4. High-harmonic generation

Transcription

1 Kenichi Ishikawa () hp://ishiken.free.fr/english/lecure.hml Advanced Plasma and Laser Science E 4. High-harmonic generaion 1

2 HARMONIC GENERATION Process whereby ineger muliples of he original ligh s frequency (fundamenal frequency) are generaed. 高調波発生 crysal, gas Linear opical effec 線形光学効果 ω Maerial response is linear in ligh inensiy 結晶等 ω Nonlinear opical effec 結晶等 非線形光学効果 ω,2,3,4,5, Nonlinear maerial response frequency conversion 物質の応答が 入射光強度に非線形に依存 波長変換 D = ε0 E + P P = ε0 [ χ (1) E + χ (2) E 2 + χ (3) E 3 +!] 非線形分極 nonlinear 線形分極 linear polarizaion 2 2次高調波(2nd harmonic) 3 3次高調波(3rd harmonic)... 2D E = µ 石川顕一

3 Even-order componens vanish for a medium wih inversion symmery E P(E) -E P(-E) = -P(E) P (E) = 0 (1) E + (2) E 2 + (3) E 3 + (4) E 4 + P ( E) = 0 (1) E + (2) E 2 (3) E 3 + (4) E 4 P (E) = 0 (1) E (2) E 2 (3) E 3 (4) E 4 (2) =0, (4) =0, 3

4 PERTURBATIVE HARMONIC GENERATION Transiion marix elemen M THG = h,i,f 3 D 1h 1 D hi 1 D ij 1 D j1 (3 1 h )(2 1 i )( 1 j )!ω!ω!ω 3!ω + 1 D 1h 3 D hi 1 D ij 1 D j1 ( 1 h)(2 1 i )( 1 j ) + 1 D 1h 1 D hi 3 D ij 1 D j1 ( 1 h)( 2 1 i )( 1 j) 1 D 1h 1 D hi 1 D ij 3 D j1 + ( 1 h)( 2 1 i )( 3 1 j ) order efficiency 4

5 (PERTURBATIVE HARMONIC GENERATION)!ω!ω!ω 3!ω!ω!ω!ω!ω!ω 5!ω order efficiency 5

6 高次高調波発生 HIGH-HARMONIC GENERATION (HHG) discovered in 1987 Inense laser pulse gas je harmonics of high orders Highly nonlinear opical process in which he frequency of laser ligh is convered ino is ineger muliples. Harmonics of very high orders are generaed.!! 新しい極端紫外 軟エックス線光源として注目される New exreme ulraviole (XUV) and sof X-ray source 6 石川顕一

7 How high orders? HARMONIC SPECTRUM 高調波スペクトル Wahlsröm e al., Phys. Rev. A 48, 4709 (1993) Takahashi e al. Takahashi e al., Appl. Phys. Le. 93, (2008) Harmonic inensiy (arb. uni) 15 W/cm FIG. 4.!Color online" Experimenally obained harmonic specra in Ar. Red and blue profile depic he specra wih #0 = 0.8!m pump and #0 = 1.4!m pump, respecively. Boh HH specra are normalized o he peak inensiy. The laser focused inensiy is adjused o generae HH under a neural condiion for boh wavelenghs. The inse shows a measured wo dimensional harmonic specrum image driven by 1.4!m pump. 800 nm, W/cm2 Only odd orders 奇数次のみ TDSE Simulaion Harmonic order = 26 was raised up o 26 mj, a maximal oupu energy exceeding 7 mj was achieved a he signal wavelengh near 1.4!m. Temporal characerizaion of amplified OPA pulses was performed using a single-sho auocorrelaion!ac" echnique. A ypical AC race is shown in he inse of Fig. 2. Assuming a Gaussian pulse shape, he pulse widh of 1.4!m nmpulse was evaluaed o be 40 fs in full widh a half maxi期待媒質は反転対称 mum!fwhm", he energy of which corresponds o he red 7 circles in Fig. 3. The solid red line depics he Fourierfilled gas is a medium of inversion symmery maching cond propagaion ax he Ar harmon cuoff energy w specrum drive magniudes low measured HH significan cuo he 0.8!m dr field generae higher energy This phoon en prediced valu In conclu sources based monic beams. pulse widh w 1.4!m. Toal #45% conver HH specrum exension exce file is almos p is aracive no he kiloelecro ergy scaling o 1 M. Henschel, R T. Brabec, P. Co 石川顕一 ure!london" 4

8 almos x-ray! Popminchev e al., Science 336, 1287 (2012) a new ype of laser- based radiaion source レーザーをベースにした新しいタイプの放射線源 8

9 Plaeau and cuoff プラトーとカットオフ - remarkable feaure of high-harmonic generaion Wahlsröm e al., Phys. Rev. A 48, 4709 (1993) Harmonic inensiy (arb. uni) plaeau cuoff 15 W/cm nm, W/cm2 plaeau cuoff TDSE Simulaion Harmonic order プラトー(plaeau) Efficiency does NOT decrease wih increasing harmonic order. 次数が上がっても強度が落ちない カットオフ(cuoff) Maximum energy of harmonic phoons e2 E Up (ev) = = 9.3 I(W/cm ) (µm) Ec Ip Up 4m 2 ポンデロモーティブエネルギー ponderomoive energy These feaures canno be undersood as perurbaive harmonic generaion. 摂動論的には解釈できない 9 石川顕一

10 3-STEP MODEL Laser field E() =E 0 cos recombinaion phoon emission (HHG) elecron unneling Semiclassical elecron moion ionizaion Paul B. Corkum, Phys. Rev. Le. 71, 1994 (1993) K. C. Kulander e al., in Super-Inense Laser-Aom Physics, NATO ASI Ser. B, Vol. 316, p. 95 (1993) Paul B. Corkum

11 3-STEP MODEL OF HHG Ionizaion a = 0 wih vanishing iniial velociy a origin 0 m z = ee 0 cos ż( 0 )=0 z( 0 )=0 Normalizaion = 0 = 0 z = E 0 2 [(cos cos 0)+( 0) sin 0 ] E kin = 2U p (sin φ sin φ 0 ) 2 Recombinaion a φ = φ re (φ 0 ), which saisfies z = 0 Laser field E() =E 0 cos recombinaion phoon emission (HHG) elecron unneling ionizaion Semiclassical elecron moion 11

12 TIME (PHASE) OF RECOMBINATION z = 0 (cos re cos 0 )+( re 0 ) sin 0 =0 (cos ) 0 = cos re cos 0 re 0 phase of ionizaion vs phase of recombinaion Phase of recombinaion (phi_r) no recombinaion for 2 < 0 < phi (degree) 240 re Phase of elecron release (phi0) 12

13 Simple explanaion of he cu-off law Elecron kineic energy (in U p ) ionizaion long shor 90 shor 180 Phase (degrees) field recombinaion 270 long Field (in E 0 ) There is he maximum kineic energy which is classically allowed. 3.17U p cu-off E c = I p +3.17U p There are wo pairs of ionizaion and recombinaion imes which conribue o he same harmonic energy. shor rajecory long rajecory 13

14 WHY DO HARMONIC SPECTRA CONSIST OF DISCRETE PEAKS? Takahashi e al., Appl. Phys. Le. 93, (2008) E() =E 0 cos Laser field recombinaion phoon emission (HHG) elecron unneling ionizaion Semiclassical elecron moion FIG. 4. Color 2 online Experimenally obained harmonic specra in Ar. Red 1 Harmonic inensiy (arb. uni) Harmonic order This is repeaed every half cycle wih an alernaing phase 14

15 Harmonic elecric field harmonic field harmonic inensiy fundamenal field Experimenally measured Nabekawa e al., Phys. Rev. Le. 97, (2006) ime [fs] Fundamenal opical cycle f() = f( / 0 ) laser cycle (2.7 fs) Specrum only consiss of odd harmonics. 15

16 SINGLE FREQUENCY COMPONENT E h () = E q cos(qω + φ q ) = E 2n+1 cos[(2n + 1)ω + φ 2n+1 ] Elecric field 電場 電場 時間 ( フェムト秒 ) Time (fs) 時間 ( フェムト秒 ) Time (fs) 16

17 MULTIPLE (ODD) HARMONIC COMPONENTS E h () = q E q cos(qω + φ q ) = q E 2n+1 cos[(2n + 1)ω + φ 2n+1 ] Elecric field 電場 aosecond pulse rain (APT) burss repeaed every half cycle of he fundamenal laser 時間 ( フェムト秒 )Time (fs) equispaced frequency componens adjacen pulses have an opposie phase rain of repeaed pulses We don need phoons o undersand harmonic generaion 17

18 DISCRETE PEAKS OF ODD HARMONICS CAN BE INTERPRETED IN TWO WAYS. Ineger number of phoon energy + inversion symmery Ligh emission repeaed every half cycle (wih alernaing phase) 18

19 Time of emission depends on harmonic order Phase of recombinaion (phi_r) Long rajecory shor rajecory W/cm 2 measuremen heory Ne Elecron energy (in Up) Higher-order componens emied laer 19 Inensiy (arb.unis) Amazing predicive power of he 3-sep model! Time (as) Mairesse e al., Science 302, 1540 (2003) Varju e al., J. Mod. Op. 52, 379 (2005)

20 Time of emission depends on harmonic order Phase of recombinaion (phi_r) Long rajecory shor rajecory TDSE simulaion Elecron energy (in Up) K. L. Ishikawa, High-harmonic generaion in Advances in Solid- Sae Lasers, ed. by M. Grishin (INTECH, 20)

21 Quanum heory of high-harmonic generaion Lewensein model 21 Lewensein e al., Phys. Rev. A 49, 2117 (1994)

22 Srong-field approximaion (SFA) The conribuion of all he excied bound saes can be negleced. The effec of he aomic poenial on he moion of he coninuum elecron can be negleced. The depleion of he ground sae can be negleced. 22

23 i (r,) = V (r)+ze() (r,) Time-dependen dipole momen x() (r,) z (r,) = i 3-sep model = i d d 3 p (r)e ii p z p + A() exp i d [p + A( )] 2 2 p + A( ) ze( ) (r)e ii p recombinaion moion in he laser field ionizaion d d 3 p (r)e ii p z p + A() exp i Dipole momen beween he recolliding wave packe and he ground sae d [p + A( )] 2 recolliding elecron wave packe 2 p + A( ) ze( ) (r)e ii p ionizaion +c.c. +c.c. x() =i semiclassical acion d d 3 p d (p + A()) exp[ is(p,,)] E( )d(p + A( )) + c.c. ransiion dipole ransiion dipole S(p,,)= d [p + A( )] 2 + I p 2 23

24 HARMONIC SPECTRUM = FOURIER TRANSFORM OF DIPOLE MOMENT ˆx( h )=i d d d 3 p d (p + A()) exp[i h is(p,,)] E( )d(p + A( )) + c.c. five-dimensional inegral saddle-poin analysis cf. pah inegral 24

25 saddle-poin analysis (SPA) Saddle-poin equaions soluions rajecories [p + A( )] 2 2 [p + A()] 2 2 = I p [p + A( )]d =0 + I p = h unneling ionizaion recombines a he locaion of ionizaion ime of ionizaion ime of recombinaion harmonic phoon energy = kineic energy a recombinaion + ionizaion poenial ˆx( h )= s + i 2 ( s s ) 3/2 i2 de S (, ) s d (p s + A( s )) exp[i h s is(p s, s, s )]E( s )d(p s + A( s )), physically corresponds o he 3-sep model 25

26 Example of saddle-poin soluions Real par (op) and imaginary par (boom) of E() =E 0 cos Ar (Ip = ev) W/cm 2 = = dashed lines: 3-sep model cuoff E c =3.17U p + gi p (g 1.3) inerpreed as unnling ime The 3-sep model is a good approximaion o he quanummechanical Lewensein model Success of he 3-sep model Lewensein 26