Particle acceleration with the axial electric field of a TEMlO mode laser beam

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Introduction
The high fields which exist in the focus region of high power lasers cannot be used directly for acceleration of unbound particles in vacuum. For a single homogeneous plane wave, the particle will always lag in phase if it is not highly relativistic and moving in parallel with the wave. But in this latter case, the fields are transverse to its direction of propagation, and thus no energy exchange occurs. Extending these thoughts to the superposition of plane waves, we cite A. Μ. Sessler [1]: The field pattem produced by any array of optical elements, provided one is not near a surface (more than a wavelength away) and not in a medium, is simply a superposition of plane waves. It is not very difficult to generalize the considerations made for a single plane wave and conclude that for a rela tivistic particle, which moves with essentially constant speed and in a straight line, there is no net acceleration.
To obtain synchronism, the phase velocity of the wave has to be slowed down to the particle velocity, i.e. the component of the wave vector k in the direction of particle motion kv has to be greater than the free space wave number, kχ>k. Due to Jd = &| + k?±, this is possible with an imaginary transverse component of k, k=]ct. These evanescent waves decay perpendicular to their direction of propagation. They exist in the immediate vicinity of dielectric or grating structures. For highly relativistic particles, this imaginary k1 has to be only small, but still the necessary distance between these surfaces and the particle beam may become inconveniently narrow.
In this paper, we will consider a laser beam propagating in z-direction, acting on a single highly relativistic charged particle propagating in parallel to it. Even though it is not guided near a surface or dielectric, such a beam has some similarity to evanescent waves: it decays exponentially in the transverse direction, it has axial fields, and there exist even straight lines parallel to the axis where vfj < c. What is the effect of the small axial electric field that exists in the laser beam on the particle?

The field of a Gaussian beam wave
The transverse electric field of a TEM. mode of a Gaussian beam wave with ψ cos lφ-dependence and linear polarization in ^-direction is given in cylindrical coordinates (p, φ, and z) by [2] = 2Ï» ∕√2-≤-∖' l!r, (2(^∖2) coslφe~lφ where w0 denotes the beam waist radius and w(z) = w0 is the beam shape with the 'confocal length' L = kw2 0, the meaning of L being the length of a confocal optical resonator supporting this beam. Lj0 denotes the generalized Laguerre poly nomial [3], and j2 = -1. I and p are the azimuthal and radial order of the mode, respective ly. The approximation is valid for infinite apertures and large spot sizes, Λ∕w0<< 1.
In Fig. 1, the transverse electric field of the TEMi0 m°de, polarized in ^-direction and with cos lφ-dependence, is plotted. The transverse coordinates are normalized to w; the length of the arrows is proportional to the electric field.

Fig. 1:
Transverse electric field of a TEM10 mode Due to the transverse variation of the transverse fields, an axial field must exist, i.e. there has to be an axial field near the center of Fig. 1 for the electric field lines to be closed. This follows directly from Maxwell's equations where k = ω|c and Z0= 377 Ω. If we write V = V x+wzJand assume again a Iinearily polarized wave, Ex = Z0By, we have The last term in each of these two equations gives rise to axial field components. The TEMψ Gaussian modes are thus not exactly 'transverse electromagnetic'; the name 'quasi TEM mode' is more appropriate. The axial fields can in good approximation be computed from the transverse variations of the transverse fields, as long as the ratio λ∕w0 is small. The axial fields are of first order in λ|w0 while correction terms for the transverse fields in the above equations are of second order.
The normalized transverse and axial electric fields of the TEM10 mode are plotted in Fig. 2 for ʃ = 0 (Note the different scales!). While the transverse field has a zero in the center, the axial field has a significant peak there. E0 is calculated from the laser beam power P by the formula πwo The ratio of the maximum axial electric field to the maximum transverse field is √r2 exp(0.5)/(kw).

Fig. 2:
Normalized transverse (solid line) and axial electric fields (dashed line) of the TEM10 mode

Lorentz transformation of the fields
The difference in the scales of the two curves in Fig. 2, kw, is usually a large number.
For a beam waist radius of some hundred λ, e.g., it is in the order of 103. The axial fields are by this factor smaller than the transverse fields. But they become important for highly rela tivistic particles due to the Lorentz transformation of the fields.
Given the fields in the laboratory frame (unprimed quantities) and the particle velocity βc and energy γ = (1 -β2Yi,'∖ we get in the rest frame of the particle (primed quantities) [4]:

ZZ ZZ
In our case of a Iinearily polarized wave travelling with the particle, this becomes for the electric field. For 1 GeV electrons, e.g., the transverse field will be decreased by a fac tor of about 4000. Thus the axial field plays in fact a dominant role.

Phase relations
Two effects lead to non-synchronism between particle and wave. First there is the devia tion of the particle velocity v from c. Let us define X≡ l-ɪ . c For a highly relativistic particle, we obtain For a 1 GeV electron, e.g., X would be in the order of 10 7.
Because both X and Y are generally positive quantities, the particle will lag in phase behind the wave, and there will be in fact no "net" acceleration, in full agreement with the statement by Sessler cited above. But on how long a distance could the particle be kept in phase? In the rest frame of the particle, the laser frequency would appear Doppler downshift ed by the factor ω √2X The nominator describes the non-relativistic Doppler shift, while the denominator accounts for the change of the proper time. After half a period, the particle would slip out of phase, this is in the particle frame Back in the laboratory frame, this time is where T is the original RF period. For the TEM10 mode, it turns out that a highly relativistic particle (X« T) can travel just the confocal length L during this time, independent of the waist radius.

Achievable energy gain
The maximum energy the particle could gain on its path is given by eVιu, where e is the charge of the particle and the total accelerating voltage is given by the integral j-c vdζ.
Focc is plotted versus z in Fig. 3 for the TEM10 mode and a highly relativistic particle moving along the axis. The maximum occurs always at z= L/2, as stated above. The maximum value is √ Z0P ∕π = E0w0∕∖r2. For a given laser power, it is indepedent of the parameters of the opti cal beam. For 1 TW, e.g., and a spotsize of 3 10~6m2 (w0= 100A, A= 10μm), E0 would be 15.5 GV/m. The maximum value of Ez would be 25 MV/m, and the maximum of Kacc would be 11 MV.

Fig. 3:
The total accelerating voltage Vacc for a highly relativistic particle moving along the axis in a TEM10-mode versus z.
For other particle trajectories and other modes, one would have sHghtly different formu lae, but the result never seems to exceed significantly the 'single-cycle acceleration', √ Z0P∕π which is small even for very high laser powers.
In [5], similar results were obtained for the single-cycle electron acceleration for elec trons passing a plane wave in an oblique angle.
The proportionality between the acceleration and the square root of the laser power (to the laser amplitude) is a remarkable difference between the acceleration scheme considered here and the one used in 'far field accelerators' [6] where the acceleration is directly propor tional to the laser power. Below a certain threshold power (which might be very high), the effect of the axial fields is greater than of the transverse fields and must not be neglected.

Refocussing
The low energy gain achieved by this scheme could possibly be increased by using a periodic structure to refocus the optical beam. The scheme sketched in Fig. 4 is not meant to be a technical solution, but just to illustrate the idea of refocussing; it might as well be some mirror arrangement. The phase shift in the lenses would have to be such that the parti cle, moving along the axis, is in the accelerating phase in each cell. In its periodicity, this structure resembles again a conventional linac. Keeping in mind that there are two types of 'near field', one in which evanescent waves exist (which normally extends only the order of λ from the structure), and a second one where the field might already be described by a super position of homogeneous plane waves (characterized by a dependence of the radiation char acteristic on the distance), the proposed scheme clearly belongs to the group of 'near field accelerators'.  Fig. 4: Refocussing of the beam by a set of lenses

Conclusions
• For a highly relativistic particle travelling along with a laser beam, the transverse fields Lorentz transform down by a factor (2y)^1, thus making the small axial fields impor tant.
• Using these axial fields in a TEM10 mode, the maximum achievable energy gain of a highly relativistic particle travelling on the optical axis is e√ Z0P∕π. This value is obtained on the 'confocal length' L and is independent of the spot size w0. For P = 1 TW and highly relativistic electrons, this would be approximately 11 MeV.
• The effect is of first order in √rP, and thus has to be considered when second order effects are investigated, for example in 'far field accelerators'.
• For the time being the axial field components of laser beams do not seem to lead to competitive particle acceleration for practically available laser powers.