強レーザー場中の原子と 高次高調波発生

強レーザー場中の原子と 高次高調波発生

Atom in an intense laser field &

high-order harmonic generation

量子ビーム発生工学特論

E

Kenichi Ishikawa (石川顕一)

http://ishiken.free.fr/english/lecture.html [email protected]

References 参考文献

Laser fundmentals, Rabi oscillation

ラビ振動

William T. Silfvast, “Laser Fundamentals”

(Cambridge University Press)

霜田光一「レーザー物理入門」（岩波書店）

Atom in an intense laser field

M. Protopapas, C.H. Keitel and P.L. Knight, “Atomic

physics with super-high intensity lasers”, Rep. Prog. Phys.

60, 389–486 (1997)

11/14

How intense is an intense laser field?

強レーザー場とは

Intensity at which the interaction with an atom becomes

non-perturbative

める強度。

Effect of laser on the electron

Effect of the nucleus on the electron

レーザー場が電子におよぼす影響 〜 原子核が電子におよ ぼす影響

Intensity

10 ¹³ 〜 10 ¹⁵ W/cm ²

3

High-field phenomena 高強度場現象

Above-threshold ionization (ATI)

Ionization upon which an atom absorbs more photons than minimum necessary.

Tunneling ionization

Ionization by the tunneling effect rather than absorption of

photons

High-harmonic generation (HHG)

Generation of harmonics of very high orders

11/14

Key concepts

キーとなる概念

Ponderomotive energy

(this week)

Quantum paths (trajectories)

(next week)

5

Why is high-field phenomena

fascinating? 高強度場現象の魅力

We can look at a common phenomenon

from various view points.

Atomic physics meets plasma physics.

11/14

１光子電離（光電効果）

I _P

基底状態

E = 0

€

 ω

€

E

=  ω − I

1905

I p : Ionization potential

Condition for ionization

€

 ω > I

Ionization rate

R I

I : Light intensity

Single-photon ionization (photoelectric effect)

Kinetic energy of the ejected electron

7

１光子電離

I _p

ground state 基底状態 ionization

電離

€

 ω

€

E

=  ω − I

€

d

dt C

(t )

= 2 π

 γ

= π

2  µ

E

€

µ

= ∫ ϕ

^z ϕ

^d

^{r = i z j}

Ionization rate (transition probability per  unit time) 単位時間当たりの遷移確率

€

ϕ

= 2e

× 1 4 π ϕ

= 2

1 − e

1

+ n ʹ′

kr 3 e

× F (i n ʹ′ + 2,4,2ikr) × 3

4 π ^cos θ

Photon energy 

(eV)

Ionization rate

Single-photon ionization 

11/14   No.

１光子電離の強度依存性

10

 W/cm

2 10

 W/cm

Ionization

€

∝ ^Intensity

optical effect)

Intensity-dependence of single-photon ionization

9

多光子電離

I _P

€

 ω

€

 ω

€

 ω I _P

基底状態

E = 0

€

 ω

€

 ω < I

強度

イオン化に必要な光子数

放出された電子の運動エネルギー

N = I _p

+ 1 E _kin = N I _p

What was believed till 1970‘s. 1970

MULTIPHOTON IONIZATION

Intensity

Ground state

Kinetic energy of the ejected electron

Number of photons necessary for ionization

Nonlinear optical phenomena

11/14   No.

例：３光子電離

I _p

€

 ω

Hydrogen atom

€

I

= 13.6 eV

非線形光学応答(nonlinear optical  effect)

Ionization

€

∝ ^Intensity

3

n-photon ionization Ionization

€

∝ ^Intensity

n

requires a bright source →  realized only with lasers

強い光源が必要 → レーザーの出現 によって初めて実現

Pulse duration  40fs

Example: 3-photon ionization

11 Peak intensity

Io n iza ti o n

Photoelectron  energy

€

E

= 3  ω − I

イオン化レートの検証

Power low confirmed for different target atoms

N

1965

1975

I < 10 ¹³ W/cm ²

R N = N N = I/

Experimental verification of the power low of ionization rate

Ionization rate

11/14

超閾電離の発見

All the previous experiments only measured the total ionization yield

Agostini et al. measured the photoelecton energy spectrum for the first time.

初めて光電子のエネルギースペクトルを測定した。

Pierre Agostini

CEA-Saclay, France

Phys. Rev. Lett. 42, 1127 (1979)

超閾電離

波長

532nm = 2.33 eV I p (Xe) = 12.1298 eV N = 6

E _kin = N I _p = 1.86 eV

A peak of energy higher than expected for 6-photon ionization

６光子電離で予想されるより高エネルギーの位置にもピークを発見

Another photon absorbed after 6- photon ionization?

６光子電離の後で もう１光子吸収？

Discovery of above-threshold ionization (ATI)

ATI

13

自由電子は光子を吸えない

Solutions exist only for n = 0 → A free electron can neither absorb nor emit photons, because the momentum cannot be conserved

解があるのは、

n=0

→

自由電子は光子を吸収も放出もできない。

Free-free transition possible only near the ion which absorbs the

momentum difference

のみ、

free-free

Does a rapidly-escaping electron have time to absorb a photon?

イオンから逃げていく電子が、光子を吸う暇があるのか？

エネルギー保存 運動量保存

p ² _i

2 + n = p ² _f 2

= c | k | p _i + n k = p f

A free electron cannot absorb photons

Energy conservation

Momentum conservation

11/14

より高強度の実験

Now certain that ATI is due to free-free transition

ATI

free-free

Kruit et al., Phys. Rev. A 28, 248 (1983) Group of FOM (Amsterdam)

MacIlrath et al., Phys. Rev. A 35, 4611 (1987) Group of AT&T Bell Lab.

E _kin = (N + S ) I _p

波長

1064 nm Xe gas

最小限必要な光子数 余分の光子数

Experiments with higher intensity

wavelength

Minimum

Extra photons

15

ATI

Comparable peak heights → non-perturbative

吸収光子数によらず、ピークの高さが同程度

→

低次の吸収ピークが消える（

peak suppression at low orders

Kruit et al., Phys. Rev. A 28, 248 (1983)

FOM (

)

MacIlrath et al., Phys. Rev. A 35, 4611 (1987) AT&T

At high intensity

Intensity dependence

11/14

高次の摂動論

i t = (H ₀ + H _I )

H _I = e

n

i=1

r _i · E(t)

H _I = e m

n

i=1

p _i · A(t) + ne ²

2m A ² (t)

LENGTH FORM VELOCITY FORM

N = 2 2e ²

0 c

N

f

M _i ^(N) _f ²

断面積 ^単位

cm ^2N s ^N-1

M _i ^(N _f ⁾ =

j ,j , ··· ,j

i | x | j j | x | j · · · j | x | f

(E _i + E _j )(E _i + 2 E _j ) · · · (E _i + (N 1) E _j )

17

High-order perturbation theory

cross section

水素原子の

(N+S)

(cm ^2(N+S) /W ^N+S /s)

最小限必要な光子数

N

N

N

N

余分の光子数

S 6 (530 nm) 8 (650 nm) 10 (910 nm) 12 (1082 nm)

0 1.39×10

1.49×10

4.51×10

3.46×10

1 2.84×10

9.85×10

7.78×10

9.81×10

2 2.92×10

2.53×10

5.35×10

1.10×10

3 2.80×10

5.84×10

2.61×10

1.08×10

4 2.66×10

1.35×10

1.89×10

9.87×10

5 2.32×10

2.75×10

1.04×10

8.91×10

Gontier and Trahin, J. Phys. B 13, 4383 (1980)

S=0

1

なる強度

(W/cm

) 4.89×10

1.51×10

5.80×10

3.53×10

非摂動論的になる強度の目安

長波長ほど低強度 ^{実験と整合}

(N+S)-photon ionization cross section of a hydrogen atom

Equal cross section for

S=0 and 2 Intensity at which the interaction becomes non-perturbative

longer wavelength → lower intensity

11/14

非摂動論的？

Why non-perturbative at much lower intensity

なぜ、これよりずっと低い強度で非摂動論的になるのか？

Why non-perturbative at lower intensity for longer

wavelength

のか？

Why low-order peaks are suppressed?

NUCLEAR COULOMBIC FORCE

LASER ELECTRIC FORCE

＝

a 0 e ² eE

4 ₀ a ² ₀

I = 3.51 10 ¹⁶ W/cm ²

？

19

Non-perturbative?

From another view point 別の観点から 見てみよう

PLASMA

プラズマ

11/14

電磁波中の荷電粒子

E(r, t) = 1

2 [E ₀ (r, t)e ^{i t} + c.c.] = | E ₀ | cos( t + ) B(r, t) = 1

2 [B ₀ (r, t)e ^{i t} + c.c.] = | B ₀ | cos( t + ) r(t) = R(t) + r(t)

Macroscopic drift motion

マクロなドリフト運動

Microscopic oscillation

ω

Slowly varying envelope

振動数

ω

r(t) = r ₀ e ^{i t} + c.c.

| r 0 · E 0 | | E 0 |

21

Charged particle in an electromagnetic wave

r(t) = R(t) + r(t)

δr 0

E 0 , B 0

r(t) = r ₀ e ^{i t} + c.c.

| r ₀ · E ₀ | | E ₀ |

| r ₀ · B ₀ | | B ₀ | v(t) = V(t) + v(t)

Non-relativistic electron velocity

電子の速度は非相対論的

V B ₀ E ₀

OSCILLATION AMPLITUDE

mass m, charge q

v(t) = v ₀ e ^{i t} + c.c.

m ˙ v = qE(r, t)

v ₀ = iqE ₀ 2m

v = ˙ r ^{r 0 =}

qE ₀ 2m ² E(r, t) = B(r, t)

t ^{B 0 =}

E ₀

i

11/14

荷電粒子に作用する力

F = q [E(r(t), t) + v(t) B(r(t), t)]

= q[E(R + r, t) + (V + v) B(R + r, t)]

q [E(R, t) + r · E(R, t) + V B(R, t) + v B(R, t)]

F q

2 ( r ₀ · E ₀ + v ₀ B ₀ + c.c.)

= q ²

4m ² [E ₀ · E ₀ + E ₀ ( E ₀ ) + c.c.] = q ²

4m ² | E ₀ | ²

F = U _p (R, t) U p (R, t) = q ² | E ₀ (R, t) | ² 4m ²

PONDEROMOTIVE POTENTIAL (ENERGY)

ポンデロモーティブポテ ンシャル（エネルギー）

23

Force acting on the charged particle

ポンデロモーティブ力（動重力）

Potential force

Proportional to the laser intensity

Independent of the sign of charge (from the beam axis to outside)

電荷の正負によらず向きが同じ（ビームの中心から外へ）

Higher energy for lighter particles (larger effect for electrons than for nuclei and ions)

A charged particle in a laser field has an energy of U

by

default.

U

ルギーを持っている。

F = U _p (R, t) U p (R, t) = q ² | E ₀ (R, t) | ² 4m ²

quasi-static so-called ponderomotive component that points against the gradient of the cycle-averaged intensity [74—76]. Although this is not always explicitly recognized, the magnetic component of the Lorentz force must be taken into account even in the non-relativistic derivation, or else the pondero- motive force may incorrectly appear to depend on the polarization of the field.

The ponderomotive force is clearly the negative of the gradient of a ponderomotive potential

V~= e2 ²

I~I2, (6.6)

4mw

which is nothing but the cycle-averaged kinetic energy in the micromotion. This same quantity was called the jitter energy

in section 4. When an electron adiabatically leaves a steady laser beam, all of its kinetic energy in the quiver motion is simply converted into translational kinetic energy by “sliding down” the potential hill, as shown in fig. 6. However, if the laser pulse duration is short enough, the potential

collapses quickly and there will be no ponderomotive acceleration.

These effects were demonstrated in recent experiments [77].A pulsed laser beam was focused in the region between a source of the electrons and an electron detector, as shown in fig. 7. The arrivals of the electrons at the detector were consistent with the predicted effects of the ponderomotive potential of the focused laser pulse.

Ponderomotive effects on a bound electron are certainly quite complicated and have never been

380J~_

247 FocusJTh

~ ~ 180 ^{J “—}

113 ^{~•____ -..._ — - ~} ~t.

I .~ 47

/ __ ^/

I /

/ /

I

I

I /

I /

~353 ________________

0.2 0.4 0.6 0.8

• eV

x

Fig. 6. Two hypothetical photoelectron trajectories under the in- Fig. 7. Direct observation of ponderomotive scattering of free dcc- fluence of ponderomotive acceleration. If the photoelectrons were trons by a light intensity gradient. The drawing shows a pulse of liberated with zero velocity, the distribution would be isotropic in the electrons approaching a pulsed-laser focus. The data show the effect x—yplane perpendicular to the laser beam axis. of changing the delay of the arrival of the laser pulse relative to the arrival of the electrons at the focus. Curves labeled 380 and 247 show no effect as the laser and electron pulses miss each other. Curves180 and 113 show that the electrons are beginning to overlap with the laser pulse and have a higher energy due to acceleration by its leading edge. Curves 47 and —20 correspond to closely timed arrival at the focus and show no electrons arriving at the detector due to strong ponderomotive scattering. The curves —87 to —187 show electron

| E

(R, t) |

I (R, t)

Ponderomotive force

PONDEROMOTIVE POTENTIAL (ENERGY)

ポンデロモーティブポテ ンシャル（エネルギー）

11/14

Ponderomotive energy from a microscopic view point

A charged particle in a laser field has an energy of Up by default.

電子（荷電粒子）は、レーザー場中にただいるだけで

U p

ミクロな視点からみた

Motion of a charge particle (mass m, charge q) in an oscillating electric field

m,

q

+ drift 並進運動

Energy of quiver motion (jitter motion)

Time average

時間平均

1

2 mv ² = q ² E ₀ ²

4m ² = U _p For an electron

U p (eV) = e ² E ₀ ²

4m ² = 9.337 10 ¹⁴ I (W/cm ² ) ² (µm) E (t) = E ₀ sin t

m v ˙ = qE 0 sin t v = qE ₀

m cos t

1

2 mv ² = q ² E ₀ ²

2m ² cos ² t

25

低次のピークがなくなるのはポンデロ モーティブシフトの効果

U p (eV) = e ² E ₀ ²

4m ² = 9.337 10 ¹⁴ I (W/cm ² ) ² (µm)

I p +U p

長波長の方が起こりやすいことも説明できる。

Lower I for longer wavelength at fixed U p

Peak suppression due to ponderomotive shift

Effective ionization potential = I p +U p

11/14

実効的なイオン化ポテンシャルが

I p +U p

J.H. Eberly et a!., Above-Threshold Ionization 351

All told, in an inhomogeneous electromagnetic field the Lorentz force acting on a free electron has a quasi-static so-called ponderomotive component that points against the gradient of the cycle-averaged intensity [74—76]. Although this is not always explicitly recognized, the magnetic component of the Lorentz force must be taken into account even in the non-relativistic derivation, or else the pondero- motive force may incorrectly appear to depend on the polarization of the field.

The ponderomotive force is clearly the negative of the gradient of a ponderomotive potential

V~= e2 ²

I~I2, (6.6)

4mw

which is nothing but the cycle-averaged kinetic energy in the micromotion. This same quantity was called the jitter energy

in section 4. When an electron adiabatically leaves a steady laser beam, all of its kinetic energy in the quiver motion is simply converted into translational kinetic energy by “sliding down” the potential hill, as shown in fig. 6. However, if the laser pulse duration is short enough, the potential

collapses quickly and there will be no ponderomotive acceleration.

These effects were demonstrated in recent experiments [77].A pulsed laser beam was focused in the region between a source of the electrons and an electron detector, as shown in fig. 7. The arrivals of the electrons at the detector were consistent with the predicted effects of the ponderomotive potential of the focused laser pulse.

Ponderomotive effects on a bound electron are certainly quite complicated and have never been

380J~_

247 FocusJTh

~ ~ 180 ^{J “—}

113 ^{~•____ -..._ — - ~} ~t.

I .~ 47

/ __ ^/

I /

/ /

I

I

I /

I /

~353 ________________

0.2 0.4 0.6 0.8

• eV

x

Fig. 6. Two hypothetical photoelectron trajectories under the in- Fig. 7. Direct observation of ponderomotive scattering of free dcc- fluence of ponderomotive acceleration. If the photoelectrons were trons by a light intensity gradient. The drawing shows a pulse of liberated with zero velocity, the distribution would be isotropic in the electrons approaching a pulsed-laser focus. The data show the effect x—yplane perpendicular to the laser beam axis. of changing the delay of the arrival of the laser pulse relative to the arrival of the electrons at the focus. Curves labeled 380 and 247 show no effect as the laser and electron pulses miss each other. Curves180 and 113 show that the electrons are beginning to overlap with the laser pulse and have a higher energy due to acceleration by its leading edge. Curves 47 and —20 correspond to closely timed arrival at the focus and show no electrons arriving at the detector due to strong ponderomotive scattering. The curves —87 to —187 show electron energy loss due to ponderomotive deceleration in the tail of the laser pulse, and curves —253 and —353 correspond to no electron-laser overlap again. [Private communication from PH. Bucksbaum; see also ref. [77].]

Number of photons necessary for ionization

E kin = [n (I p + U p )] + U p = n I p

Observed electron energy

n I _p + U _p

27

Effective ionization potential = I p +U p

Bound electrons

Negative for the ground state

→dipole trap

Positive for Rydberg atoms and free electrons

→

Lorentz oscillator model m x ¨ = eE ₀ cos t m ₀ ² x

x = x ₀ cos t

Quantum mechanically, AC-stark effect

量子力学的には：

AC

E = e ² E ₀ ² 4 _n

2 _ni | µ in | ²

2 2

ni

= 1

4 ( )E ₀ ² 2nd-order perturbation

theory

Electric dipole polarizability

E _g e ² E ₀ ² 0

4m ₀ ² I

= e ² E ₀ ²

| |

<< U p

= e ²

m( ² ₀ ² ) E = e ² E ₀ ²

4m( ² ₀ ² )

11/14

リュードベリ原子は、強レーザーパルスから逃げる

online-only Methods). Recalling the key point of our investigation, that the electron remains bound after interaction with the laser pulse, we are able to observe the centre-of-mass motion. Furthermore, by solving the coupled Lorentz equations for the electron and the ion, including the Coulomb potential, we can directly reproduce the cap- ture process into bound Rydberg orbits and the force on the centre of mass.

According to our model we can rewrite equation (1) for the centre- of-mass positionRof the neutral atom:

MRR€ð Þt ~{ 1

4mev²+j jE0² ð2Þ Here,Mandme(51 atomic unit) are the masses of the atom and the electron, respectively, andRR€ð Þt is the second derivative of the centre- of-mass positionRwith respect to time. To calculate the pondero- motive force explicitly, we assume a linearly polarized laser beam with a Gaussian spatial intensity distribution, which reads, in cylin- drical coordinates:

Ið Þr ~jE0ð Þrj²~I0 1z z z0

! "2!{1

exp{2r²

r₀² ð3Þ where r0~w0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1zðz=z0Þ² q

, w₀ is the beam waist. Evaluating the gradient in equation (2) with the intensity distribution given by equation (3), we obtain, for the radial component of the centre-of- mass position perpendicular to the laser beam direction:

€

rr tð Þ~Ið ÞR Mv²

r tð Þ

r₀² f tð Þ ð4Þ wheref(t) is the laser pulse envelope, which we assume to be of the formf(t)5exp(–t²/t²), wheretis the pulse width. From equation (4)

we find that the maximum force along the radial direction scales as r₀^{1. Similarly, one can show that it scales asz^{1₀ along the laser beam direction. Because the Rayleigh lengthz0is typically a factor of 100 larger than the beam waistr₀, the gradient and thus the ponderomo- tive force in the laser beam direction is much smaller than in the radial direction and can be neglected. (However, the situation would be very different if we used a short-pulse standing-wave laser field.

We would then obtain a strong periodic intensity gradient on the scale of the laser wavelength, and might expect to see the Kapitza–

Dirac effect for neutral atoms in an intense standing-wave laser field instead of electrons¹⁹).

To solve equation (4), we assume that the neutral atom does not move significantly during the laser pulse. Hence, we setr(t)5ron the right-hand side of the equation, which allows us to solve equation (4) analytically for any initial position of an atom in the laser beam. We will concentrate our analysis on atoms located at the half beam size r₀/2, which experience the maximum force. Solving equation (4) for these conditions by integrating over the full laser pulse, we find the maximum velocityv_max(z):

vmaxð Þz ~ I0

2Mv²w0

ffiffiffi pp

expð{0:5Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1z _z^z

0

$ %2

r 3 t ð5Þ

If we evaluate equation (5) at the focal plane for He atoms exposed to our focused laser beam at maximum intensity, we obtain a velocity of about 55 m s²¹from which accelerations of about 2310¹⁴gcan be deduced.

This exceeds the typical acceleration (deceleration) of neutral atoms²⁰or molecules in external fields^21,22. Compared to laser-cooling experiments in a continuous-wave laser field, for instance, which are

1,000 0

–60 –40 –20 0

v (m s^–1)

20 40 60

0.0 0.5

b a

c

Relative laser intensity 1.0

0 –6

–4 –2 0 2 4 6

14 7

0 r_D (mm)

z (mm)

–7

–14 5

He* yield (arbitrary units) 10 5

10

100 10 1 0.1

He* yield (arbitrary units)

Figure 1|Deflection of neutral He atoms after interaction with a focused laser beam. a, Distribution of excited He*atoms on the detector (colour scale, in number of atoms). The laser beam direction is indicated by the arrow.b, Cut through the atom distribution along the laser beam axis (zaxis) atr_D50 mm (black curve) and full projection onzaxis (dashed red curve) and intensity along thezaxis in units of the laser peak intensity

I056.9310¹⁵W cm^–2(blue curve).c, Cuts through the distribution at z50 mm (red curve) andz5 22.7 mm (black curve). The black curve shows the velocity distribution of excited neutral atoms at a position unaffected by the ponderomotive force, showing essentially the ‘natural’ velocity spread, while the red curve shows the velocity gain through the ponderomotive force.

LETTERS NATURE|Vol 461|29 October 2009

1262

Macmillan Publishers Limited. All rights reserved

©2009

E _R U _p = e ² E ₀ ²

4m ² I

質量に反比例 原子核へのポンデロモー ティブ力は無視できる

原子全体は電子に働く力 に引っ張られる

LETTERS

Acceleration of neutral atoms in strong short-pulse laser fields

U. Eichmann^1,2, T. Nubbemeyer¹, H. Rottke¹& W. Sandner^1,2

A charged particle exposed to an oscillating electric field experi- ences a force proportional to the cycle-averaged intensity gradient.

This so-called ponderomotive force¹plays a major part in a variety of physical situations such as Paul traps^2,3for charged particles, electron diffraction in strong (standing) laser fields^4–6 (the Kapitza–Dirac effect) and laser-based particle acceleration^7–9. Comparably weak forces on neutral atoms in inhomogeneous light fields may arise from the dynamical polarization of an atom^10–12; these are physically similar to the cycle-averaged forces. Here we observe previously unconsidered extremely strong kinematic forces on neutral atoms in short-pulse laser fields. We identify the ponderomotive force on electrons as the driving mechanism, leading to ultrastrong acceleration of neutral atoms with a mag- nitude as high as 10¹⁴times the Earth’s gravitational accelera- tion,g. To our knowledge, this is by far the highest observed acceleration on neutral atoms in external fields and may lead to new applications in both fundamental and applied physics.

The investigation has become possible through two recent findings concerning atomic ionization dynamics in strong laser fields. First, neutral atoms can survive a strong laser field in a (long-lived) excited state¹³, in which they can be detected directly in an atomic beam by means of a standard electron or ion detector¹⁴. Thus, any momentum transferred to the neutral atom can easily be detected. Second, according to the physical picture behind the excitation process, the excited electron behaves as a quasi-free electron during the laser pulse. More precisely, the excitation process can be viewed as a fru- strated tunnel ionization¹⁴within the three-step model for strong- field ionization¹⁵.

In the first step, the electron tunnels in the close vicinity of the maximum electric field of a laser cycle. The liberated electron is then driven by the laser field with an amplitude that slowly decreases with decreasing pulse intensity; in this way an active damping of the elec- tronic motion takes place. After the laser pulse the electron is left with a drift energy too low to overcome the Coulomb potential of the ion and is recaptured into a Rydberg state. The quivering quasi-free electron experiences the ponderomotive force during the laser pulse owing to the intensity gradient in the focused laser beam. We will show here that the quiver motion of the electron is partially converted into centre-of-mass motion of the neutral atom, leading to a sub- stantial acceleration. This results in a measurable momentum trans- fer to the atom despite the short interaction time in the femtosecond range. Remarkably, the ponderomotive effect is typically estimated to be negligible for these conditions^16,17with, however, a few excep- tions¹⁸. We note that the investigation relies on the highly selective process of excitation of neutrals in a strong laser field, where kin- ematic effects are imparted only through the gradient of the laser field.

In the experiment we excite neutral He atoms in an effusive atomic beam using a perpendicularly intersecting focused laser beam. Using

the detection technique (see the Methods) we measure the distri- bution of excited He atoms on a detector as shown in Fig. 1. If, during the laser pulse, no momentum is transferred to the atoms, we would expect a slightly enlarged projected image of the (laser-intensity- dependent) distribution of excited atoms in the laser beam on the detector, that is, a distribution that extends along the laser beam direction (zaxis), typically within the Rayleigh length, but with a very narrow radial distribution (rDaxis) of the order of the size of the laser beam waist.

In Fig. 1a, however, we see a strikingly large radial distribution of excited atoms with a strong maximum in the laser focal plane (z50) that obviously stems from a deflecting radial force during the laser pulse. In Fig. 1b the cut along thezaxis (black curve) shows two maxima at roughly half the laser peak intensityI0/2, where the net production rate of excited helium atoms He*is apparently maxi- mum, whereas the He*signal atI0shows a pronounced minimum.

However, the loss of neutral excited atoms is largely due to their radial deflection. The full projection (red dashed curve) shows only a slight decrease in signal, indicating that even at the highest inten- sities He atoms are excited. The data are taken at a low beam target pressure of^5|10^{7mbar. The radial deflection is unchanged when we increase the target pressure by more than a factor of 30.

This excludes many-particle effects based on atom density or space charge as an origin of our observations. Furthermore, we emphasize that the radial distribution is unaltered whether the linear polariza- tion of the laser beam is in the direction of the atomic beam or perpendicular to it. In this respect the intensity-dependent force very much resembles the ponderomotive force acting on charged part- icles. The question arises whether we can conclude that the ponder- omotive force is responsible for the observed centre-of-mass motion of the neutral particle.

To shed light on the underlying process we first recall that the ponderomotive forceFpon a charged particle is given by (all equa- tions are in atomic units):

Fp~{ q²

4mv²+j jE0² ð1Þ

Here,mandqare the mass and the charge of the particle, respectively, E(r,t)5E0(r,t)expivtis the electric field,vis the field frequency andE0(r,t) is the slowly varying field amplitude. Hence, in view of our frustrated tunnel ionization model, both the ionic core and the electron experience a mass-dependent ponderomotive force during the laser pulse. As a consequence of the mass dependency, however, the ionic core remains practically unaffected while the electron experiences a non-negligible ponderomotive force. This, in turn, means that to the first approximation a ponderomotive force acts directly on the centre-of-mass motion of the atom and leaves the recapturing process unaffected. This can be shown more rigorously:

we derive the centre-of-mass motion from the Lorentz force (see the

1Max-Born-Institute, Max-Born-Strasse 2a, 12489 Berlin, Germany.²Institut fu¨r Optik und Atomare Physik, Technische Universita¨t Berlin, 10632 Berlin, Germany.

Vol 461|29 October 2009|doi:10.1038/nature08481

1261 Macmillan Publishers Limited. All rights reserved

©2009

Nature 461, 1261-1264 (29 October 2009)

29

A Rydberg atom escapes from an intense laser beam

proportional to 1/m U p for the nucleus negligible Atom pulled by an electron

非摂動論的であることのめやす

Order of magnitude and trend consistent

PEAK SUPPRESSION

U _p E ₀ ² 4m ³

e ²

U

530 nm 650 nm 910 nm 1082 nm

Gontier and Trahin 4.89×10

1.51×10

5.80×10

3.53×10

8.9×10

4.8×10

1.8×10

1.0×10

A measure of non-perturbativeness

11/14

非摂動論的？

Why non-perturbative at much lower intensity

なぜ、これよりずっと低い強度で非摂動論的になるのか？

Why non-perturbative at lower intensity for longer wavelength

Why low-order peaks are suppressed?

NUCLEAR COULOMBIC FORCE

LASER ELECTRIC FORCE

＝

a 0 e ² eE

4 ₀ a ² ₀

I = 3.51 10 ¹⁶ W/cm ²

？

31

Non-perturbative?

Explained by the ponderomotive energy ポンデロモーテ ィブエネルギーでよく説明できる。

レーザー場の影響 〜 原子核の影響

U _p I _p

Xe (I

p

¹³

²

= I _p 2U _p

Keldysh parameter

ケルディッシュ パラメーター

> 1 : Multi-photon regime

1 : Tunneling regime

= 1

非摂動論的であることのめやす

A measure of non-perturbativeness

Effect of the laser field

effect of the nucleus

11/14

トンネル電離

=

電子は、光子ではなく、電界を感じてる！

レーザー電界

原子核ポテン シャル 電子

トンネル効果

V (r, t) = e ² 4 ₀

1

r + ezE (t)

33

Tunneling ionization

The electron sees a field rather than photons!

Laser electric field

Nuclear potential e ^-

Laser electric field

Tunneling

トンネル電離

レーザー電場

電子 トンネル

効果 トンネル効果とは

原子核ポテン シャル

古典力学 量子力学

11/14

Change of ionization mechanism with laser intensity

€

I > 10

W/cm

€

I > 10

W/cm

€

I > 10

W/cm

Photon 光子

35

Change of ionization mechanism with laser intensity

I > 10

W/cm

I > 10

W/cm

I > 10

W/cm

Photon 光子 Electric Field 電界

11/14   No. 37

High-harmonic generation

高次高調波発生

高調波発生 (Harmonic generation)

線形光学効果（弱い光）

非線形光学効果（強い光）

€

ω

€

ω

€

ω

€

ω ,3 ω ,5 ω ,

結晶、ガス等(crystal, gas)

Material response is linear in light intensity 物質の応答が、入射光強度に比例

物質の応答が、入射光強度に非線形に依存

€

3 ω

€

5 ω

：３次高調波(3rd harmonic)

：５次高調波(5th harmonic)

波長変換

(frequency conversion)

€

D = ε

E + P

€

P = ε

[ χ

^E + χ

^E

+ χ

^E

+  ]

反転対称な媒質では、

€

χ

= 0

線形分極 linear polarization 非線形分極 (nonlinear)

∇ × ∇ × E = −µ

∂

D

∂ t

Linear optical effect

Nonlinear optical effect

for a medium with inversion symmetry

Nonlinear material response

11/14   No.

摂動論的高調波発生

(perturbative harmonic generation)

基底状態 電離

€

 ω

€

 ω

€

 ω

仮想準位

€

3  ω

基底状態 電離

€

 ω

€

 ω

€

 ω

仮想準位

€

5  ω

€

 ω

€

 ω

３次高調波 ５次高調波

次数が高くなるほど、発生効率は減少。

39

Harmonic order ↑ Efficiency ↓

3rd harmonic 5th harmonic

Ionization Ionization

Virtual level

Virtual level

Ground state Ground state

高次高調波発生

High-harmonic generation (HHG)

新しい極端紫外・軟エックス線光源として注目される。

New extreme ultraviolet (XUV) and soft X-ray source

discovered in 1987

! !

Intense laser pulse gas jet harmonics of high orders

Highly nonlinear optical process in which the frequency of laser light is converted

into its integer multiples. Harmonics of very high orders are generated.

11/14   No.

Harmonic spectrum 高調波スペクトル

41 How high orders?

Wahlström et al., Phys. Rev. A 48, 4709 (1993)

10 ¹⁵ W/cm ²

was raised up to 26 mJ, a maximal output energy exceeding 7 mJ was achieved at the signal wavelength near 1.4 ! m.

Temporal characterization of amplified OPA pulses was performed using a single-shot autocorrelation !AC" tech- nique. A typical AC trace is shown in the inset of Fig. 2.

Assuming a Gaussian pulse shape, the pulse width of 1.4 ! m pulse was evaluated to be 40 fs in full width at half maxi-

mum !FWHM", the energy of which corresponds to the red

filled circles in Fig. 3. The solid red line depicts the Fourier- transform-limited AC trace obtained from the amplified OPA spectrum. The measured pulse width was almost transform limited and the signal pulse width was shorter than 65 fs over the entire tuning range.

Using the developed high-energy 1.4 ! m OPA pulses, we have performed a proof-of-principle experiment on soft x-ray harmonic generation from an Ar gas target under a nonionized medium condition to exhibit the performance of our developed IR source. The 1.4 ! m IR pulses were fo- cused with f =250 mm CaF

lens and delivered into the tar- get vacuum chamber through a thin CaF

window. The Ar gas target was supplied by a 2 mm synchronized gas jet op- erating at 10 Hz. We used an imaging spectrometer set 530 mm away from the Ar gas target to measure the spec- trograph of the HH beam. The blue profile in Fig. 4 shows the measured HH spectrum of Ar driven by a 1.4 ! m pulse with a backing pressure of 10 atm. The focusing intensity was fixed to be 1.5 " 10

W / cm

at the interaction region in order to use a neutral Ar gas condition. Thus, the pump en- ergy of the 1.4 ! m pulse was set at 2 mJ with a beam diam- eter of 5 mm. We have generated 105 eV harmonics in the neutral Ar gas condition. We found an intensity minimum at around 50 eV in Ar spectrum. This minimum point matches closely the minimum observed in the photon ionization cross section of Ar due to the Cooper minimum.

As shown in the inset of Fig. 4, the almost perfect Gaussian profile of the HH suggests that there is no density disturbance due to ionization in the interaction region

. The white profile in the inset indi- cates the far-field spatial profile of a 90 eV harmonic beam.

The output beam divergence was measured to be # 5 mrad FWHM. This good beam quality indicates that a phase

matching condition would be substantially satisfied on the propagation axis of the pump pulse. The red profile shows the Ar harmonic spectrum driven by a 0.8 ! m pulse of which cutoff energy was measured to be approximately 48 eV. HH spectrum driven by a 1.4 ! m pulse was roughly two order magnitudes lower than that of driven by a 0.8 ! m pulse. The measured HH spectrum driven by a 1.4 ! m pulse shows a significant cutoff extension compared with that obtained with the 0.8 ! m driving field. This result reveals that the 1.4 ! m field generates photons having approximately two times higher energy than the 0.8 ! m field with the same intensity.

This photon energy’s difference is in good agreement with a predicted value from the cutoff formula.

In conclusion, we have developed a high-energy IR sources based on OPA to generate higher photon energy har- monic beams. Output energy exceeding 7 mJ with 40 fs pulse width was achieved at a signal wavelength near 1.4 ! m. Total output energy of 12 mJ was recorded with

#45% conversion efficiency. In addition, the measured Ar HH spectrum driven by a 1.4 ! m shows a significant cutoff extension exceeding 100 eV while the harmonic spatial pro- file is almost perfectly maintained. Our developed IR source is attractive not only for extending the HHG energy toward the kiloelectronvolts region but also for examining the en- ergy scaling of HHG under the phase matching condition.

1M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec, P. Corkum, U. Heinzmanns, M. Dreschers, and F. Krausz, Na-

ture !London" 414, 509 !2001".

2T. Sekikawa, A. Kosuge, T. Kanai, and S. Watanabe, Nature !London"

432, 605!2004".

3G. Sansone, E. Benedetti, F. Calegari, C. Vozzi, L. Avaldi, R. Flammini, L.

Poletto, P. Villoresi, C. Altucci, R. Velotta, S. Stagira, S. D. Silvestri, and M. Nisoli, Science 314, 443!2006".

4P. Tzallas, D. Charalambidis, N. A. Papadogiannis, K. Witte, and G. D.

Tsakiris,Nature !London" 426, 267!2003".

5Y. Nabekawa, T. Shimizu, T. Okino, K. Furusawa, H. Hasegawa, K. Ya- manouchi, and K. Midorikawa, Phys. Rev. Lett. 96, 083901!2006".

6Y. Nabekawa, H. Hasegawa, E. J. Takahashi, and K. Midorikawa, Phys.

Rev. Lett. 94, 043001!2005".

7E. Takahashi, Y. Nabekawa, T. Otsuka, M. Obara, and K. Midorikawa, Phys. Rev. A 66, 021802 !2002".

8E. Takahashi, N. Nabekawa, and K. Midorikawa, Opt. Lett. 27, 1920

!2002".

9P. B. Corkum, Phys. Rev. Lett. 71, 1994!1993".

10V. S. Yakovlev, M. Ivanov, and F. Krausz,Opt. Express 15, 15351!2007".

11E. Constant, D. Garzella, P. Breger, M. Mevel, C. Dorrer, C. L. Blanc, F.

Salin, and P. Agostini, Phys. Rev. Lett. 82, 1668 !1999".

12J. Tate, T. Auguste, H. G. Muller, P. Salieres, P. Agostini, and L. F. Di- Mauro,Phys. Rev. Lett. 98, 013901 !2007".

13P. Colosimo, G. Doumy, C. I. Blaga, J. Wheeler, C. Hauri, F. Catoire, J.

Tate, R. Chirla, A. M. March, G. G. Paulus, H. G. Muller, P. Agostini, and L. F. DiMauro, Nat. Phys. 4, 386!2008".

14C. P. Hauri, R. B. Lopez-Martens, C. I. Blaga, K. D. Schultz, J. Cryan, R.

Chirla, P. Colosimo, G. Doumy, A. M. March, C. Roedig, E. Sistrunk, J.

Tate, J. Wheeler, L. F. DiMauro, and E. P. Power, Opt. Lett. 32, 868

!2007".

15C. Vozzi, F. Calegari, E. Benedetti, S. Gasilov, G. Sansone, G. Cerullo, M.

Nisoli, S. D. Silvestri, and S. Stagira, Opt. Lett. 32, 2957 !2007".

16T. Fuji, N. Ishii, C. Y. Teisset, X. Gu, T. Metzger, A. Baltuska, N. Forget, D. Kaplan, A. Galvanauskas, and F. Krausz, Opt. Lett. 31, 1103 !2006".

17M. Nisoli, S. D. Silvestri, V. Magni, O. Svelto, R. Danielius, A. Piskars- kas, G. Valiulis, and A. Varanavicius, Opt. Lett. 19, 1973!1994".

18J. A. R. Samson and W. C. Stolte, J. Electron Spectrosc. Relat. Phenom.

123, 265!2002".

FIG. 4. !Color online"Experimentally obtained harmonic spectra in Ar. Red and blue profile depict the spectra with #₀=0.8!m pump and#₀=1.4 !m pump, respectively. Both HH spectra are normalized to the peak intensity.

The laser focused intensity is adjusted to generate HH under a neutral con- dition for both wavelengths. The inset shows a measured two dimensional harmonic spectrum image driven by 1.4 !m pump.

041111-3 Takahashi et al. Appl. Phys. Lett. 93, 041111 !2008"

Downloaded 04 Sep 2008 to 134.160.214.76. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

Takahashi et al., Appl. Phys. Lett. 93, 041111 (2008)

800 nm, 1.6×10

W/cm

Simulation

10^-8 10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰ 10¹ 10²

Harmonic intensity (arb. unit)

50 40

30 20

10 0

Harmonic order

800÷31= 26 nm

Only odd orders