## Causality effects on accelerating light pulses |

Optics Express, Vol. 19, Issue 23, pp. 23132-23139 (2011)

http://dx.doi.org/10.1364/OE.19.023132

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### Abstract

We study accelerating and decelerating shape-preserving temporal Airy wave-packets propagating in dispersive media. We explore the effects of causality, and find that, whereas decelerating pulses can asymptotically reach zero group velocity, pulses that accelerate towards infinite group velocity inevitably break up, after a specific critical point. The trajectories and the features of causal pulses are analyzed, along with the requirements for the existence of the critical point and experimental schemes for its observation. Finally, we show that causality imposes similar effects on accelerating pulses in the presence of local Kerr-like nonlinearities.

© 2011 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 and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

3. 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]

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

## 2. Derivation and trajectories

*k”*, and negligible higher dispersion, the slowly-varying envelope approximation giveswhere

*ψ*=

*ψ(z,t)*is the (complex) envelope of the wavepacket, and

*γ = γ (|ψ|*is the nonlinear response, which throughout this paper is assumed to be a function of the pulse intensity

^{2})*|ψ|*only. From this point, we follow the derivation of our previous paper [10

^{2}10. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. **106**(21), 213903 (2011). [CrossRef] [PubMed]

**accelerating**solution of Eq. (1) aswhere

*A*is some function of

*t*(see [10

10. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. **106**(21), 213903 (2011). [CrossRef] [PubMed]

*v*is the group velocity evaluated at the carrier frequency of the pulse. On the other hand

_{0}*t*is an arbitrary shift in time. The phase term is a real function, given aswhere

_{0}*T*is a scaling parameter (say, the width of the main “lobe” in the temporal “pulse”). Figure 1a presents examples of

*T<0*(inverting it right-left would make it represent

*T>0*). Hence, for

*T*<0, generating the pulse means increasing the amplitude exponentially, then slowing the increase until the maximum is reached, and then decreasing it in an oscillatory manner – following the shape of the Airy function. As we have shown in our paper on nonlinear accelerating beams [10

10. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. **106**(21), 213903 (2011). [CrossRef] [PubMed]

*T>0*and

*T<0*determines which comes first: the exponentially decaying tail (

*T<0*) or the oscillatory tail (

*T>0*). The major difference between these two choices is easily understood when deriving the group velocity of the beam:

*T>0*, the group velocity decreases with distance (hence also in time), therefore, the wavepacket decelerates to zero group velocity, which is reached after an infinite distance (Fig. 2a ). The opposite case, of

*T<0*, has an increasing velocity (Fig. 2b). Here, the wavepacket accelerates, and surprisingly, should reach an infinite speed after a finite distance of

*z*. We call this point the “critical point”. Note that the critical point also has a meaning in the decelerating case – it is the point where the pulse slowed to exactly half of its initial velocity.

_{critical}= 2|T|^{3}/(k”^{2}v_{0})*γ = γ (|ψ|*.

^{2})## 3. Solving the mystery: what actually happens at the vicinity of the critical point?

*z*-axis, with opposite group velocities, and meet exactly at the critical point (see Figs. 1b,c, where the dashed white line marks the critical point). Figure 1b and 1c shows the propagation of the wavepacket comprised only of frequency components of the Airy pulse that have positive (negative) group velocity. What seems in Fig. 1a as causality breaking (above the dashed line) is simply the trajectory of the backward-moving pulse, because Fig. 1a is simply the superposition of Figs. 1b and 1c. The infinite slope of the trajectory at the critical point is observed as the interference between the tails of the two pulses when they reach their meeting point. Another intuitive explanation is that the wavepacket at any specific plane

*z*is composed of both forward and backward propagating waves. Mathematically, this arises from Eq. (1) being a second-order differential equation. Therefore, the seemingly causality-breaking is due to backward propagating waves that are assumed in the initial condition (as in Fig. 1c). This issue will be discussed in more details in the next section.

*z =*0 has only forward propagating frequencies components (those with positive group velocity). Hence, the physical Airy pulse is actually a “half-Airy” pulse. This way, the pulse would not interfere with a backward propagating twin at the critical point. The actual dynamics is therefore not the propagation-invariant Airy solution of Eq. (1). This peculiarity – that a single Airy pulse does not have the full dynamics of an Airy solution – leads to an interesting question: What happens at the critical point when only one pulse arrives there? We expect the Airy pulse to propagate as a shape-preserving wave structure until the critical point, because it has no interaction with the non-existing twin. But then, exactly at the critical point and beyond it, the propagation-invariant Airy dynamics should break down and the pulse should suddenly change its shape.

*z =*0, after cutting out (in the Fourier space) the backward propagating frequencies. This yields the dynamics of the “half-Airy” pulse. The results are shown in Fig. 1b, and exhibit a rapid breakup of the pulse (within ~0.1

*mm*in this example), that up to the critical point has been shape-preserving. The breakup occurs within a time frame

*T*(hence the breakup distance is

*cT*), where T, as defined above, is approximately equal to one half of the width of the main “lobe” of the Airy pulse. That is to say that the breakup distance tends to be much shorter than the dispersion length of pulses of a spectral width that can be well described within the framework of Eq. (1). To put this breakup distance, (

*cT*), on quantitative grounds, it is instructive to compare it to the 'temporal Rayleigh length' near the critical point, which is approximately equal to the dispersion length of the first lobe (

*T*). Therefore, in order to call the breakup 'abrupt', the requirement is

^{2}/k”*k”c<<T*. This condition is easily fulfilled when put together with the conditions in the next section.

## 4. Critical point: existence conditions

*z*might be too far for a realistic experimental system, or require an extremely long pulse to observe the breakup. Also, the necessary frequency spectrum of the pulse might be too wide and come too close to the carrier frequency. This section comes to show that the critical point truly exists, and to investigate the necessary and sufficient conditions for the observation of the breakup and the critical point.

_{critical}*ω*, which is found by deriving the group velocity and setting it to zero. This procedure is simple when we neglect dispersion terms above the second order, and gives

_{c}*ω*. To see the critical point, a non-negligible part of the power of the pulse should have its frequency above

_{c}= 1/(k”v_{0})*ω*. In addition, the carrier frequency

_{c}*ω*must be much larger than

_{0}*ω*, to allow the generation of such a pulse. From

_{c}*ω*, we get a condition that does not depend on the pulse itself, but depends only on the medium in which the pulse is propagating, yielding

_{0}>>ω_{c}*z*. Given a physical truncated pulse which accelerates and is shape-preserving for a finite distance z

_{critical}_{max}, let us compare z

_{max}with

*z*. For convenience of the calculation, let us use the truncation method suggested in [2

_{critical}2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

**32**(8), 979–981 (2007). [CrossRef] [PubMed]

*a*for the decay rate and multiplying the wavepacket in the linear system by

*exp(a*, we get an effective Gaussian window in the Fourier plane

^{2}t/T)*exp(-a*. This Gaussian window is drawn in Fig. 2c, along with

^{2}ω^{2}T^{2})*ω*and

_{0}*ω*which appear as vertical lines. To allow the generation of an accelerating beam which arrives at the critical point and breaks up there, the width of the spectrum,

_{c}*a*, should be larger than

^{−1}T^{−1}*ω*and at the same time much smaller than

_{c}*ω*. The former condition is to ensure that part of the power will have a negative group velocity, and the latter is to ensure that the resolution of the temporal structure does not fall close to within a single oscillation period

_{0}*2π/ω*. Altogether, the ability to observe the critical point necessitates

_{0}*a*loses its meaning, because the spectral comparison is not valid (and of course the pulse is not an Airy). To find the analogous constrains for the nonlinear case, we recall that

*a*controls the distance of acceleration. The condition (6) arises from requiring that the finite acceleration distance will be larger than the critical distance. To find the acceleration distance in a nonlinear case, we qualitatively draw the line from the end of the tail tangent to the trajectory of the peak. Figures 2a and 2b present the trajectory of the wavepacket with the tangent (black dashed line) and the farthest point of acceleration/deceleration marked as

*(t*. The point where the tangent crosses the

_{max}, z_{max})*t*axis is determined by the width of the wavepacket, hence this temporal width,

*∆t*, gives the distance of propagation: The ratio

*∆t/|T|*is qualitatively the number of lobes that the wavepacket carries, and is equal to

*a*. Now, we compare this distance to the value of the critical point and derive the same inequality as in (6).

^{−2}## 5. Physical realization

*c*in expression (5) forces

*k”*to be much larger than 800

*fsec*(otherwise the fraction of power contained in the backward-propagating frequencies is negligible). Unfortunately, “conventional” optical materials (such as glass) have

^{2}/mm*k”*which is too small by an order of magnitude. Other commonly used materials (e.g., SF10 glass at short wavelengths) have their

*k”*around a few hundreds

*fsec*- which is still not good enough, unless we work in the UV. Therefore, it would be necessary to use a medium with a tailored dispersion, where the second-order dispersion is high while the third-order dispersion is small. Such propagation medium would be, for example, photonic crystal fibers with engineered dispersion properties.

^{2}/mm*k”*=10,000

*fsec*and use

^{2}/mm,*λ*=400

*nm*,

*a*=0.1 and

*T*=20

*fsec*. Those parameters are used in the simulations creating Fig. 1a,b,c. In Fig. 1d we take

*k”'*=10,000

*fsec*, which we estimate from the relation between different orders of dispersion far from resonances. Higher values of the high-order dispersion only shift the critical point, without changing the actual phenomenon.

^{3}/mm## 6. Discussion and conclusions

## Acknowledgements

## References and links

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

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

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

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

5. | I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

6. | K.-Y. Kim, C.-Y. Hwang, and B. Lee, “Slow non-dispersing wavepackets,” Opt. Express |

7. | A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics |

8. | D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. |

9. | R. Y. Chiao and P. W. Milonni, “Fast Light, Slow Light,” Opt. Photonics News |

10. | I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. |

11. | Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express |

12. | G. P. Agrawal, |

13. | E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(320.5550) Ultrafast optics : Pulses

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: September 2, 2011

Revised Manuscript: October 11, 2011

Manuscript Accepted: October 12, 2011

Published: October 31, 2011

**Citation**

Ido Kaminer, Yaakov Lumer, Mordechai Segev, and Demetrios N. Christodoulides, "Causality effects on accelerating light pulses," Opt. Express **19**, 23132-23139 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-23132

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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 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]
- P. Saari, “Laterally accelerating airy pulses,” Opt. Express16(14), 10303–10308 (2008). [CrossRef] [PubMed]
- I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.78(4), 046605 (2008). [CrossRef] [PubMed]
- K.-Y. Kim, C.-Y. Hwang, and B. Lee, “Slow non-dispersing wavepackets,” Opt. Express19(3), 2286–2293 (2011). [CrossRef] [PubMed]
- A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics4(2), 103–106 (2010). [CrossRef]
- D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett.105(25), 253901 (2010). [CrossRef] [PubMed]
- R. Y. Chiao and P. W. Milonni, “Fast Light, Slow Light,” Opt. Photonics News13(6), 26–30 (2002). [CrossRef]
- I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett.106(21), 213903 (2011). [CrossRef] [PubMed]
- Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express19(18), 17298–17307 (2011). [CrossRef] [PubMed]
- G. P. Agrawal, Nonlinear Fiber Optics, Optics and Photonics (Academic Press, London, 2001).
- E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett.106(21), 213902 (2011). [CrossRef] [PubMed]

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