## Finite beam curvature related patterns in a saturable medium

Optics Express, Vol. 4, Issue 5, pp. 161-166 (1999)

http://dx.doi.org/10.1364/OE.4.000161

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### Abstract

We study **numerically** the effects of finite curvature and ellipticity
of the Gaussian beam on propagation through a saturating nonlinear medium. We
demonstrate generation of different types of pattern arising from the
*input phase structure* as well as the phase structure
imparted by the nonlinear medium.

© Optical Society of America

2. G.A. Swartzlander Jr. and C.T. Law, “Optical vortex solitons observed in Kerr nonlinear medium,” Phys. Rev. Lett. **69**, 2503–2506 (1992). [CrossRef] [PubMed]

3. G.A. Swartzlander Jr., D.R. Anderson, J.J. Regan, H. Yin, and A.E. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. **66**, 1583–1586 (1991). [CrossRef] [PubMed]

4. V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. **76**, 2698–2701 (1996). [CrossRef] [PubMed]

5. G. Grynberg, A. Maitre, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through rubidium cell with single feedback mirror,” Phys. Rev. Lett. **72**, 2379–2382 (1994). [CrossRef] [PubMed]

6. W.J. Firth and D.V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. **79**, 2450–2453 (1997). [CrossRef]

7. The transmission of a Gaussian beam through a Gaussian lens has been shown to yield optical vortices:L.V. Kreminskaya, M.S. Soskin, and A.I. Khizhriyak, “The Gaussian lenses give birth to optical vortices in laser beams,” Opt. Commun. **145**, 377–384 (1998). [CrossRef]

8. T. Ackeman, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. **115**, 339–346 (1995). [CrossRef]

9. J. Courtial, K. Dholakia, L. Allen, and M.J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. **144**, 210–213 (1997) [CrossRef]

*numerical study*of the propagation of a complex Gaussian beam[10]

*q*in Eq.1 can be written as

*q*

_{x,y}=

*iz*

_{Rx,y}-

*z*+

*z*

_{0}where

*z*

_{Rx,y}=

*λ*is the Rayleigh range and the beam waist is located at

*z*=

*z*

_{0}. We will throughout assume that the entry face of the nonlinear medium is at

*z*= 0. Thus for positive (negative)

*z*

_{0}the beam’s waist is inside (outside before the entry face) the medium and we have a converging (diverging) beam. The saturable nonlinearity will be the one produced by a medium modelled as a collection of two level atoms. Hence, the induced polarization is taken as [11]

**d**is the dipole matrix element for the transition with frequency

*ω*

_{0}. All frequencies have been scaled with respect to the half width

*γ*of the transition. The sign of the detuning determines whether the nonlinearity is of

*focussing*(Δ < 0) or

*defocussing*(Δ > 0) type. The parameter 2

*G*is the scaled Rabi frequency and

*n*is the density of atoms. On scaling all frequencies with respect to

*γ*and all lengths with respect to

*l*(the length of the medium), the wave equation in slowly varying envelope approximation can be written as

*α*is the absorption coefficient at line centre

*focussing or defocusing*nature of the medium as well as to the

*converging*or

*diverging*nature of the Gaussian beam. The results also depend on the

*ellipticity*of the beam. Simulations were done for propagation of a converging elliptic Gaussian beam through a focussing nonlinear medium with Δ = - 18 and

*α*= 300. The medium thickness was taken to be 7.5

*cm*. The Gaussian beam of

*λ*= 780

*nm*. and complex radius of curvatures

*q*

_{x}= .12 + 2.5

*cm*. and

*q*

_{y}= .21 + 2.5

*cm*. was propagated through the medium. To find out the proper aperture size and correct number of iterations the simulations were done with following parameters.

- The simulations were done first on a 256×256 mesh. The iteration number in each case was decided by observing the convergence of the pattern for different number of iterations. Therefore, the number of iteration for different cases varies from 60 to 100. For converging beams, it was found that the entrance aperture, of four times that of the beam size along both the axes was sufficient for the propagation of 99 percent of the total beam. In case of diverging beams, the aperture size was adjusted to allow 99 percent of the beam through it at the exit plane.
- In all cases of converging beams, the simulations were repeated with a 512×512 mesh and the aperture was five times of the beam size at the entrance plane. It was found that the results were more or less same as obtained with 256×256 mesh. Therefore, it was decided to use a 256×256 mesh with four times beam size aperture along both the axes.

*crossings*of the contours of

*Re E*= 0,

*Im E*= 0 suggesting the generation of

*vortices*[7

7. The transmission of a Gaussian beam through a Gaussian lens has been shown to yield optical vortices:L.V. Kreminskaya, M.S. Soskin, and A.I. Khizhriyak, “The Gaussian lenses give birth to optical vortices in laser beams,” Opt. Commun. **145**, 377–384 (1998). [CrossRef]

12. V. Tikhonenko, Y. S. Kivshar, V. V. Steblina, and A. A. Zozulya, “Vortex solitons in a saturable optical medium,” J. Opt. Soc. Am. B **15**, 79–86 (1998). [CrossRef]

*Kerr*medium has been studied previously [1]. For weak

*ellipticity*, the propagation through a saturating medium has also been studied [8

8. T. Ackeman, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. **115**, 339–346 (1995). [CrossRef]

5. G. Grynberg, A. Maitre, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through rubidium cell with single feedback mirror,” Phys. Rev. Lett. **72**, 2379–2382 (1994). [CrossRef] [PubMed]

*z*

_{0}≫

*z*

_{R}, the incoming beam is almost a plane wave. However the strong nonlinearity is quite sensitive to the small curvature of the wavefront.

*numerically*how the finite curvature of the input beam can generate very different kind of patterns which depend on the convergent/divergent nature of the beam and on the

*focussing or defocusing*characteristics of the medium. The ellipticity of the beam gives rise to optical vortices, which multiply as the nonlinearity of the medium increases.

## References

1. | S. Camacho-Lopez, R. Ramos-Garcia, and M.J. Damzen, “Experimental study of propagation of an apertured high intensity laser beam in Kerr active CS2,” J. Mod. Opt. |

2. | G.A. Swartzlander Jr. and C.T. Law, “Optical vortex solitons observed in Kerr nonlinear medium,” Phys. Rev. Lett. |

3. | G.A. Swartzlander Jr., D.R. Anderson, J.J. Regan, H. Yin, and A.E. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. |

4. | V. Tikhonenko, J. Christou, and B. Luther-Davies, “Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium,” Phys. Rev. Lett. |

5. | G. Grynberg, A. Maitre, and A. Petrossian, “Flowerlike patterns generated by a laser beam transmitted through rubidium cell with single feedback mirror,” Phys. Rev. Lett. |

6. | W.J. Firth and D.V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. |

7. | The transmission of a Gaussian beam through a Gaussian lens has been shown to yield optical vortices:L.V. Kreminskaya, M.S. Soskin, and A.I. Khizhriyak, “The Gaussian lenses give birth to optical vortices in laser beams,” Opt. Commun. |

8. | T. Ackeman, E. Kriege, and W. Lange, “Phase singularities via nonlinear beam propagation in sodium vapor,” Opt. Commun. |

9. | J. Courtial, K. Dholakia, L. Allen, and M.J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. |

10. | A.E. Siegman, |

11. | R.W. Boyd, |

12. | V. Tikhonenko, Y. S. Kivshar, V. V. Steblina, and A. A. Zozulya, “Vortex solitons in a saturable optical medium,” J. Opt. Soc. Am. B |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(330.0330) Vision, color, and visual optics : Vision, color, and visual optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 24, 1998

Published: March 1, 1999

**Citation**

Rakesh Kapoor and G. S. Agarwal, "Finite beam curvature related patterns in a saturable medium," Opt. Express **4**, 161-166 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-5-161

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### References

- S. Camacho-Lopez, R. Ramos-Garcia and M. J. Damzen, "Experimental study of propagation of an apertured high intensity laser beam in Kerr active CS2," J. Mod. Opt. 44, 1671-1681 (1997).
- G. A. Swartzlander Jr. and C. T. Law, "Optical vortex solitons observed in Kerr nonlinear medium," Phys. Rev. Lett. 69, 2503-2506 (1992). [CrossRef] [PubMed]
- G. A. Swartzlander Jr., D. R. Anderson, J. J. Regan, H. Yin and A. E. Kaplan, "Spatial dark-soliton stripes and grids in self-defocusing materials," Phys. Rev. Lett. 66, 1583-1586 (1991). [CrossRef] [PubMed]
- V. Tikhonenko, J. Christou and B. Luther-Davies, "Three dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium," Phys. Rev. Lett. 76, 2698-2701 (1996). [CrossRef] [PubMed]
- G. Grynberg, A. Maitre and A. Petrossian, "Flowerlike patterns generated by a laser beam transmitted through rubidium cell with single feedback mirror," Phys. Rev. Lett. 72, 2379-2382 (1994). [CrossRef] [PubMed]
- W. J. Firth and D. V. Skryabin, "Optical solitons carrying orbital angular momentum," Phys. Rev. Lett. 79, 2450-2453 (1997). [CrossRef]
- The transmission of a Gaussian beam through a Gaussian lens has been shown to yield optical vortices: L.V. Kreminskaya, M. S. Soskin and A. I. Khizhriyak, "The Gaussian lenses give birth to optical vortices in laser beams," Opt. Commun. 145, 377-384 (1998). [CrossRef]
- T. Ackeman, E. Kriege and W. Lange, "Phase singularities via nonlinear beam propagation in sodium vapor," Opt. Commun. 115, 339-346 (1995). [CrossRef]
- J. Courtial, K. Dholakia, L. Allen and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997). [CrossRef]
- A.E. Siegman, Lasers (University Science Books, Mill Valley, California); Chapter 17. Note that we adopt the convention exp(ikz - iwt) rather than the one used by engineers exp(ikz + iwt).
- R. W. Boyd, Nonlinear Optics (Academic Press, New York, 1992) p203.
- V. Tikhonenko, Y. S. Kivshar, V. V. Steblina and A. A. Zozulya," Vortex solitons in a saturable optical medium," J. Opt. Soc. Am. B 15, 79-86 (1998). [CrossRef]

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