Radiant Field Theory: A Transport Approach to Shaped Wave Transmission through Disordered Media

Arthur Goetschy; David Gaspard

REVIEW 3 major objections 4 minor 1 cited by

Radiant Field Theory: A Transport Approach to Shaped Wave Transmission through Disordered Media

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A matrix-valued radiance obeying a nonlinear transport equation determines the transmission eigenvalue distribution of a disordered waveguide across diffusive and quasiballistic regimes, with absorption and partial channel control included.

desk verdict Genuinely new, reproducible matrix transport theory that nails transmission eigenvalue statistics beyond the diffusive regime; the replica-decoupling step is the open sore, but the Letter deserves serious peer review. read the letter →

arxiv 2411.10360 v3 pith:6PUKVVNC submitted 2024-11-15 math-ph cond-mat.dis-nnmath.MPphysics.optics

David Gaspard , Arthur Goetschy This is my paper · ORCID

classification math-phcond-mat.dis-nnmath.MPphysics.optics MSC 82B4481Q2078A45

keywords transmissioneigenvaluesdisorderedwaveguidesmatrixtransportequationmatrix-valuedradiancequasiballisticabsorptionincompletechannelcontrolreplicamethod

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that the full distribution of transmission eigenvalues for coherent waves through a disordered waveguide is determined by a self-consistent matrix transport equation for a matrix-valued radiance, derived from the microscopic wave equation. This 'radiant field theory' does not assume that thin slices scatter isotropically, so it stays accurate in the quasiballistic regime where the standard DMPK transport theory fails. The same equation accommodates absorption and partial control of the input channels, both common in experiments and previously lacking an ab initio treatment. Numerical solutions of the transport equation reproduce distributions computed from the microscopic wave equation in the regimes tested. If correct, this gives a single framework covering diffusive and quasiballistic transmission, with the diffusive DMPK result recovered as a direction-averaged limit.

What carries the argument

The load-bearing object is the matrix-valued radiance $g(\boldsymbol{\Omega},\mathbf{r})$, defined by a Wigner transform of the disorder-averaged Green's function projected on the wavenumber shell $\|\mathbf{p}\|=k$. It obeys a first-order nonlinear transport equation in position and direction, with the normalized field $\tilde Q(\mathbf{r})=\int d\Omega\, g(\boldsymbol{\Omega},\mathbf{r})/S_d$ acting as a self-consistent scattering kernel, and with Heaviside factors $\Theta(m_a-\|\boldsymbol{\Omega}_\perp\|)$ that encode the numerical aperture. The boundary conditions (15) fix the incoming and outgoing angular sectors, and the constraints $g^2=1$, $\operatorname{Tr}g=0$ replace the stronger isotropy normalization of $\sigma$ models. This machinery converts the computation of $\rho(T)$ from a many-channel statistical problem into the numerical solution of a few coupled transport equations, which is what makes absorption and partial aperture straightforward to include.

What would settle it

Solve Eq. (13) for a waveguide with optical thickness $L/\ell$ fixed and gradually push the ratio $L/\xi$ toward and beyond one by increasing disorder strength or shrinking the width, then compare with recursive Green's function simulations; if the RFT prediction departs from the microscopic distributions as localization is approached, the single-replica saddle-point approximation is the point of failure.

Watch

Extended reading notes

Core claim

The central claim is that the transmission eigenvalue distribution $\rho(T)$ of a disordered waveguide is completely characterized by the matrix radiance $g(\boldsymbol{\Omega},\mathbf{r})$, which solves the nonlinear transport equation (13) with boundary conditions (15). The derivation proceeds by averaging the wave equation over a Gaussian white-noise potential, using a replica trick and a saddle-point approximation, then passing to a Wigner (semiclassical) description; the result is a transport equation whose commutator structure encodes coherent interference. The paper validates the equation against large-scale numerical solutions of the microscopic wave equation in two-dimensional waveguides, including cases with $L/\ell$ from 5 down to 0.05, with uniform absorption, and with reduced numerical aperture at the input. In the direction-averaged diffusive limit the equation reduces to the known DMPK result, and in the quasiballistic regime it agrees with numerics where DMPK deviates. The authors therefore claim a microscopic, parameter-free generalization of the standard theory that incorporates experimental realities such as absorption and incomplete channel control.

Load-bearing premise

The whole scheme depends on a saddle-point approximation in which replicas of the disordered system are treated as noninteracting; the authors state this is valid for samples shorter than the localization length but do not prove it or characterize how it fails.

Editorial extensions

If this is right

Transmission eigenvalue statistics in the quasiballistic regime ($L\lesssim\ell$) become computable from the wave equation, a regime where the standard DMPK theory is known to be inaccurate.
Absorption can be included by setting $\varepsilon=k/\ell_a$, covering the moderate-absorption range $\ell_a\le L\lesssim\xi_a$ that previously lacked an ab initio treatment.
Incomplete channel control enters through the aperture parameters $m_a,m_b$ in the transport equation, so rectangular transmission matrices are handled without the renormalized parameters of filtered random matrix theory.
The direction-averaged limit of the equation reproduces the DMPK result, showing the new theory as a strict generalization rather than a replacement.
Because the equation is derived from the microscopic Green's function, the same structure can be extended to other observables such as internal intensity and deposition, and to anisotropic or correlated disorder.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Since the boundary values of $g$ determine the generating function for $\rho(T)$, an experiment that measures angle-resolved transmission could in principle reconstruct the matrix radiance itself, although the paper does not discuss such an inversion.
If the single-replica saddle-point approximation survives beyond the stated $L\ll\xi$ regime, the framework might describe the onset of Anderson localization, including the disappearance of the $T\simeq1$ peak without a low-$T$ gap; the paper lists this as a promising direction but does not establish it.
In open geometries without waveguide side walls, the same $\boldsymbol{\Omega}\cdot\nabla_r$ transport structure could directly account for lateral diffusion, potentially replacing the fitted long-range correlations used in filtered random matrix theory with a direct calculation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: the Letter contains a genuinely new object — the matrix transport equation (13) — that extends Eilenberger/Usadel machinery to transmission eigenvalue statistics without the isotropy hypothesis, and it ships with enough numerical evidence to be taken seriously. It deserves a serious referee, not a desk rejection.

What is new: rather than assuming uniform angular scattering per slice, the theory keeps the full direction-resolved radiance, so it remains applicable in the quasiballistic regime, and it incorporates absorption and partial channel control in a natural way. The diffusive limit reduces to DMPK/Usadel, which is a good external sanity check. The Letter's own numerics — recursive Green's function simulations of the wave equation for L/ell from 0.05 to 5, with and without absorption and with reduced apertures — match the theory remarkably well, and the deviating DMPK/FRM dashed lines make the improvement visible. The code and data are on Zenodo, so the numerical part is reproducible. That is substantial credit.

The soft spot is the derivation. Eq. (13) is the central claim, but the derivation in this Letter is a sketch: replica averaging, Hubbard-Stratonovich, saddle point, Wigner transform. The load-bearing approximation is the replica decoupling ≈ Zbar^R and the single-replica saddle point, asserted to be valid for L << xi without an error estimate. All tests use W/lambda = 50.5, so N ~ 100; corrections of order 1/N or 1/(k ell) could be hidden, and the extreme tails of rho(T) are where non-perturbative effects could surface. The companion paper apparently carries the full derivation, and its acceptance in PRR gives some assurance, but from the Letter alone the central step is not independently checkable. I would not call this fatal — in the large-N nonlocalized regime the approximation is plausible and the numerics support it — but it is a real gap in the Letter itself.

The citation pattern looks fair: DMPK, sigma models, Nazarov, Brouwer, FRM, and the companion paper are all there, and the self-citation to the companion is legitimate since that is where the derivation lives.

Who is this for: people working on wavefront shaping, mesoscopic transport, or random-matrix approaches to transmission. The framework gives them a new tool with code. I would send it to peer review. A referee should push on the replica-decoupling step and on finite-N checks, but this is exactly the kind of work that merits referee time.

Referee Report

3 major / 4 minor

Summary. The manuscript presents a field-theoretic framework, 'radiant field theory' (RFT), for the distribution of transmission eigenvalues of coherent waves in a disordered waveguide. Starting from a microscopic wave equation with a white-noise random potential, the authors use a replica trick, a Hubbard-Stratonovich transformation, a saddle-point approximation, and a Wigner transform to derive a matrix transport equation for a matrix-valued radiance g(Ω,r) [Eq. (13)], with boundary conditions [Eq. (15)] and a generating function [Eq. (17)]. The theory is validated by comparing its predictions with numerical solutions of the wave equation in three settings: varying optical thickness L/ℓ, uniform absorption, and partial channel control. The authors also show that a direction-averaged limit of Eq. (13) reproduces the DMPK equation in the diffusive regime.

Significance. If the central equation (13) is valid, RFT is a substantial generalization of DMPK theory: it removes the isotropy hypothesis, extends predictions into the quasiballistic regime, and incorporates absorption and incomplete channel control in a parameter-free way. The paper is strengthened by concrete numerical evidence from the microscopic wave equation, by the exact diffusive-limit benchmark against DMPK, and by the release of two numerical codes. The main unresolved question is the rigor of the single-replica saddle-point step, which is asserted but not quantitatively controlled in the Letter.

major comments (3)

[Derivation, after Eq. (11)] The factorization ⟨Z^R⟩ ≃ (Z̄)^R, obtained by neglecting inter-replica coupling, is the step that reduces the 2R×2R saddle-point problem to the one-replica matrix transport equation (13). The Letter states that this is valid in the nonlocalized regime L ≪ ξ but gives no error estimate. Since Eq. (13) is the central object from which all predictions follow, this is a load-bearing gap. I request a quantitative justification (for instance, an estimate in powers of 1/(kℓ) or 1/N) or, failing that, an explicit numerical test of the factorization's accuracy.
[Figs. 1–3] All numerical validations use W/λ = 50.5 and L/W = 1, so only a single channel count N ≈ 100 is probed. Corrections to the replica-symmetric saddle point are generically of order 1/N, and the plotted agreement could in principle mask a finite-N sensitivity. A test at a substantially smaller width (e.g., W/λ ≈ 10) or at a different aspect ratio would materially strengthen the claim that Eq. (13) is accurate in the quasiballistic regime and is not tied to the specific large-N geometry used here.
[Eqs. (12)–(17) and companion paper [38]] The derivation of the Wigner-transport equation (13), the boundary conditions (15), and the generating function (17) is delegated to the companion paper. While this division is acceptable for a Letter, the text should state more precisely the approximations entering the on-shell Wigner reduction and their expected small parameters. In particular, the 'semiclassical' approximation is not accompanied by a statement of its error in the quasiballistic regime, so the reader cannot distinguish a controlled expansion from an uncontrolled one.

minor comments (4)

[Eq. (4) and surrounding text] The sentence 'with N a γ-independent arbitrary prefactor' is confusing because N is also used for the number of channels; please denote the prefactor by, e.g., calligraphic N or explain the notation explicitly.
[Eq. (12)] The 'dashed integral' notation is used without a definition in the main text; please define it as the Cauchy principal value consistently, or refer explicitly to the companion paper at the point of first use.
[Fig. 3 caption] There is a formatting inconsistency: 'W/λ = 50 .5' appears with a stray space; similar spacing issues occur in other figure captions (e.g., '10−1 100 101').
[Text after Eq. (19)] The claim that Eq. (19) 'exactly reduces to the DMPK solution' is stated without derivation or reference to a specific equation in the companion paper; a short derivation or precise pointer would make the benchmark check easier to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the transport equation is derived from a parameter-free microscopic model, validated against independent wave-equation numerics, and reduces to the external DMPK benchmark in the diffusive limit.

full rationale

The central output, Eq. (13), is not fitted and is not defined in terms of the predicted distribution. It follows from the Gaussian white-noise disorder model through a replica saddle point and a Wigner transform, with the detailed derivation delegated to the authors' companion paper [38]. That companion paper is self-citational but parameter-free, states its assumptions (nonlocalized regime, neglected inter-replica coupling), and does not take the target transmission-eigenvalue distribution as an input. The generating function F(γ) in Eq. (17) is evaluated from the boundary value of the same self-consistent g that solves Eq. (13), so no measured transmission eigenvalues enter the theory at any stage. The Letter also supplies external checks: Eq. (19) is shown to reduce exactly to the independent DMPK result of Ref. [5], and the figures compare against recursive Green's function simulations of the microscopic wave equation, not against the theory's own fitted values. The statement after Eq. (11) that the saddle-point approximation is well justified in the nonlocalized regime is made without a proof or error estimate in this Letter; I flag that as an omitted-support or correctness-risk point, but it is not a circular reduction because it does not presuppose the target result. No fitted parameter is renamed as a prediction, and no result is imported from a self-citation chain that itself contains the claimed conclusion.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The central equation (13) rests on the standard replica and Hubbard-Stratonovich machinery of disordered systems, plus two nontrivial approximations that are not fully proven in this Letter: the neglect of replica coupling in the nonlocalized regime and the semiclassical Wigner transform. These are not fitted parameters; they are structural assumptions. No free parameters are fitted to data in the theory.

assumptions (5)

domain assumption Gaussian white-noise disorder with ⟨U(r)U(r')⟩ = αδ(r-r')
Stated around Eq. (2); it defines the microscopic model and sets ell = k/(πνα). This is the standard model for short-range disorder.
ad hoc to paper Replica trick with R→0 and neglect of replica coupling in the nonlocalized regime
Stated in the paragraph after Eq. (11): 'coupling between replicas can be neglected in this regime. This allows us to approximate ⟨Z^R⟩ ≃ Z̄^R'. This is load-bearing; the Letter defers justification to the companion paper.
domain assumption Saddle-point approximation for the auxiliary matrix field Q(r)
Used after Eq. (10); the Letter says it is well justified for L << xi. Standard in sigma-model derivations.
domain assumption Semiclassical (Wigner) approximation: Goldstone modes vary slowly compared to the wavelength
Invoked before Eq. (12) to replace the operator equation (11) by the transport equation (13).
standard math Fisher-Lee relation connecting transmission matrix to the Green's function
Eq. (1); standard result used to express eigenvalues of t†t in terms of G+.

invented entities (1)

Matrix radiance g(Ω,r)
purpose: A 2x2 matrix-valued generalization of radiance; its boundary values encode the generating function F(γ) and hence the transmission eigenvalue distribution ρ(T).
Mathematical construct introduced in Eq. (12); it does not correspond to a directly measurable quantity but all physical predictions (ρ(T)) are derived from it. Its validity is supported by numerical comparisons but it has no falsifiable handle separate from the paper's claims.

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Cite this review

Pith. "Pith review of Radiant Field Theory: A Transport Approach to Shaped Wave Transmission through Disordered Media." pith.science (2026). https://pith.science/paper/6PUKVVNC

@misc{pith2026241110360,
  author       = {Pith},
  title        = {Pith review of: Radiant Field Theory: A Transport Approach to Shaped Wave Transmission through Disordered Media},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6PUKVVNC}},
  note         = {Machine review of arXiv:2411.10360}
}

read the original abstract

We present a field-theoretic framework to characterize the distribution of transmission eigenvalues for coherent wave propagation through disordered media. The central outcome is a transport equation for a matrix-valued radiance, analogous to the classical radiative transport equation but capable of capturing coherent effects encoded in the transmission matrix. Unlike the Dorokhov-Mello-Pereyra-Kumar (DMPK) theory, our approach does not rely on the isotropy hypothesis, which presumes uniform angular scattering by material slices. As a result, it remains valid beyond the diffusive regime, accurately describing the transmission eigenvalue distribution in the quasiballistic regime as well. Moreover, the framework is more versatile than the DMPK theory, enabling straightforward incorporation of experimental realities such as absorption and incomplete channel control. These factors are frequently encountered in wave experiments on complex media but have lacked an ab initio theoretical treatment until now. We validate our predictions through numerical simulations based on the microscopic wave equation, confirming the accuracy and broad applicability of the theory.

Figures

Figures reproduced from arXiv: 2411.10360 by the authors.

**Figure 2.** FIG. 2. Transmission eigenvalue distribution for an absorbing [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Transmission eigenvalue distribution for a disor [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Transmission eigenvalue distribution in disordered media from radiant field theory
math-ph 2024-11 conditional novelty 8.0 of 10

A matrix transport equation derived from replicated field theory yields the full transmission eigenvalue distribution in disordered media, from quasiballistic to diffusive regimes, for waveguides and infinite slabs.

Reference graph

Works this paper leans on

51 extracted references · 46 canonical work pages · cited by 1 Pith paper

[1]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Con- trolling waves in space and time for imaging and focusing in complex media , Nat. Photon. 6, 283 (2012)

work page 2012
[2]

Rotter and S

S. Rotter and S. Gigan, Light fields in complex media: Mesoscopic scattering meets wave control , Rev. Mod. Phys. 89, 015005 (2017)

work page 2017
[3]

H. Cao, A. P. Mosk, and S. Rotter, Shaping the propaga- tion of light in complex media , Nat. Phys. 18, 994 (2022)

work page 2022
[4]

O. N. Dorokhov, On the coexistence of localized and ex- tended electronic states in the metallic phase , Solid State Commun. 51, 381 (1984)

work page 1984
[5]

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

1997
[6]

Wegner, The mobility edge problem: Continuous sym- metry and a conjecture , Z

F. Wegner, The mobility edge problem: Continuous sym- metry and a conjecture , Z. Phys. B 35, 207 (1979)

work page 1979
[7]

Sch¨ afer and F

L. Sch¨ afer and F. Wegner,Disordered System with n Or- bitals per Site: Lagrange Formulation, Hyperbolic Sym- metry, and Goldstone Modes , Z. Phys. B 38, 113 (1980)

work page 1980
[8]

K. B. Efetov, Supersymmetry method in localization the- ory, Zh. Eksp. Teor. Fiz. 82, 872 (1982), [Sov. Phys. JETP 55, 514 (1982)]

work page 1982

Show all 51 references

[9]

K. B. Efetov, Supersymmetry and theory of disordered metals, Adv. Phys. 32, 53 (1983). 6

1983
[10]

K. B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997)

1997
[11]

Enrico Fermi

I. V. Lerner, Nonlinear Sigma Model for Normal and Su- perconducting Systems: A Pedestrian Approach, in Quan- tum Phenomena in Mesoscopic Systems , Proceedings of the International School of Physics “Enrico Fermi”, Vol. 151, edited by B. Altshuler, A. Tagliacozzo, and V. Togne...

2003 arXiv
[12]

Kamenev, Field Theory of Non-Equilibrium Systems , 2nd ed

A. Kamenev, Field Theory of Non-Equilibrium Systems , 2nd ed. (Cambridge University Press, Cambridge, 2023)

2023
[13]

O. N. Dorokhov, Transmission coefficient and the lo- calization length of an electron in N bound disordered chains, Pis’ma Zh. Eksp. Teor. Fiz. 36, 259 (1982), [JETP Lett. 36, 318 (1982)]

1982
[14]

P. A. Mello, P. Pereyra, and N. Kumar, Macroscopic ap- proach to multichannel disordered conductors, Ann. Phys. (NY) 181, 290 (1988)

1988
[15]

P. A. Mello and S. Tomsovic, Scattering approach to quantum transport: Toward a consistent physical picture , Phys. Rev. Lett. 67, 342 (1991)

1991
[16]

P. A. Mello and S. Tomsovic, Scattering approach to quantum electronic transport , Phys. Rev. B 46, 15963 (1992)

1992
[17]

S. F. Edwards and P. W. Anderson, Theory of spin glasses, J. Phys. F: Met. Phys. 5, 965 (1975)

1975
[18]

D. J. Thouless, Phase transitions for the one-dimensional Landau-Ginzburg system and their relation to the theory of polymers and of electrons in a random potential , J. Phys. C 8, 1803 (1975)

1975
[19]

Nitzan, K

A. Nitzan, K. F. Freed, and M. H. Cohen, Renormal- ization group and critical localization , Phys. Rev. B 15, 4476 (1977)

1977
[20]

Aharony and Y

A. Aharony and Y. Imry, The mobility edge as a spin- glass problem, J. Phys. C 10, L487 (1977)

1977
[21]

J. J. M. Verbaarschot, H. A. Weidenm¨ uller, and M. R. Zirnbauer, Grassmann integration in stochastic quantum physics: The case of compound-nucleus scattering , Phys. Rep. 129, 367 (1985)

1985
[22]

S. Iida, H. A. Weidenm¨ uller, and J. A. Zuk, Statistical scattering theory, the supersymmetry method and univer- sal conductance fluctuations , Ann. Phys. (NY) 200, 219 (1990)

1990
[23]

A. D. Mirlin, Statistics of energy levels and eigenfunc- tions in disordered systems , Phys. Rep. 326, 259 (2000)

2000
[24]

Altland, B

A. Altland, B. D. Simons, and M. R. Zirnbauer, Theories of low-energy quasi-particle states in disordered d-wave superconductors, Phys. Rep. 359, 283 (2002)

2002
[25]

Y. V. Nazarov, Limits of universality in disordered con- ductors, Phys. Rev. Lett. 73, 134 (1994)

1994
[26]

Y. V. Nazarov and Y. M. Blanter, Quantum Trans- port: Introduction to Nanoscience (Cambridge Univer- sity Press, Cambridge, 2009)

2009
[27]

L. P. Gor’kov, Microscopic derivation of the Ginzburg- Landau equations in the theory of superconductivity , Zh. Eksp. Teor. Fiz. 36, 1918 (1959), [Sov. Phys. JETP 9, 1364 (1959)]

1959
[28]

Eilenberger, Transformation of Gorkov’s equation for type II superconductors into transport-like equations , Z

G. Eilenberger, Transformation of Gorkov’s equation for type II superconductors into transport-like equations , Z. Phys. A 214, 195 (1968)

1968
[29]

A. I. Larkin and Y. N. Ovchinnikov, Quasiclassical method in the theory of superconductivity, Zh. Eksp. Teor. Fiz. 55, 2262 (1968), [Sov. Phys. JETP 28, 1200 (1969)]

1968
[30]

K. D. Usadel, Generalized Diffusion Equation for Super- conducting Alloys, Phys. Rev. Lett. 25, 507 (1970)

1970
[31]

Frahm, Equivalence of the Fokker-Planck Approach and the Nonlinear σ Model for Disordered Wires in the Unitary Symmetry Class , Phys

K. Frahm, Equivalence of the Fokker-Planck Approach and the Nonlinear σ Model for Disordered Wires in the Unitary Symmetry Class , Phys. Rev. Lett. 74, 4706 (1995)

1995
[32]

Rejaei, Equivalence of the transmission-eigenvalue density in supersymmetric and scaling theories of disor- dered wires without time-reversal symmetry , Phys

B. Rejaei, Equivalence of the transmission-eigenvalue density in supersymmetric and scaling theories of disor- dered wires without time-reversal symmetry , Phys. Rev. B 53, R13235 (1996)

1996
[33]

P. W. Brouwer and K. Frahm, Quantum transport in disordered wires: Equivalence of the one-dimensional σ model and the Dorokhov-Mello-Pereyra-Kumar equation , Phys. Rev. B 53, 1490 (1996)

1996
[34]

P. W. Brouwer, Transmission through a many-channel random waveguide with absorption , Phys. Rev. B 57, 10526 (1998)

1998
[35]

Goetschy and A

A. Goetschy and A. D. Stone, Filtering Random Matri- ces: The Effect of Incomplete Channel Control in Multi- ple Scattering, Phys. Rev. Lett. 111, 063901 (2013)

2013
[36]

S. M. Popoff, A. Goetschy, S. F. Liew, et al. , Coherent Control of Total Transmission of Light through Disor- dered Media, Phys. Rev. Lett. 112, 133903 (2014)

2014
[37]

C. W. Hsu, S. F. Liew, A. Goetschy, et al. , Correlation- enhanced control of wave focusing in disordered media , Nat. Phys. 13, 497 (2017)

2017
[38]

Gaspard and A

D. Gaspard and A. Goetschy, Transmission eigenvalue distribution in disordered media from radiant field theory, arXiv:2411.10355 [math-ph] (2024), accepted in Phys. Rev. Res. (2025)

2024 arXiv
[39]

D. S. Fisher and P. A. Lee, Relation between conductivity and transmission matrix , Phys. Rev. B 23, 6851 (1981)

1981
[40]

Kopnin, Theory of Nonequilibrium Superconductivity, International Series of Monographs on Physics No

N. Kopnin, Theory of Nonequilibrium Superconductivity, International Series of Monographs on Physics No. 110 (Oxford University Press, 2001)

2001
[41]

D. Gaspard, Ebsolve - Solving the matrix transport equa- tion for the transmission eigenvalue distribution in a dis- ordered waveguide, Zenodo (2025), https://doi.org/ 10.5281/zenodo.15090497

2025 doi
[42]

Gaspard, RecurGreen - Recursive Green method to solve the wave equation in a disordered waveguide, Zenodo (2025), https://doi.org/10.5281/zenodo

D. Gaspard, RecurGreen - Recursive Green method to solve the wave equation in a disordered waveguide, Zenodo (2025), https://doi.org/10.5281/zenodo. 15083424

2025 doi
[43]

MacKinnon and B

A. MacKinnon and B. Kramer, One-Parameter Scaling of Localization Length and Conductance in Disordered Sys- tems, Phys. Rev. Lett. 47, 1546 (1981)

1981
[44]

H. U. Baranger, D. P. DiVincenzo, R. A. Jalabert, and A. D. Stone, Classical and quantum ballistic-transport anomalies in microjunctions , Phys. Rev. B 44, 10637 (1991)

1991
[45]

Yamilov, S

A. Yamilov, S. Petrenko, R. Sarma, and H. Cao, Shape dependence of transmission, reflection, and absorption eigenvalue densities in disordered waveguides with dissi- pation, Phys. Rev. B 93, 100201 (2016)

2016
[46]

Rescanieres, R

R. Rescanieres, R. Pierrat, and A. Goetschy, Open and trapping channels in complex resonant media , arXiv:2411.19818 [physics.optics] (2024), preprint arXiv v1

2024 arXiv
[47]

H. Yu, T. R. Hillman, W. Choi, et al. , Measuring Large Optical Transmission Matrices of Disordered Me- dia, Phys. Rev. Lett. 111, 153902 (2013)

2013
[48]

Bender, A

N. Bender, A. Goetschy, C. W. Hsu, et al., Coherent en- hancement of optical remission in diffusive media , Proc. Natl. Acad. Sci. U.S.A. 119, e2207089119 (2022)

2022
[49]

Bender, A

N. Bender, A. Yamilov, A. Goetschy, et al. , Depth- 7 targeted energy delivery deep inside scattering media , Nat. Phys. 18, 309 (2022)

2022
[50]

McIntosh, A

R. McIntosh, A. Goetschy, N. Bender, et al. , Delivering broadband light deep inside diffusive media , Nat. Photon. 18, 744 (2024)

2024
[51]

Vynck, R

K. Vynck, R. Pierrat, R. Carminati, et al. , Light in cor- related disordered media , Rev. Mod. Phys. 95, 045003 (2023)

2023

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Con- trolling waves in space and time for imaging and focusing in complex media , Nat. Photon. 6, 283 (2012)

work page 2012

[2] [2]

Rotter and S

S. Rotter and S. Gigan, Light fields in complex media: Mesoscopic scattering meets wave control , Rev. Mod. Phys. 89, 015005 (2017)

work page 2017

[3] [3]

H. Cao, A. P. Mosk, and S. Rotter, Shaping the propaga- tion of light in complex media , Nat. Phys. 18, 994 (2022)

work page 2022

[4] [4]

O. N. Dorokhov, On the coexistence of localized and ex- tended electronic states in the metallic phase , Solid State Commun. 51, 381 (1984)

work page 1984

[5] [5]

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

1997

[6] [6]

Wegner, The mobility edge problem: Continuous sym- metry and a conjecture , Z

F. Wegner, The mobility edge problem: Continuous sym- metry and a conjecture , Z. Phys. B 35, 207 (1979)

work page 1979

[7] [7]

Sch¨ afer and F

L. Sch¨ afer and F. Wegner,Disordered System with n Or- bitals per Site: Lagrange Formulation, Hyperbolic Sym- metry, and Goldstone Modes , Z. Phys. B 38, 113 (1980)

work page 1980

[8] [8]

K. B. Efetov, Supersymmetry method in localization the- ory, Zh. Eksp. Teor. Fiz. 82, 872 (1982), [Sov. Phys. JETP 55, 514 (1982)]

work page 1982