Abstract

Spatial self-phase modulation of a Gaussian beam in a photopolymer generated diffraction rings that propagated over unusually long distances (Rayleigh length) in the medium. Self-phase modulation was examined under negative, positive, and infinite wavefront curvatures (R) over different pathlengths. Resulting diffraction rings exhibited previously unobserved, slow dynamics, which could be directly monitored at the sample exit face. This study complements but differs fundamentally from previous ones that were predominantly carried out in thin films (Rayleigh length) and generated static diffraction rings, which were propagated through air and imaged in the far-field. In the photopolymer, an input beam with R<0 generated diffraction rings with a dark center, which propagated through the medium while increasing in number, underwent filamentation, and finally transformed into a stable self-trapped beam. Diffraction rings generated under R>0 bore a bright center and cyclically exchanged intensity with proximal diffraction rings. Statistical analyses of self-phase modulation at R= identified diffraction rings, which (i) were superimposed with high order modes of a co-propagating self-trapped beam, (ii) resembled fingerprint-like rings, (iii) emerged sequentially, and (iv) possessed a bright center. Results were rationalized by combining self-phase modulation theory with the evolution of refractive index changes in the photopolymer. The findings expose a new facet of spatial self-phase modulation and the complex dynamics of diffraction rings that propagate over long distances.

© 2012 Optical Society of America

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  2. J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves (Taylor & Francis, 1986).
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    [CrossRef]
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  5. K. Ogusu, Y. Kohtani, and H. Shao, “Laser-induced diffraction rings from an absorbing solution,” Opt. Rev. 3, 232–234 (1996).
  6. R. G. Harrison, L. Dambly, D. Yu, and W. Lu, “A new self-diffraction pattern formation in defocusing liquid media,” Opt. Commun. 139, 69–72 (1997).
    [CrossRef]
  7. D. Yu, W. Lu, and R. G. Harrison, “Analysis of dark spot formation in absorbing liquid media,” J. Mod. Opt. 45, 2597–2606 (1998).
    [CrossRef]
  8. D. Grischkowsky, “Self-focusing of light by potassium vapor,” Phys. Rev. Lett. 24, 866–869 (1970).
  9. A. C. Tam, “Strong amplification of sidebands in self-focused laser beams in an atomic vapor,” Phys. Rev. A 19, 1971–1977 (1979).
  10. Y. H. Meyer, “Multiple conical emissions from near resonant laser propagation in dense sodium vapor,” Opt. Commun. 34, 439–444 (1980).
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    [CrossRef]
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    [CrossRef]
  17. M. Trejo-Durán, J. A. Andrade-Lucio, A. Martínez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
    [CrossRef]
  18. W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007).
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    [CrossRef]
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    [CrossRef]
  21. I. B. Burgess, W. E. Shimmell, and K. Saravanamuttu, “Spontaneous pattern formation due to modulation instability of incoherent white light in a photopolymerizable medium,” J. Am. Chem. Soc. 129, 4738–4746 (2007).
    [CrossRef]
  22. An exception is the recent observation of diffraction rings that propagate over long distances due to a self-defocusing nonlinearity in a photorefractive crystal [24].
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    [CrossRef]
  24. A. B. Villafranca and K. Saravanamuttu, “An experimental study of the dynamics and temporal evolution of self-trapped laser beams in a photopolymerizable organosiloxane,” J. Phys. Chem. C 112, 17388–17396 (2008).
    [CrossRef]
  25. K. Saravanamuttu, X. M. Du, S. I. Najafi, and M. P. Andrews, “Photoinduced structural relaxation and densification in sol-gel-derived nanocomposite thin films: implications for integrated optics device fabrication,” Can. J. Chem. 76, 1717–1729(1998).
    [CrossRef]
  26. Here, diffraction rings with the dark center were excited in a self-defocusing medium under R>0 conditions. This is theoretically equivalent to our system comprising a self-focusing medium under R<0 conditions.
  27. K. Kasala and K. Saravanamuttu, “An experimental study of the interactions of self-trapped white light beams in a photopolymer,” Appl. Phys. Lett. 93, 051111 (2008).
    [CrossRef]
  28. K. Ogusu, Y. Kohtani, and H. Shao, “Laser-induced diffraction rings from an absorbing solution,” Opt. Rev. 3, 232–234 (1996).
    [CrossRef]
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    [CrossRef]

2009

A. B. Villafranca and K. Saravanamuttu, “Diffraction rings due to spatial self-phase modulation in a photopolymerizable medium,” J. Opt. A: Pure Appl. Opt. 11, 125202–125208 (2009).
[CrossRef]

2008

A. B. Villafranca and K. Saravanamuttu, “An experimental study of the dynamics and temporal evolution of self-trapped laser beams in a photopolymerizable organosiloxane,” J. Phys. Chem. C 112, 17388–17396 (2008).
[CrossRef]

K. Kasala and K. Saravanamuttu, “An experimental study of the interactions of self-trapped white light beams in a photopolymer,” Appl. Phys. Lett. 93, 051111 (2008).
[CrossRef]

2007

I. B. Burgess, W. E. Shimmell, and K. Saravanamuttu, “Spontaneous pattern formation due to modulation instability of incoherent white light in a photopolymerizable medium,” J. Am. Chem. Soc. 129, 4738–4746 (2007).
[CrossRef]

M. Trejo-Durán, J. A. Andrade-Lucio, A. Martínez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
[CrossRef]

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007).
[CrossRef]

2006

C. M. Nascimento, A. R. C. M. Alencar, S. Chávez-Cerda, M. G. A. d. 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, 947–951 (2006).
[CrossRef]

J. Zhang and K. Saravanamuttu, “The dynamics of self-trapped beams of incoherent white light in a free-radical photopolymerizable medium,” J. Am. Chem. Soc. 128, 14913–14923 (2006).
[CrossRef]

2005

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, 409–415 (2005).
[CrossRef]

1998

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

K. Saravanamuttu, X. M. Du, S. I. Najafi, and M. P. Andrews, “Photoinduced structural relaxation and densification in sol-gel-derived nanocomposite thin films: implications for integrated optics device fabrication,” Can. J. Chem. 76, 1717–1729(1998).
[CrossRef]

1997

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

1996

K. Ogusu, Y. Kohtani, and H. Shao, “Laser-induced diffraction rings from an absorbing solution,” Opt. Rev. 3, 232–234 (1996).

K. Ogusu, Y. Kohtani, and H. Shao, “Laser-induced diffraction rings from an absorbing solution,” Opt. Rev. 3, 232–234 (1996).
[CrossRef]

1989

1988

1984

1981

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

N. F. Pilipetski, A. V. Sukhov, N. V. Tabiryan, and B. Y. Zeldovich, “The orientational mechanism of nonlinearity and the self-focusing of He-Ne-laser radiation in nematic liquid-crystal mesophase: theory and experiment,” Opt. Commun. 37, 280–284 (1981).

1980

Y. H. Meyer, “Multiple conical emissions from near resonant laser propagation in dense sodium vapor,” Opt. Commun. 34, 439–444 (1980).

S. H. Skinner and P. D. Kleiber, “Observation of anomalous conical emission from laser-excited barium vapor,” Phys. Rev. A 21, 151–156 (1980).
[CrossRef]

1979

A. C. Tam, “Strong amplification of sidebands in self-focused laser beams in an atomic vapor,” Phys. Rev. A 19, 1971–1977 (1979).

1970

D. Grischkowsky, “Self-focusing of light by potassium vapor,” Phys. Rev. Lett. 24, 866–869 (1970).

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

1967

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–104 (1967).
[CrossRef]

Alencar, A. R. C. M.

C. M. Nascimento, A. R. C. M. Alencar, S. Chávez-Cerda, M. G. A. d. 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, 947–951 (2006).
[CrossRef]

Andrade-Lucio, J. A.

M. Trejo-Durán, J. A. Andrade-Lucio, A. Martínez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
[CrossRef]

Andrews, M. P.

K. Saravanamuttu, X. M. Du, S. I. Najafi, and M. P. Andrews, “Photoinduced structural relaxation and densification in sol-gel-derived nanocomposite thin films: implications for integrated optics device fabrication,” Can. J. Chem. 76, 1717–1729(1998).
[CrossRef]

Arakelian, S. M.

Burgess, I. B.

I. B. Burgess, W. E. Shimmell, and K. Saravanamuttu, “Spontaneous pattern formation due to modulation instability of incoherent white light in a photopolymerizable medium,” J. Am. Chem. Soc. 129, 4738–4746 (2007).
[CrossRef]

Callen, W. R.

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–104 (1967).
[CrossRef]

Castaño, V. M.

M. Trejo-Durán, J. A. Andrade-Lucio, A. Martínez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
[CrossRef]

Chávez-Cerda, S.

C. M. Nascimento, A. R. C. M. Alencar, S. Chávez-Cerda, M. G. A. d. 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, 947–951 (2006).
[CrossRef]

Dabby, F. W.

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

Dai, J.-H.

Dambly, L.

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

Deng, L.

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, 409–415 (2005).
[CrossRef]

Du, X. M.

K. Saravanamuttu, X. M. Du, S. I. Najafi, and M. P. Andrews, “Photoinduced structural relaxation and densification in sol-gel-derived nanocomposite thin films: implications for integrated optics device fabrication,” Can. J. Chem. 76, 1717–1729(1998).
[CrossRef]

Durbin, S. D.

Feit, M. D.

Fleck, Jr.

Fleischer, J. W.

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007).
[CrossRef]

Grischkowsky, D.

D. Grischkowsky, “Self-focusing of light by potassium vapor,” Phys. Rev. Lett. 24, 866–869 (1970).

Gustafson, T. K.

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

Harrison, R. G.

D. Yu, W. Lu, and R. G. Harrison, “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, 69–72 (1997).
[CrossRef]

He, K.

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, 409–415 (2005).
[CrossRef]

Hickmann, J. M.

C. M. Nascimento, A. R. C. M. Alencar, S. Chávez-Cerda, M. G. A. d. 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, 947–951 (2006).
[CrossRef]

Huth, B. G.

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–104 (1967).
[CrossRef]

Jia, S.

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007).
[CrossRef]

Kasala, K.

K. Kasala and K. Saravanamuttu, “An experimental study of the interactions of self-trapped white light beams in a photopolymer,” Appl. Phys. Lett. 93, 051111 (2008).
[CrossRef]

Kelley, P. L..

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

Kleiber, P. D.

S. H. Skinner and P. D. Kleiber, “Observation of anomalous conical emission from laser-excited barium vapor,” Phys. Rev. A 21, 151–156 (1980).
[CrossRef]

Kohanzadeh, Y.

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

Kohtani, Y.

K. Ogusu, Y. Kohtani, and H. Shao, “Laser-induced diffraction rings from an absorbing solution,” Opt. Rev. 3, 232–234 (1996).
[CrossRef]

K. Ogusu, Y. Kohtani, and H. Shao, “Laser-induced diffraction rings from an absorbing solution,” Opt. Rev. 3, 232–234 (1996).

Li, C.

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, 409–415 (2005).
[CrossRef]

Lu, W.

D. Yu, W. Lu, and R. G. Harrison, “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, 69–72 (1997).
[CrossRef]

Martínez-Richa, A.

M. Trejo-Durán, J. A. Andrade-Lucio, A. Martínez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
[CrossRef]

Meneghetti, M. R.

C. M. Nascimento, A. R. C. M. Alencar, S. Chávez-Cerda, M. G. A. d. 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, 947–951 (2006).
[CrossRef]

Meyer, Y. H.

Y. H. Meyer, “Multiple conical emissions from near resonant laser propagation in dense sodium vapor,” Opt. Commun. 34, 439–444 (1980).

Najafi, S. I.

K. Saravanamuttu, X. M. Du, S. I. Najafi, and M. P. Andrews, “Photoinduced structural relaxation and densification in sol-gel-derived nanocomposite thin films: implications for integrated optics device fabrication,” Can. J. Chem. 76, 1717–1729(1998).
[CrossRef]

Nascimento, C. M.

C. M. Nascimento, A. R. C. M. Alencar, S. Chávez-Cerda, M. G. A. d. 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, 947–951 (2006).
[CrossRef]

Ogusu, K.

K. Ogusu, Y. Kohtani, and H. Shao, “Laser-induced diffraction rings from an absorbing solution,” Opt. Rev. 3, 232–234 (1996).

K. Ogusu, Y. Kohtani, and H. Shao, “Laser-induced diffraction rings from an absorbing solution,” Opt. Rev. 3, 232–234 (1996).
[CrossRef]

Pantell, R. H.

W. R. Callen, B. G. Huth, and R. H. Pantell, “Optical patterns of thermally self-defocused light,” Appl. Phys. Lett. 11, 103–104 (1967).
[CrossRef]

Pilipetski, N. F.

N. F. Pilipetski, A. V. Sukhov, N. V. Tabiryan, and B. Y. Zeldovich, “The orientational mechanism of nonlinearity and the self-focusing of He-Ne-laser radiation in nematic liquid-crystal mesophase: theory and experiment,” Opt. Commun. 37, 280–284 (1981).

Santamato, E.

Saravanamuttu, K.

A. B. Villafranca and K. Saravanamuttu, “Diffraction rings due to spatial self-phase modulation in a photopolymerizable medium,” J. Opt. A: Pure Appl. Opt. 11, 125202–125208 (2009).
[CrossRef]

A. B. Villafranca and K. Saravanamuttu, “An experimental study of the dynamics and temporal evolution of self-trapped laser beams in a photopolymerizable organosiloxane,” J. Phys. Chem. C 112, 17388–17396 (2008).
[CrossRef]

K. Kasala and K. Saravanamuttu, “An experimental study of the interactions of self-trapped white light beams in a photopolymer,” Appl. Phys. Lett. 93, 051111 (2008).
[CrossRef]

I. B. Burgess, W. E. Shimmell, and K. Saravanamuttu, “Spontaneous pattern formation due to modulation instability of incoherent white light in a photopolymerizable medium,” J. Am. Chem. Soc. 129, 4738–4746 (2007).
[CrossRef]

J. Zhang and K. Saravanamuttu, “The dynamics of self-trapped beams of incoherent white light in a free-radical photopolymerizable medium,” J. Am. Chem. Soc. 128, 14913–14923 (2006).
[CrossRef]

K. Saravanamuttu, X. M. Du, S. I. Najafi, and M. P. Andrews, “Photoinduced structural relaxation and densification in sol-gel-derived nanocomposite thin films: implications for integrated optics device fabrication,” Can. J. Chem. 76, 1717–1729(1998).
[CrossRef]

Shao, H.

K. Ogusu, Y. Kohtani, and H. Shao, “Laser-induced diffraction rings from an absorbing solution,” Opt. Rev. 3, 232–234 (1996).

K. Ogusu, Y. Kohtani, and H. Shao, “Laser-induced diffraction rings from an absorbing solution,” Opt. Rev. 3, 232–234 (1996).
[CrossRef]

Shen, Y. R.

Shimmell, W. E.

I. B. Burgess, W. E. Shimmell, and K. Saravanamuttu, “Spontaneous pattern formation due to modulation instability of incoherent white light in a photopolymerizable medium,” J. Am. Chem. Soc. 129, 4738–4746 (2007).
[CrossRef]

Silva, M. G. A. d.

C. M. Nascimento, A. R. C. M. Alencar, S. Chávez-Cerda, M. G. A. d. 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, 947–951 (2006).
[CrossRef]

Skinner, S. H.

S. H. Skinner and P. D. Kleiber, “Observation of anomalous conical emission from laser-excited barium vapor,” Phys. Rev. A 21, 151–156 (1980).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction, and Focusing of Light, Sound, and Water Waves (Taylor & Francis, 1986).

Sukhov, A. V.

N. F. Pilipetski, A. V. Sukhov, N. V. Tabiryan, and B. Y. Zeldovich, “The orientational mechanism of nonlinearity and the self-focusing of He-Ne-laser radiation in nematic liquid-crystal mesophase: theory and experiment,” Opt. Commun. 37, 280–284 (1981).

Tabiryan, N. V.

N. F. Pilipetski, A. V. Sukhov, N. V. Tabiryan, and B. Y. Zeldovich, “The orientational mechanism of nonlinearity and the self-focusing of He-Ne-laser radiation in nematic liquid-crystal mesophase: theory and experiment,” Opt. Commun. 37, 280–284 (1981).

Tam, A. C.

A. C. Tam, “Strong amplification of sidebands in self-focused laser beams in an atomic vapor,” Phys. Rev. A 19, 1971–1977 (1979).

Torruellas, W.

S. Trillo and W. Torruellas, eds., Spatial Solitons (Springer-Verlag, 2001).

Trejo-Durán, M.

M. Trejo-Durán, J. A. Andrade-Lucio, A. Martínez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
[CrossRef]

Trillo, S.

S. Trillo and W. Torruellas, eds., Spatial Solitons (Springer-Verlag, 2001).

Vera-Graziano, R.

M. Trejo-Durán, J. A. Andrade-Lucio, A. Martínez-Richa, R. Vera-Graziano, and V. M. Castaño, “Self-diffracting effects in hybrid materials,” Appl. Phys. Lett. 90, 091112 (2007).
[CrossRef]

Villafranca, A. B.

A. B. Villafranca and K. Saravanamuttu, “Diffraction rings due to spatial self-phase modulation in a photopolymerizable medium,” J. Opt. A: Pure Appl. Opt. 11, 125202–125208 (2009).
[CrossRef]

A. B. Villafranca and K. Saravanamuttu, “An experimental study of the dynamics and temporal evolution of self-trapped laser beams in a photopolymerizable organosiloxane,” J. Phys. Chem. C 112, 17388–17396 (2008).
[CrossRef]

Wan, W.

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007).
[CrossRef]

Wang, P.-Y.

Whinnery, J. R.

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

Wu, L.-A.

Yu, D.

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

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Here, diffraction rings with the dark center were excited in a self-defocusing medium under R>0 conditions. This is theoretically equivalent to our system comprising a self-focusing medium under R<0 conditions.

An exception is the recent observation of diffraction rings that propagate over long distances due to a self-defocusing nonlinearity in a photorefractive crystal [24].

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Figures (13)

Fig. 1.
Fig. 1.

Experimentally acquired temporal evolution of diffraction rings excited at R=2.20mm. 1D and 2D spatial intensity profiles were acquired at the exit face of the sample (Z=6.00mm, OPL=8.82mm). At z=0.00mm, the beam had a diameter of 35 μm and intensity of 1.6Wcm2. For clarity, each 2D profile was normalized to the peak intensity value.

Fig. 2.
Fig. 2.

Experimentally acquired temporal plot of the diameter of the dark center of diffraction rings monitored at z=6.00mm in the photopolymer. Diffraction rings were excited at R=2.20mm at 1.6Wcm2.

Fig. 3.
Fig. 3.

Calculated plot of for R=2.20mm with increasing effective beam width at (a) constant Δn and (b) increasing Δn.

Fig. 4.
Fig. 4.

Experimentally acquired temporal evolution of diffraction rings excited at R=+2.20mm with an intensity of 1.6Wcm2. 1D and 2D spatial intensity profiles were acquired at the exit face of the sample (z=6mm, OPL=8.82mm). For clarity, each 2D profile was normalized to the peak intensity value. Arrows in 1D plots indicate the flow of intensity.

Fig. 5.
Fig. 5.

Statistics of experimentally acquired diffraction ring patterns and expansion of the beam at different optical pathlengths. Rings were excited at an input intensity of 1.6Wcm2 and imaged at the exit face of the medium.

Fig. 6.
Fig. 6.

Experimentally acquired temporal evolution of diffraction rings with high-order modes excited at an intensity of 1.6Wcm2. The sample had an OPL=2.94mm; the rings were imaged in the far field at a propagation distance of 10 mm from the sample exit face. 2D and 1D spatial intensity profiles of the beam are shown. For clarity, each 2D profile has been normalized to the maximum intensity value.

Fig. 7.
Fig. 7.

Experimentally acquired relative peak intensity of self-trapped beam coexistent with diffraction rings. The experiment was carried out at an intensity of 1.6Wcm2 in a sample with OPL=2.94mm; the rings were imaged in the far field at a propagation distance of 10 mm from the sample exit face.

Fig. 8.
Fig. 8.

Beam propagation simulations of linearly polarized modes in an optical fiber (diameter=10μm, refractive index contrast=0.08) performed with BeamPROP® software.

Fig. 9.
Fig. 9.

Experimentally acquired temporal evolution of fingerprint diffraction rings excited at 1.6Wcm2 in a sample with OPL=5.58mm. 2D and 1D intensity profiles of the beam were imaged at the exit face of the sample. For clarity, each 2D profile has been normalized to the maximum intensity value.

Fig. 10.
Fig. 10.

Scheme of study of self-phase modulation at different optical pathlengths. The dark orange region at the entrance face of the sample represents the volume of refractive index changes that underlies self-phase modulation.

Fig. 11.
Fig. 11.

Experimentally acquired temporal evolution of diffraction rings excited at an intensity of 1.6Wcm2 for (a) OPL=2.94mm and (b) OPL=5.88mm. 2D and 1D spatial intensity profiles were imaged at the exit face of the sample. For clarity, each 2D profile was normalized to the maximum intensity value.

Fig. 12.
Fig. 12.

Experimentally acquired temporal evolution of diffraction rings excited at an intensity of 1.6Wcm2 for (a) OPL=11.76mm and (b) OPL=14.70mm. 2D and 1D spatial intensity profiles were imaged at the exit face of the sample. For clarity, each 2D profile has been normalized to the maximum intensity value.

Fig. 13.
Fig. 13.

Experimentally acquired temporal evolution of diffraction ring formation at an intensity of 1.6Wcm2 for (a) OPL=2.94mm and (b) OPL=5.88mm monitored in the far field (z>10mm). 2D and 1D spatial intensity profiles of the beam are shown. For clarity, each 2D profile was normalized to its maximum intensity value.

Tables (2)

Tables Icon

Table 1. Statistics of Experimentally Acquired Diffraction Rings Elicited at Early Times under R<0 and R>0 Conditions in the Photopolymer with OPL=8.82mm

Tables Icon

Table 2. Summary of Self-Phase Modulation Phenomena in the Organosiloxane Photopolymer

Equations (3)

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Δψ(r)=2πλz0z0+dΔn(r,z)dz,
ψ(r)=k0n0r22R+Δψ(r),
ψ(r)=k0n0r22R+z0z0+dΔn(r,n)dz,

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