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Generation of twisted photons by undulators filled with dispersive medium

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Abstract

We study the radiation of twisted photons by undulators filled with a homogeneous dielectric dispersive medium and obtain the general formulas for the average number of radiated twisted photons. We consider in detail the radiation of undulators in the dipole regime and the radiation of helical and planar wigglers. The selection rules for radiation of twisted photons by such undulators are found. These selection rules generalize the known ones for undulators in a vacuum. In the case of a medium with plasma permittivity, the lower undulator harmonics do not form. As a result, in such undulators, the main contribution to radiation coming from the lowest allowed harmonic possesses a nonzero projection of orbital angular momentum in the paraxial regime. We investigate the influence of the anomalous Doppler effect on the properties of radiated twisted photons and find the parameters of the system ensuring that the produced radiation, in particular, the Vavilov–Cherenkov radiation, is a pure source of twisted photons with a definite nonzero projection of orbital angular momentum. As the examples, we consider the radiation of twisted photons by beams of electrons and protons in the undulators filled with helium and xenon. The parameters are chosen so as to be achievable at the present experimental facilities.

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Notes

  1. We refer to an undulator as a wiggler when the dipole approximation for description of its radiation (see Sect. 2.1) is not applicable.

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Acknowledgements

This work was supported by the Russian Science Foundation (Project No. 17-72-20013).

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Correspondence to O. V. Bogdanov.

Appendices

Appendix A: Mode functions of twisted photons in the dielectric plate

Let the twisted photon detected by the detector possess the helicity s, the projection of the total angular momentum m to the axis 3 (the axis z), the projection of the momentum \(k_3\), and the modulus of the perpendicular component of the momentum \(k_\perp \). The energy of such a photon is

$$\begin{aligned} k_0=\sqrt{k_\perp ^2+k_3^2}, \end{aligned}$$
(A1)

and its state in the Coulomb gauge in the vacuum is characterized by the wave function [37, 48, 50, 52,53,54, 59]

$$\begin{aligned} \begin{aligned}&\psi _3(m,k_3,k_\perp ;\mathbf {x})=j_m(k_\perp x_+,k_\perp x_-) e^{ik_3x_3},\\&\quad \psi _\pm (s,m,k_3,k_\perp ;\mathbf {x})=\frac{in_\perp }{s\pm n_3}\psi _3(m\pm 1,k_3,k_\perp ;\mathbf {x}),\\&\varvec{\psi }(s,m,k_3,k_\perp ;\mathbf {x})= \frac{1}{2}\big [\psi _-(s,m,k_3,k_\perp ;\mathbf {x})\mathbf {e}_+\\&\quad +\psi _+(s,m,k_3,k_\perp ;\mathbf {x})\mathbf {e}_- \big ]+\psi _3(m,k_3,k_\perp ;\mathbf {x})\mathbf {e}_3, \end{aligned} \end{aligned}$$
(A2)

where \(n_\perp :=k_\perp /k_0, n_3:=k_3/k_0\), and \(\mathbf {e}_\pm =\mathbf {e}_1\pm i\mathbf {e}_2\). The basis unit vectors \(\{\mathbf {e}_1,\mathbf {e}_2,\mathbf {e}_3\}\) constitute a right-handed triple and the unit vector \(\mathbf {e}_3\) is directed along the z axis. Notice that we use the word detector in a somewhat abstract sense. It can be a specially designed detector of twisted photons, or an atom, or a nucleus.

The presence of the dielectric plate changes the vacuum mode functions of twisted photons. Let the dielectric plate be situated at \(z\in [-L,0]\) and the detector of twisted photons be located in the vacuum in the region \(z>0\). Then, the mode functions of the twisted photons have the form [64]:

$$\begin{aligned} a\varvec{\psi }(s,m,k_3,k_\perp ), \end{aligned}$$
(A3)

for \(z>0\);

$$\begin{aligned} a\big [b_+\varvec{\psi }'(1,m,k'_3,k_\perp ) +b_-\varvec{\psi }'(-1,m,k'_3,k_\perp ) +(k'_3\leftrightarrow -k'_3)\big ], \end{aligned}$$
(A4)

for \(z\in [-L,0]\);

$$\begin{aligned}&a\big [a_+\varvec{\psi }(1,m,k_3,k_\perp ) +d_+\varvec{\psi }(1,m,-k_3,k_\perp ) +a_-\varvec{\psi }(-1,m,k_3,k_\perp )\nonumber \\&\quad +d_-\varvec{\psi }(-1,m,-k_3,k_\perp )\big ], \end{aligned}$$
(A5)

for \(z<-L\). Here

$$\begin{aligned}&\psi '_3(m,k'_3,k_\perp )=\psi _3(m,k'_3,k_\perp ),\nonumber \\&\psi '_\pm (s',m,k'_3,k_\perp )=\frac{in_\perp }{s'\varepsilon ^{1/2}(k_0)\pm n'_3}\psi '_3(m\pm 1,k'_3,k_\perp ), \end{aligned}$$
(A6)

and

$$\begin{aligned} n'_3:=k_3'/k_0=\sqrt{\varepsilon (k_0)-n_\perp ^2}. \end{aligned}$$
(A7)

Besides,

$$\begin{aligned} \begin{aligned} b_\pm&=\frac{\varepsilon ^{1/2}\pm s}{4\varepsilon n'_3}(\pm sn'_3+\varepsilon ^{1/2}n_3),\\ a_{\pm }&=\frac{2 (1\pm s)\varepsilon n_3n'_3\cos (k'_3L)-i(\varepsilon ^{2}n_{3}^{2}+n'^2_{3}\pm s\varepsilon (n_{3}^{2}+n'^2_{3}))\sin (k'_3L)}{4\varepsilon n_3n'_3}e^{ik_3L}, \\ d_{\pm }&=-i\frac{\varepsilon ^{2}n_{3}^{2}-n'^2_{3}\pm s\varepsilon (n_{3}^{2}-n'^2_{3})}{4\varepsilon n_3n'_3}\sin (k'_3L) e^{-ik_3L}, \end{aligned} \end{aligned}$$
(A8)

where s is the helicity of the mode function (A3)–(A5). The coefficients (A8) obey the unitarity relation

$$\begin{aligned} 1+|d_+|^2+|d_-|^2=|a_+|^2+|a_-|^2, \end{aligned}$$
(A9)

for real-valued \(k_3\). The constant a is found from the normalization of the mode functions:

$$\begin{aligned} |a|^{-2}=|a_+|^2+|a_-|^2=\Big |1+\frac{1}{8}\Big [(\varepsilon ^2+1)\Big (\frac{n^2_3}{n'^2_3} +\frac{n'^2_3}{\varepsilon ^2 n^2_3}\Big )-4\Big ]\sin ^2(k'_3 L)\Big |. \end{aligned}$$
(A10)

The above formulas can be generalized to the case of the medium with absorption [64].

Appendix B: Proof of the reflection symmetry

Let us prove some general properties of the expression (92) of [64] for the average number of twisted photons produced by charged particles moving through the dielectric plate. Let \(A(s,m,k_\perp ,k_3;K]\) be the amplitude of radiation of a twisted photon by the classical current \(K_i\) entering into (92) of [64]. Then, on rotating the current \(K_i(x)\) around the detector axis by the angle of \(\varphi , K_i\rightarrow K_i^\varphi \), the amplitude transforms as

$$\begin{aligned} A(s,m,k_\perp ,k_3;K^\varphi ]=e^{im\varphi } A(s,m,k_\perp ,k_3;K], \end{aligned}$$
(B1)

i.e., it has the same transformation law as in a vacuum. Consequently, the selection rules established in Sec. 2 of [65] also hold for the radiation of twisted photons by charged particles in the presence of a dielectric plate. Furthermore, the radiation produced by the current of particles moving parallel to some plane containing the detector axis, i.e., the current being such that \(\arg K_+=const\), obeys the reflection symmetry

$$\begin{aligned} \mathrm{d}P(s,m,k_\perp ,k_3)=\mathrm{d}P(-s,-m,k_\perp ,k_3). \end{aligned}$$
(B2)

The proof of this relation is the same as it was given in [37] for the radiation from a charged particle moving along a planar trajectory in a vacuum. Selecting suitably the axes x and y, one may put \(K_+(x)=K_-(x)\). Then, performing the rotation around the z axis by the angle of \(\pi \), we obtain that \(K^\pi _+(x)=K^\pi _-(x)\) and

$$\begin{aligned}&K_3\rightarrow K^\pi _3,\qquad K_\pm \rightarrow -K^\pi _\pm ,\qquad \Phi _3(s,m)\rightarrow \Phi _3(-s,-m),\nonumber \\&\quad \Phi _\pm (s,m)\rightarrow -\Phi _\mp (-s,-m). \end{aligned}$$
(B3)

Whence

$$\begin{aligned} A(s,m,k_\perp ,k_3;K_\pi ]=A(-s,-m,k_\perp ,k_3;K]=e^{im\pi }A(s,m,k_\perp ,k_3;K]. \end{aligned}$$
(B4)

Taking the modulus squared of the both parts of the last equality, we arrive at the relation (B2).

Appendix C: Polarization of helical undulator radiation

The degree of circular polarization of radiation (34) is specified by the ratio

$$\begin{aligned} A(s):=\frac{\mathrm{d}P(s)}{\mathrm{d}P(1)+\mathrm{d}P(-1)}. \end{aligned}$$
(C1)

In the paraxial approximation and for \(m\ne 0\),

$$\begin{aligned} A(s)=\frac{\big [J'_m(\varsigma mz) -s\varsigma (n_k-1/z)J_m(\varsigma mz) \big ]^2}{2\big [J'^2_m(\varsigma mz) +(n_k-1/z)^2J^2_m(\varsigma mz)\big ]},\qquad z:=\frac{2n_k}{n_k^2+{\bar{\chi }}_c-{\bar{\chi }}}=\frac{\varsigma {\bar{k}}_0}{m}n_k,\nonumber \\ \end{aligned}$$
(C2)

where \({\bar{\chi }}_c:=1+K^{-2}\). For \(m=0\), i.e., for VC radiation, we have

$$\begin{aligned} A(s)=\frac{\big [J_1({\bar{k}}_0n_k) +s\varsigma n_kJ_0({\bar{k}}_0n_k) \big ]^2}{2\big [J^2_1({\bar{k}}_0n_k) +n_k^2J^2_0({\bar{k}}_0n_k)\big ]}. \end{aligned}$$
(C3)

As follows from (C2), all the radiated photons possess the helicity s provided that

$$\begin{aligned} J'_m(\varsigma mz)=-s\varsigma (n_k-1/z)J_m(\varsigma mz),\qquad J_1({\bar{k}}_0n_k)=s\varsigma n_kJ_0({\bar{k}}_0n_k). \end{aligned}$$
(C4)

The linear polarization appears when \(A(1)=A(-1)=1/2\), i.e.,

$$\begin{aligned} (n_k-1/z)J_m(\varsigma mz)J'_m(\varsigma mz)=0,\qquad J_0({\bar{k}}_0n_k)J_1({\bar{k}}_0n_k)=0. \end{aligned}$$
(C5)

The second equalities in (C4) and (C5) correspond to the case \(m=0\). For \(m\ne 0\), the equations (C4) and (C5) should be solved with account for the radiation spectrum (37) written in the form of the last equality in (C2). For \(m=0\), the condition of the existence of VC radiation, \({\bar{\chi }}\geqslant {\bar{\chi }}_c\), should be satisfied.

If one regards the degree of polarization (C2) as a function of \(n_k\), then the regions where the radiation of photons with definite helicity dominates are separated by the roots of the equation (C5) and the regions with the different signs of the helicity are interlaced. Developing (C2) for \(m\ne 0\) as a series in \(n_k\), it is not difficult to see that for \(n_k\rightarrow 0\), i.e., in the ultraparaxial regime, the radiation with \(s{{\,\mathrm{sgn}\,}}(m)=1\) prevails. Furthermore, the positivity of the photon energy (37) implies that, for \({\bar{\chi }}<{\bar{\chi }}_c\), the photons are radiated with \(\varsigma {{\,\mathrm{sgn}\,}}m=1\).

Fig. 9
figure 9

The function \(|z(n_k)|\) for the different values of \({\bar{\chi }}\). The undulator strength parameter \(K=1/2\). The inclined straight line is \(z=\varsigma {\bar{k}}_0 n_k/m=n_k/2\). The straight vertical line is \(n_k=\sqrt{{\bar{\chi }}-{\bar{\chi }}_c}\). The straight horizontal lines: the dotted line is \(z=1\); the dashed lines are \(z=b_{2,1}\approx 1.53, z=c_{2,1}\approx 2.57\); the thin solid line is \(z=b_{1,1}\approx 1.84\). The unique intersection point of the line \(z=\varsigma {\bar{k}}_0 n_k/m\) with the curve \(z(n_k)\) gives \(n_k\) and, thereby, \({\bar{k}}_0\) for the radiated twisted photons. Then, it is not difficult to see which polarization dominates for these parameters

For \(m\ne 0\), the graphical solution of the first equation in (C5) is given in Fig. 9. We describe it below. Let \(j_{m,p}\) and \(j'_{m,p}, p=\overline{1,\infty }\), be the positive roots of \(J_m(x)\) and \(J'_m(x)\), respectively, and

$$\begin{aligned} b_{m,p}:=j_{m,p}/|m|,\qquad c_{m,p}:=j'_{m,p}/|m|. \end{aligned}$$
(C6)

The properties of zeros of the Bessel functions (see, e.g., [105]) imply that \(b_{m,p}>1, c_{m,p}>1\) and \(b_{m,p}\rightarrow 1, c_{m,p}\rightarrow 1\) for \(|m|\rightarrow \infty \). For \({\bar{\chi }}<{\bar{\chi }}_c\), the function \(z(n_k)\) providing the solution to the first equation in (C5) has a unique maximum at the point

$$\begin{aligned} n_k=n_k^0=\sqrt{{\bar{\chi }}_c-{\bar{\chi }}},\qquad z(n_k^0)=1/n_k^0, \end{aligned}$$
(C7)

and the inflection point at \(n_k=\sqrt{3}n_k^0\). At the maximum, \(z(n^0_k)\geqslant 1\) when \({\bar{\chi }}\in [K^{-2},{\bar{\chi }}_c)\). The plots of \(z(n_k)\) for \({\bar{\chi }}\geqslant \chi _c\) are given in Fig. 9. The properties of polarization of radiation following from the plots in Fig. 9 for \(m\ne 0\) are described in Sect. 2.2.1.

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Bogdanov, O.V., Kazinski, P.O. & Lazarenko, G.Y. Generation of twisted photons by undulators filled with dispersive medium. Eur. Phys. J. Plus 135, 901 (2020). https://doi.org/10.1140/epjp/s13360-020-00924-5

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