# Effective spectrum width of the synchrotron radiation

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## Abstract

For an exact quantitative description of spectral properties of synchrotron radiation (SR), the concept of effective width of the spectrum is introduced. In the most interesting case, which corresponds to the ultrarelativistic limit of SR, the effective width of the spectrum is calculated for the polarization components, and new physically important quantitative information on the structure of spectral distributions is obtained. For the first time, the spectral distribution for the circular polarization component of the SR for the upper half-space is obtained within classical theory.

## Keywords

Synchrotron Radiation Circular Polarization Spectral Distribution Polarization Component Effective Width## 1 Introduction

The theory of synchrotron radiation (SR) is currently a quite well-developed branch of theoretical physics. Its basic elements have been stated in books (e.g., [1, 2, 3, 4, 5]) and numerous articles. As one of the most physically important features of SR, one should mention a high polarization degree of radiation and a unique structure of spectral distribution in the ultrarelativistic limit. All the theoretically predicted properties of SR have been confirmed experimentally.

The development of the SR theory makes it possible not only to predict qualitatively the peculiar features of radiation, but also to propose exact quantitative characteristics of physically important properties.

For instance, the high polarization degree of SR was predicted qualitatively by theory more than half a century ago (see, e.g., [1]), and the linear polarization was given exact quantitative characteristics; however, it was not until much later that it proved to be possible to obtain exact quantitative characteristics for circular polarization [6, 7].

In this paper, we propose new exact quantitative characteristics of the SR spectral distribution, the effective width of the spectrum. We demonstrate how to calculate this quantity theoretically in the most interesting case, which corresponds to the ultrarelativistic limit of SR, and what physically important information can be obtained using this quantity. We demonstrate that for spectral distributions of SR the well-known characteristics - the half-width of the spectrum is less informative than the proposed effective width. We place in the Appendix all necessary formulas for our purposes that describe the spectral-angular distribution of the SR. Our consideration is performed in the framework of the classical theory of SR. It is well known that such a theory is valid with high accuracy for accessible parameters of the electron beam in modern accelerators and storage rings. However, for SR in cosmic space quantum effects can be quite essential (see. Ref. [8]) and thus can significantly change classical results (which may be the subject of a separate study).

## 2 The effective width of the spectrum of SR polarization components

From a practical point of view, the most interesting case is presented by the ultrarelativistic limit (\(\beta \approx 1\), which is equivalent to \( \gamma \gg 1\)) of SR. In this case, a big part of the study of the effective width, as well as the study of other physically interesting quantitative characteristics of the spectral distribution of SR polarization components, can be done analytically.

## 3 The ultrarelativistic case

### 3.1 The effective width of the spectrum in the ultrarelativistic limit

*s*) the frequencies \(\nu <\nu _{s}^{(max)} \) will be referred to as low, and the frequencies \(\nu >\nu _{s}^{(max)}\) will be referred to as high. It is obvious that the portion \(r_{s}^{(1)}\) of the effective width of the spectrum which corresponds to the low frequencies is given by

Spectral emission characteristics in the ultra relativistic limit

0 | 2 | 3 | \(-1\) | \(+1\) | |
---|---|---|---|---|---|

\(y_{s}^{(max)}\) | 2.85812e\(-\)01 | 3.35524e\(-\)01 | 1.43921e\(-\)01 | 5.22405e\(-\)01 | 2.48583e\(-\)01 |

\(F_{s}^{(+)}\left( y_{s}^{(max)}\right) \) | 2.84696e\(-\)01 | 2.35158e\(-\)01 | 5.39423e\(-\)02 | 5.12872e\(-\)02 | 2.37335e\(-\)01 |

\(\Phi _{s}^{(+)}\left( y_{s}^{(max)}\right) \) | 7.17052e\(-\)02 | 6.90125e\(-\)02 | 6.90380e\(-\)03 | 2.26361e\(-\)02 | 5.21376e\(-\)02 |

\(\eta _{s}^{(max)}\) | 1.43410e\(-\)01 | 1.57743e\(-\)01 | 1.10461e\(-\)01 | 2.01806e\(-\)01 | 1.34433e\(-\)01 |

\(y_{s}^{(1)}\) | 3.49398e\(-\)02 | 4.87043e\(-\)02 | 1.08505e\(-\)02 | 1.22065e\(-\)01 | 2.71081e\(-\)02 |

\(F_{s}^{(+)}\left( y_{s}^{(1)}\right) \) | 1.98326e\(-\)01 | 1.67772e\(-\)01 | 3.39459e\(-\)02 | 3.84141e\(-\)02 | 1.61750e\(-\)01 |

\(\Phi _{s}^{(+)}\left( y_{s}^{(1)}\right) \) | 5.36798e\(-\)03 | 6.31493e\(-\)03 | 2.84108e\(-\)04 | 3.52777e\(-\)03 | 3.39459e\(-\)03 |

\(\eta _{s}^{(1)}\) | 1.0736e\(-\)02 | 1.44341e\(-\)02 | 4.54573e\(-\)03 | 3.14508e\(-\)02 | 8.75273e\(-\)03 |

\(y_{s}^{(2)}\) | 1.02680e\(+\)00 | 1.08939e\(+\)00 | 6.94023e\(-\)01 | 1.32370e\(+\)00 | 9.58312e\(-\)01 |

\(\Phi _{s}^{(+)}\left( y_{s}^{(2)}\right) \) | 2.55368e\(-\)01 | 2.25065e\(-\)01 | 3.15341e\(-\)02 | 5.96117e\(-\)02 | 1.97311e\(-\)01 |

\(\eta _{s}^{(2)}\) | 5.10736e\(-\)01 | 5.14434e\(-\)01 | 5.04546e\(-\)01 | 5.31451e\(-\)01 | 5.08753e\(-\)01 |

\(y_{s}^{(3)}\) | 1.10709e\(-\)02 | 1.44604e\(-\)02 | 4.90942e\(-\)03 | 3.59457e\(-\)02 | 9.26077e\(-\)03 |

\(\Phi _{s}^{(+)}\left( y_{s}^{(3)}\right) \) | 1.19916e\(-\)03 | 1.29074e\(-\)03 | 1.00954e\(-\)04 | 6.86800e\(-\)04 | 8.36754e\(-\)04 |

\(\eta _{s}^{(3)}\) | 2.39832e\(-\)03 | 2.95025e\(-\)03 | 1.61526e\(-\)03 | 6.12297e\(-\)03 | 2.15752e\(-\)03 |

\(y_{s}^{(4)}\) | 1.47628e\(+\)00 | 1.59002e\(+\)00 | 9.06361e\(-\)01 | 1.95582e\(+\)00 | 1.35291e\(+\)00 |

\(\Phi _{s}^{(+)}\left( y_{s}^{(4)}\right) \) | 3.31467e\(-\)01 | 2.96035e\(-\)01 | 3.79763e\(-\)02 | 7.97257e\(-\)02 | 2.52321e\(-\)01 |

\(\eta _{s}^{(4)}\) | 6.62933e\(-\)01 | 6.76652e\(-\)01 | 6.07621e\(-\)01 | 7.10772e\(-\)01 | 6.50593e\(-\)01 |

\(a_{s}^{(max)}\) | 4.28718e\(-\)01 | 5.03287e\(-\)01 | 2.15881e\(-\)01 | 7.83608e\(-\)01 | 3.72875e\(-\)01 |

\(a_{s}^{(1)}\) | 5.24096e\(-\)02 | 7.30564e\(-\)02 | 1.62757e\(-\)02 | 1.83097e\(-\)01 | 4.06621e\(-\)02 |

\(a_{s}^{(2)}\) | 1.54021e\(+\)00 | 1.63408e\(+\)00 | 1.04103e\(+\)00 | 1.98555e\(+\)00 | 1.43747e\(+\)00 |

\(a_{s}^{(3)}\) | 1.66063e\(-\)02 | 2.16906e\(-\)02 | 7.36413e\(-\)03 | 5.39186e\(-\)02 | 1.38912e\(-\)02 |

\(a_{s}^{(4)}\) | 2.21442e\(+\)00 | 2.38502e\(+\)00 | 1.35954e\(+\)00 | 2.93372e\(+\)00 | 2.02936e\(+\)00 |

\(b_{s}\) | 1.4878e\(+\)00 | 1.56103e\(+\)00 | 1.02476e\(+\)00 | 1.80245e\(+\)00 | 1.39681e\(+\)00 |

\(d_{s}\) | 2.19781e\(+\)00 | 2.36333e\(+\)00 | 1.35218e\(+\)00 | 2.87981e\(+\)00 | 2.01547e\(+\)00 |

\(r_{s}^{(1)}\) | 2.52929e\(-\)01 | 2.75607e\(-\)01 | 1.94782e\(-\)01 | 3.33163e\(-\)01 | 2.37837e\(-\)01 |

\(r_{s}^{(2)}\) | 1.32674e\(-\)01 | 1.43309e\(-\)01 | 1.05915e\(-\)01 | 1.70355e\(-\)01 | 1.25680e\(-\)01 |

\(r_{s}^{(3)}\) | 6.60535e\(-\)01 | 6.73701e\(-\)01 | 6.06006e\(-\)01 | 7.04649e\(-\)01 | 6.48435e\(-\)01 |

### 3.2 Another characteristics of spectral distributions in the ultrarelativistic limit

The expressions (10) and (13) make it possible to obtain exact quantitative data for interesting physical characteristics of spectral distributions. These quantitative data are presented in the following Table 1.

*s*-component of SR polarization, the portion of power radiated by the interval of frequencies \(0<\nu <\nu _{s}^{(k)}\) from the total power emitted at this polarization. It is obvious that the quantity \( 100\,\eta _{s}^{(k)}\) accordingly determines the percentage of power radiated at the spectral region \(0<\nu <\nu _{s}^{(k)}\). The order of distribution of \( \eta _{s}^{(k)}\) is identical with (23).

The numerical values \(\eta _{s}^{(max)}\) provide convincing evidence of the fact that a substantially larger part of the radiated power at each component of polarization is contributed by the region of high frequencies (the maximum percentage of radiated power at low frequencies is \(\sim \)20.2 % for \(s=-1\) and the minimum is \(\sim \)11.0 % for \(s=3\), and therefore from \(\sim \)80 % to \(\sim \)90 % of the radiated power is contributed by the high frequencies).

The numerical values \(\eta _{s}^{(1)}\) represent the portion of radiated power in the frequency range up to the beginning of the effective width of the spectrum. This portion is quite insignificant (the maximum percentage being \(\sim \)3.15 % for \(s=-1\) and the minimum being \(\sim \)0.45 % for \(s=3\)).

The values \(r_{s}^{(2)}=\eta _{s}^{(max)}-\eta _{s}^{(1)}\) determine the portion of radiated power at low frequencies that corresponds to the effective width of the spectrum (here, the maximum percentage of power radiated at low frequencies is \(\sim \)17.0 % for \(s=-1\) and the minimum percentage is \(\sim \)10.6 % for \(s=3\); there is an approximation \(r_{s}^{(1)}\sim 2r_{s}^{(2)}\), which is physically quite justified, since the radiated powers are approximately proportional to the frequency ranges).

In optics one characterizes the profiles of spectral distributions by introducing the concept of the half-width of the spectrum. Such a quantity can also be introduced in the case under consideration.

As we can see from the table, the half-width is attributed to the range from \(\sim \)60 to \(\sim \)70 % of radiated power, and the remaining part is radiated in the region of high frequencies (up to the beginning of the half-width, the radiated power is merely in the range of \(\sim \)0.16 to \(\sim \)0.61 %. Consequently, for spectral distributions of SR the concept of the half-width of the spectrum is less informative than the concept of the effective width of the spectrum that we have proposed in this article.

## 4 Brief summary

We give a new definition of the effective width for the SR spectrum and calculate the effective width for all polarization components of the SR in the ultra-relativistic limit. For the first time, the spectral distribution for the circular polarization component of the SR for the upper half-plane is obtained within classical theory. In addition, the relative radiation power emitted in some physically interesting spectral ranges is found.

## Notes

### Acknowledgments

Bagrov thanks FAPESP for its support and IF USP for its hospitality. The reported study of Gitman was partially supported by RFBR, research project No. 15- 02-00293a. Gitman thanks CNPq and FAPESP for their permanent support. The work of Loginov is partially supported by the Ministry of Science of the Russian Federation (Grant N^{ o } 2014/223), Code project 1766.

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