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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 21 — Oct. 21, 2013
  • pp: 25000–25009
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One dimensional spatial localization of polychromatic stationary wave-packets in normally dispersive media

O. Jedrkiewicz, Y.-D. Wang, G. Valiulis, and P. Di Trapani  »View Author Affiliations


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

In the last few years the localization and stationarity of optical wave packets propagating in linear or in non linear dispersive media have been the object of several studies. Many works have highlighted in the two- or three-dimensional spatial domain the intrinsic conical nature of the localized pulses which are also stationary (non diffractive and non dispersive) during propagation [1

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]

7

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 68, 016613 (2003).

]. Localized and quasi-stationary (with finite energy) three-dimensional wave-packets (WPs) have been generated also in non linear processes [8

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

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]

], with asymptotic features relating to those of non diffracting and non dispersive polychromatic Bessel beams in linear dispersive media. A general description of these linear waves has been given in [14

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

] showing how these can be identified with X-shaped or O-shaped modes of the wave equation in media with respectively normal or anomalous group velocity dispersion (GVD). Also new families of two or three dimensional Bessel X-waves have been shown to be possible in linear bidispersive systems [15

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).

].

Up to date it is not clear how linear WPs can simultaneously be tightly focused in one spatial dimension and stationary in propagation in dispersive materials. Our work proposes to reveal the mechanism for the generation of such non diffracting 1D spatial beams. A complete study highlighting the role of the material dispersion will be later presented in [20

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

] showing the 1D localization in space of stationary non trivial WPs. One example is the spatio-temporal Bessel beam (STBB) already observed and characterized in [21

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

], and analogous to a 2D version of the so-called O-wave stationary in the anomalous GVD regime [14

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

]. Note that the paper by Dallaire et al. [21

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

], being the main significant prior art to the work we present here, clearly explains how to generate non diffracting and non dispersive 2D wavepackets in media with negative GVD. They also show temporal profiles and a complete spatio-temporal characterization of their space-time Bessel pulses.

2. Theory

Note that in one spatial dimension a monochromatic wave of frequency ω0 + Ω (long pulse limit case where the temporal spectrum is described by a Dirac delta function S0(Ω, kx) = S0(kx)δ(Ω)), satisfies the quadratic equation (3) for a couple of values kx1, kx2 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) kx by k=kx2+ky2, we have for a given frequency ω0 + Ω (monochromatic wave) and setting for simplicity γ2 = 0, the spherically symmetric solution given by the monochromatic Bessel beam with k = const.

Fig. 1 Intensity profiles of the 2D stationary solution of the wave equation (with relative time integrated curves), featured by different normal dispersion spectral supports (kx = Δ(Ω)) shown in the top right corner insets.

From Eq. (5) we can see how the tail of the 2D X-type wave packet is in this case decaying as A(ξ,0)~1ξ. The numerical integration of Eq. (4) also revealed that in the case of a Gaussian spectrum S0(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.

The important role of polychromaticity in the 1D spatial localization is confirmed by the result of Fig. 1(c), obtained for a WP angular dispersion identical to that of Fig. 1(b), but with a temporal bandwidth 10 time smaller. In that case an oscillating spatial profile (due to the cosine function in Eq.(4)) appears, and the spatial localization vanishes. Figure 1(d) shows the effect of the angular dispersion line (kx = 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.

3. Experiment

Fig. 2 Sketch of the experimental setup based on a pulse shaper with a folded diffraction grating as in [21]. The reflective mask now constituted by hyperbolic branches selects the optical frequencies that are recombined on the diffraction grating. The second cylindrical lens produces the spatial Fourier transform of the mask.

In Figs. 3(a), 3(b) and 3(c), we report the spatial beam images (and in Figs. 3(d), 3(e) and 3(f), the corresponding transverse profiles along 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)).

Fig. 3 Time integrated spatial images (a,b,c) and corresponding central transverse intensity profiles (d,e,f) of the beam generated after the linear beam shaper. The mask spectrum (right) had an angular gap of 0.01 rad (case a,d), 0,02 rad (case b,e), and 0,04 rad (case c,f) respectively. The real branches thickness was 1mm (mask placed in the Fourier plane of the lens after the grating.)
Fig. 4 Time integrated spatial images, recorded in air for different z propagation positions, of the 1D spatial WP generated by means of a mask (in a)) with a gap between the branches of 0,008 rad and branches thickness 2.5 mm. In b) a neat transverse intensity profile recorded at z=0 (10mm before the exact Fourier Plane of the second cylindrical lens).

The results presented in Fig. 5 illustrate the effect of the temporal spectral selection on the 1D localization of the Bessel-like WP. In the left column are reported the spatio-temporal spectral masks used in the pulse shaper, with the red lines highlighting the effectively used internal portions of the mask (the rest being covered with black paper). As expected, the spatial localization and the quenching of the lateral decaying tails get enhanced when a broader temporal spectrum is considered.

Fig. 5 Images and transverse profiles (right column) of the beams generated by using different portions of the spatio-temporal spectral support (left column).

4. Conclusion

The results of the proof of principle experiments presented here indicate that by opportunely tailoring the spatio-temporal spectrum of the stationary wave mode in a given material, it should be possible to generate suitable 1D-spatially localized quasi-stationary pulsed beams that (after suitable beam demagnification and imaging inside the material to avoid instabilities at the air/material interface [11

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]

]) may also find applications in microfabrication experiments. Note that in order to demonstrate stationarity, full space-time measurements in propagation through a dispersive medium would be required (and are not shown in this work). Finally, a complete study of the localization properties of 2D stationary wave packets in different dispersion regimes is in progress [20

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

].

Acknowledgments

This work was supported by Cariplo foundation and Regione Lombardia. The authors thank D. Faccio and A. Parola for helpful discussions.

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. 21, 1162–1164 (1996). [CrossRef] [PubMed]

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. 22, 310–312 (1997). [CrossRef] [PubMed]

3.

M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. 26, 1364–1366 (2001). [CrossRef]

4.

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]

5.

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]

6.

M. A. Porras, S. Trillo, and C. Conti, “Paraxial envelope X waves,” Opt. Lett. 28, 1090–1092 (2003). [CrossRef] [PubMed]

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 68, 016613 (2003).

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]

9.

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]

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. 96, 193901 (2006). [CrossRef] [PubMed]

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]

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 80, 033813 (2009). [CrossRef]

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]

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, 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]

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

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

  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]
  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.22, 310–312 (1997). [CrossRef] [PubMed]
  3. M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett.26, 1364–1366 (2001). [CrossRef]
  4. 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]
  5. 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]
  6. M. A. Porras, S. Trillo, and C. Conti, “Paraxial envelope X waves,” Opt. Lett.28, 1090–1092 (2003). [CrossRef] [PubMed]
  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. E68, 016613 (2003).
  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]
  9. 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]
  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.96, 193901 (2006). [CrossRef] [PubMed]
  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]
  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. A80, 033813 (2009). [CrossRef]
  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. E73, 016608 (2006). [CrossRef]
  14. M. A. Porras and P. Di Trapani, “Localized and stationary light wave modes in dispersive media,” Phys. Rev. E69, 066606 (2004). [CrossRef]
  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, 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 Topics199, 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]
  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. Express17, 18148–18164 (2009). [CrossRef] [PubMed]

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