams with Pancharatnam-Berry metasurfaces,” Phys. Rev. B. 94, 214303 (2016).
[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] [PubMed]

Oh, J.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science,  352(6290), 1190–1194 (2016).
[Crossref] [PubMed]

Oron, D.

H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17(15), 12731–12740 (2009).
[Crossref] [PubMed]

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A. 78, 063821 (2008).
[Crossref]

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Ac. Sci. A. 44(5), 247–262 (1956).

Paparo, D.

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

Phillips, C. R.

Piccirillo, B.

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

Porat, G.

H. Suchowski, G. Porat, A. Arie, and Y. Silberberg, “Adiabatic processes in frequency conversion,” Las. Phot. Rev. 8(3), 333–367 (2014).
[Crossref]

Prabhudesai, V.

Sakurai, J. J.

J. J. Sakurai and J. Napolitano, Modern quantum mechanics, 2 edition (Addison-Wesley, 2011).

Saltzman, A. J.

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, “Nonlinear physical optics with transversely patterned quasi-phase-matching gratings,” J. Sel. Top. Quant. Elect. 8(3), 660–664 (2002).
[Crossref]

Santamato, E.

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

Schober, A. M.

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, “Nonlinear physical optics with transversely patterned quasi-phase-matching gratings,” J. Sel. Top. Quant. Elect. 8(3), 660–664 (2002).
[Crossref]

Shapira, A.

A. Shapira, L. Naor, and A. Arie, “Nonlinear optical holograms for spatial and spectral shaping of light waves,” Sci. Bull. 60(16), 1403–1415 (2015).
[Crossref]

N. Voloch Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
[Crossref]

Shemer, K.

N. Voloch Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
[Crossref]

Shi, Y.

Shiloh, R.

N. Voloch Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
[Crossref]

Silberberg, Y.

H. Suchowski, G. Porat, A. Arie, and Y. Silberberg, “Adiabatic processes in frequency conversion,” Las. Phot. Rev. 8(3), 333–367 (2014).
[Crossref]

H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17(15), 12731–12740 (2009).
[Crossref] [PubMed]

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A. 78, 063821 (2008).
[Crossref]

Slussarenko, S.

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

Solntsev, A. S.

Suchowski, H.

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

H. Suchowski, G. Porat, A. Arie, and Y. Silberberg, “Adiabatic processes in frequency conversion,” Las. Phot. Rev. 8(3), 333–367 (2014).
[Crossref]

H. Suchowski, V. Prabhudesai, D. Oron, A. Arie, and Y. Silberberg, “Robust adiabatic sum frequency conversion,” Opt. Express 17(15), 12731–12740 (2009).
[Crossref] [PubMed]

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A. 78, 063821 (2008).
[Crossref]

Sukhorukov, A. A.

Tomita, A.

A. Tomita and R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57(8), 937–940 (1986).
[Crossref] [PubMed]

Trajtenberg-Mills, S.

S. Trajtenberg-Mills and A. Arie, “Shaping light beams in nonlinear processes using structured light and patterned crystals,” Opt. Mat. Express 7(8), 2928–2942 (2017).
[Crossref]

S. Trajtenberg-Mills, I. Juwiler, and A. Arie, “On-axis shaping of second harmonic beams,” Las. Phot. Rev. 9(6), L40–L44 (2015).
[Crossref]

Tymchenko, M.

M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Advanced control of nonlinear beams with Pancharatnam-Berry metasurfaces,” Phys. Rev. B. 94, 214303 (2016).
[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] [PubMed]

Vidal, F.

Voloch Bloch, N.

N. Voloch Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
[Crossref]

Wang, K.

Wu, Y. S.

R. Y. Chiao and Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57(8), 933–936 (1986).
[Crossref] [PubMed]

Yaakobi, O.

Zhu, A. Y.

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science,  352(6290), 1190–1194 (2016).
[Crossref] [PubMed]

App. Phys. Lett (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,” App. Phys. Lett 88, 221102 (2006).
[Crossref]

J. Mod. Opt. (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34(11), 1401–1407 (1987).
[Crossref]

J. Sel. Top. Quant. Elect. (1)

J. R. Kurz, A. M. Schober, D. S. Hum, A. J. Saltzman, and M. M. Fejer, “Nonlinear physical optics with transversely patterned quasi-phase-matching gratings,” J. Sel. Top. Quant. Elect. 8(3), 660–664 (2002).
[Crossref]

Las. Phot. Rev. (2)

S. Trajtenberg-Mills, I. Juwiler, and A. Arie, “On-axis shaping of second harmonic beams,” Las. Phot. Rev. 9(6), L40–L44 (2015).
[Crossref]

H. Suchowski, G. Porat, A. Arie, and Y. Silberberg, “Adiabatic processes in frequency conversion,” Las. Phot. Rev. 8(3), 333–367 (2014).
[Crossref]

Nat. Phot. (2)

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

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

Opt. Comm. (1)

P. Mandel, P. Galatola, L. A. Lugiato, and W. Kaige, “Berry phase analogies in nonlinear optics,” Opt. Comm. 80(3), 262–266 (1991).
[Crossref]

Opt. Express (3)

Opt. Lett. (5)

Opt. Mat. Express (1)

S. Trajtenberg-Mills and A. Arie, “Shaping light beams in nonlinear processes using structured light and patterned crystals,” Opt. Mat. Express 7(8), 2928–2942 (2017).
[Crossref]

Phys. Rev. A. (1)

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A. 78, 063821 (2008).
[Crossref]

Phys. Rev. B. (1)

M. Tymchenko, J. S. Gomez-Diaz, J. Lee, N. Nookala, M. A. Belkin, and A. Alù, “Advanced control of nonlinear beams with Pancharatnam-Berry metasurfaces,” Phys. Rev. B. 94, 214303 (2016).
[Crossref]

Phys. Rev. Lett. (5)

N. Voloch Bloch, K. Shemer, A. Shapira, R. Shiloh, I. Juwiler, and A. Arie, “Twisting light by nonlinear photonic crystals,” Phys. Rev. Lett. 108, 233902 (2012).
[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] [PubMed]

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

R. Y. Chiao and Y. S. Wu, “Manifestations of Berry’s topological phase for the photon,” Phys. Rev. Lett. 57(8), 933–936 (1986).
[Crossref] [PubMed]

A. Tomita and R. Y. Chiao, “Observation of Berry’s topological phase by use of an optical fiber,” Phys. Rev. Lett. 57(8), 937–940 (1986).
[Crossref] [PubMed]

Proc. Ind. Ac. Sci. A. (1)

S. Pancharatnam, “Generalized theory of interference and its applications,” Proc. Ind. Ac. Sci. A. 44(5), 247–262 (1956).

Proc. R. Soc. Lond. A (1)

M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proc. R. Soc. Lond. A,  392(1802), 45–57 (1984).
[Crossref]

Sci. Bull. (1)

A. Shapira, L. Naor, and A. Arie, “Nonlinear optical holograms for spatial and spectral shaping of light waves,” Sci. Bull. 60(16), 1403–1415 (2015).
[Crossref]

Science (1)

M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science,  352(6290), 1190–1194 (2016).
[Crossref] [PubMed]

Other (3)

J. J. Sakurai and J. Napolitano, Modern quantum mechanics, 2 edition (Addison-Wesley, 2011).

R. W. Boyd, Nonlinear optics, 3 edition (Academic Press, 2008).

The specific value of the azimuthal angle of n^ does not play a role in our analysis.

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Figures (7)

Fig. 1
Fig. 1 Different schemes for obtaining a geometric phase of γ = −π for light. (a) The closed-trajectory Pancharatnam-Berry phase: a circularly-polarized light beam undergoes subsequent birefringence via 2 half-wavelength plates, enclosing a solid angle Ω = 2π on the Poincare sphere, and where γ = 1 2 Ω . (b) The adiabatic geometric phase for the photon helicity. A guided mode experiences an adiabatic change in direction when it propagates along a helical fiber with N = 2 windings. This path encloses a solid angle Ω = π/2 on the k-vector sphere, where now γ = −NΩ. (c) The adiabatic nonlinear optical geometric phase. A mutual beam eigenstate in the idler frequency, |ω+〉 = |ωi〉, follows a closed trajectory of the magnetic field equivalent on the SFG sphere, enclosing a solid angle Ω = 2π, where γ = 1 2 Ω . The closed trajectory is enabled through a specific modulation of the QPM parameters: the phase mismatch and poling duty cycle.
Fig. 2
Fig. 2 The control over the equivalent magnetic field is possible via the variation of the QPM parameters. The z ^ component of B is proportional to the phase mismatch Δk, the transverse radial component is given by the product of the nonlinear coupling κ and duty cycle parameter d ¯ , and the relative phase between the pump and the poling pattern, ϕ, determines the azimuthal angle.
Fig. 3
Fig. 3 Mach-Zehnder interferometric setup for measuring the geometric phase, consisting of symmetric beam splitters (BS1-2), filters (F1-2) and detectors (D1-2). The pump and idler beams mix in the different arms of the interferometer. Two identical geometric-phase-inducing nonlinear crystals are situated in each arm with oppositely oriented poling patterns with respect to the propagation axes. Due to its geometric properties, the phase difference between the two arms equals twice the geometric phase.
Fig. 4
Fig. 4 Schemes for adiabatic rotations on the SFG sphere inducing a geometric phase for the idler wave. (a) Circular adiabatic rotation: the magnetic field equivalent precesses about some unit vector n ^ ( Θ 0 ) . (b) Wedged adiabatic rotation: the trajectory of the field encloses a wedge of the unit sphere with opening angle Δϕ. (c)–(d) Normalized QPM modulation parameters corresponding to the circular (with Θ0 = π/3) and wedged (with Δϕ = π/2) rotation schemes, respectively.
Fig. 5
Fig. 5 Simulations of the normalized photon numbers of the idler and signal waves (left) and geometric phase accumulation (right) along the propagation coordinate z. (a) Results for the circular adiabatic rotation scheme with Θ0 = π/3, and thus γ+ = −π/2. (b) Results for the wedged adiabatic rotation with Δϕ = π/2, giving γ+ = −π/2.
Fig. 6
Fig. 6 Adiabatic rotations on the Bloch spheroid for η > 1. (a) For the circular adiabatic rotation, the trajectory of the field is no longer described by a simple rotation about a constant unit vector. The flux through the surface enclosed by the field’s rotation is different than for the same parametrization on the Bloch sphere (η = 1). (b) From symmetry considerations, in case of wedged adiabatic rotation the field’s trajectory encloses the same wedge as if it were on the unit sphere, retaining the value of the geometric phase.
Fig. 7
Fig. 7 A vortex beam geometric phase plate. Four GPNLCs with clockwise increasing geometric phase constitute the apparatus. An incident idler beam is expected to accumulate an OAM quantum number of m = 1 in the far-field, whereas an incident signal beam accumulates m = −1.

Equations (40)

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A i z = i 2 d ( z ) ω i 2 k i c 2 A p * A s e i Δ k 0 z
A s z = i 2 d ( z ) ω s 2 k s c 2 A p A i e i Δ k 0 z
d ( z ) = d i j m = | d m ( z ) | exp { i m [ 0 z q ( z ) d z + ϕ d ( z ) ] } ,
A i = k s ω i exp { i 2 [ Δ k 0 z 0 z q ( z ) d z ] } A ˜ i ,
A s = k i ω s exp { i 2 [ Δ k 0 z 0 z q ( z ) d z ] } A ˜ s .
i τ ( A ˜ i A ˜ s ) = σ B ( τ ) ( A ˜ i A ˜ s ) ,
B ( τ ) = B 0 ( τ ) B ^ [ θ ( τ ) , ϕ ( τ ) ] ,
B 0 ( τ ) = 1 2 k i k s κ 2 d ¯ 2 ( τ ) + Δ k 2 ( τ ) ,
cos θ ( τ ) = Δ k ( τ ) κ 2 d ¯ 2 ( τ ) + Δ k 2 ( τ ) ,
ϕ ( τ ) = ϕ p ϕ d ( τ ) .
| ω + = cos θ 2 | ω i + e i ϕ sin θ 2 | ω s ,
| ω = sin θ 2 | ω i e i ϕ cos θ 2 | ω s ,
| ψ = A ˜ + ( 0 ) e i γ + e i B 0 τ | ω + + A ˜ (0)e i γ e i B 0 τ | ω .
γ ± = 1 2 Ω .
| ψ ( T ) = exp ( i φ D + i γ + ) | ω i .
B z = 1 T 0 T Δ k ( τ ) 2 k i k s d τ
cos θ ( τ ) = sin 2 Θ 0 cos ω 0 τ + cos 2 Θ 0 ,
ϕ ( τ ) = arctan sin ω 0 τ cos Θ 0 ( 1 cos ω 0 τ ) .
Δ k ( τ ) = κ ( sin 2 Θ 0 cos ω 0 τ + cos 2 Θ 0 ) ,
D ( τ ) = 1 π arcsin 1 ( sin 2 Θ 0 cos ω 0 τ + cos 2 Θ 0 ) 2 ,
ϕ d ( τ ) = ϕ p + arctan sin ω 0 τ cos Θ 0 ( 1 cos ω 0 τ ) .
γ + = π ( 1 cos Θ 0 ) ,
φ D = B 0 T B z T = 1 2 κ L sin 2 Θ 0 .
cos θ ( τ ) = cos ω 0 τ ,
ϕ ( τ ) = H ( τ T 2 ) Δ ϕ ,
γ + = Δ ϕ .
Δ k ( τ ) = κ cos ω 0 τ
D ( τ ) = 1 π arcsin | sin ω 0 τ | ,
ϕ d ( τ ) = ϕ p H ( τ T 2 ) Δ ϕ ,
B ( τ ) = B ˜ 0 [ η sin α ( τ ) ρ ^ ( τ ) + cos α ( τ ) z ^ ] ,
A ( B ) = i ω + ; τ | B | ω + ; τ = ϕ ^ 1 cos θ 2 | B | sin θ .
γ + = A ( B ) d B = 1 2 0 T ( 1 cos θ ) ϕ ˙ d τ ,
cos θ = B z ^ | B | = cos α η 2 sin 2 α + cos 2 α .
Δ γ + ( η , Θ 0 ) = cos Θ 0 2 0 2 π f ( η , cos α ) cos α 1 + cos α d β ,
f ( η , cos α ) = 1 1 η 2 + ( 1 η 2 ) cos 2 α .
d γ + d ϵ = γ + ( 0 ) cos Θ 0 ( 1 + cos Θ 0 ) ( 3 2 cos 2 Θ 0 1 2 ) ,
| ψ ( τ ) = cos B 0 τ | ω i + i exp ( i ϕ ) sin B 0 τ | ω s .
γ + [ ϕ ( x , y ) ] = | ω i | ω s A { B [ ϕ ( x , y ) ] } d B .
1 τ 0 ω | H ˙ | ω + E + E E + .
Λ 2 π κ = π n i n s 2 | d e f f A p | k i k s

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