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X-ray Optics
Optical properties of materials are characterized by the refractive index. X-ray optics is
different from conventional optics because the refractive index is very close to (and smaller
than) unity. In addition, absorption plays a larger role than in conventional optics. This can all
be taken care of by introducing a complex refractive index, with the following common
notation:
n~  n  ik  1    i
It can be shown (Attwood 3.1) that:
na re 2 0

f1 ( )
2

na re 2 0
f 2 ( )
2
where except for the classical electron radius, re  2.82  10 15 m, the numbers are dependent
on the atomic density n a , the vacuum wavelength of the radiation  , and the atomic
scattering factors. These are tabulated and can be found on
http://www-cxro.lbl.gov/optical_constants/
Since the complex refractive index determines absorptivity, transmitivity and reflectivity,
these optical properties can be derived with this information: this is also done at this very
useful homepage.
Absorption
The absorption coefficient,  , defined in the Beer-Lambert law, which describes how the
intensity is attenuated from I 0 to I as a beam goes a distance, r , trough a material with the
density  ,
I  I 0 e  r ,
is directly related to f 20 and  :
  2na ref 20 ( ) 
4


For soft X-rays the attenuation length is some 1000Å, whereas it increases for higher
energies, where all materials eventually become transparent.
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Reflection
Reflectivity and transmitivity are governed by Fresnel’s equations, which can be derived (see
the derivation in Fresnel.doc) by considering the boundary conditions. This is relatively
simple as long as we neglect absorption (and require that only the E-field interacts with the
material). With the definitions in the figure we first get the relation of the scattering angles in
the reflection law,
i   r
and Snell’s law,
ni sin i  nt sin t
Then we define reflectivity and transmitivity as the ratio of the amplitudes:
ni
Er
Ei
i
r
Br
Bi
t
nt
Et
Bt
E 
n cosi  nt cost
r   0 r   i
 E0i  ni cosi  nt cost
E 
2ni cosi
t   0t  
 E0i  ni cosi  nt cost
ni
Er
Ei
i
Bi
t
nt
Br
r
Et
Bt
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E 
n cosi  ni cost
r//   0 r   t
 E0i  // nt cosi  ni cost
E 
2ni cos i
t //   0t  
 E0i  // nt cos i  ni cos t
From these equations it can be seen directly that reflectivity goes to zero when nt  ni . On

the other hand, already at a small deviation, ( nt  ni ), for grazing incidence (  i  ) we see
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that the magnitude of both r// and r approaches unity: Reflectivity in grazing incidence is
always 1! This is further helped by the fact that the index of refraction is smaller in the
material than in vacuum. This means that we reach total external reflection at some angle.
Already from Snell’s law we can derive this critical angle:
n sincritical  nt sin

2
 nt  1  
Defining the angle relative to the surface (as opposed to relative to surface normal)
sin(

2
 c )  cos c  1  
Since  c is a small angle
1
c2
2
1
and therefore
c  2
With a complex index of refraction, the Fresnel equations become more involved. Absorption
has mainly the effect of smearing out the variations predicted when absorption is neglected.
One can show that for the special case of normal incidence:
R r 
2
2  2
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Note that the reflectivity increases with absorptivity!
The reflectivity at grazing incidence is more complicated, but it can be shown that at the
critical angle, we have:
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2
 

2
1
 
1
R, critical  r, critical
2
Thus, the reflectivity decreases from unity as the absorption increases. These equations show
that both the real and imaginary parts of the refraction index are important to determine
reflectivity. From another perspective: absorption and reflection are connected. From a
complete absorption experiment, we can determine the imaginary part for all frequencies.
These measurements can then also be used to derive the real part, and thus predict the
reflectivity for all energies. This is done using the Kramers-Kronig relations.
Refractive Optics
As the real part of the refractive index is close to unity refractive optics is not the classical
choice. Refraction is, however, possible: This was shown early in Uppsala, and was presented
in Axel Larsson’s thesis, 1929.
Very recently, people have constructed refractive lenses for X-rays, and there is now a large
activity in this field. The schematic of the Compound Refractive Lens (CRL) and the result of
a knife-edge scan of an Au knife-edge sample at 21 keV energy using a CRL of 63 individual
Al lenses, is illustrated in the following figure, which is taken from the ESRF home page. For
further reading the following references are recommended
1. Snigirev A, Kohn V, Snigireva I , Lengeler B., Nature, 384, 49-51 (1996).
2. Lengeler B, Schroer C, Tummler J, Benner B, Richwin M, Snigirev A, Snigireva I,
Drakopoulos M., J. Synchrotron Radiation, 6, 1153-1167 (1999).
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Mirrors
Mirrors used in grazing incidence, close to the critical angle, are by far the most common
optical components. At grazing angles a concave spherical mirror focuses (almost) only in
one direction. A structure corresponding to an ordinary lens
Kirkpatrick-Baez telescope
Figure 4.1: Kirkpatrick-Baez telescope configuration (from imagine.gsfc.nasa.gov).(ps)
forming a real image was proposed by Kirkpatrick and Baez, 1948. This consisted of a set of
two orthogonal parabolas of translation as shown in Figure 4.1a. The first reflection focused
to a line, which the second reflection then focused to a point. This was necessary to avoid the
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extreme astigmatism suffered by a single. The system was attractive for the ease of
constructing the reflecting surfaces. These could be polished to a high level of accuracy as flat
plates and then mechanically bent to the required curvature. In order to increase the aperture a
number of mirrors could be nested together (figure 4.1b), This is used in telescopes to
increase intensity, but it is not needed at beamlines.
The surfaces can be bent to a figure which depends on the application. Elliptical surfaces are
used for point-to-point imaging, and spherical surfaces are used for the special case of unity
magnification. For simplicity, cost and requirements on figure errors, spherical mirrors are
often used also for other magnifications (as approximations elliptical surfaces). For point-toparallell imaging parabolic mirrors are used.
Multilayers
Multilayers can be used to increase reflectivity in such mirrors systems, as is shown in the
Figure below, which is taken from the home page of the Optics Group at the ESRF.
A schematic multilayer structure and a typical measured reflectivity spectrum. Layer A usually consists of
a strongly absorbing material (metal). Layer B is a spacer made of a low-density material.
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Focusing geometry with a multilayer mirror. Note that the layer spacing must change over the
mirror, and that such a system can be optimized only for a narrow wavelength range.
Waveguides
Again using the fact that the refractive index is less than unity, there have been efforts to
focus X-rays using glass capillaries, and two almost parallel plates. The idea is based on
multiple reflections, just like in ordinary fiber optics. (References, pictures and more details
will come)
Fig. from http://www.opticsinfobase.org/DirectPDFAccess/CA1EDE93-BDB9-137EC936B6996F9CE005_46295.pdf?da=1&id=46295&seq=0
Diffractive optics: Zone plates
Zone plates, also familiar from conventional optics are based on Fresnel diffraction. They are
often used in X-ray microfocus applications. It can be shown that they work just like a lens
with the focal distance determined by
f 
Rm2
m
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where R m is the radius of the outermost circle. This is described in Attwood 9.2.
Photon sieves
An interesting recent development in this context is that one can modify the structures in
detail to influence the ‘diffraction pattern’. Here, more of the primary intensity is lost, but one
can reduce effects of higher order diffraction. The idea is suggested in…….
Dispersive Optics
To achieve high energy resolution (or temporal coherence) a principle based on interference
has to be used. Most commonly diffraction of the radiation in periodic structures is used.
Fourier analysis sets the limit for the resolution that one can achieve. If the beam is diffracted
in a periodic structure with N periods the resolving power is:
R

 mN

where m is the order of diffraction.
When the wavelength of the radiation compares to atomic distances ( 1Å  12.4keV ) Bragg
diffraction in crystals is the obvious choice. At these wavelengths one also uses the fact that
reflectivity can be neglected: Radiation scatters at large angles only if the condition for
constructive interference between rays scattered at different scattering planes is fulfilled:
2d sin   m
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where d is the distance between the scattering planes, and the angle  is defined in the Fig
below.

d
In the hard X-ray range it is easy to find scattering plane where this condition is fulfilled.
Here one also uses that the penetration length of the photons are large, so that many crystal
planes contribute to the scattering: thus the resolving power can become very large.
At ESRF (ID16 and ID28) one performs X-ray scattering experiments with a resolution of 1
meV at an energy of around 20 keV. The limiting factor is here not fundamental, i. e. it is not
dependent on how many contributing layers, but rather practical: How well can the scattering
angles be determined. The quality of the crystals of course become crucial at some point, but
as long as one can use materials like Si, out of which large ‘perfect’ crystals can be made this
is no limitation.
A table of used crystal planes are found in the CXRO yellow booklet. The shortest 2d of any
practical crystal are the ( 50 5 2 ) planes of -quartz: 2d=1.624 Å. One of the largest 2d for any
natural crystal are the ( 10 1 0 ) planes of beryl 2d=15.954 Å. This means that one can cover,
roughly the range ( 0.2 Å    16 Å ), or in energy:(0.76 keV<E<62keV),with natural crystal.
In principle one can further extend the range towards longer wavelengths using so-called
Langmuir-Blodgett crystals and multilayers
In the soft X-ray range there are several complications. The exotic or artificial crystals have
imperfections that limit their usefulness. Absorption is also a severely limiting factor. At 760
eV a typical penetration length is around 3000Å. If d=8Å, this means that only 375 layers can
contribute to the interference maximum. This severely limits the resolving power.
For low energies, the reflection grating is much more common as a dispersive element. It can
be seen directly in the Figure that the condition for constructive interference is
d (sin b  sin i )  m
Although this equation looks very much like the equation for Bragg diffraction the
phenomenology is quite different. In the Bragg case the scattering angle is defined relative to
the lattice planes, and the scattering angle is the same as the incidence angle. Intensity is
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scattered only if the interference condition is fulfilled. At other angles (or wavelengths)
‘nothing happens’, except transmission and absorption. In the reflection grating the angles are
defined relative to the surface normal, and there is always a direction where a wavelength can
be diffracted. For m  0 , i  b and the grating acts as mirror. In principle all orders of
diffraction can be used for dispersion, and the gratings are always blazed; optimized for
certain angles, and wavelength ranges.
i
b

d
1
, cannot be much larger 2000 grooves/mm. The
d
distance between the grooves, can thus not be much smaller than 0.5 micron. One would think
that the gratings then would separate wavelengths of this length scale, but we shall see that the
we get dispersion at much smaller wavelengths. For some details about dispersion and
resolution we consider the intensity from N slits, with perfect reflectivity and incident
intensity, I 0 of monochromatic radiation:
The groove density (grating constant)
N

 sin
2
I (b)  I 0 

 sin

2







2
where, the phase difference between beams from adjacent slits is

2

d (sin b  sin i )
N
 k , for k  0,1...... The first time this happens on the side
2
of the first principal maximum is
We see that I (b)  0 when
 
2
N
Therefore the instrumental width corresponds to:
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 
4
N
Assuming that the incidence angle is fixed, we can differentiate the expression for  and
relate the phase variation to the variation in the angle b:
 
2

d cos bb 
4
N
The instrumental angular broadening thus becomes:
b 
2
Nd cos b
According to Rayleighs’s criterion for resolution two peaks are resolved when the maximum
of one coincides with the diffraction minimum of the other. This corresponds to half the
broadening:
bmin


Nd cos b
Now, the angular variations relate to the wavelength variations through the grating equation,
which we differentiate:
m min  d cos bbmin
By combining the last two equations we get, as expected:

 mN
( ) min
The best resolving power achieved with a grating in the soft X-ray range is around 0.5 meV at
60 eV) is around 120000, at BESSY-II. For first order diffraction this would imply that at
least 120000 lines of the grating must be illuminated. With 1 micron groove spacing the
grating must have been at least 120 mm long.
Compared to the resolving power achievable with crystal diffraction in the hard X-ray range,
gratings will never be able to compete. Luckily there is no competition since the two
techniques work in different wavelength ranges.
Interferometers
There are also new exotic schemes for separating wavelengths, In the X-ray range
interferometry can be performed using amplitude splitting interferometers. This is at the
moment an unusual.
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From Roland Smith, Nature 404, 345 - 347 (23 Mar 2000)
As far as I know there are today no working soft X-ray interferometric beamlines. It has been
tried at the ALS, and we are working on a wavefront splitting interferometer for spectroscopy.
Spherical Grating Monochromator (the Dragon)
The starting point for the simplest spherical grating monochromator is the geometry suggested
by Rowland in 1882. The idea is to combine the focussing properties of spherical mirror with
the diffraction properties of a grating. He found that if a source and the grating are placed on a
circle with half the radius of the grating, all wavelengths will be focussed on the same circle:
The Rowland circle. This can be shown to be a good approximation using e. g. Fermat’s
principle. The geometry takes a very simple form. If R is the radius of the grating curvature,
the distance between slit and grating is:
x  R cos i
and the distance between grating and output slit is analogously
y  R cos b
One monochromator design would be to use a fixed input slit and a spherical grating mounted
according to the first equation, and scanning the energies by moving an output slit according
to the second. Another way would be to rotate and translate the grating according to both
equations. In practice one can make small deviations from the Rowland criterion by only
rotating the grating, and keeping the slits fixed for small ranges.
In this geometry one can use the angular dispersion to calculate the spatial dispersion on the
Rowland circle. Realizing from the Fig. that a small distance along the circle relates to a small
angular change through:
z 
R
   Rb
2
the angular change relates to wavelength through the (differentiated) grating equation and we
get:
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z   Rb  
Rm

d cos b



R/2
i
b
y
x
1
have to be large. In
d
practice distances between gratings and slits are on the order of meters, whereas the slit sizes
are on the order of microns. This makes it necessary to control the angle of the grating, and
surface slope errors within microradians! It is the slit widths and slope errors which set the
limit of the resolving power! Note also that  (as defined by geometry) is independent of  .
This means that the resolving power decreases when the wavelength becomes shorter. The
absolute energy resolution becomes worse with energy:
We see that to be able to measure small wavelength differences R and

E 2 (eV )
E  E



12400
SX-700 (Pedersen) Plane Grating Monochromator
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From: R. Reininger, V. Saile, Nucl. Instr. Meth. A288 (1990) 343
The original design is based on the idea that an elliptical mirror focuses the direct source (no
input slit) on an output slit. The plane grating effectively makes a virtual monochromatic
source for this imaging. If r is the distance between the source and the grating a virtual
monochromatic source will be at a distance
2
 cos b 
2
r '  r 
  rc
cos
i


behind the grating. Scanning the monochromator by rotating the grating would then change
the distance to the virtual source. By introduction of plane pre-mirror which can also be
rotated and translated to keep the c-value fixed. This c-value can then be chosen to optimize
the resolving power (small incidence angles, large c-values) or suppress higher orders (large
angles, small c-values).
Monochromators based on Variable Line Spaced Gratings
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From Underwood and Gulliksen, JES 92 265 (1988).
With variable line spacing on the grating the focussing properties of the grating can be used to
simplify: For this set-up only rotation of the gratings are necessary to scan the energy!
Figure errors and slope errors
The fast development of synchrotron radiation optics put new demands on the optics. The
optical quality must be specified by the manufacturers. This is a typical test protocol for stateof-the-art elliptical mirrors
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The short range errors; the roughness has to be small not to lose reflectivity. Today the rms
variation is often better than 5Å.
Ray tracing
To test beamlines the optical systems are always checked by simulation of the beam: Ray
tracing. Advanced codes have been developed and the software can often be downloaded
(sites to come..) They include a simulation of the source, the influence of slope errors and
roughness, as well as deformations due to temperature.