**Fast Charged Particles and Super-Strong Magnetic Fields Generated by Intense Laser Target Interaction**

Vadim Belyaev and Anatoly Matafonov *Central Research Institute of Machine Building Russian Federation* 

## **1. Introduction**

34 Will-be-set-by-IN-TECH

86 Femtosecond-Scale Optics

[64] A. Griesmaier, J. Werner, S. Hensier, J. Stuhler, T. Pfau, Phys. Rev. Lett. 94, 160401 (2005). [65] Z. Pavlovic, R. V. Krems, R. Côté, H. R. Sadeghpour, Phys. Rev. A 71, 061402 (2005).

[61] M. Fleischhauer, A. Imamoglu, J.P. Marangos, Rev. Mod. Phys. 77, 633 (2005).

[62] A. C. Han, E. A. Shapiro, M. Shapiro, arxiv:1104.1480 (2011). [63] C. Lee, E.A. Ostrovskaya, Phys. Rev. A 72, 062321 (2005).

[66] M. Gacesa, P. Pellegrini, R. Côté, Phys. Rev. A 78, 010701(R) (2008).

The development of a new generation of solid-state lasers has resulted in unique conditions for irradiating laser targets by light pulses, with radiation intensity ranging from 1017 to 1021 W/cm2 and a duration of 20 - 1000 fs.

At such intensities, the laser pulse produces superstrong electric fields which could not be obtained earlier and considerably exceed the atomic electric field of strength *Ea* = 5.14109 V/cm. In these conditions, there arises a new physical picture of laser pulse interaction with plasma produced when the pulse leading edge or a pre-pulse affects solid targets. Laser radiation is rather efficiently transformed into fluxes of fast charged particles such as electrons and atomic ions. The latter interact with the ambient material of the target, which leads to the generation of hard X-rays, when inner atomic shells are ionized, and to various nuclear and photonuclear reactions.

One important area in investigating the interaction of sub-picosecond laser pulses with solid targets is related to the important role which arising superstrong quasistatic magnetic fields and electronic structures play in laser plasma dynamics. This area of research became most attractive after carrying out the direct measurements of quasistatic magnetic fields on the Vulcan laser system (Great Britain) (Tatarakis et al., 2002), in particular, after the pinch effect has been found experimentally in laser plasma (Beg et al., 2004).

The relativistic character of laser radiation with intensity *I* is realized at the magnitude of a dimensionless parameter *a* > 1. This parameter represents the dimensionless momentum of the electron oscillating in the electric field of linearly polarized laser radiation and can be expressed as

$$a = \frac{eE}{m c \alpha o} = 0.85 \left(\frac{\lambda}{\mu \text{m}}\right) \left(\frac{I}{10^{18} \text{W/cm}^2}\right)^{\frac{1}{2}} \tag{1}$$

$$\frac{E}{\text{V/cm}} = 27.7 \left( \frac{I}{\text{W/cm}^2} \right)^{\frac{1}{2}} \tag{2}$$

Fast Charged Particles and Super-Strong Magnetic

part of the laser plasma, we have

numerical factor.

greater than formula (3) does:

Fields Generated by Intense Laser Target Interaction 89

frequency. In contrast to the ponderomotive **vB** mechanism, vacuum heating and resonance absorption arise at nonrelativistic (substantially lower, with *a* < 1) intensities as well. In the case of the ponderomotive mechanism, the average energy of fast electrons can be estimated as the maximum energy of transverse electron oscillations in an electromagnetic field, which in the general case takes a relativistic value. In a underdense

> , <sup>2</sup> *Q I*

2

. <sup>2</sup> *<sup>e</sup> <sup>Q</sup> mc Q*

In the overdense part of the plasma, the ponderomotive heating of electrons is noticeably

In the case of vacuum heating, the maximum energy of an electron flying into the depths of a dense target is given by the formula similar to equation (3), however, with a different

There is one more mechanism for generating fast electrons in the underdense part of plasma in front of a target due to the betatron resonance in the arising magnetic field (Pukhov, 2003). In this regime, electrons are accelerated by the transverse ultrarelativistic electric field of the laser wave in the direction of wave polarization, and the azimuthal magnetic field produced by the current of fast electrons is responsible for the magnetic part of the Lorentz force. This force turns electrons in such a way that they gradually change to the opposite direction of motion. In the case of an exact betatron resonance, the reflection occurs at the instant when the transverse electric field changes its direction, so the electrons are accelerated at all times. This mechanism yields an energy of fast electrons three times

There are also further mechanisms of electron acceleration that require special experimental conditions, for example, the wake field acceleration (Esarey, 1996; Amiranoff, 1998). In the case of resonance absorption, the electric field near the plasma critical surface is much stronger than that of incident laser radiation. The result is that the heating of electrons upon

0

1 <sup>2</sup> <sup>2</sup> 0 3 . *<sup>e</sup> <sup>Q</sup> mc Q*

(4)

1 <sup>2</sup> <sup>2</sup> 0 . *<sup>e</sup> <sup>Q</sup> mc Q*

 , <sup>18</sup> <sup>2</sup> 0 2 <sup>W</sup> 1.37 10 <sup>μ</sup><sup>m</sup> cm *Q* (3)

1

By contrast, in the nonrelativistic limit *Q Q*0, we derive from formula (3) that

weaker due to a difficult penetration of the laser field into this region.

their impact with atomic ions is greater than follows from formulae (3) and (4).

<sup>2</sup> <sup>2</sup>

1 1 *<sup>e</sup> <sup>Q</sup> mc*

In the ultrarelativistic limit *Q Q*0, we hence obtain

0

*Q*

where *e* and *m* are the charge and mass of the electron, respectively, *E* is the amplitude of electric field strength (in units of V/cm) of laser radiation, is the radiation wavelength (in m), is the frequency of laser radiation, *c* is the speed of light, and *I* is the radiation intensity (in W/cm2).

Terawatt-power laser systems of moderate size can fulfill the condition a > 1, which corresponds to the electric field strength above 1010 V/cm. In such intense fields, the overbarrier ionization of atoms occurs in atomic time on the order of 10-17 s, and the electrons produced are accelerated and reach MeV-range relativistic energies during the laser pulse.

The acceleration of atomic ions in femto- and picoseconds laser plasmas constitutes a secondary process. It is caused by the strong quasistatic electric fields arising due to spatial charge separation. Such separation is related to the motion of a bunch of fast electrons. For laser radiation intensities exceeding *I* 1018 W/cm2, it is possible to obtain directed beams of high-energy ions with the energies *<sup>i</sup>* > 1 MeV.

The generation of high-energy proton and ion beams in laser plasma under the action of ultrashort pulses is a quickly developing field of investigations. This is explained, in particular, by their important applications in such fields as proton accelerators, the study of material structure, proton radiography, the production of short-living radioisotopes for medical purposes, and laser controlled fusion (Umstadter, 2003; Mourou et al., 2006). For a laser radiation intensity of *I* 1018 W/cm2, a number of nuclear reactions can be initiated that have only been realized in elementary particle accelerators (Andreev et al., 2001).

Later on, we will consider the principal mechanisms for generating fast charged particles and quasistatic magnetic fields in laser plasmas, as well as experimental results obtained both abroad and on the native laser setup NEODIM in the Central Research Institute of Machine Building (Russ. abbr. TsNIIMash) (Korolev, Moscow reg.) (Belyaev et al., 2004; Belyaev et al., 2005).

## **2. Generation of fast electrons in laser plasma**

In irradiating a target by a high-intensity ultrashort laser pulse, the radiation energy is rather efficiently converted into the energy of fast electrons which later partially transfer their energy to the atomic ions of the target. Presently, several mechanisms are being discussed concerning the generation of fast electrons when a laser pulse affects plasma with a density well above the critical value. If the laser pulse is not accompanied by a pre-pulse (the case of high contrast), then the laser radiation interacts with plasma of a solid-state density, possessing a sharp boundary. In this case, the mechanism of `vacuum heating' is realized (Brunel, 1987), as is the so-called **vB** mechanism (Wilks et al., 1992) (here, **B** is the magnetic field induction of the laser field) caused by a longitudinal ponderomotive force acting along the propagation direction of the laser pulse). This **vB** mechanism becomes substantial at relativistic intensities where the energy of electron oscillations is comparable with or exceeds the electron rest energy *mc*2 = 511 keV - that is, for the parameter *a* > 1 [see formula (1)]. In addition, fast electrons can be generated on the critical surface of plasma at a plasma resonance ( Gus'kov et al., 2001; Demchenko et al., 2001) if the vector of the laser radiation electric field has a projection along the density gradient (usually at an inclined incidence of laser radiation to target) and the laser frequency coincides with the plasma 88 Femtosecond–Scale Optics

where *e* and *m* are the charge and mass of the electron, respectively, *E* is the amplitude of

Terawatt-power laser systems of moderate size can fulfill the condition a > 1, which corresponds to the electric field strength above 1010 V/cm. In such intense fields, the overbarrier ionization of atoms occurs in atomic time on the order of 10-17 s, and the electrons produced are accelerated and reach MeV-range relativistic energies during the

The acceleration of atomic ions in femto- and picoseconds laser plasmas constitutes a secondary process. It is caused by the strong quasistatic electric fields arising due to spatial charge separation. Such separation is related to the motion of a bunch of fast electrons. For laser radiation intensities exceeding *I* 1018 W/cm2, it is possible to obtain directed beams

*<sup>i</sup>* > 1 MeV. The generation of high-energy proton and ion beams in laser plasma under the action of ultrashort pulses is a quickly developing field of investigations. This is explained, in particular, by their important applications in such fields as proton accelerators, the study of material structure, proton radiography, the production of short-living radioisotopes for medical purposes, and laser controlled fusion (Umstadter, 2003; Mourou et al., 2006). For a laser radiation intensity of *I* 1018 W/cm2, a number of nuclear reactions can be initiated that have only been realized in elementary particle accelerators (Andreev et al.,

Later on, we will consider the principal mechanisms for generating fast charged particles and quasistatic magnetic fields in laser plasmas, as well as experimental results obtained both abroad and on the native laser setup NEODIM in the Central Research Institute of Machine Building (Russ. abbr. TsNIIMash) (Korolev, Moscow reg.) (Belyaev et al., 2004;

In irradiating a target by a high-intensity ultrashort laser pulse, the radiation energy is rather efficiently converted into the energy of fast electrons which later partially transfer their energy to the atomic ions of the target. Presently, several mechanisms are being discussed concerning the generation of fast electrons when a laser pulse affects plasma with a density well above the critical value. If the laser pulse is not accompanied by a pre-pulse (the case of high contrast), then the laser radiation interacts with plasma of a solid-state density, possessing a sharp boundary. In this case, the mechanism of `vacuum heating' is realized (Brunel, 1987), as is the so-called **vB** mechanism (Wilks et al., 1992) (here, **B** is the magnetic field induction of the laser field) caused by a longitudinal ponderomotive force acting along the propagation direction of the laser pulse). This **vB** mechanism becomes substantial at relativistic intensities where the energy of electron oscillations is comparable with or exceeds the electron rest energy *mc*2 = 511 keV - that is, for the parameter *a* > 1 [see formula (1)]. In addition, fast electrons can be generated on the critical surface of plasma at a plasma resonance ( Gus'kov et al., 2001; Demchenko et al., 2001) if the vector of the laser radiation electric field has a projection along the density gradient (usually at an inclined incidence of laser radiation to target) and the laser frequency coincides with the plasma

is the frequency of laser radiation, *c* is the speed of light, and *I* is the radiation

is the radiation wavelength (in

electric field strength (in units of V/cm) of laser radiation,

m), 

laser pulse.

2001).

Belyaev et al., 2005).

intensity (in W/cm2).

of high-energy ions with the energies

**2. Generation of fast electrons in laser plasma** 

frequency. In contrast to the ponderomotive **vB** mechanism, vacuum heating and resonance absorption arise at nonrelativistic (substantially lower, with *a* < 1) intensities as well. In the case of the ponderomotive mechanism, the average energy of fast electrons can be estimated as the maximum energy of transverse electron oscillations in an electromagnetic field, which in the general case takes a relativistic value. In a underdense part of the laser plasma, we have

$$\sigma\_c = mc^2 \left[ \left( 1 + \frac{Q}{Q\_0} \right)^{\frac{1}{2}} - 1 \right], \ Q = I\lambda^2, \ Q\_0 = 1.37 \times 10^{18} \frac{\text{W}}{\text{cm}^2} \mu \text{m}^2 \tag{3}$$

In the ultrarelativistic limit *Q Q*0, we hence obtain

$$
\varepsilon\_e = mc^2 \left(\frac{Q}{Q\_0}\right)^{\frac{1}{2}}.
$$

By contrast, in the nonrelativistic limit *Q Q*0, we derive from formula (3) that

$$
\varepsilon\_e = mc^2 \frac{Q}{2Q\_0}.
$$

In the overdense part of the plasma, the ponderomotive heating of electrons is noticeably weaker due to a difficult penetration of the laser field into this region.

In the case of vacuum heating, the maximum energy of an electron flying into the depths of a dense target is given by the formula similar to equation (3), however, with a different numerical factor.

There is one more mechanism for generating fast electrons in the underdense part of plasma in front of a target due to the betatron resonance in the arising magnetic field (Pukhov, 2003). In this regime, electrons are accelerated by the transverse ultrarelativistic electric field of the laser wave in the direction of wave polarization, and the azimuthal magnetic field produced by the current of fast electrons is responsible for the magnetic part of the Lorentz force. This force turns electrons in such a way that they gradually change to the opposite direction of motion. In the case of an exact betatron resonance, the reflection occurs at the instant when the transverse electric field changes its direction, so the electrons are accelerated at all times. This mechanism yields an energy of fast electrons three times greater than formula (3) does:

$$
\varepsilon\_e = 3mc^2 \left(\frac{Q}{Q\_0}\right)^{\frac{1}{2}}.\tag{4}
$$

There are also further mechanisms of electron acceleration that require special experimental conditions, for example, the wake field acceleration (Esarey, 1996; Amiranoff, 1998). In the case of resonance absorption, the electric field near the plasma critical surface is much stronger than that of incident laser radiation. The result is that the heating of electrons upon their impact with atomic ions is greater than follows from formulae (3) and (4).

Fast Charged Particles and Super-Strong Magnetic

/2 .

action of the forces applied is always equal to


where **p** – ordinary momentum (8), **A** – vector-potential;


1 *KIN*

plasma. This value can be estimated using the energy conservation law. Omitting calculations we can use the following formula easy to keep in mind:

Substitution this formulae in expression (10) for kinetic electron energy gives:

10 *KIN <sup>J</sup> <sup>E</sup>*

of light, *B* – magnetic induction in the electron orbit. Taking electromagnetic field penetration depth

where *r* =

Considering:

length  /2, 

*,* we have *r*

where *V* – electron velocity; - use of generalized momentum

put *A* = *Br* and

So from (6) – (9) we have

Fields Generated by Intense Laser Target Interaction 91

Given the relationship between mass and velocity, the kinetic energy change due to the

0 0 () 1 *ЕKIN V m m c mc*

*e c*

*e c* **Pp A**

0 *<sup>e</sup> P m V Br*

*eBr E mc m c m c* 

To find the electron maximum kinetic energy at specified intensity of laser radiation incident on the target we need the maximum value of *B* – magnetic field induced in laser

1 2 2 2 2 0 0 2 0

1 2 2 18 <sup>9</sup> 0,5 1 0,5

 

where *mV* - relativistic mass, *m*<sup>0</sup> – electron rest mass, 2 2


2 2

to be equal to incident radiation wave

, (6)

1 1 *V c* - relativistic factor.

, (8)

, which allows to

, (10)

, (12)

**Pp A** (7)

*<sup>A</sup> B B y*

*<sup>c</sup>* (9)

*Z*

1 2 *B J MAX* [Gs] 10 [W/cm ] , (11)

Electrons are also accelerated by a transverse ponderomotive force (acting in the radial direction) due to a focal distribution of laser intensity. This acceleration leads to the maximum electron energy also expressed by formula (3) (in the underdense part of plasma) if electrons succeed in acquiring this energy moving from the focus to the periphery during the laser pulse. Thus, the duration of a laser pulse should meet the inequality *mR*/*eE* (in the nonrelativistic case). Here, *R* is the radius of the focal spot of a laser beam. This inequality holds for picosecond- and longer-duration laser light pulses with an intensity on the order of 1016 W/cm2. In fields with an intensity of 1018 W/cm2, the right-hand side of this inequality reaches dozens of femtoseconds, whereas in the overdense part of plasma this ponderomotive force is noticeably weaker.

We have discussed the above mentioned mechanisms in more detail in our article (Belyaev et al., 2008).

We suggested and investigated the new mechanism of high-energy electrons formation in ultra-high intensity laser pulse interaction with solid targets (Belyaev, 2004). This investigation is an attempt to reveal and describe, based on the model suggested, the highenergy electron formation mechanism in laser plasmas so as to derive theoretical dependences which would represent specific relations between the parameters of fast electrons, laser radiation and target substance.

Any theory can be accepted only after reliable experimental verification. The degree of reliability is determined not only by the sufficient diversity of independent experimental data, but also by the ability to choose out of these data those best representative of the overall pattern. Analysis of numerous experiments to measure energy of fast electrons formed in laser plasmas shows that with a particular laser facility, given its available radiation intensity, fast electron maximum energy can be determined most closely. Generally, it is electron maximum energy values that are most widely presented in experimental investigations. This is motivated not only by experimenters' striving to get extreme record-breaking output parameters, but also by the possibility to most closely determine the electron maximum energy around their spectrum extrapolation at specified intensity of laser radiation incident on a target. On this basis we will establish our theoretical model of the maximum-energy electron formation process for a given laser radiation intensity.

Without going into details of magnetic field generation mechanisms, it can be noted that a vortical electron structure develops eventually in plasma. Given the applied electric field (constituent of the incident laser radiation) and the dominance of tunnel ionization, a great number of electrons (practically determined by solid density) are accelerated. This current of electrons generates a magnetic field which bends their trajectory. Under certain conditions these trajectories can close at skin-layer depth within larmor-radius circle. The high electron density and, correspondingly, the circular current strength cause super-strong magnetic fields generation.

Condition for such fields generation can be written as a condition for electron movement around such a circle in the form of a balance between the centrifugal force and the Lorentz force:

$$\frac{mV^2}{r} = \frac{eVB}{c},\tag{5}$$

90 Femtosecond–Scale Optics

Electrons are also accelerated by a transverse ponderomotive force (acting in the radial direction) due to a focal distribution of laser intensity. This acceleration leads to the maximum electron energy also expressed by formula (3) (in the underdense part of plasma) if electrons succeed in acquiring this energy moving from the focus to the periphery during

(in the nonrelativistic case). Here, *R* is the radius of the focal spot of a laser beam. This inequality holds for picosecond- and longer-duration laser light pulses with an intensity on the order of 1016 W/cm2. In fields with an intensity of 1018 W/cm2, the right-hand side of this inequality reaches dozens of femtoseconds, whereas in the overdense part of plasma

We have discussed the above mentioned mechanisms in more detail in our article (Belyaev

We suggested and investigated the new mechanism of high-energy electrons formation in ultra-high intensity laser pulse interaction with solid targets (Belyaev, 2004). This investigation is an attempt to reveal and describe, based on the model suggested, the highenergy electron formation mechanism in laser plasmas so as to derive theoretical dependences which would represent specific relations between the parameters of fast

Any theory can be accepted only after reliable experimental verification. The degree of reliability is determined not only by the sufficient diversity of independent experimental data, but also by the ability to choose out of these data those best representative of the overall pattern. Analysis of numerous experiments to measure energy of fast electrons formed in laser plasmas shows that with a particular laser facility, given its available radiation intensity, fast electron maximum energy can be determined most closely. Generally, it is electron maximum energy values that are most widely presented in experimental investigations. This is motivated not only by experimenters' striving to get extreme record-breaking output parameters, but also by the possibility to most closely determine the electron maximum energy around their spectrum extrapolation at specified intensity of laser radiation incident on a target. On this basis we will establish our theoretical model of the maximum-energy electron formation process for a given laser

Without going into details of magnetic field generation mechanisms, it can be noted that a vortical electron structure develops eventually in plasma. Given the applied electric field (constituent of the incident laser radiation) and the dominance of tunnel ionization, a great number of electrons (practically determined by solid density) are accelerated. This current of electrons generates a magnetic field which bends their trajectory. Under certain conditions these trajectories can close at skin-layer depth within larmor-radius circle. The high electron density and, correspondingly, the circular current strength cause super-strong magnetic

Condition for such fields generation can be written as a condition for electron movement around such a circle in the form of a balance between the centrifugal force and the Lorentz

*r c* (5)

2 , *mV eVB*

of a laser pulse should meet the inequality

 *mR*/*eE*

the laser pulse. Thus, the duration

et al., 2008).

radiation intensity.

fields generation.

force:

this ponderomotive force is noticeably weaker.

electrons, laser radiation and target substance.

where *r* = /2, - skin-layer thickness, *e, m, V* – charge, mass, electron velocity, *c* – velocity of light, *B* – magnetic induction in the electron orbit.

Taking electromagnetic field penetration depth to be equal to incident radiation wave length *,* we have *r* /2 .

Given the relationship between mass and velocity, the kinetic energy change due to the action of the forces applied is always equal to

$$E\_{\rm KIN} = (m\_V - m\_0)c^2 = m\_0c^2 \left(\gamma - 1\right),\tag{6}$$

where *mV* - relativistic mass, *m*<sup>0</sup> – electron rest mass, 2 2 1 1 *V c* - relativistic factor. Considering:


$$\mathbf{P} = \mathbf{p} - \frac{e}{c}\mathbf{A} \tag{7}$$

where *V* – electron velocity;


$$\mathbf{P} = \mathbf{p} - \frac{e}{c}\mathbf{A} \tag{8}$$

where **p** – ordinary momentum (8), **A** – vector-potential;


put *A* = *Br* and

$$P = m\_0 V = \frac{e}{c} Br \tag{9}$$

So from (6) – (9) we have

$$E\_{\rm KIN} = m\_0 c^2 \left[ 1 + \left( \frac{eBr}{m\_0 c^2} \right)^2 \right]^{\frac{1}{2}} - m\_0 c^2 \tag{10}$$

To find the electron maximum kinetic energy at specified intensity of laser radiation incident on the target we need the maximum value of *B* – magnetic field induced in laser plasma. This value can be estimated using the energy conservation law.

Omitting calculations we can use the following formula easy to keep in mind:

$$B\_{\rm MAX} \text{ [Gs]} = 10^{-1} \sqrt{\text{J} [\text{W/cm}^2]} \,\text{.}\tag{11}$$

Substitution this formulae in expression (10) for kinetic electron energy gives:

$$E\_{\rm KIN} = 0.5 \left[ 1 + \frac{9J\lambda^2}{10^{18}} \right]^{\frac{1}{2}} - 0.5 \,\text{V} \tag{12}$$

Fast Charged Particles and Super-Strong Magnetic

meaning that the laser radiation frequency

of tunnel ionization development:

flow kept unchanged, we have

target substance atoms:

*KIN*

electrons:

structures. This process is known as a dynamic pinch.

case.

radiation frequency

Fields Generated by Intense Laser Target Interaction 93

correspondingly. In both cases electrons are accelerated under the action of an electric field. In a cyclotron, this is a periodically changing electric field applied externally. In a betatron, this is a vortex electric field occurring with axisymmetric magnetic field rise in time. In laser plasmas a magnetic field is generated giving rise to a vortex electric field accelerating electrons. Thus the laser-plasma electron acceleration mechanism resembles the betatron

Equation (10) for electron kinetic energy was derived on the assumption that the electron acceleration is governed only by the laser radiation incident on the target without considering the processes going within the target substance, specifically, ionization process. Formally, it is reflected in the fact that the skin-layer size is determined by the laser

> 

<sup>2</sup> , (16)

is an effective frequency. This assumption is

, (17)

*i i* , (18)

, (19)

< 10-12

 *c c* 

true only at the first stage of interaction with the substance when a vortical electron structure develops on skin-layer scales, its characteristic size being in accordance with (16). This structure is unstable and there is a possibility of its transformation to smaller-scale

It is demonstrated in (Belyaev & Mikhailov, 2001) that in case of laser plasmas produced by

sec) on a solid target this process is of quantum nature and can be described by the diffusion equation. Without going into the process nature, note that under tunnel ionization the vortical electron structure generated on skin-layer scales (16) transforms to another one, its characteristic size now being determined by the ionization frequency as an effective frequency at the next stage of laser radiation interaction with the substance, i.e. at the stage

> 2 *i i l c*

Assuming that the vortical electron structure transformation process goes with the magnetic

2 2 *B Bl* <sup>0</sup>

where *Bi* – magnetic field within the vortical structure, its characteristic size *li*, being determined by (17). Such a vortical structure provides the following kinetic energy to the

*кин i i* 0

Equation (19) determines the maximum energy of the small group (tail) of high-energy electrons. This dependence can be represented via the energy or ionization potential of the

*E eB l eB*

<sup>0</sup> <sup>18</sup> <sup>18</sup> 1.5 1,5

*i i*

*J JI E eB*

10 10

  2

*i*

*l* 

1 1 2 2 2 2

 

> 

[MeV], (20)

the action of high-intensity (*J* > 1016 W/cm2) laser radiation of ultrashort duration (

where intensity *J* expressed in W/cm2, – in micrometer, kinetic energy – MeV. Graph of this dependence show Fig. 1 by curve 1.

Consider limiting cases.

1. *EKIN* = *m*0*c*2 = 0,5 MeV.

Expression (12) gives this value at intensity

$$J\_R = \frac{1}{3} 10^{18} \left[ \frac{1}{\lambda \text{ [}\mu\text{m]}} \right] \frac{\text{W}}{\text{cm}^2} \text{ }^{\circ}\text{}\tag{13}$$

The intensity *JR* can be called as relativistic intensity.

2. *EKIN m*0*c*2; *J* < *JR*. In this case

$$E\_{\rm KIN} = 2.25 \frac{J\lambda^2}{10^{18}} \,\text{.}\tag{14}$$

Graph of this dependence show at Fig. 1 by curve 2.

3. *EKIN m*0*c*2; *J* > *JR*. For this case

$$E\_{\rm KIN} = \mathbf{1}\_{\prime} \mathbf{5} \left( \frac{I \lambda^2}{10^{18}} \right)^{\frac{1}{2}} \, \prime \, \tag{15}$$

and graph of this dependence show on Fig. 1 by curve 3.

Equations obtained for small (< *m*0*c*2*)* and large (> *m*0*c*2*)* values of kinetic energy agree with those in use for calculations of particle energy in a cyclotron and in a betatron, 92 Femtosecond–Scale Optics

– in micrometer, kinetic energy – MeV.

Fig. 1. Dependence of electron kinetic energy on laser radiation intensity.

*RJ*

18

1 1W <sup>10</sup> 3 [ m] cm

2 <sup>18</sup> 2,25 10 *KIN <sup>J</sup> <sup>E</sup>*

> 1 2 2

<sup>18</sup> 1,5 10 *KIN <sup>J</sup> <sup>E</sup>*

Equations obtained for small (< *m*0*c*2*)* and large (> *m*0*c*2*)* values of kinetic energy agree with those in use for calculations of particle energy in a cyclotron and in a betatron,

 

 

2

, (13)

, (15)

, (14)

where intensity *J* expressed in W/cm2,

Consider limiting cases.

1. *EKIN* = *m*0*c*2 = 0,5 MeV.

2. *EKIN m*0*c*2; *J* < *JR*.

3. *EKIN m*0*c*2; *J* > *JR*.

In this case

For this case

Expression (12) gives this value at intensity

The intensity *JR* can be called as relativistic intensity.

Graph of this dependence show at Fig. 1 by curve 2.

and graph of this dependence show on Fig. 1 by curve 3.

Graph of this dependence show Fig. 1 by curve 1.

correspondingly. In both cases electrons are accelerated under the action of an electric field. In a cyclotron, this is a periodically changing electric field applied externally. In a betatron, this is a vortex electric field occurring with axisymmetric magnetic field rise in time. In laser plasmas a magnetic field is generated giving rise to a vortex electric field accelerating electrons. Thus the laser-plasma electron acceleration mechanism resembles the betatron case.

Equation (10) for electron kinetic energy was derived on the assumption that the electron acceleration is governed only by the laser radiation incident on the target without considering the processes going within the target substance, specifically, ionization process. Formally, it is reflected in the fact that the skin-layer size is determined by the laser radiation frequency

$$
\mathcal{S} = \mathfrak{c} / \mathfrak{v} = \mathfrak{Z} \pi \mathfrak{c} / \alpha = \mathcal{X} \Big/ \tag{16}
$$

meaning that the laser radiation frequency is an effective frequency. This assumption is true only at the first stage of interaction with the substance when a vortical electron structure develops on skin-layer scales, its characteristic size being in accordance with (16). This structure is unstable and there is a possibility of its transformation to smaller-scale structures. This process is known as a dynamic pinch.

It is demonstrated in (Belyaev & Mikhailov, 2001) that in case of laser plasmas produced by the action of high-intensity (*J* > 1016 W/cm2) laser radiation of ultrashort duration ( < 10-12 sec) on a solid target this process is of quantum nature and can be described by the diffusion equation. Without going into the process nature, note that under tunnel ionization the vortical electron structure generated on skin-layer scales (16) transforms to another one, its characteristic size now being determined by the ionization frequency as an effective frequency at the next stage of laser radiation interaction with the substance, i.e. at the stage of tunnel ionization development:

2 *i i l c* , (17)

Assuming that the vortical electron structure transformation process goes with the magnetic flow kept unchanged, we have

2 2 *B Bl* <sup>0</sup>*i i* , (18)

where *Bi* – magnetic field within the vortical structure, its characteristic size *li*, being determined by (17). Such a vortical structure provides the following kinetic energy to the electrons:

$$E\_{\kappa\mu\nu} = eB\_i l\_i = eB\_0 \frac{\lambda^2}{l\_i} \, ^\prime \tag{19}$$

Equation (19) determines the maximum energy of the small group (tail) of high-energy electrons. This dependence can be represented via the energy or ionization potential of the target substance atoms:

$$E\_{\rm KIN} = eB\_0 \mathcal{L} \frac{\alpha \rho\_i}{\alpha \nu} = 1.5 \left( \frac{J \lambda^2}{10^{18}} \right)^{\frac{1}{2}} \frac{\alpha \rho\_i}{\alpha \nu} = 1,5 \left( \frac{J \lambda^2}{10^{18}} \right)^{\frac{1}{2}} \frac{I}{\hbar \alpha \nu} \text{ [MeV]}.\tag{20}$$

Fast Charged Particles and Super-Strong Magnetic

hundreds of femtoseconds.

of = .

under a high intensity of laser radiation.

**intensity laser pulses with solid targets** 

other parameters of corpuscular and electromagnetic radiations.

Fields Generated by Intense Laser Target Interaction 95

Figure 2 depicts the variations in the kinetic energy of an electron (a) and its trajectory (b) for motion with the zero initial velocity in a field with a radiation intensity of 1020 W/cm2 and a frequency that is at resonance with the cyclotron frequency (Belyaev & Kostenko, 2003). The constant magnetic field is normal to the polarization of laser radiation. One can see that an electron acquires an energy of approximately 100MeV in a time on the order of

Fig. 2. Electron kinetic energy (a) and trajectory (b) in a linearly polarized laser wave with an intensity of 1020 W/cm2 and a constant transverse magnetic field in the resonance case

Electron acceleration in the field of a circularly polarized laser wave propagating along a strong magnetic field was theoretically investigated in lectures (Pavlenko, 2002). It was shown that the relativistic factor of the electron may increase by an order of magnitude

**3. Generation of fast protons and ions in the interaction of ultrashort high-**

On the basis of the results of experimental and theoretical investigations performed in recent years, one can determine the following ranges for product plasma parameters: the electron temperature is about 1 to 10 keV; the mean energy of "fast" electrons is about 0.1 to 10 MeV (the maximum energy is as high as 300 MeV); the mean energy of fast ions ranges from several hundred keV units to a few MeV units (the maximum energy is 430 MeV); the relativistic longitudinal ponderomotive pressure of laser light is 1 to 50 Gbar; and the amplitudes of the electric field and spontaneous-magnetic-field strength range, respectively, between about 109 and 1012 V/cm and between about 1 and 500 MG (Belyaev et al., 2008; Salamin et al., 2006). Product terawatt-pulse picosecond laser plasmas appear to be some kind of a "table" pulsed "microaccelerator" and a nuclear "microreactor," which is relatively compact and cheap and on which one must not impose special radiation-safety requirements. Such a source admits a relatively simple possibility for controlling energy and

Here *J* is in W/cm2, - in μm, *I* and - in eV. This dependence is plotted in Fig. 1 (curve 4).

The equation obtained demonstrates the proportionality between the electron energy and ionization frequency, hich determines physical nature of the electron acceleration process. The physics of the electron acceleration processes as a result of high-intensity laser radiation action on a substance is closely related to the physics of the ionization processes in superatomic intensity fields.

The ionization frequency is generally one or two orders higher than the laser one. This results in the high acceleration rate and electron energy.

The process of dynamic pinch development give rises formation of high-energy tail (20) and has threshold nature. Our estimations give value of threshold 0.311018 – 3.21019 W/cm2. Threshold smearing evidences for stochastic character of the process.

The good agreement between theory and experiment (Matafonov & Belyaev, 2001; Malka & Miquel, 1996; Borodin et al., 2000; Nickles et al., 1999; Ledingham & Norreys, 1999; Cowan et al., 1999; He et al., 2004; Mangles et al., 2005) suggests the realizability of the proposed high-energy electron formation mechanism in laser plasmas.

In our article (Belyaev, Kostenko et al., 2003) we also have investigated cyclotron mechanism of electron acceleration.

The magnetic activity of picosecond laser plasma offers new mechanisms for the generation of fast electrons due to the presence of such strong quasistatic magnetic fields regardless of the mechanisms of their origin. Such a possibility is related to the emergence of cyclotron resonances when the laser frequency coincides with the Larmor gyration frequency = *eB*0/*mrc* of an electron in an external constant magnetic field with the induction *B*0 (here, *e* and *mr* are the charge and relativistic mass of the electron, respectively; *c* is the speed of

light). Indeed, the typical laser frequency (in the Hartree atomic system of units) is on the order of 0.05 a.u. and coincides with the cyclotron frequency at an induction of *B*0 = 7 a.u. ~ 100 MG. This value may become much greater with allowance made for the relativistic increase in electron mass, which is typical at laser radiation intensities on the order of 1019 - 1020 W/cm2.Hence, the generation of a constant magnetic field results in stronger interaction of laser radiation with plasma. The situation is to a certain extent similar to the radiation self-focusing effect, in which case the variations in the refraction index of the medium in the field of a laser wave influence wave propagation through the medium.

In the general relativistic case, the interaction of electrons with the field of a laser wave and with the constant magnetic field **B**0 is written out in the form

$$\frac{d\mathbf{p}}{dt} = e\left\{\mathbf{E} + \frac{1}{c}\mathbf{v} \times \left(\mathbf{B} + \mathbf{B}\_0\right)\right\}\tag{21}$$

for electrons possessing a momentum **p** and a velocity **v**.

For circular polarization, the problem is solved analytically, whereas in the general case of linear polarization the problem reduces to a system of nonlinear equations, which can only be solved numerically. The solution to these equations is specific in that there are resonances between the periodicmotion of electrons in the magnetic field and electron oscillations in the field of the laser wave. This fact leads to drastic changes in electron trajectory and energy at certain instants of time.

94 Femtosecond–Scale Optics

The equation obtained demonstrates the proportionality between the electron energy and ionization frequency, hich determines physical nature of the electron acceleration process. The physics of the electron acceleration processes as a result of high-intensity laser radiation action on a substance is closely related to the physics of the ionization processes in

The ionization frequency is generally one or two orders higher than the laser one. This

The process of dynamic pinch development give rises formation of high-energy tail (20) and has threshold nature. Our estimations give value of threshold 0.311018 – 3.21019 W/cm2.

The good agreement between theory and experiment (Matafonov & Belyaev, 2001; Malka & Miquel, 1996; Borodin et al., 2000; Nickles et al., 1999; Ledingham & Norreys, 1999; Cowan et al., 1999; He et al., 2004; Mangles et al., 2005) suggests the realizability of the proposed

In our article (Belyaev, Kostenko et al., 2003) we also have investigated cyclotron mechanism

The magnetic activity of picosecond laser plasma offers new mechanisms for the generation of fast electrons due to the presence of such strong quasistatic magnetic fields regardless of the mechanisms of their origin. Such a possibility is related to the emergence

frequency = *eB*0/*mrc* of an electron in an external constant magnetic field with the induction *B*0 (here, *e* and *mr* are the charge and relativistic mass of the electron,

order of 0.05 a.u. and coincides with the cyclotron frequency at an induction of *B*0 = 7 a.u. ~ 100 MG. This value may become much greater with allowance made for the relativistic increase in electron mass, which is typical at laser radiation intensities on the order of 1019 - 1020 W/cm2.Hence, the generation of a constant magnetic field results in stronger interaction of laser radiation with plasma. The situation is to a certain extent similar to the radiation self-focusing effect, in which case the variations in the refraction index of the medium in the

In the general relativistic case, the interaction of electrons with the field of a laser wave and

*c* 

For circular polarization, the problem is solved analytically, whereas in the general case of linear polarization the problem reduces to a system of nonlinear equations, which can only be solved numerically. The solution to these equations is specific in that there are resonances between the periodicmotion of electrons in the magnetic field and electron oscillations in the field of the laser wave. This fact leads to drastic changes in electron trajectory and energy at

<sup>0</sup>

**<sup>p</sup> E v BB** (21)

field of a laser wave influence wave propagation through the medium.

dt *e*

d 1

with the constant magnetic field **B**0 is written out in the form

for electrons possessing a momentum **p** and a velocity **v**.


coincides with the Larmor gyration

(in the Hartree atomic system of units) is on the

Here *J* is in W/cm2,

superatomic intensity fields.

of electron acceleration.

respectively; *c* is the speed of

certain instants of time.

light). Indeed, the typical laser frequency

4).


results in the high acceleration rate and electron energy.

high-energy electron formation mechanism in laser plasmas.

of cyclotron resonances when the laser frequency

Threshold smearing evidences for stochastic character of the process.

Figure 2 depicts the variations in the kinetic energy of an electron (a) and its trajectory (b) for motion with the zero initial velocity in a field with a radiation intensity of 1020 W/cm2 and a frequency that is at resonance with the cyclotron frequency (Belyaev & Kostenko, 2003). The constant magnetic field is normal to the polarization of laser radiation. One can see that an electron acquires an energy of approximately 100MeV in a time on the order of hundreds of femtoseconds.

Fig. 2. Electron kinetic energy (a) and trajectory (b) in a linearly polarized laser wave with an intensity of 1020 W/cm2 and a constant transverse magnetic field in the resonance case of = .

Electron acceleration in the field of a circularly polarized laser wave propagating along a strong magnetic field was theoretically investigated in lectures (Pavlenko, 2002). It was shown that the relativistic factor of the electron may increase by an order of magnitude under a high intensity of laser radiation.

## **3. Generation of fast protons and ions in the interaction of ultrashort highintensity laser pulses with solid targets**

On the basis of the results of experimental and theoretical investigations performed in recent years, one can determine the following ranges for product plasma parameters: the electron temperature is about 1 to 10 keV; the mean energy of "fast" electrons is about 0.1 to 10 MeV (the maximum energy is as high as 300 MeV); the mean energy of fast ions ranges from several hundred keV units to a few MeV units (the maximum energy is 430 MeV); the relativistic longitudinal ponderomotive pressure of laser light is 1 to 50 Gbar; and the amplitudes of the electric field and spontaneous-magnetic-field strength range, respectively, between about 109 and 1012 V/cm and between about 1 and 500 MG (Belyaev et al., 2008; Salamin et al., 2006). Product terawatt-pulse picosecond laser plasmas appear to be some kind of a "table" pulsed "microaccelerator" and a nuclear "microreactor," which is relatively compact and cheap and on which one must not impose special radiation-safety requirements. Such a source admits a relatively simple possibility for controlling energy and other parameters of corpuscular and electromagnetic radiations.

Fast Charged Particles and Super-Strong Magnetic

D2 were used in our experiment.

plane.

Fields Generated by Intense Laser Target Interaction 97

sectional dimensions and 1 to 6 mm in thickness and were installed at the same positions as the track detectors D1. Thus, either the track detectors D1 or the secondary activated targets

Fig. 3. Layout of the experiment: (T) target, (VC) vacuum chamber, (W) vacuum-chamber window; (M) off-axis parabolic mirror, (LR) laser radiation, (*N*) normal to the target, (D1) CR-39 track detectors equipped with aluminum filters, (D2) secondary activated targets from LiF, Cu, and Ti, (D3, D4) scintillation detectors for gamma radiation, and (D5, D6) neutron detectors on the basis of helium counters. Detectors D1–D4 and D6 lie in the *xy* 

Two scintillation detectors D3 and D4 positioned at distances of 4.3 and 3.0 m from the target, respectively, were used to record hard x-ray radiation. Lead filters 8 cm thick for D3 and 13.5 cm thick for D4 were installed in front of the detectors. The detectors D3 and D4 are scintillation detectors on the basis of plastic scintillators 510 cm in dimension. The detectors D3 and D4 were used to record hard x-ray photons of energy 0.5 to 10 MeV. The detectors D5 and D6, which are based on helium counters, were used to determine the yield of neutrons generated in (*p, n*) reactions. The detector D5 was arranged along the tangent to the target surface at a distance of 25 cm, while the D6 detector was positioned behind the target at a distance of 60 cm. The detectors D5 and D6 consisted of the following units: a block of neutron counters on the basis of three CNM-18 helium counters, a voltage transducer, a signal-selection device, and a power amplifier. The side surfaces of the

The layout of the experimental facility used to study various mechanisms of fast-proton production is displayed in Fig. 4. As targets, we employed metallic foils from titanium 30

detectors D5 and D6 were surrounded by polyethylene 2 cm thick.

At the present time, the production of high-energy proton beams in laser plasmas under the effect of ultrashort pulses is a rapidly developing field of investigations (Carrier et al., 2009; Fucuda et al., 2009; Yan et al., 2009; Willingale et al., 2009; Gonoskov et al., 2009; Psikal et al., 2010; Huang et al., 2010).

Several models that claim for explaining observed results that concern the production of directed beams of high-energy protons were proposed in theoretical investigations. One of them is based on the mechanism of proton acceleration at the front surface of the target owing to the ponderomotive pressure of a laser pulse (Sentoku et al., 2003; Maksimchuk et al., 2000). According to a different model (MacKinnon et al., 2001; Wilks et al., 2001), relativistic hot electrons produced by a laser field in a solid-state target penetrate through the target, and some electrons escape from the rear surface of the target to a distance of about the Debye radius. These electrons generate an electrostatic field at the rear surface of the target. This field, which may exceed 1012 V/cm, accelerates protons.

However, the efficiency of the proposed proton acceleration mechanisms has so far been debated (Salamin et al., 2006). In view of this, the our experimental studies were aimed at exploring various mechanisms of the acceleration of fast protons in laser plasmas under identical conditions of the irradiation of a solid-state target at a laser-radiation intensity of about 21018 W/cm2.

The experiments in question were performed at the 10-TW picosecond laser facility Neodymium (Belyaev, Vinogradov et al., 2006). This laser facility has the following laserpulse parameters: a pulse energy of up to 10 J, the wavelength of 1.055 m, and the pulse duration of 1.5 ps. Its focusing system, which is based on an off-axis parabolic mirror whose focal length is 20 cm, ensures a concentration of not less than 40% of the laser beam energy within a spot *D* = 15 m in diameter and, accordingly, an average intensity of 1018 W/cm2 at the target surface and a peak intensity of 21018 W/cm2.

Laser radiation generated by the Neodymium facility is characterized by the presence of two prepulses—one of picosecond and the other of nanosecond duration. The first prepulse appears 14 ns before the main laser pulse; it has a duration of 1.5 ps and an intensity below 10*−*8 with respect to the main pulse. The second prepulse results from amplified spontaneous emission. Its FWHM duration is 4 ns, while its intensity with respect to the main pulse is below 10*−*8.

The layout of the experiment is shown in Fig. 1. A beam of linearly polarized laser radiation of *p*-type polarization is focused by an off-axis parabolic mirror onto the surface of a solidstate target (T) at an angle of 40*◦* with respect to the normal to the target surface. For targets, we employed slabs from LiF and Cu 1 to 30 mm in thickness and the Al, Cu, and Ti foils 1 to 100 m in thickness. The targets were arranged in a vacuum chamber 30 cm in diameter and 50 cm in height. The pressure of the residual gas in the chamber was not more than 10*−*3 torr. Detectors D1 based on CR-39 track detectors of size 24 to 20 mm and equipped with aluminum filters of different thickness, from 11 to 100 m, which make it possible to cut off the energy interval 0.8–3.5 MeV for protons, were used to detect protons and to measure their energy spectrum. The detectors D1 were arranged upstream and downstream of the target at a distance of 20 mm from it along the normal.

The secondary activated targets D2, which were manufactured from LiF, Cu, and Ti and which are characterized by different threshold energies for (*p, n*) reactions (from 1.88 MeV for 7Li to 5 MeV for 48Ti), were also used to detect protons and to determine their number and maximum energy. The secondary activated targets D2 were slabs 3030 mm2 in cross96 Femtosecond–Scale Optics

At the present time, the production of high-energy proton beams in laser plasmas under the effect of ultrashort pulses is a rapidly developing field of investigations (Carrier et al., 2009; Fucuda et al., 2009; Yan et al., 2009; Willingale et al., 2009; Gonoskov et al., 2009; Psikal et al.,

Several models that claim for explaining observed results that concern the production of directed beams of high-energy protons were proposed in theoretical investigations. One of them is based on the mechanism of proton acceleration at the front surface of the target owing to the ponderomotive pressure of a laser pulse (Sentoku et al., 2003; Maksimchuk et al., 2000). According to a different model (MacKinnon et al., 2001; Wilks et al., 2001), relativistic hot electrons produced by a laser field in a solid-state target penetrate through the target, and some electrons escape from the rear surface of the target to a distance of about the Debye radius. These electrons generate an electrostatic field at the rear surface of

However, the efficiency of the proposed proton acceleration mechanisms has so far been debated (Salamin et al., 2006). In view of this, the our experimental studies were aimed at exploring various mechanisms of the acceleration of fast protons in laser plasmas under identical conditions of the irradiation of a solid-state target at a laser-radiation intensity of

The experiments in question were performed at the 10-TW picosecond laser facility Neodymium (Belyaev, Vinogradov et al., 2006). This laser facility has the following laserpulse parameters: a pulse energy of up to 10 J, the wavelength of 1.055 m, and the pulse duration of 1.5 ps. Its focusing system, which is based on an off-axis parabolic mirror whose focal length is 20 cm, ensures a concentration of not less than 40% of the laser beam energy within a spot *D* = 15 m in diameter and, accordingly, an average intensity of 1018 W/cm2 at

Laser radiation generated by the Neodymium facility is characterized by the presence of two prepulses—one of picosecond and the other of nanosecond duration. The first prepulse appears 14 ns before the main laser pulse; it has a duration of 1.5 ps and an intensity below 10*−*8 with respect to the main pulse. The second prepulse results from amplified spontaneous emission. Its FWHM duration is 4 ns, while its intensity with respect to the

The layout of the experiment is shown in Fig. 1. A beam of linearly polarized laser radiation of *p*-type polarization is focused by an off-axis parabolic mirror onto the surface of a solidstate target (T) at an angle of 40*◦* with respect to the normal to the target surface. For targets, we employed slabs from LiF and Cu 1 to 30 mm in thickness and the Al, Cu, and Ti foils 1 to 100 m in thickness. The targets were arranged in a vacuum chamber 30 cm in diameter and 50 cm in height. The pressure of the residual gas in the chamber was not more than 10*−*3 torr. Detectors D1 based on CR-39 track detectors of size 24 to 20 mm and equipped with aluminum filters of different thickness, from 11 to 100 m, which make it possible to cut off the energy interval 0.8–3.5 MeV for protons, were used to detect protons and to measure their energy spectrum. The detectors D1 were arranged upstream and downstream of the

The secondary activated targets D2, which were manufactured from LiF, Cu, and Ti and which are characterized by different threshold energies for (*p, n*) reactions (from 1.88 MeV for 7Li to 5 MeV for 48Ti), were also used to detect protons and to determine their number and maximum energy. The secondary activated targets D2 were slabs 3030 mm2 in cross-

the target. This field, which may exceed 1012 V/cm, accelerates protons.

the target surface and a peak intensity of 21018 W/cm2.

target at a distance of 20 mm from it along the normal.

2010; Huang et al., 2010).

about 21018 W/cm2.

main pulse is below 10*−*8.

sectional dimensions and 1 to 6 mm in thickness and were installed at the same positions as the track detectors D1. Thus, either the track detectors D1 or the secondary activated targets D2 were used in our experiment.

Fig. 3. Layout of the experiment: (T) target, (VC) vacuum chamber, (W) vacuum-chamber window; (M) off-axis parabolic mirror, (LR) laser radiation, (*N*) normal to the target, (D1) CR-39 track detectors equipped with aluminum filters, (D2) secondary activated targets from LiF, Cu, and Ti, (D3, D4) scintillation detectors for gamma radiation, and (D5, D6) neutron detectors on the basis of helium counters. Detectors D1–D4 and D6 lie in the *xy*  plane.

Two scintillation detectors D3 and D4 positioned at distances of 4.3 and 3.0 m from the target, respectively, were used to record hard x-ray radiation. Lead filters 8 cm thick for D3 and 13.5 cm thick for D4 were installed in front of the detectors. The detectors D3 and D4 are scintillation detectors on the basis of plastic scintillators 510 cm in dimension. The detectors D3 and D4 were used to record hard x-ray photons of energy 0.5 to 10 MeV.

The detectors D5 and D6, which are based on helium counters, were used to determine the yield of neutrons generated in (*p, n*) reactions. The detector D5 was arranged along the tangent to the target surface at a distance of 25 cm, while the D6 detector was positioned behind the target at a distance of 60 cm. The detectors D5 and D6 consisted of the following units: a block of neutron counters on the basis of three CNM-18 helium counters, a voltage transducer, a signal-selection device, and a power amplifier. The side surfaces of the detectors D5 and D6 were surrounded by polyethylene 2 cm thick.

The layout of the experimental facility used to study various mechanisms of fast-proton production is displayed in Fig. 4. As targets, we employed metallic foils from titanium 30

Fast Charged Particles and Super-Strong Magnetic

Fields Generated by Intense Laser Target Interaction 99

from the rare surface in the outward direction, and a temperature of 250 keV for protons

Fig. 5. Spectra of protons for various mechanisms of acceleration: (1) acceleration of protons

Figure 6 shows the results of our experiments aimed at determining the maximum energy of protons for aluminum targets of various thickness—from 2.5 to 100 m. These results were obtained both by using the CR-39 track detectors equipped with aluminum filters of various thickness and by using the activation procedure. From Fig. 6, one can see that there is an optimum aluminum-target thickness of 10 m, at which protons of maximum energy 5 MeV

We will now compare the experimentally measured maximum energies of protons accelerated from the front surface of the target in the inward direction (*E* ≈ 2 MeV) and protons accelerated from the rear surface of the target in the outward direction (*E* ≈ 5 MeV) with the estimates of the maximum energy of protons subjected to the effect of the

and in the case of plasma expansion into a vacuum (Zepf et al., 2003; Cowan et al., 2004;

<sup>2</sup> 2 ln . *E ZT iMAX e pi*

 

<sup>2</sup> 22 , *E a iMAX Zmc* (22)

(23)

acceleration of protons from the rear surface of the target in the outward direction (Ti, 30 m; *T* ≈ 500 keV), and (3) acceleration of protons from the front surface of the target in the inward direction (LiF, 6 m; *T* ≈ *2*50 keV). The open circles and triangles represent data from track detectors, while the closed circles, triangles, and boxes stand for data obtained on the

from the front surface of the target toward a laser ray (Ti, 30 m; *T* ≈ 180 keV); (2)

basis of the activation procedure [*p*(7Li, 7Be)*n*, *Ethr* = 1*.*88 MeV].

are produced.

Robson et al., 2007),

ponderomotive force (Pukhov, 2001),

accelerated from the front surface of the target in the inward direction.

*μ*m thick (see Figs. 4*a* and 4*b*) and a LiF plate 6 mm thick (see Fig. 4*c*). As the secondary, activated, target, we took a LiF plate 6 mm thick. In the case presented in Fig. 4*c*, the primary target serves as the activated target as well.

Fig. 4. Layout of experiments aimed at studying various mechanisms of the acceleration of fast protons: (*a*) acceleration of protons from the front surface of the target toward the laser pulse, (*b*) acceleration of protons from the rear surface of the target in the outward direction, and (*c*) acceleration of protons from the front surface of the target in the inward direction.

Taking into account the solid angle covered by the detector D5 and its efficiency, we found that the number of neutrons generated on average in the LiF secondary, activated, target over 4 sr per laser pulse is 50 in the first case (see Fig. 4*a*), about 2103 in the second case (see Fig. 4*b*), and about 2102 in the third case (see Fig. 4*c*).

The number of fast protons can be estimated by the formula *Np Yn*/(*nli*), where *Yn* is the yield of neutrons from the reaction 7Li(*p, n*)7Be, *n* is the concentration of 7Li atoms in the target, 60 mb (Youssef et al., 2006) is the cross section for the reaction 7Li(*p*, *n*)7Be at proton energies around 1.9MeV, and *li* is the proton braking length in the target. At distances longer than *li*, protons have have energies below the threshold energy of 1.88 MeV, so that the reaction 7Li(*p, n*)7Be, which leads to neutron production, cannot proceed. Under the conditions of our experiments, *li ≈* 10 m. Taking into account the yield of neutrons for the three cases considered here, we ultimately find that the number of accelerated protons from the front surface toward the laser pulse that have an energy in excess of 1.88 MeV is 107; the number of protons accelerated from the rear surface of the target in the outward direction is 4108, while the number of protons accelerated from the front surface in the inward direction is 4107. Thus, the results of our experiments have revealed that the proton-acceleration process occurs most efficiently in the case of proton acceleration from the rear surface of the target in the outward direction.

This conclusion is also confirmed by the results obtained by measuring the spectra of fast protons for various mechanisms of their acceleration.

Figures 5 shows the measured spectra of protons for various proton-acceleration mechanisms. These spectra were obtained both by using track detectors CR-39 equipped with aluminum filters of various thickness and by using the activation procedure. From these spectra, it follows that the energy distribution of fast protons corresponds to the Boltzmann distribution at a temperature of 180 keV for protons accelerated from the front surface of the target toward the laser pulse, a temperature of 500 keV for protons accelerated 98 Femtosecond–Scale Optics

*μ*m thick (see Figs. 4*a* and 4*b*) and a LiF plate 6 mm thick (see Fig. 4*c*). As the secondary, activated, target, we took a LiF plate 6 mm thick. In the case presented in Fig. 4*c*, the

Fig. 4. Layout of experiments aimed at studying various mechanisms of the acceleration of fast protons: (*a*) acceleration of protons from the front surface of the target toward the laser pulse, (*b*) acceleration of protons from the rear surface of the target in the outward direction, and (*c*) acceleration of protons from the front surface of the target in the inward direction. Taking into account the solid angle covered by the detector D5 and its efficiency, we found that the number of neutrons generated on average in the LiF secondary, activated, target over 4 sr per laser pulse is 50 in the first case (see Fig. 4*a*), about 2103 in the second case

yield of neutrons from the reaction 7Li(*p, n*)7Be, *n* is the concentration of 7Li atoms in the

proton energies around 1.9MeV, and *li* is the proton braking length in the target. At distances longer than *li*, protons have have energies below the threshold energy of 1.88 MeV, so that the reaction 7Li(*p, n*)7Be, which leads to neutron production, cannot proceed. Under the conditions of our experiments, *li ≈* 10 m. Taking into account the yield of neutrons for the three cases considered here, we ultimately find that the number of accelerated protons from the front surface toward the laser pulse that have an energy in excess of 1.88 MeV is 107; the number of protons accelerated from the rear surface of the target in the outward direction is 4108, while the number of protons accelerated from the front surface in the inward direction is 4107. Thus, the results of our experiments have revealed that the proton-acceleration process occurs most efficiently in the case of proton acceleration from

This conclusion is also confirmed by the results obtained by measuring the spectra of fast

Figures 5 shows the measured spectra of protons for various proton-acceleration mechanisms. These spectra were obtained both by using track detectors CR-39 equipped with aluminum filters of various thickness and by using the activation procedure. From these spectra, it follows that the energy distribution of fast protons corresponds to the Boltzmann distribution at a temperature of 180 keV for protons accelerated from the front surface of the target toward the laser pulse, a temperature of 500 keV for protons accelerated

60 mb (Youssef et al., 2006) is the cross section for the reaction 7Li(*p*, *n*)7Be at

*li*), where *Yn* is the

primary target serves as the activated target as well.

(see Fig. 4*b*), and about 2102 in the third case (see Fig. 4*c*).

the rear surface of the target in the outward direction.

protons for various mechanisms of their acceleration.

target, 

The number of fast protons can be estimated by the formula *Np Yn*/(*n*

from the rare surface in the outward direction, and a temperature of 250 keV for protons accelerated from the front surface of the target in the inward direction.

Fig. 5. Spectra of protons for various mechanisms of acceleration: (1) acceleration of protons from the front surface of the target toward a laser ray (Ti, 30 m; *T* ≈ 180 keV); (2) acceleration of protons from the rear surface of the target in the outward direction (Ti, 30 m; *T* ≈ 500 keV), and (3) acceleration of protons from the front surface of the target in the inward direction (LiF, 6 m; *T* ≈ *2*50 keV). The open circles and triangles represent data from track detectors, while the closed circles, triangles, and boxes stand for data obtained on the basis of the activation procedure [*p*(7Li, 7Be)*n*, *Ethr* = 1*.*88 MeV].

Figure 6 shows the results of our experiments aimed at determining the maximum energy of protons for aluminum targets of various thickness—from 2.5 to 100 m. These results were obtained both by using the CR-39 track detectors equipped with aluminum filters of various thickness and by using the activation procedure. From Fig. 6, one can see that there is an optimum aluminum-target thickness of 10 m, at which protons of maximum energy 5 MeV are produced.

We will now compare the experimentally measured maximum energies of protons accelerated from the front surface of the target in the inward direction (*E* ≈ 2 MeV) and protons accelerated from the rear surface of the target in the outward direction (*E* ≈ 5 MeV) with the estimates of the maximum energy of protons subjected to the effect of the ponderomotive force (Pukhov, 2001),

$$E\_{i\text{MAX}} \approx 2\sqrt{2} \cdot a \text{Zmc}^2,\tag{22}$$

and in the case of plasma expansion into a vacuum (Zepf et al., 2003; Cowan et al., 2004; Robson et al., 2007),

$$E\_{iMAX} \approx 2ZT\_e \ln^2 \left(\alpha\_{pi}\tau\right). \tag{23}$$

Fast Charged Particles and Super-Strong Magnetic

temperature of the fast fluoride ions is *Tfast* = 350 keV.

where *v* is the ion velocity in the observation direction, *Mv*<sup>2</sup>

fluoride ion energy distribution were also estimated theoretically.

while bottom curve refers to ions moving outward from the target.

excitation of the promising nuclear fusion reactions 6Li(*d*,

**4. Relativistic magneto-active laser plasmas** 

7Li(*p*, 

plasmas are discussed.

particles.

Fields Generated by Intense Laser Target Interaction 101

In addition, using the red shift of the Doppler profile for the *Lya* line it was found that fast ions move inwards from a target surface. In (Belyaev et al., 2005), the parameters of the

Fig. 7. Energy distribution of fast fluoride ions derived from the profile measurements of Lya line emitted by F IX ion. Top curve corresponds to ions moving towards the target,

In our paper (Belyaev et al., 2009) the results of experiments devoted to studying the

Principal results of investigations of relativistic laser plasmas are presented here. We found parameters of magnetic fields generated in laser plasma – the amplitude of the magnetic field, its lifetime, and the increment, the spatial structure. Mechanisms of acceleration of charged particles have been investigated which are related with considered magnetic fields.

1. Electrons interacting with a field of electromagnetic wave can be considered as free

2. Free electrons in relativistic laser plasmas interact only with an electromagnetic wave.

Main peculiarities that determine properties of relativistic laser plasmas are:

)4He, along with the standard reaction D(*d*, *n*)3He, in picosecond laser plasmas are presented. For the first time, it was shown that these reactions may proceed at a moderate laser-radiation intensity of 21018 W/cm2, the respective yield being 2103 to 105 per laser pulse. A brief survey of the main processes responsible for the generation of fast electrons and fast ions (protons) at the front surface of the target and for the excitation of nuclear fusion reactions is given. The calculated and experimental results on the yield from nuclear fusion reactions in picosecond laser plasmas are compared. The possibilities for optimizing the yield from the promising fusion reactions excited in femto- and picosecond laser

)4He, 3He(*d*, *p*)4He, 11B(*p*, 3

), and

0 /2 = 25 keV, and the

At *I* = 21018 W/cm2, expression (22) yields the energy value of 1.73 MeV, while expression (23) leads to 5.1 MeV. These energy values agree reasonably well with experimental data. We will now discuss the results obtained experimentally for the target-thickness dependence of the maximum energy of protons accelerated from the rare surface of the target in the outward direction. From our analysis, it follows that the reduction of the maximum proton energy at the foil thickness smaller than 10 m is due to the effect of the nanosecond prepulse because of the pulse of enhanced spontaneous emission. The nanosecond prepulse generates shock waves in the foil, which deform the rear surface of the target, and this leads to a increase in the size of the plasma inhomogeneity at the rear surface of the target and to a decrease in the energy of protons produced at the rare surface of the target. The decrease in the maximum proton energy for target thicknesses in excess of 10 m is due to the decrease in the energy of electrons as they pass through the target and to the increase of their angular spread. This in turn leads to a less efficient acceleration of protons from the rear surface of the target.

Fig. 6. Maximum proton energy *EpMAX* as a function of the aluminum-foil thickness.

In our paper (Belyaev et al., 2005), experimental data are presented on the generation of fast ions in a laser picosecond plasma at a laser radiation intensity of 21018 W/cm2. The results were obtained from Doppler spectra of hydrogen-like fluoride ions. An important peculiarity of the energy distribution of fast fluoride ions is the slow fall in ion energy to 1.4 MeV. In Fig. 7, the energy distribution of fast fluoride ions is plotted based on the results of measurements of *Lya* line profile for F IX ion. The solid curves are calculated by the formula

$$\frac{\text{dN}}{\text{dE}} \sim \exp\left[-\frac{M\left(v - v\_0\right)^2}{2T\_{\text{fast}}}\right] \tag{24}$$

100 Femtosecond–Scale Optics

At *I* = 21018 W/cm2, expression (22) yields the energy value of 1.73 MeV, while expression (23) leads to 5.1 MeV. These energy values agree reasonably well with experimental data. We will now discuss the results obtained experimentally for the target-thickness dependence of the maximum energy of protons accelerated from the rare surface of the target in the outward direction. From our analysis, it follows that the reduction of the maximum proton energy at the foil thickness smaller than 10 m is due to the effect of the nanosecond prepulse because of the pulse of enhanced spontaneous emission. The nanosecond prepulse generates shock waves in the foil, which deform the rear surface of the target, and this leads to a increase in the size of the plasma inhomogeneity at the rear surface of the target and to a decrease in the energy of protons produced at the rare surface of the target. The decrease in the maximum proton energy for target thicknesses in excess of 10 m is due to the decrease in the energy of electrons as they pass through the target and to the increase of their angular spread. This in turn leads to a less efficient acceleration of protons

Fig. 6. Maximum proton energy *EpMAX* as a function of the aluminum-foil thickness.

In our paper (Belyaev et al., 2005), experimental data are presented on the generation of fast ions in a laser picosecond plasma at a laser radiation intensity of 21018 W/cm2. The results were obtained from Doppler spectra of hydrogen-like fluoride ions. An important peculiarity of the energy distribution of fast fluoride ions is the slow fall in ion energy to 1.4 MeV. In Fig. 7, the energy distribution of fast fluoride ions is plotted based on the results of measurements of *Lya* line profile for F IX ion. The solid curves are calculated by the formula

> <sup>2</sup> <sup>d</sup> <sup>0</sup> ~ exp d 2 *fast N Mv v E T*

(24)

from the rear surface of the target.

where *v* is the ion velocity in the observation direction, *Mv*<sup>2</sup> 0 /2 = 25 keV, and the temperature of the fast fluoride ions is *Tfast* = 350 keV.

In addition, using the red shift of the Doppler profile for the *Lya* line it was found that fast ions move inwards from a target surface. In (Belyaev et al., 2005), the parameters of the fluoride ion energy distribution were also estimated theoretically.

Fig. 7. Energy distribution of fast fluoride ions derived from the profile measurements of Lya line emitted by F IX ion. Top curve corresponds to ions moving towards the target, while bottom curve refers to ions moving outward from the target.

In our paper (Belyaev et al., 2009) the results of experiments devoted to studying the excitation of the promising nuclear fusion reactions 6Li(*d*, )4He, 3He(*d*, *p*)4He, 11B(*p*, 3), and 7Li(*p*, )4He, along with the standard reaction D(*d*, *n*)3He, in picosecond laser plasmas are presented. For the first time, it was shown that these reactions may proceed at a moderate laser-radiation intensity of 21018 W/cm2, the respective yield being 2103 to 105 per laser pulse. A brief survey of the main processes responsible for the generation of fast electrons and fast ions (protons) at the front surface of the target and for the excitation of nuclear fusion reactions is given. The calculated and experimental results on the yield from nuclear fusion reactions in picosecond laser plasmas are compared. The possibilities for optimizing the yield from the promising fusion reactions excited in femto- and picosecond laser plasmas are discussed.

## **4. Relativistic magneto-active laser plasmas**

Principal results of investigations of relativistic laser plasmas are presented here. We found parameters of magnetic fields generated in laser plasma – the amplitude of the magnetic field, its lifetime, and the increment, the spatial structure. Mechanisms of acceleration of charged particles have been investigated which are related with considered magnetic fields. Main peculiarities that determine properties of relativistic laser plasmas are:


Fast Charged Particles and Super-Strong Magnetic

Fig. 8a. Vortex lines of moving potential

along the direction of propagation of electromagnetic wave:

target because of the forces of the Coulomb attraction.

vortex and its cross section

[W/cm2].

electromagnetic wave which propagates along the direction **n**:

Fields Generated by Intense Laser Target Interaction 103

relativistic velocities. The expression for the motion integral follows from the equation (26), taking into account also the Maxwell equations for an electron in the field of an

2 2

*V c* 

*<sup>c</sup> Const*

laser plasmas

2 2

11 1 1

*V a c a*

Here the quantity *a* is determined by the electromagnetic wave intensity *J*: 18 0,85

Positively charged atomic ions prevent from motion of the considered electron vortex in a

This expression is useful at the consideration of dynamics of relativistic particles in a field of an electromagnetic wave. For example, if a charged particle (for example, an electron) rotates with the velocity *V* in a circularly polarized field of an electromagnetic wave, then this particle acquires obligatory some velocity along the direction *n* of the wave propagation. When **V***/c* = 0, the expression (30) is equal to unity. This value does not change also for other velocities. Hence, one obtains the next expression for the particle velocity

**n v** (30)

Fig. 8b. Structure of magnetic field produced in

(31)

10 *<sup>J</sup> <sup>a</sup>* , *<sup>J</sup>*

1 1

3. The conservation laws and motion integrals are valid also in the range of relativistic laser intensities.

Equations describing quasi-stationary magnetic fields which are generated in laser plasmas can be derived from the conservation law for generalized momentum:

$$\mathbf{P} = m\mathbf{v}\gamma - \frac{e}{c}\mathbf{A} \tag{25}$$

Here **A** is the vector-potential of an electromagnetic wave. The relativistic equation of motion is of the form

$$\frac{d}{dt}m\mathbf{v}\gamma = e\mathbf{E} + \frac{e}{c}[\mathbf{V}\times\mathbf{B}] \tag{26}$$

Deleting the intermediate derivations, we present final equations for vortex electron structures producing magnetic field in laser plasmas:

$$
\mathbf{u} \mathbf{o} = r \mathbf{ot} \mathbf{v} \tag{27}
$$

$$
\operatorname{div} \mathbf{V} = 0 \tag{28}
$$

$$\frac{d\mathbf{u}}{dt} + r\boldsymbol{\sigma}t \left[\mathbf{u} \times \mathbf{V}\right] = 0\tag{29}$$

Here *<sup>e</sup> mc* **<sup>B</sup> ω** is a cyclotron frequency for electron rotation in the magnetic field **B**, and

2 2 1 1 *v c* is the relativistic factor.

These equations mean conservation laws for vortex electron structure: Eq. (27) is the conservation law for a generalized momentum (25); Eq. (28) is the conservation law for a number of particles, and Eq. (29) is the conservation law for a magnetic flow, or for an angular momentum.

It should be noted that these equations allow undamped solutions. In general case solution of these equations taking into account losses is a difficult mathematical problem knowing as a problem of magnetic field generation. In particular, explanation of Earth magnetism is a part of this problem.

Equations (27) – (29) coincide with equations for real potential vortexes in mechanics of continuum matter which correspond to three Helmholtz theorems (Sedov, 1983).

The potential vortex presents good description of the observed vortex. Uniform rotation is unfit for description of the observed vortex. The velocity inside the observed vortex is high and outside of it is small, while the inverse statement is valid for the case of the uniform rotation. Coincidence of equations for a magnetic field in laser plasmas and for a potential vortex results in identity of their spatial structures (see Fig. 8).

An electron vortex producing a quasi-stationary magnetic field and their analogous classical potential vortex can exist only in motion. In general case the transformation of rotational energy into a translational motion is a relativistic effect. This fact follows from requirement of relativistic invariance for motion of charged particles; it takes place also at small non102 Femtosecond–Scale Optics

3. The conservation laws and motion integrals are valid also in the range of relativistic

Equations describing quasi-stationary magnetic fields which are generated in laser plasmas

Here **A** is the vector-potential of an electromagnetic wave. The relativistic equation of

*d e*

Deleting the intermediate derivations, we present final equations for vortex electron

<sup>0</sup> *<sup>d</sup> rot*

These equations mean conservation laws for vortex electron structure: Eq. (27) is the conservation law for a generalized momentum (25); Eq. (28) is the conservation law for a number of particles, and Eq. (29) is the conservation law for a magnetic flow, or for an

It should be noted that these equations allow undamped solutions. In general case solution of these equations taking into account losses is a difficult mathematical problem knowing as a problem of magnetic field generation. In particular, explanation of Earth magnetism is a

Equations (27) – (29) coincide with equations for real potential vortexes in mechanics of

The potential vortex presents good description of the observed vortex. Uniform rotation is unfit for description of the observed vortex. The velocity inside the observed vortex is high and outside of it is small, while the inverse statement is valid for the case of the uniform rotation. Coincidence of equations for a magnetic field in laser plasmas and for a potential

An electron vortex producing a quasi-stationary magnetic field and their analogous classical potential vortex can exist only in motion. In general case the transformation of rotational energy into a translational motion is a relativistic effect. This fact follows from requirement of relativistic invariance for motion of charged particles; it takes place also at small non-

continuum matter which correspond to three Helmholtz theorems (Sedov, 1983).

vortex results in identity of their spatial structures (see Fig. 8).

**ω** is a cyclotron frequency for electron rotation in the magnetic field **B**, and

**<sup>ω</sup>**

*dt*

**v E VB**

*m e dt c*

*m*

*e*

(25)

(26)

**ω** *rot***v** (27)

*div***V** 0 (28)

**ω V** (29)

*c* **Pv A** 

can be derived from the conservation law for generalized momentum:

structures producing magnetic field in laser plasmas:

laser intensities.

motion is of the form

Here *<sup>e</sup>*

1

*mc***<sup>B</sup>**

1 *v c*

angular momentum.

part of this problem.

2 2

is the relativistic factor.

relativistic velocities. The expression for the motion integral follows from the equation (26), taking into account also the Maxwell equations for an electron in the field of an electromagnetic wave which propagates along the direction **n**:

Fig. 8a. Vortex lines of moving potential vortex and its cross section

Fig. 8b. Structure of magnetic field produced in laser plasmas

This expression is useful at the consideration of dynamics of relativistic particles in a field of an electromagnetic wave. For example, if a charged particle (for example, an electron) rotates with the velocity *V* in a circularly polarized field of an electromagnetic wave, then this particle acquires obligatory some velocity along the direction *n* of the wave propagation. When **V***/c* = 0, the expression (30) is equal to unity. This value does not change also for other velocities. Hence, one obtains the next expression for the particle velocity along the direction of propagation of electromagnetic wave:

$$\frac{V}{c} = \frac{\gamma - 1}{\gamma} = \frac{\sqrt{1 + a^2} - 1}{\sqrt{1 + a^2}} \tag{31}$$

Here the quantity *a* is determined by the electromagnetic wave intensity *J*: 18 0,85 10 *<sup>J</sup> <sup>a</sup>* , *<sup>J</sup>*

[W/cm2].

Positively charged atomic ions prevent from motion of the considered electron vortex in a target because of the forces of the Coulomb attraction.

Fast Charged Particles and Super-Strong Magnetic

than the plasma frequency.

Fig. 9a. The photo of the track detector CR-39 covered by 11 mm Al filter. Detector CR-39 shows the tracks of protons with energies *Ep* > 0.8 MeV;

1/2 » 14° (cone half angle)

instability (Weibel, 1959).

2008).

fields generated in laser plasma, term quasistationary.

Increment of the considered magnetic field is equal to the ionization rate

Fields Generated by Intense Laser Target Interaction 105

laser action and can exceed it on one-two order. For this reason the superstrong magnetic

Mechanisms of generating magnetic fields are a subject of numerous investigations performed in recent years (Tatarakis et al., 2002; Beg et al., 2004; Borghesi et al., 1998). Various mechanisms for generating a magnetic field in the interaction of intensive laser radiation with solid targets are described in a number of theoretical works (Stamper, 1991; Wilks et al., 1992; Bell et al., 1993; Buchenkov et al., 1993; Sudan, 1993; Haines, 1997; Mason & Tabak, 1998; Krainov, 2003). In particular, they predicted the origin of magnetic fields with induction of up to 1GG in the dense plasma produced during the interaction process. These fields are localized near the critical surface, where the laser energy is mainly absorbed. The arising magnetic fields noticeably affect the dynamics of laser plasma. The principal mechanisms of generating quasistatic magnetic fields were considered: (1) different directions of the temperature and plasma density gradients; (2) the flux of fast electrons accelerated by ponderomotive forces in the longitudinal and transversal directions with respect to the direction of laser pulse propagation, and (3) the collisionless Weibel

Measurements of superstrong quasistatic magnetic fields in laser plasma and their theoretical interpretation have been discussed in more detail in our article (Belyaev et al.,

Measurements of magnetic fields in plasma by various independent methods are very important for both proving the existence of such fields and determining their spatial structure (topology). For this purpose, we measured the profiles of X-ray spectral lines of hydrogen-like fluoride ions in laser plasma with a radiation intensity of 1017 W/cm2 and

Fig. 9b. The proton distribution inside the spot for detector with 11 m Al (*Ep* > 0.8 MeV). Target Cu

25 m. Protons with energy *E* < 2,5 MeV

*<sup>i</sup>*, which is larger

The requirement of quasi-neutrality results in motion of positively charged atomic ions. Omitting details of derivations and taking into account the Vlasov equations for a quasineutral two-component plasma and conservation law of the generalized momentum both for ions and for electrons, we present the final result:

Electrons and ions in relativistic laser plasmas form the one vortex structure – a potential vortex. This structure moves together with produced electromagnetic fields having the velocity of an electric drift (at **E** < **B**):

$$\mathbf{v} = c \frac{\text{[EB]}}{\text{B}^2} \tag{32}$$

Let us remark one peculiarity. The ion velocity and the ion free path are small in the process of ion motion. Ions are decelerated in a target; then new ions take their place, and finally the whole vortex structure occurs on the rear side of the target. If *l*i is the depth for ion deceleration, the last ions propagate together with electrons producing quasi-neutral potential plasma vortex.

The drift motion does not produce the electric current and charge separation, since particles with positive and negative charge drift in the same direction with the same velocity. Thus, drift produces motion of neutral plasma.

Plasma magnetization results in small divergence of these flows. It is explained by a stability of vortex quasi-neutral structures as quasi-particles

Some publications report about experimental confirmation of generation of magnetized toroidal plasma structures. Ring-shaped proton flows with small divergence were observed (Nakamura & Mima, 2008; Clark et al., 2000). The magnetic field of about 100 MG has been measured by direct spectral method on large distance (several hundreds of microns) from the target surface (Belyaev et al., 2004).

Our experiments at the peak laser intensity of 21018 W/cm2 allows us to observe on the rear side of thin (30 m) titanium target clear ring-shaped structures by the proton detector CR-39 placed on a distance of 20 mm. Photo of ring-shaped proton structure is presented in Fig. 9a, and proton distributions with the energy of 2.5 MeV are presented in Fig. 9b. The divergence of the proton beam is 1/2 14. Protons with the energy higher than 2.5 MeV present narrow collimated beam with the divergence angle of 1/2 = 3. Inside this narrow collimated beam with the divergence angle 1/2 = 3 we observed well collimated proton beams with the divergence angle of 1/2 = 0.10.3.

Note, that drift velocity can increase significantly under condition of development of pincheffect up to relativistic values. Respectively, not only electron velocity, but also the velocity of heavy positively charged atomic ions can increase up to relativistic values (Belyaev, Faenov et al., 2006).

Deleting the intermediate derivations, we present expressions for lifetime considered magnetic field:

$$T = 2\frac{\mathcal{E}}{\Delta \mathcal{E}} \Delta t \tag{33}$$

where - laser pulse energy, - losses of an energy for electron vortex structure, <sup>2</sup> *t D* , *D* – coefficient of Bohm's diffusion. This lifetime does not depend on duration of 104 Femtosecond–Scale Optics

The requirement of quasi-neutrality results in motion of positively charged atomic ions. Omitting details of derivations and taking into account the Vlasov equations for a quasineutral two-component plasma and conservation law of the generalized momentum both

Electrons and ions in relativistic laser plasmas form the one vortex structure – a potential vortex. This structure moves together with produced electromagnetic fields having the

> <sup>2</sup> *c* **EB**

Let us remark one peculiarity. The ion velocity and the ion free path are small in the process of ion motion. Ions are decelerated in a target; then new ions take their place, and finally the whole vortex structure occurs on the rear side of the target. If *l*i is the depth for ion deceleration, the last ions propagate together with electrons producing quasi-neutral

The drift motion does not produce the electric current and charge separation, since particles with positive and negative charge drift in the same direction with the same velocity. Thus,

Plasma magnetization results in small divergence of these flows. It is explained by a stability

Some publications report about experimental confirmation of generation of magnetized toroidal plasma structures. Ring-shaped proton flows with small divergence were observed (Nakamura & Mima, 2008; Clark et al., 2000). The magnetic field of about 100 MG has been measured by direct spectral method on large distance (several hundreds of microns) from

Our experiments at the peak laser intensity of 21018 W/cm2 allows us to observe on the rear side of thin (30 m) titanium target clear ring-shaped structures by the proton detector CR-39 placed on a distance of 20 mm. Photo of ring-shaped proton structure is presented in Fig. 9a, and proton distributions with the energy of 2.5 MeV are presented in Fig. 9b. The

Deleting the intermediate derivations, we present expressions for lifetime considered

*T t* 2 

, *D* – coefficient of Bohm's diffusion. This lifetime does not depend on duration of

1/2 = 0.10.3. Note, that drift velocity can increase significantly under condition of development of pincheffect up to relativistic values. Respectively, not only electron velocity, but also the velocity of heavy positively charged atomic ions can increase up to relativistic values (Belyaev,

present narrow collimated beam with the divergence angle of

**B** (32)

1/2 14. Protons with the energy higher than 2.5 MeV

1/2 = 3 we observed well collimated proton

(33)


1/2 = 3. Inside this narrow

**v**

for ions and for electrons, we present the final result:

velocity of an electric drift (at **E** < **B**):

drift produces motion of neutral plasma.

the target surface (Belyaev et al., 2004).

divergence of the proton beam is

beams with the divergence angle of

Faenov et al., 2006).

magnetic field:

where

<sup>2</sup> *t D* 

collimated beam with the divergence angle


of vortex quasi-neutral structures as quasi-particles

potential plasma vortex.

laser action and can exceed it on one-two order. For this reason the superstrong magnetic fields generated in laser plasma, term quasistationary.

Increment of the considered magnetic field is equal to the ionization rate *<sup>i</sup>*, which is larger than the plasma frequency.

Fig. 9a. The photo of the track detector CR-39 covered by 11 mm Al filter. Detector CR-39 shows the tracks of protons with energies *Ep* > 0.8 MeV; 1/2 » 14° (cone half angle)

Mechanisms of generating magnetic fields are a subject of numerous investigations performed in recent years (Tatarakis et al., 2002; Beg et al., 2004; Borghesi et al., 1998). Various mechanisms for generating a magnetic field in the interaction of intensive laser radiation with solid targets are described in a number of theoretical works (Stamper, 1991; Wilks et al., 1992; Bell et al., 1993; Buchenkov et al., 1993; Sudan, 1993; Haines, 1997; Mason & Tabak, 1998; Krainov, 2003). In particular, they predicted the origin of magnetic fields with induction of up to 1GG in the dense plasma produced during the interaction process. These fields are localized near the critical surface, where the laser energy is mainly absorbed. The arising magnetic fields noticeably affect the dynamics of laser plasma. The principal mechanisms of generating quasistatic magnetic fields were considered: (1) different directions of the temperature and plasma density gradients; (2) the flux of fast electrons accelerated by ponderomotive forces in the longitudinal and transversal directions with respect to the direction of laser pulse propagation, and (3) the collisionless Weibel instability (Weibel, 1959).

Measurements of superstrong quasistatic magnetic fields in laser plasma and their theoretical interpretation have been discussed in more detail in our article (Belyaev et al., 2008).

Measurements of magnetic fields in plasma by various independent methods are very important for both proving the existence of such fields and determining their spatial structure (topology). For this purpose, we measured the profiles of X-ray spectral lines of hydrogen-like fluoride ions in laser plasma with a radiation intensity of 1017 W/cm2 and

Fast Charged Particles and Super-Strong Magnetic

acceleration at the front and rear target surfaces.

under the pulse action of mega-ampere currents.

energy charged particles in such plasmas.

NN 09-02-00041, 10-02-01095, 10-08-00752, 11-02-12026.

**6. Acknowledgment** 

**7. References** 

radiation for medical purposes.

laser pulse duration.

Fields Generated by Intense Laser Target Interaction 107

secondary effect mainly caused by the electric fields of the spatial charge produced when fast accelerated electrons are separated from ions. The detailed distribution of such fields substantially depends on the target thickness, which makes a difference in the ion

As a whole, particle acceleration is characterized by the multifactor character of the parameters involved. Such parameters are the intensity, frequency, and duration of the laser pulse; the contrast, which determines pre-plasma parameters; the thickness and structure of the target; the presence of magnetic fields, and some other factors. By combining these parameters, one can reach the optimal (in certain limits) conditions of particle acceleration. There is a wide range of various applications of such laser-driven accelerators, starting from fundamental investigations concerning nuclear processes for isotope production, to the initiation of thermonuclear reactions using laser setups that are quite small in size compared to standard accelerators, and ending by particular applications such as sources of proton

Nevertheless, there are a sufficiently large number of problems to be solved related to particle acceleration. These are, for example, ion beam focusing and annular structures arising in the beam. In electron acceleration, the problem of forming a monoenergetic beam

As far as the generation of super-strong magnetic fields is concerned, the main problems are determination of their lifetime and topology. Experimental results definitely indicate that the lifetimes of magnetic fields are considerably longer (by orders of magnitude) than the

From our point of view, this is direct evidence that long-living magnetic configurations exist in laser plasma. This is also confirmed by investigations into the dynamics of pinch structures in irradiating wire targets by laser pulses. The topology and dynamics of such structures are, as was noted above, in surprisingly good agreement with those obtained

It is clear that the presence of high-intensity fast particles and magnetic fields in plasma, in addition to the specific features of particle acceleration mentioned above, should result in numerous instabilities arising in plasma. This is directly illustrated by the results mentioned above on measuring the profiles of spectral lines for multiply charged ions. Profile irregularity is indicative of the existence of intense electrostatic oscillations possessing definite frequencies and intensities. Thus, in view of all the specific features mentioned above, one can conclude that in the case of ultra-short laser pulses we are dealing with magneto-active turbulent plasma, numerous properties of which are not clear presently. Nevertheless, it is possible to choose sufficiently optimal conditions for generating high-

This research has been supported by the Russian Foundation for Basic Research, projects

Andreev, A.V.; Gordienko, V.M. & Savel'ev, A.B. (2001). Nuclear processes in a high-

Vol.31, No.11, (November 2001), pp. 941-956, ISSN 0018-9197

temperature plasma produced by an ultrashort laser pulse. *Quantum Electronics*,

of fast electrons with a maximum energy has not yet been solved.

pulse duration of 1 ps (Belyaev et al., 2004). The structure observed is characterized by distinct dips and peaks on the spectral line profiles (see Fig. 10). These features can be explained by invoking a conception of the strong turbulent noise that develops in the superstrong magnetic field generated in laser plasma.

Fig. 10. Comparison of experimental (thin curves) and theoretical (thick curves) profiles for the *Lya* line of F IX ion: (a) the experiment was performed at *Ilas* = 21017 W/cm2, and calculation was made at *Ti* = 100 eV, *ne* = 1020 cm-3, = 71014 s-1, *E*0 = 4108 V/cm; (b) the experimental *Ilas* = 31017 W/cm2, and the calculation was done at *Ti* = 100 eV, *ne* = 21020 cm-3, = 1015 s-1, and *E*0 = 6108 V/cm.

## **5. Conclusion**

The above-described mechanisms for accelerating electrons and ions to a greater or lesser degree comply with up-to-date concepts on the generation of fast particles in laser plasma. According to these concepts, the energy of an initial laser pulse is converted to the energy of electron motion. The mechanisms for such energy conversion are mainly related to (1) a ponderomotive potential; (2) a phase interruption of electron oscillations in the laser wave due to various mechanisms, among which the main one is electron ejection beyond the sharp boundary of a target (vacuum heating), and (3) various resonance mechanisms where the electron motion is at resonance with plasma waves (wake-field resonance absorption or acceleration) or the cyclotron or betatron oscillation of an electron in the channel produced by laser radiation in the presence of a magnetic field. Ion acceleration in this case is a 106 Femtosecond–Scale Optics

pulse duration of 1 ps (Belyaev et al., 2004). The structure observed is characterized by distinct dips and peaks on the spectral line profiles (see Fig. 10). These features can be explained by invoking a conception of the strong turbulent noise that develops in the

Fig. 10. Comparison of experimental (thin curves) and theoretical (thick curves) profiles for the *Lya* line of F IX ion: (a) the experiment was performed at *Ilas* = 21017 W/cm2, and

experimental *Ilas* = 31017 W/cm2, and the calculation was done at *Ti* = 100 eV, *ne* = 21020

The above-described mechanisms for accelerating electrons and ions to a greater or lesser degree comply with up-to-date concepts on the generation of fast particles in laser plasma. According to these concepts, the energy of an initial laser pulse is converted to the energy of electron motion. The mechanisms for such energy conversion are mainly related to (1) a ponderomotive potential; (2) a phase interruption of electron oscillations in the laser wave due to various mechanisms, among which the main one is electron ejection beyond the sharp boundary of a target (vacuum heating), and (3) various resonance mechanisms where the electron motion is at resonance with plasma waves (wake-field resonance absorption or acceleration) or the cyclotron or betatron oscillation of an electron in the channel produced by laser radiation in the presence of a magnetic field. Ion acceleration in this case is a

= 71014 s-1, *E*0 = 4108 V/cm; (b) the

superstrong magnetic field generated in laser plasma.

calculation was made at *Ti* = 100 eV, *ne* = 1020 cm-3,

= 1015 s-1, and *E*0 = 6108 V/cm.

cm-3, 

**5. Conclusion** 

secondary effect mainly caused by the electric fields of the spatial charge produced when fast accelerated electrons are separated from ions. The detailed distribution of such fields substantially depends on the target thickness, which makes a difference in the ion acceleration at the front and rear target surfaces.

As a whole, particle acceleration is characterized by the multifactor character of the parameters involved. Such parameters are the intensity, frequency, and duration of the laser pulse; the contrast, which determines pre-plasma parameters; the thickness and structure of the target; the presence of magnetic fields, and some other factors. By combining these parameters, one can reach the optimal (in certain limits) conditions of particle acceleration. There is a wide range of various applications of such laser-driven accelerators, starting from fundamental investigations concerning nuclear processes for isotope production, to the initiation of thermonuclear reactions using laser setups that are quite small in size compared to standard accelerators, and ending by particular applications such as sources of proton radiation for medical purposes.

Nevertheless, there are a sufficiently large number of problems to be solved related to particle acceleration. These are, for example, ion beam focusing and annular structures arising in the beam. In electron acceleration, the problem of forming a monoenergetic beam of fast electrons with a maximum energy has not yet been solved.

As far as the generation of super-strong magnetic fields is concerned, the main problems are determination of their lifetime and topology. Experimental results definitely indicate that the lifetimes of magnetic fields are considerably longer (by orders of magnitude) than the laser pulse duration.

From our point of view, this is direct evidence that long-living magnetic configurations exist in laser plasma. This is also confirmed by investigations into the dynamics of pinch structures in irradiating wire targets by laser pulses. The topology and dynamics of such structures are, as was noted above, in surprisingly good agreement with those obtained under the pulse action of mega-ampere currents.

It is clear that the presence of high-intensity fast particles and magnetic fields in plasma, in addition to the specific features of particle acceleration mentioned above, should result in numerous instabilities arising in plasma. This is directly illustrated by the results mentioned above on measuring the profiles of spectral lines for multiply charged ions. Profile irregularity is indicative of the existence of intense electrostatic oscillations possessing definite frequencies and intensities. Thus, in view of all the specific features mentioned above, one can conclude that in the case of ultra-short laser pulses we are dealing with magneto-active turbulent plasma, numerous properties of which are not clear presently. Nevertheless, it is possible to choose sufficiently optimal conditions for generating highenergy charged particles in such plasmas.

## **6. Acknowledgment**

This research has been supported by the Russian Foundation for Basic Research, projects NN 09-02-00041, 10-02-01095, 10-08-00752, 11-02-12026.

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**0**

**5**

<sup>1</sup>*USA* <sup>2</sup>*France*

**Physics of Quasi-Monoenergetic Laser-Plasma**

<sup>1</sup>*Department of Physics and Astronomy, University of Nebraska – Lincoln, Lincoln*

Serguei Y. Kalmykov1, Bradley A. Shadwick1,

Arnaud Beck<sup>2</sup> and Erik Lefebvre<sup>2</sup>

<sup>2</sup>*CEA, DAM, DIF, Arpajon F-91297*

**Acceleration of Electrons in the Blowout Regime**

Progress in the technology of optical pulse amplification (Herrmann et al., 2009; Ross et al., 2000; Spence et al., 1991; Strickland & Mourou, 1985) has made sub-50 fs pulse length, 0.1–10 Hz repetition rate, multi-terawatt (TW) lasers available to university-scale laboratories. These new instruments, accessible to a large community of researchers, revolutionized experiments in relativistic nonlinear optics (Mourou et al., 2006), and enabled the compact design of plasma-based particle accelerators (Esarey et al., 2009; Tajima & Dawson, 1979). Owing to continuous improvements in laser systems and gas target technology (Semushin & Malka, 2001; Spence & Hooker, 2001), stable generation of well-collimated, quasi-monoenergetic, hundred-megaelectronvolt (MeV)-scale electron beams from millimeter to centimeter-length plasmas has become experimentally routine (Brunetti et al., 2010; Faure et al., 2006; Hafz et al., 2008; Leemans et al., 2006; Maksimchuk et al., 2007; Malka et al., 2009; Mangles et al., 2007; Osterhoff et al., 2008). These beams have been used for a broad range of technical and medical physics applications – *γ*-ray radiography for material science (Glinec et al., 2005; Ramanathan et al., 2010), testing of radiation resistivity of electronic components used in harsh radiation environments (Hidding et al., 2011), efficient on-site production of radioisotopes (Leemans et al., 2001; Reed et al., 2007), and radiotherapy with tunable, high-energy electrons (DesRosiers et al., 2000; Glinec et al., 2006; Kainz et al., 2004). Their unique properties – femtosecond (fs)-scale duration and multi-kiloampere current (Buck et al., 2011; Lundh et al., 2011) – are clearly favorable for ultrafast science applications, such as high-energy radiation femtochemistry (Brozek-Pluska et al., 2005), spatio-temporal radiation biology and radiotherapy (Malka et al., 2010), and compact x-ray sources (Fuchs et al., 2009; Grüner et al., 2007; Hartemann et al., 2007; Kneip et al., 2010; Pukhov et al., 2010; Rousse et al., 2007; Schlenvoigt et al., 2008). The current record of accelerated electron energy is close to one gigaelectronvolt (GeV) (Clayton et al., 2010; Froula et al., 2009; Kneip et al., 2009; Leemans et al., 2006; Liu et al., 2011). Furthermore, ongoing introduction of sub-150 fs, compact, high repetition rate petawatt (PW) lasers (Aoyama et al., 2003; Gaul et al., 2010; Hein et al., 2006; Korzhimanov et al., 2011; Sung et al., 2010) opens possibilities beyond the GeV energy frontier (Gorbunov et al., 2005; Kalmykov et al., 2010a; Lu et al., 2007; Martins et al., 2010), enabling further steps towards practical designs of high-brightness x-

**1. Introduction**


## **Physics of Quasi-Monoenergetic Laser-Plasma Acceleration of Electrons in the Blowout Regime**

Serguei Y. Kalmykov1, Bradley A. Shadwick1, Arnaud Beck<sup>2</sup> and Erik Lefebvre<sup>2</sup> <sup>1</sup>*Department of Physics and Astronomy, University of Nebraska – Lincoln, Lincoln* <sup>2</sup>*CEA, DAM, DIF, Arpajon F-91297* <sup>1</sup>*USA* <sup>2</sup>*France*

#### **1. Introduction**

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GeV, Nanocoulomb, Collimated Proton Beam from Laser Foil Interaction at 7×1021 W/cm2. *Physical Review Letters*, Vol.103, No.13, (September 2009), pp.135001 (2009),

front surface in laser-solid interactions by neutron spectroscopy. *Physics of Plasmas*,

Norreys, P.A.; Tatarakis, M.; Wagner, U. & and Wei, M.S. (2003). Proton Acceleration from High-Intensity Laser Interactions with Thin Foil Targets. *Physical*  Progress in the technology of optical pulse amplification (Herrmann et al., 2009; Ross et al., 2000; Spence et al., 1991; Strickland & Mourou, 1985) has made sub-50 fs pulse length, 0.1–10 Hz repetition rate, multi-terawatt (TW) lasers available to university-scale laboratories. These new instruments, accessible to a large community of researchers, revolutionized experiments in relativistic nonlinear optics (Mourou et al., 2006), and enabled the compact design of plasma-based particle accelerators (Esarey et al., 2009; Tajima & Dawson, 1979). Owing to continuous improvements in laser systems and gas target technology (Semushin & Malka, 2001; Spence & Hooker, 2001), stable generation of well-collimated, quasi-monoenergetic, hundred-megaelectronvolt (MeV)-scale electron beams from millimeter to centimeter-length plasmas has become experimentally routine (Brunetti et al., 2010; Faure et al., 2006; Hafz et al., 2008; Leemans et al., 2006; Maksimchuk et al., 2007; Malka et al., 2009; Mangles et al., 2007; Osterhoff et al., 2008). These beams have been used for a broad range of technical and medical physics applications – *γ*-ray radiography for material science (Glinec et al., 2005; Ramanathan et al., 2010), testing of radiation resistivity of electronic components used in harsh radiation environments (Hidding et al., 2011), efficient on-site production of radioisotopes (Leemans et al., 2001; Reed et al., 2007), and radiotherapy with tunable, high-energy electrons (DesRosiers et al., 2000; Glinec et al., 2006; Kainz et al., 2004). Their unique properties – femtosecond (fs)-scale duration and multi-kiloampere current (Buck et al., 2011; Lundh et al., 2011) – are clearly favorable for ultrafast science applications, such as high-energy radiation femtochemistry (Brozek-Pluska et al., 2005), spatio-temporal radiation biology and radiotherapy (Malka et al., 2010), and compact x-ray sources (Fuchs et al., 2009; Grüner et al., 2007; Hartemann et al., 2007; Kneip et al., 2010; Pukhov et al., 2010; Rousse et al., 2007; Schlenvoigt et al., 2008). The current record of accelerated electron energy is close to one gigaelectronvolt (GeV) (Clayton et al., 2010; Froula et al., 2009; Kneip et al., 2009; Leemans et al., 2006; Liu et al., 2011). Furthermore, ongoing introduction of sub-150 fs, compact, high repetition rate petawatt (PW) lasers (Aoyama et al., 2003; Gaul et al., 2010; Hein et al., 2006; Korzhimanov et al., 2011; Sung et al., 2010) opens possibilities beyond the GeV energy frontier (Gorbunov et al., 2005; Kalmykov et al., 2010a; Lu et al., 2007; Martins et al., 2010), enabling further steps towards practical designs of high-brightness x-

optical diagnostics, such as second harmonic generation from the sheath (Gordon et al., 2010; Helle et al., 2010) and frequency-domain holography/shadowgraphy (Dong et al., 2010a;b), show a direct correlation between the generation of collimated electron beams and bubble formation. Also, the moment of injection can be precisely identified (Thomas et al., 2007a). Electron self-injection greatly simplifies the experimental design enabling a single-stage

Physics of Quasi-Monoenergetic Laser-Plasma Acceleration of Electrons in the Blowout Regime 115

The bubble shape and potentials are determined by the electron flow surrounding the cavitated area. The majority of electrons slip behind the bubble within a time interval close to the period of plasma oscillations, *τ<sup>p</sup>* = 2*π*/*ωpe*. This means that the bubble, on the whole, is a quasistatic structure (Lu et al., 2006; Mora & Antonsen, 1996). At the same time, the sheath electrons, exposed to the highest fields and preaccelerated to relativistic energies, stay with the bubble much longer (Kalmykov et al., 2011b). If the quasistatic bubble expands during their slippage time (satisfying semi-empirical condition discussed in section 2.1), the sheath electrons can penetrate into the bubble near its rear, synchronize with it (i.e. obtain the longitudinal momentum *<sup>p</sup>*� <sup>≈</sup> *<sup>γ</sup>*g*mec*), and then travel inside the cavity, continuously gaining energy (Kalmykov et al., 2011b). Diffraction of the driving laser pulse is usually sufficient to cause this kind of dynamic behavior (Kalmykov et al., 2009; Xu et al., 2005). Therefore, the quality of accelerated beam, sensitive to the details of self-injection process, appears to be tied

Accelerated electrons eventually outrun the slow bubble. They exit the accelerating phase

the bubble radius. Acceleration until dephasing maximizes the energy gain, yielding *E*max ≈

The condition of the force balance matches the peak vector potential of the pulse (normalized to *mec*2/|*e*|) to the bubble radius as *<sup>a</sup>*sg <sup>≈</sup> (*kpRb*/2)<sup>2</sup> � 1 (Lu et al., 2007). Even though the bubble is a natural attractor for the relativistic laser-plasma dynamics (Gordienko & Pukhov, 2005), both transient dynamics *before* the onset of self-guiding (Kalmykov et al., 2010a) and the laser evolution *during* the self-guiding (Froula et al., 2009; Kalmykov et al., 2010a; 2011b; Kneip et al., 2009; Oguchi et al., 2008) may cause unwanted additional electron injection (dark current), degrading the beam quality. The lack of balance between the light pressure and charge-separation force makes the pulse spot size oscillate (Kalmykov et al., 2010a; Oguchi et al., 2008); self-steepening (Vieira et al., 2010) and depletion (Decker et al., 1996) gradually turn the pulse into a relativistically intense piston (Kalmykov et al., 2011b); and relativistic filamentation (Andreev et al., 2007; Kalmykov et al., 2010a; Thomas et al., 2007b; 2009) distorts the transverse profile of the pulse. Resulting deformations of the bubble, to which self-injection is extremely sensitive (Kalmykov et al., 2010b; 2011b), lead to the rapid degradation of electron beam quality (Froula et al., 2009; Kalmykov et al., 2010a; 2011b; Kneip et al., 2009; Martins et al., 2010). Novel optical diagnostics, such as frequency-domain

<sup>g</sup>(*kpRb*)<sup>2</sup> MeV, where *kp* = *ωpe*/*c* (Lu et al., 2007). Numerical simulations presented in this Chapter correspond to *γ*<sup>g</sup> ≈ 16, *kpRb* ≈ 5, yielding *Ld* ≈ 1.7 mm and *E*max ≈ 550 MeV. In strongly rarefied plasmas, such as *γ*<sup>g</sup> � 3*kpRb*/4, dephasing takes many Rayleigh lengths (this estimate implies that the laser pulse spot size approximately equals to the bubble radius). To propagate the pulse over this distance in a uniform background plasma, majority of modern experiments rely on a combination of relativistic and ponderomotive self-guiding (Hafizi et al., 2000; Ralph et al., 2009). The following dynamical scenario is usually the case. Upon entering the plasma, the pulse with *P*/*P*cr � 1 and *τ<sup>L</sup>* < *τ<sup>p</sup>* self-focuses until full electron cavitation is achieved, and the charge-separation force of an electron density channel (bubble) balances the radial ponderomotive force; the pulse is then guided until depletion

<sup>g</sup> GW is the critical power for relativistic self-focusing (Sun et al., 1987)).

<sup>g</sup>*Rb* is the dephasing length, and *Rb* is

acceleration with considerable flexibility in beam parameters.

to the self-consistent optical evolution of the driver.

0.085*γ*<sup>2</sup>

(here, *P*cr = 16.2*γ*<sup>2</sup>

within a time interval *τ<sup>d</sup>* = *Ld*/*c*, where *Ld* = (2/3)*γ*<sup>2</sup>

Fig. 1. Electron density bubble driven by the 70 TW laser pulse in a plasma of density *<sup>n</sup>*<sup>0</sup> <sup>=</sup> 6.5 <sup>×</sup> 1018 cm−<sup>3</sup> (cf. Fig. 3(d.1)). The laser pulse (not shown) is centered at *<sup>z</sup>* <sup>≈</sup> *ct* and propagates in the direction indicated by the arrow. The wake bucket (plot (a)) is devoid of electrons; the peak of electron density at its rear is well above the cutoff value 10*n*0. The accelerating gradient (plot (b)) reaches 11.5 GV/cm at the rear of the bucket.

and *γ*-ray sources and compact high energy physics particle colliders (Schroeder et al., 2010). Success of these applications critically depends on high collimation and low energy spread of the multi-GeV beams. Presently, however, laser-plasma accelerators (LPAs) produce GeV-scale electrons with polychromatic energy distributions spanning from a few MeV to the maximum energy observed; sometimes with a quasi-monoenergetic feature at the high-energy end (Clayton et al., 2010; Froula et al., 2009; Kneip et al., 2009; Liu et al., 2011). In this Chapter, we explore the relationship between the electron beam quality and the nonlinear evolution of the accelerating structure – a three-dimensional (3-D) bucket of a strongly nonlinear, fully electromagnetic (EM) plasma wake – and propose dynamical scenarios to help reduce electron energy spread and suppress the poorly collimated polychromatic background.

In a modern LPA experiment, the ponderomotive force of a focused pulse produces a full cavitation of the surrounding electron fluid. All plasma electrons facing the pulse are expelled by the radiation pressure (whereas fully stripped ions remain immobile). Fields due to this charge separation attract bulk electrons to the axis, and their trajectories overshoot. The resulting closed cavity of electron density (the "bubble" (Pukhov & Meyer-ter-Vehn, 2002)) surrounded by a dense shell (sheath) of relativistic electrons encompasses the pulse and guides it over many Rayleigh lengths until depletion (Lu et al., 2007; Mora & Antonsen, 1996). Figure 1 presents one example of such fully cavitated bucket.

The bubble is a high-quality 3-D EM accelerating structure. Its longitudinally uniform but radially linear focusing gradient implies strict conservation of normalized transverse emittance. In addition, the accelerating field is radially uniform, which helps mitigate longitudinal emittance dilution (Lu et al., 2006; Rosenzweig et al., 1991). The bubble propagates with the group velocity of laser pulse in the plasma, which in a linear approximation can be expressed as *<sup>v</sup>*<sup>g</sup> <sup>=</sup> *<sup>c</sup>*(<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*−<sup>2</sup> <sup>g</sup> )1/2, where *c* is a speed of light in vacuum, and *<sup>γ</sup>*<sup>g</sup> <sup>=</sup> *<sup>ω</sup>*0/*ωpe* � 1 (here, *<sup>ω</sup>*<sup>0</sup> is the laser frequency, *<sup>ω</sup>pe* = (4*πe*2*n*0/*me*)1/2 is the Langmuir electron frequency, *me* is the electron rest mass, *n*<sup>0</sup> is the background electron density, and *e* is the electron charge). Therefore, even with the Lorentz factor *γ*g approaching 100, the bubble remains a "slow" accelerating structure capable of trapping initially quiescent electrons from the ambient plasma (Kalmykov et al., 2010a; Lu et al., 2007; Martins et al., 2010). Single-shot 2 Will-be-set-by-IN-TECH

(a) Normalized electron density (b) Accelerating gradient, −*Ez* (in GV/cm)

Fig. 1. Electron density bubble driven by the 70 TW laser pulse in a plasma of density *<sup>n</sup>*<sup>0</sup> <sup>=</sup> 6.5 <sup>×</sup> 1018 cm−<sup>3</sup> (cf. Fig. 3(d.1)). The laser pulse (not shown) is centered at *<sup>z</sup>* <sup>≈</sup> *ct* and propagates in the direction indicated by the arrow. The wake bucket (plot (a)) is devoid of electrons; the peak of electron density at its rear is well above the cutoff value 10*n*0. The

and *γ*-ray sources and compact high energy physics particle colliders (Schroeder et al., 2010). Success of these applications critically depends on high collimation and low energy spread of the multi-GeV beams. Presently, however, laser-plasma accelerators (LPAs) produce GeV-scale electrons with polychromatic energy distributions spanning from a few MeV to the maximum energy observed; sometimes with a quasi-monoenergetic feature at the high-energy end (Clayton et al., 2010; Froula et al., 2009; Kneip et al., 2009; Liu et al., 2011). In this Chapter, we explore the relationship between the electron beam quality and the nonlinear evolution of the accelerating structure – a three-dimensional (3-D) bucket of a strongly nonlinear, fully electromagnetic (EM) plasma wake – and propose dynamical scenarios to help reduce electron

In a modern LPA experiment, the ponderomotive force of a focused pulse produces a full cavitation of the surrounding electron fluid. All plasma electrons facing the pulse are expelled by the radiation pressure (whereas fully stripped ions remain immobile). Fields due to this charge separation attract bulk electrons to the axis, and their trajectories overshoot. The resulting closed cavity of electron density (the "bubble" (Pukhov & Meyer-ter-Vehn, 2002)) surrounded by a dense shell (sheath) of relativistic electrons encompasses the pulse and guides it over many Rayleigh lengths until depletion (Lu et al., 2007; Mora & Antonsen, 1996).

The bubble is a high-quality 3-D EM accelerating structure. Its longitudinally uniform but radially linear focusing gradient implies strict conservation of normalized transverse emittance. In addition, the accelerating field is radially uniform, which helps mitigate longitudinal emittance dilution (Lu et al., 2006; Rosenzweig et al., 1991). The bubble propagates with the group velocity of laser pulse in the plasma, which in a linear

and *<sup>γ</sup>*<sup>g</sup> <sup>=</sup> *<sup>ω</sup>*0/*ωpe* � 1 (here, *<sup>ω</sup>*<sup>0</sup> is the laser frequency, *<sup>ω</sup>pe* = (4*πe*2*n*0/*me*)1/2 is the Langmuir electron frequency, *me* is the electron rest mass, *n*<sup>0</sup> is the background electron density, and *e* is the electron charge). Therefore, even with the Lorentz factor *γ*g approaching 100, the bubble remains a "slow" accelerating structure capable of trapping initially quiescent electrons from the ambient plasma (Kalmykov et al., 2010a; Lu et al., 2007; Martins et al., 2010). Single-shot

<sup>g</sup> )1/2, where *c* is a speed of light in vacuum,

accelerating gradient (plot (b)) reaches 11.5 GV/cm at the rear of the bucket.

energy spread and suppress the poorly collimated polychromatic background.

Figure 1 presents one example of such fully cavitated bucket.

approximation can be expressed as *<sup>v</sup>*<sup>g</sup> <sup>=</sup> *<sup>c</sup>*(<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*−<sup>2</sup>

optical diagnostics, such as second harmonic generation from the sheath (Gordon et al., 2010; Helle et al., 2010) and frequency-domain holography/shadowgraphy (Dong et al., 2010a;b), show a direct correlation between the generation of collimated electron beams and bubble formation. Also, the moment of injection can be precisely identified (Thomas et al., 2007a). Electron self-injection greatly simplifies the experimental design enabling a single-stage acceleration with considerable flexibility in beam parameters.

The bubble shape and potentials are determined by the electron flow surrounding the cavitated area. The majority of electrons slip behind the bubble within a time interval close to the period of plasma oscillations, *τ<sup>p</sup>* = 2*π*/*ωpe*. This means that the bubble, on the whole, is a quasistatic structure (Lu et al., 2006; Mora & Antonsen, 1996). At the same time, the sheath electrons, exposed to the highest fields and preaccelerated to relativistic energies, stay with the bubble much longer (Kalmykov et al., 2011b). If the quasistatic bubble expands during their slippage time (satisfying semi-empirical condition discussed in section 2.1), the sheath electrons can penetrate into the bubble near its rear, synchronize with it (i.e. obtain the longitudinal momentum *<sup>p</sup>*� <sup>≈</sup> *<sup>γ</sup>*g*mec*), and then travel inside the cavity, continuously gaining energy (Kalmykov et al., 2011b). Diffraction of the driving laser pulse is usually sufficient to cause this kind of dynamic behavior (Kalmykov et al., 2009; Xu et al., 2005). Therefore, the quality of accelerated beam, sensitive to the details of self-injection process, appears to be tied to the self-consistent optical evolution of the driver.

Accelerated electrons eventually outrun the slow bubble. They exit the accelerating phase within a time interval *τ<sup>d</sup>* = *Ld*/*c*, where *Ld* = (2/3)*γ*<sup>2</sup> <sup>g</sup>*Rb* is the dephasing length, and *Rb* is the bubble radius. Acceleration until dephasing maximizes the energy gain, yielding *E*max ≈ 0.085*γ*<sup>2</sup> <sup>g</sup>(*kpRb*)<sup>2</sup> MeV, where *kp* = *ωpe*/*c* (Lu et al., 2007). Numerical simulations presented in this Chapter correspond to *γ*<sup>g</sup> ≈ 16, *kpRb* ≈ 5, yielding *Ld* ≈ 1.7 mm and *E*max ≈ 550 MeV. In strongly rarefied plasmas, such as *γ*<sup>g</sup> � 3*kpRb*/4, dephasing takes many Rayleigh lengths (this estimate implies that the laser pulse spot size approximately equals to the bubble radius). To propagate the pulse over this distance in a uniform background plasma, majority of modern experiments rely on a combination of relativistic and ponderomotive self-guiding (Hafizi et al., 2000; Ralph et al., 2009). The following dynamical scenario is usually the case. Upon entering the plasma, the pulse with *P*/*P*cr � 1 and *τ<sup>L</sup>* < *τ<sup>p</sup>* self-focuses until full electron cavitation is achieved, and the charge-separation force of an electron density channel (bubble) balances the radial ponderomotive force; the pulse is then guided until depletion (here, *P*cr = 16.2*γ*<sup>2</sup> <sup>g</sup> GW is the critical power for relativistic self-focusing (Sun et al., 1987)). The condition of the force balance matches the peak vector potential of the pulse (normalized to *mec*2/|*e*|) to the bubble radius as *<sup>a</sup>*sg <sup>≈</sup> (*kpRb*/2)<sup>2</sup> � 1 (Lu et al., 2007). Even though the bubble is a natural attractor for the relativistic laser-plasma dynamics (Gordienko & Pukhov, 2005), both transient dynamics *before* the onset of self-guiding (Kalmykov et al., 2010a) and the laser evolution *during* the self-guiding (Froula et al., 2009; Kalmykov et al., 2010a; 2011b; Kneip et al., 2009; Oguchi et al., 2008) may cause unwanted additional electron injection (dark current), degrading the beam quality. The lack of balance between the light pressure and charge-separation force makes the pulse spot size oscillate (Kalmykov et al., 2010a; Oguchi et al., 2008); self-steepening (Vieira et al., 2010) and depletion (Decker et al., 1996) gradually turn the pulse into a relativistically intense piston (Kalmykov et al., 2011b); and relativistic filamentation (Andreev et al., 2007; Kalmykov et al., 2010a; Thomas et al., 2007b; 2009) distorts the transverse profile of the pulse. Resulting deformations of the bubble, to which self-injection is extremely sensitive (Kalmykov et al., 2010b; 2011b), lead to the rapid degradation of electron beam quality (Froula et al., 2009; Kalmykov et al., 2010a; 2011b; Kneip et al., 2009; Martins et al., 2010). Novel optical diagnostics, such as frequency-domain

quasi-monoenergetic beam formation (Hafz et al., 2011; Kalmykov et al., 2009). This approach does not maximize electron energy and thus does not optimize the accelerator performance. However, high-quality, a few-hundred MeV electron beams with tunable parameters can be produced, which is valuable for applications. In addition, as we show in section 3, using a broad bandwidth, negatively chirped pulse may help compensate for the nonlinear red-shift and delay formation of the relativistic piston, thus reducing the amount of dark current. In this Chapter, we elucidate the intrinsic connection of electron injection with the laser pulse optical evolution and demonstrate the mechanism of monoenergetic electron beam formation. We also discuss adverse scenarios of the pulse evolution leading to the continuous injection, and propose ways to mitigate them. To examine electron injection during various stages of laser pulse evolution in a single numerical experiment, we use two complementary simulation approaches. In section 2, we explain the physics, and develop the conceptual framework of the problem using the quasistatic, cylindrically symmetric, fully relativistic PIC code WAKE (Mora & Antonsen, 1997). A fully 3-D, non-averaged, dynamic test electron tracking module incorporated in WAKE (Kalmykov et al., 2009; Malka et al., 2001) emulates the non-quasistatic response of initially quiescent electrons to a high-frequency quasi-paraxial laser field and slowly varying EM plasma wakes. In section 2.5 we validate the test-particle results in a

Physics of Quasi-Monoenergetic Laser-Plasma Acceleration of Electrons in the Blowout Regime 117

full 3-D PIC simulation using the code CALDER-Circ (Lifschitz et al., 2009).

we summarize the results and point out directions of future work.

**2. Self-injection scenarios in the blowout regime**

symmetric time-averaged (over *ω*−<sup>1</sup>

The formation of a quasi-monoenergetic electron bunch during one period of laser spot oscillation is the subject of sections 2.1, 2.2, and 2.3. By analyzing quasistatic trajectories and using the results of test electron tracking in section 2.1, we identify precisely the injection candidates, collection volume, and evaluate the minimal bubble expansion rate for the initiation of self-injection. Formation of the quasi-monoenergetic beam during one period of the bubble size oscillation is described in section 2.2. Results of section 2.3 show that self-injection and subsequent acceleration of an electron require initial reduction of its moving-frame Hamiltonian. Laser pulse self-compression and resulting continuous injection are considered in section 2.4. In section 2.5, we validate the self-injection scenarios discussed in sections 2.1 – 2.4 in a CALDER-Circ simulation. We find that the test-particle modeling correctly identifies physical processes responsible for the initiation and termination of self-injection. Using our fast simulation toolkit (WAKE with test particles) and CALDER-Circ code, we explain in section 3 that laser self-compression and concomitant continuous injection can be suppressed by using a negatively chirped, broad bandwidth driver pulse. In section 4

We study self-injection and acceleration of electrons until dephasing in a standard regime of LPA experiments at the University of Nebraska (Ramanathan et al., 2010). We examine various scenarios of injection and relate them to nonlinear dynamical processes involving the laser pulse. The quasistatic behavior of the bulk plasma makes it possible to elucidate the physics of self-injection using a conceptually simple and computationally efficient toolkit: a fully relativistic 3-D particle tracking module built into the cylindrically

WAKE uses an extended paraxial solver for the slowly varying laser pulse envelope, which preserves the group velocity dispersion in the vicinity of the carrier frequency and calculates precisely radiation absorption due to the wake excitation. Electron response to the time-averaged ponderomotive force is calculated assuming that *all* plasma electrons (macroparticles) eventually fall behind the bubble. This approach, termed the quasistatic

<sup>0</sup> ) quasistatic PIC code WAKE (Mora & Antonsen, 1997).

streak cameras (Dong et al., 2010b; Li et al., 2010; 2011), allow capturing the details of bubble evolution and associating them with the final quality of electron beam in a given experimental shot, providing necessary feedback for the real-time optimization. The combined effort of theory and experiment should thus be aimed at suppressing dark current through a proper optimization and control of laser pulse evolution without complicating the experimental design (ideally, involving a single laser pulse and a single gas target).

It has been understood that to reduce the nonlinear effects and stabilize the bubble during the pulse self-guiding, it is necessary to work with matched PW-class pulses in low-density plasmas, *γ*<sup>g</sup> > 35 (Lu et al., 2007). At even lower densities, *γ*<sup>g</sup> > 100, initial mismatching of the pulse (e.g. over-focusing) helps enforce and control the injection (Kalmykov et al., 2010b; 2011a). With *P* ∼ 10*P*cr, and *τ<sup>L</sup>* < *τp*/2, the initially over-focused laser diffracts, the bubble expands, and electrons are injected continuously. When self-guiding sets in, the bubble stabilizes, and self-injection terminates. Secondary injection into the same bucket remains suppressed, and low-energy tails do not develop in the electron spectra. Electrons injected during the period of expansion travel deep inside the bucket and are continuously accelerated. At the same time, electrons injected late are located in the region of the highest accelerating gradient. They rapidly equalize in energy with earlier injected particles. Thus, a nanocoulomb (nC), quasi-monoenergetic bunch forms long before dephasing (Kalmykov et al., 2009; 2010b; 2011a). It can be further accelerated without emittance dilution, reaching multi-GeV energy near dephasing, keeping a few percent energy spread and a few mrad divergence. By varying plasma density and/or laser focusing geometry, one can control initial mismatching, changing the system behavior during the brief initial transient stage, thus precisely controlling the charge and emittance (Kalmykov et al., 2011a). High optical quality of the PW pulse is critically important for the realization of this scheme (Kalmykov et al., 2010a; Wang et al., 2011).

With a higher power ratio, *P*/*P*cr > 10, and/or longer pulse, *τ<sup>L</sup>* > *τp*/2, the described above scheme becomes compromised (Kalmykov et al., 2010a). The length of the transient stage increases, giving rise to multiple oscillations of the laser spot size and periodic injection (Kalmykov et al., 2010a), resulting in polychromatic electron energy distributions (Martins et al., 2010). At higher plasma densities, *γ*<sup>g</sup> < 20, longitudinal deformation of the pulse becomes another source of the dark current (Kalmykov et al., 2011b). The leading edge of the pulse that pushes away plasma electrons rapidly accumulates frequency red-shift, and group velocity dispersion concurrently compresses the pulse (Fang et al., 2009; Faure et al., 2005; 2006; Pai et al., 2010; Vieira et al., 2010). Concomitant depletion of the leading edge further enhances the self-steepening effect (Decker et al., 1996; Lu et al., 2007). Initially smooth driver thus turns into a relativistically intense "piston", or a "snow-plow" that pre-accelerates and compresses the initially quiescent electron fluid. A large charge separation immediately behind the piston results in sheath electrons receiving strong longitudinal kick, increasing their inertia, and delaying their return to the axis. As a result, the bubble elongates, and massive, uninterrupted self-injection follows (Froula et al., 2009; Kalmykov et al., 2011b; Kneip et al., 2009). In spite of high injected charge, this scenario remains the same in both quasistatic and fully explicit, 3-D EM particle-in-cell (PIC) simulations (Kalmykov et al., 2011b). Beam loading (Rechatin et al., 2010; Tzoufras et al., 2009) becomes important only in the final stage of this process. Notably, in this situation, transverse matching of the pulse precludes neither periodic nor continuous injection (Martins et al., 2010), and thus does not help improve the beam characteristics. However, the dark current-free acceleration can be achieved even in these unfavorable regimes. Giving up acceleration until dephasing, and limiting the plasma length to one cycle of laser waist oscillation results in a 4 Will-be-set-by-IN-TECH

streak cameras (Dong et al., 2010b; Li et al., 2010; 2011), allow capturing the details of bubble evolution and associating them with the final quality of electron beam in a given experimental shot, providing necessary feedback for the real-time optimization. The combined effort of theory and experiment should thus be aimed at suppressing dark current through a proper optimization and control of laser pulse evolution without complicating the experimental

It has been understood that to reduce the nonlinear effects and stabilize the bubble during the pulse self-guiding, it is necessary to work with matched PW-class pulses in low-density plasmas, *γ*<sup>g</sup> > 35 (Lu et al., 2007). At even lower densities, *γ*<sup>g</sup> > 100, initial mismatching of the pulse (e.g. over-focusing) helps enforce and control the injection (Kalmykov et al., 2010b; 2011a). With *P* ∼ 10*P*cr, and *τ<sup>L</sup>* < *τp*/2, the initially over-focused laser diffracts, the bubble expands, and electrons are injected continuously. When self-guiding sets in, the bubble stabilizes, and self-injection terminates. Secondary injection into the same bucket remains suppressed, and low-energy tails do not develop in the electron spectra. Electrons injected during the period of expansion travel deep inside the bucket and are continuously accelerated. At the same time, electrons injected late are located in the region of the highest accelerating gradient. They rapidly equalize in energy with earlier injected particles. Thus, a nanocoulomb (nC), quasi-monoenergetic bunch forms long before dephasing (Kalmykov et al., 2009; 2010b; 2011a). It can be further accelerated without emittance dilution, reaching multi-GeV energy near dephasing, keeping a few percent energy spread and a few mrad divergence. By varying plasma density and/or laser focusing geometry, one can control initial mismatching, changing the system behavior during the brief initial transient stage, thus precisely controlling the charge and emittance (Kalmykov et al., 2011a). High optical quality of the PW pulse is critically important for the realization of this scheme (Kalmykov et al., 2010a; Wang et al.,

With a higher power ratio, *P*/*P*cr > 10, and/or longer pulse, *τ<sup>L</sup>* > *τp*/2, the described above scheme becomes compromised (Kalmykov et al., 2010a). The length of the transient stage increases, giving rise to multiple oscillations of the laser spot size and periodic injection (Kalmykov et al., 2010a), resulting in polychromatic electron energy distributions (Martins et al., 2010). At higher plasma densities, *γ*<sup>g</sup> < 20, longitudinal deformation of the pulse becomes another source of the dark current (Kalmykov et al., 2011b). The leading edge of the pulse that pushes away plasma electrons rapidly accumulates frequency red-shift, and group velocity dispersion concurrently compresses the pulse (Fang et al., 2009; Faure et al., 2005; 2006; Pai et al., 2010; Vieira et al., 2010). Concomitant depletion of the leading edge further enhances the self-steepening effect (Decker et al., 1996; Lu et al., 2007). Initially smooth driver thus turns into a relativistically intense "piston", or a "snow-plow" that pre-accelerates and compresses the initially quiescent electron fluid. A large charge separation immediately behind the piston results in sheath electrons receiving strong longitudinal kick, increasing their inertia, and delaying their return to the axis. As a result, the bubble elongates, and massive, uninterrupted self-injection follows (Froula et al., 2009; Kalmykov et al., 2011b; Kneip et al., 2009). In spite of high injected charge, this scenario remains the same in both quasistatic and fully explicit, 3-D EM particle-in-cell (PIC) simulations (Kalmykov et al., 2011b). Beam loading (Rechatin et al., 2010; Tzoufras et al., 2009) becomes important only in the final stage of this process. Notably, in this situation, transverse matching of the pulse precludes neither periodic nor continuous injection (Martins et al., 2010), and thus does not help improve the beam characteristics. However, the dark current-free acceleration can be achieved even in these unfavorable regimes. Giving up acceleration until dephasing, and limiting the plasma length to one cycle of laser waist oscillation results in a

design (ideally, involving a single laser pulse and a single gas target).

2011).

quasi-monoenergetic beam formation (Hafz et al., 2011; Kalmykov et al., 2009). This approach does not maximize electron energy and thus does not optimize the accelerator performance. However, high-quality, a few-hundred MeV electron beams with tunable parameters can be produced, which is valuable for applications. In addition, as we show in section 3, using a broad bandwidth, negatively chirped pulse may help compensate for the nonlinear red-shift and delay formation of the relativistic piston, thus reducing the amount of dark current.

In this Chapter, we elucidate the intrinsic connection of electron injection with the laser pulse optical evolution and demonstrate the mechanism of monoenergetic electron beam formation. We also discuss adverse scenarios of the pulse evolution leading to the continuous injection, and propose ways to mitigate them. To examine electron injection during various stages of laser pulse evolution in a single numerical experiment, we use two complementary simulation approaches. In section 2, we explain the physics, and develop the conceptual framework of the problem using the quasistatic, cylindrically symmetric, fully relativistic PIC code WAKE (Mora & Antonsen, 1997). A fully 3-D, non-averaged, dynamic test electron tracking module incorporated in WAKE (Kalmykov et al., 2009; Malka et al., 2001) emulates the non-quasistatic response of initially quiescent electrons to a high-frequency quasi-paraxial laser field and slowly varying EM plasma wakes. In section 2.5 we validate the test-particle results in a full 3-D PIC simulation using the code CALDER-Circ (Lifschitz et al., 2009).

The formation of a quasi-monoenergetic electron bunch during one period of laser spot oscillation is the subject of sections 2.1, 2.2, and 2.3. By analyzing quasistatic trajectories and using the results of test electron tracking in section 2.1, we identify precisely the injection candidates, collection volume, and evaluate the minimal bubble expansion rate for the initiation of self-injection. Formation of the quasi-monoenergetic beam during one period of the bubble size oscillation is described in section 2.2. Results of section 2.3 show that self-injection and subsequent acceleration of an electron require initial reduction of its moving-frame Hamiltonian. Laser pulse self-compression and resulting continuous injection are considered in section 2.4. In section 2.5, we validate the self-injection scenarios discussed in sections 2.1 – 2.4 in a CALDER-Circ simulation. We find that the test-particle modeling correctly identifies physical processes responsible for the initiation and termination of self-injection. Using our fast simulation toolkit (WAKE with test particles) and CALDER-Circ code, we explain in section 3 that laser self-compression and concomitant continuous injection can be suppressed by using a negatively chirped, broad bandwidth driver pulse. In section 4 we summarize the results and point out directions of future work.

## **2. Self-injection scenarios in the blowout regime**

We study self-injection and acceleration of electrons until dephasing in a standard regime of LPA experiments at the University of Nebraska (Ramanathan et al., 2010). We examine various scenarios of injection and relate them to nonlinear dynamical processes involving the laser pulse. The quasistatic behavior of the bulk plasma makes it possible to elucidate the physics of self-injection using a conceptually simple and computationally efficient toolkit: a fully relativistic 3-D particle tracking module built into the cylindrically symmetric time-averaged (over *ω*−<sup>1</sup> <sup>0</sup> ) quasistatic PIC code WAKE (Mora & Antonsen, 1997). WAKE uses an extended paraxial solver for the slowly varying laser pulse envelope, which preserves the group velocity dispersion in the vicinity of the carrier frequency and calculates precisely radiation absorption due to the wake excitation. Electron response to the time-averaged ponderomotive force is calculated assuming that *all* plasma electrons (macroparticles) eventually fall behind the bubble. This approach, termed the quasistatic

Fig. 2. Fully expanded bubble from the WAKE simulation (cf. position (2) of Fig. 3). Solid white and dashed red contours in panels (a–d) are the iso-contours of laser intensity at exp(−2) of the peak (the pulse propagates to the right). (a) Wake potential Φ = *φ* − *Az* in units *mec*2/|*e*|. (b) Focusing (−*Er* <sup>+</sup> *<sup>B</sup>θ*, top) and accelerating (−*Ez*, bottom) Lorentz forces (in GV/cm). (c) Top: the quasistatic electron density (in cm−3). Bottom: radial positions of non-quasistatic test electrons. Red markers are the test particles with *γ* > *γ*<sup>g</sup> = 16.3. (d) Trajectories *r*(*ξ*) of the quasistatic macroparticles. Green and black trajectories correspond to passing electrons. Red trajectories correspond to sheath electrons – injection candidates. (e) Normalized slippage time as a function of the impact parameter, *R*imp = *r*(*ξ* � *cτL*). (f) Longitudinal (*pz*, solid line) and transverse (*pr*, dashed line) momenta of macroparticles at the rear of the bubble (the point of trajectory crossing). Sheath electrons have the largest slippage time, and become relativistic before crossing the axis. The yellow dashed curve in panels (a–c) is a trajectory of the macroparticle with the greatest slippage time ("the innermost electron"). (g) Impact parameters of test electrons from panel (c) vs energy.

Physics of Quasi-Monoenergetic Laser-Plasma Acceleration of Electrons in the Blowout Regime 119

significantly exceeds *τ<sup>b</sup>* (here, *ζ* = −*ξ*, and *ζ* = *Lb* ≈ 18 *μ*m is the coordinate of the rear of the bubble). A yellow broken line in Figs. 2(a) – 2(c) shows the trajectory of the macroparticle with the largest slippage time, *T*slip ≈ 4.2*τb*. Figure 2(d) indicates that the sheath electrons originate from a hollow cylinder with a radius close to the laser pulse spot size. While slipping through the structure, they are exposed to the highest wakefields. At the rear of the bubble, they are strongly pre-accelerated in *both* longitudinal *and* transverse directions: Fig. 2(f) gives *pz* max ≈ 21*mec* > *γ*g*mec*, and |*pr* max| ≈ 4*mec*. The large longitudinal momentum of these electrons makes them the best injection candidates; their promotion to fully dynamic macroparticles

Longitudinal synchronization of sheath electrons, *pz* ≥ *γ*g*mec*, is necessary and, in highdensity plasmas (such as *γ*<sup>g</sup> ≈ *kpRb*), also sufficient for their injection (Kostyukov et al., 2009). In the opposite limit of strongly rarefied plasmas, *γ*<sup>g</sup> > 10*kpRb*, the sheath electrons do not

may result in their self-injection and acceleration (Morshed et al., 2010).

approximation (QSA), tremendously speeds-up the simulation, enabling serial runs and extensive parameter scans using small workstations. On the other hand, neglecting the long-term contribution of near-luminous-speed macroparticles traveling with the bubble prohibits self-consistent modeling of electron self-injection and trapping. To simulate self-injecton without compromising computational efficiency, we use test particles. The test-particle module is fully dynamic, making no assumption of cylindrical symmetry, and is not time-averaged. In particular, it takes into account the interaction of test electrons with the non-averaged, linearly polarized laser field with non-paraxial corrections (Quesnel & Mora, 1998). To capture the laser pulse interaction with non-quasistatic background electrons (and thus to model self-injection into non-stationary quasistatic wakefields), a group of quiescent test electrons is placed before the laser pulse at each time step. In this way, electron self-injection associated with bubble and driver evolution is separated from the effects brought about by the collective fields of the trapped electron bunch, i.e. from effects due to beam loading. This simulation approach allows us to fully characterize the details of self-injection process and relate them to the evolution of the laser and the bubble using a non-stationary Hamiltonian formalism (Kalmykov et al., 2010b; 2011b).

In the simulation, a transform-limited, linearly polarized Gaussian laser pulse with a full width at half-maximum in intensity *τ<sup>L</sup>* = 30 fs and central wavelength *λ*<sup>0</sup> = 0.805 *μ*m is focused at the plasma border (*z* = 0) to a spot size *r*<sup>0</sup> = 13.6 *μ*m, and propagates towards positive *z*. The plasma has a 0.5 mm linear entrance ramp followed by a 1.7 mm plateau. The length of the plateau is equal to the dephasing length. Plasma density, *<sup>n</sup>*<sup>0</sup> <sup>=</sup> 6.5 <sup>×</sup> 1018 cm−3, corresponds to *γ*<sup>g</sup> = *ω*0/*ωpe* ≈ 16.3. The laser power is 70 TW, which yields *P*/*P*cr = 16.25, a peak intensity at the focus 2.3 <sup>×</sup> 1019 W/cm2, and normalized vector potential *<sup>a</sup>*<sup>0</sup> <sup>=</sup> 3.27. The WAKE simulation uses a grid *dr* <sup>≈</sup> 0.1*k*−<sup>1</sup> *<sup>p</sup>* ≈ 0.21 *μ*m, with 30 macroparticles per radial cell, *<sup>d</sup><sup>ξ</sup>* <sup>=</sup> *dr*/3 (where *<sup>ξ</sup>* <sup>=</sup> *<sup>z</sup>* <sup>−</sup> *ct*), and time step *dt* <sup>=</sup> *dz*/*<sup>c</sup>* <sup>≈</sup> 1.325*ω*−<sup>1</sup> <sup>0</sup> .

#### **2.1 Injection candidates, collection volume, and minimal expansion rate to initiate injection**

Upon entering the plasma, the laser pulse self-focuses and reaches the highest intensity at *z* ≈ 0.8 mm. Full blowout is maintained over the entire propagation distance. Bubble expansion *and* electron injection begin soon after the laser pulse enters the density plateau. The wakefield potential, time-averaged (over *ω*−<sup>1</sup> <sup>0</sup> ) Lorentz forces, electron density, and sample trajectories of macroparticles *ri*(*ξ*) are shown in Fig. 2 for the fully expanded bubble after 1 mm of propagation (cf. position (2) of Fig. 3).

The quasistatic electron density and number density of non-quasistatic test particles in (*r*, *ξ*) space are strikingly similar in Fig. 2(c). Thus, even though the macroparticles cannot be trapped, analysis of their trajectories helps identify the injection candidates, and specify the scenario of bubble evolution favorable for injection. This analysis also provides precise estimates of the collection volume and the bubble expansion rate necessary to initiate the injection. Each macroparticle can be put into one of three clearly defined groups [color coded in Figs. 2(d) – 2(f)]. The majority of electrons, viz. those expelled by the radiation pressure (black) and those attracted from periphery to the axis (green), are passing. They fall behind the bubble roughly within a time interval *τ<sup>b</sup>* = *Lb*/*c* (where *Lb* ≈ 2*Rb* is the bubble length). The bulk plasma electrons thus obey the QSA restrictions exceptionally well, which enables precise WAKE modeling. Sheath electrons (red) are different; they may travel with the bubble over a long distance. Figure 2(e) shows that their slippage time,

$$T\_{\rm slip} = \int\_0^{L\_\theta} \frac{d\zeta}{c - v\_z} \,\tag{1}$$

6 Will-be-set-by-IN-TECH

approximation (QSA), tremendously speeds-up the simulation, enabling serial runs and extensive parameter scans using small workstations. On the other hand, neglecting the long-term contribution of near-luminous-speed macroparticles traveling with the bubble prohibits self-consistent modeling of electron self-injection and trapping. To simulate self-injecton without compromising computational efficiency, we use test particles. The test-particle module is fully dynamic, making no assumption of cylindrical symmetry, and is not time-averaged. In particular, it takes into account the interaction of test electrons with the non-averaged, linearly polarized laser field with non-paraxial corrections (Quesnel & Mora, 1998). To capture the laser pulse interaction with non-quasistatic background electrons (and thus to model self-injection into non-stationary quasistatic wakefields), a group of quiescent test electrons is placed before the laser pulse at each time step. In this way, electron self-injection associated with bubble and driver evolution is separated from the effects brought about by the collective fields of the trapped electron bunch, i.e. from effects due to beam loading. This simulation approach allows us to fully characterize the details of self-injection process and relate them to the evolution of the laser and the bubble using a non-stationary

In the simulation, a transform-limited, linearly polarized Gaussian laser pulse with a full width at half-maximum in intensity *τ<sup>L</sup>* = 30 fs and central wavelength *λ*<sup>0</sup> = 0.805 *μ*m is focused at the plasma border (*z* = 0) to a spot size *r*<sup>0</sup> = 13.6 *μ*m, and propagates towards positive *z*. The plasma has a 0.5 mm linear entrance ramp followed by a 1.7 mm plateau. The length of the plateau is equal to the dephasing length. Plasma density, *<sup>n</sup>*<sup>0</sup> <sup>=</sup> 6.5 <sup>×</sup> 1018 cm−3, corresponds to *γ*<sup>g</sup> = *ω*0/*ωpe* ≈ 16.3. The laser power is 70 TW, which yields *P*/*P*cr = 16.25, a peak intensity at the focus 2.3 <sup>×</sup> 1019 W/cm2, and normalized vector potential *<sup>a</sup>*<sup>0</sup> <sup>=</sup> 3.27. The

**2.1 Injection candidates, collection volume, and minimal expansion rate to initiate injection** Upon entering the plasma, the laser pulse self-focuses and reaches the highest intensity at *z* ≈ 0.8 mm. Full blowout is maintained over the entire propagation distance. Bubble expansion *and* electron injection begin soon after the laser pulse enters the density plateau.

sample trajectories of macroparticles *ri*(*ξ*) are shown in Fig. 2 for the fully expanded bubble

The quasistatic electron density and number density of non-quasistatic test particles in (*r*, *ξ*) space are strikingly similar in Fig. 2(c). Thus, even though the macroparticles cannot be trapped, analysis of their trajectories helps identify the injection candidates, and specify the scenario of bubble evolution favorable for injection. This analysis also provides precise estimates of the collection volume and the bubble expansion rate necessary to initiate the injection. Each macroparticle can be put into one of three clearly defined groups [color coded in Figs. 2(d) – 2(f)]. The majority of electrons, viz. those expelled by the radiation pressure (black) and those attracted from periphery to the axis (green), are passing. They fall behind the bubble roughly within a time interval *τ<sup>b</sup>* = *Lb*/*c* (where *Lb* ≈ 2*Rb* is the bubble length). The bulk plasma electrons thus obey the QSA restrictions exceptionally well, which enables precise WAKE modeling. Sheath electrons (red) are different; they may travel with the bubble

*<sup>p</sup>* ≈ 0.21 *μ*m, with 30 macroparticles per radial cell,

<sup>0</sup> ) Lorentz forces, electron density, and

, (1)

<sup>0</sup> .

Hamiltonian formalism (Kalmykov et al., 2010b; 2011b).

*<sup>d</sup><sup>ξ</sup>* <sup>=</sup> *dr*/3 (where *<sup>ξ</sup>* <sup>=</sup> *<sup>z</sup>* <sup>−</sup> *ct*), and time step *dt* <sup>=</sup> *dz*/*<sup>c</sup>* <sup>≈</sup> 1.325*ω*−<sup>1</sup>

WAKE simulation uses a grid *dr* <sup>≈</sup> 0.1*k*−<sup>1</sup>

The wakefield potential, time-averaged (over *ω*−<sup>1</sup>

after 1 mm of propagation (cf. position (2) of Fig. 3).

over a long distance. Figure 2(e) shows that their slippage time,

*T*slip =

 *Lb* 0

*dζ c* − *vz*

Fig. 2. Fully expanded bubble from the WAKE simulation (cf. position (2) of Fig. 3). Solid white and dashed red contours in panels (a–d) are the iso-contours of laser intensity at exp(−2) of the peak (the pulse propagates to the right). (a) Wake potential Φ = *φ* − *Az* in units *mec*2/|*e*|. (b) Focusing (−*Er* <sup>+</sup> *<sup>B</sup>θ*, top) and accelerating (−*Ez*, bottom) Lorentz forces (in GV/cm). (c) Top: the quasistatic electron density (in cm−3). Bottom: radial positions of non-quasistatic test electrons. Red markers are the test particles with *γ* > *γ*<sup>g</sup> = 16.3. (d) Trajectories *r*(*ξ*) of the quasistatic macroparticles. Green and black trajectories correspond to passing electrons. Red trajectories correspond to sheath electrons – injection candidates. (e) Normalized slippage time as a function of the impact parameter, *R*imp = *r*(*ξ* � *cτL*). (f) Longitudinal (*pz*, solid line) and transverse (*pr*, dashed line) momenta of macroparticles at the rear of the bubble (the point of trajectory crossing). Sheath electrons have the largest slippage time, and become relativistic before crossing the axis. The yellow dashed curve in panels (a–c) is a trajectory of the macroparticle with the greatest slippage time ("the innermost electron"). (g) Impact parameters of test electrons from panel (c) vs energy.

significantly exceeds *τ<sup>b</sup>* (here, *ζ* = −*ξ*, and *ζ* = *Lb* ≈ 18 *μ*m is the coordinate of the rear of the bubble). A yellow broken line in Figs. 2(a) – 2(c) shows the trajectory of the macroparticle with the largest slippage time, *T*slip ≈ 4.2*τb*. Figure 2(d) indicates that the sheath electrons originate from a hollow cylinder with a radius close to the laser pulse spot size. While slipping through the structure, they are exposed to the highest wakefields. At the rear of the bubble, they are strongly pre-accelerated in *both* longitudinal *and* transverse directions: Fig. 2(f) gives *pz* max ≈ 21*mec* > *γ*g*mec*, and |*pr* max| ≈ 4*mec*. The large longitudinal momentum of these electrons makes them the best injection candidates; their promotion to fully dynamic macroparticles may result in their self-injection and acceleration (Morshed et al., 2010).

Longitudinal synchronization of sheath electrons, *pz* ≥ *γ*g*mec*, is necessary and, in highdensity plasmas (such as *γ*<sup>g</sup> ≈ *kpRb*), also sufficient for their injection (Kostyukov et al., 2009). In the opposite limit of strongly rarefied plasmas, *γ*<sup>g</sup> > 10*kpRb*, the sheath electrons do not

Fig. 4. Phase space rotation and formation of a quasi-monoenergetic bunch (Kalmykov et al., 2011b). (a) Phase space rotation of injected test electrons. Longitudinal phase space is shown at the positions (1)–(3) of Fig. 3(a). (1) Injection begins. (2) The bubble is fully expanded, injection stops, and phase space rotation begins. (3) The bucket slightly contracts. Electrons injected lately equalize in energy with those injected earlier. Quasi-monoenergetic bunch forms. Test electrons are color coded according to *HMF* < 0 (red), 0 < *HMF* < 1 (green), *HMF* > 1 (black). (b) Axial line-outs of the accelerating gradient (in GV/cm). (c) *HMF* vs

Physics of Quasi-Monoenergetic Laser-Plasma Acceleration of Electrons in the Blowout Regime 121

where the parameter Δsh has to be found empirically from simulations. For the fully expanded bubble of Fig. 2, Δ*Lb* ≈ 3Δsh ≈ 0.5 *μ*m (the estimate obtained using Fig. 3(a)), which is sufficient to maintain the injection. Indeed, a group of test particles accelerated to *γ* > *γ*<sup>g</sup> (red markers) can be seen at the base of the bubble in Fig. 2(c). Figure 2(g) indicates that, in agreement with earlier studies (Pukhov et al., 2010; Tsung et al., 2006; Wu et al., 2009), only electrons with impact parameters such that they enter the sheath are collected and accelerated. And, the condition (2) always holds, sometimes rather closely, when self-injection occurs in

All accelerated electrons in Figs. 2(c,g) are collected from a cylindrical shell with thickness Δ*R*coll ≈ 2 *μ*m and radius close to the laser spot size, *R*coll ≈ 8 *μ*m. The length of this hollow cylinder equals to the interval of bubble expansion from Fig. 3(a), Δ*z*exp ≈ 400 *μ*m. This collection volume contains 2.6 <sup>×</sup> 1011 electrons – injection candidates (42 nC charge). To calculate the injected charge correctly, one has to take into account the self-fields of both sheath and recently trapped electrons near the base of the bubble, making resort to the fully kinetic simulations (Kalmykov et al., 2011b; Morshed et al., 2010). 3-D PIC simulation of section 2.5 shows that only 0.5% of particles from the collection volume are actually injected. Injection from a very narrow range of impact parameters, together with low collection efficiency in realistic PIC modeling, makes massive self-injection during the slippage time very unlikely. Injected particles thus remain the minority, and their contribution to the bubble evolution is insignificant. This justifies the quasistatic treatment of the plasma electrons making up the

accelerating structure, and validates the test-particle model of self-injection process.

in the non-uniform plasma is *Lb*(*z*) = *κ*(2*π*/*kp*0)/

The background plasma is never perfectly uniform in the laboratory experiment. Localized density depressions, which may naturally occur in gas jet targets, can also cause electron self-injection (Hemker et al., 2002; Suk et al., 2001). Even if the driver evolution is negligible (as in the beam-driven case (Lu et al., 2006; Rosenzweig et al., 1991)), the bubble crossing a density down-ramp necessarily expands; if the ramp is longer than a slippage distance, *L*ramp � *cT*slip, then the bubble evolution is slow, and Eq. (2) applies. The bubble length

*<sup>n</sup>*˜*e*(*z*), where the parameter *<sup>κ</sup>* <sup>∼</sup> 1 is

energy gain for the fully expanded (2) and contracted (3) bubble.

low-density plasmas, *γ*<sup>g</sup> > 65 (Kalmykov et al., 2009; 2010a;b; 2011a).

Fig. 3. Electron injection into the oscillating bubble. (a) Peak laser intensity (red) and length of the accelerating phase *L*acc.ph. (black) vs propagation distance; *L*acc.ph. is the distance between the positions of zero and peak accelerating gradient on axis – see Fig. 4(b). Panels (b) and (c): energy of test electrons vs their initial longitudinal (b) and radial (c) positions after one period of bubble size oscillation (cf. position (3) of panel (a)). (d.1) – (d.3) Quasistatic electron density in cm−<sup>3</sup> (color map) and number density of non-quasistatic test particles (dots) corresponding to the positions (1) – (3) of the plot (a) (labeled accordingly). Red contour is an iso-contour of laser intensity at exp(−2) of the peak. Data from positions (1) – (3) are used to describe the process of monoenergetic electron bunch formation in Fig. 4.

synchronize with (and thus cannot be injected into) a non-evolving bubble (i.e. depending on variables *r* and *ξ* only); in this case, evolution of the bucket is vital for self-injection (Kalmykov et al., 2009; 2010a;b; 2011a). In the intermediate regime with *γ*<sup>g</sup> ≈ 3.5*kpRb* (which is the focus of this Chapter), longitudinal synchronization appears to be insufficient for injection. The sheath electrons crossing the axis have relativistic transverse momenta, and thus tend to exit the cavity in the radial direction. Such electrons have been earlier observed in the laboratory in the absence of any noticeable trapping (Helle et al., 2010; Kaganovich et al., 2008). To be injected (i.e. return to axis), the injection candidates must be confined into a focusing cavity *after* crossing the axis. To this effect, the bubble must continuously *expand*, changing its size by an appreciable fraction during the electron transit time *T*slip. Energetic sheath electrons can then outrun the boundary of the bubble and stay inside long enough to *both* synchronize longitudinally *and* make a U-turn transversely. To separate the most energetic electrons from the sheath, elongation of the bubble over the slippage time has to exceed the sheath thickness Δsh at the rear of the bucket (Kalmykov et al., 2010b; 2011b),

$$
\Delta L\_b = L\_b(t + T\_{\text{slip}}) - L\_b(t) \ge \Delta\_{\text{sh}} \tag{2}
$$

8 Will-be-set-by-IN-TECH

Fig. 3. Electron injection into the oscillating bubble. (a) Peak laser intensity (red) and length of the accelerating phase *L*acc.ph. (black) vs propagation distance; *L*acc.ph. is the distance between the positions of zero and peak accelerating gradient on axis – see Fig. 4(b). Panels (b) and (c): energy of test electrons vs their initial longitudinal (b) and radial (c) positions after one period of bubble size oscillation (cf. position (3) of panel (a)). (d.1) – (d.3)

Quasistatic electron density in cm−<sup>3</sup> (color map) and number density of non-quasistatic test particles (dots) corresponding to the positions (1) – (3) of the plot (a) (labeled accordingly). Red contour is an iso-contour of laser intensity at exp(−2) of the peak. Data from positions (1) – (3) are used to describe the process of monoenergetic electron bunch formation in Fig. 4.

synchronize with (and thus cannot be injected into) a non-evolving bubble (i.e. depending on variables *r* and *ξ* only); in this case, evolution of the bucket is vital for self-injection (Kalmykov et al., 2009; 2010a;b; 2011a). In the intermediate regime with *γ*<sup>g</sup> ≈ 3.5*kpRb* (which is the focus of this Chapter), longitudinal synchronization appears to be insufficient for injection. The sheath electrons crossing the axis have relativistic transverse momenta, and thus tend to exit the cavity in the radial direction. Such electrons have been earlier observed in the laboratory in the absence of any noticeable trapping (Helle et al., 2010; Kaganovich et al., 2008). To be injected (i.e. return to axis), the injection candidates must be confined into a focusing cavity *after* crossing the axis. To this effect, the bubble must continuously *expand*, changing its size by an appreciable fraction during the electron transit time *T*slip. Energetic sheath electrons can then outrun the boundary of the bubble and stay inside long enough to *both* synchronize longitudinally *and* make a U-turn transversely. To separate the most energetic electrons from the sheath, elongation of the bubble over the slippage time has to exceed the sheath thickness Δsh at the rear of the bucket (Kalmykov et al., 2010b; 2011b),

Δ*Lb* = *Lb*(*t* + *T*slip) − *Lb*(*t*) ≥ Δsh, (2)

Fig. 4. Phase space rotation and formation of a quasi-monoenergetic bunch (Kalmykov et al., 2011b). (a) Phase space rotation of injected test electrons. Longitudinal phase space is shown at the positions (1)–(3) of Fig. 3(a). (1) Injection begins. (2) The bubble is fully expanded, injection stops, and phase space rotation begins. (3) The bucket slightly contracts. Electrons injected lately equalize in energy with those injected earlier. Quasi-monoenergetic bunch forms. Test electrons are color coded according to *HMF* < 0 (red), 0 < *HMF* < 1 (green), *HMF* > 1 (black). (b) Axial line-outs of the accelerating gradient (in GV/cm). (c) *HMF* vs energy gain for the fully expanded (2) and contracted (3) bubble.

where the parameter Δsh has to be found empirically from simulations. For the fully expanded bubble of Fig. 2, Δ*Lb* ≈ 3Δsh ≈ 0.5 *μ*m (the estimate obtained using Fig. 3(a)), which is sufficient to maintain the injection. Indeed, a group of test particles accelerated to *γ* > *γ*<sup>g</sup> (red markers) can be seen at the base of the bubble in Fig. 2(c). Figure 2(g) indicates that, in agreement with earlier studies (Pukhov et al., 2010; Tsung et al., 2006; Wu et al., 2009), only electrons with impact parameters such that they enter the sheath are collected and accelerated. And, the condition (2) always holds, sometimes rather closely, when self-injection occurs in low-density plasmas, *γ*<sup>g</sup> > 65 (Kalmykov et al., 2009; 2010a;b; 2011a).

All accelerated electrons in Figs. 2(c,g) are collected from a cylindrical shell with thickness Δ*R*coll ≈ 2 *μ*m and radius close to the laser spot size, *R*coll ≈ 8 *μ*m. The length of this hollow cylinder equals to the interval of bubble expansion from Fig. 3(a), Δ*z*exp ≈ 400 *μ*m. This collection volume contains 2.6 <sup>×</sup> 1011 electrons – injection candidates (42 nC charge). To calculate the injected charge correctly, one has to take into account the self-fields of both sheath and recently trapped electrons near the base of the bubble, making resort to the fully kinetic simulations (Kalmykov et al., 2011b; Morshed et al., 2010). 3-D PIC simulation of section 2.5 shows that only 0.5% of particles from the collection volume are actually injected. Injection from a very narrow range of impact parameters, together with low collection efficiency in realistic PIC modeling, makes massive self-injection during the slippage time very unlikely. Injected particles thus remain the minority, and their contribution to the bubble evolution is insignificant. This justifies the quasistatic treatment of the plasma electrons making up the accelerating structure, and validates the test-particle model of self-injection process.

The background plasma is never perfectly uniform in the laboratory experiment. Localized density depressions, which may naturally occur in gas jet targets, can also cause electron self-injection (Hemker et al., 2002; Suk et al., 2001). Even if the driver evolution is negligible (as in the beam-driven case (Lu et al., 2006; Rosenzweig et al., 1991)), the bubble crossing a density down-ramp necessarily expands; if the ramp is longer than a slippage distance, *L*ramp � *cT*slip, then the bubble evolution is slow, and Eq. (2) applies. The bubble length in the non-uniform plasma is *Lb*(*z*) = *κ*(2*π*/*kp*0)/ *<sup>n</sup>*˜*e*(*z*), where the parameter *<sup>κ</sup>* <sup>∼</sup> 1 is

according to *dHMF*/*dt* = *∂HMF*/*∂t*. For a test electron moving away from the bubble, *HMF* <sup>=</sup> *<sup>γ</sup><sup>e</sup>* <sup>+</sup> <sup>Φ</sup> <sup>−</sup> *pz* <sup>→</sup> <sup>1</sup> <sup>+</sup> **<sup>p</sup>**<sup>2</sup> <sup>−</sup> *pz* <sup>&</sup>gt; 0. Hence, the electron is confined inside the bucket at all times (trapped) if the *HMF* remains *negative* in the course of interaction. As soon as the bubble stabilizes, *HMF* is conserved. All test electrons can be then divided into 3 groups: (1) *HMF* < 0 — trapped; (2) 0 < *HMF* < 1 — injected (accelerated); and (3) *HMF* > 1. All the three groups are represented in Fig. 4(a), where the phase space of test electrons is shown at the stationary points of full expansion (labeled (2)) and full contraction of the bubble (labeled (3)). Electron phase space for the fully expanded bubble shows that the bubble expansion causes a reduction in *HMF* (Kalmykov et al., 2009; 2010b). The condition *HMF* < 1 is thus *necessary* for injection and initial acceleration. For instance, it can be used for promotion of test electrons into the non-quasistatic electron beam particles in order to self-consistently incorporate beam loading into the model. Conversely, even minimal bubble contraction may raise *HMF* significantly. Figure 4(c) shows that electrons with 0 < *HMF* < 2 are accelerated as effectively as those which are formally trapped. Hence, the natural evolution of the structure may result in violation of the sufficient trapping condition; this, however, does not disrupt

Physics of Quasi-Monoenergetic Laser-Plasma Acceleration of Electrons in the Blowout Regime 123

acceleration with good collimation and low energy spread (Kalmykov et al., 2011b).

Although a monoenergetic electron bunch forms early, the general experimental trend is to push the accelerator efficiency to the limit and use the entire dephasing length. Electron beam quality, however, can be compromised in this pursuit. The driver pulse evolves continuously, which may cause uninterrupted electron injection and emittance growth. Understanding the physical mechanism of continuous injection will help control the beam quality by limiting injection via a judicious choice of laser-plasma interaction geometry and target design (Kalmykov et al., 2011a) or by manipulating the phase and envelope of the incident pulse. Running the simulation until the nonlinear dephasing limit, we find two distinct stages of the system evolution. Stage I, discussed above, corresponds to a single oscillation of the laser spot size and produces a monoenergetic electron bunch. Stage II is characterized by a gradual increase of laser intensity (up to 2 <sup>×</sup> 1020 W/cm2) and a steady elongation of the bubble. Figure 5 shows that bubble elongation is accompanied by continuous injection and growth of the energy spread. At the end of the run, the number of continuously injected test electrons (*γ* > *γ*g) is factor of 7.5 larger than the number of electrons in the leading quasi-monoenergetic bunch. Similar development of self-injection process has been reported elsewhere (Froula et al., 2009; Kneip et al., 2009). Figures 5(c.1) – 5(d.3) show that the asymmetric growth (elongation) of the bubble is accompanied by self-compression of the driver pulse from 25 to roughly 5.5 fs, and simultaneous 5-fold increase in intensity. The compressed pulse (relativistic piston) acts as a snow-plow. The ponderomotive push of its front pre-accelerates plasma electrons to *γ* > *γ*<sup>g</sup> and, as is seen in Figs. 5(c.2) and 5(c.3), creates a strongly compressed electron slab an order of magnitude denser than the ambient plasma. As a result of this strong charge separation, plasma electrons entering the sheath (as can be seen in Figs. 6(b) and 6(d)) are exposed to the positive electric field a factor 2.25 higher than in the case of smooth driver (Figs. 6(a) and 6(c)). Hence, upon passing the piston, sheath electrons receive a large kick in the backward direction, and quickly become relativistic, *pz* ≈ −1.65*mec* (in contrast to −0.55*mec* in the smooth driver case). Therefore, in the piston case, it takes nearly twice as long for the sheath electron to reach the point of return, *pz* = 0, and to start getting accelerated; reaching the axis also takes longer time, which explains the

**2.4 Continuous self-injection caused by self-compression of the driving pulse**

bubble elongation.

determined empirically from simulations. Assuming a power-law density profile, *n*˜*e*(*z*) ≡ *ne*(*z*)/*n*<sup>0</sup> = 1 − *A*[(*z* − *z*in)/*L*ramp] *<sup>α</sup>*, where *n*˜*e*(*z* < *z*in) = 1, and *n*˜*e*(*z* > *z*in + *L*ramp) = 1 − *A* < 1, and substituting *z* − *z*in = *cT*slip into Eq. (2), we find the relation between the length of the ramp *L*ramp and density depression *A* necessary to incur injection: *L*ramp < *cT*slip[*Aκπ*/(*kp*0Δsh)]1/*α*. Using parameters of the bubble from Fig. 2, we find that a linear down-ramp with a 10% density depression may produce self-injection if *L*ramp < 0.47 mm. If the plasma density is more homogeneous than that, the self-injection into the bubble can be enforced by the pulse evolution only.

#### **2.2 Self-injection into an oscillating bubble: formation of quasi-monoenergetic collimated electron beam**

The first period of bubble size oscillations is displayed in Fig. 3. Figure 3(d.1) shows the bubble and the laser pulse at the beginning of bubble expansion (cf. position (1) of Fig. 3(a)); Fig. 3(d.2) at the end of expansion (cf. position (2) of Fig. 3(a)); and Fig. 3(d.3) at the end of oscillation period (cf. position (3) of Fig. 3(a)). The laser pulse entering the density plateau at *z* = 0.5 mm is longer than one-half of the electron plasma period, *τ<sup>L</sup>* ≈ 0.7*τp*. Its head, residing in an incompletely evacuated nonlinear channel, remains guided with an almost invariant spot size, *r*(*ξ* ≈ 0) ≈ 6.5 *μ*m, whereas the tail, confined within an evacuated bubble, is strongly mismatched. Beating of the mismatched tail, causing alternating expansion and contraction of the bubble, are clearly seen in progression from Fig. 3(d.1) to 3(d.3).

Figure 3(a) shows that bubble expansion starts near the edge of the density plateau and continues until *z* ≈ 1 mm. The bubble expands by 14% of its size over a 400 *μ*m distance (∼ 20 bubble lengths). Injection of non-quasistatic test electrons continues uninterrupted during this stage, and, as is clear from Fig. 4(a), their momentum distribution is continuous. Contraction of the bubble between *z* = 1 and 1.25 mm extinguishes injection and truncates the bunch: electrons injected at the very end of expansion are expelled. Particles remaining in the bucket are further accelerated. At this stage, the bunch becomes quasi-monoenergetic. According to Fig. 4(b), the longitudinally nonuniform, co-moving accelerating gradient changes insignificantly during the contraction. The tail of the bunch, constantly exposed to the highest gradient, equalizes in energy with earlier injected electrons (cf. position (3) of Fig. 4(a)). This rotation of longitudinal phase space, responsible for the formation of a quasi-monoenergetic bunch long before dephasing, is clearly different from that discussed in literature (Tsung et al., 2006). According to Figs. 3(b) and (c), electrons remaining in the bucket at the end of bubble contraction are collected during the interval of bubble expansion from a cylindrical shell with the radius close to the laser spot size. They form the bunch with the energy *E* = 360+<sup>40</sup> <sup>−</sup><sup>20</sup> MeV and 4.3 mrad divergence. *Therefore, limiting the plasma length to a single period of the bubble size oscillations gives a quasi-monoenergetic, collimated electron beam* (Hafz et al., 2011; Kalmykov et al., 2009).

#### **2.3 Evolution of test electron Hamiltonian during injection**

WAKE calculates all potentials directly, which makes the Hamiltonian analysis of test particle tracking straightforward. Using the definitions of normalized momentum **p** ≡ **p**/(*mec*), wake potential <sup>Φ</sup> <sup>=</sup> <sup>|</sup>*e*|(*<sup>ϕ</sup>* <sup>−</sup> *Az*)/(*mec*2) (where *<sup>φ</sup>* is a scalar potential, and *Az* is the longitudinal component of vector potential), envelope of the laser vector potential **<sup>a</sup>** ≡ |*e*|**a**/(*mec*2), and *γ<sup>e</sup>* = (1 + **p**<sup>2</sup> + **a**2/2)1/2, we introduce the normalized time-averaged moving-frame (MF) Hamiltonian *HMF*(*r*, *z*, *ξ*) = *γ<sup>e</sup>* + Φ − *pz*. For the quasistatic macroparticles, *HMF* ≡ 1 (Mora & Antonsen, 1997). Test electrons (which are not assumed to be quasistatic) move in explicitly time-dependent potentials; hence, *HMF* changes in the course of propagation 10 Will-be-set-by-IN-TECH

determined empirically from simulations. Assuming a power-law density profile, *n*˜*e*(*z*) ≡

1 − *A* < 1, and substituting *z* − *z*in = *cT*slip into Eq. (2), we find the relation between the length of the ramp *L*ramp and density depression *A* necessary to incur injection: *L*ramp < *cT*slip[*Aκπ*/(*kp*0Δsh)]1/*α*. Using parameters of the bubble from Fig. 2, we find that a linear down-ramp with a 10% density depression may produce self-injection if *L*ramp < 0.47 mm. If the plasma density is more homogeneous than that, the self-injection into the bubble can be

**2.2 Self-injection into an oscillating bubble: formation of quasi-monoenergetic collimated**

The first period of bubble size oscillations is displayed in Fig. 3. Figure 3(d.1) shows the bubble and the laser pulse at the beginning of bubble expansion (cf. position (1) of Fig. 3(a)); Fig. 3(d.2) at the end of expansion (cf. position (2) of Fig. 3(a)); and Fig. 3(d.3) at the end of oscillation period (cf. position (3) of Fig. 3(a)). The laser pulse entering the density plateau at *z* = 0.5 mm is longer than one-half of the electron plasma period, *τ<sup>L</sup>* ≈ 0.7*τp*. Its head, residing in an incompletely evacuated nonlinear channel, remains guided with an almost invariant spot size, *r*(*ξ* ≈ 0) ≈ 6.5 *μ*m, whereas the tail, confined within an evacuated bubble, is strongly mismatched. Beating of the mismatched tail, causing alternating expansion and contraction

Figure 3(a) shows that bubble expansion starts near the edge of the density plateau and continues until *z* ≈ 1 mm. The bubble expands by 14% of its size over a 400 *μ*m distance (∼ 20 bubble lengths). Injection of non-quasistatic test electrons continues uninterrupted during this stage, and, as is clear from Fig. 4(a), their momentum distribution is continuous. Contraction of the bubble between *z* = 1 and 1.25 mm extinguishes injection and truncates the bunch: electrons injected at the very end of expansion are expelled. Particles remaining in the bucket are further accelerated. At this stage, the bunch becomes quasi-monoenergetic. According to Fig. 4(b), the longitudinally nonuniform, co-moving accelerating gradient changes insignificantly during the contraction. The tail of the bunch, constantly exposed to the highest gradient, equalizes in energy with earlier injected electrons (cf. position (3) of Fig. 4(a)). This rotation of longitudinal phase space, responsible for the formation of a quasi-monoenergetic bunch long before dephasing, is clearly different from that discussed in literature (Tsung et al., 2006). According to Figs. 3(b) and (c), electrons remaining in the bucket at the end of bubble contraction are collected during the interval of bubble expansion from a cylindrical shell with the radius close to the laser spot size. They form the bunch with

*a single period of the bubble size oscillations gives a quasi-monoenergetic, collimated electron beam*

WAKE calculates all potentials directly, which makes the Hamiltonian analysis of test particle tracking straightforward. Using the definitions of normalized momentum **p** ≡ **p**/(*mec*), wake potential <sup>Φ</sup> <sup>=</sup> <sup>|</sup>*e*|(*<sup>ϕ</sup>* <sup>−</sup> *Az*)/(*mec*2) (where *<sup>φ</sup>* is a scalar potential, and *Az* is the longitudinal component of vector potential), envelope of the laser vector potential **<sup>a</sup>** ≡ |*e*|**a**/(*mec*2), and *γ<sup>e</sup>* = (1 + **p**<sup>2</sup> + **a**2/2)1/2, we introduce the normalized time-averaged moving-frame (MF) Hamiltonian *HMF*(*r*, *z*, *ξ*) = *γ<sup>e</sup>* + Φ − *pz*. For the quasistatic macroparticles, *HMF* ≡ 1 (Mora & Antonsen, 1997). Test electrons (which are not assumed to be quasistatic) move in explicitly time-dependent potentials; hence, *HMF* changes in the course of propagation

<sup>−</sup><sup>20</sup> MeV and 4.3 mrad divergence. *Therefore, limiting the plasma length to*

of the bubble, are clearly seen in progression from Fig. 3(d.1) to 3(d.3).

*<sup>α</sup>*, where *n*˜*e*(*z* < *z*in) = 1, and *n*˜*e*(*z* > *z*in + *L*ramp) =

*ne*(*z*)/*n*<sup>0</sup> = 1 − *A*[(*z* − *z*in)/*L*ramp]

enforced by the pulse evolution only.

**electron beam**

the energy *E* = 360+<sup>40</sup>

(Hafz et al., 2011; Kalmykov et al., 2009).

**2.3 Evolution of test electron Hamiltonian during injection**

according to *dHMF*/*dt* = *∂HMF*/*∂t*. For a test electron moving away from the bubble, *HMF* <sup>=</sup> *<sup>γ</sup><sup>e</sup>* <sup>+</sup> <sup>Φ</sup> <sup>−</sup> *pz* <sup>→</sup> <sup>1</sup> <sup>+</sup> **<sup>p</sup>**<sup>2</sup> <sup>−</sup> *pz* <sup>&</sup>gt; 0. Hence, the electron is confined inside the bucket at all times (trapped) if the *HMF* remains *negative* in the course of interaction. As soon as the bubble stabilizes, *HMF* is conserved. All test electrons can be then divided into 3 groups: (1) *HMF* < 0 — trapped; (2) 0 < *HMF* < 1 — injected (accelerated); and (3) *HMF* > 1. All the three groups are represented in Fig. 4(a), where the phase space of test electrons is shown at the stationary points of full expansion (labeled (2)) and full contraction of the bubble (labeled (3)). Electron phase space for the fully expanded bubble shows that the bubble expansion causes a reduction in *HMF* (Kalmykov et al., 2009; 2010b). The condition *HMF* < 1 is thus *necessary* for injection and initial acceleration. For instance, it can be used for promotion of test electrons into the non-quasistatic electron beam particles in order to self-consistently incorporate beam loading into the model. Conversely, even minimal bubble contraction may raise *HMF* significantly. Figure 4(c) shows that electrons with 0 < *HMF* < 2 are accelerated as effectively as those which are formally trapped. Hence, the natural evolution of the structure may result in violation of the sufficient trapping condition; this, however, does not disrupt acceleration with good collimation and low energy spread (Kalmykov et al., 2011b).

#### **2.4 Continuous self-injection caused by self-compression of the driving pulse**

Although a monoenergetic electron bunch forms early, the general experimental trend is to push the accelerator efficiency to the limit and use the entire dephasing length. Electron beam quality, however, can be compromised in this pursuit. The driver pulse evolves continuously, which may cause uninterrupted electron injection and emittance growth. Understanding the physical mechanism of continuous injection will help control the beam quality by limiting injection via a judicious choice of laser-plasma interaction geometry and target design (Kalmykov et al., 2011a) or by manipulating the phase and envelope of the incident pulse.

Running the simulation until the nonlinear dephasing limit, we find two distinct stages of the system evolution. Stage I, discussed above, corresponds to a single oscillation of the laser spot size and produces a monoenergetic electron bunch. Stage II is characterized by a gradual increase of laser intensity (up to 2 <sup>×</sup> 1020 W/cm2) and a steady elongation of the bubble. Figure 5 shows that bubble elongation is accompanied by continuous injection and growth of the energy spread. At the end of the run, the number of continuously injected test electrons (*γ* > *γ*g) is factor of 7.5 larger than the number of electrons in the leading quasi-monoenergetic bunch. Similar development of self-injection process has been reported elsewhere (Froula et al., 2009; Kneip et al., 2009). Figures 5(c.1) – 5(d.3) show that the asymmetric growth (elongation) of the bubble is accompanied by self-compression of the driver pulse from 25 to roughly 5.5 fs, and simultaneous 5-fold increase in intensity. The compressed pulse (relativistic piston) acts as a snow-plow. The ponderomotive push of its front pre-accelerates plasma electrons to *γ* > *γ*<sup>g</sup> and, as is seen in Figs. 5(c.2) and 5(c.3), creates a strongly compressed electron slab an order of magnitude denser than the ambient plasma. As a result of this strong charge separation, plasma electrons entering the sheath (as can be seen in Figs. 6(b) and 6(d)) are exposed to the positive electric field a factor 2.25 higher than in the case of smooth driver (Figs. 6(a) and 6(c)). Hence, upon passing the piston, sheath electrons receive a large kick in the backward direction, and quickly become relativistic, *pz* ≈ −1.65*mec* (in contrast to −0.55*mec* in the smooth driver case). Therefore, in the piston case, it takes nearly twice as long for the sheath electron to reach the point of return, *pz* = 0, and to start getting accelerated; reaching the axis also takes longer time, which explains the bubble elongation.

Fig. 6. Bubble elongation due to formation of a relativistic piston: quasistatic analysis. Physical quantities are shown before (left column) and after (right column) the piston

(3) of Fig. 5(a). (a), (b) Longitudinal electric field *Ez* (in GV/cm). Black lines are the

continuous injection.

step *dt* = 0.1244*ω*−<sup>1</sup>

formation. The left column corresponds to position (1), and the right column – to the position

Physics of Quasi-Monoenergetic Laser-Plasma Acceleration of Electrons in the Blowout Regime 125

innermost electron trajectories, *r*in(*ξ*), plotted until the point of return, *pz* = 0. Blue contours are iso-contours of laser intensity at 5 <sup>×</sup> 1018 W/cm−2. (c), (d) Longitudinal electric field in the points of the innermost electron trajectory, *Ez*(*r*in(*ξ*)), from plots (a) and (b), respectively. (e), (f) Longitudinal momentum of the innermost electron along its trajectory, *pz*(*r*in(*ξ*)).

large bandwidth explains pulse compression to roughly two cycles. The red shift of central frequency together with the continuous front etching additionally slows down the pulse and the bubble and provides another reason for the occurrence of continuous injection (Fang et al., 2009). We show in section 3 that *negative chirp* of the incident pulse may compensate for the gradually accumulating red-shift, thus delaying the pulse contraction and partly suppressing

**2.5 Validation of self-injection scenarios in full 3-D PIC simulation: role of beam loading** Collective fields of the electron beam, neglected in the test-particle treatment, are known to change the shape of the sheath and thus reduce accelerating gradient, eventually terminating self-injection (Tzoufras et al., 2009). In this section, we verify the test-particle results by running a fully explicit 3-D PIC simulation with the identical set of initial conditions. We use the quasi-cylindrical code CALDER-Circ (Lifschitz et al., 2009), which preserves realistic geometry of interaction, and accounts for the axial asymmetry by decomposing EM fields (laser and wake) into a set of poloidal modes (whereas the particles remain in full 3-D). Well preserved cylindrical symmetry during the interaction enables us to use just the two lowest order modes and thus reduce a 3-D problem to an essentially 2-D one. We suppress sampling noise by using a large number of macroparticles (45 per cell) and high resolution in the direction of propagation, *dz* = 0.125*c*/*ω*0. The aspect ratio *dr*/*dz* = 15.6, and the time

underlying approximations, the WAKE simulation correctly captures all relevant physics of plasma wake evolution and dynamics of electron self-injection. In addition, CALDER-Circ having fully self-consistent macroparticle dynamics yields the complete electron phase space,

and thus calculates precisely injected charge and beam emittance.

<sup>0</sup> . Figure 8 shows that despite a much coarser grid, larger time step and

Fig. 5. Pulse self-compression and continuous injection (Kalmykov et al., 2011b). (a) Peak laser intensity (red) and the length of the accelerating phase vs propagation length (black). (b) Energy of test electrons vs their initial positions at *z* = 2.2 mm. The leading quasimonoenergetic bunch forms during Stage I (one period of bubble size oscillation). Bubble expansion during Stage II causes continuous injection with broad energy spectrum. The bubble and injected test electrons at the positions (1) – (3) of panel (a) are shown in panels (c.1) – (c.3). Grayscale: electron density (in cm−3); red dots: test electrons with *γ* > *γ*g. (d.1) – (d.3) Normalized laser intensity, |*a*| 2, at the positions (1) – (3) of panel (a); red contour: iso-contour of an incident pulse intensity at exp(−2) of the peak. Self-steepening of the pulse (formation of a relativistic piston) causes elongation of the bubble and continuous injection.

This physical interpretation, deduced from the analysis of quasistatic electron trajectories, is validated in section 2.5 in fully explicit 3-D PIC simulations. It suggests that the root cause of continuous injection is the pulse self-steepening. The steepening is partly caused by depletion due to the wake excitation (∼ 33% at *z* = 2.2 mm) (Decker et al., 1996; Fang et al., 2009; Lu et al., 2007), and is partly a nonlinear optical effect (Faure et al., 2005; Pai et al., 2010; Vieira et al., 2010). Figures 7(a) and 7(b) show axial lineouts of normalized intensity and of the nonlinear index of refraction. The pulse leading edge witnesses the *index down-ramp* at all times. Hence, the laser frequency *red-shifts* in the region of index gradient. At the same time, the tail traveling inside the bubble remains unshifted. With lower frequencies temporally leading higher frequencies, the pulse acquires a *positive chirp*. The anomalous group velocity dispersion of plasma compresses the positively chirped pulse: the red-shifted leading edge slows down with respect to the non-shifted tail, building up the field amplitude first in the pulse head (Fig. 7(a)), and later near the pulse center (Fig. 7(b)). The chirp, as shown in Fig. 7(c), broadens the laser spectrum towards *ω* = *ωpe*; envelope oscillations in Fig. 7(b) result from the strong reduction of the pulse central frequency. The 12 Will-be-set-by-IN-TECH

Fig. 5. Pulse self-compression and continuous injection (Kalmykov et al., 2011b). (a) Peak laser intensity (red) and the length of the accelerating phase vs propagation length (black). (b) Energy of test electrons vs their initial positions at *z* = 2.2 mm. The leading quasimonoenergetic bunch forms during Stage I (one period of bubble size oscillation). Bubble expansion during Stage II causes continuous injection with broad energy spectrum. The bubble and injected test electrons at the positions (1) – (3) of panel (a) are shown in panels (c.1) – (c.3). Grayscale: electron density (in cm−3); red dots: test electrons with *γ* > *γ*g. (d.1) –

iso-contour of an incident pulse intensity at exp(−2) of the peak. Self-steepening of the pulse (formation of a relativistic piston) causes elongation of the bubble and continuous injection.

This physical interpretation, deduced from the analysis of quasistatic electron trajectories, is validated in section 2.5 in fully explicit 3-D PIC simulations. It suggests that the root cause of continuous injection is the pulse self-steepening. The steepening is partly caused by depletion due to the wake excitation (∼ 33% at *z* = 2.2 mm) (Decker et al., 1996; Fang et al., 2009; Lu et al., 2007), and is partly a nonlinear optical effect (Faure et al., 2005; Pai et al., 2010; Vieira et al., 2010). Figures 7(a) and 7(b) show axial lineouts of normalized intensity and of the nonlinear index of refraction. The pulse leading edge witnesses the *index down-ramp* at all times. Hence, the laser frequency *red-shifts* in the region of index gradient. At the same time, the tail traveling inside the bubble remains unshifted. With lower frequencies temporally leading higher frequencies, the pulse acquires a *positive chirp*. The anomalous group velocity dispersion of plasma compresses the positively chirped pulse: the red-shifted leading edge slows down with respect to the non-shifted tail, building up the field amplitude first in the pulse head (Fig. 7(a)), and later near the pulse center (Fig. 7(b)). The chirp, as shown in Fig. 7(c), broadens the laser spectrum towards *ω* = *ωpe*; envelope oscillations in Fig. 7(b) result from the strong reduction of the pulse central frequency. The

2, at the positions (1) – (3) of panel (a); red contour:

(d.3) Normalized laser intensity, |*a*|

Fig. 6. Bubble elongation due to formation of a relativistic piston: quasistatic analysis. Physical quantities are shown before (left column) and after (right column) the piston formation. The left column corresponds to position (1), and the right column – to the position (3) of Fig. 5(a). (a), (b) Longitudinal electric field *Ez* (in GV/cm). Black lines are the innermost electron trajectories, *r*in(*ξ*), plotted until the point of return, *pz* = 0. Blue contours are iso-contours of laser intensity at 5 <sup>×</sup> 1018 W/cm−2. (c), (d) Longitudinal electric field in the points of the innermost electron trajectory, *Ez*(*r*in(*ξ*)), from plots (a) and (b), respectively. (e), (f) Longitudinal momentum of the innermost electron along its trajectory, *pz*(*r*in(*ξ*)).

large bandwidth explains pulse compression to roughly two cycles. The red shift of central frequency together with the continuous front etching additionally slows down the pulse and the bubble and provides another reason for the occurrence of continuous injection (Fang et al., 2009). We show in section 3 that *negative chirp* of the incident pulse may compensate for the gradually accumulating red-shift, thus delaying the pulse contraction and partly suppressing continuous injection.

#### **2.5 Validation of self-injection scenarios in full 3-D PIC simulation: role of beam loading**

Collective fields of the electron beam, neglected in the test-particle treatment, are known to change the shape of the sheath and thus reduce accelerating gradient, eventually terminating self-injection (Tzoufras et al., 2009). In this section, we verify the test-particle results by running a fully explicit 3-D PIC simulation with the identical set of initial conditions. We use the quasi-cylindrical code CALDER-Circ (Lifschitz et al., 2009), which preserves realistic geometry of interaction, and accounts for the axial asymmetry by decomposing EM fields (laser and wake) into a set of poloidal modes (whereas the particles remain in full 3-D). Well preserved cylindrical symmetry during the interaction enables us to use just the two lowest order modes and thus reduce a 3-D problem to an essentially 2-D one. We suppress sampling noise by using a large number of macroparticles (45 per cell) and high resolution in the direction of propagation, *dz* = 0.125*c*/*ω*0. The aspect ratio *dr*/*dz* = 15.6, and the time step *dt* = 0.1244*ω*−<sup>1</sup> <sup>0</sup> . Figure 8 shows that despite a much coarser grid, larger time step and underlying approximations, the WAKE simulation correctly captures all relevant physics of plasma wake evolution and dynamics of electron self-injection. In addition, CALDER-Circ having fully self-consistent macroparticle dynamics yields the complete electron phase space, and thus calculates precisely injected charge and beam emittance.

Fig. 8. Continuous injection in quasistatic (WAKE with test particles) and full 3-D PIC (CALDER-Circ) simulations (Kalmykov et al., 2011b). (a)–(c) Electron density from CALDER-Circ (top half) and WAKE (bottom) runs. Yellow dots are the test electrons with *γ* > *γ*g. (d)–(f) Electron energy spectrum (CALDER-Circ). (g)–(i) Longitudinal phase space (colormap – CALDER-Circ; test electrons – yellow dots). Panels (a), (b), (c) are counterparts

Physics of Quasi-Monoenergetic Laser-Plasma Acceleration of Electrons in the Blowout Regime 127

continuous injection discussed in section 2.4. Continuous injection can be thus associated solely with frequency red-shift and self-compression of the driver pulse, causing the growth

We have learned in the last section that ploughing through the electron fluid significantly reduces the frequency of the pulse leading edge. Subsequent self-steepening and compression of the driver to a few cycles causes gradual expansion of the bubble, bringing about massive continuous injection, degrading the beam quality. This physical scenario prevents effective use of the entire dephasing length for high-energy quasi-monoenergetic electron acceleration. Using a *negatively chirped* pulse may naturally alleviate this issue. If the pulse has a sufficiently large frequency bandwidth (corresponding to a few-cycle transform-limited duration), the blue shift of the leading edge would largely compensate for a gradually accumulating nonlinear red-shift. This flattening of the phase would then reduce the pulse self-steepening

**3. Suppression of continuous injection using negatively chirped driving pulse**

of Figs. 5(c.1), 5(c.2) and 5(c.3).

of the quasistatic bubble.

Fig. 7. Pulse red-shifting and formation of the relativistic piston (Kalmykov et al., 2011b). (a) Axial lineouts of normalized intensity (red) and nonlinear refractive index (black) at the position (1) of Fig. 5(a). (b) Same for the position (3) of Fig. 5(a). (c) Laser frequency spectra (radially integrated). Blue corresponds to panel (a), red – to panel (b), black – to the incident pulse. The co-moving index gradient causes frequency red shifting and spectral broadening. The spectrally broadened pulse in panel (b) is compressed to approximately two cycles.

In spite of the great difference in the algorithms and physics content, both codes demonstrate the same correlation between the laser and bubble evolution. Self-injection begins, terminates, and resumes at exactly the same positions along the propagation axis in both runs. Figures 8(a), 8(d), and 8(g) show the result of Stage I — formation of quasi-monoenergetic electron bunch before dephasing. Self-fields of the bunch are unable to prevent the bucket contraction and partial de-trapping of electrons. The bunch phase space has a characteristic "U"-shape produced by the phase space rotation. The bunch has 8% energy spread around 245 MeV and a charge *Q*mono ≈ 230 pC (in addition, electrons from the second bucket produce a separate, rather diffuse peak around 150 MeV in Fig. 8(d)). The bunch duration, *tb* = 10 fs, is the same as in the test-particle simulation; whereas divergence, 13 mrad, is three times higher. The normalized transverse emittance in the plane of polarization is *εN*,*<sup>x</sup>* = (*mec*)−<sup>1</sup> �Δ*x*2��Δ*p*<sup>2</sup> *<sup>x</sup>*�−�Δ*x*Δ*px*�<sup>2</sup>1/2 <sup>≈</sup> 8.74*<sup>π</sup>* mm mrad. It was understood that the absence of beam self-fields in the test particle model leads to strong underestimation of emittance, which validates the importance of 3-D PIC simulations for precise calculation of the phase space volume of self-injected electrons (Kalmykov et al., 2011b).

The difference between the phase spaces of WAKE test electrons and CALDER-Circ macroparticles, clearly seen in Fig. 8(g), can be attributed to the effect of beam loading which reduces the accelerating gradient along the bunch and slows down phase space rotation, ultimately reducing the bunch energy by 30%. Using the formalism of (Tzoufras et al., 2009), we find that the repulsive EM fields of electron bunch are not high enough to prevent sheath electrons crossing the axis and are thus unable to prevent further injection (Kalmykov et al., 2011b).

Figures 8(a)–8(c) show that continuous injection develops in both CALDER-Circ and WAKE runs in exactly the same fashion, which validates the physical origin and continuous injection scenario inferred from the analysis of quasi-static electron trajectories in section 2.4. Near dephasing, continuously injected charge in CALDER-Circ simulation reaches *Q*cont ≈ 1.06 nC; the beam divergence is 36 mrad. The ratio *Q*cont/*Q*mono ≈ 4.6 is lower than the test particle result of section 2.4 on account of beam loading. Figure 8(c) shows that, in spite of the high injected charge, the bubble shape at the dephasing point is almost unaffected by the presence of the electron beam; this observation rules out beam loading as a cause of continuous injection. The bubble is still not fully loaded and injection in the CALDER-Circ simulation continues beyond the dephasing point. Therefore, apart from slight reduction of the accelerating gradient, beam loading brings no new physical features into the scenario of

14 Will-be-set-by-IN-TECH

−10 0 10

0.998

*<sup>x</sup>*�−�Δ*x*Δ*px*�<sup>2</sup>1/2 <sup>≈</sup> 8.74*<sup>π</sup>* mm mrad. It was understood that

0.5 1 1.5

<sup>ω</sup> (ω<sup>0</sup> ) −3

0

3

log10 S(

ω) (a.u.)

nnl

1.0

(a) (b) (c)

z − ct (μm)

Fig. 7. Pulse red-shifting and formation of the relativistic piston (Kalmykov et al., 2011b). (a) Axial lineouts of normalized intensity (red) and nonlinear refractive index (black) at the position (1) of Fig. 5(a). (b) Same for the position (3) of Fig. 5(a). (c) Laser frequency spectra (radially integrated). Blue corresponds to panel (a), red – to panel (b), black – to the incident pulse. The co-moving index gradient causes frequency red shifting and spectral broadening. The spectrally broadened pulse in panel (b) is compressed to approximately two cycles.

In spite of the great difference in the algorithms and physics content, both codes demonstrate the same correlation between the laser and bubble evolution. Self-injection begins, terminates, and resumes at exactly the same positions along the propagation axis in both runs. Figures 8(a), 8(d), and 8(g) show the result of Stage I — formation of quasi-monoenergetic electron bunch before dephasing. Self-fields of the bunch are unable to prevent the bucket contraction and partial de-trapping of electrons. The bunch phase space has a characteristic "U"-shape produced by the phase space rotation. The bunch has 8% energy spread around 245 MeV and a charge *Q*mono ≈ 230 pC (in addition, electrons from the second bucket produce a separate, rather diffuse peak around 150 MeV in Fig. 8(d)). The bunch duration, *tb* = 10 fs, is the same as in the test-particle simulation; whereas divergence, 13 mrad, is three times higher. The normalized transverse emittance in the plane of polarization is

the absence of beam self-fields in the test particle model leads to strong underestimation of emittance, which validates the importance of 3-D PIC simulations for precise calculation of

The difference between the phase spaces of WAKE test electrons and CALDER-Circ macroparticles, clearly seen in Fig. 8(g), can be attributed to the effect of beam loading which reduces the accelerating gradient along the bunch and slows down phase space rotation, ultimately reducing the bunch energy by 30%. Using the formalism of (Tzoufras et al., 2009), we find that the repulsive EM fields of electron bunch are not high enough to prevent sheath electrons crossing the axis and are thus unable to prevent further injection (Kalmykov et al., 2011b). Figures 8(a)–8(c) show that continuous injection develops in both CALDER-Circ and WAKE runs in exactly the same fashion, which validates the physical origin and continuous injection scenario inferred from the analysis of quasi-static electron trajectories in section 2.4. Near dephasing, continuously injected charge in CALDER-Circ simulation reaches *Q*cont ≈ 1.06 nC; the beam divergence is 36 mrad. The ratio *Q*cont/*Q*mono ≈ 4.6 is lower than the test particle result of section 2.4 on account of beam loading. Figure 8(c) shows that, in spite of the high injected charge, the bubble shape at the dephasing point is almost unaffected by the presence of the electron beam; this observation rules out beam loading as a cause of continuous injection. The bubble is still not fully loaded and injection in the CALDER-Circ simulation continues beyond the dephasing point. Therefore, apart from slight reduction of the accelerating gradient, beam loading brings no new physical features into the scenario of

the phase space volume of self-injected electrons (Kalmykov et al., 2011b).

0.998

1.0

−10 0 10

z − ct (μm)

0

*εN*,*<sup>x</sup>* = (*mec*)−<sup>1</sup>

�Δ*x*2��Δ*p*<sup>2</sup>

20


40

Fig. 8. Continuous injection in quasistatic (WAKE with test particles) and full 3-D PIC (CALDER-Circ) simulations (Kalmykov et al., 2011b). (a)–(c) Electron density from CALDER-Circ (top half) and WAKE (bottom) runs. Yellow dots are the test electrons with *γ* > *γ*g. (d)–(f) Electron energy spectrum (CALDER-Circ). (g)–(i) Longitudinal phase space (colormap – CALDER-Circ; test electrons – yellow dots). Panels (a), (b), (c) are counterparts of Figs. 5(c.1), 5(c.2) and 5(c.3).

continuous injection discussed in section 2.4. Continuous injection can be thus associated solely with frequency red-shift and self-compression of the driver pulse, causing the growth of the quasistatic bubble.

## **3. Suppression of continuous injection using negatively chirped driving pulse**

We have learned in the last section that ploughing through the electron fluid significantly reduces the frequency of the pulse leading edge. Subsequent self-steepening and compression of the driver to a few cycles causes gradual expansion of the bubble, bringing about massive continuous injection, degrading the beam quality. This physical scenario prevents effective use of the entire dephasing length for high-energy quasi-monoenergetic electron acceleration. Using a *negatively chirped* pulse may naturally alleviate this issue. If the pulse has a sufficiently large frequency bandwidth (corresponding to a few-cycle transform-limited duration), the blue shift of the leading edge would largely compensate for a gradually accumulating nonlinear red-shift. This flattening of the phase would then reduce the pulse self-steepening

Fig. 10. Reduction of dark current using the negatively chirped driver pulse. Panels on the left correspond to the non-chirped driver, and on the right to the chirped pulse. (a,b) Phase space of test electrons showing signatures of continuous injection. (c,d) Accelerating gradient. Shaded area covers the accelerating phase (*Ez* < 0). (e,f) Energy spectrum of test electrons (number of test particles per spectrometer energy bin). The data are extracted from WAKE simulations at *z* = 2.2 mm (cf. position (3) of Fig. 9(b)). Chirp of the driver reduces expansion of the bubble (cf. panel (d) vs (c)); early dephasing is thus prevented (cf. (b) vs (a)). Injected electrons, exposed to higher accelerating gradient (cf. (d) vs (c)), gain higher energy.

Physics of Quasi-Monoenergetic Laser-Plasma Acceleration of Electrons in the Blowout Regime 129

The amount of continuously injected charge is significantly reduced (cf. (f) vs (e)).

in progression from Fig. 9(d.1) to 9(d.3).

see below, results in higher final electron energy.

It is during the Stage II (*z* > 1.2 mm) that the chirp changes the pulse behavior most drastically. From Fig. 9(a), the chirped pulse stabilizes after one spot size oscillation and propagates until *z* = 2.1 mm with almost invariable intensity. As expected, the front steepening remains almost unnoticeable till the end of the run: the chirped pulse envelopes in Figs. 9(e.1) and 9(e.2) show just minimal longitudinal compression. As a result, according to Fig. 9(b), the bubble driven by the chirped pulse expands by roughly 12% between positions (1) and (3), in contrast to 35% in the non-chirped driver case. This stabilizing effect of the pulse chirp is the most vividly seen

Figure 9(c) shows 22% depletion of the chirped pulse, in contrast to 33% in the non-chirped driver case. Therefore, the chirped pulse energy is transferred to the plasma wake less effectively. The chirp not only compensates for the nonlinear frequency shift and concomitant pulse self-compression (nonlinear optical effects unrelated to the pump depletion), but also reduces the pulse front etching due to the local pump depletion (Decker et al., 1996). As a result, the chirped pulse propagates with a higher group velocity. The peak of the chirped pulse in Figs. 9(e.1) – 9(e.3) is temporally advanced with respect to its non-chirped counterpart. Larger velocity of the structure extends the dephasing length and, as we shall

Despite all promising tendencies, continuous injection is not fully shut down. As shown in Figs. 10(a) and 10(b), longitudinal phase space at the end of simulation (*z* = 2.2 mm) consists

Fig. 9. Effect of the negative chirp on the evolution of driver pulse and plasma bubble. Red lines in panels (a) – (c) correspond to the chirped pulse; black lines to the non-chirped pulse. (a) Peak intensity, (b) length of the accelerating phase, and (c) pulse energy vs propagation distance. Positions (1) – (3) are the same as in Fig. 5(a). Panels (d.1) – (d.3) show electron density and test electrons (red and blue dots) with *γ* > *γ*<sup>g</sup> at the positions (1) – (3) (labeled accordingly). Grayscale is linear with a cutoff at *ne* <sup>=</sup> 3.25 <sup>×</sup> 1019 cm−3. (e.1) – (e.3) Normalized laser intensity, |*a*| 2, at the positions (1) – (3). Negative chirp prevents rapid self-compression of the pulse, slowing down bubble expansion and reducing the number of injected test electrons.

and delay the relativistic piston formation. Elongation of the bubble, vividly demonstrated in Fig. 8, would be thus reduced, and concomitant continuous injection partly suppressed. We first verify this scenario in a WAKE simulation with test particles. We take the 30 fs, 70 TW Gaussian pulse with the parameters specified at the beginning of section 2 and introduce a linear chirp, temporally advancing higher frequencies. The center of this negatively chirped pulse corresponds to the carrier frequency, and the bandwidth corresponds to a transform-limited 5 fs duration (viz. the relativistic piston of the last section). Multi-Joule amplification systems delivering such broad-bandwidth pulses are not available yet; their development, however, is being actively pursued (Herrmann et al., 2009). As in section 2.4, we run the WAKE simulation until *z* = 2.2 mm. Figures 9 and 10 show that the negative chirp profoundly changes the pulse evolution, and, hence, the dynamics of self-injection. Figure 9(a) indicates that the initial stage of laser evolution, corresponding to the Stage I of Fig. 5(a), remains almost unaltered. Relativistic self-focusing appears to be quite insensitive to the frequency chirp. Figure 9(b) shows that the bubble experiences one pulsation between *z* = 0.6 and 1.3 mm. The maximal bubble expansion around *z* = 1 mm appears to be 25% smaller than in the non-chirped driver case, with a proportionally smaller number of injected

test particles.

16 Will-be-set-by-IN-TECH

Fig. 9. Effect of the negative chirp on the evolution of driver pulse and plasma bubble. Red lines in panels (a) – (c) correspond to the chirped pulse; black lines to the non-chirped pulse. (a) Peak intensity, (b) length of the accelerating phase, and (c) pulse energy vs propagation distance. Positions (1) – (3) are the same as in Fig. 5(a). Panels (d.1) – (d.3) show electron density and test electrons (red and blue dots) with *γ* > *γ*<sup>g</sup> at the positions (1) – (3) (labeled

self-compression of the pulse, slowing down bubble expansion and reducing the number of

and delay the relativistic piston formation. Elongation of the bubble, vividly demonstrated in Fig. 8, would be thus reduced, and concomitant continuous injection partly suppressed. We first verify this scenario in a WAKE simulation with test particles. We take the 30 fs, 70 TW Gaussian pulse with the parameters specified at the beginning of section 2 and introduce a linear chirp, temporally advancing higher frequencies. The center of this negatively chirped pulse corresponds to the carrier frequency, and the bandwidth corresponds to a transform-limited 5 fs duration (viz. the relativistic piston of the last section). Multi-Joule amplification systems delivering such broad-bandwidth pulses are not available yet; their development, however, is being actively pursued (Herrmann et al., 2009). As in section 2.4, we run the WAKE simulation until *z* = 2.2 mm. Figures 9 and 10 show that the negative chirp

2, at the positions (1) – (3). Negative chirp prevents rapid

accordingly). Grayscale is linear with a cutoff at *ne* <sup>=</sup> 3.25 <sup>×</sup> 1019 cm−3. (e.1) – (e.3)

profoundly changes the pulse evolution, and, hence, the dynamics of self-injection.

Figure 9(a) indicates that the initial stage of laser evolution, corresponding to the Stage I of Fig. 5(a), remains almost unaltered. Relativistic self-focusing appears to be quite insensitive to the frequency chirp. Figure 9(b) shows that the bubble experiences one pulsation between *z* = 0.6 and 1.3 mm. The maximal bubble expansion around *z* = 1 mm appears to be 25% smaller than in the non-chirped driver case, with a proportionally smaller number of injected

Normalized laser intensity, |*a*|

injected test electrons.

test particles.

Fig. 10. Reduction of dark current using the negatively chirped driver pulse. Panels on the left correspond to the non-chirped driver, and on the right to the chirped pulse. (a,b) Phase space of test electrons showing signatures of continuous injection. (c,d) Accelerating gradient. Shaded area covers the accelerating phase (*Ez* < 0). (e,f) Energy spectrum of test electrons (number of test particles per spectrometer energy bin). The data are extracted from WAKE simulations at *z* = 2.2 mm (cf. position (3) of Fig. 9(b)). Chirp of the driver reduces expansion of the bubble (cf. panel (d) vs (c)); early dephasing is thus prevented (cf. (b) vs (a)). Injected electrons, exposed to higher accelerating gradient (cf. (d) vs (c)), gain higher energy. The amount of continuously injected charge is significantly reduced (cf. (f) vs (e)).

It is during the Stage II (*z* > 1.2 mm) that the chirp changes the pulse behavior most drastically. From Fig. 9(a), the chirped pulse stabilizes after one spot size oscillation and propagates until *z* = 2.1 mm with almost invariable intensity. As expected, the front steepening remains almost unnoticeable till the end of the run: the chirped pulse envelopes in Figs. 9(e.1) and 9(e.2) show just minimal longitudinal compression. As a result, according to Fig. 9(b), the bubble driven by the chirped pulse expands by roughly 12% between positions (1) and (3), in contrast to 35% in the non-chirped driver case. This stabilizing effect of the pulse chirp is the most vividly seen in progression from Fig. 9(d.1) to 9(d.3).

Figure 9(c) shows 22% depletion of the chirped pulse, in contrast to 33% in the non-chirped driver case. Therefore, the chirped pulse energy is transferred to the plasma wake less effectively. The chirp not only compensates for the nonlinear frequency shift and concomitant pulse self-compression (nonlinear optical effects unrelated to the pump depletion), but also reduces the pulse front etching due to the local pump depletion (Decker et al., 1996). As a result, the chirped pulse propagates with a higher group velocity. The peak of the chirped pulse in Figs. 9(e.1) – 9(e.3) is temporally advanced with respect to its non-chirped counterpart. Larger velocity of the structure extends the dephasing length and, as we shall see below, results in higher final electron energy.

Despite all promising tendencies, continuous injection is not fully shut down. As shown in Figs. 10(a) and 10(b), longitudinal phase space at the end of simulation (*z* = 2.2 mm) consists

Fig. 11. Acceleration of electrons in the wake of negatively chirped pulse: suppression of dark current (CALDER-Circ simulation). (a.1), (a.2), (b.1), (b.2) Longitudinal phase space; (a.3), (b.3) electron energy spectra. Panel (a.1) – dephasing point of electrons accelerated with the *non-chirped* driver – is the same as Fig. 8(i). The leading bunch with the energy 515 MeV and relative spread 8.5% is followed by a continuous component carrying 4.6 times higher charge. (a.2), (a.3) Electrons accelerated with the chirped driver achieve 515 MeV energy earlier, and with much weaker energy tail. (b.2) Dephasing point of electrons accelerated with the *chirped* driver. Leading bunch with the energy 660 MeV and 5.8% relative spread is the dominating spectral feature in the energy range *E* > 100 MeV (cf. panels (b.2) and (b.3)). At this point, electrons accelerated with the non-chirped pulse are completely dominated by

Physics of Quasi-Monoenergetic Laser-Plasma Acceleration of Electrons in the Blowout Regime 131

In conclusion, 3-D PIC simulations successfully support the idea of suppressing the dark current using negative chirp of the driving pulse. Even though complete elimination of dark current is hard to achieve in high-density plasmas (*γ*<sup>g</sup> = 10 – 15), strong reduction of the charge in the poorly collimated, continuous low-energy tail is useful for applications. Subsequent manipulations with the beam using permanent magnets may further improve its

A time-varying electron density bubble created by the radiation pressure of a tightly focused laser pulse guides the pulse through a uniform, rarefied plasma, traps ambient plasma electrons and accelerates them to GeV-level energy. Natural pulse evolution (nonlinear focusing and self-compression) *is in most cases sufficient* to initiate and terminate self-injection. Bubble dynamics and the self-injection process are governed primarily by the driver evolution. Expansion of the bubble facilitates injection, whereas stabilization and contraction extinguishes injection and suppresses the low-energy background. Simultaneously, longitudinal non-uniformity of the accelerating gradient causes rapid phase space rotation.

the dark current (cf. panels (b.1) and (b.3)).

quality (Weingartner et al., 2011).

**4. Conclusion**


Table 1. Parameters of the quasi-monoenergetic electron bunch (CALDER-Circ simulation, *z* = 2.2 mm). *Q*mono is the charge in nC; *E*mono is the central energy in MeV; Δ*E*mono is the absolute energy spread (FWHM) in MeV; Δ*E*/*E*mono is the normalized energy spread; *εN*,*<sup>x</sup>* and *εN*,*<sup>y</sup>* are the root-mean-square (RMS) normalized transverse emittances (in mm mrad) in and out of the laser polarization plane; �Δ*α*�mono is the RMS divergence in mrad.


Table 2. Parameters of the low-energy background with continuous spectrum, 50 MeV < *E* < *E*max (CALDER-Circ simulation, *z* = 2.2 mm). *Q*cont is the charge in nC; *E*max is the highenergy cutoff (in MeV); �Δ*α*�cont is the RMS divergence in mrad.

of two distinct components: the leading (quasi-monoenergetic) bunch and long continuous tail. In the non-chirped driver case (cf. Fig. 10(a)), the leading bunch has reached dephasing. In the chirped driver case, however, the electrons are still located deeply inside the accelerating phase and continue gaining energy (cf. Fig. 10(b)). Owing to very slow expansion of the bubble, these particles have been exposed to a higher gradient, gaining additional 150 MeV energy over the same acceleration distance. Comparison of Figs. 10(e) and 10(f) also shows that the dark current is much lower in the chirped driver case, on account of much slower bubble expansion during the second half of the simulation.

We quantify the effect of dark current suppression running a CALDER-Circ simulation with the same initial conditions. We find that the electron beam components – leading quasi-monoenegretic bunch and a polychromatic background – are affected by the pulse chirp differently. Data presented in Tables 1 and 2 show that negative chirp reduces the charge of the background more than twice, and noticeably improves the leading bunch quality, increasing its energy by 20% and reducing the energy spread from 8.5% to 4.7%.

Poor collimation of the continuously injected electrons, �Δ*α*�cont ≈ 3�Δ*α*�mono, together with their distribution over the energy range 10-15 times broader than the absolute energy spread of the leading bunch, reduce dramatically the brightness of the energy tail. These poorly collimated beam components can be dispersed in vacuum using miniature magnetic quadrupole lenses, further improving the beam collimation and reducing the energy spread (Weingartner et al., 2011).

More details on electron acceleration in the wake of negatively chirped pulse are shown in Fig. 11. First, comparison of Figs. 11(a.1) and 11(a.2) indicates that the chirped driver can accelerate the leading bunch to a given energy (515 MeV in our case) in a shorter plasma, and with much weaker background. Figures 11(b) also demonstrate that using the negatively chirped driver improves the LPA efficiency, increasing the dephasing length and final electron energy without compromising beam quality. Indeed, Figs. 11(b.2) and 11(b.3) show that, in the chirped driver case, the leading bunch with 181 pC charge, energy 660 MeV and 5.8% relative energy spread remains the dominating spectral feature in the energy range *E* > 100 MeV. In contrast, according to Fig. 11(b.1), electrons accelerated in the wake of non-chirped driver are completely dominated by the dark current by this point.

Fig. 11. Acceleration of electrons in the wake of negatively chirped pulse: suppression of dark current (CALDER-Circ simulation). (a.1), (a.2), (b.1), (b.2) Longitudinal phase space; (a.3), (b.3) electron energy spectra. Panel (a.1) – dephasing point of electrons accelerated with the *non-chirped* driver – is the same as Fig. 8(i). The leading bunch with the energy 515 MeV and relative spread 8.5% is followed by a continuous component carrying 4.6 times higher charge. (a.2), (a.3) Electrons accelerated with the chirped driver achieve 515 MeV energy earlier, and with much weaker energy tail. (b.2) Dephasing point of electrons accelerated with the *chirped* driver. Leading bunch with the energy 660 MeV and 5.8% relative spread is the dominating spectral feature in the energy range *E* > 100 MeV (cf. panels (b.2) and (b.3)). At this point, electrons accelerated with the non-chirped pulse are completely dominated by the dark current (cf. panels (b.1) and (b.3)).

In conclusion, 3-D PIC simulations successfully support the idea of suppressing the dark current using negative chirp of the driving pulse. Even though complete elimination of dark current is hard to achieve in high-density plasmas (*γ*<sup>g</sup> = 10 – 15), strong reduction of the charge in the poorly collimated, continuous low-energy tail is useful for applications. Subsequent manipulations with the beam using permanent magnets may further improve its quality (Weingartner et al., 2011).

## **4. Conclusion**

18 Will-be-set-by-IN-TECH

Chirp 0.181 637 30 0.047 8.1*π* 7.4*π* 10.6 No chirp 0.230 515 44 0.085 8.74*π* 8.5*π* 12.9 Table 1. Parameters of the quasi-monoenergetic electron bunch (CALDER-Circ simulation, *z* = 2.2 mm). *Q*mono is the charge in nC; *E*mono is the central energy in MeV; Δ*E*mono is the absolute energy spread (FWHM) in MeV; Δ*E*/*E*mono is the normalized energy spread; *εN*,*<sup>x</sup>* and *εN*,*<sup>y</sup>* are the root-mean-square (RMS) normalized transverse emittances (in mm mrad) in

Chirp 0.46 500 33 No chirp 1.06 400 36 Table 2. Parameters of the low-energy background with continuous spectrum, 50 MeV < *E* < *E*max (CALDER-Circ simulation, *z* = 2.2 mm). *Q*cont is the charge in nC; *E*max is the high-

of two distinct components: the leading (quasi-monoenergetic) bunch and long continuous tail. In the non-chirped driver case (cf. Fig. 10(a)), the leading bunch has reached dephasing. In the chirped driver case, however, the electrons are still located deeply inside the accelerating phase and continue gaining energy (cf. Fig. 10(b)). Owing to very slow expansion of the bubble, these particles have been exposed to a higher gradient, gaining additional 150 MeV energy over the same acceleration distance. Comparison of Figs. 10(e) and 10(f) also shows that the dark current is much lower in the chirped driver case, on account of much slower

We quantify the effect of dark current suppression running a CALDER-Circ simulation with the same initial conditions. We find that the electron beam components – leading quasi-monoenegretic bunch and a polychromatic background – are affected by the pulse chirp differently. Data presented in Tables 1 and 2 show that negative chirp reduces the charge of the background more than twice, and noticeably improves the leading bunch quality, increasing

Poor collimation of the continuously injected electrons, �Δ*α*�cont ≈ 3�Δ*α*�mono, together with their distribution over the energy range 10-15 times broader than the absolute energy spread of the leading bunch, reduce dramatically the brightness of the energy tail. These poorly collimated beam components can be dispersed in vacuum using miniature magnetic quadrupole lenses, further improving the beam collimation and reducing the energy spread

More details on electron acceleration in the wake of negatively chirped pulse are shown in Fig. 11. First, comparison of Figs. 11(a.1) and 11(a.2) indicates that the chirped driver can accelerate the leading bunch to a given energy (515 MeV in our case) in a shorter plasma, and with much weaker background. Figures 11(b) also demonstrate that using the negatively chirped driver improves the LPA efficiency, increasing the dephasing length and final electron energy without compromising beam quality. Indeed, Figs. 11(b.2) and 11(b.3) show that, in the chirped driver case, the leading bunch with 181 pC charge, energy 660 MeV and 5.8% relative energy spread remains the dominating spectral feature in the energy range *E* > 100 MeV. In contrast, according to Fig. 11(b.1), electrons accelerated in the wake of non-chirped driver are

and out of the laser polarization plane; �Δ*α*�mono is the RMS divergence in mrad.

energy cutoff (in MeV); �Δ*α*�cont is the RMS divergence in mrad.

bubble expansion during the second half of the simulation.

completely dominated by the dark current by this point.

(Weingartner et al., 2011).

its energy by 20% and reducing the energy spread from 8.5% to 4.7%.

*Q*mono *E*mono Δ*E*mono Δ*E*/*E*mono *εN*,*<sup>x</sup> εN*,*<sup>y</sup>* �Δ*α*�mono

*Q*cont *E*max �Δ*α*�cont

A time-varying electron density bubble created by the radiation pressure of a tightly focused laser pulse guides the pulse through a uniform, rarefied plasma, traps ambient plasma electrons and accelerates them to GeV-level energy. Natural pulse evolution (nonlinear focusing and self-compression) *is in most cases sufficient* to initiate and terminate self-injection. Bubble dynamics and the self-injection process are governed primarily by the driver evolution. Expansion of the bubble facilitates injection, whereas stabilization and contraction extinguishes injection and suppresses the low-energy background. Simultaneously, longitudinal non-uniformity of the accelerating gradient causes rapid phase space rotation.

Buck, A., Nicolai, M., Schmid, K., Sears, C. M. S., Sävert, A., Mikhailova, J. M., Krausz,

Physics of Quasi-Monoenergetic Laser-Plasma Acceleration of Electrons in the Blowout Regime 133

Clayton, C. E., Ralph, J. E., Albert, F., Fonseca, R. A., Glenzer, S. H., Joshi, C., Lu, W., Marsh,

DesRosiers, C., Moskvin, V., Bielajew, A. F. & Papiez, L. (2000). 150-250 meV electron beams

Dong, P., Reed, S. A., Yi, S. A., Kalmykov, S., Shvets, G., Downer, M. C., Matlis, N. H., Leemans

Dong, P., Reed, S. A., Yi, S. A., Kalmykov, S., Li, Z. Y., Shvets, G., Matlis, N. H., McGuffey,

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Although beam loading reduces the accelerating gradient and slows down phase space rotation, a quasi-monoenergetic, well collimated electron bunch forms long before dephasing. At the same time, extending the acceleration length to the dephasing limit without compromising electron beam quality is not straightforward. In modern experiments, phase self-modulation and frequency red-shift due to the wake excitation cause gradual compression of the driver pulse, turning it into a relativistic piston. This process causes bubble elongation and massive continuous secondary injection (dark current). Compensation of the nonlinear frequency shift by negatively chirping the pulse is one way to delay the piston formation. As a result, nearly 60% reduction of the dark current is observed in our 3-D PIC simulations. The same set of simulations also shows that higher stability of the broad bandwidth negatively chirped pulse in plasma leads to a 20% increase in the central energy, a 50% reduction of relative energy spread, and a 20% emittance reduction of the quasi-monoenergetic, high-energy component of electron beam.

The reported results highlight the importance of reduced physics models. Reduced models not only lower the the computational cost of simulations (sometimes by many orders of magnitude), but also allow for the identification of the underlying physical processes responsible for the observed phenomena. The self-injection dynamics and its relation to the nonlinear optical evolution of the driver was understood using especially simple simulation tools (cylindrical quasistatic PIC code with fully 3-D dynamic test particle module). In practical terms, this means that the system performance (electron beam duration, mean energy, energy spread, and, very roughly, divergence) can be approximately assessed without recourse to computationally intensive 3-D PIC simulations. It appears, however, that calculation of the beam charge and transverse emittance still needs a 3-D fully kinetic simulation. Clarifying the nature of self-consistent effects affecting the phase space volume of self-injected electrons in various numerical models, and establishing the true physical origin of these effects is the subject of ongoing work.

## **5. Acknowledgements**

S. Y. K. is grateful to S. A. Yi, V. Khudik, and M. C. Downer for many stimulating discussions and keen interest in the work presented in this Chapter. The work is partly supported by the U. S. Department of Energy grant DE-FG02-08ER55000. A. B. and E. L. acknowledge the support of LASERLAB-EUROPE/LAPTECH through EC FP7 contract no. 228334. The authors acknowledge CCRT at CEA (www-ccrt.cea.fr) for providing computing resources.

## **6. References**


20 Will-be-set-by-IN-TECH

Although beam loading reduces the accelerating gradient and slows down phase space rotation, a quasi-monoenergetic, well collimated electron bunch forms long before dephasing. At the same time, extending the acceleration length to the dephasing limit without compromising electron beam quality is not straightforward. In modern experiments, phase self-modulation and frequency red-shift due to the wake excitation cause gradual compression of the driver pulse, turning it into a relativistic piston. This process causes bubble elongation and massive continuous secondary injection (dark current). Compensation of the nonlinear frequency shift by negatively chirping the pulse is one way to delay the piston formation. As a result, nearly 60% reduction of the dark current is observed in our 3-D PIC simulations. The same set of simulations also shows that higher stability of the broad bandwidth negatively chirped pulse in plasma leads to a 20% increase in the central energy, a 50% reduction of relative energy spread, and a 20% emittance reduction of the

The reported results highlight the importance of reduced physics models. Reduced models not only lower the the computational cost of simulations (sometimes by many orders of magnitude), but also allow for the identification of the underlying physical processes responsible for the observed phenomena. The self-injection dynamics and its relation to the nonlinear optical evolution of the driver was understood using especially simple simulation tools (cylindrical quasistatic PIC code with fully 3-D dynamic test particle module). In practical terms, this means that the system performance (electron beam duration, mean energy, energy spread, and, very roughly, divergence) can be approximately assessed without recourse to computationally intensive 3-D PIC simulations. It appears, however, that calculation of the beam charge and transverse emittance still needs a 3-D fully kinetic simulation. Clarifying the nature of self-consistent effects affecting the phase space volume of self-injected electrons in various numerical models, and establishing the true physical origin

S. Y. K. is grateful to S. A. Yi, V. Khudik, and M. C. Downer for many stimulating discussions and keen interest in the work presented in this Chapter. The work is partly supported by the U. S. Department of Energy grant DE-FG02-08ER55000. A. B. and E. L. acknowledge the support of LASERLAB-EUROPE/LAPTECH through EC FP7 contract no. 228334. The authors acknowledge CCRT at CEA (www-ccrt.cea.fr) for providing computing resources.

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**6** 

 *Norway* 

**Time-Resolved Laser Spectroscopy** 

**Processes and Methods of Analysis** 

T. Brudevoll1, A. K. Storebo1, O. Skaaring2, C. N. Kirkemo3,

*4Department of Physics, Norwegian University of Science and Technology, Trondheim* 

Time-resolved laser spectroscopy has become an important method for extracting optical and transport parameters of semiconductors and semiconductor nanostructures. In many of the spectroscopic techniques using lasers the material is brought into a state far away from thermal equilibrium. Interpreting the non-equilibrium state of the material often constitutes a considerable challenge. Many different types of quasiparticles are involved, free electrons and different flavors of holes as well as excitons in many variants, all coupled to lattice modes or hybrid lattice/charge-carrier modes. Spin and magnetic systems are included among the cases considered important. A simplification that can be made under low-level laser irradiation using long pulses is to assume that carrier and lattice temperatures are equal with no excess sample heating. Once this 'isothermal' assumption is broken, sophisticated analysis tools must be used, even for seemingly simple problems and

There is now a vast literature dealing with the results from time-resolved laser spectroscopy experiments. In particular, the book by Shah (Shah, 1999) covers the fundamentals of the subject. However, this field is in very rapid progress, and therefore the text will mainly emphasize later works. We describe new developments and survey the current state of the art regarding methods used for analyzing experiments, assuming some basic à priori knowledge of key experimental techniques. We shall do this survey by taking particular examples from the literature which in our opinion defines some of the main trends. By extracting the essence of selected papers and adding some of our own recent results, we hope to provide a useful guide for those interested in the subject of laser-matter interactions. Obviously, this type of interaction lies at the heart of many future technologies. Because of its central position, laser spectroscopy has become a mixture of old and new theoretical

**1. Introduction** 

regardless of laser pulse length.

**of Semiconductors - Physical** 

O. C. Norum4, O. Olsen4 and M. Breivik5

*5Department of Electronics and Telecommunications* 

*2Kongsberg Defence and Areospace, Kjeller 3Department of Physics, University of Oslo, Oslo* 

*1FFI (Norwegian Defence Research Establishment), Kjeller* 

 *Norwegian University of Science and Technology, Trondheim* 

