# Direct Electron Acceleration with Radially Polarized Laser Beams

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Radially Polarized Laser Beams

_{p}

_{1}beam—is characterized by p + 1 rotationally symmetric concentric rings [24,25]. There exists several ways to produce RPLBs (see Appendix A). In particular, the lowest-order RPLB (TM

_{01}) is often represented as a superposition of two Hermite-Gaussian modes with orders (0, 1) and (1, 0) [26]. As the name says, the electric field oscillations of RPLBs are radially polarized (in the weak focusing limit). This feature confers to this beam family very particular properties. The most striking characteristic is the central dark intensity region that turns into a bright spot of sub-wavelength diameter under tight focusing conditions [27–29]. This behaviour is explained by the particular beam symmetry that favors a strong axial longitudinal electric field component [30].

_{0}t + jϕ

_{0})], where ω

_{0}is the frequency of maximum spectral amplitude and ϕ

_{0}is a constant phase delay), the field components of the lowest-order RPLB can be written in the very compact form [25]:

_{0}is a normalization constant, k

_{0}= ω

_{0}/c is the wave number, ${j}_{n}({k}_{0}\tilde{R})$ is the order-n spherical Bessel function of the first kind, and ${P}_{2}(\mathrm{cos}\tilde{\theta})=\frac{1}{4}[1+3\mathrm{cos}(2\tilde{\theta})]$ is the Legendre polynomial of degree 2. The complex coordinates $(\tilde{R},\tilde{\theta})$ are defined as $\tilde{R}={[{x}^{2}+{y}^{2}+{(z+j{z}_{0})}^{2}]}^{1/2}$ and $\tilde{\theta}=(z+j{z}_{0})/\tilde{R}$, respectively. The confocal parameter z

_{0}is related to the Gaussian beam waist size w

_{0}by ${z}_{0}={w}_{0}{[1+{(\frac{1}{2}{k}_{0}{w}_{0})}^{2}]}^{1/2}$. All other field components $\left({\tilde{E}}_{\theta},{\tilde{B}}_{r},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{\tilde{B}}_{z}\right)$ are zero. The beam waist corresponds to z = 0.

_{0}z

_{0}≫ 1), the electromagnetic field components can be expanded into infinite series [30,32,33] whose leading terms are:

_{R}is the Rayleigh distance [34]. In the paraxial limit, the parameter A

_{0}is related to the peak value E

_{0}(in V/m) of the ${\tilde{E}}_{r}^{(0)}$ field component by ${A}_{0}=-j\left(\sqrt{2}/4\right){k}_{0}^{3}{w}_{0}^{3}\mathrm{exp}(1/2){E}_{0}$.

_{0}= μ

_{0}c is the impedance of free space. In the paraxial limit:

_{0}z

_{0}= 0, the power defined at Equation (4) is identically zero. It corresponds to the case where an RPLB is focused uniformly over 4π steradians [37,38]. In an ideal setup, the inward and outward energy flows would then be perfectly balanced, resulting in a null Poynting vector.

_{0}z

_{0}≫ 1), RPLBs appear as the characteristic doughnut-shape profile shown in Figure 1(a). In this case, most of the electric energy is concentrated in the transverse component (W

_{e}∝ |E|

^{2}≃ |E

_{r}|

^{2}). Nevertheless, as the beam spot size is decreased, part of it is transferred to the longitudinal component E

_{z}that is maximum at r = 0. The dark center then gradually disappears and, eventually, the longitudinal component of the electric field dominates [see Figure 1(b)–(c)]. In the limit of extreme focusing (k

_{0}z

_{0}→ 0), most of the electric energy is concentrated at the center of the beam that now appears as a bright symmetric spot whose diameter is smaller than the dominant laser wavelength [see Figure 1(d)]. This particular transition from paraxial to subwavelength focusing, where rotational symmetry of the beam profile is preserved, was predicted [25,28,30,33,39] and observed [27,40–43] by different authors. Because of their field symmetries and strong axial longitudinal electric field component, RPLBs are appealing for electron acceleration.

## 3. Electron Acceleration with Radially Polarized Laser Beams

#### 3.1. Free-Space Electron Acceleration in Laser Beams

**p**= γm

**v**, and W = γmc

^{2}, respectively, with γ = (1 − v

^{2}/c

^{2})

^{−1}

^{/}

^{2}. E and B are the electromagnetic field components of the laser beam [e.g., from Equations (1) or (2)]. For a non-relativistic motion in a weak harmonic electric field of angular frequency ω

_{0}:

_{0}λ

_{0}/w

_{0}), including it into calculations changes the result from vanishingly small to considerable acceleration [54,55,57]. This can be explained by the fact that an electron moving longitudinally can possibly stay in phase with the longitudinal electric field over longer distances.

#### 3.2. Direct Longitudinal Electron Acceleration with Radially Polarized Laser Beams

_{0}≫ λ

_{0}and p = 0, for simplicity. Because of the velocity mismatch, a particle moving in the beam will inevitably drift with respect to the carrier oscillations. Within a given time interval Δt, the net energy gain is optimal if that drift is less than half the laser wavelength, i.e., if Δt|υ

_{phase}− υ

_{z}| ≲ λ

_{0}/2, where υ

_{z}is the particle’s longitudinal velocity. During that same time interval, the particle has traveled over a distance

_{z}≃ c), the dephasing length is:

_{dph}≃ (πz

_{R})/2]. This sets a fundamental limit on the energy that can be transferred from the laser field to particles (see also [14]).

_{i}to z

_{f}is

_{z}is the complex envelope of the field component given at Equation (2b). Above, −e is the electron charge, ${E}_{z0}=[2\sqrt{2}\mathrm{exp}(1/2)/({k}_{0}{w}_{0})]{E}_{0}$ is the amplitude of the longitudinal electric field component, A(z) = [w

_{0}/w(z)]

^{2}, and B(z) = z

_{R}/R(z). Respectively, w(z) = w

_{0}[1 + (z/z

_{R})

^{2}]

^{1}

^{/}

^{2}and $R(z)=z+{z}_{R}^{2}/z$ are the beam waist size and radius of curvature of the wavefront at z. According to Equation (11b), there are two situations where the energy gain is optimal. One is when the acceleration occurs between two axial positions where the wavefront curvature radius is minimum but of opposite sign. This happens when [z

_{i}, z

_{f}] = [−z

_{R}, z

_{R}]. The other situation is when acceleration takes place between two positions where the beam spot size is minimal and infinite, respectively. This corresponds to [z

_{i}, z

_{f}] = [0, ±∞]. For these two cases ΔW = ez

_{R}E

_{z}

_{0}. This defines the following theoretical limit to the energy gain (see also [14,67,68]):

_{R}≲ z ≲ z

_{R}. However, the proposed schemes require optical materials close to the high field intensity regions. Such a configuration is inevitably limited by the destruction of the device and does not take full advantage of the high peak power delivered by actual ultraintense lasers.

_{lim}comes from acceleration outside the Rayleigh zone, i.e., between z

_{R}and ∞. In this region, the longitudinal electric field is weaker than at focus but the Gouy phase evolves much more slowly (which considerably increases the dephasing time).

_{lim}represents ideal acceleration scenarios. Rigorous numerical simulations show that, instead, the maximum energy gain is much less unless the beam parameters are carefully optimized [67,68,73–75]. We will see in Section 4 that exceptional conditions are provided by ultrashort and tightly focused pulses.

#### 3.3. Threshold for Sub-Cycle Acceleration and Attosecond Bunching

_{0}is split into radial (a

_{r}) and axial (a

_{z}) components. These two new parameters are defined as follows:

_{r}|

_{peak}and | Ẽ

_{z}|

_{peak}represent the peak values of the field envelope of the radial and longitudinal components, respectively.

## 4. Acceleration by Ultrashort and Tightly Focused Radially Polarized Laser Pulses

_{0}z

_{0}≫ 1) [18,67,68], modeling the dynamics under nonparaxial and ultrashort pulse conditions is now necessary to bridge the gap between theory and current experiments (see Section 5).

_{01}beams [see Equations (1)] with different frequencies but identical confocal parameter z

_{0}[77]. If we weight each frequency component by the spectral amplitude function F (ω) in Fourier space and then calculate the inverse Fourier transform, we obtain the following field components in complex notation for a TM

_{01}pulsed beam [75,77]:

_{0}), the fields given in Equations (1) for a TM

_{01}beam of frequency ω

_{0}are recovered [77].

_{0}is the frequency of maximum spectral amplitude. Γ(s + 1) is the Gamma function, and H(ω) is the Heaviside step function that ensures the absence of components with negative frequencies. The constant s is a real and positive parameter that controls the shape of the spectrum and the corresponding pulse. The inverse Fourier transform of Equation (17) is

_{01}pulse. Note that as s increases, the width of the spectrum F(ω) decreases while the pulse duration increases. The spectrum F(ω) and the corresponding temporal profile of the E

_{z}field component are shown in Figure 5 for different values of s.

_{01}pulsed beam was investigated by some of us in a recent contribution [75]. More specifically, the laser power dependence of the maximum final kinetic energy that an electron initially at rest at r = 0 can acquire was studied for different pulse durations and degrees of focusing. The maximum energy gain, denoted ΔW

_{max}, is calculated numerically by optimizing the electron’s initial position on the optical axis and the laser pulse phase such that the final kinetic energy of the particle is maximal (see also [68]).

_{max}with the laser power for different combinations of k

_{0}z

_{0}and s. Figure 6(a), in which ΔW

_{max}is expressed as a fraction of the theoretical energy gain limit ΔW

_{lim}[see Equation (12)] shows that for constant values of s, the power above which significant acceleration occurs is greatly reduced as k

_{0}z

_{0}decreases, i.e., as the focusing is made tighter. According to Figure 6(b), MeV energy gains may be reached under tight focusing conditions (k

_{0}z

_{0}~ 1) with laser peak powers as low as 15 GW. In contrast, previous works based on the paraxial approximation suggested that powers three orders of magnitude higher would be required to reach MeV kinetic energies [68]. At high peak power, Figure 6(a) shows that shorter pulses yield a more efficient acceleration, with a ratio ΔW

_{max}/ΔW

_{lim}reaching 80% for single-cycle (s = 1) pulses. This is mainly a consequence of the fact that shorter pulses allow the electron to move close to the pulse peak; in longer pulses, the electron is trapped and accelerated by the front edge of the pulse. As explained in [75], where additional details about the results discussed in this paragraph may be found, the data shown in Figure 6 is completely independent of the dominant wavelength λ

_{0}of the TM

_{01}laser pulse.

_{0}z

_{0}= 100 and s = 100 is shown alone in Figure 6(c). At low laser power, ΔW

_{max}scales as P

^{2}. This is a consequence of the ponderomotive force, which is proportional to ${F}_{p}\sim {A}_{0}^{2}\sim P$ and consequently leads to an energy gain that increases as P

^{2}. As the power is further increased, the dynamics undergoes a transition to the relativistic regime where sub-cycle acceleration begins. The maximum final kinetic energy of the electron increases rapidly since it is now allowed to copropagate with the laser pulse over longer distances. Eventually, the energy that the electron is able to extract from the pulse saturates toward a constant fraction of the theoretical energy gain limit ΔW

_{lim}[see Figure 6(a)]. This marks the onset of the ultrarelativistic regime, which is characterized by the scaling relation ΔW

_{max}~ P

^{1}

^{/}

^{2}, in agreement with Equation (12). For ultrashort pulses (s ≲ 50), ΔW

_{max}does not scale as P

^{2}in the nonrelativistic regime. This shows that the ponderomotive force model is not appropriate to describe the interaction of electrons with pulses of a few-optical-cycle duration.

## 5. Experimental Observation of Electron Acceleration with Tightly Focused Radially Polarized Laser Beams

#### 5.1. Method for Generating Tightly Focused Ultrashort RPLPs

_{00}) transverse mode at a repetition rate of 100 Hz [80]. The central wavelength is 1.8 μm and the laser pulse energy stability is of the order of 2.5% rms. The s parameter associated with these IR pulses is approximately s = 125.

_{z}|

^{2}≪ |E

_{r}|

^{2}). To reach the longitudinal acceleration threshold $({a}_{z}^{2}\sim 1)$, the few-cycle RPLP was focused to a sub-wavelength focal spot with an on-axis high NA (0.7) parabolic mirror.

_{0}/4 at λ

_{0}= 675 nm with a diode laser source. Performances were extrapolated to 1.8 μm. The beam waist is estimated to w

_{0}≈ 0.6λ

_{0}, corresponding to k

_{0}z

_{0}≈ 8 and a transverse beam profile somewhere in between those shown in Figure 1(b) and 1(c) (|E

_{z}|

^{2}≳ |Er|

^{2}). The energy measured after the parabola is 550 μJ per pulse. This results in a peak power of 36 GW and an intensity of 7.2 × 10

^{17}W/cm

^{2}, approximately. According to Figure 2, this corresponds to a normalized parameter ${a}_{z}^{2}\simeq 1.7$. However, a comparison with Figure 6 suggests that this value of ${a}_{z}^{2}$ overestimates the real strength of the interaction. In the light of the predictions made in Section 4 and the experimental results presented below, it is most likely that electron acceleration proceeded in the early phase of the relativistic regime where the longitudinal electric field is dominant but too weak to create attosecond electron pulses.

#### 5.2. Electron Acceleration Measurements

^{17}cm

^{−3}in the focal region at the peak of the laser pulse.

_{0}was fluctuating randomly between −π and π from shot to shot. The average transverse profile of the electron beam that propagates through the hole of the reflective mirror is shown in Figure 9(a). The measured divergence of the electron beam, from the focal plane to the detector, is 37 mrad (half-angle at 1/e). This includes probable space-charge effects and scattering.

_{01}mode) that generates a maximum longitudinal field in the focal plane. With a horizontal input polarization (0 and 180°), the output beam has an azimuthal polarization (TE

_{01}mode), without a longitudinal field in the focal plane. Figure 9(b) shows the electron signal measured by the PMT while shifting gradually from a TE

_{01}mode with a purely transverse electric field to a TM

_{01}mode with a dominant longitudinal electric field. It is observed that the signal peaks when the effective mode approaches the ideal TM

_{01}mode, where the longitudinal electric field is the strongest. On the other hand, no signal is observed with the TE

_{01}or TEM

_{00}mode (when removing the PSC). This is consistent with the RPLB acceleration scheme described in Section 3 where the electrons are accelerated along the propagation axis by the longitudinal electric field.

^{6}electrons per laser shot (~ 0.2 pC per electron pulse). A linear dependence between the number of charges per shot and the pressure is observed between 50 mTorr and 1 Torr. This suggests that electron pulses with considerably more charges could be produced at higher pressures, or with solid-density targets. However, it should be noted that in these conditions space-charge effects and scattering will also be more important.

_{z}/c ≃ 0.3, we can thus expect a duration in the 20 fs range, close to the beam waist. We recall that non-relativistic electron pulses will broaden considerably during propagation due to velocity dispersion and space-charge effects.

## 6. Discussion and Conclusions

## Appendix

## A. Challenges for Producing High-Power Radially Polarized Laser Beams

_{01}and TE

_{01}modes, a laser cavity can be made to selectively oscillate in only one of the two by introducing a birefringent [84–86] or diffractive element [87]. Outside the cavity, the complementary TM

_{01}and TE

_{01}modes can be converted into each other by rotating the transverse electric field component by 90° [84,85,88]. Amplification of radially polarized laser beams is also possible using specially cut Ti:sapphire crystals [89]. This offers the possibility to create dedicated ultrafast and ultra-intense laser systems tailored to specific needs.

## B. Modelling Tightly Focused Ultrafast Laser Beams in Vacuum

## Acknowledgments

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**Figure 1.**Transverse distribution of the average electric energy density W

_{e}= ϵ

_{0}|E|

^{2}/2 of a focused RPLB at beam waist. The energy density associated with the individual electric field components is shown to emphasize the contribution of the longitudinal field when the beam spot size is comparable to or below the wavelength. In (

**a**) k

_{0}z

_{0}= 350; (

**b**) k

_{0}z

_{0}= 10; (

**c**) k

_{0}z

_{0}= 5; and (

**d**) k

_{0}z

_{0}= 1. Insets show the intensity distribution (the scale is that of the corresponding transverse coordinate axis).

**Figure 2.**Overview of electron acceleration in RPLBs. The far right of the graph corresponds to the paraxial limit, where the dynamics is dominated by the transverse electromagnetic field. In this limit, the sub-cycle longitudinal acceleration regime is reached only if the average beam power is in the PW range (see also [18,67]). Alternatively, the beam can be tightly focused to increase the peak intensity and lower the threshold down to the TWs and GWs [68,75]. The sub-cycle regime characterized by ${a}_{z}^{2}>1$ can itself be split into two sub-regimes corresponding to cases where the strength of the radial component is above threshold [Sub-cycle (1)] or not [Sub-cycle (2)].

**Figure 3.**Single-electron model of longitudinal acceleration in a 12-fs (FWHM) RPLP (k

_{0}z

_{0}~ 500 with λ

_{0}= 800 nm). The electron is initially at rest at z = 0. (

**a**) At low intensity $({a}_{z}^{2}=0.1)$, the electron experiences a quasi-harmonic motion, with slightly longer excursions in accelerating half field-cycles; (

**b**) At the longitudinal acceleration threshold $({a}_{z}^{2}=0.1)$, this phenomenon is stronger but the final kinetic energy remains relatively small (in the keV range); (

**c**) Above threshold $({a}_{z}^{2}=0.1)$, the electron escapes the laser pulse with a relativistic longitudinal momentum. During the sub-cycle acceleration represented in (

**c**), the electron stays locked to the phase of the pulse carrier.

**Figure 4.**In the sub-cycle regime, a collection of electrons initially at rest at the waist of an ultra-intense 12-fs (FWHM) RPLP (λ

_{0}= 800 nm) is bunched to form an attosecond pulse. The initial electron positions (•) followed a 100-nm spherical Gaussian distribution centered at (r, z) = (0, 0). Here, a magnified view reveals the extreme longitudinal compression experienced by the electrons during sub-cycle acceleration. The position of the leading edge of the accelerated electron distributions is about 6 mm away from beam waist. It was translated along z for a direct comparison of the relative durations. The size of the characters used to represent the particles gives the impression that the electron pulses are longer than they really are. For (+), we estimate that the duration along the longitudinal axis is as short as 170 zs. Simulation parameters were the following: (×) P = 100 TW (k

_{0}z

_{0}≃ 280, $({k}_{0}{z}_{0}\simeq 280,\phantom{\rule{0.2em}{0ex}}{a}_{z}^{2}\simeq 4.7)$), (+) P = 1 PW $({k}_{0}{z}_{0}\simeq 850,\phantom{\rule{0.2em}{0ex}}{a}_{z}^{2}\simeq 5.1)$. The carrier phase ϕ

_{0}was effectively optimized for the shortest durations.

**Figure 5.**(

**a**) Poisson spectrum F(ω) for different values of s with ϕ

_{0}= 0; (

**b**) Temporal profile of the on-axis longitudinal electric field component at z = 0 of the corresponding TM

_{01}pulsed beam with λ

_{0}= 0.8 μm and k

_{0}z

_{0}= 10.

**Figure 6.**Maximum (

**a**) normalized and (

**b**) absolute final energy gain of an electron initially at rest on the optical axis versus the laser pulse power for different combinations of k

_{0}z

_{0}and s; (

**c**) A close-up on the case k

_{0}z

_{0}= 100 and s = 100 illustrates the transition from the non-relativistic (NR) to the relativistic (R) and ultra-relativistic (UR) dynamical regimes discussed in Table 1.

**Figure 7.**Method to generate the lowest-order RPLB. (

**a**) Transverse profile of the incident linearly polarized Gaussian (TEM

_{00}) beam; (

**b**) The polarization state converter (PSC) composed of 4 sections of IR achromatic half-wave plates with the fast axis orientation of 0° (top), +45° (right), +90° (bottom), −45° (left); (

**c**) The near field beam profile and relative electric field direction at the output of the PSC; (

**d**) The far field image of the weakly focused RPLB. (

**a**) to (

**c**) are theoretical representations while (

**d**) is an experimental measurement.

**Figure 8.**RPLP electron acceleration and measurement set-up. The 25-mm diameter RPLP is reflected off a mirror placed at 45° and sent toward the focusing on-axis parabola. The resulting longitudinal field at focus is strong enough to ionize the oxygen molecules from the ambient gas (800 mTorr) and accelerate the photo-electrons along the propagation axis. The electron beam created passes through a small aperture (4 mm) in the reflecting mirror and is collected by different detectors positioned 10 cm away from the focal plane.

**Figure 9.**Experimental observation of direct longitudinal electron acceleration in tightly-focused RPLPs. (

**a**) Electron beam profile averaged over 10

^{4}shots as observed by the camera (the circular shadow is the projected small aperture of the reflecting mirror); (

**b**) Electron signal measurement from the photomultiplier tube while moving gradually from the TE

_{01}mode (0° and 180°) to the TM

_{01}mode (90°). The normalized longitudinal electric field intensity is also indicated; (

**c**)–(

**d**) Electron spectrum measurement through the magnetic coil (average of 10

^{4}shots); (

**c**) Signal with the magnetic coil residual field of 2mT; (

**d**) Signal with a magnetic field of 10mT. Both measurements confirm the production of 23-keV electrons. Figure adapted from [20].

**Table 1.**The three dynamical regimes associated with the motion of a charged particle in a laser beam in terms of the normalized field parameter a

_{0}= (q/mcω

_{0})E

_{0}.

Regime | ${a}_{0}^{2}$ |
---|---|

Non-relativistic | ≪ 1 |

Relativistic | ~ 1 |

Ultra-relativistic | ≫ 1 |

**Table 2.**Comparison between different electron acceleration scenarios in RPLBs. In the first column is the interaction range, followed by the corresponding change in energy calculated with Equation (11b), and finally a short description (scenario) with references.

[z_{i}, z_{f}] | $\mathrm{\Delta}W{|}_{{z}_{i}}^{{z}_{f}}$ | Scenario |
---|---|---|

[−∞, ∞] | 0 | Lawson-Woodward [14] |

[−z_{R}, z_{R}] | ΔW_{lim} | Limited interaction [14,15,69] |

[0, ∞] | ΔW_{lim} | Single pulse [19,64,72] |

[z_{R}, ∞] | ΔW_{lim}/2 | Pump-probe [16,18,67] |

© 2013 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Varin, C.; Payeur, S.; Marceau, V.; Fourmaux, S.; April, A.; Schmidt, B.; Fortin, P.-L.; Thiré, N.; Brabec, T.; Légaré, F.;
et al. Direct Electron Acceleration with Radially Polarized Laser Beams. *Appl. Sci.* **2013**, *3*, 70-93.
https://doi.org/10.3390/app3010070

**AMA Style**

Varin C, Payeur S, Marceau V, Fourmaux S, April A, Schmidt B, Fortin P-L, Thiré N, Brabec T, Légaré F,
et al. Direct Electron Acceleration with Radially Polarized Laser Beams. *Applied Sciences*. 2013; 3(1):70-93.
https://doi.org/10.3390/app3010070

**Chicago/Turabian Style**

Varin, Charles, Stéphane Payeur, Vincent Marceau, Sylvain Fourmaux, Alexandre April, Bruno Schmidt, Pierre-Louis Fortin, Nicolas Thiré, Thomas Brabec, François Légaré,
and et al. 2013. "Direct Electron Acceleration with Radially Polarized Laser Beams" *Applied Sciences* 3, no. 1: 70-93.
https://doi.org/10.3390/app3010070