## Azimuthal modulation instability for a cylindrically polarized wave in a nonlinear Kerr medium

Optics Express, Vol. 14, Issue 11, pp. 4757-4764 (2006)

http://dx.doi.org/10.1364/OE.14.004757

Acrobat PDF (195 KB)

### Abstract

Inhomogeneously polarized optical waves form a class of nonlinear vector wave propagation that has not been widely studied in the literature. We find a modulation instability only when the wave has nonzero ellipticity in a medium where the Kerr nonlinearity possesses opposite handness. Under the modulation instability the wave develops an azimuthally periodic shape with two or four peaks.

© 2006 Optical Society of America

## 1. Introduction

1. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. **21**, 1948–1950 (1996). [CrossRef] [PubMed]

9. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D. **32**, 1455–1461 (1999). [CrossRef]

11. H. Wang and W. She, “Nonparaxial optial Kerr vortex soliton with radial polarization,” Opt. Express **14**, 1590–1595 (2006). [CrossRef] [PubMed]

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

13. D. Rozas, Z. S. Sacks, and G. A. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. **79**, 3399–3402 (1997). [CrossRef]

14. J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A **45**, 3168–3175 (1992). [CrossRef] [PubMed]

16. D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Azimuthal instability of spinning spatiotemporal solitons,” Phys. Rev. E **62**, R1505–R1508 (2000). [CrossRef]

17. D. V. Petrov, L. Torner, J. Martorell, R. Vilseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulation instability and formation of patterns of optical solitons in quadratic nonlinear crystal,” Opt. Lett **23**, 1444–1447 (1998). [CrossRef]

## 2. Simulations

*α*denotes

*x*or

*y*. The first term on the right hand side has a transverse Laplacian operator that describes diffraction. The transverse field amplitude, longitudinal and transverse coordinates are scaled. The transverse coordinates are scaled to the width,

*w*, of the initial Gaussian envelope function; the longitudinal coordinate is scaled by the diffraction length

*z*

_{d}

*=2w*

^{2}

*k*

_{0}, where

*k*

_{0}is the wavenumber in the medium. The field is scaled to the Kerr nonlinearity denoted in Boyd [19] by A, so that

*E*denotes the “physical” field in Gaussian units and ω is the angular frequency of the wave. The equations use the paraxial approximation and assume pulse dispersion is negligible. Temporal propagation effects are included when the longitudinal coordinate is assumed to be in the co-propagating reference frame.

^{phys}*B*coefficient, which is scaled to the

*A*coefficient. B introduces circular birefringence to the nonlinear propagation effects [19] and vector propagation effects for homogeneous polarizations were experimentally studied [20

20. M. S. Malcuit, D. J. Gauthier, and R. W. Boyd, “Vector phase conjugation by two-photon-resonant degenerate four-wave mixing,” Opt. Lett. **13**, 663–665 (1988). [CrossRef] [PubMed]

21. D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,”Phys. Rev. Lett. **64**, 1721–1724 (1990). [CrossRef] [PubMed]

*B*coefficient has values

*B*=

*0*for an electrostrictive nonlinearity,

*B*=

*1*for a nonresonant electronic nonlinearity, and

*B*=

*6*for a molecular orientational nonlinearity [18

18. P. D. Maker and R. W. Terhune, “Study of Optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. **137**, A801 (1965). [CrossRef]

22. J. A. Fleck, J. R. Morris, and M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. **10**, 129–160 (1976). [CrossRef]

*x=y=0*for the radial polarization case. We start with a cylindrical vector beam that passes through a wave plate with retardance

*φ*. For the Cartesian components of the field we have

*E*

_{0}is the field amplitude variable. The angle

*θ*is the rotation of the local polarization orientation vector from radial polarization (

*θ=0*) to tangential polarization state (

*θ=π/2*) [4

4. Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express **10**, 324–331 (2002). [PubMed]

*φ*is the phase retardance due to the phase-plate element, for a quarterwave plate

*φ=π/2*and for a half-wave plate

*φ=π*. When

*φ=0*the beam is cylindrically polarized.

## 3. Results

### 3.1. Homogeneous linear polarization case

23. K. D. Moll, A. Gaeta, and G. Fibich, “Self-similar optical wave collapse: Observation of the Townes Profile,” Phys. Rev. Lett. **90**, 203902 (2003). [CrossRef] [PubMed]

*E*

_{0}

*=1.5*and

*B=6*. The initial intensity is close to the critical power threshold for self-focusing collapse for this case. After propagating a distance

*z=2*the profile takes on a symmetric shape with a peak centered at

*r=0*. The appearance of the central maximum is a consequence of linear diffraction effects and it is well-known as the Poisson spot phenomenon. The Poisson spot phenomenon can be avoided by adding a phase rotation around the origin, i.e. topological charge, which is identical to the optical vortex soliton case, except for a nonlinear coefficient having the opposite sign.

### 3.2. Cylindrically polarized vector wave

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

13. D. Rozas, Z. S. Sacks, and G. A. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. **79**, 3399–3402 (1997). [CrossRef]

*z*. The amplitude of

*E*

_{0}

*=2*is chosen.

*E*

_{0}

*=2.2*, as shown in Fig. 5 where the maximum intensity normalized to the initial intensity is plotted for five initial amplitudes spanning the range (2.1, 2.5). The fields undergo an initial rise due to the modulation instability, but below a critical power the intensity increase is dominated by a subsequent diffraction dominated regime. The data indicates that the critical power lies in the range of the field amplitudes between 2.3 (dotted) and 2.4 (dashed), and it is a function of the retardance angle.

*φ*close to zero; for most values the modulation instability is dominated by two initial peaks form on the circle where the intensity is maximum; they are angularly diametrically opposed. The angle they appear at depends on the value of

*θ. At φ=π/2*four equal peaks angularly spaced by

*π/2*are formed. The animation in Fig. 6 demonstrates the modulation instability profile’s dependence on the retardance angle for an initial amplitude of 2. The propagation distance is

*z=2*.

*θ*. For

*θ=0*the peaks lie on the x axis and for

*θ=π/2*the peaks lie on the y axis.

## 4. Conclusion

*S*

_{3}≠0; a wave plate is imposed on the initial field.

*θ=0*) to tangential (

*θ=π/2*) rotates the angle of the intensity peaks by the same angle. The peaks appear at initially zero phase retardance points. As the phase-plate retardance is increased, the initial instability creates two peaks of equal height, which are positioned opposite to one another on a circle, at the radius of the intensity maximum.

*π/2*from the initial two peaks. For a phase-plate retardance of

*π/2*four equal height peaks grow from the initial intensity profile. For higher intensities the profiles undergo self-focusing collapse at an intensity that is higher than for the scalar case.

^{2}intensity range.

*φ=0*collapses uniformly above the critical power and this could be useful for optical traps that would confine the specimens more tightly and the beam would taper its cross-sectional area. Optical damage with cylindrical beams in materials could controllably write two or four damage spots to design coupled optical waveguiding and the longitudinal intensity component of a collapsing wave could be defined within a smaller area yielding greater resolution.

## References and links

1. | M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. |

2. | R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. |

3. | Q. Zhan, “Radiation forces on a dielectric sphere produced by a highly focused cylindrical vector beam,” J. Opt. A: Pure and Appl. Opt. |

4. | Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express |

5. | L. E. Helseth, “Roles of polarization, phase and amplitude in solid immersion lens system,” Opt. Commun. |

6. | A. Bouhelier, J. Renger, M. R. Beversluis, and L. Novotny, “Plasmon-coupled tip enhance near-field optical microscopy,” J. of Microsc. |

7. | Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express , |

8. | Y. Q. Zhao, Q. Zhan, Y. L. Zhang, and Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. |

9. | V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D. |

10. | Y. Kivshar and G. P. Agrawal, |

11. | H. Wang and W. She, “Nonparaxial optial Kerr vortex soliton with radial polarization,” Opt. Express |

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

13. | D. Rozas, Z. S. Sacks, and G. A. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. |

14. | J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, “Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium,” Phys. Rev. A |

15. | A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: spatially modulated vortex solitons,” Phys. Rev. Lett. |

16. | D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, “Azimuthal instability of spinning spatiotemporal solitons,” Phys. Rev. E |

17. | D. V. Petrov, L. Torner, J. Martorell, R. Vilseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulation instability and formation of patterns of optical solitons in quadratic nonlinear crystal,” Opt. Lett |

18. | P. D. Maker and R. W. Terhune, “Study of Optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. |

19. | R.W. Boyd, |

20. | M. S. Malcuit, D. J. Gauthier, and R. W. Boyd, “Vector phase conjugation by two-photon-resonant degenerate four-wave mixing,” Opt. Lett. |

21. | D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams,”Phys. Rev. Lett. |

22. | J. A. Fleck, J. R. Morris, and M. D. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. |

23. | K. D. Moll, A. Gaeta, and G. Fibich, “Self-similar optical wave collapse: Observation of the Townes Profile,” Phys. Rev. Lett. |

**OCIS Codes**

(190.3100) Nonlinear optics : Instabilities and chaos

(190.3270) Nonlinear optics : Kerr effect

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 21, 2006

Revised Manuscript: May 19, 2006

Manuscript Accepted: May 19, 2006

Published: May 29, 2006

**Citation**

Joseph W. Haus, Zasim Mozumder, and Qiwen Zhan, "Azimuthal modulation instability for a cylindrically polarized wave in a nonlinear Kerr medium," Opt. Express **14**, 4757-4764 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-11-4757

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### References

- M. Stalder and M. Schadt, "Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters," Opt. Lett. 21, 1948-1950 (1996). [CrossRef] [PubMed]
- R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, "The formation of laser beams with pure azimuthal or radial polarization," Appl. Phys. Lett. 77, 3322-3324 (2000). [CrossRef]
- Q. Zhan, "Radiation forces on a dielectric sphere produced by a highly focused cylindrical vector beam," J. Opt. A: Pure and Appl. Opt. 5, 229-232 (2003). [CrossRef]
- Q. Zhan and J. R. Leger, "Focus shaping using cylindrical vector beams," Opt. Express 10, 324-331 (2002). [PubMed]
- L. E. Helseth, "Roles of polarization, phase and amplitude in solid immersion lens system," Opt. Commun. 191, 161-172 (2001). [CrossRef]
- A. Bouhelier, J. Renger, M. R. Beversluis, and L. Novotny, "Plasmon-coupled tip enhance near-field optical microscopy," J. of Microsc. 210, 220-224 (2003). [CrossRef]
- Q. Zhan, "Trapping metallic Rayleigh particles with radial polarization," Opt. Express, 12, pp. 3377-3382, (2004). [CrossRef] [PubMed]
- Y. Q. Zhao, Q. Zhan, Y. L. Zhang, and Y. P. Li, "Creation of a three-dimensional optical chain for controllable particle delivery," Opt. Lett. 30, 848-850 (2005). [CrossRef] [PubMed]
- V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D. 32, 1455-1461 (1999). [CrossRef]
- Y. Kivshar and G. P. Agrawal, Opitcal solitons: from fiber to photonic crystals (Elsevier, Amsterdam, 2003).
- H. Wang and W. She, "Nonparaxial optial Kerr vortex soliton with radial polarization," Opt. Express 14,1590-1595 (2006). [CrossRef] [PubMed]
- G. A. Swartzlander and C. T. Law, "Optical vortex solitons observed in Kerr nonlinear media," Phys. Rev. Lett. 69, 2503-2506 (1992). [CrossRef] [PubMed]
- D. Rozas, Z. S. Sacks, and G. A. Swartzlander, "Experimental observation of fluidlike motion of optical vortices," Phys. Rev. Lett. 79, 3399-3402 (1997). [CrossRef]
- J. M. Soto-Crespo, E. M. Wright, and N. N. Akhmediev, "Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beam in a saturable self-focusing medium," Phys. Rev. A 45, 3168-3175 (1992). [CrossRef] [PubMed]
- A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, "Azimuthons: spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005). [CrossRef] [PubMed]
- D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, and F. Lederer, "Azimuthal instability of spinning spatiotemporal solitons," Phys. Rev. E 62,R1505-R1508 (2000). [CrossRef]
- D. V. Petrov, L. Torner, J. Martorell, R. Vilseca, J. P. Torres, and C. Cojocaru, "Observation of azimuthal modulation instability and formation of patterns of optical solitons in quadratic nonlinear crystal," Opt. Lett 23, 1444-1447 (1998). [CrossRef]
- P. D. Maker and R. W. Terhune, "Study of Optical effects due to an induced polarization third order in the electric field strength," Phys. Rev. 137, A801 (1965). [CrossRef]
- R.W. Boyd, Nonlinear Optics (Academic Press, San Diego, CA, 1992).
- M. S. Malcuit, D. J. Gauthier, and R. W. Boyd, "Vector phase conjugation by two-photon-resonant degenerate four-wave mixing," Opt. Lett. 13, 663-665 (1988). [CrossRef] [PubMed]
- D. J. Gauthier, M. S. Malcuit, A. L. Gaeta, and R. W. Boyd, "Polarization bistability of counterpropagating laser beams,"Phys. Rev. Lett. 64, 1721-1724 (1990). [CrossRef] [PubMed]
- J. A. Fleck, J. R. Morris, and M. D. Feit, "Time dependent propagation of high energy laser beams through the atmosphere," Appl. Phys. 10, 129-160 (1976). [CrossRef]
- K. D. Moll, A. Gaeta, and G. Fibich, "Self-similar optical wave collapse: Observation of the Townes Profile," Phys. Rev. Lett. 90, 203902 (2003 [CrossRef] [PubMed]

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