## One dimensional spatial localization of polychromatic stationary wave-packets in normally dispersive media |

Optics Express, Vol. 21, Issue 21, pp. 25000-25009 (2013)

http://dx.doi.org/10.1364/OE.21.025000

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### Abstract

In this paper we illustrate how the localization of the stationary two-dimensional solution of the propagation equation strongly depends on the features of its spatio-temporal spectral bandwidth. We especially investigate the role of the ultra-broad temporal support and of the spatial bandwidth of the spectrum on the high localization in one spatial dimension of ”Bessel-like” or ”blade-like” beams, quasi-stationarily propagating in normally dispersive materials, and potentially interesting for microfabrication applications.

© 2013 OSA

## 1. Introduction

1. 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]

8. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously Generated X-Shaped Light Bullets,” Phys. Rev. Lett. **91**, 093904 (2003). [CrossRef] [PubMed]

13. A. Couairon, E. Gaiauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E **73**, 016608 (2006). [CrossRef]

14. M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E **69**, 066606 (2004). [CrossRef]

16. C.R. Phipps, *Laser ablation and its Applications* (Springer, 2007). [CrossRef]

17. M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. **97**, 081102 (2010). [CrossRef]

18. M.K. Bhuyan, F. Courvoisier, H.S. Phing, O. Jedrkiewicz, S. Recchia, P. Di Trapani, and J.M. Dudley, “Laser micro- and nanostructuring using femtosecond Bessel beams,” Eur. Phys. J. Special Topics **199**, 101–110 (2011); and see references therein. [CrossRef]

19. A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. **101**, 071110 (2012). [CrossRef]

21. M. Dallaire, N. McCarthy, and M. Piché, “Spatiotemporal bessel beams: theory and experiments,” Opt. Express **17**, 18148–18164 (2009). [CrossRef] [PubMed]

14. M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E **69**, 066606 (2004). [CrossRef]

21. M. Dallaire, N. McCarthy, and M. Piché, “Spatiotemporal bessel beams: theory and experiments,” Opt. Express **17**, 18148–18164 (2009). [CrossRef] [PubMed]

21. M. Dallaire, N. McCarthy, and M. Piché, “Spatiotemporal bessel beams: theory and experiments,” Opt. Express **17**, 18148–18164 (2009). [CrossRef] [PubMed]

**17**, 18148–18164 (2009). [CrossRef] [PubMed]

*and*localization requires at least 2 dimensions, this being also valid in the anomalous GVD case. More precisely, in this paper we present the explanation for the existence of stationary 1D spatial WPs, which in the normal dispersion regime and with a suitable tailoring could spatially resemble 1D ”Bessel-like” beams featured by a lateral-only flowing energy (in fact truly 2D X-type WPs in the space and time domain). In contrast to the standard conical WPs (Bessel beams, 3D X-waves, etc...), such pulses spatially focused in 1D might become interesting for their potential application to high quality machining or cutting of transparent materials even in single shot.

## 2. Theory

*simultaneous*features of wave stationarity and localization indeed require infinite energy and at least two ”dimensions”; i.e. if a 2D spatially localized beam (i.e. Bessel Beam) can be monochromatic, a 1D spatially localized stationary beam must have a polychromatic spectral support. The conditions leading to spatial localization of a WP can be understood by analyzing the stationary solutions of the scalar wave equation

*E*(

*t*,

*r*) being the electric field with

*r*= (

*x*,

*y*,

*z*) the spatial coordinates,

*D*(

*t*,

*r*) the displacement operator, and

*t*and featured by one transverse coordinate

*x*, propagating along the

*z*axis. In the coordinate frame moving with the velocity

*u*

_{0}and setting

*E*=

*A*(

*η*,

*x*,

*z*)exp{

*i*(

*ω*

_{0}

*t*−

*k*

_{0}

*z*)} +

*c.c.*, where the complex amplitude

*A*(

*η*,

*x*,

*z*) reads as where Ω =

*ω*−

*ω*

_{0}(

*ω*

_{0}being the carrier frequency).

*S*

_{0}(Ω,

*k*) is the initial spatio-temporal spectrum of the WP, and the effect of dispersion and diffraction is described by where

_{x}*k*(

*ω*

_{0}+ Ω) is the dispersion relation function depending on the material properties. In general, the WP stationarity requires the function

*G*(Ω,

*k*) to be linear [1

_{x}1. 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]

*G*(Ω,

*k*) =

_{x}*γ*

_{1}Ω +

*γ*

_{2}

*k*+

_{x}*γ*where

*γ*

_{1},

*γ*

_{2}and

*γ*are free parameters, so that the field amplitude profile can be written as

*A*(

*η*,

*x*,

*z*) =

*A*

_{0}(

*η*−

*γ*

_{1}

*z*,

*x*−

*γ*

_{2}

*z*)exp(−

*iγz*); thus using Eq.(2) the following relation must be satisfied:

*ω*

_{0}+ Ω (long pulse limit case where the temporal spectrum is described by a Dirac delta function

*S*

_{0}(Ω,

*k*) =

_{x}*S*

_{0}(

*k*)

_{x}*δ*(Ω)), satisfies the quadratic equation (3) for a couple of values

*k*

_{x}_{1},

*k*

_{x}_{2}of the transverse wavevector. Thus, in this case the only stationary solution is a couple of plane waves: no localized solutions are possible. In two spatial dimensions instead, if we replace in Eq. (3)

*k*by

_{x}*ω*

_{0}+ Ω (monochromatic wave) and setting for simplicity

*γ*

_{2}= 0, the spherically symmetric solution given by the monochromatic Bessel beam with

*k*

_{⊥}=

*const*.

*one spatial dimension*we must allow for a non monochromatic solution, and thus a temporal bandwidth is needed. The stationarity condition expressed by Eq. (3) shows that the spatial localization and the temporal localization are not independent, since it sets Ω =

*f*(

*k*). As a consequence, spatial localization, which implies having a bandwidth of spatial frequencies (from FT properties), necessary implies having a bandwidth of temporal frequencies as well. Thus a 2D linear WP (1D spatial and 1D temporal) can be stationary propagating

_{x}*and*localized when its space and time coordinates become entangled via angular dispersion, i.e. when the different temporal frequencies are distributed at different propagating angles. Such a WP can be described by setting its spatio-temporal spectrum for instance as

*S*

_{0}(Ω,

*k*) = 2

_{x}*πS*

_{0}(

*k*)

_{x}*δ*(Ω − Δ(

*k*)) or (as in the case we shall consider from now on)

_{x}*S*

_{0}(Ω,

*k*) = 2

_{x}*πS*

_{0}(Ω)

*δ*(

*k*− Δ(Ω)). The function Δ(Ω) defines the transverse wave-vector

_{x}*k*=

_{x}*k*sin(

*θ*(Ω)) = Δ(Ω) and consequently the angular dispersion

*θ*(Ω) of the stationary WP in the dispersive medium such that

*G*(Ω, Δ(Ω)) =

*α*Ω +

*β*, where

*α*and

*β*are free parameters. Indeed for the longitudinal wave-vector

*k*= ±Δ(Ω)) travels with group velocity

_{x}*v*=

_{ph}*ω*

_{0}/(

*k*

_{0}+

*β*): The transverse dispersion relation can have different characteristics depending on the material dispersion properties, and thus can lead to various real space solutions as it will be described in great detail in [20]. In particular a 2D X-like WP can be obtained if its spatio-temporal spectrum

*S*(Ω,

*k*) is featured by the characteristic hyperbolic curves described in the paraxial approximation by

_{x}*S*

_{0}(Ω) (with the same width as the rectangular spectrum considered above), the tails decay occurs even faster. Note however, that in both cases illustrated in Fig. 1(a) and Fig. 1(b), the infinite energy of the theoretical solution supports the non-decaying tails in the

*time-integrated*intensity profiles, shown just below the space-time profiles.

*k*= Δ(Ω)) ”thickness” which acts as a beam apodization effect, thus leading to a time-integrated profile with decaying tails (in this case the solution is energy limited and thus quasi-stationary). A suitable tailoring of the spatio-temporal spectral properties of a 2D WP may therefore lead to a quasi-stationary beam highly localized in one spatial dimension, potentially interesting for many microfabrication applications. Indeed in this context a good high contrast spatial shape of the time-integrated profile is the essential feature needed, for an effective radiation absorption close to the beam core where the pulse duration is shorter, and thus where the intensity, the multi-photon absorption and the energy transfer to matter are larger.

_{x}## 3. Experiment

**17**, 18148–18164 (2009). [CrossRef] [PubMed]

14. M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E **69**, 066606 (2004). [CrossRef]

*S*(Ω,

*k*)- or equivalently

_{x}*S*(

*λ*,

*θ*)- space), was engraved (this mask is different from the ring-shaped mask of [21

_{x}**17**, 18148–18164 (2009). [CrossRef] [PubMed]

*mm*width along the horizontal direction. As shown in Fig. 2, the temporal inverse FT of the signal first dispersed by a grating and then reflected back by the mask was performed along the horizontal axis by means of a first cylindrical lens (in our case of focal length 100 mm). A second cylindrical lens of focal length 300 mm, was then used to perform the inverse FT in the spatial domain (along the vertical axis). The dispersion of our system was minimized by adjusting the distance between the grating and the first cylindrical lens, and by slightly pre-chirping the laser input pulse. Our principal aim here was the spatial characterization of the resulting beam observed in the Fourier plane of the second cylindrical lens. This has been done by recording the temporally integrated spatial profile of the beams, featured by different spectral supports, by means of a 14 bit CCD camera (Sony ICX205AL, WincamD).

*x*taken at the beam center) of the generated WPs recorded in air for three different values of the gap between the two branches of the spectrum. We observe, in accordance with theoretical predictions, that when the gap increases, the interference process of the different spatio-temporal bandwidth portions, leads to an increase of the number of lateral decaying fringes (similar to Fig. 1c). The angular gap has an analogy with the cone angle of the standard 2D spatial Bessel beam. Indeed the bigger the angle, and the smaller the core size of the beam. In our case this is reflected in the transverse size of the fringes characterizing the 1D ”Bessel-like” beam generated, that decreases when the spectral angular gap increases. On the other hand by increasing the thickness of the spectral branches, we have observed a better 1D localization in space (along the x axis) with the corresponding quenching of the lateral tails. Indeed a broader spatial spectral support in the transverse wave vector domain naturally corresponds to a stronger apodization of the resulting WP in the real space domain (see Fig. 4, Fig. 1(d)).

*μm*(FWHM) has been verified. In Fig. 4(a) we have reported the spatial evolution of the beam in air along the propagation direction

*z*, showing a quasi stationary behaviour of the central blade (constant size) over a length of at least 20 mm (5 times the Rayleigh range of a Gaussian beam of similar dimensions). Moreover the ratio between the central peak intensity and the intensity of the lateral lobes of the transverse intensity profile shown in Fig. 4(b) remains constant along 15 mm of propagation (data not shown). Note that while the limitations in the dimension and maximum intensity of the blade-like beam are given by the input beam apodization and the energy for what concerns the major section, the limitations in the minimum dimension (perpendicular direction) and in the duration are dictated by the spatio-temporal spectral bandwidth support. At z=50mm the far field structure of the 1D beam appears. The generated WP has also been launched in rectangular slabs of glass with different lengths, allowing us to check its spatial localization for distances up to 30–40mm. Note that cross-correlation measurements performed by mixing in a second order non linear crystal the 1D-like spatial beam with a portion of the laser beam have shown that the pulses generated with our set up are featured by a total duration of 350 fs full width at half maximum.

## 4. Conclusion

11. P. Polesana, A. Couairon, D. Faccio, A. Parola, M. A. Porras, A. Dubietis, A. Piskarskas, and P. Di Trapani, “Observation of Conical Waves in Focusing, Dispersive, and Dissipative Kerr Media,” Phys. Rev. Lett. **99**, 223902 (2007). [CrossRef]

## Acknowledgments

## References and links

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

2. | 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. |

3. | M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. |

4. | M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersive broadening of light pulses with Bessel-Gauss beams,” Opt. Commun. |

5. | S. Orlov, A. Piskarskas, and A. Stabinis, “Localized optical subcycle pulses in dispersive media,” Opt. Lett. |

6. | M. A. Porras, S. Trillo, and C. Conti, “Paraxial envelope X waves,” Opt. Lett. |

7. | M. A. Porras, G. Valiulis, and P. Di Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. Lett. E |

8. | P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously Generated X-Shaped Light Bullets,” Phys. Rev. Lett. |

9. | M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic Nonlinear X Waves for Femtosecond Pulse Propagation in Water,” Phys. Rev. Lett. |

10. | D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical Emission, Pulse Splitting, and X-Wave Parametric Amplification in Nonlinear Dynamics of Ultrashort Light Pulses,” Phys. Rev. Lett. |

11. | P. Polesana, A. Couairon, D. Faccio, A. Parola, M. A. Porras, A. Dubietis, A. Piskarskas, and P. Di Trapani, “Observation of Conical Waves in Focusing, Dispersive, and Dissipative Kerr Media,” Phys. Rev. Lett. |

12. | O. Jedrkiewicz, M. Clerici, E. Rubino, and P. Di Trapani, “Generation and control of phase-locked conical wave packets in type-I seeded optical parametric amplification,” Phys. Rev. A |

13. | A. Couairon, E. Gaiauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E |

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

15. | D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, “Bessel X waves in two- and three-dimensional bidispersive optical systems,” Opt. Lett. 29, 1446–1448 (2004). |

16. | C.R. Phipps, |

17. | M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett. |

18. | M.K. Bhuyan, F. Courvoisier, H.S. Phing, O. Jedrkiewicz, S. Recchia, P. Di Trapani, and J.M. Dudley, “Laser micro- and nanostructuring using femtosecond Bessel beams,” Eur. Phys. J. Special Topics |

19. | A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. |

20. | G. Valiulis, O. Jedrkiewicz, Y.-D. Wang, and P. Di Trapani, in preparation. |

21. | M. Dallaire, N. McCarthy, and M. Piché, “Spatiotemporal bessel beams: theory and experiments,” Opt. Express |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(140.3300) Lasers and laser optics : Laser beam shaping

(190.0190) Nonlinear optics : Nonlinear optics

(320.5540) Ultrafast optics : Pulse shaping

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: July 18, 2013

Revised Manuscript: September 3, 2013

Manuscript Accepted: September 8, 2013

Published: October 11, 2013

**Citation**

O. Jedrkiewicz, Y.-D. Wang, G. Valiulis, and P. Di Trapani, "One dimensional spatial localization of polychromatic stationary wave-packets in normally dispersive media," Opt. Express **21**, 25000-25009 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-25000

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### References

- 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]
- 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]
- M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett.26, 1364–1366 (2001). [CrossRef]
- M. A. Porras, R. Borghi, and M. Santarsiero, “Suppression of dispersive broadening of light pulses with Bessel-Gauss beams,” Opt. Commun.206, 235–241 (2002). [CrossRef]
- S. Orlov, A. Piskarskas, and A. Stabinis, “Localized optical subcycle pulses in dispersive media,” Opt. Lett.27, 2167–2169 (2002); 27, 2103–2105 (2002). [CrossRef]
- M. A. Porras, S. Trillo, and C. Conti, “Paraxial envelope X waves,” Opt. Lett.28, 1090–1092 (2003). [CrossRef] [PubMed]
- M. A. Porras, G. Valiulis, and P. Di Trapani, “Unified description of Bessel X waves with cone dispersion and tilted pulses,” Phys. Rev. Lett. E68, 016613 (2003).
- P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously Generated X-Shaped Light Bullets,” Phys. Rev. Lett.91, 093904 (2003). [CrossRef] [PubMed]
- M. Kolesik, E. M. Wright, and J. V. Moloney, “Dynamic Nonlinear X Waves for Femtosecond Pulse Propagation in Water,” Phys. Rev. Lett.92, 253901 (2004). [CrossRef] [PubMed]
- D. Faccio, M. A. Porras, A. Dubietis, F. Bragheri, A. Couairon, and P. Di Trapani, “Conical Emission, Pulse Splitting, and X-Wave Parametric Amplification in Nonlinear Dynamics of Ultrashort Light Pulses,” Phys. Rev. Lett.96, 193901 (2006). [CrossRef] [PubMed]
- P. Polesana, A. Couairon, D. Faccio, A. Parola, M. A. Porras, A. Dubietis, A. Piskarskas, and P. Di Trapani, “Observation of Conical Waves in Focusing, Dispersive, and Dissipative Kerr Media,” Phys. Rev. Lett.99, 223902 (2007). [CrossRef]
- O. Jedrkiewicz, M. Clerici, E. Rubino, and P. Di Trapani, “Generation and control of phase-locked conical wave packets in type-I seeded optical parametric amplification,” Phys. Rev. A80, 033813 (2009). [CrossRef]
- A. Couairon, E. Gaiauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E73, 016608 (2006). [CrossRef]
- M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E69, 066606 (2004). [CrossRef]
- D. N. Christodoulides, N. K. Efremidis, P. Di Trapani, and B. A. Malomed, “Bessel X waves in two- and three-dimensional bidispersive optical systems,” Opt. Lett.29, 1446–1448 (2004).
- C.R. Phipps, Laser ablation and its Applications (Springer, 2007). [CrossRef]
- M. K. Bhuyan, F. Courvoisier, P. A. Lacourt, M. Jacquot, R. Salut, L. Furfaro, and J. M. Dudley, “High aspect ratio nanochannel machining using single shot femtosecond Bessel beams,” Appl. Phys. Lett.97, 081102 (2010). [CrossRef]
- M.K. Bhuyan, F. Courvoisier, H.S. Phing, O. Jedrkiewicz, S. Recchia, P. Di Trapani, and J.M. Dudley, “Laser micro- and nanostructuring using femtosecond Bessel beams,” Eur. Phys. J. Special Topics199, 101–110 (2011); and see references therein. [CrossRef]
- A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: Femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett.101, 071110 (2012). [CrossRef]
- G. Valiulis, O. Jedrkiewicz, Y.-D. Wang, and P. Di Trapani, in preparation.
- M. Dallaire, N. McCarthy, and M. Piché, “Spatiotemporal bessel beams: theory and experiments,” Opt. Express17, 18148–18164 (2009). [CrossRef] [PubMed]

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