## Abstract

In this work we present a simple model that can be used to calculate the far field intensity distributions when a Gaussian beam cross a thin sample of nonlinear media but the response can be nonlocal.

© 2010 OSA

Full Article | PDF Article**Optics Express**- Vol. 18,
- Issue 21,
- pp. 22067-22079
- (2010)
- •https://doi.org/10.1364/OE.18.022067

In this work we present a simple model that can be used to calculate the far field intensity distributions when a Gaussian beam cross a thin sample of nonlinear media but the response can be nonlocal.

© 2010 OSA

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- W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused
light,” Appl. Phys. Lett. 11(3), 103–105 (1967).

[Crossref] - F. W. Dabby, T. K. Gustafson, J. R. Whinnery, and Y. Kohanzadeh, “Thermally self-induced phase modulation of
laser beam,” Appl. Phys. Lett. 16(9), 362–365 (1970).

[Crossref] -
S. D. Durbin, S. M. Arakelian, and Y. R. Shen, “Laser-induced diffraction rings from a
nematic-liquid-crystal film,” Opt. Lett. 6(9), 411–413 (1981).

[PubMed] -
E. Santamato and Y. R. Shen, “Field-curvature effect on the diffraction
ring pattern of a laser beam dressed by spatial self-phase modulation in a nematic
film,” Opt. Lett. 9(12), 564–566 (1984).

[Crossref] [PubMed] - R. G. Harrison, L. Dambly, D. Yu, and W. Lu, “A new self-diffraction pattern formation in
defocusing liquid media,” Opt. Commun. 139(1-3), 69–72 (1997).

[Crossref] - A. Shevchenko, S. C. Buchter, N. V. Tabiryan, and M. Kaivola, “Creation of a hollow laser beam using
self-phase modulation in a nematic liquid crystal,” Opt. Commun. 232(1-6), 77–82 (2004).

[Crossref] - D. Yu, W. Lu, R. G. Harrison, and N. N. Rosanov, “Analysis of dark spot formation in absorbing
liquid media,” J. Mod. Opt. 45, 2597–2606 (1998).

[Crossref] - S. Brugioni and R. Meucci, “Self-phase modulation in a nematic liquid
crystal film induced by a low-power CO2 laser,” Opt. Commun. 206(4-6), 445–451 (2002).

[Crossref] - L. Lucchetti, S. Suchand, and F. Simoni, “Fine structure in spatial self-phase
modulation patterns: at a glance determination of the sign of optical nonlinearity
in highly nonlinear films,” J. Opt. A, Pure Appl.
Opt. 11(3), 034002 (2009).

[Crossref] - L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field
diffraction patterns of divergent and convergent Gaussian beams passing through
self-focusing and self-defocusing media,” J. Opt.
A, Pure Appl. Opt. 7(8), 409–415 (2005).

[Crossref] - C. M. Nascimento, M. Alencar, S. Chávez-Cerda, M. Da Silva, M. R. Meneghetti, and J. M. Hickmann, “Experimental demonstration of novel effects
on the far-field diffraction patterns of a Gaussian beam in a Kerr
medium,” J. Opt. A, Pure Appl. Opt. 8(11), 947–951 (2006).

[Crossref] - F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead
glasses,” Appl. Phys. Lett. 13(8), 284–286 (1968).

[Crossref] - M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive
media,” Phys. Rev. Lett. 68(7), 923–926 (1992).

[Crossref] [PubMed] - D. Suter and T. Blasberg, “Stabilization of transverse solitary waves
by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48(6), 4583–4587 (1993).

[Crossref] [PubMed] - W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact
solutions,” Phys. Rev. E Stat. Phys. Plasmas
Fluids Relat. Interdiscip. Topics 63(1), 016610 (2000).

[Crossref] - Y. R. Shen, The principles of nonlinear optics (Wiley classics library, 2003). Chap. 17.

- M. Born, and E. Wolf, Principles of Optics (Oxford:Pergamon, 1980). Chap. 8.

- W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, ““Modulational instability, solitons
and beam propagation in spatially nonlocal nonlinear
media,” J. Opt. B Quantum Semiclass. Opt. 6, S288–S294 (2004).

- E. W. Van Stryland, and D. Hagan, “Measuring nonlinear refraction and its dispersion” in Self-focusing: past and present, R.W. Boyd, S.G. Lukishova Bad, Y.R. Shen, Eds. (Springer, 2009) 573–591.

- S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser
beams,” JEPT Lett. 6, 38 (1967).

L. Lucchetti, S. Suchand, and F. Simoni, “Fine structure in spatial self-phase
modulation patterns: at a glance determination of the sign of optical nonlinearity
in highly nonlinear films,” J. Opt. A, Pure Appl.
Opt. 11(3), 034002 (2009).

[Crossref]

C. M. Nascimento, M. Alencar, S. Chávez-Cerda, M. Da Silva, M. R. Meneghetti, and J. M. Hickmann, “Experimental demonstration of novel effects
on the far-field diffraction patterns of a Gaussian beam in a Kerr
medium,” J. Opt. A, Pure Appl. Opt. 8(11), 947–951 (2006).

[Crossref]

L. Deng, K. He, T. Zhou, and C. Li, “Formation and evolution of far-field
diffraction patterns of divergent and convergent Gaussian beams passing through
self-focusing and self-defocusing media,” J. Opt.
A, Pure Appl. Opt. 7(8), 409–415 (2005).

[Crossref]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, ““Modulational instability, solitons
and beam propagation in spatially nonlocal nonlinear
media,” J. Opt. B Quantum Semiclass. Opt. 6, S288–S294 (2004).

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, and M. Kaivola, “Creation of a hollow laser beam using
self-phase modulation in a nematic liquid crystal,” Opt. Commun. 232(1-6), 77–82 (2004).

[Crossref]

S. Brugioni and R. Meucci, “Self-phase modulation in a nematic liquid
crystal film induced by a low-power CO2 laser,” Opt. Commun. 206(4-6), 445–451 (2002).

[Crossref]

W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact
solutions,” Phys. Rev. E Stat. Phys. Plasmas
Fluids Relat. Interdiscip. Topics 63(1), 016610 (2000).

[Crossref]

D. Yu, W. Lu, R. G. Harrison, and N. N. Rosanov, “Analysis of dark spot formation in absorbing
liquid media,” J. Mod. Opt. 45, 2597–2606 (1998).

[Crossref]

R. G. Harrison, L. Dambly, D. Yu, and W. Lu, “A new self-diffraction pattern formation in
defocusing liquid media,” Opt. Commun. 139(1-3), 69–72 (1997).

[Crossref]

D. Suter and T. Blasberg, “Stabilization of transverse solitary waves
by a nonlocal response of the nonlinear medium,” Phys. Rev. A 48(6), 4583–4587 (1993).

[Crossref]
[PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive
media,” Phys. Rev. Lett. 68(7), 923–926 (1992).

[Crossref]
[PubMed]

E. Santamato and Y. R. Shen, “Field-curvature effect on the diffraction
ring pattern of a laser beam dressed by spatial self-phase modulation in a nematic
film,” Opt. Lett. 9(12), 564–566 (1984).

[Crossref]
[PubMed]

S. D. Durbin, S. M. Arakelian, and Y. R. Shen, “Laser-induced diffraction rings from a
nematic-liquid-crystal film,” Opt. Lett. 6(9), 411–413 (1981).

[PubMed]

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, and Y. Kohanzadeh, “Thermally self-induced phase modulation of
laser beam,” Appl. Phys. Lett. 16(9), 362–365 (1970).

[Crossref]

F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead
glasses,” Appl. Phys. Lett. 13(8), 284–286 (1968).

[Crossref]

S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser
beams,” JEPT Lett. 6, 38 (1967).

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused
light,” Appl. Phys. Lett. 11(3), 103–105 (1967).

[Crossref]

S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser
beams,” JEPT Lett. 6, 38 (1967).

[Crossref]

[PubMed]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, ““Modulational instability, solitons
and beam propagation in spatially nonlocal nonlinear
media,” J. Opt. B Quantum Semiclass. Opt. 6, S288–S294 (2004).

W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact
solutions,” Phys. Rev. E Stat. Phys. Plasmas
Fluids Relat. Interdiscip. Topics 63(1), 016610 (2000).

[Crossref]

[Crossref]
[PubMed]

[Crossref]

A. Shevchenko, S. C. Buchter, N. V. Tabiryan, and M. Kaivola, “Creation of a hollow laser beam using
self-phase modulation in a nematic liquid crystal,” Opt. Commun. 232(1-6), 77–82 (2004).

[Crossref]

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused
light,” Appl. Phys. Lett. 11(3), 103–105 (1967).

[Crossref]

[Crossref]

[Crossref]
[PubMed]

[Crossref]

F. W. Dabby, T. K. Gustafson, J. R. Whinnery, and Y. Kohanzadeh, “Thermally self-induced phase modulation of
laser beam,” Appl. Phys. Lett. 16(9), 362–365 (1970).

[Crossref]

F. W. Dabby and J. R. Whinnery, “Thermal self-focusing of laser beams in lead
glasses,” Appl. Phys. Lett. 13(8), 284–286 (1968).

[Crossref]

[Crossref]

[Crossref]

[PubMed]

[Crossref]
[PubMed]

[Crossref]

D. Yu, W. Lu, R. G. Harrison, and N. N. Rosanov, “Analysis of dark spot formation in absorbing
liquid media,” J. Mod. Opt. 45, 2597–2606 (1998).

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]
[PubMed]

[Crossref]
[PubMed]

[Crossref]
[PubMed]

[PubMed]

[Crossref]

[Crossref]

[Crossref]

[Crossref]
[PubMed]

[Crossref]

[Crossref]

[Crossref]

[Crossref]
[PubMed]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[Crossref]

[PubMed]

[Crossref]
[PubMed]

[Crossref]
[PubMed]

[Crossref]

[Crossref]
[PubMed]

Y. R. Shen, The principles of nonlinear optics (Wiley classics library, 2003). Chap. 17.

M. Born, and E. Wolf, Principles of Optics (Oxford:Pergamon, 1980). Chap. 8.

E. W. Van Stryland, and D. Hagan, “Measuring nonlinear refraction and its dispersion” in Self-focusing: past and present, R.W. Boyd, S.G. Lukishova Bad, Y.R. Shen, Eds. (Springer, 2009) 573–591.

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Far field intensity profiles (upper row) and cross sections (lower row)
obtained for a sample with _{0}

Far field intensity profiles (upper row) and cross sections (lower row)
obtained for a sample with _{0}

Far field intensity profiles (upper row) and cross sections (lower row)
obtained for a sample with _{0}

Far field intensity profiles (upper row) and cross sections (lower row)
obtained for a sample with _{0}

Far field intensity profiles (upper row) and cross sections (lower row)
obtained for a sample with _{0}

Far field intensity profiles (upper row) and cross sections (lower row)
obtained for a sample with _{0}

Far field intensity profiles (upper row) and cross sections (lower row)
obtained for a sample with

Far field intensity profiles (upper row) and cross sections (lower row)
obtained for a sample with

Far field intensity profiles (upper row) and cross sections (lower row)
obtained for a sample with _{0}

Far field intensity profiles (upper row) and cross sections (lower row)
obtained for a sample with _{0}

Far field intensity profiles (upper row) and cross sections (lower row)
obtained for a sample with _{0}

Far field intensity profiles (upper row) and cross sections (lower row)
obtained for a sample with _{0}

Far field intensity profiles (upper row) and cross sections (lower row)
obtained for a sample with _{0}

Far field intensity profiles (upper row) and cross sections (lower row)
obtained for a sample with _{0}

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

$$E\left(r,z\right)={A}_{0}\frac{{w}_{0}}{w\left(z\right)}\mathrm{exp}\left[-\frac{{r}^{2}}{w{\left(z\right)}^{2}}\right]\mathrm{exp}\left[-ikz-ik\frac{{r}^{2}}{2R\left(z\right)}+i\epsilon \left(z\right)\right],$$

$$w\left(z\right)={w}_{0}{\left[1+{\left(\raisebox{1ex}{$z$}\!\left/ \!\raisebox{-1ex}{${z}_{0}$}\right.\right)}^{2}\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.},$$

$$R\left(z\right)=z\left[1+{\left(\raisebox{1ex}{${z}_{0}$}\!\left/ \!\raisebox{-1ex}{$z$}\right.\right)}^{2}\right],$$

$$\epsilon \left(z\right)={\mathrm{tan}}^{-1}\left(\raisebox{1ex}{$z$}\!\left/ \!\raisebox{-1ex}{${z}_{0}$}\right.\right),$$

$${E}_{out}=E\left(r,z\right)\mathrm{exp}\left(-i\Delta \phi \left(r\right)\right),$$

$$\Delta \phi \left(r\right)\approx \Delta {\phi}_{0}\mathrm{exp}\left(-\raisebox{1ex}{$2{r}^{2}$}\!\left/ \!\raisebox{-1ex}{${w}^{2}$}\right.\right),$$

$$\Delta \phi \left(r\right)\approx \Delta {\phi}_{0}\mathrm{exp}\left(-\raisebox{1ex}{$m{r}^{2}$}\!\left/ \!\raisebox{-1ex}{${w}^{2}$}\right.\right)=\Delta {\phi}_{0}\mathrm{exp}\left(-\raisebox{1ex}{${r}^{2}$}\!\left/ \!\raisebox{-1ex}{${\left(w/\sqrt{m}\right)}^{2}$}\right.\right),$$

$$N(I)=s{\displaystyle \int R(\xi -r)I(\xi ,z)d\xi ,}$$

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