## Programmable ultrashort-pulsed flying images

Optics Express, Vol. 17, Issue 9, pp. 7465-7478 (2009)

http://dx.doi.org/10.1364/OE.17.007465

Acrobat PDF (975 KB)

### Abstract

We report the generation of programmable two-dimensional arrangements of ultrashort-pulsed fringe-less Bessel-like beams of extended depth of focus (referred to as needle beams) without truncating apertures. A sub-20-fs Ti:sapphire laser and a liquid-crystal-on-silicon spatial light modulator (LCoS-SLM) of high-fidelity temporal transfer in phase-only operation mode were used in the experiments. Axicon profiles with ultra-small conical angles were approximated by adapted gray scale distributions. It was demonstrated that digitized image information encoded in amplitude-phase maps of the needle beams is propagated over considerably large distances at minimal cross talk without the need for additional relay optics. This experiment represents a physical realization of Saari’s proposal of spatio-temporally nondiffracting “flying images” on a few-femtosecond time scale.

© 2009 Optical Society of America

## 1. Introduction

## 2. Pseudo-nondiffracting needle beams

9. J. Durnin, “Exact solutions for nondiffracting beams I. The scalar theory,” J. Opt. Soc. Am A **4**, 651–654 (1986). [CrossRef]

10. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**, 1499–1501 (1987). [CrossRef] [PubMed]

11. J. H. McLeod, “The axicon: A new type of optical element,” J. Opt. Soc. Am. **44**, 592–597 (1954). [CrossRef]

12. K. Reivelt and P. Saari, “Bessel-Gauss pulse as an appropriate mathematical model for optically realizable localized waves,” Opt. Lett. **29**, 1176–1178 (2004). [CrossRef] [PubMed]

16. R. M. Herman and T. A. Wiggins, “Propagation and focusing of Bessel-Gauss, generalized Bessel-Gauss, and modified Bessel-Gauss beams,” J. Opt. Soc. Am. A **18**, 170–176 (2001). [CrossRef]

17. S. Huferath-von Luepke, V. Kebbel, M. Bock, and R. Grunwald, “Noncollinear autocorrelation with radially symmetric nondiffracting beams,” Proc. SPIE , Vol. **7063**, 706311 (2008). [CrossRef]

18. Z. Bouchal, J. Wagner, and M. Chlup, Self-reconstruction of a distorted nondiffracting beam, Opt. Commun. **151**, 207–211 (1998). [CrossRef]

19. C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Opt. Soc. Am. A **18**, 2594–2600 (2001). [CrossRef]

22. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. **79**, 4135–4138 (1997). [CrossRef]

23. R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E.T.J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wavepackets,” Phys. Rev. A **67**, 063820 (2003). [CrossRef]

24. 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]

7. R. Grunwald, U. Neumann, U. Griebner, G. Steinmeyer, G. Stibenz, M. Bock, and V. Kebbel, “Self-reconstruction of pulsed optical X- waves,” in
M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa (eds.), *Localized Waves, Theory and experiments* (Wiley & Sons, New York, 2008), pp. 299–313. [CrossRef]

25. R. Grunwald, M. Bock, V. Kebbel, S. Huferath, U. Neumann, G. Steinmeyer, G. Stibenz, J.-L. Néron, and M. Piché, “Ultrashort-pulsed truncated polychromatic Bessel-Gauss beams,” Opt. Express **16**, 1077–1089 (2008). [CrossRef] [PubMed]

7. R. Grunwald, U. Neumann, U. Griebner, G. Steinmeyer, G. Stibenz, M. Bock, and V. Kebbel, “Self-reconstruction of pulsed optical X- waves,” in
M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa (eds.), *Localized Waves, Theory and experiments* (Wiley & Sons, New York, 2008), pp. 299–313. [CrossRef]

25. R. Grunwald, M. Bock, V. Kebbel, S. Huferath, U. Neumann, G. Steinmeyer, G. Stibenz, J.-L. Néron, and M. Piché, “Ultrashort-pulsed truncated polychromatic Bessel-Gauss beams,” Opt. Express **16**, 1077–1089 (2008). [CrossRef] [PubMed]

^{2}(maximum achievable about 700:1) were realized. In this sense, the propagation behavior can be regarded as being “pseudo-nondiffracting”. Foci of comparable depth and simultaneously comparable spectral and temporal stability can not be generated with Gaussian beams because of the inverse wavelength dependence of the beam parameters. The main disadvantage of type-I-needle beam shaping is that large arrays of high-quality needle beams can hardly be generated because of the critical adjustment of many apertures (in particular with non-planar wavefronts).

26. R. Grunwald, S. Huferath, M. Bock, U. Neumann, and S. Langer, “Angular tolerance of Shack-Hartmann wavefront sensors with microaxicons,” Opt. Lett. **32**, 1533–1535 (2007). [CrossRef] [PubMed]

7. R. Grunwald, U. Neumann, U. Griebner, G. Steinmeyer, G. Stibenz, M. Bock, and V. Kebbel, “Self-reconstruction of pulsed optical X- waves,” in
M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa (eds.), *Localized Waves, Theory and experiments* (Wiley & Sons, New York, 2008), pp. 299–313. [CrossRef]

25. R. Grunwald, M. Bock, V. Kebbel, S. Huferath, U. Neumann, G. Steinmeyer, G. Stibenz, J.-L. Néron, and M. Piché, “Ultrashort-pulsed truncated polychromatic Bessel-Gauss beams,” Opt. Express **16**, 1077–1089 (2008). [CrossRef] [PubMed]

## 3. Spatial light modulators for shaping of ultrashort pulses in spatial domain

27. G. D. Love, S. R. Restaino, G. C. Loos, and A. Purvis, “Wave-front control using a 64 × 64 pixel liquid crystal array,” in *Adaptive Optics in Astronomy*, Ealey and
F. Merkle (eds.), Proc. SPIE , **Vol. 2201**, 1068–1072 (1994). [CrossRef]

28. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable femtosecond pulse shaping by use of a multielement liquid-crystal phase modulator,” Opt. Lett. **15**, 226–228 (1990). [CrossRef]

1. J. C. Vaughan, T. Feurer, and K. A. Nelson, “Automated two-dimensional femtosecond pulse shaping,” J. Opt. Soc. Am. B **19**, 2489–2495 (2002). [CrossRef]

29. R. M. Koehl, T. Hattori, and K. A. Nelson, “Automated spatial and temporal shaping of femtosecond pulses,” Opt. Commun. **157**, 57–61 (1998). [CrossRef]

2. K. B. Hill, K. G. Purchase, and D. J. Brady, “Pulsed-image generation and detection,” Opt. Lett. **20**, 1201–1203 (1995). [CrossRef] [PubMed]

30. M. C. Nuss and R. L. Morrison, “Time-domain images,” Opt. Lett. **20**, 740–742 (1995). [CrossRef] [PubMed]

31. T. Brixner and G. Gerber, “Femtosecond polarization pulse shaping,” Opt. Lett. **26**, 557–559 (2001). [CrossRef]

32. E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murnane, G. Mourou, and H. Kapteyn, “Pulse compression by use of deformable mirrors,” Opt. Lett. **24**, 493–495 (1999). [CrossRef]

33. M. Bock, S. K. Das, R. Grunwald, S. Osten, P. Staudt, and G. Stibenz, “Spectral and temporal response of liquid-crystal-on-silicon spatial light modulators,“ Appl. Phys. Lett. **92**, 151105 (2008). [CrossRef]

^{2}). Most of the loss energy is transformed in parasitic diffraction orders which do not affect the zero order beam. To overcome the limitations given by the maximum phase, a Fresnel zone lens approach can be applied to create flat axicons which are referred to as “Fresnel axicons” or “Fraxicons” [34

34. I. Golub, “Fresnel axicon,” Opt. Lett. **31**, 1890–1892 (2006). [CrossRef] [PubMed]

33. M. Bock, S. K. Das, R. Grunwald, S. Osten, P. Staudt, and G. Stibenz, “Spectral and temporal response of liquid-crystal-on-silicon spatial light modulators,“ Appl. Phys. Lett. **92**, 151105 (2008). [CrossRef]

## 4. Experimental techniques

26. R. Grunwald, S. Huferath, M. Bock, U. Neumann, and S. Langer, “Angular tolerance of Shack-Hartmann wavefront sensors with microaxicons,” Opt. Lett. **32**, 1533–1535 (2007). [CrossRef] [PubMed]

*three basic geometrical arrangements*of gray value addressed phase axicons were realized for the following situations: (i) programming single needle beams (to totally exclude any cross talk effects), (ii) programming complex ray patterns consisting of reconfigurable arrays of needle beams, and (iii) arrays like above but with an additional background management. Whereas the background was chosen to be unstructured (i.e. with homogeneous phase) in cases (i) and (ii), a checkered phase pattern acting as a diffractive cross grating was programmed in the spaces between the programmed binary phase axicons (to study the influence of surrounding high-spatial-frequency textures and, in particular, to separate unshaped light from the main propagation direction of the shaped sub-beams).

*r*= 170 μm corresponds to an average conical angle <α(r)> of 2.1 mrad. To demonstrate the propagation of nondiffracting ultrashort-pulsed images, we arranged 15 identical axicons to form a letter “E”, as presented in Fig. 2(b) as a special case of type-(ii)-shaping. To directly generate the pattern without a magnifying telescope and thus to avoid additional dispersion in lenses or aberrations from off-axis illumination, we had to accept certain restrictions. The useable area was limited to 2.0 mm

_{max}^{2}(input beam diameter 2w

_{0}of 4 mm). Consequently, it was necessary to shrink the interval between adjacent elements and to reduce the number of lines to a minimum. A special overlapping algorithm was introduced to preserve the gray level distribution of the individual axicons. The outer parts of each axicon gray value map were corrected depending on distance and diameter of the surrounding axicons. Horizontal and vertical periods were adjusted to a value of 324 μm (40 pixels). To reduce the diffraction at hard edges, profile functions with slight deviations from a linear cone were programmed. For this experiment, the above described conical axicon with a radius of

*r*= 170 μm (blue line in Fig. 2(a)) was replaced by an adapted distribution (red symbols in Fig. 2(a)) which can be described part by part by fitting Gaussian functions (black line: fit of the tailing part with a radius of 55 μm in terms of the standard deviation). The geometrical arrangement (iii) of diffractive checkered gray value patterns is shown in the inset in Fig. 2(b). The chosen minimum period was 2 pixels (1 white, 1 black) corresponding to a grating period 16.2 μm.

_{max}## 5. Results and discussion

### 5.1. Propagation of ultrashort-pulsed needle beams

*K*(kurtosis) of central cuts through measured profiles in transversal direction was calculated to extract specific information about peakedness (positive value) or flatness (negative value). This enables us to evaluate also non-uniform shapes with high sensitivity. The propagation dependent kurtosis of the transversal shape functions in Fig. 4 is nearly identical for single (blue curve) and arrayed needle beams (green curve) if no additional grating is programmed in the dead space. However, the presence of such a phase structure (black curve) leads to significant deviations over most parts but a convergence between

*z*= 38 and 50 mm. The abrupt changes of the kurtosis from positive to negative values can be attributed to the peculiarities of the formation of realistic broadband Bessel-like beams of finite energy in contrast to idealized Bessel beams.

*J*

_{0}

^{2}are a measure of the non-conicity of the generating wave. On the basis of this criterion, we found the solitary needle beam and the sub-beams of needle beam arrays with homogeneous background in the experiment to well approximate

*J*

_{0}

^{2}between

*z*= 19 mm and

*z*= 50 mm. In the case of a checkered phase pattern in the dead space between the axicons,

*J*

_{0}

^{2}is approximated from larger distances (about

*z*= 38 mm).

*z*(distance at which the cross sectional beam area has doubled) and the beam waist radius

_{0}*w*(1/e

_{0}^{2}of the transversal intensity distribution) we define an aspect ratio of

*z*/

_{0}*w*as a measure for the depth of the focal zone of the needle beams. The aspect ratio of solitary needle beams and sub-beams of needle beam arrays was found in all cases to exceed 300:1 (see Tab. 1).

_{0}^{rd}order polynomial. In the case of a non-uniform background, the beam radius increased significantly in the proximity of the SLM where a transient superposition of needle beam and the radiation diffracted at the checkered phase area is expected. For z > 25 mm, the axial dependence of the radius within the Rayleigh range was found to be nearly identical again for all curves including the simulation. That implies that the cross-talk between adjacent optical channels remains low over large propagation distances.

### 5.2. Ultrashort-pulsed flying images

*C*= (

*I*

_{max}-

*I*

_{min})/(

*I*

_{max}+

*I*

_{min}) appeared to be clearly higher if the SLM was operated with structured background (

*C*= 0.66) whereas it was found to be lower in the other case (

*C*= 0.59). Because of the spatial envelope of the beam, the contrast undergoes local variations. Nevertheless, the qualitative behavior could be verified at all positions in a similar way. In the presence of a checkered phase background, the contrast increases towards larger axial distance whereas it is depleted in the near zione. The spatial fine structure of a selected sub-beam propagation zone can be recognized in the intensity map of radially averaged cuts determined along the propagation path for a checkered phase background (Fig. 8). The Rayleigh range was

*z*-

_{2}*z*= 27.2 mm in this case (compared to a value of 23 mm without structured background). The reduction factor for the axial scale in Fig. 8 is still about 1:17 but the needle shape can well be imagined.

_{1}### 5.3. Spectral analysis of flying images with statistical moments

**16**, 1077–1089 (2008). [CrossRef] [PubMed]

26. R. Grunwald, S. Huferath, M. Bock, U. Neumann, and S. Langer, “Angular tolerance of Shack-Hartmann wavefront sensors with microaxicons,” Opt. Lett. **32**, 1533–1535 (2007). [CrossRef] [PubMed]

*S*, kurtosis

*K*and center of gravity

*C*were determined as well. Spatially averaged values are listed in Tab. 2.

_{o}G*S*) and kurtosis (

*K*) values indicate the symmetry and a certain flattening of the spectrum of the sub-beam. It has to be mentioned that the bandwidth can differ from sub-beam to sub-beam because of the laser properties. The quantitative interpretation has to take into account that the spectral FWHM can directly be derived from the standard deviation (by applying the well-known factor of 2.35) only if the deviation from a Gaussian distribution (where

*S*= 0,

*K*= 0) is negligible. The spectral data were confirmed by autocorrelation measurements for a selected sub-beam (central peak in Fig. 7) of a flying image. At a distance of

*z*= 50 mm from the SLM (distance from oscillator: 1.5 m), a pulse duration of 23 fs was determined.

## 6. Conclusions

**16**, 1077–1089 (2008). [CrossRef] [PubMed]

## Appendix: Self-truncation condition for the generation of needle beams with conical axicons

*D*is illuminated with a plane wave in normal incidence without an additional truncating aperture, the (in our case programmable) phase depth

*h*has to be chosen properly so that no outer fringes of a Bessel beam appear. This can be regarded as a “self-truncation” where the finite extension of the illuminated area replaces the spatially filtering aperture (as applied, e.g., in the self-apodizing truncation setup described in Ref. [25

**16**, 1077–1089 (2008). [CrossRef] [PubMed]

*λ*= center wavelength,

_{0}*n*= refractive index of air, 2

*θ*= total conical beam angle) If the maximum diameter of the overlapping zone (D/2) at an axial distance

*z*(center of the focal zone) equals the diameter ∧ of the central lobe it follows

_{c}*h*= structure depth). Here, a factor 2 is due to the reflective geometry. From Eq. (A2), we obtain the self-truncation condition for the maximum acceptable concial beam angle

*D*= 500 μm and

*λ*

_{0}= 0.8 μm, one ontains a theoretical aspect ration A

^{*}> 620:1. For a more realistic desription, however, a detailed simulation of the beam propagation including diffraction and initial beam divergence is inevitable.

## Acknowledgments

## References and links

1. | J. C. Vaughan, T. Feurer, and K. A. Nelson, “Automated two-dimensional femtosecond pulse shaping,” J. Opt. Soc. Am. B |

2. | K. B. Hill, K. G. Purchase, and D. J. Brady, “Pulsed-image generation and detection,” Opt. Lett. |

3. | J. Turunen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A |

4. | V. Kettunen and J. Turunen, “Propagation-invariant spot arrays,” Opt. Lett. |

5. | Z. Bouchal, “Controlled spatial shaping of nondiffrqacting patterns and arrays,” Opt. Lett. |

6. | P. Saari, Spatially and temporally nondiffracting ultrashort pulses, in:
O. Svelto, S. De Silvestri, and G. Denardo (Eds.): |

7. | R. Grunwald, U. Neumann, U. Griebner, G. Steinmeyer, G. Stibenz, M. Bock, and V. Kebbel, “Self-reconstruction of pulsed optical X- waves,” in
M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa (eds.), |

8. | R. Grunwald, |

9. | J. Durnin, “Exact solutions for nondiffracting beams I. The scalar theory,” J. Opt. Soc. Am A |

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

11. | J. H. McLeod, “The axicon: A new type of optical element,” J. Opt. Soc. Am. |

12. | K. Reivelt and P. Saari, “Bessel-Gauss pulse as an appropriate mathematical model for optically realizable localized waves,” Opt. Lett. |

13. | P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E |

14. | R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A |

15. | F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. |

16. | R. M. Herman and T. A. Wiggins, “Propagation and focusing of Bessel-Gauss, generalized Bessel-Gauss, and modified Bessel-Gauss beams,” J. Opt. Soc. Am. A |

17. | S. Huferath-von Luepke, V. Kebbel, M. Bock, and R. Grunwald, “Noncollinear autocorrelation with radially symmetric nondiffracting beams,” Proc. SPIE , Vol. |

18. | Z. Bouchal, J. Wagner, and M. Chlup, Self-reconstruction of a distorted nondiffracting beam, Opt. Commun. |

19. | C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Opt. Soc. Am. A |

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

21. | M. A. Porras, R. Borghi, and M. Santarsiero, “Few-optical-cycle Bessel-Gauss pulsed beams in free space,” Phys. Rev. E |

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

23. | R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E.T.J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wavepackets,” Phys. Rev. A |

24. | 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. |

25. | R. Grunwald, M. Bock, V. Kebbel, S. Huferath, U. Neumann, G. Steinmeyer, G. Stibenz, J.-L. Néron, and M. Piché, “Ultrashort-pulsed truncated polychromatic Bessel-Gauss beams,” Opt. Express |

26. | R. Grunwald, S. Huferath, M. Bock, U. Neumann, and S. Langer, “Angular tolerance of Shack-Hartmann wavefront sensors with microaxicons,” Opt. Lett. |

27. | G. D. Love, S. R. Restaino, G. C. Loos, and A. Purvis, “Wave-front control using a 64 × 64 pixel liquid crystal array,” in |

28. | A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable femtosecond pulse shaping by use of a multielement liquid-crystal phase modulator,” Opt. Lett. |

29. | R. M. Koehl, T. Hattori, and K. A. Nelson, “Automated spatial and temporal shaping of femtosecond pulses,” Opt. Commun. |

30. | M. C. Nuss and R. L. Morrison, “Time-domain images,” Opt. Lett. |

31. | T. Brixner and G. Gerber, “Femtosecond polarization pulse shaping,” Opt. Lett. |

32. | E. Zeek, K. Maginnis, S. Backus, U. Russek, M. Murnane, G. Mourou, and H. Kapteyn, “Pulse compression by use of deformable mirrors,” Opt. Lett. |

33. | M. Bock, S. K. Das, R. Grunwald, S. Osten, P. Staudt, and G. Stibenz, “Spectral and temporal response of liquid-crystal-on-silicon spatial light modulators,“ Appl. Phys. Lett. |

34. | I. Golub, “Fresnel axicon,” Opt. Lett. |

35. | P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. |

36. | Volker Kebbel, “Untersuchungen zur Erzeugung und Propagation ultrakurzer optischer Bessel-Impulse”, Doctoral thesis, University Bremen, 2004 (BIAS Verlag Bremen, in German). |

37. | R. Grunwald, R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H.-J. Hartmann, and W. Jüptner, “Generation of femtosecond Bessel beams with micro-axicon arrays,” Opt. Lett. |

38. | O. Brzobohatý, T. Cižmár, and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express |

**OCIS Codes**

(320.0320) Ultrafast optics : Ultrafast optics

(320.5540) Ultrafast optics : Pulse shaping

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: February 27, 2009

Revised Manuscript: April 16, 2009

Manuscript Accepted: April 20, 2009

Published: April 21, 2009

**Citation**

M. Bock, S. K. Das, and R. Grunwald, "Programmable ultrashort-pulsed flying images," Opt. Express **17**, 7465-7478 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7465

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### References

- J. C. Vaughan, T. Feurer, and K. A. Nelson, "Automated two-dimensional femtosecond pulse shaping," J. Opt. Soc. Am. B 19, 2489-2495 (2002). [CrossRef]
- K. B. Hill, K. G. Purchase, and D. J. Brady, "Pulsed-image generation and detection," Opt. Lett. 20, 1201-1203 (1995). [CrossRef] [PubMed]
- J. Turunen, A. Vasara, and A. T. Friberg, "Propagation invariance and self-imaging in variable-coherence optics," J. Opt. Soc. Am. A 8, 282-289 (1991). [CrossRef]
- V. Kettunen and J. Turunen, "Propagation-invariant spot arrays," Opt. Lett. 23, 1247-1249 (1998). [CrossRef]
- Z. Bouchal, "Controlled spatial shaping of nondiffrqacting patterns and arrays," Opt. Lett. 27, 1376-1378 (2002). [CrossRef]
- P. Saari, Spatially and temporally nondiffracting ultrashort pulses, in: O. Svelto, S. De Silvestri, and G. Denardo (Eds.): Ultrafast Processesin Spectroscopy, (Plenum Press, New York, 1996), pp. 151-156.
- R. Grunwald, U. Neumann, U. Griebner, G. Steinmeyer, G. Stibenz, M. Bock, and V. Kebbel, "Self- reconstruction of pulsed optical X- waves," in M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa (eds.), Localized Waves, Theory and experiments (Wiley & Sons, New York, 2008), pp. 299-313. [CrossRef]
- R. Grunwald, Thin film microoptics - new frontiers of spatio-temporal beam shaping (Elsevier, Amsterdam, 2007).
- J. Durnin, "Exact solutions for nondiffracting beams I. The scalar theory," J. Opt. Soc. Am A 4, 651-654 (1986). [CrossRef]
- J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
- J. H. McLeod, "The axicon: A new type of optical element," J. Opt. Soc. Am. 44, 592-597 (1954). [CrossRef]
- K. Reivelt and P. Saari, "Bessel-Gauss pulse as an appropriate mathematical model for optically realizable localized waves," Opt. Lett. 29, 1176-1178 (2004). [CrossRef] [PubMed]
- P. Saari and K. Reivelt, "Generation and classification of localized waves by Lorentz transformations in Fourier space," Phys. Rev. E 69, 036612 (2004). [CrossRef]
- R. M. Herman, T. A. Wiggins, "Production and uses of diffractionless beams," J. Opt. Soc. Am. A 8, 982-942 (1991). [CrossRef]
- F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 247-249 (1987).
- R. M. Herman and T. A. Wiggins, "Propagation and focusing of Bessel-Gauss, generalized Bessel-Gauss, and modified Bessel-Gauss beams," J. Opt. Soc. Am. A 18, 170-176 (2001). [CrossRef]
- S. Huferath-von Luepke, V. Kebbel, M. Bock, and R. Grunwald, "Noncollinear autocorrelation with radially symmetric nondiffracting beams," Proc. SPIE, Vol. 7063, 706311 (2008). [CrossRef]
- Z. Bouchal, J. Wagner, and M. Chlup, Self-reconstruction of a distorted nondiffracting beam, Opt. Commun. 151, 207-211 (1998). [CrossRef]
- C. J. R. Sheppard, "Bessel pulse beams and focus wave modes," J. Opt. Soc. Am. A 18, 2594-2600 (2001). [CrossRef]
- Z. Y. Liu and D. Y. Fan, "Propagation of pulsed zeroth order Bessel beams," J. Mod. Opt. 45, 17-22 (1998). [CrossRef]
- M. A. Porras, R. Borghi, and M. Santarsiero, "Few-optical-cycle Bessel-Gauss pulsed beams in free space," Phys. Rev. E 62, 5729-5730 (2000). [CrossRef]
- P. Saari and K. Reivelt, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135-4138 (1997). [CrossRef]
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