advanced plasma and laser science - kenichi...

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/[email protected]

Advanced Plasma and Laser Scienceプラズマ・レーザー特論E

4. High-harmonic generation高次高調波発生

1

石川顕一


2

HARMONIC GENERATION高調波発生

€

ω

€

ω

€

ω

crystal, gas

Material response is linear in light intensity

：3次高調波(3rd harmonic)

frequency conversion

€

D = ε0E +P

€

P = ε0 χ(1)E + χ (2)E2 + χ (3)E 3 +![ ]

線形分極　linear polarization　　非線形分極　nonlinear

€

∇×∇×E = −µ0∂2D∂t2

Linear optical effect

Nonlinear optical effect

Nonlinear material response�, 2�, 3�, 4�, 5�, · · ·

2�：2次高調波(2nd harmonic)3�

.....

Process whereby integer multiples of the original light’s frequency (fundamental frequency) are generated.

線形光学効果

非線形光学効果

結晶等

結晶等

波長変換物質の応答が、入射光強度に非線形に依存

石川顕一


Even-order components vanish for a medium with inversion symmetry

3

E

P(E)

-E

P(-E) = -P(E)

P (E) = �0��(1)E + �(2)E2 + �(3)E3 + �(4)E4 + · · ·

�

P (�E) = �0��(1)E + �(2)E2 � �(3)E3 + �(4)E4 � · · ·

�

�P (E) = �0��(1)E � �(2)E2 � �(3)E3 � �(4)E4 � · · ·

�

�(2) = 0,�(4) = 0, · · ·

反転対称な物質では、偶数次はゼロ

石川顕一


摂動論的高調波発生PERTURBATIVE HARMONIC GENERATION

次数が高くなるほど、発生効率は減少。

基底状態

電離

€

!ω

€

!ω

€

!ω仮想準位

€

3!ω

３次高調波発生の概念図Ionization

Virtual level

Ground state

4

MTHG =�

h,i,f

��3 · D1h�1 · Dhi�1 · Dij�1 · Dj1

(3�1 � �h)(2�1 � �i)(�1 � �j)

+�1 · D1h�3 · Dhi�1 · Dij�1 · Dj1

(��1 � �h)(2�1 � �i)(�1 � �j)

+�1 · D1h�1 · Dhi�3 · Dij�1 · Dj1

(��1 � �h)(�2�1 � �i)(��1 � �j)

+�1 · D1h�1 · Dhi�1 · Dij�3 · Dj1

(��1 � �h)(�2�1 � �i)(�3�1 � �j)

遷移行列要素Transition matrix element3rd harmonic

order ↑ efficiency ↓

石川顕一


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

5

3rd harmonic 5th harmonic

Ionization Ionization

Virtual levelVirtual level

Ground state Ground state

次数が高くなるほど、発生効率は減少。

order ↑ efficiency ↓

石川顕一


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

6

discovered in 1987

OZSF$:$@E< O]/khHd7<;3=>

ZF$:$KG-/YQJQ��-0"'O]OUh(*MN,1#

mfP.OUh/<C9>E

�� !��

BD>$6=>5A

! BD>$ %\[/][*jZS-g^&! 6=>5A %XO][/g^&

nW_P.i`edV-0bR-+.)TcIT/ML

inW_P.i`edV/lNkhH"O4?E8$"2>a

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.

石川顕一


HARMONIC SPECTRUM 高調波スペクトルHow high orders?

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

1015 W/cm2

was raised up to 26 mJ, a maximal output energy exceeding7 mJ was achieved at the signal wavelength near 1.4 !m.

Temporal characterization of amplified OPA pulses wasperformed 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 !mpulse was evaluated to be 40 fs in full width at half maxi-mum !FWHM", the energy of which corresponds to the redfilled circles in Fig. 3. The solid red line depicts the Fourier-transform-limited AC trace obtained from the amplified OPAspectrum. The measured pulse width was almost transformlimited and the signal pulse width was shorter than 65 fsover the entire tuning range.

Using the developed high-energy 1.4 !m OPA pulses,we have performed a proof-of-principle experiment on softx-ray harmonic generation from an Ar gas target under anonionized medium condition to exhibit the performance ofour developed IR source. The 1.4 !m IR pulses were fo-cused with f =250 mm CaF2 lens and delivered into the tar-get vacuum chamber through a thin CaF2 window. The Argas target was supplied by a 2 mm synchronized gas jet op-erating at 10 Hz. We used an imaging spectrometer set530 mm away from the Ar gas target to measure the spec-trograph of the HH beam. The blue profile in Fig. 4 showsthe measured HH spectrum of Ar driven by a 1.4 !m pulsewith a backing pressure of 10 atm. The focusing intensitywas fixed to be 1.5"1014 W /cm2 at the interaction region inorder 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 theneutral Ar gas condition. We found an intensity minimum ataround 50 eV in Ar spectrum. This minimum point matchesclosely the minimum observed in the photon ionization crosssection of Ar due to the Cooper minimum.18 As shown in theinset of Fig. 4, the almost perfect Gaussian profile of the HHsuggests that there is no density disturbance due to ionizationin the interaction region7. 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 mradFWHM. This good beam quality indicates that a phase

matching condition would be substantially satisfied on thepropagation axis of the pump pulse. The red profile showsthe Ar harmonic spectrum driven by a 0.8 !m pulse of whichcutoff energy was measured to be approximately 48 eV. HHspectrum driven by a 1.4 !m pulse was roughly two ordermagnitudes lower than that of driven by a 0.8 !m pulse. Themeasured HH spectrum driven by a 1.4 !m pulse shows asignificant cutoff extension compared with that obtained withthe 0.8 !m driving field. This result reveals that the 1.4 !mfield generates photons having approximately two timeshigher energy than the 0.8 !m field with the same intensity.This photon energy’s difference is in good agreement with apredicted value from the cutoff formula.

In conclusion, we have developed a high-energy IRsources based on OPA to generate higher photon energy har-monic beams. Output energy exceeding 7 mJ with 40 fspulse width was achieved at a signal wavelength near1.4 !m. Total output energy of 12 mJ was recorded with#45% conversion efficiency. In addition, the measured ArHH spectrum driven by a 1.4 !m shows a significant cutoffextension exceeding 100 eV while the harmonic spatial pro-file is almost perfectly maintained. Our developed IR sourceis attractive not only for extending the HHG energy towardthe kiloelectronvolts region but also for examining the en-ergy scaling of HHG under the phase matching condition.7

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, andM. 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, andL. 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. Redand blue profile depict the spectra with #0=0.8 !m pump and #0=1.4 !mpump, 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 dimensionalharmonic 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, 1.6×1014 W/cm2

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Harm

onic

inte

nsity (

arb

. unit)

50403020100

Harmonic order

800÷31= 26 nm

Only odd orders 奇数次のみ

7

gas is a medium of inversion symmetry

TDSE Simulation

期待媒質は反転対称

石川顕一


8

Even up to 1.6 keV, > 5000 ordersalmost x-ray!

a new type of laser-‐‑‒based radiation sourceレーザーをベースにした新しいタイプの放射線源

Popmintchev et al., Science 336, 1287 (2012)

石川顕一


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

1015 W/cm2 TDSE Simulation

プラトー(plateau)：Efficiency does NOT decrease with increasing harmonic order. 次数が上がっても強度が落ちない。

カットオフ(cutoff)：Maximum energy of harmonic photons

摂動論的には解釈できない

plateau

cutoffplateau

cutoff

ポンデロモーティブエネルギーUp(eV) =

e2E20

4m�2= 9.3� 10�14I(W/cm2)�2(µm)

800 nm, 1.6×1014 W/cm2

9

Ec � Ip + 3.17Up

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Harm

onic

inte

nsity (

arb

. unit)

50403020100

Harmonic order

Plateau and cutoff プラトーとカットオフ- remarkable feature of high-harmonic generation -

ponderomotive energyThese features cannot be understood as perturbative harmonic generation.

石川顕一


高次高調波発生のメカニズム３ステップモデル 3-STEP MODEL

Paul B. Corkum, Phys. Rev. Lett. 71, 1994 (1993)

10

K. C. Kulander et al., in Super-Intense Laser-Atom Physics, NATO ASI Ser. B, Vol. 316, p. 95 (1993)

Paul B. Corkum

→

tunneling ionization

recombination

photon emission (HHG)

Laser field レーザー電場

Semiclassical electron motion

electron

E(t) = E0 cos �t

トンネル電離電場中の古典的加速

再結合

発光（高次高調波）

石川顕一


3-STEP MODEL OF HHG高次高調波発生の３ステップモデル

Ekin = 2Up(sinφ − sinφ0)2

再衝突

11

時刻でイオン化。原点に初速ゼロで出現t0

mz = �eE0 cos �t z(t0) = 0 z(t0) = 0

Normalization 規格化 �0 = �t0� = �t

z =E0

�2[(cos �� cos �0) + (�� 0) sin�0]

Ionization at with vanishing initial velocity at origint = t0

Recombination at z = 0φ = φret(φ0) , which satisfies

→


recombination




electron

E(t) = E0 cos �t

トンネル電離

電場中の古典的加速

再結合


石川顕一


TIME (PHASE) OF RECOMBINATION再衝突時刻

350

300

250

200

150

100

50

0

Phas

e of

reco

mbi

natio

n (p

hi_r

)

150100500Phase of electron release (phi0)

12

z = 0

(cos �)�|�0 =cos �ret � cos �0

�ret � �0

(cos �ret � cos �0) + (�ret � �0) sin�0 = 0

-1.0

-0.5

0.0

0.5

1.0

36032028024020016012080400

phi (degree) �ret�0

イオン化時刻と再衝突時刻の関係

の場合再結合できない

�

2< �0 < �

no recombination for

phase of ionization vs phase of recombination

石川顕一


Simple explanation of the cut-off lawカットオフ則のシンプルな説明

3

2

1

0Ele

ctr

on k

inetic e

nerg

y (

in U

p)

360270180900

Phase (degrees)

-1

0

1

Fie

ld (in

E0 )

ionization recombination

short

lon

g

short

lon

g

field

同じ高調波次数（光子エネルギー）に対応するイオン化時刻と再結合時刻のペアは２つある。

short trajectorylong trajectory

13

Ec = Ip + 3.17Up

再結合時の運動エネルギーの最大値

cut-off カットオフ3.17Up

There is the maximum kinetic energy which is classically allowed.

There are two pairs of ionization and recombination times which contribute to the same harmonic energy.

石川顕一


WHY DO HARMONIC SPECTRA CONSIST OF DISCRETE PEAKS? なぜ、高次高調波スペクトルは離散的なのか？

FIG. 4. !Color online" Experimentally obtained harmonic spectra in Ar. Red

トンネル電離と高調波の発生は、レーザーの半周期ごとに起こる。

14

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

Harm

onic

inte

nsity (

arb

. unit)

50403020100

Harmonic order

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

→


recombination




electron

E(t) = E0 cos �t

トンネル電離電場中の古典的加速

再結合


This is repeated every half cycle with an alternating phase

石川顕一


the emission of a pulse of a high-order harmonic field everyhalf-cycle period. In this model, the trajectory of theelectron, moving along the direction of the electric fieldof the fundamental laser, determines the phase of theemitted harmonic field. Therefore, our finding of the!-flipped phase in the attosecond pulse train verifies thatharmonic pulses are certainly from electrons detachedfrom opposite sides at every half-cycle period of the fun-damental laser field.

Note that the clear observation of phase locking in theactual experiment is due to the small number of interfer-ence fringes emerging from near single-cycle pulses in theAPT. We estimated the duration of the pulse by comparingthe calculated results of the interferometric autocorrelationof the harmonic fields with group-delay dispersion (GDD)and its Fourier transform to the measured results. Details ofthis calculation will be reported elsewhere. We plot thecalculated autocorrelation trace and its magnitude squareof Fourier amplitude as solid blue curves in Figs. 2(a) and2(b), respectively. Setting the GDD to 1:3! 10"32 #s2$,which is the measured numeric value with the separateexperiment [11], there is fairly good agreement betweenthe experimental observation and the calculated result ofboth autocorrelation and Fourier transform. The increasedGDD to 2:0! 10"32 #s2$ reduced the height of the enve-lope with the remaining interference fringe in the autocor-relation so that the intensity at 2"f in the Fourier transformis almost equal to that at 11"f. Therefore, 1:3! 10"32 #s2$is the reasonable GDD in practice, although we might takeinto account the intrinsic response of N2 to two-photonabsorption, as was mentioned in Refs. [27,28] for rare-gasatoms, in order to determine GDD in more detail.

The envelope and optical field of the attosecond pulsetrain, evaluated from the measured intensity ratio of theharmonic fields from the 9th to 19th orders and the esti-mated GDD, are shown in Fig. 3. The duration of the pulseenvelope is 320 attoseconds in full width at half maximum,which corresponds to only an %1:3-cycle period of theprincipal carrier frequency of the 11th harmonic field.Thus, we can clearly see the relative phase of interferencefringes to pulse envelopes.

In conclusion, we have shown clear evidence of thelocked phase in an attosecond pulse train by interferomet-ric autocorrelation. The detection of fringes that emergedfrom the interference of a pulse sequence in the XUV rangeenables us to determine the phase difference betweenpulses, so that we would obtain the instantaneous shapeof the optical field in the attosecond pulse train by combin-ing this result with those obtained by other techniques formeasuring the chirp [9,11,29] and by using a driving laserfield whose absolute phase is stabilized [1,2]. Thus, anoptical synthesizer of XUV light comes into the realm ofreality.

We thank Dr. E. J. Takahashi at RIKEN for his support ingenerating intense harmonic fields. This study was finan-cially supported by MEXT through a Grant-in-Aid forScientific Research on Priority Areas, for YoungScientists (A) and for Young Scientists (B).

*Electronic address: [email protected][1] A. Baltuska et al., Nature (London) 421, 611 (2003).[2] R. Kienberger et al., Nature (London) 427, 817 (2004).[3] J. Reichelt et al., Opt. Commun. 172, 59 (1999).[4] H. R. Telle et al., Appl. Phys. B 69, 327 (1999).[5] D. H. Jones et al., Science 288, 635 (2000).[6] C. Gohle et al., Nature (London) 436, 234 (2005).[7] R. J. Jones et al., Phys. Rev. Lett. 94, 193201 (2005).[8] P. Antoine et al., Phys. Rev. Lett. 77, 1234 (1996).[9] P. M. Paul et al., Science 292, 1689 (2001).

[10] P. Tzallas et al., Nature (London) 426, 267 (2003).[11] Y. Nabekawa et al., Phys. Rev. Lett. 96, 083901 (2006).[12] T. Sekikawa et al., Nature (London) 432, 605 (2004).[13] K. Varju et al., Phys. Rev. Lett. 95, 243901 (2005).[14] E. J. Takahashi et al., Opt. Lett. 27, 1920 (2002).[15] E. Takahashi et al., Phys. Rev. A 68, 023808 (2003).[16] Y. Nabekawa et al., Phys. Rev. Lett. 94, 043001 (2005).[17] H. Hasegawa et al., Laser Phys. 15, 812 (2005).[18] E. J. Takahashi et al., Opt. Lett. 29, 507 (2004).[19] E. J. Takahashi et al., IEEE J. Sel. Top. Quantum Electron.

10, 1315 (2004).[20] K. Hoshina et al., J. Phys. B 39, 813 (2006).[21] T. E. Glover et al., Phys. Rev. Lett. 76, 2468 (1996).[22] E. W. Thulstrup and A. Andersen, J. Phys. B 8, 965 (1975).[23] R. W. Wetmore and R. K. Boyd, J. Phys. Chem. 90, 5540

(1986).[24] In fact, the fringes corresponding to the carrier frequencies

of the harmonic fields in the IAC trace should originatefrom the term ~S&'(&!2&N'1(; 0(~S&'() &!2&N'1(; #(, where themode-resolved correlation amplitude ~S&'(&!2&N'1(; #( wasdefined as Eq. (2) in Ref. [11]. The factor ei&2n'1(!f# in~S&'(&!2&N'1(; #( necessarily yields the odd parity of thefringes to the translation of the delay with &q' 1=2(Tf.

[25] P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993).[26] M. Lewenstein et al., Phys. Rev. A 49, 2117 (1994).[27] T. Nakajima and S. Watanabe, Phys. Rev. A 70, 043412

(2004).[28] L. A. A. Nikolopoulos et al., Phys. Rev. Lett. 94, 113905

(2005).[29] T. Sekikawa et al., Phys. Rev. Lett. 88, 193902 (2002).

-3 -2 -1 0 1 2 3

time [fs]

FIG. 3 (color online). Estimated intensity profile of the atto-second pulse train (dark-green curve with hatched area) and theoptical field (dark-blue curve).

PRL 97, 153904 (2006) P H Y S I C A L R E V I E W L E T T E R S week ending13 OCTOBER 2006

153904-4

Nabekawa et al., Phys. Rev. Lett. 97, 153904 (2006)

実験的にも観測されている高調波の電場波形の概念図

f(t) = �f(t� �/�0)基本波の奇数倍の周波数成分のみを含む。

15

-2 -1 0 1 2Fundamental optical cycle

harmonic field harmonic intensity fundamental field

laser cycle レーザーの１周期(2.7 fs)

Harmonic electric fieldExperimentally measured

Spectrum only consists of odd harmonics.

石川顕一


１つの次数のみが存在するときの光電界

16

Eh(t) = Eq cos(qω + φq) = E2n+1 cos[(2n + 1)ω + φ2n+1]

1.0

0.5

0.0

-0.5

-1.0

電場

-2 -1 0 1 2時間（フェムト秒）

1.0

0.5

0.0

-0.5

-1.0

電場

-0.40 -0.30 -0.20時間（フェムト秒）

Continuous wave (no pulse)連続波（パルスではない）

Elect

ric fi

eldSINGLE FREQUENCY COMPONENT

Time (fs) Time (fs)

石川顕一


train of repeated pulses

複数の次数（奇数次）が混在するときの光電界

17

Eh(t) =!

q

Eq cos(qω + φq) =!

q

E2n+1 cos[(2n + 1)ω + φ2n+1]

等間隔の周波数成分等間隔のパルス列

MULTIPLE (ODD) HARMONIC COMPONENTS

• attosecond pulse train (APT)アト秒パルス列になっている

• bursts repeated every half cycle of the fundamental laserパルスの間隔は、基本波の半周期

• adjacent pulses have an opposite phase隣り合うパルスは位相が反転

1.0

0.5

0.0

-0.5

-1.0

電場

-2 -1 0 1 2時間（フェムト秒）Time (fs)

Elec

tric

fiel

d

equispaced frequency components

We don’t need “photons” to understand harmonic generation

石川顕一


DISCRETE PEAKS OF ODD HARMONICS CAN BE INTERPRETED IN TWO WAYS.高次高調波の奇数次のみを含む離散的なピークは、二通りに解釈できる。

18

•Integer number of photon energy + inversion symmetry光子エネルギーの整数倍＋反転対称性

•Light emission repeated every half cycle (with alternating phase)基本波の半サイクルごとに、反対の位相で光放出

石川顕一


次数によって高調波の発生時刻が異なる

19

低次が先に高次が後で発生する。

ポジティブチャープ

350

300

250

200

150

100

Phas

e of

reco

mbi

natio

n (p

hi_r

)

3.02.52.01.51.00.50.0Electron energy (in Up)

short trajectory

Long trajectory

Mairesse et al., Science 302, 1540 (2003)

ショートトラジェクトリーの場合

where !O is the selected spectral bandwidth. The attosecond pulses correspondingto the first trajectory are thus positively chirped (the low orders are emitted beforethe high orders), while those corresponding to the second trajectory exhibit anegative chirp.

Figure 4. XFROGmeasurements [20]. The images on the right and left sides of the figurerepresent the photoelectron signal as a function of the delay between pump and probeand energy of the photoelectron (vertical). On the left, several sidebands with order22, 20, 18, 16, 14 from the top to the bottom obtained for a Fourier-transform-limitedlaser pulse are shown. On the right, the sideband 18 is shown for three fundamentalchirps: bfund ¼ 1:2, 0, "0:8# 1027 s"2. The graph in the middle shows the measured(filled symbols) and simulated (open symbols) chirp rates as a function of order inthe three different fundamental chirp cases (see text). The black straight line is a fit tothe simulated points; the blue and red lines are obtained from the latter by usingequation (7).

0 500 1000 1500 20000

10

20

30

40

Inte

nsity

(ar

b.un

its)

Time (as)

Figure 5. Attosecond pulse train corresponding to the superposition of groups of 5consecutive harmonics generated in neon at 3:8# 1014 Wcm"2: harmonics 25 to 33(red), 35 to 43 (green), 45 to 53 (blue), 55 to 63 (purple). The train correspondingto the full available spectral width (25 to 69) is shown by the yellow-filled curve.Dots represent the absolute value of the laser electric field [21].

Frequency chirp of harmonic and attosecond pulses 387

Ne

測定

理論

Varju et al., J. Mod. Opt. 52, 379 (2005)

3.8×1014 W/cm2

高調波放出時刻

Time of emission depends on harmonic order

measurement

theory

Positive chirp

Higher-order components emitted later

For the case of short trajectory

Amazing predictive power of the 3-step model!

石川顕一


20

350

300

250

200

150

100

Phas

e of

reco

mbi

natio

n (p

hi_r

)

3.02.52.01.51.00.50.0Electron energy (in Up)

short trajectory

Long trajectory

K. L. Ishikawa, “High-harmonic generation” in Advances in Solid-State Lasers, ed. by M. Grishin (INTECH, 2010) 439-464

次数によって高調波の発生時刻が異なるTime of emission depends on harmonic order

TDSE simulation

Positive chirpfor the short trajectory

for the long trajectory

Negative chirp


高次高調波発生の量子論

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Lewenstein et al., Phys. Rev. A 49, 2117 (1994)

Quantum theory of high-harmonic generation

Lewenstein model

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高強度場近似

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•励起状態の寄与を無視　The contribution of all the excited bound

states can be neglected.

•連続状態に対する原子のポテンシャルの効果を無視（連続状態を平面波で近似）The effect of the atomic potential on

the motion of the continuum electron can be neglected.

•基底状態の減少を無視　The depletion of the ground state can be

neglected.

Strong-field approximation (SFA)

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双極子モーメント Time-dependent dipole moment

i��(r, t)

�t=

��1

2�2 + V (r) + zE(t)

��(r, t)

x(t) � ��(r, t) | z | �(r, t)�

x(t) = i

� t

��dt�

�d3p d�(p + A(t)) · exp[�iS(p, t, t�)] · E(t�)d(p + A(t�)) + c.c.

= i

� t

��dt�

�d3p ��(r)eiIpt|z|p + A(t)� exp

��i

� t

t�dt��

�[p + A(t��)]2

2

��p + A(t�)|zE(t�)|�(r)eiIpt�� + c.c.

イオン化再結合レーザー場中での運動

= i

� t

��dt�

�d3p ��(r)eiIpt|z|p + A(t)� exp

��i

� t

t�dt��

�[p + A(t��)]2

2

��p + A(t�)|zE(t�)|�(r)eiIpt�� + c.c.

イオン化再衝突電子波束

遷移双極子行列要素transition dipole半古典的作用積分 S(p, t, t�) =

� t

t�dt��

�[p + A(t��)]2

2+ Ip

�

semiclassical action

recombination motion in the laser field ionization3-step model

ionizationrecolliding electron wave packetDipole moment between the recolliding wave packet and the ground state

再衝突電子波束と基底状態との間の双極子モーメント

transition dipole

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高調波スペクトル＝双極子モーメントのフーリエ変換

x(�h) = i

� �

��dt

� t

��dt�

�d3p d�(p + A(t)) · exp[i�ht� iS(p, t, t�)] · E(t�)d(p + A(t�)) + c.c.

５重積分

鞍点解析cf. path integral 経路積分

five-dimensional integral

saddle-point analysis

HARMONIC SPECTRUM= FOURIER TRANSFORM OF DIPOLE MOMENT

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saddle-point analysis (SPA)

➡ physically corresponds to the 3-step model

[p + A(t�)]2

2= �Ip

[p + A(t)]2

2+ Ip = �h

tunneling ionization� t

t�[p + A(t��)]dt�� = 0

harmonic photon energy= kinetic energy at recombination + ionization potential

recombines at the location of ionization

Saddle-point equations

x(�h) =�

s

��

� + i2 (ts � t�s)

�3/2i2��

det S��(t, t�)|sd�(ps + A(ts))

� exp[i�hts � iS(ps, ts, t�s)]E(t�s)d(ps + A(t�s)),

solutions 解 → trajectories トラジェクトリーt’ time of ionizationt time of recombinationトンネル電離

イオン化と再結合の位置が同じ

高調波の光子エネルギー＝再結合時の運動エネルギー＋イオン化ポテンシャル

３ステップモデルに物理的に対応

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Example of saddle-point solutions 鞍点解の例

• ３ステップモデルは、量子力学的なLewensteinモデルのよい近似になっている。→ ３ステップモデルの成功の理由

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E(t) = E0 cos �t Ar (Ip = 15.7596 eV) 1.6� 1014 W/cm2

破線は３ステップモデルの解

「トンネル時間」に対応すると解釈されている

Ec = 3.17Up + gIp

(g � 1.3)

cutoff カットオフ

の実部（上）と虚部（下）� = �t�� = �t�Real part (top) and imaginary part (bottom) of

dashed lines: 3-step model

interpretted as “tunnling time”

‣ The 3-step model is a good approximation to the quantum-mechanical Lewenstein model → Success of the 3-step model

advanced plasma and laser science - kenichi...

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