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.

© 1999 Optical Society of America

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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 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, 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 - iωt) rather than the one used by engineers exp(-ikz + iωt).
  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]

1998 (2)

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]

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]

1997 (3)

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

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).

1996 (1)

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]

1995 (1)

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

1994 (1)

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]

1992 (1)

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

1991 (1)

G.A. Swartzlander, 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]

Ackeman, T.

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

Allen, L.

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]

Anderson, D.R.

G.A. Swartzlander, 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]

Boyd, R.W.

R.W. Boyd, Nonlinear Optics (Academic Press, New York, 1992) p203.

Camacho-Lopez, S.

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).

Christou, J.

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]

Courtial, J.

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]

Damzen, M.J.

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).

Dholakia, K.

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]

Firth, W.J.

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

Grynberg, G.

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]

Kaplan, A.E.

G.A. Swartzlander, 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]

Khizhriyak, A.I.

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]

Kivshar, Y. S.

Kreminskaya, L.V.

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]

Kriege, E.

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

Lange, W.

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

Law, C.T.

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

Luther-Davies, B.

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]

Maitre, A.

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]

Padgett, M.J.

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]

Petrossian, A.

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]

Ramos-Garcia, R.

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).

Regan, J.J.

G.A. Swartzlander, 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]

Siegman, A.E.

A.E. Siegman, Lasers (University Science Books, Mill Valley, California); Chapter 17. Note that we adopt the convention exp(ikz - iωt) rather than the one used by engineers exp(-ikz + iωt).

Skryabin, D.V.

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

Soskin, M.S.

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]

Steblina, V. V.

Swartzlander, G.A.

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

G.A. Swartzlander, 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]

Tikhonenko, V.

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]

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]

Yin, H.

G.A. Swartzlander, 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]

Zozulya, A. A.

J. Mod. Opt. (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).

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

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]

Phys. Rev. Lett. (5)

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

G.A. Swartzlander, 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]

Other (2)

A.E. Siegman, Lasers (University Science Books, Mill Valley, California); Chapter 17. Note that we adopt the convention exp(ikz - iωt) rather than the one used by engineers exp(-ikz + iωt).

R.W. Boyd, Nonlinear Optics (Academic Press, New York, 1992) p203.

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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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