## Generation of Airy solitary-like wave beams by acceleration control in inhomogeneous media |

Optics Express, Vol. 19, Issue 17, pp. 16448-16454 (2011)

http://dx.doi.org/10.1364/OE.19.016448

Acrobat PDF (2951 KB)

### Abstract

We investigate the propagation of Airy beams in linear gradient index inhomogeneous media. We demonstrate that by controlling the gradient strength of the medium it is possible to reduce to zero their acceleration. We show that the resulting Airy wave beam propagates in straight line due to the balance between two opposite effects, one due to the inhomogeneous medium and the other to the diffraction of the beam, in a similar way as a solitary wave in a nonlinear inhomogeneous medium. Going even further we were able to invert the sign of the acceleration of the beam.

© 2011 OSA

## 1. Introduction

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

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]

8. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**(1), 44–46 (2004). [CrossRef] [PubMed]

9. C. López-Mariscal, M. Bandres, J. Gutiérrez-Vega, and S. Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express **13**(7), 2364–2369 (2005). [CrossRef] [PubMed]

10. D. Marcuse, “TE modes of graded index slab waveguides,” IEEE J. Quantum Electron. **9**(10), 1000–1006 (1973). [CrossRef]

12. D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett. **21**(18), 1460–1462 (1996). [CrossRef] [PubMed]

16. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. **36**(7), 1164–1166 (2011). [CrossRef] [PubMed]

## 2. Propagation in inhomogeneous media

### I. Fermat's ray theory

*n*=

*n*(

*x,y,z*) obtained from Fermat's variational principle is [17],where

**r**is the position vector of a point on a ray function of the length of arc

*s*of the ray. For a material with a linear gradient index (GRIN)

*n*

^{2}(

*x*) =

*n*

_{0}

^{2}±

*n*

_{1}

*x*, with

*n*

_{1}<<

*n*

_{0}, and paraxial rays this equation yields

*z*, follow parabolas whose branches extend in the direction of the gradient. This is, if the gradient is negative the rays bend towards the negative

*x*-axis and vice versa, see Figs. 1a ) and 1b). Seen in another way, the rays bend in the direction of the increasing of the refractive index (red dotted line). For each case, the corresponding Airy differential equation has also the same sign of the gradient. This is relevant since the Airy differential equation with the positive sign,

*u*” +

*xu*= 0, is rarely quoted in the literature and its solution is a mirror reflection of the solution of the Airy equation with the negative sign. By looking at the corresponding Airy mode, we notice that if it was to propagate in free space it would move in the opposite direction as that of the rays, see Fig. 1c) and 1d) where the modulus squared of the amplitude is plotted. Then, from this it can be deduced that the gradient index of the medium presents an opposite effect to the propagating Airy beam that can be associated to the opposite force described by Berry and Balazs [7

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

### II. Airy beam propagation in linear gradient index media

*u*” −

*ξu*= 0 that is one of the two forms of the Airy differential equation [11]. This is a good reason to use the Airy function as the initial condition below.

18. There exist several forms for producing a solution of this equation, for instance by using the Zassenhaus formula [19], or by simplifying the differential equation via a transformation (see for instance [20] for a quadratic term). The operators to disentangle the exponential of the sum of two operators can also be given in different orderings of the exponentials involved. Of course, they are all equivalent.

20. H. Moya-Cessa and M. Fernández Guasti, “Coherent states for the time dependent harmonic oscillator: the step function,” Phys. Lett. A **311**(1), 1–5 (2003). [CrossRef]

*e*[

^{χA}Be^{-χA}=B+χ*A,B*]

*+*(

*χ*)[

^{2}/2!*A,*[

*A,B*]]

*+*…, we simplify the problem of the linear GRIN paraxial wave equation to find the solution ofthat can immediately be integrated to give

*u*(

*x*,0) =

*Ai*(

*k*) for the paraxial wave Eq. (3), it is transformed by substitution in Eq. (4) as

_{x}x*k*(

_{x}*k*

_{1}−

*k*

_{x}^{3}/2)/2. This is shown in Fig. 2 in which this factor is negative, zero and positive, respectively. A relevant case is that shown in Fig. 2 b) that corresponds to the solitary-like wave propagation of the Airy beam. This behavior is determined by the balance between the free space parabolic wave propagation of Airy beams, characterized by

*k*, and the inhomogeneous medium that induces the optical rays in the beam to travel along parabolic trajectories in the opposite direction determined by the GRIN strength through

_{x}*k*

_{1}. This is demonstrated experimentally in the next section.

## 3. Experiment

24. W. M. Strouse, “Bouncing light beams,” Am. J. Phys. **40**(6), 913–914 (1972). [CrossRef]

25. D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori, “Bouncing light beams and the Hamiltonian analogy,” Eur. J. Phys. **18**(4), 284–289 (1997). [CrossRef]

*Ai*represents the Airy beam,

*k*is the conjugate variable (spatial frequency) corresponding to the spatial coordinate

*x*, and

*q*=[ α/(6π)]

^{1/3}. The phase optical field Ψ(

*x*) is generated with a phase SLM [26], and the Fourier transform of the phase field Ψ(

*x*) [in Eq. (12)] is implemented with a lens of positive power. The experimental setup, depicted in Fig. 3 , shows a beam expander (BE) that conditions the beam of a solid state laser (verdi V8, 532 nm) to illuminate the total active area of the SLM (≅1cm

^{2}). The illumination on the SLM is oblique, avoiding the necessity of a beam splitter. The lens (L) that produces the Fourier transform of the cubic phase has a focal distance f = 50cm. The dimensions of the solution container (SC) were 60 cm (length), 2.5 cm (height) and 5 cm (width). The back focal plane of the lens, where the Airy beam is obtained, coincides with the front end of the SC. The parameter α=1.287x10

^{10}μm

^{3}, of the generated phase modulation Ψ(x), was chosen to achieve an Airy beam deflection of approximately 2 mm in a propagation distance of 600mm, when no sugar is added to the water. We employed 3 different concentrations of sugar: C

_{1}(with no added sugar), C

_{2}(with 6.5 gr of sugar) and C

_{3}(with 13 gr of sugar). To obtain the concentrations C

_{2}and C

_{3}, the water with the added sugar is gently stirred for approximately 15 seconds. After that, the solution is allowed to stabilize before capturing images of the propagated beams.

*x*/Δ

*z*

^{2}= (

*n*

_{1}/2

*n*

_{0}

^{2}) where Δ

*x*is the vertical deviation of the beam and Δz is the length of the container. Notice that from this expression we can obtain the gradient index strength coefficient

*n*

_{1}. The estimated accelerations for the experimentally generated Airy beams, in Fig. 4 (b,d,f), are 5.5×10-6 mm-1, 0, and −13.8×10-6 mm-1, respectively.

12. D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett. **21**(18), 1460–1462 (1996). [CrossRef] [PubMed]

16. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. **36**(7), 1164–1166 (2011). [CrossRef] [PubMed]

## 4. Conclusions

## Acknowledgements

## References and links

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

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

3. | J. Durnin, J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

4. | J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

5. | D. DeBeer, S. R. Hartmann, and R. Friedberg, “Comment on “Diffraction-free beams”,” Phys. Rev. Lett. J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Durnin, Miceli, and Eberly reply,” Phys. Rev. Lett. |

6. | S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. |

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

8. | M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. |

9. | C. López-Mariscal, M. Bandres, J. Gutiérrez-Vega, and S. Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express |

10. | D. Marcuse, “TE modes of graded index slab waveguides,” IEEE J. Quantum Electron. |

11. | C.-L. Chen, |

12. | D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett. |

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

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

15. | I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. 35(10), 1581–1583 (2010). Ye, Zhuoyi; Liu, Sheng; Lou, Cibo; Zhang, Peng; Hu, Yi; Song, Daohong; Zhao, Jianlin; Chen, Zhigang, Quantum Electronics and Laser Science Conference (QELS) 2011 paper: JTuI32, OSA Technical Digest (CD). |

16. | W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. |

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

18. | There exist several forms for producing a solution of this equation, for instance by using the Zassenhaus formula [19], or by simplifying the differential equation via a transformation (see for instance [20] for a quadratic term). The operators to disentangle the exponential of the sum of two operators can also be given in different orderings of the exponentials involved. Of course, they are all equivalent. |

19. | R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. |

20. | H. Moya-Cessa and M. Fernández Guasti, “Coherent states for the time dependent harmonic oscillator: the step function,” Phys. Lett. A |

21. | W. H. Louissel, |

22. | H. Moya-Cessa and F. Soto-Eguibar, |

23. | G. N. Watson, |

24. | W. M. Strouse, “Bouncing light beams,” Am. J. Phys. |

25. | D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori, “Bouncing light beams and the Hamiltonian analogy,” Eur. J. Phys. |

26. | SLM 512, Boulder nonlinear systems. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(080.1510) Geometric optics : Propagation methods

(080.5692) Geometric optics : Ray trajectories in inhomogeneous media

(070.7345) Fourier optics and signal processing : Wave propagation

(260.2710) Physical optics : Inhomogeneous optical media

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 15, 2011

Revised Manuscript: July 28, 2011

Manuscript Accepted: July 28, 2011

Published: August 11, 2011

**Citation**

Sabino Chávez-Cerda, Ulises Ruiz, Victor Arrizón, and Héctor M. Moya-Cessa, "Generation of Airy solitary-like wave beams by acceleration control in inhomogeneous media," Opt. Express **19**, 16448-16454 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16448

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### References

- G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]
- 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]
- J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]
- J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987). [CrossRef]
- D. DeBeer, S. R. Hartmann, and R. Friedberg, “Comment on “Diffraction-free beams”,” Phys. Rev. Lett. 59(22), 2611 (1987).J. Durnin, J. J. Miceli, and J. H. Eberly, “Durnin, Miceli, and Eberly reply,” Phys. Rev. Lett. 59(22), 2612 (1987). [CrossRef] [PubMed]
- S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).
- M. V. Berry and N. L. Balazs, “Nonspreading wave-packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]
- M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004). [CrossRef] [PubMed]
- C. López-Mariscal, M. Bandres, J. Gutiérrez-Vega, and S. Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express 13(7), 2364–2369 (2005). [CrossRef] [PubMed]
- D. Marcuse, “TE modes of graded index slab waveguides,” IEEE J. Quantum Electron. 9(10), 1000–1006 (1973). [CrossRef]
- C.-L. Chen, Foundations of guided-wave optics (Wiley, New Jersey 2006), Ch 3.
- D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett. 21(18), 1460–1462 (1996). [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]
- 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]
- I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. 35(10), 1581–1583 (2010). Ye, Zhuoyi; Liu, Sheng; Lou, Cibo; Zhang, Peng; Hu, Yi; Song, Daohong; Zhao, Jianlin; Chen, Zhigang, Quantum Electronics and Laser Science Conference (QELS) 2011 paper: JTuI32, OSA Technical Digest (CD).
- W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. 36(7), 1164–1166 (2011). [CrossRef] [PubMed]
- M. Born and E. Wolf, Principles of Optics, Seventh Ed., (Cambridge University Press, Cambridge 1999), Ch. 3.
- There exist several forms for producing a solution of this equation, for instance by using the Zassenhaus formula [19], or by simplifying the differential equation via a transformation (see for instance [20] for a quadratic term). The operators to disentangle the exponential of the sum of two operators can also be given in different orderings of the exponentials involved. Of course, they are all equivalent.
- R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8(4), 962–982 (1967). [CrossRef]
- H. Moya-Cessa and M. Fernández Guasti, “Coherent states for the time dependent harmonic oscillator: the step function,” Phys. Lett. A 311(1), 1–5 (2003). [CrossRef]
- W. H. Louissel, Quantum Statistical Properties of Radiation (Wiley-Interscience, New York 1990), Ch. 3.
- H. Moya-Cessa and F. Soto-Eguibar, Differential equations: an operational approach, (Rinton Press, New Jersey 2011), Ch. 2.
- G. N. Watson, A treatise on the theory of Bessel functions, Ch. VI, Cambridge University Press, Cambridge 1944).
- W. M. Strouse, “Bouncing light beams,” Am. J. Phys. 40(6), 913–914 (1972). [CrossRef]
- D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori, “Bouncing light beams and the Hamiltonian analogy,” Eur. J. Phys. 18(4), 284–289 (1997). [CrossRef]
- SLM 512, Boulder nonlinear systems.

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