High-Field Laser Physics

ETH Zürich

Spring Semester 2010

H. R. Reiss

OUTLINE

I. STRONG-FIELD ENVIRONMENTA. Atomic unitsB. The intensity-frequency diagramC. Inadequacy of the electric field as an intensity measureD. Failure of perturbation theory and its consequencesE. Measures of intensityF. Upper and lower frequency limits on the dipole approximation

II. S MATRIXA. Physically motivated derivation of the S matrixB. Perturbation theory from the S matrixC. Nonperturbative analytical approximationsD. The A2(t) term; classical-quantum difference

III. NONPERTURBATIVE METHODSA. TunnelingB. Strong-field approximation

IV. BASIC ELECTRODYNAMICSA. Fundamental Lorentz invariants: essential field identifiersB. Longitudinal and transverse fieldsC. Lorentz forceD. Gauges and their special importance in strong fields

2

OUTLINE (continued)

V. APPLICATIONSA. RatesB. SpectraC. Momentum distributionsD. Angular distributionsE. Higher harmonic generation (HHG)F. Short pulses

VI. EFFECTS OF POLARIZATIONVII. STABILIZATIONVIII. RELATIVISTIC EFFECTS

A. Relativistic notationB. Dirac equation and Dirac matricesC. Negative energy solutionsD. Relativistic quantum mechanics (RQM) vs. relativistic quantum field theory (RQFT)E. Relativistic rates and spectra

IX. OVERVIEWA. Very low frequenciesB. Compendium of misconceptions

3

INTENSITY-FREQUENCY DIAGRAM

An intensity-frequency diagram is very informative; it will be used repeatedly.

Some of the axis units are in a.u.More complete details on atomic units will follow.For now:

• 1 a.u. intensity = 3.511016 W/cm2 |F| = 1 a.u. applied electric field = Coulomb field at 1 a.u. from a Coulomb center.• 1 a.u. length = a0 = Bohr radius• 1 a.u. frequency = 1 a.u. energy = 2R = 27.2 eV = 45.56 nm

NOTE: The electric field will be designated by F rather than E to avoid confusion with the energy E.

4

5

10-3 10-2 10-1 100 101

Field frequency (a.u.)

10-3

10-2

10-1

100

101

102

103

104

105

106

107In

ten

sity (

a.u

.)104 103 102 101

Wavelength (nm)

1014

1015

1016

1017

1018

1019

1020

1021

1022

1023

Inte

nsity (

W/c

m2)

10.6 m

CO2

800 nm

Ti:sapph

100 eV

E=1a.u.=3.51e16W/cm2F

6

10-3 10-2 10-1 100 101

Field frequency (a.u.)

10-3

10-2

10-1

100

101

102

103

104

105

106

107In

ten

sity (

a.u

.)104 103 102 101

Wavelength (nm)

1014

1015

1016

1017

1018

1019

1020

1021

1022

1023

Inte

nsity (

W/c

m2)

10.6 m

CO2

800 nm

Ti:sapph

100 eV

E=1 a.u.

U p=

FFF

7

10-3 10-2 10-1 100 101

Field frequency (a.u.)

10-3

10-2

10-1

100

101

102

103

104

105

106

107In

ten

sity (

a.u

.)104 103 102 101

Wavelength (nm)

1014

1015

1016

1017

1018

1019

1020

1021

1022

1023

Inte

nsity (

W/c

m2)

10.6 m

CO2

800 nm

Ti:sapph

100 eV

E=1 a.u.

U p=

2U p=mc

2

FFFF

FF

F

8

10-3 10-2 10-1 100 101

Field frequency (a.u.)

10-3

10-2

10-1

100

101

102

103

104

105

106

107In

ten

sity (

a.u

.)104 103 102 101

Wavelength (nm)

1014

1015

1016

1017

1018

1019

1020

1021

1022

1023

Inte

nsity (

W/c

m2)

10.6 m

CO2

800 nm

Ti:sapph

100 eV

E=1 a.u.

U p=

2U p=mc

2

U. Laval

1988

F = 1

THE ELECTRIC FIELD IS NOT A GOOD INTENSITY MEASURE

The experiments at Université Laval (Canada) in 1988 used a CO2 laser at 10.6 m to ionize Xe at a peak intensity of 1014 W/cm2.

9000 photons were required just to supply the ponderomotive energy Up.

This was a very-strong-field experiment.

The maximum electric field was only |F| = 0.05 a.u.

Notice also that relativistic effects can be reached at low frequencies and small electric fields.

9

WHY IS THE INTENSITY LIMIT ON PERTURBATION THEORY SO IMPORTANT?

When using any power series expansion, it may not be obvious from qualitative behavior that the radius of convergence has been surpassed.

However, when the expansion is used beyond the radius of convergence, the result may be drastically in error.

A simple example can illustrate this.Consider the geometric series:

1|x|

...xxx1x1

1)x(F 32

To simplify matters, suppose x is positive: x> 0.

1/(1-x) changes sign when x > 1, but the expansion is a sum of positive terms!

Perturbation theory (PT) in quantum mechanics does not fail so spectacularly, but PT can be completely misleading.

10

0.2 0.4 0.6 0.8 1.0

x

-20

-10

0

10

20

1/(1-x)

1+x+x2+x3+x4+x5

Radius of convergence

11

ATOMIC UNITS

Define 3 basic quantities to be of unit value. This will set values for e2, ħ, m.

Unit of length:

Unit of velocity:

Unit of time:

m1029177249.5.u.a1me

a 11

2

2

0

s

m1018769142.2.u.a1

ev

2

meR

2

mvSet 6

2

02

42

0

s1041888433.2.u.a1meemev

aSet 17

4

3

22

2

0

00

These 3 definitions give:

The only solution is:

(Note: some of the choices, such as mv02 /2 = R were selected to give ħ = 1, m = 1, e2 = 1.)

43222 me,e,me

1e,1m,1 2

12

ATOMIC UNITS, continued

Derived quantities:

Unit of energy:

Unit of frequency:

Unit of electric field strength:

Unit of energy flux:

eV2113962.27.u.a1me

R2E2

4

0

Hz1013413732.4me

.u.a11 16

3

4

0

0

m/V1014220826.5.u.a1

cetandis

volts

length

1

e

voltselectronensionlessdim

mc

e

mc

c

emc

e

1me

ea

R2F

11

3

23

2

2

2

2

4

0

0

88

.u.a8

cthen.,u.a1Fcesin

2/

1mc

8F

8

c

.u.a

0

.u.a

00

2

e

2252

00

III

I

II

I

I

13

ATOMIC UNITS, continued 2

Convenient conversions:

Wavelength to frequency:

example: 800 nm (a.u.) = 45.5633526 / 800 = 0.0570 a.u.

Atomic units and mc2 units:

nm

5633526.45.u.a

25 mc1032513625.5.u.a1

.u.a1087788622.1unitmc1 42

14

15

PONDEROMOTIVE ENERGY

The electric field strength is not a useful indicator of the strength of a plane-wave field.(However, it is the only strength indicator for a quasistatic electric (QSE) field. Much more to be said later.)

The limit for perturbation theory: Up < Up = ponderomotive energyIndex for relativistic behavior: Up = O(mc2)

Ponderomotive energy is a useful guide to the coupling between an electron and a plane-wave field.

An electron in a plane-wave field oscillates (“quivers”, “wiggles”, “jitters”) in consequence of the electromagnetic coupling between the field and the electron. The energy of this interaction in the frame of reference in which it is minimal is the ponderomotive energy.

In a.u.:

onpolarizaticircularF

2

1

onpolarizatilinear2

F

2

F

2U

2

0

2

0

2

2

2p

I

16

k

F

FIGURE-8 MOTION OF A FREE ELECTRON IN A

LINEARLY POLARIZED PLANE-WAVE FIELD

CIRCULAR MOTION OF A FREE ELECTRON IN ACIRCULARLY POLARIZED PLANE-WAVE FIELD

F

B

17

MEASURES OF INTENSITY

Dimensionless measures are always to be preferred.

Up is basic, but it can be compared to 3 other relevant energies in strong-field problems: energy of a single photon of the field EB binding energy of an electron in a bound systemmc2 electron rest energy (actually, just c2 in a.u.; the m is retained because of its familiarity)

This leads to 3 dimensionless measures of intensity: z = Up / non-perturbative intensity parameter (note that z 1 /3 ) z1 = 2 Up / EB bound-state intensity parameter (z 1 /2 ) zf = 2 Up / mc2 free-electron intensity parameter (z 1 /2 )

A commonly used measure is the Keldysh parameter K = 1/ z11/2

K is the ratio of tunneling time to wave period; a measure that implies a tunneling process, which is not appropriate to ionization by a plane wave. It is generally the only parameter employed, which is not sufficient.

18

MEASURES OF INTENSITY, continued

z = Up /

This is called the non-perturbative intensity parameter because a channel closing is non-analytical behavior that limits the radius of convergence of a perturbation expansion. An ionized electron must have at least the energy Up + EB to be a physical particle. If Up , then at least one additional photon must participate to satisfy energy conservation. If the minimum number of photons n0 that must participate is increased from n0 to n0 +1, this is called a channel closing, and perturbation theory fails.

z1 = 2 Up / EB

This ratio compares the interaction energy of an electron with the field to the interaction energy of an electron with a binding potential. If Up > EB the field is dominant, if Up < EB then the binding potential is dominant. Validity of the Strong-Field Approximation (SFA) requires that the field should be dominant.

zf = 2 Up / mc2

This is the primary intensity parameter for free electrons.