Making hard x-ray micro-focus beam and imaging microscope with Fresnel zone plate optics-SPring-8 summer school textSeptember 2008
Revised June 2009,
Revised October 2009 (Imaging microscope)
Revised Sep. 2010 (minor revisions)Final Revision September 2011

Yoshio Suzuki, JASRI/SPring-8

Contents

1. What is Fresnel zone plate for x-rays?
1.1. A simple and intuitional explanation of Fresnel zone plate (FZP)

3

1.2. More precise treatment

6

1.3. Diffraction efficiency

8

1.4. Actual structure of FZP for hard x-rays, fabrication of zone plates

9

2. Making microfocus beam with Fresnel zone plate
2.1. How small is the limit of focused beam size?
Diffraction limit and geometrical limit of resolution

12

2.2. Depth of focus and chromatic aberrations

15

2.3. Preservation of emittance and brilliance16

2.4. General description of focusing optics

17

3. Application of FZP optics to imaging microscopy and microtomography
3.1. X-ray imaging microscopy

19

3.2. Order selection by spatial filter and illuminating optics for FZP objective

19

3.3. Spatial resolution of FZP microscopy with illuminating optics

21

3.4. Depth of focus

22

3.5. Chromatic aberration and spherical aberration

25

3.6. Off-axis aberration theory based on wave optics

26

3.7. Typical example of imaging microtomography with FZP objective in SPring-829

4. References

32

Basic exercises before the experiments

33

Practice at the beamline(s)34Advanced exercises

36

2

1. What is Fresnel zone plate for hard x-rays?

1.1. Introduction - a simple and intuitional explanation of Fresnel zone plate (FZP) It is difficult to use refractive optics in the hard x-ray regions, while the refractive lens is mostpopular optics for visible light. This is because the index for refraction in the x-ray region is very near to
the index for refraction of vacuum. The refractive index of vacuum is defined as unity. The discrepancy ofrefractive index from unity is only 10-5-10-6 for any materials. Therefore it is very difficult to deflect the
x-ray beam by refraction at the interface of optical media, and any optical devices that are used in visible
light optics, lens or prism, cannot be used for x-rays, as it is. Although, some refractive lenses or prisms
are developed in the hard x-ray region, they are still minority. At present, the Fresnel zone plate (FZP) is a
widely used and practical optics device in the x-ray regions, the x-ray energy around 10 keV or higher
energy regions. The micro-focus beam size of about 25 nm is already achieved with the FZP optics in thehard x-ray region.
The Fresnel zone plate is a concentric transmission grating with radially decreasing grating
period, as schematically shown in Fig. 1.1, coarse grating period at central part, and finer pitch at theouter (marginal) area. An optical micrograph of FZP is shown in Fig. 1.2. This is a real Fresnel zone plateused as micro-focusing optics or image forming optics at the SPring-8 beamlines.

Fig.1.1. Schematic drawing of FZP structure. The radius of n-th ring is defined by rn = √(nλf). λ is x-ray
wavelength and f is focal length.

3

Fig. 1.2. An optical micrograph of FZP used at the SPring-8 imaging beamlines as microfocus lens and/or
microscope objective lens. The diameter of FZP is 100 µm and the outermost zone pitch (period) is 0.5µm. The focal length of the FZP is designed to be 248 mm for 1.0 Å (12.4 keV) x-rays. The zone
structure is made of 1 µm-thick tantalum deposited on a 2 µm-thick silicon nitride membrane. The FZP isfabricated by electron beam lithography and reactive dry-etching process at the NTT Advanced
Technology, Japan. This FZP will be used in our experimental course.

In order to understand the FZP optics intuitionally, it is useful to start from two-slit optics, so
called Young’s double-slit. The two-slit is a most simple and basic tool in wave optics. As shown in Fig.
1.3, when a pair of slit is placed with a distance of d, intensity distribution on a screen far from slit hasperiodic structure. When the observation angle θ satisfy an equation, nλ = d sin θ, the intensity is
maximum, and minimum is observed at (n + 1/2)λ = d sin θ. Here, n is an integer number called order ofdiffraction. If the incident radiation is perfectly coherent and the slit width is negligibly narrow, theobserved intensity distribution must be sinusoidal! But the perfect coherence is not realistic in generalexperimental condition. Therefore, the actual interference pattern becomes weak far from optical axis, as
shown in Fig. 1.3.

θｄ

λ=2d sinθIntensity
distribution

Two-slit

Fig. 1.3. Schematic diagram of Young’s two-slit experiment and its interference pattern. Gaussian typespatial coherence is assumed for numerical calculation of interference pattern.

Next, let us consider diffraction by grating: a periodic slit array. As shown in Fig. 1.4, the beam
deflection angle at a grating with a period of d is described as sin θ = nλ/d, where n = 0, ±1, ±2, ±3,,,,,.
The integer n is also called order of diffraction. The n = 0 is undiffracted beam.

4n = +3

θn = +1

d

n=0
n =-1

λ

n = -3λ = d sin θ
Fig. 1.4. Linear grating and diffraction pattern by the grating. The intensity distribution (diffraction
pattern of grating) is given by the Laue function as sin2(πNdsinθ)/sin2(πdsinθ), where N is number ofgrating period. Even order diffractions, n = 2, 4, 6,,, are not allowed for 1:1 ratio of obstacle andtransparent area, and higher order diffractions are generally weaker than the fundamental diffraction.

When the grating period varies as d = λ/sinθ, as shown in Fig. 1.4, all the diffracted rays arefocused at a point, where θ is determined by tanθ = r/f, and the r is radius of circular grating, and f is a
specific distance from zone plate.

rθfFig. 1.5. Diffraction at Fresnel zone plate (schematic diagram), and focus of x-ray beam.

The specific length f is called focal length. Assuming θ << 1 (this condition is usually satisfied in
the hard x-ray region), diffraction condition, dθ ~ λ, for n-th ring can be written as(rn+1 - rn-1) rn / f ~ λ,where, θ ∼ rn /f, and d ~ (rn+1 - rn-1). Then, regarding the above equation as a differential equation of2∂r(n)/∂n = λf / r(n).

5By solving the above equation, we can get an equation for Fresnel zone plate structure as

rn2 ~ nλf + Constant,generally Constant = 0, and rn = (nλf)1/2.

This is only an approximation, because the above discussion is valid only for large n and small diffraction
angle. However, this intuitional explanation is very useful for understanding the geometricalcharacteristics of FZP optics.
As is understandable from these considerations, there are positive and negative orders ofdiffraction. Thus, the zone plate acts as a convex lens for visible light, and it simultaneously works as aconcave lens with negative order diffraction. Furthermore, 0th order and higher order diffractions usuallyexist. Therefore, a kind of spatial filtering is indispensable in FZP optics to work as an optical lens. A
typical method of selecting a required order of diffraction is putting a small diaphragm near the focal
point. When the diameter of aperture is sufficiently smaller than the FZP diameter, the undesired order ofdiffractions including direct beam are suppressed enough. If the suppression is not sufficient, an
additional beam stop disc may be used for diffraction order selection as shown in the figure. This beamstop disc is usually added in fabrication process of FZP, and it is called center stop, or center obstacle.
The aperture near the focus is called order-selecting-aperture (OSA), or sometimes order-sorting-aperture.
When a central beam stop of diameter r0 is used, the zone plate structure can be modified as

rn 2 ~ n λ f + r0 2 .

Above formula is widely used for general FZP that is fabricated together with central beam stop disc.
Then, the first zone stars from r0, and the inner radius of second zone radius is r1 = (λf + r02) 1/2.
Order-selecting Apperture
Center Beam Stop

Fresnel Zone Plate

Focus

Fig. 1.6. Diffraction-order-selection in FZP microbeam optics.

1. 2. More precise treatment of FZP optics

6

The exact equation of FZP structure is obtained by optical path difference and expansion of
Fermat’s principle. The Fermat’s principle is that an actual optical path between any two points should beshorter than any other paths that connect the two points. By considering the finite wavelength of λ, the nth zone boundary of a Fresnel zone plate that focuses the spherical wave emitted from point A to the point
B is expressed by the following equation,(Ra + Rb) = nλ/2 + (a + b),

as shown in Fig. 1.7, and(a2 + rn2) 1/2 + (b2 + rn2) 1/2 – (a +b ) = nλ/2,(1.1)

where a is a distance from point A to the FZP, b is a distance between B and FZP, and rn is n-th boundaryof zone structure, and n is integer. This formula is a basic and exact equation of Fresnel zone structure.
Expanding with rn and neglecting higher order terms, the equation (1.1) is rewritten as

a + rn2/2a + b + rn2/2b – (a + b) = nλ/2.

(1.2)

Then,

rn2/a + rn2/b = nλ.

(1.3)

Again, by defining 1/a + 1/b = 1/f, the equation (1.3) is rewritten as

rn2 = nλf.

(1.4)

The equation, 1/a + 1/b = 1/f, is well-known Newton’s lens equation. Thus, an FZP whose zone boundary
is defined by the above equation works as a lens with a focal length of f.

When the incident beam is plane wave, the equation (1.1) is simplified as(f2 + rn2) 1/2 – f = nλ/2,

(1.5)

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Then, the zone structure is represent by

rn2 = nλf + n2λ2/4.

(1.6)

Comparing with eq.(1.4), there is an additional term, n2λ2/4, which represents spherical aberration. The
most of FZPs are generally designed as the spherical aberration is negligible for f >> nλ/2. Then, thenumber of Fresnel zone is restricted by this condition.

Ra

Rb

rn

A

B

a

b

Fig. 1.7. Optical path description of Fresnel zone plate

1.3. Diffraction efficiency of FZPThe diffraction efficiency is an important characteristic of diffraction-based optics, because theintensity of focused beam is mainly determined by the diffraction efficiency, and the diffractive optics
such as FZP or grating can gather only a portion of incident radiation. The diffraction efficiency of FZP isequivalent to the linear grating. Therefore, we can calculate it by solving one-dimensional Kirchhoff’s
integral for single period of gating. For instance, first order diffraction of opaque and transparent zone
structure with 1:1 zone ratio, the intensity of diffracted beam is given by

I = |E|2, and E = ∫ C * exp (ik x sinθ) dx,

Here, the integral is taken for one cycle, d, of grating, and 2π/k = λ = d sin θ. The constant C = 1 fortransparent area, and C = 0 for opaque zone. Then, for the first-order diffraction,

E = ∫ exp (2πi/d) dx = 2

By normalizing with the incident beam intensity of (2π)2, I = 4/(2π) 2 (=1/π2 ~ only 10% for 1st-orderdiffraction). The general form for n-th order diffraction can be easily calculated by the similar manner. It
is known that the n-th order diffraction efficiency for transparent and opaque zone, i.e., black and white8

zone, with even zone width (1:1 zone ratio) is 1/(nπ) 2, where n = ±1, ±3, ±5,,,,. The even order diffraction
does not exit for the grating with 1:1 groove width.
More efficient grating is realized by the phase-modulated structure that is a utilization of phaseshift through the zone, instead of stopping the beam with opaque zone. The maximum diffraction
efficiency is attained at a optical path difference of half wavelength (phase shift of π). The efficiency of nth order diffraction for ideal phase grating of 1:1 zone ratio is 4/(nπ) 2 (n = ±1, ±3, ±5,,,,. ). So, the firstorder diffraction has an efficiency of about 40% in the case of ideal phase-modulated zone plate.
However, in the x-ray region, pure phase medium without absorption dose not exist. All the optical media
are complex of phase and absorption effect.
More efficient FZP can be made by using so-called kinoform structure, that is recently introducedin hard x-ray regions. Optimized kinoform FZP can give nearly 100% efficiency, if the absorption loss
can be ignored. Real diffraction efficiency is lower than 100%, but the diffraction efficiency of higher
than 50% is already achieved in the hard x-ray region with quasi-kinoform structure.

Fig. 1.8. Schematic diagram of kinoform zone plate. The structure and characteristics is essentially thesame as Fresnel lens in the visible light region.

1.4. Fabrication of zone plates:
Most of zone plate for x-ray microscope is fabricated by using a state-of-art technology developed
in semiconductor technology, i.e., electron-beam lithography technique. Recent ULSI technology makes
it possible to fabricate sub-100 nm microstructure on a silicon tip. This is also a key technology in x-ray
microscopy. Major vender of x-ray FZP is NTT Advanced Technology (http://www.ntt-at.com/prdsvc/xray_euv_opticssystem.html) and Xradia (http://www.xradia.com/). At present, x-ray FZPs with anoutermost zone width of about 25 nm are commercially supplied by either company. Schematic drawingof cross sectional structure is shown in Fig. 1.9. The zone plate pattern is drawn by electron-beam to theresist on thin membrane on silicon wafer. The thin membrane is usually a few µm-thick silicon nitride orsilicon carbide. The pattern transfer from photo-resist to zone material is done by reactive dry-etchingprocess or by wet electroplating process (usually, dry-etching is used for tantalum, and electroplating is

9

employed for gold pattern). Finally, the silicon wafer of pattering area is removed by chemical etchingprocess, as shown in the figure.

Zone Plate
Supporting Membrane
Silicon Wafer

Fig. 1.9. Structure of Fresnel zone plate. Cross-sectional view and SEM micrograph.

The difficulty on fabrication of x-ray zone plate is high aspect ratio (zone-height/zone-width) ofzone structure at the marginal zone area. In the hard x-ray region, the phase shift is much greater than theabsorption contrast even for the high-Z elements. Therefore, most of the x-ray FZPs is designed as aphase-modulation zone plate. Assuming a free electron approximation, the real part of index for refraction,n, for x-rays is written as

Re(n ) = 1 – δ,

δ = 1.35 x 10-6 ρ(g/cm3) λ (Å)2,

where ρ is density of material, and Z/A = 1/2 is also assumed (Z: atomic number, A: atomic weight). The
optimum thickness of zone plate, t, is equivalent to the optical path difference of λ/2, that is calculated bytδ = λ/2. Then, high dens material is preferable for zone plate. Considering the manufacturing process of
microstructure, gold or tantalum is usually chosen as a zone material for hard x-ray FZP. The optimized
thickness and δ for tantalum zone plate, for instance, is 2.9 µm and 1.71 x 10-5 at an x-ray wavelength of
1.0 Å, respectively. Then, the aspect ratio should be 29 for the FZP having 100 nm-zone-width. The
required aspect ratio is still challenging even in the present nano-fabrication technology. The resolution

10

limit of FZP (width of outermost zone) is being improved year by year, and the finest zone structure ofabout 20 nm is possible at present. However, the thickness of these high resolution FZPs is far from ideal
value for hard x-rays.

112. Making microfocus beam with Fresnel zone plate

2.1. How large is the limit of focused beam size? Diffraction limit and geometrical limit ofresolution
The microbeam optics is to demagnify the light source image with an optical lens or mirror.
Therefore, the generated microbeam size is firstly determined by the magnification ratio and source size,i.e. geometrical definition. However, there is another significant limitation on beam focusing, that islimitation due to diffraction of wavefield.
Ultimate spatial resolution (focused beam size limit) of imaging optics is determined by thediffraction of light. The diffraction-limited resolution is deduced from the uncertainty principle inquantum mechanics. If a photon is squeezed within a small space of <Δx>, the momentum uncertainty ofphoton, <Δp>, is given by following formula,<Δx> <Δp> ≥ h/2π,

where h is the Plank constant. Consequently, larger spread of momentum is required to focus (localize) a
photon in a smaller area. This is a fundamental limitation of microfocus beam size. The momentum ofphoton is given by the formula, p = h/λ, and the uncertainty of momentum is related to the angular spreadof momentum. As shown in the figure, when a parallel incident beam is focused by a lens, the angularspread of focused-photon’s momentum becomes (h/λ)sinθ, and momentum uncertainty becomes Δp = ±(h/λ)sinθ. Then the limit of focused beam size, Δx, can be written as

Δx ≥ 2h/(2πΔp) ~ λ/(πsinθ),

The term, sinθ is known as numerical aperture (NA) of lens. The F-number defined by F = 1/tanθ isusually used in optical instruments instead of NA. In the hard x-ray region, the NA << 1, and sinθ isapproximately equal to rN/f, where the rN is radius of outermost zone.
This discussion is one-dimensional condition, and we cannot get an exact form of intensity
distribution. The lens optics, including FZP optics, is an axis-symmetric system. For the axisymmetric
optical systems, the formula of resolution limit is slightly deformed as∆x = 0.61 λ/NA,

12This definition is well known Rayleigh’s criterion for spatial resolution of microscope with an objective
lens of circular aperture. This formula was obtained by a classical diffraction theory for electromagnetic
wave, far before the birth of quantum physics, but essentially the same as that derived from quantum
mechanics. This relation comes from a definition of quantum photon momentum, p = h/λ.
Lens
NA = sinθθ
P = h/λΔP= P sinθ = h/λ sinθFig. 2.1. Numerical aperture of objective lens.

The diffraction limit can also be explained by a simple diffraction theory for single slit, as
described below. When a parallel and monochromatic beam is incident on a slit with an opening of w (onedimensional case), the propagating wave through the slit is not a parallel wave anymore.
Using the Kirchhoff-Huygens’ principle, the electromagnetic wave field is given by an integral formula ofE(θ) = ∫ exp (ik x sinθ) dx

Here, k is a wave number defined by k = 2π/λ. Taking a integral over the slit opening, the amplitude ofelectromagnetic wave far from the slit is given by the following function.

E = sin (kw/2 sinθ) / (kw/2 sinθ).

The function (sinX)/X is the sinc function which frequently appears in the diffraction theory. The beam
intensity far from slit takes maxima on the optical axis (θ = 0), and first zero (local minimum) at(kw/2 sinθ) = π,or

λ = w sinθ ~ wθ,

because w is usually much greater than wavelength in the hard x-ray region. Thus, the angular spread(blurring) of the propagating wave is broadened by passing a slit. It is equivalent that a finite angularconvergence of incident beam is required to focus the beam within a limited area. Saying differently,
focusing a photon beam is always accompanied by a finite divergence of outgoing wave. This result of the13diffraction theory of electromagnetic wave is essentially the same as the uncertainty principle in thequantum physics.

θw

Intensity distribution

Fig. 2.2. Diffraction by a single slit. Radiation through the slit is broadened by propagating in a freespace.

It is also apparent from the diffraction by a grating that the sinθ is equal to λ/d, where d is theoutermost zone period of FZP. Then, for the circular grating (FZP without center stop), the diffractionlimited resolution is defined as

Δx = 0.61d.

A formula of Δx = 1.22 drN is frequently used instead of above equation, where drN is a half pitch of theoutermost zone period (or width of outermost zone). Then, the ultimate limit of the focused beam size isnearly equal to the width of outermost Fresnel’s zone of the objective FZP. This relation can also bederived from the Fresnel zone plate equation and the formula of diffraction-limited resolution.
However, the actual focus is generally limited by the quality of incident beam. We have implicitly
assumed a parallel beam for explaining how the FZP works as a beam-focusing device. But, in any lightsources, the emitted radiation is neither a perfect plane wave, nor a perfect spherical-wave. The radiation
is emitted from a finite source area, and is emitted in random direction, a chaotic source! In this case, thebeam focusing should be considered as generation of a demagnified image of the light source. This
limitation of focused beam size is known as the geometrical optics limit. This operation of microfocusing
optics is similar to that of telescope optics. As shown in Fig. 2.3, the light source image is formed at focus
by a magnification factor of b/a according to the Newton’s equation. Then the focused image size, Sb, isgiven by

Sb = Sa b/a,14

where Sa is source dimension, a is source to lens distance, and b is lens to focus distance. Usually a >> b,then b ~ f. A long distance from source point or a small source size is necessary to achieve small (micro)
focus beam. The factor Sa/a is sometimes called angular size of light source. In most synchrotron
radiation facilities, the angular size is not small enough to generate microfocus beam, and the distance islimited by beamline length. So, we need to makeup a virtual source by putting a small pinhole (or slit) at
the upstream section of the beamline. The slits equipped at beamline, front-end slit or transport-channel
slit, are usually used as the virtual source for micro-focusing experiment. It is also important that there are
no disturbances of wave front by the optical elements in the beamline. The monochromator crystals,
mirrors, and vacuum windows usually tend to give serous deformation of the x-ray beam direction and the
wavefront shape.
f
Sa
Sb
a

b

Fig. 2.3. Geometrical description of beam focusing optics

2.2. Depth of focus and chromatic aberrations
If the wavelength is slightly different from designed value, the focus slightly moves on the opticalaxis. That means different wavelength x-rays are focused at a different position. The focal length f varies
inversely as the wavelength of x-rays. These phenomena are called chromatic aberrations. Therefore,
monochromatic x-ray beam is required for FZP microbeams. Saying differently, the FZP microbeam
optics can be used as a monochromator or spectrometer. From a practical point of view, an importantfeature of microbeam optics is a tolerance of focal point, e.g. the distance where the beam size is nealy
equal to that at the exact focus. This tolerance is named depth of focus.
Geometrical beam size at a distance of Δf from the exact focus can be calculated to be 2NA Δf, as
shown in Fig. 2.4. This value should be smaller than the diffraction-limited resolution in order to achieve
the full-performance of FZP optics. Then the condition can be written as

0.61 λ/NA ≥ 2NA Δf,Considering both side from focus, the depth of focus is written as15

Δf ~ ±0.3 λ/(NA)2.

or using the FZP equation, we can get a simple and useful formula, as

0.61/(2N) ≥ Δf/f.

Here, N is number of Fresnel zone.
The chromatic aberration is also estimated by similar manner, because Δf/f = -Δλ/λ from the FZP
equation. So, the chromatic aberration is characterized by the number of Fresnel zone, N.

NA = sin θ
θ

2NAΔf
Δf

Fig. 2.4. Depth of focus

2.3. Preservation of emittance and brilliance
An important law, emittance preservation rule, in beam-focusing optics is described in this section.
As shown in Fig. 2.5, when a magnified (or demagnified) image of light source is formed by an opticalelement, there is a relation between the image dimension, Sa, Sb, and angular spread, θa, θb as

Sa θa = Sb θb .

The product of beam dimension and angular spread, Sθ, is called the beam emittance, and the above
formula represents the emittance preservation rule. This emittance preservation is deduced by thegeometrical optics. The efficiency of optical system is not 100%, even if a perfect lens (without any
aberrations) is used. This is because the focus beam size is limited by the diffraction. Therefore, theequation should be rewritten as

Sa θa ≤ Sb θb , for general optics.

If the absorption loss at optical system is ignorable, the flux of focused beam, Ib, is equal to the flux of the16incident radiation, Ia, as

Ib ≤ Ia
and Ib = Ia for ideal case.

Then, flux, I, divided by the source dimension, S, and the angular divergence, θ, is constant as

Ib, / (Sb θb) = Ia / (Sa θa), for the ideal optics.

This relation is called brilliance preservation rule, and the flux density limit of focused beam isdetermined by this equation. The flux density of the focused beam is generally higher than that of theunfocused beam, where the unfocused beam is a microbeam generated simple by putting a micro-pinholein front of sample. Although, the beam intensity seems to increase by focusing, there is no real gain ofbeam brilliance. The brilliance is always preserved in any ideal optical systems, and the actual brilliancealways decreases by passing through the optical element, because the throughput, reflectivity of mirror ordiffraction efficiency of gating and FZP, is always less than 100%. Remember that any optical systems
decrease the brilliance, and increase the emittance. The highest brilliance is obtained only at the initialsource point! This is an important principle of beam focusing optics.

Lens

Light Source
Sa

θa

Focus
θb

a

Sb

b

Fig. 2.5. Schematic diagram of emittance preservation rule2.4. General description of focusing optics
In the above sections, we focused only on the Fresnel zone plate optics for x-ray focusing.
Even in the x-ray regions, various types of beam focusing, or imaging, devices are developed, such as
total reflection mirrors, multilayer mirrors, Bragg Fresnel lenses, multi-layer Laue lenses, etc. These
optical elements are, however, able to be described by a simple form of extended Fermat’s principle.
His (Fermat’s) idea is this: That out of all possible paths that it might take to get from one point toanother, light takes the path which requires the shortest time (the least possible time),
And an application of the Fermat’s principle is that,17

We want to bring all the light back to what we call a focus. How? If the light always takes the path ofleast time, then certainly it should not want to go over all these other paths. The only way that the light
can be perfectly satisfied to take several adjacent paths is to make those times exactly equal! (Lectures on
Physics by R. Feynman) Then, with wavelength λ of light, all the different paths with optical path
difference of nλ are also allowed as shown in Fig. 2.6.
The Pn represents ellipse with foci F1 and F2. The ellipse is an ensemble of points that havethe property that the sum of the distances from two points is a constant. A plane ellipse is used for onedimensional focusing, and ellipse of rotation (ellipsoid) for 2D focusing. A single ellipse or ellipsoid , forexample, represents total-reflection-mirror optics. All kinds of beam focusing optical system can beunderstood using the formula ofPn = P0 + n λ ,where n is integer, and λ is wavelength. P0 is an ellipse with foci F1 and F2. Now, the Fresnel zone plate
can be regarded as an approximation of ellipsoids that consists of only the pints on a plane S1 that isnormal to the optical axis. The S2 and S3 represent Bragg Fresnel lens. The S2 also represents anapproximation of self-focusing grating. The total reflection mirror system is represented by a shell of
ellipse or ellipsoid. The multi-layer Laue lenses and multi-layer mirrors can also be understood as a threedimensional structure that consists of ensemble of ellipses or ellipsoids. Therefore, all the optical devices
utilizing reflection or diffraction, except for refractive lens, are only an approximation of the nesting
ellipse shown in the figure. The ellipse and ellipsoid should be replaced by parabola and paraboloid forparallel beam or a point source at infinite distance, and hyperbola and hyperboloid for convergent beams.

Fig. 2.6. General description of focusing optics.

18

3. Application of FZP optics to imaging microscopy and microtomography

3.1. X-ray imaging microscopy
The limitation of spatial resolution for conventional imaging system such as projection imaging is
usually determined by spatial resolution of imaging detector. The development of high-resolution
detectors, spatial resolution of µm – nm, is very difficult task in the hard x-ray region. Although, highresolution photo-resists for x-ray lithography have spatial resolution better than 100 nm, they cannot beapplied to general use, because linearity of the photo-resist is insufficient, and real-time image-processing
is usually required in microscopy.
There is another limitation of projection imaging. Geometrical optics approximation is assumed inprojection imaging and contact microscopy. This assumption is generally valid for the medical imaging
instruments because of short wavelength (~0.2 Å) and low spatial resolution (~mm). However, even in
the hard x-ray region, diffraction at the object cannot be ignored for high-resolution imaging. The spatial
resolution of measured image is deteriorated by the diffraction of x-rays that occurs within the object
itself. Image blurring due to Fresnel diffraction of x-rays cannot be neglected in the high-resolution
imaging. The amount of blur for Fresnel diffraction is approximately (λL)1/2, where λ is x-ray wavelength.
The L represents the distance between sample and detector in the case of contact microscopy, and it
corresponds to the distance from source point to sample in the case of projection imaging with a sphericalwave from small point source, i.e. cone beam projection imaging with high magnification. When λ = 1 Åand L = 1 cm, the image blur becomes 1 µm. Therefore, high-resolution imaging is difficult, if simple
projection imaging scheme is used. Microscopy in combination with the holographic image
reconstruction is provably only an exception.
The use of x-ray imaging optics is one of the best ways to overcome this problem. By using the xray high-magnification imaging-objective optics, the influence of Fresnel diffraction and limitation ofimaging detector can be solved. Then, the ultimate limit of spatial resolution is thickness of specimen.
Image blurring caused by the beam deflection inside of the object is now expressed as (λL)1/2, where the L
now represents the thickness of specimen. This limitation is related to the depth of focus of objective lens.
The zone plate optics is widely used in the hard x-ray microscopy, and spatial resolution of better than 50
nm is already achieved in the hard x-ray region with the FZP objective. The imaging property of FZP
objective is very suitable for hard x-ray microscopy, because off-axis aberrations of FZP optics areusually negligible in hard x-ray region..

3.2. Order selection by spatial filter and illuminating optics for FZP objective19

As described in the previous section, the FZP cannot be used as conventional refractive lensbecause of existence of undesirable diffractions. A simple way of diffraction order selection is shown in
Fig. 3.1. The object and objective is illuminated by a planar wave with a cross-section limited by adiaphragm, and only a half of FZP objective is illuminated. Then, positive order diffraction and negative
order one is well separated at the image plane, and the direct beam appears at the boundary between thepositive diffraction and negative diffraction. The optical system shown in Fig. 3.1 is called off-axis
illumination. The configuration shown in the figure cannot separate higher order diffraction. Additional
spatial filters are needed for perfect order-selection. However, higher order diffraction has much low
efficiency and large magnification at the image plane. Therefore, higher order diffraction is usuallyignorable. The serious problem of off-axial illumination is non-uniformity of imaging property, i.e. strongedge-enhancement in the marginal region of image field. Therefore, illumination and spatial filtering
combining with condenser optics is commonly used in imaging microscopy with FZP objective. There are
typically two types of illumination optics for imaging microscopy, critical illumination and Köhler’s
illumination.
The schematic diagrams of critical illumination and Köhler illumination are shown in Fig. 3.2a and3.2b, respectively. The critical illumination, however, is not suitable for synchrotron radiation light source,
because the synchrotron radiation light source has small source dimension and narrow divergent angle. In
the critical illumination, a demagnified image of the source is formed at the object plane by the condenser
lens. Then, the typical field of view becomes only a few µm – submicron in the general synchrotron
radiation beamlines. Therefore, in the critical illumination, additional technique, such as scanning thecondenser FZP during the exposure time, is employed for attaining uniform field of view. On the other
hand, the condenser optics for the Köhler’s illumination does not generate the demagnified image. A
constant angle deflection is done by the condenser lens optics, and the deflected beams are merged
together in the field of view of objective lens. The Köhler illumination results in a hollow cone beam
behind the objective, as shown in the figure. The condenser optics for Köhler’s illumination can berealized by using a constant period zone plate with a center beam stop (not a Fresnel zone plate) or byusing a rotating double-mirror mechanism. It is important that, using the hollow cone illumination, theorder selecting is also achieved simultaneously. Only the first order diffraction appears in the center circle
defined by the hollow cone beam.

20

-1st diffraction

Direct
FZP+1st diffraction

Fig. 3.1. Order-selecting for plane wave illumination. Only a half of FZP aperture is illuminated.

Beam stop
ObjectCondenser lens

Image

Objective FZPFig. 3.2a Critical illumination with condenser zone plate and objective zone plate. Hollow cone
illumination is used for diffraction order-selecting.

Beam stop
ObjectImage

Objective FZPCondenser lensFig. 3.2b Köhler’s illumination with condenser zone plate. Hollow cone illumination is also achieved by
combination with a center beam-stop and an aperture in front of object.

3.3. Spatial resolution of FZP microscopy with illuminating optics
Spatial resolution of imaging optics is essentially determined by the diffraction of light. That is thesame as that of microfocusing optics. The diffraction-limited resolution for microscope with an objective

21

lens of circular aperture (axis-symmetric optics) and incoherent illumination (for example, spontaneouslight-emitting object) is expressed by the well-known Rayleigh’s criterion as∆ = 0.61 λ/NA,(3.1)

Where NA is so-called numerical aperture of objective lens defined as NA = sinθ. Here, the θ is the angle
between marginal ray and optical axis, as shown in Fig. 3.3. It is also apparent from the diffraction of
linear grating that the sinθ is equal to the λ/d, where d is the outermost zone period of FZP. Then,∆ = 0.61d.

A formula of Δ = 1.22 drN is frequently used, where drN is a half pitch of outermost zone period (or widthof outermost zone). This is the same as that of micro-focusing optics.
Thus, the spatial resolution is, in principle, limited by the numerical aperture of objective lens, andthe limitation is simply described by the diffraction theory as same as that of microbeam optics. However,when condenser lens optics is used, the spatial resolution is determined not only by the NA of objective
but also by the convergent angle of illuminating beam. This is called matching of numerical aperture.
When the convergent angle is equivalent to the NA of objective lens, the spatial resolution is equal to thatdefined by the equation (3.1). The best resolution is achieved, when the NA of condenser is 1.5 times thatof the objective lens. In this condition, the spatial resolution becomes ~0.57λ/NA. When the condenser
optics is not used, i.e. in the case of plane wave illumination (coherent illumination), the spatial resolution
is written as 0.82λ/NA instead of 0.61λ/NA, as well.

NA = sin θθFig. 3.3. Numerical aperture of objective lens.

3.4. Depth of focus
Hard x-ray microscopy has an advantage that thick samples can be observed in high spatial

22

resolution for its high tansmissivity and depth of focus. Although the depth of focus in hard x-ray
microscope is generally much greater than that of optical microscope, the restriction by depth of focus issometimes not negligible. The depth of focus can be evaluated by the geometrical optics, as shown in Fig.
3.4. When a point source is placed at a distance of D from the exact object plane, the image size extracted
at the object plane is expressed as 2DNA. The amount of defocusing, 2DNA, should be less than or equal
to the diffraction-limited resolution in order to neglect the influence to spatial resolution. By taking theboth side of object plane into account, the depth of focus, 2D, is expressed by

2D = 0.61λ/NA2.

(3.2)

Differently speaking, the NA of objective determines the ratio of field of view and spatial resolution. For
an example, when λ = 1 Å, and NA = 10-3, the diffraction-limited resolution is 61 nm, and the limit ofsample thickness is 61 µm in diameter. It should be noted that the restriction on the depth of focus
expressed by the equation (3.2) is essentially the same as the ultimate limitation on spatial resolution for
finite sample thickness, ∆ ~ (λL)1/2, where L is thickness of sample.

NA = sin θ
θ
2NAD
D

Fig. 3.4 Depth of focus. Black line represents primary focus, and blue line is marginal path of defocusedrays.

The depth of focus is also defined by using Rayleigh’s quarter wavelength criterion. The λ/4 rule isthat, when the optical path discrepancy from the primary path is within one quarter of wavelength, theinfluence of wave-front aberration to the spatial resolution may be ignored. As shown in Fig. 3.5, when an
initial focus is ideal, the wave front at objective is exact sphere with a radius of f. Then, appling cosine
law to the marginal ray track,

L2 = f2 + D2 + 2Df cosθ.

(3.3)

23

Eq (3.3) can be written as

L2 – (f + D) 2 = 2Df (cosq –1).

And, then{L – (D + f)}{L + (D + f)} = 2Df (cosθ –1).

Ignoring higher order, {L + (D + f)} ~ 2f, and (cosθ – 1) ~ -θ2/2. Therefore, eq (3.3) can be rewritten as

L – (D + f) ~ -Dθ2/2.

(3.4)

The length (D +f) is optical path length of primary ray, and L is optical length of marginal ray for the
defocus point at a distance D from initial focus. Therefore, the λ/4 rule is described as|L – (D + f)| ~ Dθ2/2 < λ/4.

The angle θ is approximately equal to the numerical aperture NA, because θ << 1 in the x-ray region.
Therefore, by taking the both side of object plane into account, the depth of focus, 2D, is now expressedby

2D ~ λ/NA2.

(3.5)

This evaluation of depth of focus is slightly different from the value given by eq (3.3). However, thesedefinitions are essentially the same. The definition of eq (3.5) is generally used in microscopy optics.

24

L
θ
f
D
NA = sin θ

Objective Lens

Fig. 3.5 Depth of focus defined by Rayleigh’s quarter wavelength criterion. Red line indicates ideal wave
front from point image at focus. Black lines represent primary focus, and blue lines are marginal path ofdefocused rays.

3.5. Chromatic aberration and spherical aberration
In above discussion, perfectly monochromatic x-ray beam is assumed. However, real x-ray beam
has some bandwidth, even if the crystal monochromator is used. The influence of the finite bandwidth ofincident x-ray spectra is called chromatic aberration. The defocusing effect by the chromatic aberration isevaluated by the similar manner as that of depth of focus, asΔf NA < 0.61λ/NA.

(3.3)

Here, Δf is a displacement of focus caused by chromatic aberration. The focal length is defined by rn2 =nλf. Here, the rn is constant for given FZP, and the focal length, f, is proportional to 1/λ. Then, thedifference of focal length Δf is written asΔf /f = Δλ/λ,(3.4)

where Δλ is bandwidth of incident radiation. Then, the equation (3.3) can be rewritten as

Δλ/λ f NA < 0.61λ/NA.

(3.5)

By using approximation of NA ~ rN/f and rN2 = Nλf, where N is the zone number of outermost zone, a25

useful formula is deduced as follows,

Δλ/λ < 0.61/N.

This formula gives a tolerance on bandwidth of incident x-ray beam for FZP microscope.

The spherical aberration comes from the approximation used to derive the equation (1.2). By using
second higher order term, the equation is approximately rewritten as

rn2/2a – 1/8rn4/a3 + rn2/2b –1/8rn4/b3 = nλ/2.

(3.6)

In the case of imaging microscope, a << b, and a ~ f. Then, the equation (3.6) can be written as

rn2/2a + rn2/2b –1/8rn4/f 3 = nλ/2.

(3.7)

In order to evaluate the wave-front aberration, a Rayleigh’s quarter wavelength criterion can be used. By
comparing the equation (1.2) and equation (3.7), the λ/4 rule for the spherical aberration is expressed by

1/8rN4/f 3 < λ/4.

(3.8)

Using the fundamental structure of Fresnel zone plate, rN2 = Nλf, the relation is rewritten as

1/2N 2 < f / λ.

(3.9)

This formula also gives a limitation on the number of outermost zone as well as the chromatic aberration.
Although the spherical aberration can be perfectly erased by using the exact formula of equation (1.1),
these zone plates can be used only at the fixed wavelength (λ) and fixed optical distances (a and b). TheseFZPs do not have flexibility. The FZPs defined by the conventional formula of equation (1.4) are muchconvenient for practical use.

3.6. Off-axis aberration theory based on wave optics
In this section, off-axis imaging property of FZP microscope is discussed. The optical pathequation is written as26

{(ra – rn cosφ)2 + (rn sinφ) 2 + a2}1/2 + {(rb + rn cosφ) 2 + (rn sinφ) 2 + b2}1/2= nλ/2 + (a2 + ra2 ) 1/2 + (b2 + rb2) 1/2,

(3.10)

where ra/a = rb/b, as shown in the figure, and rb/ra (= b/a) represents the geometrical magnification of
microscope. The φ is angle at a point of Fresnel zone plate objective as shown in Fig. 3.5. The influence
to spatial resolution is evaluated by the similar manner as that on the spherical aberration. By using thesecond order approximation, the equation (3.10) is rewritten as

1/2(ra 2 – 2 ra rn cosφ + rn 2)/a - 1/8(ra 2 – 2 ra rn cosφ + rn 2) 2/a3
+ 1/2(rb 2 + 2 rb rn cosφ + rn 2)/b - 1/8(rb 2 + 2 rb rn cosφ + rn 2) 2/b3
= nλ/2 + 1/2(ra 2/a) - 1/8(ra 4/a3) + 1/2(rb 2/b) - 1/8(rb 4/b3).

(3.11)

Then, the wavefront aberrations are analyzed by the optical path equation (3.11), and the Rayleigh’s
quarter wavelength rule is expressed as|{- ra rn cosφ/a + rb rn cosφ/b} +1/2{ rn 2/a + rn 2/b –nλ}-1/8{(ra 2 – 2 ra rn cosφ + rn 2) 2/a3 - ra 4/a3 + (rb 2 + 2 rb rn cosφ + rn2) 2/b3 - rb 4/b3}|< λ/4.

(3.12)

The first term represents the magnification equation of ra/a = rb/b, and the second term corresponds to thebasic zone plate equation of rn2 = nλf. Then, the residual aberration is rewritten as|1/8{(-4 ra3 rn cosφ/a3 + 2 ra2 rn2/a3 + (2 ra rncosφ)2/a3 – 4 ra rn3cosφ)/a3 + rn4/a3)+ (4 rb3 rn cosφ/b3 + 2 rb2 rn2/b3 + (2 rb rncosφ)2/b3 + 4 rb rn3cosφ)/b3 + rn4/b3) }|< λ/4.

(3.13)

The aberration that includes first order of rN can also be ignored, because the residual aberration is written
as|-1/8{-4 ra3 rN cosφ/a3 + 4 rb3 rN cosφ/b3}| < λ/4.

This term is always zero, because ra /a = rb /b.

27(3.14)

The residual off-axis aberration that consists of square of rN is extracted from the (3.13) as|1/8{(2 ra rNcosφ)2/a3 + 2 ra2 rN2/a3 + (2 rb rNcosφ)2/b3 + 2 rb2 rN2/b3)| < λ/4.

(3.15)

By using –1≤ cosφ ≤ +1, and a << b and a ~ f, the following expression is derived.

ra2rN2/f 3 < λ/3.

(3.16)

Concerning to the third order of rN, following formula is extracted from the (3.13).

|1/8{(– 4 ra rN3cosφ)/a3 + (4 rb rN3cosφ)/b3}| < λ/4.

(3.17)

This expression may also be reduced as

ra rN3/f 3 < λ/2.

(3.18)

Concerning the fourth order term of rN, the residual aberration is reduced as|1/8(rN4/a3 + rN4/b3)| < λ/4.

(3.19)

This formula is also approximated as

1/8rN4/f 3 < λ/4This is the spherical aberration discussed in the previous section.
These formulas give a restriction on the maximum field of view for a given FZP to achieve a diffractionlimited resolution. By comparing (3.16) and (3.18) with the relation (3.2), it is apparent that the above
conditions are generally well satisfied, when the field of view is less than the depth of focus. Therefore,
from the point of view of three-dimensional observation, the Fresnel zone plate can be used as an
objective lens for x-ray microscope without considering the off-axis aberrations, and the FZP imaging
microscopy is applied to imaging microtomography in SPring-8

28

B
rb

b

a
φ

rn

ra

A
Fig. 3.5 Schematic diagram of off-axis optical path for FZP imaging optics.

3.7. Typical example of imaging microtomography with Fresnel zone plate objective in SPring-8

Typical experimental setup is schematically shown in Fig. 3.6. This microscopy/micro-tomography
setup is installed at the beamline 47XU of the SPring-8. The light source is an in-vacuum planar
undulator with permanent magnet array. The undulator radiation is monochromatized by liquid-nitrogencooling silicon 111 double crystal monochromator. The x-ray energy of 8 keV is selected in thisexperiment. The bandwidth of monochromatic x-ray beam through the crystal monochromator isestimated be ∆λ/λ ~10-4. Therefore, the monochromaticity is sufficient for the zone plate objective.
Condenser optics is a specially designed zone plate with a constant grating pitch of 400 nm. Thezone material is 1 µm-thick tantalum, and the diameter of zone plate is 500 µm. This condenser zone plateis used in combination with a center beam stop with a diameter of 300 µm in order to achieve hollowcone illumination. The circle of least confusion is located at about 500 mm downstream of the condenser
zone plate. Then, by placing the object at the circle of least confusion, a quasi-Köhler illumination
condition is satisfied. In front of the object, a pinhole with a diameter of 100 µm was placed as an orderselecting-aperture for the condenser zone plate. The field of view is actually defined by this aperture. A
beam diffuser made of charcoal powder (grain size of a few tens µm, and rotating at 600 rpm) is also
placed upstream of the condenser zone plate for the purpose of reducing strong speckle noise due to highcoherence of incident x-ray beam.
The objective zone plate is also made of tantalum with a thickness of 1 µm. The outermost zone29

width is 100 nm and the focal length is 100 mm at 8 keV. The diameter of object zone plate is 155 µm,
and the number of zone is 388. Although the numerical aperture (convergent angle) of condenser optics isa half of that of objective lens, the influence to the spatial resolution is not serous in this case. An X-ray
imaging detector is used to measure the three dimensional data at single scan. The imaging detector
consists of a fine powder phosphor screen (P43, Gd2O2S:Tb), relay lens and cooled CCD camera(C4880-41S, Hamamatsu Phtotonis KK, Japan). A high-precision rotating stage made at Kohzu Precision
Co. is used for sample rotation in tomographic measurement. The wobbling of rotation axis is measured
to be ±70 nm within 360º rotation.

Rotating diffuser for Speckle Noise Reduction
Center stop
Diaphragm

Objective FZP

X-ray
Condenser lens Sample rotation(constant-pitch zone plate)
Beam stop

Imaging detector
Image

Diaphragm
ObjectCondenser lensObjective FZP

1/f = 1/a + 1/bfbaFig. 3.6. Schematic diagram of experimental setup for imaging microtomography.

30diffuser
Beam stop

pinhole

FZP

X-ray
CZP

sampleFig. 3.7. Experimental setup. Condenser optics, sample stage and objective FZP are shown in the figure.
The imaging detector is located 7 m from the objective FZP, right-hand side out of the figure.

31

Useful references for general understanding of x-ray optics
R. Feynman, R. Leighton and M. Sands: Lecture on Physics (Addison-Wesley),
M. Born and E. Wolf, Principles of Optics, (Cambridge University Press)
For specific topics on Fresnel zone plate optics (in soft x-ray regions),
D. Attwood, Soft X-rays and Extreme Ultraviolet Radiation, Principles and Applications (Cambridge
University Press).
A. G. Michette, Optical Systems for Soft X-rays (Plenum Press).

Other references,N. Kamijo, et al., Proc. of 8th Int. Conf. X-ray Microscopy, IPAP Conf. Series 7 (2006) 97-99.
B. Nieman, et al., AIP CP507 (2000) 440-405.
Y. Suzuki, et al., Jpn. J. Appl. Phys. 40 (2001) 1508-1510.
Y. Suzuki, et al., SPIE Proceedings 4499, (2001) 74-83.
Y. Suzuki, et al., Journal de Physique IV France 104 (2003) 35-40.
Y. Suzuki, Jpn. J. Appl. Phys. 43 (2004) 7311-7314.
Y. Suzuki, et al., Jpn. J. Appl. Phys. 44 (2005) 1994-1998.
Y. Suzuki and H. Toda, Advanced Tomographic Methods in Materials Research and Engineering, ed. John
Banhart, Oxford University Press, (2008), Section 7.1 Fresnel zone plate microscopy andmicrotomography.
Y. Suzuki, et al., X-Ray Optics and Instrumentation Volume 2010 (2010), Article ID 824387.
H. Takano, Y. Suzuki, and A. Takeuchi, Jpn. J. Appl. Phys. 43 (2003) L132-L134.
A. Takeuchi, et al., Jpn. J. Appl. Phys. 40 (2001) 1499-1503.
A. Takeuchi, Y. Suzuki, and H. Takano, J. Synchrotron Radiation 9 (2002) 115-118.
A. Takeuchi, et al., Rev. Sci. Instrum. 73 (2002) 4246-4249.
A. Takeuchi, K. Uesugi, and Y. Suzuki, Journal of Physics: Conference Series 186 (2009) 012020.
S. Tamura, et al., J. Synchrotron Radiation (2002) 9, 154-159.
S. Tamura, et al., Vacuum 83 (2008) 691-694.
K. Uesugi, A. Takeuchi, and Y. Suzuki, (2006), Proc. of SPIE 6318, 63181F.
D. Weiß, et al., Ultramicroscopy 84 (2000) 185-197.

32

Basic exercises, before the experiment
1. Calculating the radius of some rings of the FZP in Fig. 1.2, and total number of zone.
2. Calculate the focal length of the FZP in Fig. 1.2 for 8 keV x-rays.
3. Calculate the exact focal point for a given source to FZP distance (depending on the beamline optics).
--> For example, source to FZP distance is 46 m (BL47XU)
4. Calculate the geometrical spot size, assuming a beamline where you will do the microfocusing
experiment.
--> Source size of SPring-8 is ~ 40 µm vertically and ~ 600 µm horizontally.
5. The monochromaticity of x-ray beam is sufficient for the FZP in Fig. 1.2 at your beamline (with aconventional crystal monochromator)?
--> Look at the optics of beamline from WEB or ask to beamline staff.
6. Calculate the depth of focus for the FZP shown in Fig. 1.2.

7. Calculate the optimized thickness of zone for 8 keV x-rays. Assume the zone material of gold or
tantalum.
8. Drawing the sinc function and Laue function for a given slit and grating.
--> 10 µm slit, 1 Å wavelength, and 100 line grating.
9. Introducing a relation between required monochromaticity (λ/Δλ) and number of zone (N), using theLaue function for N-period grating.
10. Which kind of pin-hole should be used as the order-selecting-aperture? Diameter, material andthickness, considering work distance of optics.
--> Once, use 0.5 center stop disc.
11. Calculate the zone plate parameter of the FZP in Fig. 1.2 with a center stop of 50 µm-diameter.

33

Practice at the beamlines – microfocus beam
1. General understanding on the beamline optics.
Configuration of beamline 20XU is shown below as an example.

Front-end Cross-slit (virtual source point)
MonochromatorFZP
OSA (20-30 µm)
Si 111
E/ΔE ~ 7000
~ 210 m
245 m from source pointf

Experimental Setup of Micro-focusing at BL20XU

2. Finding FZP and order-selecting-aperture, using x-ray imaging detectors.
3. Alignment of FZP and OSA on optical axis.
4. Searching for the fundamental (first-order) diffraction.
5. Finding the focal point by means of Foucault’s knife-edge test.
Far-field image varies as shown below.
OSA

OSA
Knife-edge

Knife-edge

Focus

Focus

FZP

Upstream of Focus

Downstream of Focus

34

6. Measurement of beam size by differential knife-edge scanning method, and comparison to theoreticalvalue.

7. Changing the front-end slit size, dependence of focus beam size on spatial coherence condition.

8. Only at BL47XU, measurement of stigmatism for light source, and elimination of the stigmatism byanti-bending of pre-mirror.

Practice at beamline – imaging microscopy with parallel beam illumination

1. Setting the instruments
Spatial filter in front of sample, FZP, and imaging detector,
2. Searching for the FZP. Observing the diffraction from FZP, and separating the positive, negative anddirect beams by the spatial filter.
Beam Stop
SampleFZP
Imaging
Detector

Saptial Filter(Slit Blade)

3. Put a He path between FZP and imaging detector, and setting a direct beam stop in front of the imaging
detector.
4. Observing some test objects, and how we can adjust the focus?
5. Imaging properties. Change the beam coherence by using a diffuser in front of object.

35

Advanced exercises,
1. Calculate the diffraction efficiency, and comparison to the experimental result.
2. Calculating and experimentally testing of chromatic aberrations, how is the tolerance of wavelengthand bandwidth?
3. Theoretical estimation and experimental test of the depth of focus by changing the position of knifeedge.
4. Influence of mask in front of FZP, shading half of FZP aperture. Both vertical and horizontal focusing
properties. This experiment must be done for the diffraction-limited resolution case.

5. Influence of mask in front of FZP, center obstacle. This experiment also requires diffraction-limited
resolution.

6. Inserting something in the optical path, how serious is the disturbance of wave-front, for instance,
putting a sheet of paper, Kapton foil, aluminum foils, etc?
7. Inclination of FZP and focusing properties, how is the tolerance for angular misalignment?
8. Focusing properties of higher order diffractions, how is the diffraction-limited resolution, anddiffraction efficiencies?
9. The longitudinal source size of SPring-8 undulator is 4.5 m. This length of light source gives someproblem on focusing properties?
10. Focusing properties for off-axis undulator radiation. How large is the effective angular size ofundulator radiation? and its relation to inclination to optical axis (electron orbit). This experiment should
be done at the short beamlines. The BL47XU is preferable.
11. At the undulator beamlines, the x-ray source is very long. SPring-8 standard undulator has alongitudinal length of 4.5 m. When a demagnified image of source is generated by an optical lens, thefront end and the rear end of undulator, for example, may yield different foci. It is actually problem?

36