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Towards anti-causal Green’s function for three-dimensional sub-diffraction focusing


In causal physics, the causal Green’s function describes the radiation of a point source. Its counterpart, the anti-causal Green’s function, depicts a spherically converging wave. However, in free space, any converging wave must be followed by a diverging one. Their interference gives rise to the diffraction limit that constrains the smallest possible dimension of a wave’s focal spot in free space, which is half the wavelength. Here, we show with three-dimensional acoustic experiments that we can realize a stand-alone anti-causal Green’s function in a large portion of space up to a subwavelength distance from the focus point by introducing a near-perfect absorber for spherical waves at the focus. We build this subwavelength absorber based on membrane-type acoustic metamaterial, and experimentally demonstrate focusing of spherical waves beyond the diffraction limit.

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Fig. 1: Formation of a focal spot.
Fig. 2: An absorber for converging spherical waves.
Fig. 3: Focal spot in the presence of the absorber.
Fig. 4: Improving resolution with absorbers.


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G.M. and P.S. acknowledge the support of the Hong Kong Research Grants Council (grant no. AoE/P-02/12). J.d.R and M.F. acknowledge LABEX WIFI (Laboratory of Excellence within the French Program Investments for the Future) under references ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL*.

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Authors and Affiliations



M.F., G.M. and P.S. supervised the research. G.M. designed the experiment with the help of J.d.R. X.F. and G.M. carried out the experiment. F.M. performed experiments at the early stage of this project. All authors were involved in discussion and analysis of data. G.M., P.S. and M.F. prepared the manuscript.

Corresponding authors

Correspondence to Guancong Ma, Ping Sheng or Mathias Fink.

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The authors declare no competing interests.

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Supplementary Information

Supplementary Note, Supplementary Figures 1–5, References

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Ma, G., Fan, X., Ma, F. et al. Towards anti-causal Green’s function for three-dimensional sub-diffraction focusing. Nature Phys 14, 608–612 (2018).

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