I About Direct Laser Acceleration

redirmigician
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How does this Lawson–Woodward theorem work. I read on the wiki that the particles cannot be accelerated by lasers. But I do see acceleration of electrons with free space. I wonder how this is done.
https://rdcu.be/c0fRw
http://dx.doi.org/10.1103/PhysRevAccelBeams.19.021303
In addition, I have seen some designs that utilize the center-focusing force in the ion channel to realize the design of short-wavelength undulators, and combine them with plasma accelerators to realize compact free-electron lasers. Can it be achieved in direct laser acceleration, generating a center-pointing focused field as a short-wavelength undulator.
 
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The short internet blurb I read on the internet said no combination of far fields can accelerate a charged particle. This seems a reasonable statement since far field EM fields are predominantly transverse. There are no longitudinal electric fields to affect the acceleration.

laser induced particle accelerators use periodic structures like diffraction gratings to produce the needed longitudinal electric fields near the grating. This is analogous to the RF resonators used in linacs.
 
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I'm following this paper by Kitaev on SL(2,R) representations and I'm having a problem in the normalization of the continuous eigenfunctions (eqs. (67)-(70)), which satisfy \langle f_s | f_{s'} \rangle = \int_{0}^{1} \frac{2}{(1-u)^2} f_s(u)^* f_{s'}(u) \, du. \tag{67} The singular contribution of the integral arises at the endpoint u=1 of the integral, and in the limit u \to 1, the function f_s(u) takes on the form f_s(u) \approx a_s (1-u)^{1/2 + i s} + a_s^* (1-u)^{1/2 - i s}. \tag{70}...
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