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.

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Corrections

Rakesh Kapoor and G. S. Agarwal, "Finite beam curvature related patterns in asaturable medium: errata," Opt. Express 4, 229-230 (1999)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-4-7-229

References

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  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. 44, 1671-1681 (1997).
  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]
  10. 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).
  11. R. W. Boyd, Nonlinear Optics (Academic Press, New York, 1992) p203.
  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]

Other (12)

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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Figures (9)

Figure 1.
Figure 1.

Patterns obtained with propagation of a converging circular-Gaussian beam through a focusing nonlinear medium. The medium was placed at several positions before the beam waist. Δ = -18, α = 300, G 0 = 30, and q = (z 0 + .21i) cm. The successive frames (from left to right) are for z 0 = 3.0, 3.5, 4.0 and 5.0 cm.

Figure 2.
Figure 2.

Patterns obtained with propagation of a converging elliptic-Gaussian beam through a focusing nonlinear medium. The medium was placed at several positions before the beam waist. Δ = -18, α = 300, G 0 = 30, and q = [z 0+.12i, z 0+.21i] cm. The successive frames (from left to right) are for z 0 = 2.5, 3.0, 3.5, 4.0 and 5.0 cm.

Figure 3.
Figure 3.

Patterns obtained with propagation of a converging elliptic-Gaussian beam through a focusing nonlinear medium. The medium was placed at before the beam waist. Δ = -18, q = [z 0 + .12i,z 0 + .21i] cm. Patterns were generated for different values of a and G 0. For different frames the value of α = 300, 400, and 500 (from top row to bottom row) and the value of G 0 = 30, 40, and 50 (from left column to right column)

Figure 4.
Figure 4.

Patterns obtained with propagation of a diverging circular-Gaussian beam through a focusing nonlinear medium. The medium was placed at several positions after the beam waist. Δ = -18, α = 300, G 0 = 30 and q = (z 0 + .21i) cm. Aperture was 8 times of the input beam size along both the axes. The successive frames (from left to right) are for z 0 = -3.0, -3.5, -4.0 and -5.0 cm.

Figure 5.
Figure 5.

Patterns obtained with propagation of a diverging elliptic-Gaussian beam through a focusing nonlinear medium. The medium was placed at several positions after the beam waist, Δ = -18, α = 300, G 0 = 30, and q = [z 0+.12i, z 0+.21i] cm. Aperture was 8 times of the input beam size along both the axes. The successive frames (from left to right) are for z 0 = -2.5, -3.0, -3.5, -4.0 and -5.0 cm.

Figure 6.
Figure 6.

Patterns obtained with propagation of a diverging elliptic-Gaussian beam through a focusing nonlinear medium. The medium was placed at 2.5cm after the beam waist. Δ = -18, and q = [-2.5 + .12i, -2.5 + .21i] cm. Patterns were generated for different values of α and G 0. Aperture was 8 times of the input beam size along both the axes. For top row frames α = 300, and for the bottom row frames α = 400. The value of G 0 = 30, 40, 50 (from left to right)

Figure 7.
Figure 7.

Patterns obtained with propagation of a diverging elliptic-Gaussian beam through a defocusing nonlinear medium. The medium was placed at several positions after the beam waist. Δ = -18, α = 300, G 0 = 30, and q = [z 0 + .12i,z 0 + .21i] cm. Aperture was 12 times of the input beam size along both the axes. The successive frames (from left to right) are for z 0 = -3.0, -4.0 and -5.0 cm.

Figure 8.
Figure 8.

Patterns obtained with propagation of a converging elliptic-Gaussian beam through a defocusing nonlinear medium. The medium was placed at several positions before the beam waist. Δ = -18, α = 300, G 0 = 30, and q = [z 0 + .12i,z 0 + -21i] cm. Aperture was 12 times of the input beam size along both the axes. The successive frames (from left to right) are for z 0 = 3.0, 4.0 and 5.0 cm.

Figure 9.
Figure 9.

Intensity profiles (left) and zero lines (right) of the real (red line) and imaginary (blue line) part of an elliptic beam propagating through a focusing nonlinear medium. The input beam was a converging beam as the medium was placed at 2.5 cm before the waist of the beam. Δ = -18, α = 300, G 0 = 30, and q = [2.5 + .12i, 2.5 + .21i] cm. Aperture was 4 times of the input beam size along both the axes. The successive frames were recorded at different positions inside the medium. From top to bottom z = 4.5, 6.75 and 7.5 cm.

Equations (8)

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E = E 0 exp ( ik x 2 2 q x ik y 2 2 q y ) e ikz iωt
P = n d ( Δ + i Δ 2 + 1 + 2 G 2 ) G
Δ = ( ω 0 ω ) γ and G = d · E ħγ ,
G ζ = i 2 kl 2 G + iαl 2 ( Δ + i Δ 2 + 1 + 2 G 2 ) G ,
2 = ( 2 x 0 2 + 2 y 0 2 ) ,
ζ = z l , x 0 = x l , y 0 = y l ,
α = 4 πn d 2 ω ħcγ .
G = G 0 exp ( ik x 0 2 l 2 2 q x ik y 0 2 l 2 2 q y ) ,

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