ナノデザイン特論１

ナノデザイン特論１（石川顕一）学内向け講義資料

 12/5 No.

ナノデザイン特論１

•Attosecond science (simulation)

downloadable from

http://ishiken.free.fr/lecture.html

2006

12

5

1

石川顕一（東京大学内部向け講義資料）

Attosecond double- and

triple-slit experiment

Photoionization by attosecond soft X-ray double pulse

Temporal version of “Young’s double slit experiment”

1.0x10^-6

0.8

0.6

0.4

0.2

Photoelectron probability density (a.u.) 0.0

500 400

300 200

100 0

Distance from the nucleus (a.u.) 1.0x10⁴

0.8 0.6 0.4 0.2 0.0

Spectrum (arb. unit)

50 40

30 20

10 Photoelectron energy (eV) Single pulse Double pulse

Lack of “which-way” information Interference fringes

How can we control the “which-way information” ?

∆ E · τ = h

∆ E = 2¯ hω

-1.5x10^-2 -1.0 -0.5 0.0 0.5 1.0 1.5

Soft X-ray electric field (a.u.)

-3 -2 -1 0 1 2 3

Time (fs)

τ = π/ω

H23, H25, H27, H29, H31 of

¯

hω = 1 . 55 eV ( λ = 800 nm)

cf. Wollenhaupt et al. (2002), Lindner et al. (2005)

石川顕一（東京大学内部向け講義資料）

Laser-field-induced energy shear (streaking)

Goulielmakis et al.,

Science 305, 1267 (2004)

4

A

(t) = − (E

(t)/ω) sin(ωt + φ) E

(t) = E

(t) cos(ωt + φ)

∆ p ( t

) = eA

( t

)

The energy shear carries information of the release time.

Retrieval of the which-way information W _f (t _r ) ≈ W _i − !

2e ² W _i /m A _L (t _r ) cos θ

Double-slit scheme

-1.5x10^-2

-1.0 -0.5 0.0 0.5 1.0 1.5

Soft X-ray electric field (a.u.)

-3 -2 -1 0 1 2 3

Time (fs)

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

Laser vector potential (a.u.)

Soft X-ray Laser (phi = 0) Laser (phi = pi/2)

A

(t) = − (E

(t)/ω) sin(ωt + φ)

How is the photoelectron energy spectrum modified ?

1.0x10⁴ 0.8 0.6 0.4 0.2 0.0

Spectrum (arb. unit)

50 40

30 20

10

Photoelectron energy (eV)

w/o laser

?

∆W = −

! 2e

W

/m A

(t

) ∝ − A

(t

) at θ = 0

石川顕一（東京大学内部向け講義資料）

Double-slit scheme

6

-1.5x10^-2

-1.0 -0.5 0.0 0.5 1.0 1.5

Soft X-ray electric field (a.u.)

-3 -2 -1 0 1 2 3

Time (fs)

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

Laser vector potential (a.u.)

Soft X-ray Laser (phi = 0) Laser (phi = pi/2)

1.0x10

0.8 0.6 0.4 0.2 0.0

Spectrum (arb. unit)

50 40

30 20

10

Photoelectron energy (eV)

w/o laser phi = pi/2 phi = 0

which-way info lost interference fringes which-way info retrieved Fringes disappear.

ωt

+ φ = 0

ωt

+ φ = − π/ 2

A

(t) = − (E

(t)/ω) sin(ωt + φ)

∆W = −

! 2e

W

/m A

(t

) ∝ − A

(t

) at θ = 0

Fringe visibility can be controlled by the CEP of the laser pulse.

45 40

35 30

25 20

15 10

Photoelectron energy (eV) 360

315 270 225 180 135 90 45 0

Laser carrier envelope phase (degrees)

E

(t) = E

(t) cos(ωt + φ)

Variation of photoelectron energy spectra as a function of CEP φ

石川顕一（東京大学内部向け講義資料）

Triple-slit scheme

8 2.5x10⁴

2.0 1.5 1.0 0.5 0.0

Spectrum (arb. unit)

50 40

30 20

10

Photoelectron energy (eV)

w/o laser phi = 0

which-way info lost interference fringes

∆W = −

! 2e

W

/m A

(t

) ∝ − A

(t

) at θ = 0

-1.5x10^-2

-1.0 -0.5 0.0 0.5 1.0 1.5

Soft X-ray electric field (a.u.)

-3 -2 -1 0 1 2 3

Time (fs)

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

Laser vector potential (a.u.)

Soft X-ray Laser (phi = 0)

Triple-slit scheme

2.5x10⁴ 2.0 1.5 1.0 0.5 0.0

Spectrum (arb. unit)

50 40

30 20

10

Photoelectron energy (eV)

w/o laser phi = -pi/2

2nd pulse 1st and 3rd pulses

Co-presence of a single-slit and a double-slit scheme !

∆W = −

! 2e

W

/m A

(t

) ∝ − A

(t

) at θ = 0

-1.5x10^-2

-1.0 -0.5 0.0 0.5 1.0 1.5

Soft X-ray electric field (a.u.)

-3 -2 -1 0 1 2 3

Time (fs)

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

Laser vector potential (a.u.)

Soft X-ray

Laser (phi = -pi/2)

石川顕一（東京大学内部向け講義資料）

Triple-slit scheme

The same electron encounters

 single slit

 double slit simultaneously.

The results of both schemes recorded as a single

photoelectron energy spectrum

 in the same direction

 by the same detector

10

2.5x10⁴ 2.0 1.5 1.0 0.5 0.0

Spectrum (arb. unit)

50 40

30 20

10

Photoelectron energy (eV) w/o laser phi = -pi/2

2nd pulse

1st and 3rd pulses

-1.5x10^-2

-1.0 -0.5 0.0 0.5 1.0 1.5

Soft X-ray electric field (a.u.)

-3 -2 -1 0 1 2 3

Time (fs)

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

Laser vector potential (a.u.)

Soft X-ray Laser (phi = -pi/2) Laser (phi = 0)

A_L(t) =−(E⁰(t)/ω) sin(ωt+φ)

Spacing between adjacent interference fringes

-1.5x10^-2

-1.0 -0.5 0.0 0.5 1.0 1.5

Soft X-ray electric field (a.u.)

-3 -2 -1 0 1 2 3

Time (fs)

-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

Laser vector potential (a.u.)

Soft X-ray

Laser (phi = -pi/2) Laser (phi = 0)

2.5x10⁴ 2.0 1.5 1.0 0.5 0.0

Spectrum (arb. unit)

50 40

30 20

10

Photoelectron energy (eV)

w/o laser phi = -pi/2 phi = 0

∆ E · τ = h

τ = π/ω τ = π/ω

∆ E = ¯ hω

∆ E = 2¯ hω

石川顕一（東京大学内部向け講義資料）

Fringe visibility can be controlled by the CEP of the laser pulse.

12

E

(t) = E

(t) cos(ωt + φ) Variation of photoelectron energy spectra as a function of CEP φ

45 40

35 30

25 20

15 10

Photoelectron energy (eV) 360

315 270 225 180 135 90 45 0

Laser carrier envelope phase (degrees)

Conclusions

Theoretical analysis of photoionization by attosecond soft X-ray pulses as a temporal version of the double-slit and triple-slit experiment

Control of the visibility of interference fringes

(=discrete peaks in the photoelectron energy spectrum)

Variation of the magnitude of which-way information

Momentum change with a phase-controlled laser pulse

石川顕一（東京大学内部向け講義資料）

Two-Photon Ionization of He with ultrashort soft X-ray pulses

14

0.33 µJ @ λ = 29.6 nm (Ti:Sapphire H27)

10

W/cm

Soft x-ray

focused to an area of 10µm² by a mirror Assuming the pulse duration < 30 fs

 Above-threshold ionization  (ATI) of He

 Sekikawa et al., Nature 432,  605 (2004)

 h ν  = 27.9 eV

 ATI & Two-photon double  ionization (TPDI) of He

 Hasegawa et al., Phys. Rev. 

A 71, 023407 (2005)

 h ν  = 41.8 eV

Intense XUV and soft X-ray source

Mashiko et al., Opt. Lett. 29, 1927 (2004)

Soft X-ray XUV

15

石川顕一（東京大学内部向け講義資料）

Time-dependent Schrödinger equation

€

i ∂

∂ t ψ (r

,r

,t ) = [ H

+ ( z

+ z

) ^{E(t )} ] ^{ψ (r}

^,r

^{,t )}

€

H

= − 1

2 ∇

− 1

2 ∇

− 2 r

− 2

r

+ 1 r

€

1

r

= 4 π 2 λ +1

r

r

Y

( ˆ r

) Y

( ˆ r

)

q=−λ λ

∑

λ=0

∞

∑

€

ψ (r

,r

,t ) = P

1l₂

L

(r

,r

,t )

r

r

Λ

(

l₁,l₂

∑

L

∑ ^{r ˆ}

^{, ˆ r}

⁾

€

Λ

( ˆ r

, ˆ r

) = l

ml

− m L 0 Y

1m

( ˆ r

)

m

∑ ^Y

^{( ˆ r}

⁾ Coupled spherical harmonics

€

r

→ j

− 1 2



  

  Δr

€

r

→ j

− 1 2



  

  Δr

€

P

(r

,r

,t ) ⇒ P

(t )

16

€

P

1l₂

L

(r

,r

,t )

€

(r

,r

)

ATI of He

0 25 50 75 100 125 150 0

25 50 75 100 125 150

r1 (a.u.)

r2 (a.u.)

0.0e+00 2.0e-08 4.0e-08 wf202_d_094208_dat 0 25 50 75 100 125 150

0 25 50 75 100 125 150

r1 (a.u.)

r2 (a.u.)

0.0000 0.0001 0.0002 0.0003 0.0004 wf101_d_094208_dat

€

L = 1,l

= 0,l

= 1

€

L = 2,l

= 0,l

= 2

27次高調波(41.85 eV, 29.6 nm), 10

W/cm

, 450 as

7 6 5 4 3 2 1 0

x10-6

80 70 60 50 40

E2 (eV)

L=2, l1=0, l2=2

STOCK60bis

１光子電離 ２光子電離

(Above-threshold ionization)

€

σ

= 5.4 ×10

cm

⋅ s

石川顕一（東京大学内部向け講義資料）

Two-photon ionization of He by Ti:S 27th harmonic pulses (42eV)

 Above-threshold ionization (ATI)

 Hasegawa et al., Phys. Rev. A 71, 023407 (2005)

10 80

30 20 70 60 50 40

0 110 100 90

24.6 79.0

He He

He

120

Energy (eV)

1s

2s, 2p nl

18

19

Two-photon double ionization (TPDI)

0 25 50 75 100 125 150 0

25 50 75 100 125 150

r1 (a.u.)

r2 (a.u.)

0.0e+00 2.0e-08 4.0e-08 wf202_d_094208_dat

€

L = 2,l

= 0,l

= 2

27次高調波(41.85 eV, 29.6 nm), 10

W/cm

, 450 as

7 6 5 4 3 2 1 0

x10-6

80 60

40 20

0

E2 (eV)

Total 1s 2s 3s 4s L=2, l1=0, l2=2

２光子電離

 Ionization + Core excitation

 TPDI

€

σ = 3.0 ×10

cm

⋅ s

石川顕一（東京大学内部向け講義資料）

Two-photon ionization of He by Ti:S 27th harmonic pulses (42eV)

 Two-photon double ionization (TPDI)

 Nabekawa et al., Phys. Rev. Lett.

94, 043001 (2005)

10 80

30 20 70 60 50 40

0 110 100 90

24.6 79.0

He He

He

120

Energy (eV)

1s

2s, 2p nl

20

Attosecond soft X-ray source

 TU Wien & MPI Garching (F. Krausz)

 250 attoseconds @ 13.3 nm (Ti:S H57)

 Direct measurement of light waves

Kienberger et al., Nature 427, 817 (2004)

Goulielmakis et al., Science 305, 1267 (2004)

21

石川顕一（東京大学内部向け講義資料）

HHG soft X-ray sources

 High intensity (<10 ¹⁴ W/cm ² )

 Ultrashort pulse duration (>250 as)

What happens when these two are united?

22

Two-photon double ionization (TPDI)

of He by an intense attosecond 59th harmonic pulse (hν=91.5 eV, λ=13.6 nm)

1st ionization potential : 24.6 eV 2nd ionization potential : 54.4 eV

can be doubly ionized by a single photon

Above-Threshold Double Ionization (ATDI)

  91.5−24.6=67 eV & 91.5−54.4=37 eV

Usual sequential TPDI sharp peaks at

10 80

30 20 70 60 50 40

0 110 100 90

24.6 79.0

He He

He

67 eV 37 eV

120

Energy (eV)

1s 2s, 2p nl

37eV 67eV

23

石川顕一（東京大学内部向け講義資料）

Questions

 How does an ultrashort pulse duration modify the electron energy spectrum?

 Does non-sequential two-photon double ionization take place?

37eV 67eV

?

24

石川顕一（東京大学内部向け講義資料）

Electron energy distribution

450 as

59th harmonic (h ν =91.5 eV, λ =13.6 nm)   10 ¹⁵ W/cm ² Gaussian temporal profile

0255075 100 125 150 175 2000255075100125150175200colrow0.0e+005.0e-111.0e-101.5e-102.0e-102.5e-103.0e-103.5e-10wf211_d_098304_dat 0255075 100 125 150 175 2000255075100125150175200colrow0.0e+005.0e-111.0e-101.5e-102.0e-102.5e-103.0e-103.5e-10wf211_d_098304_dat

0 25 50 75 100 125 150 175 200 0

25 50 75 100 125 150 175 200

col

row

0.0e+00 1.0e-10 2.0e-10 3.0e-10 4.0e-10 wf211_d_098304_dat

r

(a.u.) r

(a.u.)

Probability density Electron energy distribution

€

P

1l₂

L

(E

, E

) = R

1l₁

(r

)R

2l₂

(r

) Λ

( ˆ r

, ˆ r

) ψ (r

,r

,t )

L=2, l

=l

=1

€

R

(r) : Coulomb wave function

25

石川顕一（東京大学内部向け講義資料）

Electron energy distribution

450 as 225 as 150 as

59th harmonic (h ν =91.5 eV, λ =13.6 nm)   10 ¹⁵ W/cm ²

usual sequential double ionization

Anomalous component between the peaks !

26

Core relaxation takes time.

60

55

50

45

40

35

2nd ionization potential (eV)

1000 800

600 400

200 0

Time after ionization (as)

He

(1s)

Ip

-Ip

=29.8 eV Correlation time = 22 as He

*

7x10^-6 6 5 4 3 2 1 0

Spectrum (1/eV)

100 80 60 40 20 0

Electron energy (eV) 450 asec

€

P( Δ t ) = 2 2 ln 2 T π ^e

−2 ln 2Δt² T²

10x10^-3

8

6

4

2

0

Probability distribution

1000 800

600 400

200 0

Ionization interval (as)

150 as

450 as 225 as

Second ionization during core relaxation

Instantaneous ionization potential Distribution of ionization interval

Ip₁+Ip₂=79.0 eV 79.0 2=39.5 eV

- 39.5 eV

- 54.4 eV

ionization

core relaxation

27

石川顕一（東京大学内部向け講義資料）

Semi-classical model

 Ionization interval distributed as

  Ionization potential

  Electron energy

 Folding with the peak shape function

60

55

50

45

40

35

2nd ionization potential (eV)

1000 800 600 400 200 0

Time after ionization (as)

He⁺(1s) He⁺*

€

P( ω ) ∝ e

∫

e

1 + erf ( t − i ω /2 )

^dt

70 65 60 55 50 45 40 35

Electron energy (eV)

1000 800 600 400 200 0

Ionization interval (as)

Electron 1 Electron 2

10x10^-3

8

6

4

2

0

Probability distribution

1000 800 600 400 200 0

Ionization interval (as)

150 as 450 as 225 as

Second ionization during core relaxation

16 14 12 10 8 6 4 2 0

x103

100 80 60 40 20 0

Electron energy (eV)

150 as

28

The semi-classical model well reproduces the electron energy spectrum.

 Attosecond pulse duration

 Finite core relaxation time

 Anomalous component (sequential ATDI)

 Displacement of the peaks

7x10^-6 6 5 4 3 2 1 0

Spectrum (1/eV)

100 80 60 40 20 0

Electron energy (eV) 450 asec ^{TDSE Fitting}

Semi-classical model

1.0x10^-6 0.8 0.6 0.4 0.2 0.0

Spectrum (1/eV)

100 80 60 40 20 0

Electron energy (eV) 225 asec ^TDSE _Fitting

Semi-classical model

350x10^-9 300 250 200 150 100 50 0

Spectrum (1/eV)

100 80 60 40 20 0

Electron energy (eV) 150 asec TDSE

Fitting Semi-classical model

35

30

25

20

15

Difference between the two peaks (eV) ^{5004003002001000}

Pulse width T (as)

TDSE

Semi-classical model

Electron correlation Second ionization during core relaxation

29