research papers
Universal function for the
of undulator radiation considering the energy spread effect^{a}RIKEN SPring8 Center, Koto 111, Sayo, Hyogo 6795148, Japan
^{*}Correspondence email: ztanaka@spring8.or.jp
Angular and spatial profiles of undulator radiation have been investigated to derive a universal function that evaluates the
of undulator radiation and takes into account the effects of electron beam emittance and energy spread. It has been found that the effects of energy spread on the angular divergence and source size can be expressed by simple analytic expressions, and a universal function has been derived by convolution with the electron beam distribution functions. Comparisons with numerical results have been carried out to show the validity and applicability of the universal function.Keywords: undulator radiation; universal function; brilliance; emittance; energy spread.
1. Introduction
which is usually defined by the density in the phase space spanned by position and angle, is a common figure of merit to specify the performance of a synchrotron radiation (SR) source. It specifies the number of photons in the phase space, and thus is strongly associated with the throughput of a SR beamline.
Strictly speaking, ) as a function in fourdimensional phase space (x, y, x′, y′). The distribution should then be convoluted with the electron beam distribution function in order to take into account the effects of the emittance and energy spread. If the experimental users of SR need to know the exact profile of in the fourdimensional phase space, the above method should be adopted. In practice, however, the is usually evaluated only onaxis, i.e. at the observation point x = y = x′ = y′ = 0 (onaxis hereinafter simply called brilliance), just to represent the performance of a SR source.
should be calculated by the method based on the theory of Wigner (Kim, 1986It is well known that the ; Chubar & Elleaume, 1998; Tanaka & Kitamura, 2001) to compute the of various kinds of SR sources. Although the can be precisely evaluated by means of these codes, it takes some time to obtain the computation results. In addition, these codes do not help the SR users understand the mechanism of how the electron beam quality contributes to the degradation.
is quite sensitive to the quality of the electron beam: lower beam quality results in significant degradation of especially in the case of undulator radiation (UR). It is therefore important to evaluate the effects of the electron beam quality, such as the emittance and energy spread. This actually requires a complicated numerical analysis using a large number of parameters to specify the SR source and accelerator performance, which is usually bulky for those who are not familiar with numerical methods or accelerator physics. This is the reason why a number of computer codes have been developed (Walker & Diviacco, 1991Instead of the numerical analysis described above, a simple method can be applied, in which the ^{−8} m rad and an energy spread of 10^{−3}.
is obtained by dividing the total by the effective optical emittance obtained by convolution between the natural optical emittance determined by the diffraction limit, and the electron beam emittance. This method enables the calculation only with computation of the Bessel functions of the first kind and several elementary functions, and thus has usually been used to evaluate the performance of a SR facility. It should be noted, however, that the effect of the energy spread is assumed to be negligible in the above method. This assumption is valid if the effect of the beam emittance is much larger than that of the energy spread, which is the case for a typical thirdgeneration SR facility with a beam emittance of 10Although the above conventional method can be applied in many cases, care should be taken if the undulator periodic number is large and/or the harmonic number of UR is high. Furthermore, the effect of the energy spread can be comparable with or larger than that of the finite emittance in SR facilities that accommodate accelerators to generate a highquality electron beam, which will soon be available thanks to the evolution of accelerator theories and technologies, such as ultralowemittance storage rings or energy recovery linacs. In a worst case, the conventional method can overestimate
in such SR facilities by up to two orders of magnitude. Thus the conventional formula should be improved to take into account the effects of the energy spread as well as the emittance for more precise evaluation of brilliance.In this paper, a new analytic expression is derived as a universal function to evaluate the
of UR. Then the universal function is compared with the conventional method and numerical analysis in order to investigate its validity and applicability.2. Analytical method
2.1. Basic formulae on UR
Let us first start with the complex amplitude of radiation emitted by a single electron, which is given by the temporal Fourier transform of the electric field (Chubar & Elleaume, 1998; Tanaka & Kitamura, 2001),
with
where ∊_{0} is the of vacuum, c is the speed of light, e is the ω is the photon energy, r = (X, Y, Z) is the position of observation, and r(t′) = (x, y, z) and β(t′) = (β_{x}, β_{y}, β_{z}) specify the position and velocity of the electron at the retarded time t′. Using the complex amplitude F_{ω}, the spatial at r is calculated as
where α is the finestructure constant.
Under the farfield approximation, i.e. r r′ and R ≃ r = a constant, equation (1) is simplified to
with
where γ is the Lorentz factor of the electron and we have introduced the observation angles θ_{x,y} and changed the integration variable from t′ to z/c.
Now let us consider the complex amplitude of radiation for the planar undulator with the number of periods N. Assuming that the magnetic field is ideal, i.e. purely sinusoidal in the vertical plane, the relative velocity is given by
where λ_{u} is the periodic length and K is the deflection parameter of the undulator.
Substituting the above formulae into equation (3) and expanding into a Fourier series, we have
with
where ω_{n} is the peak energy of the nth harmonic radiation.
Equation (4) gives the complex amplitude observed at a position far from the source point and thus gives the angular distribution. The spatial distribution at the same position is obtained by the spatial Fourier transform of F_{ω}. In order to evaluate the however, we need to know the distribution functions at the source point, i.e. the beam waist position, which can be found by propagating the radiation back to the source point. From the theory of wave optics, it is found that such backward propagation is given by the mathematical operation (see, for example, Born & Wolf, 1984)
where we have assumed that the distance from the source point to the observer equals Z, which is valid because the source point of UR exists at the midpoint of the undulator. Substituting equation (4) into the above formula, we have
The function F_{ω,o} gives the angular distribution of the complex amplitude at the source point, while its spatial Fourier transform gives the spatial distribution.
Now let us consider the case when the electron has an energy offset Δγ compared with a reference energy γ_{0}. Then the complex amplitude observed at the energy ω_{n}(γ_{0}, 0), which is the nth harmonic energy of UR for the reference electron with γ_{0}, is given by
with
where we have assumed that Δγ γ_{0}. Apart from the constant, the function F_{ω,o} is given by the product of two factors, i.e. f_{n} and the sinc function. The former function denotes the complex amplitude of the nth harmonic radiation within one undulator period and thus is a slowly varying function of θ_{x,y}, independent of the number of periods N, while the latter function represents the effects of the coherent summation of radiation emitted over the total undulator length, and is strongly associated with N. Under the condition N 1, which is satisfied in most SR beamlines with undulators as light sources, the angular profile is thus dominated by the sinc function, and equation (6) can be simplified to
Note that this simplification gives rise to an error in
evaluation especially for higher harmonics, as discussed later.Equations (7)–(9) are the basic formulae for studying the effects of electronbeam energy spread on which will be discussed in the following sections. Note that these equations are valid for any types of undulators, such as the helical, elliptical and figure8 undulators, except that f_{n} should be recalculated according to the electron trajectory in the concerned undulator, and K^{2} should be replaced by K_{x}^{2} + K_{y}^{2}, where K_{x} and K_{y} are the horizontal and vertical deflection parameters, respectively. In addition, a reservation should be made for even harmonics of planar undulators and for higher harmonics (n > 1) of helical undulators, because f_{n} vanishes onaxis (θ_{x,y} = 0).
2.2. Angular profile
Substituting equation (9) into (2), we have the angular profile of the at the photon energy ω_{n}(γ_{0},0),
where the function d^{2}N_{0}/dΩ(dω/ω) denotes the on axis (Θ = 0), which is given by the well known expression on UR (see, for example, Kim, 1989),
with
where J_{m} is the mthorder Bessel function of the first kind.
Let us consider the effects of the energy spread of the electron beam. Assuming the energy distribution function to be a Gaussian function with the standard deviation σ_{E}, we have
with
where we have introduced the normalized energy spread defined as
Fig. 1(a) shows the graphical plot of P_{a} for different values of . It is found that the peak value at Θ = 0 decreases and the peak is broadened as increases.
Let us approximate the function P_{a} by the twodimensional Gaussian function with the standard deviation , i.e.
Integrating over Θ, we have
with
Now let us show that S_{a} is independent of and is equal to π^{2}/2. First let us modify the above equation as follows,
Thus, S_{a} is the area of obtained by integrating from 0 to . The function is obtained by convoluting with the Gaussian function having the total area (obtained by integrating from to ) of unity. From the theory of convolution, it is therefore obvious that the total area of is the same as that of , i.e. π. In addition, is an even function because both and the Gaussian function are even functions. Thus we obtain S_{a} = π^{2}/2.
As for , the integration can be performed analytically to give
with
being the Gauss error function.
Now we can calculate the angular divergence of UR with the effects of the energy spread taken into account. Substituting the above results into equation (14) and considering equation (7), we have
with
where λ_{n} is the wavelength of the nth harmonic radiation and L is the total length of the undulator. The parameter , which is the angular divergence of UR at = 0, is the well known expression found in textbooks on SR (see, for example, Kim, 1989). Equation (16) indicates that the angular divergence grows according to the energy spread by the factor Q_{a}.
Fig. 2(a) shows a graphical plot of the growth factor Q_{a} = as a function of the normalized energy spread . Note that Q_{a} has a minimum value of 1 at = 0, and becomes greater for larger values. For example, Q_{a} reaches 2 around = 5, meaning that the angular divergence is doubled owing to the effects of the energy spread.
2.3. Spatial profile
Next, let us investigate the spatial profile of the F_{ω,o},
at the source point, which is given by taking the square of the spatial Fourier transform ofwith
being the normalized radial coordinate. Now we have the spatial profile of the
at the source point,with
If the Gaussian approximation is possible, which has been used to obtain the angular divergence in the previous section, the standard deviation of P_{s}, i.e. the normalized source size Σ_{R}, can be determined by
with
Then the source size of UR, which is denoted by σ_{r}, is given by
with
where σ_{r0} is the well known expression for the source size of UR at = 0 (see, for example, Kim, 1989).
Fig. 1(b) shows a graphical plot of P_{s} for difference values of . It is found that the peak value at R = 0 increases and the peak shrinks as increases. It should be noted, however, that a longrange tail exists together with the narrow peak around R = 0 for larger values of . Owing to this tail, the Gaussian approximation is not valid and thus equation (20) is not available for larger values of . Thus we have to modify the above equation in order to utilize the Gaussian approximation as follows.
First, we convolute P_{s} with the Gaussian function with a standard deviation Σ_{ref}. If Σ_{ref} is large enough, the resultant spatial profile can be well approximated by the Gaussian function. Then the normalized source size , which is determined by , is obtained by applying the same method as in the previous section. Namely,
with
Because is regarded to be the convolution between Σ_{ref} and Σ_{R}, we have
The value of Σ_{ref} should be determined to meet two requirements. Firstly, the convoluted profile can be well approximated by the Gaussian function. Secondly, is not significantly different from Σ_{R}. In order to meet the two requirements, we have repeated numerical analysis and found the reasonable condition that Σ_{ref} = 1/π^{3/2}. It is interesting to calculate the source size σ_{ref} that corresponds to the normalized source size Σ_{ref}, namely,
As shown later, 2σ_{r0} is found to be the source size of UR at = 0.
Let us define the function as
which is regarded to be the growth factor of the source size according to the energy spread.
Fig. 2(b) shows a graphical plot of obtained by the numerical analysis described above. It should be noted that Q_{s}(0) = 2, meaning that the source size of UR obtained with the current analysis is larger than the well known expression by a factor of 2. This discrepancy is attributable to the difference in the method of analysis. The well known expression is derived based on the assumption that both the spatial and angular profiles are given by the Gaussian function and thus UR is like a singlemode laser, which is not necessarily valid. Although we also use the Gaussian approximation to determine the angular divergence and source size, the spatial profile has been derived by the spatial Fourier transform of the angular distribution of the complex amplitude, which is more reasonable. The degradation of UR owing to the nonGaussian angular profile has also been pointed out by Kim (1986). In addition, the discrepancy may become more pronounced at the detuned photon energy ω = ω_{n} (Coisson, 1988), which is out of scope in this paper.
It is worth noting that the function Q_{s} can be well fitted using the function Q_{a},
The above fitting function is plotted in Fig. 2(b) to be compared with the results obtained by the numerical analysis, where we find a good agreement between the two. Hereinafter, we use the above fitting function to calculate Q_{s}.
2.4. Evaluation of brilliance
Having determined the angular divergence and source size with the effects of the energy spread taken into account, let us now evaluate the
which is given bywhere F is the total of radiation emitted over the whole solid angle, and and are the angular divergence and source size of the photon beam in the horizontal and vertical directions, respectively. Using the formulae derived in the former sections, we have
and similar equations for y, where and σ_{x} are the angular divergence and beam size of the electron beam in the horizontal direction, respectively. Note that we have omitted the argument of the function Q_{a} and Q_{s} for simplicity. Equations (13), (17) and (24)–(28) form a universal function to evaluate the of undulator radiation, which takes into account the finite emittance and energy spread of the electron beam. This is the conclusion of the discussions so far. Note that the expression on by the conventional method is obtained by substituting Q_{a,s} = 1 into equations (27) and (28). This means that the conventional formula has been derived with an assumption that the UR is a Gaussian beam and the effects of the energy spread are negligible.
Assuming that the position of the electron beam waist is placed at the source point (midpoint of the undulator) and that the dispersion function is zero, the above equation is simplified to
with
where β_{x} is the horizontal betatron value at the midpoint of the undulator, and E_{x} can be regarded as the emittance of the photon beam with the effects of the electron beam emittance and energy spread (effective optical emittance), while ∊_{λ} is the optical emittance of the Gaussian beam at a wavelength of λ_{n}.
The factor G(β_{x}/β_{o}) has a minimum value of 1 when β_{x} = β_{o}, then E_{x} is simplified to
Thus the effective optical emittance is given by a simple summation of the electron beam emittance (∊_{x}), and the natural optical emittance (∊_{λ}) multiplied by the growth factor (Q_{a}Q_{s}) determined by the energy spread, if the betatron function is optimized.
3. Comparison with numerical results
In order to examine the validity of the expressions derived so far, comparisons with numerical results have been made using SPECTRA (Tanaka & Kitamura, 2001), the computer code for numerical analysis on SR characteristics. For this purpose, the code has been revised to implement the function to calculate the spatial profile at the source point with the effects of the energy spread taken into account, which was not implemented in the former version. The is then evaluated by dividing the angular by the source size, which is obtained from the spatial profile at the source point.
The accelerator and undulator parameters of the thirdgeneration SR facility SPring8 have been used for the . The length of the standard straight section (S) is around 5 m, in which invacuum undulators with L = 4.5 m and λ_{u} = 32 mm are usually installed for Xray beamlines. In addition to S, SPring8 has four long straight sections (LSs) of length 27 m, and a 25 m invacuum undulator has been installed in one of them. Because of the large number of periods, the available with the long undulator in a LS is expected to be sensitive to the energy spread, and thus is well suited for examination of the calculation method.
calculation, and are summarized in Table 1

The i.e. the numerical method with SPECTRA, the universal function derived in the former sections, and the conventional formula.
calculations have been carried out for harmonics between the first and 11th by three different methods,Figs. 3(a) and 3(b) show the results of calculations in the case of S and LS undulators, respectively. It is found that the conventional formula overestimates the under all the conditions, and the overestimation depends on the harmonic number and the undulator type, i.e. the undulator length. In other words, the evaluation with the conventional formula is not reliable for large values of the normalized energy spread . On the other hand, we find a good agreement between the results of the numerical analysis and universal function when the harmonic number is less than, for example, 7. For larger harmonic numbers, the universal function overestimates the to some extent.
In order to understand the overestimation of instead of its simplified form (9), we have
at higher harmonics, let us derive the angular profile of the without approximation. By using equation (6)with
Note that the universal function has been derived under the approximation
It has been shown in §2.3 that integration of P_{a} over Θ is constant and thus the total F obtained by integrating (29) is found to be independent of the normalized energy spread as long as the approximation (30) is valid.
In order to examine the applicability of (30), we have calculated the horizontal angular profiles of ρ_{n} for two different harmonic numbers, 1 and 11, which are plotted in Fig. 4. It is found that ρ_{1} is a slowly varying function of θ_{x}, while ρ_{11} oscillates rapidly. We now find that F decreases more rapidly as for larger n, which is easily understood by looking to the angular profiles of the two factors P_{a} and ρ_{n} shown in Figs. 1(a) and 4, respectively. In other words, the approximation (30) becomes less accurate for larger n and N, and the universal function tends to overestimate the which coincides with the results shown in Fig. 3.
4. Summary
New analytical expressions have been derived as a universal function to evaluate the
of UR, and have been examined by comparing with numerical analysis. It has been found that the can be evaluated precisely by the universal function, although care should be taken on its use for higher harmonics.The universal function contains two special functions, i.e. the Bessel function of the first kind and the Gauss error function. The former function is also contained in the conventional formula, while the latter function can be well approximated by a combination of elementary functions (see, for example, Stoer & Bulirsch, 1991). Thus the universal function is easily utilized not only by experts on SR but also by those who are not familiar with the numerical method on SR. Although the can be evaluated precisely by the numerical codes as already mentioned in the Introduction, the universal function derived in this paper is useful for many applications. For example, it can be easily implemented not only in generalpurpose computer software for investigating the performance of SR sources but also in specialized codes written for a variety of purposes.
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