Analytical evaluation of the X-ray scattering contribution to imaging
  degradation in grazing-incidence X-ray telescopes
Astronomy & Astrophysics manuscript no. 7228 c© ESO 2022
March 1, 2022
Analytical evaluation of the X-ray scattering contribution
to imaging degradation in grazing-incidence X-ray telescopes
D. Spiga
INAF/Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807, Merate (LC) - Italy
Received 02 February 2007 / Accepted 27 March 2007
ABSTRACT
Aims. The focusing performance of X-ray optics (conveniently expressed in terms of HEW, Half Energy Width) strongly depend on
both mirrors deformations and photon scattering caused by the microroughness of reflecting surfaces. In particular, the contribution
of X-ray Scattering (XRS) to the HEW of the optic is usually an increasing function H(E) of the photon energy E. Therefore, in
future hard X-ray imaging telescopes of the future (SIMBOL-X, NeXT, Constellation-X, XEUS), the X-ray scattering could be the
dominant problem since they will operate also in the hard X-ray band (i.e. beyond 10 keV). In order to ensure the imaging quality at
all energies, clear requirements have to be established in terms of reflecting surfaces microroughness.
Methods. Several methods were proposed in the past years to estimate the scattering contribution to the HEW, dealing with the surface
microroughness expressed in terms of its Power Spectral Density (PSD), on the basis of the well-established theory of X-ray scattering
from rough surfaces. We faced that problem on the basis on the same theory, but we tried a new approach: the direct, analytical
translation of a given surface roughness PSD into a H(E) trend, and – vice versa – the direct translation of a H(E) requirement into a
surface PSD. This PSD represents the maximum tolerable microroughness level in order to meet the H(E) requirement in the energy
band of a given X-ray telescope.
Results. We have thereby found a new, analytical and widely applicable formalism to compute the XRS contribution to the HEW
from the surface PSD, provided that the PSD had been measured in a wide range of spatial frequencies. The inverse problem was also
solved, allowing the immediate evaluation of the mirror surface PSD from a measured function H(E). The same formalism allows
establishing the maximum allowed PSD of the mirror in order to fulfill a given H(E) requirement. Practical equations are firstly
developed for the case of a single-reflection optic with a single-layer reflective coating, and then extended to an optical system with
N identical reflections. The results are approximately valid also for multilayer-coated mirrors to be adopted in hard X-rays. These
results will be extremely useful in order to establish the surface finishing requirements for the optics of future X-ray telescopes.
Key words. Telescopes – Methods: analytical – Instrumentation: high angular resolution
1. Introduction
The adoption of grazing-incidence optics in X-ray telescopes in
the late 70s allowed a great leap forward in X-ray astronomy
because they endowed the X-ray instrumentation with imag-
ing capabilities in the soft X-ray band (E < 10 keV). The
excellent performances of the soft X-ray telescopes ROSAT
(Aschenbach 1988), Chandra (Weisskopf 2003) and Newton-
XMM (Gondoin et al. 1998) are well known.
To date, the utilized technique to focus soft X-rays consists
in systems of double-reflection mirrors with a single layer coat-
ing (Au, Ir) in total external reflection at shallow grazing inci-
dence angles. In this case, the incidence angle θi (as measured
from the mirror surface) cannot exceed the critical angle for total
reflection, otherwise the mirror reflectivity would be very low.
The critical angle is inversely proportional to E, the energy of
the photons to be focused. Using Au coatings, for instance, the
incidence angle cannot exceed ∼ 0.4 deg for photon energies
E ≈ 10 keV.
An extension of this technique to the hard X-ray energy band
(E > 10 keV) can be pursued by combining long focal lengths
(> 10 m), very small incidence angles (0.1 ÷ 0.25 deg), and
wideband multilayer coatings to enhance the reflectance of the
mirrors at high energies (Joensen et al. 1995; Tawara et al. 1998).
A very long focal length is hardly managed using a single
Send offprint requests to: daniele.spiga@brera.inaf.it
spacecraft, therefore the optics and the focal plane instruments
should be carried by two separate spacecrafts in formation-
flight configuration. This is the baseline for the future X-ray
telescopes SIMBOL-X (Pareschi & Ferrando 2006) and XEUS
(Parmar et al. 2004). Other hard X-ray imaging telescopes of
the future are NeXT (Ogasaka et al. 2006) and Constellation-X
(Petre et al. 2006).
The focusing and reflection efficiency of X-ray optics can be
tested and calibrated on ground by means of full-illumination
X-ray facilities like PANTER (Bräuninger et al. 2004;
Freyberg et al. 2006), successfully utilized in the last years to
calibrate the optics of a number of soft X-ray telescopes. The
PANTER X-ray facility now allows testing in soft (0.2÷ 10 keV)
and hard (15 ÷ 50 keV) X-rays multilayer-coated optics pro-
totypes for future X-ray telescopes (Pareschi et al. 2005;
Romaine et al. 2005). The source distance finiteness causes
some departures of the optic performances, with respect to
the case with the source at astronomical distance: effective
area loss, different incidence angles on paraboloid and hyper-
boloid, focal length displacement, a slight focal spot blurring
(Van Speybroeck & Chase 1972). However, there effects can
be quantified and subtracted from experimental data. After this
treatment, the focusing-concentration performances of the optic
can be experimentally characterized as a function of the inci-
dent photon energy, in terms of Half-Energy Width (HEW) and
Effective Area (EA).
http://arxiv.org/abs/0704.1612v2
2 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . .
The focusing performance, in particular, is altered by mirror
deformations that may arise in the manufacturing, handling, in-
tegration, positioning processes. The consequent imaging degra-
dation can be calculated from the measured departures of the
mirrors from the nominal profile, by means of a ray-tracing pro-
gram. As long as the geometrical optics approximation can be
applied, the effect is independent of the photon energy. The fig-
ure errors contribution to the HEW can be also directly measured
using a highly collimated beam of visible/UV light in a precision
optical bench. In this case, however, the light diffraction has to
be carefully estimated and subtracted.
Another drawback is the X-ray scattering (XRS) caused by
the microroughness of reflecting surfaces (Church et al. 1979;
Stearns et al. 1998; Stover 1995; and many others). The XRS
spreads a variable fraction of the reflected beam intensity in the
surrounding directions: the result is the effective area loss in the
specular direction (i.e. in the focus) and a degradation of the
imaging quality. The XRS is an increasing function of the pho-
ton energy; due to the impact that the XRS can have on astro-
nomical X-ray images quality, the height fluctuations rms of the
mirror surface should not exceed few angströms. Loss of effec-
tive area is also caused by interdiffusion of layers in multilayer
coatings, which enhances the X-ray transmission and absorption
throughout the stack. On the other hand, an uniform interdiffu-
sion does not cause X-ray scattering (Spiller 1994), hence it does
not contribute to the focusing degradation.
The microroughness of an X-ray mirror can be measured on
selected samples using several metrological instruments, each of
them sensitive to a definite interval of spatial scales l̂: Long Trace
Profilometers (10 cm > l̂ > 0.5 mm: Takács et al. 1999), opti-
cal interference profilometers (5 mm > l̂ > 10 µm) and Atomic
Force Microscopes (100 µm > l̂ > 5 nm) can be suitable in-
struments to provide a detailed profile characterization of X-ray
mirrors surface. It is convenient to present the deviation of sur-
face from the ideality in terms of Power Spectral Density (PSD),
because its values do not depend on the measurement technique
in use (see ISO 10110 Standard). In addition, the XRS diagram,
and consequently the HEW, can be immediately computed from
the PSD at any photon energy (Church et al. 1979).
In the past years, several approaches were elaborated to
relate a mirror PSF (Point Spread Function) to the PSD
of its surface. Among a wealth of works, we can cite
(De Korte et al. 1981) the assumption of a Lorentzian model
for the PSD to fit the mirror PSFs at some photon energies,
allowing the derivation of two parameters (roughness rms and
correlation length) of the model PSD. Christensen et al. (1988)
perform a fit of experimental high-resolution XRS data dealing
with the surface correlation function. Harvey et al. (1988) re-
late the PSF of Wolter-I optics to the parameters of an exponen-
tial self-correlation function along with a transfer function-based
approach. Willingale (1988) derived the surface PSD of a mirror
from the wings of a few PSFs, measured at PANTER at some
soft X-ray photon energies. O’Dell et al. (1993) interpret the PSF
of a focusing mirror on the basis of surface roughness and partic-
ulate contamination. Zhao & Van Speybroeck (2003) construct
from the PSD of a focusing mirror a model surface and compute
the X-ray scattering PSF from the Fraunhofer diffraction theory.
In the present work that problem is faced in a new and differ-
ent way, looking for a general and simple link between measured
roughness and mirror HEW. More precisely, we considered the
following question: for an X-ray grazing-incidence optic, what
is the maximum acceptable PSD of the surface that fulfills the
angular resolution (HEW) requirements of the telescope, in all
the energy band of sensitivity?
In this work we shall give a definite answer to this question.
In the sect. 2 we shall summarize the causes of imaging degra-
dation. In the sect. 3 we show how to evaluate H(E), the XRS
contribution to the HEW of a focusing mirror at the photon en-
ergy E, from any surface microroughness PSD, measured over a
very wide range of spatial frequencies. We shall see in the sect. 4
that for the special class of fractal surfaces we can even relate
the power-law indexes of PSD and HEW, and in the sect. 5 we
see how to treat the other cases. Then we prove in the sect. 6 that
the formalism can be reversed, providing thereby an independent
evaluation of the surface PSD from an analytical calculation over
H(E), and in the sect. 7 we extend the results to focusing mirrors
with more than one reflection. Finally, an example of computa-
tion is provided in the sect. 8.
2. Contributions to the imaging degradation
We shall henceforth indicate with λ the wavelength of photons
impinging on the mirror, and we shall consider the HEW as a
function of λ instead of the photon energy E. For isotropical re-
flecting surfaces in grazing incidence, the X-ray scattering dis-
tribution lies essentially in the incidence plane, so we denote the
incidence angle on the mirror as θi and the scattering angle as θs,
both measured from the surface plane (a schematic of the scatter-
ing geometry is drawn in fig. 1). If we do not consider the optic
roundness errors, the longitudinal deviations from the nominal
profile of a focusing mirror can be classified on the basis of their
typical length l̂. According to De Korte et al. (1981), they are:
1. Power errors: errors with l̂ equal to the mirror length L. They
consist in a single-concavity deformation of the profile with
respect to the nominal one.
2. Regularity errors: errors in the spatial range from 0.1 L <
l̂ < 0.5 L.
3. Surface roughness: surface defects with l̂ < 0.1 L.
However, other criteria were also formulated to separate fig-
ure errors from roughness. Consider a single Fourier component
of the surface profile with wavelength l̂ and root mean square σ.
That Fourier component is dominated by figure error if it fulfills
the condition (Aschenbach 2005)
4π sin θiσ > λ. (1)
Otherwise, it is dominated by microroughness. In other words,
surface defects within the smooth-surface approximation can be
mainly considered as microroughness. To understand the impor-
tance of this approximation, we write the optical path difference
∆s of X-rays reflected by two points of the surface with a hori-
zontal spacing l̂ and vertical spacing σ̂ = 2
2σ (for optically-
polished surfaces, σ is a increasing function of l̂, and usually
σ≪ 10−3l̂) as
∆s = l̂(cos θs − cos θi) + σ̂(sin θi + sin θs) (2)
that, for small incidence angles, becomes
∆s = l̂ sin θi(θs − θi) + σ̂(θi + θs). (3)
If that component is responsible for X-ray scattering, it has
to be ∆s ≈ λ, to cause the diffraction from surface features with
a l̂ spacing and σ̂ height. Conversely, the ”figure errors”, which
are treated with the methods of the geometrical optics, should
D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 3
Fig. 1. The geometry of X-ray scattering: the strictly speaking
”reflected” rays (i.e. in the focus direction) are characterized by
the equality θs = θi, the others are scattered apart. The rough sur-
face is a simulated one, assuming a PSD with power-law index
n = 2.4 (see sect. 4).
be characterized by the inequality ∆s ≫ λ. Note that this condi-
tion becomes similar to the eq. 1 in the limit |θi − θs | → 0. The
application of this criterion and of the subsequent X-ray scatter-
ing theory requires the incident radiation to be spatially coher-
ent over the spatial scale l̂, so that the properties of the reflected
wavefront are determined only by the coherence properties of
the mirror surface. This in turn requires the angular diameter of
the source φS to fulfill the inequality (Holý et al. 1999)
l̂ sin θi
. (4)
This equation sets a maximum to the values of l̂ that can be used
in the application of the results presented in this work. The limi-
tation can affect X-ray sources at finite distance, like those used
for X-ray optics calibrations in full-illumination setup. For very
distant astronomical X-ray sources, the condition 4 is met even
for larger l̂, up to l̂ ≈ L.
It is worth pointing out that, for a given reflecting surface,
the separation of figure errors from microroughness is strongly
affected by the incidence/scattering angles. In fact, even for large
l̂, ∆s can become comparable with λ, if θi and θs are sufficiently
small: thus, the spatial wavelength window of interest for X-ray
scattering can shift to the large l̂ domain (or, equivalently, to the
range of low spatial frequencies f = 1/l̂), provided that the con-
dition 4 is fulfilled.
Let us now consider how to separate the figure and scattering
terms in HEW data. In absence of XRS, the mirror PSF would
be independent of the energy and due only to figure errors (i.e.
in the approximation of the geometrical optics). The resulting
HEW would be also constant. Instead, due to the XRS, the figure
PSF is convolved with the X-ray scattering PSF to return the
PSF(λ) being measured (Willingale 1988; Stearns et al. 1998;
and many others),
PS F(λ) = PS Ffig ⊗ PS FXRS(λ). (5)
The resulting HEW will depend on the photon wavelength,
as it does the PSF. In order to isolate the scattering term from the
total PSF a deconvolution should be carried out, provided that
the PSFfig is known. However, if we assume that the XRS and
the mirror deformations are statistically independent, the total
HEW can be approximately calculated as the squared sum of the
two contributions:
HEW2(λ) ≈ HEW2fig + H
2(λ). (6)
An estimation of HEWfig can be obtained:
1. from the application of a ray-tracing code to several mea-
surements of the mirror profile,
2. from reliable extrapolation of the HEW(λ) curve to E → 0,
in absence of low-energy diffraction effects like dust contam-
ination, studied in detail by O’Dell et al. (1993)
3. from a direct measurement of the HEW in visible/UV light,
provided that the diffraction at the mirror edges can be reli-
ably calculated and subtracted.
Once known the measured HEW(λ) experimental trend and
the HEWfig term, the eq. 6 can be used to isolate the scattering
contribution from the experimental HEW trend: we shall prove
in the next section that the H(λ) function is immediately related
to the reflecting surface 1D Power Spectral Density (PSD) P( f )
P( f ) =
z(x)e−2πi f dx
where z(x) is a height profile (of length L) of the mirror, mea-
sured in any direction (Stover 1995): the surface is assumed to
be isotropic, and the spectral properties of the profile to be rep-
resentative of the whole surface. The PSD is often measured in
nm3 units, and for optically-polished surfaces it is usually a de-
creasing function of the frequency f .
PSD measurements have always a finite extent [ fmin, fmax],
determined by the length and the spatial resolution of the mea-
sured profile. As well known, the surface rms σ is simply com-
puted from the PSD by integration over the spatial frequencies f :
∫ fmax
P( f ) d f (8)
note that the integration range should always be specified.
3. Estimation of H(λ) for single-reflection focusing
mirrors
3.1. Single-layer coatings
Firstly, we suppose the mirror to be plane and single-layer
coated. For a surface with roughness rms σ, the specular beam
intensity obeys the well-known Debye-Waller formula
R = RF exp
16π2σ2 sin2 θi
, (9)
here RF is the reflectivity at the grazing incidence angle θi, as
calculated from Fresnel’s equations (zero roughness). However,
it should be noted in the eq. 9 that neither the spatial frequen-
cies range where the PSD should be integrated is specified, nor
the separation between reflected and scattered ray is clearly in-
dicated: these ambiguities can be solved as follows.
Let us derive the total scattered intensity Is from the con-
servation of the energy: for smooth surfaces, i.e. fulfilling the
inequality 2σ sin θi ≪ λ, we can approximate
Is = I0RF
1 − exp
16π2σ2 sin2 θi
≈ I0RF
16π2σ2 sin2 θi
.(10)
4 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . .
In grazing incidence, X-ray scattering lies mainly in the inci-
dence plane. Moreover, the normalized scattered intensity per ra-
dian at the scattering angle θs (either θs > θi or θs < θi) is related
to the PSD along with the well-known formula at first-order ap-
proximation (Church et al. 1979; Church & Takács 1986), valid
for smooth, isotropic surfaces and for scattering directions close
to the specular ray (i. e. |θs − θi| ≪ θi),
sin3 θiRFP( f ) (11)
where P( f ) is the Power Spectral Density of the surface (eq. 7)
and I0 is the flux intensity of the incident X-rays. If the scattered
intensity is evaluated at the scattering angle θs, the PSD can be
immediately evaluated as a function of the spatial frequency f :
f = l̂−1 =
cos θi − cos θs
sin θi(θs − θi)
. (12)
In the eq. 12 the approximation was justified by the assumption
|θs − θi| ≪ θi and the negative frequencies are conventionally
assumed to scatter at θs < θi: the assumed approximations make
the XRS diagram symmetric, because the PSD is an even func-
tion.
For a single-reflection mirror shell, the extension of the for-
mulae above-mentioned is straightforward by regarding |θs − θi|
as the angular distance at which the PSF is evaluated. The fo-
cal image is the superposition of many identical XRS diagrams
on the image plane, generated by every meridional section of the
mirror shell: since a π angle rotation of every meridional plane of
the shell sweeps the whole image plane, the scattered intensity is
spread over a π angle. The integration on circular coronae used
to compute the mirror PSF (at positive angles) compensates this
factor multiplying the XRS diagram by 2π (De Korte et al. 1981).
The remaining 2-fold factor accounts for the negative frequen-
cies in the surface PSD. We shall henceforth suppose that the fac-
tor 2 is embedded in the PSD definition. Therefore, the eqs. 11
and 12 can be used to describe the XRS contribution to the PSF.
We are now interested in the scattered power at angles larger
than a definite angle αmeasured from the focus. Due to the steep
fall of scattering intensity for increasing angles, the integral has
a finite value
I [|θs − θi| > α] =
∫ π−θi
dθs. (13)
Combining eqs. 11 and 13, one obtains:
I [|θs − θi| > α] = I0RF
16π2 sin3 θi
∫ π−θi
P( f ) dθs (14)
with respect to the definition used in the eqs. 7 and 11, a factor
2 was included in the PSD. The upper integration limit corre-
sponds to a photon back-scattering: at first glance, this seems
to violate our small-scattering angle assumption (eqs. 11 and
12), but it should be remembered that only the angles close to θi
contribute significantly to the integral in eq. 13: hence its value
should not be significantly affected by a particular choice of the
upper integration limit. After a variable change from θs to f
(eq. 12), the eq. 14 becomes (approximating cos θi ≈ 1 in the
upper integration limit):
I [|θs − θi| > α] = I0RF
16π2 sin2 θi
P( f ) d f (15)
where f0 = α sin θi/λ is the spatial frequency corresponding to
the scattering at the angle α. As expected, this equation equals
the integrated scattering according to the eq. 10, provided that
we identify I [|θs − θi| > α] with Is, and the squared roughness
rms with
P( f ) d f . (16)
The eq. 16 is in agreement with the eq. 8, but it states clearly the
window of spatial frequencies involved in the XRS. Therefore,
for a definite angular limit α the ”reflected beam” intensity can
be simply calculated by using the Debye-Waller formula, pro-
vided that σ2 is computed from the PSD integration beyond the
frequency f0, which corresponds to an X-ray scattering at α. The
upper integration limit is a very high frequency (close to 1/Å):
hence, the atomic structure of the surface is not important in the
integral of the eq. 16. Moreover, considering that the PSD trend
for optically-polished surfaces decreases steeply for increasing
f , the largest contribution to the integral should be given by the
frequencies close to f0.
Now we can evaluate H(λ), the scattering term of the HEW.
For simplicity, in the following we will suppose that the HEW
is obtained from the collection of all the reflected/scattered pho-
tons: this allows us to avoid problems related to the finite size of
the detector, and to extend the surface roughness PSD up to very
large spatial frequencies. By definition, H(λ) is twice the angu-
lar distance from focus at which the integrated scattered power
halves the total reflected intensity:
I [|θs − θi| > α] =
I0RF (17)
we immediately derive, from the eq. 9,
16π2σ2 sin2 θi
, (18)
where σ2 has now the meaning as per the eq. 16. Solving the
eq. 18 for σ2 and equating to the integral of the PSD,
P( f ) d f =
λ2 ln 2
16π2 sin2 θi
, (19)
once known the PSD from topography measurements over a
wide range of spatial frequencies, the PSD numerical integra-
tion in the eq. 19 allows to recover f0. In turn, f0 is related to
H(λ) through the eq. 12, that we write in the following form
H(λ) =
2λ f0
sin θi
, (20)
where H is measured in radians. Note that the condition
H(λ) ≪ θi is very important, for the eq. 20 to hold. Small
scattering angles and grazing incidence are also very important
for the considerations that follow.
3.2. Multilayer coatings
The obtained result (eq. 19) can be extended to mirrors with
multilayer coatings, used to enhance the grazing incidence re-
flectivity of mirrors in hard X-rays (E > 10 keV). In general,
the multilayer cannot be characterized by means of a single
PSD, due to the evolution of the roughness throughout the stack
(Spiller et al. 1993; Stearns et al. 1998). Moreover, due to the
interference of scattered waves at each multilayer interface, the
final scattering pattern is more structured than eq. 11, with peaks
whose height depends on the phase coherence of the interfaces
D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 5
Fig. 2. Dependence of the spectral exponents for different in-
dexes n of a power-law PSD, for a single-reflection focusing
mirror. In the forbidden region (n > 3) γ would be negative.
(Kozhevnikov 2003). The HEW term can be computed numeri-
cally from the XRS diagram.
In order to extend the eq. 19 to mirrors coated with a graded
multilayer, we have to assume the additional requirements:
1. the PSD is constant and completely coherent throughout
the multilayer stack: i.e., the deposition process does not
cause additional roughness and replicates simply the profile
of the substrate. Therefore, all the PSDs and all the cross-
correlation between interface profiles equal the PSD mea-
sured at the multilayer surface. This is often observed in
the l̂ > 10 µm regime (Canestrari et al. 2006), where most
of frequencies f0 fall when the incidence angle is less than
0.5 deg. Most of microroughness growth, indeed, takes place
for 10 µm> l̂ > 0.1 µm.
2. the multilayer reflectivity Rλ(θi) at the photon wavelength λ
changes gradually over angular scales of H(λ). Ideally, this
condition should be fulfilled by wideband multilayer coat-
ings for astronomical X-ray mirrors.
Under these hypotheses, a quite tedious calculation reported
in appendix A shows that the eq. 19 can be approximately ap-
plied also with multilayer coatings. The following developments
also apply in that case.
4. H(λ) for a fractal surface
We apply now the equations 19 and 20 to the typical (monodi-
mensional) PSD model for optically-polished surfaces, a power-
law (Church 1988)
P( f ) =
, (21)
where the power-law index n is a real number in the interval
1 < n < 3 and Kn is a normalization factor. A power-law PSD is
typical of a fractal surface, and it represents the high-frequency
regime of a K-correlation model PSD (Stover 1995). This model
exhibits a saturation for f → 0 that avoids the PSD divergence.
In practice, the fractal behavior dominates in almost all spatial
frequencies of interest for X-ray optics.
There are interesting reasons for which n can take val-
ues on the interval (1:3). In fact, for a surface in the 3D
space, n is related to its Hausdorff-Besicovitch dimension D
(Barabási & Stanley 1995) along with the equation n = 7 − 2D
(see Church 1988; Gouyet 1996). The restriction 1 < n < 3 for
a fractal surface is therefore necessary to have 3 > D > 2.
A power-law PSD is particularly interesting because the in-
tegral on left-hand side of the eq. 19 can be explicitly calculated:
f 1−n0 −
n − 1
λ2 ln 2
16π2 sin2 θi
. (22)
As 1 − n < 0, in grazing incidence the (2/λ)1−n term can be
neglected with respect to f 1−n0 . By isolating the frequency f0 and
using the eq. 20 to derive H(λ), we obtain after some algebra,
for the scattering term of the HEW,
H(λ) = 2
16π2Kn
(n − 1) ln 2
sin θi
. (23)
This equation states that:
1. The H(λ) function for a power-law PSD has a power-law
dependence on the photon energy E ∝ 1/λ, i.e., H(E) ∝ Eγ.
The power-law index γ is related to the PSD power-law index
n through the simple equation:
3 − n
n − 1
. (24)
As 1 < n < 3, γ is positive, i.e. H is an increasing function
of the photon energy. For a fixed value of Kn, the HEW di-
verges quickly for n ≈ 1 but very slowly for n ≈ 3: a PSD
power-law index close to 2-3 would hence be preferable in
order to reduce the degradation of focusing performances for
increasing energies.
2. H(λ) depends on the sine of the incidence angle at the γth
power. In other words, the HEW depends only on the ratio
sin θi/λ: this scaling relation shows that for a given power-
law PSD (with n < 3) at a given photon wavelength λ we can
reduce the HEW by decreasing the incidence angle.
3. H(λ) increases with the PSD normalization Kn, as expected:
the dependence is also a power law with spectral index
n − 1
. (25)
As for γ(n), the closeness of n to the maximum allowed value
for fractal surfaces makes less severe the roughness effect on
imaging degradation.
The functions β and γ are plotted in fig. 2. For instance, if
n = 2, γ = β = 1, and H(E) increases linearly with both pho-
ton energy and Kn coefficient. The divergence of indexes β, γ for
n ≈ 1 makes apparent the importance of obtaining steep PSDs
in the optical polishing of X-ray mirrors. Finally, it is worth not-
ing that for n > 3 there is the theoretical possibility of a slight
decrease of H(E) for increasing energy because γ(n) becomes
negative.
To clarify the dependence of the HEW on the power-law in-
dex n and the incidence angle, we depict in fig. 3 and 4 some ex-
amples of H(E) simulations (single reflection) for some power-
law PSDs in the photon energy range 0.1-50 keV. The H(E)
curves were computed using the eq. 23. In fig. 3 the incidence
angle θi is fixed at 0.5 deg and the index n is variable; a constant
n = 1.8 and a variable θi is instead assumed in the simulations of
fig. 4. Note in fig. 3 the slower H(E) increase for larger n and the
common intersection point, determined by the particular choice
of the incidence angle and the σ = 4 Å value in the window of
spatial wavelengths [100 ÷ 0.01 µm].
6 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . .
Fig. 3. H(E) simulations assuming power-law PSDs with con-
stant σ = 4 Å in the spatial wavelengths range [100 ÷ 0.01µm],
but variable power-law index n. The incidence angle is fixed at
θi = 0.5 deg.
Fig. 4. H(E) simulations assuming a power-law PSD with
power-law index n = 1.8 and with σ = 4 Å in the spatial wave-
lengths range [100 ÷ 0.01 µm], but variable incidence angle θi.
5. Numerical integration of the PSD
A power-law PSD is a modelization that can be used for
optically-polished surfaces. If the polishing process is not op-
timized or a reflecting layer is grown onto a optically polished
substrate, several deviations from a power-law trend can be ob-
served. A typical ”bump”, for instance, can be present in the PSD
of multilayer coatings, often in the range of spatial wavelengths
[10 ÷ 0.1 µm], as a result of the replication of the substrate to-
pography and of fluctuations intrinsically related to the random
deposition process (Spiller et al. 1993; Stearns et al. 1998). If
the PSD deviates significantly from a power-law, the eq. 23 can-
not be used. However, if the surface PSD has been extensively
measured over a wide range of spatial frequencies [ fm, fM] (wide
enough to have fm < f0(λ) for all λ), the HEW scattering term
H(λ) can be computed by numerical integration (eqs. 19 and 20),
on condition that the following approximation is valid:
P( f ) d f ≈
P( f ) d f . (26)
The condition above is usually satisfied when f0 ≪ fM i.e. when
the following inequality holds:
H(λ)≪
2λ fM
sin θi
. (27)
As we are also interested in computing H(λ) in hard X-rays
(small λ), there is the possibility that the two integrals in the
eq. 26 differ by a significant factor. In this case the integral can
be corrected by adding the remaining term
P( f ) d f =
P( f ) d f +
P( f ) d f , (28)
that can be evaluated, in principle, by measuring the mirror re-
flectivity within an angular acceptance corresponding to the spa-
tial frequency fM, and using the Debye-Waller formula to derive
σ2; then, the importance of measuring the PSD in a very wide
frequencies interval becomes apparent. The value of f0 depends
strongly on both incidence angle and photon energy: for soft X-
rays (< 10 keV) and very small angles (< 0.2 deg) the character-
istic spatial wavelength l̂ = 1/ f0 often falls in the millimeter or
centimeter range.
It should be noted that, if the detector is small, a fraction of
the scattered photons can be lost; to account for the finite an-
gular radius of the detector d (as seen from the optic principal
plane), one should integrate the PSD over the smaller interval
[ f0, d sin θi/λ] to recover the measured H(λ) trend. As an alter-
native method, one can compare the theoretical predictions of
eqs. 19 and 20 with the experimental H(λ) values, as calculated
from the Encircled Energy normalized to the photon count fore-
seen by the Fresnel equations (i.e. with zero roughness), rather
than to the maximum of the measured Encircled Energy func-
tion.
6. Computation of the PSD from the H(λ) trend
If the approach described above can be used to simulate the
HEW trend from a measured surface PSD, the reverse problem,
i.e. the derivation of surface PSD from the measured HEW trend
is also possible. This requires that the figure error contribution
had been reliably measured, in order to isolate the scattering
term function H(λ) using the eq. 6.
This problem is interesting for three reasons at least:
1. it is a quick, non-destructive surface characterization method
in terms of its PSD.
2. The measurement is extended to a large portion of the illu-
minated optic, hence local surface features are averaged and
ruled out from the PSD.
3. For a given HEW(λ) requirement in the telescope sensitiv-
ity energy band, it allows establishing the maximum allowed
In order to find an analytical expression for the PSD, we note
that the spatial frequency f0 that scatters at an angular distance
H/2 from the specular beam is a function only of λ, along with
the eq. 20. Solving for f0, we have
f0(λ) ≈ H(λ)
sin θi
. (29)
We suppose that all scattered photons are collected, so we can
assume the eq. 19 as valid. By deriving both sides of eq. 19 with
respect to λ, we have
P( f ) d f
8π2 sin2 θi
λ, (30)
that is,
P( f0) =
8π2 sin2 θi
λ, (31)
D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 7
and, using the eq. 29 to compute the derivative of f0:
+ f0P( f0)
sin θi
dH(λ)
P( f0) =
8π2 sin2 θi
λ. (32)
Now remember that, in grazing incidence, f0 ≪ 2λ−1 by several
orders of magnitude. Even if P( f ) is not a power-law, it is always
a steeply decreasing function of f . Moreover, it should have over
[ f0, 2λ
−1] an average PSD index ñ > 1, for the reasons explained
in the sect. 4. This means that
P( f0)
P(2λ−1)
, (33)
therefore, in practical cases the 2λ−1P(2λ−1) term in the eq. 32 is
negligible with respect to f0P( f0). Consequently, we can neglect
the first term of eq. 31: then we have
P( f0) ≈
8π2 sin2 θi
λ. (34)
Combining this with the eq. 29 and collecting the constants, we
obtain the final result
P( f0)
4π2 sin3 θi
≈ 0. (35)
The eq. 35 enables the computation of the PSD (at the spatial
frequency given by the eq. 29) along with the derivative of the
ratio H(λ)/λ with respect to λ.
The obtained equation shows that P( f ) is inversely propor-
tional to the derivative of H(λ)/λ. This result seems strange at
first glance, because by decreasing H(λ) one would obtain a
larger P( f ) (a rougher surface). One should remember, indeed,
that by reducing H(λ) we increase P( f0), but f0 is shifted to-
wards the low frequencies domain, where P( f0) is expected to
be higher. In fact, the ”rough” or ”smooth” feature of the surface
depends on whether f0 or P( f0) varies more rapidly, i.e. on the
overall H(λ) trend.
We can also check the correctness of the eq. 35 by computing
the PSD for the particular case of the H(λ) derived from the inte-
gration of a power-law PSD (the eq. 23, derived under the same
approximation, the eq. 33). If the results are correct, the substi-
tution of the HEW trend of the eq. 23 in the eq. 35 should return
the original PSD (eq. 21). The straightforward, but lengthy cal-
culation (carried out in appendix B) shows that the substitution
returns
P( f0) =
as expected.
The eq. 35 should be approximately valid also for graded
multilayers with a slowly-decreasing reflectivity (see sect. 3.2),
however, due to the approximations needed to extend the eq. 19
to the multilayers, the resulting PSD should be considered a
”first guess” in this case. Then, the matching of the PSD to the
required HEW trend should be checked by means of a detailed
computation of the XRS PSF(λ).
7. Extension to X-ray mirrors with multiple
reflections
The formalism exposed in the previous sections can be extended
to a double-reflection optic (like a Wolter-I one). In this op-
tical configuration, photons are firstly reflected by a parabolic
surface and subsequently by a hyperbolic one. If the smooth-
surface condition is satisfied, multiple scattering is often negli-
gible (Willingale 1988) and the scattering diagrams of the two
reflecting surfaces can be simply summed (De Korte et al. 1981;
Stearns et al. 1998). The source is assumed to be at infinite dis-
tance, then X-rays impinge on the two surfaces at the same an-
gle θi. If the surface PSDs are the same for both reflections, the
scattering diagram will be simply doubled. Thus, the integrated
scattered intensity is also doubled:
Is = 2I0R
1 − exp
16π2σ2 sin2 θi
. (37)
The RF factor is squared in the eq. 37 because each ray is re-
flected twice: in absence of scattering the reflected power would
be I0R
F, so the half-power scattering angle condition reads
|θs − θi| >
F, (38)
and, combining the eqs. 37 and 38, we obtain
16π2σ2 sin2 θi
. (39)
Solving for σ2, and using the eq. 16,
P( f ) d f =
λ2 ln(4/3)
16π2 sin2 θi
, (40)
that differs from the eq. 19 only in the factor ln(4/3) instead of
ln 2 on right-hand side. Consequently, the corresponding differ-
ential equation is
P( f0)
ln(4/3)
4π2 sin3 θi
≈ 0. (41)
Similar equations can be derived for an optical system with
an arbitrary number of reflections N: to compute the H(λ) from
the PSD,
P( f ) d f =
16π2 sin2 θi
2N − 1
. (42)
If the PSD is a power-law P( f ) = Kn/ f
n we can generalize the
eq. 23:
H(λ) = 2
2N − 1
16π2Kn
(n − 1)
sin θi
, (43)
note the divergence of the logaritmic factor for increasing N,
due to the negative exponent 1/(1 − n). This indicates that H(λ)
increases rapidly with the number of reflections, as expected.
Finally, we can also generalize the differential eq. 35 to an
arbitrary number of reflections,
P( f0)
4π2 sin3 θi
≈ 0. (44)
In the eqs. 42 and 44, f0 is always related to H(λ) by the eq. 29.
8 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . .
Fig. 5. an hypothetical PSD with reasonable values and a PSD
break around a 100 µm spatial wavelength (dashed line). This
PSD is adopted to compute the corresponding HEW trends for
1,2,3 reflections at a 0.3 deg grazing incidence angle (fig. 6). The
achieved HEW trends were used to re-calculate the respective
PSDs (solid line). For clarity, we do not plot the single PSDs,
but just their overlap.
8. An example
As an application of the equations reported above, we shall
make use of a simulated surface PSD with reasonable values,
that is not a power-law. The PSD (see fig. 5, dashed line) is ex-
tended from 105 µm down to a 0.01 µm spatial wavelength, with
a break around 100 µm: at the lowest frequencies the PSD is
steep (n ≈ 2.3), whereas at the largest frequencies it is smoother
(n ≈ 1.3). From the discussion in sect. 4 concerning the relation
between the exponents of the PSD and the HEW (eq. 24), we
should expect that the PSD break causes a slope change in the
function H(λ): however, as the actual PSD is not a power-law,
the H(λ) function should be computed by means of the eqs. 20
and 42. Before carrying out the integration, we can remark qual-
itatively that, as we increase the photon energy, the highest fre-
quencies in the PSD (where the PSD index becomes smaller)
become important; hence, we can expect a steeper increase of
the HEW at the highest energies.
The analysis is made quantitative in fig. 6, where we show
the calculated HEW trends from the PSD in fig. 5 (the dashed
line) by means of the eqs. 20 and 42, assuming 1,2,3 reflections
at the same grazing incidence angle (0.3 deg). The approxima-
tion of eq. 26 was adopted. In addition to the scattering term, 15
arcsec of HEW due to figure errors were added in quadrature.
The HEW increases slowly (concave downwards) at low ener-
gies, corresponding to a frequency f0 in the steeper part of the
PSD. Then it increases more steeply (concave upwards) when
the energy becomes large enough to set f0 in the portion of the
spectrum with n ≈ 1.3. By increasing the number of reflections,
the HEW values also increase, and the ”turning point” where the
HEW starts to diverge (arrows in fig. 6) shifts at lower X-ray
energies. All the calculation is based on the assumption that the
contribution of the PSD over the maximum measured frequency
fM = 0.01 µm is negligible. Otherwise, the computed HEW val-
ues will be underestimated (see sect. 5).
In addition to the general trend of the HEW, there are oscil-
lations due to small irregularities in the adopted PSD: the cal-
culation is, in fact, very sensitive to small variations of the PSD
values. Notice that for a definite energy all the frequencies larger
than f0 contribute to the HEW value, even if the largest contri-
Fig. 6. the HEW trend computed from the PSD for 1,2,3 reflec-
tions, plus 15 arcsec of HEW due to figure errors. The HEW
trends were used to compute back the PSD (the solid line in
fig. 5) to verify the reversibility of the calculation. The energy
at which the concavity change takes place is also indicated (ar-
rows).
bution comes from frequencies near f0: this is a consequence of
the steeply decreasing trend of the PSD.
We checked the reversibility of the result by computing the
PSD from the HEW trends (after subtracting in quadrature 15
arcsec figure error) by means of the eq. 44 with the respec-
tive value of N. The resulting PSDs (the solid line in fig. 5)
were overplotted to the initial PSD, with a perfect superposition.
Each obtained PSD has, indeed, an extent of spatial frequencies
smaller than the initial one: the overall PSD ranges from 104
to 11 µm (vs. the initial 105 ÷ 0.01 µm), and the smaller wave-
lengths could be computed from the HEW trend with N = 3. The
limitation in spatial frequency ranges occurs for two reasons:
1. small f − large l̂: all the power scattered by the lowest fre-
quencies is found at angles less than 1/2 HEW even for the
lowest energies being considered: therefore, that part of the
spectrum is not necessary to compute the HEW in the energy
range of interest;
2. large f − small l̂: the PSD is computed from a derivative,
therefore the information concerning the absolute magnitude
of the HEW is substantially lost. This information was in-
cluded in the integral of the PSD (eq. 42) for the maximum
considered energy.
Therefore, from the integral in the eq. 42 we cannot re-
cover the PSD over the minimum computed spatial wavelength
(11 µm, using the HEW trend with N = 3), but we can at
least calculate the value of σ at spatial wavelengths smaller than
11 µm. Substituting the incidence angle and the minimum pho-
ton wavelength being considered (λ = 0.24 Å) in the eq. 42
with N = 3 and with the approximation of the eq. 26, we ob-
tain σ = 1.6 Å, in perfect agreement with the value computed
from the original PSD.
Summing up, for a given incidence angle the H(λ) function
in a definite photon energy range is equivalent to the PSD in
a corresponding range of spatial frequencies f (or equivalently,
spatial wavelengths l̂), plus the integral of the PSD beyond the
maximum frequency being computed. Therefore, requirements
of a definite HEW(λ) function in designing an X-ray optical sys-
tem can be translated in terms of PSD in a frequencies range
[ fmin, fmax] plus the surface rms at frequencies beyond fmax. The
usefulness of such a relationship is apparent.
D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . . 9
9. Conclusions
In the previous pages we have developed useful equations to
compute the contribution of the X-ray scattering to the HEW of
a grazing incidence X-ray optic, by means of a simple integra-
tion. The formalism has been inverted in order to derive the PSD
of the surface from the function H(λ), and it can be extended
to an arbitrary number of reflections at the same incidence an-
gle. The equations are valid for a single-layer coating mirror, but
they can be approximately applied to a multilayer-coated mir-
ror. This approach is particularly useful in order to establish the
surface finishing level needed to keep the X-ray scattering HEW
of X-ray optics within the limits fixed by the X-ray telescope
requirements.
It should be remarked that the reasoning was developed for
the Half-Energy Width, but it can be extended to any angular
diameter including a fraction η of the energy spread around the
focal point. To do this, it is sufficient to substitute the logarithmic
factors in equations 42, 43, 44,
2N − 1
N − 1 + η
, (45)
and for instance, to compute the 90%-energy diameter for a dou-
ble reflection mirror, simply substitute η = 0.9 and N = 2. The
proof is straightforward: however, one should always keep in
mind that the energy diameters computed with this method can
be considered valid only if they are much smaller than the inci-
dence angle θi.
Notice that in the development of the exposed formalism we
have supposed, in addition to the smooth-surface condition, two
additional hypotheses:
1. the source is at infinite distance from the mirror
2. the X-ray detector is large enough to collect all the scattered
photons.
In order to apply the mentioned equations to experimental
calibrations of X-ray optics at existing facilities (like MPE-
PANTER), where the source is at a finite distance and the de-
tector has a finite size, some corrections should be taken into
account. We will deal with their quantification in a subsequent
paper.
Appendix A: Extension to multilayer coatings
Here we provide with a plausibility argument to extend the for-
malism of sect. 3.1 to mirror shells with multilayer coatings (see
sect. 3.2). The intensity of a scattered wave at each interface is
proportional to its PSD as per the eq. 11, and the overall scatter-
ing diagram will be their coherent interference. To simplify the
notation, we neglect the X-ray refraction and we suppose that
the incidence angle is beyond the critical angles of the multilayer
components. The electric field scattered by the kth interface can
be written as
Ek = E0TkrkXk( f ) exp(−iφk), (A.1)
where rk is the single-boundary amplitude reflectivity, E0 the
incident electric field amplitude, the weights Tk are the rel-
ative amplitudes of the electric field in the stack (in scalar,
single-scattering approximation), and account for the extinction
of the incident X-rays due to gradual reflection and absorption.
Xk( f ) is the single-boundary scattering power (proportional to
the PS D( f ) amplitude), and φk is the phase of the scattered wave
at θs by the k
th interface
φk = 2π
sin θi + sin θs
zk, (A.2)
where zk is the depth of the k
th interface with respect to the outer
surface of the multilayer. Now, the measured intensity is
|Escatt|2 =
= |E0|2|Xk( f )|2
rkTk exp(−iφk)
. (A.3)
Now, |E0|2 = I0, the incident X-ray flux intensity, and |Xk( f )|2 is
proportional to the interfacial PSD P( f ), which is independent
of k by hypothesis. Assuming the proportionality factor of eq. 11
for |Xk( f )|2, we obtain for the scattering diagram
sin3 θiP( f )
rkTk exp(−iφk)
(A.4)
and if we set
Kλ(θi, θs) =
rkTk exp(−iφk)
, (A.5)
the eq. A.4 becomes analogous to the eq. 11, with Kλ(θi, θs)
playing the role of RF. Note that Kλ(θi, θi) = Rλ(θi), the mul-
tilayer reflectivity in single reflection approximation. As before,
we write the scattering diagram for a mirror with axial symmetry
as a function of the angular distance from the focus α = |θi − θs|
averaging the contributions of negative and positive frequencies
sin3 θiP( f )[Kλ(θi, θi − α) + Kλ(θi, θi + α)]. (A.6)
For a single reflection optic, we can calculate the scattered power
over H/2, where H is the scattering term of optic Half-Energy
Width:
Is[α > H/2] =
I0Rλ(θi). (A.7)
Now, the steps 18 and 19 can be repeated:
Kλ(θi, θi + α) + Kλ(θi, θi − α)
Rλ(θi)
P( f ) d f =
λ2 ln 2
8π2 sin2 θi
(A.8)
where f0 is still defined by the eq. 20. For small scattering angles
(α ≪ θi), since we assumed a slow variation of Rλ over angular
scales of H/2 (and the same occurs for Kλ), we can approximate
Kλ(θi, θi ± α) ≈ Rλ(θi) ± α
∂Kλ(θi, θs)
θs=θi
. (A.9)
Substituting in the eq. A.8, we obtain
P( f ) d f ≈
λ2 ln 2
16π2 sin2 θi
(A.10)
because the two derivatives have opposite sign and cancel out.
This is the same equation found for the case of a single-layer
coating (eq. 19).
10 D. Spiga: Analytical evaluation of X-ray scattering contribution to imaging degradation . . .
Appendix B: Derivation of the PSD from the HEW
for a fractal surface (single reflection)
We recall here the H(λ) trend for a power-law PSD (eq. 23):
H(λ) = 2
16π2Kn
(n − 1) ln 2
sin θi
(B.1)
we verify that it returns a power-law PSD if substituted in the
differential eq. 35:
P( f0)
4π2 sin3 θi
= 0. (B.2)
To simplify the notation, we write simply H instead of H(λ): by
carrying out the derivation,
n − 1
16π2Kn
(n − 1) ln 2
(sin θi)
n−1 λ
n−1−3. (B.3)
Using again the eq. B.1:
n − 1
−3 (B.4)
hence, the related PSD is
P( f0) = −
4π2 sin3 θi
λ3 ln 2
4π2H sin3 θi
n − 1
. (B.5)
Now, we can derive (n − 1)/2 from the eq. B.1,
n − 1
4π2HKn
)−n ( sin θi
(B.6)
and combining the eqs. B.5-B.6, one obtains
P( f0) = Kn
H sin θi
, (B.7)
that is, by recalling the eq. 29,
P( f0) =
(B.8)
i.e., the expected power-law PSD.
Acknowledgements. Many thanks to G. Pareschi, O. Citterio, R. Canestrari, S.
Basso, F. Mazzoleni, P. Conconi, V. Cotroneo (INAF/OAB) for support and
useful discussions. The author is indebted to MIUR (the Italian Ministry for
Universities) for the COFIN grant awarded to the development of multilayer
coatings for X-ray telescopes.
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	Introduction
	Contributions to the imaging degradation
	Estimation of H() for single-reflection focusing mirrors
	Single-layer coatings
	Multilayer coatings
	H() for a fractal surface
	Numerical integration of the PSD
	Computation of the PSD from the H() trend
	Extension to X-ray mirrors with multiple reflections
	An example
	Conclusions
	Extension to multilayer coatings
	Derivation of the PSD from the HEW for a fractal surface (single reflection)
