Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Anti-reflection structure for perfect transmission through complex media

Nature volume 607, pages 281–286 (2022)Cite this article

Subjects

Abstract

The scattering of waves when they propagate through disordered media is an important limitation for a range of applications, including telecommunications1, biomedical imaging2, seismology3 and material engineering4,5. Wavefront shaping techniques can reduce the effect of wave scattering, even in opaque media, by engineering specific modes—termed open transmission eigenchannels—through which waves are funnelled across a disordered medium without any back reflection6,7,8,9. However, with such channels being very scarce, one cannot use them to render an opaque sample perfectly transmitting for any incident light field. Here we show that a randomly disordered medium becomes translucent to all incoming light waves when placing a tailored complementary medium in front of it. To this end, the reflection matrices of the two media surfaces facing each other need to satisfy a matrix generalization of the condition for critical coupling. We implement this protocol both numerically and experimentally for the design of electromagnetic waveguides with several dozen scattering elements placed inside them. The translucent scattering media we introduce here also have the promising property of being able to store incident radiation in their interior for remarkably long times.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Prices vary by article type

from$1.95

to$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Illustration of the concept.
Fig. 2: Fully transmitting waveguide with disorder.
Fig. 3: Numerical and experimental results.
Fig. 4: Energy stored within the sample.

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.

References

  1. Basar, E. et al. Wireless communications through reconfigurable intelligent surfaces. IEEE Access 7, 116753–116773 (2019).

    Article  Google Scholar 

  2. Kubby, J., Gigan, S. & Cui, M. (eds.) Wavefront Shaping for Biomedical Imaging (Cambridge University Press, 2019).

  3. Campillo, M. & Paul, A. Long-range correlations in the diffuse seismic coda. Science 299, 547–549 (2003).

    Article  CAS  PubMed  ADS  Google Scholar 

  4. Chen, H.-T. et al. Antireflection coating using metamaterials and identification of its mechanism. Phys. Rev. Lett. 105, 073901 (2010).

    Article  PubMed  ADS  Google Scholar 

  5. Molesky, S. et al. Inverse design in nanophotonics. Nat. Photonics 12, 659–670 (2018).

    Article  CAS  ADS  Google Scholar 

  6. Dorokhov, O. N. On the coexistence of localized and extended electronic states in the metallic phase. Solid State Commun. 51, 381–384 (1984).

    Article  CAS  ADS  Google Scholar 

  7. Gérardin, B., Laurent, J., Derode, A., Prada, C. & Aubry, A. Full transmission and reflection of waves propagating through a maze of disorder. Phys. Rev. Lett. 113, 173901 (2014).

    Article  PubMed  ADS  Google Scholar 

  8. Sarma, R., Yamilov, A. G., Petrenko, S., Bromberg, Y. & Cao, H. Control of energy density inside a disordered medium by coupling to open or closed channels. Phys. Rev. Lett. 117, 086803 (2016).

    Article  PubMed  ADS  Google Scholar 

  9. Jeong, S. et al. Focusing of light energy inside a scattering medium by controlling the time-gated multiple light scattering. Nat. Photonics 12, 277–283 (2018).

    Article  CAS  ADS  Google Scholar 

  10. Pai, P., Bosch, J., Kühmayer, M., Rotter, S. & Mosk, A. P. Scattering invariant modes of light in complex media. Nat. Photonics 15, 431–434 (2021).

    Article  CAS  ADS  Google Scholar 

  11. Florescu, M., Torquato, S. & Steinhardt, P. J. Designer disordered materials with large, complete photonic band gaps. Proc. Natl Acad. Sci. USA 106, 20658–20663 (2009).

    Article  CAS  PubMed  PubMed Central  ADS  Google Scholar 

  12. Horsley, S. A. R., Artoni, M., & La Rocca, G. C. Spatial Kramers–Kronig relations and the reflection of waves. Nat. Photonics 9, 436–439 (2015).

    Article  CAS  ADS  Google Scholar 

  13. Rivet, E. et al. Constant-pressure sound waves in non-Hermitian disordered media. Nat. Phys. 14, 942–947 (2018).

    Article  CAS  Google Scholar 

  14. Leseur, O., Pierrat, R. & Carminati, R. High-density hyperuniform materials can be transparent. Optica 3, 763–767 (2016).

    Article  ADS  Google Scholar 

  15. Spinelli, P., Verschuuren, M. A. & Polman, A. Broadband omnidirectional antireflection coating based on subwavelength surface Mie resonators. Nat. Commun. 3, 692 (2012).

    Article  CAS  PubMed  ADS  Google Scholar 

  16. Im, K., Kang, J.-H. & Park, Q. H. Universal impedance matching and the perfect transmission of white light. Nat. Photonics 12, 143–149 (2018).

    Article  CAS  ADS  Google Scholar 

  17. Baranov, D. G., Krasnok, A., Shegai, T., Alù, A. & Chong, Y. Coherent perfect absorbers: linear control of light with light. Nat. Rev. Mater. 2, 17064 (2017).

    Article  CAS  ADS  Google Scholar 

  18. Beenakker, C. W. J. Random-matrix theory of quantum transport. Rev. Mod. Phys. 69, 731 (1997).

    Article  CAS  ADS  Google Scholar 

  19. Resisi, S., Viernik, Y., Popoff, S. M. & Bromberg, Y. Wavefront shaping in multimode fibers by transmission matrix engineering. APL Photonics 5, 036103 (2020).

    Article  CAS  ADS  Google Scholar 

  20. Dinsdale, N. J. et al. Deep learning enabled design of complex transmission matrices for universal optical components. ACS Photonics 8, 283–295 (2021).

    Article  CAS  Google Scholar 

  21. Jensen, J. S. & Sigmund, O. Topology optimization for nano-photonics. Laser Photonics Rev. 5, 308–321 (2011).

    Article  CAS  ADS  Google Scholar 

  22. So, S., Badloe, T., Noh, J., Bravo-Abad, J. & Rho, J. Deep learning enabled inverse design in nanophotonics. Nanophotonics 9, 1041–1057 (2020).

    Article  Google Scholar 

  23. Liu, V., Jiao, Y., Miller, D. A. B. & Fan, S. Design methodology for compact photonic-crystal-based wavelength division multiplexers. Opt. Lett. 36, 591–593 (2011).

    Article  PubMed  ADS  Google Scholar 

  24. Shen, B., Wang, P., Polson, R. & Menon, R. An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 m2 footprint. Nat. Photonics 9, 378–382 (2015).

    Article  CAS  ADS  Google Scholar 

  25. Riboli, F. et al. Engineering of light confinement in strongly scattering disordered media. Nat. Mater. 13, 720–725 (2014).

    Article  CAS  PubMed  ADS  Google Scholar 

  26. Mohammadi Estakhri, N., Edwards, B. & Engheta, N. Inverse-designed metastructures that solve equations. Science 363, 1333–1338 (2019).

    Article  MathSciNet  CAS  PubMed  MATH  ADS  Google Scholar 

  27. Ambichl, P. et al. Focusing inside disordered media with the generalized Wigner–Smith operator. Phys. Rev. Lett. 119, 033903 (2017).

    Article  PubMed  ADS  Google Scholar 

  28. Horodynski, M. et al. Optimal wave fields for micromanipulation in complex scattering environments. Nat. Photonics 14, 149–153 (2020).

    Article  CAS  ADS  Google Scholar 

  29. Bouchet, D., Rotter, S. & Mosk, A. P. Maximum information states for coherent scattering measurements. Nat. Phys. 17, 564–568 (2021).

    Article  CAS  Google Scholar 

  30. del Hougne, P., Yeo, K. B., Besnier, P. & Davy, M. Coherent wave control in complex media with arbitrary wavefronts. Phys. Rev. Lett. 126, 193903 (2021).

    Article  PubMed  ADS  Google Scholar 

  31. Chéron, É., Félix, S. & Pagneux, V. Broadband-enhanced transmission through symmetric diffusive slabs. Phys. Rev. Lett. 122, 125501 (2019).

    Article  PubMed  ADS  Google Scholar 

  32. Shi, Z. & Genack, A. Z. Dynamic and spectral properties of transmission eigenchannels in random media. Phys. Rev. B 92, 184202 (2015).

    Article  ADS  Google Scholar 

  33. Davy, M., Shi, Z., Wang, J., Cheng, X. & Genack, A. Z. Transmission eigenchannels and the densities of states of random media. Phys. Rev. Lett. 114, 033901 (2015).

    Article  PubMed  ADS  Google Scholar 

  34. Davy, M., Shi, Z., Park, J., Tian, C. & Genack, A. Z. Universal structure of transmission eigenchannels inside opaque media. Nat. Commun. 6, 6893 (2015).

    Article  CAS  PubMed  ADS  Google Scholar 

  35. Durand, M., Popoff, S. M., Carminati, R. & Goetschy, A. Optimizing light storage in scattering media with the dwell-time operator. Phys. Rev. Lett. 123, 243901 (2019).

    Article  CAS  PubMed  ADS  Google Scholar 

  36. Shi, Z. & Genack, A. Z. Diffusion in translucent media. Nat. Commun. 9, 1862 (2018).

    Article  PubMed  PubMed Central  ADS  Google Scholar 

  37. Bonnet-Ben Dhia, A.-S., Chesnel, L. & Pagneux, V. Trapped modes and reflectionless modes as eigenfunctions of the same spectral problem. Proc. R. Soc. A 474, 20180050 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  38. Sweeney, W. R., Hsu, C. W. & Stone, A. D. Theory of reflectionless scattering modes. Phys. Rev. A 102, 063511 (2020).

    Article  MathSciNet  CAS  ADS  Google Scholar 

  39. Krasnok, A., Baranov, D. G., Generalov, A., Li, S. & Alù, A. Coherently enhanced wireless power transfer. Phys. Rev. Lett. 120, 143901 (2018).

    Article  CAS  PubMed  ADS  Google Scholar 

  40. Park, J.-H., Sun, W. & Cui, M. High-resolution in vivo imaging of mouse brain through the intact skull. Proc. Natl Acad. Sci. USA 112, 9236–9241 (2015).

    Article  CAS  PubMed  PubMed Central  ADS  Google Scholar 

  41. Pichler, K. et al. Random anti-lasing through coherent perfect absorption in a disordered medium. Nature 567, 351–355 (2019).

    Article  CAS  PubMed  ADS  Google Scholar 

  42. Chen, L., Kottos, T. & Anlage, S. M. Perfect absorption in complex scattering systems with or without hidden symmetries. Nat. Commun. 11, 5826 (2020).

    Article  CAS  PubMed  PubMed Central  ADS  Google Scholar 

  43. del Hougne, P., Yeo, K. B., Besnier, P. & Davy, M. On-demand coherent perfect absorption in complex scattering systems: time delay divergence and enhanced sensitivity to perturbations. Laser Photonics Rev. 15, 2000471 (2021).

    Article  Google Scholar 

  44. Schöberl, J. NETGEN an advancing front 2D/3D-mesh generator based on abstract rules. Comput. Visual. Sci. 1, 41–52 (1997).

    Article  MATH  Google Scholar 

  45. Schöberl, J. C++11 implementation of Finite Elements in NGSolve ASC Report, Institute for Analysis and Scientific Computing (Vienna Univ. of Technology, 2014).

  46. Carminati, R., Sáenz, J. J., Greffet, J.-J. & Nieto-Vesperinas, M. Reciprocity, unitarity, and time-reversal symmetry of the S matrix of fields containing evanescent components. Phys. Rev. A 62, 012712 (2000).

    Article  ADS  Google Scholar 

  47. Beenakker, C. & Brouwer, P. Distribution of the reflection eigenvalues of a weakly absorbing chaotic cavity. Physica E 9, 463–466 (2001).

    Article  ADS  Google Scholar 

  48. Kuang, Z., Zhang, L. & Miller, O. D. Maximal single-frequency electromagnetic response. Optica 7, 1746–1757 (2020).

    Article  ADS  Google Scholar 

  49. Davy, M., Ferise, C., Chéron, É., Félix, S. & Pagneux, V. Experimental evidence of enhanced broadband transmission in disordered systems with mirror symmetry. Appl. Phys. Lett. 119, 141104 (2021).

    Article  CAS  ADS  Google Scholar 

Download references

Acknowledgements

We acknowledge useful discussions with D. B. Phillips and P. del Hougne. We also acknowledge P. E. Davy for his help in rendering of Figs. 1a,b and 2a. This publication was supported by the European Union through the European Regional Development Fund (ERDF), by the French region of Brittany and Rennes Métropole through the CPER Project SOPHIE/STIC & Ondes, and by the Austrian Science Fund (FWF) through project P32300 (WAVELAND). C.F. acknowledges funding from the French ‘Ministère de la Défense, Direction Générale de l’Armement’. M.D. acknowledges the Institut Universitaire de France. The computational results presented were achieved using the Vienna Scientific Cluster (VSC).

Author information

Authors and Affiliations

  1. Institute for Theoretical Physics, Vienna University of Technology (TU Wien), Vienna, Austria

    Michael Horodynski, Matthias Kühmayer & Stefan Rotter

  2. Université de Rennes, CNRS, IETR (Institut d’Électronique et des Technologies du numéRique), UMR–6164, Rennes, France

    Clément Ferise & Matthieu Davy

Authors
  1. Michael Horodynski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthias Kühmayer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Clément Ferise
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Stefan Rotter
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Matthieu Davy
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

M.D. proposed the project. Numerical simulations were carried out by M.H and M.K. under the supervision of S.R. Measurements and data evaluation were carried out by C.F. and M.D. M.H., S.R. and M.D. wrote the manuscript with input from all authors.

Corresponding authors

Correspondence to Stefan Rotter or Matthieu Davy.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review information

Peer review information

Nature thanks Alex Krasnok, Roarke Horstmeyer, Q-Han Park and Fenghan Lin for their contribution to the peer review of this work.

Additional information

Extended data

is available for this paper at https://doi.org/10.1038/s41586-022-04843-6.

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

Extended data figures and tables

Extended Data Fig. 1 Effects of absorption.

a, Transmission (blue) and reflection (orange) plotted over the imaginary part of the global refractive index for the geometry optimized for 11.2 GHz (solid lines) and an empty waveguide (dashed lines). b, Transmission (blue) and reflection (orange) spectrum of the sample with 49 scatterers. The spectra without absorption are depicted by a solid line, while transmission and reflection curves with added absorption are indicated with dashed lines.

Extended Data Fig. 2 Analysis of the optimization procedure.

a, Probability of successfully designing a fully transmitting medium as a function of the fixed disorders transmission. Success is defined as a transmission \(T\,\ge 0.999\) (blue line) or \(T\,\ge 0.99\) (orange line). Note that we are using binned data here with the bin edges indicated by the grey dashed lines. The number of inverse design processes in each bin is given by the green line. The total number of configurations is 220. b, Plot of the critical coupling condition’s Frobenius norm, \(||{r}_{L}^{{\prime} }-{r}_{R}^{\dagger }|{|}_{F}\), versus the total transmission for each step of the optimization process. The progress in the inverse design is marked by the colourmap transition from dark blue to bright yellow.

Extended Data Fig. 3 Steep angles of incidence.

Poynting vector distribution in an empty waveguide (a, b) and a waveguide filled with a fully transmitting disorder (c, d). The incoming wave is either the highest order mode (out of 4) (a, c) or the state optimized for a steep angle of incidence (b, d). The white circles indicate the position of the scatterers.

Extended Data Fig. 4 Experimental inaccuracies.

Transmission spectrum of the optimized sample with 52 scatterers (black solid line), where the vertical dashed line marks the frequency at which the optimization has been performed. Increasing the waveguide width by 1% of the initial width W (blue solid line) causes the peak to shift to lower frequencies, where a global shift \({\Delta }_{x}^{({\rm{opt}})}\) in the negative/positive longitudinal direction of half a scatterer radius r of only the optimized scatterers results in a shift to lower/higher frequencies (orange dashed/dotted line). In both cases, the peaks are also lowered owing to the deviation from the optimized configuration. Applying small random displacements \({\Delta }_{x,y}^{({\rm{i}})}\) in the range \([-r/4,r/4]\) in x- and y-direction to every single scatterer also results in a reduction of the peak height and a shift (green solid line). Performing full vectorial 3D simulations, we also find that using cylindrical scatterers with a height \({h}_{{\rm{scat}}}=7.98\,{\rm{mm}}\) smaller than the waveguide height h = 8 mm also lowers and shifts the peak to higher frequencies (red solid line).

Extended Data Fig. 5 Probability for randomly sampling a fully transmitting disorder.

Histogram of the transmission eigenvalues of 2,500 random configurations composed of 49 scatterers. Note that we have scanned every sample between 10.7 and 11.7 GHz with a resolution of 501 data points within this frequency window.

Extended Data Fig. 6 Comparison of our anti-reflection structure to hyperuniform media.

Structure factor (a) for a hyperuniform medium and (b) for a fully translucent disorder resulting from our design protocol. The hyperuniform medium consists of 100 scatterers, while our inverse designed medium features 52 scatterers, corresponding to the stronger variant disorder presented in the main text. c, Comparison to mirror media. Histogram of the distance of 2,000 random disorders to a mirror symmetric disorder (see text). A distance of 0 would mean that we have perfectly mirror symmetric medium, while larger distances signify that we move away from mirror symmetry. The orange line shows the distance for the optimized medium.

Extended Data Fig. 7 Transmission matrix for an empty waveguide.

a, b, Experimental intensity of the elements of the transmission \(|{t}_{mn}^{0}{|}^{2}\) in the basis of waveguide modes for an empty waveguide at 7 and 11.2 GHz. At these frequencies, the waveguide supports N = 4 and N = 7 modes, respectively.

Extended Data Fig. 8 Experimental result for the bimodal distribution of transmission eigenvalues.

Experimental transmission eigenvalue histogram for a waveguide supporting four modes compared to the bimodal law \({P}_{0}(\tau )\). The random disorder is composed of 6 aluminum cylinders and 34 Teflon cylinders.

Extended Data Fig. 9 Complete transmission of individual waveguide modes.

a, Experimental intensity of the elements of the transmission \(|{t}_{mn}{|}^{2}\) in the basis of waveguide modes at 7 GHz for a sample of complete transmission with 52 scatterers. b, Spectrum of the transmission of each mode through the waveguide (dotted lines). The average transmission for the four modes is represented with the blue line.

Extended Data Fig. 10 Complete transmission through a multichannel cavity.

a, Photography of the cavity. The top plate has been removed to see the interior of the cavity. Four transition-to-coax antennas are placed at the left and right side of the cavity. Measurement of the transmission matrix between these two arrays is carried out with a vector network analyser. Fifteen metallic cylinders are placed at the positions determined numerically for perfect transmission. b, Total transmission \({T}_{n}(\nu )={\Sigma }_{m}|{t}_{mn}(\nu ){|}^{2}\) for the four incoming channels (dashed lines) and the average transmission \(T(\nu )=({\Sigma }_{n}{T}_{n}(\nu ))/N\) over incoming channels (blue line). The placement of the cylinders corresponds to positions optimized numerically for perfect transmission at v0 = 8.4 GHz. Deviations from the maximal transmission value 1 are primarily due to absorption in the cavity.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Horodynski, M., Kühmayer, M., Ferise, C. et al. Anti-reflection structure for perfect transmission through complex media. Nature 607, 281–286 (2022). https://doi.org/10.1038/s41586-022-04843-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-022-04843-6

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing