## Ultrashort highly localized wavepackets |

Optics Express, Vol. 20, Issue 11, pp. 12563-12578 (2012)

http://dx.doi.org/10.1364/OE.20.012563

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### Abstract

The recently introduced concept of radially non-oscillating, temporally stable ultrashort-pulsed Bessel-like beams we referred to as needle beams is generalized to a particular class of highly localized wavepackets (HLWs). Spatio-temporally quasi-nondiffracting pulses propagating along extended zones are shaped from Ti:sapphire oscillator radiation with a spatial light modulator and characterized with spatially resolved second order autocorrelation. Few-cycle wavepackets tailored to resemble circular disks, rings and bars of light represent the closest approximation of linear-optical light bullets known so far. By combining multiple HLWs, complex pulsed nondiffracting patterns are obtained.

© 2012 OSA

## 1. Introduction

1. P. Saari, “How small a packet of photons can be made?” Laser Phys. **16**(4), 556–561 (2006). [CrossRef]

3. P. Saari, M. Menert, and H. Valtna, “Photon localization barrier can be overcome,” Opt. Commun. **246**(4-6), 445–450 (2005). [CrossRef]

4. B. Piglosiewicz, D. Sadiq, M. Mascheck, S. Schmidt, M. Silies, P. Vasa, and C. Lienau, “Ultrasmall bullets of light-focusing few-cycle light pulses to the diffraction limit,” Opt. Express **19**(15), 14451–14463 (2011). [CrossRef] [PubMed]

*superresolution*). Another objective, however, is the localization of free propagating optical fields in the angular domain (

*supercollimation*). Perfect supercollimation would mean that a wavepacket of finite diameter would propagate without any spread in transversal direction so that it behaves quasi particle-like. In nonlinear optics, this is approximated in media by steady-state self-trapping (needle solitons) [5]. Self-induced spectral reshaping via conical refraction [6–8

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**(9), 093904 (2003). [CrossRef] [PubMed]

9. P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. **5**(2), 93–96 (2006). [CrossRef] [PubMed]

39. 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**(2), 1077–1089 (2008). [CrossRef] [PubMed]

41. M. Bock, S. K. Das, and R. Grunwald, “Programmable ultrashort-pulsed flying images,” Opt. Express **17**(9), 7465–7478 (2009). [CrossRef] [PubMed]

39. 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**(2), 1077–1089 (2008). [CrossRef] [PubMed]

## 2. Spatial localization

39. 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**(2), 1077–1089 (2008). [CrossRef] [PubMed]

41. M. Bock, S. K. Das, and R. Grunwald, “Programmable ultrashort-pulsed flying images,” Opt. Express **17**(9), 7465–7478 (2009). [CrossRef] [PubMed]

*D*of a monochromatic needle beam at an axial position

*z*depends on the wavelength

*λ*and the conical angle

*θ*(half angle against the optical axis):(

*n*= refractive index of air). For polychromatic beams with sufficiently narrow or symmetric spectral profiles,

*λ*can be replaced by the center-of-gravity wavelength

*λ*

_{0}. The scaling factor

*f*mainly depends on the divergence of the illuminating beam. Further modifications arise from the diffraction at the edges of the programmed axicons. At small conical angles, the vanishing “contact angle” at the rim leads to a phase apodization which reduces the diffraction. The decay of intensity in the SLM plane along the radius

*r*results from the coherent superposition (spectral interference) of all conical contributions. In the center of the zone of constructive interference at the distance

*z*

_{0}, it is proportional to the square of the first-kind, zero-order Bessel function

*J*within the limits of the first zero (first dark ring): (

_{0}^{2}(r)*r*

_{1}= radius of the first minimum of

*J*). The axial extension of the nondiffracting propagation zone (“confocal parameter”) can be defined by two characteristic distances

_{0}^{2}*z*

_{1}and

*z*

_{2}where, in analogy to the Gaussian beam description, the beam area is doubled and the intensity at the axis is reduced by a factor of 2 (FWHM extension). A more comprehensive theoretical description of the beam quality of localized beams (e.g. by means of the Wigner function) would have to take into account that inside the zone of constructive interference, each point in space is the origin of a bundle of rays (corresponding to a local wavefront ambiguity).

## 3. Spatio-temporal localization

*P*[54

^{2}54. G. Rousseau, N. McCarthy, and M. Pichãé, “Description of pulse propagation in a dispersive medium by use of a pulse quality factor,” Opt. Lett. **27**(18), 1649–1651 (2002). [CrossRef] [PubMed]

*σ*and

_{ν}*σ*are the variances in spectral and temporal domain, respectively. Thus, a dimensionless

_{t}*spatio-temporal localization parameter L*

^{2}of an ultrashort-pulsed wavepacket can be defined by the root of the product of

*M*

^{2}and

*P*

^{2}[55]:The quality parameters are spatially (

*M*

^{2}) and temporally integrated (

*P*

^{2}) so that this approximation is valid only in the paraxial case (small angles, negligible travel time effects). For non-paraxial propagation (large angles),

*L*cannot be applied because spatial and temporal features are not fully separable. In the most experiments reported here, the paraxial case was well approximated by working at extremely small conical angles and can serve as a figure of merit for the localization of wavepackets in space and time. It has to be noted, however, that the localization of HLW cannot completely be described by the used approach. Improved models require to include temporal changes of spatial and angular parameters (e.g. in frame of the Wigner function). For pulses with pulse durations beyond the area of validity of the slowly varying envelope approximation, however, the determination of the temporal and spectral statistical parameters requires a more sophisticated analysis.

^{2}## 4. The class of highly localized wavepackets (HLWs)

## 5. Experimental realization of HLWs

### 5.1. Experimental techniques

62. I. Golub, “Fresnel axicon,” Opt. Lett. **31**(12), 1890–1892 (2006). [CrossRef] [PubMed]

### 5.2. Few-cycle pulsed needle beams

*p*= 80 µm and

*p*= 500 µm and different geometrical arrangements (hexagonal, rectangular) were programmed in different types of LCoS-SLMs (HoloEye) with maximum phase steps between π and 2π at the center wavelength of 800 nm [63

63. R. Grunwald and M. Bock, “Programmable microoptics for ultrashort pulses,” Proc. SPIE **7716**, 77160P, 77160P-8 (2010). [CrossRef]

59. M. Bock, S. K. Das, C. Fischer, M. Diehl, P. Börner, and R. Grunwald, “Reconfigurable wavefront sensor for ultrashort pulses,” Opt. Lett. **37**(7), 1154–1156 (2012). [CrossRef] [PubMed]

**16**(2), 1077–1089 (2008). [CrossRef] [PubMed]

41. M. Bock, S. K. Das, and R. Grunwald, “Programmable ultrashort-pulsed flying images,” Opt. Express **17**(9), 7465–7478 (2009). [CrossRef] [PubMed]

*z*= 18.5 mm and

_{1}*z*= 138.5 mm (Figs. 4(b) and 4(c), respectively). The double beam structure enables to quantitatively evaluate specific multi-beam effects like cross-talk and interference.

_{2}59. M. Bock, S. K. Das, C. Fischer, M. Diehl, P. Börner, and R. Grunwald, “Reconfigurable wavefront sensor for ultrashort pulses,” Opt. Lett. **37**(7), 1154–1156 (2012). [CrossRef] [PubMed]

*w*at the chosen distance of

_{0}*z*= 100 mm was found to be 150 µm. The distances of neighboring maxima indicate the cycles of the electric field (2.7 fs at 800 nm central wavelength).

*M*

^{2}was determined to be 1.8.

### 5.3. Few-cycle nondiffracting light rings

63. R. Grunwald and M. Bock, “Programmable microoptics for ultrashort pulses,” Proc. SPIE **7716**, 77160P, 77160P-8 (2010). [CrossRef]

*r*and

_{i}*R*are the inner radius (center-to-peak) and outer radius (center to rim) (see Fig. 7(c)). The symmetry factors were

*SF*= 0.50 and 1.00 in Figs. 7(a), 7(b) and in the curves with red dots and blue squares in Fig. 7(c), respectively.

*SF*values of 0.5, 0.7 and 0.9, respectively. The best overall contrast was obtained at a distance of

*z*= 8 mm whereas inner and outer contrast

*C*= 84% and

_{i}*C*= 55% were nearly constant for all values of

_{a}*SF*. The hollow beam diameter (peak-to-peak distance for central cuts) depends, in good approximation, linearly on the

*SF*(Fig. 8(b)). An intensity map of a part of the hexagonal beam array detected at a distance of 8 mm is shown in Fig. 9(b). A 3D-plot of the hollow beam array (see Fig. 9(a)) was reconstructed using all measured depth information. The period was about 430 µm after slightly correcting for elliptical distortion introduced by the deviation from normal angle of incidence (20°). The procedure was similar to the correction of aberrations described in a recent paper on adaptive wavefront sensing of ultrashort pulses with needle beams in a reflective setup [59

59. M. Bock, S. K. Das, C. Fischer, M. Diehl, P. Börner, and R. Grunwald, “Reconfigurable wavefront sensor for ultrashort pulses,” Opt. Lett. **37**(7), 1154–1156 (2012). [CrossRef] [PubMed]

### 5.4. Few-cycle nondiffracting light blades and patterns composed of linear elements

**17**(9), 7465–7478 (2009). [CrossRef] [PubMed]

63. R. Grunwald and M. Bock, “Programmable microoptics for ultrashort pulses,” Proc. SPIE **7716**, 77160P, 77160P-8 (2010). [CrossRef]

## 6. Conclusions

*light bullets*” which do not require any nonlinear effects to compensate the diffraction over significantly extended depth of focus.

## Acknowledgments

## References and links

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**OCIS Codes**

(320.0320) Ultrafast optics : Ultrafast optics

(320.5540) Ultrafast optics : Pulse shaping

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: March 19, 2012

Revised Manuscript: May 4, 2012

Manuscript Accepted: May 4, 2012

Published: May 18, 2012

**Citation**

M. Bock, S. K. Das, and R. Grunwald, "Ultrashort highly localized wavepackets," Opt. Express **20**, 12563-12578 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-12563

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### References

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