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# Anti-reflection structure for perfect transmission through complex media

## 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.

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## 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.

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## 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

### 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.

### Extended data

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

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## 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.

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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

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

• ### Shaping the propagation of light in complex media

• Hui Cao
• Allard Pieter Mosk
• Stefan Rotter

Nature Physics (2022)