From this we can see that for a 1 degree beamline at 1 Å, the mirror roughness σ
should be on the order of a few Å—about 1 atomic layer. The tolerances are
obviously more forgiving for longer wavelengths or shallower glancing angles.
The effects of increasing σ on mirror performance can also be seen in Fig. 4.4.
Since β increases sharply above an X-ray absorption edge (Chap. 6), the reflectivity of a mirror will drop as the X-ray energy passes through that edge. Despite
these losses, the larger δ helps enhance the reflectivity at higher energies. These
effects are both illustrated in Fig. 4.5.
4.3.3 Mirror Shapes and Aberrations
Grazing incidence X-ray mirrors are often shaped to collimate or focus the X-ray
beam. Just as for visible region optics, spherical shapes are easier to produce than
aspheres such as ellipsoids and paraboloids. However, because of the grazing
incidence geometries, the “astigmatism” aberrations for spherical mirrors are severe,
as shown in Fig. 4.6.
Astigmatism means that the sagittal rays in the horizontal plane come to a
different focus than do the tangential rays in the vertical plane. It can be reduced
by (a) using two different mirrors with different radii for horizontal and vertical
focusing, (b) using one mirror with two different radii of curvature, or (c) employing
two different elliptical mirrors to achieve focusing in the horizontal and vertical
planes (Fig. 4.6).
The last of these combinations is called a Kirkpatrick-Baez pair (Fig. 4.7) [90]. It
has the wonderful property that near-perfect focusing can be achieved in both
horizontal and vertical planes. Furthermore, it is “achromatic”—its focal length
does not change with the X-ray wavelength. Elliptical shapes were originally
Fig. 4.5 Reflectivity vs.
energy for an SiO 2 (glass)
mirror at 0.4
and a goldcoated mirror at 0.5
. Notice
the structure resulting from
the Si K-edge and Au
M-edges, as well as how the
Au coating extends the
useful range to nearly
10 keV, calculated using
CXRO website
76
4 X-ray Optics and Synchrotron Beamlines
should be on the order of a few Å—about 1 atomic layer. The tolerances are
obviously more forgiving for longer wavelengths or shallower glancing angles.
The effects of increasing σ on mirror performance can also be seen in Fig. 4.4.
Since β increases sharply above an X-ray absorption edge (Chap. 6), the reflectivity of a mirror will drop as the X-ray energy passes through that edge. Despite
these losses, the larger δ helps enhance the reflectivity at higher energies. These
effects are both illustrated in Fig. 4.5.
4.3.3 Mirror Shapes and Aberrations
Grazing incidence X-ray mirrors are often shaped to collimate or focus the X-ray
beam. Just as for visible region optics, spherical shapes are easier to produce than
aspheres such as ellipsoids and paraboloids. However, because of the grazing
incidence geometries, the “astigmatism” aberrations for spherical mirrors are severe,
as shown in Fig. 4.6.
Astigmatism means that the sagittal rays in the horizontal plane come to a
different focus than do the tangential rays in the vertical plane. It can be reduced
by (a) using two different mirrors with different radii for horizontal and vertical
focusing, (b) using one mirror with two different radii of curvature, or (c) employing
two different elliptical mirrors to achieve focusing in the horizontal and vertical
planes (Fig. 4.6).
The last of these combinations is called a Kirkpatrick-Baez pair (Fig. 4.7) [90]. It
has the wonderful property that near-perfect focusing can be achieved in both
horizontal and vertical planes. Furthermore, it is “achromatic”—its focal length
does not change with the X-ray wavelength. Elliptical shapes were originally
Fig. 4.5 Reflectivity vs.
energy for an SiO 2 (glass)
mirror at 0.4
and a goldcoated mirror at 0.5
. Notice
the structure resulting from
the Si K-edge and Au
M-edges, as well as how the
Au coating extends the
useful range to nearly
10 keV, calculated using
CXRO website
76
4 X-ray Optics and Synchrotron Beamlines