Semiclassical Description of Undulator Radiation

Abstract

We present a semiclassical approximation for treating the radiation from classical currents. In particular, we present exact quantum states of the quantized electromagnetic field interacting with classical currents. These states are used to calculate a probability of many-photon radiation from the vacuum initial state of the electromagnetic field. In this manner, in the present article, we study characteristics of electromagnetic radiation of a planar undulator. We find the total radiated energy and its spectral-angular distribution. We compare our results with ones obtained in the framework of classical electrodynamics, discussing differences introduced by accurate accounting for the quantum nature of electromagnetic radiation and present results of some numerical calculations that confirm, in particular, the latter discussion. In Appendix we present the calculation of the radiated energy using an alternative parametrization of the trajectory of electrons moving in a planar undulator.

Appendices

APPENDIX

Calculating Radiation of Planar Undulator in Semiclassical Approximation Using an Alternative Parametrization of Trajectory

In [15] a special trajectory for electrons moving in a plane undulator were used to calculate their radiation. It is supposed that the trajectory of electrons is plane and symmetrical relatively to the axis x, and consists of circular arcs of length l and radius R. This representation provides an alternative parametrization of the electron trajectory in plane undulator. In this appendix we apply the semiclassical method to calculate the radiation from electrons moving in such a trajectory.

Consider electrons moving in a periodic magnetic field parallel to the axis z, such that in each period the magnetic field is homogeneous and constant. The undulator is assumed to be of infinite length. A length l of each individual arc is related to the effective radius of curvature R via the so-called injection angle α = l/R, 0 < α < π. The velocity of the electrons is ${v}$ = cβ = ωR, where ω is the angular velocity. The electrons move along the axis x with an average velocity ${{{v}}_{0}}$ = c$\bar {\beta }$, and perform periodic oscillations along the axes x and y. This implies

$$\begin{gathered} \bar {\beta } = \beta {\text{sinc}}(\alpha {\text{/}}2),\quad T = 2\pi \omega _{0}^{{ - 1}} = 2\alpha {{\omega }^{{ - 1}}}, \\ {{\omega }_{0}} = \pi \omega {{\alpha }^{{ - 1}}} = \pi \beta c{{l}^{{ - 1}}}. \\ \end{gathered} $$

(29)

The trajectory at the time interval (0, T) can be presented as [15]

$$\begin{gathered} x(t) = \left\{ \begin{gathered} R[\sin (\alpha {\text{/}}2) + \sin (\omega t - \alpha {\text{/}}2)],\quad t \in {{T}_{1}} \hfill \\ R[3\sin (\alpha {\text{/}}2) + \sin (\omega t - \alpha {\text{/}}2)],\quad t \in {{T}_{2}}, \hfill \\ \end{gathered} \right. \\ y(t) = \left\{ \begin{gathered} R[\cos (\omega t - \alpha {\text{/}}2) - \cos (\alpha {\text{/}}2)],\quad t \in {{T}_{1}} \hfill \\ R[\cos (\alpha {\text{/}}2) - \cos (\omega t - \alpha {\text{/}}2)],\quad t \in {{T}_{2}}, \hfill \\ \end{gathered} \right. \\ \end{gathered} $$

(30)

where the time intervals T₁ and T₂ are defined as

$${{T}_{1}} = [0,T{\text{/}}2];\quad {{T}_{2}} = (T{\text{/}}2,T].$$

(31)

The current j ⁱ(x) formed by electrons moving in the trajectory (30) has the form

$$\begin{gathered} {{j}^{i}}(x) = e{{{v}}^{i}}(t)\delta (x - x(t))\delta (y - y(t))\delta (z - z(t)), \\ {{{v}}^{i}}(t) = {{{\dot {r}}}^{i}}(t) = (\dot {x}(t),\dot {y}(t),0), \\ \dot {x}(t) = \left\{ \begin{gathered} \omega R\cos [\alpha {\text{/}}2 - \omega t],\quad t \in {{T}_{1}} \hfill \\ \omega R\cos [3\alpha {\text{/}}2 - \omega t],\quad t \in {{T}_{2}}, \hfill \\ \end{gathered} \right. \\ \dot {y}(t) = \left\{ \begin{gathered} \omega R\sin [\alpha {\text{/}}2 - \omega t],\quad t \in {{T}_{1}} \hfill \\ - \omega R\sin [3\alpha {\text{/}}2 - \omega t],\quad t \in {{T}_{2}}. \hfill \\ \end{gathered} \right. \\ \end{gathered} $$

(32)

Let us calculate the radiation energy during a single period T. The functions y_kλ(t) at the interval t = T can be calculated as

$${{y}_{{{\mathbf{k}}\lambda }}}(T) = {{y}_{{{\mathbf{k}}\lambda }}}({{T}_{1}}) + {{y}_{{{\mathbf{k}}\lambda }}}({{T}_{2}}).$$

(33)

Using the definitions (4) for the wave vector k and polarization vectors ${{\epsilon }_{{{\mathbf{k}}\lambda }}}$, we obtain

$$\begin{gathered} {{y}_{{{\mathbf{k}}1}}}({{T}_{j}}) = {{z}_{j}}{{e}^{{i{{\phi }_{j}}}}}\cos \theta \int\limits_{\tau _{j}^{{{\text{in}}}}}^{\tau _{j}^{{{\text{out}}}}} {{{\omega }^{{ - 1}}}d{{\tau }_{j}}} \\ \, \times [c\bar {\beta }\cos \varphi + \omega R\cos {{\tau }_{j}}]\exp [i\kappa ({{\tau }_{j}})], \\ {{y}_{{{\mathbf{k}}2}}}({{T}_{j}}) = - {{z}_{j}}{{e}^{{i{{\phi }_{j}}}}}\int\limits_{\tau _{j}^{{{\text{in}}}}}^{\tau _{j}^{{{\text{out}}}}} {{{\omega }^{{ - 1}}}d{{\tau }_{j}}} \\ \, \times [c\bar {\beta }\sin \varphi + {{( - 1)}^{{j - 1}}}\omega R\sin {{\tau }_{j}}]\exp [i\kappa ({{\tau }_{j}})], \\ \end{gathered} $$

(34)

where we used the notation

$$\begin{gathered} \kappa ({{\tau }_{j}}) = c{{k}_{0}}{{\omega }^{{ - 1}}}{{\tau }_{j}} - \xi \sin {{\tau }_{j}},\quad \xi = R{{k}_{0}}\sin \theta , \\ {{z}_{j}} = ie\exp [ - i\xi {{f}_{j}}(\varphi ,\alpha )]{{[\hbar c{{k}_{0}}{{(2\pi )}^{2}}]}^{{ - 1/2}}},\quad j = 1,2, \\ {{f}_{1}}(\varphi ,\alpha ) = \sin (\alpha {\text{/}}2)\cos \varphi - \cos (\alpha {\text{/}}2)\sin \varphi , \\ \end{gathered} $$

$$\begin{gathered} {{f}_{2}}(\varphi ,\alpha ) = \cos (\alpha {\text{/}}2)\sin \varphi + 3\sin (\alpha {\text{/}}2)\cos \varphi , \\ {{\tau }_{1}} = \omega t - \alpha {\text{/}}2 + \varphi ,\quad {{\tau }_{2}} = \omega t - 3\alpha {\text{/}}2 - \varphi , \\ \end{gathered} $$

(35)

$$\begin{gathered} \tau _{1}^{{{\text{in}}}} = \varphi - \alpha {\text{/}}2,\quad \tau _{1}^{{{\text{out}}}} = \varphi + \alpha {\text{/}}2, \\ \tau _{2}^{{{\text{in}}}} = - \varphi - \alpha {\text{/}}2,\quad \tau _{2}^{{{\text{out}}}} = \alpha {\text{/}}2 - \varphi , \\ {{\phi }_{1}} = {{\omega }^{{ - 1}}}\kappa (\alpha {\text{/}}2 - \varphi ),\quad {{\phi }_{2}} = {{\omega }^{{ - 1}}}\kappa (\varphi + 3\alpha {\text{/}}2). \\ \end{gathered} $$

Using the known expansions of trigonometric functions in terms of Bessel functions,

$$\exp ( - i\xi \sin \tau ) = \sum\limits_{n = - \infty }^{ + \infty } {{{J}_{n}}(\xi )\exp ( - in\tau ),} $$

$$\sin \tau \exp ( - i\xi \sin \tau ) = i\sum\limits_{n = - \infty }^{ + \infty } {J_{n}^{'}(\xi )\exp ( - in\tau ),} $$

(36)

$$\cos \tau \exp ( - i\xi \sin \tau ) = \sum\limits_{n = - \infty }^{ + \infty } {\frac{n}{\xi }{{J}_{n}}(\xi )\exp ( - in\tau ),} $$

we rewrite (34) as

$$\begin{gathered} {{y}_{{{\mathbf{k}}1}}}({{T}_{j}}) = {{z}_{j}}{{e}^{{i{{\phi }_{j}}}}}\sum\limits_{n = - \infty }^{ + \infty } {\cos \theta [n{{{({{k}_{0}}\sin \theta )}}^{{ - 1}}}} \\ \, + {{\omega }^{{ - 1}}}c\bar {\beta }\cos \varphi ]{{J}_{n}}(\xi ){{K}_{n}}({{T}_{j}}), \\ {{y}_{{{\mathbf{k}}2}}}({{T}_{j}}) = - {{z}_{j}}{{e}^{{i{{\phi }_{j}}}}}\sum\limits_{n = - \infty }^{ + \infty } {[{{\omega }^{{ - 1}}}c\bar {\beta }\sin \varphi {{J}_{n}}(\xi )} \\ \, + {{( - 1)}^{{j - 1}}}iRJ_{n}^{'}(\xi )]{{K}_{n}}({{T}_{j}}), \\ \end{gathered} $$

(37)

where functions K_n(T_j) have the form

$$\begin{gathered} {{K}_{n}}({{T}_{j}}) = \int\limits_{\tau _{j}^{{{\text{in}}}}}^{\tau _{j}^{{{\text{out}}}}} {d{{\tau }_{j}}\exp [i({{\omega }^{{ - 1}}}c{{k}_{0}} - n){{\tau }_{j}}]} \\ = \exp [{{( - 1)}^{j}}i({{\omega }^{{ - 1}}}c{{k}_{0}} - n)\varphi ]\frac{{\sin [\alpha ({{\omega }^{{ - 1}}}c{{k}_{0}} - n){\text{/}}2]}}{{({{\omega }^{{ - 1}}}c{{k}_{0}} - n){\text{/}}2}}. \\ \end{gathered} $$

(38)

The functions p_kλ(T) on the time interval T have the form

$${{p}_{{{\mathbf{k}}\lambda }}}(T) = {\text{|}}{{y}_{{{\mathbf{k}}\lambda }}}(T){{{\text{|}}}^{2}} = {\text{|}}{{y}_{{{\mathbf{k}}\lambda }}}({{T}_{1}}) + {{y}_{{{\mathbf{k}}\lambda }}}({{T}_{2}}){{{\text{|}}}^{2}}.$$

(39)

Substituting expressions (37) into (39), we obtain

$$\begin{gathered} {{p}_{{{\mathbf{k}}1}}}(T)\, = \,{{\left| {\sum\limits_{n = - \infty }^{ + \infty } {[n{{{({{k}_{0}}{\text{sin}}\theta )}}^{{ - 1}}}\, + \,{{\omega }^{{ - 1}}}c\bar {\beta }{\text{cos}}\varphi ]{\text{cos}}\theta {{J}_{n}}(\xi ){{S}_{1}}} } \right|}^{2}}, \\ {{p}_{{{\mathbf{k}}2}}}(T) = {{\left| {\sum\limits_{n = - \infty }^{ + \infty } {[{{\omega }^{{ - 1}}}c\bar {\beta }\sin \varphi {{J}_{n}}(\xi ){{S}_{1}} + iRJ_{n}^{'}(\xi ){{S}_{2}}]} } \right|}^{2}}, \\ {{S}_{1}} = {{z}_{1}}{{e}^{{i{{\phi }_{1}}}}}{{K}_{n}}({{T}_{1}}) + {{z}_{2}}{{e}^{{i{{\phi }_{2}}}}}{{K}_{n}}({{T}_{2}}), \\ {{S}_{2}} = {{z}_{1}}{{e}^{{i{{\phi }_{1}}}}}{{K}_{n}}({{T}_{1}}) - {{z}_{2}}{{e}^{{i{{\phi }_{2}}}}}{{K}_{n}}({{T}_{2}}). \\ \end{gathered} $$

(40)

The total radiated energy W(T) takes the form

$$W(T) = \int\limits_0^\infty {k_{0}^{2}d{{k}_{0}}\int\limits_0^\pi {{\text{sin}}\theta d\theta } } \int\limits_0^{2\pi } {d\varphi [{{p}_{{{\mathbf{k}}1}}}(T) + {{p}_{{{\mathbf{k}}2}}}(T)]} ,$$

(41)

where p_k1(T) and p_k2(T) are given by Eq. (40).