Copyright

by

Alexandre S Maltsev

2004

Above Threshold Ionization with Ultrahigh Intensity

Laser Light

by

Alexandre S Maltsev, B.S.

THESIS

Presented to the Faculty of the Graduate School of

The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

MASTER OF ARTS

THE UNIVERSITY OF TEXAS AT AUSTIN

December 2004

Above Threshold Ionization with Ultrahigh Intensity

Laser Light

APPROVED BY

SUPERVISING COMMITTEE:

Todd Ditmire , Supervisor

To my special people...

Acknowledgments

First of all I would like to thank my supervisor Prof. Todd Ditmire

who made this work possible. Todd’s style of managing the group is abso-

lutely great and I couldn’t wish for better. Watching him I learnt what a good

team leader should be like.

I am grateful to Anatoly Maksimchuk who was always helping me with

the experiment. He also showed me how persistent a good scientist should be

- as a matter of fact he was so persistent trying to make my experiment work

that he got his family upset because of spending too much time in the lab.

Special thanks go to Galina Kalintchenko and Vladimir Chvykov who spent

long hours in the Hercules lab providing us with laser light.

Finally I want to thank Prof. Michael Downer who was so kind to be

the co-reader of this thesis.

v

Above Threshold Ionization with Ultrahigh Intensity

Laser Light

Alexandre S Maltsev, M.A.

The University of Texas at Austin, 2004

Supervisor: Todd Ditmire

This document has the form of a “fake” doctoral dissertation in order

to provide an example of such, but it is actually a copy of Miguel Lerma’s doc-

umentation for the Mathematics Department Computer Seminar of 25 March

1998 updated in July 2001 and following by Craig McCluskey to meet the

March 2001 requirements of the Graduate School.

This document and its source file show to write a Doctoral Dissertation

using LATEX and the utdiss2 package.

vi

Table of Contents

Acknowledgments v

Abstract vi

List of Figures viii

Chapter 1. Theory 1

1.1 Ionization by Laser Light . . . . . . . . . . . . . . . . . . . . . 1

1.2 Calculating Ion Yields . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Exact fields in the focus . . . . . . . . . . . . . . . . . . . . . 10

1.4 Non-Relativistic Above Threshold Ionization (ATI) . . . . . . 14

1.5 Relativistic ATI . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.6 Simulations of Relativistic ATI . . . . . . . . . . . . . . . . . . 27

Bibliography 30

Vita 32

vii

List of Figures

1.1 BSI: a) Coulomb potential and laser potential combine b) to setelectron free. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Electron tunnels through the potential barrier. . . . . . . . . . 4

1.3 Dependence of ion yields on intensity for neon. . . . . . . . . . 10

1.4 Fields in a focused beam: a) paraxial approximation, b) com-plete description. . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Electrons ”slide off the potential hill”. . . . . . . . . . . . . . 17

1.6 Shape of the electron trajectory upon ionization at ultrahighintensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.7 Electron surfing a laser wave. . . . . . . . . . . . . . . . . . . 21

1.8 Dependence of the electron energy on the position in the beam. 26

1.9 Simulation results for 1×1018 W/cm2. Dependence of the ejec-tion angle on the electron energy for (a) paraxial approximation,(b) longitudinal fields included. . . . . . . . . . . . . . . . . . 28

1.10 Simulation results for 5×1021 W/cm2. Dependence of the ejec-tion angle on the electron energy for (a) paraxial approximation,(b) longitudinal fields included. . . . . . . . . . . . . . . . . . 29

viii

Chapter 1

Theory

Some theory goes next.

1.1 Ionization by Laser Light

Ionization by light can be described in different terms depending on the

intensity of the laser and on the ionization potential of a given atom or ion.

At lower intensities the process is most adequately described as a multiphoton

ionization:

An+ + N~ ω → A(n+1)+ + e−,

where N~ ω > Ip (see also section Non-Relativistic Above Threshold Ionization

(ATI)). However, we are interested in higher intensities, in which case the

process of ionization is more appropriately described as a tunnelling process.

These two regimes can be more quantitatively distinguished using the Keldysh

tunnelling parameter [1]:

γ = (Ip/2Φpond)1/2, (1.1)

where Φpond is the ponderomotive potential of the laser and Ip is the ionization

potential of the atom or ion. The ponderomotive potential is given by

Φpond [eV] = e2E2/4meω

2 = 9.33× 10−14Iλ2, (1.2)

1

where E is the electric field of the laser, I is the corresponding intensity in

W/cm2, and λ is the wavelength in µm.

Typically, in laser ionization experiments the measured quantity is the number

of ions of a particular ionic species per shot. A popular way of estimating the

minimum intensity at which the ionic species is observed (the ”appearance

intensity”) is using the barrier suppression ionization (BSI) mechanism. BSI

was demonstrated to predict the appearance intensities quite well by Augst et

al. in [2]. The idea is to consider a system of an electron and a charged core,

and describe this system using the simple Coulomb potential. The field of

the laser is considered as uniform and static, which is reasonable because the

spatial variations in a laser beam happen at scales much larger than the size of

an atom, and the laser oscillation period is much larger than the characteristic

atomic time scale. As shown in Fig. 1.1 a sufficiently strong electric field of

the laser can completely suppress the Coulomb barrier and let the electron

escape. Calculating the appearance intensities through BSI is quite simple.

The total potential can be written as

U(r) = −Ze2/r − eEr. (1.3)

To find the maximum of the potential we use dU/dr = Ze2/r2 − eE = 0,obtaining

rmax =√

Ze/E. (1.4)

We then assume that the ionization potential is equal to the maximum of the

combined potential, U(rmax) = −Ip. This gives us the electric field required

2

Figure 1.1: BSI: a) Coulomb potential and laser potential combine b) to setelectron free.

for the barrier suppression:

Ethreshold =I2p

4e3Z. (1.5)

The corresponding threshold intensity is found from Ithreshold = (c/8π)E2threshold,

and transforming to the practical units:

Ithreshold [W/cm2] ≈ 4× 109 Ip [eV]

Z2. (1.6)

The Z in this equation should refer to an ”effective charge” of the core, which

should take into account the deviation of the actual potential, experienced by

the electron in the atom or ion, from the assumed simple Coulomb potential.

Very often, however, Z is taken to be simply the charge of the ion created in

the ionization process. Appendix .. lists ionization potentials and calculated

BSI thresholds for some important species.

Quantum mechanics, of course, permits the electron to get ionized even when

3

the Coulomb barrier is not completely suppressed - through tunnelling (Fig.

1.2). There are several theories calculating the probability rate for the elec-

Figure 1.2: Electron tunnels through the potential barrier.

tron to tunnel ionize at a given intensity. A detailed comparison of the rates

predicted by different theories with experimental results at intensities from

mid-1013 W/cm2 to mid-1016 W/cm2 was performed by Augst et al. in [3].

They showed that the species dependence of the ionization rates is best de-

scribed by the theory developed by Ammosov, Delone and Krainov [4]. In ADK

model the initial atomic or ionic state is described by the effective principal

quantum number n∗ (n∗ = Z/(2Ip)1/2, where Z is the charge of the created

ion), the orbital angular momentum, and the magnetic quantum numbers l

and m. The ionization rate is given by

W = ωAC2n∗lf(l, m) Ip

(3E

π(2Ip)3/2

)1/2 (2

E(2Ip)

3/2

)2n∗−|m|−1(1.7)

× exp(− 2

3E(2Ip)

3/2

),

4

where ωA is the atomic unit of frequency (ωA = 4.134 × 1016 sec−1), E is theelectric field in atomic units (EA = 5.142 × 109 V/cm), Ip is the ionizationpotential in hartrees (1 hartree = 27.2 eV) and the factors f and C are given

by

f(l,m) =(2l + 1)(l + |m|)!

2|m|(|m|)!(l − |m|)! , (1.8)

Cn∗l =(

2e

n∗

)n∗1

(2πn∗)1/2, (1.9)

where e is the base of natural logarithm. The rate shown in Eq. (1.7) is an

approximation that is valid for n∗ À 1, E ¿ 1, and ω ¿ Ip. Thus, the valid-ity of ADK improves as n∗ increases, i.e. for heavier atoms and higher charge

states.

I would like to make a remark that normally we consider ionization of multi-

electron atoms or ions as a sequential process, i.e. the electrons are torn away

one by one. This description seems to be reasonable because the time scale

for atomic processes is on the order of 10−17 sec, while the characteristic laser

time is on the order of 10−15 sec. But this reasoning is only perfectly true

when the ionized electron cannot affect the remaining electrons anymore. It

was experimentally shown that the process of non-sequential (NS) ionization

can also take place. Two mechanisms for NS ionization were proposed, that

use a simple two-step picture. In the first step the electron passes over or tun-

nels through the Coulomb barrier. In the second step the electron’s motion

is dominated by the action of the laser fields. At non-relativistic intensities

the electron gets accelerated away from the core and then accelerated back

5

towards the core during the next half cycle. In the mechanism proposed by

Corkum [5] the electron can collisionally ionize the second electron (e → 2e)when it revisits the core (rescattering). Fittinghoff et al. [6] suggested that

the second electron can get ionized because of the sudden loss of screening due

to the rapid removal of the first electron (shakeoff). Walker et al. performed

measurements of NS ionization rates over 12 orders of magnitude in ion signal

[7]. At relativistic intensities, however, the rescattering process is suppressed

because the v×B forces do not allow the first ionized electron to get back tothe core. This phenomenon of NS ionization suppression was experimentally

verified by Chowdhury and Walker in [8].

1.2 Calculating Ion Yields

It was mentioned in the previous chapter that in laser ionization ex-

periments the measured quantity is the number of ions of a particular species.

Thus, in order to compare the predictions of a chosen model of ionization with

the experiment one needs to calculate the ion yields for different species. In an

actual optical system the intensity distribution in the focal region can be very

complex. Even though that could probably be taken into account in numerical

calculations, usually the situation is simplified by assuming an ideal Gaussian

focusing. In this case it is possible to find exact equations of the ”isointensity

surfaces”, i.e. the 3D-surfaces of constant intensity. An isointensity surface

is characterized by the parameter I0/I, where I0 is the peak intensity in the

focus, and I is an intensity of interest. The volume of space limited by an

6

isointensity surface can be calculated analytically, and is given by

V

(I0I

)=

π2w40λ

(2

9Z3 +

4

3Z − 4

3tan−1 Z

), (1.10)

where Z =√

I0/I − 1, w0 is the beam waist (1/e2 radius for intensity), and λ -the laser wavelength. Using the knowledge about the isointensity surfaces one

can split the focal volume into ”shells” - volumes limited by two neighboring

isointensity surfaces. Shells can be made thin enough such that in the process

of computations one could claim that all the volume in one shell is experiencing

approximately the same intensity. In this case the ionization probability is the

same for all the atoms or ions in one shell. There can be different methods

of choosing the thicknesses of shells such that they could be considered thin.

The intensity of the beam exactly in the focal plane is given by

I(r) = I0 exp(−r2/w20).

And let’s say we consider a shell thin if the difference of intensities between

the two limiting isointensity surfaces is ∆I = αI0, where α is a small num-

ber. The difference in intensities can be approximated by ∆I(r) ≈ I ′(r)∆r.Differentiating intensity

I ′(r) = I0 · (−2r/w20) exp(−r2/w20).

There are two methods for splitting into shells that seem most natural. First,

one can choose to have shells of different thickness ∆r and keep |I ′(r)∆r| =|∆I(r)| = αI0, which gives

(2r/w20) exp(−r2/w20)∆r = α =⇒ ∆r1 = α (w20/2r) exp(r2/w20). (1.11)

7

Second method is to have the shells of the same thickness ∆r = const,

but thickness small enough so that (∆I)max = |I ′(r)|max ∆r = αI0. Wecan, of course, find |I ′(r)|max by finding the second derivative of the inten-sity and equating it to zero. Ultimately we get |I ′(r)|max = |I ′(w0/

√2)| =

I0(√

2/w0) exp(−1/2), so that

∆r2 = α

√e

2w0 ≈ α · 1.166 w0. (1.12)

The first method is a little more difficult to realize, so at the time of writing

the code for calculating ion yields I chose the second method. In general,

the evolution of the different charge states in time is described by a series of

first-order coupled ordinary differential equations [9]

dnjdt

=

j−1∑i=0

Wijni −Zmax∑

k=j+1

Wjknj(r, t), (1.13)

where Wij is the ionization rate from species i to j, n(r, t) is the number

density of species i at a specific point in space r at a time t, and Zmax is the

maximum charge state. We, however, consider only sequential ionization, so

we get a much simpler expression

dnjdt

= Wj−1, j nj−1 −Wj, j+1 nj. (1.14)

Notice that we can integrate these series of equations for every shell separately,

as ions presumably don’t have time to move much within the focus during

the laser pulse propagation. The shape of the laser pulse is significant as it

determines the rates Wij at a given time. So the description of the algorithm

8

for calculating ion yields is:

1) Initialize each shell by setting the number of atoms proportional to the

volume of the shell, and setting the numbers of all other ionization states to

zero.

2) Solve the equations (1.14) for each shell separately to get the numbers

(nKi )final, where K is the shell number and i is the ionization state.

3) Sum the ion populations for different species over all the shells to get the

total ion yields

ntotali =Kmax∑K=0

(nKi )final. (1.15)

In order to make a comparison with a theory one needs to calculate the ion

yields dependence on the peak intensity of the laser, so the three steps of the

above algorithm must be repeated for every peak intensity in the region of

interest. The ultimate result could look like Fig. 1.3.

It is easy to notice in the figure that above certain intensity the curve

turns into a straight line. This is a saturation effect that takes place when

I0/Im À 1, where Im is the appearance intensity for ionization state m. Itcan be seen from Eq. (1.10) that in this case Z ≈ (I0/Im)1/2 and the Z3 termdominates. Thus a shell volume ∆V = V (Ii) − V (Ij) ∝ I3/20 , for arbitraryvalues of Ii and Ij (as long as they satisfy I0/I À 1). The saturation resultsfrom the fact that the intensity is so high that all the atoms/ions within the

shell get ionized and the yield becomes simply proportional to the volume of

the shell.

9

Figure 1.3: Dependence of ion yields on intensity for neon.

1.3 Exact fields in the focus

For most applications the electromagnetic fields in a focused laser beam

can be described using the so-called ”paraxial approximation”. Paraxial ap-

proximation assumes that the electric and magnetic fields are exactly perpen-

dicular to the direction of propagation of the beam and are mutually perpen-

dicular (Fig. 1.4(a)). However if one tries to imagine the picture of propagation

of a focused beam, it becomes clear that the paraxial approximation can only

hold well for ”soft focusing”, that is only for systems with large f-numbers.

This point can be made using some simple physical considerations. A flow of

electromagnetic energy in a point in space far from focus can be characterized

by the local Pointing vector S =1

µE×B. Tighter focusing thus implies that

10

Pointing vectors make greater angles with the beam axis. But from the defi-

nition of the Pointing vector it follows that the local plane of vectors E and

B makes a sharper angle with the beam axis (Fig. 1.4(b)).

Figure 1.4: Fields in a focused beam: a) paraxial approximation, b) completedescription.

The exact fields of a focused beam can be calculated. For my work I

used results by Quesnel and Mora [10]. As the starting point of their derivation

they assume a Gaussian profile with beam waist w0 for the transverse electric

field exactly at the focal plane, i.e. Ex(x, y, z = 0) = E0 exp(−(x2 + y2)/w20).They then derive integral expressions for all the components of both electric

and magnetic fields, which were not of particular interest for me because they

would require too much computational time if used in simulations. However,

11

the authors proceed to derive the expansion of the integral expressions in

terms of the small parameter ε = 1/kw0, where k is the wave number of

the laser light. Turns out that the transverse field components (along x and

y) contain only even powers of ε, while the longitudinal ones (along z) only

contain odd powers. Thus the second term in every expansion is of the order

ε2 as compared to the first term, and is not of great significance. Even though

the components Ey and Bx are second order in ε, it might not be reasonable to

neglect them completely from the very beginning, as one wouldn’t want to lose

any interesting effects when doing simulations. But after I got some insight

into the physics of electron acceleration by laser light it became evident that

the most interesting effects come from the longitudinal electric field, as will

be demonstrated later in this text, and small transverse components are of no

significance. Following are the first order formulas that I used in my work:

Ex = E0w0w

exp(− r2

w2) sin ϕG (1.16a)

Ez = 2E0εxw0w2

exp(− r2

w2) cos ϕ

(1)G (1.16b)

By = Ex/c (1.16c)

Bz = 2E0c

εyw0w2

exp(− r2

w2) cos ϕ

(1)G (1.16d)

Ey = Bx = 0 (1.16e)

where ϕG = ωt− kz +tan−1(z/zR)− zr2/zRw2−ϕ0, ϕ(1)G = ϕG +tan−1(z/zR);w = w0

√1 + z2/z2R is the beam waist at longitudinal position z, zR = kw

20/2

is the Rayleigh length, and ϕ0 is an arbitrary constant.

It may be of some interest to see how the characteristic numbers in these formu-

12

las (w0, zR, ε) depend on the optical system used for focusing, i.e. ultimately

on the f-number of the system - f# = F/D, where F is the characteristic

focal length, and D is the characteristic aperture. For example, when using a

parabolic mirror as a focusing element, F is the effective focal length of the

mirror and D is the beam diameter (if it’s a flat-top beam and it can reflect

completely off the mirror). I’d like to point out that for a tightly focusing

parabolic mirror the definition of f# becomes somewhat vague.

It is known that for a perfectly focused Gaussian beam the beam waist is given

by w0 =2

πλf#. Consequently we have:

zR =kw202

=4

πλ(f#)2, ² =

1

kw0=

1

4f#.

For a system with f# = 2 and λ=800 nm, which is close to the experimental

conditions described later, we have: w0 ≈ 1 µm, zR ≈ 4 µm, ² = 0.125. Thisfocusing is already very tight and ² is quite large, though ²2 ≈ 0.01 is stillsmall.

Usually in the lab the quality of focusing is characterized by the full width at

half-maximum (FWHM) of the intensity distribution in the focal plane. At

the same time w0 is 1/e radius for the electric field distribution. Using I ∝ E2

to relate the two distributions, we find ρFWHM = w0√

2 ln 2 ≈ 1.177w0 for anideal Gaussian beam.

13

1.4 Non-Relativistic Above Threshold Ionization (ATI)

Albert Einstein received the Nobel prize in physics in 1921 for explain-

ing the photoelectric effect in 1905. The photoelectric effect refers to the

emission, or ejection, of electrons from the surface of, generally, a metal in

response to incident light. If using the classical Maxwell wave theory of light,

one would expect ejected electrons to have higher kinetic energy for higher

intensity of the incident light. It was experimentally found, however, that

the energies of the emitted electrons were independent of the intensity of the

incident radiation. Einstein resolved this paradox by using quantum descrip-

tion of light. Photoelectric effect is very close in nature to ionization of single

atoms by light (photoionization). Einstein’s theory suggests that an atom can

absorb a photon whose energy is higher than the atom’s ionization potential

and the electron emitted in this process will have a kinetic energy given by

K = ~ω− Ip, where ω is the photon’s frequency and Ip - the ionization poten-tial.

In 1931 Goeppert-Mayer first considered two-photon transitions. Experimen-

tal verification of his theory required very intense sources of EM energy. Such

sources in radiofrequency domain were developed in 1950s and allowed for

observation of multiphoton transitions between bound states of atoms and

molecules. The first lasers in the early 1960s provided an opportunity to ob-

serve multiphoton ionization.

Intensity of light characterizes the ”concentration of photons” in a beam. Mul-

tiphoton ionization is a probabilistic process requiring interaction of N photons

14

with one bound electron. The probability that there is a photon in a vicin-

ity of a particular electron is proportional to the concentration of photons -

thus proportional to intensity. Therefore the N -photon ionization rate should

vary as σNIN , where σN is a generalized N -photon ionization cross-section.

Though σN decreases with increasing N , an N -photon process can be observed

at sufficiently high intensity.

Initially for multiphoton processes a simple extrapolation of Einstein’s picture

was suggested, i.e. after absorbing N photons an electron was expected to

have kinetic energy K = N~ω − Ip. However, in 1979 Agostini et al. [11] ob-served that the energy spectrum of electrons produced in 6-photon ionization

of xenon atoms consisted of two peaks corresponding to absorption of 6 and

7 photons. Results produced in other labs for various atomic species, intensi-

ties and laser wavelengths showed spectra consisting of multiple peaks evenly

spaced by the photon energy. This phenomenon was named above threshold

ionization (ATI). As is well known, a free electron cannot absorb a photon

because that would break the law of conservation of momentum. So it was

suggested that during the process of ionization electron stays in the neighbor-

hood of its parent ion for long enough to absorb additional photons, the ion

serving the function of a momentum sink.

An important result was obtained in 1983 in 11-photon ionization of xenon

at 1064 nm with a laser intensity of about 1013 W/cm2 (Kruit et al. [12]).

This experiment revealed that the first peak of the electron spectrum was

smaller than the peaks corresponding to absorption of larger numbers of pho-

15

tons. Moreover, the first peak was decreasing with increasing intensity. Later,

in experiments conducted in helium with the intensity of about 1015 W/cm2,

disappearance of 30 peaks was demonstrated. This effect can in principle be

explained by considering multiple stimulated scattering events, with photons

in a focused beam carrying momenta of different directions. It is much more

reasonable for high intensities, however, to describe the laser light classically.

In the non-relativistic case when the maximum velocity achieved by the elec-

tron is small (vmax ¿ c) the motion of the electron is completely dominatedby the electric field of the laser light. This motion will be simply oscillation

with laser frequency. In case of a plane wave and infinite pulse duration the

electron would be oscillating around a single point. Now if we look at more

realistic situations, the electron motion should be affected by both temporal

effects (intensity changing with time) and spatial effects (intensity gradient in

a focused beam). The most widely accepted description of spatial effects is

that using the concept of ”ponderomotive potential”. Ponderomotive poten-

tial is given by:

Up(r) =e2E2(r)

4meω2.

Also can be introduced the concept of ”ponderomotive force” as the gradient

of ponderomotive potential. Basically ponderomotive potential describes the

averaged motion of the electron, i.e. the motion of the point, around which

16

the electron oscillates, as

d2 < r >

dt2= − 1

me∇Up(r).

In the limit of a very long pulse the electron has time to make a lot of oscil-

lations and ”slide off the potential hill” (Fig. 1.5). In this case the pondero-

motive energy gets converted completely into kinetic energy and the electron

is registered outside the interaction region with the energy Etot = Up + Etrans,

where Etrans is the kinetic energy given to the electron in the ionization process

itself (Freeman et al. [13]). Respectively, for very short pulses there is almost

no conversion of ponderomotive energy into kinetic energy and electrons are

registered with the energy they had upon ionization, which depends on the

position of a parent atom (or ion) within the beam.

Figure 1.5: Electrons ”slide off the potential hill”.

There is an interesting point to make concerning ponderomotive forces.

17

For an ideal beam they would be evidently axially symmetric and independent

of polarization. At the same time it would seem natural to suggest that direc-

tion of polarization should be somehow preferential. Indeed, this symmetry of

ponderomotive forces cannot be explained if we consider the beam in paraxial

approximation. This symmetry is a consequence of the fact that in a laser

beam ∇·E = 0 and ∇·B = 0. And if we look at the components of the fieldsit turns out that the symmetry arises thanks to the longitudinal component of

the magnetic field Bz, which produces electron motion perpendicular to both

the propagation direction and the electric field polarization.

1.5 Relativistic ATI

First, let us consider ATI in the weakly relativistic case, which corre-

sponds to intensities of ∼ 1018−1020 W/cm2. In this case the electron achievesa velocities that is comparable with the speed of light. Recall that the motion

of the electron is determined by the Lorentz equation:

ṗe = −e(E + 1c

v ×B). (1.17)

It means that as v becomes greater the effect of the magnetic fields becomes

larger. Let’s think in terms of paraxial approximation for a moment and

imagine the beginning of electron’s acceleration. If electron was produced

via ionization initially at rest then it would start accelerating along E (the

direction of polarization). It would continue pretty much along this line until

18

its speed becomes significantly large and there appears a component of force

perpendicular to E due to the magnetic field. This force would tend to bend

the trajectory towards the direction of propagation k. Let θ be the angle

between k and the momentum of the electron pe. It can be shown (e.g. Moore

et al. [14]) that, assuming paraxial approximation and electrons born at rest,

there is an exact relation between the electron energy and its θ:

tan θ =p⊥pz

=

√p2 − p2zpz

=

√γ2 − 1− (γ − 1)2

γ − 1 =√

2

γ − 1 . (1.18)

This relation holds at any point of the electron’s trajectory. In experiments by

Meyerhofer et al. [15] it was demonstrated that this relation holds quite well

for ultimate momenta of the electrons at the intensity of 1018 W/cm2. They

registered electrons with energies of up to 140 keV at the corresponding angle

θ ≈ 68o.Ionization provides an opportunity to ”inject” electrons into laser pulse around

its peak intensity. This is very different from simply propagating a high inten-

sity pulse through a medium of free electrons, because free electrons would be

expelled from the beam by lower intensities and would never see the peak of the

pulse. Hu et al. [16] showed by simulations that a linearly polarized Gaussian

beam focused to the peak intensity of 8× 1021 W/cm2 with the beam waist ofw0 = 10 µm can accelerate the electrons produced from ionization of V

22+ to

energies of up to 2 GeV. However in their simulations they used the paraxial

approximation for the fields while later in this chapter it will be demonstrated

that it is absolutely necessary to use more exact description of the fields at

these ultrahigh intensities. Their results for energy-angle distribution of the

19

electrons were, naturally, consistent with Eq. (1.18).

Good way of thinking about the electron acceleration by the ultrahigh inten-

sity laser is to imagine it ”surfing the pulse”. Indeed, upon ionization the

electron acquires the energy very quickly so that its relativistic gamma-factor

goes into hundreds within a small part of the wave period. It results in v ≈ cand the electric and magnetic fields affecting the electron’s trajectory equally

strongly. Consequently the electron starts moving nearly parallel to the direc-

tion of propagation of the beam (Fig. 1.6). It means that at the electron’s

position the phase of the light would be changing much slower than it would

for the electron standing still (Fig. 1.7). This slowing of the rate of change

would be determined by the electron’s gamma-factor and the angle between

its trajectory and the direction of propagation of light.

Figure 1.6: Shape of the electron trajectory upon ionization at ultrahigh in-tensity.

Let us try to describe the electron motion in a more formal way (the

following part of this section is an extended version of the paper [17] by Todd

20

Figure 1.7: Electron surfing a laser wave.

Ditmire and myself). Upon ionization the electron energy changes according

to

dE

dt=

d

dt(γmc2) = −eE · v. (1.19)

Thus the electron would be gaining energy from the laser light as long as E ·vstays negative, say for a period of time δτ - the ”dephasing time”.

Let us first consider an ultra-relativistic electron (γ À 1) moving in the fieldof an infinite monochromatic plane wave, such that the electron’s momentum

makes a small angle θ with the wave vector. In this case the dephasing time

is completely determined by the difference between the z component of the

electron velocity and the light phase velocity c.

γ =

(1− v

2

c2

)−1/2=⇒ v = c

√1− 1

γ2≈ c

(1− 1

2γ2

)

vz = v cos θ ≈ v(

1− θ2

2

)≈ c

(1− 1

2γ2− θ

2

2

)

If we presume that the electron is born at the peak of the field then the

dephasing time will correspond to the time it takes for the light phase at the

21

electron’s position to change by π/2. For the sake of convenience we say that

the electron was born at z = 0. Another assumption that is going to be used

several times in this section is:

θ ¿ 1γ, (1.20)

which is not necessarily very meaningful for a real ionization experiment. It

might, however, represent an artificially realizable situation with preacceler-

ated electrons ejected into the beam (an experiment of this kind was conducted

at mildly relativistic intensities by Malka et al. [18]). We use this assumption

because it gives an opportunity to compare different physical models of laser-

electron interaction with potentially different resulting values of θ, using only

the energetic parameter γ. With this said, we have:

π/2 = ωt− kz = k(ct− z) = kt(c− vz) ≈ kct(

1

2γ2− θ

2

2

)≈ kct/2γ2

=⇒ δτplane ≈ πγ2/kc. (1.21)

Let us now consider the motion of an electron in a focused beam. In Eqs.

(1.16) t = 0, z = 0 corresponds to the peak field strength if ϕ0 = π/2. For

simplicity we consider an electron that is born via ionization at t = 0 and

x, y, z = 0, which is realistic because it corresponds to the peak of the field

where the ionization probability is highest. Up to the first order in θ and 1/γ

we have:

z ≈ vzt ≈ ct(

1− 12γ2

− θ2

2

)≈ ct (1.22)

x ≈ vxt = (vz tan θ) t ≈ vzt θ ≈ zθ ≈ θct. (1.23)

22

At t = 0 the right hand side of Eq. (1.19) is positive and the electron is

gaining energy from the field. The electron will start losing energy right after

the moment when E · v = 0. Using Eqs. (1.16a) and (1.16b) we have:

E · v = Exvx + Ezvz ∝ vx sin ϕG + vz · 2² xw

cos ϕ(1)G

≈ vx(

sin ϕG + 2²ct

wcos ϕ

(1)G

). (1.24)

Let us consider the paraxial approximation first, which corresponds to ² = 0 in

Eq. (1.24). Thus, we need to solve the equation sin ϕG = 0, which is equivalent

to ϕG = 0 because ϕG(t = 0) = −π/2 and ϕG increases with time. For theparaxial approximation and γ À 1 the dephasing distance is much greater thanthe Rayleigh length (z ¿ zR). We then have the following approximations:

tan−1 z/zR ≈ π/2− zR/z

w2 = w20 (1 + z2/z2R) ≈ w20 z2/z2R

Using these and Eqs. (1.22), (1.23) up to the second order in θ and 1/γ we

ultimately get

δτparaxial ≈ γw0c

. (1.25)

This dephasing time scales linearly with electron energy and is clearly much

smaller than in the plane wave case, where it scales as the square of the

electron energy. This can be explained by the fact that the field undergoes

a Guoy phase shift as the electron propagates out of the focus, leading to

reversal of the field at the electron much sooner.

Finally, we consider the more complete description of the laser fields, i.e. ² 6= 0

23

in Eq. (1.24). Under our assumptions about the time and position of the

ionization event, right after the ionization we have sin ϕG ≈ −1 and cos ϕ(1)G ≈cos ϕG > 0. It means that Eq. (1.24) contains two competing terms of opposite

signs. This holds true for an extended period of time until ϕ(1)G becomes

positive. Thus we can say that the longitudinal component Ez of the electric

field serves to decelerate the electron even before the time δτparax. It is not

possible to solve the equation E · v = 0 analytically in this case. However,during some preliminary numerical trajectory calculations it was noticed that

the electrons would start decelerating around z ≈ zR. Knowing this fact, weTaylor expanded the expression in parentheses with z near zR, and found that

the dephasing time in the properly treated Gaussian focus is

δτGaussian ≈ zRc

(1− kzR

2γ2

). (1.26)

For γ À 1 we can ignore the second term in parentheses in Eq. (1.26) andmake the following comparison:

δτGaussian/δτparaxial ≈ kw0/2γ,

meaning that the effect from the inclusion of the longitudinal components is

larger for higher intensities (larger γ) and for tighter focusing (smaller w0, or,

equivalently, larger ²).

Now we can roughly estimate the energy pickup by an electron during its

first accelerating cycle using Eq. (1.19). It is not possible to analytically

calculate the energy pickup for ² 6= 0, that’s why we choose a different tactics.We estimate the energy pickup for ² = 0 assuming the average field strength

24

during acceleration Ex = Emax/2, and then compare the resulting energies

using the different dephasing times.

γmax = − emec2

∫E · v dt ∝

∫Exvx dt =

∫Emax/2√1 + z2/z2R

θc dt

≈ Emaxθ2

zmax∫

0

dz√1 + z2/z2R

=EmaxθzR

2

zmax∫

0

d(z/zR)√1 + (z/zR)2

.

Making substitution z/zR = sinh s, we have√

1 + (z/zR)2 =√

1 + sinh2 s =

cosh s and d(z/zr) = cosh s · ds. Thus:

γmax ∝ EmaxθzR2

∫cosh s ds

cosh s=

(EmaxθzR s

2

)z=zmaxz=0

⇒ γmax ≈ emec2

Emax2

θzR sinh−1

(cδτ

zR

). (1.27)

Using Eqs. (1.25), (1.26) and (1.27) we get

γparaxialmax /γGaussianmax ≈ ln(4γ/kw0). (1.28)

Eq. (1.27) suggests that for the peak intensity of 5 × 1021 W/cm2, w0 = 5µm and θ = 3o the electron would acquire a maximum γ of about 950 (cor-

responding to the energy of about 500 MeV). While in case of using paraxial

approximation the maximum energy should be about 5 times higher, accord-

ing to Eq. (1.28).

To verify in a more exact way the significance of proper treatment of longitu-

dinal fields, suggested by our very approximate theory, we conducted a set of

computer simulations described in detail in the next section.

One final remark I would like to make concerns the possibility of applying

25

the concept of ponderomotive force to describe electron’s motion at ultrahigh

intensities. There have been attempts to derive a relativistic version of the

ponderomotive force. For example, Quesnel and Mora [10] analyze some pre-

vious publications and give the following expression:

dpedt

= − e2meγ

∇Ã2⊥, (1.29)

where the overlines mean averaging over the fast time scale, and Ã⊥ is the

quickly varying component of the vector potential perpendicular to the direc-

tion of propagation of the beam. However, this expression is only valid under

the condition 1−vz/c À ², which does not hold for the ultrarelativistic regime,especially when the focusing is tight.

Figure 1.8: Dependence of the electron energy on the position in the beam.

Also, Eq. (1.29) involves averaging over the fast time scale, i.e. it pre-

sumes that the electron makes many oscillations on its way out of the beam.

However, as can be seen from Fig. 1.8, which depicts a typical computed

26

dependence of the electron energy on its position in the beam, the electron

practically does not oscillate. It means that the idea of describing the elec-

tron motion through any averaged quantity does not make much sense in the

ultrarelativistic case.

1.6 Simulations of Relativistic ATI

Computer simulations were performed primarily to verify the effects

of inclusion of the longitudinal fields [17]. Here is a brief description of the

algorithm used:

1) A position within the focal region is chosen randomly and ion of a certain

charge state is put into that position.

2) A laser pulse is propagated through the ion. Ionization can occur in two

ways: a) when the intensity becomes higher than the BSI threshold intensity

for the current charge state; b) when the intensity is below the BSI threshold,

the ADK ionization probability is calculated as p = WADKδt, where δt is the

simulation time step; the probabilistic nature is modelled by generating a ran-

dom number b in the range (0, 1) and comparing it to p - if b < p ionization

occurs.

3) Once ionization occurred in step 2 an electron, initially at rest, is placed

in the ion position. Then the relativistic equation of motion (1.17) is numer-

ically integrated to find the vector of momentum ultimately acquired by the

electron. The time of ionization serves as one of the initial conditions in the

integration. The information about ultimate momentum is saved in a file.

27

4) Steps 1 to 3 are repeated until the maximum ionization stage is reached, or

until the laser pulse has propagated far enough so it clearly can not result in

further ionization.

5) Steps 1 to 4 are repeated for a desired number of ions, large enough to

collect a representative statistics.

The first set of simulations was performed for the weakly relativistic intensity

of 1 × 1018 W/cm2 for ionization of atomic neon. Laser pulse width was 30fs, wavelength 800 nm. The results are shown in Fig. 1.9 with the green line

corresponding to the dependence (1.18) tan2 θ = 2/(γ − 1). In (a) the longi-tudinal components were deliberately excluded and in (b) they were properly

included.

Figure 1.9: Simulation results for 1×1018 W/cm2. Dependence of the ejectionangle on the electron energy for (a) paraxial approximation, (b) longitudinalfields included.

It can be seen that at this intensity the angle-energy dependence is

28

close to (1.18) in agreement with the experiment by Meyerhofer et al. [15].

The maximum achieved energy does not change with inclusion of longitudinal

components.

The other set of simulations was performed rather closely following the param-

eters of simulations by Hu and Starace [16]. We calculated ionization of Ar17+

ions by an 800 nm, 30 fs laser pulse focused to w0 = 5 µm and the intensity of

5×1021 W/cm2. The results are shown in Fig. 1.10. Clearly, at this ultrarela-tivistic intensity introduction of the longitudinal fields has drastic effects: the

dependence (1.18) doesn’t hold anymore, and the maximum achieved energy

decreases approximately 2.5 times. According to the previous section, one

could expect that these effects are even stronger for tighter focussing.

Figure 1.10: Simulation results for 5×1021 W/cm2. Dependence of the ejectionangle on the electron energy for (a) paraxial approximation, (b) longitudinalfields included.

29

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31

Vita

Alexander Sergeevich Maltsev was born in Cheboksary, Russia on 5

March 1980, the son of Tatyana Nikolaevna Maltseva and Sergey Vsevolodovich

Maltsev. Received Bachelor of Science degree in Applied Mathematics and

Physics from Moscow Institute of Physics and Technology in June 2001. Started

graduate studies in physics program of the University of Texas at Austin in Fall

of 2001. In January 2002 started work as a Research Assistant for professor

Todd Ditmire.

Permanent address: Russia, Cheboksary 428027Egersky blvd, 43-153

This thesis was typeset with LATEX† by the author.

†LATEX is a document preparation system developed by Leslie Lamport as a specialversion of Donald Knuth’s TEX Program.

32