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838 M. Howells et al.

Much of modern X-ray microscopy centers on the exploitation of X-

ray absorption edges. X-ray absorption edges arise when the X-ray

photon reaches the threshold energy needed to completely remove an

electron from an inner-shell orbital. The energy at which this occurs is

approximately given by the Bohr model as E

= (13.6 eV)(Z-z

shield

)

where Z is the atomic number, z

shield

approximates the partial screening

of the nucleus’ charge by other inner-shell electrons (z

shield

≈ 1 for K

edges), and n is the principal quantum number (n = 1 for K edges, 2

for L edges, and so on). This produces the step-like rise in the cross-

section for photoelectric absorption that can be seen in the plots of f

in Figure 13–2. If one takes one image I

at an energy E

just below an

element’s absorption edge where the incident ﬂ ux is I

, and a similar

image I

at an energy just above an absorption edge, one can recover

the mass per area m

/A of the element x from (Engström, 1946)

EE II I I

(

)

(

)

−

(

)

−

(

)

µµ

101 202

2112

ln ln

(1)

This approach works well for mass concentrations greater than about

1%. Another way in which X-ray absorption edges are exploited is by

means of the “water window.” At X-ray energies between the carbon

and oxygen absorption edges at 290 and 540 eV, respectively, organic

materials show strong absorption contrast while water layers up to

several µm thick are reasonably transmissive (Wolter, 1952); this is

particularly valuable for imaging hydrated biological and environmen-

tal science specimens.

For those elements which have absorption edges below the energy

of incident X-rays so that inner-shell ionization occurs, the aftermath

of absorption involves the emission of either a ﬂ uorescent photon or

an Auger electron of characteristic energy. The energy of these ﬂ uores-

cent photons, and the ﬂ uorescence yield (Krause, 1979) (the fraction of

events which result in ﬂ uorescence rather than Auger electron emis-

sion), are both shown in Figure 13–3. At X-ray energies below 1 keV,

0.20

020406080

Atomic number Z

0.001

0.01

0.10

1.00

Fluorescence yield Y

100

Fluorescence energy (keV)

0.1

0.2

0.5

0.50

0.05

0.02

0.005

0.002

10 30 50 70 90

020406080

Atomic number Z

10 30 50 70 90

Figure 13–3. Energies (left) and ﬂ uorescence yields (right) for K and L edge emission. (Krause,

1979.)

Chapter 13 Principles and Applications of Zone Plate X-Ray Microscopes 839

Auger emission dominates, and scanning photoemission microscopes

(SPEM) use electron spectrometers to exploit these electrons for surface

studies (Ade et al., 1990a; Günther et al., 1997; Warwick et al., 1997; Ko

et al., 1998). At higher energies, the ﬂ uorescence signal dominates and

detection of these characteristic X-rays provides information on the

concentration of various elements in the specimen. Most scanning

ﬂ uorescence X-ray microprobes (SFXM) (Horowitz and Howell, 1972;

Sparks, 1980) use energy dispersive detectors where the number of

electron-hole pairs created by each ﬂ uorescent photon is used to

measure its energy, though crystal-based wavelength dispersive spec-

trometers can also be used. Exact quantitation of the elemental concen-

tration requires accurate knowledge of a number of factors, including

the solid angle acceptance of the detector and its quantum efﬁ ciency,

the degree to which ﬂ uorescent photons are reabsorbed in the speci-

men, and other factors, so that in most cases comparison is made with

standards with known elemental concentration and matrix concentra-

tion similar to that of the specimen under study. When compared with

electron microprobes, X-ray microprobes do not suffer from expansion

of the probe beam due to electron scattering, or a large continuum

background, so that the sensitivity to trace elements is often in the 100

parts per billion range.

Because X-ray interactions are well understood and do not involve

signiﬁ cant complications due to multiple scattering at energies below

about 10 keV, reliable predictions of image contrast can be made. If we

have a normalized signal I

from a feature-containing pixel and I

from

a background region, the signal to noise ratio obtained with N illumi-

nating photons is (Glaeser, 1971; Sayre et al., 1977a)

SNR

Signal

Noise

−

(2)

where we have used the Gaussian approximation to Poisson statistics

(which is quite good for NI greater than about 10) and the assumption

that there are no other noise sources with signiﬁ cant ﬂ uctuations. The

contrast parameter Θ is different from the usual deﬁ nition of contrast

due to the square root in the denominator. With this deﬁ nition, the

number of photons required to see a feature with a desired signal to

noise ratio SNR is given by N = (SNR)

/Θ

, and a common choice for

the minimum detectable signal to noise ratio is the Rose criterion of

SNR = 5 (Rose, 1946). Using this approach, Sayre et al. showed that

“water window” X-ray microscopes are able to image organic speci-

mens in micrometer-thick water layers with greatly reduced radiation

dose compared to electron microscopy (Sayre, 1977b; Sayre, 1977a).

This conclusion remains true even when modern energy-ﬁ ltered elec-

tron microscopes are considered (Grimm et al., 1998; Jacobsen et al.,

1998) (see Figure 13–4). Other investigators have extended the same

approach to include the effects of phase contrast (Rudolph et al., 1990;

Gölz, 1992) (see Figure 13–5) and the reduction of modulation transfer

at high spatial frequencies (Schneider, 1998), while Kirz et al. have used

this approach to compare elemental mapping using both differential

absorption and X-ray ﬂ uorescence (Kirz et al., 1978, 1980a, 1980b).

840 M. Howells et al.

1.3 Focusing Optics

Microscopes require focusing optics, or some other means to provide

a magniﬁ ed view of the object. X-rays reﬂ ect well from single refractive

interfaces only at grazing angles of incidence less than a critical angle

θδ

≈

which is typically in the range of 1–5º for soft X-rays. (Once

a particular angle has been selected, this same relationship gives a

critical energy above which the reﬂ ectivity becomes low; this can be

used to low-pass-ﬁ lter the energy spectrum from a radiation source).

While a number of labs have explored the use of axially symmetric

paraboloid or hyperboloid optics (Wolter, 1952; Aoki, 1994), most present

efforts center on the use of two orthogonal cylindrical grazing mirrors

in the Kirkpatrick-Baez geometry (Kirkpatrick and Baez, 1948). Advan-

tageous characteristics of these optics include their relatively long focal

Protein in ice: 30 nm resolution

(100% efficient optics, detectors)

Electron microscopy

(zero loss)

100 kV

300 kV

phase contrast

X-ray microscopy

at 520 eV

3000 kV

absorption contrast

-1

300 kV electrons: e

/Å

051015 20

Ice thickness (µm)

Dose in Gray

(x-ray cryo mass loss)

Energy (eV)

500200 2000 50001000 10,000

Skin dose (Gray)

5 µm

ice

25 µm

ice

125 µm

ice

Phase

contrast

Absorption

contrast

5 µm

25 µm

125 µm

5 µm

25 µm

125 µm

Feature: 20 nm protein

(x-ray cryo mass loss)

Figure 13–4. Estimated radiation dose required for imaging 30-nm-protein

features as a function of ice thickness for 200 keV electron microscopes and

520 eV soft X-ray microscopes. (Reprinted from Jacobsen et al., 1998, with per-

mission from Springer Science+Business Media.)

Figure 13–5. Estimated radiation dose required for imaging 20-nm-thick

protein features in various ice thicknesses as a function of X-ray energy; 100%

efﬁ cient optics and detectors are assumed.

Chapter 13 Principles and Applications of Zone Plate X-Ray Microscopes 841

length (several centimeters is typical) and their low chromaticity, so

that the incident beam energy can be tuned for spectroscopy without

any need to adjust the focus on the specimen. Optics of this sort have

recently achieved better than 100 nm resolution probe sizes using 12 keV

X-rays (Hignette et al., 2003; Yamamura et al., 2003; Mimura

et al., 2004), although the proﬁ le of the focus always has some degree

of “tail” outside of the geometrical image of the source due to scattering

from the residual surface roughness of even the best available mirrors.

Synthetic multilayer X-ray mirrors (Spiller, 1972; Barbee et al., 1981) can

increase the incidence angle well beyond θ

for narrow-bandwidth

radiation, and can achieve good reﬂ ection efﬁ ciencies for normal inci-

dence reﬂ ection at photon energies below about 200 eV. This approach

has seen rapid improvements due to the development of EUV projec-

tion lithography at 95 eV. However, notwithstanding recent progress in

mirror manufacture, it is important to recognize that even a perfectly

made Kirkpatrick-Baez mirror system still suffers from aberrations,

especially obliquity of ﬁ eld, which severely restrict its ﬁ eld of view and

therefore its performance as a microscope. On the other hand, it is still

well-able to focus points on or near the optical axis, which has led to a

resurgence in its popularity for microprobes and relay mirrors that are

imaging small sources such as synchrotrons.

When Röntgen discovered X-rays, he immediately tried to focus

them using refractive lenses but without success. The reason for this

is now well known: the focal length for a planoconvex lens with radius

of curvature R

is given by f

= −R

/δ, so that at 10 keV a glass lens with

= 1 cm would have a focal length of about 2 km. This does not pre-

clude the usefulness of refractive optics, however; a series of lenses

with small R

can be placed together to produce a signiﬁ cant net focus-

ing effect. One simple way to achieve this result in 1D is to drill a series

of holes in a solid block (Snigirev et al., 1996), and more recent work

using parabolic optics has demonstrated a resolution of about 100 nm

for hard X-ray imaging (Lengeler et al., 2002) with theoretical promise

for sub-10 nm resolution imaging (Schroer and Lengeler, 2005). Because

the ratio of phase shift to absorption increases with increasing X-ray

energy, these optics work primarily at energies above about 5 keV, and

at higher energies one will ultimately need to consider the contribu-

tions of inelastic scattering to the image due to the overall thickness of

the optic. Still, this approach is of interest especially since these optics

can be easily water cooled for high power applications.

The third way to focus X-rays is to use diffraction. While bent crys-

tals can provide focused beams of Bragg or Laue diffracted X-rays,

most work in X-ray microscopy centers on the use of microfabricated

diffractive optics in the form of Fresnel zone plates. Efforts in X-ray

microscopy using zone plate optics date back nearly a half century

(Baez, 1960, 1961), and X-ray Fresnel zone plates are now beneﬁ ting

from a high degree of development. Apart from detailed literature that

we will cite in Section 2. general reviews can be found in the books by

(Michette, 1986; Attwood, 1999). Due to their popularity as high resolu-

tion optics for X-ray microscopy, the properties of Fresnel zone plates

are described in some detail below.

842 M. Howells et al.

1.4 Overview and Recent Trends

We now review some of the general trends of X-ray microscopy tech-

nology and how they might affect the planning of a new X-ray micros-

copy program today. In this section, the number of relevant references

is essentially unlimited so we cite instead the subsections of this article

where many of the references can be found.

Modern X-ray microscopy was based on the twin ideas of the water

window (Section 1.1) and microfabricated zone-plate lenses (Section 2).

These led initially to life-science experiments consisting of 2D imaging

of wet samples in room-temperature air with “naturalness” as the

unique capability of the technique. Early versions of both the TXM

(Section 3.1.1) and STXM (Section 3.1.3) were capable of such imaging

and this style of X-ray microscopy has proved to be good ﬁ t to

many of the needs of the polymer (Section 4.3) and environmental

research communities (Section 4.2) among others. However, the room-

temperature approach could not be easily adapted to 3D imaging of

radiation-sensitive samples and this was something for which there

was, and still is, considerable demand. On the other hand the STXM,

even in its earliest realizations, was ready to begin the development of

spectromicroscopy (Sections 3.5 and 4). This development has contin-

ued over the last two decades or so and spectromicroscopy is now quite

a mature ﬁ eld that spans a wide range of application areas.

The step which has brought 3D imaging of biological samples within

reach has been the recent move toward X-ray microscopy of hydrated

biological samples frozen to cryogenic temperatures. For this type of

sample, X-ray microscopy ﬁ lls an important gap in the range of resolu-

tion values and sample thicknesses which can be covered by other

methods. The use of cryogenic temperatures is the key step in provid-

ing enough radiation-damage protection (see Figure 13–5) to enable

tomographic (three-dimensional) imaging (see Section 3.4). However,

3D cryomicroscopy requires signiﬁ cant technical additions to the X-ray

microscope including either the use of a vacuum sample chamber or a

gas-stream approach to keeping the sample cold plus the mechanical

devices required for recording a tilt series.

The interest in 3D imaging is also high in materials science and

engineering where there is a similar gap in the coverage by other

methods and where generally the samples have higher atomic num-

ber and have nonaqueous background materials (microcircuits for

example). For these samples the water window has no advantage and

X-ray microscopy at higher X-ray energies allows examination of bigger

samples and has in fact been going on for some time. Even in biology

there are factors pushing in the same direction. Ice is equally transpar-

ent at 1.5 keV as in the water window. Moreover, the 1.5–3.0 keV region

gives less absorption and more phase contrast so there is only a moder-

ate increase of the required radiation dose compared to the water

window (see Figure 13–5). In this energy range zone plates also have

longer focal lengths (important for sample tilting) and greater depth

of focus (important for 3D reconstruction (Section 3.4.2)). Overall,

multi-keV 3D X-ray microscopy has great promise but it poses some-

Chapter 13 Principles and Applications of Zone Plate X-Ray Microscopes 843

what different challenges with respect to the resolution/efﬁ ciency

capabilities of the zone plates and the provision of a suitable

condenser.

When possible we have cited publications in widely available jour-

nals rather than conference proceedings. However, good snapshots of

the ﬁ eld are provided at three-year intervals by the series of conference

proceedings dating back to 1984 (Section 1.1). As of the time of writing

of this chapter the most recent one (of which the proceedings

(Kagoshima et al., 2006) have not yet been published) was in Himeji,

Japan, in 2005. We have in some cases cited papers in the proceedings

of this conference because they have not appeared yet elsewhere. This

is especially true for papers on multi-keV X-ray microscopy which was

a growth subject at Himeji. Reports on the high-aspect-ratio zone plates

needed for multi-keV X-ray microscopy (Section 2.4.6) were particu-

larly promising. Zone plates of 50 nm outer zone width for 3–10 keV

and 30 nm for 1–3 keV were reported with efﬁ ciencies of at least 10%

(Section 2.4.6). Several of the former are operational in synchrotrons.

Equally signiﬁ cantly, images were shown using a 50 nm zone plate in

3rd order to get sub-30-nm resolution (Section 2.4.6).

The resolution values of the best water-window and multi-keV three-

dimensional images are both currently around 60 nm. The underlying

reasons for this value probably include the depth of focus effect (3.4b)

for water-window images and the zone plates for multi-keV ones. If

this is true, then the above reports suggest that the tools are now in

place for improvements to the 60 nm limit.

On condensers the news was also good. It was announced publicly

for the ﬁ rst time that single-reﬂ ection monocapillary mirrors have been

in use for some time as condensers for multi-keV TXMs and that exten-

sive 3D imaging has already been done using them. The condenser

system (Section 2.4.3) has traditionally been the Achilles heel of the

TXM and this new generation of condensers is delivering an elegant

solution which is described in more detail in Section 2.4.4. Apart from

removing various limitations of the condenser-zone-plate, the most

important innovation is energy-tunability. This combined with its

multiplexed data collection could make the TXM competitive in the

spectromicroscopy arena for specimens that are tolerant of its

higher dose.

The TXM and STXM have always been seen as complementary

devices; roughly equally popular (Table 13–3) and with important

advantages on both sides. We do not believe that that general percep-

tion is likely to change in the near future. The STXM will always have

the advantage in trace element mapping, the ability to instantly switch

from high to low magniﬁ cation, the ability to hold constant magniﬁ ca-

tion even when the X-ray energy is changing, the ability to image at

close to the theoretical minimum dose and so forth. On the other hand,

the TXM is going through a period of technical enhancement. It has

been moving into the multi-keV region and into cryomicroscopy which

together with its traditional main asset, multiplexed data collection, is

strengthening its capability in tomography which appears to us to be

its natural home.

844 M. Howells et al.

2 Fresnel Zone Plates

2.1 Introduction

A Fresnel zone plate is a circular diffraction grating that can be made

to focus light waves in the manner of a lens. It consists of a series of

concentric, usually metal, rings alternating with circular slots. Typi-

cally the rings are about equal in width to the slots and are fabricated

on a thin membrane. The design is based on the idea, that by blocking,

say, the even-numbered Fresnel half-period zones (Born and Wolf,

1999), the wavelets from the remaining (odd-numbered) zones will add

constructively. To see this quantitatively, consider plane-wave illumi-

nation of a zone plate with n zones with radius r

, outer zone width

∆r

, and a focal length f at wavelength λ (Figure 13–6). To get a ﬁ rst

order diffracted beam in which the signals from all the open zones

reinforce at the focus, we need a path difference λ between neighboring

open zones. In other words the optical path

−λ

should

equal f. Expanding the square root and neglecting terms above fourth

order, we obtain

22 8

=− +

(3)

Evidently the focusing condition is

= nλf (4)

to second order and

rnf

=+λ

(5)

to fourth order. In view of Eq. (7), we can neglect the fourth order

(spherical-aberration) term of Eq. (3) if the numerical aperture (NA) <<

1 which is often the case for X-ray zone plates. If the fourth-order term

is signiﬁ cant, then the zone plate can be made according to (5) and will

be corrected for spherical aberration but the correction will only apply

near the chosen wavelength and conjugate distances (⬁ and f). If it can

be neglected, we have

rnf

=λ, and this deﬁ nes a zone plate that will

zone plate

lens

focus

Figure 13–6. Geometry to calculate the radius of the nth half-period zone of

a zone plate illuminated by parallel light. The path from the nth zone to the

focus must be equal to f plus n/2 waves.

Chapter 13 Principles and Applications of Zone Plate X-Ray Microscopes 845

focus well for a range of wavelengths, although the focal length will

vary inversely with wavelength. Thus the chromatic aberration of a

zone-plate lens is much larger than that of a refractive lens and, to get

a good focus, the zone plate needs to be illuminated by monochromatic

light. The required degree of monochromaticity for achievement of the

diffraction-limited resolution is roughly given by ∆λ/λ ≤ 1/n (Thieme,

1988).

Some useful quantities follow from the fundamental zone-plate

Eq. (4). First we can take the difference between the nth and (n − 1)th

equations to get the outer zone width ∆r

∆r

. (6)

This allows the conclusion that all of the zones have equal area and

also gives us the numerical aperture

≡=

2∆

, (7)

and thence the Rayleigh resolution

Rayleigh n

r==

061

122

. ∆

. (8)

Thus we see that a given zone plate can be speciﬁ ed by its r

and ∆r

from which the resolution (which is independent of wavelength) and

the focal length and numerical aperture at any given wavelength,

follow from Eq. (8), Eq. (6) and Eq. (7) respectively. So far we have been

discussing the ﬁ rst-order focus but in general, beams of all integral

orders may be produced. Thus there is a zero-order (unfocused) beam,

a series of positive-order converging beams with focal distance f/m and

a series of negative-order diverging beams with focal distance −f/m

(Figure 13–7). In mth order the numerical aperture is m times larger

(m=-5)

(m=-3)

(m =-1)

(m = -5)

(m=-3)

(m=-1)

OSA

5nλ

3nλ

nλ

f +

Figure 13–7. Diagram showing orders number

−5, −3, −1, 1, 3, 5 of a zone plate. Compared to

the incident beam, the negative orders diverge,

the positive orders converge and the zero order

(omitted for clarity) has the same shape. For

plane-wave illumination, the virtual foci of the

negative order beams and the real foci of the

positive order beams are at distances | f |/m from

the zone plate where | f | is the focal distance of

sion of Cambridge University Press.)

846 M. Howells et al.

and hence the resolution is m times smaller (better) than in ﬁ rst order.

As we will see, if the open and opaque zones are of equal width, then

the even orders are missing.

2.2 Zone Plate Image Quality

2.2.1 Optical Path Function Analysis

We consider the imaging of a general point A (x,y,z) by a planar zone

plate lying in the y − z plane (Figure 13–8) (Kamiya, 1963). We will use

the method of the optical path function so we start by calculating the

optical path from A to a general point B (x′,y′,z′) that we will later iden-

tify as the Gaussian image point. Without loss of generality we set y =

y′ = 0. We calculate the path APB where P (0,w,l) is a general point in

the zone plate. The expression for the optical path will be a power

series in the aperture coordinates, w and l, and the ﬁ eld angle, z/x and

each term in the series will represent a speciﬁ c aberration. Evidently

AP =

++−xw zl

22 2

()

(9)

so expanding the square root and keeping terms up to fourth order,

we have

AP =+

+−− +

()





{

−

wl z

zl z

22 4

22 3322222

222lwlz wlzl++

()

−+

()

]

}



(10)

There is an identical series for PB except that, for PB, the x, y and z are

replaced by x′, y′, and z′ (Figure 13–8). We are now in a position to write

down the optical path function, F. Before doing so we drop terms

Figure 13–8. Notation for the optical-path-function analysis of a zone plate.

Chapter 13 Principles and Applications of Zone Plate X-Ray Microscopes 847

which do not depend on w, l or z/x because they do not represent aber-

rations, and introduce the term −nmλ/2 as we did to get Eq. (1). We

choose to analyze the case of a parabolic zone plate (that is one built

according to Eq. (4)) so initially, we use Eq. (4) to write the optical path

function up to third order only, as follows.

xx fm

=+− =+−

′

−













−+

′

AP PB AP PB

11 1

′′















(11)

Specializing to the case when B is the Gaussian image point, the ﬁ rst

(defocus) term vanishes and by considering the ray AOB we obtain

z/x = −z′/x′ (12)

so that the second term also vanishes. This leaves only the ﬁ ve fourth-

order terms

xfm

=−

()

′

(

)

−++

′









−

(

))

()

′

(

)

xfm

xx x



(13)

These are the ﬁ ve Seidel aberrations; spherical aberration, astigmatism,

distortion, ﬁ eld curvature and coma respectively. Because of Eq. (12),

the distortion term vanishes identically which is a useful property of

zone-plate lenses and we can therefore turn our attention to the remain-

ing aberrations.

2.2.2 Ray Aberrations

We need to know the ray pattern delivered by the zone plate for a given

point object. That is, we want the ray aberrations ∆y′ and ∆x′ relative

to the Gaussian image point (coordinates identiﬁ ed by subscript zero).

For a normal-incidence optic these are given by the following expres-

sion (Born and Wolf, 1999)

∆∆

′

′′

′

∂

(14)

We now apply this to the Seidel aberrations individually.

2.2.3 Spherical Aberration

We can rearrange the spherical aberration term using the magniﬁ cation

M = x′

sp ab

=−

()

1()

(15)

The last term of this expression, which we will denote by Θ, approaches

unity in the cases of interest to us namely M large (a microscope) or

M small (a microprobe). If we consider the case that Θ does approach

unity, then Eq. (15) reduces to the fourth-order term of Eq. (3). This is

expected because we are reverting to the conjugates used to derive (3)