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.
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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, θopt, 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 θopt, 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 S5; at a wavelength of 3.5nm, the
exact value of θopt is 5.32°, whereas the value obtained from the fifth order polynomial
approximation is 5.44°. Since Rsinθ varies slowly close to θopt the performance of the
mirror is not significantly degraded by using the approximate value. The included
angle φ = π−2θopt 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
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 ) + Fab(u, v, ρs, ρm)
(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, ρs and ρm, in the aberration term
Fab 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 ρs and ρm 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 ∂Fab
,
cosβ ∂lz
∆Y = v
∂Fab
∂ly
(5)
where lz and ly are, respectively, the coordinates running perpendicular to and along
the grating lines; (ly, lz) = (0,0) at the centre of the grating. The corresponding
wavelength error, which determines the spectral resolution λ /∆λ, of the grating is
∆λ =
d ∂Fab d cosβ
=
∆X
m ∂lz
m v
(6)
The best values of ρs and ρm 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 ρs and ρm 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 ρm. Similarly, minimising ∆Y gives two values
of ρs. For each corner it is necessary to iterate between ∆X and ∆Y until the values of
ρm and ρs 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 169.36°.
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]
900
1.11
–1
3.5
83.707
85.653
2.5
8
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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 = ρm cosα, v = ρm 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]
ρm [m]
ρs [cm]
∆X [µm]
∆Y [µm]
∆λ [pm]
(a) 1.000
1.000
10.440
9.27
29.2
37.2
2.46
(b) 1.144
0.791
10.436
8.67
28.9
27.6
3.08
(c) 1.552
0.761
12.113
9.48
24.9
33.8
2.75
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
2.5
83.96
85.34
27.9
367
3.5
83.71
85.65
33.8
1273
4.5
83.43
85.93
39.9
708
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.
Condenser Optics for Dark Field X-Ray Microscopy
600
a)
Non-dispersive direction [µm]
Non-dispersive direction [µm]
60
40
20
0
-20
-40
-60
-100
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-80
-60
-40
-20
Dispersive direction [µm]
0
b)
400
200
0
-200
-400
-600
-600
-400 -200
0
200 400
Dispersive direction [µm]
600
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 (532003.G143) and by the King’s College Research Strategy Fund.
References
1.
2.
3.
C.J. Buckley, N. Khaleque, S.J. Bellamy and X. Zhang, these proceedings.
J.B. West and H.A. Padmore, in Handbook on Synchrotron Radiation, vol. 2,
ed. G.V. Marr (Elsevier, Amsterdam, 1987).
C. Welnak, G.J. Chen and F. Cerrina, Nucl. Inst. Methods Phys. Res. A347, 344 (1994).