Generation and Applications of High Harmonics

First Asian Summer School on Aug. 9, 2006 Generation and Applications of High Harmonics Chang Hee NAM Dept. of Physics & Coherent X-ray Research Center Korea Advanced Institute of Science and Technology

Contents 1. Basic physics of high harmonic generation - single atom response - propagation effects 2. Generation of strong harmonics - long gas jet - two-color laser field 3. Applications - soft x-ray interferometry - attosecond physics 4. Conclusion

High Harmonic Generation (HHG) r e m High harmonics h c S l o o t a i d m a u R S d spectrometer Soft X-ray n Gasijet n a n a s O t s A n o i t 1 a r Signal processor e l ce Ultrashort coherent soft c x-rays! A a m s a l P r e s La Intense fs laser X-ray filter electron s n io X-ray CCD Laser atom X-ray H95 80 85 H85 90 95 H75 100 H65 105 110 115 120 125 130 High-order harmonics (Å)

Typical spectral structure of high harmonics Plateau Cutoff Perturbative HG Characteristics ly odd harmonics Sharp dropping plateau cutoff Plateau structure: HHG is a nonperturbative process. Ar; τ=36 ps; Ι= 3x10 13 W/cm 2 Li et al., Phys. Rev. A 39, 5751 (1989) Atomic unit of intensity = 6x10 15 W/cm 2

1. Basic Physics of HHG Single atom response - semi-classical three-step model - electron paths of HHG - quantum mechanical models: TDSE and SFA Propagation effects - elements of propagation effect - full simulation of HHG

Semi-classical Three-step Model d(t) = - d(t+t/2) = d(t+t) due to inversion symmetry only odd harmonics Cutoff order ω = I + KEmax = I + 3.17 U, c p p p Corkum, Phys. Rev. Lett. 71, 1994 (1993) U p U p : ponderomotive potential

Electron Paths of HHG Ne; τ=30 fs; λ=820 nm; I=9x10 14 W/cm 2 ω = I + KEmax = I + 3.17 U c p p p

Quantum Mechanical Approach Schrödinger eq. 2 i Ψ (,) x t = + V() x + x E() t Ψ(,) x t t 2 Atomic dipole (or dipole acceleration) is the source of radiation. d() t = Ψ() t x Ψ() t 2 d da () t = Ψ() t x Ψ () t = Ψ() t V( x) + E() t Ψ() t 2 dt Ehrenfest s theorem ce atomic dipole is given, the harmonic field can be calculated. Harmonic field Harmonic spectrum Time-frequency analysis E I h h () t d () t a ( d t ) 2 ( ω ) FT ( ) in atomic unit (m e =ћ= e =1) Wigner distribution or short-time Fourier transform a

a Strong field approximation (SFA) model Assumptions: - Hydrogenlike atom with ground state only - ce the electron is ionized and in continuum state, it is driven solely by the intense laser field; the Coulomb field is neglected (strong field approximation). Calculation of dipole acceleration t ( ) ( ) Lewenstein et al., Phys. Rev. A 49, 2117 (1994) Becker et al., Phys. Rev. A 56, 645 (1997) * is ( p, t, t ) d () t = i dt d p da p+ A() t E( t ) d p+ A( t ) e + CC.. Recombination at t Ionization at t 1 t S( p, t, t ) = ( ( t )) 2 d t Ip ( t t ) 2 p+ A + t da( p) = p V( x) g, d( p) = p x g Traveling in continuum between t and t -> wavepacket diffusion Ionization can be incorporated by inserting the remaining probability amplitude calculated with ADK or PPT formula.

Elements of Propagation Effects Focusing space-dependent laser intensity phase distribution (Gouy phase) Ionization - space-time-dependent laser intensity and laser phase due to self-phase modulation (defocusing and chirp modulation). - depletion of neutral atoms 2 2 en( ) (, ) e x, t ωp x t Δ npl( x, t) = = 2 2 2ε m ω 2ω In experiments, these effects appear together. To incorporate these effects, a numerical simulation is performed. 0 e

Laser pulse propagation Full Simulation of HHG 2 2 2 1 E1 ω1 2 E1 = (1 n ) 2 2 2 eff E1 c t c n (,,) rzt = n(,,) rzt + nirzt (,,) ω (,,)/2 rzt ω eff 2 2 0 2 p 1 Harmonic pulse propagation E ( r, z, t) P r z t c t t P( r, z, t) = [ N N ( r, z, t)] x( r, z, t) 2 2 2 1 h (,, ) E h ( r, z, t) = μ 2 2 0 2 0 e P( r, z, t) N e ( r, z, t) E1 ( r, z, t) Priori et. al., Phys. Rev A 61, 063801 (2000) Takahashi et al., Phys. Rev. A 68, 063808 (2004) Atomic response (SFA & ADK) 3/2 π isst (, t τ ) xt () = i dτ d( pst(, tτ) + At ()) e 0 ε + iτ /2 E( t τ) id( p( t, τ) + At ( τ)) at ( τ) at ( ) + cc.. E h N e st ( r, z, t) Ionization is calculated using ADK or PPT formula. ( r, z, t)

2. Generation of Strong Harmonics Long gas jet - long gas jet for strong HHG - plasma image and laser beam profile - Guiding and profile flattening - harmonic optimization Two-color laser field - experimental setup - dramatic signal enhancement - dependence on relative phase - electron paths in two-color field

f=1.2 m Long Gas Jet for Strong HHG 5 mj, -42 fs Filter CCD 1 9-mm gas nozzle z<0 z>0 z=0 Image relay CCD 2 Attenuator A long gas jet provides a long high-density gas medium with simple target alignment.

Plasma Image and Laser Beam Profile I 0 =2x10 15 W/cm 2 (at z=0), E=5 mj, τ=-42 fs, Ne (40 Torr), slit nozzle (L=9 mm) Plasma image (CCD1) 0 1 2 3 4 5 6 7 8 9 Laser beam profile (CCD2) Entrance z=0 Exit L (mm) Entrance (z=0) (z=-18 mm) z=-18 mm Exit

Self-guiding and Profile-flattening I r z 0 ionizing medium z=0 z<0 z>0 r z k < r k = r k > r 0 0 0 Radial wave vector k r z Medium at z < 0 Profile flattening k r >0 : diverging k r <0 : converging k r =0 : boundary of profile flattening Refractive index rapidly changes at k r =0 Creating waveguide Self-guiding of laser pulses

Intensity (10 14 W/cm 2 ) Intensity (10 14 W/cm 2 ) 3D Propagation Calculation I 0 =2x10 15 W/cm 2 (at z=0), E=5 mj, τ=-42 fs, Ne (40 Torr), slit nozzle (L=9 mm) 12 10 8 6 4 2 0 1 When medium is at z=0 When medium is at z= -18 mm 3 5 7 9 150 100 50 0 5 4 3 2 1 0 1 3 5 7 150 100 50 0 r (μm) L (mm) r (μm) L (mm) defocusing by plasma focusing by lens + defocusing by plasma = profile flattening 9

Harmonic Optimization in Space I 0 =2x10 15 W/cm 2 (at z=0), E=5 mj, τ=-42 fs, Ne (40 Torr), slit nozzle (L=9 mm) H61 Spatial distribution of harmonics Beam divergence: 0.5 mrad 115 120 125 130 135 140 145 150 Wavelength (Å) Applying self-guided and profile-flattened laser pulses, bright harmonics with low beam divergence were obtained. 1 0-1 y (mm)

Harmonic Optimization in Time: Coherent Control I 0 =2x10 15 W/cm 2 (at z=0), E=5 mj, z=-18 mm, Ne (40 Torr), slit nozzle (L=9 mm) Intensity (arb. units) 0.07 nm -77 100 120 140 160 180 Wavelength (Å ) +86 +41 27-42 Pulse duration (fs) Chirp control was performed with self-guided laser pulses.

Spatiotemporal optimization of HHG Coherent control of guided and profile-flattened laser pulses leads to strong harmonic generation. Spatially uniform laser beam focusing by lens + defocusing by plasma guiding and profile flattening Coherent control of harmonic generation SPM-induced positive laser chirp in the leading edge + negative harmonic chirp of short path components + chirp control of laser pulse compensation of harmonic chirp H. T. Kim et al., Phys. Rev. A 69, 031805(R) (2004) Tosa et al., Phys. Rev. A 71, 063807 and 063808 (2005)

Strong harmonic generation in a Two-color Laser Field k y Atoms θ E 2ω x E ω New degrees of freedom - intensity - polarization - relative phase - time delay Use the new degrees of freedom to control HHG at microscopic level.

Two-color High Harmonic Generation Focusing mirror Wave plate 2ω 1ω 2ω 1ω BBO Wave plate Glass plate z <0 1000 n1ω : 1.45298 800 600 400 200 0 0 2 4 6 8 10 12 Rotation angle (degree) z >0 n 2ω : 1.46907 for fused silica Relative phase (degree) (7.3, 360) Femtosecond laser 2.8 mj, 26fs Gas jet Glass plate Flat-field XUV spectrometer Al filter X-ray CCD I J. Kim et al., Phys. Rev. Lett. 94, 243901 (2005) (10.3, 720)

Dramatic Signal Enhancement He (ρ=950 torr), circular nozzle (L=0.5 mm), Al filter 1.5μm, 2.8 mj, τ= 26 fs, z=-12 mm

Dependence on Relative Phase (a) Orthogonal polarization π-periodic modulation is clearly seen. (b) Parallel polarization At 38 th order (21.6 nm) conversion efficiency : 5 10-5, photon energy : 150 nj signal enhancement: 600 I J. Kim et al., Phys. Rev. Lett. 94, 243901 (2005)

Electron Paths in Two-color Field Ne, 30 fs, ortho. pol., I w =6x10 14 W/cm 2, I 2w =3x10 14 W/cm 2 (a) φ=0, long paths only; (e) φ=0.5π, short paths only C. M. Kim et al., J. Phys. B 39, 3199 (2006), Phys. Rev. A 72, 033817 (2005)

3. Applications Soft x-ray interferometry - spatial coherence of harmonic beam - point diffraction interferometry Attosecond physics - generation of attosecond pulses - temporal characterization of attosecond pulses - compensation of attosecond pulse chirp

Double-Pinhole Interferometry Double-pinhole (Φ=10μm) plate Harmonic x-ray (centered at 30 nm) Beam size of 130 μm (FWHM) divergence = 0.9 mrad d Spatial Coherence Measurement X-ray CCD Intensity (arb. units) Ti:sapphire laser pulse 1.0 0.8 0.6 0.4 0.2 X-ray filter & Pinhole Gas-filled hollow tube Harmonic X-ray beam 0.0 24 26 28 30 32 34 36 Wavelength (nm) Harmonic spectrum H27 > 68%

100 80 60 40 20 0 d = 100 μm Visibility = 1 7 8 9 10 11 12 13 x (mm) Pinhole position d = 100 μm d = 200 μm 800 600 400 200 0 d = 200 μm 7 8 9 10 11 12 13 x (mm) Visibility > 0.8 Intensity (arb. units) Intensity (arb. units)

Pinhole plate (Φ=10 μm) Harmonic X-ray (Centered at 30 nm) 4 3.5 3 2.5 2 1.5 1 0.5 Harmonic beam 0.5 1 1.5 2 2.5 3 3.5 4 Lee et al., Opt. Lett. 28, 480 (2003) Point Diffraction Interferometry 4 3.5 3 2.5 2 1.5 1 0.5 X-ray filter 0.5 1 1.5 2 2.5 3 3.5 4 X-ray detector Point-diffracted beam 4 3.5 3 2.5 2 1.5 1 0.5 Interferogram 0.5 1 1.5 2 2.5 3 3.5 4

Intensity (arb. units) 4 3.5 3 2.5 2 1.5 1 0.5 8000 6000 4000 2000 Wavefront analysis of high harmonics 0.5 1 1.5 2 2.5 3 3.5 4 Horizontal profile 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 X (mm) Degree 25 20 15 10 5 Wave-front error OPD error Measurement error due to the CCD pixel size of 24μm 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Radial distance (mm) 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 The wave-front phase of the harmonic x-ray beam is nearly spherical within the error of λ/12. λ

Generation of Attosecond Pulses Phase-locked high harmonics can behave just like a modelocked laser, generating an attosecond pulse train. when phase-locked τ xuv To generate a single attosecond pulse, a continuum radiation with appropriate bandwidth is needed. > 24ω 0 ex) For 50 as, bandwidth > 24 ω 0 T 2N xuv laser harmonics Δωτ 1 2

RABBIT φ Temporal characterization of attosecond pulses q FROG-CRAB ( ) φ ω 2ω 0 Frequency Frequency A q ( ) A ω Assume Freq. comb Average phase ( ) = cos( ω0 + φ ) E t A q t XUV q q q= 2n+ 1 EXUV ( t) = A( ω) cos( ωt+ φ( ω) ) dω T 0 /2 Time Time

RABITT Reconstruction of Attosecond Beating by Interference of Two-photon Transition ψ f ωq 1 ω q+ 1 ω L ω L ω L ψ i Sideband interference f ( 1 + 1 ( atomic) ) A cos 2 ϕ + ϕ ϕ +Δ ϕ, with ϕ = ω τ f L q q L L Amplitude modulation Relative phases of harmonics ω L ω q 1 ω q + 1 ω L ω Photoelectron signal ( t) Aqcos( ωqt ϕq) Ψ = + Paul et al., Science 292, 1689 (2001) q L ω L

Attosecond Pulse Chirp Attosecond pulses are either positively or negatively chirped, depending on which electron paths contribute to harmonic generation. In the harmonic generation process, several electron paths contribute. They form positively chirped as pulse when short path harmonics are superposed. Chirp compensation needed Negative GDD material can compensate for the positive chirp!!

Dispersion and Absorption n ( ω) 1 δ iβ; Looking for negative GDD = + Δ filter = ( Δ filter ) Single absorption 2 2 φ nωt/ c; GDD= d φ / dω Square-well transmission If an X-ray filter has square-well transmission, a negative GDD region exists at the lower frequency part. This X-ray filter can compress the attosecond pulse! (Zr, Ag, In, Sn ) GDD<0 Tin x-ray filter

Chirp compensation with a negative GDD material Attosecond pulse compression using x-ray filter material x-ray filter -GDD K. T. Kim et al., Phys. Rev. A 69, 051805(R) (2004)

30fs, 815nm Ti:sapphire Laser 1 F=60cm Experimental Setup for Attosecond Physics Ar Gas Cell Time delay 200nm Al x-ray filter Ti:sapphire Laser 2 Gold-coated Toroidal mirror TOF electron spectrometer TOF 1 khz, 30 fs Ti:S Laser He

Self-compression of attosecond harmonic pulses A certain harmonic generation medium has negative GDD! 20 torr 30 torr Argon filled gas cell Generation, Compression, Filtering 40 torr 206-as pulse from 40-torr Ar (RABITT measurement)

Experimental result Photoelectron Energy ( τ) FROG-CRAB FROG for Complete Reconstruction of Attosecond Burst Transition amplitude a from 0> v> iφ () t iw ( Ip ) t a v, = i + dt e d E t e +, φ Gate p () X ( τ t ) Signal + ( t) = dt ( t ) + 2 ( t ) v A A t /2 Time delay (optical cycle)

I XUV (arb. units) Complete Reconstruction of Attosecond Pulse Train Reconstructed XUV 0.8 0.4 0.0 60 62 64 66 68 Time (optical cycle) 100 2100 4100 12000 Reconstructed IR 300-as pulse train with 10-fs envelope Mairesse et al., Phys. Rev. A 71, 011401(R) (2005) K. T. Kim et al. (CLEO 06 postdeadline paper) A 2 (arb. units) 0.5 0.0-0.5-1.0-1.5-2.0 0 5 10 15 20 25 30 35 40 45 50 55 Time (optical cycle)

Time-resolved Atomic Inner-shell Spectroscopy dn dw W h W 1 W 2 W kin 0 W bind Drescher et al, Nature 419, 803 (2002) Streak images Kr, M(3d 5/2 ) τ h =7.9 fs Δt Photo emission τ x - duration of X-ray pulse Auger emission τ h - lifetime of core hole

Conclusion High harmonics are an ultrashort coherent soft x-ray source for various applications. In HHG itself, the following topics will be pursued. - Enhancement of energy ( > μj for a single order) - Extension of wavelength region (down to the water window) In applications of HHG, the following topics will be pursued. - Generation and characterization of strong attosecond pulses - Ultrafast atomic and molecular dynamics - Soft x-ray interferometry and microscopy High harmonic x-ray source is a light source for EXTREME SCIENCE in space nano metrology & time attosecond science!!

Acknowledgement This work has been supported by Korea Science and Engineering Foundation through the Creative Research Initiative Program.