## Abstract

In this work, scattering of an incident electric field from a moving atom is reexamined classically in two steps: the time-dependent current density created by the field inside the atom is first calculated under the electric-dipole approximation, and is then used to calculate the field scattered from the atom. Unlike the conventional frame-hopping method, the present method does not need to treat the Doppler effect as an effect separated from the scattering process, and it derives instead of simply uses the Doppler effect.

© 2012 Optical Society of America

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### Equations (10)

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(1)
$$\mathrm{j\u20d7}(\mathrm{r\u20d7},t)={\mathrm{j\u20d7}}_{0}(t)\delta (\mathrm{r\u20d7}-\mathrm{v\u20d7}t),$$
(2)
$$m\frac{{d}^{2}}{d{t}^{2}}(\mathrm{v\u20d7}t+\mathrm{R\u20d7})+m{\omega}_{0}^{2}\mathrm{R\u20d7}=e{\mathrm{E\u20d7}}_{0}{e}^{i\mathrm{k\u20d7}\xb7(\mathrm{v\u20d7}t+\mathrm{R\u20d7})-i\omega t},$$
(3)
$$\frac{{d}^{2}\mathrm{R\u20d7}}{d{t}^{2}}+{\omega}_{0}^{2}\mathrm{R\u20d7}=\frac{e}{m}{\mathrm{E\u20d7}}_{0}{e}^{-i\omega (1-\hat{k}\xb7\mathrm{v\u20d7}{c}^{-1})t}\equiv \frac{e}{m}{\mathrm{E\u20d7}}_{0}{e}^{-i\overline{\omega}t},$$
(4)
$$\mathrm{R\u20d7}=\frac{e}{m({\omega}_{0}^{2}-{\overline{\omega}}^{2})}{\mathrm{E\u20d7}}_{0}{e}^{-i\overline{\omega}t}$$
(5)
$$\mathrm{j\u20d7}(\mathrm{r\u20d7},t)=i\frac{{e}^{2}\overline{\omega}}{m({\overline{\omega}}^{2}-{\omega}_{0}^{2})}{\mathrm{E\u20d7}}_{0}{e}^{-i\overline{\omega}t}\delta (\mathrm{r\u20d7}-\mathrm{v\u20d7}t).$$
(6)
$${\nabla}^{2}\mathrm{A\u20d7}-\frac{1}{{c}^{2}}\frac{{\partial}^{2}\mathrm{A\u20d7}}{\partial {t}^{2}}=-\frac{4\pi}{c}\mathrm{j\u20d7}.$$
(7)
$$\mathrm{A\u20d7}(\mathrm{r\u20d7},t)=\frac{1}{c}\int \mathrm{d}{\mathrm{r\u20d7}}_{1}\int \mathrm{d}{t}_{1}\frac{\mathrm{j\u20d7}({\mathrm{r\u20d7}}_{1},{t}_{1})}{|\mathrm{r\u20d7}-{\mathrm{r\u20d7}}_{1}|}\delta ({t}_{1}+\frac{|\mathrm{r\u20d7}-{\mathrm{r\u20d7}}_{1}|}{c}-t)\phantom{\rule{0ex}{0ex}}=\frac{i{e}^{2}\overline{\omega}}{mc({\overline{\omega}}^{2}-{\omega}_{0}^{2})}{\mathrm{E\u20d7}}_{0}\int \mathrm{d}{t}_{1}\frac{{e}^{-i\overline{\omega}{t}_{1}}}{|\mathrm{r\u20d7}-\mathrm{v\u20d7}{t}_{1}|}\delta ({t}_{1}+\frac{|\mathrm{r\u20d7}-\mathrm{v\u20d7}{t}_{1}|}{c}-t),$$
(8)
$$\mathrm{A\u20d7}(\mathrm{r\u20d7},t)\simeq \frac{i{e}^{2}\overline{\omega}}{mc({\overline{\omega}}^{2}-{\omega}_{0}^{2})|\mathrm{r\u20d7}|}{\mathrm{E\u20d7}}_{0}\int \mathrm{d}{t}_{1}{e}^{-i\overline{\omega}{t}_{1}}{(1-\hat{r}\xb7\mathrm{v\u20d7}{c}^{-1})}^{-1}\phantom{\rule{0ex}{0ex}}\delta ({t}_{1}+\frac{|\mathrm{r\u20d7}|{c}^{-1}-t}{1-\hat{r}\xb7\mathrm{v\u20d7}{c}^{-1}})\phantom{\rule{0ex}{0ex}}=\frac{i{e}^{2}\overline{\omega}}{mc({\overline{\omega}}^{2}-{\omega}_{0}^{2})(1-\hat{r}\xb7\mathrm{v\u20d7}{c}^{-1})|\mathrm{r\u20d7}|}{\mathrm{E\u20d7}}_{0}{e}^{-i\tilde{\omega}t+i\tilde{k}|\mathrm{r\u20d7}|},$$
(9)
$${\mathrm{E\u20d7}}_{s}(\mathrm{r\u20d7},t)=\frac{i}{\tilde{k}}\nabla \times \nabla \times \mathrm{A\u20d7}(\mathrm{r\u20d7},t)\phantom{\rule{0ex}{0ex}}={\tilde{k}}^{2}(\hat{r}\times {\mathrm{E\u20d7}}_{s}^{\prime})\times \hat{r}\frac{{e}^{i\tilde{k}|\mathrm{r\u20d7}|}}{|\mathrm{r\u20d7}|}\phantom{\rule{0ex}{0ex}}+[3\hat{r}(\hat{r}\xb7{\mathrm{E\u20d7}}_{s}^{\prime})-{\mathrm{E\u20d7}}_{s}^{\prime}](\frac{1}{{|\mathrm{r\u20d7}|}^{3}}-\frac{i\tilde{k}}{{|\mathrm{r\u20d7}|}^{2}}){e}^{i\tilde{k}|\mathrm{r\u20d7}|}\phantom{\rule{0ex}{0ex}}\simeq {\tilde{k}}^{2}(\hat{r}\times {\mathrm{E\u20d7}}_{s}^{\prime})\times \hat{r}\frac{{e}^{i\tilde{k}|\mathrm{r\u20d7}|}}{|\mathrm{r\u20d7}|},$$
(10)
$$\overline{\omega}=\omega \frac{1-\hat{k}\xb7\mathrm{v\u20d7}{c}^{-1}}{1-\hat{r}\xb7\mathrm{v\u20d7}{c}^{-1}},$$