Abstract

Anti-diffraction is a theoretically predicted nonlinear optical phenomenon that occurs when a light beam spontaneously focalizes independently of its intensity. We observe anti-diffracting beams supported by the peak-intensity-independent diffusive nonlinearity that are able to shrink below their diffraction-limited size in photorefractive lithium-enriched potassium-tantalate-niobate (KTN:Li).

© 2014 Optical Society of America

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References

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    [Crossref]
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2014 (1)

2013 (3)

2012 (4)

J. Parravicini, C. Conti, A. J. Agranat, and E. DelRe, “Programming scale-free optics in disordered ferroelectrics,” Opt. Lett. 37, 2355–2357 (2012).
[Crossref] [PubMed]

J. Parravicini, A. J. Agranat, C. Conti, and E. DelRe, “Equalizing disordered ferroelectrics for diffraction cancellation,” Appl. Phys. Lett. 101, 111104 (2012).
[Crossref]

V. Folli, E. DelRe, and C. Conti, “Beam Instabilities in the Scale-Free Regime,” Phys. Rev. Lett. 108, 033901 (2012).
[Crossref] [PubMed]

Z. G. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75, 086401 (2012).
[Crossref] [PubMed]

2011 (4)

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photonics 5, 39–42 (2011).
[Crossref]

C. Conti, A. J. Agranat, and E. DelRe, “Subwavelength optical spatial solitons and three-dimensional localization in disordered ferroelectrics: Toward metamaterials of nonlinear origin,” Phys. Rev. A 84, 043809 (2011).
[Crossref]

J. Parravicini, F. Di Mei, C. Conti, A.J. Agranat, and E. DelRe, “Diffraction cancellation over multiple wavelengths in photorefractive dipolar glasses,” Opt. Express 19, 24109–24114 (2011).
[Crossref] [PubMed]

A. Gumennik, Y. Kurzweil-Segev, and A. J. Agranat, “Electrooptical effects in glass forming liquids of dipolar nano-clusters embedded in a paraelectric environment,” Opt. Mat. Express 1, 332–343 (2011).
[Crossref]

2009 (1)

O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, “Elimination, reversal and directional bias of optical diffraction,” Nat. Phys. 5, 665–668 (2009).
[Crossref]

2006 (2)

K. Staliunas and R. Herrero, “Nondiffractive propagation of light in photonic crystals,” Phys. Rev. E 73, 016601 (2006).
[Crossref]

A. A. Bokov and Z. -G. Ye, “Recent progress in relaxor ferroelectrics with perovskite structure,” J. Mater. Sci 41, 31–52 (2006).
[Crossref]

2000 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. A. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[Crossref] [PubMed]

1999 (3)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys Lett. 74, 1212–1214 (1999).
[Crossref]

B. Crosignani, A. Degasperis, E. DelRe, P. DiPorto, and A. J. Agranat, “Nonlinear optical diffraction effects and solitons due to anisotropic charge-diffusion based self-interaction,” Phys. Rev. Lett. 82, 1664–1667 (1999).
[Crossref]

E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83, 1954–1957 (1999).
[Crossref]

1998 (1)

1996 (1)

1992 (1)

Agranat, A.

Agranat, A. J.

D. Pierangeli, J. Parravicini, F. DiMei, GB Parravicini, A. J. Agranat, and E. DelRe, “Photorefractive light needles in glassy nanodisordered KNTN,” Opt. Lett. 391657–1660 (2014).
[Crossref] [PubMed]

J. Parravicini, D. Pierangeli, F. DiMei, A. J. Agranat, C. Conti, and E. DelRe, “Aging solitons in photorefractive dipolar glasses,” Opt. Express 21, 30573–30579 (2013).
[Crossref]

J. Parravicini, A. J. Agranat, C. Conti, and E. DelRe, “Equalizing disordered ferroelectrics for diffraction cancellation,” Appl. Phys. Lett. 101, 111104 (2012).
[Crossref]

J. Parravicini, C. Conti, A. J. Agranat, and E. DelRe, “Programming scale-free optics in disordered ferroelectrics,” Opt. Lett. 37, 2355–2357 (2012).
[Crossref] [PubMed]

A. Gumennik, Y. Kurzweil-Segev, and A. J. Agranat, “Electrooptical effects in glass forming liquids of dipolar nano-clusters embedded in a paraelectric environment,” Opt. Mat. Express 1, 332–343 (2011).
[Crossref]

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photonics 5, 39–42 (2011).
[Crossref]

C. Conti, A. J. Agranat, and E. DelRe, “Subwavelength optical spatial solitons and three-dimensional localization in disordered ferroelectrics: Toward metamaterials of nonlinear origin,” Phys. Rev. A 84, 043809 (2011).
[Crossref]

E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83, 1954–1957 (1999).
[Crossref]

B. Crosignani, A. Degasperis, E. DelRe, P. DiPorto, and A. J. Agranat, “Nonlinear optical diffraction effects and solitons due to anisotropic charge-diffusion based self-interaction,” Phys. Rev. Lett. 82, 1664–1667 (1999).
[Crossref]

Agranat, A.J.

Aitchison, J. A.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. A. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[Crossref] [PubMed]

Bokov, A. A.

A. A. Bokov and Z. -G. Ye, “Recent progress in relaxor ferroelectrics with perovskite structure,” J. Mater. Sci 41, 31–52 (2006).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Chang, Y-C.

Chen, Z. G.

Z. G. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75, 086401 (2012).
[Crossref] [PubMed]

Christodoulides, D. N.

Z. G. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75, 086401 (2012).
[Crossref] [PubMed]

D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett. 21, 1460–1462 (1996).
[Crossref] [PubMed]

Conti, C.

J. Parravicini, D. Pierangeli, F. DiMei, A. J. Agranat, C. Conti, and E. DelRe, “Aging solitons in photorefractive dipolar glasses,” Opt. Express 21, 30573–30579 (2013).
[Crossref]

J. Parravicini, A. J. Agranat, C. Conti, and E. DelRe, “Equalizing disordered ferroelectrics for diffraction cancellation,” Appl. Phys. Lett. 101, 111104 (2012).
[Crossref]

J. Parravicini, C. Conti, A. J. Agranat, and E. DelRe, “Programming scale-free optics in disordered ferroelectrics,” Opt. Lett. 37, 2355–2357 (2012).
[Crossref] [PubMed]

V. Folli, E. DelRe, and C. Conti, “Beam Instabilities in the Scale-Free Regime,” Phys. Rev. Lett. 108, 033901 (2012).
[Crossref] [PubMed]

J. Parravicini, F. Di Mei, C. Conti, A.J. Agranat, and E. DelRe, “Diffraction cancellation over multiple wavelengths in photorefractive dipolar glasses,” Opt. Express 19, 24109–24114 (2011).
[Crossref] [PubMed]

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photonics 5, 39–42 (2011).
[Crossref]

C. Conti, A. J. Agranat, and E. DelRe, “Subwavelength optical spatial solitons and three-dimensional localization in disordered ferroelectrics: Toward metamaterials of nonlinear origin,” Phys. Rev. A 84, 043809 (2011).
[Crossref]

E. DelRe and C. Conti, “Scale-Free Optics,” Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer, New York, 2012).
[Crossref]

Coskun, T. H.

Cowley, J. M.

J. M. Cowley, Diffraction Physics, 3rd ed. (Elsevier Science B.V., Amsterdam, 1995).

Crosignani, B.

B. Crosignani, A. Degasperis, E. DelRe, P. DiPorto, and A. J. Agranat, “Nonlinear optical diffraction effects and solitons due to anisotropic charge-diffusion based self-interaction,” Phys. Rev. Lett. 82, 1664–1667 (1999).
[Crossref]

B. Crosignani, E. DelRe, P. Di Porto, and A. Degasperis, “Self-focusing and self-trapping in unbiased centrosymmetric photorefractive media,” Opt. Lett. 23, 912–914 (1998).
[Crossref]

Davidson, N.

O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, “Elimination, reversal and directional bias of optical diffraction,” Nat. Phys. 5, 665–668 (2009).
[Crossref]

Degasperis, A.

B. Crosignani, A. Degasperis, E. DelRe, P. DiPorto, and A. J. Agranat, “Nonlinear optical diffraction effects and solitons due to anisotropic charge-diffusion based self-interaction,” Phys. Rev. Lett. 82, 1664–1667 (1999).
[Crossref]

B. Crosignani, E. DelRe, P. Di Porto, and A. Degasperis, “Self-focusing and self-trapping in unbiased centrosymmetric photorefractive media,” Opt. Lett. 23, 912–914 (1998).
[Crossref]

Della Pergola, R.

E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83, 1954–1957 (1999).
[Crossref]

DelRe, E.

D. Pierangeli, J. Parravicini, F. DiMei, GB Parravicini, A. J. Agranat, and E. DelRe, “Photorefractive light needles in glassy nanodisordered KNTN,” Opt. Lett. 391657–1660 (2014).
[Crossref] [PubMed]

J. Parravicini, D. Pierangeli, F. DiMei, A. J. Agranat, C. Conti, and E. DelRe, “Aging solitons in photorefractive dipolar glasses,” Opt. Express 21, 30573–30579 (2013).
[Crossref]

J. Parravicini, A. J. Agranat, C. Conti, and E. DelRe, “Equalizing disordered ferroelectrics for diffraction cancellation,” Appl. Phys. Lett. 101, 111104 (2012).
[Crossref]

V. Folli, E. DelRe, and C. Conti, “Beam Instabilities in the Scale-Free Regime,” Phys. Rev. Lett. 108, 033901 (2012).
[Crossref] [PubMed]

J. Parravicini, C. Conti, A. J. Agranat, and E. DelRe, “Programming scale-free optics in disordered ferroelectrics,” Opt. Lett. 37, 2355–2357 (2012).
[Crossref] [PubMed]

J. Parravicini, F. Di Mei, C. Conti, A.J. Agranat, and E. DelRe, “Diffraction cancellation over multiple wavelengths in photorefractive dipolar glasses,” Opt. Express 19, 24109–24114 (2011).
[Crossref] [PubMed]

C. Conti, A. J. Agranat, and E. DelRe, “Subwavelength optical spatial solitons and three-dimensional localization in disordered ferroelectrics: Toward metamaterials of nonlinear origin,” Phys. Rev. A 84, 043809 (2011).
[Crossref]

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photonics 5, 39–42 (2011).
[Crossref]

E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83, 1954–1957 (1999).
[Crossref]

B. Crosignani, A. Degasperis, E. DelRe, P. DiPorto, and A. J. Agranat, “Nonlinear optical diffraction effects and solitons due to anisotropic charge-diffusion based self-interaction,” Phys. Rev. Lett. 82, 1664–1667 (1999).
[Crossref]

B. Crosignani, E. DelRe, P. Di Porto, and A. Degasperis, “Self-focusing and self-trapping in unbiased centrosymmetric photorefractive media,” Opt. Lett. 23, 912–914 (1998).
[Crossref]

E. DelRe and C. Conti, “Scale-Free Optics,” Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer, New York, 2012).
[Crossref]

Di Mei, F.

Di Porto, P.

DiMei, F.

DiPorto, P.

B. Crosignani, A. Degasperis, E. DelRe, P. DiPorto, and A. J. Agranat, “Nonlinear optical diffraction effects and solitons due to anisotropic charge-diffusion based self-interaction,” Phys. Rev. Lett. 82, 1664–1667 (1999).
[Crossref]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. A. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[Crossref] [PubMed]

Firstenberg, O.

O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, “Elimination, reversal and directional bias of optical diffraction,” Nat. Phys. 5, 665–668 (2009).
[Crossref]

Folli, V.

V. Folli, E. DelRe, and C. Conti, “Beam Instabilities in the Scale-Free Regime,” Phys. Rev. Lett. 108, 033901 (2012).
[Crossref] [PubMed]

Gumennik, A.

A. Gumennik, Y. Kurzweil-Segev, and A. J. Agranat, “Electrooptical effects in glass forming liquids of dipolar nano-clusters embedded in a paraelectric environment,” Opt. Mat. Express 1, 332–343 (2011).
[Crossref]

Herrero, R.

K. Staliunas and R. Herrero, “Nondiffractive propagation of light in photonic crystals,” Phys. Rev. E 73, 016601 (2006).
[Crossref]

Hoffman, R. C.

Hofmeister, R.

Jona, F.

F. Jona and G. Shirane, Ferroelectric Crystals (Dover, New York1993).

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys Lett. 74, 1212–1214 (1999).
[Crossref]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys Lett. 74, 1212–1214 (1999).
[Crossref]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys Lett. 74, 1212–1214 (1999).
[Crossref]

Kurzweil-Segev, Y.

A. Gumennik, Y. Kurzweil-Segev, and A. J. Agranat, “Electrooptical effects in glass forming liquids of dipolar nano-clusters embedded in a paraelectric environment,” Opt. Mat. Express 1, 332–343 (2011).
[Crossref]

London, P.

O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, “Elimination, reversal and directional bias of optical diffraction,” Nat. Phys. 5, 665–668 (2009).
[Crossref]

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. A. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[Crossref] [PubMed]

Mott, A. G.

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys Lett. 74, 1212–1214 (1999).
[Crossref]

Parravicini, GB

Parravicini, J.

Pierangeli, D.

Ron, A.

O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, “Elimination, reversal and directional bias of optical diffraction,” Nat. Phys. 5, 665–668 (2009).
[Crossref]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys Lett. 74, 1212–1214 (1999).
[Crossref]

Segev, M.

Z. G. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75, 086401 (2012).
[Crossref] [PubMed]

E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83, 1954–1957 (1999).
[Crossref]

Shirane, G.

F. Jona and G. Shirane, Ferroelectric Crystals (Dover, New York1993).

Shuker, M.

O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, “Elimination, reversal and directional bias of optical diffraction,” Nat. Phys. 5, 665–668 (2009).
[Crossref]

Silberberg, Y.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. A. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
[Crossref] [PubMed]

Spinozzi, E.

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photonics 5, 39–42 (2011).
[Crossref]

Staliunas, K.

K. Staliunas and R. Herrero, “Nondiffractive propagation of light in photonic crystals,” Phys. Rev. E 73, 016601 (2006).
[Crossref]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys Lett. 74, 1212–1214 (1999).
[Crossref]

Tamburrini, M.

E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83, 1954–1957 (1999).
[Crossref]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys Lett. 74, 1212–1214 (1999).
[Crossref]

Wang, C.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Yariv, A.

Ye, Z. -G.

A. A. Bokov and Z. -G. Ye, “Recent progress in relaxor ferroelectrics with perovskite structure,” J. Mater. Sci 41, 31–52 (2006).
[Crossref]

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York1984).

Yin, S.

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H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys Lett. 74, 1212–1214 (1999).
[Crossref]

Appl. Phys. Lett. (1)

J. Parravicini, A. J. Agranat, C. Conti, and E. DelRe, “Equalizing disordered ferroelectrics for diffraction cancellation,” Appl. Phys. Lett. 101, 111104 (2012).
[Crossref]

J. Mater. Sci (1)

A. A. Bokov and Z. -G. Ye, “Recent progress in relaxor ferroelectrics with perovskite structure,” J. Mater. Sci 41, 31–52 (2006).
[Crossref]

Nat. Photonics (1)

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photonics 5, 39–42 (2011).
[Crossref]

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A. Gumennik, Y. Kurzweil-Segev, and A. J. Agranat, “Electrooptical effects in glass forming liquids of dipolar nano-clusters embedded in a paraelectric environment,” Opt. Mat. Express 1, 332–343 (2011).
[Crossref]

Phys. Rev. A (1)

C. Conti, A. J. Agranat, and E. DelRe, “Subwavelength optical spatial solitons and three-dimensional localization in disordered ferroelectrics: Toward metamaterials of nonlinear origin,” Phys. Rev. A 84, 043809 (2011).
[Crossref]

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K. Staliunas and R. Herrero, “Nondiffractive propagation of light in photonic crystals,” Phys. Rev. E 73, 016601 (2006).
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[Crossref]

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[Crossref]

E. DelRe and C. Conti, “Scale-Free Optics,” Nonlinear Photonics and Novel Optical Phenomena, Z. Chen and R. Morandotti, eds. (Springer, New York, 2012).
[Crossref]

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York1984).

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Figures (4)

Fig. 1
Fig. 1 Anti-diffraction setup. A He-Ne laser operating at 633 nm is enlarged through lenses L1 and L2 and focused down to an 8 μm spot at the input facet of the KTN:Li sample, rotated with the respect to the propagation axis z by a variable angle θ and brought through a temperature cycle T (t). (Front-view) The input and output facets are imaged through lens L4 onto a CCD camera. (Top-view) Scattered light is captured above the sample and imaged, through a microscope, onto a second CCD camera.
Fig. 2
Fig. 2 Strong anti-diffraction for TD = 283K. The input 800 μW 8 μm Gaussian beam (a) diffracts to 38 μm at TA = 303K (b). It then shrinks after 15 s to a waist of 5 μm (c), before relaxing once again into a strongly spreading beam. (d)–(i) Top-view images captured through a high-aperture microscope of the stray light emitted by the beam showing the transition, in time, from a diffracting (d) to an anti-diffracting beam (g), and once again to a diffracting one (i). (j) Intensity profiles of the input beam compared to the anti-diffracting beam at t = 15s.
Fig. 3
Fig. 3 Weak anti-diffraction for TD = 286K. The input 7.8 μm beam (a) diffracts as in the previous case (b) and shrinks to 7 μm after 10 s (c). (d) Profiles of input and anti-diffracting beams (at t = 10s).
Fig. 4
Fig. 4 Time sequence of the anti-diffraction. Output intensity distributions at different instants of time showning the decay of the anti-diffracting regime and the formation of transient spatial patterns in the cases of strong (Top) and weak (Bottom) anti-diffraction.

Equations (7)

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2 i k A z + 2 A L 2 λ 2 ( | A | 2 2 | A | 2 ) 2 A = 0 ,
2 i k α z + 2 α x 2 L 2 λ 2 ( x | α | 2 ) 2 4 | α | 4 α = 0 .
α ( x , z ) = α 0 w x ( z ) e x 2 w x 2 ( z ) + i [ ϕ 0 ( z ) + 1 2 ϕ 2 ( z ) x 2 ]
ϕ 0 ( z ) = 1 k w 0 x 2 tan 1 ( a z ) a
ϕ 2 ( z ) = a z 1 + a z 2 .
w ( z ) = w 0 1 + 4 k 2 w 0 4 [ 1 ( L 2 λ 2 ) ] z 2 .
z c = n π w 0 2 λ 1 ( L / λ ) 2 1 ,

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