Download Partially coherent image formation with x

Partially coherent image
formation with x-ray microscopes
L. Jochum and W. Meyer-Ilse
Image formation with partially coherent radiation is evaluated with the Hopkins formula and then
applied to x-ray microscopy. Image characteristics expected from instruments with circular and annular
pupils in partially coherent conditions are considered for two-point objects and a knife-edge object. The
theoretically expected values for image characteristics that are easy accessible by an experiment, such as
the width of a knife edge, are given for various x-ray microscopes.
Key words: Image formation, partially coherent imaging, x-ray microscopy, resolution limit.
1.
Introduction
X-ray microscopy is a unique technique that extends
visible light microscopy to higher resolution and
makes use of unique contrast mechanisms. It does
not compete with techniques such as electron microscopy in terms of resolution but rather offers unique
advantages including the ability to image samples in
an aqueous environment. X-ray microscopy yields
information with high resolution from thick samples
that cannot be obtained by any other technique
including methods such as atomic force microscopy or
near-field microscopy. Applications of x-ray microscopy are in several fields including biology and material science. The interaction of x rays with matter
provides unique elemental and chemical contrast.
Construction of an x-ray microscopy resource center is
planned at the Advanced Light Source 1ALS2 in Berkeley, Calif.1,2 that will provide a broad user community
with the necessary instrumentation to use this technique optimally.
There are two basic types of x-ray microscope,
conventional x-ray microscopes (XM’s) and scanning
x-ray microscopes (SXM’s). XM’s are similar to
their visible light counterparts with critical illumination.
Sometimes they are also called imaging or
full-field x-ray microscopes. XM’s with zone plates
were pioneered at the University of Gottingen.3 The
most prominent instruments of each kind are the
Göttingen XM at the Berlin electron storage ring4
The authors are with the Center for X-ray Optics, Lawrence
Berkeley Laboratory, 1 Cyclotron Road, Berkeley, California 94720.
Received 14 July 1994; revised manuscript received 3 February
1995.
0003-6935@95@224944-07$06.00@0.
r 1995 Optical Society of America.
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APPLIED OPTICS @ Vol. 34, No. 22 @ 1 August 1995
1BESSY2 and the STXM operated at the Brookhaven
National Laboratory 1BNL2.5 Because these two microscope types are complementary, there are plans to
build them at the ALS. An XM that uses bending
magnet radiation is completed and has been operational since the Fall of 1994.
The resolution of x-ray microscopes is currently not
limited by the wavelength of the imaging light but
rather by the ability to fabricate optical elements that
provide high-resolution images. The highest resolution for both types of microscope is achieved with zone
plate lenses. So far, the numerical aperture 1NA2 of
the zone plates is typically below NA 5 0.1. For
microscopes operating in the water window 12.4
nm # l # 4.5 nm2, a resolution approximately five
times better than with visible light is commonly
achieved.
Unlike conventional refractive lenses, the focusing
properties of zone plates are based on diffraction.
This implies that the focal length of a zone plate lens
strongly depends on the wavelength and that multiple diffraction orders exist. In most cases only the
first diffraction order is used. The diffraction efficiency in the first diffraction order of objective zone
plates is currently in the range of 10%, with improvements expected in the future. If operated with quasimonochromatic radiation, and if the number of zones
is large, zone plate lenses obey the same laws as
refractive lenses.6 In the scope of this paper we
assume that the above-mentioned assumptions are
fulfilled and treat zone plates as conventional lenses.
Therefore the conclusions in this paper also hold for
microscopes other than x-ray microscopes.
2.
X-Ray Microscopes
A schematic of the optical layout of an XM like the
one at BESSY and the one at the ALS is shown in Fig.
Fig. 1.
1a2 Schematic optical layout of an XM. 1b2 Schematic optical layout of an SXM.
11a2. Polychromatic synchrotron radiation is focused
onto the sample by a condenser zone plate, and an
objective zone plate forms an enlarged high-resolution image of the object. This image is detected with
an x-ray CCD camera. Because zone plates are
diffraction optics, they have a strong chromatic error,
and the image quality depends on the monochromaticity of the image-forming light. To provide the required monochromaticity, a pinhole is put in close
proximity to the sample plane. This pinhole together with the condenser zone plate acts as a linear
monochromator. With D as the diameter of the zone
plate and d as the pinhole diameter, the spectral
bandwidth of the illuminating light is given by7; Dl@l
5 2d@D. The center wavelength can be tuned by
changing the distance between the condenser zone
plate and the pinhole.
A schematic of an SXM like the one at the BNL is
shown in Fig. 11b2. A high-resolution objective zone
plate is spatially coherent illuminated and forms a
diffraction-limited x-ray spot. This spot is scanned
over the sample area while the transmitted x-ray
intensity is recorded by a detector. Because only the
spatially coherent fraction of the x-ray source can be
used, such an instrument would preferably be built at
an undulator light source to achieve reasonably short
exposure times. A central stop on the objective zone
plate and an order-sorting aperture between the
objective zone plate and the sample are used to block
out zero-order radiation.
The main advantage of an SXM compared with an
XM is the absence of an objective zone plate downstream of the sample. The low efficiency of the
objective zone plate contributes to the transmission
losses in an XM. Therefore an SXM requires a
smaller radiation dose on the sample to obtain the
same signal-to-noise ratio as an XM.
To take advantage of the increased resolution
compared with visible light microscopy, microscopy
with x-ray-excited visible light luminescence8,9 requires an SXM.
3.
Image Formation in X-Ray Microscopes
Microscopes use a probe to determine the distribution
of one or more physical quantities in the sample with
high spatial resolution. In x-ray microscopy the
probe consists of x rays. For low-energy x rays as
used in x-ray microscopy, the main interactions are
photoelectric absorption and elastic scattering,
whereas inelastic scattering 1Compton scattering2
may be neglected.10 In a single-atom model, the
interaction of any material with x rays can be described by the complex atomic scattering factor f 5
f1 1 if2; values of f1 and f2 are listed in Ref. 10. On a
macroscopic scale the interaction of x rays with
matter is characterized by the complex index of
refraction n 5 1 2 1l@2p2h 2 i1l@4p2µ, where µ is the
linear absorption coefficient and h is the phase shift
coefficient of the wave with respect to its propagation
in vacuum. The atomic scattering factors are related to µ and h as µ 5 2r0lnA f 2 and h 5 r0lnAf1 with
r0 being the classical electron radius and nA the
number of atoms per unit volume.10 In x-ray microscopes the sample is imaged in transmission. This
image contains the information about the spatial
distribution of the complex index of refraction n1x, y, z2
in the sample. For the low-energy x rays used in
x-ray microscopy, where inelastic scattering processes can be neglected, the image formation for both
types of x-ray microscope 1SXM and XM2 may be
described by the same formalism.11,12 The equivalence of scanning and conventional microscopes is
valid even for imaging thick objects. The differences
between the two instruments are purely technical.
Therefore the conclusions and results in this paper
apply for both types of x-ray microscope, although the
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4945
parameters below have different meanings, depending on the type of instrument.
The microscopes considered here detect the intensity distribution in the detector plane, so that only one
two-dimensional 12D2 section of the sample can be
seen at a time. To record the complex index of
refraction n1x, y, z2 in all three spatial dimensions
with such an instrument, one needs to take several
images with different views of the sample. To gather
three-dimensional 13D2 information in one exposure
by using only one detector plane, the recording of the
amplitude and the phase of the wave transmitted by
the sample is required, as in holography. With only
one image the intensity recorded in the detector plane
of a microscope from a 3D object cannot be distinguished from a 2D object that produces the identical
wave field in a plane downstream of the object.
Below we briefly review the theory of partially coherent image formation and with that background describe the properties of a 2D view of different samples
in an x-ray microscope.
4.
Partially Coherent Imaging
The theory of partially coherent image formation was
first developed by Hopkins.13,14 If 1x0, y02 and 1x1, y12
are Cartesian coordinates in the object plane and in
the image plane, respectively, the intensity profile
I1x1, y12 in the image plane of an isoplanatic microscope is
I1x1, y12 5
`
`
`
`
2`
2`
2`
2`
eeee
J1x0 2 x08, y0 2 y082
3 F1x0, y02F*1x08, y082K1x1 2 x0, y1 2 y02
3 K*1x1 2 x08, y1 2 y082dx0dy0dx08dy08.
112
In Eq. 112 J1x, y2 is the mutual intensity in the object
plane, F1x, y2 is the complex transmission of the object,
and K1x, y2 is the Fourier transform of the complex
transmission of the objective lens. Any objective
lens aberrations or effects of defocusing can be included in K1x, y2.
The physical meaning of the mutual intensity is
that it characterizes the phase correlation between
any two points in the object plane. If the illuminating light is perfectly incoherent, the phases vary
statistically so that the mutual intensity equals zero
unless the two points are identical. Mathematically,
this behavior is described by the Dirac d function, and
Eq. 112 reduces to the well-known linear relation
between the object and the image intensity with the
intensity point-spread function as the linear filter.
For totally coherent illumination the phase correlation and thus the mutual intensity is constant for any
two object points. In this case Eq. 112 describes a
linear transformation of the complex amplitude from
the object to the image plane with the amplitude
point-spread function as the linear filter. To make
use of this linearity, the detector must record the
amplitude and phase in the image plane as it is done
in holography.
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APPLIED OPTICS @ Vol. 34, No. 22 @ 1 August 1995
The results of investigations of the resolution in
x-ray microscopy for the limiting cases of coherent
and incoherent imaging have been published in Ref.
15. In general, the image formation in a microscope
is partially coherent. As we show below, the effects
of partial coherence are most important for feature
sizes near the limit of resolution.
When imaging with a circular symmetric aberrationfree objective lens that might have a central stop of
relative radius 0 # ek , 1, the objective lens function
K1x, y2 is given by
K1x, y2 5
J11n2
2
23 n
2
11 2 ek
2
ekJ11ekn2
n
4
with n 5 2p1x2 1 y221@2.
122
J1 is the Bessel function of first order and must not be
mistaken for the mutual intensity J1x, y2. The spatial dimensions of x and y are measured in units of
l@NAo, where l is the wavelength and NAo is the
numerical aperture of the objective lens. In these
normalized units and if ek 5 0, the first minimum of
K1x, y2 is at a distance of 0.61 from the central
maximum. In an XM there is normally no central
stop on the objective zone plate. That the innermost
zone of a zone plate may act as a central stop can be
neglected, since it results in an effective ek of 1@Œn
with n being the total number of zones. The n is of
the order of 102 for objective zone plates and 105 for
condenser zone plates. In an SXM, however, a central stop on the objective zone plate is normally used
to remove zero-order radiation 1ek . 02.
Assuming that the mutual intensity in the object
plane is caused by a circular symmetric aberrationfree aperture that might have a central stop, we
obtain
J1x, y2 5
J11mn2
2
11 2 e 2 3 mn
2
2
j
ejJ11ejmn2
4
mn
with n 5 2p1x2 1 y221@2,
132
with ej describing the relative radius of the central
stop and m the width of the central maximum of J1x, y2
relative to the one of K1x, y2. Equation 132 includes
the limiting cases of coherent and incoherent image
formation as discussed above. If m = 0, J1x, y2 is
constant over the sample area leading to a coherent
image formation, and if m = `, J1x, y2 can be approximated by a d function that corresponds to an incoherent image formation. In an XM, ej would be the
radius of the central stop of the condenser zone plate
divided by the radius of the condenser zone plate.
A central stop is necessary in an XM to remove
zero-order radiation, so that normally ej . 0 with ej 5
1@3 as a typical number.
In unnormalized coordinates of the XM, the coherence parameter m would be
m5
lo@NAo
lc@NAc
,
142
with lc and lo being the wavelengths that pass the
condenser and objective lens and NAc and NAo being
the numerical apertures of the condenser and objective lens, respectively. As for soft x rays, where
inelastic scattering can be neglected, lc equals lo, the
coherence parameter m is given by the ratio of the
numerical apertures of the condenser and the objective lens. Equation 142 also holds for an SXM if NAc is
replaced by the sine of the half angle of the acceptance
cone of a circular detector.
If either the object itself or its Fourier spectrum is
given by the sum of a finite number of Dirac d
functions, an analytical solution for the image intensity in Eq. 112 can be found, but for arbitrary objects a
numerical evaluation is necessary. We have chosen
a two-point object and a knife edge as an object to
investigate the effects of partial coherence and central stops on the properties of the optical system.
5.
Partially Coherent Imaging of a Two-Point Object
A two-point amplitude object with a point distance of
d is described by
F1x, y2 5 d121⁄2 d 2 x, y2 1 d11⁄2 d 2 x, y2.
152
With the above object function the four integrations in
Eq. 112 have been evaluated. The result is the image
intensity of the two-point object. We calculated the
image contrast of a two-point object as a function of
the parameters m, ej, and ek for point distances of 0 #
d # 1.22. As mentioned above, the linear dimensions are measured in units of l@NAo. The figures
show examples of how the variation of the parameters
m, ej, and ek affects the image contrast.
Figure 2 shows the image contrast of a two-point
object as a function of the inverse point distance 1@d
for different relative numerical apertures m, indicating different degrees of coherence in the object plane.
In the range of 0 # m # 1 the image contrast at a
fixed object point distance generally increases with
increasing m. If we compare the image contrast for
an incoherent imaging system 1m = `2 with the case
of matching numerical apertures 1m 5 12, the image
and thus the contrast is identical only for a point
separation of d 5 0.61. Two points, which are
unresolved in a coherent imaging system 1m 5 02, can
Fig. 2. Image contrast of a two-point object imaged with ej 5 ek 5
0 and coherence parameters m of 0, 0.5, 1, and infinity; d is in units
of l@NAo.
Fig. 3. Influence of m on the image contrast of a two-point object
for four selected point distances d imaged with circular pupils
1ej 5 ek 5 02; d is in units of l@NAo.
be imaged with a contrast of more than 40% with a
partially coherent imaging system and an appropriate coherence parameter m. For a fixed point distance and increasing m, the image contrast steadily
increases to a maximum between m 5 1 and m 5 1.5,
which is shown in Fig. 3. As m approaches infinity,
the contrast is slightly oscillating around the contrast
obtained with an incoherent imaging system 1m = `2.
Figure 4 and 5 show the influence of different
central stops on the image contrast. The coherence
parameter is set to m 5 1 for these examples. The
mutual intensity J1x, y2 in the object plane varies with
the parameter ej, whereas ek describes a central stop
on the objective lens and thus modifies K1x, y2. For a
two-point object a central stop on the objective lens
1ek . 02 has a greater effect on the image contrast
than a change of the mutual intensity from a similar
stop on the condenser lens 1ej . 02.
Fig. 4. Image contrast of a two-point object versus the inverse of
the point distance d 1in units of l@NA2 for m 5 1 and 1a2 ek 5 0 for
different mutual intensities 10 # ej # 0.92, 1b2 ej 5 0 for different
central stops on the objective lens 10 # ek # 0.92.
1 August 1995 @ Vol. 34, No. 22 @ APPLIED OPTICS
4947
Fig. 5. Image contrast of a two-point object for four selected point
distances d imaged with m 5 1: 1a2 ek 5 0, different mutual
intensities 10 # ej # 0.92; 1b2 ej 5 0, different central stops on the
objective lens 10 # ek # 0.92.
Fig. 6. Distance L between two points that form an image with
15.3% contrast as a function of the coherence parameter m at 1a2
ek 5 0, different mutual intensities 10 # ej # 0.92, 1b2 ej 5 0, different
central stops on the objective lens 10 # ek # 0.92.
For any point distance d an increase in ek results in
a higher image contrast. The contrast enhancement
by changing ek is less significant for larger point
distances, which can be seen as different slopes of the
curves in Fig. 51b2.
If m 5 1 and ej 5 ek 5 0, a point separation of d 5
0.61 corresponds to the resolution limit according to
the Rayleigh criterion 1Ref. 14, p. 3332. Applying the
Rayleigh criterion to an incoherent imaging system
with circular pupils leads to an image contrast of
15.3% for a two-point object. By changing ek from 0
to 0.9 at a point distance of d 5 0.61, the image
contrast increases from 15.3% to almost 70% 3see Fig.
51b24. Let L be the distance of two object points at
which the image contrast is 15.3%. In unnormalized
coordinates this distance is dx 5 Ll@NAo. Figure 6
shows L as a function of m for various ej and ek. In
the range of 0 # m # 1, which is relevant for x-ray
microscopy, an increase of ej as well as ek results in a
decreasing L. This can be interpreted as an improved resolution, but one must remember that these
results refer only to the imaging of a two-point object.
Figure 7 shows two examples of an edge response
and the differentiated response. The imaging parameters in Fig. 71a2 represent a typical layout for the
XM-1 at the ALS, and in Fig. 61b2 are typical parameters for an SXM.
The full width at half-maximum 1FWHM2 of the
differentiated image from an edge is often used to
determine the resolution of a system experimentally.
6.
Partially Coherent Imaging of a Knife Edge
After examining the behavior of a partially coherent
imaging system in the case of a two-point object, we
investigated the same system with a 2D knife edge as
an object. Analyzing the image of a knife edge is a
very common and convenient method to check the
image quality experimentally. To simulate the partially coherent imaging of a knife edge, Eq. 112 was
evaluated numerically with the program SPLAT,16
developed by Toh at the Department of Electrical
Engineering and Computer Sciences, University of
California.
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APPLIED OPTICS @ Vol. 34, No. 22 @ 1 August 1995
Fig. 7. Response from imaging an edge with partially coherent
light and the differentiated response at 1a2 m 5 0.54, ej 5 0.33, and
ek 5 0 1same as XM-1, 30 nm in Table 12; 1b2 m 5 1, ej 5 0, and ek 5
0.45 1same as SXM in Table 12.
Table 1.
Optical
Setup
Incoherent
SXM-lum
SXM
XM-2
XM-1, 30 nm
XM-1, 25 nm
Coherent
Image Characteristics of a Knife-Edge Object and a Two-Point Object for Different Optical Setups
m
ej
ek
Different Edge
Response FWHM
1l@NA2
`
`
0
0
0
0
0.33
0.33
0
0
0.45
0.45
0
0
0
0
0.49
0.47
0.45
0.47
0.45
0.45
0.42
1
1
0.54
0.45
0
Edge Response
25–75%
1l@NA2
Edge Response
10–90%
1l@NA2
L 5 Two-Point
Distance for
15.3% Contrast
Two-Point
Contrast at
d 5 0.61 1%2
0.31
0.42
0.40
0.29
0.22
0.21
0.19
0.62
1.32
1.31
0.59
0.39
0.37
0.34
0.61
0.55
0.57
0.61
0.70
0.73
0.82
15.3
29.5
29.5
15.3
0
0
0
The indicated value for the FWHM in Fig. 7 is what
can be theoretically expected from that particular
instrumental configuration. FWHM values of the
differentiated edge response for other instrument
configurations are tabulated in Table 1. Although
the intensity profiles of the edge response in Fig. 7
look quite different, they result in the same FWHM of
the differentiated response. This is because the
FWHM, like the image contrast of a two-point object,
is mainly characterized by the width of the central
maximum of K1x, y2. By introducing a central stop in
the objective lens, the central maximum of K1x, y2
becomes narrower, but the higher-order fringes around
the maximum increase in intensity, as is shown in
Fig. 8. As previously shown, the narrower central
maximum results in a higher image contrast in the
case of a two-point object, especially for small point
distances. When a knife edge is imaged with the
same optical system, the increased intensity in the
higher-order fringes blur the image, as seen in Fig.
71b2, and tend to reduce the overall contrast of an
arbitrary object. When looking at the distance of a
90–10% intensity drop in the edge response, which is
the corresponding quantity to the Rayleigh resolution
limit of an incoherent imaging system, we see that
this distance is significantly larger in Fig. 71b2 than in
Fig. 71a2. Other then the FWHM, which is the same
for both cases, this distance is sensitive to the changed
parameters of the optical system and thus is a more
likely candidate to characterize the quality and performance of a particular imaging system.
Fig. 8. Point images for the parameters in Fig. 71a2 1m 5 0.54,
ej 5 0.33, and ek 5 0, continuous curve2 and Fig. 71b2 1m 5 1,
ej 5 0, and ek 5 0.45, dashed curve2.
7.
Summary
The properties of the image formation with partially
coherent light have been described for x-ray microscopes. The results are not restricted to x-ray microscopy; they also apply to other microscopes including
visible light and transmission electron microscopes.
The results of our evaluations are summarized in
Table 1. It shows the image characteristics of a
knife-edge object and a two-point object for different
combinations of m, ej, and ek. Each row represents a
particular optical setup. The first and the last rows
show the theoretical cases of a totally incoherent and
a totally coherent system, respectively. All the other
optical setups shown in Table 1 are likely candidates
for the microscopes planned at the ALS.
SXM-lum stands for a scanning x-ray microscope
operating with the x-ray excitation of visible light
luminescence. Because the luminescence light is
incoherent, the coherence parameter m is set to
infinity. An unobstructed detector and a central stop
on the objective zone plate define the parameters ej
and ek, respectively. The incoherent image formation implies that the image formation is linear in
intensity, even beyond the diffraction limit of resolution. This enables techniques that use inverse filtering to overcome the diffraction limit 1Ref. 17, p. 133ff 2.
The SXM in Table 1 is the same instrument but not
in the luminescence mode, so that m is given by the
numerical aperture of the objective zone plate and
half of the acceptance angle of the detector. XM-2 is
a conventional microscope. The SXM and the XM-2
are to operate at a short-period undulator source.
The optical design of XM-2 will be such that it has a
circular entrance and exit pupils and matching numerical apertures of the illuminating and imaging
optics. XM-1 is the conventional microscope that
operates with bending magnet radiation. In Table 1
its imaging properties are shown for two different
objective zone plates, one with a 30-nm and one with a
25-nm outermost zone width. In both cases it has a
large diameter condenser zone plate with a 55-nm
outermost zone width, so that these two setups have a
different coherence parameter m.
Comparing the first two rows in Table 1 or the SXM
and the XM-2 cases, one can see how an annular exit
pupil improves the two-point resolution but increases
1 August 1995 @ Vol. 34, No. 22 @ APPLIED OPTICS
4949
the width of the edge response at the same time. For
circular exit pupils 1ek 5 02, the edge width decreases
so that it appears sharper with a decreasing coherence parameter m, whereas the two-point resolution
is reduced at the same time.
This object dependence of the image formation is
caused by the nonlinearity of Eq. 112. The fact that
knife-edge images produced with partially coherent
radiation appear sharper compared with the incoherent case is important for the characterization of
instruments near their diffraction limit. Looking at
the numbers in the first three rows of Table 1, one can
see that a central stop on the objective zone plate,
which is necessary in a SXM, affects the image
characteristics significantly compared with the influence of the coherence parameter. In a knife-edge
test an instrument such as the SXM in Table 1 should
produce a 75–25% image intensity drop at a distance
of 0.4 l@NAo or a FWHM of 0.45 l@NAo, to be
considered as diffraction limited.
We thank Derek Leed from the Department of
Electrical Engineering and Computer Sciences, University of California, for helpful support with the
program SPLAT, which was used to calculate the edge
response data. This research is supported by the
United States Department of Energy, Office of Basic
Energy Sciences, and the Office of Health and Environmental Research under contract DE-AC0376SF00098.
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