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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.


  1. 1.

    Feynman, R. P., Leighton, R. B. & Sands, M. The Feynman Lectures on Physics Vol. II (Basic Books, New York, NY, 2011).

  2. 2.

    de Rosny, J. & Fink, M. Overcoming the diffraction limit in wave physics using a time-reversal mirror and a novel acoustic sink. Phys. Rev. Lett. 89, 124301 (2002).

    ADS  Article  Google Scholar 

  3. 3.

    Noh, H., Popoff, S. M. & Cao, H. Broadband subwavelength focusing of light using a passive sink. Opt. Express 21, 17435–17446 (2013).

    ADS  Article  Google Scholar 

  4. 4.

    Maznev, A. A. & Wright, O. B. Upholding the diffraction limit in the focusing of light and sound. Wave Motion 68, 182–189 (2017).

    MathSciNet  Article  Google Scholar 

  5. 5.

    Pendry, J. B. Negative refraction makes a perfect lens. Phys. Rev. Lett. 85, 3966 (2000).

    ADS  Article  Google Scholar 

  6. 6.

    Kaina, N., Lemoult, F., Fink, M. & Lerosey, G. Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials. Nature 525, 77–81 (2015).

    ADS  Article  Google Scholar 

  7. 7.

    Li, J., Fok, L., Yin, X., Bartal, G. & Zhang, X. Experimental demonstration of an acoustic magnifying hyperlens. Nat. Mater. 8, 931–934 (2009).

    ADS  Article  Google Scholar 

  8. 8.

    Shen, C. et al. Broadband acoustic hyperbolic metamaterial. Phys. Rev. Lett. 115, 254301 (2015).

    ADS  Article  Google Scholar 

  9. 9.

    Lemoult, F., Lerosey, G., de Rosny, J. & Fink, M. Resonant metalenses for breaking the diffraction barrier. Phys. Rev. Lett. 104, 203901 (2010).

    ADS  Article  Google Scholar 

  10. 10.

    Zhu, J. et al. A holey-structured metamaterial for acoustic deep-subwavelength imaging. Nat. Phys. 7, 52–55 (2011).

    Article  Google Scholar 

  11. 11.

    Fink, M. Time reversed acoustics. Phys. Today 50, 34–40 (2008).

    Article  Google Scholar 

  12. 12.

    Yon, S., Tanter, M. & Fink, M. Sound focusing in rooms: The time-reversal approach. J. Acoust. Soc. Am. 113, 1533–1543 (2003).

    ADS  Article  Google Scholar 

  13. 13.

    Jackson, J. D. Classical Electrodynamics (Wiley, New York, NY, 2007).

  14. 14.

    Cummer, S. A., Christensen, J. SpringerAmpamp; Alù, A. Controlling sound with acoustic metamaterials. Nat. Rev. Mater. 1, 16001 (2016).

    ADS  Article  Google Scholar 

  15. 15.

    Ma, G. & Sheng, P. Acoustic metamaterials: From local resonances to broad horizons. Sci. Adv. 2, e1501595 (2016).

    ADS  Article  Google Scholar 

  16. 16.

    Ma, G., Yang, M., Xiao, S., Yang, Z. & Sheng, P. Acoustic metasurface with hybrid resonances. Nat. Mater. 13, 873–878 (2014).

    ADS  Article  Google Scholar 

  17. 17.

    Mei, J. et al. Dark acoustic metamaterials as super absorber for low-frequency sound. Nat. Commun. 3, 756 (2012).

    ADS  Article  Google Scholar 

  18. 18.

    Li, Y. & Assouar, B. M. Acoustic metasurface-based perfect absorber with deep subwavelength thickness. Appl. Phys. Lett. 108, 063502 (2016).

    ADS  Article  Google Scholar 

  19. 19.

    Yang, M., Chen, S., Fu, C. & Sheng, P. Optimal sound-absorbing structures. Mater. Horiz. 4, 673–680 (2017).

    Article  Google Scholar 

  20. 20.

    Cai, X., Guo, Q., Hu, G. & Yang, J. Ultrathin low-frequency sound absorbing panels based on coplanar spiral tubes or coplanar Helmholtz resonators. Appl. Phys. Lett. 105, 121901 (2014).

    ADS  Article  Google Scholar 

  21. 21.

    Leroy, V. et al. Superabsorption of acoustic waves with bubble metascreens. Phys. Rev. B 91, 020301 (2015).

    ADS  Article  Google Scholar 

  22. 22.

    Fink, M. et al. Time-reversed acoustics. Rep. Prog. Phys. 63, 1933–1995 (2000).

    ADS  Article  Google Scholar 

  23. 23.

    Leonhardt, U. Perfect imaging without negative refraction. New J. Phys. 11, 093040 (2009).

    ADS  Article  Google Scholar 

  24. 24.

    Ma, Y. G., Sahebdivan, S., Ong, C., Tyc, T. & Leonhardt, U. Evidence for subwavelength imaging with positive refraction. New J. Phys. 13, 033016 (2011).

    Article  Google Scholar 

  25. 25.

    Tyc, T. & Zhang, X. Perfect lenses in focus. Nature 480, 42–43 (2011).

    ADS  Article  Google Scholar 

  26. 26.

    Fink, M., de Rosny, J., Lerosey, G. & Tourin, A. Time-reversed waves and super-resolution. C. R. Phys. 10, 447–463 (2009).

    ADS  Article  Google Scholar 

  27. 27.

    Draeger, C. & Fink, M. One-channel time reversal of elastic waves in a chaotic 2D-silicon cavity. Phys. Rev. Lett. 79, 407–410 (1997).

    ADS  Article  Google Scholar 

  28. 28.

    Lerosey, G. et al. Time reversal of electromagnetic waves. Phys. Rev. Lett. 92, 193904 (2004).

    ADS  Article  Google Scholar 

  29. 29.

    Przadka, A. et al. Time reversal of water waves. Phys. Rev. Lett. 109, 064501 (2012).

    ADS  Article  Google Scholar 

  30. 30.

    Chong, Y. D., Ge, L., Cao, H. & Stone, A. D. Coherent perfect absorbers: time-reversed lasers. Phys. Rev. Lett. 105, 053901 (2010).

    ADS  Article  Google Scholar 

  31. 31.

    Wan, W. et al. Time-reversed lasing and interferometric control of absorption. Science 331, 889–892 (2011).

    ADS  Article  Google Scholar 

  32. 32.

    Li, S. et al. Broadband perfect absorption of ultrathin conductive films with coherent illumination: Superabsorption of microwave radiation. Phys. Rev. B 91, 220301 (2015).

    ADS  Article  Google Scholar 

  33. 33.

    Meng, C., Zhang, X., Tang, S. T., Yang, M. & Yang, Z. Acoustic coherent perfect absorbers as sensitive null detectors. Sci. Rep. 7, 43574 (2017).

    ADS  Article  Google Scholar 

  34. 34.

    Pirruccio, G., Martín Moreno, L., Lozano, G. & Gómez Rivas, J. Coherent and broadband enhanced optical absorption in graphene. ACS Nano 7, 4810–4817 (2013).

    Article  Google Scholar 

  35. 35.

    Brunet, T. et al. Soft 3D acoustic metamaterial with negative index. Nat. Mater. 14, 384–388 (2014).

    ADS  Article  Google Scholar 

  36. 36.

    Assous, F., Kray, M., Nataf, F. & Turkel, E. Time-reversed absorbing condition: application to inverse problems. Inverse Probl. 27, 065003 (2011).

    ADS  MathSciNet  Article  Google Scholar 

  37. 37.

    Kuttruff, H. Room Acoustics (Taylor & Francis, New York, NY, 2009).

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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*.

Author information




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 or 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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