STUDIA UNIVERSITATIS BABEŞ-BOLYAI, PHYSICA, SPECIAL ISSUE, 2001
HIGH-ORDER HARMONICS IN THE FEW-OPTICAL-CYCLE
REGIME
C. ALTUCCI1,3, R. BRUZZESE3, C. DE LISIO3, V. TOSA2, M. NISOLI4,
S. STAGIRA4, G. CERULLO4, S. DE SILVESTRI4, O. SVELTO4,
P. BARBIERO4, L. POLETTO5, G. TONDELLO5, P. VILLORESI5
1Dipartimento
di Chimica, Università della Basilicata (Italy), INFM Unità di Napoli (Italy)
Institute for Isotopic and Molecular Technology, Cluj-Napoca (Romania)
3Dipartimento di Scienze Fisiche, Università di Napoli (Italy), INFM Unità di Napoli (Italy)
4Centro di Elettronica Quantistica – CNR, Dipartimento di Fisica, Politecnico di Milano
(Italy), INFM Unità di Milano (Italy)
5Università di Padova (Italy), INFM Unità di Padova (Italy)
2National
ABSTRACT. The beam divergence of high-order harmonics generated in Helium by an
ultra-short Ti:sapphire laser (7 fs and 30 fs) is experimentally characterized by means of a
flat-field, high-resolution spectrometer. The harmonic beam divergence is also analysed as a
function of the gas-jet position relative to the laser beam waist. Results, which are partly
different from previous measurements performed at longer laser pulse duration, are discussed.
1. INTRODUCTION
High-order harmonics as a coherent XUV radiation source exhibit unique
properties, such as high brightness, due to good temporal and spatial coherence [1,2],
and extremely short pulse duration [3]. Thanks to these unique features, high-order
harmonic sources lend themselves as interesting candidates for applications in physics,
chemistry, and biochemistry.
The availability of extremely short pulse duration solid state lasers [4,5] has
allowed efficient generation of harmonic pulses with duration in the range 30-5 fs, and the
emission of coherent radiation down to the water window spectral region (4.4-2.3 nm)
[6,7].
Moreover, the advent of the above mentioned laser sources makes two exciting
perspectives much closer: i) the use of sub-10 fs laser pulses opens the way to single
attosecond x-ray pulses; ii) the availability of intense soft-x-ray pulses in a neardiffraction-limited beam, which should be focusable to peak intensities of the order of
1013 Wcm-2, promises to speed up the field of x-ray nonlinear optics [8].
In the present paper we concentrate on the characterisation of the spatial mode
of harmonic radiation generated, in the spectral region 30-8 nm, both in Neon and
Helium by Ti:sapphire laser pulses with duration of 30 fs or  7 fs. In particular, we
show the measured harmonic beam divergence as a function of the harmonic order and
of the distance, z, of the gas jet relative to the laser beam waist. One of the most
relevant features of our observations is that the harmonic beam divergence generally
increases with harmonic order. This behaviour is somehow different from that of
previous measurements performed at longer laser pulse duration [9].
C. ALTUCCI ET AL.
It is worth stressing that, in spite of the relatively small number of photons per
harmonic pulse (typically 105 at 31.8 nm which corresponds to the 25th harmonic of
the fundamental), the spectral brightness of harmonic radiation can be very high, due to
the extremely short pulse duration and to the good spatial coherence. Correspondingly,
an estimate of the spectral brightness for the 25th harmonic leads to 1022-1023 photons/
(Å·s·srad).
2. DESCRIPTION OF THE EXPERIMENT AND RESULTS
The Ti:sapphire laser system used for the experiment delivers 30 fs, up to 1 mJ
laser pulses (centred at 795 nm) at a 1 kHz repetition rate. Sub-10-fs pulses are generated
by the hollow fiber compression technique [4]. In our set-up, the laser pulse duration
can be set to 30 fs or to 5-7 fs, depending on whether the hollow fiber is filled in with
Argon or not Amplitude and phase of the compressed pulses have been monitored by
Spectral Phase Interferometry for Direct Electric field Reconstruction (SPIDER) [10].
Typical pulses have duration of 5-7 fs and energy of 0.3 mJ, while the spectral phase is
generally constant over the pulse spectrum. Laser pulses are focussed onto the gas jet
by a 25 cm focal length silver mirror, which leads to a confocal parameter of about 6 mm.
The laser-gas interaction length is 1 mm, with a gas pressure of 50 Torr. The position z
of the gas jet along the propagation axis can be easily varied by translating the
electromagnetic valve. The harmonic radiation is spectrally dispersed by a grazing
incidence, flat field spectrometer (Fig.1) consisting of a toroidal mirror (incidence
angle 86, and radii of 6500 mm in the tangential plane and 14.7 mm in the sagittal
plane) followed by a Pt-coated spherical grating (radius of 5649 mm, average groove
density of 1200 mm-1). Harmonics are finally detected by means of an intensified
microchannel plate (MCP), followed by a charge coupled device (CCD) camera. The
sensitive element of the MCP is located 235 mm far from the grating.
A typical spectrum, obtained in He for 7 fs laser pulses and at laser peak
intensity at focus of 1015 Wcm-2, is shown in Fig.2. The harmonic spectrum develops
along the horizontal direction, and harmonics from the 23rd (rightmost) up to the 67th
are clearly observable. The vertical direction contains information on the spatial profile
of the harmonic beams. The spectrum of Fig.2 is measured when the gas-jet is placed
about 1 mm before the laser focus in the converging beam: in these conditions a good
compromise between conversion efficiency and harmonic beam divergence is reached,
in order to maximize the harmonic spectral brightness. As Fig.2 clearly depicts, the
harmonic beam divergence increases almost linearly with the harmonic order. The
corresponding quantitative analysis is reported in the insert of Fig.2, where the
harmonic beam divergence angle (FWHM) is plotted against the harmonic order. The
harmonic beam divergence increases, in this case, almost three times, from 1.2 mrad
for the lowest order harmonic observed, namely H23, up to more than 3 mrad for
harmonics at the end of the plateau, as H65. A similar behaviour is observed also for
30 fs laser pulses, though, in this second case, the harmonic beam divergence is
generally larger, increasing by a factor of two, from 2.3 mrad for the lowest order
harmonics up to 4.6 mrad for harmonics at the end of the plateau.
156
H43
HIGH-ORDER HARMONICS IN THE FEW-OPTICAL-CYCLE REGIME
Figure 1. Layout of the experimental apparatus: detail of the interaction chamber, the
XUV flat-field spectrometer, and the detector.

Figure 2. Spectrum obteained in 30 Torr of He for 7 fs laser pulses and for a laser peak
intensity of 1015 Wcm—2 at focus. The gas jet is placed, in this case, about 1 mm before
the laser focus, in the converging beam. The insert shows the harmonic divergence,
evaluated as FWHM of the spatial profile, vs the harmonic order.
In figure 3 we report the spatial profiles of harmonics belonging to different
regions of the spectrum, namely H25 and H43, respectively in the central and final part
of the plateau, and H61 in the cutoff region. Apart from the increasing divergence
157
C. ALTUCCI ET AL.
, Nph, B (normalized)
, Nph, B (normalized)
Intensity (arb. un.)
angle, it is worth stressing that they present a bell-shape without any structure. This is a
general behaviour, observed for all the investigated ranges of the experimental
parameters.
The harmonic beam divergence
has also been investigated as a
1.0
function of, z. In general, the
H61
largest harmonic beam divergence
0.8
H43
has been observed about 1 mm
after the laser focus, in the divergH25
0.6
ing beam, while the highest conversion efficiency has been ob0.4
tained just around the laser
focus. Such a surprising feature,
0.2
widely reproducible and observed
in a large number of experimental
0.0
conditions (both for 30 fs and 7
-4
-3
-2
-1
0
1
2
3
4
fs pulses and for various gases),
 (mrad)
is in contrast with previous measFigure 3. Spatial intensity profile of h25, H43, H61 in the
urements performed at longer
same conditions as Fig.2
laser pulse duration [9] and is
1.0
still under study.
b)
H43
Figure 4 illustrates the above men0.8
tioned behaviour. Beam divergence, photon number, and spec0.6
tral brightness are plotted, each
0.4
one normalized to its maximum
value, as a function of z, for H25
0.2
(31.8 nm), case a), and H43
(18.5 nm), case b), in He. In the
0.0
-3
-2
-1
0
1
2
figure, the value z = 0 corre1.0
a)
H25
sponds to the gas jet at focus,
0.8
and the laser propagates from
z<0 to z>0. Both harmonics
0.6
exhibit the same characteristics:
the highest conversion efficiency,
0.4
i.e. the largest photon number
0.2
Nph, is obtained approximately
at the laser focus, whereas the
0.0
beam divergence, , is maximal
-3
-2
-1
0
1
2
z (mm)
about 1 mm after the laser focus
(z=1 mm). Thus, the spectral
Figure 4. Beam divergence,  (squares), number of photons, Nph
(circles), and spectral brightness, B (up triangles), all normalized,
brightness, BNph/2, is maximal
vs the gas jet position, z, along the propagation axis, for H25 (a)
at z=-1 mm, namely about 1 mm
and H43 (b) in 30 Torr of He for a laser pulse duration of 7 fs.
before the laser focus, in the
158
HIGH-ORDER HARMONICS IN THE FEW-OPTICAL-CYCLE REGIME
converging beam. No satisfactory explanation has been found yet for such a behaviour:
additional experimental investigations and theoretical analysis, based on a 3-D
propagation model [11] which also accounts for the nonadiabatic single atom response,
are planned in order to understand this feature.
We remark that in our experimental conditions the maximum brightness is
generally obtained at z  -1 mm, i.e. in the converging laser beam. It is worth noticing
that the peak value of the brightness, B, of Fig.4 is 1022-1023 photons/Å·s·srad. It has
been estimated by the formula:
B
meas
N ph
 TOT
     h
,
where Nphmeas is the measured number of harmonic photons, TOT is the overall
conversion efficiency of the whole apparatus,  is the solid angle subtended by the
harmonic beam,  the harmonic bandwidth, and h the harmonic pulse duration. All
the above quantities but h, have been measured, whereas h has been assumed to be 3
fs. However, this estimate is reasonable only if nondispersive elements are employed
for harmonic selection or if harmonic pulse recompression is implemented. In our case,
the use of a diffraction grating introduces pulse lengthening.
3. A PHYSICAL MODEL FOR THE GENERATION AND
PROPAGATION OF THE HIGH ORDER HARMONICS
To understand better the above experimental results we developed a computer
code based on a physical model [11] for the generation and the propagation of high
order harmonics. It includes a physical model for the single atom response in intense
electric field, and the propagation equations for both the laser field and for the
generated radiation.
a) Single atom response:
The nonlinear dipole moment dnl(t) in the Strong Field Approximation formalism
is:
3/ 2
 t





d (t )  2 Re i  dt ' 
d * p (t ' , t )  A(t ) d p (t ' , t )  A(t ' ) 
nl
st
st
      i (t  t ' ) / 2 
 t


 exp[ iS (t ' , t )]E (t ' )  exp   w(t ' )dt '
st
1
  





where the stationary value of momentum and action are
1 t
p (t ' , t ) 
 A(t" )dt"
st
t  t' t'
159
C. ALTUCCI ET AL.
t
1
S (t ' , t )  (t  t ' ) I  p 2 (t ' , t )(t  t ' )  12  A2 (t" )dt"
p 2 st
t'
and the dipole matrix element for transitions from the ground state to a continuum state
characterized by a momentum p is:
7
d ( p)  i
2 2 (2 I p )

5
4
p
( p  2 I p )3
2
The tunnel ionization rate from the ground state is written in Amosov Delone
Krainov (ADK) model as:
2n * 1
4I
2 4I p 

p
p
w(t ) 
C
exp 


 n*
 wt 
 3wt
I



b) Propagation of the fundamental field E1:
The 3-D wave equation for the propagation of the fundamental field, takes into
account the presence of a plasma generated by the laser in the gas. If ne is the electron
density one can write an equation of the form:
2
2
1  E1(r , z, t ) e ne (r , z, t )
2
 E (r , z, t ) 

E (r , z, t )
1
1
c2
t 2
c2
which eventually can be written in the moving frame (z’=z, t’=t-z/c) as:
2
2
2  E1(r , z ' , t ' ) e ne (r , z ' , t ' )
2
 E (r , z ' , t ' ) 

E (r , z ' , t ' )
 1
1
c
t ' z '
c2
where the paraxial approximation (
 2 E1
z 2
 0 ) has been used. The temporal derivative
can be eliminated by performing a Fourier transform (FT), yielding the parabolic type
equation:
 e2ne (r, z ' , t ' )

2i E1(r , z ' ,  )
2
 E (r , z ' ,  ) 
 FT
E (r , z ' , t ' )
 1
1
 c2

c
z '


Due to the complex nature of the quantities involved, this is in fact a system of
two equations which has been solved numerically using the Cranck-Nicolson method.
c) Propagation of the harmonic field Eh:
Harmonic field (Eh) propagation obeys the 3-D wave equation similar to that
written for the harmonic field. A spatial polarisation Pnl(r,z,t) is present, due to the laser
induced dipoles of the atoms which did not ionise, and plays the role of the source:
160
HIGH-ORDER HARMONICS IN THE FEW-OPTICAL-CYCLE REGIME
2
 2 P (r , z, t )
1  E h (r , z , t )
2
nl
 E (r , z, t ) 

h
0
c2
t 2
t 2
As above, it can be written in the moving frame (z’=z, t’=t-z/c) as:
2
 2 P (r , z ' , t ' )
2  Eh (r , z ' , t ' )
2
nl
 E (r , z ' , t ' ) 

 h
0
c
t ' z '
t 2
By performing a Fourier transform (FT), the above equation is transformed in
system of two parabolic type equations:
2i Eh (r , z ' ,  )
 2 E (r , z ' ,  ) 
  2 P (r , z ' ,  )
 h
0 0 nl
c
z '
which can be soved numerically. Note that equations were derived without performing
the slowly varying envelope approximation in time.
After solving numerically the equations for E1 and Eh, the program compute
the power spectrum of the harmonics.
3. CONCLUSIONS
The beam divergence of high-order harmonics generated in He by a 7fs
Ti:sapphire laser has been extensively investigated. The main and somehow
unexpected features of our results are: the harmonic beam divergence generally
increases with the harmonic order, and, its trend as a function of z, presents a
maximum when the gas jet is positioned in the diverging beam, whereas in previous
measurements at rather longer laser pulse duration (100 fs) [9] the maximum of the
harmonic divergence angle occurred in the converging laser beam. This result, very
well reproducible and observed also for 30 fs laser pulses, seems to indicate that
different phase-matching conditions for the harmonic propagation process occur in the
case of very short laser pulse duration. The dipole phase, which could be very much
affected by nonadiabatic effects due to the extremely short laser pulse duration, might
play a different role in the phase-matching conditions when compared to harmonic
generation by longer laser pulses.
REFERENCES
[1] Bellini M., Lyngå C., Gaarde M.B., Hänsch T.W., L'Huillier A., and Wahlström C.-G.,
Phys. Rev. Lett. 81 (1998) 297.
[2] Zerne R., Altucci C., Bellini M., Gaarde M.B., Hänsch T.W., L'Huillier A., Lyngå C., and
Wahlström C.-G., Phys. Rev. Lett. 79 (1997) 1006.
[3] Sartania S., Cheng Z., Lenzner M., Tempea G., Spielmann C., Krausz F., and Ferencz K.,
Opt. Lett. 22 (1997) 1562.
161
C. ALTUCCI ET AL.
[4] Nisoli M., De Silvestri S., Svelto O., Szipöcs R., Ferencz K., Spielmann C., Sartania S.,
and Krausz F., Opt. Lett. 22 (1997) 522.
[5] Spielmann C., Burnett N.H., Sartania S., Koppitsch R., Schnürer M., Kan C., Lenzner M.,
Wobrauschek P., Krausz F., Science 78 (1997) 661.
[6] Chang Z., Rundquist A., Wang H., Murnane M.M.. Kapteyn H.C., Phys. Rev. Lett. 79
(1997) 2967.
[7] Villoresi P., Ceccherini P., Poletto L., Tondello G., Altucci C., Bruzzese R., de Lisio C.,
Nisoli M., Stagira S., Cerullo G., De Silvestri S., Svelto O., Phys. Rev. Lett. 85 (2000)
2494.
[8] Salières P., Le Dèroff L., Auguste T., Monot P., d'Oliveira P., Campo D., Hergott J.-F.,
Merdji H., Carrè B., Phys. Rev. Lett 83 (1999) 5483.
[9] Salières P., Ditmire T., Perry M.D., L’Huillier A., and Lewenstein M., J. Phys. B 29
(1996) 4771.
[10] Iaconis C. and Walmsley I.A., IEEE J. Quantum Electron. 35 (1999) 501.
[11] Priori E., Cerulllo G., Nisoli M. Stagira S., De Silvestri S., Villoresi P., Poletto L.,
Ceccherini P., Altucci C., Bruzzese R., and de Lisio C., Phys. Rev. A 61 (2000) 063801.
162