## Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis |

Optics Express, Vol. 20, Issue 3, pp. 2196-2205 (2012)

http://dx.doi.org/10.1364/OE.20.002196

Acrobat PDF (2116 KB)

### Abstract

Analytical propagation expression of an Airy beam in uniaxial crystals orthogonal to the optical axis is derived. The ballistic dynamics of an Airy beam in uniaxial crystals is also investigated. The Airy beam propagating in uniaxial crystals orthogonal to the optical axis mainly depends on the ratio of the extraordinary refractive index to the ordinary refractive index. As an example, the propagation of an Airy beam in the positive uniaxial crystals orthogonal to the optical axis is demonstrated. The acceleration of an Airy beam in the transversal direction along the optical axis is more rapidly than that in the other transversal direction. With increasing the ratio of the extraordinary refractive index to the ordinary refractive index, the acceleration of the Airy beam in the transversal direction along the optical axis speeds up and the acceleration of the Airy beam in the other transversal direction slows down. The Airy beam propagating in uniaxial crystals orthogonal to the optical axis follows a ballistic trajectory. The effective beam size of the Airy beam in the transversal direction along the optical axis is always larger than that in the other transversal direction.

© 2012 OSA

## 1. Introduction

1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**(3), 264–267 (1979). [CrossRef]

2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef] [PubMed]

3. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

4. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express **16**(17), 12880–12891 (2008). [CrossRef] [PubMed]

5. H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express **16**(13), 9411–9416 (2008). [CrossRef] [PubMed]

6. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. **33**(3), 207–209 (2008). [CrossRef] [PubMed]

7. R. Chen and C. Ying, “Beam propagation factor of an Airy beam,” J. Opt. **13**(8), 085704 (2011). [CrossRef]

8. R. P. Chen, H. P. Zheng, and C. Q. Dai, “Wigner distribution function of an Airy beam,” J. Opt. Soc. Am. A **28**(6), 1307–1311 (2011). [CrossRef] [PubMed]

9. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express **18**(8), 8440–8452 (2010). [CrossRef] [PubMed]

10. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics **2**(11), 675–678 (2008). [CrossRef]

11. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics **3**(7), 395–398 (2009). [CrossRef]

*ABCD*optical system [12

12. M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express **15**(25), 16719–16728 (2007). [CrossRef] [PubMed]

13. P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. **103**(12), 123902 (2009). [CrossRef] [PubMed]

14. S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. **104**(25), 253904 (2010). [CrossRef] [PubMed]

15. R. Chen, C. Yin, X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A **82**(4), 043832 (2010). [CrossRef]

16. X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. **36**(14), 2701–2703 (2011). [CrossRef] [PubMed]

17. A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A **19**(4), 792–796 (2002). [CrossRef] [PubMed]

23. J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D **57**(3), 419–425 (2010). [CrossRef]

## 2. Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis

*z*-axis is taken to be the propagation axis. The optical axis of the uniaxial crystal coincides with the

*x*-axis. The input plane is

*z*=0 and the observation plane is

*z*. The ordinary and extraordinary refractive indices of the uniaxial crystal are

*n*and

_{o}*n*, respectively. The relative dielectric tensor of the uniaxial crystal reads as

_{e}*x*-direction and is incident on a uniaxial crystal in the plane

*z*=0. The Airy beam in the input plane

*z*=0 takes the form as [2

2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef] [PubMed]

3. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

*w*

_{0}is the transverse scale.

*Ai*(⋅) is the Airy function.

*a*is the modulation parameter. Within the framework of the paraxial approximation, the propagation of the Airy beam in uniaxial crystals orthogonal to the optical axis obeys the following equations [24

24. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A **20**(11), 2163–2171 (2003). [CrossRef] [PubMed]

*j*=

*x*or

*y*(hereafter). In the paraxial approximation, the longitudinal component of the field can be neglected as long as the transversal beam radius

*w*

_{0}is larger compared with the wavelength [26

26. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A **19**(2), 404–412 (2002). [CrossRef] [PubMed]

*x*component of the optical field is only a superposition of extraordinary plane waves and the

*y*component uniquely contains ordinary plane waves [24

24. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A **20**(11), 2163–2171 (2003). [CrossRef] [PubMed]

24. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A **20**(11), 2163–2171 (2003). [CrossRef] [PubMed]

*λ*being the optical wavelength. Inserting Eq. (2) into Eq. (6), we can obtainwith

*e*=

*n*/

_{e}*n*is the ratio of the extraordinary refractive index to the ordinary refractive index. We recall that the Airy function can be written in terms of the integral representation:

_{o}*n*=

_{e}*n*= 1, Eq. (23) reduces to the familiar propagation formula of Airy beams in free space.

_{o}*x*-

*z*and

*y*-

*z*planes that are described by the following parabolas

*g*and

_{x}*g*play the role of “gravity”. As

_{y}*e*>1 holds for the positive uniaxial crystals, the displacement in the

*x*-direction is always larger than that in the

*y*-direction, and

*g*is always larger than

_{x}*g*. When

_{y}*n*=

_{e}*n*= 1, Eqs. (25)-(28) reduce to the ballistic dynamics in free space, which are same as those in Ref [6

_{o}6. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. **33**(3), 207–209 (2008). [CrossRef] [PubMed]

*x*- and

*y*-directions of the observation plane is defined as [28]:where

*j*is the centre of gravity of the Airy beam in uniaxial crystals orthogonal to the optical axis and is given by

_{C}6. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. **33**(3), 207–209 (2008). [CrossRef] [PubMed]

*x*- and

*y*-directions of the observation plane turn out to be whereare the effective beam sizes in the input plane.

*z*and

_{rx}*z*represent the Rayleigh length in the

_{ry}*x*- and

*y*-directions of the Airy beam in uniaxial crystals and are given by

## 3. Numerical calculations and analyses

*λ*= 0.53μm,

*w*

_{0}= 100μm,

*a*= 0.1, and

*n*= 2.616. Figure 1 represents the contour graph of the normalized intensity distribution of an Airy beam propagating in the uniaxial crystals at several observation planes. The normalized intensity is given by

_{o}*e*= 1.1 and

*e*= 1.5, respectively. The two observation planes are

*z*= 0.1

*z*

_{0}and

*z*= 5

*z*

_{0}. Upon propagation in the uniaxial crystals orthogonal to the optical axis, the acceleration of the Airy beam in the

*y-*direction is far slower than that in the

*x-*direction, which is caused by the anisotropic effect of the crystals. In free space, the Airy beam remains almost invariant up to a certain distance, and it accelerates in the same manner along the 45° axis in the

*x*-

*y*plane [3

3. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

*x*-

*y*plane. The angle along which the Airy beam accelerates in the

*x*-

*y*plane of the uniaxial crystal is equal to

*e*, we can find that the acceleration of the Airy beam in the

*x-*direction speeds up and the acceleration of the Airy beam in the

*y-*direction slows down. When

*e*= 1.1,

*θ*= 34.33°.

*θ*decreases with increasing the value of

*e*.

*x*and

*y*variables can be separated from each other. To further reveal the accelerating properties of the Airy beam in uniaxial crystals, the normalized intensity distribution in

*x*- and

*y*-directions of an Airy beam propagating in the uniaxial crystals are shown in Figs. 2 and 3 . When propagating in the uniaxial crystals orthogonal to the optical axis, the Airy beam in the

*x*- and

*y*-directions accelerates, which is more evident than that in Fig. 1. With increasing the value of

*e*, the acceleration of Airy beam in the

*x-*direction quickens up, and the acceleration of Airy beam in the

*y-*direction retards. The above exotic features only belong to the Airy beam. Equations (25) and (26) denote that the displacements in the

*x*- and

*y*-directions are proportional to the square of the axial propagation distance. When the observation plane is far from the input plane, therefore, the displacement deflection augments with increasing the value of

*e*.

*x*-

*z*and

*y*-

*z*planes are shown in Fig. 4 . The top and bottom rows correspond to

*e*= 1.1 and

*e*= 1.5, respectively. The Airy beam in uniaxial crystals orthogonal to the optical axis follows a ballistic trajectory. With increasing the value of

*e*,

*g*increases and

_{x}*g*decreases, which is indicated by Eqs. (27) and (28). Therefore, the parabolic deflection of the normalized intensity in the

_{y}*x*-

*z*plane augments with increasing the value of

*e*. However, the parabolic deflection of the normalized intensity in the

*y*-

*z*plane decreases with increasing the value of

*e*, which is open and shut in Fig. 4. Comparison between the right and left parts of Fig. 4 denotes that the acceleration of the Airy beam in the

*x-*direction is far faster than that in the

*y-*direction under the same condition.

*z*are shown in Fig. 5 . With increasing the value of

*e*, the effective beam size in the

*x*-direction increases and the effective beam size in the

*y*-direction decreases. The reason is that

*n*/

_{e}*n*of the anisotropic crystal is larger than unity. The effective beam size in the

_{o}*x*-direction is always larger than that in the

*y*-direction at any given propagation distance.

## 4. Conclusions

*n*, the ratio of the extraordinary refractive index to the ordinary refractive index

_{o}*e*, the axial propagation distance

*z*, and the beam parameters

*w*

_{0}and

*a*. As an example, the influence of the positive uniaxial crystals on the propagation properties of an Airy beam is numerically investigated. Upon propagation in the uniaxial crystals, the Airy beam in the two transversal directions accelerates. Moreover, the acceleration of the Airy beam in the transversal direction orthogonal to the optical axis is far slower than that in the transversal direction along the optical axis. The Airy beam in uniaxial crystals orthogonal to the optical axis follows a ballistic trajectory. With increasing the value of

*e*, the acceleration of the Airy beam in the transversal direction along the optical axis speeds up, and the acceleration of the Airy beam in the transversal direction orthogonal to the optical axis slows down. The dependence of the effective beam size of the Airy beam in uniaxial crystals on the value of

*e*is also examined. With increasing the value of

*e*, the effective beam size in the transversal direction along the optical axis increases and that in the other transversal direction orthogonal to the optical axis decreases. At any given propagation distance, the effective beam size in the transversal direction along the optical axis is always larger than that in the other transversal direction.

## Acknowledgments

## References and links

1. | M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. |

2. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. |

3. | G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. |

4. | J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express |

5. | H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express |

6. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. |

7. | R. Chen and C. Ying, “Beam propagation factor of an Airy beam,” J. Opt. |

8. | R. P. Chen, H. P. Zheng, and C. Q. Dai, “Wigner distribution function of an Airy beam,” J. Opt. Soc. Am. A |

9. | Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express |

10. | J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics |

11. | T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics |

12. | M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express |

13. | P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. |

14. | S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. |

15. | R. Chen, C. Yin, X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A |

16. | X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. |

17. | A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A |

18. | B. Lü and S. Luo, “Propagation properties of three-dimensional flatted Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. |

19. | D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. |

20. | D. Liu and Z. Zhou, “Generalized stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. |

21. | B. Tang, “Hermite-cosine-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A |

22. | C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. |

23. | J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D |

24. | A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A |

25. | M. Born and E. Wolf, |

26. | C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A |

27. | I. S. Gradshteyn and I. M. Ryzhik, |

28. | N. Hodgson and H. Weber, |

**OCIS Codes**

(260.1180) Physical optics : Crystal optics

(260.1960) Physical optics : Diffraction theory

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: November 23, 2011

Revised Manuscript: December 29, 2011

Manuscript Accepted: January 10, 2012

Published: January 17, 2012

**Citation**

Guoquan Zhou, Ruipin Chen, and Xiuxiang Chu, "Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis," Opt. Express **20**, 2196-2205 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2196

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### References

- M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264–267 (1979). [CrossRef]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007). [CrossRef] [PubMed]
- G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32(8), 979–981 (2007). [CrossRef] [PubMed]
- J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express16(17), 12880–12891 (2008). [CrossRef] [PubMed]
- H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express16(13), 9411–9416 (2008). [CrossRef] [PubMed]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett.33(3), 207–209 (2008). [CrossRef] [PubMed]
- R. Chen and C. Ying, “Beam propagation factor of an Airy beam,” J. Opt.13(8), 085704 (2011). [CrossRef]
- R. P. Chen, H. P. Zheng, and C. Q. Dai, “Wigner distribution function of an Airy beam,” J. Opt. Soc. Am. A28(6), 1307–1311 (2011). [CrossRef] [PubMed]
- Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express18(8), 8440–8452 (2010). [CrossRef] [PubMed]
- J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008). [CrossRef]
- T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics3(7), 395–398 (2009). [CrossRef]
- M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express15(25), 16719–16728 (2007). [CrossRef] [PubMed]
- P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett.103(12), 123902 (2009). [CrossRef] [PubMed]
- S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett.104(25), 253904 (2010). [CrossRef] [PubMed]
- R. Chen, C. Yin, X. Chu, and H. Wang, “Effect of Kerr nonlinearity on an Airy beam,” Phys. Rev. A82(4), 043832 (2010). [CrossRef]
- X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett.36(14), 2701–2703 (2011). [CrossRef] [PubMed]
- A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A19(4), 792–796 (2002). [CrossRef] [PubMed]
- B. Lü and S. Luo, “Propagation properties of three-dimensional flatted Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol.36(1), 51–56 (2004). [CrossRef]
- D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun.281(2), 202–209 (2008). [CrossRef]
- D. Liu and Z. Zhou, “Generalized stokes parameters of stochastic electromagnetic beams propagating through uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt.11(6), 065710 (2009). [CrossRef]
- B. Tang, “Hermite-cosine-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A26(12), 2480–2487 (2009). [CrossRef] [PubMed]
- C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt.57(5), 375–384 (2010). [CrossRef]
- J. Li, Y. Chen, Y. Xin, and S. Xu, “Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D57(3), 419–425 (2010). [CrossRef]
- A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A20(11), 2163–2171 (2003). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics, 7th ed. (Pergamon, Oxford, 1999).
- C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A19(2), 404–412 (2002). [CrossRef] [PubMed]
- I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic Press, New York, 1980).
- N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer Press, New York, 2005).

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