Abstract: We propose a scheme for producing attosecond pulses withsophisticated spatio-spectral waveforms. The profile of a seed attosecondpulse is modified and its central frequency is up-converted through inter-action with an infrared pump pulse. The transverse profile of the infraredbeam and a spatiotemporal shift between the seed and infrared pulses areused for manipulating the spatio-spectral waveform of the generated pulsebeam. We present several examples of sophisticated isolated attosecondpulse beam generation, including spatio-spectral Airy beam that exhibitsprismatic self-bending effect and a beam undergoing auto-focusing to asub-micron spot without the need of a focusing lens or nonlinearity.
References and links1. P. H. Bucksbaum, “The future of attosecond spectroscopy,” Science317,766–769 (2007).2. M. Drescher, M. Hentschel, R. Kienberger, M. Uiberacker, V. Yakovlev, A. Scrinzi, T. Westerwalbesloh, U.
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6. F. Calegari, C. Vozzi, M. Negro, G. Sansone, F. Frassetto, L. Poletto, P. Villoresi, M. Nisoli, S. De Silvestri, andS. Stagira, “Efficient continuum generation exceeding 200 eV by intense ultrashort two-color driver,” Opt. Lett.34,3125–3127 (2009).
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11. F. Ferrari, F. Calegari, M. Lucchini, C. Vozzi, S. Stagira, G. Sansone, and M. Nisoli, “High-energy isolatedattosecond pulses generated by above-saturation few-cycle fields,” Nat. Photonics4, 875–879 (2010).
12. X. Feng, S. Gilbertson, H. Mashiko, H. Wang, S. D. Khan, M. Chini, Y. Wu, K. Zhao, and Z. Chang, “Generationof isolated attosecond pulses with 20 to 28 femtosecond lasers,” Phys. Rev. Lett.103,183901 (2009).
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1. Introduction
Ongoingdevelopment in isolated attosecond pulse generation pushes forward the field of at-tosecond science where ultrafast dynamics in atoms molecules and solids are investigated and
#153743 - $15.00 USD Received 31 Aug 2011; revised 25 Sep 2011; accepted 25 Sep 2011; published 19 Oct 2011(C) 2011 OSA 24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21740
controlled in the time domain [1, 2]. Isolated attosecond pulses are produced by temporal gat-ing and spectral filtering of high-order harmonics. Several gating processes have been demon-strated, including amplitude gating using one [3,4] or multiple colors driving fields [5,6], polar-ization gating [7,8], ionization-induced phase-mismatch gating [9,10], and ionization-inducedsaturation [11] as well as combinations of the schemes [12]. Research in attosecond pulse gen-eration has been focusing on decreasing the pulse-width of the generated pulse [13]. While ascheme for generation of temporally-shaped attosecond pulses has been recently proposed [14],the control over spatial properties of attosecond pulses has not been investigated yet. In thefemtosecond regime, on the other hand, there has been a lot of research on shaping the spatialproperties of the beam, which facilitates new applications of femtosecond pulses. A fascinat-ing example is the recently discovered optical Airy beam in which the transverse electric fieldprofile is described by an Airy function [15]. Airy beams exhibit intrigue properties includingnon-diffraction, self-bending and self-healing [15–17] that facilitate new applications [18]. Forexample, curved plasma filaments that are induced by the self-bending beams allow longitu-dinal resolution in remote spectroscopy applications [19]. To-date, Airy beams were producedby using non-trivial phase masks [16] or through second harmonic generation with intricatequasi-phase matching pattern in the visible and infrared spectral regions [20].
Here, we propose a scheme for producing attosecond pulses with sophisticated spatio-spectral waveforms. Spatial and spectral profiles of a seed attosecond pulse are shaped andits central frequency is up-converted through interaction with an infrared pump laser pulse ina gas of atoms or ions with large binding potential. The pump field, which is too weak to re-lease electrons to the continuum, amplifies the energy of electrons that were ionized by the seedpulse through single photon absorption. An x-ray attosecond pulse is emitted when the ener-getic electrons recombine with their parent ions. At maximum amplification, the x-ray emissiondisplays Airy spectrum with huge bandwidth which corresponds to an attosecond pulse with aflat-top pulse-shape. Generation of attosecond pulses with fast leading and trailing edges arealso shown. We also present settings for generation of attosecond pulse beams with sophisti-cated spatial waveforms. When the source pulse is focused to a region in which the intensity ofthe pump beam has a linear slope, the produced pulse exhibits Airy profiles in both spectrumand the transverse coordinates. This newly discovered “spatio-spectral Airy beam” displaysprismatic effect where each spectral component propagates along a different curved path. Thiseffect can be utilized for fine spectral tuning of the attosecond pulse by a slit that is located upstream of the nonlinear medium. In addition, we show that a pump beam with a trough intensityprofile induces a spatial parabolic phase profile in the produced attosecond pulse beam. As aresult the beam auto focuses to a nano-scale hot spot at a controlled propagation distance.
Isolated attosecond pulses are produced through the process of high harmonic generation(HHG). In HHG, an electron is first excited to the continuum through tunneling ionization, thenpropagates under the influence of the driving laser field, and finally re-combines with its parention while emitting a high-energy photon [21]. The source for the short-wavelength radiationis associated with the electron acceleration to high kinetic energy during its re-collision path.The re-collision energy is proportional to the ponderomotive energy of the driving laserUp ∝Iλ 2, whereI and λ are the laser intensity and wavelength, respectively [21]. In most HHGexperiments, the ionization and propagation steps are driven by a single laser field. As a result,the high frequency emission is dominantly produced by the spatio-temporal maxima in the fieldof the driving laser pulse. This coupling complicates any attempt to engineer the waveform ofthe produced attosecond pulses.
We propose to generate sophisticated attosecond pulses by employing a temporal or spa-tiotemporal gating in the ionization step of the HHG process. The gating can be realized by aseed attosecond pulse [22–24]. In this process, the seed pulse induces a transition to an excited
#153743 - $15.00 USD Received 31 Aug 2011; revised 25 Sep 2011; accepted 25 Sep 2011; published 19 Oct 2011(C) 2011 OSA 24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21741
level from which tunneling ionization by the optical field becomes significant. Subsequently,themotion of the electrons in the continuum is dominantly determined by the optical field. Theelectrons may recombine with their parent ions while emitting HHG radiation in the same fash-ion as in HHG that is driven by a single field. Importantly, the enhancement in the ionizationrate also dramatically increases the efficiency of HHG as was demonstrated experimentally inseveral works [23, 25–27]. In this work, we implement the ionization gating by a seed attosec-ond pulse for generating spatially, temporally and spectrally sophisticated attosecond pulses.In our technique, a seed attosecond pulse confines the ionization step, and therefore also thegeneration process, into a limited region in space and time. In this way, the properties of thegenerated attosecond pulse are controlled by the spatiotemporal profile of the mid-IR beam atthat region. For example, the phase front of the generated pulse beam resembles the transverseintensity profile of the mid-IR beam. This correspondence results from the fact that in HHG,the emitted phase of a high-order harmonic is approximately linear with the intensity of thelaser that accelerates the electron during its trajectory in the continuum [28, 29]. This relationwas used for manipulating the quadratic phase fronts of diverging HHG beams [30, 31]. Theionization by a single photon from the EUV attosecond pulse induces de-coupling between thefirst and second steps of the HHG process which enables us to significantly extend this ma-nipulation. The usage of mid-IR pulses with non-standard (non bell-shape) intensity profilesin the region of significant ionization can lead to generation of attosecond pulse beams withsophisticated waveforms.
2. Scheme for producing sophisticated attosecond pulses
We first present an example for generation of temporally-sophisticated attosecond pulses. We
consider a mid-IR driving fieldEIR = f IR (t)cos(
2πcλ IR t
)
and an extreme ultraviolet (EUV) iso-
lated attosecond pulseEEUV = f EUV (t − τ)cos(
2πcλ EUV t
)
that are jointly focused onto a medium
of singly ionized helium ions (ionization potential is 54.4eV) (Fig. 1a). Hereλ IR = 2µm,λ atto = 30nm and c is the speed of light. The temporal envelopes,f IR (t) and f EUV (t − τ)are Gaussians with full width at half maximum (FWHM) of 50f s and 250as, respectively,and peak intensities of 7.4 · 1014W · cm−2 and 1012W · cm−2, respectively. The parameterτ denotes the time delay between the pulses which in the first example is set to−3.06 f s(+15o after a peak of the mid-IR field). This time delay was chosen so electrons that areionized by the attosecond pulse would emit the highest photon-energy upon recombination(the cut-off radiation). The motion of the electron under the influence of the mid-IR andEUV fields is calculated by solving the one dimensional time dependent Schrodinger equa-tion for the potentialV (x, t) = Vbind(x) + x
(
EIR +EEUV)
. The ion binding potential is mod-
eled byVbind(x) = −54.4/
√
(x/a0)2 +0.5 eV, having an ionization potential of 54.4eV (a0
is Bohrs radius) [32, 33]. Ionization rate is obtained by calculating the low frequency out-going flux. Importantly, the parameters are chosen such that ionization yield by the mid-IRfield alone is very low (10−7) whereas ionization is enhanced by the EUV field (Fig. 1a). Thehigh-order radiation is calculated by the acceleration expectation value, using Ernfest theorema(t) ∝
⟨
ψ (t)∣
∣
(
ddxV (x, t)
)∣
∣ψ (t)⟩
, whereψ is the electronic wavefunction. As shown in Fig. 1b,
the radiated spectral field highly resembles an Airy function,E (ω)∼Ai[
(ω −ωcut-off)/ 3√
α/2]
with α = 1600f s−3 is the chirp quadratic coefficient andωcut off = 930eV is the cut-off fre-quency as predicted by the classical model [21]. While it is known that the spectral fieldat the cut-off region exhibits an Airy profile [34], the ionization gating by a seed attosec-ond pulse leads to generation of an Airy spectral field over a very large bandwidth. Thecorresponded emitted field exhibits a flat-top amplitude and cubic phase at some time inter-
#153743 - $15.00 USD Received 31 Aug 2011; revised 25 Sep 2011; accepted 25 Sep 2011; published 19 Oct 2011(C) 2011 OSA 24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21742
−8 −6 −4 −2 0 2 4
−0.1
−0.05
0
0.05
0.1
Time [fs]
EE
UV+
Em
id−
IR
[ at.
u.]
Ioniz
ati
on
rate
[×10−
6s−
1]
a)
0
2
4
850 900 9500
0.1
0.2
0.3
0.4
hω [eV]
∣ ∣ ∣E
(ω)∣ ∣ ∣
[arb
.u.]
b)
−1
−0.5
0
0.5
1
Spec
tralphase
φ(ω
)/π
0
0.5
1
1.5
Spec
tralin
tensi
ty[a
rb.u
.]
800 850 900 9500
0.2
0.4
0.6
0.8
1
hω [eV]
puls
eF
WH
M[fs]
c)
0.5 1 1.5 20
0.5
1
1.5
2
2.5
Time [fs]
inte
nsi
ty[a
rb.u
.]
d)
−10
−5
0
5
10
φ(t
)/π
0.5 1 1.5 20
0.5
1
1.5
2
2.5
Time [fs]
inte
nsi
ty[a
rb.u
.]
−5o delayf)
−10
−5
0
5
10
φ(t
)/π
0.5 1 1.5 20
0.5
1
1.5
2
2.5
Time [fs]
inte
nsi
ty[a
rb.u
.]
+5o delaye)
−10
−5
0
5
10
φ(t
)/π
Fig. 1. (color online) Frequency up-conversion of attosecond pulses. (a) Driving field (solidline) which consists of a mid-IR (λIR = 2µm) laser field and an EUV pulse of 250 attosec-ond duration (λEUV = 30nm). When the joint field interacts with He+ medium, ionization(dashed blue) is gated by the attosecond pulse since the mid-IR field is too weak to inde-pendentlyionize the medium. (b) Emitted spectral field amplitude (blue) and phase (green)asa function of photon energy. The spectral field almost perfectly matches an Airy function(dashed black). x-ray attosecond pulses are produced when the lower part of the spectrumis filtered out. (c) Pulse width of the produced x-ray attosecond pulse versus the bottomlimit of the high pass filter.Blue, greenandred circles mark920eV, 900eV and800eVspectralfilters respectively. (d) x-ray attosecond pulses produced byhigh-pass filtration at920eV (blue),900eV (green) and800eV (red), where multiple spectral lobes result in aflat-toppulse (phase is marked bydashed red). (e-f) Delaying the EUV pulse by+5o and−5o with respect to the mid-IR driver, result in a trailing edge and leading edge pulses,respectively.
val E(t) = F−1[
E (ω)]
˜E0exp[
i(
ωcut-off (t − tcut-off)−α6 (t − tcut-off)
3)]
. tcut-off corresponds
to the emission time of the cut-off radiation, and the amplitude,E0, is time-independent be-cause the seed EUV pulse populates uniformly the quantum trajectories that recombine aroundthe cut-off time. Notably, the spectral field,E (ω), is real (Fig. 1b) because the emitted field intime domain obeysE (t − tcut-off) = E∗ (tcut-off− t) in the vicinity of tcut-off.
Figures 1c and 1d present results concerning the temporal shape of the generated pulse whenthe spectral Airy field (Fig. 1b) is filtered by a high-pass filter. A 250 attoseconds (FWHM)transform limited pulse corresponds to the single (and largest) lobe of the Airy spectrum at
#153743 - $15.00 USD Received 31 Aug 2011; revised 25 Sep 2011; accepted 25 Sep 2011; published 19 Oct 2011(C) 2011 OSA 24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21743
X [µm]
hω
[eV
]
578 eV
a)55 60 65 70 75
450
500
550
600
650
−100 0 100X[µm]
55 60 65
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
X [µm]
E(x
)[a
rb.u
.]
Gaussian envelopedAiry function
E(x)
b)
X [µm]
Pro
pagati
on
dis
tance
[mm
]
θ
c)60 80 100 120
0
2
4
6
8
10
540 560 580 600 620 640 660
0.24
0.26
0.28
0.3
0.32
Init
ialdiff
ract
ion
angle
[o]
hω [eV]
1.5
2
2.5
3
3.5
Curv
atu
re·1
0−
4[m
m−
1]
50
55
60
65
70
Init
ialposi
tion
[µm
]
d)
Fig. 2. (color online) Spatio-spectral Airy beam. (Inset of plot a) The source EUV at-tosecondpulse is transversely shifted relative to the peak intensity of the mid-IR beam.(a) Emission exhibits Airy spectrum and Airy intensity profile in the transverse axis. (b)Single spectral component of the emitted field,E578eV
(x) , versus the transverse coordinate(solid blue) compared to a Gaussian enveloped Airy function (dash red). (c) Propagation ofE578 eV
(x) . The peak of the beam exhibits parabolic propagation path. (d) Initial propagationangle (θ ), parabolic curvature coefficient, and initial position of the main lobe as a functionof photon-energy.
the cut-off (∼930eV). Figure 1c shows that the pulse-width increases when more lobs are in-cluded. This results from theπ phase jump between adjacent lobs of the Airy field. Moreover,as the number of spectral lobs increases, the pulse temporal shape becomes squarish (flat-top)(Fig. 1d). Finally, figures 1e and 1f show the pulse shapes for additional +0.1 and -0.1 fem-toseconds time-delays, respectively (+5o and−5o shift in the phase of the mid-IR wave). Forthe large bandwidth case, the pulse shape exhibit a triangle (leading or trailing edges) pulses.The asymmetry emerges from uneven population of the quantum trajectories set to recombinebefore and aftertcut-off. The square and triangle pulses demonstrate a simple form for attosec-ond pulse shaping. More complicated pulses can be produced by using complex structures ofionizing pulses.
Next, we present production of spatially-sophisticated attosecond pulses that are controlledby the intensity profile of the mid-IR beam. We calculated the time and transverse (x-coordinate) dependence of the emitted field by solving the Schrodinger equation separately ineach point of the x-grid. The accuracy of the transverse sampling of the generated radiation isguaranteed by using a grid pixel size of 100nmwhich is much smaller than the transverse varia-tion scales of the incoming laser fields (several microns). Subsequently, we simulated the propa-gation of the produced pulse beam along the propagation axis, z, by solving the linear (refractiveindex is 1), paraxial, and absorption-free Helmholtz equation for each narrow spectral compo-nent (intensity is too weak for nonlinear effects and propagation distance is much smaller thanthe absorption length). Coherent super position of the spectral components gives the space-timedependence of the pulse at propagation distance z:E (x, t,z) =∑
nE (ωn,x,z)exp[iωn · t]∆ω.
The first example, which we term spatio-spectral Airy pulse beam, is presented in figure 2. In
#153743 - $15.00 USD Received 31 Aug 2011; revised 25 Sep 2011; accepted 25 Sep 2011; published 19 Oct 2011(C) 2011 OSA 24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21744
Spectra
lfiltra
tion
by
1µm
slit
λ1
λ2
λ3
d)
slit center [µm]
hω
[eV
]
7.6 eV
e)80 90 100 110
450
500
550
600
650
Tim
e[fs]
X[µm]
c)
548 eV
0.43 [fs]
1.4 [µm]
80 85 90 95 100 105
−0.8
−0.4
0
0.4
0.8
Tim
e[fs]
b)
578 eV
0.51 [fs]
2.1 [µm]−0.8
−0.4
0
0.4
0.8
Tim
e[fs]
a)
608 eV
0.52 [fs]
2.5 [µm]−0.8
−0.4
0
0.4
0.8
Fig. 3. (color online) Prism effect of spatio-spectral Airy beam. Propagation of attosecondpulsesof 10eV bandwidth centered at (a) 608eV (b) 578eV and (c) 548eV . Plots showthe spatio-temporal intensity after 5mmpropagation is of attosecond scale duration andµmwidth. (d) Schematic illustration for enhanced prism effect utilization by a slit. (e) A 1µmslit located 5mmafter the emission plane act as a source of 7.6eV bandwidth attosecondpulse, with linearx(ω) dependence.
this example, the mid-IR and EUV fields of Fig 1a possess transverse (x) profiles of Gaussianswith FWHM of 10µm and 150µm respectively. A lateral shift ofx0 = 63.7µm between thepeaks of the pulses was introduced such that the EUV pulse resides on an approximately linearmid-IR intensity slope (zero 2nd derivative) (inset of Fig. 2a). Consequently, in this geometrythe local cut-off frequency also grows linearly withx. Thus, the spectral Airy of Fig. 1 becomes
x-dependent:E (ω,x) = Ai[
(ω −ω0 +β (x−x0))/3√
α/2]
. Hereω0 is the cut-off atx0 andβis the cut-off lateral slope. Indeed, the calculated intensity exhibits spatial and spectral Airyprofile (Fig. 2a). In this pulse, each spectral component forms a spatial Airy beam (Fig. 2b) thatpropagates along a curved path (Fig. 2c). The dependences of the curvature, initial propaga-tion angle and initial position of the peak on the photon-energy (Fig. 2d) disperse the spectralcomponents in space. The consequence of this prismatic effect is shown in figure 3. Figures3(a-c) show spatiotemporal profiles after 5mmpropagation, formed by a 10eV band pass filtercentered at 608eV, 578eV and 548eV, respectively. As shown, each spectral region undergoesa different lateral shift. As a result, a narrow slit that is located downstream can be used as atunable spectral filter of the attosecond pulse (Fig. 3d). A linear dependence of the spectral in-tensity vs. the slit central-position is shown in Fig. 3e. Figure 3 shows that spatio-spectral Airypulse beam may be used for tunable manipulation and control of the spectrum of attosecondpulses or for spectrometer-less attosecond spectroscopy, where different segments of a sampleare irradiated by an attosecond pulse of varying spectrum.
We now move to a second example of isolated attosecond pulse production that exhibit so-phisticated spatial characteristic. In this example, an isolated attosecond pulse with parabolicphase front is generated where the source EUV pulse overlaps a trough profile of the mid-IR pulse. Consequently, the produced isolated attosecond pulse beam gets focused down-stream. Figure 4 shows a concrete example where the EUV (30nm) source pulse is 10µmwide and centered at the mid-IR (2µm) trough (inset of Fig. 4a). The mid-IR intensity is given
by Imid-IRtrough (x) = I IR
[
Awexp(
−(x/σw )2)
−Anexp(
−(x/σn )2)]
whereAw = 1.4, An = 0.4,
σw = 90µm, σw = 30µm, andI IR is the mid-IR intensity previously discussed. Figures 4a
#153743 - $15.00 USD Received 31 Aug 2011; revised 25 Sep 2011; accepted 25 Sep 2011; published 19 Oct 2011(C) 2011 OSA 24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21745
hω
[eV
]X [µm]
|E(x, ω)|2
a)
−10 −5 0 5 10
820
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940
−100 0 100X[µm]
X [µm]
hω
[eV
]
φ(x, ω)
−10 −5 0 5 10
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φ(x
,ω)/
π
−0.5
0
0.5
φ(x) @929eVb)
X [µm]
Pro
pagati
on
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tance
[mm
]
7µm FWHM
450 nmFWHM
hω =929 eV
2.5
mm
c)
−4 −2 0 2 40
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e[fs] 0.23 [fs]
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d)
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Fig. 4. (color online) Generation of auto-focusing x-ray attosecond pulses. (Inset of plota) The source attosecond pulse beam (blue) is located at a trough of the mid-IR beam(red). Emitted spatial-spectral (a) intensity, and (b) phase exhibiting Airy spectrumwithparabolic variation in the transverse coordinate. (b-inset show the phase profile along thedashed line). (c) Propagation of the 929eV spectral component of the generated radiationdemonstrates focusing effect due to spatial quadratic phase. (d) Spatio-temporal spot ofgenerated attosecond pulse at the focal point. The calculation consists of 250eV bandwidthcentered athω = 929eV.
and 4b show the intensity and phase of the radiation, clearly exhibiting the parabolic profiles.The parabolic phase front leads to focusing of the produced pulse after 5mm. Figure 4c showsthe propagation and focusing of the 929eV photon-energy spectral component. The intricatefocusing spot details and the extended Rayleigh range result from theπ phase jumps at thebeam margins (see inset in Fig. 4b). Figure 4d shows the spatio-temporal profile of the focusedpulse at the focal point. The pulse consists of a 250eV spectral bandwidth that is centered at929eV and exhibit a sub-micron attosecond focal spot. The focusing distance and width can bemanipulated by the profile of the mid-IR beam.
We emphasize that the results presented in this paper are a direct consequence of universalfeatures of HHG at the single-atom level, namely, the cut-off relation, the quadratic chirp atcutoff, and the fact that the intrinsic phase grows linearly with the intensity of the driving laser.Nonetheless, propagation effects can influence the process. First, phase matching is crucialfor obtaining efficient macroscopic generation. When the wave-mixing process is not phasematched, the generated attosecond pulse builds up only over a propagation distance where therelative phase between the high-order polarization and attosecond pulse slip by radians; theso-called coherence length. Coherence lengths of very high-order harmonics in pre-ionizedplasma, as are the numerical examples in this paper, are in the order of several microns [35]. Inthis regime, grating assisted phase matching can be used for significantly extending the effectiveregion that contributes to the produced signal [36,37]. A potential approach for phase-matchingthe process is to implement pressure tuning phase matching in initially neutral or partially pre-ionized nonlinear medium [38]. As shown in Ref. 38, mid-IR pump can be phase matched
#153743 - $15.00 USD Received 31 Aug 2011; revised 25 Sep 2011; accepted 25 Sep 2011; published 19 Oct 2011(C) 2011 OSA 24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21746
with very short-wavelength HHG light through detailed balance between the atomic, plasmaandmodal dispersions. The second important propagation effect is volume averaging which, inprinciple, can average out the sophisticated details of the produced isolated attosecond pulse.Attosecond pulses with sophisticated transvers profiles can exhibit non-trivial dynamics (e.g.the parabolic propagation path of Airy beams). That is, the transverse profiles of these pulsesare approximately conserved for only a limited propagation distance (this “shape preserved”length is in the order of 0.4 millimeter for the spatio-spectral Airy beam of Fig. 2). Thus, inorder to obtain the aimed sophisticated pulse at the exit of the nonlinear medium, its effectivethickness (i.e. coherence length) should not exceed the “shape preserved” propagation length.
3. Conclusions
In this work we propose the first scheme for producing attosecond pulses with sophisticatedspatio-temporal and spatio-spectral waveforms. In this scheme, a seed attosecond pulse gatesthe ionization step of high harmonic generation process, temporally and spatially, and a shapedmid-IR field which accelerates the electrons in the continuum, controls the spatial dependenceof the emitted attosecond pulses. Using the proposed scheme, we demonstrated numerically,all for the first time, i) HHG emission with very broad Airy spectrum which corresponds to aflat-top attosecond pulse, ii) spatio-spectral Airy beam which exhibits prismatic self-bendingpropagation, and iii) an auto-focusing isolated attosecond pulse which focuses to a sub-micronspot without the need of a focusing lens or nonlinearity.
Acknowledgments
This work was supported by Legacy Heritage fund of Israel Science Foundation (ISF), theMarie Curie International Reintegration Grant (IRG) and USA–Israel Binational Science Foun-dation (BSF).
#153743 - $15.00 USD Received 31 Aug 2011; revised 25 Sep 2011; accepted 25 Sep 2011; published 19 Oct 2011(C) 2011 OSA 24 October 2011 / Vol. 19, No. 22 / OPTICS EXPRESS 21747
