High Harmonic Generation of Coherent EUV/SXR Radiationattwood/sxr2009/... · Prof. David Attwood / UC Berkeley EE213 & AST210 / Spring 2009 14_HHG_2009.ppt The HHG Process 1.A high

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#('!


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


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).


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


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


EUV High Harmonics:


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


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


v =


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



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 =


= m dv/dt

since v(ti) = 0


• 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


Electron trajectory


• Most electrons don’t have opportunity to recollide

• Transverse “spread” of electron wavefunction further

reduces recollisions

Electron trajectories


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


ω

ω


Electronneverreturns

Electron bornbefore peak of pulse


ω

ω

ω




Electron bornafter peak of pulse


Maximumkinetic energyat φ = 18° & 198°

φ = –10°

φ = 0°

φ = +15°800 nmE-field

For λ = 800 nmI = 5 1014 W/cm2


–5

–4

–3

–2

–1

0

1

90° 180° 270° 360° 450°0

Dis

tanc

e fr

om Io

n (n

m)


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


–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



ω

ω

ω




Electron bornafter peak 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”)


–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


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


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


80

60

40

20

0

120

100

180° 360° 540° 720° 900° 1080° 1260°0

Em

itted

pho

ton

ener

gy (

eV)

E-field


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).

Why Use Hollow Fibers?

WhyHollowFibers.ai

Documents

High Harmonic Generation of Coherent EUV/SXR Radiationattwood/sxr2009/... · Prof. David Attwood / UC Berkeley EE213 & AST210 / Spring 2009 14_HHG_2009.ppt The HHG Process 1.A high