High Harmonic Generation of Coherent EUV/SXR Radiation David Attwood University of California, Berkeley Prof. David Attwood / UC Berkeley EE213 & AST21 / Spring 29 14_HHG_29.ppt
HHG: Extreme nonlinear optics Ti: sapphire laser 8 nm 3 fsec > 1 14 W/cm 2 EUV/SXR Gas jet High harmonics of the intense 8 nm (1.55 ev) laser pulse Photon energies throughout the EUV, extending to SXR Spatially and temporally coherent Femtosecond pulse duration, recently to attoseconds HHG_XtremeNonllinOpt.ai
High-order Harmonic Generation (HHG) High-order harmonics Intense fs laser X-ray filter electron,,,,,,,,,x-ray,,,,,,,,,x-ray CCD Soft X-ray spectrometer Gas jet %!&'(!"#$ )*(!+ Signal process 6/3 -. -3 6-3 /. /3 683.. 673.3. 3 1. 13 2. High-order harmonics,4!5 harmonics,4! 9#:("'&+,#;,<(#;'&&#(,9=!>?,6'',@!$A,BCDEF,G>HI'(&H"+A,J!'K'#>A,B#('! CXRC
The physics of High Harmonic Generation (HHG) U( x,t) laser field h! cutoff = I p + 3.2U p I p electron (Kulander et al, Corkum et al) Ion EUV/SXR Ionization potential of atom U p " I # 2 Cycle averaged energy of an oscillating electron (ponderomotive energy or potential) electron Courtesy of Professor Henry Kapteyn, U. Colorado, Boulder Prof. David Attwood / UC Berkeley EE213 & AST21 / Spring 29 14_HHG_29.ppt
The HHG Process 1. A high electric field of a coherent, intense laser pulse liberates a core electron from the atom. 2. The electron is accelerated in the laser field. 3. The electron recombines with the atom (ion) in a very short interaction time, emitting relatively high energy photons, also of short duration. 4. The process simultaneously involves many electron-ion pairs, emitting photons in phase with the coherent laser pulse. 5. The coherence of the incident laser field is effectively transferred to the emitted EUV/SXR radiation. 6. The process occurs twice per cycle of the incident laser pulse, at well defined phases, resulting in harmonic emissions (odd only). Prof. David Attwood / UC Berkeley EE213 & AST21 / Spring 29 14_HHG_29.ppt
Ultrashort light pulses: Ti: sapphire lasers, 8 nm wavelength The current state of the art! 5-1 femtoseconds High-power amplifier systems: 15-25 fs 1 fs light pulse: c = 3 nm/fsec $x = 3 micrometers Courtesy of Professor Henry Kapteyn, U. Colorado, Boulder Prof. David Attwood / UC Berkeley EE213 & AST21 / Spring 29 14_HHG_29.ppt
High Harmonic Generation (HHG) of Femtosecond IR laser pulses into the EUV Photon signal (mv) 47 57 67 77 87 97 1 1 1 Intensity (arbitrary units) Harmonic signal (arbitrary units) 1 2 15 1 125 135 145 155 12 16 2 24 28 (5.2 nm) Wavelength (nm) Harmonic order Wavelength (nm) 15 1 5 H81 H61 H39 L Huillier and Balcou, Phys.Rev.Lett.7, 774 (1993) Neutral neon at 4 torr 1.53 m, 1.5 1 15 W/cm 2 1 ps duration n = 135 Z. Chang, A. Rundquist, H. Wang, M. Murnane, H. Kapteyn, Phys.Rev.Lett.79, 2967 (1997) Neutral neon at 8 torr 8 nm, 6 1 15 W/cm 2 26 fs duration n = 155 (n = 211 in helium) D. Schultze, M. Dörr, G. Sommerer, J. Ludwig, P. Nickles, T. Schlegel, W. Sandner, M. Drescher, U. Kleineberg, U. Heinzmann, Phys.Rev.A 57, 33 (1998) Neutral neon 1.53 nm, 5 1 14 W/cm 2 7 fs duration n = 81 (polarization confirmed) Ch6_F34VG.ai
EUV High Harmonics: Prof. David Attwood / UC Berkeley EE213 & AST21 / Spring 29 14_HHG_29.ppt
Typical HHG spectrum using argon: Only odd-order harmonics are generated Plateau Cutoff 17 25 29 39 45 Harmonic order J. Zhou, J. Peatross, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, PRL 76, 752 (1996) Prof. David Attwood / UC Berkeley EE213 & AST21 / Spring 29 14_HHG_29.ppt
Some HHG equations Speed of light c = 3 nm/fsec Duration of one cycle f λ = c 1 λ 8 nm 8 = = = fsec f c 3 nm/fsec 3 Bohr orbit time (n = 1) 2πa τ = 2πa = v αc 2πa τ = = 152 asec c/137 (Tipler, Modern Physics, eq. 4-28) Uncertainty Principle E FWHM τ FWHM = 1.82 ev fsec (152 asec pulse requires a 12 ev bandwidth) Electron energy in an oscillating field m( iω)v = ee 1 e 2 E 2 mv 2 = 2 2mω 2 ; I = µ E 2 1 e 2 I mv 2 = 2 2mc ω 2 ; r e = e 2 /4π mc 2 cycle averaged energy: 1 r e Iλ 2 2 mv 2 = πcr e I/ω 2 = = U 4πc p U p = 9.33 1 14 I(W/cm 2 )[λ(µm)] 2 ev HHG_equations.ai
Energetics Ponderomotive potential (cycle averaged kinetic energy of a free electron in an electric field E and frequency! ). F = ma = ee o e!i"t = m dv dt ee o # e!i"t dt = ee o v = m!i"m e!i"t U p = [ K.E. ] time avg = 1 2 mv2 = e2 E 2 o 2m" 2 e!i"t 2 [ ] time avg = e2 E o 2 4m" 2 Courtesy of Professor Henry Kapteyn, U. Colorado, Boulder Prof. David Attwood / UC Berkeley EE213 & AST21 / Spring 29 14_HHG_29.ppt
Energetics (continued) Using (ch 2) I =! o µ o E 2 We obtain ( ) $ µm U p = e2 E 2 4m! 2 = 9.33 "1 #14 I W cm 2 [ ( )] 2 ( ev) = 6 ev @ 1 15 W/cm 2, 8 nm Energy scale I p + 3.2 Up = 24.6 ev + 192 ev 22 ev of HHG in He Courtesy of Professor Henry Kapteyn, U. Colorado, Boulder Prof. David Attwood / UC Berkeley EE213 & AST21 / Spring 29 14_HHG_29.ppt
Assume: atom Electron trajectory e - Electron is suddenly, completely free Electron is released at rest K. C. Kulander, K. J. Schafer, and J. L. Krause, in Super-intense laser-atom physics, vol. 316, NATO Advanced Science Institutes Series p. 95 (1993); P. B. Corkum, PRL 71, 1994 (1993). F = ma = ee o e!i"t = m dv dv/dt dt ee o # e!i"t $ dt = ee o v = m %& ( ) = since v(ti) v t i =!i"m e!i"t ' t * () ti = ee o [ t! e!i"t ] i = dx dt!i"m e!i" * Courtesy of Professor Henry Kapteyn, U. Colorado, Boulder Prof. David Attwood / UC Berkeley EE213 & AST21 / Spring 29 14_HHG_29.ppt
Solve for trajectory: Electron trajectory dx dt = ee o!i"m e!i" t #! e!i"t i x = [ ] ee o t! e!i"t i $ = t f t i [ ] dt'!i"m e!i" # % ee o!" 2 m e!i" t #! e!i"t i ' & [ ] ( * ) t f t i Electron released at atom: x(t i ) = Electron trajectory ends at atom, for HHG : x(t f ) = Solve for t f Find v(t f ) Find return energy of electron E= 1 / 2 mv 2 Courtesy of Professor Henry Kapteyn, U. Colorado, Boulder Prof. David Attwood / UC Berkeley EE213 & AST21 / Spring 29 14_HHG_29.ppt
Electron trajectories Most electrons don t have opportunity to recollide Transverse spread of electron wavefunction further reduces recollisions Courtesy of Professor Henry Kapteyn, U. Colorado, Boulder Prof. David Attwood / UC Berkeley EE213 & AST21 / Spring 29 14_HHG_29.ppt
HHG electron trajectories and return energies depend on time of liberation vis à vis the driving electric field 1 8 nm E-field Electron liberated ( born ) at peak of pulse ω Zero kinetic energy upon return Distance from Ion (nm) 1 2 3 4 5 9 φ = For λ = 8 nm I = 5 1 14 W/cm 2 9 18 27 36 45 Time (phase of E-field) HHG_YWL_1a.ai
HHG electron trajectories and return energies depend on time of liberation vis à vis the driving electric field Electron born before peak of pulse Electron never returns 1 8 nm E-field ω Electron liberated ( born ) at peak of pulse ω Zero kinetic energy upon return Distance from Ion (nm) 1 2 3 φ = φ = 1 4 5 9 For λ = 8 nm I = 5 1 14 W/cm 2 9 18 27 36 45 Time (phase of E-field) HHG_YWL_1b.ai
HHG electron trajectories and return energies depend on time of liberation vis à vis the driving electric field Electron born before peak of pulse Electron never returns 1 8 nm E-field φ = +15 ω Electron liberated ( born ) at peak of pulse ω Zero kinetic energy upon return Distance from Ion (nm) 1 2 3 φ = φ = 1 Electron born after peak of pulse ω Maximum kinetic energy at φ = 18 & 198 4 5 9 Courtesy of Dr. Yanwei Liu, UC Berkeley and Lawrence Berkeley National Lab For λ = 8 nm I = 5 1 14 W/cm 2 9 18 27 36 45 Time (phase of E-field) HHG_YWL_1c.ai
HHG electron trajectories and return energies depend on time of liberation vis à vis the driving electric field Electron born before peak of pulse Electron never returns 1 8 nm E-field Recombination kinetic energy 83 ev at φ = 15 46 ev at φ = 5 φ = +15 φ = +5 ω Electron liberated ( born ) at peak of pulse ω Zero kinetic energy upon return Distance from Ion (nm) 1 2 3 φ = φ = 1 Electron born after peak of pulse ω Maximum kinetic energy at φ = 18 & 198 4 5 9 φ = 45 For λ = 8 nm I = 5 1 14 W/cm 2 9 18 27 36 45 Time (phase of E-field) Courtesy of Dr. Yanwei Liu, UC Berkeley and Lawrence Berkeley National Lab HHG_YWL_1d.ai
Different electron trajectories (times of birth) result in varied return energies, different path lengths, and thus different times of emission causing a chirp (photon energy vs time) 1 Short trajectories Maximum return energy (HHG cutoff ) Distance from Ion (nm) 1 2 3 λ = 8 nm I = 5 1 14 W/cm 2 Neon (I p = 21.6 ev) 48 47 ev 69 ev 94 ev 16 ev 85 ev 52 ev 38 28 18 8 3 Long trajectories 9 9 18 27 36 Time (phase of E-field) Courtesy of Dr. Yanwei Liu, UC Berkeley and Lawrence Berkeley National Lab HHG_YWL_2.ai
Electron return energy as a function of liberation time vis à vis the driving electric field 3 No HHG φ = 18 φ = 198 No HHG Return energy, E/Up 2 1 Long trajectory E-field Short trajectory 1 9 9 18 27 Time (phase of E-field) HHG_YWL_3.ai
Perfectly periodic emissions generate only odd harmonics 12 λ = 8 nm, I = 5 1 14 W/cm 2 Neon (I p = 21.6 ev) 1 Emitted photon energy (ev) 8 6 4 2 E-field 18 36 54 72 9 18 126 Time (phase of E-field) Courtesy of Dr. Yanwei Liu, UC Berkeley and Lawrence Berkeley National Lab HHG_YWL_4.ai
High Harmonic Generation (HHG) Provides Coherent, Femtosecond Pulses 15 µm Fiber with 3 Torr Argon Ultrafast laser beam (76 nm, 25 fs) y (mm) 3 Filter EUV beam Pinholes Lineout EUV CCD R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, C. Jacobsen, Science 297, 376 (19 July 22). 36 nm 5 5 x (mm) P 1 µw 2 1 12 ph/sec @ 36 nm (n = 21; 34 ev) Courtesy of Professors Margaret Murnane and Henry Kapteyn, Univ. Colorado, Boulder, and Dr. Yanwei Liu, U. California, Berkeley, and LBL. HHG_cohFempto_Mar29.ai
Why Use Hollow Fibers? WhyHollowFibers.ai