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868 M. Howells et al.

The beam then re-enters a vacuum environment where the objective

zone plate and the image detector are located. At energies below a few

keV, the most common detector is a backside-thinned CCD which is

directly illuminated by the X-ray beam; at higher energies, phosphor

screens imaged by a visible light lens onto a CCD detector are com-

monly used. Because of the desire to deliver 10–50 nm resolution using

detectors with 1–20 µm pixel size, the distance from the zone plate

objective to the detector is often in the range 1–2 m to give acceptably

high optical magniﬁ cation.

The approach described above is commonly used with bending

magnet and laboratory sources. Particular challenges arise when the

source phase space area is dramatically smaller than desired, which is

the case for undulator sources on low emittance storage rings. As an

example, Niemann has studied a variety of solutions for the condenser

of the current Berlin TXM which is illuminated by a BESSY II undulator

(Niemann, 1998). The adopted solution (Niemann et al., 2000) involves

a zone plate segment, and three ﬂ at mirrors. The magniﬁ cation of the

zone plate is chosen to fully illuminate the object ﬁ eld. The ﬁ rst mirror

is ﬁ xed but must be tilted for a change of wavelength. The other two

mirrors are mounted in a structure that rotates and delivers an incoher-

ent hollow-cone beam of which the inner-to-outer angular difference

(∆ϑ say) is determined by phase-space matching and the outer angle

(the NA) is determined by the last-mirror reﬂ ection angle. Thus the

illuminated area and the NA can both be chosen, no wobbling is

required and Liouville’s theorem (which states that the phase space

area of an optical beam is a conserved quantity) is respected by allow-

ing ∆ϑ to ﬂ oat. This system represents an elegant optical solution,

though it is mechanically quite complex. Other strategies to expand the

phase space of an XUV beam have been explored by the microfabrica-

tion community who are concerned about “fringing” in XUV lithogra-

phy (Murphy et al., 1993; White et al., 1995). One such approach is to

design a pseudorandom diffractive optic speciﬁ cally to “spoil” the

phase space of a beam and match the object size and the NA (David

et al., 2003); such optics must meet the challenge of evenly ﬁ lling both

the object plane and the back focal plane with light.

3.1.2 TXM Phase Contrast Layout

As noted above, phase contrast plays an important role in X-ray micros-

copy, particularly at higher photon energies. In order for phase varia-

tions at the specimen plane to produce intensity variations at the

detector, some method of mixing the wave diffracted by the specimen

with an undiffracted phase-reference wave must be employed. The

most common approach in X-ray microscopes is that of Zernike. In

light microscopes, Köhler illumination is provided by using a relay

lens to image the source on to the front focal plane of the condenser.

Points at this front focal plane deliver parallel beams to the object plane

which (if undeviated by the object) are focused on to a phase ring at

the back focal plane of the objective where they are phase shifted

usually by ±π/2. Thus the ring aperture at the front focal plane of the

condenser provides a narrow, hollow cone of illumination of the speci-

Chapter 13 Principles and Applications of Zone Plate X-Ray Microscopes 869

men, and is conjugate to the phase ring. At the same time, light origi-

nating from a point scatterer in the object is focused to the detector,

where it interferes with the phase-shifted unscattered light (see Figure

13–17).

In X-ray microscopes, it is more difﬁ cult to use a relay optic to work

in the Köhler illumination condition; instead, the front focal plane ring

aperture is illuminated by nearly parallel light from the source (i.e.,

critical illumination) so that a much smaller area of the condenser is

illuminated. Because the light from the front aperture is much more

collimated than would have been the case with Köhler illumination,

the longitudinal location of the aperture and its corresponding phase

ring is much less critical than it is with visible light microscopes so the

Figure 13–17. Two illustrations of a Zernike phase contrast optical system. The upper one shows the

classical scheme used in light microscopes based on Köhler’s illumination (Born and wolf, 1999). Light

from each point of the annular aperture, placed in the front focal plane of the condenser, is delivered

to the object as a parallel beam. Two object points are shown receiving example rays from the source.

The rays may be undeviated by the object, in which case they are seen to pass through the phase ring

on their way to the detector. On the other hand the rays may be deviated (diffracted) by the object in

which case they reach the detector without passing through the phase ring. Interferences between

these two types of optical signal result in a mapping of the object phase variations into an intensity

pattern on the detector. The lower diagram shows a practical synchrotron radiation implementation

of a Zernike-phase-contrast TXM at the 4.1 keV beam line on ID21 at the European Synchrotron Radi-

ation Facility in Grenoble. An undulator X-ray source is followed by a crystal monochromator illu-

minating a condenser zone plate (which can be small since it does not have to act as a linear

monochromator). The condenser provides critical illuminated to the sample rather than Köhler and,

due to the good collimation of the incoming beam, the illuminated area of the condenser is projected

on to a phase ring in the back-focal plane of the objective zone plate so as to provide phase contrast.

The tradeoffs involved in choosing the illuminated area of the condense pupil are discussed in Section

3.3.6 (Courtesy J. Susini, ESRF.)

870 M. Howells et al.

primary alignment requirement is transverse to the X-ray beam direc-

tion. For computations it is convenient to consider the two rings as

built into the lens pupil functions. Evidently when the source is well-

collimated, as in the case of a synchrotron, there is no need for the two

rings to be exactly conjugate. The use of the ring aperture obviously

makes the illumination more coherent. The question of the best choice

of width and radius for the two rings or, equivalently, how much coher-

ence to have, we defer until later (Section 3.3.6). The phase ring itself

is constructed out of a material with a large phase shift per absorption

length f

/(2f

), such as any material used at energies just below an

absorption edge.

Zernike phase contrast X-ray microscopy was pioneered by Schmahl

and Rudolph (1987) and Ruldoph et al. (1990), and the Göttingen group

has shown impressive results in phase contrast for water-window

imaging of biological specimens. The ring aperture and phase ring

used in these experiments could be rapidly inserted or retracted

(Schmahl et al., 1994, 1995; Schneider, 1998). Phase contrast is arguably

even more important at higher X-ray energies where it is the dominant

contrast mechanism. For example a hard X-ray (4 keV) phase-contrast

microscope, illuminated by a system using a crystal monochromator

followed by a condenser zone plate, is operating at the ID21 beam line

at the European Synchrotron Radiation Facility in Grenoble, France

(see Figure 13–17). Imaging of functioning integrated circuits at 60 nm

resolution has been demonstrated (Neuhäusler, 2003); see Section 4 for

more information on this. In another example of this conﬁ guration, the

National Synchrotron Radiation Research Center in Hsinchu, Taiwan

has installed a TXM built by Xradia. This instrument (alluded to in

Sections 1.4 and 2.4.4) has been used to demonstrate the long-existing

idea of using zone plate higher focal orders for imaging. In water-

window instruments which typically have focal lengths on the order

of 1–2 mm in ﬁ rst order (and therefore 1/3–2/3 mm in third order), the

idea has not been readily adopted due to practical considerations of

working distance. However, in the Taiwan experiment (Tang et al.,

2006) a phase contrast image of a fabricated test object was made at

8 keV using a 50 nm outer-zone-width zone plate in third order. Lines

of minimum width 30 nm were imaged clearly and the authors esti-

mate a resolution below 25 nm. This is evidently a most important

development (see Section 5).

3.1.3 Scanning Ttransmission X-Ray Microscope (STXM) Layout

Scanning transmission X-ray microscopes (STXMs) typically use a

zone plate to demagnify a pinhole source to a small focus spot through

which the specimen is scanned. While initial demonstrations using

synchrotron radiation used pinhole optics (Horowitz and Howell, 1972;

Rarback et al., 1980), the use of zone plate optics in scanning micro-

scopes was pioneered by Rarback et al. (1984) and later by (Niemann,

1987; Niemann et al., 1988). (Normal incidence optics with synthetic

multilayer reﬂ ective coatings have also been used in the 50–120 eV

range (Haelbich, 1980a; Haelbich et al., 1980b; Ng et al., 1990)). Since

scanning microscopes require coherent illumination to reach their

Chapter 13 Principles and Applications of Zone Plate X-Ray Microscopes 871

maximum resolution, they have often used undulators as high bright-

ness sources (Rarback et al., 1988; Kenney et al., 1989; Morrison et al.,

1989b) though excellent performance has also been obtained using

bending magnet sources on low emittance storage rings (Kilcoyne

et al., 2003). While a large number of STXMs are now in operation, we

describe here the characteristics of the most recent in a series (Rarback

et al., 1988; Jacobsen et al., 1991; Feser et al., 1998, 2000) of undulator-

based scanning microscopes built at Stony Brook University for opera-

tion at the National Synchrotron Light Source at Brookhaven National

Laboratory in New York. A soft X-ray undulator plus spherical grating

monochromator with an energy resolution that can be as good as

0.06 eV at 290 eV (Winn et al., 2000) is used to deliver soft X-rays to a

2D exit slit which can limit the beam size in the range 25–120 µm in

both x and y. This slit then serves as a secondary radiation source for

zone plates of either 80 or 160 µm diameter and zone widths of 30–

45 nm (Spector et al., 1997; Tennant et al., 2000), producing a focal spot

of 36–54 nm Rayleigh resolution. The beam emerges from the ultra high

vacuum synchrotron beam line into an atmospheric pressure environ-

ment by passing through a 100 nm thick Si

window. The zone plate

includes a central stop of about half the zone plate diameter; this stop

must be made quite thick (0.3 µm gold is common for soft X-ray applica-

tions) so that the undiffracted light transmitted through the large

central stop is kept to a very small level compared to the ﬂ ux in the

focused X-ray beam. The zone plate is then followed by an order sorting

or selecting aperture (OSA) so that a pure ﬁ rst-order focal spot is

obtained.

While steering mirrors are used to scan the beam in visible light

scanning microscopes, it is easier to maintain signal uniformity by

keeping the beam and zone plate ﬁ xed and scanning the specimen

through the focal spot. This is accomplished using an X-Y-Z stack of

stepping motor stages for large motion with 1 µm precision, and a piezo

scanning stage for 50–100 µm range and nanometer precision. Because

piezos have nonlinearities and hysteresis in their response to scan volt-

ages, some form of closed-loop feedback is generally used, based on

position signals such those provided by linear voltage differential

transformers (Kenney et al., 1985), capacitance micrometers (Jacobsen

et al., 1991), or laser interferometers (Shu et al., 1988; Kilcoyne et al.,

2003); the latest Stony Brook STXM allows the user to choose between

capacitive or laser interferometer feedback. The specimen is then fol-

lowed by a high efﬁ ciency X-ray detector; common choices include the

use of gas-based proportional counters which offer extremely high

efﬁ ciency of detection for those X-rays that make it through a thin

entrance window (Rarback et al., 1980; Kenney et al., 1985; Feser et al.,

2000) but which suffer from a count-rate limit of about 1 MHz. Alterna-

tives are phosphor-coated screens followed by photomultipliers to

detect the resulting visible light (Maser et al., 2000), and solid state

detectors which are capable of signiﬁ cantly higher signal rates (Barrett

et al., 1998; Wiesemann et al., 2000; Feser et al., 2001, 2003; Guttmann

et al., 2001). In the Stony Brook STXM, the user can choose between

proportional counter and segmented silicon detectors, and a visible

872 M. Howells et al.

light microscope is also placed on the detector stage with X-Y-Z motor-

ized motion so as to pre-locate desired regions of the specimen. Another

approach, which works for either a TXM or a STXM, is co-indexing of

off-line light microscopes with the X-ray microscope (Meyer-Ilse et al.,

1994, 2001; Kilcoyne et al., 2003).

Scanning microscopes offer different characteristics than full-ﬁ eld

imaging systems do. These include the ability to quickly change from

scanning very large areas at low resolution to taking high resolution,

small ﬁ eld scans, and reduced radiation dose because the 5–20% efﬁ -

cient zone plate is located upstream of the specimen rather than down-

stream. Because of the need to mechanically scan the specimen in most

present microscopes, and the need for coherent illumination, imaging

times are generally longer (in the range of one or a few minutes, rather

than seconds in the case of many TXMs). At the same time, the require-

ment for coherent illumination means that the etendue or phase space

that the monochromator must accept is greatly reduced, so that aber-

rations are reduced and it is relatively easy to obtain very high spectral

resolution. These characteristics make scanning transmission X-ray

microscopes especially well suited to low-dose spectromicroscopy

applications, as will be described below.

Phase contrast has historically seen less use in scanning transmis-

sion X-ray microscopes. However, refractive and diffractive effects by

the specimen lead to a redistribution of signal on the detector which

can be interpreted to give phase contrast images (as will be discussed

below). The ultimate approach is to use a 2D detector (such as a CCD

camera) to detect the entire intensity distribution at each pixel of a

scanned image; Chapman has used this in an impressive demonstra-

tion of Wigner deconvolution microscopy to recover the phase and

magnitude distribution of the specimen as well as the zone plate objec-

tive (Chapman, 1996a), while Morrison et al. have used this to obtain

ﬁ rst moment images which reveal the dominant phase gradient at each

pixel location (Eaton et al., 2000; Morrison et al., 2002). Coupled with

the potential power of these approaches are signiﬁ cant challenges: the

readout time of large pixel detectors is often not in the few or sub

millisecond pixel timescale required for fast scanning, and the result-

ing 4D data ﬁ les are quite large. More fundamental is the question of

statistical signiﬁ cance in each pixel of a large array detector when

radiation dose to the specimen must be considered; in some cases it

may be preferable to divide a weaker signal into fewer detector seg-

ments. This approach has been used by Feser et al. (Figure 13–18) who

have used a detector with only 8 segments to obtain quantitative phase

contrast images while operating at per-pixel acquisition times of a few

milliseconds and producing data ﬁ les of manageable size (Feser et al.,

2003) (see Figure 13–18). Additional approaches to obtaining phase

contrast in scanning microscopes include the use of zone plate dou-

blets (Kaulich et al., 2002) or phase modiﬁ ers (Polack et al., 2000) to

produce differential interference contrast. Undoubtedly different

experiments will involve different choices in the tradeoff of the ﬁ ne-

ness of segmentation of scanning microscope detectors, but in any case

Chapter 13 Principles and Applications of Zone Plate X-Ray Microscopes 873

it is clear that phase contrast plays an interesting role in scanning

transmission X-ray microscopy as well as in microprobes as will be

noted below.

3.1.4 Scanning Fluorescence X-Ray Microprobe (SFXM) Layout

Scanning ﬂ uorescence X-ray microprobes (SFXM) use a focused X-ray

beam to stimulate the emission of characteristic ﬂ uorescence X-rays

from speciﬁ c elements in the specimen. When linearly polarized radia-

tion (such as is usually obtained from synchrotron sources) is used, a

ﬂ uorescence detector placed 90º to the beam in the polarization plane

will detect a minimum of coherent scattering signal; this detector must

then have some means of discriminating between different X-ray emis-

sion energies. Energy-dispersive detectors accomplish this by measur-

ing the number of electron-hole pairs created by each X-ray in a

semiconductor material, while wavelength-dispersive detectors use a

crystal optic or a grating to separate the X-ray energies. Energy disper-

sive detectors generally have large solid angle collection, and multi-

element detectors can be used to overcome the ∼50 kHz count rate limit

determined by charge readout time, while wavelength dispersive

detectors offer better separation between nearby spectral lines and

larger dynamic range for detecting low concentration elements amongst

other ﬂ uorescing elements of higher concentration. There is a long and

rich history of synchrotron-based microprobes (Horowitz and Howell,

1972; Sparks, 1980; Rivers et al., 1988; Thompson et al., 1988; Hayakawa

et al., 1989), and a variety of optical approaches including the use of

compound refractive lenses and Kirkpatrick-Baez mirror optics are

now achieving submicron resolution. We outline here some of the

characteristics of microprobes using zone plate optics (Barrett et al.,

1998; Yun et al., 1998a; Suzuki et al., 2001; Kamijo et al., 2003) by

Figure 13–18. Amplitude (left) and phase (right) contrast images of a germanium test pattern imaged

using a scanning transmission X-ray microscope with a segmented detector. Also shown is a schematic

view of the detector. The undeﬂ ected beam from the ﬁ rst order focus is directed into bright ﬁ eld seg-

ments 1, 2 and 3 (these segments also allow differential interference contrast). The deﬂ ected beam is

detected in the angular segments 4, 5, 6, and 7 for dark ﬁ eld imaging and differential phase contrast.

Zero order

OSA

First

order

focus

1 µm

874 M. Howells et al.

considering the example of the 2-ID-E microprobe at the Advanced

Photon Source at Argonne National Laboratory near Chicago.

This microprobe operates using a side-deﬂ ecting crystal monochro-

mator to transfer an off-axis part of the central cone produced by a

hard X-ray undulator. In the vertical direction, the objective zone plate

images the source directly onto the specimen, while in the horizontal

direction the variable width monochromator exit slit is imaged. Astig-

matism effects are avoided in the resulting focused beam by the fact

that the depth of focus is much larger than the difference between

the positions of the horizontal and vertical foci of the zone plate (in

addition, the zone plate can be tilted to compensate for more severe

source astigmatism). Zone plates of diameter 160–320 µm and outer-

most zone width of 100 nm are typically used, giving focal lengths of

12–25 cm at 10 keV. While the probe size can be as small as 150 nm, a

larger horizontal source size is often chosen to give more ﬂ ux at the

cost of resolution. The specimen is mounted at 15º to the incident

beam to provide access to both the incident X-ray beam and the ﬂ uo-

rescence detector, and it is scanned by motor-driven stages with 0.1 µm

step size. A multi-element germanium ﬂ uorescence detector is used

to collect the ﬂ uorescent signal; one can either record the signal in a

limited number of pre-deﬁ ned energy windows for rapid analysis

with modest data ﬁ le size, or record the full ﬂ uorescence spectrum

per pixel for improved quantitation of elements with closely spaced

ﬂ uorescence energies. The region consisting of the specimen and

detector is located inside a glovebox which can be purged with helium

to eliminate ﬂ uorescence from argon in air which would otherwise

obscure a number of low-Z elements, and to reduce the absorption of

low-Z ﬂ uorescence signals by air. Per-pixel dwell times are on the

order of one second, so that the experimenter must be judicious in the

choice of scan area (the use of common position indexing between a

visible light microscope and the microprobe aids in rapid specimen

location).

Zone plates of diameter 50–100 µm and outermost zone width of

100–300 nm are typically used, giving focal lengths of 5–30 mm. While

the probe size can be as small as the Rayleigh resolution of 1.22 times

the outermost zone width, a larger virtual source size is often chosen

to give more ﬂ ux at the cost of resolution. The specimen is mounted at

45º to the incident beam to provide access to both the incident X-ray

beam and the ﬂ uorescence detector, and it is scanned by motor-driven

stages with 0.1 µm step size. A multi-element germanium ﬂ uorescence

detector is used to collect the ﬂ uorescent signal; one can either record

the signal in a limited number of pre-deﬁ ned energy windows for

rapid analysis with modest data ﬁ le size, or record the full ﬂ uorescence

spectrum per pixel for improved quantitation of elements with closely

spaced ﬂ uorescence energies. The region consisting of the zone plate,

specimen, and detector is all located inside a glovebox which can be

purged with helium to eliminate ﬂ uorescence from argon in air which

would otherwise obscure a number of low-Z elements. Per-pixel dwell

times are on the order of one second, so that the experimenter must be

judicious in the choice of scan area (the use of common position index-

Chapter 13 Principles and Applications of Zone Plate X-Ray Microscopes 875

ing between a visible light microscope and the microprobe aids in

rapid specimen location).

Trace element mapping by ﬂ uorescence detection with sensitivities

down to about 100 parts per billion, or about 10

−17

grams of iron within

a (200 nm)

spot, represents the majority of microprobe applications.

However, X-ray microprobes can be used in a number of other ways

as well, including measurements of crystal strain in small regions

(Rebonato et al., 1989; Cai et al., 1999; Soh et al., 2002) and differential-

aperture measurements of microstructure and strain (Larson et al.,

2002). The phase contrast methods described above for STXM are

equally applicable in SFXM, and offer a much-needed way to image

the overall mass and ultrastructure of specimens while simultaneously

forming trace element or strain maps.

3.2 Fundamentals of Contrast in the TXM

It is useful to have an analytical treatment that provides insight into

the way a microscope produces contrast and at the same time allows

simple calculations to assess experimental plans. This was provided

by Rudolph and coworkers (1990) in a form that allows amplitude-

contrast, Zernike-phase-contrast and dark-ﬁ eld imaging, to be included

in a uniﬁ ed description, that is largely independent of the microscope

design. Assuming only that we have an imaging microscope, we con-

sider ﬁ rst the Zernike phase-contrast TXM.

We are interested in the contrast C or the contrast parameter Q (see

Section 1) between an interesting feature F and a background feature

B generated via the phase shifter S. F and B are deﬁ ned to have the same

thickness but in reality the background material (water for example)

may be thicker than the feature so we allow for that by adding a layer

L. If we deﬁ ne the complex transmission factors of F, B, S and L as a

⋅ p

≡ [exp{−2πβ

/l}] ⋅ [exp{2πiδ

/l}] etc where 1–δ–iβ is the refractive

index and t is the thickness, then following Rudolph et al. (1990) we can

obtain the image intensities (I

and I

) and thence C and Q

Iaaaaa ppp

FBSFBS FBS

−

(

)

(

)

−

(

)

2Re









−









{

+−









}

222

aa p

aaa ppaa

BS S

FFB FBBL

Re .

aaa

The dose D (the energy deposited per unit mass of sample) needed to

detect a feature of area d

, thickness t

= d and density ρ with signal-

to-noise ratio S/N can now be calculated (Rudolph et al., 1990) as

follows

(

)

−

λρ Θ

where hc = 1240 eV-nm represents the product of Planck’s constant and

the velocity of light. The above relations are convenient because, in

876 M. Howells et al.

addition to phase contrast, they also describe amplitude-contrast (t

0) and dark-ﬁ eld (t

= large) experiments. The formula for D yields dose

plots like Figure 13–5 and also tells us that the number of X-rays (of

energy E) per unit area required to make the measurement with the

given resolution and signal-to-noise ratio is Dρ/(µE), where µ is the X-

ray absorption coefﬁ cient.

In the multi-keV X-ray energy range, the phase contrast is substan-

tially larger than the absorption contrast for suitable choices of the

thickness of the phase shifter. The best result is typically achieved by

attenuating the direct beam by the phase plate so that its amplitude

is comparable to that of the scattered signal, resulting in an interfer-

ence of two beams of similar amplitude. The available choices of

phase plate thickness to optimize this are positive phase contrast

(phase shift = π/2, 5π/2, . . .) or negative phase contrast (phase shift =

3π/2, 7π/2, . . .). Figure 13–19 shows contrast plots of some of these

possibilities. Although these plots are useful for providing compara-

tive information, they represent a considerable idealization; the phase

shift is assumed to be applied to 100% of the undiffracted light and

0% of the diffracted light, the thickness of the phase shifter is chosen,

at each energy, to give the stated phase shift and the optical system

is assumed 100% efﬁ cient. Under these assumptions, the dark-ﬁ eld

contrast is identically equal to one. This suggests that dark-ﬁ eld has

a dose advantage that will be dependent on the practical value of the

nominally zero signal due to the undiffracted light and on the strength

of the dark-ﬁ eld signal (Chapman et al., 1996c; Vogt et al., 2001a).

3.3 Partial Coherence

3.3.1 History

The resolution of microscopes, including X-ray microscopes, depends

on the angular widths of the light beams delivered to, and collected

100 1,000 10,000

Energy (eV)

Phase 7π/2

Amplitude

-1

-2

-3

-4

Fractional contrast

100 1,000

10,000 100,000

Energy (eV)

Amplitude

Phase 3π/2

Phase 7π/2

-1

-2

-3

Fractional contrast

Protein in water

Glass

Figure 13–19. Intrinsic amplitude and Zernike phase contrast for two types of sample of thickness

30 nm relative to a background material of the same thickness: protein in water (left) and vacuum in

glass, representing a crack (right). The protein is modeled assuming a density of 1.35 g/cm

, composi-

tion of H

S and a phase ring made of copper. Glass is modeled assuming a density of 2.5 g/cm

composition Si

and a phase ring made of gold. It is noteworthy that the phase

contrast can be much greater than the amplitude contrast even in the 290–540 eV water window.

Chapter 13 Principles and Applications of Zone Plate X-Ray Microscopes 877

from, the sample . The analysis of this effect was pioneered in the 1950s

by Hopkins, Wolf and others and was part of a movement to apply the

linear-systems ideas, widely used by the engineering community, in

the optical arena. This work has been reviewed by Hopkins (1957),

Thompson (1969), and in various texts (Wilson and Sheppard, 1984;

Goodman, 1985; Born and Wolf, 1999). The main point is that the ﬁ nest

features (highest spatial frequencies) in the sample diffract the illumi-

nating beam by the largest angles θ. The best geometry to include such

large deﬂ ection angles is therefore one that has a wide-angle beam

both inward to, and outward from, the sample. This implies broadly

that in a TXM, a large-area source providing spatially incoherent illumi-

nation gives better resolution than a point source giving coherent illu-

mination (Figure 13–20), although such a comparison is not as simple

as it sounds (Goodman, 1968). It would be wrong to conclude from this

that STXMs which use coherent illumination have intrinsically worse

resolution than TXMs. In fact, scanning microscopes with large area

detectors and transmission microscopes with large illumination angles

both deliver incoherent bright-ﬁ eld images with the same resolution

and transfer function provided only that they use objective lenses of

equal resolution.

The ﬁ rst application of linear-systems concepts in X-ray microscopy

was in the analysis of STXM images (Jacobsen et al., 1991; Zhang et al.,

d/2

Figure 13–20. Schematic showing why the transfer function for incoherent

imaging extends to twice the spatial frequency of coherent imaging for a given

optic numerical aperture. For coherent imaging (top) the marginal ray is

deviated by an angle theta due to diffraction by the sample periodicity d. For

incoherent imaging (bottom) some light is deviated by 2θ due to the sample

periodicity d/2. Both TXM and STXM with large area detector deliver incoher-

ent bright-ﬁ eld images. (After Jacobsen et al., 1992b.)