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

1. Introduction and motivation

Diffraction causes light beams to spread out, losing spatial definition and intensity [1, 2]. This forms a limit to the spatial resolution of optical imaging systems based on far-field optics, such as a standard wide-area microscope. In nonlinear materials, self-focusing can change this spreading, but the effect is intrinsically peak-intensity dependent [3]. When self-focusing exactly balances beam spreading caused by diffraction, something that imposes a precise peak-intensity beam-width relationship, stable non-spreading beams in the form of spatial solitons appear [4, 5].

Experiments in waveguide arrays and photonic crystals have shown that interference can cause beams in specific directions to suffer a cancelled diffraction [68]. In electro-magnetic-induced transparency experiments, interference can even lead to inverted (or negative) diffraction [9]. Based on interference, this modified diffraction occurs along specific directions and for beams with a small angular spectrum. A more general effect would be the observation of beams that literally ”anti-diffract” as they propagate in a substance. In such a system, beams will naturally converge instead of spreading, irrespective of direction of propagation and for a wide range of beam sizes, even with a considerable angular spectrum. In distinction to self-focusing, that depends on intensity and generally becomes stronger as beams shrink, anti-diffraction should be intensity-independent.

Studies in nanodisordered photorefractive crystals have shown that the diffusive nonlinearity in paraelectric samples [1013] can strongly reduce natural diffraction, ultimately cancelling it, a phenomenon known as scale-free optics [1417].

In this paper we theoretically predict anti-diffraction supported by the diffusive nonlinearity and report its first observation in lithium-enriched potassium-tantalate-niobate (KTN:Li).

2. Theoretical

In a photorefractive crystal, light absorbed by deep in-band impurities diffuses and gives rise to a static electric field Edc = −(kBT/q)∇I/I, where kB is the Boltzmann constant, T the crystal temperature, q the elementary charge, I = |A|2 the optical intensity, and A the optical field amplitude [1013]. When the crystal is a disordered ferroelectric above its peak temperature Tm [18], the electro-optic response of the mesoscopic dipoles (polar-nanoregions - PNRs) [19] gives rise to a scalar change Δn=(n03/2)gε02χPNR2|Edc|2 in the background index of refraction n0 [20], where χPNR is the PNR low-frequency susceptibility, g is the electro-optic coefficient, and L=4πn02ε0gχPNR(kBT/q) [14, 15]. In the paraxial approximation, the slowly varying optical amplitude A obeys the equation

2ikAz+2AL2λ2(|A|22|A|2)2A=0,
where k = k0n0, k0 = 2π/λ, z is the propagation axis, ∇ ≡ (∂x, ∂y), and λ is the optical wavelength. Separating the variables, A(x, y, z) = α(x, z)β (y, z), α must obey
2ikαz+2αx2L2λ2(x|α|2)24|α|4α=0.
The same equation holds for β replacing x with y. Eq. (2) is satisfied by the solution
α(x,z)=α0wx(z)ex2wx2(z)+i[ϕ0(z)+12ϕ2(z)x2]
with
ϕ0(z)=1kw0x2tan1(az)a
and
ϕ2(z)=az1+az2.
Here a(1L2/λ2)/k2w0x4, wox is the initial beam in the x–direction, and α0 is a constant. For a round launch beam with wox = woy = w0, the waist in two transverse dimensions along the propagation direction z is given by
w(z)=w01+4k2w04[1(L2λ2)]z2.
For L > λ, Eq. (6) foresees beams that shrink into a point-like focus at a characteristic ”collapse length”
zc=nπw02λ1(L/λ)21,
independently of intensity.

3. Experimental

To experimentally demonstrate diffusive anti-diffraction described by Eq. (6) we use the setup illustrated in Fig. 1. A 0.8 mW (before L3) He-Ne laser operating at λ = 632.8nm is expanded and subsequently focused down to a spot with an w0 = 7.8μm (intensity full-width-at-half-maximum of Δx = Δy ≃ 9.4μm) at the input face of a sample of lithium-enriched potassium-tantalate-niobate (KTN:Li). The composite ferroelectric is grown through the top-seeded solution method so as to have a peak dielectric maximum Tm at room temperature and high optical quality [21]. Our specific crystal is a zero-cut 2.6 × 3.0 × 6.0 mm sample with a composition of K1−xTa1−yNbyO3:Lix with x = 0.003, y = 0.36. Cu impurities (approximately 0.001 atoms per mole) support photorefraction in the visible, whereas focusing and cross-polarizer experiments give n0 = 2.2 and g = 0.14m4C−2. The beam is polarized in the x direction and propagates inside the crystal for a distance of Lz ≃ 3.0mm. The crystal is rotated to a desired angle θ in the x, z plane. The output intensity distribution of the beam is imaged by a CCD camera through an imaging lens (NA≃ 0.35). Light scattered in the vertical y direction is captured by a second CCD camera placed above the sample in the y direction through a high aperture microscope (NA≃ 0.8) positioned so as to image the plane of propagation.

 figure: 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.

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We are able to achieve L > λ during a transient by operating near Tm = 287.5K, identified through dielectric constant measurements, and enacting a non-monotonic temperature trajectory T (t) [2227]. In fact, considering the values of n0, g, and kBT/q ≃ 25 mV, Lλ for χPNR~λ/(4πn02ε0g(kBT/q))105, i.e., an anomalously large value of susceptibility only observable in proximity of the dielectric peak. In each anti-diffraction experiment we enacted the following procedure: the crystal was first cleaned of photorefractive space-charge by illuminating it with a fully powered microscope illuminator placed at approximately 0.1 m above the crystal for over 10 minutes. Using a temperature controller that drives the current of a Peltier junction placed directly below the crystal in the y direction, we brought the sample to thermalize at TA = 303K. The sample is then cooled from TA = 303K at the rate of 0.07 K/s to a temperature TD (that is fixed to different values in experiments, see below), where it is kept for 60 s. Then the sample is heated once again at a rate of 0.2 K/s to the operating temperature (> TD) TB = 290K. The strong transient response is observed to have a characteristic response time of 10–30 s, with measured values of collapse length zc = 3.9 − 6.8mm that depend on the actual value of TD used. This regime is not otherwise accessible with our apparatus by a standard rapid cooling (i.e., from TA directly to TB). Once TB is reached, the temperature cycle T (t) is complete and we switched on the laser beam, recording top-view and front view images of the captured intensity distribution. All intervals of time t are indicated such that the laser is turned on at t = 0.

4. Results

In Fig. 2 we show a condition of strong anti-diffraction observed when TD = 283K. As shown in Fig. 2(a–c), the w0 = 7.8μm input beam diffracts to 38 μm as it propagates to the output facet at the initial TA = 303K. After the cooling/heating cycle, the output beam shrinks to 5 μm (L ≃ 0.643μm). Snapshots of the top-view scattered light illustrate the transition from the diffracting Fig. 2(d–f) to the shrinking beam condition Fig. 2(g), and ultimately to the once again spreading phase Fig. 2(h–i) with strongly reduced scattering. In this case, the crystal is rotated by θ = 11°. The beam profiles of the input and output distributions (at t = 15s) are compared in Fig. 2(j). From Eqs. (67) we deduce a value of zc = 3.9mm. To confirm the approximate intensity-independent and angle-independent nature of the effect, we repeated the experiment with different levels of beam power and propagation angles. We found same levels of anti-diffraction repeating experiments with 8, 30, 240, 800 μW beams and for launch angles θ = 5° − 11°. For example, at a fixed angle θ = 11°, increasing the beam power from 30 μW and 240 μW, alters the minimum waist by less than 12%. In turn, at θ = 5°, for beam powers from 30 μW and 240 μW, the minimum waist of the antidiffracting beams varies by less than 14 %. The only relevant systematic effect associated with different beam powers was a lenghtening of the anti-diffraction response time, as expected for the cumulative nature of the photorefractive response.

 figure: 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.

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In Fig. 3 we show a condition of weaker anti-diffraction from 7.8 to 7 microns when TD = 286K, (L ≃ 0.636μm). Here from Eqs. (67) zc = 6.8mm, and the maximum anti-diffraction occurs after 10 s from the end of the thermal cycle.

 figure: 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).

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In Fig. 4 we show the time sequence for the two reported cases of Fig. 2 and Fig. 3. The transverse intensity distribution is shown for different intervals of time t from the completion of the temperature cycle and the launching of the laser beam, highlighting the transient nature of the anti-diffraction.

 figure: 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.

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5. Conclusion

Anti-diffraction is a new nonlinear intensity-independent wave phenomenon that can possibily lead to new ideas in imaging techniques. From a purely fundamental perspective, we note that our paraxial theory will break down if Lzzc, where the strong-focusing requires a fully nonparaxial treatment, so that future experiments with shorter zc or longer Lz may hold further novel effects. Moreover, one phenomenological aspect that already at this stage of anti-diffraction merits discussion is the formation of transient patterns after the strong anti-diffraction stage, as reported in Fig. 4. The patterns are more evident as the value of zc decreases and are not strongly dependent on θ for the range θ = 5 − 22° we scanned. Since this excludes the possible influence of ferroelectric domains, which are pinned to the principal axes of the crystal in its nominal paraelectric m3m phase [28, 29], these patterns appear an effect of the nonlinear (but peak-intensity-independent) propagation itself.

Acknowledgments

Funding from grants PRIN 2012BFNWZ2, Sapienza Ricerca 2012 and Award 2013 are acknowledged. A.J.A. acknowledges the support of the Peter Brojde Center for Innovative Engineering.

References and links

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

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

3. R. W. Boyd, S. G. Lukishova, and Y. R. Shen, eds. Self-Focusing: Past and Present, (Springer, New York2009). [CrossRef]  

4. S. Trillo and W. Torruellas, eds. Spatial Solitons (Springer, Berlin, 2001). [CrossRef]  

5. 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]  

6. 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]  

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

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

9. 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]  

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

11. 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]  

12. 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]  

13. 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]  

14. 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]  

15. 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]  

16. 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]  

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

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

19. 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]  

20. 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]  

21. A. Agranat, R. Hofmeister, and A. Yariv, “Characterization of a new photorefractive material K1−yLyT1−xNx,” Opt. Lett. 17, 713–715 (1992). [CrossRef]   [PubMed]  

22. Y-C. Chang, C. Wang, S. Yin, R. C. Hoffman, and A. G. Mott, “Giant electro-optic effect in nanodisordered KTN crystals,” Opt. Lett. 38, 4574–4577 (2013). [CrossRef]   [PubMed]  

23. 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]  

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

25. Y-C. Chang, C. Wang, S. Yin, R. C. Hoffman, and A. G. Mott, “Kovacs effect enhanced broadband large field of view electro-optic modulators in nanodisordered KTN crystals,” Opt. Express 21, 17760–17768 (2013). [CrossRef]   [PubMed]  

26. 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]  

27. 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]  

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

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

References

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  1. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).
  2. J. M. Cowley, Diffraction Physics, 3rd ed. (Elsevier Science B.V., Amsterdam, 1995).
  3. R. W. Boyd, S. G. Lukishova, and Y. R. Shen, eds. Self-Focusing: Past and Present, (Springer, New York2009).
    [Crossref]
  4. S. Trillo and W. Torruellas, eds. Spatial Solitons (Springer, Berlin, 2001).
    [Crossref]
  5. 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]
  6. 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]
  7. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. A. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000).
    [Crossref] [PubMed]
  8. K. Staliunas and R. Herrero, “Nondiffractive propagation of light in photonic crystals,” Phys. Rev. E 73, 016601 (2006).
    [Crossref]
  9. 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]
  10. D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett. 21, 1460–1462 (1996).
    [Crossref] [PubMed]
  11. 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]
  12. 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]
  13. 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]
  14. 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]
  15. 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]
  16. 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]
  17. V. Folli, E. DelRe, and C. Conti, “Beam Instabilities in the Scale-Free Regime,” Phys. Rev. Lett. 108, 033901 (2012).
    [Crossref] [PubMed]
  18. A. A. Bokov and Z. -G. Ye, “Recent progress in relaxor ferroelectrics with perovskite structure,” J. Mater. Sci 41, 31–52 (2006).
    [Crossref]
  19. 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]
  20. 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]
  21. A. Agranat, R. Hofmeister, and A. Yariv, “Characterization of a new photorefractive material K1−yLyT1−xNx,” Opt. Lett. 17, 713–715 (1992).
    [Crossref] [PubMed]
  22. Y-C. Chang, C. Wang, S. Yin, R. C. Hoffman, and A. G. Mott, “Giant electro-optic effect in nanodisordered KTN crystals,” Opt. Lett. 38, 4574–4577 (2013).
    [Crossref] [PubMed]
  23. 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]
  24. J. Parravicini, A. J. Agranat, C. Conti, and E. DelRe, “Equalizing disordered ferroelectrics for diffraction cancellation,” Appl. Phys. Lett. 101, 111104 (2012).
    [Crossref]
  25. Y-C. Chang, C. Wang, S. Yin, R. C. Hoffman, and A. G. Mott, “Kovacs effect enhanced broadband large field of view electro-optic modulators in nanodisordered KTN crystals,” Opt. Express 21, 17760–17768 (2013).
    [Crossref] [PubMed]
  26. 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]
  27. 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]
  28. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York1984).
  29. F. Jona and G. Shirane, Ferroelectric Crystals (Dover, New York1993).

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.

Appl. Phys Lett. (1)

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]

Nat. Phys. (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]

Opt. Express (3)

Opt. Lett. (6)

Opt. Mat. Express (1)

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]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (4)

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

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

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]

Rep. Prog. Phys. (1)

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]

Other (7)

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

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

R. W. Boyd, S. G. Lukishova, and Y. R. Shen, eds. Self-Focusing: Past and Present, (Springer, New York2009).
[Crossref]

S. Trillo and W. Torruellas, eds. Spatial Solitons (Springer, Berlin, 2001).
[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).

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

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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)

Equations on this page are rendered with MathJax. Learn more.

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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