Gamma-ray vortices emitted from nonlinear inverse Thomson scattering of a two-wavelength laser beam

We develop a classical theory of nonlinear inverse Thomson scattering of a two-wavelength laser beam, which is valid for laser beams with linear or circular polarization and arbitrary intensity and wavelength. We reveal that an electron inside a circularly polarized two-wavelength laser field undergoes a cycloid motion in the transverse plane and radiates an electromagnetic wave forming a spiral phase structure. Its photon energy is proportional to a linear combination of the product between the initial laser photon energies $\hslash {\omega }_{01}$ and $\hslash {\omega }_{02}$, and their harmonic numbers $n_1$ and $n_2$, namely, $n_1\hslash {\omega }_{01}+n_2\hslash {\omega }_{02}$. The orbital angular momentum of a photon due to the spiral phase structure is given by $(n_1+n_2\pm 1)\hslash$. We show that a combination of two wavelengths differing by one order of magnitude or more is advantageous for producing gamma rays carrying a large orbital angular momentum of $\sim 10\hslash$. Moreover, this work indicates that a helical undulator with two magnetic periods is capable of producing photons with a large orbital angular momentum. This radiation process plays an important role in the development of optical vortex beams and even in astrophysical environments.

Figure 1

(a) Coordinate system used to calculate NITS for a two-wavelength laser beam. (b) Typical transverse electron motion induced by a two-wavelength laser beam circularly polarized in the $x\text{−}y$ plane.

Figure 2

(a)–(c) Energy spectra of gamma rays emitted by NITS with two circularly polarized laser beams at angle $\theta =0.5/{\gamma }_0$ and calculated using Eq. (27). Red and black arrows show the fundamental scattered energy calculated using $\hslash \omega (n_1=1,n_2=0)=4{\gamma }_0^2\hslash {\omega }_{01}/\{1+{\gamma }_0^2{\theta }^2+(a_{01}^2+a_{02}^2)/2\}$ and $\hslash \omega (n_1=0,n_2=1)=4{\gamma }_0^2\hslash {\omega }_{02}/\{1+{\gamma }_0^2{\theta }^2+(a_{01}^2+a_{02}^2)/2\}$, respectively. (d)–(f) The number of gamma-ray photons integrated up to ${\theta }_c=3/{\gamma }_0$ and calculated using Eq. (29). The harmonic number $n_2=0$. (g)–(i) Same as (d)–(f) but $n_2=1$. (j)–(l) Projection of normalized transverse electron trajectories inside two circularly polarized laser beams for $-5{\lambda }_{01}<\eta <5{\lambda }_{01}$, as calculated using Eqs. (13, 14, 15). The calculation parameters are ${\gamma }_0=2000$, ${\lambda }_{01}=10\mu \text{m}$, $a_{01}=1$, $a_{02}=1$, $N_{01}=150$, and $N_e={10}^9$ electron ${\mathrm{s}}^{-1}$. The wavelength of the second laser ${\lambda }_{02}$ is indicated above each panel.

Figure 3

(a)–(d) Line distributions of the Stokes parameter for gamma rays emitted by NITS for two circularly polarized laser beams with the same helicity. The Stokes parameter has cylindrical symmetry because Eq. (26) is not a function of $\varphi$. (e)–(h) Spatial distributions of the radiation energy of gamma rays calculated using Eq. (27). Each spatial distribution is normalized by the maximum value of $n_1=0$. The calculation parameters are ${\gamma }_0=2000$, ${\lambda }_{01}=10\mu \mathrm{m}$, ${\lambda }_{02}=1\mu \mathrm{m}$, $a_{01}=1$, $a_{02}=1$, and $n_2=1$. The harmonic number $n_1$ is indicated above the panels. Note that the scale for the horizontal axis in (a) and (e) is different from that in the other panels.

Figure 4

(a) OAM of gamma-ray vortices calculated using $n_1+n_2-1$ for positive helicity as a function of harmonic number $n_1$. (b) The number of gamma-ray photons integrated for a solid angle up to ${\theta }_c$, which contains more than 90% of the circular polarization with positive helicity. (c) Maximum energy of gamma-ray vortices ($\theta =0$) calculated using Eq. (28). The calculation parameters are ${\gamma }_0=2000$, ${\lambda }_{01}=10\mu \mathrm{m}$, ${\lambda }_{02}=1\mu \mathrm{m}$, $a_{01}=1$, $a_{02}=1$, $N_{01}=150$, $N_{02}=1500$, $n_2=1$, and $N_e={10}^9$ electron ${\mathrm{s}}^{-1}$.

Figure 5

Number of gamma-ray photons integrated over the solid angle up to ${\theta }_c$, which contains more than 90% of the circular polarization with positive helicity. The calculation parameters are ${\gamma }_0=2000$, ${\lambda }_{01}=10\mu \mathrm{m}$, $a_{01}=1$, $N_{01}=150$, $n_2=1$, and $N_e={10}^9$ electron ${\mathrm{s}}^{-1}$. The other parameters are (a) $a_{02}=1$ and $n_1=5$, (b) $a_{02}=1$ and $n_1=13$, and (c) ${\lambda }_{02}=1\mu \mathrm{m}$ and $n_1=5$.

Figure 6

Spatial distribution of the radiation energy of gamma rays emitted by NITS with two linearly polarized laser beams, calculated from Eq. (27). Each spatial distribution is normalized by the maximum values of $n_1=5$ and $n_2=1$. The calculation parameters are ${\gamma }_0=2000$, ${\lambda }_{01}=10\mu \mathrm{m}$, ${\lambda }_{02}=1\mu \mathrm{m}$, $a_{01}=1$, and $a_{02}=1$. The harmonic numbers $n_1$ and $n_2$ are indicated above each panel. Note that the scales of the horizontal axes in (g) and (h) are different from those of the other panels.