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Optical polarization analogue in free electron beams

Nature Physics volume 17, pages 598–603 (2021)Cite this article

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Abstract

Spectromicroscopy techniques with fast electrons can quantitatively measure the optical response of excitations with unrivalled spatial resolution. However, owing to their inherently scalar nature, electron waves cannot access the polarization-related quantities. Despite promising attempts based on the conversion of concepts originating from singular optics (such as vortex beams), the definition of an optical polarization analogue for fast electrons has remained an open question. Here we establish such an analogue using the dipole transition vector of the electron between two well-chosen singular wave states. We show that electron energy loss spectroscopy allows the direct measurement of the polarized electromagnetic local density of states. In particular, in the case of circular polarization, it directly measures the local optical spin density. This work establishes electron energy loss spectroscopy as a quantitative technique to tackle fundamental issues in nano-optics, such as super-chirality, local polarization of dark excitations or polarization singularities at the nanoscale.

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Fig. 1: Comparison of polarized LES and polarized EELS.
Fig. 2: Comparison of linearly and circularly polarized LES and non-spatially resolved pEELS experiments on simple plasmonic nanostructures.
Fig. 3: Spatially resolved pEELS.

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Source data are available for this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

We acknowledge J. Verbeeck for introduction to this field and thank J. Verbeeck, D. Ugarte, F. Houdellier and G. Guzzinati for insightful discussions. H.L.-M. thanks T. R. Harvey for insightful discussions. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 823717-ESTEEM3 and no. 101017720-EBEAM, from the French state managed by the National Agency for Research under the programme of future investment EQUIPEX TEMPOS-CHROMATEM with the reference ANR-10-EQPX-50 and ANR-17-CE24-0039 (2D-CHIRAL).

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Authors and Affiliations

  1. University of Göttingen, IV. Physical Institute, Göttingen, Germany

    Hugo Lourenço-Martins

  2. Light, nanomaterials & nanotechnologies (L2n), CNRS-ERL 7004, Université de Technologie de Troyes, Troyes, France

    Davy Gérard

  3. Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, Orsay, France

    Mathieu Kociak

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  1. Hugo Lourenço-Martins
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Contributions

M.K. and H.L.-M. developed the theory. H.L.-M. wrote the simulation codes and performed the numerical simulations. M.K., H.L.-M. and D.G. developed the analogy between the Bloch and Poincaré spheres. All the authors discussed the results and contributed to the writing of the manuscript.

Corresponding author

Correspondence to Mathieu Kociak.

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The authors declare no competing interests.

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

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Extended data

Extended Data Fig. 1 Examples of transitions between higher order HG states possessing a non-zero OPA.

Transitions a and b possess a dipole moment along the x direction and therefore measure Ex(ω, qz). Transitions c and d possess a dipole moment along the y direction and therefore measure Ey(ω, qz).

Extended Data Fig. 2 Illustration of the effect of the summation over the final states.

a, Wavefunction ψi and plasmonic potential ϕm of the four first modes m of a 100nm * 15nm silver nano-antenna. The relative position and scale of the wavefunction and potential are respected. Scales bar is 50 nm. b, pEELS spectrum as a function of the collection angle. The peaks of the four modes shown in a. are marked with arrows. Below θω, the second and fourth peak are suppressed due to the selection rule reminiscent of formula (184) of the SI. Above θω,all the peaks are present and the resulting spectrum is essentially classical.

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Supplementary discussion and extended demonstrations.

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Lourenço-Martins, H., Gérard, D. & Kociak, M. Optical polarization analogue in free electron beams. Nat. Phys. 17, 598–603 (2021). https://doi.org/10.1038/s41567-021-01163-w

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