Calculation of the Bragg angle for synthetic multilayer x-ray reflectors. Kenneth D. Shaw and Allen S. Krieger. Radiation Science, Inc., P.O. Box 293, Belmont, ...
Calculation of the Bragg angle for synthetic multilayer x-ray reflectors Kenneth D. Shaw and Allen S. Krieger Radiation Science, Inc., P.O. Box 293, Belmont, Massa chusetts 02178. Received 23 June 1988. 0003-6935/89/061052-03$02.00/0. © 1989 Optical Society of America. An expression is derived for the Bragg angle of synthetic multilayer x-ray reflectors. It displays the explicit depen dences on not only the wavelength of the radiation and the period of the multilayer, but also on the ratio of the thick nesses of the two layers comprising one period. Absorptive dependence is also taken into account. In recent years, there has been increasing interest in thin film multilayer structures for use as Bragg reflectors for low to medium energy x rays. These multilayers consist of alter nating layers of high and low Z materials; the low Z layers, which are practically transparent to radiation in the energy regime of interest, act as spacers for the high Z layers which play the same role as do the crystal planes in the more familiar phenomenon of Bragg scattering from a crystal. In designing such a reflector, it is important to be able to calcu late the Bragg angle as a function of the various parameters which characterize the multilayer. These are: (1) the re fractive indices of the high and low Z materials which com prise the multilayer, (2) the period of the structure (one period consists of one high Z and one low Z layer), (3) the ratio of the thicknesses of the two layers comprising one period, and (4) the total number of periods in the multilayer stack. To accomplish this, other workers1-3 have employed the formula for the Bragg angle familiar from x-ray crystallogra phy4:
whereθbis the Bragg angle, λ is the wavelength in vacuum of the incident radiation, δ is the decrement in the real part of the complex refractive index, and d is the period of the structure. To apply this to the multilayer, it is customary to treat the multilayer as if it were a crystal of refractive index n = 1 — δ + iβ, where the value used for δ is a weighted average of the values for each of the two components comprising a 1052
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period, with different weighting formulas being employed by various workers. 5 The absorptive term β in the refractive index is neglected, as it is generally assumed that it gives a negligible contribution. 1 However, at least one group 6 has shown that under certain circumstances this term can have a significant effect. The validity of the various weighting schemes can be checked experimentally by measuring the angular location of the Bragg peak and then using this in Eq. (1) to solve for δ. The agreement of the values thus determined with those predicted by the different weighting procedures varies sig nificantly from one method to another, particularly at ener gies below 1 keV. 5 Therefore, it would be advantageous to have an expression for θb which does not depend on such approximations, which permits the inclusion of absorptive effects, and which better illuminates the physics of the observed dependence of the Bragg angle on the ratio of the thicknesses of the high and low Z layers. To derive such an expression, we calculate the phase dif ference Δφ between rays reflected from two successive high Z layers (refer to Fig. 1). If the thicknesses of the high and low Z layers are d1 and d2, respectively, and their respective complex refractive indices are n1 and n 2 , this is given by
Fig. 1. Plane wave impinging on a multilayer at a grazing angle equal to the Bragg angle θb. The high and low Z layers have refrac tive indices n1 and n2 and thicknesses d1 and d2, respectively.
where θi is the angle that the wavevector ki makes with the normal to the multilayer, i = 1,2, and k0 is the magnitude of the wavevector in vacuum. Using Snell's law, θ2 can be eliminated to give
The Bragg condition is that the path length difference related to Δφ by
is an integral multiple of the wavelength of the radiation. Applying this condition to Eqs. (3) and (4) yields
where m is an integer. If we are interested in first-order diffraction, we set m = 1 and then solve Eq. (5) for cosθ1. For d1 ≠ d2 we obtain
and for
The Bragg angle (measured, as is customary, with respect to the surface of the entrance layer) is then given by
For d1 = d2 = d/2 and n1 = n2 = n = 1 – δ, Eqs. (7) and (8) com bined reduce to Eq. (1) as is to be expected. As a test of the validity of these expressions, we have performed a numerical calculation of the reflectivity at 1.54 Å of a typical tungsten-silicon multilayer as a function of grazing angle. The reflector comprises sixty-four periods,
Fig. 2. Reflectivity at 1.54 Å as a function of grazing angle for a tungsten-silicon multilayer with d1 = 5 Å and d2 = 30 Å. The value of θb calculated using Eqs. (6) and (8) is 22.5804 mrad, while the peak reflectivity occurs at 22.4336 mrad.
with d1 = 5 Å and d2 = 30 Å. Since this calculation does not depend on the value of θb, the equations just derived may be checked by comparing the calculated Bragg angle with the angular position of the peak of the reflectivity curve. The results are displayed in Fig. 2, with the calculated value of θb indicated by an arrow. The agreement can be seen to be quite close, with the calculated angle differing from the location of the peak by only 0.65%. We have derived an expression for the Bragg angle for multilayer reflectors and shown that its yields a value which agrees quite closely with that which is physically expected. The dependence of θb on d1 and d2 appears explicitly in the equations in contrast to the procedure usually employed. The absorptive dependence is also explicit, since it is includ ed in the complex refractive indices n t . This work was supported by the Innovative Science & Technology Office of the Strategc Defense Initiative Organi zation and managed by the Army Strategic Defense Com mand. 15 March 1989 / Vol. 28, No. 6 / APPLIED OPTICS
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References 1. B. L. Henke, J. Y. Uejio, H. T. Yamada, and R. E. Tackaberry, "Characterization of Multilayer X-Ray Analyzers: Models and Measurements," Opt. Eng. 25, 937 (1986). 2. J. H. Underwood and T. W. Barbee, Jr., "Synthetic Multilayers as Bragg Diffractors for X-rays and Extreme Ultraviolet: Calculations of Performance," AIP Conf. Proc. 75, 170 (1981). 3. T. W. Barbee, Jr., "Multilayers for X-Ray Optics," Opt. Eng. 25, 898 (1986). 4. A. H. Compton and S. K. Allison, X-Rays in Theory and Experiment (Van Nostrand, New York, 1935), p. 674. 5. T. W. Barbee, Jr., "Sputtered Layered Synthetic Microstructure (LSM) Dispersion Elements," AIP Conf. Proc. 75, 131 (1981). 6. A. E. Rosenbluth and J. M. Forsyth, "The Reflecting Properties of Soft X-ray Multilayers," AIP Conf. Proc. 75, 280 (1981).
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