Numerical analysis of the location of spectral maxima of polarization components of synchrotron radiation in the classical theory

TL;DR

This paper numerically investigates how the spectral maxima of polarization components in synchrotron radiation shift with increasing particle energy, focusing on higher harmonics up to the 100th.

Contribution

It provides a detailed numerical analysis of the spectral maxima shifts in polarization components of synchrotron radiation within classical theory.

Findings

Higher harmonic peaks shift with increasing particle energy

Numerical calculations extend up to the 100th harmonic

Results enhance understanding of spectral behavior in synchrotron radiation

Abstract

Shift of higher harmonic peak in the spectrum of synchrotron radiation with increasing energy of the radiating particle is investigated by numerical method. Calculations were carried out for the range till 100 harmonics for all polarization components.

Tables

Table 1. Table 1: Boundary values of energy (corresponding to β ν ( s ) subscript superscript 𝛽 𝑠 𝜈 \beta^{(s)}_{\nu} ) for spectral maximum shift

$\nu$	${\beta }_{\nu }^{\left(2\right)}$	${\gamma }_{\nu }^{\left(2\right)}$	${\beta }_{\nu }^{\left(3\right)}$	${\gamma }_{\nu }^{\left(3\right)}$	${\beta }_{\nu }^{\left(0\right)}$	${\gamma }_{\nu }^{\left(0\right)}$	${\beta }_{\nu }^{\left(-1\right)}$	${\gamma }_{\nu }^{\left(-1\right)}$	${\beta }_{\nu }^{\left(1\right)}$	${\gamma }_{\nu }^{\left(1\right)}$
\k

Equations

$$\gamma =\frac{E}{m_0{c}^2}=\frac{1}{\sqrt{1-{\beta }^2}},\beta =\frac{v}{c},$$

$$W_s=W\sum _{\nu =1}^{\infty }{\int }_0^{\pi }{f}_s(\beta ;\nu ,\theta )\sin \theta d\theta .$$

$$W=\frac{2}{3}\frac{c{e}^2}{R^2}({\gamma }^2-1{)}^2=\frac{2e^4{H}^2({\gamma }^2-1)}{3m_0^2{c}^3},$$

$$f_2(\beta ;\nu ,\theta )=\frac{3{\nu }^2}{2{\gamma }^4}{J}_{\nu }^{\prime }{}^2(x);f_3(\beta ;\nu ,\theta )=\frac{3{\nu }^2}{2{\gamma }^4}\frac{{\cos }^2\theta }{{\beta }^2{\sin }^2\theta }{J}_{\nu }^2(x);$$

$$f_{\pm 1}(\beta ;\nu ,\theta )=\frac{3{\nu }^2}{4{\gamma }^4}{\left[J_{\nu }^{\prime }(x)\pm \epsilon \frac{\cos \theta }{\beta \sin \theta }{J}_{\nu }(x)\right]}^2;x=\nu \beta \sin \theta ;\epsilon =-\frac{e}{|e|};$$

$$f_0(\beta ;\nu ,\theta )=f_2(\beta ;\nu ,\theta )+f_3(\beta ;\nu ,\theta )=f_1(\beta ;\nu ,\theta )+f_{-1}(\beta ;\nu ,\theta ).$$

$$W_s=W_s^{(+)}+W_s^{(-)},W_s^{(\pm )}=W\sum _{\nu =1}^{\infty }{F}_s^{(\pm )}(\beta ;\nu );$$

$$F_s^{(+)}(\beta ;\nu )={\int }_0^{\pi /2}{f}_s(\beta ;\nu ,\theta )\sin \theta d\theta ,F_s^{(-)}(\beta ;\nu )={\int }_{\pi /2}^{\pi }{f}_s(\beta ;\nu ,\theta )\sin \theta d\theta ,$$

$$F_s^{(-)}(\beta ;\nu )=F_s^{(+)}(\beta ;\nu ),s=0,2,3;$$

$$F_1^{(-)}(\beta ;\nu )=F_{-1}^{(+)}(\beta ;\nu ),F_{-1}^{(-)}(\beta ;\nu )=F_1^{(+)}(\beta ;\nu ).$$

$$F_2^{(+)}(\beta ;\nu )=\frac{3\nu }{4\gamma {}^4{\beta }^3}\left[2{\beta }^2{J}_{2\nu }^{\prime }(2\nu \beta )+{\beta }^2{\int }_0^{2\nu \beta }{J}_{2\nu }(x)dx-2\nu \beta {\int }_0^{2\nu \beta }\frac{J_{2\nu }(x)}{x}dx\right],$$

$$F_3^{(+)}(\beta ;\nu )=\frac{3\nu }{4\gamma {}^4{\beta }^3}\left[2\nu \beta {\int }_0^{2\nu \beta }\frac{J_{2\nu }(x)}{x}dx-{\int }_0^{2\nu \beta }{J}_{2\nu }(x)dx\right],$$

$$F_0^{(+)}(\beta ;\nu )=F_2^{(+)}(\beta ;\nu )+F_3^{(+)}(\beta ;\nu )=\frac{3\nu }{4\gamma {}^4{\beta }^3}\left[2{\beta }^2{J}_{2\nu }^{\prime }(2\nu \beta )-(1-{\beta }^2){\int }_0^{2\nu \beta }{J}_{2\nu }(x)dx\right],$$

$$F_{\pm 1}^{(+)}(\beta ;\nu )=\frac{1}{2}{F}_0^{(+)}(\beta ;\nu )\pm \frac{3\nu J{}_{\nu }^2(\nu \beta )}{4\gamma {}^4{\beta }^2}.$$

$$F_s^{(+)}(\beta ;1)=F_s^{(+)}(\beta ;2).$$

$$F_s^{(+)}(\beta ;2)=F_s^{(+)}(\beta ;3).$$

$$F_s^{(+)}(\beta ;\nu )=F_s^{(+)}(\beta ;\nu +1),$$

$${\beta }_{\nu -1}^{(s)}⩽\beta <{\beta }_{\nu }^{(s)}\left(\text{mbox}respectively{\gamma }_{\nu -1}^{(s)}⩽\gamma <{\gamma }_{\nu }^{(s)}\right)$$

$${\gamma }_{\nu }^{(3)}>{\gamma }_{\nu }^{(1)}>{\gamma }_{\nu }^{(0)}>{\gamma }_{\nu }^{(2)}>{\gamma }_{\nu }^{(-1)}\left(\text{mbox}respectively{\beta }_{\nu }^{(3)}>{\beta }_{\nu }^{(1)}>{\beta }_{\nu }^{(0)}>{\beta }_{\nu }^{(2)}>{\beta }_{\nu }^{(-1)}\right).$$

$${\nu }_s^{(max)}\approx a_s{\gamma }^3,$$

$$a_3\approx 0,215881,a_1\approx 0,372875,a_0\approx 0,428718,a_2\approx 0,503287,a_{-1}\approx 0,783608.$$

Full text

Numerical analysis of the location of spectral maxima of polarization components of synchrotron radiation in the classical theory

A. S. Loginov , A. D. Saprykin Tomsk State University, Russia. E-mail: as_loginov@phys.tsu.ru Tomsk State University, Russia. E-mail: alexx.saprykin.phys@gmail.com

Abstract

Shift of higher harmonic peak in the spectrum of synchrotron radiation with increasing energy of the radiating particle is investigated by numerical method. Calculations were carried out for the $\nu^{(max)}\leqslant 100$ for all polarization components.

1 Introduction

At present, the theory of synchrotron radiation is a fairly well developed area of theoretical physics. Its main elements are described in monographs (e.g. [1, 2, 3, 4, 5, 6, 7, 8]) and numerous articles

Spectral distribution of SR has one physically important feature, namely increasing energy of radiating particle leads to shift of spectral maximum to higher harmonics. This feature was predicted in works [9, 10] and has been confirmed by numerous experiments.

It’s convenient to represent energy $E$ of radiating particle by relativistic factor $\gamma\,(\gamma\geqslant 1)$, or quantity $\beta\,(0\leqslant\beta<1)$

[TABLE]

where $m_{0}$ – the charge rest mass, $v$ is the charge orbital motion speed, $c$ – the speed of lite. With respect to notation of rotation frequency of radiating particle by $\omega_{0}$, spectrum of SR has frequencies $\omega=\omega_{0}\nu$, where whole number $\nu=1,\,2,\,3,...$ – number of radiated harmonic.

For nonrelativistic particle which corresponds to the condition $\beta\ll 1$ (equivalently $\gamma\sim 1$) maximum of SR spectrum falls on first harmonics $\nu^{(max)}=1$. With increasing energy of the particle spectral maximum shifts sequentially to 2,3,4… harmonics and for ultra-relativistic particle (corresponding to $\gamma\gg 1$) asymptotic estimate $\nu^{(max)}\sim\gamma^{3}$ holds (It was shown in [9, 10]). This estimate is correct for polarization component of SR.

Exact values of $\beta$ (or $\gamma$) which lead to shift of maximum in SR spectral distribution from harmonic $\nu$ to harmonic $\nu+1$ can be carried out by numerical calculations. within the framework of classical theory such calculations have been carried out in the work [11] for $\nu\leqslant 15$ for the total radiated power and the linear polarization component of SR. This paper presents the results of numerical analysis for $\nu\leqslant 100$ and all polarization component of SR.

2 Spectral distribution of SR polarization component in upper half-space

To put a numerical analysis of the problem considered in this article we present here some well-known expression of the physical characteristics SR in the classical theory which can be found in [1, 2, 3, 4, 5, 6, 7, 8].

The spectral - angular distribution of the radiation power of the SR polarization component can be represented by

[TABLE]

Here, the following notation is used: $\theta$ is the angle between the control magnetic field strength and the radiation field pulse, $W$ – the total radiated power of unpolarized radiation, which can be written as

[TABLE]

where $e$ – the particle charge;; $R$ – orbit radius;; $H$ – is the control field strength. The index $s$ numbers the polarization components:: $s=2$ corresponds to $\sigma$ - component of linear polarization; $s=3$ corresponds to $\pi$ - component of linear polarization; $s=1$ corresponds to right-hand circular polarization; $s=-1$ – corresponds to left-hand circular polarization;; $s=0$ corresponds to the power of unpolarized radiation. The functions $f_{s}(\beta;\,\nu,\,\theta)$ have the form

[TABLE]

[TABLE]

[TABLE]

Here, $J_{\nu}(x)$ and $J_{\nu}^{\prime}(x)$ are the Bessel functions and their derivatives. In what follows, the case of an electron is considered, which corresponds to $\varepsilon=1$.

It is well known [1, 2, 3, 4, 5, 6, 7] that the angle range $0\leqslant\theta<\pi/2$ (this range will be called the upper half-space) is dominated by right-hand circular polarization, and the angle range $\pi/2<\theta\leqslant\pi$ (this range will be called the lower half-space) is dominated by left-hand circular polarization (exact quantitative characteristics of SR properties were first obtained in [12, 13, 14, 15]). However, if we integrate in (2. 1)) over $\theta\ \ (0\leqslant\theta\leqslant\pi)$ then the differences in the spectral distribution of right- and left-hand circular polarizations disappear. To reveal these differences, a new form of expression (2. 1)) has been proposed in [16] It can be represented as

[TABLE]

[TABLE]

and it suffices to study the properties of functions $F_{s}^{(+)}(\beta;\,\nu)$ due to the evident relations

[TABLE]

[TABLE]

Integration over $\theta$ in the upper half-space $0\leqslant\theta\leqslant\pi/2$ in (2. 4) can be carried out exactly, yielding the expressions

[TABLE]

[TABLE]

[TABLE]

[TABLE]

3 Numerical analysis of the shift of the maximum in the SR spectrum with increasing the energy of the radiating particle

Expressions ( ref B6) make possible to translate into formal language the task of analyzing the shift of SR spectral maximum at change of radiating particle energy $E$.

As is known, in the non-relativistic limit (which corresponds to $\beta\rightarrow 0$ and $\gamma\rightarrow 1$) maxima of any components $W_{s}^{(+)}$ in the spectral distribution occur in first harmonic ($\nu^{(max)}_{s}=1$). With increasing the energy of particle (increase of $\beta$ or, respectively, $\gamma$) maximum in the spectrum of $W_{s}^{(+)}$ goes to the second harmonic.

Obviously, the value of $\beta=\beta^{(s)}_{1}$, which is carried out at the maximum shift in the spectrum of $W_{s}^{(+)}$ to the second harmonic is the root of the equation

[TABLE]

So for $0\leqslant\beta<\beta^{(s)}_{1}$ (or $1\leqslant\gamma<\gamma^{(s)}_{1}$) the maximum in spectrum of $W_{s}^{(+)}$ falls on the first harmonic

With increasing $\beta$ (or $\gamma$) maximum in spectrum of $W_{s}^{(+)}$ shift to second harmonic. The value $\beta=\beta^{(s)}_{2}$, corresponding to this shift, is root of equation

[TABLE]

So for $\beta^{(s)}_{1}\leqslant\beta<\beta^{(s)}_{2}$ (or $\gamma^{(s)}_{1}\leqslant\gamma<\gamma^{(s)}_{2}$) maximum in spectrum of $W_{s}^{(+)}$ falls on second harmonic.

In general, for a fixed $s$ determining the sequence of numbers $\beta=\beta^{(s)}_{\nu}$ as the roots of equations

[TABLE]

It can be found that in energy range

[TABLE]

maximum in spectrum of $W_{s}^{(+)}$ falls on harmonic $\nu$.

The following Table for $\nu\leqslant$ 100 for each polarization $s$ $\beta^{(s)}_{\nu}$ and $\gamma^{(s)}_{\nu}$ are given. It should be noted that following inequalities are correct

[TABLE]

It is also easy to see from Table, that since $\nu^{(max)}_{-1}(\beta)=8$ inequalities, obtained in [16] for ultra-relativistic particle, hold.

In the ultra-relativistic case ($1\ll\gamma$) for $\nu^{(max)}_{s}=1$ asymptotic formulas are known.

[TABLE]

where $a_{s}$ – numbers, approximate values of ones are equal (see [16]) to

[TABLE]

If $\nu^{(max)}_{s}=$ 100 value corresponding $\gamma$ determined by the asymptotic formulas (3. 6), they may differ from the table of values in no more than fourth place.

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Acknowledgements

The work of Loginov and Saprykin is supported by Tomsk State University Competitiveness Improvement Program.