Prof. David Attwood / UC Berkeley EE213 & AST210 / Spring 2009 14_HHG_2009.ppt
High Harmonic Generation
of
Coherent EUV/SXR Radiation
David Attwood
University of California, Berkeley
HHG_XtremeNonllinOpt.ai
HHG: Extreme nonlinear optics
• High harmonics of the intense 800 nm (1.55 eV) laser pulse• Photon energies throughout the EUV, extending to SXR• Spatially and temporally coherent• Femtosecond pulse duration, recently to attoseconds
Gas jet
EUV/SXRTi: sapphire laser800 nm30 fsec > 1014 W/cm2
CXRC
High-order Harmonic Generation (HHG)
High-order harmonicsHigh-order harmonics
Intense Intense fs fs laserlaser
Gas jetGas jet
!"#$!"#$
electron
%!&'(%!&'(
)*(!+
Signal processSignal process
Soft X-ray spectrometerSoft X-ray spectrometer
X-ray filterX-ray filter,,,,,,,,,,,,,,,,,,X-ray CCDX-ray CCD
-.-. /./. 0..0.. 00.00. 01.01. 02.02.0130130030030.30.3/3/3-3-3
High-order harmonicsHigh-order harmonics,4,4!!55
6736836-36/3
9#:("'&+,#;,<(#;'&&#(,9=!>?,6'',@!$A,BCDEF,G>HI'(&H"+A,J!'K'#>A,B#('!
Prof. David Attwood / UC Berkeley EE213 & AST210 / Spring 2009 14_HHG_2009.ppt
h!cutoff = Ip + 3.2Up
The physics of High Harmonic Generation (HHG)
U(x,t)
Ip electron
laser field
(Kulander et al, Corkum et al)
Ion electronEUV/SXR
Ionization
potential
of atom Up " I #2
Cycle averaged energy
of an oscillating electron
(ponderomotive energy
or potential)
Courtesy of Professor Henry Kapteyn, U. Colorado, Boulder
Prof. David Attwood / UC Berkeley EE213 & AST210 / Spring 2009 14_HHG_2009.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 & AST210 / Spring 2009 14_HHG_2009.ppt
Ultrashort light pulses:
• Ti: sapphire lasers, 800 nm wavelength
• The current state of the art ! 5-10 femtoseconds
• High-power amplifier systems: 15-25 fs
10 fs light pulse:
$x = 3 micrometers
c = 300 nm/fsec
Courtesy of Professor Henry Kapteyn, U. Colorado, Boulder
Ch06_F34VG.ai
1000
100
10
0
5
10
15
1 020 15Phot
on s
igna
l (m
V)
Inte
nsity
(arb
itrar
y un
its)
Har
mon
ic s
igna
l(a
rbitr
ary
units
)
10 125 135 145 155 12 16
H81
H61
H39
20 24 28
47 57 67 77 87 97
Wavelength (nm) Wavelength (nm)Harmonic order(5.2 nm)
L’Huillier and Balcou,Phys.Rev.Lett.70, 774 (1993)Neutral neon at 40 torr1.053 m, 1.5 1015 W/cm2
1 ps durationn = 135
Z. Chang, A. Rundquist,H. Wang, M. Murnane,H. Kapteyn,Phys.Rev.Lett.79, 2967 (1997)Neutral neon at 8 torr800 nm, 6 1015 W/cm2
26 fs durationn = 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, 3003 (1998)Neutral neon1.053 nm, 5 1014 W/cm2
700 fs durationn = 81 (polarization confirmed)
High Harmonic Generation (HHG) ofFemtosecond IR laser pulses into the EUV
Prof. David Attwood / UC Berkeley EE213 & AST210 / Spring 2009 14_HHG_2009.ppt
Typical HHG spectrum using argon:
Only odd-order harmonics are generated
25 29 39 45
Harmonic order
“Plateau”
“Cutoff”
J. Zhou, J. Peatross, M. M. Murnane, H. C. Kapteyn, and I. P. Christov, PRL 76, 752 (1996)
17
f λ = c
= = = fsec
HHG_equations.ai
Some HHG equations
1f
83
800 nm300 nm/fsec
λc
c = 300 nm/fsec
1212
12
0µ0
e2E2
2mω2
reIλ2
4πc
Iω2
e2
2mc0
∆EFWHM ∆τFWHM = 1.82 eV fsec
(152 asec pulse requiresa 12 eV bandwidth)
Uncertainty Principle
τ = =
τ = = 152 asec
(Tipler, Modern Physics, eq. 4-28)
Bohr orbit time (n = 1)
2πa0v
2πa0
c/137
2πa0αc
m(–iω)v = –eE
mv2 = ; I = E2
mv2 =
cycle averaged energy:
mv2 = πcreI/ω2 = = Up
Up = 9.33 10–14 I(W/cm2)[λ(µm)]2eV
Electron energy in an oscillating field
; re = e2/4π0mc2
Duration of one cycle
Speed of light
Prof. David Attwood / UC Berkeley EE213 & AST210 / Spring 2009 14_HHG_2009.ppt
Energetics
Ponderomotive potential (cycle averaged kinetic energy of a
free electron in an electric field E0 and frequency !0).
F = ma = eEoe!i"t = mdvdt
v =eEo
m# e
!i"tdt =
eEo
!i"me!i"t
Up = K.E.[ ]time avg =1
2mv
2 =e
2Eo
2
2m"2e!i"t[ ]
time avg=e
2Eo
2
4m"2
2
Courtesy of Professor Henry Kapteyn, U. Colorado, Boulder
v =
Prof. David Attwood / UC Berkeley EE213 & AST210 / Spring 2009 14_HHG_2009.ppt
Energetics (continued)
– Using (ch 2)
– We obtain
– Energy scale Ip + 3.2 Up = 24.6 eV + 192 eV
! 220 eV of HHG in He
I =!o
µoE
2
Up =e2E2
4m!2= 9.33 "10
#14I W
cm2( ) $ µm( )[ ]2
eV( )
= 60 eV @ 1015 W/cm2, 800 nm
Courtesy of Professor Henry Kapteyn, U. Colorado, Boulder
Prof. David Attwood / UC Berkeley EE213 & AST210 / Spring 2009 14_HHG_2009.ppt
Electron trajectory
• Assume:
– 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).
e-
atom
F = ma = eEoe!i"t = m dv
dt
v =eEo
m# e
!i"tdt =
eEo
!i"me!i"t$
% & '
( ) ti
* t
=eEo
!i"me!i" * t ! e
!i"ti[ ] =dx
dt
since v ti( ) = 0
v =
Courtesy of Professor Henry Kapteyn, U. Colorado, Boulder
= m dv/dt
since v(ti) = 0
Prof. David Attwood / UC Berkeley EE213 & AST210 / Spring 2009 14_HHG_2009.ppt
• Solve for trajectory:
– Electron released at atom: x(ti) = 0
– Electron trajectory ends at atom, for HHG : x(tf) = 0
• Solve for tf
• Find v(tf)
• Find return energy of electron E= 1/2mv2
dx
dt=
eEo
!i"me!i" # t ! e
!i"ti[ ]
x =eEo
!i"me!i" # t ! e
!i"ti[ ]dt'ti
t f$ =eEo
!"2me!i" # t ! e
!i"ti[ ]%
& '
(
) * ti
t f
Courtesy of Professor Henry Kapteyn, U. Colorado, Boulder
Electron trajectory
Prof. David Attwood / UC Berkeley EE213 & AST210 / Spring 2009 14_HHG_2009.ppt
• Most electrons don’t have opportunity to recollide
• Transverse “spread” of electron wavefunction further
reduces recollisions
Electron trajectories
Courtesy of Professor Henry Kapteyn, U. Colorado, Boulder
ω
Zero kineticenergy upon return
Electron liberated (“born”) atpeak of pulse
φ = 0°
800 nmE-field
For λ = 800 nmI = 5 1014 W/cm2
Time (phase of E-field)
–90°–5
–4
–3
–2
–1
0
1
90° 180° 270° 360° 450°0
Dis
tanc
e fr
om Io
n (n
m)
HHG_YWL_1a.ai
HHG electron trajectories and return energies depend on time of liberation vis à vis the driving electric field
φ = –10°
φ = 0°
800 nmE-field
For λ = 800 nmI = 5 1014 W/cm2
Time (phase of E-field)–90°
–5
–4
–3
–2
–1
0
1
90° 180° 270° 360° 450°0
Dis
tanc
e fr
om Io
n (n
m)
HHG_YWL_1b.ai
HHG electron trajectories and return energies depend on time of liberation vis à vis the driving electric field
ω
ω
Zero kineticenergy upon return
Electronneverreturns
Electron bornbefore peak of pulse
Electron liberated (“born”) atpeak of pulse
ω
ω
ω
Zero kineticenergy upon return
Electronneverreturns
Electron bornbefore peak of pulse
Electron bornafter peak of pulse
Electron liberated (“born”) atpeak of pulse
Maximumkinetic energyat φ = 18° & 198°
φ = –10°
φ = 0°
φ = +15°800 nmE-field
For λ = 800 nmI = 5 1014 W/cm2
Time (phase of E-field)–90°
–5
–4
–3
–2
–1
0
1
90° 180° 270° 360° 450°0
Dis
tanc
e fr
om Io
n (n
m)
HHG electron trajectories and return energies depend on time of liberation vis à vis the driving electric field
HHG_YWL_1c.ai
Courtesy of Dr. Yanwei Liu, UC Berkeley and Lawrence Berkeley National Lab
φ = –45°
φ = –10°
φ = 0°
φ = +5°
φ = +15°
800 nmE-field
Recombinationkinetic energy83 eV at φ = 15°46 eV at φ = 5°
For λ = 800 nmI = 5 1014 W/cm2
Time (phase of E-field)–90°
–5
–4
–3
–2
–1
0
1
90° 180° 270° 360° 450°0
Dis
tanc
e fr
om Io
n (n
m)
HHG_YWL_1d.ai
HHG electron trajectories and return energies depend on time of liberation vis à vis the driving electric field
Courtesy of Dr. Yanwei Liu, UC Berkeley and Lawrence Berkeley National Lab
ω
ω
ω
Zero kineticenergy upon return
Electronneverreturns
Electron bornbefore peak of pulse
Electron bornafter peak of pulse
Electron liberated (“born”) atpeak of pulse
Maximumkinetic energyat φ = 18° & 198°
λ = 800 nmI = 5 1014 W/cm2
Neon (Ip = 21.6 eV)
Shorttrajectories
Longtrajectories
Maximum return energy(HHG “cutoff”)
Time (phase of E-field)–90°
–3
–2
–1
0
1
90° 180° 270° 360°0
Dis
tanc
e fr
om Io
n (n
m)
48°
47 eV 69 eV 94 eV 106 eV 85 eV 52 eV
38°
28°
18°
8°
3°
Different electron trajectories (times of birth) result in variedreturn energies, different path lengths, and thus differenttimes of emission – causing a “chirp” (photon energy vs time)
HHG_YWL_2.ai
Courtesy of Dr. Yanwei Liu, UC Berkeley and Lawrence Berkeley National Lab
HHG_YWL_3.ai
Electron return energy as a function of liberationtime vis à vis the driving electric field
No HHG No HHG
E-field
Time (phase of E-field)
Shorttrajectory
Longtrajectory
–90°–1
0
1
2
3
90° 180° 270°0
Ret
urn
ener
gy, E
/Up
φ = 18° φ = 198°
λ = 800 nm, I = 5 1014 W/cm2
Neon (Ip = 21.6 eV)
Perfectly periodic emissions generate only odd harmonics
HHG_YWL_4.ai
Time (phase of E-field)
80
60
40
20
0
120
100
180° 360° 540° 720° 900° 1080° 1260°0
Em
itted
pho
ton
ener
gy (
eV)
E-field
Courtesy of Dr. Yanwei Liu, UC Berkeley and Lawrence Berkeley National Lab
HHG_cohFempto_Mar2009.ai
High Harmonic Generation (HHG)Provides Coherent, Femtosecond Pulses
Courtesy of Professors Margaret Murnane and Henry Kapteyn, Univ. Colorado, Boulder,and Dr. Yanwei Liu, U. California, Berkeley, and LBL.
150 µm Fiber
with 30 Torr Argon
Ultrafast laser beam
(760 nm, 25 fs)
Filter
EUV
beam
EUV CCD
Pinholes
x (mm)Li
neou
t
y (m
m)
0 5–5
–3 –0
0
36 nm
P 10 µW → 2 × 1012 ph/sec @ 36 nm (n = 21; 34 eV)
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 2002).