Experimental observation of three-dimensional Anderson localization of electromagnetic waves

Alexey Yamilov; Antton Go\"icoechea; Cl\'ement Ferise; Hui Cao; Matthieu Davy; Sergey E. Skipetrov

arxiv: 2606.04897 · v1 · pith:3XC3OGICnew · submitted 2026-06-03 · ❄️ cond-mat.dis-nn

Experimental observation of three-dimensional Anderson localization of electromagnetic waves

Antton Go\"icoechea , Alexey Yamilov , Cl\'ement Ferise , Sergey E. Skipetrov , Hui Cao , Matthieu Davy This is my paper

Pith reviewed 2026-06-28 03:02 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn

keywords Anderson localizationthree-dimensionalelectromagnetic wavesmicrowavesdisordered metal aggregatesscaling analysiswave localizationinterference effects

0 comments

The pith

Three-dimensional Anderson localization of microwaves is observed in disordered metal aggregates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes experimental evidence for Anderson localization of microwaves in three-dimensional disordered metal aggregates. Samples with different metal volume fractions display a distinction between diffusive transport and localized behavior. The localized regime is identified through scaling of the transmitted beam width, which matches theoretical and numerical expectations. A reader would care because prior experiments on light localization in 3D were inconclusive due to artifacts, and this result confirms wave trapping by interference in disorder.

Core claim

The central claim is an unambiguous experimental proof of three-dimensional Anderson localization of microwaves in disordered metal aggregates. Studying samples with different metal volume fractions reveals a clear difference between diffusive and localized behaviors. The localized behavior is confirmed by a scaling analysis of transmitted beam width in excellent agreement with theoretical and numerical results.

What carries the argument

Scaling analysis of transmitted beam width performed across samples with varying metal volume fractions, used to identify the localized regime and separate it from diffusion.

If this is right

Varying metal volume fraction separates diffusive from localized regimes in the experiment.
Transmitted beam width scaling matches theory and numerics only in the localized case.
This setup overcomes artifacts that affected all earlier 3D light localization attempts.
The result enables both fundamental studies and practical applications of the localization effect.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The volume fraction could serve as a tunable parameter to explore the localization transition in other 3D disordered systems.
If the same scaling approach works, analogous metal-aggregate samples might be adapted for optical or acoustic waves.
Control of localization via volume fraction might allow engineered materials that trap specific microwave frequencies.

Load-bearing premise

The scaling analysis of transmitted beam width on samples with different metal volume fractions is sufficient to distinguish true Anderson localization from diffusive transport or experimental artifacts.

What would settle it

A measurement in which transmitted beam width fails to show the predicted scaling for localization or shows no difference between volume fractions would falsify the claim of three-dimensional Anderson localization.

read the original abstract

A prominent phenomenon in contemporary condensed matter physics is Anderson localization -- suppression of wave propagation in disordered systems as a result of interference effects. Despite being observed with various types of waves over the years, all prior attempts to reach Anderson localization of light in three-dimensional systems have been hampered by experimental artifacts. Here, we report an unambiguous experimental proof of three-dimensional Anderson localization of microwaves in disordered metal aggregates. By studying samples with different metal volume fractions, we show a clear difference between diffusive and localized behaviors, and the latter is confirmed by a scaling analysis of transmitted beam width in excellent agreement with theoretical and numerical results. Our demonstration opens avenues for both fundamental studies and practical applications of this extraordinary phenomenon.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper reports what appears to be the first clear experimental distinction of 3D Anderson localization for microwaves via beam-width scaling in metal aggregates, but absorption in the samples could still mimic the signature.

read the letter

The main takeaway is that this work uses disordered metal aggregates at varying volume fractions and tracks how the transmitted beam width scales to separate diffusive from localized regimes, with the localized case matching theoretical predictions. That scaling approach and the choice of metallic scatterers are the concrete experimental steps forward from earlier attempts that ran into artifacts.

What stands out is the direct comparison across samples that shows a clear behavioral shift, plus the agreement between the measured scaling and both theory and numerics. The abstract frames this as resolving a long-standing experimental gap for electromagnetic waves in 3D, and the method itself looks like a practical way to tune the disorder strength.

The soft spot is the material choice. Metal aggregates introduce frequency-dependent absorption and scattering losses that can produce exponential decay and beam narrowing on their own. The stress-test note flags exactly this: without explicit checks such as a lossless limit, frequency scaling of the localization length, or side-by-side numerics with and without dissipation, the volume-fraction transition could reflect changing dissipation rather than interference-driven localization. The paper claims the scaling rules out prior artifacts, but that claim rests on how thoroughly those loss mechanisms were modeled or subtracted.

This is the kind of result that matters for the disordered-media community. A reader working on wave localization or microwave experiments would get value from the data and the scaling protocol even if the interpretation needs tightening. The work is coherent on its own terms and engages the literature directly, so it deserves a serious referee rather than a desk rejection. I would send it out for review with the expectation that the authors supply the loss checks and raw scaling plots.

Referee Report

1 major / 0 minor

Summary. The manuscript reports an experimental demonstration of three-dimensional Anderson localization of microwaves in disordered metal aggregates. By varying metal volume fractions, the authors distinguish diffusive from localized regimes via a scaling analysis of transmitted beam width, claiming excellent agreement with theory and numerics and an unambiguous proof that overcomes prior experimental artifacts.

Significance. If the central experimental claim holds, this would constitute a significant advance in disordered media physics by realizing 3D Anderson localization for electromagnetic waves, a long-sought result with implications for fundamental wave interference studies and potential applications in wave control.

major comments (1)

[Abstract / scaling analysis] Abstract and scaling analysis section: The distinction between localized and diffusive behavior rests on the scaling of transmitted beam width across volume fractions. However, this scaling is not shown to isolate interference-driven localization from frequency-dependent absorption and scattering losses inherent to metallic samples, which can produce exponential intensity decay mimicking the expected localized regime (as raised by the stress-test concern). Explicit comparison to loss-inclusive vs. lossless numerics or frequency dependence of the localization length is needed to support the claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract / scaling analysis] Abstract and scaling analysis section: The distinction between localized and diffusive behavior rests on the scaling of transmitted beam width across volume fractions. However, this scaling is not shown to isolate interference-driven localization from frequency-dependent absorption and scattering losses inherent to metallic samples, which can produce exponential intensity decay mimicking the expected localized regime (as raised by the stress-test concern). Explicit comparison to loss-inclusive vs. lossless numerics or frequency dependence of the localization length is needed to support the claim.

Authors: The scaling analysis uses the transverse beam width, which evolves differently in localized versus diffusive regimes due to interference and is largely insensitive to uniform absorption (which reduces total intensity but does not alter the width scaling with thickness or volume fraction in the same way). Our numerical results are based on the standard lossless Anderson model and match the experimental scaling across volume fractions, supporting that the observed behavior arises from localization rather than losses. We agree that an explicit comparison would strengthen the manuscript and will add loss-inclusive versus lossless numerical comparisons, along with discussion of the relevant frequency range, in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: experimental measurements compared to external theory

full rationale

The paper is an experimental report. Its central claim rests on direct measurements of transmitted beam width for samples with varying metal volume fractions, with scaling behavior compared to independent theoretical and numerical results. No derivation chain, fitted parameters renamed as predictions, or self-citation load-bearing steps are described. The scaling analysis is presented as external confirmation rather than a self-referential construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Experimental observation paper; central claim rests on validity of sample fabrication, microwave transmission measurements, and interpretation of scaling analysis as localization. No free parameters, axioms, or invented entities are introduced in the abstract.

pith-pipeline@v0.9.1-grok · 5662 in / 1020 out tokens · 36439 ms · 2026-06-28T03:02:14.660441+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

45 extracted references · 35 canonical work pages

[1]

P. W. Anderson, Absence of Diffusion in Certain Random Lattices.Phys. Rev.109(5), 1492– 1505 (1958), doi:10.1103/PhysRev.109.1492

work page doi:10.1103/physrev.109.1492 1958
[2]

John, Electromagnetic Absorption in a Disordered Medium near a Photon Mobility Edge

S. John, Electromagnetic Absorption in a Disordered Medium near a Photon Mobility Edge. Phys. Rev. Lett.53, 2169–2172 (1984), doi:10.1103/PhysRevLett.53.2169,https://link. aps.org/doi/10.1103/PhysRevLett.53.2169

work page doi:10.1103/physrevlett.53.2169 1984
[3]

Anderson, The question of classical localization A theory of white paint?Phil

P. Anderson, The question of classical localization A theory of white paint?Phil. Mag. B52(3), 505–509 (1985)

1985
[4]

Kramer, A

B. Kramer, A. MacKinnon, Localization: Theory and Experiment.Rep. Prog. Phys.56(12), 1469–1564 (1993), doi:10.1088/0034-4885/56/12/001

work page doi:10.1088/0034-4885/56/12/001 1993
[5]

Imada, A

M. Imada, A. Fujimori, Y. Tokura, Metal-Insulator Transitions.Rev. Mod. Phys.70(4), 1039– 1263 (1998), doi:10.1103/RevModPhys.70.1039

work page doi:10.1103/revmodphys.70.1039 1998
[6]

Billy, V

J. Billy,et al., Direct Observation of Anderson Localization of Matter Waves in a Controlled Disorder.Nature453(7197), 891–894 (2008), doi:10.1038/nature07000

work page doi:10.1038/nature07000 2008
[7]

Jendrzejewski, A

F. Jendrzejewski,et al., Three-Dimensional Localization of Ultracold Atoms in an Optical Disordered Potential.Nat. Phys.8(5), 398–403 (2012), doi:10.1038/nphys2256

work page doi:10.1038/nphys2256 2012
[8]

Guazzelli, E

E. Guazzelli, E. Guyon, B. Souillard, On the Localization of Shallow Water Waves by a Random Bottom.J. Phyique Lett.44(20), 837–841 (1983), doi:10.1051/jphyslet:019830044020083700

work page doi:10.1051/jphyslet:019830044020083700 1983
[9]

A. A. Chabanov, M. Stoytchev, A. Z. Genack, Statistical Signatures of Photon Localization. Nature404(6780), 850–853 (2000), doi:10.1038/35009055

work page doi:10.1038/35009055 2000
[10]

Schwartz, G

T. Schwartz, G. Bartal, S. Fishman, M. Segev, Transport and Anderson Localization in Disordered Two-Dimensional Photonic Lattices.Nature446(7131), 52–55 (2007), doi: 10.1038/nature05623

work page doi:10.1038/nature05623 2007
[11]

Segev, Y

M. Segev, Y. Silberberg, D. N. Christodoulides, Anderson Localization of Light.Nat. Photonics 7(3), 197–204 (2013), doi:10.1038/nphoton.2013.30. 11

work page doi:10.1038/nphoton.2013.30 2013
[12]

Weaver, Anderson Localization of Ultrasound.Wave Motion12(2), 129–142 (1990), doi: 10.1016/0165-2125(90)90034-2

R. Weaver, Anderson Localization of Ultrasound.Wave Motion12(2), 129–142 (1990), doi: 10.1016/0165-2125(90)90034-2

work page doi:10.1016/0165-2125(90)90034-2 1990
[13]

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, B. a. van Tiggelen, Localization of Ultrasound in a Three-Dimensional Elastic Network.Nat. Phys.4(12), 945–948 (2008), doi: 10.1038/nphys1101

work page doi:10.1038/nphys1101 2008
[14]

L. A. Cobus, W. K. Hildebrand, S. E. Skipetrov, B. A. Van Tiggelen, J. H. Page, Transverse Con- finement of Ultrasound through the Anderson Transition in Three-Dimensional Mesoglasses. Phys. Rev. B98(21), 214201 (2018), doi:10.1103/PhysRevB.98.214201

work page doi:10.1103/physrevb.98.214201 2018
[15]

Delmotte, T

F. Delmotte, T. Brunet, J. Leng, J. H. Page, 3D Anderson Localization of Classical Scalar Waves.Science Advances12(9), eaed1319 (2026), doi:10.1126/sciadv.aed1319

work page doi:10.1126/sciadv.aed1319 2026
[16]

Abrahams, P

E. Abrahams, P. W. Anderson, D. C. Licciardello, T. V. Ramakrishnan, Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions.Phys. Rev. Lett.42(10), 673–676 (1979), doi:10.1103/PhysRevLett.42.673

work page doi:10.1103/physrevlett.42.673 1979
[17]

A. F. Ioffe, A. R. Regel, Non-Crystalline, Amorphous, and Liquid Electronic Semiconductors. Prog. Semicond.4, 237–291 (1960)

1960
[18]

D. S. Wiersma, P. Bartolini, A. Lagendijk, R. Righini, Localization of Light in a Disordered Medium.Nature390(6661), 671–673 (1997), doi:10.1038/37757

work page doi:10.1038/37757 1997
[19]

Scheffold, R

F. Scheffold, R. Lenke, R. Tweer, G. Maret, Localization or Classical Diffusion of Light? Nature398(6724), 206–207 (1999), doi:10.1038/18347

work page doi:10.1038/18347 1999
[20]

D. S. Wiersma, J. G. Rivas, P. Bartolini, A. Lagendijk, R. Righini, Localization or Classical Diffusion of Light?Nature398(6724), 207–207 (1999), doi:10.1038/18350

work page doi:10.1038/18350 1999
[22]

Sperling, W

T. Sperling, W. B¨ uhrer, C. M. Aegerter, G. Maret, Direct Determination of the Transition to Localization of Light in Three Dimensions.Nat. Photonics7(1), 48–52 (2013), doi:10.1038/ nphoton.2012.313

2013
[23]

Scheffold, D

F. Scheffold, D. Wiersma, Inelastic Scattering Puts in Question Recent Claims of Anderson Localization of Light.Nat. Photonics7(12), 934–934 (2013), doi:10.1038/nphoton.2013.210

work page doi:10.1038/nphoton.2013.210 2013
[24]

Sperling,et al., Can 3D Light Localization Be Reached in ‘White Paint’?New J

T. Sperling,et al., Can 3D Light Localization Be Reached in ‘White Paint’?New J. Phys. 18(1), 013039 (2016), doi:10.1088/1367-2630/18/1/013039

work page doi:10.1088/1367-2630/18/1/013039 2016
[25]

S. E. Skipetrov, J. H. Page, Red Light for Anderson Localization.New J. Phys.18(2), 021001 (2016), doi:10.1088/1367-2630/18/2/021001

work page doi:10.1088/1367-2630/18/2/021001 2016
[26]

Yamilov,et al., Anderson localization of electromagnetic waves in three dimensions.Nature Physics19(9), 1308–1313 (2023), doi:10.1038/s41567-023-02091-7

A. Yamilov,et al., Anderson localization of electromagnetic waves in three dimensions.Nature Physics19(9), 1308–1313 (2023), doi:10.1038/s41567-023-02091-7

work page doi:10.1038/s41567-023-02091-7 2023
[28]

Rezvani Naraghi, A

R. Rezvani Naraghi, A. Dogariu, Phase Transitions in Diffusion of Light.Phys. Rev. Lett. 117(26), 263901 (2016), doi:10.1103/PhysRevLett.117.263901

work page doi:10.1103/physrevlett.117.263901 2016
[29]

B. A. van Tiggelen, S. E. Skipetrov, Longitudinal Modes in Diffusion and Localization of Light.Phys. Rev. B103(17), 174204 (2021), doi:10.1103/PhysRevB.103.174204

work page doi:10.1103/physrevb.103.174204 2021
[30]

V. M. Shalaev, Electromagnetic Properties of Small-Particle Composites.Phys. Rep.272(2-3), 61–137 (1996), doi:10.1016/0370-1573(95)00076-3

work page doi:10.1016/0370-1573(95)00076-3 1996
[31]

A. K. Sarychev, V. A. Shubin, V. M. Shalaev, Anderson Localization of Surface Plasmons and Nonlinear Optics of Metal-Dielectric Composites.Phys. Rev. B60(24), 16389–16408 (1999), doi:10.1103/PhysRevB.60.16389

work page doi:10.1103/physrevb.60.16389 1999
[32]

Lalanne, W

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, J.-P. Hugonin, Light Interaction with Photonic and Plasmonic Resonances.Laser Photonics Rev.12(5), 1700113 (2018), doi:10.1002/lpor. 201700113. 13

work page doi:10.1002/lpor 2018
[33]

A. Z. Genack, N. Garcia, Observation of Photon Localization in a Three-Dimensional Disor- dered System.Phys. Rev. Lett.66(16), 2064–2067 (1991), doi:10.1103/PhysRevLett.66.2064

work page doi:10.1103/physrevlett.66.2064 2064
[34]

Cherroret, S

N. Cherroret, S. E. Skipetrov, B. A. van Tiggelen, Transverse Confinement of Waves in Three- Dimensional Random Media.Phys. Rev. E82(5), 056603 (2010), doi:10.1103/PhysRevE.82. 056603

work page doi:10.1103/physreve.82 2010
[35]

Materials and methods are available as supplementary material
[36]

Ishimaru,Wave Propagation and Scattering in Random Media(Academic Press, New York) (1978)

A. Ishimaru,Wave Propagation and Scattering in Random Media(Academic Press, New York) (1978)

1978
[37]

J. H. Page, H. P. Schriemer, A. E. Bailey, D. A. Weitz, Experimental test of the diffusion approximation for multiply scattered sound.Phys. Rev. E52, 3106–3114 (1995), doi:10.1103/ PhysRevE.52.3106,https://link.aps.org/doi/10.1103/PhysRevE.52.3106

work page doi:10.1103/physreve.52.3106 1995
[38]

W ¨olfle, D

P. W ¨olfle, D. Vollhardt, Self-consistent theory of Anderson localization: General formalism and applications.Int. J. Mod. Phys. B.24(12n13), 1526–1554 (2010)

2010
[39]

B. A. van Tiggelen, A. Lagendijk, D. S. Wiersma, Reflection and Transmission of Waves near the Localization Threshold.Phys. Rev. Lett.84, 4333–4336 (2000), doi:10.1103/PhysRevLett. 84.4333,https://link.aps.org/doi/10.1103/PhysRevLett.84.4333

work page doi:10.1103/physrevlett 2000
[40]

Cherroret, S

N. Cherroret, S. E. Skipetrov, Microscopic derivation of self-consistent equations of Anderson localization in a disordered medium of finite size.Phys. Rev. E77, 046608 (2008)

2008
[41]

Go ¨ıcoechea,et al., Experimental observation of three-dimensional Anderson localization of electromagnetic waves, Repository name (2026), doi:10.1000.dataset-DOI

A. Go ¨ıcoechea,et al., Experimental observation of three-dimensional Anderson localization of electromagnetic waves, Repository name (2026), doi:10.1000.dataset-DOI

2026
[42]

Jofre, L

M. Jofre, L. Jofre, L. Jofre-Roca, On the Wireless Microwave Sensing of Bacterial Mem- brane Potential in Microfluidic-Actuated Platforms.Sensors21(10), 3420 (2021), doi: 10.3390/s21103420

work page doi:10.3390/s21103420 2021
[43]

Palacios, M

C. Palacios, M. Jofre, L. Jofre, J. Romeu, L. Jofre-Roca, Superheterodyne Microwave System for the Detection of Bioparticles With Coplanar Electrodes on a Microfluidic Platform.IEEE Trans. Instrum. Meas.71, 1–10 (2022), doi:10.1109/TIM.2022.3165790. 14

work page doi:10.1109/tim.2022.3165790 2022
[44]

Sheng,Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, no

P. Sheng,Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, no. 88 in Springer Series in Materials Science (Springer, Berlin ; New York), 2nd ed ed. (2006)

2006
[45]

R. C. Haskell,et al., Boundary conditions for the diffusion equation in radiative transfer.J. Opt. Soc. Am. A11(10), 2727–2741 (1994), doi:10.1364/JOSAA.11.002727,https://opg. optica.org/josaa/abstract.cfm?URI=josaa-11-10-2727

work page doi:10.1364/josaa.11.002727 1994
[46]

Surface wave suppression

J. B. Pendry, L. Mart ´ın-Moreno, F. J. Garcia-Vidal, Mimicking Surface Plasmons with Struc- tured Surfaces.Science305(5685), 847–848 (2004), doi:10.1126/science.1098999. Acknowledgments In memory of Bart Van Tiggelen, whose pioneering spirit continues to inspire our work. We thank Laurent Cronier for his help in preparing the samples. Funding:This work i...

work page doi:10.1126/science.1098999 2004
[47]

Field values in with PEC regions are zero before smoothing

spatially integrates the field, we apply spatial convolution with a Gaussian function of full-width-at-half-maximum (FWHM)=2.4 cm to the intensity distribution𝐼(𝑥, 𝑦, 𝑧= 𝐿−1.5 cm, 𝑡). Field values in with PEC regions are zero before smoothing. The smoothed intensity 𝐼smooth(𝑥, 𝑦, 𝑡)is azimuthally averaged about the beam axis to obtain𝐼 smooth(𝜌, 𝑡), and u...

[1] [1]

P. W. Anderson, Absence of Diffusion in Certain Random Lattices.Phys. Rev.109(5), 1492– 1505 (1958), doi:10.1103/PhysRev.109.1492

work page doi:10.1103/physrev.109.1492 1958

[2] [2]

John, Electromagnetic Absorption in a Disordered Medium near a Photon Mobility Edge

S. John, Electromagnetic Absorption in a Disordered Medium near a Photon Mobility Edge. Phys. Rev. Lett.53, 2169–2172 (1984), doi:10.1103/PhysRevLett.53.2169,https://link. aps.org/doi/10.1103/PhysRevLett.53.2169

work page doi:10.1103/physrevlett.53.2169 1984

[3] [3]

Anderson, The question of classical localization A theory of white paint?Phil

P. Anderson, The question of classical localization A theory of white paint?Phil. Mag. B52(3), 505–509 (1985)

1985

[4] [4]

Kramer, A

B. Kramer, A. MacKinnon, Localization: Theory and Experiment.Rep. Prog. Phys.56(12), 1469–1564 (1993), doi:10.1088/0034-4885/56/12/001

work page doi:10.1088/0034-4885/56/12/001 1993

[5] [5]

Imada, A

M. Imada, A. Fujimori, Y. Tokura, Metal-Insulator Transitions.Rev. Mod. Phys.70(4), 1039– 1263 (1998), doi:10.1103/RevModPhys.70.1039

work page doi:10.1103/revmodphys.70.1039 1998

[6] [6]

Billy, V

J. Billy,et al., Direct Observation of Anderson Localization of Matter Waves in a Controlled Disorder.Nature453(7197), 891–894 (2008), doi:10.1038/nature07000

work page doi:10.1038/nature07000 2008

[7] [7]

Jendrzejewski, A

F. Jendrzejewski,et al., Three-Dimensional Localization of Ultracold Atoms in an Optical Disordered Potential.Nat. Phys.8(5), 398–403 (2012), doi:10.1038/nphys2256

work page doi:10.1038/nphys2256 2012

[8] [8]

Guazzelli, E

E. Guazzelli, E. Guyon, B. Souillard, On the Localization of Shallow Water Waves by a Random Bottom.J. Phyique Lett.44(20), 837–841 (1983), doi:10.1051/jphyslet:019830044020083700

work page doi:10.1051/jphyslet:019830044020083700 1983

[9] [9]

A. A. Chabanov, M. Stoytchev, A. Z. Genack, Statistical Signatures of Photon Localization. Nature404(6780), 850–853 (2000), doi:10.1038/35009055

work page doi:10.1038/35009055 2000

[10] [10]

Schwartz, G

T. Schwartz, G. Bartal, S. Fishman, M. Segev, Transport and Anderson Localization in Disordered Two-Dimensional Photonic Lattices.Nature446(7131), 52–55 (2007), doi: 10.1038/nature05623

work page doi:10.1038/nature05623 2007

[11] [11]

Segev, Y

M. Segev, Y. Silberberg, D. N. Christodoulides, Anderson Localization of Light.Nat. Photonics 7(3), 197–204 (2013), doi:10.1038/nphoton.2013.30. 11

work page doi:10.1038/nphoton.2013.30 2013

[12] [12]

Weaver, Anderson Localization of Ultrasound.Wave Motion12(2), 129–142 (1990), doi: 10.1016/0165-2125(90)90034-2

R. Weaver, Anderson Localization of Ultrasound.Wave Motion12(2), 129–142 (1990), doi: 10.1016/0165-2125(90)90034-2

work page doi:10.1016/0165-2125(90)90034-2 1990

[13] [13]

H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, B. a. van Tiggelen, Localization of Ultrasound in a Three-Dimensional Elastic Network.Nat. Phys.4(12), 945–948 (2008), doi: 10.1038/nphys1101

work page doi:10.1038/nphys1101 2008

[14] [14]

L. A. Cobus, W. K. Hildebrand, S. E. Skipetrov, B. A. Van Tiggelen, J. H. Page, Transverse Con- finement of Ultrasound through the Anderson Transition in Three-Dimensional Mesoglasses. Phys. Rev. B98(21), 214201 (2018), doi:10.1103/PhysRevB.98.214201

work page doi:10.1103/physrevb.98.214201 2018

[15] [15]

Delmotte, T

F. Delmotte, T. Brunet, J. Leng, J. H. Page, 3D Anderson Localization of Classical Scalar Waves.Science Advances12(9), eaed1319 (2026), doi:10.1126/sciadv.aed1319

work page doi:10.1126/sciadv.aed1319 2026

[16] [16]

Abrahams, P

E. Abrahams, P. W. Anderson, D. C. Licciardello, T. V. Ramakrishnan, Scaling Theory of Localization: Absence of Quantum Diffusion in Two Dimensions.Phys. Rev. Lett.42(10), 673–676 (1979), doi:10.1103/PhysRevLett.42.673

work page doi:10.1103/physrevlett.42.673 1979

[17] [17]

A. F. Ioffe, A. R. Regel, Non-Crystalline, Amorphous, and Liquid Electronic Semiconductors. Prog. Semicond.4, 237–291 (1960)

1960

[18] [18]

D. S. Wiersma, P. Bartolini, A. Lagendijk, R. Righini, Localization of Light in a Disordered Medium.Nature390(6661), 671–673 (1997), doi:10.1038/37757

work page doi:10.1038/37757 1997

[19] [19]

Scheffold, R

F. Scheffold, R. Lenke, R. Tweer, G. Maret, Localization or Classical Diffusion of Light? Nature398(6724), 206–207 (1999), doi:10.1038/18347

work page doi:10.1038/18347 1999

[20] [20]

D. S. Wiersma, J. G. Rivas, P. Bartolini, A. Lagendijk, R. Righini, Localization or Classical Diffusion of Light?Nature398(6724), 207–207 (1999), doi:10.1038/18350

work page doi:10.1038/18350 1999

[21] [22]

Sperling, W

T. Sperling, W. B¨ uhrer, C. M. Aegerter, G. Maret, Direct Determination of the Transition to Localization of Light in Three Dimensions.Nat. Photonics7(1), 48–52 (2013), doi:10.1038/ nphoton.2012.313

2013

[22] [23]

Scheffold, D

F. Scheffold, D. Wiersma, Inelastic Scattering Puts in Question Recent Claims of Anderson Localization of Light.Nat. Photonics7(12), 934–934 (2013), doi:10.1038/nphoton.2013.210

work page doi:10.1038/nphoton.2013.210 2013

[23] [24]

Sperling,et al., Can 3D Light Localization Be Reached in ‘White Paint’?New J

T. Sperling,et al., Can 3D Light Localization Be Reached in ‘White Paint’?New J. Phys. 18(1), 013039 (2016), doi:10.1088/1367-2630/18/1/013039

work page doi:10.1088/1367-2630/18/1/013039 2016

[24] [25]

S. E. Skipetrov, J. H. Page, Red Light for Anderson Localization.New J. Phys.18(2), 021001 (2016), doi:10.1088/1367-2630/18/2/021001

work page doi:10.1088/1367-2630/18/2/021001 2016

[25] [26]

Yamilov,et al., Anderson localization of electromagnetic waves in three dimensions.Nature Physics19(9), 1308–1313 (2023), doi:10.1038/s41567-023-02091-7

A. Yamilov,et al., Anderson localization of electromagnetic waves in three dimensions.Nature Physics19(9), 1308–1313 (2023), doi:10.1038/s41567-023-02091-7

work page doi:10.1038/s41567-023-02091-7 2023

[26] [28]

Rezvani Naraghi, A

R. Rezvani Naraghi, A. Dogariu, Phase Transitions in Diffusion of Light.Phys. Rev. Lett. 117(26), 263901 (2016), doi:10.1103/PhysRevLett.117.263901

work page doi:10.1103/physrevlett.117.263901 2016

[27] [29]

B. A. van Tiggelen, S. E. Skipetrov, Longitudinal Modes in Diffusion and Localization of Light.Phys. Rev. B103(17), 174204 (2021), doi:10.1103/PhysRevB.103.174204

work page doi:10.1103/physrevb.103.174204 2021

[28] [30]

V. M. Shalaev, Electromagnetic Properties of Small-Particle Composites.Phys. Rep.272(2-3), 61–137 (1996), doi:10.1016/0370-1573(95)00076-3

work page doi:10.1016/0370-1573(95)00076-3 1996

[29] [31]

A. K. Sarychev, V. A. Shubin, V. M. Shalaev, Anderson Localization of Surface Plasmons and Nonlinear Optics of Metal-Dielectric Composites.Phys. Rev. B60(24), 16389–16408 (1999), doi:10.1103/PhysRevB.60.16389

work page doi:10.1103/physrevb.60.16389 1999

[30] [32]

Lalanne, W

P. Lalanne, W. Yan, K. Vynck, C. Sauvan, J.-P. Hugonin, Light Interaction with Photonic and Plasmonic Resonances.Laser Photonics Rev.12(5), 1700113 (2018), doi:10.1002/lpor. 201700113. 13

work page doi:10.1002/lpor 2018

[31] [33]

A. Z. Genack, N. Garcia, Observation of Photon Localization in a Three-Dimensional Disor- dered System.Phys. Rev. Lett.66(16), 2064–2067 (1991), doi:10.1103/PhysRevLett.66.2064

work page doi:10.1103/physrevlett.66.2064 2064

[32] [34]

Cherroret, S

N. Cherroret, S. E. Skipetrov, B. A. van Tiggelen, Transverse Confinement of Waves in Three- Dimensional Random Media.Phys. Rev. E82(5), 056603 (2010), doi:10.1103/PhysRevE.82. 056603

work page doi:10.1103/physreve.82 2010

[33] [35]

Materials and methods are available as supplementary material

[34] [36]

Ishimaru,Wave Propagation and Scattering in Random Media(Academic Press, New York) (1978)

A. Ishimaru,Wave Propagation and Scattering in Random Media(Academic Press, New York) (1978)

1978

[35] [37]

J. H. Page, H. P. Schriemer, A. E. Bailey, D. A. Weitz, Experimental test of the diffusion approximation for multiply scattered sound.Phys. Rev. E52, 3106–3114 (1995), doi:10.1103/ PhysRevE.52.3106,https://link.aps.org/doi/10.1103/PhysRevE.52.3106

work page doi:10.1103/physreve.52.3106 1995

[36] [38]

W ¨olfle, D

P. W ¨olfle, D. Vollhardt, Self-consistent theory of Anderson localization: General formalism and applications.Int. J. Mod. Phys. B.24(12n13), 1526–1554 (2010)

2010

[37] [39]

B. A. van Tiggelen, A. Lagendijk, D. S. Wiersma, Reflection and Transmission of Waves near the Localization Threshold.Phys. Rev. Lett.84, 4333–4336 (2000), doi:10.1103/PhysRevLett. 84.4333,https://link.aps.org/doi/10.1103/PhysRevLett.84.4333

work page doi:10.1103/physrevlett 2000

[38] [40]

Cherroret, S

N. Cherroret, S. E. Skipetrov, Microscopic derivation of self-consistent equations of Anderson localization in a disordered medium of finite size.Phys. Rev. E77, 046608 (2008)

2008

[39] [41]

Go ¨ıcoechea,et al., Experimental observation of three-dimensional Anderson localization of electromagnetic waves, Repository name (2026), doi:10.1000.dataset-DOI

A. Go ¨ıcoechea,et al., Experimental observation of three-dimensional Anderson localization of electromagnetic waves, Repository name (2026), doi:10.1000.dataset-DOI

2026

[40] [42]

Jofre, L

M. Jofre, L. Jofre, L. Jofre-Roca, On the Wireless Microwave Sensing of Bacterial Mem- brane Potential in Microfluidic-Actuated Platforms.Sensors21(10), 3420 (2021), doi: 10.3390/s21103420

work page doi:10.3390/s21103420 2021

[41] [43]

Palacios, M

C. Palacios, M. Jofre, L. Jofre, J. Romeu, L. Jofre-Roca, Superheterodyne Microwave System for the Detection of Bioparticles With Coplanar Electrodes on a Microfluidic Platform.IEEE Trans. Instrum. Meas.71, 1–10 (2022), doi:10.1109/TIM.2022.3165790. 14

work page doi:10.1109/tim.2022.3165790 2022

[42] [44]

Sheng,Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, no

P. Sheng,Introduction to Wave Scattering, Localization, and Mesoscopic Phenomena, no. 88 in Springer Series in Materials Science (Springer, Berlin ; New York), 2nd ed ed. (2006)

2006

[43] [45]

R. C. Haskell,et al., Boundary conditions for the diffusion equation in radiative transfer.J. Opt. Soc. Am. A11(10), 2727–2741 (1994), doi:10.1364/JOSAA.11.002727,https://opg. optica.org/josaa/abstract.cfm?URI=josaa-11-10-2727

work page doi:10.1364/josaa.11.002727 1994

[44] [46]

Surface wave suppression

J. B. Pendry, L. Mart ´ın-Moreno, F. J. Garcia-Vidal, Mimicking Surface Plasmons with Struc- tured Surfaces.Science305(5685), 847–848 (2004), doi:10.1126/science.1098999. Acknowledgments In memory of Bart Van Tiggelen, whose pioneering spirit continues to inspire our work. We thank Laurent Cronier for his help in preparing the samples. Funding:This work i...

work page doi:10.1126/science.1098999 2004

[45] [47]

Field values in with PEC regions are zero before smoothing

spatially integrates the field, we apply spatial convolution with a Gaussian function of full-width-at-half-maximum (FWHM)=2.4 cm to the intensity distribution𝐼(𝑥, 𝑦, 𝑧= 𝐿−1.5 cm, 𝑡). Field values in with PEC regions are zero before smoothing. The smoothed intensity 𝐼smooth(𝑥, 𝑦, 𝑡)is azimuthally averaged about the beam axis to obtain𝐼 smooth(𝜌, 𝑡), and u...