Exact Solution of the Landau-Lifshitz Equation in a Plane Wave

Author of paper: A Di Piazza

Citation: A Di Piazza, Lett Math Phys 83 (2008)

Link to paper: https://link.springer.com/article/10.1007/s11005-008-0228-9

Complementary reading: Link to Fabien Niel's PhD thesis: https://theses.hal.science/tel-03714678/

Abstract: The Landau–Lifshitz (The Classical Theory of Fields. Elsevier, Oxford 1975) form of the Lorentz–Abraham–Dirac equation in the presence of a plane wave of arbitrary shape and polarization is solved exactly and in closed form. The explicit solution is presented in the particular, paradigmatic cases of a constant crossed field and of a monochromatic wave with circular and with linear polarization.

Repository by: Bernardo Barbosa and Óscar Amaro

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Tricks:

• structure of PW (in particular, the orthogonality of the f_j^munu functions) leads to a finite number of Piccard iterations

Exercises:

• prove eq 7
• prove higher order terms from Piccard series of eq 10 are identically zero
• prove eq 11
• prove eq 11 satisfies (u u) = 1
• prove eq 11 reduces to LL in the limit alpha->0
• confirm estimates of classical and quantum parameters before conclusions

FAQ:

• Why do we need to decompose eq 3 into 2 terms if the $\psi$ s can already be arbitrary functions of phase $\phi$? The $\psi$ s are scalars. The two terms represent different polarization directions. This choice is general, and immediately includes LP and CP as particular cases.
• Why general function of phase if it's already a PW? PW means that the EM field is invariant in transverse coordinates. It can still have a temporal (longitudinal) envelope, and also more complicated structure, like a two-color laser (2 frequencies with some dephasing between them).

TODO:

• place particle within the Plane Wave with the correct initial momenta (such that it would perform a "standard" trajectory if RR was absent)
• implement analytical expressions of momentum (either following DiPiazza's paper or Fabien Niel's PhD thesis)
• compare with osiris PIC simulation (CRR, not QRR), show agreement
• compile functions in a .py file (easier to share and to be used in further projects)
• improve comments and introduction in the Jupyter notebook for easy reading