## Abstract

A similarity theory is developed for ultra-relativistic laser–plasmas. It is used to compare and optimize possible regimes of three-dimensional wake field acceleration. The optimal scalings for laser wake field electron acceleration are obtained analytically. The main message of the present work is that the bubble acceleration regime [see Pukhov A. & Meyer-ter-Vehn J. 2002. *Appl. Phys. B* **74**, 355] satisfies these optimal scalings.

## 1. Introduction

The concept of laser–plasma electron acceleration has the decisive advantage over conventional accelerators: plasma supports electric fields orders of magnitude higher than the breakdown-limited field in radio-frequency cavities of conventional linacs. It is expected that the relativistic laser–plasma will finally lead to a compact high energy accelerator (Katsouleas 2004; Malka 2004). The very first experiments already have delivered high quality electron beams in the energy range 70–170 MeV (Faure *et al*. 2004; Gedder *et al*. 2004; Mangles *et al*. 2004). Yet, the way to a real laser–plasma accelerator that generates a high-energy electron beam with superior properties is long and full of problems which have to be solved. The main obstacle is that the experiments depend on too many parameters. Often, this makes the interpretation of experimental results ambiguous. At the same time, theoretical models suffer from a similar drawback. The system of kinetic equations describing the problem is both strongly non-linear and contains many parameters. As a result, the quest of searching for new perspective acceleration regimes is challenging and the physics of electron acceleration in plasma is often rather obscure.

The scientific difficulties just listed are neither new nor unique. Quite analogous problems are encountered with classical (magneto-)hydrodynamics. One of the most powerful theoretical tools in such situations is the similarity theory (Birkhoff 1960; Sedov 1993). The similarity allows engineers to scale the behaviour of a physical system from a laboratory acceptable size to the size of practical use.

To the best of our knowledge, no similarity theory has been applied to relativistic laser–plasma interactions. This situation is surprising and unnatural, because the power of similarity theory for the magnetic confinement was recognized in the late 1970s and the similarity theory (Kadomtsev 1975; Connor & Taylor 1977; Lackner *et al*. 1994) has been in use for design of large devices (tokamaks, stellarators) ever thereafter.

The aim of this article is to fill this theory gap. For the first time, we develop a similarity theory for laser–plasma interactions in the ultra-relativistic limit. Using a fully kinetic approach, we show that the similarity parameter *S*=*n*_{e}/*a*_{0}*n*_{c} exists, where *a*_{0}=*eA*_{0}/*m*_{e}*c*^{2} is the relativistically normalized laser amplitude, *n*_{e} is the plasma electron density and is the critical density for a laser with the carrier frequency *ω*_{0}. The basic ultra-relativistic similarity states that laser–plasma interactions with different *a*_{0} and *n*_{e}/*n*_{c} are similar as soon as the similarity parameter *S*=*n*_{e}/*a*_{0}*n*_{c}=const. for these interactions.

The basic *S*-similarity is valid for both over- and underdense plasmas. In the present work, we are interested in the special limit *S*≪1 of relativistically underdense plasmas as it is important for the high energy electron acceleration. In this case, *S* can be considered as a small parameter and quite general scalings for laser–plasma interactions can be found. It follows from the theory that in the optimal configuration the laser pulse has the focal spot radius and the duration *τ*≤*R*/*c*. Here, *k*_{p}=*ω*_{p}/*c* is the plasma wavenumber and is the plasma frequency. This corresponds to the ‘bubble’ acceleration regime (Pukhov & Meyer-ter-Vehn 2002).

The central result of our theory is that the bubble regime of electron acceleration is stable, scalable and the scaling for the maximum energy *E*_{mono} of the monoenergetic peak in the electron spectrum is(1.1)Here, is the laser pulse power, GW is the natural relativistic power unit, and *λ*=2*πc*/*ω*_{0} is the laser wavelength. The scaling (1.1) assumes that the laser pulse duration satisfies the condition *cτ*<*R*. The scaling for the number of accelerated electrons *N*_{mono} in the monoenergetic peak is(1.2)where *r*_{e}=*e*^{2}/*m*_{e}*c*^{2} is the classical electron radius, and *k*_{0}=2*π*/*λ*. The acceleration length *L*_{acc} scales as(1.3)where is the Rayleigh length.

The parametric dependencies in the scalings (1.1)–(1.3) follow from the analytical theory. The numerical pre-factors are taken from three-dimensional PIC simulations. These pre-factors may change depending on the particular shape of the pulse envelope. However, as soon as the envelope of the incident laser pulse is defined, the pre-factors are fixed. The parametric dependencies are universal and do not depend on the particular pulse shape.

## 2. *S*-similarity

We consider collisionless laser–plasma dynamics and neglect the ion motion. The electron distribution function *f*(*t*, ** r**,

**) is described by the Vlasov equation(2.1)where**

*p***=**

*p**m*

_{e}

*γ*

**and self-consistent fields**

*v***and**

*E***satisfy the Maxwell equations (Jackson 1999).**

*B*We suppose that the laser pulse vector potential at the time *t*=0 short before entering the plasma is , where *k*_{0}=*ω*_{0}/*c* is the wavenumber, *R* is the focal spot radius and *τ* is the pulse duration. If one fixes the laser envelope ** a**(

*r*_{⊥},

*x*), then the laser–plasma dynamics depends on four dimensionless parameters: the laser amplitude

*a*

_{0}=max|

*e*

**/**

*a**m*

_{e}

*c*

^{2}|, the focal spot radius

*k*

_{0}

*R*, the pulse duration

*ω*

_{0}

*τ*, and the plasma density ratio

*n*

_{e}/

*n*

_{c}, where is the critical density.

Now, we are going to show that in the ultra-relativistic limit when *a*_{0}≫1, the number of independent dimensionless parameters reduces to three: *k*_{0}*R*, *ω*_{0}*τ* and *S*, where the similarity parameter *S* is(2.2)

Let us introduce the new dimensionless variables(2.3)and the new distribution function defined as(2.4)where and .

The normalized distribution function is a universal one describing the interaction of the given laser pulse with a fixed plasma profile. It satisfies the equations(2.5)(2.6)where is the electron velocity, , and the initial condition for the vector potential is(2.7)with the slow envelope such that .

Equation (2.5) together with the initial condition (2.7) still depend on the four dimensionless parameters , , *S* and *a*_{0}. However, the parameter *a*_{0} appears only in the expression for the electron velocity. In the limit *a*_{0}≫1, one can write(2.8)Consequently, for the ultra-relativistic amplitude *a*_{0}≫1, the laser–plasma dynamics does not depend separately on *a*_{0} and *n*_{e}/*n*_{c}. Rather, they converge into the single similarity parameter *S*.

The ultra-relativistic similarity means that for different interaction cases with *S*=const, plasma electrons move along similar trajectories. The number of these electrons *N*_{e}, their momenta ** p**, and the plasma fields scale as(2.9)(2.10)for

*ω*

_{0}

*τ*=const.,

*k*

_{0}

*R*=const. and

*S*=const.

The ultra-relativistic similarity is valid for arbitrary *S*-values. The *S* parameter appears only in the initial condition (2.7) so that *S*^{−1/2} plays the role of the laser frequency. It separates the relativistically overdense plasmas with *S*≫1 from the underdense ones with *S*≪1.

Let us discuss shortly the applicability area of the similarity theory. The only approximation made is expressed by the formula (2.8). It states that all the electron velocities are equal to the vacuum light velocity *c*. Is this approximation well grounded in the context of the laser wake field acceleration (LWFA)? We stress that although the electron velocities are always *c*, their longitudinal components can be arbitrary depending on the particular direction of the three-dimensional electron momentum.

What matters in the LWFA is the electron trapping and dephasing. Both these phenomena are correctly included in our similarity theory. The trapping and dephasing depend on the *longitudinal* component of the electron velocity:(2.11)When , one can neglect the term *m*_{e}^{2}*c*^{2} under the square root in the denominator of the expression (2.11). According to the similarity theory the transverse electron momentum *p*_{⊥} scales as *a*_{0}, see (2.9). Thus, the approximation (2.8) is valid in the *three dimensional* geometry if *a*_{0}≫1.

Another question that might arise is whether the approximation (2.8) is consistent, as clearly a large part of the electron distribution outside and in front of the laser pulse is non-relativistic. Indeed, electrons outside of the laser pulse do not participate in the interaction: their momenta remain zero. Consequently, they produce no currents and can be simply discarded. Electrons ahead of the laser pulse also have zero momenta initially. However, they must pass through the weakly relativistic precursor of the laser pulse. Our similarity theory is accurate as long as one can neglect this transient interaction with the low-intensity precursor.

## 3. Optimal scalings for LWFA

From now on, we concentrate on the special case of underdense plasma, particularly the bubble acceleration regime (Pukhov & Meyer-ter-Vehn 2002). In this paper, we provide a *heuristic* derivation for the bubble similarity. A mathematically accurate and detailed derivation will be published elsewhere (Gordienko & Pukhov 2005).

If we fix the laser pulse envelope, then the laser–plasma dynamics depends on the three dimensionless parameters: the pulse radius *k*_{0}/*R*, its aspect ratio *Π*=*cτ*/*R* and the parameter *S*. In the case of tenious plasmas, *S*≪1, one can develop all the universal functions around *S*≈0 and obtain additional similarities. An additional similarity means that the number of truly independent dimensionless parameters decreases, i.e. one of the parameters *S*, *k*_{0}*R*, or *Π* can be expressed as a function of the remaining two. We choose *S* and *Π* as the independent parameters:(3.1)where *α*_{1} are unknown powers and *γ*_{0}(*Π*), *R*_{0}(*Π*), *L*_{0}(*Π*) depend only on the dimensionless parameter *Π*. In this notation, *N* is the number of trapped electrons and *L* is the acceleration length.

It follows from the Maxwell equations that the accelerating bubble field scales as(3.2)simply because the bubble is free from the background electrons. This field accelerates electrons to the energy(3.3)where *ϰ* is the acceleration efficiency.

The laser energy is deposited in plasma in the form of the bubble field (3.2). We introduce the depletion factor *θ*:(3.4)where is the laser pulse energy. Analogously, energy of the electron bunch is(3.5)where *η* is the overall energy conversion efficiency. The similarity demands that(3.6)Equations (3.1)–(3.6) relate the unknown powers *α*_{i} and the *Π*-dependent functions:(3.7)The first three equations in (3.7) contain seven unknown variables and are insufficient to define all the powers *α*_{1}. This is not unexpected, because we have used the energy relations only and did not discuss details of the acceleration mechanism. Although, it is reasonable to assume that the mechanism of laser energy depletion (3.4) does not depend on *S* and to set *α*_{θ}=0. This leads immediately to *α*_{R}+*α*_{L}=−2. Consequently, one can introduce a fundamental similarity length , which follows from the energy considerations only.

Further, we mention that the acceleration rate cannot be parametrically faster than that defined by the field *E*_{bubble}. Thus, the expansion (3.6) for *ϰ* must converge for small *S*, and we have *α*_{ϰ}>0. The fastest acceleration is reached for *α*_{ϰ}=0. Substituting into equation (3.7), one obtains *α*_{γ}=−1, i.e. .

Because of its physical meaning, the energy conversion efficiency *η*≤1. Consequently, *α*_{η}≥0. The most efficient energy transformation corresponds to *α*_{η}=0. This, in turn, leads to the scaling , i.e. .

It follows from the bubble physics (Pukhov & Meyer-ter-Vehn 2002; Gordienko & Pukhov 2005) that if the aspect ratio *Π*<1, the laser pulse fits into the cavity. In this case, it is reasonable to expect that the dependence of the efficiencies *ϰ*_{0}, *θ*_{0}, *η*_{0} on *Π* is weak. For simplicity, we neglect this weak dependence. In this case, the last three equations in (3.7) claim that the dependence of *N*_{0} on *Π* is also weak. At the same time, *γ*_{0}(*Π*) is simply proportional to *Π*. Summarizing, we write(3.8)We emphasize once more that these scalings describe the optimal acceleration regime in the limit of small *S*, because we have chosen the largest physically allowed parametric dependencies for the accelerating force and the energy transformation efficiency. Even if one might find better acceleration regimes for *S*≈1, these regimes will not be scalable towards *S*≪1.

To obtain further scalings on the radius *R* and the length *L*, one needs additional information on the accelerating structure. It follows from the Maxwell equations (Gordienko & Pukhov 2005) that all the potentials scale simultaneously, i.e. the bubble potential *φ* scales together with the laser potential *a*_{0}. Because , we obtain from (3.2) that and that the dependence *R*_{0}(*Π*) is weak. Finally, we obtain for the acceleration length(3.9)and for the similarity length . The presence of these two different lengths leads to the so-called ‘ladder’ similarity as discussed in detail in Gordienko & Pukhov (2005).

Adding dimensional factors to the scalings (3.8)–(3.9) and comparing with numerical simulations we come to the formulae (1.1)–(1.3).

The acceleration length (3.9) is defined by the pulse depletion. In the bubble regime, the depletion length is always shorter than the so-called dephasing length *L*_{deph}. The dephasing length is a propagation distance where the accelerated electrons start to enter the decelerating field region. The expression for the dephasing length is derived in Gordienko & Pukhov (2005), where it was called ‘wave breaking distance’. It scales as(3.10)Thus, *L*_{deph}>*L*_{acc} if the pulse aspect ratio *Π*<1. As a result, the acceleration length (3.9) is limited by the pulse depletion.

## 4. Simulations with three-dimensional PIC code VLPL

To check the analytical scalings, we use three-dimensional Particle-in-Cell simulations with the code VLPL (Virtual Laser–Plasma Laboratory) (Pukhov 1999). In the simulations, we use a circularly polarized laser pulse with the envelope , which is incident on a plasma with uniform density *n*_{e}. We used grid steps *h*_{x}=0.07*λ*, *h*_{y}=*h*_{z}=0.5*λ*, and four particles per cell.

First, we check the basic ultra-relativistic similarity with *S*=const. We choose the laser pulse duration *τ*=8.2*π*/*ω*_{0}. The laser radius is *R*=8*λ*, where *λ*=2*πc*/*ω*_{0} is the laser wavelength. The laser pulse aspect ratio *cτ*/*R*=1 in this case.

We fix the basic similarity parameter to the value *S*^{i}=10^{−3} and perform a series of four simulations with (i) *a*_{0}^{i}=10, *n*_{e}^{i}=0.01*n*_{c}; (ii) *a*_{0}^{ii}=20, *n*_{e}^{ii}=0.02*n*_{c}; (iii) *a*_{0}^{iii}=40, *n*_{e}^{iii}=0.04*n*_{c}; (iv) *a*_{0}^{iv}=80, *n*_{e}^{iv}=0.08*n*_{c}. Assuming the laser wavelength *λ*=800 nm, one can calculate the laser pulse energies in these four cases: *W*^{i}=6 J; *W*^{ii}=24 J; *W*^{iii}=96 J; *W*^{iv}=384 J. These simulation parameters correspond to the bubble regime of electron acceleration (Pukhov & Meyer-ter-Vehn 2002), because the laser pulse duration *τ* is shorter than the relativistic plasma period . We let the laser pulses propagate the distance *L*_{b}^{i}=1000*λ* through plasma in the all four cases. At this distance, the laser pulses are depleted, the acceleration ceases and the wave breaks.

Figure 1(i)–(iv) shows evolution of electron energy spectra for these four cases. One sees that the energy spectra evolve quite similarly. Several common features can be identified. First, a monoenergetic peak appears after the acceleration distance *L*≈200*λ*. Later, after the propagation distance *L*≈600*λ*, the single monoenergetic peak splits into two peaks. One peak continues the acceleration towards higher energies, while another peak decelerates and finally disappears. Comparing the axes scales in figure 1, we conclude that the scalings (2.9) hold with a good accuracy.

Now, we are going to check the general scalings (3.8)–(3.9) for the variable *S*-parameter. We choose the laser amplitude *a*_{0}^{v}=80 and the plasma density *n*_{e}^{v}=0.02*n*_{c}. This corresponds to *S*^{v}=2.5×10^{−4} and the laser energy *W*^{v}≈1.5 kJ. In this case, the initial laser radius and duration must be increased by the factor . Thus, we use the laser pulse with *R*^{v}=16*λ* and *τ*^{v}=16.2*π*/ω_{0}. This case gives the pure density scaling when compared with the case (iv), or the pure laser amplitude scaling when compared with the case (ii). We let the laser run *L*_{acc}^{v}=8000*λ* through the plasma. At this distance, the energy of the laser pulse is completely depleted and the wave breaks. The change of the depletion length coincides with the scaling (3.9).

The electron spectrum evolution obtained in this simulation is shown in figure 2. The energy of the monoenergetic peak continuously grows up to some 12 GeV at the end. Between the control points, where the spectra in figure 2 have been taken, the laser pulse propagated the distance *L*=800*λ*. This distance is *S*^{i}/*S*^{v}=4 times larger than that in the cases (i)–(iv). One sees that the first five electron spectra in figure 2 are similar to those in figure 1. However, the last four spectra in figure 2 are new. This corresponds to the ladder similarity. The geometrical similarity is illustrated in figure 3.

## 5. Conclusions

In this section, we recapitulate the most important results of the developed similarity theory for ultra-relativistic laser–plasmas and provide some simple ‘engineering’ scalings for electron acceleration in the bubble regime.

Ultra-relativistic laser–plasma interactions with different

*a*_{0}and*n*_{e}/*n*_{c}are similar as soon as the similarity parameter*S*=*n*_{e}/*a*_{0}*n*_{c}=const. for these interactions. In this case, electron energies scale as*E*_{e}∝*a*_{0}and the number of accelerated electrons also scale as*N*_{tr}∝*a*_{0}.The present work states that the bubble acceleration regime is stable and scalable.

Starting with the Maxwell equations on the fields and the relativistic Vlasov equation on the electron distribution function we were able to derive ‘engineering’ scalings for the electron acceleration.

Here, we list the most important scalings once again. The similarity theory tells us that the optimal radius *R* of the laser pulse is(5.1)For the acceleration to be efficient and to generate structured (quasi-monoenergetic) electron energy spectra, the pulse duration *τ* must be

Looking at the scalings (1.1)–(1.2) one mentions that the plasma density does not appear explicitly. Rather, everything is defined by the laser pulse parameters: the power and the duration *τ*. This is the consequence of the pulse radius scaling (5.1) corresponding to the bubble regime. With a given laser pulse, the bubble regime is achievable only in some density range *n*_{1}<*n*_{e}<*n*_{2}. The lower density limit *n*_{1} is defined by the condition that the laser pulse is still ultra-relativistic, *a*_{1}>1 when focused to the corresponding focal spot . The upper density limit is defined by the condition that the corresponding focal spot equals the pulse duration: . This density range can be expressed via the laser parameters:(5.2)The density range (5.2) exists only when the laser power is large enough:(5.3)The condition (5.3) can be considered as the threshold power needed to reach the bubble regime for a laser pulse with the given duration *τ*. In practical units, this threshold power is(5.4)

In conclusion, we hope that the developed non-trivial similarity theory will help to make another step towards a practical realization of a compact high-energy laser–plasma electron accelerator.

## Acknowledgments

This work was supported in parts by the Transregio project TR-18 of DFG (Germany) and by RFFI 04-02-16972, NSH-2045.2003.2 (Russia).

## Footnotes

One contribution of 15 to a Discussion Meeting Issue ‘Laser-driven particle accelerators: new sources of energetic particles and radiation’.

- © 2006 The Royal Society