research papers
Effects of energy spread on
and coherence of undulator sources^{a}European XFEL, Schenefeld, Germany, and ^{b}Deutsches Elektronen Synchrotron, Hamburg, Germany
^{*}Correspondence email: gianluca.geloni@xfel.eu
The (spectral)
for partially transverse coherent sources such as synchrotron radiation and freeelectron laser sources can be defined as the maximum of the Wigner distribution. Then, the includes information on both coherence and wavefront characteristics of the radiation field. For undulator sources, it is customary to approximate the singleelectron electric field at resonance with a Gaussian beam, leading to great simplifications. Attempts to account for the modified spatial and angular profile of the undulator radiation in the presence of detuning due to energy spread, currently build on the simplified expression derived under the assumption of Gaussian beams. The influence of energy spread on undulator radiation properties is becoming important in view of diffractionlimited rings with ultralow emittance coming online. Here the effects of energy spread on the of undulator radiation at resonance are discussed, as well as relevant relations with coherence properties.Keywords: synchrotron radiation; brightness; Wigner distribution; undulator radiation; energy spread.
1. Introduction
The concept of (spectral) ). Radiometry treats radiation within the framework of geometrical optics and characterizes sources in terms of radiance, that is the maximum density in phase space, measured as a spectral per unit area per unit projection solid angle. Other quantities of interest can be derived by computing the marginals of the photon phase space distribution. A particularly attractive feature of the radiance is that, for nondissipative systems where the Liouville theorem holds, this quantity is an invariant along the beamline. Therefore, it is strictly related to the maximum spectral that can be obtained at the sample position, assuming an ideal optical system.
which is used as a figure of merit for synchrotron radiation and freeelectron laser (FEL) sources, is historically rooted in radiometry (Born & Wolf, 1999Starting from the pioneering works (Coisson & Walker, 1986; Kim, 1986, 1987a,b), a lot of literature is available, which deals with the generalization of the concept of radiance to the case of partially transverse coherent sources as synchrotron radiation and FEL sources (Coisson & Walker, 1986; Kim, 1986 1987a,b; Hulbert & Weber, 1992; Howells & Kincaid, 1994; Bahrdt, 1997; Hulbert & Williams, 1992, 1998; Bosch, 1999; Attwood, 1999; Bosch, 2000; Ciocci et al., 2000; Duke, 2000; Thomson & Vaugham, 2001; Wiedemann, 2002; Onuki & Elleaume, 2003; Hofmann, 2004; Clarke, 2004; Talman, 2006; Williams, 2006; Tanaka & Kitamura, 2009; Bazarov, 2012; Huang, 2013; Tanaka, 2014; Geloni et al., 2015; Vartanyants & Singer, 2018). This generalization process required changing the working framework from pure geometrical optics to wave optics, backed up by statistical optics. This led to the substitution of the phase space of optical rays in geometrical optics with a Wigner distribution that, as is well known, is a quasiprobability distribution, not everywhere positive definite.
As underlined by Bazarov (2012), this generalization process naturally includes a strong analogy with quantum mechanics in position representation, where wavefunctions are analogous to spatial field distributions and the classical concept of phase space is substituted by a Wigner distribution, which can assume negative values related to the ability of wavefunctions (and electric fields) to interfere. In quantum mechanics (or in wave optics) one often deals with random processes so that it becomes necessary to describe the state of the system (or the electric field) in terms of a density matrix, which assumes the form of a correlation function in position representation. In the case of synchrotron radiation and FEL sources the underlying, fundamental stochastic process is the shotnoise in the electron beam. As is well known, in statistical optics the spatial field correlation function at a given frequency takes the name of crossspectral density. The overall degree of transverse coherence is just analogous to the trace of the square of the density matrix representing the statistical operator for a quantum mixture, and can therefore be expressed in terms of integrals of the crossspectral density. It is interesting to remark here that the trace of a matrix is invariant with respect to a basis transformation. This fact is well known in statistical quantum mechanics, where a mixed state can be thought of as a mixture of pure states that diagonalize the statistical operator with weights given by its eigenvalues. The same fact is similarly well known in statistical optics, where the coherent mode decomposition theorem allows a crossspectral density to be written as a sum of uniquely defined statistically independent contributions, obviously leaving the overall degree of coherence unvaried.
The relation between crossspectral density (or density matrix) and Wigner distribution is a simple Fourier transformation. In other words, they carry the same identical amount of information. The ) for a review].
can be seen as a figure of merit that is extracted from the Wigner distribution. There are several recipes for doing so. One defines the in terms of integrals of the Wigner function and of its square. Another defines it as the Wigner function onaxis [see, for example, Bazarov (2012However, as noted by Geloni et al. (2015), there is a correspondence principle between wave and geometrical optics, exactly as there is a correspondence principle between quantum and classical mechanics. In particular, there is a special class of sources, called quasihomogeneous sources, for which the Wigner distribution function factorizes as
where and can be identified, respectively, with the source intensity distribution and with the angular distribution of radiation intensity. Then, W is the product of two positive quantities, and, being positivedefinite everywhere, can be identified with a phase space. In this limit, the must strictly correspond to the radiance, and is the maximum of the Wigner distribution function. It is therefore natural to define the for any source as the maximum of the Wigner distribution.
With this last definition, the
includes information on both coherence and wavefront characteristics of the electric field, in contrast to the case where it is defined in terms of integrals of the Wigner function, and only information on the degree of coherence is present.The previous discussion is meant to be a quick summary of the strict relations between coherence properties and
which are important to keep in mind when discussing radiation properties from FELs and storagering sources, and are becoming more important for storageringbased sources, in view of the coming online of many stateoftheart diffractionlimited rings.For the case of storage rings, one can approximate the transverse electron phase space with a Gaussian function. Moreover, for undulator sources, it is customary to approximate the singleelectron electric field at resonance with a Gaussian beam. In contrast to the real undulator field, Gaussian functions are separable, and a simplified expression for the ),
results in this case (Kim, 1986where F indicates the total per unit spectral bandwidth, while , are effective source size and divergences, calculated by summing in quadrature the sizes and divergences of the electron beam and of the singleelectron radiation.
Equation (2), derived under the Gaussian beam approximation, does not include detuning or energyspread effects on the radiation beam. However, for diffractionlimited rings, studying the influence of energy spread of undulator radiation properties is becoming more and more important, because of the ultralow electron emittance.
In the paper by Tanaka & Kitamura (2009), an attempt is reported where the authors account for the modified spatial and angular profile of the undulator radiation in the presence of detuning. However, the approximate formula for the that they obtain still builds on equation (2), that is based on the Gaussian beam approximation in the first place.
It is therefore interesting to study energy spread effects on the
of undulator radiation by avoiding to rely on the Gaussian beam approximation from the very beginning, and defining the as the maximum of the Wigner distribution. Moreover, given the strict relation between coherence and highlighted above, one should complement a study on the effects on the with a study on the effects on coherence.Here we will discuss the effects of energy spread on both ).
and coherence of undulator radiation at resonance. We will first introduce basic quantities and notations. Then, using a simple model, we will demonstrate a very counterintuitive fact. In the limit for a vanishing small emittance the from an undulator is not influenced by the electron beam energy spread, in the case of a symmetrical distribution around the nominal energy. Further on, with the help of semianalytical calculations, we will discuss the impact of energy spread on the coherence properties of undulator radiation. We will illustrate our results with examples compatible with modern diffractionlimited sources, discussing similarities and differences with respect to the approach by Tanaka & Kitamura (20092. Basic quantities and notations
We follow notations similar to those used by Geloni et al. (2008, 2015). We write the fundamental wavelength of a planar undulator with periods as = , where is the undulator period, L_{u} = , k_{u} = and K is the maximum undulator parameter. Likewise, the fundamental frequency is = . denotes the Fourier transform of the electric field, and we define with = the slowly varying envelope of the field in the frequency domain, which we will refer to simply as `the field'.
Consider an electron that enters the undulator at a small angle and offset with energy fixed by γ that can be different from the nominal value . The farfield angular distribution seen at a distance from the middle of the undulator and at frequency ω such that (where the resonance approximation applies) depends on the parameters and , and is given by^{1}
Here, A_{JJ} = J_{0}[K^{ 2}/(4+2K^{ 2})]J_{1}[K^{ 2}/(4+2K^{ 2})] is the coupling strength for the first harmonic under the resonance approximation. Our considerations can be easily extended to odd harmonics. For even harmonics one should consider, instead, a different position of the maximum of the Wigner function. Note that under the resonance approximation the field is linearly polarized, hence is a scalar quantity. An expression for the field at the virtual position z = 0, i.e. in the middle of the undulator and for any position after the undulator exit at perfect resonance, can be found in equations (34) and (35) of Geloni et al. (2007). However, to the authors' knowledge there is no analytical expression for the field at z = 0 at finite detuning, which should be calculated propagating equation (3).
Following the references above we use normalized units defined as
so that it is natural to define
Roughly speaking, this amounts to normalizing angles to the diffraction angle of singleelectron emission, sizes to the diffraction size, fractional energy deviation to the undulator resonant bandwidth, and times to inverse radiation frequency. Moreover, here are the r.m.s. of the electron beam dimensions in phase space, and we assume for simplicity that at z = 0, i.e. in the middle of the undulator, the electron beam phase space can be factorized as
with
where we defined the various Gaussian distributions in terms of the variances N_{x,y}, D_{x,y}, and and we introduced and only for completeness, because in this paper we deal with spontaneous radiation and therefore these quantities are not used. The farzone field in normalized units can be written as
where = z/L_{u}. As discussed above, one may calculate the analogous field at the virtual source as
In the top panel of Fig. 1 we plot , i.e. the well known farfield profile for = 0 and = 0 (which is azimuthalsymmetric) as a function of for different values of the detuning , while in the bottom panel we plot the field at the virtual source (which has a plane wavefront) for the same choices of the detuning parameter. For comparison, in the top and bottom panels of Fig. 2 we plot, respectively, and that are the corresponding intensity distributions.
It can be shown that for negative values of the maximum intensity at the source increases and tends to `saturate' for large negative values, while it remains constant in the far zone. We will discuss the consequences of this fact later on.
Also, even at = 0, the intensity distribution at the virtual source and in the far zone are not Gaussian. Therefore, any Gaussian approximation relies on a fitting procedure. In this regard it is important to remark that the intensity distribution in the far zone and at the virtual source are related by the laws of field propagation in freespace, basically a Fourier transformation. One may fit the intensity at the virtual source with a Gaussian, but in that case the real intensity in the farzone does not match the propagated Gaussian beam. One may fit the intensity in the far zone with the Gaussian, but in that case the intensity at the virtual source does not match the backpropagated Gaussian beam. In other words, there is some freedom when it comes to applying the Gaussian approximation. Many different choices can be found in the literature [see, for example, Kim (1986, 1987a), Lindberg & Kim (2015)]. One of the possible choices (Kim, 1987a; Tanaka & Kitamura, 2009) is to fix, for the singleelectron intensity distribution, = and = , corresponding to the (strictly speaking we cannot define a except in those cases when the Wigner distribution is positive definite, and the Gaussian approximation is one of those cases) = = . In our normalized units, they amount to = and = . The corresponding FWHM values are obtained multiplying by ≃ 2.35 and read = 0.664 and = 4.17, to be compared with the corresponding FWHM values for the actual intensities at = 0, which are found to be = 1.36 and = 4.72.
Having discussed the singleelectron field and intensity profiles, we now introduce the crossspectral density in normalized units,
where the brackets indicate averaging over an ensemble realizations, is the vector position at which a twopinholes system is introduced to probe coherence, and is the vector describing the separation between the two pinholes, see equation (4). Clearly, and may have different directions. We remind that the spectral degree of coherence is defined as
and the fringe visibility of an interference experiment is given by
Finally, the Wigner distribution in normalized units is
Following the same formalism as Geloni et al. (2015), the corresponding result in dimensional units is found to be linked to equation (13) by the constant
with = the Introduction, for which equation (1) is valid. The defined by us as the maximum of the Wigner distribution, is therefore given by
This result follows from the correspondence principle for quasihomogeneous sources discussed in theHere we underline the fact that, while this is often the case, in the most general case the maximum of the Wigner function may not be onaxis, i.e. may not be at = 0 and = 0. Choosing the maximum of the Wigner function for defining the assures that the correspondence principle discussed in the Introduction is consistently applied.
Substitution of equation (8) into equation (10) gives the following explicit expression for the crossspectral density in the case of undulator radiation around the fundamental harmonic (or, with simple changes, for odd harmonics),
Note that the singleelectron spectralangular intensity distribution has, in our case, its maximum at resonance onaxis. Then, for a Gaussian distribution of energy spread, divergence and size of the electron beam, the maximum of the Wigner distribution must be at = 0 and = 0, and therefore
the integral extending over all the plane spanned by the vector .
3. Effects of energy spread on the brightness
Let us first consider the simplest case of a beam with vanishing emittance. Equation (16) simplifies accordingly, and substitution into equation (17) gives the following expression for the brightness^{2},
where we used the fact that in the limit for zero emittance is azimuthal symmetric. Now we note that
where
where = is the sine integral function.
By definition, the function is proportional to the angleintegrated spectral
from a single electron, and therefore the is proportional to the singleelectron angleintegrated spectral averaged over the energy spread distribution.Moreover, the function has the property that = independently of the value of the real number . We conclude that for zero emittance and symmetric energy spread distribution we cannot have any effect of the energy spread on the
that can in fact be written asIn order to make our argument clearer, we calculated the SPECTRA (Tanaka & Kitamura, 2001). Both cases refer to parameters compatible with the PETRA IV project, with an energy of 6 GeV, and a planar undulator with period = 65.6 mm and a length of 5 m, corresponding to 76 periods. We set zero emittance and discuss two singleelectron cases with resonant energies at 580 eV and 4000 eV. The results are plotted in the top panel of Fig. 3 as a function of the detuning = , where we show the divided by the value at zero detuning. In the bottom panel of the same figure we plot the function . By comparing the two plots one can see, as expected, a very similar behaviour. The only difference is that the computed with SPECTRA (which is not based on the resonant approximation used for the analytical calculations) has its maximum around ≃ −4, while the analytical calculation shows that the function continues to grow for values of below that. This last fact can be seen as a consequence of the fact that, at the source, the maximum of the intensity profile is increasing for negative detuning values (see Fig. 1, bottom panel), as previously discussed. Note that, in any case, the is roughly antisymmetrical with respect to the point = 0 also for large detuning values, and this reinforces our conclusion that for zero emittance and symmetric energy spread distribution we cannot have any effect of the energy spread on the in agreement with the analysis of equations (18) and (20).
for two specific cases using the codeIt is interesting to compare our results with those of Tanaka & Kitamura (2009). As discussed in the Introduction, Tanaka & Kitamura (2009) proposed an approximated formula for the which is derived starting from equation (2), that is the usual expression for the based on Gaussian approximation but includes the impact of a modified spatial and angular profile of the undulator radiation in the presence of detuning. In our notations, setting for simplicity and , this formula reads
where
and the subscript `A' stands for `Approximated'.
We considered the previously discussed parameters compatible with the PETRA IV project and we analysed the case of zero emittance as well as the case for a finite emittance = 10 pm, equal betatron functions = 1 m, and no dispersion. A comparison of the , as calculated using our formulas, SPECTRA and equation (22). The cases of nonzero emittance correspond to N_{x,y} = 0.0059 and D_{x,y} = 0.15 for the case of 580 eV, and N_{x,y} = 0.04 and D_{x,y} = 1.01 for the case of 4000 eV. The main parameters are summarized in Table 1. Note that the detuning parameter depends linearly on the harmonic number. If we consider N_{u} ≃ 100 and = 10^{3}, one immediately sees that the normalized r.m.s. energy spread parameter is about 1.3 for the first harmonic, but, since it scales linearly with the harmonic number, for the fifth it would be already about 6.3. Therefore, we chose to present plots up to values of = 5.
as a function of the energy spread for zero and nonzero emittance at the two different resonance photon energies of 580 eV and 4000 eV is shown in Fig. 4

Looking at Fig. 4 we see that there is a factor of four difference between equation (22) in the limit for no emittance and energy spread and equation (21). In Tanaka & Kitamura (2009) this seems to be explained as due to the fact that while the Gaussian approximation was used `to determine the angular divergence and source size, the spatial profile' was `derived by the spatial Fourier transform of the angular distribution of the complex amplitude', leading to a factor of two in the source size. We argue that this procedure should not lead to any difference in the in the case of zero emittance and energy spread, because in that limit one must have (Kim, 1987a)
as is confirmed by equation (21) and (see Fig. 4) by direct calculations with the code SPECTRA.
Aside for the factor of four discrepancy, we note that equation (22) approximates the influence of energy spread by summing emittancerelated parameters (N and D) with powers of the function Q_{a} that depend on the energy spread. Therefore, in the limit for zero emittance, energy spread effects dominate the In contrast to this, equation (18) is completely independent of the energy spread. This behaviour is exemplified in Fig. 4.
Our conclusion is that, while equation (22) may constitute a good approximation in some region of the parameter space, when it comes to the limit for a diffractionlimited beam with nonnegligible energy spread, a more detailed study is needed. In particular, when one is well within the diffraction limit, there is no region where the is dominated by energyspread effects.
Clearly, the above considerations are valid only for a vanishing emittance of the electron beam, i.e. in the limit for and . In fact, even for vanishing offsets , if we cannot assume the expression for the includes the integrated spectral for electrons with different angles, and the sum of contributions with positive an negative detuning is now depending on the detuning value, at difference with the case above where + = 8π, independently of .
4. Effects of energy spread on coherence
It is interesting to discuss possible effects of the energy spread on the coherence properties of undulator radiation. As before, we will first consider the case for zero emittance.
Clearly, the phase of the field in equation (8) only depends on the electron offset, and is fully independent of , i.e. of γ. However, we note that the magnitude and, most importantly, the sign of the field depend on . Let us discuss the impact of this sign on the spectral degree of coherence. We write explicitly a simplified expression in the case of zero emittance as
This equation has been found on the basis of equation (16), where we took the limit for zero emittance and we assumed, for simplicity, that the two vectors and are directed along the same direction. This simplification does not deprive our model of any useful physics, but it makes all arguments scalars, and hence easier to consider. Further on, since, as remarked above, the phase of the field in equation (8) only depends on the electron offset, we factorize g in the product of and of the phase factor . Note that is still allowed to assume negative values.
It is easy to see by inspection of equation (25) that, when , G is different from unity, but = everywhere. Moreover, onaxis, i.e. for = 0, one has g = = 1, while offaxis, i.e. for , one has jumps of from +1 to −1 at all those values of where and differ by an odd multiple of π.
Let us consider the case of nonzero energy spread. If we look onaxis at = 0, from equation (25) we see directly that g = = 1. However, offaxis, an interesting phenomenon takes place. The field from different electrons with different detuning experiences a change in sign at different values of . This means that different electrons generate radiation with different wavefronts, and coherence is therefore decreased. This effect is encoded in the function , while the phase factor cannot change. To our understanding, this mechanism was not discussed before and is at the basis of any possible coherence deterioration related to energy spread effects. It is important to underline that it is present only offaxis, while energy spread alone cannot influence coherence properties onaxis. In the presence of a finite emittance, one must include the effect of different angles in equation (16). Then, even onaxis, different electrons generate radiation with different wavefronts, and coherence deteriorates.
In order to illustrate our statements and to estimate the importance of the effects of energy spread on coherence we performed semianalytical calculations for the case of zero emittance. We fixed different values of and plot the crossspectral density (a real function, in our case), the spectral degree of coherence, and the visibility calculated above in the farzone as a function of for different values of the energy spread.
Fig. 5 presents results for = 0.5 and = 1. We remind the reader that the definition of our dimensionless units is given in equation (4). The normalization factor is of the order of the angular size of the central cone. Therefore, it does not make too much sense to consider values of larger than unity. As one immediately sees from the plots, even at = 1 the effects of energy spread on coherence deterioration is very small. This is because the first change in sign for happens at = (and the second would be at = ). Obviously there is little interest in going at such distance from the axis, and our conclusion is that the effect of energy spread on the deterioration of coherence is usually negligible in the far zone.
However, the situation changes if the optics images, at the sample position, the virtual source in the middle of the undulator. In this case the previous analysis must be repeated using the quantities defined as before, but considering equation (9) instead of equation (8).
Fig. 6 presents results for = 0.5 and = 1. This time, the normalization factor in equation (4) is of the order of the transverse size of the central cone at the virtual source and, analogously as in the far zone, we limit ourselves to values of up to unity. The same remarks made for the far zone hold for the values of the energy spread parameter. Inspection of Fig. 6 shows an important effect of the energy spread on coherence properties at the virtual source position. While we do not possess a simple expression as equation (8) at the virtual source position, the mechanism that leads to coherence degradation is the same: namely, there is a change in the sign of the field (see Fig. 1). This happens, however, for smaller values of , which leads to degradation already for small values of , as seen from Fig. 6.
It should be noted that, although the shape of the spectral degree of coherence is different when we compare the source with the far zone, the overall degree of coherence ζ remains unchanged. As discussed in the Introduction, the overall degree of coherence is analogous to the trace of the square of the density matrix representing the statistical operator for a quantum mixture, the statistical operator being just, in our case, the crossspectral density. The same degree of coherence can be expressed in terms of the Wigner distribution, because it is related to the cross spectral density by a simple Fourier transform,
Since the freespace propagation of the Wigner function is given by
a simple change of integration variables = shows that ζ is invariant for freespace propagation.^{3}
We now complicate the situation by introducing finite emittance effects, corresponding to the two previously discussed cases for 580 eV and 4000 eV, respectively. In particular we consider again the two settings N_{x} = N_{y} = 0.0059, D_{x} = D_{y} = 0.15, corresponding to a resonant energy of 580 eV, and N_{x} = N_{y} = 0.04, D_{x} = D_{y} = 1.01, corresponding to a resonant energy of 4000 eV. We set = 0.5 in the far zone and = 0.5 at the virtual source position, and we plot the three functions , and V at the virtual source and in the far zone. Results are shown in Figs. 7 and 8. Comparing Fig. 6 with Figs. 7 and 8 we see how the effects of emittance become more and more important and finally dominate over energy spread effects. One can see coherence degradation already at zero energy spread, both at the virtual source and in the far zone. As is to be expected from the previous discussion, energy spread effects are more visible at the virtual source, while in the far zone they are much less important.
It should be underlined once more that none of the degradation effects on coherence has an impact on the . No degradation was found in the case for zero emittance. However, as is obvious, in the case of nonzero emittance degrades. Equation (17) can, once more, be used to investigate the degradation.
when the beam has zero emittance. We checked this fact by using the expression for to evaluate the according to equation (17)The fact that the i.e. at = 0 and = 0. As one can see from the previous analysis, at = 0 and = 0 there is no coherence degradation, whatever the energy spread parameter chosen. Intuitively speaking, the is strictly related to the ability to focus a radiation beam on a sample. It can be spoiled by a decrease in spectral by degradation of coherence or by wavefront distortions. In the previous parts of this paper we showed that, for a vanishing emittance and in the case of a symmetric energy spread distribution, one has a constant spectral over a large region of the energy spread parameter. However, we have seen here that there is an offaxis decrease of coherence. It is difficult to imagine that this has no effect on the ability to focus radiation. The dependence of the on the onaxis Wigner function (where no coherence degradation takes place) seems, in this case, in contradiction with intuition. However, the decrease of offaxis coherence is only given by changes in sign of the field, happening at different transverse positions for particles with different energies. The ability to focus the field cannot depend on a change in sign, because it only introduces a trivial wavefront distortion: only trivial phase changes of π are introduced by changing the energy, as in Fig. 1. As a result, the remains unvaried even though the coherence properties offaxis are degraded.
cannot be affected by the energy spread alone, whereas the energy spread alone has an impact on the coherence properties of the radiation, seems paradoxical. However, one should remember that in our case the according to our definition, is the Wigner distribution onaxis,5. Conclusions
In this article we noted that changes in the π phasedifference) by the presence of energy spread, meaning that there cannot be any detrimental effect to the related with wavefront distortions. Finally, equation (20) shows no effects on the so we concluded that the cannot be affected, in this case, by the energy spread. The same conclusion was reached by a direct calculation of the equation (21). We studied the situation by means of semianalytical calculations in §2.
can be determined, roughly speaking, by influences related to the spectral to the coherence or to the wavefront. These three quantities can influence the because they influence the ability to focus radiation onto the sample. Consider vanishing emittance and a symmetrical energy spread distribution. We have discussed a mechanism for degradation of coherence offaxis, while we have seen that, onaxis, coherence is preserved. Moreover, the field wavefront is not influenced (aside for aIn §3 we extended our considerations to the case of a finite emittance. First, we confirmed our previous semianalytical results. Then we increased the emittance and we studied its impact on coherence and showing how it degrades for parameters compatible with diffractionlimited storage rings of the next generation.
We conclude that there is no `energyspread dominated' regime: when the emittance decreases, so does also the influence of the energy spread on coherence properties and brightness.
The spectral degree of coherence is seen, instead, to decrease offaxis: this result is in agreement with our conclusion concerning the
We illustrated our statements with simulation results, complementing them with remarks for the case of finite emittance.Footnotes
^{1}Note the minus sign in the first term under the sinc function. If, for example, we fix but our electron has , then the resonance frequency for that electron is higher than , effectively corresponding to a negative detuning .
^{2}Mathematically speaking, here we take limit for and .
^{3}In principle one may directly show this fact in terms of integrals involving the spectral degree of coherence. However, carrying our the calculation explicitly would require extending the tabulation of Figs. 5 and 6 to very large values of and . We therefore prefer to give a synthetic, and more general, demonstration of the invariance of ζ.
Acknowledgements
We thank Ivan Vartaniants (DESY) for carefully reading the manuscript and for useful discussions on the subject.
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