* Corresponding author. Tel.: #7-095-132-6112; fax: #7- 095-135-7880. E-mail address: ivk@sci.lebedev.ru (I.V. Kozhevnikov). Nuclear Instruments and Methods in Physics Research A 460 (2001) 424}443 Design of X-ray supermirrors Igor V. Kozhevnikov*, Inna N. Bukreeva, Eric Ziegler Lebedev Physical Institute, Leninsky prospect 53, Moscow 117934, Russia European Synchrotron Radiation Facility, BP 220, 38043 Grenoble cedex, France Received 7 September 2000; accepted 21 September 2000 Abstract A new approach is proposed for the design of wide band-pass multilayer optical elements operating in the hard X-ray spectral region. The method, based on the combination of analytical and numerical methods, solves the inverse problem consisting of inferring the composition pro"le of a depth-graded multilayer coating. The key feature of our approach consists in using an analytical expression for the depth-distribution of the period as initial solution for direct computer calculations. This allows a global minimization of the merit function and a many-fold decrease of the computer run time. Simulations of two particular cases are presented: a constant re#ectivity over a wide spectral range and a complicated re#ectivity pro"le. The best choice of material pairs for composing a depth-graded multilayer structure is discussed from the viewpoint of maximum achievable re#ectivity and least number of bi-layers. Features of depth-graded multilayer mirrors, which are distinctive from conventional periodic mirrors, are examined. The factors in#uencing the optical quality of broad-band multilayers are also considered.  2001 Elsevier Science B.V. All rights reserved. Keywords: Multilayer; Supermirror; Hard X-ray optics; Synchrotron optics 1. Introduction The recent operation of high energy, third- generation, synchrotron sources has opened up new scienti"c opportunities in the use of high- energy X-ray di!raction. Exciting applications in- clude the study of small volumes of materials such as, buried interfaces of composite materials [1], materials under high pressure [2], and medical imaging. In the same time hard X-ray telescopes are being developed for the observation of the sky, e.g. of supernovae, at X-ray energies above 20 keV. Perfect crystals have historically been the optical components of choice in high-energy X-ray experi- ments. However, their very narrow energy band- width consumes a substantial fraction of the incident #ux, which can be a handicap. In some cases, such as in angiography experiments [3], the use of bent crystals in the Laue geometry could circumvent this problem. More recently periodic structure multilayers were shown to be a competi- tive alternative [4], their bandpass being more than 10 times larger than the one of a bent crystal and about 100 times larger than the one of a #at perfect crystal [5]. When deposited on either "gured or bent mirrors large beam demagni"cation could be obtained [1,6]. For a number of applications multilayer mirrors with a spectral band of re#ection as wide as 0168-9002/01/$-see front matter  2001 Elsevier Science B.V. All rights reserved. PII:S0168-9002(00)01079-2