Abstract

In focusing Kerr media, small-scale filamentation is the major obstacle to imaging at high light intensities. In this article, we experimentally and numerically demonstrate a method based on statistical averaging to reduce the detrimental effects of filamentation on the reconstructed images. The experiments are performed with femtosecond optical pulses propagating through a nonlinear liquid (toluene). We use digital holography to capture the transmitted optical image. The reverse propagation of the captured field is numerically performed using a numerical solution of the nonlinear Schrödinger equation. Because of their intrinsic sensitivity to measurement noise, filaments fail to propagate back on their initial trajectories and parasitic filaments form. The principle of the method is the introduction of artificial perturbations on the measurement, which spatially displace the parasitic filaments. By averaging the reconstruction over many realizations of the artificial perturbation, we show that the reconstruction improves the quality of the images. Finally, in order to identify the different regimes of optical power for which the filaments are time reversible, we also derive an analytical estimate for the condition number of the nonlinear propagator.

© 2015 Optical Society of America

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References

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    [Crossref]
  4. A. Goy and D. Psaltis, “Digital reverse propagation in focusing Kerr media,” Phys. Rev. A 83(3), 031802 (2011).
    [Crossref]
  5. C. Barsi and J. W. Fleischer, “Nonlinear Abbe theory,” Nat. Photonics 7(8), 639–643 (2013).
    [Crossref]
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    [Crossref]
  7. R. W. Boyd, Nonlinear Optics (Academic, 2008).
  8. G. A. Askar’yan, “Self-trapping of optical beams,” Sov. Phys. JETP 15, 1088–1090 (1962).
  9. R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
    [Crossref]
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    [Crossref] [PubMed]
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  13. M. Tsang, D. Psaltis, and F. G. Omenetto, “Reverse propagation of femtosecond pulses in optical fibers,” Opt. Lett. 28(20), 1873–1875 (2003).
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    [Crossref]
  18. S. Kedenburg, M. Vieweg, T. Gissibl, and H. Giessen, “Linear refractive index and absorption measurements of nonlinear optical liquids in the visible and near-infrared spectral region,” Opt. Mater. Express 2(11), 1588–1611 (2012).
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2014 (1)

2013 (2)

C. Barsi and J. W. Fleischer, “Nonlinear Abbe theory,” Nat. Photonics 7(8), 639–643 (2013).
[Crossref]

A. Goy and D. Psaltis, “Imaging in focusing Kerr media using reverse propagation,” Phot. Res. 1(2), 96–101 (2013).
[Crossref]

2012 (2)

2011 (1)

A. Goy and D. Psaltis, “Digital reverse propagation in focusing Kerr media,” Phys. Rev. A 83(3), 031802 (2011).
[Crossref]

2010 (2)

D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nat. Photonics 4(5), 323–328 (2010).
[Crossref]

L. Barletti and G. Busoni, “Deterministic effective equations for the propagation of expectation in noisy nonlinear optical fibers,” Math. Methods Appl. Sci. 33(10), 1221–1227 (2010).
[Crossref]

2009 (1)

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics 3(4), 211–215 (2009).
[Crossref]

2006 (1)

J. Garnier, “Statistical analysis of noise-induced multiple filamentation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(4), 046611 (2006).
[Crossref] [PubMed]

2005 (1)

M. Centurion, Y. Pu, M. Tsang, and D. Psaltis, “Dynamics of filament formation in a Kerr medium,” Phys. Rev. A 71(6), 063811 (2005).
[Crossref]

2003 (2)

S. Courisa, M. Renarda, O. Fauchera, B. Lavorela, R. Chauxa, E. Koudoumasb, and X. Michautb, “An experimental investigation of the nonlinear refractive index (n2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369(3-4), 318–324 (2003).
[Crossref]

M. Tsang, D. Psaltis, and F. G. Omenetto, “Reverse propagation of femtosecond pulses in optical fibers,” Opt. Lett. 28(20), 1873–1875 (2003).
[Crossref] [PubMed]

2001 (1)

2000 (1)

C. Cecchi-Pestellini, L. Barletti, S. Aiello, and A. Belleni-Morande, “Mathematical methods for photon transport in random media,” J. of Quant. Spec. and Rad. Trans. 65(6), 835–851 (2000).
[Crossref]

1999 (1)

C. Ziehmann, L. A. Smith, and J. Kurths, “The bootstrap and Lyapunov exponents in deterministic chaos,” Physica D 126(1-2), 49–59 (1999).
[Crossref]

1988 (1)

1966 (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” ZhETF Pis’ma 3(11), 471–476 (1966).

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[Crossref]

1962 (1)

G. A. Askar’yan, “Self-trapping of optical beams,” Sov. Phys. JETP 15, 1088–1090 (1962).

Aiello, S.

C. Cecchi-Pestellini, L. Barletti, S. Aiello, and A. Belleni-Morande, “Mathematical methods for photon transport in random media,” J. of Quant. Spec. and Rad. Trans. 65(6), 835–851 (2000).
[Crossref]

Askar’yan, G. A.

G. A. Askar’yan, “Self-trapping of optical beams,” Sov. Phys. JETP 15, 1088–1090 (1962).

Barletti, L.

L. Barletti and G. Busoni, “Deterministic effective equations for the propagation of expectation in noisy nonlinear optical fibers,” Math. Methods Appl. Sci. 33(10), 1221–1227 (2010).
[Crossref]

C. Cecchi-Pestellini, L. Barletti, S. Aiello, and A. Belleni-Morande, “Mathematical methods for photon transport in random media,” J. of Quant. Spec. and Rad. Trans. 65(6), 835–851 (2000).
[Crossref]

Barsi, C.

C. Barsi and J. W. Fleischer, “Nonlinear Abbe theory,” Nat. Photonics 7(8), 639–643 (2013).
[Crossref]

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics 3(4), 211–215 (2009).
[Crossref]

Belleni-Morande, A.

C. Cecchi-Pestellini, L. Barletti, S. Aiello, and A. Belleni-Morande, “Mathematical methods for photon transport in random media,” J. of Quant. Spec. and Rad. Trans. 65(6), 835–851 (2000).
[Crossref]

Berti, N.

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” ZhETF Pis’ma 3(11), 471–476 (1966).

Bramati, A.

Busoni, G.

L. Barletti and G. Busoni, “Deterministic effective equations for the propagation of expectation in noisy nonlinear optical fibers,” Math. Methods Appl. Sci. 33(10), 1221–1227 (2010).
[Crossref]

Cecchi-Pestellini, C.

C. Cecchi-Pestellini, L. Barletti, S. Aiello, and A. Belleni-Morande, “Mathematical methods for photon transport in random media,” J. of Quant. Spec. and Rad. Trans. 65(6), 835–851 (2000).
[Crossref]

Centurion, M.

M. Centurion, Y. Pu, M. Tsang, and D. Psaltis, “Dynamics of filament formation in a Kerr medium,” Phys. Rev. A 71(6), 063811 (2005).
[Crossref]

Chauxa, R.

S. Courisa, M. Renarda, O. Fauchera, B. Lavorela, R. Chauxa, E. Koudoumasb, and X. Michautb, “An experimental investigation of the nonlinear refractive index (n2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369(3-4), 318–324 (2003).
[Crossref]

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[Crossref]

Chinaglia, W.

Courisa, S.

S. Courisa, M. Renarda, O. Fauchera, B. Lavorela, R. Chauxa, E. Koudoumasb, and X. Michautb, “An experimental investigation of the nonlinear refractive index (n2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369(3-4), 318–324 (2003).
[Crossref]

Di Trapani, P.

Dylov, D. V.

D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nat. Photonics 4(5), 323–328 (2010).
[Crossref]

Ettoumi, W.

Fauchera, O.

S. Courisa, M. Renarda, O. Fauchera, B. Lavorela, R. Chauxa, E. Koudoumasb, and X. Michautb, “An experimental investigation of the nonlinear refractive index (n2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369(3-4), 318–324 (2003).
[Crossref]

Feit, M. D.

Fibich, G.

B. Shim, S. E. Schrauth, A. L. Gaeta, M. Klein, and G. Fibich, “Loss of phase of collapsing beams,” Phys. Rev. Lett. 108(4), 043902 (2012).
[Crossref] [PubMed]

Fleck, J. A.

Fleischer, J. W.

C. Barsi and J. W. Fleischer, “Nonlinear Abbe theory,” Nat. Photonics 7(8), 639–643 (2013).
[Crossref]

D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nat. Photonics 4(5), 323–328 (2010).
[Crossref]

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics 3(4), 211–215 (2009).
[Crossref]

Gaeta, A. L.

B. Shim, S. E. Schrauth, A. L. Gaeta, M. Klein, and G. Fibich, “Loss of phase of collapsing beams,” Phys. Rev. Lett. 108(4), 043902 (2012).
[Crossref] [PubMed]

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[Crossref]

Garnier, J.

J. Garnier, “Statistical analysis of noise-induced multiple filamentation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(4), 046611 (2006).
[Crossref] [PubMed]

Giessen, H.

Gissibl, T.

Goy, A.

A. Goy and D. Psaltis, “Imaging in focusing Kerr media using reverse propagation,” Phot. Res. 1(2), 96–101 (2013).
[Crossref]

A. Goy and D. Psaltis, “Digital reverse propagation in focusing Kerr media,” Phys. Rev. A 83(3), 031802 (2011).
[Crossref]

Kasparian, J.

Kedenburg, S.

Klein, M.

B. Shim, S. E. Schrauth, A. L. Gaeta, M. Klein, and G. Fibich, “Loss of phase of collapsing beams,” Phys. Rev. Lett. 108(4), 043902 (2012).
[Crossref] [PubMed]

Koudoumasb, E.

S. Courisa, M. Renarda, O. Fauchera, B. Lavorela, R. Chauxa, E. Koudoumasb, and X. Michautb, “An experimental investigation of the nonlinear refractive index (n2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369(3-4), 318–324 (2003).
[Crossref]

Kurths, J.

C. Ziehmann, L. A. Smith, and J. Kurths, “The bootstrap and Lyapunov exponents in deterministic chaos,” Physica D 126(1-2), 49–59 (1999).
[Crossref]

Lavorela, B.

S. Courisa, M. Renarda, O. Fauchera, B. Lavorela, R. Chauxa, E. Koudoumasb, and X. Michautb, “An experimental investigation of the nonlinear refractive index (n2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369(3-4), 318–324 (2003).
[Crossref]

Michautb, X.

S. Courisa, M. Renarda, O. Fauchera, B. Lavorela, R. Chauxa, E. Koudoumasb, and X. Michautb, “An experimental investigation of the nonlinear refractive index (n2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369(3-4), 318–324 (2003).
[Crossref]

Minardi, S.

Omenetto, F. G.

Psaltis, D.

A. Goy and D. Psaltis, “Imaging in focusing Kerr media using reverse propagation,” Phot. Res. 1(2), 96–101 (2013).
[Crossref]

A. Goy and D. Psaltis, “Digital reverse propagation in focusing Kerr media,” Phys. Rev. A 83(3), 031802 (2011).
[Crossref]

M. Centurion, Y. Pu, M. Tsang, and D. Psaltis, “Dynamics of filament formation in a Kerr medium,” Phys. Rev. A 71(6), 063811 (2005).
[Crossref]

M. Tsang, D. Psaltis, and F. G. Omenetto, “Reverse propagation of femtosecond pulses in optical fibers,” Opt. Lett. 28(20), 1873–1875 (2003).
[Crossref] [PubMed]

Pu, Y.

M. Centurion, Y. Pu, M. Tsang, and D. Psaltis, “Dynamics of filament formation in a Kerr medium,” Phys. Rev. A 71(6), 063811 (2005).
[Crossref]

Renarda, M.

S. Courisa, M. Renarda, O. Fauchera, B. Lavorela, R. Chauxa, E. Koudoumasb, and X. Michautb, “An experimental investigation of the nonlinear refractive index (n2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369(3-4), 318–324 (2003).
[Crossref]

Schrauth, S. E.

B. Shim, S. E. Schrauth, A. L. Gaeta, M. Klein, and G. Fibich, “Loss of phase of collapsing beams,” Phys. Rev. Lett. 108(4), 043902 (2012).
[Crossref] [PubMed]

Shim, B.

B. Shim, S. E. Schrauth, A. L. Gaeta, M. Klein, and G. Fibich, “Loss of phase of collapsing beams,” Phys. Rev. Lett. 108(4), 043902 (2012).
[Crossref] [PubMed]

Smith, L. A.

C. Ziehmann, L. A. Smith, and J. Kurths, “The bootstrap and Lyapunov exponents in deterministic chaos,” Physica D 126(1-2), 49–59 (1999).
[Crossref]

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” ZhETF Pis’ma 3(11), 471–476 (1966).

Townes, C. H.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[Crossref]

Tsang, M.

M. Centurion, Y. Pu, M. Tsang, and D. Psaltis, “Dynamics of filament formation in a Kerr medium,” Phys. Rev. A 71(6), 063811 (2005).
[Crossref]

M. Tsang, D. Psaltis, and F. G. Omenetto, “Reverse propagation of femtosecond pulses in optical fibers,” Opt. Lett. 28(20), 1873–1875 (2003).
[Crossref] [PubMed]

Vieweg, M.

Wan, W.

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics 3(4), 211–215 (2009).
[Crossref]

Wolf, J.-P.

Ziehmann, C.

C. Ziehmann, L. A. Smith, and J. Kurths, “The bootstrap and Lyapunov exponents in deterministic chaos,” Physica D 126(1-2), 49–59 (1999).
[Crossref]

Chem. Phys. Lett. (1)

S. Courisa, M. Renarda, O. Fauchera, B. Lavorela, R. Chauxa, E. Koudoumasb, and X. Michautb, “An experimental investigation of the nonlinear refractive index (n2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369(3-4), 318–324 (2003).
[Crossref]

J. of Quant. Spec. and Rad. Trans. (1)

C. Cecchi-Pestellini, L. Barletti, S. Aiello, and A. Belleni-Morande, “Mathematical methods for photon transport in random media,” J. of Quant. Spec. and Rad. Trans. 65(6), 835–851 (2000).
[Crossref]

J. Opt. Soc. Am. B (1)

Math. Methods Appl. Sci. (1)

L. Barletti and G. Busoni, “Deterministic effective equations for the propagation of expectation in noisy nonlinear optical fibers,” Math. Methods Appl. Sci. 33(10), 1221–1227 (2010).
[Crossref]

Nat. Photonics (3)

C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics 3(4), 211–215 (2009).
[Crossref]

D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nat. Photonics 4(5), 323–328 (2010).
[Crossref]

C. Barsi and J. W. Fleischer, “Nonlinear Abbe theory,” Nat. Photonics 7(8), 639–643 (2013).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Opt. Mater. Express (1)

Phot. Res. (1)

A. Goy and D. Psaltis, “Imaging in focusing Kerr media using reverse propagation,” Phot. Res. 1(2), 96–101 (2013).
[Crossref]

Phys. Rev. A (2)

A. Goy and D. Psaltis, “Digital reverse propagation in focusing Kerr media,” Phys. Rev. A 83(3), 031802 (2011).
[Crossref]

M. Centurion, Y. Pu, M. Tsang, and D. Psaltis, “Dynamics of filament formation in a Kerr medium,” Phys. Rev. A 71(6), 063811 (2005).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

J. Garnier, “Statistical analysis of noise-induced multiple filamentation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(4), 046611 (2006).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

B. Shim, S. E. Schrauth, A. L. Gaeta, M. Klein, and G. Fibich, “Loss of phase of collapsing beams,” Phys. Rev. Lett. 108(4), 043902 (2012).
[Crossref] [PubMed]

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13(15), 479–482 (1964).
[Crossref]

Physica D (1)

C. Ziehmann, L. A. Smith, and J. Kurths, “The bootstrap and Lyapunov exponents in deterministic chaos,” Physica D 126(1-2), 49–59 (1999).
[Crossref]

Sov. Phys. JETP (1)

G. A. Askar’yan, “Self-trapping of optical beams,” Sov. Phys. JETP 15, 1088–1090 (1962).

ZhETF Pis’ma (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” ZhETF Pis’ma 3(11), 471–476 (1966).

Other (3)

R. W. Boyd, S. G. Lukishova, G. Svetlana, and Y. R. Shen, eds., Self-Focusing: Past and Present (Springer, 2009).

R. W. Boyd, Nonlinear Optics (Academic, 2008).

A. Quarteroni, R. Sacco, and F. Saleri, Numerical Mathematics (Springer, 2000).

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Figures (7)

Fig. 1
Fig. 1 (a) An object field u0 is projected onto the input window of a nonlinear medium and propagates through it. A field u L is formed at the output window. The operator F ^ L represents the physical nonlinear operator that transforms u 0 into u L . (b) The object is digitally reconstructed using operator F L 1 , which represents the numerical model of F ^ L 1 . Operator F L 1 is applied to the measurement u L +δ u L and produces a reconstruction u 0 +δ u 0 , where δ u L represents the experimental noise and δ u 0 the induced distortion in the reconstruction. Any other perturbation η L added on top of the experimental measurement gives rise to an additional term η 0 in the reconstruction.
Fig. 2
Fig. 2 (a) Example of linear and nonlinear imaging experiments. The nonlinear experiment illustrates the effect of filaments on the reconstructed image. (b) Simulation illustrating the relationship between the occurrence of filaments in a focusing Kerr medium and the growth of perturbations with propagation. The image labeled “z = 0” shows the intensity of the object at z = 0mm. The other images show the intensity with increasing propagation distances in millimeters. Graph at the bottom: The dotted line shows the maximum intensity of the field in the corresponding z = d plane. The solid line shows the maximum (over 100 realizations of the noise) perturbation amplification factor γ of the reconstruction from the field recorded at distance z = d. The vertical bars on the graph indicate the positions of the corresponding images. The surges in the curves are correlated with the occurrence of filaments.
Fig. 3
Fig. 3 Diagram of the experimental apparatus. The laser source was a regenerative amplifier the output compressor of which has been detuned in order to produced a temporally chirped pulse of 1.5ps. A variable neutral density filter (NDF) is placed at the laser output to adjust the power. Beam splitter BS1 sampled the laser beam and sends part of it to a photodiode (PD) to monitor the pulse energy. Beam splitter BS2 reflects 10% of the beam power to the compressor on the left (BG: blazed grating, VRR: vertical retroreflector). The distance between the gratings was adjusted to produce a transform-limited pulse of 150fs. This pulse served as a reference for the interferometric measurement. The reference is passed through a spatial filter (SF) to clean up the spatial profile. The pulse that was transmitted through BS2 is spatially reduced in order to reach the required intensity range for the nonlinear experiment. An absorptive object was placed at the input window of a silica cell filled with a nonlinear liquid (toluene). The output window of the cell was imaged by a 4f lens system in a plane 22cm far from the CCD detector plane. The image and reference beams are recombined through beam splitter BS3 before the CCD detector.
Fig. 4
Fig. 4 Example of experimental image reconstructions. The gray levels represent the intensity of the field and are rescaled for better visualisation. (a) Intensity of the optical field measured at the output of the cell for the linear (low intensity) experiment. (b) Linear reconstruction from the field shown in (a) using linear back-propagation (Fresnel diffraction). (c) Output intensity of the nonlinear experiment. The arrow indicates the position of the filament. (d) Reconstruction form the field shown in (c) using the nonlinear reverse propagation.
Fig. 5
Fig. 5 (a) Pseudo-object (obtained from the linear reconstruction of the low intensity control experiment). The gray levels represent the intensity of the field and are all mapped to the same scale, except the image in (a) which is a linearly scaled version of the linear image of the object. (b) Optical field intensity after 100mm propagation in a positive Kerr medium. The position of the filament is indicated by the arrow. (c) Nonlinear reconstruction from the output field (b). (d) One instance of the random perturbation η L used for averaging. In this example, the power of the perturbation is 40% of the power of the signal. (e) Digital reconstruction from one realization of the perturbation on the data shown in (b). (f) Intensity of the average reconstruction over 50 realizations of the perturbation.
Fig. 6
Fig. 6 (a) Error in the reconstruction for the complex field as a function of the number of realizations. The horizontal solid line is the error of the unperturbed nonlinear reconstruction. The horizontal dashed line is the error of the linear reconstruction. The thick dotted line is the error for the average. The thin dashed-dotted line is the error (resp. condition number) of the current reconstruction. (b) Natural logarithm of the condition number. The horizontal line is the condition number of the unperturbed nonlinear reverse propagation. The dashed line is the condition number of the reverse propagation for a particular realization of the perturbation. The dotted line is the condition number of the average reverse propagation.
Fig. 7
Fig. 7 (a) Nonlinear reconstruction with no artificial perturbation. (b) Example of nonlinear reconstruction with a specific realization of the perturbation η. The arrows indicate the theoretical characteristic size of the filamentation pattern of 0.27mm for the experiment presented in the main article. (c) Amplitude of the Fourier transform of image (a). The dashed circle shows the spatial frequency corresponding to the characteristic defect size. (d) Amplitude of the Fourier transform of image (b).

Equations (12)

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i u z + 1 2k ( 2 u x 2 + 2 u y 2 )+ k 0 n ˜ 2 |u | 2 u=0,
K L = sup δ u 0 0 γ(δ u 0 ), with γ(δ u 0 )= δ u L / u L δ u 0 / u 0 ,
ln( K ^ L )=k 0 L | n ˜ 2 | max x,y |u(x,y,z) | 2 dz=k 0 L Δ n NL, max dz,
K L = sup δ u 0 0 γ(δ u 0 ), with γ(δ u 0 )= δ u L / u L δ u 0 / u 0 ,
K L = sup δ u 0 0 δ u L δ u 0 .
F L ( u 0 +δ u 0 ) F L ( u 0 )+J( u 0 )δ u 0 ,
δ u L = F L ( u 0 +δ u 0 ) F L ( u 0 )Jδ u 0 .
J= lim N m=1 N e AΔ z m e B m Δ z m .
K L = sup δ u 0 0 δ u L δ u 0 = sup δ u 0 0 Jδ u 0 δ u 0 = J = = lim N m=1 N e M m Δ z m lim N m=1 N ( e M m Δ z m )= m=1 ( max n σ n )= K ^ L ,
K ^ L = lim N m=1 N e M m Δ z m = lim N m=1 N e AΔ z m e B m Δ z m lim N m=1 N ( e AΔ z m e B m Δ z m ) = = lim N m=1 N ( e B m Δ z m ) =lim N m=1 N ( e k 0 n ˜ 2 | u m | 2 Δ z m ) ,
K ^ L =exp[ lim N m=1 N Δ z m ( k 0 Δ n NL, max ) m ]=exp[ k 0 0 L Δ n NL, max dz ].
ln( K ^ L )= k 0 0 L | n ˜ 2 | I max dz= k 0 0 L Δ n NL, max dz,

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