Generation of Optical Vortices by Nonlinear Inverse Thomson Scattering at Arbitrary Angle Interactions

The motion of an electron inside a circularly polarized electromagnetic wave has been treated using classical electrodynamics in several previous studies (Gunn & Ostriker 1971; Stewart 1972; Ride et al. 1995). In this paper, we follow the notation described in Ride et al. (1995). The coordinate systems used in the calculation are shown in Figure 1. Here, we assume that the circularly polarized strong electromagnetic wave is propagating in the -z direction. The normalized vector potential of the circularly polarized electromagnetic wave is expressed as

$$\begin{eqnarray}&&{\boldsymbol{a}}=\displaystyle \frac{{a}_{0}}{\sqrt{2}}\{\cos ({k}_{0}\eta ){{\boldsymbol{e}}}_{x}+\sin ({k}_{0}\eta ){{\boldsymbol{e}}}_{y}\}.\end{eqnarray} \tag{ 2 }$$

Here, ${k}_{0}(=2\pi /{\lambda }_{0})$ and λ₀ are the wave number and the wavelength of the initial electromagnetic wave, respectively, $\eta =z+{ct}$ is an independent variable, t is time, and ${{\boldsymbol{e}}}_{x}$ and ${{\boldsymbol{e}}}_{y}$ are the unit vectors along the x- and y-axes, respectively.

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The electron is initially moving in the x–z plane at an angle ζ₀ with respect to the z-axis with a normalized velocity ${{\boldsymbol{\beta }}}_{0}={{\boldsymbol{v}}}_{0}/c={\beta }_{x0}{{\boldsymbol{e}}}_{x}+{\beta }_{z0}{{\boldsymbol{e}}}_{z}$, where ${{\boldsymbol{v}}}_{0}$ is the initial electron velocity, ${\beta }_{x0}={\beta }_{0}\sin {\zeta }_{0}$, ${\beta }_{z0}={\beta }_{0}\cos {\zeta }_{0}$ (Figure 1(a)), and ${{\boldsymbol{e}}}_{z}$ is a unit vector along the z-axis. The electron momentum normalized by the electron rest energy during the electron and electromagnetic wave interaction is written as ${\boldsymbol{u}}=\gamma {\boldsymbol{\beta }}$, where γ and ${\boldsymbol{\beta }}$ are the Lorentz factor and the normalized velocity of the electron, respectively. The vector components of ${\boldsymbol{u}}$ in the coordinate system $(x,y,z)$ can be expressed as follows by solving the relativistic Lorentz equations:

$$\begin{eqnarray}&&{u}_{x}=\displaystyle \frac{{a}_{0}}{\sqrt{2}}\cos ({k}_{0}\eta )+{\gamma }_{0}{\beta }_{x0},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{u}_{y}=\displaystyle \frac{{a}_{0}}{\sqrt{2}}\sin ({k}_{0}\eta ),\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{u}_{z}=\displaystyle \frac{1}{2{\gamma }_{0}(1+{\beta }_{z0})}\{{\gamma }_{0}^{2}{(1+{\beta }_{z0})}^{2}-1-({u}_{x}^{2}+{u}_{y}^{2})\}.\end{eqnarray} \tag{ 5 }$$

Here, ${\gamma }_{0}=1/\sqrt{1-{\beta }_{0}^{2}}$. The averages of the components over the electromagnetic wave period are given by the following:

$$\begin{eqnarray}&&{\overline{u}}_{x}={\gamma }_{0}{\beta }_{x0},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{\overline{u}}_{y}=0,\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\overline{u}}_{z}={\gamma }_{0}{\beta }_{z0}-\displaystyle \frac{{a}_{0}^{2}}{4{\gamma }_{0}(1+{\beta }_{z0})}.\end{eqnarray} \tag{ 8 }$$

Even though the average electron momenta along the x- and y-axes conserve the initial momenta, the momentum along the z-axis is reduced by ${a}_{0}^{2}/\{4{\gamma }_{0}(1+{\beta }_{z0})\}$. The decrease in the average electron momentum is due to the ponderomotive force of a strong electromagnetic wave (Ride et al. 1995). Consequently, the propagation direction of an electron within a strong electromagnetic wave is slightly changed from the initial direction. We define the electron as moving along the z'-axis within the electromagnetic wave (Figure 1(b)). The angle between the z- and z'-axes is given by $\zeta ={\cos }^{-1}({\overline{u}}_{z}/\overline{u})\,={\sin }^{-1}({\overline{u}}_{x}/\overline{u})$, where $\overline{u}=\sqrt{{\overline{u}}_{x}^{2}+{\overline{u}}_{z}^{2}}$. Note that the photon is scattered into a narrow cone centered about the z'-axis. Therefore, to see the phase structure of the scattered photon, we should consider the electron trajectory in the coordinate system $({x}^{{\prime} },{y}^{{\prime} },{z}^{{\prime} })$.

First, we consider the electron orbits, ${{\boldsymbol{r}}}_{e}={x}_{e}{{\boldsymbol{e}}}_{x}+{y}_{e}{{\boldsymbol{e}}}_{y}+{z}_{e}{{\boldsymbol{e}}}_{z}$, inside the circularly polarized electromagnetic wave in the coordinate system $(x,y,z)$. These orbits are derived using the relation $d{{\boldsymbol{r}}}_{e}/d\eta ={\boldsymbol{u}}/\{{\gamma }_{0}(1+{\beta }_{z0})\}$ and are given by the following:

$$\begin{eqnarray}&&{x}_{e}(\eta )={r}_{1}\sin ({k}_{0}\eta )+{\beta }_{1x}\eta +{x}_{0},\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{y}_{e}(\eta )=-{r}_{1}\cos ({k}_{0}\eta )+{y}_{0},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{z}_{e}(\eta )=-{r}_{1}{\beta }_{1x}\sin ({k}_{0}\eta )+{\beta }_{1z}\eta +{z}_{0}.\end{eqnarray} \tag{ 11 }$$

Here, x₀, y₀, and z₀ are the initial positions of the electron along the x-, y-, and z-axes, respectively, r₁ is the radius of the electron motion given by

$$\begin{eqnarray}&&{r}_{1}=\displaystyle \frac{{a}_{0}}{\sqrt{2}{\gamma }_{0}(1+{\beta }_{z0}){k}_{0}},\end{eqnarray} \tag{ 12 }$$

and ${\beta }_{1x}$ and ${\beta }_{1z}$ are given by

$$\begin{eqnarray}&&{\beta }_{1x}=\displaystyle \frac{{\beta }_{x0}}{1+{\beta }_{z0}}=\displaystyle \frac{{\overline{u}}_{x}}{{\gamma }_{0}(1+{\beta }_{z0})},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{\beta }_{1z}=\displaystyle \frac{{\beta }_{z0}}{1+{\beta }_{z0}}-\displaystyle \frac{{a}_{0}^{2}}{4{\gamma }_{0}^{2}{(1+{\beta }_{z0})}^{2}}=\displaystyle \frac{{\overline{u}}_{z}}{{\gamma }_{0}(1+{\beta }_{z0})}.\end{eqnarray} \tag{ 14 }$$

Therefore, ${\beta }_{1x}/{\beta }_{1z}=\tan \zeta$.

The unit vectors in the two coordinate systems $(x,y,z)$ and $({x}^{{\prime} },{y}^{{\prime} },{z}^{{\prime} })$ are transformed by the following relations:

$$\begin{eqnarray}&&{{\boldsymbol{e}}}_{x}=\cos \zeta {{\boldsymbol{e}}}_{{x}^{{\prime} }}+\sin \zeta {{\boldsymbol{e}}}_{{z}^{{\prime} }},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{{\boldsymbol{e}}}_{y}={{\boldsymbol{e}}}_{{y}^{{\prime} }},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{{\boldsymbol{e}}}_{z}=-\sin \zeta {{\boldsymbol{e}}}_{{x}^{{\prime} }}+\cos \zeta {{\boldsymbol{e}}}_{{z}^{{\prime} }}.\end{eqnarray} \tag{ 17 }$$

Therefore, the electron orbits, ${{\boldsymbol{r}}}_{e}^{{\prime} }={x}_{e}^{{\prime} }{{\boldsymbol{e}}}_{{x}^{{\prime} }}+{y}_{e}^{{\prime} }{{\boldsymbol{e}}}_{{y}^{{\prime} }}+{z}_{e}^{{\prime} }{{\boldsymbol{e}}}_{{z}^{{\prime} }}$, in the coordinate system $({x}^{{\prime} },{y}^{{\prime} },{z}^{{\prime} })$ are given by

$$\begin{eqnarray}{x}_{e}^{{\prime} }(\eta ) & = & {r}_{1}(\cos \zeta +{\beta }_{1x}\sin \zeta )\sin ({k}_{0}\eta )\\ & & +{x}_{0}\cos \zeta -{z}_{0}\sin \zeta ,\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&{y}_{e}^{{\prime} }(\eta )=-{r}_{1}\cos ({k}_{0}\eta )+{y}_{0},\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}{z}_{e}^{{\prime} }(\eta ) & = & {r}_{1}(\sin \zeta -{\beta }_{1x}\cos \zeta )\sin ({k}_{0}\eta )\\ & & +\displaystyle \frac{{\beta }_{1z}}{\cos \zeta }\eta +{x}_{0}\sin \zeta +{z}_{0}\cos \zeta .\end{eqnarray} \tag{ 20 }$$

In general, Equations (18)–(20) do not represent a helical circular motion. We can rewrite Equations (18)–(20) by introducing a quantity ${\rm{\Delta }}\equiv ({a}_{0}^{2}+2)/(4{\gamma }_{0}^{2})$, which is much smaller than unity under the conditions of ${a}_{0}\ll {\gamma }_{0}$ and ${\gamma }_{0}\gg 1$. The coefficients of $\sin ({k}_{0}\eta )$ in Equations (18) and (20) can then be expressed as

$$\begin{eqnarray}&&\cos \zeta +{\beta }_{1x}\sin \zeta \simeq 1-\displaystyle \frac{1-{\beta }_{z0}}{1+{\beta }_{z0}}{\rm{\Delta }},\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&\sin \zeta -{\beta }_{1x}\cos \zeta \simeq {\beta }_{1x}(1+{\rm{\Delta }}).\end{eqnarray} \tag{ 22 }$$

Here, we used the relation

$$\begin{eqnarray}&&\overline{u}\simeq {\gamma }_{0}-\displaystyle \frac{2+2{\beta }_{z0}+{\beta }_{z0}{a}_{0}^{2}}{4{\gamma }_{0}(1+{\beta }_{z0})}.\end{eqnarray} \tag{ 23 }$$

To derive Equations (21)–(23), we assumed the following condition:

$$\begin{eqnarray}&&\displaystyle \frac{2+2{\beta }_{z0}+{\beta }_{z0}{a}_{0}^{2}}{2{\gamma }_{0}^{2}(1+{\beta }_{z0})}\ll 1.\end{eqnarray} \tag{ 24 }$$

Therefore, Equations (18)–(20) can be rewritten as

$$\begin{eqnarray}{x}_{e}^{{\prime} }(\eta ) & = & {r}_{1}\left(1-\displaystyle \frac{1-{\beta }_{z0}}{1+{\beta }_{z0}}{\rm{\Delta }}\right)\sin ({k}_{0}\eta )\\ & & +{x}_{0}\cos \zeta -{z}_{0}\sin \zeta ,\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{y}_{e}^{{\prime} }(\eta )=-{r}_{1}\cos ({k}_{0}\eta )+{y}_{0},\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}{z}_{e}^{{\prime} }(\eta ) & = & {r}_{1}{\beta }_{1x}(1+{\rm{\Delta }})\sin ({k}_{0}\eta )+\displaystyle \frac{{\beta }_{1z}}{\cos \zeta }\eta \\ & & +{x}_{0}\sin \zeta +{z}_{0}\cos \zeta .\end{eqnarray} \tag{ 27 }$$

When ${\rm{\Delta }}\ll 1$, Equations (25)–(27) indicate that the electron motion is well represented by a helical circular trajectory in the coordinate system $({x}^{{\prime} },{y}^{{\prime} },{z}^{{\prime} })$, in which the electron moves counterclockwise when the observer is facing the oncoming electron. We call this electron motion positive helicity.

The Fourier component of the electric field emitted by a single electron can be calculated from the Lienard–Wiechert potentials (Section 14.5 in (Jackson 1999)):

$$\begin{eqnarray}{\boldsymbol{E}} & = & -i\displaystyle \frac{{ek}}{\sqrt{2\pi }}\displaystyle \frac{{e}^{{ikR}}}{R}\\ & & \times {\displaystyle \int }_{-\infty }^{\infty }{dt}\{{{\boldsymbol{n}}}^{{\prime} }\times ({{\boldsymbol{n}}}^{{\prime} }\times {\boldsymbol{\beta }})\}{e}^{{ik}\{{ct}-{{\boldsymbol{n}}}^{{\prime} }\cdot {{\boldsymbol{r}}}_{e}(t)\}},\end{eqnarray} \tag{ 28 }$$

where $k=\omega /c$ and ω are the wave number and the angular frequency of the emitted photon, respectively, ${\epsilon }_{0}$ is the permittivity of a vacuum, R is the distance from the origin to the observation point, ${{\boldsymbol{n}}}^{{\prime} }$ is a unit vector pointing from the origin to the observation point as shown in Figure 1(b), and ${{\boldsymbol{r}}}_{e}$ is the electron orbit described in Equations (9)–(11). Equation (28) can be calculated in the spherical coordinate system $({r}^{{\prime} },{\theta }^{{\prime} },{\phi }^{{\prime} })$ with unit vectors $({{\boldsymbol{e}}}_{{r}^{{\prime} }},{{\boldsymbol{e}}}_{{\theta }^{{\prime} }},{{\boldsymbol{e}}}_{{\phi }^{{\prime} }})$ using the following relation:

$$\begin{eqnarray}&&{{\boldsymbol{n}}}^{{\prime} }\times ({{\boldsymbol{n}}}^{{\prime} }\times {\boldsymbol{\beta }})=-({\boldsymbol{\beta }}\,\cdot \,{{\boldsymbol{e}}}_{{\theta }^{{\prime} }}){{\boldsymbol{e}}}_{{\theta }^{{\prime} }}-({\boldsymbol{\beta }}\,\cdot \,{{\boldsymbol{e}}}_{{\phi }^{{\prime} }}){{\boldsymbol{e}}}_{{\phi }^{{\prime} }},\end{eqnarray} \tag{ 29 }$$

where

$$\begin{eqnarray}&&{\boldsymbol{\beta }}=\displaystyle \frac{1}{c}\displaystyle \frac{d{{\boldsymbol{r}}}_{e}}{{dt}}=\displaystyle \frac{1}{c}\left(\displaystyle \frac{{{dz}}_{e}}{{dt}}+c\right)\displaystyle \frac{d{{\boldsymbol{r}}}_{e}}{d\eta }.\end{eqnarray} \tag{ 30 }$$

Each component of the electric field in the spherical coordinate system is expressed as follows:

$$\begin{eqnarray}&&{E}_{{\theta }^{{\prime} }}=i\displaystyle \frac{{ek}}{\sqrt{2\pi }c}\displaystyle \frac{{e}^{{ikR}}}{R}{\int }_{-{\eta }_{0}}^{{\eta }_{0}}d\eta \displaystyle \frac{d{{\boldsymbol{r}}}_{e}}{d\eta }\cdot {{\boldsymbol{e}}}_{{\theta }^{{\prime} }}{e}^{i\psi },\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&{E}_{{\phi }^{{\prime} }}=i\displaystyle \frac{{ek}}{\sqrt{2\pi }c}\displaystyle \frac{{e}^{{ikR}}}{R}{\int }_{-{\eta }_{0}}^{{\eta }_{0}}d\eta \displaystyle \frac{d{{\boldsymbol{r}}}_{e}}{d\eta }\cdot {{\boldsymbol{e}}}_{{\phi }^{{\prime} }}{e}^{i\psi }.\end{eqnarray} \tag{ 32 }$$

Here, $\psi \equiv k\{{ct}-{{\boldsymbol{n}}}^{{\prime} }$·${{\boldsymbol{r}}}_{e}(t)\}$ and ${\eta }_{0}={N}_{0}{\lambda }_{0}/2$, where N₀ is the number of periods of the initial electromagnetic wave interacting with the single electron. The variable ${\eta }_{0}$ will change according to the initial interaction angle. In a head-on collision (ζ₀ = 0 rad), ${\eta }_{0}$ corresponds to half the longitudinal pulse width of the initial electromagnetic wave. In a 90° collision (ζ₀ = π/2 rad), ${\eta }_{0}$ corresponds to the transverse radius of the initial electromagnetic wave if the longitudinal pulse width is longer than the transverse diameter and it corresponds to the half the longitudinal pulse width of the initial electromagnetic wave if the transverse diameter is longer than the longitudinal pulse width (Ride et al. 1995).

The unit vectors $({{\boldsymbol{e}}}_{{r}^{{\prime} }},{{\boldsymbol{e}}}_{{\theta }^{{\prime} }},{{\boldsymbol{e}}}_{{\phi }^{{\prime} }})$ are given by

$$\begin{eqnarray}{{\boldsymbol{e}}}_{{r}^{{\prime} }} & = & {{\boldsymbol{n}}}^{{\prime} }\\ & = & \sin {\theta }^{{\prime} }\cos {\phi }^{{\prime} }{{\boldsymbol{e}}}_{{x}^{{\prime} }}+\sin {\theta }^{{\prime} }\sin {\phi }^{{\prime} }{{\boldsymbol{e}}}_{{y}^{{\prime} }}+\cos {\theta }^{{\prime} }{{\boldsymbol{e}}}_{{z}^{{\prime} }}\\ & = & (\sin {\theta }^{{\prime} }\cos {\phi }^{{\prime} }\cos \zeta +\cos {\theta }^{{\prime} }\sin \zeta ){{\boldsymbol{e}}}_{x}\\ & & +\sin {\theta }^{{\prime} }\sin {\phi }^{{\prime} }{{\boldsymbol{e}}}_{y}\\ & & -(\sin {\theta }^{{\prime} }\cos {\phi }^{{\prime} }\sin \zeta -\cos {\theta }^{{\prime} }\cos \zeta ){{\boldsymbol{e}}}_{z},\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}{{\boldsymbol{e}}}_{{\theta }^{{\prime} }} & = & \cos {\theta }^{{\prime} }\cos {\phi }^{{\prime} }{{\boldsymbol{e}}}_{{x}^{{\prime} }}+\cos {\theta }^{{\prime} }\sin {\phi }^{{\prime} }{{\boldsymbol{e}}}_{{y}^{{\prime} }}-\sin {\theta }^{{\prime} }{{\boldsymbol{e}}}_{{z}^{{\prime} }}\\ & = & (\cos {\theta }^{{\prime} }\cos {\phi }^{{\prime} }\cos \zeta -\sin {\theta }^{{\prime} }\sin \zeta ){{\boldsymbol{e}}}_{x}\\ & & +\cos {\theta }^{{\prime} }\sin {\phi }^{{\prime} }{{\boldsymbol{e}}}_{y}\\ & & -(\cos {\theta }^{{\prime} }\cos {\phi }^{{\prime} }\sin \zeta +\sin {\theta }^{{\prime} }\cos \zeta ){{\boldsymbol{e}}}_{z},\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}{{\boldsymbol{e}}}_{{\phi }^{{\prime} }} & = & -\sin {\phi }^{{\prime} }{{\boldsymbol{e}}}_{{x}^{{\prime} }}+\cos {\phi }^{{\prime} }{{\boldsymbol{e}}}_{{y}^{{\prime} }}\\ & = & -\sin {\phi }^{{\prime} }\cos \zeta {{\boldsymbol{e}}}_{x}+\cos {\phi }^{{\prime} }{{\boldsymbol{e}}}_{y}+\sin {\phi }^{{\prime} }\sin \zeta {{\boldsymbol{e}}}_{z}.\end{eqnarray} \tag{ 35 }$$

The phase term in Equations (31) and (32) can be written in the form

$$\begin{eqnarray}&&\psi ={\psi }_{0}+{kg}\eta -{b}_{1}\sin ({k}_{0}\eta -{\phi }^{{\prime} })+{b}_{2}\sin ({k}_{0}\eta ),\end{eqnarray} \tag{ 36 }$$

where

$$\begin{eqnarray}{\psi }_{0} & \equiv & -{{kx}}_{0}(\sin {\theta }^{{\prime} }\cos {\phi }^{{\prime} }\cos \zeta +\cos {\theta }^{{\prime} }\sin \zeta )\\ & & -{{ky}}_{0}\sin {\theta }^{{\prime} }\sin {\phi }^{{\prime} }\\ & & -{{kz}}_{0}(1-\sin {\theta }^{{\prime} }\cos {\phi }^{{\prime} }\sin \zeta +\cos {\theta }^{{\prime} }\cos \zeta ),\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}&&g\equiv 1-{\beta }_{1z}\left(1+\displaystyle \frac{\cos {\theta }^{{\prime} }}{\cos \zeta }\right),\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&{b}_{1}\equiv {r}_{1}k\sin {\theta }^{{\prime} },\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}{b}_{2} & \equiv & {r}_{1}k\left\{\displaystyle \frac{1-{\beta }_{z0}}{1+{\beta }_{z0}}{\rm{\Delta }}\sin {\theta }^{{\prime} }\cos {\phi }^{{\prime} }\right.\\ & & -{\beta }_{1x}(\cos {\theta }^{{\prime} }-1+{\rm{\Delta }}\cos {\theta }^{{\prime} })\}.\end{eqnarray} \tag{ 40 }$$

The inner products in Equations (31) and (32) are expressed as

$$\begin{eqnarray}\displaystyle \frac{d{{\boldsymbol{r}}}_{e}}{d\eta }\cdot \,{{\boldsymbol{e}}}_{{\theta }^{{\prime} }} & = & -{\beta }_{1z}\displaystyle \frac{\sin {\theta }^{{\prime} }}{\cos \zeta }+{r}_{1}{k}_{0}\cos {\theta }^{{\prime} }\cos ({k}_{0}\eta -{\phi }^{{\prime} })\\ & & -{r}_{1}{k}_{0}\cos ({k}_{0}\eta )\left\{\displaystyle \frac{1-{\beta }_{z0}}{1+{\beta }_{z0}}{\rm{\Delta }}\cos {\theta }^{{\prime} }\cos {\phi }^{{\prime} }\right.\\ & & +{\beta }_{1x}(1+{\rm{\Delta }})\sin {\theta }^{{\prime} }\},\end{eqnarray} \tag{ 41 }$$

$$\begin{eqnarray}\displaystyle \frac{d{{\boldsymbol{r}}}_{e}}{d\eta }\cdot {{\boldsymbol{e}}}_{{\phi }^{{\prime} }} & = & {r}_{1}{k}_{0}\sin ({k}_{0}\eta -{\phi }^{{\prime} })\\ & & +{r}_{1}{k}_{0}\displaystyle \frac{1-{\beta }_{z0}}{1+{\beta }_{z0}}{\rm{\Delta }}\sin {\phi }^{{\prime} }\cos ({k}_{0}\eta ).\end{eqnarray} \tag{ 42 }$$

Finally, Equations (31) and (32) can be written in the limit ${\rm{\Delta }}\ll 1$ as follows:

$$\begin{eqnarray}{E}_{{\theta }^{{\prime} }} & = & \displaystyle \sum _{n=-\infty }^{\infty }i\displaystyle \frac{{ek}{\lambda }_{0}{N}_{0}}{\sqrt{2\pi }c}\displaystyle \frac{\sin (\overline{k}{\eta }_{0})}{\overline{k}{\eta }_{0}}\\ & & \times \left(\displaystyle \frac{{{nk}}_{0}\cos {\theta }^{{\prime} }}{k\sin {\theta }^{{\prime} }}-{\beta }_{1z}\displaystyle \frac{\sin {\theta }^{{\prime} }}{\cos \zeta }\right){J}_{n}({b}_{1})\displaystyle \frac{{e}^{i{\psi }_{0}+{ikR}+{in}{\phi }^{{\prime} }}}{R}\\ & \equiv & \displaystyle \sum _{n=-\infty }^{\infty }{{iC}}_{{\theta }^{{\prime} }}\displaystyle \frac{{e}^{i{\psi }_{0}+{ikR}+{in}{\phi }^{{\prime} }}}{R},\end{eqnarray} \tag{ 43 }$$

$$\begin{eqnarray}{E}_{{\phi }^{{\prime} }} & = & \displaystyle \sum _{n=-\infty }^{\infty }(-)\displaystyle \frac{{ek}{\lambda }_{0}{N}_{0}}{\sqrt{2\pi }c}\displaystyle \frac{\sin (\overline{k}{\eta }_{0})}{\overline{k}{\eta }_{0}}\\ & & \times {r}_{1}{k}_{0}{J}_{n}^{{\prime} }({b}_{1})\displaystyle \frac{{e}^{i{\psi }_{0}+{ikR}+{in}{\phi }^{{\prime} }}}{R}\\ & \equiv & \displaystyle \sum _{n=-\infty }^{\infty }(-){C}_{\phi }\displaystyle \frac{{e}^{i{\psi }_{0}+{ikR}+{in}{\phi }^{{\prime} }}}{R}.\end{eqnarray} \tag{ 44 }$$

Here, n is an integer and J_n and ${J}_{n}^{{\prime} }$ are the Bessel function of the first kind and its derivative, respectively. The details of the derivation of Equations (43) and (44) are described in the Appendix. Equations (43) and (44) peak at a wave number given by

$$\begin{eqnarray}&&\overline{k}\equiv {kg}-{{nk}}_{0}=0.\end{eqnarray} \tag{ 45 }$$

This equation indicates that n is the harmonic number of the scattered photon, as described in Equation (58).

To show the phase structure in the transverse plane against the z'-axis, we express the electric field in the Cartesian coordinate system $({x}^{{\prime} },{y}^{{\prime} },{z}^{{\prime} })$, as shown below:

$$\begin{eqnarray}&&{E}_{{x}^{{\prime} }}={E}_{{\theta }^{{\prime} }}\cos {\theta }^{{\prime} }\cos {\phi }^{{\prime} }-{E}_{{\phi }^{{\prime} }}\sin {\phi }^{{\prime} },\end{eqnarray} \tag{ 46 }$$

$$\begin{eqnarray}&&{E}_{{y}^{{\prime} }}={E}_{{\theta }^{{\prime} }}\cos {\theta }^{{\prime} }\sin {\phi }^{{\prime} }+{E}_{{\phi }^{{\prime} }}\cos {\phi }^{{\prime} },\end{eqnarray} \tag{ 47 }$$

$$\begin{eqnarray}&&{E}_{{z}^{{\prime} }}=-{E}_{{\theta }^{{\prime} }}\sin {\theta }^{{\prime} }.\end{eqnarray} \tag{ 48 }$$

Therefore, the electric field is

$$\begin{eqnarray}{\boldsymbol{E}} & = & {E}_{{x}^{{\prime} }}{{\boldsymbol{e}}}_{{x}^{{\prime} }}+{E}_{{y}^{{\prime} }}{{\boldsymbol{e}}}_{{y}^{{\prime} }}+{E}_{{z}^{{\prime} }}{{\boldsymbol{e}}}_{{z}^{{\prime} }}\\ & = & \displaystyle \frac{{E}_{{x}^{{\prime} }}-{{iE}}_{{y}^{{\prime} }}}{\sqrt{2}}{{\boldsymbol{e}}}_{+}+\displaystyle \frac{{E}_{{x}^{{\prime} }}+{{iE}}_{{y}^{{\prime} }}}{\sqrt{2}}{{\boldsymbol{e}}}_{-}+{E}_{{z}^{{\prime} }}{{\boldsymbol{e}}}_{{z}^{{\prime} }}\\ & = & \displaystyle \sum _{n=-\infty }^{\infty }\left\{\displaystyle \frac{i({C}_{{\theta }^{{\prime} }}\cos {\theta }^{{\prime} }+{C}_{{\phi }^{{\prime} }})}{\sqrt{2}}\displaystyle \frac{{e}^{i{\psi }_{0}+{ikR}+i(n-1){\phi }^{{\prime} }}}{R}{{\boldsymbol{e}}}_{+}\right.\\ & & +\displaystyle \frac{i({C}_{{\theta }^{{\prime} }}\cos {\theta }^{{\prime} }-{C}_{{\phi }^{{\prime} }})}{\sqrt{2}}\displaystyle \frac{{e}^{i{\psi }_{0}+{ikR}+i(n+1){\phi }^{{\prime} }}}{R}{{\boldsymbol{e}}}_{-}\\ & & \left.-{{iC}}_{{\theta }^{{\prime} }}\sin {\theta }^{{\prime} }\displaystyle \frac{{e}^{i{\psi }_{0}+{ikR}+{in}{\phi }^{{\prime} }}}{R}{{\boldsymbol{e}}}_{{z}^{{\prime} }}\right\}.\end{eqnarray} \tag{ 49 }$$

Here, the electric field is expressed by the complex orthogonal unit vectors ${{\boldsymbol{e}}}_{\pm }\equiv ({{\boldsymbol{e}}}_{{x}^{{\prime} }}\pm i{{\boldsymbol{e}}}_{{y}^{{\prime} }})/\sqrt{2}$ whose helicities of the circularly polarized electric fields are positive and negative, respectively. For the positive helicity, the rotation of the electric field is counterclockwise when the observer is facing the oncoming radiation that is denoted as right-hand circular polarization (Hamaker & Bregman 1996; Trippe 2014). Conversely, for the negative helicity, the rotation of the electric field is clockwise and the radiation is denoted as left-hand circular polarization.

In the paraxial approximation (θ' ≪ 1), the longitudinal component of the electric field along the z'-axis is much smaller than the transverse component. With the assumption of $({x}_{0},{y}_{0},{z}_{0})=(0,0,0)$ and R ∼ z₁ (z₁ is the distance along the z'-axis from the origin to the observation point), the electric field can be expressed as follows:

$$\begin{eqnarray}{\boldsymbol{E}} & = & \displaystyle \sum _{n=-\infty }^{\infty }\left\{\displaystyle \frac{i({C}_{{\theta }^{{\prime} }}+{C}_{{\phi }^{{\prime} }})}{\sqrt{2}}\displaystyle \frac{{e}^{{{ikz}}_{1}+i(n-1){\phi }^{{\prime} }}}{{z}_{1}}{{\boldsymbol{e}}}_{+}\right.\\ & & \left.+\displaystyle \frac{i({C}_{{\theta }^{{\prime} }}-{C}_{{\phi }^{{\prime} }})}{\sqrt{2}}\displaystyle \frac{{e}^{{{ikz}}_{1}+i(n+1){\phi }^{{\prime} }}}{{z}_{1}}{{\boldsymbol{e}}}_{-}\right\}.\end{eqnarray} \tag{ 50 }$$

The degree of circular polarization of the emitted photon can be represented by the Stokes parameter, which is expressed as ((Hamaker & Bregman 1996; Trippe 2014), Section 7.2 in (Jackson 1999))

$$\begin{eqnarray}\displaystyle \frac{V}{I} & = & \displaystyle \frac{{({C}_{{\theta }^{{\prime} }}\cos {\theta }^{{\prime} }+{C}_{{\phi }^{{\prime} }})}^{2}-{({C}_{{\theta }^{{\prime} }}\cos {\theta }^{{\prime} }-{C}_{{\phi }^{{\prime} }})}^{2}}{{({C}_{{\theta }^{{\prime} }}\cos {\theta }^{{\prime} }+{C}_{{\phi }^{{\prime} }})}^{2}+{({C}_{{\theta }^{{\prime} }}\cos {\theta }^{{\prime} }-{C}_{{\phi }^{{\prime} }})}^{2}}\\ & = & \displaystyle \frac{2{C}_{{\theta }^{{\prime} }}{C}_{{\phi }^{{\prime} }}\cos {\theta }^{{\prime} }}{{C}_{{\theta }^{{\prime} }}^{2}{\cos }^{2}{\theta }^{{\prime} }+{C}_{{\phi }^{{\prime} }}^{2}}.\end{eqnarray} \tag{ 51 }$$

The Bessel function of the first kind can be expressed as (Section 2.11 in (Watson 1962))

$$\begin{eqnarray}&&{J}_{n}({b}_{1})=\displaystyle \frac{1}{n!}{\left(\displaystyle \frac{{b}_{1}}{2}\right)}^{n}(1+\mu ),\end{eqnarray} \tag{ 52 }$$

where

$$\begin{eqnarray}&&\mu \equiv \exp \left\{\displaystyle \frac{1}{n+1}{\left(\displaystyle \frac{{b}_{1}}{2}\right)}^{2}\right\}-1.\end{eqnarray} \tag{ 53 }$$

When $\mu \ll 1$, the Bessel function of the first kind and its derivative can be expressed as

$$\begin{eqnarray}&&{J}_{n}({b}_{1})=\displaystyle \frac{1}{n!}{\left(\displaystyle \frac{{b}_{1}}{2}\right)}^{n},\end{eqnarray} \tag{ 54 }$$

$$\begin{eqnarray}&&{J}_{n}^{{\prime} }({b}_{1})=\displaystyle \frac{1}{2(n-1)!}{\left(\displaystyle \frac{{b}_{1}}{2}\right)}^{n-1}.\end{eqnarray} \tag{ 55 }$$

By substituting Equations (54) and (55) into Equation (51), Equation (51) will become, in the limit of θ' ≪ 1,

$$\begin{eqnarray}&&\displaystyle \frac{V}{I}=\displaystyle \frac{2\alpha }{{\alpha }^{2}+1},\end{eqnarray} \tag{ 56 }$$

where

$$\begin{eqnarray}&&\alpha \equiv \displaystyle \frac{1-{\gamma }_{0}^{2}{\theta }^{{\prime} 2}+{a}_{0}^{2}/2}{1+{\gamma }_{0}^{2}{\theta }^{{\prime} 2}+{a}_{0}^{2}/2}.\end{eqnarray} \tag{ 57 }$$

Therefore, when $\mu \ll 1$, the Stokes parameter only depends on the Lorentz factor of the initial electron, the scattering angle of the photon, and the field strength parameter of the initial electromagnetic wave. The Stokes parameter does not depend on the wavelength of the initial electromagnetic wave, the harmonic number of the scattered photon, or the initial interaction angle.

The energy of the nth harmonic scattered photon can be calculated from Equation (45). In the limit θ' ≪ 1, the energy will be expressed as

$$\begin{eqnarray}{\omega }_{n} & = & \displaystyle \frac{n{\omega }_{0}}{1-{\beta }_{1z}(1+\cos {\theta }^{{\prime} }/\cos \zeta )}\\ & \simeq & \displaystyle \frac{2n(1+{\beta }_{z0}){\gamma }_{0}^{2}{\omega }_{0}}{1+{\gamma }_{0}^{2}{\theta }^{{\prime} 2}+{a}_{0}^{2}/2}.\end{eqnarray} \tag{ 58 }$$

The radiation energy per unit angular frequency, ω, and the solid angle, Ω', is expressed as (Section 14.5 in (Jackson 1999))

$$\begin{eqnarray} & & \displaystyle \frac{{d}^{2}I}{d\omega d{{\rm{\Omega }}}^{{\prime} }}=\displaystyle \frac{c}{2\pi }{R}^{2}{| {\boldsymbol{E}}| }^{2}\\ & = & \displaystyle \frac{{e}^{2}{k}^{2}{\lambda }_{0}^{2}{N}_{0}^{2}}{4{\pi }^{2}c}\\ & & \times \left[{\left\{\displaystyle \sum _{n=1}^{\infty }\displaystyle \frac{\sin (\overline{k}{\eta }_{0})}{\overline{k}{\eta }_{0}}\left(\displaystyle \frac{{{nk}}_{0}\cos {\theta }^{{\prime} }}{k\sin {\theta }^{{\prime} }}-{\beta }_{1z}\displaystyle \frac{\sin {\theta }^{{\prime} }}{\cos \zeta }\right){J}_{n}({b}_{1})\right\}}^{2}\right.\\ & & \left.+{\left\{\displaystyle \sum _{n=1}^{\infty }\displaystyle \frac{\sin (\overline{k}{\eta }_{0})}{\overline{k}{\eta }_{0}}{r}_{1}{k}_{0}{J}_{n}^{{\prime} }({b}_{1})\right\}}^{2}\right].\end{eqnarray} \tag{ 59 }$$

Here, $d{{\rm{\Omega }}}^{{\prime} }=d{\phi }^{{\prime} }d{\theta }^{{\prime} }\sin {\theta }^{{\prime} }$. This spatial radiation energy distribution is independent of the wavelength of the initial electromagnetic wave because the variables ${{nk}}_{0}/k$, ${\beta }_{1z}/\cos \zeta$, b₁, and ${r}_{1}{k}_{0}$ are independent of the wavelength.

The number of nth harmonic photons emitted per second toward the scattering angle between ${\theta }_{1}^{{\prime} }$ and ${\theta }_{2}^{{\prime} }$ can be approximately calculated from Equation (59):

$$\begin{eqnarray}N & = & \displaystyle \frac{{N}_{e}{\rm{\Delta }}\omega }{{\hslash }{\omega }_{n}}{\displaystyle \int }_{0}^{2\pi }d{\phi }^{{\prime} }{\displaystyle \int }_{{\theta }_{1}^{{\prime} }}^{{\theta }_{2}^{{\prime} }}d{\theta }^{{\prime} }\sin {\theta }^{{\prime} }\displaystyle \frac{{d}^{2}I}{d\omega d{{\rm{\Omega }}}^{{\prime} }}\\ & = & \displaystyle \frac{{N}_{e}}{n{\hslash }{N}_{0}}{\displaystyle \int }_{0}^{2\pi }d{\phi }^{{\prime} }{\displaystyle \int }_{{\theta }_{1}^{{\prime} }}^{{\theta }_{2}^{{\prime} }}d{\theta }^{{\prime} }\sin {\theta }^{{\prime} }\displaystyle \frac{{d}^{2}I}{d\omega d{{\rm{\Omega }}}^{{\prime} }}\\ & = & {N}_{e}F.\end{eqnarray} \tag{ 60 }$$

Here, F is the emission rate,

$$\begin{eqnarray}&&F\equiv \displaystyle \frac{1}{n{\hslash }{N}_{0}}{\int }_{0}^{2\pi }d{\phi }^{{\prime} }{\int }_{{\theta }_{1}^{{\prime} }}^{{\theta }_{2}^{{\prime} }}d{\theta }^{{\prime} }\sin {\theta }^{{\prime} }\displaystyle \frac{{d}^{2}I}{d\omega d{{\rm{\Omega }}}^{{\prime} }},\end{eqnarray} \tag{ 61 }$$

and N_e is the number of electrons interacting with the electromagnetic wave per second. Here, we made the calculation by approximating $\overline{k}=0$ and the bandwidth of the scattered photons arising from the finite number of periods of the initial electromagnetic wave interacting with the electron, ${\rm{\Delta }}\omega /{\omega }_{n}=1/({{nN}}_{0})$ (Esarey et al. 1993). The number of photons and the emission rate are proportional to N₀.

Equation (50) indicates that the electric field of the emitted photon is elliptically polarized and can be decomposed into circularly polarized components with positive and negative helicities. Each component carries $+{\hslash }$ SAM and $-{\hslash }$ SAM, respectively. The positive helicity component possesses the phase term $\exp \{i(n-1){\phi }^{{\prime} }\}$, which means that the fundamental component is not an optical vortex but that the higher harmonics ($n\geqslant 2$) are vortices carrying OAM equal to $(n-1){\hslash }$. Conversely, the negative helicity component possesses the phase term $\exp \{i(n+1){\phi }^{{\prime} }\}$ and all harmonics are vortices carrying OAM equal to $(n+1){\hslash }$.

The degree of circular polarization of the emitted photon is expressed by the Stokes parameter. In the regions of $0\lt {\gamma }_{0}{\theta }^{{\prime} }\lt 15$ and $0\lt {a}_{0}\lt 10$, μ of Equation (53) is less than unity up to n = 3. Therefore, the Stokes parameter is a function of the Lorentz factor, the scattering angle of the photon, and the field strength parameter. The spatial distributions of the Stokes parameter calculated from Equation (56) are shown in Figures 2(a)–(d). We used the parameters ${\gamma }_{0}=2000$ and a₀ = 0.1–10. Stokes parameters of $V/I=1$ and −1 indicate 100% circular polarization with positive and negative helicity, respectively. The maximum scattering angle giving a circular polarization higher than 90% with positive helicity increases as the field strength parameter increases, as shown in Figure 2(e). Around the z'-axis in the scattering angles of ${\theta }^{{\prime} }\lt 0.6/{\gamma }_{0}$ and ${\theta }^{{\prime} }\lt 4.4/{\gamma }_{0}$, the degree of circular polarization is higher than 90% with positive helicity when a₀ = 1 and 10, respectively. Note that the nth higher harmonic photons emitted inside this scattering angle carry OAM equal to $(n-1){\hslash }$. The degree of circular polarization decreases as the scattering angle increases. The polarization changes from circular polarization with positive helicity to linear polarization and then back again to circular polarization with negative helicity at large scattering angles. All harmonic photons emitted in the area of circular polarization with negative helicity carry OAM equal to $(n+1){\hslash }$.

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The spatial radiation energy distributions of the emitted photons for each harmonic number are shown in Figures 3 and 4. We used the parameters ${\gamma }_{0}=2000$, n = 1, 2, and 3, and a₀ = 1 and 10 for Figures 3 and 4, respectively, and calculated the 90° interaction (ζ₀ = π/2 rad). A single peaked intensity distribution is observed only for the fundamental harmonic photons (Figures 3(a) and 4(a)). An annular shape with zero intensity at the center axis can be seen only for the higher harmonics (Figures 3(b), (c), 4(b), and (c)). Note that this feature in the higher harmonics corresponds to a typical characteristic of an optical vortex. The majority of the emitted radiation energy is concentrated around the central axis where the helicity is positive. As already described, the contour of the spatial radiation energy distribution does not depend on the wavelength of the initial electromagnetic wave.

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Figure 5 shows the emission rate of NITS calculated from Equation (61). The emission rate is calculated by integrating the solid angle up to ${\theta }_{2}^{{\prime} }$, where circular polarization with positive helicity is obtained, as shown in Figure 2(e). We used the parameters ${\gamma }_{0}=2000$, a₀ = 0.1–10, ζ₀ = π/2 rad, and N₀ = 100. Here, we assume that the transverse diameter of the initial electromagnetic wave is equal to or longer than the longitudinal pulse width and ${\eta }_{0}$ corresponds to half the longitudinal pulse width. The emission rate as a function of the harmonic numbers and the field strength parameters are shown in Figures 5(a) and (b), respectively. The emission rate of higher harmonic photons becomes close to that of the fundamental photons as the field strength parameter increases. One can see a saturation of the emission rate in the range of ${a}_{0}\gt 2$. Therefore, the emission rates for both the fundamental and the second harmonics do not significantly change in this range. In addition, the emission rate is independent of the Lorentz factor of the initial electron, as shown in Figure 5(c). This is because the emission rate is integrated over the angle normalized by the Lorentz factor.

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As seen in Equation (58), maximum photon energy can be obtained at the center axis (${\theta }^{{\prime} }=0$). Figure 6(a) shows the maximum photon energy of the second harmonics as a function of the initial injection angle. The photon energy is normalized by $2{\gamma }_{0}^{2}{\omega }_{0}$. The calculation parameters can be seen in the figure. Here, we assume ${\beta }_{0}\sim 1$. The photon energy at the 90° interaction is half that of the head-on interaction, similar to normal inverse Thomson scattering (Taira et al. 2011). This feature of inverse Thomson scattering will enable us to produce energy tunable X-ray and gamma-ray vortices in the laboratory.

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The energy of a photon decreases as its scattering angle increases, as shown in Figure 6(b). In addition, the energy drastically decreases in the range of ${a}_{0}\gt 1$, as shown in Figure 6(c). This is due to a decrease in the longitudinal electron velocity and the growth of the transverse helical motion, as indicated in Equations (25) and (26). The photon energy is almost constant and equal to that of normal inverse Thomson scattering in the range of ${a}_{0}\lt 0.1$. In addition, as shown in 5(b), the emission rate of the second harmonic photons is significantly smaller than that of the fundamental photons. Therefore, when ${a}_{0}\lt 0.1$, the scattering process can be considered as ordinary inverse Thomson scattering.