Pith. sign in

REVIEW 2 major objections 4 minor 66 references

Stochastic Dynamics of Incoherent Branched Flow

T0 review · 2 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Closed-form equations for the scintillation index of coherent and incoherent branched flow show that, for speckled initial fields, the peak intensity fluctuation depends only on $X_o=\sigma^{2/3}\rho_o/\alpha^{1/3}$.

desk verdict New closed-form statistics for incoherent branched flow, backed by no-parameter numerics; the main weakness is an asserted diffusion limit with no convergence check or error bars, but the paper deserves serious refereeing. read the letter →

arxiv 2502.07028 v2 pith:TUTWLFS3 submitted 2025-02-10 math.AP physics.flu-dyn

Josselin Garnier , Antonio Picozzi , Theo Torres This is my paper · ORCID
classification math.APphysics.flu-dyn MSC 35Q4135R6060F1778A45 PACS 42.25.Dd05.40.-a
keywords branchedflowscintillationindexparaxialwaveequationWignertransformdiffusionapproximationpartialcoherenceintensitycorrelationfunctionrandommedia
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

Branched flow, the focusing of waves into branching filaments by weak smooth disorder, is usually studied with fully coherent waves, but recent experiments use incoherent light. This paper develops a stochastic theory of the paraxial wave equation that covers coherent plane waves, coherent speckle fields, and partially coherent speckle fields. Its central result is a closed-form evolution equation for the intensity correlation function and the scintillation index, from which the value and location of the maximum intensity fluctuation can be read off. The theory shows that for speckled initial fields the maximal scintillation index depends only on the dimensionless parameter $X_o=\sigma^{2/3}\rho_o/\alpha^{1/3}$ and not on the medium correlation length. Direct numerical simulations match the predictions without adjustable parameters.

What carries the argument

The argument runs through the Wigner transform, the phase-space representation of the field's two-point correlation, which obeys a scaled Vlasov equation in the regime where the initial correlation radius sits between the wavelength and the medium correlation length. The characteristics of this kinetic equation are the ray equations (14), and a diffusion-approximation theorem (App. D) shows that, in the $\varepsilon\to0$ limit, the joint ray process converges to a Markov diffusion with generator $L^{(n)}=\sum_{j=1}^n 2\alpha K_j\partial_{X_j}+\frac12\sum_{j,j'=1}^n\Gamma(X_j-X_{j'})\partial_{K_j}\partial_{K_{j'}}$, with $\Gamma(x)=\int_{-\infty}^{\infty}\mathbb{E}[\partial_x V(0,0)\partial_x V(z,x)]\,dz$. Combining this generator with the Gaussian fourth-moment factorization of the initial speckle field reduces the fourth-order intensity statistics to the closed equation (17) for $\tilde\Pi$. The dimensionless parameter $X_o$ appears only through the initial condition and controls the peak scintillation.

What would settle it

Fix $\sigma$, $\rho_o$, and $\alpha$, and vary the medium correlation length $\ell_c$: the theory predicts that the maximum of the scintillation index is unchanged, so a statistically significant change in the peak value would falsify the $X_o$-only dependence. A second check is the predicted relation $S_z^{(c)}=2S_z^{(pc)}+1$ between coherent-speckle and partially coherent settings with identical parameters.

Watch

Extended reading notes

Core claim

For an initial speckle field with Gaussian statistics and correlation radius $\rho_o$, the scintillation index is $S_z^{(pc)}=\tilde\Pi_{z/z_c}(0,0)-1$ in the partially coherent case and $S_z^{(c)}=2\tilde\Pi_{z/z_c}(0,0)-1$ in the coherent-speckle case, where $\tilde\Pi$ solves the closed PDE (17); the only memory of the initial field enters through the dimensionless parameter $X_o=\sigma^{2/3}\rho_o/\alpha^{1/3}$. From this the authors derive that the maximum of the scintillation index depends only on $X_o$ and increases with it, while the distance at which the maximum occurs scales as $z_c=\ell_c/(2\sigma^{2/3}\alpha^{2/3})$. They also obtain the two-scale intensity correlation function $C_z^{I,(c)}(x)=\tilde\Pi_{z/z_c}(x/\ell_c,0)+\tilde\Pi_{z/z_c}(x/\ell_c,X_o x/\rho_o)-1$ for coherent speckle and $C_z^{I,(pc)}(x)=\tilde\Pi_{z/z_c}(x/\ell_c,0)-1$ for partially coherent fields, and verify all predictions against parameter-free simulations.

Load-bearing premise

The whole closed-form theory rests on the assumption that, in the limit where the wavelength is much smaller than the medium correlation length, the random ray motion converges to a Markov diffusion; if that convergence fails or is too slow in the parameter regime of interest, the predicted scintillation indices and their $X_o$-only dependence would not hold.

Editorial extensions

If this is right

  • For speckled initial fields, the maximal intensity fluctuation is set by the initial correlation radius $\rho_o$ and the medium strength $\sigma$, but not by the medium correlation length $\ell_c$; two media with the same $\sigma$ and $\rho_o$ produce the same peak scintillation even if their disorder is smoother or rougher.
  • A coherent-speckle source and a partially coherent source with the same parameters are linked by $S_z^{(c)}=2S_z^{(pc)}+1$, so a diffuser switched between static and rotating modes gives a direct experimental test of coherence effects.
  • In the coherent-speckle case the intensity correlation function has a two-scale structure: rapid decorrelation on the initial speckle scale $\rho_o$ and slow, medium-induced variations on the scale $\ell_c$; a partially coherent source shows only the slow scale and satisfies the energy-conservation identity $\int C_z^{(pc)}(x)\,dx=0$.
  • Early propagation is universal: the partially coherent scintillation index grows as $(\tilde\gamma_4/6)(z/z_c)^3$, independent of $\rho_o$ and of the initial correlation function, matching the plane-wave early growth.
  • For large propagation distances the field statistics relax to Gaussian, and the scintillation index tends to 1, so branched-flow intensity enhancements are a finite-distance phenomenon.

Reading between the lines

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

  • The same closed-form structure should apply to any paraxial system, so ocean-wave statistics over random currents could be predicted from measured effective parameters $\Gamma$ and $X_o$ without resolving individual caustics.
  • For non-Gaussian initial fields, the Gaussian fourth-moment factorization fails; the first visible signature should be a distortion of the small-scale term $\tilde\Pi(x/\ell_c,X_o x/\rho_o)$ in the coherent-speckle correlation function, making that term a measurable probe of non-Gaussian speckle statistics.
  • Feeding the predicted intensity correlation function into a nonlinear Schr\"odinger simulation would provide a quantitative test of the authors' suggestion that linear branched-flow focusing seeds extreme nonlinear events such as freak waves.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper develops a stochastic theory of branched flow for waves governed by the two-dimensional paraxial wave equation, focusing on incoherent (speckled) initial fields. It derives approximate PDEs for the intensity correlation function and the scintillation index in two regimes: a coherent plane-wave initial condition and a partially coherent or coherent speckled initial field. The central results are the expressions S^{(pc)}_z = Pi(0,0)-1 and S^{(c)}_z = 2Pi(0,0)-1, where Pi solves the closed PDE (17), and the prediction that the maximal scintillation index depends only on the dimensionless parameter X_o = sigma^(2/3) rho_o / alpha^(1/3), not on the medium correlation length. The theoretical predictions are compared with direct numerical simulations of the paraxial equation and are reported to agree quantitatively without adjustable parameters.

Significance. If the central asymptotic results are rigorous, the paper makes a substantial contribution to the theory of branched flow: it provides a tractable stochastic description of incoherent branched flow, identifies a striking non-trivial dependence of the scintillation-index maximum on a single combination of parameters, and extends the authors' established fourth-moment program to partially coherent sources. The absence of fitted parameters in the comparisons with simulations is a clear strength, as is the consistency of the simulations across variations of rho_o, sigma^2, and l_c. The computational validation is, however, confined to a single modest scale separation, and the main technical step is the asserted diffusion-approximation limit in the supplement, so the significance and the strength of the validation both depend on that limit being correct.

major comments (2)
  1. [Supplementary App. D] The convergence of the ray process (X^epsilon_z, K^epsilon_z) solving Eq. (14) to the Markov diffusion with generator (D1) is asserted by reference to standard diffusion-approximation theory, but no theorem statement, hypothesis check, or error estimate is provided for this specific scaled system. This limit is load-bearing: all closed-form results in the main text, including Eqs. (16)-(19) and the X_o-only dependence of the maximal scintillation index, follow from it. The reader cannot verify that the smooth, stationary, mixing assumptions on V are sufficient for the joint convergence needed in the n=2 application, nor can the size of the finite-epsilon error be assessed. Please state the precise convergence theorem used, verify its hypotheses for Eq. (14), and either give a proof of the generator (D1) or cite a reference where this exact result is established; an estimate of the convergence rate would also address the validation concern raised below.
  2. [Main text, Figs. 3-4 and App. H] The numerical validation of the limit formulas is performed at a single scale-separation value, epsilon = lambda/l_c = 0.01 (l_c/lambda = 100), with rho_o/l_c = 0.1, and no error bars are reported. The paper's own estimate in App. B shows an O(epsilon^{(5d-1)/2}) remainder in the Vlasov equation, which is about 0.03 for d = 0.5, and the diffusion-approximation error is uncontrolled. Consequently, the reported 'excellent quantitative agreement' supports the theory at one modest separation of scales, but it does not by itself establish that the epsilon -> 0 limit is the operative mechanism. Please add a convergence study in epsilon (for example, l_c/lambda = 50, 100, 200, with other parameters adjusted to keep X_o fixed) and report the deviation of the simulation curves from the theory as a function of epsilon, or provide a quantitative bound on the expected finite-epsilon corrections.
minor comments (4)
  1. [Main text, Fig. 3 caption] The caption of Fig. 3(b) lists only rho_o/lambda and sigma^2 lambda^2 and does not state that l_c is the parameter being varied; please specify the varying parameter and its values explicitly.
  2. [Main text, Eq. (16)] The notation Pi_z(x,y) is used in Eq. (16) before the PDE (17) is introduced, and the reader must infer the scaling; please define Pi_z and its arguments (or the dimensionless variables) explicitly in the text preceding Eq. (16).
  3. [Supplementary App. D, Eq. (D1)] Please add a consistency check for the normalization of Gamma(x) in the generator (D1) and the Fokker-Planck equation (D7), since the factors of 1/2 and 2 in (D1), (D8), and (E8) are easy to get wrong and a reader would benefit from an explicit verification that the final PDE (17) follows with the stated coefficients.
  4. [Abstract] The phrase 'closed-form equations' may be misread as explicit analytic solutions; the main results are a closed coupled PDE system plus explicit small-propagation-distance expansions. Consider rephrasing to 'a closed system of equations' for precision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the incoherent branched-flow formulas are obtained from a genuine asymptotic limit and are checked by parameter-free simulations.

full rationale

The paper's central results are not equivalent to their inputs. Starting from the paraxial wave equation (1), the scaling (10), and the Wigner transform (12), the authors derive the Vlasov equation (13) in App. B, the exact ray representation (15) in App. C, and then apply a diffusion-approximation theorem in App. D to obtain the generator (D1). Solving the resulting Fokker-Planck equations and using Isserlis' theorem for the Gaussian initial speckle field yields the closed PDE (17), whose solution gives Eqs. (16), (18), and (19). At no point is a target quantity (e.g., the scintillation index or its maximum) inserted as an input; the parameter Xo enters as a scaling variable in the initial condition, and the claim that the maximal scintillation index depends only on Xo follows from the structure of the PDE, not by definition. The simulations solve the original equation (1) with independently generated realizations of V and of the initial speckle, with no adjustable parameters, so the agreement is not a fit renamed as a prediction. The self-citations used in the paper are to prior peer-reviewed results, namely the white-noise paraxial theory [50] and the diffusion-approximation theorem [S7, Chapter 6]; these theorems have general assumptions that do not include the present target formulas, and they are therefore independent support rather than load-bearing circularity. The main weakness is that the finite-epsilon convergence is imported from the cited theory and the Vlasov remainder is dropped without a quantitative error estimate; this is an asymptotic-validity concern, not a circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central formulas (16)-(19) are derived without any free parameters. The derivation rests on standard diffusion-approximation theorems for random ODEs (Fouque et al., 2007) and on modeling assumptions: the initial field is Gaussian with correlation radius between wavelength and medium correlation length, and the medium potential is smooth, stationary, mean-zero with integrable covariance. No invented entities are introduced.

assumptions (3)
  • standard math Diffusion approximation for random ODEs with rapidly varying coefficients (Fouque et al., 2007, Ch. 6)
    Invoked in App. D to obtain the Markov diffusion generator (D1) for the ray process; this is the foundation of the Fokker-Planck equations used to derive Eqs. (16)-(19).
  • domain assumption Initial field has Gaussian statistics with correlation radius between wavelength and medium correlation length
    Used in App. E with Isserlis' theorem to express fourth-order intensity moments as products of second-order moments; this yields the factor 2 between S(c) and S(pc). The experimental setups described (static or rotating diffusers) are argued to produce such fields.
  • domain assumption Scaling regime d in (1/5,1), b=1-d, c=3(1-d)/2 with eps small
    Defines the white-noise scaling limit (Eq. 10) in which the Vlasov equation (13) is derived; the simulations use parameters that place them in this regime, though with limited scale separation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Stochastic Dynamics of Incoherent Branched Flow." pith.science (2026). https://pith.science/paper/TUTWLFS3

@misc{pith2026250207028,
  author       = {Pith},
  title        = {Pith review of: Stochastic Dynamics of Incoherent Branched Flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TUTWLFS3}},
  note         = {Machine review of arXiv:2502.07028}
}
read the original abstract

Waves propagating through weakly disordered smooth linear media undergo a universal phenomenon called branched flow. Branched flow has been observed and studied experimentally in various systems by considering coherent waves. Recent experiments have reported the observation of optical branched flow by using an incoherent light source, thus revealing the key role of coherent phase-sensitive effects in the development of incoherent branched flow. By considering the paraxial wave equation as a generic representative model, we elaborate a stochastic theory of both coherent and incoherent branched flow. We derive closed-form equations that determine the evolution of the intensity correlation function, as well as the value and the propagation distance of the maximum of the scintillation index, which characterize the dynamical formation of incoherent branched flow. We report accurate numerical simulations that are found in quantitative agreement with the theory without free parameters. Our theory highlights the important impact of coherence and interference on branched flow, thereby providing a framework for exploring branched flow in nonlinear media, in relation with the formation of freak waves in oceans.

Figures

Figures reproduced from arXiv: 2502.07028 by the authors.

Figure 2
Figure 2. FIG. 2: Coherent initial plane wave with a medium with Gaussian [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. FIG. 4: Incoherent initial wave with a medium with Gaus [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. FIG. 5: Coherent initial plane wave. Example of a congruence [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figures from the paper (1 more)
Figure 6
Figure 6. Figure 6: FIG. 6: Incoherent initial wave. (a) Evolution of the BF obtained [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages

  1. [1]

    We will first consider the coherent case in which the initial field is a plane wave: ψo(x) = 1. The measured intensity is |ψz(x)|2, the mean intensity is E |ψz(x)|2 , and the scintil- lation index (i.e., the relative variance of the intensity) is Sz(x) = E |ψz(x)|4 − E |ψz(x)|2 2 E [|ψz(x)|2]2 . (2)

    work page

  2. [2]

    We will then consider in detail the situation in which the initial field is a coherent or partially coherent speckled field. We will consider the two following situations: (c) ψo is a coherent speckled field, which will be modeled as a stationary random field with Gaussian statistics and corre- lation radius ρo (the width of the field correlation function...

    work page

  3. [3]

    Topinka, B.J

    M.A. Topinka, B.J. LeRoy, R.M. Westervelt, S.E.J. Shaw, R. Fleischmann, E.J. Heller, K.D. Maranowski, and A.C. Gos- sard, Coherent branched flow in a two-dimensional electron gas, Nature 410, 183–186 (2001)

    work page 2001

  4. [4]

    Aidala, et al

    K.E. Aidala, et al. Imaging magnetic focusing of coherent elec- tron waves, Nat. Phys. 3, 464–468 (2007)

    work page 2007

  5. [5]

    Jura, M.A

    M.P. Jura, M.A. Topinka, L. Urban, A. Yazdani, H. Shtrik- man, L.N. Pfeiffer, K.W. West, and D. Goldhaber-Gordon, Unexpected features of branched flow through high-mobility two-dimensional electron gases, Nat. Phys. 3, 841–845 (2007)

    work page 2007

  6. [6]

    Maryenko, et al., How branching can change the conduc- tance of ballistic semiconductor devices, Phys

    D. Maryenko, et al., How branching can change the conduc- tance of ballistic semiconductor devices, Phys. Rev. B 85, 195329 (2012)

    work page 2012

  7. [7]

    Liu and E.J

    B. Liu and E.J. Heller, Stability of branched flow from a quan- tum point contact, Phys. Rev. Lett. 111, 236804 (2013)

    work page 2013

  8. [8]

    H¨ ohmann, U

    R. H¨ ohmann, U. Kuhl, H.-J. St¨ ockmann, L. Kaplan, and E.J. Heller, Freak waves in the linear regime: a microwave study. Phys. Rev. Lett. 104, 093901 (2010)

    work page 2010

Show all 66 references
  1. [9]

    Barkhofen, J.J

    S. Barkhofen, J.J. Metzger, R. Fleischmann, U. Kuhl, and H.-J. St¨ ockmann, Experimental observation of a fundamental length scale of waves in random media, Phys. Rev. Lett. 111, 183902 (2013)

    2013

  2. [10]

    Heller, R

    E.J. Heller, R. Fleischmann, and T. Kramer, Branched flow, Physics Today 74(12), 44–51 (2021)

    2021

  3. [11]

    Green and R

    G. Green and R. Fleischmann, Branched flow and caustics in nonlinear waves, New J. Phys. 21, 083020 (2019)

    2019

  4. [12]

    Sun and X

    Z.-Y. Sun and X. Yu, Nonlinear Schr¨ odinger waves in a disor- dered potential: Branched flow, spectrum diffusion, and rogue waves, Chaos 32, 023108 (2022)

    2022

  5. [13]

    Jiang, T.W

    K. Jiang, T.W. Huang, R. Li, M.Y. Yu, H.B. Zhuo, S.Z. Wu, C.T. Zhou, and S.C. Ruan, Branching of high-current relativistic electron beam in porous materials, Phys. Rev. Lett. 130, 185001 (2023)

    2023

  6. [14]

    Mattheakis and G.P

    M. Mattheakis and G.P. Tsironis, Quodons in mica. In Ex- treme Waves and Branched Flows in Optical Media 425–454 (Springer International Publishing, 2015)

    2015

  7. [15]

    Mattheakis, I.J

    M. Mattheakis, I.J. Pitsios, G.P. Tsironis, and S. Tzortza- kis, Extreme events in complex linear and nonlinear photonic media. Chaos Solitons Fractals 84, 73–80 (2016)

    2016

  8. [16]

    White and B

    B.S. White and B. Fornberg, On the chance of freak waves at sea, J. Fluid Mech. 355, 113–138 (1998)

    1998

  9. [17]

    Berry, Tsunami asymptotics, New J

    M.V. Berry, Tsunami asymptotics, New J. Phys. 7, 129 (2005)

    2005

  10. [18]

    Berry, Focused tsunami waves, Proc

    M.V. Berry, Focused tsunami waves, Proc. R. Soc. A 463, 3055–3071 (2007)

    2007

  11. [19]

    Heller, L

    E.J. Heller, L. Kaplan, and A. Dahlen, Refraction of a Gaus- sian Seaway, J. Geophys. Res. 113, C09023 (2008)

    2008

  12. [20]

    L.H. Ying, Z. Zhuang, E.J. Heller, and L. Kaplan, Linear and nonlinear rogue wave statistics in the presence of random currents. Nonlinearity 24, R67 (2011)

    2011

  13. [21]

    Degueldre, J.J

    H. Degueldre, J.J. Metzger, T. Geisel, and R. Fleischmann, Random focusing of tsunami waves. Nat. Phys. 12, 259–262 (2016)

    2016

  14. [22]

    Wolfson and S

    M.A. Wolfson and S. Tomsovic, On the stability of long-range 6 sound propagation through a structured ocean. J. Acoust. Soc. Am. 109, 2693–2703 (2001)

    2001

  15. [23]

    Mattheakis, G.P

    M. Mattheakis, G.P. Tsironis, and E. Kaxiras, Emergence and dynamical properties of stochastic branching in the elec- tronic flows of disordered Dirac solids. EPL122, 27003 (2018)

    2018

  16. [24]

    K. Jose, N. Ferguson, and A. Bhaskar, Branched flows of flexural waves in non-uniform elastic plates Comm. Phys. 5, 152 (2023)

    2023

  17. [25]

    Brandst¨ otter, A

    A. Brandst¨ otter, A. Girschik, P. Ambichl, and S. Rotter, Shaping the branched flow of light through disordered media, Proc. Natl. Acad. Sci. USA 116(27), 13260–13265 (2019)

    2019

  18. [26]

    St’avina and P

    J. St’avina and P. Bokes, Quantum and classical branching flow in space and time, Phys. Rev. A 106, 052215 (2022)

    2022

  19. [27]

    Daza, E.J

    A. Daza, E.J. Heller, A.M. Graf, and E. R¨ as¨ anen, Propaga- tion of waves in high Brillouin zones: Chaotic branched flow and stable superwires, Proc. Natl. Acad. Sci. USA 118(40), e2110285118 (2021)

    2021

  20. [28]

    Wagemakers, A

    A. Wagemakers, A. Hartikainen, A. Daza, E. R¨ as¨ anen, and M.A.F. Sanju´ an, Chaotic dynamics creates and destroys branched flow, Phys. Rev. E 111, 014214 (2025)

    2025

  21. [29]

    Mok and R

    K.H. Mok and R. Fleischmann, Branched flows in active ran- dom walks and the formation of ant trail patterns, Phys. Rev. Res. 5, 043299 (2023)

    2023

  22. [30]

    Patsyk, U

    A. Patsyk, U. Sivan, M. Segev, and M.A. Bandres, Observa- tion of branched flow of light, Nature 583, 60–65 (2020)

    2020

  23. [31]

    Chang, K.-H

    S. Chang, K.-H. Wu, S. Liu, Z.-K. Lin, J. Wu, S. Ge, L.-J. Chen, P. Chen, W. Hu, Y. Xu, H. Chen, D. He, D.-Q. Yang, J.-H. Jiang, Y. Lu, and J. Chen, Electrical tuning of branched flow of light, Nature Comm. 15, 197 (2024)

    2024

  24. [32]

    Rotter and S

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

    2017

  25. [33]

    Cao, A.P

    H. Cao, A.P. Mosk, and S. Rotter, Shaping the propaga- tion of light in complex media, Nature Physics 18, 994–1007 (2022)

    2022

  26. [34]

    Kulkarny and B.S

    V. Kulkarny and B.S. White, Focusing of waves in turbulent inhomogeneous media, Phys. Fluids 25, 1770–1784 (1982)

    1982

  27. [35]

    Zwillinger and B.S

    D.I. Zwillinger and B.S. White, Propagation of initially plane waves in the region of random caustics, Wave Motion 7, 207– 227 (1985)

    1985

  28. [36]

    Berry and C

    M.V. Berry and C. Upstill, IV catastrophe optics: morpholo- gies of caustics and their diffraction patterns, Progress in Op- tics 18, 257–346 (1980)

    1980

  29. [37]

    Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations, CRC Press, 1999

    J.F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations, CRC Press, 1999

    1999

  30. [38]

    Kaplan, Statistics of branched flow in a weak correlated random potential, Phys

    L. Kaplan, Statistics of branched flow in a weak correlated random potential, Phys. Rev. Lett. 89, 184103 (2002)

    2002

  31. [39]

    Metzger, R

    J.J. Metzger, R. Fleischmann, and T. Geisel, Universal statis- tics of branched flows, Phys. Rev. Lett. 105, 020601 (2010)

    2010

  32. [40]

    Metzger, R

    J.J. Metzger, R. Fleischmann, and T. Geisel, Statistics of extreme waves in random media, Phys. Rev. Lett.112, 203903 (2014)

    2014

  33. [41]

    Pradas, A

    M. Pradas, A. Pumir, and M. Wilkinson, Uniformity transi- tion for ray intensities in random media, J. Phys. A51, 155002 (2018)

    2018

  34. [42]

    Metzger, R

    J.J. Metzger, R. Fleischmann, and T. Geisel, Intensity fluc- tuations of waves in random media: What is the semiclassical limit?, Phys. Rev. Lett. 111, 013901 (2013)

    2013

  35. [43]

    Berry, Elementary branching: Waves, rays, decoher- ence, J

    M.V. Berry, Elementary branching: Waves, rays, decoher- ence, J. Opt. 22, 115608 (2020)

    2020

  36. [44]

    Patsyk, Y

    A. Patsyk, Y. Sharabi, U. Sivan, and M. Segev, Incoherent branched flow of light, Phys. Rev. X 12, 021007 (2022)

    2022

  37. [45]

    Andrews and R.L

    L.C. Andrews and R.L. Phillips, Laser Beam Propagation through Random Media, SPIE-International Society for Opti- cal Engineering, 2005

    2005

  38. [46]

    Tatarskii, Waves Propagation in a Turbulent Medium, McGraw-Hill, 1961

    V.I. Tatarskii, Waves Propagation in a Turbulent Medium, McGraw-Hill, 1961

    1961

  39. [47]

    Crosignani, B

    B. Crosignani, B. Daino, and P. Di Porto, Light scattering by a rotating disk, J. Appl. Phys. 42, 399–403 (1971)

    1971

  40. [48]

    Goodman, Speckle Phenomena in Optics: Theory and Applications, 2nd ed

    J.W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 2nd ed. (SPIE Press, 2020)

    2020

  41. [49]

    Erkmen and J.H

    B.I. Erkmen and J.H. Shapiro, Unified theory of ghost imag- ing with Gaussian-state light, Phys. Rev. A77, 043809 (2008)

    2008

  42. [50]

    Shapiro and R.W

    J.H. Shapiro and R.W. Boyd, The physics of ghost imaging, Quantum Inf. Process. 11, 949–993 (2012)

    2012

  43. [51]

    Ishimaru, Wave Propagation and Scattering in Random Media, Academic Press, 1978

    A. Ishimaru, Wave Propagation and Scattering in Random Media, Academic Press, 1978

    1978

  44. [52]

    Garnier and K

    J. Garnier and K. Sølna, Scintillation in the white-noise paraxial regime, Communications in Partial Differential Equa- tions 39, 626–650 (2014)

    2014

  45. [53]

    Garnier and K

    J. Garnier and K. Sølna, Fourth-order moments analysis for partially coherent electromagnetic beams in random media, Waves in Random and Complex Media 33, 1346–1365 (2023)

    2023

  46. [54]

    This results from the fact that, for z large, the correlation function has the form exp −γ2zy 2/2 , where γ2 = −∂2 xγ(0) = σ2ℓ−1 c ˜γ2, ˜γ2 = −∂2 ˜x˜γ(0), is always positive because it is propor- tional to the power spectral density of the stationary process ∂xV (z, 0) at frequency 0

  47. [55]

    Strang, On the construction and comparison of difference schemes, SIAM J

    G. Strang, On the construction and comparison of difference schemes, SIAM J. Num. Anal. 5, 506–517 (1968)

    1968

  48. [56]

    Garnier and K

    J. Garnier and K. Sølna, Fourth-moment analysis for beam propagation in the white-noise paraxial regime, Archive on Rational Mechanics and Analysis 220, 37–81 (2016)

    2016

  49. [57]

    Bal and A

    G. Bal and A. Nair, Complex Gaussianity of long-distance random wave processes, arXiv:2402.17107

  50. [58]

    Bal and A

    G. Bal and A. Nair, Long distance propagation of light in ran- dom media with partially coherent sources, arXiv:2406.05252

    arXiv

  51. [59]

    See Supplementary Material for the theoretical developments and details about the numerical simulations

  52. [60]

    Foley and M.S

    J.T. Foley and M.S. Zubairy, The directionality of Gaus- sian Schell-model beams, Optics Communications 26, 297–300 (1978)

    1978

  53. [61]

    Friberg and R.J

    A.T. Friberg and R.J. Sudol, Propagation parameters of Gaussian Schell-model beams, Optics Communications 41, 383–387 (1982)

    1982

  54. [62]

    Fouque, J

    J.-P. Fouque, J. Garnier, G. Papanicolaou, and K. Sølna, Wave Propagation and Time Reversal in Randomly Layered Media, Springer, 2007

    2007

  55. [63]

    Dudley, F

    J.M. Dudley, F. Dias, M. Erkintalo, and G. Genty, Insta- bilities, breathers and rogue waves in optics, Nat. Photon. 8, 755–764 (2014)

    2014

  56. [64]

    Safari, R

    A. Safari, R. Fickler, M.J. Padgett, and R.W. Boyd, Gener- ation of caustics and rogue waves from nonlinear instability, Phys. Rev. Lett. 119, 203901 (2017)

    2017

  57. [65]

    Dudley, G

    J.M. Dudley, G. Genty, A. Mussot, A. Chabchoub, and F. Dias, Rogue waves and analogies in optics and oceanography, Nature Reviews Physics 1, 675 (2019)

    2019

  58. [66]

    Jiang, T.W

    K. Jiang, T.W. Huang, C.N. Wu, M. Y. Yu, H. Zhang, S.Z. Wu, H.B. Zhuo, A. Pukhov, C.T. Zhou, and S.C. Ruan, Non- linear branched flow of intense laser light in randomly uneven media, Matter Radiat. Extremes 8, 024402 (2023). 7 Supplementary Material on the Article Appendix A: ...

    2023

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.