Harmonic spin–orbit angular momentum cascade in nonlinear optical crystals



Optical angular momentum-based photonic technologies demonstrate the key role of the optical spin–orbit interaction that usually refers to linear optical processes in spatially engineered optical materials1. Re-examining the basics of nonlinear optics of homogeneous crystals under circularly polarized light2,3, we report experiments on the enrichment of the spin–orbit angular momentum spectrum of paraxial light. The demonstration is made within the framework of second-harmonic generation using a crystal with three-fold rotational symmetry. Four spin–orbit optical states for the second harmonic field are predicted from a single fundamental state owing to the interplay between linear spin–orbit coupling and nonlinear wave mixing; three of these states are experimentally verified. Besides representing a spin-controlled nonlinear route to orbital angular multiplexing4, modal vortex light sources5,6, high-dimensional parametric processes7 and multi-state optical magnetization8, our findings suggest that the fundamentals of nonlinear optics are worth revisiting through the prism of the spin–orbit interaction of light.

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Fig. 1: Spin-selective second-harmonic generation along the optical axis of a BBO crystal.
Fig. 2: Spin–orbit angular momentum cascade along the optical axis of a BBO crystal with weakly focused pumping light.
Fig. 3: Three-step description of the spin–orbit angular momentum cascade for the second-harmonic generation process.
Fig. 4: Spin–orbit tomography of the second-harmonic field for weakly focused pumping light.

Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors on reasonable request.


  1. 1.

    Bliokh, K. Y., Rodríguez-Fortuño, F. J., Nori, F. & Zayats, A. V. Spin–orbit interactions of light. Nat. Photon. 9, 796–808 (2015).

    ADS  Article  Google Scholar 

  2. 2.

    Simon, H. J. & Bloembergen, N. Second-harmonic light generation in crystals with natural optical activity. Phys. Rev. 171, 1104–1114 (1968).

    ADS  Article  Google Scholar 

  3. 3.

    Patel, C. K. N. & Van Tran, N. Phase-matched nonlinear interaction between circularly polarized waves. Appl. Phys. Lett. 15, 189–191 (1969).

    ADS  Article  Google Scholar 

  4. 4.

    Wang, J. et al. Terabit free-space data transmission employing orbital angular momentum multiplexing. Nat. Photon. 6, 488–496 (2012).

    ADS  Article  Google Scholar 

  5. 5.

    Omatsu, T., Miyamoto, K. & Lee, A. J. Wavelength-versatile optical vortex lasers. J. Opt. 19, 123002 (2017).

    ADS  Article  Google Scholar 

  6. 6.

    Sroor, H. et al. High-purity orbital angular momentum states from a visible metasurface laser. Nat. Photon. 14, 498–503 (2020).

    Article  Google Scholar 

  7. 7.

    Fleischer, A., Kfir, O., Diskin, T., Sidorenko, P. & Cohen, O. Spin angular momentum and tunable polarization in high-harmonic generation. Nat. Photon. 8, 543–549 (2014).

    ADS  Article  Google Scholar 

  8. 8.

    Lin, S. et al. All-optical vectorial control of multistate magnetization through anisotropy-mediated spin-orbit coupling. Nanophotonics 8, 2177–2188 (2019).

    Article  Google Scholar 

  9. 9.

    Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C. & Woerdman, J. P. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Phys. Rev. A 45, 8185–8189 (1992).

    ADS  Article  Google Scholar 

  10. 10.

    Shen, Y. et al. Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities. Light Sci. Appl. 8, 90 (2019).

    ADS  Article  Google Scholar 

  11. 11.

    Volyar, A. V., Fadeeva, T. A. & Egorov, Yu. A. Vector singularities of Gaussian beams in uniaxial crystals: optical vortex generation. Tech. Phys. Lett. 28, 70–77 (2002).

    Article  Google Scholar 

  12. 12.

    Ciattoni, A., Cincotti, G. & Palma, C. Circularly polarized beams and vortex generation in uniaxial media. J. Opt. Soc. Am. A 20, 163–171 (2003).

    ADS  Article  Google Scholar 

  13. 13.

    Biener, G., Niv, A., Kleiner, V. & Hasman, E. Formation of helical beams by use of Pancharatnam-Berry phase optical elements. Opt. Lett. 27, 1875–1877 (2002).

    ADS  Article  Google Scholar 

  14. 14.

    Marrucci, L., Manzo, C. & Paparo, D. Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media. Phys. Rev. Lett. 96, 163905 (2006).

    ADS  Article  Google Scholar 

  15. 15.

    Li, G. et al. Spin-enabled plasmonic metasurfaces for manipulating orbital angular momentum of light. Nano Lett. 13, 4148–4151 (2013).

    ADS  Article  Google Scholar 

  16. 16.

    Karimi, E. et al. Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface. Light Sci. Appl. 3, e167 (2014).

    MathSciNet  Article  Google Scholar 

  17. 17.

    Basistiy, I. V., Bazhenov, V. Y., Soskin, M. S. & Vasnetsov, M. V. Optics of light beams with screw dislocations. Opt. Commun. 103, 422–428 (1993).

    ADS  Article  Google Scholar 

  18. 18.

    Dholakia, K., Simpson, N. B., Padgett, M. J. & Allen, L. Second-harmonic generation and the orbital angular momentum of light. Phys. Rev. A 54, R3742–R3745 (1996).

    ADS  Article  Google Scholar 

  19. 19.

    Beržanskis, A., Matijošius, A., Piskarskas, A., Smilgevičius, V. & Stabinis, A. Sum-frequency mixing of optical vortices in nonlinear crystals. Opt. Commun. 150, 372–380 (1998).

    ADS  Article  Google Scholar 

  20. 20.

    Bloch, N. V. et al. Twisting light by nonlinear photonic crystals. Phys. Rev. Lett. 108, 233902 (2012).

    ADS  Article  Google Scholar 

  21. 21.

    Liu, S. et al. Nonlinear wavefront shaping with optically induced three-dimensional nonlinear photonic crystals. Nat. Commun. 10, 3208 (2019).

    ADS  Article  Google Scholar 

  22. 22.

    Wei, D. et al. Efficient nonlinear beam shaping in three-dimensional lithium niobate nonlinear photonic crystals. Nat. Commun. 10, 4193 (2019).

    ADS  Article  Google Scholar 

  23. 23.

    Li, G., Zhang, S. & Zentgraf, T. Nonlinear photonic metasurfaces. Nat. Rev. Mater. 2, 17010 (2017).

    ADS  Article  Google Scholar 

  24. 24.

    Li, G., Zentgraf, T. & Zhang, S. Rotational Doppler effect in nonlinear optics. Nat. Phys. 12, 736–740 (2016).

    Article  Google Scholar 

  25. 25.

    Chen, C., Wu, B., Jiang, A. & You, G. A new-type ultraviolet SHG crystal: β-BaB2O4. Sci. Sin. Ser. B 28, 235–243 (1985).

    Google Scholar 

  26. 26.

    Bekshaev, A. Y., Soskin, M. S. & Vasnetsov, M. V. Transformation of higher-order optical vortices upon focusing by an astigmatic lens. Opt. Commun. 241, 237–247 (2004).

    ADS  Article  Google Scholar 

  27. 27.

    Loussert, C. & Brasselet, E. Efficient scalar and vectorial singular beam shaping using homogeneous anisotropic media. Opt. Lett. 35, 7–9 (2010).

    ADS  Article  Google Scholar 

  28. 28.

    Brasselet, E. et al. Dynamics of optical spin-orbit coupling in uniaxial crystals. Opt. Lett. 34, 1021–1023 (2009).

    ADS  Article  Google Scholar 

  29. 29.

    Langford, N. K. et al. Measuring entangled qutrits and their use for quantum bit commitment. Phys. Rev. Lett. 93, 053601 (2004).

    ADS  Article  Google Scholar 

  30. 30.

    Bhagavantam, S. & Chandrasekhar, P. Harmonic generation and selection rules in nonlinear optics. Proc. Ind. Acad. Sci. A 76, 13–20 (1972).

    Article  Google Scholar 

  31. 31.

    Belyi, V., Khilo, N., Forbes, A. & Ryzhevich, A. Generation and propagation of high-order Bessel vortices in linear and non-linear crystals. In Proc. SPIE 7430, Laser Beam Shaping X 74300F (SPIE, 2009).

  32. 32.

    Sato, S. & Kozawa, Y. Radially polarized annular beam generated through a second-harmonic-generation process. Opt. Lett. 34, 3166–3168 (2009).

    ADS  Article  Google Scholar 

  33. 33.

    Belyi, V., Khilo, N., Kazak, N., Ryzhevich, A. & Forbes, A. Propagation of high-order circularly polarized Bessel beams and vortex generation in uniaxial. Opt. Eng. 50, 059001 (2011).

    ADS  Article  Google Scholar 

  34. 34.

    Shao, G.-H., Wu, Z.-J., Chen, J.-H., Xu, F. & Lu, Y.-Q. Nonlinear frequency conversion of fields with orbital angular momentum using quasi-phase-matching. Phys. Rev. A 88, 063827 (2013).

    ADS  Article  Google Scholar 

  35. 35.

    Chen, S. et al. Symmetry-selective third-harmonic generation from plasmonic metacrystals. Phys. Rev. Lett. 113, 033901 (2014).

    ADS  Article  Google Scholar 

  36. 36.

    Konishi, K. et al. Polarization-controlled circular second-harmonic generation from metal hole arrays with threefold rotational symmetry. Phys. Rev. Lett. 112, 135502 (2014).

    ADS  Article  Google Scholar 

  37. 37.

    Buono, W. T. et al. Polarization-controlled orbital angular momentum switching in nonlinear wave mixing. Opt. Lett. 43, 1439–1442 (2018).

    ADS  Article  Google Scholar 

  38. 38.

    Wang, K. et al. Quantum metasurface for multiphoton interference and state reconstruction. Science 361, 1104–1108 (2018).

    ADS  Article  Google Scholar 

  39. 39.

    Stav, T. et al. Quantum entanglement of the spin and orbital angular momentum of photons using metamaterials. Science 361, 1101–1104 (2018).

    ADS  Article  Google Scholar 

  40. 40.

    Rego, L. et al. Generation of extreme-ultraviolet beams with time-varying orbital angular momentum. Science 364, eaaw9486 (2019).

    ADS  Article  Google Scholar 

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G.L. is financially supported by the National Natural Science Foundation of China (grant numbers 91950114 and 11774145), a Guangdong Provincial Innovation and Entrepreneurship Project (2017ZT07C071) and the Qiu Shi Science & Technologies Foundation.

Author information




E.B. and G.L. proposed the idea and designed the experiment. Y.T., K.L., J.D., X.Z. and G.L. conducted the nonlinear optical measurements. E.B., G.L., Y.T. and J.D. wrote the manuscript. All authors participated in the data analysis and discussions. G.L. and E.B. supervised the project.

Corresponding authors

Correspondence to Guixin Li or Etienne Brasselet.

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Supplementary Information

Supplementary Figs. 1–21, Tables 1–8 and discussion.

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Tang, Y., Li, K., Zhang, X. et al. Harmonic spin–orbit angular momentum cascade in nonlinear optical crystals. Nat. Photonics 14, 658–662 (2020). https://doi.org/10.1038/s41566-020-0691-0

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