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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 9 — Apr. 27, 2009
  • pp: 7465–7478
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Programmable ultrashort-pulsed flying images

M. Bock, S. K. Das, and R. Grunwald  »View Author Affiliations


Optics Express, Vol. 17, Issue 9, pp. 7465-7478 (2009)
http://dx.doi.org/10.1364/OE.17.007465


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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

Adaptive microoptical systems for encoding and robust transfer of complex, ultrashort-pulsed image information are important building blocks for applications in ultrafast optical processing, free-space communication, coherent control, nonlinear spectroscopy, metrology and microstructuring. Femtosecond laser pattern shaping with reflective mode spatial light modulators (SLMs) in conjunction with imaging components was reported [1

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]

]. Photorefractive volume reflection holograms were utilized for pulsed-image generation at pulse durations about 200 fs [2

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

]. This method, however, is limited with respect to larger bandwidths because of the diffractive functionality and requires an imaging cross-correlator setup with moving parts as a detecting system. The approximation of propagation invariant transverse intensity distributions by diffractive elements and transforming optics was shown in theoretical and experimental studies with random spots and spot arrays [3-5

3. 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]

] but is essentially limited by diffraction and losses at narrow slits of Fourier filters as well. An alternative, more promising approach is based on the combination of wavefront division multiplexing of ultrashort pulses with the generation of Bessel-type pseudo-nondiffracting beams each carrying the information of one pixel. This concept called “flying images” was proposed in 1996 by Peeter Saari [6

6. 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.

] and first demonstrated by the authors in collaboration with other co-workers [7

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]

,8

8. R. Grunwald, Thin film microoptics - new frontiers of spatio-temporal beam shaping (Elsevier, Amsterdam, 2007).

]. In these experimental studies, an amplitude mask was programmed into an SLM and the encoded beam was subsequently transformed into multiple Bessel-like sub-beams by a separate microaxicon array.

Here we report on an advanced concept based on a more compact, flexible and robust arrangement for generating flying images where the basic functionalities of microoptical beam shaping and spatial addressing are integrated in a single reflective SLM. By replacing Bessel beams by fringe-free sub-beams (“needle beams”) with very extended focal zones and of high-fidelity pulse transfer in few-femtosecond range, the free-space projection of high-power laser ultrafast pixelated flying images without the need of any additional imaging optics is well enabled. We demonstrate that a high-accuracy phase control at ultrasmall conical angles and a defined background texturing can be exploited to minimize the cross-talk of multiple needle beams.

2. Pseudo-nondiffracting needle beams

Idealized Bessel-beams consist of concentric fringes with a Bessel function as their radial field profile and represent propagation invariant solutions of the Helmholtz wave equation. They result from the constructive interference of conical partial waves. Because of propagating unchanged over infinite distances, these kinds of theoretical Bessel beams are often considered “diffraction-free” [9

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

,10

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

] or “nondiffracting”. By Fourier-transforming waves transmitted through (lossy) annular slits or by deflecting light with conical, refractive or reflective axicons [11

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

], nondiffracting beams can physically be approximated in finite but (e.g. in comparison to Gaussian beams) extended focal zones [12-14

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]

]. The axial oscillations of such zones can be modified or even depleted by certain amplitude and/or phase functions of the illuminating beams [15

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

,16

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]

]. Spatial frequencies and contrast of the radial oscillations (Bessel fringes) depend on spectrum, angular distribution and temporal coherence [17

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]

]. A further advantage of Bessel-like beams is their robustness against distortions (self-reconstruction) [18

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

]. If broadband ultrashort laser pulses are applied to generate pulsed Bessel-like beams or Bessel pulses [19-21

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

], the interplay of spectral bandwidth and angular spectrum in the linear regime (low intensities, free-space) with conical beam-shapers [22

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

,23

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]

] as well as in the nonlinear regime (high intensities, nonlinear media) with spontaneous self-focusing [24

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]

] leads to more complex, in particular x-shaped spatio-temporal interference patterns. Such “X-pulses” also display self-reconstruction properties in spatial and temporal domain [7

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]

]. The time-integrated spectral analysis of the fringes of ultrashort pulsed Bessel beams shows that the spectral bandwidth of outer fringes is significantly reduced whereas the central maximum remains nearly unchanged [25

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]

]. This spectrally nondiffracting characteristics of the central lobe corresponds to a temporally nondiffracting characteristics (i.e. constant pulse duration). In this context it seems to be justified to use the term “supercollimation” (which is mostly applied to particle beams of narrow angular spectra) to emphasize the unique propagation properties of these particular types of optical wavepackets. There are two different options to produce arrays of such ultrashort-pulsed fringe-less beams (in the following referred to as “needle beams” [7

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]

,8

8. R. Grunwald, Thin film microoptics - new frontiers of spatio-temporal beam shaping (Elsevier, Amsterdam, 2007).

]) of excellent pulse transfer which are closely connected with specific advantages and disadvantages: (I) spatial filtering, (II) small angles. In case (I), the concentric fringe patterns of Bessel-like beams have to be spatially filtered by truncating apertures. To minimize the diffraction at the rims of the apertures, the first minimum has to coincide with the radius of the aperture (self-apodization) [25

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]

]. In this way, a higher spatial resolution can be obtained but the energy of outer fringes is waisted. Recently it was shown by experiments that needle beams of extremely high aspect ratio (i.e. the ratio of Rayleigh range to the initial radius) can be shaped while the temporal pulse characteristics remains unchanged over focal zones > 10 cm for 10-fs pulses of a Ti:sapphire laser oscillator. Aspect ratios of a few 102 (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).

The alternative solution (II) is to work at sufficiently small angles so that only those partial beams interfere which create the central lobe. This setup works without external truncation (in “self-truncation”) and thus can be made more compact and less complicated to adjust. Another advantage of type-II-needle beams is that larger Rayleigh ranges can be obtained for the price of larger waist diameters. By writing ultraflat axicon profiles with conical angles far below 1° into the phase of the SLM corresponding to option (b), arrays of needle beams can easily be obtained as we will show later on. In addition to this, small-angle operation is advantageous for reflective setups because of the extraordinarily high angular tolerance of ultraflat axicons so that an off-axis design is not necessary [26

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]

]. It has further to be mentioned that both types of needle beams were found to behave self-reconstructing like Bessel beams [7

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]

,8

8. R. Grunwald, Thin film microoptics - new frontiers of spatio-temporal beam shaping (Elsevier, Amsterdam, 2007).

,25

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]

]. This property can be utilized for stabilizing the image transfer under the influence of distortions.

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

For encoding amplitude and/or phase maps in reconfigurable arrays of ultrashort wavepackets, SLMs have to fulfill specific requirements: (a) high spatial resolution, (b) excellent pulse transfer, and (c) high damage resistance. Acousto-optic modulators (AOMs) are proven tools for controlling amplitude and phase in temporal shaping but neither reach the necessary spatial resolution nor enable for a spatial wavefront manipulation. With pixelated liquid crystal displays (LCDs), high-definition wavefront control can be realized by changing the phase via birefringence [27

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]

]. Such devices are operated as temporal pulse shapers where the spectral phase is manipulated in the Fourier-plane of a zero-dispersion setup [28

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]

]. Because of the parallel processing of spectral channels, temporal transfer and damage are uncritical and a linear array of phase elements is sufficient. A first step toward a spatio-temporal programming was realized by individual temporal shaping along one spatial coordinate [1

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

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]

]. Non-adaptive spatial shaping was achieved with volume holograms [2

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

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

]. Polarization properties of LCDs were also exploited [31

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

]. Flexible mirrors [32

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]

] and micro-electro-mechanical systems (MEMS) are attractive for femtosecond applications because of low losses and minimal dispersion. Under realistic conditions, however, limitations by slow heat transfer, dynamic distortions and spatial resolution have still to be overcome.

An acceptable compromise are liquid crystal on silicon spatial light modulators (LCoS-SLM) which combine high resolution, large fill factor and moderate damage threshold with phase shifts of up to 2π in near infrared. Such reflective microdisplays contain a thin LC-layer and have a refractive index close to that of the alignment/conducting layer. They show an excellent spectral transfer because of being designed for RGB-projection [33

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]

]. With well adapted 2D gray level maps, arbitrary phase functions (aspheric, toroidal, convex, concave etc.) can be approximated even for white light sources. For a proper spatial shaping of ultrashort pulses, in particular at few or single optical cycles and up to octave-spanning spectral bandwidths, however, specific problems arise from spatio-temporal coupling and spectrally dependent propagation effects (chromatic aberration, dispersion, diffraction).

If the SLMs are applied to video projection, gray value maps are transferred to amplitude maps by filtering of a linearly polarized input beam with an external analyzer. To approximate the phase-only operation mode, however, SLMs have to be driven without additional polarizing element. In case of an optimized input polarization, the beam shaping can be realized by phase profiles only at unchanged intensity. Although the total efficiency is typically in the range of 60%, the majority of losses are non-absorptive and the damage resistance is high (peak intensities of few GW/cm2). 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]

]. At ultrashort pulse durations, the large spectral bandwidth and spatio-temporal coupling significantly complicate the Fraxicon design. Therefore, our investigations were focused on continuous profiles which can be controlled with acceptable phase accuracy in selected types of vertical aligned nematic (VAN) and parallel aligned nematic (PAN) SLMs [33

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

Beam shaping experiments were performed with a high-definition LCoS-SLM (HoloEye) which was carefully calibrated with respect to its phase characteristics (phase as a function of gray value and wavelength). The setup is shown schematically in 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).

The SLM contains a 3 μm thick VAN-type LC layer of negative dielectric anisotropy and reaches a maximum phase shift of about π at a wavelength of 800 nm. The spatial resolution is given by the matrix of 1920 × 1200 square-shaped pixels (8.1 μm pitch, fill factor > 0.9). The phase resolution (minimum phase step at a given wavelength) resulting from the dynamic range is π/255 (corresponding to a wavelength of 800 nm). The LCoS microdisplay was illuminated with sub-20 fs pulses of a Ti-sapphire laser oscillator (Femtosource, 4 nJ pulse energy, 75.3 MHz repetition rate, center wavelength 800 nm). Time-integrated spatial patterns shaped with the LCoS-SLM were detected with a 4 million pixel CCD-camera (Vosskuehler) after magnifying with a zoom lens and a microscope objective (10x magnification). For generating focal zones in close proximity to the SLM, an incident angle of 20 degree had to be chosen. Advantageously, working with off-axis illumination at such a relatively large angle was supported by the aforementioned tilt tolerance of flat conical phase profiles [26

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]

]. To write axicon phase profiles in the LCoS-SLM, a 2D matrix of gray values was encoded by using a graphics processor unit (GPU) with generic image processing software. Gray level images were transferred from the GPU via digital visual interface (DVI) to the controller unit which further translates the information into a voltage signal changing the local alignment of the LCs, thus modifying their effective refractive index pattern and hence the entire phase map. To study the propagation conditions including spatio-spectral and spatio-temporal contrast ratio and cross-talk, 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).

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).

In the example, the radius of rmax = 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 mm2 (input beam diameter 2w0 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 rmax = 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.

The following section concerns with the results of simulations as well as experiments on the propagation of spatially structured femtosecond pulses (flying images), adaptively shaped with above addressed techniques in all three modes of operation.

5. Results and discussion

5.1. Propagation of ultrashort-pulsed needle beams

Under the assumption of a concial phase profile, a Gaussian-shaped spectrum, and a homogeneous illumination over the axicon area, the evolution of pulsed Bessel beams (PBB) can be simulated analytically in a similar way as described in Refs. [35

35. P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).

,36

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

]. The signal measured by a time-integrating photodetector (e.g. camera pixel) can be expressed as

JPBB2=t1t2Re{Z·exp[τa2+(t+td)22τ02]·exp[i(t+td)·2π·cλ0]·J0}2dt
(1)

where

Z=1+i(t+td)·λ02π·τ02
(2)
td=tc·[2α(r')]r'tc
(3)
τa=rcsin(θ')
(4)
τ0=τFWHM2ln2
(5)
J0=1π0πcos[Z·2π·(sin(θ')λ0)·sin(q)]·dq
(6)
θ'=2[<α(r)>(1tan(δ)·F)]
(7)

The measured and simulated propagation of a solitary needle beam generated with a uniform phase background and a smoothed transition between minimum and maximum phase is plotted in Figs. 3a,b. The corresponding experimentally determined and Gaussian fit spectra can be found in Fig. 3(c),(d) respectively.

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.

The measured and the calculated propagation zones agree relatively well in their qualitative features. A detailed analysis, however, reveals certain deviations which are caused by unavoidable parasitic diffraction at sharp sub-structures (SLM pixels), non-Gaussian fine structure of the real spectrum and chirp effects. The measured needle beam shows a notably extended region of approximately constant intensity caused by the broader angular spectrum compared to the narrow one of a conical axicon. In the important central region of the needle beam, the simulated transversal beam diameter was found to be in excellent agreement with the experimental results (a quantitative analysis is given later on). Although the interference pattern should disappear at a distance of approximately 40 mm (zmax = rmax/<2θ>), the observed depth of the focal zone was significantly larger. This effect is explained by the smoothed Gaussian-like edges of the axicon phase function which induce flatter conical angles and consequently lead not only to an apodization but also to an axially asymmetric, more extended interference zone.

A central aspect of the propagation analysis is the evolution of the radial shape function. Its mathematical description should be compact and selective but keeping track of significant physical information. Therefore, well-known tools of higher order statistical moments were applied. In particular, the fourth order statistical moment 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.

The deviations from the kurtosis value of −0.53 of the central maximum of an ideal squared mathematical Bessel function 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).

For a further quantitative comparison of the propagation of different types of needle beams, we introduce a set of characteristic beam parameters in analogy to the description of Gaussian beams. With the Rayleigh length z0 (distance at which the cross sectional beam area has doubled) and the beam waist radius w0 (1/e2 of the transversal intensity distribution) we define an aspect ratio of z0/w0 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).

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. 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).

The comparison of relevant beam parameters in Tab. 1 indicates no significant dependence of the Rayleigh range on the background phase structure. This was proved by switching from a solitary needle beam to an array of needle beams with non-textured background. The position of the beam waists, however, was found to be slightly shifted towards the CCD detector if the checkered background phase pattern was applied. Axially-dependent beam radii extracted from measured and simulated data are shown in Fig. 5. Each waist radius was determined by sectional fitting the progressing curve by a 3rd 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.

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

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5.2. Ultrashort-pulsed flying images

In a proof-of-principle study on ultrashort-pulsed flying images we tested the performance of our approach by programming well-recognizable patterns (letter “E”) into the gray value map of an LCoS-SLM (for the principle, see Fig. 2(b)). The minimum period was chosen to be 324 μm. The detected intensity maps show the evolution of the pattern in axial direction with background structuring (animation (Media1) in Fig. 6(a)). To better visualize the beam propagation in 3D, the data were further processed by means of a tomographic-based software (animation (Media 2) in Fig. 6(b)).

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. 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. 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).

The contrast 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 z2-z1 = 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.

5.3. Spectral analysis of flying images with statistical moments

The spectral properties of flying images (letter “E”) were analyzed with high resolution. To realize a 2D spectral mapping by acquiring complete spectra at each position, we replaced the camera by a fiber-based spectrometer [25

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]

,26

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]

]. The spatial resolution was limited by an aperture of 15 μm diameter in front of the fiber. The pattern was scanned in transversal steps of 30 μm at distances of 50 mm and 60 mm, respectively. By compressing the data to a set of four statistical moments and peak signal, an unambiguous reconstructing essential spectral features and a quantitative comparison were enabled. Exemplary, the spatial and axial dependences of the standard deviation of a selected sub-beam (central peak in Fig. 7) are shown in Fig. 9 for two distances and allow to recognize the transversal structure even with a noisy background. Other statistical moments like skewness S, kurtosis K and center of gravity CoG were determined as well. Spatially averaged values are listed in Tab. 2.

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.

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Within the area of interest, the spectral bandwidth was obviously transferred over a distance of 10 mm (> 10 array periods) with low changes. A comparison of the averaged statistical moments in Tab. 2 confirms the broadband spectral transfer of needle beams generated by this LCoS-SLM. The averaged skewness (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.

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.

6. Conclusions

To conclude, the concept of Saari’s flying images was realized with programmable arrays of ultrashort-pulsed pseudo-nondiffracting fringe-free beams (“needle beams”). The shaping of a Ti:sapphire laser oscillator pulse in spatial domain was performed by a phase-only, reflective liquid-crystal-on-silicon spatial light modulator (LCoS-SLM) of proved high-fidelity transfer. Needle beams were obtained by self-truncation, i.e., by adjusting the foot-to-foot diameter of the radial intensity distribution to remain within the first minima of a Bessel function. The smooth rims of the intensity distribution lead, at the same time, to a self-apodization effect similar to the case of self-apodized truncation [25

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]

] (see Appendix). To study the time-integrated beam propagation, solitary needle beams as well as discrete images (letter “E”) composed of multiple needle beams were detected with high spatial resolution. Aspect ratios of > 300:1 were achieved. The effective aspect ratios are significantly influenced by the laser spectrum, divergence, wavefront errors and diffracting substructures (in particular for pixelated SLMs). A detailed study of the specific limitations will be a subject of future investigations. It was demonstrated that the contrast can be essentially improved by spatially separating the background light via controlled diffraction. For this purpose, an additional, high-spatial-frequency binary grating (checkered phase pattern) was programmed in the gaps between the axicons. Furthermore, statistical moments were applied to evaluate the spatially resolved spectrum of a propagating pulsed needle beam. By comparing spectral moments of four orders at different distances it was verified that no significant spatio-spectral changes emerge. The approach is robust against distortions and fluctuating initial conditions. Criteria for a reconstruction of relevant image information far behind the Fresnel domain will be a subject of continuing work. A more compact image shaper design based on radially asymmetric phase axicons is currently under investigation.

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

A condition for the generation of needle beams with conical axicons will be derived in a simplistic ray-optics picture (neglecting for diffraction). If a small-angle reflecting axicon of a finite diameter 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

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]

]). The geometrical situation is schematically drawn in 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 J02 (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).

The diameter ∧ of the central lobe of a polychromatic Bessel beam with a symmetric spectrum (first zero) is approximately given by

Λ=λ02nsinθ
(A1)

(λ0 = center wavelength, n = refractive index of air, 2θ = total conical beam angle) If the maximum diameter of the overlapping zone (D/2) at an axial distance zc (center of the focal zone) equals the diameter ∧ of the central lobe it follows

D=λ0nsinθ
(A2)

with

θ=2arctan(2hD)
(A3)

(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

θmax=arcsin(λ0Dn)
(A4)

and the corresponding minimum length of the needle beam of

Δzmin=D2tanθmax
(A5)

The last two relationships are useful for a rough design consideration. A geometric-optic estimation of a maximum aspect ratio can be found with the help of Eqs. (A5) and (A1):

A*=ΔzminΛ=Dncosθmaxλ0
(A6)

For example, for 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

The authors thank T. Elsaesser, G. Steinmeyer, P. Saari, V. Kebbel, S. Huferath-von-Luepke, S. Langer, M. Piché, M. Fortin, E. Recami, M. Besieris, S. Trillo and C. Conti for stimulating discussions as well as HoloEye AG (Berlin) and APE GmbH for a close collaboration. We appreciate the assistance of C. Poppe and M. Tischer. The work was supported in part by DFG project no. GR 1782/7-1.

References and links

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]

2.

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

3.

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]

4.

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

5.

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

6.

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.

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]

8.

R. Grunwald, Thin film microoptics - new frontiers of spatio-temporal beam shaping (Elsevier, Amsterdam, 2007).

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]

13.

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004). [CrossRef]

14.

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

15.

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

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]

20.

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

21.

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]

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]

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]

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]

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]

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]

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]

34.

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

35.

P. Saari and H. Sõnajalg, “Pulsed Bessel beams,” Laser Phys. 7, 32–39 (1997).

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. 25, 981–983 (2000). [CrossRef]

38.

O. Brzobohatý, T. Cižmár, and P. Zemánek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express 16, 12688–12700 (2008). [PubMed]

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

  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]
  2. K. B. Hill, K. G. Purchase, and D. J. Brady, "Pulsed-image generation and detection," Opt. Lett. 20, 1201-1203 (1995). [CrossRef] [PubMed]
  3. 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]
  4. V. Kettunen and J. Turunen, "Propagation-invariant spot arrays," Opt. Lett. 23, 1247-1249 (1998). [CrossRef]
  5. Z. Bouchal, "Controlled spatial shaping of nondiffrqacting patterns and arrays," Opt. Lett. 27, 1376-1378 (2002). [CrossRef]
  6. 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.
  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]
  8. R. Grunwald, Thin film microoptics - new frontiers of spatio-temporal beam shaping (Elsevier, Amsterdam, 2007).
  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]
  13. P. Saari and K. Reivelt, "Generation and classification of localized waves by Lorentz transformations in Fourier space," Phys. Rev. E 69, 036612 (2004). [CrossRef]
  14. R. M. Herman, T. A. Wiggins, "Production and uses of diffractionless beams," J. Opt. Soc. Am. A 8, 982-942 (1991). [CrossRef]
  15. F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 247-249 (1987).
  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]
  20. Z. Y. Liu and D. Y. Fan, "Propagation of pulsed zeroth order Bessel beams," J. Mod. Opt. 45, 17-22 (1998). [CrossRef]
  21. 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]
  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]
  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]
  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]
  27. G. D. Love, S. R. Restaino, G. C. Loos, and A. Purvis, "Wave-front control using a 64 × 64 pixel liquid crystal array," Proc. SPIE, 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]
  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]
  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]
  34. I. Golub, "Fresnel axicon," Opt. Lett. 31, 1890-1892 (2006). [CrossRef] [PubMed]
  35. P. Saari and H. Sõnajalg, "Pulsed Bessel beams," Laser Phys. 7, 32-39 (1997).
  36. Volker Kebbel, "Untersuchungen zur Erzeugung und Propagation ultrakurzer optischer Bessel-Impulse," Doctoral thesis, University Bremen, 2004 (BIAS Verlag Bremen, in German).
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