# Characteristics of Photon Beam and Preservation of Coherence in Fourth-Generation Light Sources
## 5 Preservation of Coherence  
As emittance approaches the diffraction limit, the coherence of SR increases in the transverse dimension. To quantify the degree of coherence, the fraction of coherence, defined in Equation (6) was evaluated (Figure 7) to be about 100 times higher, on average, in Korea-4 GSR than in PLS-II. Figure 6: Phase space photon beam distribution from four different types of light source. Horizontal phase space @ 10-keV photon beam from (**a**) wiggler in PLS-II, (**b**) wiggler in Korea-4 GSR. Vertical phase space @ 10-keV photon beam from (**c**) wiggler in PLS-II, (**d**) wiggler in Korea-4 GSR. Horizontal phase space @ 10-keV photon beam from (**e**) bending magnet in PLS-II, (**f**) bending magnet in Korea-4 GSR. Vertical phase space @ 10-keV photon beam from (**g**) bending magnet in PLS-II, (**h**) bending magnet in Korea-4 GSR. Horizontal phase space @ 8-keV photon beam from (**i**) in-vacuum undulator in PLS-II, (**j**) in-vacuum undulator in Korea-4 GSR. Vertical phase space @ 8-keV photon beam from (**k**) in-vacuum undulator in Korea-4 GSR. Horizontal phase space @ 0.2-keV photon beam from (**m**) out-vacuum undulator in PLS-II, (**n**) out-vacuum undulator in Korea-4 GSR. Vertical phase space @ 0.2-keV photon beam from (**o**) out-vacuum undulator in PLS-II, (**p**) out-vacuum undulator in Korea-4 GSR. To evaluate the coherence fraction along a wide spectrum range, an out-vacuum undulator and an in-vacuum undulator were considered as SR sources. The out-vacuum undulator generates SR up to 1\(\sim\)3 keV, and the in-vacuum undulator generates SR \(>\) 1 keV (Figure 8). The coherence fraction beyond 10 keV is at most a few percent coherent. To satisfy high coherency at 10 keV, electron emittance \(<\)10 pm should be realized. However, currently, the storage ring yields a partially coherent beam at 10 keV. Coherence also depends on the SR source, as well as on the lattice function at the source point. The undulator is the strongest coherent SR source due to the small electron-beam path variation in the SR region. A comparison of coherence among the undulator, bending magnet, and wiggler at the given parameter of the storage ring demonstrates that an undulator must be used for experiments that require high coherence (Figure 8). Figure 8: Comparison of the transverse coherent fraction of Korea-4 GSR among undulator, bending magnet (2T central dipole), and wiggler. Figure 7: Comparison of the transverse coherent fraction between Korea-4 GSR and PLS-II. An out-vacuum undulator (OVU) is applied below about 1 keV, and an in-vacuum undulator (IVU) is applied above about 2 keV. The improvements in photon beam coherence caused by a reduction in electron beam emittance will be amplified by matching the phase-space distributions of the electron beam and photon beam. Proper matching of the phase space distribution of the electron beam to the photon beam requires tuning of the betatron function. The optimal betatron function that provides maximum coherence and brilliance is [19; 20; 21]. \[\beta_{x,y}^{opt}\ \sim\ L/\pi \tag{7}\]  
where \(L\) is the undulator length. To investigate the optimal conditions of each SR source for Korea-4 GSR, the coherences and brilliances of each SR source were quantified as a function of the betatron function. At optimal conditions along the betatron function, the coherence of the undulator is increased to 30% by reducing the betatron function to 1 m (Figure 9). Coherence in the wiggler source does not depend on the betatron function, because coherence is dominantly determined by large electron-beam path variation rather than by phase-matching in the wiggler source. The coherence of the photon beam from a bending magnet is strongly dependent on the betatron function, and less dependent on the undulator. The effect of error of focusing by the mirror was quantified to identify the tolerance range of such errors. The errors of the focusing mirror are caused by vibration of the mirror (slop error) and error of the mirror surface (roughness).