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Dependence of effective spectrum width of synchrotron radiation on particle energy

For an exact quantitative description of spectral properties in the theory of synchrotron radiation, the concept of effective spectral width is introduced. In the classical theory, numeric calculations of effective spectral width (using an effective width not exceeding 100 harmonics) for polarization components of synchrotron radiation are carried out. The dependence of the effective spectral width and initial harmonic on the energy of a radiating particle is established.


Introduction
As one of the major quantitative characteristics of spectral distributions for electromagnetic radiation, one commonly uses the concept of spectral half-width. For spectral distributions having a sharp maximum, spectral half-width is the most informative physical characteristic.
However, once a spectral distribution has no pronounced maximum, spectral half-width ceases to be an adequate quantitative characteristic. In particular, this is exactly the case of spectral distributions for synchrotron radiation (SR), and therefore SR spectral half-width has neither been calculated theoretically, nor measured experimentally.
Instead of spectral half-width, the present study proposes to introduce a new precise quantitative characteristic of SR spectral distributions: effective spectral width. It is shown how this quantity can be calculated theoretically, and which physically relevant information can be obtained using this quantity.
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]) and numerous articles. Among the most important SR physical features, one should take into account a high polarization degree of radiation and a unique structure of spectral distributions. All theoretically predictable SR properties have been confirmed by experiment. The development of SR theory allows one not only to predict radiation characteristics qualitatively, but also to offer exact quantitative characteristics of physically important properties. For example, the high degree of SR polarization was qualitatively predicted by theory more than half a century ago (see, e.g., [1]); precise quantitative characteristics for linear polarization were also obtained, whereas such quantitative characteristics for circular polarization were obtained much later [8][9][10][11].
In order to set up the problem, we now present some well-known expressions of the classical SR theory for the physical characteristics of synchrotron radiation, which can be found in [1][2][3][4][5][6][7].
The spectral-angular distribution for radiation power of SR polarization components can be written as (1. 1) Here, the following notation is used: θ is the angle between the control magnetic field strength and the radiation field pulse; ν is the number of an emitted harmonic; the charge orbital motion rate is v = cβ, where c is the speed of light; W is the total radiated power of unpolarized radiation, which can be written as where e is the particle charge; R is the orbit radius; H is the control field strength; m 0 is the charge rest mass; γ is the relativistic factor. The index s numbers the polarization components: s = 2 corresponds to the σ-component of linear polarization; s = 3 corresponds to the πcomponent 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 (β; ν, θ) have the form [1][2][3][4][5][6][7] Here, J ν (x) and J ′ ν (x) are the Bessel functions and their derivatives. In what follows, the case of an electron is considered, which corresponds to ε = 1.

Spectral distribution for polarization components of synchrotron radiation in the upper half-space
It is well known [1][2][3][4][5][6][7] that the angle range 0 θ < π/2 (this range will be called the upper half-space) is dominated by right-hand circular polarization, and the angle range π/2 < θ π (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 [8][9][10][11]). However, if we integrate in (1. 1) over θ (0 θ π), then the differences in the spectral distribution of right-and left-hand circular polarizations disappear. To reveal these differences, the expressions (1. 1) can be represented as and it suffices to study the properties of functions F (+) s (β; ν) (respectively, the properties of functions Φ (+) s (β)), due to the evident relations Integration over θ in the upper half-space 0 θ π/2 in (2. 1) can be carried out exactly, yielding the expressions (2. 3) The sums over the harmonics ν in (2. 1) can also be calculated exactly, yielding the expressions The function χ 1 (β) introduced above was defined and studied in [8]. In particular, it was shown [8] that in the segment 0 β 1 the function χ 1 (β) is finite and decreases monotonously; at the end of this segment, it takes the following values: 3 Effective spectral width for polarization components of synchrotron radiation As one of the quantitative characteristics of physical properties for spectral distributions of SR polarization components, it is proposed to introduce the concept of effective spectral width Λ s (β). Let us define Λ s (β) as follows.
For each fixed value of β, we examine the quantities The following relations are obvious: Obviously, such values ν (1) , ν (2) do exist for any β (for example, ν (1) = 1 necessarily yields such a finite value ν (2) ). It is equally obvious that condition (3. 3) alone is generally insufficient to determine the pair of values ν (1) , ν (2) . Let us now choose such ν s (β), as well as the minimum of the non-negative value The effective spectral width Λ s (β) is defined by the expression Consequently, effective spectral width is the minimum spectral range that accounts for at least half of all the radiated power of a given polarization component. The harmonics ν (1) s (β) and ν (2) s (β) determine the beginning and the end of this minimum spectral range. A definition equivalent to the one presented above for effective spectral width can be given using the concept of partial contributions P s (β; ν) for individual spectral harmonics, introduced in [12]. Namely, we suppose, according to [12], .
(3. 6) We choose some values ν (1) s (β) and ν (2) s (β) such that the minimum difference ν Introducing Λ s (β) in accordance with (3. 5), we arrive at the following equivalent definition: effective spectral width is the minimum spectral range at which the sum of partial contributions for individual harmonics is not less than 1/2. In practice, the most interesting case is the ultra-relativistic limit (β ≈ 1, equivalent to γ ≫ 1). In this case, the analytical study of effective spectral width and other physically interesting quantitative characteristics for spectral distributions of SR polarization components can be significantly extended. This study was carried out in [13].
Given a particular value of β (or γ), it is a purely computational task to obtain the exact values of Λ s (β) and ν (1) s (β). In this article, we present a numerical study of the region 1 Λ s (β) 100. It is essential to observe the following. The effective width Λ s (β) is a positive integer, so there exists a range of β (corresponding to a range of γ; hereinafter, we only indicate γ) in which Λ s (β) is constant.

Analysis of numerical results for effective spectral width of synchrotron radiation
The main results of a numerical study for effective spectral width of SR polarization components are given by our Table. The numerical study is carried out as follows. For each type of polarization s, we examine the sequences of integers Λ s = 1 , 2 , 3 ... and ν (1) and determine the regions of values γ s for which the condition (3. 8) (equivalent to (3. 4)) is satisfied. It is clear that the boundary points of possible regions for γ s can be found, according to (3. 8), as solutions of the equations Let us consider in more detail the calculation method and the results pertaining to the σcomponent of linear polarization. These results are given in the column s = 2 of our Table. We denote ν At the first step, we examine the smallest possible value Λ 2 = 1. In this case, equations (4. 1) have the form According to the results of [12], all partial contributions at k 2 are such that P 2 (β; k) < 1/2 for all the values of γ, whereas in the first harmonic (k = 1) such a region of values γ does exist, and equation (4. 2) at k = 1 has a single root, γ 2 = γ 2 (1 , 1), shown in the Table. Consequently, in the region of values γ 2 andγ 2 for Λ = 1, which is also indicated in the Table. Next, we examine the value Λ 2 = 2. In this case, equations (4. 1) have the form Since at the previous step it has been established that in the region 1 γ γ 2 (1 , 1) the effective width is Λ 2 = 1, it is required to examine the solutions of equations (4. 4) only in the region γ > γ 2 (1 , 1) , (4. 5) Analysis of equations (4. 4) shows that these equations have a unique solution, γ 2 (2 , 1), only for k = 1, where γ 2 (1 , 1) < γ 2 (2 , 1). Consequently, in the region of values γ At the next step, we examine the value Λ 2 = 3. Equations (4. 1) at Λ 2 = 3 have the form P 2 (β; k) + P 2 (β; k + 1) + P 2 (β; k + 2) − 1 2 = 0 , (4. 7) and we should only be concerned with solutions of these equations that belong to the region since at the previous steps it has been found that the effective spectral width at 1 γ γ 2 (2 , 1) is such that Λ 2 2. It turns out that under restriction (4. 8) equations (4. 7) possess the solutions γ 2 (3 , 1) at k = 1, and γ 2 (3 , 2) at k = 2. These solutions obey the inequalities γ 2 (2 , 1) < γ 2 (3 , 2) < γ 2 (3 , 1).
In general, the column s = 2 of the  Consequently, in the column s = 2 we indicate the regions of values γ 2 (Λ 2 )), (4. 15) for which the effective spectral width for the σ-component of SR linear polarization equals to Λ 2 . We also indicate the initial points of the effective spectral width. These points are not determined uniquely. For the smallest initial value ν 2 , the range of values (4. 15) taken by γ is the largest one, while this region is the smallest one for the largest possible valueν 2 .
For the other polarization components, the results of calculation are given in the respective columns of the Table. In particular, the Table shows that for equal values Λ s the corresponding values γ s obey the inequalities (4. 16) At a fixed energy γ, the corresponding values of Λ s are restricted by In this way, for each polarization component of synchrotron radiation we have found energy regions at which the effective spectral width equals to Λ s , and the initial harmonic of this effective width is determined. Numeric calculations have been carried out in the case Λ s 100.
In the ultrarelativistic case, the corresponding results have been obtained in [13].