* Corresponding author. Tel.: #81-426-77-1111 (ext.3256); fax: #81-426-77-2483. E-mail address: takayama@phys.metro-u.ac.jp (Y. Takayama) Nuclear Instruments and Methods in Physics Research A 441 (2000) 565}576 Spatial coherence of undulator radiation beyond the van Cittert}Zernike theorem Yasuhiro Takayama*, Noriko Takaya, Tsuneaki Miyahara, Susumu Kamada, Wataru Okamoto, Tadashi Hatano, RenZhong Tai, Yasushi Kagoshima Department of Physics, Faculty of Science, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji-shi, Tokyo 192-0397, Japan KEK, 1-1 Oho, Tsukuba-shi, Ibaraki 305-0801, Japan Research Institute for Scientixc Measurements, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan The Graduate University for Advanced Studies, 1-1 Oho, Tsukuba-shi, Ibaraki 305-0801, Japan Department of Physics, Faculty of Science, Himeji Institute of Technology, 1479-1 Kanaji, Kamigori, Ako, Hyogo 678-1297, Japan Received 23 March 1999; received in revised form 13 August 1999; accepted 4 September 1999 Abstract A novel approximation for calculating the "rst-order spatial coherence of undulator radiation is presented. Since the far-"eld approximation is not used in the calculation, it is applicable even in the near-"eld region. Compared with numerical results based on the "rst principles, the approximation is proved to be quite reasonable. The spatial coherence measured at the Photon Factory, KEK is analyzed using the formula and the results are found to be consistent with the design value within experimental errors.  2000 Elsevier Science B.V. All rights reserved. PACS: 41.60.m; 41.60.Ap; 41.50.#h; 42.50.Ar Keywords: Undulator; Spatial coherence; Van Cittert}Zernike theorem; Emittance 1. Introduction Owing to recent developments in accelerator technology, the horizontal emittance for a typical third-generation storage ring has been reduced to several nm rad and highly coherent X-rays are obtained with an undulator. A quantitative esti- mate of spatial coherence is an important theme for various experiments which make use of the coher- ence. With respect to diagnostics of accelerators, a measurement of the spatial coherence for SR (synchrotron radiation) gives information about the electron-beam emittance and the Twiss param- eters [1]. For these reasons, some measurements of the spatial coherence have been performed in soft and hard X-ray regions [2}6]. In order to theoret- ically estimate the spatial coherence of SR, the van Cittert}Zernike theorem, which states that the spa- tial coherence is given by Fourier transformation of intensity distribution on a light source, has been widely used [7}9]. If the van Cittert}Zernike the- orem is applicable to SR, the electron-beam size can be easily known with the spatial coherence. 0168-9002/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 0 9 8 8 - 2