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

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## Abstract

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## 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

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**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.

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**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