Condenser Optics for Dark Field X-Ray Microscopy

Transcription

1 Condenser Optics for Dark Field X-Ray Microscopy S. J. Pfauntsch, A. G. Michette, C. J. Buckley Centre for X-Ray Science, Department of Physics, King s College London, Strand, London WC2R 2LS, UK Abstract. The optical system for a dark field x-ray microscope is described, in particular the optics for condensing the x-rays onto the specimen. The microscope is designed for two types of source, line and continuum, using a toroidal mirror or a toroidal grating to perform simultaneous focusing and dispersion. The design criteria for these optics are discussed and a procedure for minimising the aberrations at the specimen plane is described; this results in good spectral resolution and minimises loss of flux due to non-dispersive aberration of the grating. 1 Introduction The dark field microscope (figure 1) consists of a toroidal condenser, a post specimen zone plate which forms an image using x-rays scattered by the specimen, a central stop to prevent directly transmitted x-rays from reaching the detector, and a microchannel plate / phosphor / CCD combination detector. The condenser may either be a mirror for use with a line source or a grating for use with a continuous source. One pulse of the laser plasma source at the Lasers for Science Facility (LSF), Rutherford Appleton Laboratory, should be sufficient to form an image. Planned uses of the dark field microscope include the location of variety of deposits with elemental and, possibly, chemical specificity using XANES [1]. The ability of a dark field system to detect features below the nominal resolution limit of the optics will allow smaller deposits to be detected than with other forms of x-ray microscopy. Fig. 1. Layout of the dark field x-ray microscope. A method for selecting the optimum reflection angle of the toroidal mirror is presented in section 2. This is followed in section 3 by a description of the optimisation procedure for the grating condenser. The predicted performance of the grating, in respect of the planned uses of the microscope, is discussed in section 4.

2 IV - 94 S. J. Pfauntsch et al. 2 Selection of the Optimum Reflection Angle The condenser is used at grazing incidence, and thus the aperture of the optic is reduced by a factor sinθ compared to a normal-incidence system. For a curved surface the grazing incidence angle, θ, and hence the reflectivity, vary along the surface. In order to collect the maximum flux it is necessary to maximise the product of the reflectivity and sinθ integrated over the range of incidence angles. In practice, since the meridional radii of curvature of x-ray mirrors are very large (typically 10m) compared to the lengths (typically 25 50mm), the incidence angle varies by only 1 2% across the mirror and so a good approximation is given by maximising A = R sinθ (1) where R is the reflectivity determined by the Fresnel equations. The reflectivity depends on the complex refractive index of the mirror material as well as the grazing incidence angle. Differentiating equation (1) with respect to S = sinθ and setting the result equal to zero gives an equation for the maximum value of A of the form A/ S = N(S)/D(S) = 0 (2) The optimum value, θ opt, of the grazing incidence angle is thus given by N(S) = 0. However, this leads to a 28th order polynomial in S and hence there is no analytic solution. It is therefore necessary to use graphical or numerical techniques to obtain an exact value of θ opt, but a good approximation can be obtained as the higher order terms of N(S) are very small, which makes the problem more tractable to analytic solution. For nickel in the water window, for example, an accuracy of better than 5% is obtained by ignoring all terms of order higher than S 5 ; at a wavelength of 3.5nm, the exact value of θ opt is 5.32, whereas the value obtained from the fifth order polynomial approximation is Since Rsinθ varies slowly close to θ opt the performance of the mirror is not significantly degraded by using the approximate value. The included angle φ =π 2θ opt of the mirror should be maintained when it is replaced by the grating, so that the deviation of the x-ray beam remains constant. 3 Optimisation of the Grating Condenser Once a suitable included angle has been determined the angle of incidence, α, and the angle of diffraction, β, of the grating can be found from the grating equation sinα + sinβ = mλ/d (3) where m is the diffraction order, d is the grating period and α+β=φ. Here, the negative first order (m = 1, lying between the zero order and the grating surface) is used here as, for a givenφ, it has a smaller value of α than the positive order does, so that the grating presents a larger effective aperture to the beam. For a point source the optical

3 Condenser Optics for Dark Field X-Ray Microscopy IV - 95 path length function, which describes the path of a ray from the source via a point on the grating surface to the image plane, can be written as [2] F = u + v + mλ /d (sinα + sinβ ) + F ab (u, v, ρ s, ρ m ) (4) where u is the distance from the source to the centre of the grating, and v is the distance from the centre of the grating to the Gaussian image point. The shape of the surface is determined by the two radii of curvature, ρ s and ρ m, in the aberration term F ab which describes defocus, astigmatism, coma, astigmatic coma and spherical aberration. The design requirement is to minimise the sum of the aberrations by selecting the best values of ρ s and ρ m for given values of u and v. The deviation of the aberrated ray from the Gaussian image point may be written in terms of components X in the dispersive direction and Y in the non-dispersive direction [2], X v F Y v F ab =, = cosβ l l z ab y (5) where l z and l y are, respectively, the coordinates running perpendicular to and along the grating lines; (l y, l z ) = (0,0) at the centre of the grating. The corresponding wavelength error, which determines the spectral resolution λ / λ, of the grating is d Fab λ = = m l z d m cos β X (6) v The best values of ρ s and ρ m are determined from the minima of X and Y. As the aberrations are largest for rays via the corners of the grating, X and Y should be determined as functions of ρ s and ρ m at each corner. However, because the grating is symmetrical about its centre only two corners need to be considered. Since the dispersive aberration is primarily dependent on the meridional radius, minimising X at the two corners gives two values of ρ m. Similarly, minimising Y gives two values of ρ s. For each corner it is necessary to iterate between X and Y until the values of ρ m and ρ s converge. Average values of the radii are then used to determine the aberrations for the grating as a whole. If the resulting values of X ( Y) at the two corners have the same sign, the aberration is given by the largest; if they are of opposite sign the sum of their modulii must be used. Table 1. Parameters of the grating to be optimised, for an included angle of Number of lines per mm period d [µm] order m design wavelength [nm] incidence angle α diffraction angle β grating length [cm] grating width [mm]

4 IV - 96 S. J. Pfauntsch et al. 4 Calculated Grating Performance The parameters of the grating to be optimised are shown in table 1. Table 2 shows results of the optimisation (a) with u = v = 1m, (b) for the corresponding Rowland circle geometry with u = ρ m cosα, v = ρ m cosβ and (c) a solution which minimises the sum of the aberrations within the geometrical constraints of the microscope layout. Solution (c) is used for the further analysis described below. Table 2. Values of the meridional and sagittal radii and the aberrations of toroidal gratings at the design wavelength of 3.5nm for dark field x-ray microscopy. u [m] v [m] ρ m [m] ρ s [cm] X [µm] Y [µm] λ [pm] (a) (b) (c) The same grating may be used at different wavelengths by varying the angles αandβ, with constant included angle. The aberrations will not be minimised, except at the design wavelength, but can be kept tolerable as shown in table 3, which gives the non-dispersive aberrations and wavelength errors for the water window x-rays. Table 3. Performance of grating (c) of Table 2 over a range of wavelengths. λ [nm] α β Y [µm] λ/ λ The results summarised in tables 2 and 3 are for a point source, and it would be complicated to take into account the effects of finite source size using the analysis described above. It is more appropriate to employ ray-tracing techniques, which can also be used to verify the solutions obtained from the analysis of section 3. The raytracing program SHADOW [3] was used, initially for a point source. The computed ray positions at the image plane for the design wavelength of 3.5nm are shown in figure 2a. The majority of the rays fall within the bounds given in table 2, confirming that the aberrations are minimised. For a finite source the aberration terms increase, as the optic is used off axis. For the dark field x-ray microscope the main effect will be to decrease the spectral resolution, and this has been investigated using SHADOW for a source diameter of 60µm. The LSF source has a diameter less than half of this. The source was assumed to be emitting x-rays of equal intensity at two wavelengths with a separation of 0.06nm. Figure 2b shows that the two wavelengths do not overlap for imaging across the oxygen, calcium and carbon absorption edges.

5 Condenser Optics for Dark Field X-Ray Microscopy IV - 97 Non-dispersive direction [µm] Dispersive direction [µm] a) Non-dispersive direction [µm] Dispersive direction [µm] b) Fig. 2. The spread of points in the image plane, obtained using the ray-tracing program SHADOW a) for a point source and grating (c) of table 2, b) for a finite source emitting x-rays at two wavelengths with a separation of 0.06nm either side of the calcium L edge (blue), the carbon K edge (green) and the oxygen K edge (red). 5 Conclusions The above analysis has shown that a toroidal grating used in a dark field x-ray microscope can give a spectral resolution, λ/ λ, of several hundred in the waterwindow. Typical objective zone plates will have diameters of ~100µm and outer zone widths of about 50nm, leading to spectral resolutions of ~250. The resolving power of the condenser, when the finite source size is taken into consideration, can be smaller than this necessitating the use of a slit at the image plane of the optic. For a spectral resolution of 250, equation (6) shows that the required slit width ranges from about 80µm at the short wavelength end of the water window through about 130µm at the design wavelength to about 175µm at the long wavelength end. Comparing these widths with figure 2b shows that the slit transmits about 25% of the available flux at the oxygen K edge, essentially 100% at the calcium L edge and about 90% at the carbon K edge. A more severe loss of flux away from the design wavelength is caused by the spread in the non-dispersive direction, and in future it will be desirable to have a range of gratings optimised for different wavelengths. Acknowledgments The dark field x-ray microscopy project is supported by The Paul Instrument Fund of the Royal Society ( G143) and by the King s College Research Strategy Fund. References 1. C.J. Buckley, N. Khaleque, S.J. Bellamy and X. Zhang, these proceedings. 2. J.B. West and H.A. Padmore, in Handbook on Synchrotron Radiation, vol. 2, ed. G.V. Marr (Elsevier, Amsterdam, 1987). 3. C. Welnak, G.J. Chen and F. Cerrina, Nucl. Inst. Methods Phys. Res. A347, 344 (1994).