Material / 講義資料 - 石川顕一

Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Kenichi Ishikawa (石川顕一)
http://ishiken.free.fr/english/lecture.html
[email protected]
Advanced Plasma and Laser Science
プラズマ・レーザー特論E
Attosecond Science (2)
アト秒科学(2)
2014/5/27
1
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
★
FROG-CRAB
★
分子軌道トモグラフィー
2014/5/27
2
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
How to measure (analyze) attosecond pulses
FROG-CRAB
(Frequency-Resolved Optical Gating - Complete
Reconstruction of Attosecond Bursts)
2014/5/27
3
80
0.5
80
Photoelectron energ
Photoelectron energ
tion. Indeed,
filtering radiatio
Advanced
ISHIKAWA) for internal use only (Univ.
of Tokyo)
70 Plasma and Laser Science (Kenichi
70
60
0.4
60
0.3
depicted by the dashed-and-d
to isolate XUV radiation with
energy delivered in a single a
range of CE phases as broad a
1B). In contrast, with few-cy
generation resulting in isola
pulses over only a relatively
CE phase near ϕ ≈ 0° (3), si
appears to permit robust isola
for a variety of driver wave
near-cosine– to sine-shaped
order-of-magnitude variatio
probability within a single wa
We used phase-controlled
pulses carried at a waveleng
720 nm (19) to generate X
neon gas jet up to photon e
(fig. S1). The emerging XU
a spectral filtering through
and-dotted line in Fig. 1A)
foils and a Mo/Si multilaye
subsequently propagates, alo
er wave, through a second j
which the XUV pulse ioniz
presence of the NIR field.
with initial momenta directe
field vector of the linearly po
collected and analyzed with
trometry (17).
The variation of the mea
spectra versus CE phase sho
with the predictions of our
A and B). Figure 2, C to
electron spectra correspondi
How to measure (analyze) attosecond
pulses アト秒パルスはどうやって測る？
50
40
30
0
Delay (fs)
2
4
0.1
−4
C
4
0.8
τx= 80 ± 5 as
0.6
2
0.4
1
0.2
-300
-200
-100
0
100
Time (as)
200
300
1.0
−2
0
2
Delay (fs)
4
D
0.8
0
Goulielmakis
et al. (2008)
0.4
phase (rad)
3
XUV spectral intensity (arb.u.)
−2
phase (rad)
XUV intensity (arb.u.)
0.2
40
30
−4
1.0
50
0.6
0.2
-3
φ″=(1.5 ± 0.2)×10 as
3
40
50
60
2
70 80 90 100 110
Photon energy (eV)
Fig. 3. Sub-100-as XUV pulse retrieval. (A) Measured ATR spectrogram compiled from 126 energy
spectra of photoelectrons launched by an XUV pulse with a bandwidth of ~28 eV (FWHM) and recorded
at delay settings increased in steps of 80 as. Here, a positive delay corresponds to the XUV pulse
arriving before the NIR pulse. The high flux of the XUV source allows this spectrogram to be recorded
within ~30 min. (B) ATR spectrogram reconstructed after ~103 iterations of the FROG algorithm (17).
(C) Retrieved temporal intensity profile and spectral phase of the XUV pulse. The intrinsic chirp of the
XUV emission (Fig. 4B) is almost fully compensated by a 300-nm-thick Zr foil introduced into the XUV
beam between the attosecond source and the ATR measurement. Arrows indicate the temporal FWHM of
the XUV pulse. (D) XUV spectra evaluated from the measurement of the XUV-generated photoelectron
spectrum in the absence of the NIR streaking field (blue dashed line) and from the ATR retrieval (blue
solid line). The black dotted line indicates the retrieved spectral phase.
www.sciencemag.org
SCIENCE
VOL 320
20 JUNE 2008
2014/5/27
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Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Most people think of acoustic waves
in terms
of a musical score.
Typical
representation
of a sound wave
Intensity
ff
pp
frequency
pp
time
Plot of frequency vs. time
information on top about the intensity
2014/5/27
5
80
80
0.5
Photoelectron energ
Photoelectron energ
tion. Indeed,
filtering radiatio
Advanced
ISHIKAWA) for internal use only (Univ.
of Tokyo)
70 Plasma and Laser Science (Kenichi
70
depicted by the dashed-and-d
to isolate XUV radiation with
60
60
0.3
energy delivered in a single a
50
50
range of CE phases as broad a
0.2
40
1B). In contrast, with few-cy
40
generation resulting in isola
0.1
30
30
pulses over only a relatively
CE phase near ϕ ≈ 0° (3), si
−4
−2
0
2
4
−4
−2
0
2
4
appears to permit robust isola
Delay (fs)
Delay (fs)
for a variety of driver wave
near-cosine– to sine-shaped
1.0 D
1.0 C
order-of-magnitude variatio
4
probability within a single wa
0.8
0.8
0
We used phase-controlled
τx= 80 ± 5 as
3
pulses
carried at a waveleng
0.6
0.6
Goulielmakis
720 nm (19) to generate X
2
et
al.
(2008)
neon gas jet up to photon e
0.4
0.4
(fig. S1). The emerging XU
a spectral filtering through
1
0.2
0.2
-3
and-dotted line in Fig. 1A)
φ″=(1.5 ± 0.2)×103 as2
foils and a Mo/Si multilaye
A mathematically rigorous form
of apropagates, alo
subsequently
-300 -200 -100
0
100 200 300
40 50 60 70 80 90 100 110
er wave, through a second j
Time (as)
energy
musical Photon
score
is (eV)
the “spectrogram.”
which the XUV pulse ioniz
Fig. 3. Sub-100-as XUV pulse retrieval. (A) Measured ATR spectrogram compiled from 126 energy
presence of the NIR field.
spectra
launched
an XUV pulse
with a bandwidth of ~28 eV (FWHM) and recorded
If ofa photoelectrons
spectrogram
ofbyFROG
trace
at delay settings increased in steps of 80 as. Here, a positive delay corresponds to the XUV pulse with initial momenta directe
2 to be recorded field vector of the linearly po
arriving before the NIR pulse. The high flux of the XUV source allows this spectrogram
3
within ~30 min. (B) ATR spectrogram reconstructed after ~10 iterations ofi the
t FROG algorithm (17). collected and analyzed with
, profile
) = and spectralG(t)E(t
)e Thedtintrinsic chirp of the trometry (17).
(C) Retrieved temporalS(
intensity
phase of the XUV pulse.
The variation of the mea
XUV emission (Fig. 4B) is almost fully compensated by a 300-nm-thick Zr foil introduced into the XUV
spectra
versus CE phase sho
beam between the attosecond source and the ATR measurement. Arrows indicate the temporal FWHM of
with the
predictions of our
is measured
for different
of delay
, the fieldphotoelectron
E(t) and gate
G(t)
the XUV
pulse. (D) XUV spectra
evaluated fromvalues
the measurement
of the XUV-generated
spectrum in the absence of the NIR streaking field (blue dashed line) and from the ATR retrieval (blue A and B). Figure 2, C to
can be reconstructed (principal component generalized projections
electron spectra correspondi
solid line). The black dotted line indicates the retrieved spectral phase.
0.4
algorithm).
XUV spectral intensity (arb.u.)
phase (rad)
Principle
phase (rad)
XUV intensity (arb.u.)
How to measure (analyze) attosecond
pulses アト秒パルスはどうやって測る？
widely used to
analyze
laser
www.sciencemag.org
SCIENCE
VOL 320
20 pulses
JUNE 2008
2014/5/27
6
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
高調波とレーザー光を遅延
時間を持たせて照射し、光
電子スペクトルを測定。
Irradiate an atom with an
attosecond pulse and
laser pulse with delay,
and
measure
a
photoelectron spectrum
Lewenstein model
photoelectron momentum spectrum
a(p, )
action 作用積分
a(p, ) =
i
exp[ iS(t)]EX (t
)dp+A(t) dt =
attosecond pulse electric field
アト秒パルスの電場波形
i
ei
(t)
dp+A(t) EX (t
(t) =
t
)ei(p
2
/2+Ip )t
dt
p · A(t ) + A2 (t )/2 dt
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Elec
Photo
C
70
50
60
40
Spectrogram or CRAB trace
(t) =
t
0
30
20 40 60 80 100 120 140 160
1 Carrier-envelope phase (deg)
D
ϕ = 130°
Fig. 2. Control of bandpass-filtered XUV emission with the
(A) and simulated (B) (17) photoelectron spectra versus CE ph
~11° ( p/ 16 rad). (C to E) Spectra measured at the CE phase
2
Fig. 1A. The2zero
of the CE phase scale in (A) was set to yiel
0
spectrai(p
in (B)./2+Ip )t
B
70
e60i (t) dp+A(t) EX (t
)e
50
40
E
1
0
p · A(t ) + A2 (t )/2 dt
0
90
dt
1
E
ϕ = 170°
A
1
80
0
30
20 40 60 80 100 120 140 16070
Carrier-envelope phase (deg)
40
50
60
70
Photoelectron energy (eV)
60
Photoelectron energy ( eV)
|a(p, )|2 =
40
Photoelectron energy ( eV)
photoelectron
momentum spectrum
50
0
1
ϕ = 70°
0
Electron counts (arb. u.)
Photoelectron energy (eV)
A
B
90
80
80
70
60
2 emission with the waveform of monocycle light. Measured
Fig. 2. Control of bandpass-filtered XUV
50
50
RAPID COMMUNICATIONS
i t
(A) and simulated (B) (17) photoelectron
spectra versus CE phase, with the delay increased in steps of
40 CE phase setting closest to the values selected
40
in
~11° ( p/ 16 rad). (C to E) Spectra measured at the
PHYSICAL REVIEW A 71, 011401#R$ #2005$
Fig. 1A. The zero of the CE phase scale in (A) was
30 set to yield the best agreement with the modeled
30
spectra in (B).
)e
G(t)E(t
dt
A
80
70
60
50
40
0.8
80
0.6
0.4
2
4
−4
4
τx= 80 ± 5 as
70
3
60
2
50
1
40
C
−2
0
Delay (fs)
2
4
-300
-200
−4
-100
0
100
200
−2
2
Time (as)0
Delay (fs)
300
4
1.00.6D
0.80.5
0.4
0.6
0.3
0.4
0.2
0.2
0.1
40
Fig. 3. Sub-100-as XUV pulse retrieval. (A) Measured ATR
spectraet
of photoelectrons
launched by an XUV pulse with a ban
Goulielmakis
al.
(2008)
1.0 D
(arb.u.)
.u.)
1.0
0.8
0
Delay (fs)
30
−4
FIG. 1. #a$ CRAB trace of a single 315 as pulse !full width at
half maximum #FWHM$ of intensity", having second- and third-
B
C
90
0.2
30
−2
1.0
Photoelectron energy ( eV)
90
XUV intensity (arb.u.)
Photoelectron energy ( eV)
−4
XUV spectral intensity (arb.u.)
S( , ) =
phase (rad)
n its
: the
rally
roded in
ned.
hase
eady
ngle
pericket.
elecarach the
ments
antly
1
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal
only (Univ. of Tokyo)
60 use
at delay settings increased in steps of 80 as. Here, a positi
2014/5/27
4
arriving
before the NIR pulse. The high flux of the8XUV sourc
0.8
Approach to questions at the heart of
chemistry
Matter is made up of molecules.
Hydrogen
atom (H)
Carbon atom
(C)
Oxygen atom
(O)
A molecule is made up of atoms.
But how do atoms
link to each other
at all?
What is chemical
bond?
Hydrogen
molecules (H2)
Water (H2O)
Methane
(CH4)
Oxygen
molecules (O2)
Carbon
dioxide (CO2)
Molecular orbital theory
Bonds in general are a mixing or sharing of the electrons from different atoms.
Nitrogen atom (N)
Nitrogen molecule (N2)
Well, in atoms and molecules, electrons are not point particles but spread out
like a cloud or wave...
How the electron wave is shared by atoms is
described by molecular orbital (wave
function).
Theory first developed in late 1920s, Nobel
prize in 1966
Now, basis for the understanding of
molecular structures and chemical reaction.
But, how do the molecular orbital really
looks?
Now, thanks to attosecond technology, we
can see molecular orbitals!
Molecular orbital tomography
Not just the electron density
Experiment
| (r)|2
But the wave function itself
(r)
Theory
Nitrogen molecule (N2)
Jiro Itatani et al., Nature (2004)
is measured.
The wave function can be
measured!
Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
高次高調波発生の３ステップモデル
3-step model of high-harmonic generation
Laser field
E(t) = E0 cos ωt
レーザー電場 recombination
再結合→
発光 photon emission (HHG)
electron 電子
トンネル
電離
tunneling ionization
電場中の古典
的運動
Semiclassical
electron motion
Quantum mechanical theory
2014/5/27
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Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Lewenstein model
M. Lewenstein et al., Phys. Rev. A 49(3), 2117
Time-dependent Schrödinger equation
i
(r, t)
=
t
1
2
2
+ V (r) + zE(t)
(r, t),
Strong-field approximation (SPA)
• The contribution of all the excited bound states can be
neglected.
• The effect of the atomic potential on the motion of the
continuum electron can be neglected.
• The depletion of the ground state can be neglected.
2014/5/27
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Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Lewenstein model
Within this approximation ...
Time-dependent dipole moment
(r, t) | z | (r, t)
x(t)
transition dipole matrix element
x(t) = i
t
dt
d3 p d (p + A(t)) · exp[ iS(p, t, t )] · E(t )d(p + A(t )) + c.c.
recombination
Semiclassical action
作用積分
exp[ iS(p, t, t )]
propagation
t
S(p, t, t ) =
dt
t
ionization
[p + A(t )]2
+ Ip
2
phase of a path (in the spirit of path integral)
経路の位相（ファインマンの経路積分に関連）
Clear physical picture corresponding to the three-step model
2014/5/27
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Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Lewenstein model
Time-dependent dipole moment
x(t) = i
t
x(t)
(r, t) | z | (r, t)
d3 p d (p + A(t)) · exp[ iS(p, t, t )] · E(t )d(p + A(t )) + c.c.
dt
recombination
propagation
ionization
Fourier transform フーリエ変換
x
ˆ(
h)
=i
t
dt
dt
d3 p d (p + A(t)) · exp[i
iS(p, t, t )] · E(t )d(p + A(t )) + c.c..
ht
4
10
Harmonic spectrum
|ˆ
x(
2
h )|
2
10
0
10
-2
10
-4
10
-6
10
-8
10
-10
10
-12
10
-14
10
0
20
40
60
80
100
Harmonic order
120
140
2014/5/27
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Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Lewenstein model
From another viewpoint 別の視点から
t
x(t) = i
d3 p d (p + A(t)) · exp[ iS(p, t, t )] · E(t )d(p + A(t )) + c.c.
dt
=
= a(k, t) amplitude of the
recolliding wave
ground state 基底状態
packet 再結合電子波
g (r)
k = p + A(t)
x
ˆ(
h)
a(k)
g (r) | r | e
g (r)
|ˆ
x(
2
h )|
|r|e
|
|r|e
ik·r
束の振幅
a(k, t)e
ik·r
ik·r
=i
2
ˆ
k g (k)|
ik·r
ˆ
k g (k)
electron 電子
The harmonic spectrum contains the information of the spatial Fourier
transform of the ground-state wave function 高次高調波スペクトルは、原
子や分子の波動関数の空間フーリエ変換の情報を含んでいる。
2014/5/27
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Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Molecular orbital tomography
|ˆ
x(
k
2
h )|
|
2
ˆ
k g (k)|
Laser polarization レーザーの偏光方向
Molecular axis 分子軸
1. Align molecules
2. Measure harmonic spectrum
3. Repeat for different alignment angles
4
10
2
10
0
10
-2
10
-4
10
-6
10
-8
10
-10
10
-12
10
-14
10
0
20
40
60
80
100
Harmonic order
120
140
Reconstruction of
g (r)
2014/5/27
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Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Computed tomography (CT)
X-ray absorption
measurement from
different angles. 様々な
方向からエックス線を照
射し、吸収測定
Reconstruction 再構成
3D image 三次元画像が得られる。
2014/5/27
18
dimensional Fourier transform F of the object. This is the basis of
tomography (Kenichi
based on the inverse
Radon transform.
Our
Advanced Plasma and computed
Laser Science
ISHIKAWA)
for internal
use only (Univ. of Tokyo)
Calibrating the continuum wave packet
As discussed above, the harmonic spectrum is an experimental
evaluation of the dipole, d(q). If we could evaluate the plane-wave
amplitude a[k(q)]
independently, then our measurement would
R
determine w g(r)(er)exp[ik(q)x]dr—that is, the spatial Fourier
components of rw g(r). One way to do this is to perform the same
experiment with a reference atom.
Argon is very similar to N2 in its response to strong laser fields,
having nearly the same ionization potential and intensity-dependent ionization probability6. This is confirmed by the dependence of
the instantaneous ionization rates29 for atoms, and for different
orientations of N2 (ref. 30). That means that the first, critical, step in
the three-step high harmonic generation process is the same.
Because the laser field dominates wave packet motion in the
direction of the laser field, the second step, which determines the
chirp of the re-colliding wave packets seen by Ar or N2, will be
the same. Thus, a[k(q)] will be the same.
The continuum wave packet will also be similar for Ar and N2.
The narrow saddle point through which the electron tunnels acts as
a spatial filter that removes much of the structure of the orbital from
the continuum wave packet. This can be seen in numerical simulations31. By measuring the ellipticity dependence of the high
harmonic signal24 produced by N2 and argon, we confirmed that
the lateral spread of the wave packets is similar. The ionization rate
of N2 is angle-dependent30,32, but is readily measured from the ion
yield, and varies only by 25% for N2 (ref. 33). This variation is
almost cancelled by the angular dependence of the wave-packet
dipole is the Fourier transform of a projection of the wavefunction,
and so can be inverted.
We describe the mathematical details of the tomographic reconstruction in the Methods section. This procedure can reconstruct
orbital shapes with symmetries such as jg, pg and pu, using
harmonics 17–51 of an 800-nm laser field and 25 angles from 08
to 1808 (fewer angles are needed for symmetric molecules).
A complete inversion of a general orbital requires knowledge of
the relative phase and amplitude of each harmonic for two ortho-
Molecular orbital tomography
|ˆ
x(
2
h )|
|
2
ˆ
k g (k)|
k
Experiment
Theory (Hartree-Fock)
Reconstruction
再構成
Nitrogen molecule (N2)
Jiro Itatani et al., Nature (2004)
Figure 3 High harmonic spectra were recorded for N2 molecules aligned at 19 different
angles between 0 and 908 relative to the polarization axis of the laser. For clarity, only
some of the angles have been plotted above. The high harmonic spectrum from argon is
also shown; argon is used as the reference atom. Clearly the spectra depend on both the
Figure 4 Molecular orbital wavefunction of N2. a, Reconstructed wavefunction of the
HOMO of N2. The reconstruction is from a tomographic inversion of the high harmonic
spectra taken at 19 projection angles. Both positive and negative values are present, so
this is a wavefunction, not the square of the wavefunction, up to an arbitrary phase. b, The
shape of the N2 2p jg orbital from an ab initio calculation. The colour scales are the same
for both images. c, Cuts along the internuclear axis for the reconstructed (dashed) and
2014/5/27
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Advanced Plasma and Laser Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)
Wave function can be measured!
波動関数は測定できる！
Experiment
Wave function is measured!
= | | exp(i )
Theory (Hartree-Fock)
Nitrogen molecule (N2)
Jiro Itatani et al., Nature (2004)
X-ray diffraction, electron
microscope, STM etc.
measures just electron
density.
= | |2
2014/5/27
20

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