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

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    [PubMed]

2008

2007

2006

2004

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]

2003

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]

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]

2002

2001

2000

1999

1998

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

V. Kettunen and J. Turunen, "Propagation-invariant spot arrays," Opt. Lett. 23, 1247-1249 (1998).
[CrossRef]

Z. Y. Liu and D. Y. Fan, "Propagation of pulsed zeroth order Bessel beams," J. Mod. Opt. 45, 17-22 (1998).
[CrossRef]

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

1997

P. Saari and H. Sõnajalg, "Pulsed Bessel beams," Laser Phys. 7, 32-39 (1997).

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

1995

1991

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]

R. M. Herman, T. A. Wiggins, "Production and uses of diffractionless beams," J. Opt. Soc. Am. A 8, 982-942 (1991).
[CrossRef]

1990

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]

1987

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 247-249 (1987).

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

1986

J. Durnin, "Exact solutions for nondiffracting beams I. The scalar theory," J. Opt. Soc. Am A 4, 651-654 (1986).
[CrossRef]

1954

Backus, S.

Bock, M.

Borghi, R.

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]

Bouchal, Z.

Z. Bouchal, "Controlled spatial shaping of nondiffrqacting patterns and arrays," Opt. Lett. 27, 1376-1378 (2002).
[CrossRef]

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

Brady, D. J.

Brixner, T.

Brzobohatý, O.

Chlup, M.

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

Cižmár, T.

Conti, C.

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]

Das, S. K.

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]

Di Trapani, P.

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]

Durnin, J.

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

J. Durnin, "Exact solutions for nondiffracting beams I. The scalar theory," J. Opt. Soc. Am A 4, 651-654 (1986).
[CrossRef]

Eberly, J. H.

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

Fan, D. Y.

Z. Y. Liu and D. Y. Fan, "Propagation of pulsed zeroth order Bessel beams," J. Mod. Opt. 45, 17-22 (1998).
[CrossRef]

Feurer, T.

Fortin, M.

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]

Friberg, A. T.

Gerber, G.

Golub, I.

Gori, F.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 247-249 (1987).

Griebner, U.

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]

Grunwald, R.

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]

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]

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]

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]

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. 25, 981-983 (2000).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 247-249 (1987).

Hattori, T.

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

Herman, R. M.

Hill, K. B.

Huferath, S.

Jedrkiewicz, O.

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]

Kapteyn, H.

Kebbel, V.

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]

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]

Kettunen, V.

Koehl, R. M.

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

Kummrow, A.

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]

Langer, S.

Leaird, D. E.

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]

Liu, Z. Y.

Z. Y. Liu and D. Y. Fan, "Propagation of pulsed zeroth order Bessel beams," J. Mod. Opt. 45, 17-22 (1998).
[CrossRef]

Maginnis, K.

McLeod, J. H.

Miceli, J. J.

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

Morrison, R. L.

Mourou, G.

Murnane, M.

Nelson, K. A.

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]

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

Néron, J.-L.

Neumann, U.

Nibbering, E.T.J.

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]

Nuss, M. C.

Osten, S.

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]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 247-249 (1987).

Patel, J. S.

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]

Piché, M.

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]

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]

Piskarskas, A.

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]

Porras, M. A.

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]

Purchase, K. G.

Reivelt, K.

P. Saari and K. Reivelt, "Generation and classification of localized waves by Lorentz transformations in Fourier space," Phys. Rev. E 69, 036612 (2004).
[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, "Evidence of X-shaped propagation-invariant localized light waves," Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

Rini, M.

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]

Rousseau, G.

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]

Russek, U.

Saari, P.

P. Saari and K. Reivelt, "Generation and classification of localized waves by Lorentz transformations in Fourier space," Phys. Rev. E 69, 036612 (2004).
[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 H. Sõnajalg, "Pulsed Bessel beams," Laser Phys. 7, 32-39 (1997).

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

Santarsiero, M.

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]

Sheppard, C. J. R.

C. J. R. Sheppard, "Bessel pulse beams and focus wave modes," J. Opt. Soc. Am. A 18, 2594-2600 (2001).
[CrossRef]

Sõnajalg, H.

P. Saari and H. Sõnajalg, "Pulsed Bessel beams," Laser Phys. 7, 32-39 (1997).

Staudt, P.

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Supplementary Material (2)

» Media 1: AVI (2357 KB)     
» Media 2: AVI (4044 KB)     

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Figures (10)

Fig. 1.
Fig. 1.

Experimental setup for the generation of addressable beam arrays shown for a hexagonal matrix of sub-beams (schematically). A femtosecond pulsed laser source is shining directly onto the LCoS-SLM, where a phase map of an ultraflat axicon is encoded. After being attenuated by a neutral glass filter the beam is magnified and imaged by a microscope objective and zoom lens on a high-resolution, high-dynamic-range CCD camera (4 MPixels, 12 bit).

Fig. 2.
Fig. 2.

Beam shaping with programmable phase elements and diffractive background control: (a) adapted distribution (red symbols) and a Gaussian function fitting the rim (black line) with a radius of 55 μm representing the standard deviation; (b) Gray scale map of fifteen elements programmed in the LCoS-SLM to approximate smoothed axicons corresponding to the black line in the left picture (diameter: 62 pixels, period: 40 pixels, pixel pitch: 8.1 μm). Shape errors were taken into account by applying an overlap-correction algorithm. The generated needle beams propagate the letter “E”. Inset: checkered phase pattern programmed in the gaps for contrast improvement (period: 2 pixels).

Fig. 3.
Fig. 3.

Intensity propagation of a solitary pseudo-nondiffracting needle beam: (a) measured propagation zone (20-fs pulse of a Ti:sapphire laser oscillator, center wavelength 800 nm), and (b) simulated propagation for realistic parameters (Gaussian spectrum) over an axial depth from z = 0 mm to z = 77 mm in steps of 1 mm; (c) measured spectrum, (d) Gaussian fit of the spectral data used as synthetic spectrum for the simulation.

Fig. 4.
Fig. 4.

Measured kurtosis of the radial shape depending on the distance for solitary needle beams (ex-SNB), array with homogeneous background (ex-NBA-HB) and array with checkered-phase background (ex-NBA-CB). The kurtosis for an ideal squared Bessel function is marked with the red horizontal line.

Fig. 5.
Fig. 5.

Averaged beam radii as a function of the distance measured for all three types of beams in comparison to the simulated evolution of a solitary needle beam (sim-SNB). Arrows indicate the positions of maximum intensity (colors of the related curves).

Fig. 6.
Fig. 6.

Ultrashort-pulsed flying image (letter “E”) composed of addressable needle beams (initial pulse duration 20 fs, minimum period: 324 μm). (a) and (b): Intensity map in a transversal plane at a distance of z = 50 mm., (Media 1),(b) 3D visualization of the propagation over an axial depth range of Δ z = 30 mm starting from z = 20 mm, The pattern dimensions (related to the centers of sub-beams) are Δx = 972 μm, Δy = 1620 μm). (Media 2)

Fig. 7.
Fig. 7.

Intensity contrast of vertical cuts through the second column of a flying pattern “E” (minimum array period 324 μm) at a distance of z = 50 mm; with homogeneous phase background (HB, blue cicles) and with checkered phase background (CB, black squares),. The contrast enhancement in case CB is clearly indicated at this distance.

Fig. 8.
Fig. 8.

Central cut through a selected needle beam generated with checkered phase background. The intensity was nonlinearly processed to enhance the visibility. Note the different scales for z and r (1:17).

Fig. 9.
Fig. 9.

Spectral mapping of a sub-beam of a flying image represented by 2D-resolved second order spectral moment (as a measure of bandwidth) in two planes perpendicular to the optical axis: (a) z = 50 mm, (b) z = 60 mm. Over the propagation path of 10 mm, the bandwidth remains fairly constant.

Fig. A1.
Fig. A1.

Geometrical conditions for the self-truncating generation of needle beams as fringe-free Bessel beams (schematically). A reflective axicon is illuminated in normal incidence. Dashed line: side lobes of the squared Bessel function J 0 2 (Λ = diameter of central lobe). The conical beam angle has to confine the distribution to match the first nodes (z = propagation axis, D = axicon diameter, z * = center of focal zone, θmax = maximum allowed conical beam angle, Δz min = corresponding length of the needle beam).

Tables (2)

Tables Icon

Table 1. Propagation parameters of sub-beams of different types of beam arrays compared to a solitary needle beam.

Tables Icon

Tab. 2. Spatially averaged values for the center of gravity (CoG), standard deviation (StDev), skewness (S), kurtosis (K) and spectral peak (P) of the spectrum of a selected sub-beam (central peak in Fig. 7) of a pulsed flying image.

Equations (13)

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J PBB 2 = t 1 t 2 Re { Z · exp [ τ a 2 + ( t + t d ) 2 2 τ 0 2 ] · exp [ i ( t + t d ) · 2 π · c λ 0 ] · J 0 } 2 dt
Z = 1 + i ( t + t d ) · λ 0 2 π · τ 0 2
t d = t c · [ 2 α ( r ' ) ] r ' t c
τ a = r c sin ( θ ' )
τ 0 = τ FWHM 2 ln 2
J 0 = 1 π 0 π cos [ Z · 2 π · ( sin ( θ ' ) λ 0 ) · sin ( q ) ] · dq
θ ' = 2 [ < α ( r ) > ( 1 tan ( δ ) · F ) ]
Λ = λ 0 2 n sin θ
D = λ 0 n sin θ
θ = 2 arctan ( 2 h D )
θ max = arcsin ( λ 0 Dn )
Δ z min = D 2 tan θ max
A * = Δ z min Λ = Dn cos θ max λ 0

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