Advanced Laser and Photon 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 Laser and Photon Science
 レーザー・光量子科学特論E





Attosecond Science (2)
 アト秒科学(2) 1

Advanced Laser and Photon Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

★FROG-CRAB ★分子軌道トモグラフィー
 (molecular orbital tomography)

2

Advanced Laser and Photon 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)

3

80

0.5

80

Photoelectron energ

Photoelectron energ

tion. Indeed, filtering radiati Advanced ISHIKAWA) for internal use only (Univ. of Tokyo) 70 Laser and Photon 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 range of CE phases as broad 1B). In contrast, with few-c generation resulting in isol pulses over only a relatively CE phase near ϕ ≈ 0° (3), s appears to permit robust isola for a variety of driver wave near-cosine– to sine-shaped order-of-magnitude variatio probability within a single w We used phase-controlle pulses carried at a waveleng 720 nm (19) to generate X neon gas jet up to photon (fig. S1). The emerging XU a spectral filtering through and-dotted line in Fig. 1A) foils and a Mo/Si multilay subsequently propagates, alo er wave, through a second which the XUV pulse ioni presence of the NIR field. with initial momenta direct field vector of the linearly po collected and analyzed with trometry (17). The variation of the me spectra versus CE phase sh with the predictions of our A and B). Figure 2, C to electron spectra correspond

How to measure (analyze) attosecond 50 40 30

50

0.2

40

0.1

pulses アト秒パルスはどうやって測る？ 30

0 Delay (fs)

2

4

−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

phase (rad)

XUV intensity (arb.u.)

1.0

−2

XUV spectral intensity (arb.u.)

−4

0.6

0.2

-3 φ″=(1.5 ± 0.2)×103 as2 40

50

60

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

4

Advanced Laser and Photon 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

5

80

80

0.5

Photoelectron energ

Photoelectron energ

tion. Indeed, filtering radiati Advanced ISHIKAWA) for internal use only (Univ. of Tokyo) 70 Laser and Photon Science (Kenichi 70

depicted by the dashed-and-d to isolate XUV radiation with 60 60 0.3 energy delivered in a single 50 50 range of CE phases as broad 0.2 40 1B). In contrast, with few-c 40 generation resulting in isol 0.1 30 30 pulses over only a relatively CE phase near ϕ ≈ 0° (3), s −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 w 0.8 0.8 0 We used phase-controlle 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 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 multilay A mathematically rigorous form of a propagates, alo subsequently -300 -200 -100 0 100 200 300 40 50 60 70 80 90 100 110 er wave, through a second Time (as) energy musical Photon score is (eV) the “spectrogram.” which the XUV pulse ioni 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 direct 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 i the t FROG algorithm (17). collected and analyzed with within ~30 min. (B) ATR spectrogram reconstructed after ~10 iterations of S( , ) = G(t)E(t )e dt (C) Retrieved temporal intensity profile and spectral phase of the XUV pulse. The intrinsic chirp of the trometry (17). The variation of the me 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 spectra versus CE phase sh is measured for different values of delay , the field photoelectron E(t) and gate with G(t) the predictions of our the XUV pulse. (D) XUV spectra evaluated from 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 correspond solid line). The black dotted line indicates the retrieved spectral phase. 0.4

How to measure (analyze) attosecond

algorithm).

XUV spectral intensity (arb.u.)

phase (rad)

Principle

phase (rad)

XUV intensity (arb.u.)

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 6

Advanced Laser and Photon 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 S(p, t) =

Z ✓

◆ [p + A(t)]2 + Ip dt 2

i

ei

(t)

dp+A(t) EX (t

(t) = t

)ei(p

2

/2+Ip )t

dt

p · A(t ) + A2 (t )/2 dt

7

Ele

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 scale in (A) was set to yie Fig. 1A. The20zero of the CE phase i(p /2+I )t p spectra in (B).

B 70

e60i (t) dp+A(t) EX (t

)e

p · A(t ) + A2 (t )/2 dt 0

dt

1

50 40

E

1

0

E

ϕ = 170°

A

90 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

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 2 with the delay increased in steps of (A) and simulated (B) (17) photoelectron spectra versus CE phase, 40 CE phase setting closestpto 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

! = p /2 + I

A

80 70 60 50 40 30

0.8

0.8

80

0.6

2

4

−4

4 τx= 80 ± 5 as

70

3

60

0.4

50

0.2

40

C

−2

0 Delay (fs)

2

4

-300

2 1

-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

Goulielmakis al. spectraet of photoelectrons launched by an XUV pulse with a ba 1.0 (2008) D arb.u.)

.u.)

1.0

B C 90

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-

−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 60 use Advanced Laser and Photon Science (Kenichi ISHIKAWA) for internal only (Univ. of Tokyo)

0

at delay settings increased in steps of 80 as. Here, a posit 8 4 arriving before the NIR pulse. The high flux of the XUV sourc 0.8

SpectrogramRAPID COMMUNICATIONS PHYSICAL REVIEW A 71, 011401#R$ #2005$

Musical score

Most people think of acoustic waves in terms of a musical score. Intensity

pp

ff

pp

frequency

on its ": the orally proded in fined. phase ready ingle periacket. oelecarach the ments antly ng on interan be at the wave

Advanced Laser and Photon Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

time

FIG. 1. #a$ CRAB trace of a single 315 as pulse !full width at half maximum #FWHM$ of intensity", having second- and thirdorder spectral phases #Fourier limit= 250 as$, gated by a Fourierlimited 6-fs 800 nm laser pulse, of 0.5 TW / cm2 peak intensity. The electrons are collected around % = 0 with an acceptance angle of ±30°. #b$, #c$ A comparison of the exact as pulse and the laserinduced gate phase !#t$ #full line$ with the corresponding reconstructions #dots$ obtained from the CRAB trace after 100 iterations

9

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 Laser and Photon 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

13

Advanced Laser and Photon 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.

14

Advanced Laser and Photon 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

15

Advanced Laser and Photon 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

16

Advanced Laser and Photon 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

|

k

|r|e

ik·r

束の振幅 a(k, t)e

ik·r

ik·r

=i

ˆg (k)|2

k

ik·r

ˆ (k) g

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。 17

Advanced Laser and Photon Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo)

Molecular orbital tomography |ˆ x(

k

2 h )|

|

k

ˆg (k)|2

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)

18

Advanced Laser and Photon 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。 19

dimensional Fourier transform F of the object. This is the basis of computed tomography based on the inverse Radon transform. Our Advanced Laser and Photon Science (Kenichi ISHIKAWA) for internal dipole is the Fourier transform of a projection of the wavefunction, Calibrating the continuum wave packet As discussed above, the harmonic spectrum is an experimental and so can be inverted. We describe the mathematical details of the tomographic reconevaluation of the dipole, d(q ). If we could evaluate the plane-wave struction in the Methods section. This procedure can reconstruct amplitude a[k(q )] independently, then our measurement would R orbital shapes with symmetries such as jg, p g and p u, using determine w g(r)(er)exp[ik(q )x]dr—that is, the spatial Fourier harmonics 17–51 of an 800-nm laser field and 25 angles from 08 components of rw g(r). One way to do this is to perform the same to 1808 (fewer angles are needed for symmetric molecules). experiment with a reference atom. A complete inversion of a general orbital requires knowledge of Argon is very similar to N2 in its response to strong laser fields, the relative phase and amplitude of each harmonic for two orthohaving nearly the same ionization potential and intensity-depen2 2 6 dent ionization probability . This is confirmed by the dependence of h k g 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. Experiment 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

use only (Univ. of Tokyo)

Molecular orbital tomography |ˆ x(

)|

|

ˆ (k)|

k

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

20

Advanced Laser and Photon 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

=| |

21