# Electron Dynamics and Thomson Scattering for Ultra-Intense Lasers: Elliptically Polarized and OAM Beams

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. General Formulation

#### 2.1. Classical Equations of Motion for the Electron

#### 2.2. Asymptotic Liénard–Wiechert Retarded Radiation Fields

#### 2.3. Spectral Asymptotic Liénard–Wiechert Retarded Radiation Fields: Integral Representations

## 3. Elliptically Polarized Radiation: Parametric Representation of Dynamical Variables

#### 3.1. Representation of an Arbitrary Non-Monochromatic Plane Wave through the Vector Potential ${\mathbf{A}}_{i}(\xi )$

#### 3.2. The Vector Potential and the Solution of the Electron Dynamics for Monochromatic Elliptically Polarized Radiation

#### 3.3. Magnitudes Derived from the Parametric Representation of Trajectories

## 4. Spectral Representation for Elliptically Polarized Incoming Radiation: The Integrand and Doppler Frequencies

## 5. TS for an Incoming Laser Beam with OAM

#### 5.1. Incoming Laser Beam with OAM: Vector Potential and Electromagnetic Fields

#### 5.2. Approximating the Electron Dynamics in OAM Beams by the Plane-Wave Solution: Numerical Computations

## 6. Conclusions and Final Comments

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Miscellaneous Symbol Definitions and Formulas

## Appendix B. The Function ${\mathit{J}}_{\mathit{G},\mathit{l}}$ ($\mathit{x},\mathit{y};{\mathit{\phi}}_{\mathbf{2}\mathit{s},\mathit{E}}-{\mathit{\phi}}_{\mathbf{1}\mathit{s},\mathit{E}}$) and Some Properties

## Appendix C. Monochromatic Polarized Radiation with φ_{0} ≠ 0: Linearly and Circularly

#### Appendix C.1. Linear Polarization

#### Appendix C.2. Circular Polarization

## Appendix D. OAM

## References

- Froula, D.H.; Glenzer, S.H.; Luhmann, N.C., Jr.; Sheffield, J. Plasma Scattering of Electromagnetic Radiation: Theory and Measurement Techniques; Academic Press, Elsevier: Amsterdam, The Netherlands, 2011. [Google Scholar]
- Hutchison, I. Principles of Plasma Diagnostics; Cambridge Univ. Press: Cambridge, UK, 2006. [Google Scholar]
- Strickland, D.; Mourou, G. Compression of amplified chirped optical pulses. Opt. Comm.
**1985**, 55, 447–449. [Google Scholar] [CrossRef] - Danson, C.N.; Haefner, C.; Bromage, J.; Butcher, T.; Chanteloup, J.-C.F.; Chowdhury, E.A.; Galvanauskas, A.; Gizzi, L.A.; Hein, J.; Hillier, D.I.; et al. Petawatt and exawatt class lasers worldwide. High Power Laser Sci. Eng.
**2019**, 7, e54. [Google Scholar] [CrossRef] - Rohrlich, F. Classical Charged Particles, 3rd ed.; World Scientific: Singapore, 2007. [Google Scholar]
- Allen, L.; Barnett, S.M.; Padgett, M.J. Optical Angular Momentum; IOP Publishing: London, UK, 2003. [Google Scholar]
- Torres, J.P.; Torner, L. (Eds.) Twisted Photons: Applications of Light with Orbital Angular Momentum; Wiley-VCH: Weinheim, Germany, 2011. [Google Scholar]
- Karimi, E.; Schulz, S.A.; De Leon, I.; Qassim, H.; Upham, J.; Boyd, R.W. Generating optical orbital angular momentum at visible wavelengths using a plasmonic metasurface. Light Sci. Appl.
**2014**, 3, e167. [Google Scholar] [CrossRef][Green Version] - Noyan, M.A.; Kikkawa, J.M. Time-resolved orbital angular momentum spectroscopy. Appl. Phys. Lett.
**2015**, 107, 032406. [Google Scholar] [CrossRef] - Persuy, D.; Ziegler, M.; Crégut, O.; Kheng, K.; Gallart, M.; Honerlage, B.; Gilliot, P. Four-wave mixing in quantum wells using femtosecond pulses with Laguerre–Gauss modes. Phys. Rev. B
**2015**, 92, 115312. [Google Scholar] [CrossRef] - Schmiegelow, C.T.; Schulz, J.; Kaufmann, H.; Ruster, T.; Poschinger, U.G.; Schmidt-Kaler, F. Transfer of optical orbital angular momentum to a bound electron. Nat. Commun.
**2016**, 7, 12998. [Google Scholar] [CrossRef] - Seghilani, M.S.; Myara, M.; Sellahi, M.; Legratiet, L.; Sagnes, I.; Beaudoin, G.; Lalanne, P.; Garnache, A. Vortex Laser based on III-V semiconductor metasurface: Direct generation of coherent Laguerre–Gauss modes carrying controlled orbital angular momentum. Sci. Rep.
**2016**, 6, 38156. [Google Scholar] [CrossRef][Green Version] - Shigematsu, K.; Yamane, K.; Morita, R.; Toda, Y. Coherent dynamics of exciton orbital angular momentum transferred by optical vortex pulses. Phys. Rev. B
**2016**, 93, 045205. [Google Scholar] [CrossRef][Green Version] - Picón, A.; Benseny, A.; Mompart, J.; Vázquez de Aldana, J.R.; Plaja, L.; Calvo, G.F.; Roso, L. Transferring orbital and spin angular momenta of light to atoms. New J. Phys.
**2010**, 12, 083053. [Google Scholar] [CrossRef] - Longman, A.; Salgado, C.; Zeraouli, G.; Apinariz, J.I.; Pérez-Hernández, J.A.; Eltahlawy, M.K.; Volpe, L.; Fedosejevs, R. Off axis spiral phase mirrors for generating high-intensity optical vortices. Opt. Lett.
**2020**, 45, 2187. [Google Scholar] [CrossRef] [PubMed][Green Version] - Pastor, I.; Alvarez-Estrada, R.F.; Roso, L.; Castejon, F.; Guasp, J. Nonlinear relativistic electron Thomson scattering for laserradiation with orbital angular momentum. J. Phys. Commun.
**2020**, 4, 065010. [Google Scholar] [CrossRef] - He, F.; Lau, Y.Y.; Umstadter, D.P.; Strickler, T. Phase dependence of Thomson scattering in an ultraintense laser field. Phys. Plasmas
**2002**, 9, 4325. [Google Scholar] [CrossRef][Green Version] - Evans, D.E.; Katzenstein, J. Laser light scattering in laboratory plasmas. Rep. Prog. Phys.
**1969**, 32, 207. [Google Scholar] [CrossRef] - Mattioli, M. Incoherent Light Scattering from High Temperature Plasmas; Report DPh-PFC-SPP (EUR-CEA-FC) 752; EURATOM-CEA: Fontenay-aux-Roses, France, 1974. [Google Scholar]
- Matoba, T.; Itagaki, T.; Yamauchi, T.; Funahashi, A. Analytical Approximations in the Theory of Relativistic Thomson Scattering for High Temperature Fusion Plasma. Jpn. J. Appl. Phys.
**1979**, 18, 1127. [Google Scholar] [CrossRef] - Weyssow, B. Motion of a single charged particle in electromagnetic fields with cyclotron resonances. J. Plasma Phys.
**1990**, 43, 119. [Google Scholar] [CrossRef] - Naito, O.; Yoshida, H.; Matoba, T. Analytic formula for fully relativistic Thomson scattering spectrum. Phys. Fluids B Plasma Phys.
**1993**, 5, 4256. [Google Scholar] [CrossRef] - Beausang, K.V.; Prunty, S.L. An analytic formula for the relativistic Thomson scattering spectrum for a Maxwellian velocity distribution. Plasma Phys. Control. Fusion
**2008**, 50, 095001. [Google Scholar] [CrossRef] - Walsh, M.J.; Beurskens, M.; Carolan, P.G.; Gilbert, M.; Loughlin, M.; Morris, A.W.; Riccardo, V.; Xue, Y. Design challenges and analysis of the ITER core LIDAR Thomson scattering system. Rev. Sci. Instrum.
**2006**, 77, 10E525. [Google Scholar] [CrossRef] - Ross, J.S.; Glenzer, S.H.; Palastro, J.P.; Pollock, B.B.; Price, D.; Divol, L.; Tynan, G.R.; Froula, D.H. Observation of Relativistic Effects in Collective Thomson Scattering. Phys. Rev. Lett.
**2010**, 104, 105001. [Google Scholar] [CrossRef][Green Version] - Palastro, J.P.; Ross, J.S.; Pollock, B.; Divol, L.; Froula, D.H.; Glenzer, S.H. Fully relativistic form factor for Thomson scattering. Phys. Rev. E
**2010**, 81, 036411. [Google Scholar] [CrossRef] - Landau, L.D.; Lifchitz, E.M. The Classical Theory of Fields, 4th ed.; Pergamon Press: New York, NY, USA, 1975. [Google Scholar]
- Sarachik, E.S.; Schappert, G.T. Classical Theory of the Scattering of Intense Laser Radiation by Free Electrons. Phys. Rev. D
**1970**, 1, 2738. [Google Scholar] [CrossRef] - Esarey, E.; Ride, S.K.; Sprangle, P. Nonlinear Thomson scattering of intense laser pulses from beams and plasmas. Phys. Rev. E
**1993**, 48, 3003. [Google Scholar] [CrossRef] - Ride, S.K.; Esarey, E.; Baine, M. Thomson scattering of intense lasers from electron beams at arbitrary interaction angles. Phys. Rev. E
**1995**, 52, 5425. [Google Scholar] [CrossRef] [PubMed] - Brau, C.A. Modern Problems in Classical Electrodynamics; Oxford Univ. Press: Oxford, UK, 2004. [Google Scholar]
- Avetissian, H. Relativistic Nonlinear Electrodynamics; Springer Series in Optical Sciences; Springer: New York, NY, USA, 2006. [Google Scholar]
- Yang, J.H.; Craxton, R.S.; Haines, M.G. Explicit general solutions to relativistic electron dynamics in plane-wave electromagnetic fields and simulations of ponderomotive acceleration. Plasma Phys. Control. Fusion
**2011**, 53, 125006. [Google Scholar] [CrossRef] - Panofsky, W.K.H.; Phillips, M. 1955 Classical Electricity and Magnetism; Addison-Wesley: Reading, MA, USA, 1965. [Google Scholar]
- Pastor, I.; Guasp, J.; Alvarez-Estrada, R.F.; Castejon, F. Monte Carlo approach to Thomson scattering in relativistic fusion plasmas with allowance for ultraintense laser radiation. Nucl. Fusion
**2011**, 51, 04011. [Google Scholar] [CrossRef] - Alvarez-Estrada, R.F.; Pastor, I.; Guasp, J.; Castejon, F. Nonlinear relativistic single-electron Thomson scattering power spectrum for incoming laser of arbitrary intensity. Phys. Plasmas
**2012**, 19, 062302. [Google Scholar] [CrossRef][Green Version] - Jackson, J.D. Classical Electrodynamics, 2nd ed.; John Wiley and Sons: New York, NY, USA, 1974. [Google Scholar]
- Duke, P.J. Synchrotron Radiation: Production and Properties; Oxford University Press: Oxford, UK, 2000. [Google Scholar]
- Loetstedt, E.; Jentschura, U.D. Recursive algorithm for arrays of generalized Bessel functions: Numerical access to Dirac-Volkov solutions. Phys. Rev. E
**2009**, 79, 026707. [Google Scholar] [CrossRef][Green Version] - Calvo, G.F.; Picón, A.; Bagan, E. Quantum field theory of photons with orbital angular momentum. Phys. Rev. A
**2006**, 73, 013805. [Google Scholar] [CrossRef][Green Version] - Olver, F.W.J. Bessel functions of integer order. In Handbook of Mathematical Functions; Abramowitz, M., Stegun, I.A., Eds.; Dover: New York, NY, USA, 1965. [Google Scholar]

**Figure 1.**Fundamental scattered frequency for a scattering unit vector ${\mathbf{n}}_{0}=\mathbf{j}$, as a function of initial phase angle ${\phi}_{0}$ for ${E}_{kin0}=0$ (upper panel) and ${E}_{kin0}=200.0$ keV (lower panel). The reference electric field in the laser is in both cases ${E}_{r}=5.0\times {10}^{12}$ V/m. Three different polarizations are represented, keeping the sum ${E}_{0}^{2}+{E}_{1}^{2}={E}_{r}^{2}$ constant. Red curves represent the case of linear polarization, blue curves are for circular polarization (solid lines, ${\theta}_{s}=\pi /4$; dashed lines, ${\theta}_{s}=-\pi /4$) and green curves are for elliptical polarization with ${\theta}_{s}=\pi /8$ (solid lines) and ${\theta}_{s}=-\pi /8$ (dashed lines).

**Figure 2.**The fundamental period of the trajectory, as a function of initial phase angle ${\phi}_{0}$, is given for three different polarizations. The red curve represents the case of linear polarization; blue curves are for circular polarization (solid line, ${\theta}_{s}=\pi /4$; dashed line, ${\theta}_{s}=-\pi /4$); and green curves are for elliptical polarization with ${\theta}_{s}=\pi /8$ (solid line) and ${\theta}_{s}=-\pi /8$ (dashed line). Laser and electron parameters are given in the main text.

**Figure 3.**“Figure of eight” of variables ${x}_{1}^{\prime}\equiv {x}^{\prime}$, ${x}_{3}^{\prime}\equiv {z}^{\prime}$ in the Lorentz frame with velocity equal to the drift velocity defined in the main text.

**Figure 4.**Dependency of the maximum of $\gamma $ factor on initial laser phase ${\phi}_{0}$ for the conditions given in the main text.

**Figure 5.**Comparing the dynamics and radiated field at the detector, OAM laser beam (full lines) vs. plane-wave solution (open symbols). Normalized velocity components are given in panel (

**a**), panel (

**b**) gives the relativistic gamma factor, and panel (

**c**) gives the x and z components of radiated electric field at detection point. See the main text for the details.

**Figure 6.**Comparing the dynamics and radiated field at the detector, OAM laser beam (full lines) vs. plane-wave solution (open symbols). Normalized velocity components are given in panel (

**a**), panel (

**b**) gives the relativistic gamma factor, and panel (

**c**) gives the x and z components of radiated electric field at detection point. See the main text for the details.

**Figure 7.**Comparing the dynamics and radiated field at the detector, OAM laser beam (full lines) vs. plane-wave solution (open symbols). Normalized velocity components are given in panel (

**a**), panel (

**b**) gives the relativistic gamma factor, and panel (

**c**) gives the x and z components of radiated electric field at detection point. See the main text for the details.

**Figure 8.**Comparing the dynamics and radiated field at the detector, OAM laser beam (full lines) vs. plane-wave solution (open symbols). Normalized velocity components are given in panel (

**a**), panel (

**b**) gives the relativistic gamma factor, and panel (

**c**) gives the x and z components of radiated electric field at detection point. See the main text for the details.

**Figure 9.**An example of a trapped trajectory for an OAM beam with $(l,p)=(1,0)$. The upper panel shows the normalized (to ${k}_{0}$) trajectory components, the lower one the normalized (to c) velocity components. From the lower panel, the existence of two clearly different time scales is apparent, one close to the optical frequency and a much slower one related to the finite motion in the X-Y plane. See the main text for the details.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pastor, I.; Álvarez-Estrada, R.F.; Roso, L.; Guasp, J.; Castejón, F. Electron Dynamics and Thomson Scattering for Ultra-Intense Lasers: Elliptically Polarized and OAM Beams. *Photonics* **2021**, *8*, 182.
https://doi.org/10.3390/photonics8060182

**AMA Style**

Pastor I, Álvarez-Estrada RF, Roso L, Guasp J, Castejón F. Electron Dynamics and Thomson Scattering for Ultra-Intense Lasers: Elliptically Polarized and OAM Beams. *Photonics*. 2021; 8(6):182.
https://doi.org/10.3390/photonics8060182

**Chicago/Turabian Style**

Pastor, Ignacio, Ramón F. Álvarez-Estrada, Luis Roso, José Guasp, and Francisco Castejón. 2021. "Electron Dynamics and Thomson Scattering for Ultra-Intense Lasers: Elliptically Polarized and OAM Beams" *Photonics* 8, no. 6: 182.
https://doi.org/10.3390/photonics8060182