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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 15 — Jul. 16, 2012
  • pp: 17220–17229
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Temporal focusing of ultrashort pulsed Bessel beams into Airy–Bessel light bullets

Peeter Piksarv, Heli Valtna-Lukner, Andreas Valdmann, Madis Lõhmus, Roland Matt, and Peeter Saari  »View Author Affiliations


Optics Express, Vol. 20, Issue 15, pp. 17220-17229 (2012)
http://dx.doi.org/10.1364/OE.20.017220


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Abstract

We present measurements of the impulse response of a circular phase diffraction grating in dependence of the field point location behind it. These measurements were carried out using a white-light spectral interferometry set-up, which employs photonic crystal fibers in both the signal and reference arms, and achieves a few micron spatial and almost one-wave-cycle temporal resolution. Our study shows that the grating as a simple and robust single-element optical device (i) suppresses the material-induced spread of ultrashort pulses, (ii) thereby generates the Airy–Bessel light bullets, and (iii) enables temporal focusing of the pulses at the prescribed propagation depth.

© 2012 OSA

1. Introduction

For numerous applications in non-linear optics, imaging, lithography, communications, ultrafast photonics, etc. high localization of pulses—both in space and time—is desired. Laser pulses lasting only about one optical cycle have been accomplished [1

1. S. Baker, I. A. Walmsley, J. W. G. Tisch, and J. P. Marangos, “Femtosecond to attosecond light pulses from a molecular modulator,” Nat. Photonics 5, 664–671 (2011). [CrossRef]

4

4. A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007). [CrossRef]

]. In order to achieve such ultrashort pulses close to the fundamental wave-optical limits, whose spectrum spans over the whole visible region, every optical element is a challenge of its own and requires careful pulse dispersion compensation and compression as well as appropriate measurement methods (see, e.g., reviews [5

5. R. Trebino, “Measuring the seemingly immeasurable,” Nat. Photonics 5, 189–192 (2011). [CrossRef]

, 6

6. I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1, 308–437 (2009). [CrossRef]

]). Here, we show how temporal focusing of a broadened pulse can be accomplished for certain propagation depths by a single circular phase diffraction grating. We propose a version of spatio-spectral interferometry technique employing supercontinuum laser source and photonic crystal fibers for full spatio-temporal measurement of impulse responses of optical systems with almost one-wave-cycle temporal resolution. Our results open up new possibilities in the characterization of ultrafast optical processes, devices, and in their applications.

Optical pulse compression with a pair of diffraction gratings is well established ever since the pioneering work by Treacy [7

7. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5, 454–458 (1969). [CrossRef]

]. However, when an ultrashort broadband pulse diffracts from a grating, it broadens due to the acquired chirp, which is rather substantial in the reality [8

8. P. Saari, J. Aaviksoo, A. Freiberg, and K. Timpmann, “Elimination of excess pulse broadening at high spectral resolution of picosecond duration light emission,” Opt. Commun. 39, 94–98 (1981). [CrossRef]

, 9

9. R. Imhof and D. Birch, “Distortion of Gaussian pulses by a diffraction grating,” Opt. Commun. 42, 83–86 (1982). [CrossRef]

]. The properties of circular diffraction gratings have been first considered by Dyson [10

10. J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. Lond. A 248, 93–106 (1958). [CrossRef]

]. The suppression of temporal spread for 30...200fs duration pulses has been accomplished with Bessel pulses generated by circular phase elements [11

11. H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. 22, 310–312 (1997). [CrossRef] [PubMed]

], and in a set-up of two refractive axicons [12

12. M. Clerici, D. Faccio, E. Rubino, A. Lotti, A. Couairon, and P. D. Trapani, “Space–time focusing of Bessel-like pulses,” Opt. Lett. 35, 3267–3269 (2010). [CrossRef] [PubMed]

].

High-resolution spatio-temporal measurement of light pulses is essential for understanding complex time and space couplings they exhibit over propagation [18

18. S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. 12, 093001 (2010). [CrossRef]

]. The recent advancements in the ultrashort pulse characterization techniques which enable the measurement of the complex light-field simultaneously in both spatial and temporal domain include SEA TADPOLE [19

19. P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral interferometry: a simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in real time,” Opt. Express 14, 11892–11900 (2006). [CrossRef] [PubMed]

, 20

20. P. Bowlan, P. Gabolde, M. A. Coughlan, R. Trebino, and R. J. Levis, “Measuring the spatiotemporal electric field of ultrashort pulses with high spatial and spectral resolution,” J. Opt. Soc. Am. B 25, A81–A92 (2008). [CrossRef]

], STARFISH [16

16. O. Mendoza-Yero, B. Alonso, O. Varela, G. Mínguez-Vega, Í. J. Sola, J. Lancis, V. Climent, and L. Roso, “Spatio-temporal characterization of ultrashort pulses diffracted by circularly symmetric hard-edge apertures: theory and experiment,” Opt. Express 18, 20900–20911 (2010). [CrossRef] [PubMed]

, 21

21. B. Alonso, Í. J. Sola, Ó. Varela, J. Hernández-Toro, C. Méndez, J. S. Román, A. Zaïr, and L. Roso, “Spatiotemporal amplitude-and-phase reconstruction by fourier-transform of interference spectra of high-complex-beams,” J. Opt. Soc. Am. B 27, 933–940 (2010). [CrossRef]

], SEA-SPIDER [6

6. I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1, 308–437 (2009). [CrossRef]

, 22

22. T. Witting, F. Frank, C. A. Arrell, W. A. Okell, J. P. Marangos, and J. W. G. Tisch, “Characterization of high-intensity sub-4-fs laser pulses using spatially encoded spectral shearing interferometry,” Opt. Lett. 36, 1680–1682 (2011). [CrossRef] [PubMed]

], and others. For example, superluminal propagation of the Bessel-X pulse and an ultrashort counterpart of Arago spot have been exquisitely demonstrated [15

15. P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34, 2276–2278 (2009). [CrossRef] [PubMed]

, 23

23. H. Valtna-Lukner, P. Bowlan, M. Lõhmus, P. Piksarv, R. Trebino, and P. Saari, “Direct spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets,” Opt. Express 17, 14948–14955 (2009). [CrossRef] [PubMed]

25

25. P. Piksarv, P. Bowlan, M. Lõhmus, H. Valtna-Lukner, R. Trebino, and P. Saari, “Diffraction of ultrashort Gaussian pulses within the framework of boundary diffraction wave theory,” J. Opt. 14, 015701 (2012). [CrossRef]

].

2. Pulsed Bessel beam

The Bessel beam can be formed from plane waves in various ways—either by a planar circular diffraction grating, or by an annular slit in conjunction with a collimating lens, or by a conical lens (refractive axicon) or mirror. While in the monochromatic case it is all the same, under ultrashort pulsed illumination the grating generates what is named “Bessel pulse” or “pulsed Bessel beam” [13

13. A. M. Shaarawi and I. M. Besieris, “On the superluminal propagation of X-shaped localized waves,” J. Phys. A. 33, 7227–7254 (2000). [CrossRef]

, 26

26. Z. Liu and D. Fan, “Propagation of pulsed zeroth-order Bessel beams,” J. Mod. Opt. 45, 17–21 (1998). [CrossRef]

28

28. M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002). [CrossRef]

]. Figuratively speaking, it is like a more or less thin slice cut from a Bessel beam. In distinction from the superluminal Bessel-X pulse [14

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

,15

15. P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34, 2276–2278 (2009). [CrossRef] [PubMed]

], the spatial profile of the Bessel pulse has no X-like section, its group velocity is subluminal but not the same over its spectrum—the effect of group velocity dispersion (GVD) takes place—and therefore the pulse is not propagation-invariant but broadens along the axis z in the course of propagation.

Let us take a closer look at a pulsed Bessel beam generator comprising a circular phase grating under broadband illumination. If the grating constant is K = 2π/d, d being the groove spacing, then for a given diffraction order m the normally incident light acquires a radial wave vector component k = m × K. The resulting pulsed Bessel beam can be expressed in the form
ΨPBB(ρ,z,t)J0(kρ)dωA(ω,z)eiφ0(ω)eikzzeiωt,
(1)
where kz2=ω2/c2k2, and φ0 (ω) is the initial spectral phase.

The complex spectral amplitude of the pulse is
A(ω,z)S0(ω)I0(ω,z)η(ω),
(2)
S0 (ω) being the spectrum of input pulse and η(ω) the grating efficiency. I0 (ω, z) describes the on-axis intensity dependence for a monochromatic wave component, which for an incident Gaussian beam with a waist w reads [29

29. V. Jarutis, R. Paškauskas, and A. Stabinis, “Focusing of Laguerre–Gaussian beams by axicon,” Opt. Commun. 184, 105–112 (2000). [CrossRef]

],
I0k2cωzexp[2(kcωzw)2].
(3)
For a square profile phase grating with a duty cycle of κ and groove depth of δ, the grating efficiency is given by [30

30. D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).

],
η(ω)=|κsinc(κm)+(1κ)sinc((1κ)m)eiπmeiϕ(ω,δ)|2,
(4)
where ϕ(ω, δ) = ωδ(n(ω) − 1)/c and n(ω) is the refractive index of the grating substrate.

The grating induced negative chirp may partially compensate for the positive chirp present due to the material dispersion at certain propagation distances. Figure 1 shows the calculated evolution of the on-axis amplitude of the electric field of the pulsed Bessel beam with various initial spectral phases corresponding to a dispersion induced by 0, 0.92 and 1.57mm thick fused silica samples. The simulations agree with the initial assumptions for temporal focusing. In the propagation range near the temporal focus the pulses can be described as the Airy–Bessel wave packets [31

31. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010). [CrossRef]

, 32

32. Z. Ren, H. Jin, Y. Shi, J. Xu, W. Zhou, and H. Wang, “Spatially induced spatiotemporally nonspreading airy–bessel wave packets,” J. Opt. Soc. Am. A 29, 848–853 (2012). [CrossRef]

]. It is evident that shorter temporal focus region can be obtained by using higher diffraction order or higher density grating. In addition, the more dispersive the grating, the smaller the lateral size of the maximum is obtained.

Fig. 1 The axial evolution of pulsed Bessel beam. Evaluation of Eq. (1) with different initial phase φ0 (ω) : (a) an uniform phase, i.e. no initial chirp, (b) a phase corresponding to a propagation through a fused silica sample with a thickness of 0.92mm, and (c) 1.57mm. Color shows the amplitude of the electric field, temporal position t = 0fs corresponds to a plane wave traveling at c.

3. Methods and experimental set-up

The underlying principle of our technique for measuring spatiotemporally resolved impulse responses is the well known spectrally resolved white-light interferometry (see, e.g., [33

33. J. Piasecki, P. Colombeau, M. Vampouille, C. Froehly, and J. A. Arnaud, “Nouvelle méthode de mesure de la réponse impulsionnelle des fibres optiques,” Appl. Opt. 19, 3749–3755 (1980). [CrossRef] [PubMed]

36

36. A. Börzsönyi, A. Kovács, M. Görbe, and K. Osvay, “Advances and limitations of phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun. 281, 3051–3061 (2008). [CrossRef]

]). However, since the output field of an optical element generally depends not only on time and/or the longitudinal coordinate, but also on the transversal coordinates, the interferometry has to be generalized into three dimensions as follows.

Let E0 (t) be the electric field of the homogeneous, transversally coherent light beam in the input plane z = 0 of the optical system. The field E (r,t) in the output half-space z > 0, which generally has acquired a 3D dependence on the field point coordinates r = (x,y,z) and a specific temporal dependence in every point, is given as temporal convolution
E(r,t)=0h(r,t)E0(tt)dt,
(5)
where h(r,τ) is the impulse response of the optical system for a location r behind it.

Let us suppose that somehow we have measured the cross-correlation function K (r,τ) between the input and output fields K (r,τ) ≡ 〈E (r,t)E0 (tτ)〉. Thereby we have also found out the impulse response function sought, since
K(r,τ)=0h(r,t)E0(tt)dtE0(tτ)==0h(r,t)E0(tt)E0(tτ)dt==const×h(r,τ),
(6)
as far as the temporal dependence of the input field can be considered as a delta-correlated stationary noise in which case 〈E0 (tt′)E0 (tτ)〉 ∝ δ(t′ − τ) and the integration over t′ drops out. In a general case, K (r,τ) is the convolution of h(r,τ) with the autocorrelation function of the input laser field and the impulse response can be recovered by a deconvolution method. As a matter of fact, the proportionality relation h(r,t) ∝ K (r,t) practically holds always whenever the spectrum of the input field is sufficiently “white”, i.e., wider than the spectrum of h(r,t)—in other words, wider than the support of the transfer function.

Our set-up follows the scheme of SEA TADPOLE, where the cross-correlation K (r,τ) has been obtained through three procedures: (i) quadratic detection of the sum E (r,t) + E0 (tτ), where the τ-dependence appears in the overlap region of the two beams tilted with respect to each other; (ii) temporal averaging due to the “slowness” of the detector (a CCD camera); (iii) spectral-domain separation of the cross-correlation term from the two non-oscillating autocorrelation terms. The spatial dependence of h(r,t) ≡ h(r,τ) is obtained by simply scanning the light-collecting tip of the signal-carrying fiber in the output half-space.

It should be pointed out that Eq. (6) in the case of time averaging is actually insensitive to the nature of the temporal dependence of the input light—be it deterministic ultrashort pulses or stationary noise or periodical bursts of temporally incoherent noise (which is the case of our set-up with a supercontinuum fiber laser). This favorable independence from the spectral phase of the light source follows from the basic principle of the spectrally resolved white-light interferometry, or—more generally—from the properties of time-averaged responses of linear systems. The resulting equivalence between the short pulse duration and the short coherence length of the illuminating light was used already in the 1970-ies for the holographic imaging of impulse responses [37

37. N. Abramson, “Light-in-flight recording by holography,” Opt. Lett. 3, 121–123 (1978). [CrossRef] [PubMed]

], and is widely exploited nowadays in the optical coherence tomography (see, e.g., [38

38. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991). [CrossRef] [PubMed]

40

40. M. Akiba, K. P. Chan, and N. Tanno, “Full-field optical coherence tomography by two-dimensional heterodyne detection with a pair of CCD cameras,” Opt. Lett. 28, 816–818 (2003). [CrossRef] [PubMed]

]).

Therefore the measurement part of our experimental set-up (Fig. 2) is nothing but a version of that of SEA TADPOLE technique [19

19. P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral interferometry: a simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in real time,” Opt. Express 14, 11892–11900 (2006). [CrossRef] [PubMed]

, 20

20. P. Bowlan, P. Gabolde, M. A. Coughlan, R. Trebino, and R. J. Levis, “Measuring the spatiotemporal electric field of ultrashort pulses with high spatial and spectral resolution,” J. Opt. Soc. Am. B 25, A81–A92 (2008). [CrossRef]

]. However, for accommodating the challenges the ultra-broadband spectrum of the supercontinuum laser poses, all optical components were carefully chosen. For example, the single mode operation of the light collecting fibers turned out to be achievable in the whole usable spectral range only by introducing photonic crystal fibers (PCF) [41

41. J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003). [CrossRef] [PubMed]

].

Fig. 2 Experimental set-up. The beam from the laser L is expanded by a reflective beam expander BE and then sampled into the reference arm of the interferometer by an UV fused silica window BS1, directed by mirror M1 through a beam splitter dispersion compensation plate BS2 and a variable delay line D, and focused by a spherical mirror SM1 into photonic crystal fiber PCF1. On the measurement arm the light is directed with mirrors M2–4 onto circularly symmetric diffraction grating G and the resulting field is sampled by the fiber PCF2. The output of the fibers is placed on the entrance slit of an imaging spectrometer with a few mm separation, so that light enters the spectrometer under a small angle 2θ perpendicular to the frequency axis ω. Spectrometer consists of a spherical mirror SM2, reflecting half-prism P, uncoated FS cylindrical lens CL and a CCD camera. Spatial interference pattern on the CCD camera is processed to retrieve the spectral phase and amplitude information of the field correlation function. Optical fibers allow effortless redesign of measurement set-up while keeping the most alignment-critical part—the spectrometer—intact.

The spectral range of the spectrometer determines the available temporal resolution of the method. In the current configuration the spectral amplitude and phase response is measurable from 450nm to 1020nm, which means that a resolution of 2.7fs, i.e. 1.4 optical cycles, can be achieved. The typical spectral and impulse response function of the measurement system is shown in Fig. 3. As being a linear method, this serves as a calibration and can be subtracted from subsequent measurements [20

20. P. Bowlan, P. Gabolde, M. A. Coughlan, R. Trebino, and R. J. Levis, “Measuring the spatiotemporal electric field of ultrashort pulses with high spatial and spectral resolution,” J. Opt. Soc. Am. B 25, A81–A92 (2008). [CrossRef]

]. The spectral range is limited by the diminishing quantum efficiency of Si-based CCD camera in the near-infrared, and by the increasing dispersion of the glass prism in the blue. Spectral resolution, which determines the maximum pulse length, ranges between δω=1.50.6radps, resulting that pulses up to 4ps duration and with a time-bandwidth product of 1500 could be measured. A method similar to MUD TADPOLE [42

42. J. Cohen, P. Bowlan, V. Chauhan, P. Vaughan, and R. Trebino, “Measuring extremely complex pulses with time-bandwidth products exceeding 65,000 using multiple-delay crossed-beam spectral interferometry,” Opt. Express 18, 24451–24460 (2010). [CrossRef] [PubMed]

], could be applied to enhance the technique for even longer temporal range. The spatial resolution—currently 3μm—is determined by the mode size of the fibers.

Fig. 3 Instrument function of the set-up. Left: typical spectral intensity and phase response with a linear spectral phase component subtracted, and right: temporal intensity and phase. The blue dots on the right mark data points, and the light blue line is spline interpolation revealing a FWHM of 3.8fs of the instrument function.

In our set-up we have used Fianium ultra-broadband supercontinuum fiber laser SC-400-2-PP. For single mode pulse delivery NKT Photonics polarization-maintaining endlessly single-mode photonic crystal fibers LMA-PM-5 were used, which have an almost constant mode field diameter ∼ 4.1μm from 400nm to 1200nm. Although polarization-maintaining properties were not used in the current experiments, they may prove useful for measuring the polarization response of optical systems. In the measurement set-up shown on Fig. 2, an additional broadband polarizer with a negligible wave-front distortion was placed in the spectrometer right before the CCD camera (Allied Vision Technologies Stingray F-504B). As the sensitivity of the camera abruptly decays at longer wavelengths than 1000nm, the whole spectrum of the laser source was not used, resulting in some loss of temporal resolution. The non-uniform coupling of all angular-spectral components into fiber has the same result. However, as these circumstances affect the two arms equally, they do not cause measurement errors or artifacts. For more detailed consideration of the effects of numerical aperture of fibers in SEA TADPOLE set-up see Ref. [20

20. P. Bowlan, P. Gabolde, M. A. Coughlan, R. Trebino, and R. J. Levis, “Measuring the spatiotemporal electric field of ultrashort pulses with high spatial and spectral resolution,” J. Opt. Soc. Am. B 25, A81–A92 (2008). [CrossRef]

].

Circularly symmetric binary phase gratings have the groove spacing of d = 20μm and were manufactured in Fraunhofer Heinrich Hertz Institute. The groove depth varies in the range δ = 940...945nm in a 1.57mm thick fused silica substrate was initially optimized for illumination around 800nm. Under electron microscope the duty cycle of the grating was measured to be κ = 47.5%, therefore yielding weak even diffraction orders.

High-precision UV fused silica windows were used as beam splitters. They have been put into the beam path at an almost normal incidence in order to minimize unnecessary wavelength dependent polarization effects and spatial chirp.

For measurement series “a”, the dispersion of grating substrate was compensated by another part cut out from the same substrate. In order to obtain partial dispersion compensation in “b” series, a 6mm thick fused silica substrate was placed in the reference arm of the interferometer and two fused silica substrates about 2mm and 3mm thick were placed in the measurement arm which resulted an effective thickness of 0.92mm dispersion uncompensated grating substrate. For series “c”, the dispersion from grating substrate was entirely uncompensated.

In order to make measurements at different propagation distances z after the diffraction grating, the grating was moved away from the fiber tip, while keeping the lengths of the interferometer arms constant. The lateral scans were performed with 1μm step. The FWHM temporal duration of pulses was measured from cubic spline interpolation of data points. For data smoothing purposes zero filling was used by adding 2870 zeros to both ends of retrieved spectral amplitude and phase.

The simulations were carried out by evaluating the spectral phase and amplitude of pulsed Bessel beam in Eq. (1), which was Fourier transformed into time-domain. Additional spectral term that accounts for the numerical aperture of fibers was also introduced in the simulations. In the calculations only the zeroth to fifth diffraction orders were taken into account. Experimentally measured spectral phase for given configurations was used in the simulations.

4. Results

The measurement and simulation results on the impulse response of a circular phase grating with a period of 20μm and an effective fused silica substrate thickness of 0.92mm on several distances behind the grating are compared in Fig. 4. It shows that initially long pulse focuses in time for some propagation distance z and thereafter spreads out again. The theoretical predictions show good agreement with the experimental results.

Fig. 4 Temporal focusing of the pulsed Bessel beam. Temporally broadened pulse shortens and thereupon spreads out again after propagating through the circularly symmetric phase grating with a substrate thickness of 0.92mm. Top row: results of measurements performed at different propagation depths; bottom row: simulation based on Eq. (1). Color shows the amplitude of the electric field. Temporal origin t = 0fs on the horizontal axis corresponds to a plane wave front traveling at c and its position relative to the measured pulse reveals subluminal velocity of the latter (cf. the superluminality of the Bessel-X pulse [14, 15]). Vertical axis shows the scan through the radial coordinate revealing the propagation of the pulsed Bessel beam with no lateral spread. At the closest distance the pulse corresponding to the 3rd diffraction order with a fringe spacing of 3.3μm was clearly resolved.

The small discrepancy in the pulse intensities of different diffraction orders and in the z dependence of intensity arises from the non-Gaussian pulse profile in the experiments as the Gaussian radial profile was assumed in the simulations. Also neither the finite radius of the grating nor the small radial spectral variation of the laser pulse were accounted for. The small tilt in the tx plane for the pulses registered at higher z values might be due to the asymmetric intensity profile of the laser beam.

To further study the temporal focusing of diffraction gratings the FWHM duration of the pulse intensities were calculated from each measurement. The results are shown in Fig. 5. The prevailing 1st diffraction order with a transform limited duration of 3.7fs, which initially after propagation through 0.92mm substrate had a duration of 27.8fs achieves average duration of 5.9fs for propagation distances z = 20...65mm. For the fully uncompensated grating substrate with a thickness of 1.57mm we can observe temporal focusing from 44.8fs down to an average of 7.0fs within the propagation range z = 40...110mm. In conjunction with the non-diffracting properties of the Airy–Bessel wave packets from the theoretical study [32

32. Z. Ren, H. Jin, Y. Shi, J. Xu, W. Zhou, and H. Wang, “Spatially induced spatiotemporally nonspreading airy–bessel wave packets,” J. Opt. Soc. Am. A 29, 848–853 (2012). [CrossRef]

], the temporal focusing effect was achieved in a quite long propagation extent.

Fig. 5 Measured on-axis intensities of pulsed Bessel beams. (a) Pulsed Bessel beam generated by circularly symmetric binary phase grating with dispersion compensated grating substrate. Initial pulse with a measured FWHM of 4.1fs broadens to 91.7fs over 10cm of propagation length. (b) Partially dispersion-compensated grating substrate: 27.8fs pulse focuses to 5.1fs and then broadens to a duration of 52.7fs. (c) Dispersion-uncompensated grating substrate: 44.8fs pulse achieves a minimum duration of 6.1fs by the end of its propagation length. Vertical scale shows normalized intensity In (z,t), color shows relative maximum intensity within a given measurement series. Pulses with the shortest temporal duration for each series have been highlighted with red border line and in the temporal focal region (series “b” and “c”) they exhibit a quasi-Airy profile. The latter is formed by a residual uncompensated cubic phase (cf. theoretical analysis of this effect in Ref. [32]).

5. Conclusions

We have studied the temporal broadening and focusing effect of ultrashort pulsed Bessel beams generated by a circularly symmetric binary phase grating. We have shown that a 7-fold decrease in the temporal spread of ultrashort pulse can be achieved and the Airy–Bessel wave packet with a prescribed propagation depth formed by an extremely simple device consisting only of one optical element—a diffractive axicon. Our white-light spectral interferometry set-up for characterization of ultrashort impulse responses of optical devices was built following the SEA TADPOLE scheme adopted for ultra-wide (> 1 octave) passband and with its present realization we achieved sub-3fs temporal resolution. Using incoherent supercontinuum laser source exhibits advantages over conventional sources of few-femtosecond pulses in terms of cost, simplicity of the set-up and ease of operation. Generally, the set-up should be especially useful for examining the underlying physics of light field transformation by optical systems or in designing and selecting suitable optical elements for obtaining prescribed ultrashort light-fields.

Acknowledgments

The authors acknowledge Pamela Bowlan and Rick Trebino for valuable insights into the SEA TADPOLE pulse measurement method. This work has been supported by the Estonian Science Foundation grant 7870 and by the European Regional Development Fund under project 3.2.0101.11-0037. P. P. has been partially supported by the European Social Fund under project 1.2.0401.09-0079. This work was carried out in part in the High Performance Computing Center of the University of Tartu.

References and links

1.

S. Baker, I. A. Walmsley, J. W. G. Tisch, and J. P. Marangos, “Femtosecond to attosecond light pulses from a molecular modulator,” Nat. Photonics 5, 664–671 (2011). [CrossRef]

2.

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T. Kobayashi, J. Liu, and K. Okamura, “Applications of parametric processes to high-quality multicolour ultrashort pulses, pulse cleaning and cep stable sub-3fs pulse,” J. Phys. B. 45, 074005 (2012). [CrossRef]

4.

A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys. 9, 242 (2007). [CrossRef]

5.

R. Trebino, “Measuring the seemingly immeasurable,” Nat. Photonics 5, 189–192 (2011). [CrossRef]

6.

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1, 308–437 (2009). [CrossRef]

7.

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5, 454–458 (1969). [CrossRef]

8.

P. Saari, J. Aaviksoo, A. Freiberg, and K. Timpmann, “Elimination of excess pulse broadening at high spectral resolution of picosecond duration light emission,” Opt. Commun. 39, 94–98 (1981). [CrossRef]

9.

R. Imhof and D. Birch, “Distortion of Gaussian pulses by a diffraction grating,” Opt. Commun. 42, 83–86 (1982). [CrossRef]

10.

J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. Lond. A 248, 93–106 (1958). [CrossRef]

11.

H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. 22, 310–312 (1997). [CrossRef] [PubMed]

12.

M. Clerici, D. Faccio, E. Rubino, A. Lotti, A. Couairon, and P. D. Trapani, “Space–time focusing of Bessel-like pulses,” Opt. Lett. 35, 3267–3269 (2010). [CrossRef] [PubMed]

13.

A. M. Shaarawi and I. M. Besieris, “On the superluminal propagation of X-shaped localized waves,” J. Phys. A. 33, 7227–7254 (2000). [CrossRef]

14.

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

15.

P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34, 2276–2278 (2009). [CrossRef] [PubMed]

16.

O. Mendoza-Yero, B. Alonso, O. Varela, G. Mínguez-Vega, Í. J. Sola, J. Lancis, V. Climent, and L. Roso, “Spatio-temporal characterization of ultrashort pulses diffracted by circularly symmetric hard-edge apertures: theory and experiment,” Opt. Express 18, 20900–20911 (2010). [CrossRef] [PubMed]

17.

M. Lõhmus, P. Bowlan, P. Piksarv, H. Valtna-Lukner, R. Trebino, and P. Saari, “Diffraction of ultrashort optical pulses from circularly symmetric binary phase gratings,” Opt. Lett 37, 1238–1240 (2012). [CrossRef] [PubMed]

18.

S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt. 12, 093001 (2010). [CrossRef]

19.

P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral interferometry: a simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in real time,” Opt. Express 14, 11892–11900 (2006). [CrossRef] [PubMed]

20.

P. Bowlan, P. Gabolde, M. A. Coughlan, R. Trebino, and R. J. Levis, “Measuring the spatiotemporal electric field of ultrashort pulses with high spatial and spectral resolution,” J. Opt. Soc. Am. B 25, A81–A92 (2008). [CrossRef]

21.

B. Alonso, Í. J. Sola, Ó. Varela, J. Hernández-Toro, C. Méndez, J. S. Román, A. Zaïr, and L. Roso, “Spatiotemporal amplitude-and-phase reconstruction by fourier-transform of interference spectra of high-complex-beams,” J. Opt. Soc. Am. B 27, 933–940 (2010). [CrossRef]

22.

T. Witting, F. Frank, C. A. Arrell, W. A. Okell, J. P. Marangos, and J. W. G. Tisch, “Characterization of high-intensity sub-4-fs laser pulses using spatially encoded spectral shearing interferometry,” Opt. Lett. 36, 1680–1682 (2011). [CrossRef] [PubMed]

23.

H. Valtna-Lukner, P. Bowlan, M. Lõhmus, P. Piksarv, R. Trebino, and P. Saari, “Direct spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets,” Opt. Express 17, 14948–14955 (2009). [CrossRef] [PubMed]

24.

P. Saari, P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, and R. Trebino, “Basic diffraction phenomena in time domain,” Opt. Express 18, 11083–11088 (2010). [CrossRef] [PubMed]

25.

P. Piksarv, P. Bowlan, M. Lõhmus, H. Valtna-Lukner, R. Trebino, and P. Saari, “Diffraction of ultrashort Gaussian pulses within the framework of boundary diffraction wave theory,” J. Opt. 14, 015701 (2012). [CrossRef]

26.

Z. Liu and D. Fan, “Propagation of pulsed zeroth-order Bessel beams,” J. Mod. Opt. 45, 17–21 (1998). [CrossRef]

27.

W. Hu and H. Guo, “Ultrashort pulsed Bessel beams and spatially induced group-velocity dispersion,” J. Opt. Soc. Am. A 19, 49–53 (2002). [CrossRef]

28.

M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E 65, 026606 (2002). [CrossRef]

29.

V. Jarutis, R. Paškauskas, and A. Stabinis, “Focusing of Laguerre–Gaussian beams by axicon,” Opt. Commun. 184, 105–112 (2000). [CrossRef]

30.

D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).

31.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010). [CrossRef]

32.

Z. Ren, H. Jin, Y. Shi, J. Xu, W. Zhou, and H. Wang, “Spatially induced spatiotemporally nonspreading airy–bessel wave packets,” J. Opt. Soc. Am. A 29, 848–853 (2012). [CrossRef]

33.

J. Piasecki, P. Colombeau, M. Vampouille, C. Froehly, and J. A. Arnaud, “Nouvelle méthode de mesure de la réponse impulsionnelle des fibres optiques,” Appl. Opt. 19, 3749–3755 (1980). [CrossRef] [PubMed]

34.

C. Sainz, J. E. Calatroni, and G. Tribillon, “Refractometry of liquid samples with spectrally resolved white light interferometry,” Meas. Sci. Technol. 1, 356–361 (1990). [CrossRef]

35.

A. P. Kovács, K. Osvay, Z. Bor, and R. Szipöcs, “Group-delay measurement on laser mirrors by spectrally resolved white-light interferometry,” Opt. Lett. 20, 788–790 (1995). [CrossRef] [PubMed]

36.

A. Börzsönyi, A. Kovács, M. Görbe, and K. Osvay, “Advances and limitations of phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun. 281, 3051–3061 (2008). [CrossRef]

37.

N. Abramson, “Light-in-flight recording by holography,” Opt. Lett. 3, 121–123 (1978). [CrossRef] [PubMed]

38.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991). [CrossRef] [PubMed]

39.

M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Laser Eng. 37, 631–641 (2002). [CrossRef]

40.

M. Akiba, K. P. Chan, and N. Tanno, “Full-field optical coherence tomography by two-dimensional heterodyne detection with a pair of CCD cameras,” Opt. Lett. 28, 816–818 (2003). [CrossRef] [PubMed]

41.

J. C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003). [CrossRef] [PubMed]

42.

J. Cohen, P. Bowlan, V. Chauhan, P. Vaughan, and R. Trebino, “Measuring extremely complex pulses with time-bandwidth products exceeding 65,000 using multiple-delay crossed-beam spectral interferometry,” Opt. Express 18, 24451–24460 (2010). [CrossRef] [PubMed]

43.

P. Bowlan and R. Trebino, “Using phase diversity for the measurement of the complete spatiotemporal electric field of ultrashort laser pulses,” J. Opt. Soc. Am. B 29, 244–248 (2012). [CrossRef]

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(320.5550) Ultrafast optics : Pulses
(320.7100) Ultrafast optics : Ultrafast measurements

ToC Category:
Ultrafast Optics

History
Original Manuscript: May 25, 2012
Revised Manuscript: July 1, 2012
Manuscript Accepted: July 1, 2012
Published: July 12, 2012

Citation
Peeter Piksarv, Heli Valtna-Lukner, Andreas Valdmann, Madis Lõhmus, Roland Matt, and Peeter Saari, "Temporal focusing of ultrashort pulsed Bessel beams into Airy–Bessel light bullets," Opt. Express 20, 17220-17229 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-17220


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References

  1. S. Baker, I. A. Walmsley, J. W. G. Tisch, and J. P. Marangos, “Femtosecond to attosecond light pulses from a molecular modulator,” Nat. Photonics5, 664–671 (2011). [CrossRef]
  2. 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. Express19, 14451–14463 (2011). [CrossRef] [PubMed]
  3. T. Kobayashi, J. Liu, and K. Okamura, “Applications of parametric processes to high-quality multicolour ultrashort pulses, pulse cleaning and cep stable sub-3fs pulse,” J. Phys. B.45, 074005 (2012). [CrossRef]
  4. A. L. Cavalieri, E. Goulielmakis, B. Horvath, W. Helml, M. Schultze, M. Fieß, V. Pervak, L. Veisz, V. S. Yakovlev, M. Uiberacker, A. Apolonski, F. Krausz, and R. Kienberger, “Intense 1.5-cycle near infrared laser waveforms and their use for the generation of ultra-broadband soft-x-ray harmonic continua,” New J. Phys.9, 242 (2007). [CrossRef]
  5. R. Trebino, “Measuring the seemingly immeasurable,” Nat. Photonics5, 189–192 (2011). [CrossRef]
  6. I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon.1, 308–437 (2009). [CrossRef]
  7. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron.5, 454–458 (1969). [CrossRef]
  8. P. Saari, J. Aaviksoo, A. Freiberg, and K. Timpmann, “Elimination of excess pulse broadening at high spectral resolution of picosecond duration light emission,” Opt. Commun.39, 94–98 (1981). [CrossRef]
  9. R. Imhof and D. Birch, “Distortion of Gaussian pulses by a diffraction grating,” Opt. Commun.42, 83–86 (1982). [CrossRef]
  10. J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. Lond. A248, 93–106 (1958). [CrossRef]
  11. H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett.22, 310–312 (1997). [CrossRef] [PubMed]
  12. M. Clerici, D. Faccio, E. Rubino, A. Lotti, A. Couairon, and P. D. Trapani, “Space–time focusing of Bessel-like pulses,” Opt. Lett.35, 3267–3269 (2010). [CrossRef] [PubMed]
  13. A. M. Shaarawi and I. M. Besieris, “On the superluminal propagation of X-shaped localized waves,” J. Phys. A.33, 7227–7254 (2000). [CrossRef]
  14. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett.79, 4135–4138 (1997). [CrossRef]
  15. P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett.34, 2276–2278 (2009). [CrossRef] [PubMed]
  16. O. Mendoza-Yero, B. Alonso, O. Varela, G. Mínguez-Vega, Í. J. Sola, J. Lancis, V. Climent, and L. Roso, “Spatio-temporal characterization of ultrashort pulses diffracted by circularly symmetric hard-edge apertures: theory and experiment,” Opt. Express18, 20900–20911 (2010). [CrossRef] [PubMed]
  17. M. Lõhmus, P. Bowlan, P. Piksarv, H. Valtna-Lukner, R. Trebino, and P. Saari, “Diffraction of ultrashort optical pulses from circularly symmetric binary phase gratings,” Opt. Lett37, 1238–1240 (2012). [CrossRef] [PubMed]
  18. S. Akturk, X. Gu, P. Bowlan, and R. Trebino, “Spatio-temporal couplings in ultrashort laser pulses,” J. Opt.12, 093001 (2010). [CrossRef]
  19. P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, “Crossed-beam spectral interferometry: a simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in real time,” Opt. Express14, 11892–11900 (2006). [CrossRef] [PubMed]
  20. P. Bowlan, P. Gabolde, M. A. Coughlan, R. Trebino, and R. J. Levis, “Measuring the spatiotemporal electric field of ultrashort pulses with high spatial and spectral resolution,” J. Opt. Soc. Am. B25, A81–A92 (2008). [CrossRef]
  21. B. Alonso, Í. J. Sola, Ó. Varela, J. Hernández-Toro, C. Méndez, J. S. Román, A. Zaïr, and L. Roso, “Spatiotemporal amplitude-and-phase reconstruction by fourier-transform of interference spectra of high-complex-beams,” J. Opt. Soc. Am. B27, 933–940 (2010). [CrossRef]
  22. T. Witting, F. Frank, C. A. Arrell, W. A. Okell, J. P. Marangos, and J. W. G. Tisch, “Characterization of high-intensity sub-4-fs laser pulses using spatially encoded spectral shearing interferometry,” Opt. Lett.36, 1680–1682 (2011). [CrossRef] [PubMed]
  23. H. Valtna-Lukner, P. Bowlan, M. Lõhmus, P. Piksarv, R. Trebino, and P. Saari, “Direct spatiotemporal measurements of accelerating ultrashort Bessel-type light bullets,” Opt. Express17, 14948–14955 (2009). [CrossRef] [PubMed]
  24. P. Saari, P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, and R. Trebino, “Basic diffraction phenomena in time domain,” Opt. Express18, 11083–11088 (2010). [CrossRef] [PubMed]
  25. P. Piksarv, P. Bowlan, M. Lõhmus, H. Valtna-Lukner, R. Trebino, and P. Saari, “Diffraction of ultrashort Gaussian pulses within the framework of boundary diffraction wave theory,” J. Opt.14, 015701 (2012). [CrossRef]
  26. Z. Liu and D. Fan, “Propagation of pulsed zeroth-order Bessel beams,” J. Mod. Opt.45, 17–21 (1998). [CrossRef]
  27. W. Hu and H. Guo, “Ultrashort pulsed Bessel beams and spatially induced group-velocity dispersion,” J. Opt. Soc. Am. A19, 49–53 (2002). [CrossRef]
  28. M. A. Porras, “Diffraction effects in few-cycle optical pulses,” Phys. Rev. E65, 026606 (2002). [CrossRef]
  29. V. Jarutis, R. Paškauskas, and A. Stabinis, “Focusing of Laguerre–Gaussian beams by axicon,” Opt. Commun.184, 105–112 (2000). [CrossRef]
  30. D. C. O’Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2003).
  31. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics4, 103–106 (2010). [CrossRef]
  32. Z. Ren, H. Jin, Y. Shi, J. Xu, W. Zhou, and H. Wang, “Spatially induced spatiotemporally nonspreading airy–bessel wave packets,” J. Opt. Soc. Am. A29, 848–853 (2012). [CrossRef]
  33. J. Piasecki, P. Colombeau, M. Vampouille, C. Froehly, and J. A. Arnaud, “Nouvelle méthode de mesure de la réponse impulsionnelle des fibres optiques,” Appl. Opt.19, 3749–3755 (1980). [CrossRef] [PubMed]
  34. C. Sainz, J. E. Calatroni, and G. Tribillon, “Refractometry of liquid samples with spectrally resolved white light interferometry,” Meas. Sci. Technol.1, 356–361 (1990). [CrossRef]
  35. A. P. Kovács, K. Osvay, Z. Bor, and R. Szipöcs, “Group-delay measurement on laser mirrors by spectrally resolved white-light interferometry,” Opt. Lett.20, 788–790 (1995). [CrossRef] [PubMed]
  36. A. Börzsönyi, A. Kovács, M. Görbe, and K. Osvay, “Advances and limitations of phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun.281, 3051–3061 (2008). [CrossRef]
  37. N. Abramson, “Light-in-flight recording by holography,” Opt. Lett.3, 121–123 (1978). [CrossRef] [PubMed]
  38. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science254, 1178–1181 (1991). [CrossRef] [PubMed]
  39. M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, “Geometric phase-shifting for low-coherence interference microscopy,” Opt. Laser Eng.37, 631–641 (2002). [CrossRef]
  40. M. Akiba, K. P. Chan, and N. Tanno, “Full-field optical coherence tomography by two-dimensional heterodyne detection with a pair of CCD cameras,” Opt. Lett.28, 816–818 (2003). [CrossRef] [PubMed]
  41. J. C. Knight, “Photonic crystal fibres,” Nature424, 847–851 (2003). [CrossRef] [PubMed]
  42. J. Cohen, P. Bowlan, V. Chauhan, P. Vaughan, and R. Trebino, “Measuring extremely complex pulses with time-bandwidth products exceeding 65,000 using multiple-delay crossed-beam spectral interferometry,” Opt. Express18, 24451–24460 (2010). [CrossRef] [PubMed]
  43. P. Bowlan and R. Trebino, “Using phase diversity for the measurement of the complete spatiotemporal electric field of ultrashort laser pulses,” J. Opt. Soc. Am. B29, 244–248 (2012). [CrossRef]

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