Study of the whispering gallery effect on a spherical surface in the hard X-ray region
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ACE AND THIN FILMS
Study of the Whispering Gallery Effect on a Spherical Surface in the Hard X-Ray Region I. V. Yakimchuka, B. S. Roshchina, I. V. Kozhevnikova, V. E. Asadchikova, and Zh. Wangb a
Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskiœ pr. 59, Moscow, 119333 Russia e-mail: [email protected] b Tongji University, Shanghai, China Received June 3, 2008
Abstract—The whispering gallery effect in the hard X-ray region (0.15–0.3 nm) has been theoretically and experimentally investigated using a concave spherical mirror with a 25-cm radius of curvature and a diameter of 6 cm. A spherical mirror finds and captures the rays incident on its surface in the whispering gallery mode; there is no need to perform an exact alignment of the mirror with respect to the incident X-ray beam in order to observe the effect. To analyze the experimental data, we used the ray tracing method. It is shown that the results of the experiment and computer simulation are in good agreement. PACS numbers: 07.85.Fv DOI: 10.1134/S1063774508060229
INTRODUCTION The essence of the whispering gallery effect, which is well-known in acoustics and radiophysics [1], is as follows: an acoustic or electromagnetic wave, tangentially incident on a concave surface, slides along it due to successive reflections (Fig. 1). It was proposed in [2, 3] that this effect could be used to stear X-ray beams. The results of the theoretical investigations of the whispering gallery effect in the short-wavelength range were reviewed in [4–6]. The experimental investigations of this effect in soft and hard X-rays were described in [7–11] and [12, 13], respectively. The efficiency of rotating an X-ray beam tangentially incident on a concave surface with an opening angle ψ is determined by the expression R 0 ( ψ ) = exp [ – 2ψ Im ( 1 – ε )
– 1/2
3/2
] ≈ exp ( – ψγ /δ );
ε = 1 – δ + iγ ,
(1)
where ε is the complex permittivity of the mirror material; the last approximate formula is valid only for materials with low absorption (γ/δ 1). Formula (1) can be obtained both in the geometrical-optics approximation and by solving the wave equation [4, 5]. The characteristic features of expression (1) are as follows. First, the rotation efficiency is maximum at low X-ray absorption in the reflecting coating material rather than at its large polarizability, as it is in conventional grazing-incidence optics. Second, the rotation efficiency is independent of both the X-ray path length along the concave surface (radius of curvature) and its specific shape; it is determined only by the angle of rotation ψ. When the correct reflecting coating material
is chosen, the efficiency (1) is several tens of percent at X-ray beam rotation by 90°. These specific features of the whispering gallery effect are demonstrated in Fig. 2, which shows the dependence of the efficiency (1) on the angle of beam rotation for hard X rays (λ = 0.154 nm) and a quartz concave mirror (curve 1). For comparison, Fig. 2 shows also the efficiency of beam rotation in the case of a singl
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