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Second-harmonic generation and the conservation of spatiotemporal orbital angular momentum of light


Light with spatiotemporal orbital angular momentum (ST-OAM) is a recently discovered type of structured and localized electromagnetic field. This field carries characteristic space–time spiral phase structure and transverse intrinsic OAM. Here, we present the generation and characterization of the second harmonic of ST-OAM pulses. We uncover the conservation of transverse OAM in a second-harmonic generation process, where the space–time topological charge of the fundamental field is doubled along with the optical frequency. Our experiment thus suggests a general ST-OAM nonlinear scaling rule, analogous to that in conventional OAM of light. Furthermore, we observe that the topology of a second-harmonic ST-OAM pulse can be modified by complex spatiotemporal astigmatism, giving rise to multiple phase singularities separated in space and time. Our study opens a new route for nonlinear conversion and scaling of light carrying ST-OAM, with the potential for driving other secondary ST-OAM sources of electromagnetic fields and beyond.

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Fig. 1: Second-harmonic ST-OAM pulse generation and characterization.
Fig. 2: Experimentally reconstructed amplitude and phase of ST-OAM pulses.
Fig. 3: Modal decomposition of the ST-OAM pulses from experiments.
Fig. 4: Experimentally extracted spatiotemporal momentum density and energy density flux of ST-OAM pulses.

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Data availability

The datasets utilized to prepare the data presented in this manuscript are available free of charge from the corresponding author under reasonable request.


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We acknowledge funding from an AFOSR MURI grant (FA9550-16-1-0121). N.J.B. acknowledges support from the National Science Foundation Graduate Research Fellowships (grant no. DGE-1650115).

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Authors and Affiliations



C.-T.L. conceived the project. G.G. conducted and designed the experiment. C.-T.L. and G.G. both analysed the data. M.M.M. and H.C.K. proposed the research thrust, supervised the research, developed the generation and measurement capabilities, and applications. All authors contributed to the discussion and writing of the manuscript.

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Correspondence to Chen-Ting Liao.

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Competing interests

M.M.M. and H.C.K. have a financial interest in KMLabs. The other authors declare no competing interests.

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Peer review information Nature Photonics thanks the anonymous reviewers for their contribution to the peer review of this work.

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Supplementary Figs. 1 and 2 and Discussion.

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Gui, G., Brooks, N.J., Kapteyn, H.C. et al. Second-harmonic generation and the conservation of spatiotemporal orbital angular momentum of light. Nat. Photon. 15, 608–613 (2021).

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