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Scattering invariant modes of light in complex media

An Author Correction to this article was published on 26 April 2021

This article has been updated


Random scattering of light in disordered media is an intriguing phenomenon of fundamental relevance to various applications1. Although techniques such as wavefront shaping and transmission matrix measurements2,3 have enabled remarkable progress in advanced imaging concepts4,5,6,7,8,9,10,11, the most successful strategy to obtain clear images through a disordered medium remains the filtering of ballistic light12,13,14. Ballistic photons with a scattering-free propagation are, however, exponentially rare and no known method has been able to increase their proportion. To address these limitations, we introduce and experimentally implement a new set of optical states that we term scattering invariant modes, whose transmitted field pattern is the same, irrespective of whether they scatter through a disordered sample or propagate ballistically through a homogeneous medium. We observe scattering invariant modes that are only weakly attenuated in dense scattering media, and show in simulations that their correlations with the ballistic light can be used to improve imaging inside scattering materials.

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Fig. 1: Illustration of the concept.
Fig. 2: Experimentally transmitted SIMs and their statistics.
Fig. 3: Experimental realization of a sparse SIM.
Fig. 4: Simulations of imaging using SIMs.

Data availability

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

Code availability

The code used to generate simulated data and plots is available from the corresponding authors on reasonable request.

Change history


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We acknowledge helpful discussions with P. Ambichl, D. Bouchet, S. Faez, D. van Oosten, P. Jurrius, D. Killian, C. R. de Kok, F. Salihbegovic and S. Steinhauer. Financial support was provided by the Austrian Science Fund (FWF) under project WAVELAND (grant number P32300 to S.R.) and by the Netherlands Organization for Scientific Research NWO (grant number Vici-68047618 to A.P.M.). The computational results presented in this paper were achieved using the Vienna Scientific Cluster (VSC).

Author information

Authors and Affiliations



The experiments were designed by A.P.M., P.P. and J.B. and implemented by P.P. and J.B. The 2D full-wave numerical simulations and the theoretical analysis were carried out by M.K. and the 3D calculations by A.P.M. S.R. proposed the idea and supervised the theoretical research. All authors analysed the results and contributed to the writing of the manuscript.

Corresponding authors

Correspondence to Stefan Rotter or Allard P. Mosk.

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The authors declare no competing interests.

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Peer review information Nature Photonics thanks Roarke Horstmeyer and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–19, discussion and methods.

Supplementary Video 1

Evolution of SIM eigenvalues in the complex plane corresponding to Supplementary Fig. 18 for a scatterer filling fraction of 0.05.

Supplementary Video 2

Evolution of SIM eigenvalues in the complex plane corresponding to Supplementary Fig. 18 for a scatterer filling fraction of 0.10.

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Pai, P., Bosch, J., Kühmayer, M. et al. Scattering invariant modes of light in complex media. Nat. Photonics 15, 431–434 (2021).

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