## Observation of accelerating parabolic beams

Optics Express, Vol. 16, Issue 17, pp. 12866-12871 (2008)

http://dx.doi.org/10.1364/OE.16.012866

Acrobat PDF (616 KB)

### Abstract

We report the first observation of accelerating parabolic beams. These accelerating parabolic beams are similar to the Airy beams because they exhibit the unusual ability to remain diffraction-free while having a quadratic transverse shift during propagation. The amplitude and phase masks required to generate these beams are encoded onto a single liquid crystal display. Experimental results agree well with theory.

© 2008 Optical Society of America

## 1. Introduction

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

2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Physical Review Letters **99**, 213901 (2007). [CrossRef]

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

3. I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. **32**, 2447–2449 (2007). [CrossRef] [PubMed]

5. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” American Journal of Physics **47**, 264–267 (1979). [CrossRef]

*x*

^{-1/2}while pure Airy beams as

*x*

^{-1/4}as

*x*→-∞) and might be more suited for practical applications.

3. I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. **32**, 2447–2449 (2007). [CrossRef] [PubMed]

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

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

## 2. Accelerating parabolic beams

*η*,

*ξ*) are parabolic coordinates defined as

*κ*is a transverse scale and

*k*is the wave number. The functions Θ

*(*

_{n}*η*), where

*n*=0,1,2, …, correspond to square integrable eigensolutions of the quartic oscillator equation [8

8. K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, “The anharmonic oscillator,” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences **360**, 575–586 (1978). [CrossRef]

*n*zeros and definite parity given by the parity of

*n*. At

*z*=0 the parameter

*a*controls the exponential aperture function, exp(

*ax*/

*κ*), that ensures the containment of the accelerating parabolic beams. Therefore for finite-energy beams it is necessary that Re(

*a*) >0.

*a*≪1 the accelerating parabolic beams have a quasidiffraction-less behavior over long distances that increase as

*a*decreases. The case

*a*=0 corresponds to the pure accelerating parabolic beams that are perfectly diffraction-free but have infinite energy. From Eq. (2) we can see that during propagation the accelerating parabolic beams exhibit a transverse shift given by

*z*=0 is given by [6]

*k*,

_{x}*k*) are the spatial frequency coordinates. The parameter

_{y}*κ*controls the cubic phase that characterizes the Fourier spectrum as well as the Gaussian amplitude factor. Note that this parameter also controls the transverse shift as seen in Eq. (3).

*n*={0,3,6} with

*a*=0.02. There are two critical differences between the Fourier spectrum of the accelerating parabolic beams and the two-dimensional Airy beams. First, the phase patterns are much more complicated for large values of

*n*. More importantly, the amplitude modulation is not a simple Gaussian as with the Airy beams. Not only it is not circularly symmetric, but there is an extra modulation in the

*k*-axis given by Θ

_{y}*(√2*

_{n}*k*). The Gaussian factor exp[-

_{y}κ*aκ*

^{3}(

*k*

^{2}

*+*

_{x}*k*

^{2}

*)] in the Fourier spectrum assures that the beam carries finite energy if Re(*

_{y}*a*)>0.

## 3. Experiment

2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Physical Review Letters **99**, 213901 (2007). [CrossRef]

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

9. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. **38**, 5004–5013 (1999). [CrossRef]

10. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. **31**, 649–651 (2006). [CrossRef] [PubMed]

11. J. A. Davis, C. S. Tuvey, O. López-Coronado, J. Campos, M. J. Yzuel, and C. Iemmi, “Tailoring the depth of focus for optical imaging systems using a Fourier transform approach,” Opt. Lett. **32**, 844–846 (2007). [CrossRef] [PubMed]

*M*(

_{p}*x*,

*y*) and phase

*ϕ*(

_{p}*x*,

*y*) and written as

*M*(

_{p}*x*,

*y*)exp[

*iϕ*(

_{p}*x*,

*y*)]. In this approach, we combine the phase pattern with a vertically oriented linear phase grating

*ϕ*(

_{G}*y*)=2

*πy*/

*d*with period

*d*as

*M*(

_{p}*x*,

*y*) exp[

*iϕ*(

_{p}*x*,

*y*)+

*iϕ*(

_{G}*y*)]. The total phase is the sum of the parabolic phase term with the grating phase. Amplitude information is then encoded by spatially modulating the phase pattern with the amplitude portion of the desired pattern [9

9. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. **38**, 5004–5013 (1999). [CrossRef]

10. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. **31**, 649–651 (2006). [CrossRef] [PubMed]

11. J. A. Davis, C. S. Tuvey, O. López-Coronado, J. Campos, M. J. Yzuel, and C. Iemmi, “Tailoring the depth of focus for optical imaging systems using a Fourier transform approach,” Opt. Lett. **32**, 844–846 (2007). [CrossRef] [PubMed]

*iM*′

*(*

_{p}*x*,

*y*) [

*ϕ*(

_{p}*x*,

*y*)+

*ϕ*(

_{G}*y*)]. The intensity that is diffracted into the first order varies spatially and reproduces the desired amplitude and phase distribution. There is a slight distortion that is corrected by modifying the amplitude term

*M*′

*(*

_{p}*x*,

*y*) as described in Ref. [9

9. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. **38**, 5004–5013 (1999). [CrossRef]

*ϕ*=

*ϕ*(

_{p}*x*,

*y*)+

*ϕ*(

_{G}*y*) in the range [-

*π*,

*π*] while the amplitude is defined in the range [0≤

*M*′

*(*

_{p}*x*,

*y*)≤1]. The zero order diffraction produces an undesired term that can be spatially filtered.

11. J. A. Davis, C. S. Tuvey, O. López-Coronado, J. Campos, M. J. Yzuel, and C. Iemmi, “Tailoring the depth of focus for optical imaging systems using a Fourier transform approach,” Opt. Lett. **32**, 844–846 (2007). [CrossRef] [PubMed]

*n*=0 and where

*κ*=68.5

*µ*m for both beams.

*n*={0,3,6},

*a*=0.02 and

*κ*=68.5

*µ*m, at propagation distances of

*z*={0,15,30,45,60} cm. The width of the image plane corresponds to ≈4 mm. As expected, the beams remain almost diffraction free for ≈45 cm and have a transverse quadratic right shift during propagation. The direction of the transverse shift can be reversed using the complex conjugate of the phase mask. The theoretical behavior is in good agreement with the experimental results and therefore is not shown.

*a*of the beams. In Fig. 4 we show the experimental results for the propagation of the accelerating parabolic beams with

*n*=4,

*a*={0.01,0.05,0.1} and

*κ*=68.5

*µ*m, at propagation distances

*z*={0,15,30,45,60} cm. These experimental results show clearly that, as the parameter

*a*decreases, the beams propagate almost diffraction-free over longer distances. For example the beams with

*a*={0.01,0.05,0.1} start diffracting significantly at

*z*≈{60,45,30} cm, respectively, as can be seen from Fig. 4.

*n*=0 accelerating parabolic beam with

*a*=0.01 and

*κ*=68.5

*µ*m from

*z*=0 to 70 cm. We show three curves with different initial launch angles of velocities using values of

*θ*=0,+2.1,-4.2 mrad.

*µ*m. The data fits the quadratic depencence on propagation distance from Eq. (5) using the parameter

*κ*=68.6

_{exp}*µ*m in excellent agreement with the theoretical value. Beams with the same parameters,

*κ*and

*a*, but different orders,

*n*, propagate in exactly the same way and therefore they are not shown.

## 4. Conclusion

## Acknowledgments

## 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,” Physical Review Letters |

3. | I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. |

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

5. | M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” American Journal of Physics |

6. | M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett., doc. ID 96139 (posted 20 June 2008, in press). |

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

8. | K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, “The anharmonic oscillator,” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences |

9. | J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. |

10. | J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. |

11. | J. A. Davis, C. S. Tuvey, O. López-Coronado, J. Campos, M. J. Yzuel, and C. Iemmi, “Tailoring the depth of focus for optical imaging systems using a Fourier transform approach,” Opt. Lett. |

12. | J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Optical Engineering |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(100.5090) Image processing : Phase-only filters

(140.3300) Lasers and laser optics : Laser beam shaping

(230.6120) Optical devices : Spatial light modulators

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 10, 2008

Revised Manuscript: July 9, 2008

Manuscript Accepted: July 13, 2008

Published: August 8, 2008

**Citation**

Jeffrey A. Davis, Mark J. Mintry, Miguel A. Bandres, and Don M. Cottrell, "Observation of accelerating parabolic beams," Opt. Express **16**, 12866-12871 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-12866

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### References

- G. A. Siviloglou and D. N. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 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, 213901 (2007). [CrossRef]
- I. M. Besieris and A. M. Shaarawi, "A note on an accelerating finite energy Airy beam," Opt. Lett. 32, 2447-2449 (2007). [CrossRef] [PubMed]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Ballistic dynamics of Airy beams," Opt. Lett. 33, 207-209 (2008). [CrossRef] [PubMed]
- M. V. Berry and N. L. Balazs, "Nonspreading wave packets," Am. J. Phys. 47, 264-267 (1979). [CrossRef]
- M. A. Bandres, "Accelerating parabolic beams," Opt. Lett., doc. ID 96139 (posted 20 June 2008, in press).
- M. A. Bandres and J. C. Gutierrez-Vega, "Airy-Gauss beams and their transformation by paraxial optical systems," Opt. Express 15, 16719-16728 (2007). [CrossRef] [PubMed]
- K. Banerjee, S. P. Bhatnagar, V. Choudhry, and S. S. Kanwal, "The anharmonic oscillator," Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 360, 575-586 (1978). [CrossRef]
- J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, "Encoding amplitude information onto phase-only filters," Appl. Opt. 38, 5004-5013 (1999). [CrossRef]
- J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutierrez-Vega, "Generation of helical Ince-Gaussian beams with a liquid-crystal display," Opt. Lett. 31, 649-651 (2006). [CrossRef] [PubMed]
- J. A. Davis, C. S. Tuvey, O. Lopez-Coronado, J. Campos, M. J. Yzuel, and C. Iemmi, "Tailoring the depth of focus for optical imaging systems using a Fourier transform approach," Opt. Lett. 32, 844-846 (2007). [CrossRef] [PubMed]
- J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, "Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects," Opt. Eng. 38, 1051-1057 (1999). [CrossRef]

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