## Finite-energy, accelerating Bessel pulses

Optics Express, Vol. 16, Issue 24, pp. 19807-19811 (2008)

http://dx.doi.org/10.1364/OE.16.019807

Acrobat PDF (198 KB)

### Abstract

We numerically investigate the possibility to generate freely accelerating or decelerating pulses. In particular it is shown that acceleration along the propagation direction *z* may be obtained by a purely spatial modulation of an input Gaussian pulse in the form of finite-energy Bessel pulses with a cone angle that varies along the radial coordinate. We discuss simple practical implementations of such accelerating Bessel beams.

© 2008 Optical Society of America

## 1. Introduction

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

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

4. P. Saari, “Laterally accelerating airy pulses,” Opt. Express **16**, 10303–10308 (2008) [CrossRef] [PubMed]

*et al.*[5

5. J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**, 1499 (1987). [CrossRef] [PubMed]

7. V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature (London) **419**, 145–147 (2002). [CrossRef]

8. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. **46**, 15–28 (2005). [CrossRef]

*z*. However this requires infinite energy so that practical implementations rely on finite-energy Bessel beams with a spatial apodization of some kind, usually Gaussian (giving the so-called Bessel-Gauss beam). We may therefore call the propagation “sub-diffractive” in the sense that the central intensity peak diffracts at a much slower rate if compared to the diffraction of a Gaussian beam with the same width. A fundamental property of Bessel beams is the conical distribution of wave-vectors such that all of the beam wave-vectors form an identical angle,

*θ*, with respect to the propagation axis. This leads to a superluminal phase velocity

*ν*

_{ϕ}(

*ω*

_{0})=

*c*/(

*n*(

*ω*

_{0})cos

*θ*), where

*c*is the velocity of light and

*n*(

*ω*

_{0}) is the medium refractive index at the carrier frequency

*ω*

_{0}. In the pulsed regime we should consider also the group velocity along the propagation direction

*z*,

*ν*

*. This will depend, among other parameters, on the specific method by which the Bessel pulse is obtained. For example a Bessel pulse generated by an axicon (conical lens) will have*

_{g}*ν*

*=*

_{g}*ν*

*/cos*

_{G}*θ*(Bessel-X pulse) [9

9. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. **79**, 4135 (1997) [CrossRef]

10. H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. **22**, 310–312 (1997) [CrossRef] [PubMed]

11. Z. Liu and D. Fan, “Propagation of pulsed zeroth-order Bessel beams, ” J. Mod. Opt. **45**, 17–22 (1998). [CrossRef]

*ν*

*=*

_{g}*ν*

*G*cos

*θ*, where

*νG*=1/(

*dk*/

*dω*|

*ω*

_{0}) is the Gaussian-pulse group velocity determined by the material first order dispersion evaluated at

*ω*

_{0}. Different dependencies on the Bessel cone angle

*θ*or even negative group velocities may be obtained by modifying the pulse front tilt [13

13. C. J. Zapata-Rodriguez and M. A. Porras, “X-wave bullets with negative group velocity in vacuum,” Opt. Lett. **31**, 3532–3534 (2006). [CrossRef] [PubMed]

*z*may be varied by choosing the appropriate axicon profile. Therefore these pulses exhibit arbitrary acceleration rates whilst maintaining the sub-diffractive nature of the Bessel beam.

14. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. **A 21**, 53–58 (2004). [CrossRef]

17. U. Fuchs, U. D. Zeitner, and A. Tünnermann, “Ultra-short pulse propagation in complex optical systems,” Opt. Express **13**, 3852–3861 (2005). [CrossRef] [PubMed]

## 2. Decelerating Bessel pulses

17. U. Fuchs, U. D. Zeitner, and A. Tünnermann, “Ultra-short pulse propagation in complex optical systems,” Opt. Express **13**, 3852–3861 (2005). [CrossRef] [PubMed]

*z*with a larger angle than the paraxial rays (see Fig. 1(a)) and this will lead to distortion of the spatio-temporal profile of the propagating pulse. In particular a leading Gaussian pulse will be formed, followed by a trailing Bessel-X pulse [15, 16, 17

17. U. Fuchs, U. D. Zeitner, and A. Tünnermann, “Ultra-short pulse propagation in complex optical systems,” Opt. Express **13**, 3852–3861 (2005). [CrossRef] [PubMed]

*ν*

*is largest. As the pulse moves toward the paraxial focus the angles decrease and the Bessel-X pulse gradually slows down until it reaches the group velocity vG of the leading Gaussian component. Figure 1(b) shows the numerically evaluated*

_{g}*ν*

*=*

_{g}*ν*

*/cos*

_{G}*θ*of the Bessel-X pulse generated between the marginal and paraxial focii of an f=43 mm lens (

*θ*=

*θ*(

*z*) is evaluated by ray-tracing and then substituted into the equation for the group velocity). Figure 1(c) shows the numerically evaluated spatio-temporal pulse profile at a distance z=3.2 cm after the lens when this is illuminated by a ring so as to eliminate the paraxial rays and thus create an isolated decelerating Bessel-X pulse. Indeed, by changing the input beam shape, i.e. by using a ring-shaped illumination and by changing the profile of the ring (e.g. apodization and radius) it is possible to select the illuminated portion of the propagation axis and therefore the properties, such as the maximum and minimum

*ν*

*values and the intensity evolution versus*

_{g}*z*, of the Bessel-X pulse. On the other hand, by changing the lens characteristics, e.g. focal length, it is possible to tune the deceleration rate versus

*z*.

## 3. Acelerating Bessel pulses

*c*to ~4/3

*c*in a few cms. In Fig. 2(c) we consider in detail the linearly accelerating Bessel-X pulse which is generated by illuminating with a 10 cm radius (at 1/e) Gaussian pulse, a nonlinear axicon approximated by a ninth-power polynomial,

*r*=∑

*a*

_{i}*z*

^{i}with

*i*=0…9 and

*a*

_{0}=1.59810

^{2},

*a*

_{1}=-1.128,

*a*

_{2}=6.05710

^{-3},

*a*

_{3}=-4.52810

^{-4},

*a*

^{4}=1.73510

^{-5},

*a*

_{5}=-3.80410

^{-7},

*a*

_{6}=4.94910

^{-9},

*a*

_{7}=-3.77410

^{-11},

*a*

_{8}=1.55710

^{-13},

*a*

_{9}=-2.68610

^{-16}. The spatio-temporal profile is clearly that of a Bessel-X pulse and it maintains the same profile over the whole Bessel zone reaching a maximum group velocity of

*ν*

*≃3.9×10*

_{g}^{8}m/s. This is in stark contrast with the Airy pulse that substantially modifies its profile during propagation. However it is also true that the acceleration described here is fruit of the peculiar conical flux within the Bessel pulse and therefore requires at least two dimensions, i.e. the temporal dimension and at least one transverse spatial dimension, whereas the Airy pulse is possible in just a single dimension, e.g. in a fiber.

*c*/

*ω*)×2.4048/sin

*θ*. So, for example, the accelerating Bessel pulse shown in Fig. 2(c) has an initial cone angle (at the axicon tip) of 8 deg and a beam peak diameter of 7

*µ*m. The final cone angle at the end of the 5 cm Bessel zone is 40 deg and the peak diameter is 1.5

*µ*m: this Bessel zone is huge if compared to the ~250

*µ*m Rayleigh range of a Gaussian pulse with the same initial diameter.

18. D. Fischer, C. Harkrider, and D. Moore, “Design and manufacture of a gradient-index axicon,” Appl. Opt. **39**, 2687–2694 (2000). [CrossRef]

*z*explains why the peak acceleration takes place.

## 4. Conclusions

20. H. Sonajalg and P. Saari, “Suppression of temporal spread of ultrashort pulses in dispersive media by bessel beam generators,” Opt. Lett. **21**, 1162–1164 (1996). [CrossRef] [PubMed]

## Acknowledgment

## References and links

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

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

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

4. | P. Saari, “Laterally accelerating airy pulses,” Opt. Express |

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

6. | A. Averchi, D. Faccio, R. Berlasso, M. Kolesik, J. V. Moloney, A. Couairon, and P. Di Trapani, “Phase matching with pulsed bessel beams for high-order harmonic generation,” Phys. Rev. |

7. | V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature (London) |

8. | D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. |

9. | P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. |

10. | H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. |

11. | Z. Liu and D. Fan, “Propagation of pulsed zeroth-order Bessel beams, ” J. Mod. Opt. |

12. | S. Longhi, D. Janner, and P. Laporta, “Propagating pulsed Bessel beams in periodic media,” J. Opt. |

13. | C. J. Zapata-Rodriguez and M. A. Porras, “X-wave bullets with negative group velocity in vacuum,” Opt. Lett. |

14. | M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. |

15. | M. Kempe and W. Rudolph, “Femtosecond pulses in the focal region of lenses,” Phys. Rev. |

16. | Z. Horvath and Z. Bor, “Diffraction of short pulses with boundary diffraction wave theory,” Phys. Rev. |

17. | U. Fuchs, U. D. Zeitner, and A. Tünnermann, “Ultra-short pulse propagation in complex optical systems,” Opt. Express |

18. | D. Fischer, C. Harkrider, and D. Moore, “Design and manufacture of a gradient-index axicon,” Appl. Opt. |

19. | M. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. |

20. | H. Sonajalg and P. Saari, “Suppression of temporal spread of ultrashort pulses in dispersive media by bessel beam generators,” Opt. Lett. |

**OCIS Codes**

(080.3630) Geometric optics : Lenses

(320.5540) Ultrafast optics : Pulse shaping

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: August 25, 2008

Revised Manuscript: September 28, 2008

Manuscript Accepted: October 16, 2008

Published: November 14, 2008

**Citation**

M. Clerici, D. Faccio, A. Lotti, E. Rubino, O. Jedrkiewicz, J. Biegert, and P. Di Trapani, "Finite-energy, accelerating Bessel pulses," Opt. Express **16**, 19807-19811 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-24-19807

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### References

- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating airy beams," Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]
- G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Ballistic dynamics of airy beams," Opt. Lett. 33, 207-209 (2008). [CrossRef] [PubMed]
- G. A. Siviloglou and D. N. Christodoulides, "Accelerating finite energy airy beams," Opt. Lett. 32, 979-981 (2007). [CrossRef] [PubMed]
- P. Saari, "Laterally accelerating airy pulses," Opt. Express 16, 10303-10308 (2008) [CrossRef] [PubMed]
- J. Durnin, J. Miceli, and J. H. Eberly,"Diffraction-free beams," Phys. Rev. Lett. 58, 1499 (1987). [CrossRef] [PubMed]
- A. Averchi, D. Faccio, R. Berlasso, M. Kolesik, J. V. Moloney, A. Couairon, and P. Di Trapani, "Phase matching with pulsed bessel beams for high-order harmonic generation," Phys. Rev. A 77, 021802 (2008).
- V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature (London) 419, 145-147 (2002). [CrossRef]
- D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005). [CrossRef]
- P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135 (1997) [CrossRef]
- H. Sonajalg, M. Ratsep and P. Saari, "Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium," Opt. Lett. 22, 310-312 (1997) [CrossRef] [PubMed]
- Z. Liu and D. Fan, "Propagation of pulsed zeroth-order Bessel beams, " J. Mod. Opt. 45, 17-22 (1998). [CrossRef]
- S. Longhi, D. Janner and P. Laporta, "Propagating pulsed Bessel beams in periodic media," J. Opt. B 6, 477-481 (2004)
- C. J. Zapata-Rodriguez and M. A. Porras, "X-wave bullets with negative group velocity in vacuum," Opt. Lett. 31, 3532-3534 (2006). [CrossRef] [PubMed]
- M. Guizar-Sicairos and J. C. Gutierrez-Vega, "Computation of quasi-discrete hankel transforms of integer order for propagating optical wave fields," J. Opt. Soc. Am. A 21, 53-58 (2004). [CrossRef]
- M. Kempe and W. Rudolph, "Femtosecond pulses in the focal region of lenses," Phys. Rev. A 48, 4721 (1993).
- Z. Horvath and Z. Bor, "Diffraction of short pulses with boundary diffraction wave theory," Phys. Rev. E 63, 026601 (2001).
- U. Fuchs, U. D. Zeitner, and A. Tunnermann, "Ultra-short pulse propagation in complex optical systems," Opt. Express 13, 3852-3861 (2005). [CrossRef] [PubMed]
- D. Fischer, C. Harkrider, and D. Moore, "Design and manufacture of a gradient-index axicon," Appl. Opt. 39, 2687-2694 (2000). [CrossRef]
- M. Porras and P. Di Trapani, "Localized and stationary light wave modes in dispersive media," Phys. Rev. E 69, 066606 (2004).
- H. Sonajalg and P. Saari, "Suppression of temporal spread of ultrashort pulses in dispersive media by bessel beam generators," Opt. Lett. 21, 1162-1164 (1996). [CrossRef] [PubMed]

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