nonlinear frequency conversion: ω q = qω 800, q >> 1

Size: px
Start display at page:

Download "nonlinear frequency conversion: ω q = qω 800, q >> 1"

Transcription

1 ABCs of high harmonics and attoseconds Louis DiMauro nonlinear frequency conversion: ω q = qω 800, q >> 1 helium 800 nm 40-5 fs 1 PW/cm 2 H51 H79 Saclay, Chicago, wavelength (nm) SF physics, the semi-classical view and sub-cycle response ahosecond formaion by Fourier synthesis measurement methods unique informaion from applicaions

2 first international attosecond school March 19-28, 2017 Majorana Center, Erice, Sicily:

3 what is & why are attosecond interesting? as 25 as as s 1 as n as 1me is defined by physical processes modified from LCLS/SLAC website

4 challenges of breaking the fs barrier sub-femtosecond short wavelengths (XUV, x-rays) single-cycle 25 as pulse 7.5 nm ( 165 ev photon energy) uncertainty principle: Δν*Δt 1 need bandwidth!! single-cycle 25 as pulse Hz frequency spread control the phases of the field, i.e. mode-locking ahosecond metrology

5 challenges of breaking the fs barrier sub-femtosecond short wavelengths (XUV, x-rays) single-cycle 25 as pulse 7.5 nm ( 165 ev photon energy) uncertainty Q0: can principle: I produce Δν*Δt an attosecond 1 need EM bandwidth!! pulse single-cycle 25 as in pulse the optical regime? Hz frequency spread control the phases of the field, i.e. mode-locking ahosecond metrology

6 basic measurements and observables log(electrons) electrons 1 PW/cm 2 ions argon charge states energy (ev) 0.8 µm He time-of-flight (µs) Lompre et al. (Saclay) log(photons) H51 photons H wavelength (nm) 1.6 ev non-linear non-resonant non-perturbaive high harmonic generaion L Huillier et al. (Lund)

7 what is a strong-field?: the atomic gauge r = cm - Coulomb law: E= q/r 2 ~ V/cm 1 au what laser intensity gives an equivalent field strength? I p ε 2 = o ce 2 p W/cm 2

8 motion of the free electron 2α classical equa1ons: F = mx = -eε sin(ωt + φ o ) mx = (eε/ω) cos(ωt + φ o ) + mv o cycle-averaged energy U p <E k > = 1/T ½mx 2 dt = (eε) 2 /4mω 2 ponderomo1ve or quiver energy: U p λ 2 Ι /4 displacement: α λ 2 Ε for 0.8 µm (red) laser at W/cm 2 (0.18 au field) U p = 60 ev α ~ 50 au (25 A) think in ponderomo1ve units!!!

9 the simpleman s picture of ionization Gallagher, PRL 61, 2304 (1988) Van Linden van den Heuvell & Muller, in MulIphoton Processes (1988) Corkum, BurneH & Brunel, PRL 62, 1259 (1989) Field amplitude φ o π Time 2π electric field E = E o sinωt velocity v(t) = E o /ω[cosωt - cosφ o ] + v o quiver drift for tunneling, v o =0

10 simpleman prediction: electron distribution v(t) = E o /ω[cosωt - cosφ o ] quiver dri. in the experiment, we detect the drift energy not quiver!! E(t) = ½v 2 = 2U p cos 2 φ o V electron counts electron counts x tunneling e E E/Up maximum drift energy = 2U p. x V ac-tunneling: Ammosov, Delone & Krainov, Sov. Phys. JETP 64, 1191 (1986)

11 simpleman prediction: electron distribution quantum analogy: strong-field approxima1on or KFR theory: Keldysh V S, Sov. Phys. JETP 20,1307(1965) Faisal F H J, Phys. B 6,L89 (1973) Reiss H R, Phys. Rev. A 22, 1786 (1980 ) V electron counts electron counts x tunneling e E E/Up maximum drift energy = 2U p. x V ac-tunneling: Ammosov, Delone & Krainov, Sov. Phys. JETP 64, 1191 (1986)

12 simpleman s comparison to experiments E/U p electron counts energy (ev) xenon 30 TW/cm 2 U p = 3 ev bad news? γ = 1.5 (multiphoton) electron counts U p energy (ev) helium 1 PW/cm 2 U p = 50 ev good news! γ = 0.5 (tunneling)

13 2 nd approximation: semi-classical rescattering posiion versus iniial phase: x(t) = E o /ω 2 (sinωt - sinφ o + (φ o - ωt) cos φ o ) Position Time π 3π 5π π 3π 5π Position Position Time Time

14 2 nd approximation: semi-classical rescattering Schafer, Yang, DiMauro & Kulander, PRL, 70, 1599 (1993) Corkum, PRL, 71, 1994 (1993) position collision optical cycles ü electron-core interacion ~ ½ cycle ü electron gains field energy, max 3.17U p ü physics is inherently sub-cycle (ahosecond!)

15 semi-classical description of harmonic emission The electron tunnels through the I distorted Coulomb barrier e - in Coulomb + laser fields II III The free electron is accelerated by the field, and may return to the atomic core The electron recombines with the atom, emitting its energy as a photon

16 rescattering picture: the cartoon Schafer, Yang, DiMauro & Kulander, PRL, 70, 1599 (1993) Corkum, PRL, 71, 1994 (1993) classical trajectories have different release Ime (phase), propagaion Ime and return Imes & energy. control of wave packet dynamics

17 Fourier synthesis of harmonic comb FFT odd harmonics separated by 2ω log(photons) H51 photons H wavelength (nm)

18 Fourier synthesis of harmonic comb FFT odd harmonics separated by 2ω log(photons) H51 photons H79 Q1: What does the harmonic spectrum look like? wavelength (nm)

19 electron counts Fourier synthesis of electron comb FFT above-threshold ionization (ATI) peaks separated by ω E/U p energy (ev)

20 Fourier synthesis of electron comb FFT above-threshold ionization (ATI) peaks separated by ω electron counts E/U p Q2: What does the ATI spectrum look like? energy (ev)

21 origin of single-atom dispersion: attochirp electron posi1on x x(t i )=0 v(t i )= me (T L ) return energy short traj. long traj. chirp > 0 chirp < 0 0 emission 1me (t e ) *modified from talk by B. Mahieu (CEA)

22 e initial phase ( ω t ) return energy & harmonic cutoff return energy: E(t = t r ) = 2U p (cos 2 φ r + cos 2 φ o - 2cos φ o cos φ r ) 0.5 ¼ cycle U p cutoff e return energy (E/Up) Helium, 800 nm optical cycle cutoff energy ~ 3.17U p + I P 3U p

23 high energy electrons & photons helium 0.8 µm, 1 PW/cm 2 U p = 50 ev, γ 0.5 elasic scahering yields energeic (10U p ) electrons recombinaion (1-photon matrix element) with iniial state produces high harmonics inelasic (e, 2e) scahering muliple electron ejecion B. Walker, B. Sheehy, K.C. Kulander & L.F. DiMauro, PRL 77, 5031 (1996)

24 high energy electrons & photons helium 0.8 µm, 1 PW/cm 2 U p = 50 ev, γ 0.5 Q3: What type of collision produces 10U p? (a) forward scattering (b) back scattering the short-range physics is important wave packet spread reduces the effecive rescahering the recollision occurs in less than an opical cycle B. Walker, B. Sheehy, K.C. Kulander & L.F. DiMauro, PRL 77, 5031 (1996)

25 elastic rescattering: intensity scaling ponderomotive potential: U p λ 2 Ι normalized counts electron counts helium, 0.8 µm γ = 1 γ = 0.5 PW/cm PW/cm 2 (U p ) (10 ev) 0.4 (20 ev) 1 (50 ev) electron E/Up energy (ev) low I high I in scaled energy, distributions look similar! Q4: Why is it that as intensity increases, rescattering is less efficient?

26 σ elastic rescattering: intensity scaling Rutherford (Coulomb) scattering φ v o b dσ dω = ε o sin φ ( / 2) PW/cm higher return energies produces less rescattered electrons since ε o 3U p Ι Up (ev)

27 HHG frequency comb for different λ, constant Ι argon HHG, neon PES detector 0.8 µm 110 TW/cm µm 2 µm λ 2 -scaling produces sufficient number of photons ~200 ev

28 attosecond Fourier synthesis monochroma1c model for harmonics I ( t) = q odd A e q i ( ) 2 qω t+ ϕ 0 q Δ +2 ϕ = ϕ q ϕ q Δϕ = 0 train at T/2 and pulses are sub-cycle (attosecond)

29 attosecond Fourier synthesis monochroma1c model for harmonics I ( t) = q odd A e q i ( ) 2 qω t+ ϕ 0 q Δ +2 ϕ = ϕ q ϕ q Δϕ = const Q5: What happens in the time domain for this phase? constant phase difference results in transform-limited pulses

30 attosecond Fourier synthesis monochroma1c model for harmonics I ( t) = q odd A e q i ( ) 2 qω t+ ϕ 0 q Δ +2 ϕ = ϕ q ϕ q Δϕ q need a control and method to measure this!

31 attosecond engineering step 1: spectral filtering harmonic spectrum temporal profile long τ 2 short τ 1 intrinsic time-structure is dominated by the beating between the strong low-order harmonics select the plateau region by spectral filtering

32 attosecond engineering step 2: select trajectory harmonic spectrum temporal profile contributions from 2 dominant trajectories long select the short trajectory by spatial filtering short Bellini et al. PRL (1998)

33 attosecond pulse train generation frequency time intense ultra-fast laser µm generation argon jet Al filter aperture to experiment selec1on in argon: harmonic order 13 to 31

34 isolated attosecond pulse generation confine emission to a half-cycle using CEP few-cycle pulses polarization gating amplitude gating bichromatic fields see review paper by Popmintchev et al. Nature Photonics 2010

35 metrology: RABBITT measurement ReconstrucIon of AHosecond BeaIng By Interference of Two-photon TransiIons H. Muller, Appl. Phys. B, 74, S17 (2002) sideband signal: SB ( ω τ ) SB Iq+ 1 cos 2 IR Φq+ 1 sideband phase: Φ SB atomic q+ 1 = ϕq+ 2 ϕq Δϕq+ 1 Δϕ q+1 intense optical field absorption & emission energy degenerate one-photon absorption energy photoelectron spectrum I p q q+2

36 metrology: RABBITT measurement intense opical field absorpion & emission energy degenerate I p Y. Mairesse et al., Science 302, 1540 (2003) P. Johnsson et al., PRL 95, (2005) q q+4 q+8 in combination with the harmonic amplitudes, the average attosecond pulse can be reconstructed

37 OSU attosecond beamline/end-station HHG arm 2π magnebc bocle e-spectrometer toroidal mirror Ar HHG, Ne PES detector 1.3 µm 2 µm RABBITT spectrogram probe arm BS 1-2 µm

38 RABBITT: λ-dependence of the attochirp 2 µm 1.3 µm 0.8 µm the ahochirp decreases with λ at constant intensity the spectral phase is more complex than classical predicions can produce 100 as pulses from ev

39 streaking metrology: isolated attosecond pulses attosecond pulse duration less than optical period attosecond pulse releases a photoelectron with some initial energy the photoelectron s energy is shifted and broadened by the optical field

40 MPQ streaking for isolated pulse metrology PE energy (ev) delay (fs) p f = p i + A(φ) atto pulse 0.8 µm (2.6 fs) from Drescher et al., Nature 419, 803 (2002)

41 attosecond spectroscopy: scattering phase δx a slow free paricle (k = 0.5 a.u.) the delay is ahoseconds the phase shit depends on momentum the paricle s energy Q6: Does the particle s change? 0,5 0-0,5-1 -1,5-2 x -2,5-3 -3, x [a.u]

42 atomic delay: Wigner delay Wigner, Eisenbud, Smith (1950) the phase shit cannot be measured the relaive delay can be measured by comparison, e.g. valence & core compared the s and p photoionizaion for Ne (MPQ) and Ar (Lund) and measured 20 and 100 as, respecively. RABBITT measures a group delay τ RABBITT = τ chirp + τ atomic ahochirp (source term) τ atomic = τ Wigner + τ cc measurement term

43 attosecond spectroscopy: Wigner delay effective Wigner delay (as) aboseconds pulses can measure detailed electronic informa1on source terms are removed by pairwise comparison the atomic delay of Ne, Ar and Kr are referenced to accurate He calculaion measure the energy dependence of the Wigner delay delay decreases with increasing energy delay is sensiive to many-body effects, e.g. resonances collaborators: M. Dahlström (Stockholm) A. Kheifets (Canberra)

44 HHG spectroscopy: Cooper minimum in argon in 1962, Cooper predicted a minimum in the photoabsorpion caused by a sign change (π-phase jump) in the dipole matrix element the Cooper minimum has been extensively studied in the frequency domain, paricularly in argon the phase jump is not measured but inferred from other observables

45 HHG spectroscopy: Cooper minimum in argon in HHG, the field-driven electron wave packet probes the Cooper structure via the complex recombinaion dipole HHG gives you access to the phase collaborators: M. Gaarde (LSU) K. Schafer (LSU)

PIs: Louis DiMauro & Pierre Agostini

PIs: Louis DiMauro & Pierre Agostini Interaction of Clusters with Intense, Long Wavelength Fields PIs: Louis DiMauro & Pierre Agostini project objective: explore intense laser-cluster interactions in the strong-field limit project approach:

More information

time is defined by physical processes

time is defined by physical processes frontiers in attosecond science Louis F. DiMauro as 100 as as as n as 10-18 s 25 as 1 as 10-18 s 1 as n as modified from LCLS/SLAC website time is defined by physical processes a history of ultra-fast:

More information

Looking into the ultrafast dynamics of electrons

Looking into the ultrafast dynamics of electrons Looking into the ultrafast dynamics of electrons G. Sansone 1,2,3 1) Dipartimento di Fisica Politecnico Milano, Italy 2) Institute of Photonics and Nanotechnology, CNR Politecnico Milano Italy 3) Extreme

More information

Generation and Applications of High Harmonics

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

More information

High Harmonic Generation of Coherent EUV/SXR Radiation. David Attwood University of California, Berkeley

High Harmonic Generation of Coherent EUV/SXR Radiation. David Attwood University of California, Berkeley 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

More information

XUV attosecond pulses

XUV attosecond pulses XUV attosecond pulses D. Charalambidis / Univ. of Crete chara@iesl.forth.gr E. Benis E. Goulielmakis E. Hert L. Nikolopoulos N.A. Papadogiannis P. Tallas In collaboration with G. Tsakiris P. Tallas K.

More information

Nonlinear Optics (WiSe 2015/16) Lecture 12: January 15, 2016

Nonlinear Optics (WiSe 2015/16) Lecture 12: January 15, 2016 Nonlinear Optics (WiSe 2015/16) Lecture 12: January 15, 2016 12 High Harmonic Generation 12.1 Atomic units 12.2 The three step model 12.2.1 Ionization 12.2.2 Propagation 12.2.3 Recombination 12.3 Attosecond

More information

Electron dynamics in a strong laser field

Electron dynamics in a strong laser field Available online at www.worldscientificnews.com WSN 35 (2016) 1-16 EISSN 2392-2192 Electron dynamics in a strong laser field C. C. Gunatilaka, K. A. I. L. Wijewardena Gamalath* Department of Physics, University

More information

Models for Time-Dependent Phenomena

Models for Time-Dependent Phenomena Models for Time-Dependent Phenomena I. Phenomena in laser-matter interaction: atoms II. Phenomena in laser-matter interaction: molecules III. Model systems and TDDFT Manfred Lein p. Outline Phenomena in

More information

Attosecond laser systems and applications

Attosecond laser systems and applications Attosecond laser systems and applications Adrian N. Pfeiffer Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA 8th Annual Laser Safety Officer Workshop September

More information

Models for Time-Dependent Phenomena. I. Laser-matter interaction: atoms II. Laser-matter interaction: molecules III. Model systems and TDDFT

Models for Time-Dependent Phenomena. I. Laser-matter interaction: atoms II. Laser-matter interaction: molecules III. Model systems and TDDFT Models for Time-Dependent Phenomena I. Laser-matter interaction: atoms II. Laser-matter interaction: molecules III. Model systems and TDDFT Manfred Lein, TDDFT school Benasque 22 p. Outline Laser-matter

More information

Overview: Attosecond optical technology based on recollision and gating

Overview: Attosecond optical technology based on recollision and gating Overview: Attosecond optical technology based on recollision and gating Zenghu Chang Kansas State University Team members Kansas State University Zenghu Chang (Dept. of Phys.) Lew Cocke (Dept. of Phys.)

More information

Part II. Interaction with Single Atoms. Multiphoton Ionization Tunneling Ionization Ionization- Induced Defocusing High Harmonic Generation in Gases

Part II. Interaction with Single Atoms. Multiphoton Ionization Tunneling Ionization Ionization- Induced Defocusing High Harmonic Generation in Gases - Part II 27 / 115 - 2-28 / 115 Bohr model recap. At the Bohr radius - a B = the electric field strength is: 2 me 2 = 5.3 10 9 cm, E a = e ab 2 (cgs) 5.1 10 9 Vm 1. This leads to the atomic intensity:

More information

AMO physics with LCLS

AMO physics with LCLS AMO physics with LCLS Phil Bucksbaum Director, Stanford PULSE Center SLAC Strong fields for x-rays LCLS experimental program Experimental capabilities End-station layout PULSE Ultrafast X-ray Summer June

More information

Coherent Electron Scattering Captured by an Attosecond Quantum Stroboscope

Coherent Electron Scattering Captured by an Attosecond Quantum Stroboscope 1 Coherent Electron Scattering Captured by an Attosecond Quantum Stroboscope J. Mauritsson 1, P. Johnsson 1, E. Gustafsson 1, M. Swoboda 1, T. Ruchon 1, A. L Huillier 1 & K. J. Schafer 2 1 Department of

More information

High-Harmonic Generation II

High-Harmonic Generation II Soft X-Rays and Extreme Ultraviolet Radiation High-Harmonic Generation II Phasematching techniques Attosecond pulse generation Applications Specialized optics for HHG sources Dr. Yanwei Liu, University

More information

Generation of ultrashort XUV femtosecond to attosecond pulses Katalin Varjú ELI-ALPS. 2nd MOLIM Training School 6 10 March, 2017 Paris-Saclay

Generation of ultrashort XUV femtosecond to attosecond pulses Katalin Varjú ELI-ALPS. 2nd MOLIM Training School 6 10 March, 2017 Paris-Saclay Generation of ultrashort XUV femtosecond to attosecond pulses Katalin Varjú ELI-ALPS 2nd MOLIM Training School 6 10 March, 2017 Paris-Saclay Characteristic times Krausz: RevModPhys 81, 163 (2009) Fs light

More information

HHG Sub-cycle dynamics

HHG Sub-cycle dynamics Quantum Optics and Laser Science Group Blackett Laboratory, Imperial College London HHG Sub-cycle dynamics 1. Chirp of electron recollision 2. Measuring ultra-fast intramolecular proton motion 3. Controlling

More information

attosecond laser pulse

attosecond laser pulse Kenichi Ishikawa ( ) http://ishiken.free.fr/english/lecture.html ishiken@atto.t.u-tokyo.ac.jp Advanced Plasma and Laser Science E attosecond laser pulse 1 attosecond pulse train (APT) isolated attosecond

More information

Chapter 13. High Harmonic Generation

Chapter 13. High Harmonic Generation Chapter 13 High Harmonic Generation High harmonic generation (HHG) is a technique for producing spatially and temporally coherent extreme-ultraviolet (EUV) light, as well as light pulses as short as hundred

More information

C. D. Lin Kansas State U.

C. D. Lin Kansas State U. Dynamic Imaging of molecules using laser-induced Highorder harmonics and High-energy photoelectrons Goal: probing time-dependent structural changes Example: Isomerization of C 2 H 2 C. D. Lin Kansas State

More information

1 Mathematical description of ultrashort laser pulses

1 Mathematical description of ultrashort laser pulses 1 Mathematical description of ultrashort laser pulses 1.1 We first perform the Fourier transform directly on the Gaussian electric field: E(ω) = F[E(t)] = A 0 e 4 ln ( t T FWHM ) e i(ω 0t+ϕ CE ) e iωt

More information

WP-3: HHG and ultrafast electron imaging

WP-3: HHG and ultrafast electron imaging WORKPACKAGE WP-3: HHG and ultrafast electron imaging Coordinators: P. Salières (CEA), A. Assion (FEMTO, Spectra Physics Vienna) Period: Start Month 4 End Month 48 Leading Participants (Orange in the picture):

More information

Strong Field Physics with Mid-infrared Lasers

Strong Field Physics with Mid-infrared Lasers Journal of Modern Optics Vol. 00, No. 00, DD Month 200x, 1 8 Strong Field Physics with Mid-infrared Lasers K. D. Schultz 1, C. I. Blaga 2, R. Chirla 1, P. Colosimo 2, J. Cryan 1, A. M. March 2, C. Roedig

More information

Efficient isolated attosecond pulse generation from a multi-cycle two-color laser field

Efficient isolated attosecond pulse generation from a multi-cycle two-color laser field Efficient isolated attosecond pulse generation from a multi-cycle two-color laser field Wei Cao, Peixiang Lu, Pengfei Lan, Xinlin Wang, and Guang Yang Wuhan National Laboratory for Optoelectronics and

More information

High order harmonic generation and applications

High order harmonic generation and applications High order harmonic generation and applications E. CONSTANT Centre Laser Intenses et Applications H39 H69 ELI & Hilase Summer School 2016 1 21 26 August 2016 Introduction Laser are unique light sources:

More information

High-energy collision processes involving intense laser fields

High-energy collision processes involving intense laser fields High-energy collision processes involving intense laser fields Carsten Müller Max Planck Institute for Nuclear Physics, Theory Division (Christoph H. Keitel), Heidelberg, Germany EMMI Workshop: Particle

More information

ATTOSECOND AND ANGSTROM SCIENCE

ATTOSECOND AND ANGSTROM SCIENCE ADVANCES IN ATOMIC, MOLECULAR AND OPTICAL PHYSICS, VOL. 54 ATTOSECOND AND ANGSTROM SCIENCE HIROMICHI NIIKURA 1,2 and P.B. CORKUM 1 1 National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario,

More information

Photoelectron Spectroscopy using High Order Harmonic Generation

Photoelectron Spectroscopy using High Order Harmonic Generation Photoelectron Spectroscopy using High Order Harmonic Generation Alana Ogata Yamanouchi Lab, University of Tokyo ABSTRACT The analysis of photochemical processes has been previously limited by the short

More information

Introduction to intense laser-matter interaction

Introduction to intense laser-matter interaction Pohang, 22 Aug. 2013 Introduction to intense laser-matter interaction Chul Min Kim Advanced Photonics Research Institute (APRI), Gwangju Institute of Science and Technology (GIST) & Center for Relativistic

More information

Multiphoton transitions for delay-zero calibration in attosecond spectroscopy arxiv: v1 [physics.atom-ph] 12 Jun 2014

Multiphoton transitions for delay-zero calibration in attosecond spectroscopy arxiv: v1 [physics.atom-ph] 12 Jun 2014 Multiphoton transitions for delay-zero calibration in attosecond spectroscopy arxiv:1406.3137v1 [physics.atom-ph] 1 Jun 014 J Herrmann 1, M Lucchini 1, S Chen, M Wu, A Ludwig 1, L Kasmi 1, K J Schafer,

More information

Resonant Structures in the Low-Energy Electron Continuum for Single Ionization of Atoms in the Tunneling Regime

Resonant Structures in the Low-Energy Electron Continuum for Single Ionization of Atoms in the Tunneling Regime Resonant Structures in the Low-Energy Electron Continuum for Single Ionization of Atoms in the Tunneling Regime A. Rudenko, K. Zrost, C.D. Schröter, V.L.B. de Jesus, B. Feuerstein, R. Moshammer, J. Ullrich

More information

Brief, Incomplete Summary of Some Literature on Ionization

Brief, Incomplete Summary of Some Literature on Ionization Page 1 Brief, Incomplete Summary of Some Literature on Ionization Regimes of Photo Ionization There are two limiting regimes for ionization in strong optical fields. From reference [1]. The ratio γ of

More information

A.J. Verhoef, A.V. Mitrofanov, D. Kartashov, A. Baltuska Photonics Institute, Vienna University of Technology. E.E. Serebryannikov, A.M.

A.J. Verhoef, A.V. Mitrofanov, D. Kartashov, A. Baltuska Photonics Institute, Vienna University of Technology. E.E. Serebryannikov, A.M. A.J. Verhoef, A.V. Mitrofanov, D. Kartashov, A. Baltuska Photonics Institute, Vienna University of Technology E.E. Serebryannikov, A.M. Zheltikov Physics Department, International Laser Center, M.V. Lomonosov

More information

Exact field ionization rates in the barrier suppression-regime. from numerical TDSE calculations. Abstract

Exact field ionization rates in the barrier suppression-regime. from numerical TDSE calculations. Abstract Exact field ionization rates in the barrier suppression-regime from numerical TDSE calculations D. Bauer and P. Mulser Theoretical Quantum Electronics (TQE), Darmstadt University of Technology, Hochschulstr.

More information

Wavelength scaling of high-order harmonic yield from an optically prepared excited state atom

Wavelength scaling of high-order harmonic yield from an optically prepared excited state atom Wavelength scaling of high-order harmonic yield from an optically prepared excited state atom J. Chen 1, 3, Ya Cheng 2,, and Zhizhan Xu 2, 1 Institute of Applied Physics and Computational Mathematics,

More information

stabilized 10-fs lasers and their application to laser-based electron acceleration

stabilized 10-fs lasers and their application to laser-based electron acceleration Carrier-envelope envelope-phase-stabilized stabilized sub-10 10-fs lasers and their application to laser-based electron acceleration L. Veisz, E. Goulielmakis, A. Baltuška, and F. Krausz Vienna University

More information

Electron dynamics in a strong laser field with Gaussian Potential well

Electron dynamics in a strong laser field with Gaussian Potential well Available online at www.worldscientificnews.com WSN 40 (2016) 265-284 EISSN 2392-2192 Electron dynamics in a strong laser field with Gaussian Potential well C. C. Gunatilaka, K. A. I. L. Wijewardena Gamalath*

More information

Rescattering of Ultra Low-Energy Electrons for Single Ionization of Ne in the Tunneling Regime

Rescattering of Ultra Low-Energy Electrons for Single Ionization of Ne in the Tunneling Regime 1 Rescattering of Ultra Low-Energy Electrons for Single Ionization of Ne in the Tunneling Regime R. Moshammer 1*, J. Ullrich 1, B. Feuerstein 1, D. Fischer 1, A. Dorn 1, C.D. Schröter 1, J.R. Crespo Lopez-Urrutia

More information

4. High-harmonic generation

4. High-harmonic generation Advanced Laser and Photn Science (Kenichi ISHIKAWA) for internal use only (Univ. of Tokyo) Kenichi Ishikawa () http://ishiken.free.fr/english/lecture.html ishiken@n.t.u-tokyo.ac.jp Advanced Laser and Photon

More information

High-contrast pump-probe spectroscopy with high-order harmonics

High-contrast pump-probe spectroscopy with high-order harmonics UVX 2008 (2009) 107 111 C EDP Sciences, 2009 DOI: 10.1051/uvx/2009017 High-contrast pump-probe spectroscopy with high-order harmonics Y. Mairesse 1,W.Boutu 2, P. Breger 2, E. Constant 1,D.Descamps 1, N.

More information

Multiple Ionization in Strong Laser Fields

Multiple Ionization in Strong Laser Fields in: Advances in Atomic and Molecular Physics ed.: B. Bederson and H. Walther, 2002 Multiple Ionization in Strong Laser Fields R. Dörner 1,Th. Weber 1,M.Weckenbrock 1, A. Staudte 1, M. Hattass 1, R. Moshammer

More information

Ideal laser waveform construction for the generation of super-bright attosecond pulses

Ideal laser waveform construction for the generation of super-bright attosecond pulses Home Search Collections Journals About Contact us My IOPscience Ideal laser waveform construction for the generation of super-bright attosecond pulses This content has been downloaded from IOPscience.

More information

Quantum mechanical approach to probing the birth of attosecond pulses using a twocolour field

Quantum mechanical approach to probing the birth of attosecond pulses using a twocolour field Quantum mechanical approach to probing the birth of attosecond pulses using a twocolour field Dahlström, Marcus; Lhuillier, A; Mauritsson, Johan Published in: Journal of Physics B: Atomic, Molecular and

More information

Wavelength Effects on Strong-Field Single Electron Ionization

Wavelength Effects on Strong-Field Single Electron Ionization Adv. Studies Theor. Phys., Vol. 2, 2008, no. 6, 271-279 Wavelength Effects on Strong-Field Single Electron Ionization J. Wu and Chunlei Guo The Institute of Optics, University of Rochester Rochester, NY

More information

The Lund Attosecond Science Centre in the MEDEA network PER THE MEDEA KICK-OFF MEETING, BERLIN, JANUARY 2015

The Lund Attosecond Science Centre in the MEDEA network PER THE MEDEA KICK-OFF MEETING, BERLIN, JANUARY 2015 The Lund Attosecond Science Centre in the MEDEA network PER JOHNSSON @ THE MEDEA KICK-OFF MEETING, BERLIN, JANUARY 2015 Lund University Founded in 1666 47 700 students (individuals) 7 500 employees - 840

More information

High-order harmonics with fully tunable polarization by attosecond synchronization of electron recollisions

High-order harmonics with fully tunable polarization by attosecond synchronization of electron recollisions High-order harmonics with fully tunable polarization by attosecond synchronization of electron recollisions,, Ofer Kfir, Zvi Diskin, Pavel Sidorenko and Oren Cohen Department of Physics and Optical Engineering,

More information

Provided for non-commercial research and educational use only. Not for reproduction, distribution or commercial use

Provided for non-commercial research and educational use only. Not for reproduction, distribution or commercial use Provided for non-commercial research and educational use only. Not for reproduction, distribution or commercial use This chapter was originally published in the book Advances in Atomic, Molecular, and

More information

On the angular dependence of the photoemission time delay in helium

On the angular dependence of the photoemission time delay in helium On the angular dependence of the photoemission time delay in helium I A Ivanov 1,2, J M Dahlström 3,4,5, E Lindroth 3 and A S Kheifets 1 1 Research School of Physics and Engineering, The Australian National

More information

Atomic and macroscopic measurements of attosecond pulse trains

Atomic and macroscopic measurements of attosecond pulse trains Atomic and macroscopic measurements of attosecond pulse trains Dahlström, Marcus; Fordell, Thomas; Mansten, Erik; Ruchon, T.; Swoboda, Marko; Klünder, Kathrin; Gisselbrecht, Mathieu; Lhuillier, A; Mauritsson,

More information

Time-dependent density functional theory

Time-dependent density functional theory Time-dependent density functional theory E.K.U. Gross Max-Planck Institute for Microstructure Physics OUTLINE LECTURE I Phenomena to be described by TDDFT LECTURE II Review of ground-state DFT LECTURE

More information

Lecture on: Multiphoton Physics. Carsten Müller

Lecture on: Multiphoton Physics. Carsten Müller Lecture on: Multiphoton Physics Carsten Müller Institut für Theoretische Physik I, Heinrich-Heine-Universität Düsseldorf Max-Planck-Institut für Kernphysik, Heidelberg IMPRS-QD Annual Event, MPIK, Heidelberg,

More information

Mid-Infrared Strong Field Ionization Angular Distributions 1

Mid-Infrared Strong Field Ionization Angular Distributions 1 ISSN 154-66X, Laser Physics, 29, Vol. 19, No. 8, pp. 1574 158. Pleiades Publishing, Ltd., 29. Original Russian Text Astro, Ltd., 29. ABOVE-THRESHOLD IONIZATION Mid-Infrared Strong Field Ionization Angular

More information

Measurement and control of the frequency chirp rate of high-order harmonic pulses

Measurement and control of the frequency chirp rate of high-order harmonic pulses Measurement and control of the frequency chirp rate of high-order harmonic pulses Mauritsson, Johan; Johnsson, Per; Lopez, Rodrigo; Varju, Katalin; Kornelis, W; Biegert, J; Keller, U; Gaarde, MB; Schafer,

More information

REFERENCES I. N. M. Allinson et al., Nucl. Instrum. & Methods Phys. Res. Sect. A 266, 592 (1988).

REFERENCES I. N. M. Allinson et al., Nucl. Instrum. & Methods Phys. Res. Sect. A 266, 592 (1988). LLE REVIEW, Volume 50 Conclusions Photodiode arrays provide the experimentalist with an x-ray detection medium that is more sensitive than film and has a larger dynamic range than film. They are appropriate

More information

Ionization of Rydberg atoms in Intense, Single-cycle THz field

Ionization of Rydberg atoms in Intense, Single-cycle THz field Ionization of Rydberg atoms in Intense, Single-cycle THz field 4 th year seminar of Sha Li Advisor: Bob Jones Dept. of Physics, Univ. of Virginia, Charlottesville, VA, 22904 April. 15 th, 2013 Outline

More information

Optical Spectroscopy of Advanced Materials

Optical Spectroscopy of Advanced Materials Phys 590B Condensed Matter Physics: Experimental Methods Optical Spectroscopy of Advanced Materials Basic optics, nonlinear and ultrafast optics Jigang Wang Department of Physics, Iowa State University

More information

Lecture Notes: March C.D. Lin Attosecond X-ray pulses issues:

Lecture Notes: March C.D. Lin Attosecond X-ray pulses issues: Lecture Notes: March 2003-- C.D. Lin Attosecon X-ray pulses issues: 1. Generation: Nee short pulses (less than 7 fs) to generate HHG HHG in the frequency omain HHG in the time omain Issues of attosecon

More information

Circularly-polarized laser-assisted photoionization spectra of argon for attosecond pulse measurements

Circularly-polarized laser-assisted photoionization spectra of argon for attosecond pulse measurements Circularly-polarized laser-assisted photoionization spectra of argon for attosecond pulse measurements Z. X. Zhao, Zenghu Chang, X. M. Tong and C. D. Lin Physics Department, Kansas State University, Manhattan,

More information

Ultrafast XUV Sources and Applications

Ultrafast XUV Sources and Applications Ultrafast XUV Sources and Applications Marc Vrakking Workshop Emerging Sources Lund, June 12th 2007 Overview: Attosecond Science from a user perspective What do we want? What do we use as our starting

More information

SUPPLEMENTARY INFORMATION

SUPPLEMENTARY INFORMATION SUPPLEMENTARY INFORMATION doi:10.1038/nature10820 S.1. Method for extracting bond length from laser-induced electron diffraction images The laser-induced electron diffraction (LIED) method aims at probing

More information

Construction of a 100-TW laser and its applications in EUV laser, wakefield accelerator, and nonlinear optics

Construction of a 100-TW laser and its applications in EUV laser, wakefield accelerator, and nonlinear optics Construction of a 100-TW laser and its applications in EUV laser, wakefield accelerator, and nonlinear optics Jyhpyng Wang ( ) Institute of Atomic and Molecular Sciences Academia Sinica, Taiwan National

More information

Strongly Dispersive Transient Bragg Grating for High Harmonics

Strongly Dispersive Transient Bragg Grating for High Harmonics SLAC-PUB-14092 Strongly Dispersive Transient Bragg Grating for High Harmonics J. P. Farrell, 1,2 L. S. Spector, 1,2 M. B. Gaarde, 1,3 B. K. McFarland 1,2, P. H. Bucksbaum, 1,2 and Markus Gühr 1,2 1 Stanford

More information

Ultrafast Laser Physics!

Ultrafast Laser Physics! Ultrafast Laser Physics! Ursula Keller / Lukas Gallmann ETH Zurich, Physics Department, Switzerland www.ulp.ethz.ch Chapter 10: Ultrafast Measurements Ultrafast Laser Physics ETH Zurich Ultrafast laser

More information

SUPPLEMENTARY INFORMATION

SUPPLEMENTARY INFORMATION Fig. S1: High-Harmonic Interferometry of a Chemical Reaction A weak femtosecond laser pulse excites a molecule from its ground state (on the bottom) to its excited state (on top) in which it dissociates.

More information

COHERENT X-ray sources are attractive because of their

COHERENT X-ray sources are attractive because of their 266 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 4, NO. 2, MARCH/APRIL 1998 Generation of Coherent, Femtosecond, X-Ray Pulses in the Water Window Zenghu Chang, Andy Rundquist, Haiwen Wang,

More information

Revival Structures of Linear Molecules in a Field-Free Alignment Condition as Probed by High-Order Harmonic Generation

Revival Structures of Linear Molecules in a Field-Free Alignment Condition as Probed by High-Order Harmonic Generation Journal of the Korean Physical Society, Vol. 49, No. 1, July 2006, pp. 337 341 Revival Structures of Linear Molecules in a Field-Free Alignment Condition as Probed by High-Order Harmonic Generation G.

More information

Models for Time-Dependent Phenomena

Models for Time-Dependent Phenomena Models for Time-Dependent Phenomena I. Phenomena in laser-matter interaction: atoms II. Phenomena in laser-matter interaction: molecules III. Model systems and TDDFT Manfred Lein p.1 Outline Phenomena

More information

Non-sequential and sequential double ionization of NO in an intense femtosecond Ti:sapphire laser pulse

Non-sequential and sequential double ionization of NO in an intense femtosecond Ti:sapphire laser pulse J. Phys. B: At. Mol. Opt. Phys. 30 (1997) L245 L250. Printed in the UK PII: S0953-4075(97)80013-2 LETTER TO THE EDITOR Non-sequential and sequential double ionization of NO in an intense femtosecond Ti:sapphire

More information

XUV frequency comb development for precision spectroscopy and ultrafast science

XUV frequency comb development for precision spectroscopy and ultrafast science XUV frequency comb development for precision spectroscopy and ultrafast science R. Jason Jones (PI) College of Optical Sciences, University of Arizona email: rjjones@optics.arizona.edu Collaborators Graduate

More information

2.B Laser Ionization of Noble Gases by Coulomb Barrier Suppression

2.B Laser Ionization of Noble Gases by Coulomb Barrier Suppression ADVANCED TECHNOLOGY DEVELOPMENTS 2.B Laser Ionization of Noble Gases by Coulomb Barrier Suppression Introduction The ionization of atoms and ions in high-intensity laser fields allows a study of atomic

More information

Ion momentum distributions for He single and double ionization in strong laser fields

Ion momentum distributions for He single and double ionization in strong laser fields PHYSICAL REVIEW A 66, 053415 2002 Ion momentum distributions for He single and double ionization in strong laser fields J. Chen and C. H. Nam Department of Physics and Coherent X-Ray Research Center, Korea

More information

Connecting Attosecond Science and XUV FEL Research

Connecting Attosecond Science and XUV FEL Research Connecting Attosecond Science and XUV FEL Research Marc Vrakking Attosecond Workshop Imperial College, May 13th 2008 Overview Present status of attosecond science - recent example: electron localization

More information

Laser heating of noble gas droplet sprays: EUV source efficiency considerations

Laser heating of noble gas droplet sprays: EUV source efficiency considerations Laser heating of noble gas droplet sprays: EUV source efficiency considerations S.J. McNaught, J. Fan, E. Parra and H.M. Milchberg Institute for Physical Science and Technology University of Maryland College

More information

High-Harmonic Generation

High-Harmonic Generation High-Harmonic Generation Kenichi L. Ishikawa Photon Science Center, Graduate School of Engineering, University of Tokyo Japan 1. Introduction We present theoretical aspects of high-harmonic generation

More information

Supplementary Material for In situ frequency gating and beam splitting of vacuum- and extreme-ultraviolet pulses

Supplementary Material for In situ frequency gating and beam splitting of vacuum- and extreme-ultraviolet pulses Supplementary Material for In situ frequency gating and beam splitting of vacuum- and extreme-ultraviolet pulses Rajendran Rajeev, Johannes Hellwagner, Anne Schumacher, Inga Jordan, Martin Huppert, Andres

More information

Attosecond optics and technology: progress to date and future prospects [Invited]

Attosecond optics and technology: progress to date and future prospects [Invited] Review Vol. 33, No. 6 / June 2016 / Journal of the Optical Society of America B 1081 Attosecond optics and technology: progress to date and future prospects [Invited] ZENGHU CHANG, 1, *PAUL B. CORKUM,

More information

Attosecond time delay in photoionization

Attosecond time delay in photoionization Attosecond time delay in photoionization J. Marcus Dahlström PhD: [LTH] -> Post-doc: [SU] -> Guest res.: [CFEL/MPG] -> Researcher [SU] ----------------------* http://www.medea-horizon2020.eu/training/webinar/

More information

Author(s): Niikura, Hiromichi; Wörner, Hans Jakob; Villeneuve, David M.; Corkum, Paul B.

Author(s): Niikura, Hiromichi; Wörner, Hans Jakob; Villeneuve, David M.; Corkum, Paul B. Research Collection Journal Article Probing the Spatial Structure of a Molecular Attosecond Electron Wave Packet Using Shaped Recollision Trajectories Author(s): Niikura, Hiromichi; Wörner, Hans Jakob;

More information

arxiv: v1 [cond-mat.mes-hall] 24 May 2013

arxiv: v1 [cond-mat.mes-hall] 24 May 2013 arxiv:35.56v [cond-mat.mes-hall] 4 May 3 Effects of excitation frequency on high-order terahertz sideband generation in semiconductors Xiao-Tao Xie Department of Physics, The Chinese University of Hong

More information

Abstract The development of intense, ultrashort, table-top lasers operating in the mid-infrared spectral region, offers many

Abstract The development of intense, ultrashort, table-top lasers operating in the mid-infrared spectral region, offers many IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 18, NO. 1, JANUARY/FEBRUARY 2012 419 Scaling of High-Order Harmonic Generation in the Long Wavelength Limit of a Strong Laser Field Anthony

More information

Recollision processes in strong-field QED

Recollision processes in strong-field QED Recollision processes in strong-field QED Antonino Di Piazza Program on Frontiers of Intense Laser Physics Santa Barbara, California, August 21st 2014 Outline Introduction to recollision processes in atomic

More information

As-few-as-possible-body physics (laser-induced nonsequential multiple ionization)

As-few-as-possible-body physics (laser-induced nonsequential multiple ionization) As-few-as-possible-body physics (laser-induced nonsequential multiple ionization) Wilhelm Becker Max Born-Institut, Berlin 614th Wilhelm und Else Heraeus Seminar, Bad Honnef, 18.-20. April 2016 Collaborators

More information

Wavelength Frequency Measurements

Wavelength Frequency Measurements Wavelength Frequency Measurements Frequency: - unit to be measured most accurately in physics - frequency counters + frequency combs (gear wheels) - clocks for time-frequency Wavelength: - no longer fashionable

More information

Dynamics of Laser Excitation, Ionization and Harmonic Conversion in Inert Gas Atoms

Dynamics of Laser Excitation, Ionization and Harmonic Conversion in Inert Gas Atoms UCRGJC122277 PREPRINT Dynamics of Laser Excitation, Ionization and Harmonic Conversion in Inert Gas Atoms Kenneth C. Kulander This was pre ared for submittal to the Progress in Crysta P Growth and Characterizations

More information

Strong-field double ionization of rare gases

Strong-field double ionization of rare gases Strong-field double ionization of rare gases J. L. Chaloupka, R. Lafon, L. F. DiMauro Chemistry Department, Brookhaven National Laboratory, Upton, NY 11973 USA jcha@bnl.gov P. Agostini SPAM, Centre d'etudes

More information

PLEASE SCROLL DOWN FOR ARTICLE

PLEASE SCROLL DOWN FOR ARTICLE This article was downloaded by: [Ohio State University Libraries] On: 23 October 2008 Access details: Access Details: [subscription number 788843122] Publisher Taylor & Francis Informa Ltd Registered in

More information

Attosecond time-delays in photoionisation

Attosecond time-delays in photoionisation Attosecond time-delays in photoionisation J. Marcus Dahlström 2018-08-31 ELISS Szeged, Hungary. Outline of lecture: Review of attosecond pulse characterization Simple models based on SFA How large is the

More information

A Coincidence Study of Strong Field Ionization of Rare. Gases

A Coincidence Study of Strong Field Ionization of Rare. Gases A Coincidence Study of Strong Field Ionization of Rare Gases By James Cryan In Fulfillment of the Requirements for Graduation with Distinction from The Ohio State University 2007 Abstract Single and double

More information

Effects of driving laser jitter on the attosecond streaking measurement

Effects of driving laser jitter on the attosecond streaking measurement Effects of driving laser jitter on the attosecond streaking measurement Shiyang Zhong, 1 Xinkui He, 1, Peng Ye, 1 Minjie Zhan, 1 Hao Teng 1 and Zhiyi Wei 1,* 1 Beijing National Laboratory for Condensed

More information

Electron Impact Ionization in the Presence of a Laser Field: A kinematically Complete (nγ e,2e) Experiment. Abstract

Electron Impact Ionization in the Presence of a Laser Field: A kinematically Complete (nγ e,2e) Experiment. Abstract Electron Impact Ionization in the Presence of a Laser Field: A kinematically Complete (nγ e,2e) Experiment C. Höhr, A. Dorn, B. Najjari, D. Fischer, C.D. Schröter, and J. Ullrich Max-Planck-Institut für

More information

Ultrafast nanoscience with ELI ALPS

Ultrafast nanoscience with ELI ALPS Ultrafast nanoscience with ELI ALPS Péter Dombi Wigner Research Centre for Physics, Budapest & Max Planck Institute of Quantum Optics, Garching Overview ultrafast (femtosecond/attosecond) dynamicsin metal

More information

Abnormal pulse duration dependence of the ionization probability of Na atoms in intense laser fields

Abnormal pulse duration dependence of the ionization probability of Na atoms in intense laser fields INSTITUTE OF PHYSICSPUBLISHING JOURNAL OFPHYSICSB: ATOMIC, MOLECULAR AND OPTICAL PHYSICS J. Phys. B: At. Mol. Opt. Phys. 36 (2003) 1121 1127 PII: S0953-4075(03)53141-8 Abnormal pulse duration dependence

More information

Lecture 22 Ion Beam Techniques

Lecture 22 Ion Beam Techniques Lecture 22 Ion Beam Techniques Schroder: Chapter 11.3 1/44 Announcements Homework 6/6: Will be online on later today. Due Wednesday June 6th at 10:00am. I will return it at the final exam (14 th June).

More information

1 Multielectron High Harmonic Generation: simple man on complex plane

1 Multielectron High Harmonic Generation: simple man on complex plane Chap. 1 2012/4/5 14:16 page 1 1 1 Multielectron High Harmonic Generation: simple man on complex plane Olga Smirnova and Misha Ivanov 1.1 Introduction Attosecond Science has emerged with the discovery of

More information

Molecular alignment, wavepacket interference and Isotope separation

Molecular alignment, wavepacket interference and Isotope separation Molecular alignment, wavepacket interference and Isotope separation Sharly Fleischer, Ilya Averbukh and Yehiam Prior Chemical Physics, Weizmann Institute Yehiam.prior@weizmann.ac.il Frisno-8, Ein Bokek,

More information

Attosecond Science (1) (1)

Attosecond Science (1) (1) Kenichi Ishikawa ( ) http://ishiken.free.fr/english/lecture.html ishiken@atto.t.u-tokyo.ac.jp Advanced Plasma and Laser Science E Attosecond Science (1) (1) 1 m n f a 3 10 8 (m/s) 30 10 15 (s) = 9 10 6

More information

Electron detachment from negative ions in a short laser pulse

Electron detachment from negative ions in a short laser pulse PHYSICAL REVIEW A 8, 9 () Electron detachment from negative ions in a short laser pulse S. F. C. Shearer, * M. C. Smyth, and G. F. Gribakin School of Mathematics and Physics, Queen s University Belfast,

More information

Tunneling Time in Ultrafast Science is Real and Probabilistic

Tunneling Time in Ultrafast Science is Real and Probabilistic Tunneling Time in Ultrafast Science is Real and Probabilistic Authors: Alexandra Landsman 1*, Matthias Weger *, Jochen Maurer, Robert Boge, André Ludwig, Sebastian Heuser, Claudio Cirelli, Lukas Gallmann,

More information

3 Controlling Coherent Quantum Nuclear Dynamics in LiH by Ultra Short IR Atto Pulses Astrid Nikodem, R.D. Levine and F. Remacle 3.

3 Controlling Coherent Quantum Nuclear Dynamics in LiH by Ultra Short IR Atto Pulses Astrid Nikodem, R.D. Levine and F. Remacle 3. Contents 1 Strong-Field S-Matrix Series with Coulomb Wave Final State... 1 F.H.M. Faisal 1.1 Introduction... 1 1.2 Three-Interaction Formalism... 2 1.3 Coulomb Volkov Hamiltonian and Propagator... 6 1.4

More information