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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 23 — Nov. 7, 2011
  • pp: 23132–23139
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Causality effects on accelerating light pulses

Ido Kaminer, Yaakov Lumer, Mordechai Segev, and Demetrios N. Christodoulides  »View Author Affiliations


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

In an intriguing paper from 1979, Berry and Balazs found a shape-preserving self-accelerating solution to the free-particle Schrödinger equation, in the form of an Airy function [1

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

]. Almost 30 years later, Siviloglou and Christodoulides used the analogy between the Schrödinger equation and the optical paraxial wave equation to predict and demonstrate Airy self-accelerating beams [2

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

,3

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]

]. Near the end of their paper [2

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

], they also discussed temporal accelerating wave-packets, utilizing the mathematical equivalence between pulse propagation in dispersive optical fibers and the paraxial wave equation describing the propagation of optical beams in homogenous media. However, there is an important physical difference between spatial and temporal accelerations: although both have the Airy shape, a spatial accelerating beam bends its trajectory in space, whereas only the temporal accelerating pulse truly changes its actual group velocity. Additionally, there is another fundamental difference between the two cases of accelerating wave-packets: the temporally accelerating pulse must obey causality, whereas the spatial Airy beam can bend either way, depending only on its shape at the launch plane.

Here we study the propagation of accelerating and decelerating shape-preserving temporal wave-packets in linear media and in media exhibiting instantaneous and spatially-local Kerr-like nonlinearities. The requirements for the existence of such pulses are analyzed, and a setup with realistic parameters is proposed. We find two regimes. For decelerating pulses, we find that the velocity of the wavepacket begins with the group-velocity of the pulse, and decreases all the way to zero, which it would reach asymptotically after an infinite propagation distance. For accelerating pulses, we show that the velocity of the wavepacket begins with the group velocity of the pulse, and seems to diverge even after a finite propagation distance, where we find that the wavepacket breaks apart. This phenomenon can be observed even in the presence of high order dispersion terms, as it is caused solely because casualty is imposed on the initial conditions. We investigate this point of “diverging speed” and find what happens to the pulse before and after the point. We conclude with applications arising from the abrupt breakup of the Airy pulse.

2. Derivation and trajectories

With this in mind, the choice between 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:

v=dzdt=11v0+k''2z2T3=v01+k''2z2T3v0
(4)

For 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
Fig. 2 Schematic trajectories of (a) a decelerating and (b) an accelerating Airy wavepacket. The dashed tangent lines mark the point where acceleration stops due to the finite extent of the pulse. (c) The spectrum of an Airy pulse truncated at the critical frequency, and compared to the carrier frequency.
). 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 zcritical = 2|T|3/(k”2v0). 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.

We emphasize that, up to this point, the formulation is completely general, without any need to specify the nonlinearity, except for stating that the nonlinear refractive index change is a local function of the optical intensity γ = γ (|ψ|2).

These findings are a direct outcome of the exact solution. They immediately raise the question, how can that be that an exact solution to a physical propagation equation yields wavepackets that could accelerate to infinite velocities? One could argue that perhaps the problem is with the equation, which uses dispersion to second-order only, whereas in all physical systems the dispersion curve is some analytic function, and the second order is merely the leading term in a Taylor expansion [12

12. G. P. Agrawal, Nonlinear Fiber Optics, Optics and Photonics (Academic Press, London, 2001).

]. In other words, would adding third-order dispersion necessarily leads to some bound on the maximum velocity of an accelerating pulse? What happens to causality, how does it enter the evolution equation and how does it affect the solutions? The purpose of this article is to address this kind of questions.

3. Solving the mystery: what actually happens at the vicinity of the critical point?

Let us first handle the linear case of Eq. (1). As shown above, the analytic solution for accelerating shape-preserving solutions of Eq. (1) yields a critical point. At such point, the trajectory of the main lobe (Fig. 1a) has infinite slope, which one may naively interpret as infinite velocity (dashed line). But even more bizarre is the fact that the trajectory continues after this critical point backwards in time (curve above the dashed line). It seems that not only the speed diverges, but also causality is violated. How can this be? What actually happens at the vicinity of the critical point?

In experiments, one launches only one Airy-shaped pulse from only one side of the medium. i.e., the wavepacket at the initial plane 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.

The maximal intensity of the pulse remains constant during propagation until it gets close to the critical point, where the maximal intensity drops. The pulse continues to propagate after the critical point, but the pulse broadens and its maximum intensity decays. Figure 1c displays the dynamics of the opposite pulse, where the forward-propagating frequencies were cut out, hence simulating only the backward-moving part. The superposition of the two counterpropagating pulses is mathematically equivalent to Fig. 1a.

4. Critical point: existence conditions

One might be tempted to suggest that perhaps the critical point is non-physical, but rather an artifact of the approximations involved, such as the fact that the dispersion we use is second-order only. That is, one could justly question whether high-order dispersion could interfere with the breakup phenomenon and prevent it from happening. Moreover, in practical terms, the distance zcritical 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.

To handle the issue of higher-order dispersion, we simulate the propagation of the (non-causal) Airy solution of Eq. (1), in the presence of strong third-order dispersion. The result is presented in Fig. 1d. We find that adding the third-order dispersion indeed changes the propagation dynamics (compare Fig. 1d to Fig. 1a), but the rapid breakup behavior of the pulse is clearly apparent nevertheless.

As mentioned above, part of the spectrum of the Airy pulse gives rise to backward propagating waves, as shown in the example of Fig. 2c. Interestingly, the division between positive and negative group velocities is not symmetric. Let us explain the asymmetry. Denoting the transition point between the two parts of the spectrum as the “critical frequency” ωc, 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 ωc = 1/(k”v0). To see the critical point, a non-negligible part of the power of the pulse should have its frequency above ωc. In addition, the carrier frequency ω0 must be much larger than ωc, to allow the generation of such a pulse. From ω0>>ωc, 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

ω0v0k''>>1
(5)

But this condition is insufficient to guarantee the phenomena associated with the rapid pulse breakup: the actual width of the Airy pulse (~number of lobes) must be also considered. Physically, the Airy wavepakcet cannot have an infinitely-long tail. Hence, just like the Airy beams in space, which are always launched from a finite aperture and hence are diffraction-free only for a finite propagation distance, also for temporal pulses the tail must be truncated at some point. Hence, the propagation-invariant Airy dynamics of the temporal pulse lasts only for a finite distance. The concept of the critical point arises from causality, not from the truncation of the pulse. Hence, to observe the critical point, the truncation must be such that the propagation-invariant dynamics should last for a distance much larger than zcritical. Given a physical truncated pulse which accelerates and is shape-preserving for a finite distance zmax, let us compare zmax with zcritical. For convenience of the calculation, let us use the truncation method suggested in [2

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

], where the initial Airy beam was multiplied by an exponential tail, to yield a “finite energy Airy pulse”. As in [2

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

], using the parameter a for the decay rate and multiplying the wavepacket in the linear system by exp(a2t/T), we get an effective Gaussian window in the Fourier plane exp(-a2ω2T2). This Gaussian window is drawn in Fig. 2c, along with ω0 and ωc 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, a−1T−1, should be larger than ωc and at the same time much smaller than ω0. 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 2π/ω0. Altogether, the ability to observe the critical point necessitates

Tω0>>1a>Tk''v0
(6)

In the case of nonlinear propagation, the decay-rate parameter 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 (tmax, zmax). The point where the tangent crosses the 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−2. Now, we compare this distance to the value of the critical point and derive the same inequality as in (6).

Note that, we find numerically that this “tangent” technique, whose logic is equivalent to caustics methods, works reasonably well also in nonlinear media, even though caustics assume linear rays. The underlying reason why this method works in the nonlinear case relates to the conservation of momentum in Eq. (1).

5. Physical realization

We would like to end this article with a specific design for experimental observation of the effects driven by causality on accelerating pulses. Namely, we ask what realistic system will exhibit the causality-driven pulse breakup? Substituting the wavelength of visible light, and group velocity close to c in expression (5) forces k” to be much larger than 800 fsec2/mm (otherwise the fraction of power contained in the backward-propagating frequencies is negligible). Unfortunately, “conventional” optical materials (such as glass) have 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 fsec2/mm - 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.

For example, to fulfill the condition in (5), we look for a medium with k”=10,000 fsec2/mm, and use λ=400nm, a=0.1 and T=20fsec. Those parameters are used in the simulations creating Fig. 1a,b,c. In Fig. 1d we take k”'=10,000 fsec3/mm, 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.

6. Discussion and conclusions

Acknowledgements

This work was supported by an Advanced Grant from the European Research Council (ERC), by the Israel-USA Binational Science Foundation (BSF), and by the Israel Science Foundation.

References and links

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]

4.

P. Saari, “Laterally accelerating airy pulses,” Opt. Express 16(14), 10303–10308 (2008). [CrossRef] [PubMed]

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. 78(4), 046605 (2008). [CrossRef] [PubMed]

6.

K.-Y. Kim, C.-Y. Hwang, and B. Lee, “Slow non-dispersing wavepackets,” Opt. Express 19(3), 2286–2293 (2011). [CrossRef] [PubMed]

7.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]

8.

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]

9.

R. Y. Chiao and P. W. Milonni, “Fast Light, Slow Light,” Opt. Photonics News 13(6), 26–30 (2002). [CrossRef]

10.

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

11.

Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express 19(18), 17298–17307 (2011). [CrossRef] [PubMed]

12.

G. P. Agrawal, Nonlinear Fiber Optics, Optics and Photonics (Academic Press, London, 2001).

13.

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]

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

  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]
  4. P. Saari, “Laterally accelerating airy pulses,” Opt. Express16(14), 10303–10308 (2008). [CrossRef] [PubMed]
  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.78(4), 046605 (2008). [CrossRef] [PubMed]
  6. K.-Y. Kim, C.-Y. Hwang, and B. Lee, “Slow non-dispersing wavepackets,” Opt. Express19(3), 2286–2293 (2011). [CrossRef] [PubMed]
  7. 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]
  8. 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]
  9. R. Y. Chiao and P. W. Milonni, “Fast Light, Slow Light,” Opt. Photonics News13(6), 26–30 (2002). [CrossRef]
  10. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett.106(21), 213903 (2011). [CrossRef] [PubMed]
  11. Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express19(18), 17298–17307 (2011). [CrossRef] [PubMed]
  12. G. P. Agrawal, Nonlinear Fiber Optics, Optics and Photonics (Academic Press, London, 2001).
  13. 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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