Quantum synchrotron radiation in the case of a field with finite extension

The semiclassical operator method of Baier and Katkov allows one to obtain the spectrum of synchrotron radiation in a way similar to the classical derivation but which is fully valid also in the quantum case of very strong electromagnetic fields. In the usual calculation the extension of the field is taken to be infinite. In this paper we apply a numerical routine based on the semiclassical operator method to the case of a constant field but with a finite extension. For large extensions of the field one obtains the usual result of quantum synchrotron radiation, while in the limit of small extension of the field one obtains a spectrum resembling that of bremsstrahlung. We derive a formula for the radiation spectrum in this limit. In the transition toward shorter field extensions one finds that the power-spectrum increases for soft photons and slightly diminishes for harder photons. It is found that in the classical case the total power emitted decreases as the field extension decreases while in the quantum case the total power emitted is first increased and then decreases. Such an effect could be important for future $e^{+}{e}^{-}$ colliders such as the ILC or CLIC where the dominant energy and luminosity loss is due to synchrotron radiation by an $e^{-}/e^{+}$ in the field of the opposing bunch, often termed “beamstrahlung.” In this paper we also discuss how these effects, in the quantum case could be measured in an experiment using thin aligned single crystals and high energy electrons available at e.g. the CERN SPS North Area, and in the classical case could already be relevant at existing accelerators with conventional magnets providing the electromagnetic field.

Figure 1

A plot of the power spectra for the case $\chi =0.1$, $ϵ=1.5\mathrm{TeV}$ and different values of $\eta$. QS formula is the formula of quantum synchrotron radiation seen in Eq. (31). SF formula is the short field formula of Eq. (51) for the case of $\eta =0.1$.

Figure 2

A plot of the power spectra for the case $\chi =10$, $ϵ=1.5\mathrm{TeV}$ and different values of $\eta$. QS formula is the formula of quantum synchrotron radiation seen in Eq. (31). SF formula is the short field formula of Eq. (51) for the case of $\eta =1/4$.

Figure 3

A plot of the total power emitted in units of $P_0$ as function of $1/\eta$. In this case $\chi =0.1$. The red curve is obtained by the numerical procedure described in the text and the green dashed curve is a plot of Eq. (52).

Figure 4

A plot of the total power emitted in units of $P_0$ as function of $1/\eta$. In this case $\chi =10$. The red curve is obtained by the numerical procedure described in the text and the green dashed curve is a plot of Eq. (52).

Figure 5

A plot of the numerical values obtained for the power in the case of $\eta =2\sqrt{2}$ where the power is at a maximum as function of $\chi$ and a plot of an analytical fit valid for large values of $\chi$.