Exact solution of Hartemann–Luhmann equation of motion for a charged particle interacting with an intense electromagnetic wave/pulse

Abstract

We report an exact solution of the Hartemann–Luhmann equation of motion for a charged particle interacting with an intense electromagnetic wave/pulse. It is found that the radiation reaction force has a significant affect on the charged particle dynamics and the particle shows, on average, a net energy gain over a period of time. Further, using a MATHEMATICA based single particle code, the net energy gained by the particle is compared with that obtained using Landau–Lifshitz and Ford–O’Connell equation of motion, for different polarizations of the electromagnetic wave. It is found that the average energy gain is independent of both the chosen model equation and polarization of the electromagnetic wave. Our results thus show that the simpler and hence analytically tractable Hartemann–Luhmann equation of motion (as compared to Landau–Lifshitz and Ford–O’Connell equation of motion) is adequate for calculations of practical use (for e.g. energy calculation).

Analytical expressions for particle momentum and position

The integrals ${I}_{1}$, $\mathbf {I}_{2}$ and expressions for momentum and position of a particle, starting from origin with zero initial momentum and interacting with an elliptically polarized electromagnetic wave train, are given by

$$\begin{aligned} I_{1}&=\frac{1}{4} a_{0}^{2}\left[ -2\phi +(2\delta ^{2}-1)\sin (2\phi )\right] \end{aligned}$$

(30)

$$\begin{aligned} I_{2x}&=\frac{1}{24}a_{0}^{3}\delta \nonumber \\&\qquad \Big [(15-6\delta ^{2})\sin (\phi )+(2\delta ^{2}-1)\sin (3\phi )-12 \phi \cos (\phi )\Big ] \end{aligned}$$

(31)

$$\begin{aligned} I_{2y}&=\frac{1}{24} a_{0}^{3}g\sqrt{(1-\delta ^{2})}\Big [-3(2\delta ^{2}+3)\cos (\phi ) \nonumber \\&\quad -(2\delta ^{2}-1)(\cos (3\phi )+8)-12 \phi \sin (\phi )\Big ] \end{aligned}$$

(32)

$$\begin{aligned} {p}_{x}&= {\varDelta } \Big [ - a_{0} \delta (\cos (\phi ) -1) -\frac{1}{24} a_{0}^{3} \tau _{0}\delta \Big \{(15-6\delta ^{2})\sin (\phi ) \nonumber \\&\quad +(2\delta ^{2}-1)\sin (3\phi ) -12 \phi \cos (\phi )\Big \} \Big ] \end{aligned}$$

(33)

$$\begin{aligned} {p}_{y}&= \varDelta \Big [ - a_{0} g\sqrt{(1-\delta ^{2})} \sin (\phi ) -\frac{1}{24} a_{0}^{3} \tau _{0} g\sqrt{(1-\delta ^{2})} \nonumber \\&\quad \times \Big \{-3(2\delta ^{2}+3)\cos (\phi )-(2\delta ^{2}-1) \nonumber \\&(\cos (3\phi )+8) -12 \phi \sin (\phi )\Big \}\Big ] \end{aligned}$$

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$$\begin{aligned} p_{z}&= {\varDelta }\Bigg [ - \frac{a_{0}^{2}}{2}\Big (\sqrt{1-\delta ^{2}} \sin (\phi )^{2} + \delta ^{2} (\cos (\phi )-1)^{2}\Big ) \nonumber \\&\quad + \frac{1}{24} {\tau _{0}} a_{0}^{4} \Big \{(\delta ^{2} (\cos (\phi )-1))\Big \{(15-6\delta ^{2})\sin (\phi ) \nonumber \\&\quad +(2\delta ^{2}-1)\sin (3\phi )-12 \phi \cos (\phi )\Big \} \nonumber \\&\quad + (1-\delta ^{2}) \sin (\phi )\Big \{-3(2\delta ^{2}+3)\cos (\phi ) \nonumber \\&\quad -(2\delta ^{2}-1) (\cos (3\phi )+8) -12 \phi \sin (\phi )\Big \}\Big \}+ \frac{1}{1152} \nonumber \\&\quad \tau _{0}^{2} a_{0}^{6}\Big \{ \delta ^{2} \Big ((15-6\delta ^{2})\sin (\phi )+(2\delta ^{2}-1)\sin (3\phi )\nonumber \\&\quad -12 \phi \cos (\phi )\Big )^{2} + (1-\delta ^{2}) \Big (-3(2\delta ^{2}+3)\cos (\phi ) \nonumber \\&\quad -(2\delta ^{2}-1)(\cos (3\phi )+8) -12 \phi \sin (\phi )\Big )^{2} \Big \} \Bigg ] \end{aligned}$$

(35)

$$\begin{aligned} x(\phi )&=a_{0}\delta \big (\phi -\sin (\phi )\big ) \nonumber \\&\quad +\frac{\tau _{0}a_{0}^{2}\delta }{72}\Big [(-81+18\delta ^{2})\cos (\phi )+(1-2\delta ^{2})\cos (3\phi ) \nonumber \\&\quad +\Big \{80-16\delta ^{2} +9\phi \sin (\phi )\Big \}\Big ] \end{aligned}$$

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$$\begin{aligned} y(\phi )&=a_{0}\sqrt{1-\delta ^{2}}\big (1-\cos (\phi )\big ) \nonumber \\&\quad +\frac{\tau _{0}a_{0}^{2}}{72}\sqrt{1-\delta ^{2}} \Big \{24(1-\delta ^{2})\phi +36\phi \cos (\phi ) \nonumber \\&\quad -9(7+2\delta ^{2}) \sin (\phi ) +(1-2\delta ^{2})\sin (\phi )\Big \} \end{aligned}$$

(37)

$$\begin{aligned} z(\phi )&=\frac{1}{13824} \nonumber \\&\times \Bigg [13824\Bigg \{ \frac{a_{0}^{2}(2(1\!+\!2\delta ^{2})\phi -8\delta ^{2}\sin (\phi )\!+\!(-1\!+\!2\delta ^{2})\sin (\phi )}{8}\Bigg \} \nonumber \\&\quad +24\tau _{0}a_{0}^{3}\Big [a_{0} \Big \{-99+460\delta ^{2} +52\delta ^{4}+72\phi ^{2} \nonumber \\&\quad -24(-8+27\delta ^{2}+2\delta ^{4})\cos (\phi )+96(-1+2\delta ^{2})\cos (2\phi ) \nonumber \\&\quad +8\delta ^{2}(1-2\delta ^{2})\cos (3\phi ) +(1-2\delta ^{2})^{2}\cos (4\phi ) \nonumber \\&\quad -288\delta ^{2}\phi \sin (\phi )+72(-1+2\delta ^{2})\phi \sin (2\phi )\Big \}\Big ] \nonumber \\&\quad -a_{0}^{6}\tau _{0}^{2}\Big \{-12\phi (105+188\delta ^{2} - 188\delta ^{4} \nonumber \\&\quad -64\delta ^{6}+24\phi ^{2})+2304(-1+\delta ^{4})\phi \cos (\phi )+1152 \nonumber \\&\quad \times (1-2\delta ^{2})\phi \cos (2\phi )-36(1-2\delta ^{2})^{2}\phi \cos (4\phi ) \nonumber \\&\quad -576(-7-2\delta ^{2}+7\delta ^{4} +2\delta ^{6})\sin (\phi ) +9(-1+2\delta ^{2}) \nonumber \\&\quad \times (85-20\delta ^{2}+20\delta ^{4}-48\phi ^{2})\sin (2\phi )-64(1-2\delta ^{2}-\delta ^{4} \nonumber \\&\quad +2\delta ^{6})\sin (3\phi ) +36(1-2\delta ^{2})^{2}\sin (4\phi ) \nonumber \\&\quad +(-1+2\delta ^{2})^{3}\sin (6\phi )\Big \}\Bigg ] \end{aligned}$$

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