## Diffraction cancellation over multiple wavelengths in photorefractive dipolar glasses |

Optics Express, Vol. 19, Issue 24, pp. 24109-24114 (2011)

http://dx.doi.org/10.1364/OE.19.024109

Acrobat PDF (1029 KB)

### Abstract

We report the simultaneous diffraction cancellation for beams of different wavelengths in out-of-equilibrium dipolar glass. The effect is supported by the photorefractive diffusive nonlinearity and scale-free optics, and can find application in imaging and microscopy.

© 2011 OSA

## 1. Introduction

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

*λ*

_{1}= 532 nm and a red

*λ*

_{2}= 633 nm laser beam of same size and intensity. The phenomenon is supported by the scale-free diffusive nonlinearity [11

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]

13. E. DelRe, B. Crosignani, and P. Di Porto, “Photorefractive Solitons and Their Underlying Nonlocal Physics,” Prog. Optics **53**, 153–200 (2009) [CrossRef]

14. G. Samara, “The relaxational properties of compositionally disordered ABO_{3} perovskites,” J. Phys.: Condens. Matter **15**, R367–R411 (2003) [CrossRef]

16. P. Ben Ishai, A. J. Agranat, and Y. Feldman, “Confinement kinetics in a KTN : Cu crystal: Experiment and theory,” Phys. Rev. B **73**, 104104 (2006) [CrossRef]

*T*[1

_{C}1. E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photon. **5**, 39–42 (2011). [CrossRef]

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

## 2. Model

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

*n*is the electro-optic tensor,

_{ij}*n*

_{0}is the crystal unperturbed index of refraction,

*g*the effective quadratic electro-optic coefficient,

*δ*the Kronecker symbol, and

_{ij}**P**the low frequency polarization of the glass. When the dipolar glass is photorefractive, photoexcited charges diffuse and give rise to the steady-state electric field [1

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

*k*is the Boltzmann constant,

_{B}*T*the crystal temperature,

*q*the charge of the diffusing ions, and

*I*is the local optical intensity. When the dipolar glass is brought far from equilibrium the PNR become highly polarizable and the result is a giant electro-optic response: low frequency polarization

**P**=

*ɛ*

_{0}

*χ*

_{PNR}**E**

*is dominated by the PNR susceptibility*

_{d}*χ*≫

_{PNR}*χ*, where

*χ*is the equilibrium low-frequency susceptibility, and

*χ*depends strongly on the previous thermal history of the nanodisordered ferroelectric. Inserting Eq. (1) and Eq. (2) into the parabolic equation that describes the scalar paraxial propagation of a single linear polarization component of the slowly-varying optical field

_{PNR}*A*(|

*A*|

^{2}=

*I*is the optical intensity) we obtain [1

**5**, 39–42 (2011). [CrossRef]

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

*L*is the characteristic length scale introduced by the diffusive response,

*k*= 2

*πn*

_{0}/

*λ*, and Δ

*n*= Δ

*n*, assuming

_{xx}*x*is the direction of the linear polarization of the beam. In general, in Eq. (3) the Δ

*n*does not introduce a spatial scale. Examples of this are the guiding index of refraction patterns of optical waveguides, or the self-focusing responses of Kerr-like media. In turn, the specific form of the nonlinearity that emerges from Eq. (1) and Eq. (2) introduces a new spatial scale

*L*(see Eq. (5)), and this profoundly transforms the transverse dynamics of the propagation. In the second and third terms of Eq. (4), i.e., in those terms that drive the transverse dynamics of the beam,

*λ*only appears normalized to

*L*. If we neglect the weak dispersion in

*L*due to the material dispersion, that is, the dependence of

*n*

_{0}and

*χ*on

_{PNR}*λ*,

*L*is fixed through the supercooling of the dipolar glass and forms a threshold value for the optical spectrum [1

**5**, 39–42 (2011). [CrossRef]

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

*λ*=

*L*will form Gaussian beams that do not diffract; those with

*λ*<

*L*will also form beams that have a cancelled diffraction, but with a shape that is slowly morphed from the original Gaussian beam typical of laser modes, to a more exotic diamond-like shape that intervenes for

*λ*≪

*L*[1

**5**, 39–42 (2011). [CrossRef]

**84**, 043809 (2011). [CrossRef]

*λ*>

*L*components will diffract. Accordingly, diffraction cancellation supported by scale-free optics should hold for all beams with

*λ*≤

*L*, to the point that fixing a large enough

*L*extends this conditions to a large portion of the visible spectrum, and all these wavelengths can be subject to scale-free propagation simultaneously.

## 3. Single-wavelength experiments

**5**, 39–42 (2011). [CrossRef]

_{00}beams, respectively from a doubled Nd-Yagg laser (

*λ*

_{1}= 532 nm) and a He-Ne laser (

*λ*

_{2}= 633 nm) are first expanded an focused down onto the input facet of the crystal. The two beams have independent optical paths and are recombined through a standard beam-splitter right before the crystal, so that both their size, intensity, polarization, and focusing can be fixed separately. In our basic experiment, we fix both beams to an approximately circular intensity distribution of equal intensity Full-Width-at-Half-Maximum (FWHM) Δ

*x*≃ Δ

*y*≃ 12

*μ*m, and equal peak intensity of

*I*

_{p}_{1}≃

*I*

_{p}_{2}≃ 50 W/cm

^{2}, copropagating in parallel and with the same

*x*linear polarization.

*α*

_{1}≃ 0.11°C/s (see the thermal cooling process in Fig. 1(d)) we observe the output intensity distribution to pass from the spread out shape caused by diffraction (Δ

*x*≃ Δ

*y*≃ 30

*μ*m, Fig. 1(a,b)) to the non-diffracting case (Fig. 1(c)), where the input beam size is held throughout the

*L*= 3mm propagation in the sample (this being the threshold value of

_{z}*χ*associated to the value of

_{PNR}*α*

_{1}for which

*L*=

*L*

_{1}=

*λ*

_{1}).

*α*

_{2}≃ 0.13°C/s (see Fig. 2(d)), for which hence

*L*=

*L*

_{2}=

*λ*

_{2}. We note that the threshold cooling rates for the dipolar glass scale

*α*

_{1}/

*α*

_{2}≃ 0.84 ≃

*λ*

_{1}/

*λ*

_{2}, as expected from the scale-free model, being that

*χ*∝

_{PNR}*α*(and

*L*∝

*χ*, see Eq. (5)).

_{PNR}## 4. Dual-wavelength experiments

*μ*m (so as to not undergo interaction) in the horizontal direction. We select

*α*= 0.13°C/s, so that

*L*=

*λ*

_{2}is the critical value for scale-free optics. In these conditions we expect the

*λ*

_{1}beam to be above the scale-free propagation threshold

*L*>

*λ*

_{1}. What we observe is reported in Fig. 3. Remarkably, as shown in Fig. 3(a), the two beams propagate in the very same manner, suffering no chromatic effects, even though one is green and the other is red, in accordance with what expected from Eq. (4). Interestingly, the same conditions of scale-free propagation are observed for a wide range of peak intensities, also for

*I*

_{p}_{1}≠

*I*

_{p}_{2}, as expected from the intensity-independence of the scale-free model (see Fig. 4).

## 5. Discussion

*χ*and the dipolar glass cooling rate,

_{PNR}*α*, i.e.,

*L*(

*α*) ∝

*α*. Third, results indicate that scale-free optics can be implemented for wide spectral ranges, this being of utmost importance for its use in imaging and microscopy, where collected fluorescence is intrinsically non-monochormatic.

## Acknowledgments

## References and links

1. | E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photon. |

2. | C. Conti, A. J. Agranat, and E. DelRe, “Subwavelength optical spatial solitons and three-dimensional localization in disordered ferroelectrics: towards metamaterials of nonlinear origin,” Phys. Rev. A |

3. | D. Marcuse, |

4. | A. Yariv, |

5. | S. Trillo and W. Torruellas (eds.), |

6. | D. Kip, C. Anastassiou, E. Eugenieva, D. Christodoulides, and M. Segev, “Transmission of images through highly nonlinear media by gradient-index lenses formed by incoherent solitons,” Opt. Lett. |

7. | J. K. Yang, P. Zhang, M. Yoshihara, Y. Hu, and Z. G. Chen, “Image transmission using stable solitons of arbitrary shapes in photonic lattices,” Opt. Lett. |

8. | O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, “Elimination, reversal and directional bias of optical diffraction,” Nat. Phys. |

9. | D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nat. Photon. |

10. | D. B. Murphy, |

11. | B. Crosignani, E. DelRe, P. Di Porto, and A. Degasperis, “Self-focusing and self-trapping in unbiased centrosymmetric photorefractive media,” Opt. Lett. |

12. | B. Crosignani, A. Degasperis, E. DelRe, P. Di Porto, and A. J. Agranat, “Nonlinear optical diffraction effects and solitons due to anisotropic charge-diffusion-based self-interaction,” Phys. Rev. Lett. |

13. | E. DelRe, B. Crosignani, and P. Di Porto, “Photorefractive Solitons and Their Underlying Nonlocal Physics,” Prog. Optics |

14. | G. Samara, “The relaxational properties of compositionally disordered ABO |

15. | A. A. Bokov and Z. -G. Ye, “Recent progress in relaxor ferroelectrics with perovskite structure,” J. Mater. Sci |

16. | P. Ben Ishai, A. J. Agranat, and Y. Feldman, “Confinement kinetics in a KTN : Cu crystal: Experiment and theory,” Phys. Rev. B |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(160.2260) Materials : Ferroelectrics

(160.5320) Materials : Photorefractive materials

(190.0190) Nonlinear optics : Nonlinear optics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 16, 2011

Revised Manuscript: October 13, 2011

Manuscript Accepted: October 14, 2011

Published: November 10, 2011

**Citation**

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)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-24109

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### References

- E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nanodisordered ferroelectrics,” Nat. Photon.5, 39–42 (2011). [CrossRef]
- C. Conti, A. J. Agranat, and E. DelRe, “Subwavelength optical spatial solitons and three-dimensional localization in disordered ferroelectrics: towards metamaterials of nonlinear origin,” Phys. Rev. A84, 043809 (2011). [CrossRef]
- D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, New York, 1974).
- A. Yariv, Quantum Electronics, 3rd Edition (Wiley, New York, 1989).
- S. Trillo and W. Torruellas (eds.), Spatial solitons (Springer-Verlag, Berlin, 2001).
- D. Kip, C. Anastassiou, E. Eugenieva, D. Christodoulides, and M. Segev, “Transmission of images through highly nonlinear media by gradient-index lenses formed by incoherent solitons,” Opt. Lett.26, 524–526 (2001). [CrossRef]
- J. K. Yang, P. Zhang, M. Yoshihara, Y. Hu, and Z. G. Chen, “Image transmission using stable solitons of arbitrary shapes in photonic lattices,” Opt. Lett.36, 772–774 (2011) [CrossRef] [PubMed]
- 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]
- D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nat. Photon.4, 323–328 (2010) [CrossRef]
- D. B. Murphy, Fundamentals of light microscopy and electronic imaging (Wiley, New York, 2001)
- 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]
- B. Crosignani, A. Degasperis, E. DelRe, P. Di Porto, 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, B. Crosignani, and P. Di Porto, “Photorefractive Solitons and Their Underlying Nonlocal Physics,” Prog. Optics53, 153–200 (2009) [CrossRef]
- G. Samara, “The relaxational properties of compositionally disordered ABO3 perovskites,” J. Phys.: Condens. Matter15, R367–R411 (2003) [CrossRef]
- A. A. Bokov and Z. -G. Ye, “Recent progress in relaxor ferroelectrics with perovskite structure,” J. Mater. Sci41, 31–52 (2006) [CrossRef]
- P. Ben Ishai, A. J. Agranat, and Y. Feldman, “Confinement kinetics in a KTN : Cu crystal: Experiment and theory,” Phys. Rev. B73, 104104 (2006) [CrossRef]

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