4. High-harmonic generation
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1 Advanced Laser and Photn Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) Kenichi Ishikawa () Advanced Laser and Photon Science E 4. High-harmonic generation 1
2 HARMONIC GENERATION Linear optical effect ω Nonlinear optical effect ω, 2, 3, 4, 5, D = ε 0 E + P [ ] P = ε 0 χ (1) E + χ (2) E 2 + χ (3) E 3 +! E = µ 0 2 D t 2 Process whereby integer multiples of the original light s frequency (fundamental frequency) are generated. ω
3 Even-order components vanish for a medium with inversion symmetry 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 Transition matrix element 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 5!ω!ω!ω order efficiency 5
6 HIGH-HARMONIC GENERATION (HHG) 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. 6
7 How high orders? HARMONIC SPECTRUM Wahlström et al., Phys. Rev. A 48, 4709 (1993) Takahashi et al., Appl. Phys. Lett. 93, (2008) W/cm 2 Harmonic intensity (arb. unit) nm, W/cm Harmonic order = 26 nm FIG. 4. Color online Experimentally obtained harmonic spectra in Ar. Red Only odd orders gas is a medium of inversion symmetry 7
8 almost x-ray! Popmintchev et al., Science 336, 1287 (2012) 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 8
9 Plateau and cutoff - remarkable feature of high-harmonic generation - Wahlström et al., Phys. Rev. A 48, 4709 (1993) W/cm Harmonic intensity (arb. unit) nm, W/cm Harmonic order E c U p (ev) = e2 E0 2 I p +3.17U p 4m 2 = I(W/cm 2 ) 2 (µm) ponderomotive energy These features cannot be understood as perturbative harmonic generation.
10 3-STEP MODEL Laser field E(t) =E 0 cos recombination t photon emission (HHG) electron tunneling Semiclassical electron motion ionization Paul B. Corkum, Phys. Rev. Lett. 71, 1994 (1993) K. C. Kulander et al., in Super-Intense Laser-Atom Physics, NATO ASI Ser. B, Vol. 316, p. 95 (1993) Paul B. Corkum 10
11 3-STEP MODEL OF HHG Ionization at t = t 0 with vanishing initial velocity at origin t 0 m z = ee 0 cos t ż(t 0 )=0 z(t 0 )=0 Normalization = t 0 = t 0 z = E 0 2 [(cos cos 0)+( 0) sin 0 ] E kin = 2U p (sin φ sin φ 0 ) 2 φ = φ ret (φ 0 ), which satisfies Recombination at z = 0 Laser field E(t) =E 0 cos t recombination photon emission (HHG) electron tunneling ionization Semiclassical electron motion 11
12 TIME (PHASE) OF RECOMBINATION z = 0 (cos ret cos 0 )+( ret 0 ) sin 0 =0 (cos ) 0 = cos ret cos 0 ret 0 phase of ionization vs phase of recombination Phase of recombination (phi_r) no recombination for 2 < 0 < phi (degree) 240 ret Phase of electron release (phi0) 12
13 Simple explanation of the cut-off law Electron kinetic energy (in U p ) ionization long short 90 field recombination short long Phase (degrees) Field (in E 0 ) There is the maximum kinetic energy which is classically allowed. 3.17U p cut-off E c = I p +3.17U p There are two pairs of ionization and recombination times which contribute to the same harmonic energy. short trajectory long trajectory 13
14 WHY DO HARMONIC SPECTRA CONSIST OF DISCRETE PEAKS? Takahashi et al., Appl. Phys. Lett. 93, (2008) E(t) =E 0 cos Laser field recombination t photon emission (HHG) FIG. 4. Color 10 2 online Experimentally obtained harmonic spectra in Ar. Red 10 1 Harmonic intensity (arb. unit) Harmonic order electron tunneling ionization Semiclassical electron motion This is repeated every half cycle with an alternating phase 14
15 Harmonic electric field harmonic field harmonic intensity fundamental field Experimentally measured Nabekawa et al., Phys. Rev. Lett. 97, (2006) Fundamental optical cycle f(t) = f(t / 0 ) time [fs] laser cycle (2.7 fs) Spectrum only consists of odd harmonics. 15
16 SINGLE FREQUENCY COMPONENT E h (t) = E q cos(qω + φ q ) = E 2n+1 cos[(2n + 1)ω + φ 2n+1 ] Electric 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 (t) = q E q cos(qω + φ q ) = q E 2n+1 cos[(2n + 1)ω + φ 2n+1 ] Electric field 電場 attosecond pulse train (APT) bursts repeated every half cycle of the fundamental laser 時間 ( フェムト秒 )Time (fs) equispaced frequency components adjacent pulses have an opposite phase train of repeated pulses We don t need photons to understand harmonic generation 17
18 DISCRETE PEAKS OF ODD HARMONICS CAN BE INTERPRETED IN TWO WAYS. Integer number of photon energy + inversion symmetry Light emission repeated every half cycle (with alternating phase) 18
19 Time of emission depends on harmonic order Phase of recombination (phi_r) Long trajectory short trajectory W/cm 2 measurement theory Ne Electron energy (in Up) Higher-order components emitted later 19 Intensity (arb.units) Amazing predictive power of the 3-step model! Time (as) Mairesse et al., Science 302, 1540 (2003) Varju et al., J. Mod. Opt. 52, 379 (2005)
20 Time of emission depends on harmonic order Phase of recombination (phi_r) Long trajectory short trajectory TDSE simulation Electron energy (in Up) K. L. Ishikawa, High-harmonic generation in Advances in Solid- State Lasers, ed. by M. Grishin (INTECH, 2010)
21 Quantum theory of high-harmonic generation Lewenstein model 21 Lewenstein et al., Phys. Rev. A 49, 2117 (1994)
22 Strong-field approximation (SFA) 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. 22
23 i (r,t) t = V (r)+ze(t) (r,t) Time-dependent dipole moment x(t) (r,t) z (r,t) = i 3-step model = i t t dt d 3 p (r)e ii pt z p + A(t) exp i dt d 3 p (r)e ii pt z p + A(t) exp i Dipole moment between the recolliding wave packet and the ground state t t dt [p + A(t )] 2 t t dt 2 [p + A(t )] 2 2 p + A(t ) ze(t ) (r)e ii pt recombination motion in the laser field ionization recolliding electron wave packet p + A(t ) ze(t ) (r)e ii pt ionization +c.c. +c.c. x(t) =i t semiclassical action dt d 3 p d (p + A(t)) exp[ is(p,t,t)] E(t )d(p + A(t )) + c.c. transition dipole transition dipole S(p,t,t)= t t dt [p + A(t )] 2 + I p 2 23
24 HARMONIC SPECTRUM = FOURIER TRANSFORM OF DIPOLE MOMENT ˆx( h )=i dt t dt d 3 p d (p + A(t)) exp[i h t is(p,t,t)] E(t )d(p + A(t )) + c.c. five-dimensional integral saddle-point analysis cf. path integral 24
25 saddle-point analysis (SPA) Saddle-point equations [p + A(t )] 2 t t 2 [p + A(t)] 2 2 = I p [p + A(t )]dt =0 + I p = h solutions trajectories tunneling ionization recombines at the location of ionization t time of ionization t time of recombination harmonic photon energy = kinetic energy at recombination + ionization potential ˆx( h )= s + i 2 (t s t s ) 3/2 i2 det S (t, t ) s d (p s + A(t s )) exp[i h t s is(p s,t s,t s )]E(t s )d(p s + A(t s )), physically corresponds to the 3-step model 25
26 Example of saddle-point E(t) =E 0 cos t Ar (Ip = ev) W/cm 2 solutions Real part (top) and imaginary part (bottom) of = t = t dashed lines: 3-step model cutoff E c =3.17U p + gi p (g 1.3) interpretted as tunnling time The 3-step model is a good approximation to the quantummechanical Lewenstein model Success of the 3-step model Lewenstein 26
27 attosecond pulse train (APT) isolated attosecond pulse (IAP) 27
28 High-order harmonics are generated as attosecond bursts repeated each half cycle of the fundamental laser (attosecond pulse train) Paul et al., Science 292, 1689 (2001) Nabekawa et al., Phys. Rev. Lett. 97, (2006) time [fs] Optical cycle(2.7 fs) Only one burst Isolated attosecond pulse (IAP) 28
29 Isolated attosecond pulse generation by a few-cycle laser pulse Baltuska et al. Nature 421, 611 (2003) Hentschel et al. Nature 414, 509 (2001) X-ray intensity (arbitrary units) Energy (ev) Time (fs) 530 as τ x = 530 as Laser electric field (arbitrary units) 5fs Light emission takes place only once. 29 Zhao et al. (2012) Attosecond (10-18 sec) pulse more details in Quantum Beam Engineering
30 10-15 sec sec Molecular rotation Molecular vibration Electronic dynamics Pulse duration (fs) Year increase in intensity Single cycle at 800 nm (courtesy of Prof. J. Itatani) 30
31 0.1attosecond!
32 supplementary materials 32
33 How to generate IAP 33 K. L. Ishikawa
34 Isolated attosecond pulse generation by a few-cycle laser pulse Baltuska et al. Nature 421, 611 (2003) Hentschel et al. Nature 414, 509 (2001) X-ray intensity (arbitrary units) Energy (ev) Time (fs) 530 as τ x = 530 as Laser electric field (arbitrary units) 5fs XUV intensity (arb.u.) Ne C 80 as τ x = 80 ±5 as phase (rad) Goulielmakis et al. Science 320, 1614 (2008) Light emission takes place only once. Fig Time (as) A Attosecond (10-18 sec) pulse 34 K. L. Ishikawa
35 IONIZATION SHUTTER domain, and the autocorrelation functions were then calculated. HHG is suppressed when neutral atoms are depleted density of neutral Ar atoms 9th harmonic (of 400 nm) = 27.9 ev fundamental field envelope (400 nm) 35 tion traces were 1.3 ^ 0.1 and 1.8 ^ 0.1 fs, resulting in pulse durations of 950 ^ 90 as and 1.3 ^ 0.1 fs, respectively. In the 950-as pulse, however, bumps appeared around the main peak and the gaussian function does not seem to be appropriate to describe the pulse shape. To check the validity of the experimental results, the spectra of the ninth harmonic (Fig. 3c) were Fouriertransformed with an assumption of a flat phase in the frequency The results are shown by the blue lines in Fig 3a, b. Both the autocorrelation trace of the 1.3-fs pulse and that of 8.3-fs pulse are reproduced well. The bumps are therefore attributable to the spectrum shape. Consequently, no other pulses were observed within the scanned time range of 20 fs, showing the isolated single 950 as from 8.3 fs 1.3 fs from 12 fs Isolated sub-fs pulse generation from a ~10 fs pulse Ar Sekikawa et al., Nature 432, 605 (2004) K. L. Ishikawa The spec Ti:sapphire around 800 amplifier of with two pe duration, al spectra are m For furth use of a mu generate hig earlier than duration. H attosecond p duration is the tempor (650 as). Th to induce n Finally, w two-photon volume V ( ¼ cm cross-sectio the pulse du were 7.8 was set to e electrons pe efficiencies, Methods Driving laser Blue laser pulses pulses to obtain the laser pulse, b spectrum comp pulse energies o durations were system. The opt configuration fo tilt and phase m coherence of th pulses. The puls and were found Autocorrelatio In the present produced by s conventional a
36 POLARIZATION GATING (PG) FOCUS REVIEW ARTICLE HHG is suppressed when circular polarization is used counter-rotating circularly polarized pulses with a delay b Circularly polarized laser field Linearly polarized laser field EUV intensity Ar 130 as Phase (rad) L = 0.8 µm Contributing subcycle Time (as) Sansone et al., Science 314, 443 (2006) 5 36 K. L. Ishikawa
37 e information of encoded in I ωl, pulse are guessed DOUBLE OPTICAL GATING (DOG) Polarization gating + two-color gating PRL 100, (2008) week ending 14 MARCH 2008 PHYSICAL REVIEW LETTERS 2 2 Egate "t# $ E0 "e'2ln2&"t%td =2'T0 =4# ="! ( 2 2 ' e'2ln2&"t'td =2'T0 =4# ="! ( #sin"!0 t % CE #; (2) where E0 is the amplitude of the circularly polarized fundamental laser field with carrier frequency!0 (period T0 ), pulse duration "!, and CE phase CE. Td is the time delay between the two circular pulses. The delay, T0 =4, between the gating and the driving fields is introduced by the quarter-wave plate. #!;2! is the relative phase between the fundamental and second harmonic pulses. The duration of the SH pulse is "2!. Finally, a represents the strength of the second harmonic field relative to the fundamental field. Figure 2(a) shows harmonic spectra of argon for onecolor (linearly polarized fundamental field only, Td $ 0, a $ 0), two-color (a second harmonic field added to a fundamental field polarized in the same direction, Td $ 0), conventional PG (a $ 0), and DOG fields. Notice that +2 Ne with secondharmonic field Fig. 3. (Color online) Characterization of a 67 as XUV pulse. (a) Streaked photoelectron spectrogram obtained experimentally. (b) Filtered I ωl trace (left) from the spectrogram in (a) and the retrieved I ωl trace (right). (c) Photoelectron specnerated by DOG in trum obtained experimentally (thick solid) and retrieved specal., PRL and 2008,FROG-CRAB (2008) e gas cell is 1 mm. tra and spectral phases from Mashiko PROOFet(solid) Zhao etprofiles al., Opt. Lett (2012) polarization gate is FIG. 1 (color). (dashed). (d) Retrieved temporal and 37, phases from The driving filed components for PG correspond to (a) without and (b) with the second harmonic field, PROOF (solid) (dashed). respectively. The driving field is shown as theand red line. FROG-CRAB The two 37 K. L. Ishikawa IAP generation from a ~10 fs pulse vertical lines represent the gate width. Here, the filled curves are
38 GENERALIZED DOUBLE OPTICAL GATING (GDOG) = 0.8 µm Elliptical instead of circular polarization L c Laser field Bi-colour field with shaped polarization L HHG bursts = 0.8 µm Single HHG burst EUV intensity 1.0 Ar as Time (as) Phase (rad) L = 0.8 µm IAP generation from a v > 20 fs pulse without L = c initial need E L of carrierenvelope E L final stabilization initial E X-ray Gilbertson et al., PRL 105, (2010) final Gilbertson et al., PRA 81, (2010) E X-ray v X-ray = c 38 1 L = 2.0 µm K. L. Ishikawa
39 2 INFRARED TWO-COLOR SYNTHESIS mix [arb. units] nm nm two-color driving field 800 nm 800 nm nm (a) ( 10 3 a. ( 10 3 a.u.) Time [fs] Takahashi et al., PRL 104, (2010) autocorrelation trace Xe 500 as 29 ev Δt (fs) Takahashi et al., Nat. Commun. 4, 2691 (2013) e 3 Measured AC traces of an IAP obtained from the side peak of N þ ion signals. The time resolutio 8 and 28 as, respectively. The error bars show the s.d. of each data point. The grey solid profiles are AC tr High-energy (1.3 micro J), high-power (2.6 GW) IAP more than 100 times more energetic than previously reported NATURE COMMUNICATIONS 4:2691 DOI: /nco 39 & 2013 Macmillan Publishers Limited. All rights K. reserved. L. Ishikawa
40 FROM FEMTOSECOND TO ATTOSECOND Molecular rotation Molecular vibration Electronic dynamics Pulse duration (fs) Year increase in intensity Single cycle at 800 nm (courtesy of Prof. J. Itatani) 40 K. L. Ishikawa
41 Quest for higher photon energy (shorter wavelength) cutoff E c = I p +3.17U p U p (ev) = e2 E 2 0 4m 2 = I(W/cm 2 ) 2 (µm) Longer fundamental wavelength is advantageous Optical parametric chirped-pulse amplification (OPCPA) 41 K. L. Ishikawa
42 WATER-WINDOW HHG spectral range between the K-absorption edges of C (284 ev) and O (543 ev) absorbed by biological samples but not by water attractive for high-contrast biological imaging 0 =1.55 µm I = W/cm µm He HHG [arb. units] He Space Carbon K edge Photon energy Transmission of Mylar filter Photon energy [ev] Takahashi et al., PRL 101, (2008) K. L. Ishikawa
43 kev HHG almost x-ray! 0 =3.9 µm Popmintchev et al., Science 336, 1287 (2012) a new type of laser-based radiation source 43 K. L. Ishikawa
44 ATTOSECOND SCIENCE atomic unit of time = 24 attoseconds Orbital period of the Bohr electron mω 2 r = 1 4πϵ 0 e 2 r 2 T = 2! =2 r 4 0 mr 3 real-time observation and time-domain control of atomic-scale electron dynamics e 2 = 152 as = 2 a.u. 44 K. L. Ishikawa
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