A multi-prism lens for hard X-Rays

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A Multi-Prism Lens for Hard X-Rays

Bj¨ orn Cederstr¨ om

AKADEMISK AVHANDLING

som med tillst˚ and av Kungl Tekniska H¨ ogskolan framl¨ agges till oﬀentlig granskning f¨ or vinnande av teknologie doktorsexamen fredagen den 8 november 2002

kl. 9:15i Kollegiesalen, Administrationsbyggnaden, KTH, Valhallav¨ agen 79, Stockholm

Avhandlingen f¨ orsvaras p˚ a engelska

Kungl Tekniska H¨ ogskolan

Stockholm 2002

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Bj¨ orn Cederstr¨ om: A Multi-Prism Lens for Hard X-Rays

ABSTRACT

This thesis describes a new type of refractive lens for hard x-rays. It is shown that a linear array of prisms, slightly inclined with respect to the optical axis, will form a line focus at a certain distance from the lens. Hence, the name multi-prism lens.

These lenses are free from spherical aberration and are similar to planar parabolic compound refractive lenses in terms of performance. However, they distinguish themselves in that only planar surfaces need to be fabricated. A special feature is that the focal length can be easily varied by adjusting the inclination angle.

Theoretical calculations, based on geometrical and physical optics, are used to characterize the lenses. Aberrations are discussed, as well as the sensitivity to fabrication imperfections, and insufficient flatness is identified as a potential problem. Ray-tracing is used to test the approximations and assumptions used in the theory. Applications in x-ray microscopy and mammography are discussed.

Lenses have been made of beryllium, silicon, epoxy and diamond using different methods. Results from measurements of surface roughness and figure error show that the imperfections of the silicon and epoxy lenses should have a small impact, while the beryllium lenses should suffer from strong scattering. Experiments were performed at the European Synchrotron Radiation Facility and sub-µm focal line widths, close to theoretical expectations, were measured for silicon and epoxy lenses at 30 keV and 14 keV, respectively. Insertion gains up to 40 were reached. Two crossed lenses were used to obtain focusing in two dimensions and a point focus.

The smallest measured focal spot size was 1.0 µm by 5 .4 µm, and an insertion gain exceeding 100 was achieved using epoxy lenses.

The diamond lenses suﬀered from voids in the material formed in the chemical vapor deposition process, but nevertheless provided focal lines less than 2 µm in width, albeit at at relatively low insertion gain of 13. Due to their excellent ther- mal properties, these lenses are put forward as candidates for optics at the next generation ultra-high-intensity synchrotron beams and x-ray free electron lasers.

Descriptors: x-ray, optics, refractive, lens, mammography, synchrotron.

Bj¨orn Cederstr¨om 2002c ISBN 91-7283-385-8 ISSN 0280-316X

ISRN KTH/FYS/--02:43--SE TRITA-FYS-2002:43

Printed by Universitetsservice US AB 2002

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5 .7.2 Experimental evaluation . . . . 5 2 5 .7.3 Conclusions . . . . 5 3 6 Diffraction and Resolution 55 6.1 Scalar diffraction theory . . . . 5 5 6.2 Kirchhoff’s diffraction theory . . . . 5 5 6.3 Gaussian aperture . . . . 5 6 6.4 Fraction of a Gaussian aperture . . . . 5 7 6.5Point spread function and image formation . . . . 5 8 6.6 Diffraction limited focusing . . . . 5 8 6.6.1 Minimal focal line width . . . . 5 9 6.6.2 Maximal insertion gain . . . . 60

6.6.3 Dependence on focal length, material and energy . . . . 60

7 Imperfections of the Multi-Prism Lens 63 7.1 Surface roughness and scatter . . . . 63

7.2 Aberrations . . . . 66

7.2.1 Lens bow . . . . 66

7.2.2 Aberration due to discrete lens proﬁle . . . . 68

8 Material Selection and Lens Fabrication 73 8.1 Introduction . . . . 73

8.2 Fabrication of beryllium lenses . . . . 74

8.3 Fabrication of silicon masters . . . . 75

8.4 Molding of epoxy replicas . . . . 77

8.5 CVD diamond lenses . . . . 77

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CONTENTS 3

9 Lens Metrology79

9.1 Flatness measurements . . . . 79

9.1.1 Silicon lenses . . . . 79

9.1.2 Epoxy lenses . . . . 79

9.1.3 Beryllium lenses . . . . 80

9.1.4 Diamond lenses . . . . 81

9.1.5 Conclusions . . . . 82

9.2 Surface roughness measurements . . . . 83

9.2.1 Silicon lenses . . . . 83

9.2.2 Epoxy lenses . . . . 84

9.2.3 Beryllium lenses . . . . 84

9.2.4 Conclusions . . . . 85

10 Synchrotron Beam Measurements 87 10.1 Lens attachment and alignment . . . . 87

10.2 Experimental setup and methods . . . . 88

10.3 Measurement results: 2001 run . . . . 89

10.3.1 CCD measurements . . . . 89

10.3.2 Knife-edge measurements . . . . 92

10.3.3 Conclusions . . . . 93

10.4 Measurement results: 2002 run . . . . 93

10.4.1 Vertical focusing . . . . 93

10.4.2 Horizontal focusing . . . . 94

10.4.3 2-D focusing . . . . 95

10.4.4 Comments and conclusions . . . . 98

11 Application to Mammographic Imaging 101 11.1 Introduction . . . 101

11.2 Breast composition and geometry . . . 101

11.3 Assessment of the delivered dose . . . 102

11.4 Optimal energy for a monochromatic beam . . . 102

11.5 Spectral eﬃciency . . . 103

11.6 Improved spectral eﬃciency with the lens . . . 106

11.7 Acquisition time reduction . . . 106

11.8 Conclusions . . . 107

12 Sub-Diﬀraction Focusing 109 12.1 Towards Fresnel x-ray lenses . . . 109

12.2 Intensity equalization . . . 109

12.2.1 Calculation of lens proﬁles . . . 110

12.2.2 Calculation of transmission loss . . . 111

12.2.3 Calculation of resolution improvement . . . 111

12.2.4 Numerical example and practical considerations . . . 112

12.3 Conclusions . . . 113

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13 Conclusions 115

13.1 Comparison to other methods . . . 115

13.2 Current state of refractive x-ray optics . . . 116

13.3 Final conclusions and outlook . . . 116

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Chapter 1

Introduction

1.1 Background

The present work was started within a project on digital mammography with par- ticipation from the Royal Institute of Technology in Stockholm, Sweden and the Lawrence Berkeley National Laboratory in Berkeley CA, U.S.A. The goal is to replace existing analog screen-film technology with a completely digital system, counting the x-ray photons one-by-one. This is achieved by placing silicon strip de- tectors in an edge-on geometry to provide sufficient depth of interaction. Each strip is wire-bonded to high-speed parallel-processing application-specific integrated cir- cuits (ASICs), which are readout from a personal computer. The collimated beam is scanned across the breast to obtain a full-field image. Such a system over- comes many of the shortcomings of the existing analog techniques, such as high background from scattered radiation, low absorption efficiency and noise from the photon-to-light conversion process [1].

The obvious drawback of a scanning system is the poor photon-economy, i.e.

the ratio of photons actually used in the image formation versus photons created in the x-ray tube. This results in high tube load and/or long image acquisition time. Since there is a time constraint in mammography, due to patient discomfort during breast compression and movement artifacts, the solution is to use a multi-slit system. This, of course, adds to complexity and cost.

A way to improve the photon economy would be to increase the x-ray ﬂux on the detector by focusing the beam. However, before refractive focusing was demonstrated there was no obviously viable way of achieving this. The focusing geometry is shown in Figure 1.1 with approximate dimensions. Note that focusing only takes place in the vertical direction and that the beam is unaﬀected in the horizontal direction. The focal length should be of the order of a few decimeters.

5

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50 µm 100 µm

90 cm

Focal line X-ray source Lens

Figure 1.1: Outline of the focusing geometry relevant for the scanned-slit mam- mography system. Dimensions are approximate.

1.2 Purpose

The initial objective of this thesis was to evaluate if refractive x-ray optics could be used in a scanning x-ray imaging system for digital mammography. This work led to the design and fabrication of a parabolic compound refractive lens. Sub- sequently, the idea for a new type of refractive lens, the multi-prism lens (MPL), was developed. Thus, the objective was shifted towards more basic x-ray optics re- search. This includes theoretical calculations, design, fabrication and experimental veriﬁcation of these new lenses. In addition, the original purpose has been retained.

1.3 Methods

A theoretical framework for calculating the focusing properties of this new type of lens has been developed. These calculations are based on geometrical and physical optics. Ray-tracing programs have been developed to investigate aberrations and approximations made in the analytical calculations. Parabolic compound refractive lenses and multi-prism lenses have been fabricated in plastic substrates, beryllium, silicon and diamond. The proﬁle and surface characteristics have been measured with confocal microscopy, interferometric methods and scanning needle proﬁlome- try, from which predictions can be made about the x-ray optical performance.

An experimental setup to measure the insertion gain has been built. This comprises a micro-focus x-ray tube, mechanical stages and a cadmium-zinc-telluride solid-state detector connected to a multi-channel analyzer, all of which is controlled by a personal computer. Furthermore, the lenses have been tested at a dedicated optics testing beam line at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France.

Analytical calculations and Monte-Carlo results are used to evaluate the lens in a mammography context.

1.4 Organization ofthe material

There are at least two sensible ways to organize the material in this thesis. The

most obvious one is to make the presentation as coherent and logical as possible,

group all related material and progress gradually, beginning with simple schematics

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1.5. CONTRIBUTION BY OTHERS 7

and idealized conditions and thereafter reﬁne the analysis and include more details.

However, there are several ways of doing this. Should the different kinds of lenses be the top hierarchical structure or should they be sub-groups of different parts like theory, fabrication and measurements? The second way, a more chronological arrangement, also has its virtues. The gradual progress in fabrication is easier to follow, as well as the refinement of the experimental setup and conditions. These two principles work hand in hand most of the time, although sometimes they collide.

I have tried to follow the former as much as possible, but in some occasions the latter has taken priority. I may not have found the optimal way of presenting this material. However, I think this is rather a question of personal taste than scientiﬁc practice.

The thesis has the following organization. The ﬁrst chapter you are already familiar with. Chapter 2 describes some historic background to the subject, the discovery of R¨ ontgen, brief notes on the development of x-ray science in the ﬁrst decades of the 20

^th

century and the ﬁrst attempts to refract and focus x-rays. In ad- dition a background on the modern development of refractive x-ray optics is given.

The topic of Chapter 3 is x-ray interaction with matter, and the concept of a refrac- tive index is derived from Maxwell’s equations. Chapter 4 describes the parabolic refractive x-ray lens in all aspects, including fabrication and measurements in a laboratory setup. The main topic of this thesis, the multi-prism lens, is introduced in Chapter 5. This includes basic theory and ray-tracing. The first vinyl LP lens is also fully included in this chapter, since both manufacturing and the measurements are very different than for the lenses made later. In Chapter 6 we go beyond the limits of geometrical optics and use physical optics to calculate diffraction effects and resolution. A closer look at aberrations, both intrinsic and caused by fabrica- tion imperfections, is taken in Chapter 7. Surface roughness and scattering are also discussed. The selection of lens materials and manufacturing issues are dealt with in Chapter 8 and the following chapter describes measurements of lens profiles and surface roughness. The impact from these imperfections on the lens performance is calculated and discussed. In Chapter 10 the experimental evaluation at a syn- chrotron beam line is described in detail and all results are summarized. The next chapter concerns a possible applications - mammography. In Chapter 12 an idea to improve the resolution of refractive x-ray lenses is described. Chapter 13 closes the thesis with some notes on comparison to other kinds of x-ray optics, including the current state of refractive optics, conclusions and outlook.

1.5 Contribution by others

Where not otherwise stated, the material in this thesis is contributed by the author.

I would like to point out some exceptions and acknowledgments though. The idea to split the parabolic compound refractive lens in two halves to facilitate manufac- turing stems from Prof. David Nygren at Lawrence Berkeley National Laboratory.

The etch-process used to fabricate most of the multi-prism lenses was proposed and

performed by Carolina Ribbing at Uppsala University. She also carried out most of

the proﬁlometry measurements. The two rounds of measurements at the European

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Synchrotron Radiation Facility were done by Mats Lundqvist, Carolina Ribbing

and the author.

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Chapter 2

Historic Background

2.1 R¨ ontgen’s discovery

X-rays were discovered in 1895by the German physicist Wilhelm Conrad R¨ ontgen [2], a discovery that awarded him the ﬁrst Nobel prize in physics in 1901 [3]. As many great discoveries, it was accidental in nature, although it took a man of R¨ ontgen’s intellect to understand the importance of it and study it systematically.

While experimenting with cathode rays (which we now know as electrons), R¨ ontgen discovered that a paper plate covered on one side with barium platinocyanide placed in the path of the rays became ﬂuorescent even when it was as far as two meters from the discharge tube

¹

. Since the cathode ray tube was sealed in a light-absorbing material, and even blocking the path with thick materials would not stop the ﬂuo- rescence, he concluded that this was a new type of radiation. He named it “x-rays”, due to its unknown nature. In many languages (including Swedish and German) the radiation has come to bear his name. R¨ ontgen, who was a very modest and humble man and never became accustomed to his fame, did not want this honor, however.

R¨ ontgen’s discovery quickly became front-page news all over the world, due to the spectacular shadowgraphs of parts of the human body that were routinely taken only months after his ﬁrst presentation. Not an advocate of popular science in general, R¨ ontgen was displeased of the coverage and the wide-spread obsession of the anatomical shadowgraphs. For him, this was just a means to investigate the true physical nature of the new rays.

The medical community realized the importance of x-rays immediately. Already in 1896, the ﬁrst angiography was performed, and the same year saw the new ﬁeld of x-ray therapy be born. Not until a few years into the 20

^th

century were the harmful aspects of x-rays realized. For many pioneers in the ﬁeld, this was too late, and several suﬀered burns, amputations or death.

In his systematic investigations of the x-rays, R¨ ontgen performed several re- fraction experiments. From the failure to see any eﬀect from various prisms, he

1Most material in this section taken from Refs. [4, 5].

9

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concluded that the index of refraction of x-rays must deviate from unity with less than 0.05(he actually writes “...could not be more than 1.05” [2], since the idea of a negative deviation was probably alien to him). Glass and hard rubber lenses also proved ineﬀective. R¨ ontgen also interpreted the failure to see any scattering from pulverized metals as evidence that the reﬂectance of x-rays was very low. He concludes that “...x-rays are not identical with cathode rays, but they are produced by the cathode rays in the glass wall of the discharge apparatus”.

2.2 Understanding ofx-rays takes form

The importance of the development of the understanding of x-rays is evident from the many awards to the field in the first decades of Nobel Prize history. 1914’s prize was awarded to Max von Laue for his work on x-ray diffraction by crystals [6]. The following year the laureates were W. H. Bragg and W. L. Bragg (father and son) for determination of crystal structure by x-ray diffraction [7]. C. G. Barkla discovered the characteristic radiation of x-rays from elements (Nobel Prize 1917) and verified experimentally the polarization of x-rays. It is worth mentioning that Barkla and Bragg had a long-standing debate whether x-rays were waves or corpuscles.

Moseley managed to bring order to the ﬂora of characteristic lines and set up his famous formula in 1914. In so doing he “discovered” several unknown elements, which were missing in his plots. He would certainly had shared the prize with Barkla, had he not been killed in battle in the ﬁrst World War.

The great experimentalist Manne Siegbahn was awarded the Nobel Prize in 1924 for his refinement of x-ray spectroscopy [8]. Stenstr¨ om found a small deviation from Bragg’s law when studying reflections from crystals [9]. He interpreted this as a refractive effect in the surface of the crystal. To explain the data, the index of refraction would have to be smaller than unity. This would lead to total external reflection of x-rays for small grazing angles, which was also experimentally verified by A. H. Compton shortly after Stenstr¨ om’s measurements [10]. With knowledge of the magnitude of the refractive index, the group of Siegbahn was the first to succeed in refracting x-rays with a prism [11].

Compton discovered in 1922 the important eﬀect that bears his name (Nobel Prize 1927). This was conceptually troublesome at a time when most physicists had been convinced of the wave-theory of x-rays. de Broglie solved this elegantly two years later when he postulated the particle-wave duality in his doctoral thesis [12] and the rapid development of quantum mechanics in the 1920’s laid things on a solid theoretical foundation.

2.3 Early thoughts on refractive x-ray optics

Refractive x-ray focusing was again considered in the literature by Kirkpatrick in

1949, but was dismissed as being impracticable and ineﬃcient [13]. The author

photographed the K-line spectra of molybdenum with a converging prism having a

concave curvature of the emergence face. Few details are given, however. He con-

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2.4. THE COMPOUND REFRACTIVE LENS 11

cluded that because of the disadvantages of high absorption and strong chromatic aberration, x-ray lenses would probably be generally inferior to mirror systems.

It lasted almost half a century before the issue was revitalized, when Suehiro et al. proposed to use a single concave lens to focus the x-ray beam from the new generation high-brilliance synchrotron facilities [14]. Such a device would have a focal length of 50 to 100 m. The approach was immediately ruled out by Michette who gave three reasons against it [15]:

1. The diﬀraction-limited resolution is poor due to the small possible aperture.

2. Design is diﬃcult, since the optical constants are not known well enough.

3. The lens has to be fairly short to avoid absorption losses, which limits the achievable spatial resolution

²

.

A thorough treatise on the theory of Fresnel and refractive lenses was published by Yang [16], in which he concludes about refractive lenses that “...its application is severely limited by available fabrication capabilities today”, and continues to promote Fresnel lenses instead.

2.4 The compound refractive lens

The turning-point came in 1996 by the introduction of the Compound Refractive Lens (CRL) by Snigirev et al. [17]. The idea was to multiply the refractive eﬀect of a single lens by simply putting several in a row. This was realized by drilling 30 holes with a radius of 300 µm in a block of aluminum, see drawing to the left in Fig.

2.1. The focal length was thus reduced by a factor of 30 to give the feasible focal length of 1.8 m. A 150 µm wide source 36 m away from the CRL was demagniﬁed to 8 µm full width at half maximum (FWHM), close to the theoretical demagniﬁcation factor s

i

/s

o

= 1/20. The measured profile is shown to the right in Fig. 2.1. A gain of flux of about 3 was achieved. This was significantly lower than the theoretical value due to large absorption in the bridges between the holes.

This marks the beginning of feasible refractive x-ray optics and it is also where this thesis departs.

2This actually seems to be identical to the ﬁrst paragraph.

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Figure 2.1: Left: Schematic diagram showing the principle of the Compound

Refractive Lens. Right: Measured proﬁle of the beam at the focus. The theoretical

width is 7.5 µm. (Reprinted from Ref. [17].)

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Chapter 3

Hard X-Ray Interaction with Matter

This chapter includes the main steps of arriving at an index of refraction for mate- rials. It is of course valid for electro-magnetic radiation of any wave-length, but we will examine the peculiarities of hard x-ray energies.

¹

In this energy region, Comp- ton scatter must also be included. Approximate formulas for the total cross-section and real part of the index of refraction are ﬁtted to tabulated values.

3.1 X-ray interactions from Maxwell’s equations

A semi-classical description of the interactions of x-rays with matter takes its stand from Maxwell’s equations, applicable to any kind of electromagnetic radiation. We will not go into detail and lengthy calculations, but emphasize the main results and the reasoning behind them. For details, see [18, 19, 20]. By a few basic operations on the Maxwell equations, the vector wave equation can be formed:

∂

²

∂t

²

− c

²

∇

²

E(r, t) = − 1

0

∂J(r, t)

∂t + c

²

∇ρ(r, t)

, (3.1)

where E, J and ρ are electric ﬁeld, current density and charge density, respectively.

c ≡ 1/√

0

µ

0

is the phase velocity of an electromagnetic wave in vacuum. Solv- ing this equation with appropriate source terms will eventually give us a complex refractive index, with a refractive and an absorptive part. Consider now a bound electron undergoing harmonic oscillation, for which the equation of motion can be

1There is no generally accepted definition of soft vs. hard x-rays, and the distinction is somewhat arbitrary. The line tends to vary from at least 1 to 10 keV among authors, and seems to be somewhat dependent on field. To the flora of definitions the author would like to add the following: Hard x-rays are x-rays for which refractive optics can be utilized. This definition is certainly not more arbitrary than many others. The disadvantage is that many statements in this thesis (including the title) would be simply tautological.

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written

m d

²

x

dt

²

+ mγ dx

dt + mω

_s²

x = −eE, (3.2)

where γ is the damping and ω

s

the resonant frequency. For a sinusoidal driving force, the solution is

x(r, t) = e m

1 ω

²

− ω

s²

+ iγω E(r, t) (3.3) The current density is a sum of the velocity of all electrons in the atom times the charge and atomic number density

J

0

(r, t) = −en

a

X

s

g

s

v

s

(r, t) = − e

²

n

a

m

X

s

g

s

ω

²

− ω

s²

+ iγω

∂E(r, t)

∂t , (3.4) where

P

s

g

s

= Z and g

s

is called the oscillator strengh. This we can regard as the number of electrons in each orbital in this simple model. Now we can insert the partial derivative of the current density with respect to t into Eq. 3.1 and move some terms

"

1 − e

²

n

a

0

m

X

s

g

s

ω

²

− ω

s²

+ iγω

!

∂

²

∂t

²

− c

²

∇

²

#

E

_T

(r, t) = 0. (3.5)

This can be rearranged into the standrad form of a wave-equation

∂

²

∂t

²

− c

²

n

²

(ω) ∇

²

E

T

(r, t) = 0, (3.6)

and we identify

n(ω) =

"

1 − e

²

n

a

0

m

X

s

g

s

ω

²

− ω

s²

+ iγω

#1/2

= 1 − e

²

n

a

2

0

m

X

s

g

s

ω

²

− ω

²s

+ iγω , (3.7) where √

1 + x ≈ 1 + x/2, x 1 was used. This is a very good approximation for hard x-rays. We introduce the atomic scattering factor

f

⁰

(ω) =

X

s

g

s

ω

²

ω

²

− ω

s²

+ iγω ≡ f

1⁰

(ω) − if

2⁰

(ω) (3.8) Superscript 0 indicates that this is the scattering factor in the forward direction.

Since the wave-length is of the same order as the distance between the scatter- ing electrons, the angular dependence of scattering is generally complicated, but for forward scattering this does not concern us. We can now write the index of refraction

n(ω) = 1 − n

a

r

e

λ

²

2π f

₁⁰

− if

2⁰

≡ 1 − δ + iβ, (3.9)

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3.2. ATTENUATION LENGTH 15

where the constants were also changed using the classical electron radius r

e

= e

²

/(4π

0

mc

²

). We can now identiy the real (refractive) and imaginary (absorptive) part

δ = n

a

r

e

λ

²

2π f

₁⁰

(ω), β = n

a

r

e

λ

²

2π f

₂⁰

(ω). (3.10)

The special circumstances for the energies and materials considered here is that the wave frequency (energy) is always greater than the resonant frequencies (binding energies) of all electrons. This is the reason why the real part of the index of refraction is smaller than one, which can be seen by looking at the expression for the real part of the scattering factor (Eq. 3.8). Multiply with the complex conjugate and we obtain

f

1⁰

(ω) =

X

s

g

s

ω

²

(ω

²

− ω

²s

)

(ω

²

− ω

s²

)

²

+ γ

²

ω

²

(3.11) Also, if the energy is much greater than all binding energies, the expression simpliﬁes to

f

₁⁰

(ω) ≈

^X

s

g

s

= Z, ω ω

s

, (3.12)

i.e. the electrons are behaving as they were free. f

₁⁰

(ω) can be considered the eﬀective number of scattering electrons of the atom.

The index of refraction changes the wave propagation vector to k = k

0

n and the wave can be written

E(x, t) = E

0

e

^{i(kx ωt)}

= E

0

e

i[(2π/λ)x(1 δ+iβ) ωt]

= (3.13)

= E

0

e

i[(2π/λ)x ωt]

· e

^i(2πδ/λ)x

· e

^(2πβ/λ)x

. (3.14) Changing to time-averaged intensity we get

I(x) = K · |E(x, t)|

²

t

= K

⁰

e

^(4πβ/λ)x

. (3.15) This can be compared to the macroscopic attenuation expression

I(x) = I

0

e

^µx

= I

0

e

^x/l

, (3.16) where µ and l are the attenuation coeﬃcient and attenuation length, respectively.

We thus have

µ = 1 l = 4πβ

λ = 2n

a

r

e

λf

₂⁰

(ω). (3.17)

3.2 Attenuation length

Values of the attenuation length were taken from Ref. [21]. However, for analytical calculation of various dependencies on energy and material, the following ﬁtted function for the cross-section per atom was used (σ in barns, E in keV) [22].

σ

P hoto+Compton

= 24.2·Z

^4.2

E

³

+ 0.56·Z. (3.18)

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The attenuation coeﬃcient can be calculated from

µ = n

a

σ, (3.19)

where the atomic number density, n

a

, is given by n

a

= N

A

·ρ/M. N

A

is Avogadro’s number and M is the molar mass. The attenuation length is plotted in Fig. 3.1 together with the tabulated values for a few materials. The ﬁtted expresion is correct within 5% in the interval Z ∈ [4, 14] and E ∈ [10, 60] keV.

0 10 20 30 40 50 60

10⁻³ 10⁻² 10⁻¹ 10⁰ 10¹

E (keV)

Attenuation length (cm)

Beryllium Diamond Silicon

Figure 3.1: Comparison of tabulated attenuation length from Ref. [21] (markers) and values from Eq. 3.18 (solid lines). Deviation is less than 5% above 10 keV.

From Eq. 3.18 can be derived that the photo-absorption and Compton cross- sections are equal at E/[keV] ≈ 3.5 · Z. In Fig. 3.2 the regions in E-Z space are shown where the respective cross-section is dominating. This we will need when we look at energy and material dependencies later on. We can already now note the following:

Photo-absorption is dominating (ﬁve times larger) if E/[keV] < 2.1 · Z. We then have

µ(Z, E, ρ) ∝ ρZ

^3.2

E

³

. (3.20) Compton scattering is dominating (ﬁve times larger) if E/[keV] > 6.0 · Z, in which case

µ(Z, E, ρ) ∝ ρ, (3.21)

i.e. attenuation is independent on both Z and E. The last two equations should

be used with care. They are only approximate and extreme cases, but nevertheless

provide good qualitative guidance.

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3.3. INDEX OF REFRACTION 17

4 6 8 10 12 14

0 10 20 30 40 50 60

Z

E (keV)

5σ_ph.=σ_C

σ_ph.=σ_C

σ_ph.=5σ_C

Strong photo−absorption domination

Strong Compton domination

Figure 3.2: Plot showing the regions in E-Z space where Compton scattering and photo-absorption is dominating, respectively.

3.3 Index ofrefraction

For a free-electron gas, the real part of the atomic scattering factor f

₁⁰

= Z. For solids, where electrons are bound, f

₁⁰

< Z. However, for the rather high energies (compared to the binding energies) considered here, it is a very good approximation to treat the electrons as free. We can thus write

δ = n

a

r

e

λ

²

2π f

₁⁰

(ω) = r

e

h

²

c

²

2π E

²

ρN

A

M Z ≈ r

e

h

²

c

²

2π E

²

· ρN

A

2Z Z, (3.22) and we note that

δ ∝ ρE

²

. (3.23)

This is a good rule of thumb to remember, but the approximation A ≈ 2Z is

unnecessary, and not good for low-Z elements such as beryllium. Thus, a function

of the form δ = C · E

²

was ﬁtted to tabulated values of δ from Ref. [23]. Table 3.1

shows the resulting expressions. Using these expressions, the decrement of the

index of refraction as a function of energy is plotted in Fig. 3.3 and compared to

tabulated values. The deviation is less than 0.5% above 10 keV.

(22)

Material δ (E in keV) ρ (g/cm

³

) Beryllium 3.41·10

⁴

· E

²

1.85 PMMA 2.68·10

⁴

· E

²

1.19 PVC 3.00·10

⁴

· E

²

1.41 CVD diamond 7.28·10

⁴

· E

²

3.5 Aluminum 5.43·10

⁴

· E

²

2.70 Silicon 4.84 ·10

⁴

· E

²

2.33 Table 3.1: Expressions used to calculate the decrement of the index of refraction from unity (δ) for diﬀerent materials.

5 10 15 20 25 30

10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴

E (keV)

δ

Beryllium Diamond Silicon

Figure 3.3: Comparison of tabulated decrement of index of refraction from Ref. [23]

(markers) and values the expressions in Table. 3.1 (lines). Deviation is less than

0.5% between 10 keV and 30 keV.

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Chapter 4

A Parabolic Refractive X-Ray Lens

4.1 Introduction

The objective of the Parabolic Refractive X-ray Lens was to overcome a few short- comings of the cylindrical compound refractive lens. Essentially there were three problems:

1. Spherical aberration rather than absorption determined the resolution and aperture (and thus gain).

2. A certain amount of material ( ≈ 25 µm) was needed between the holes, oth- erwise the drilling process would destroy the brittle bridges. This of course leads to unnecessary absorption.

3. Aluminum is not a very good lens material since the absorption/refraction- ratio is rather high. Generally this ratio is lower for materials with lower atomic number.

Thus the ideal lens should avoid spherical aberration, have no “land” between holes and be made of a low-Z material. A schematic of an ideal lens is shown in Figure 4.1.

For clarity only 10 out of a few hundred holes are shown.

4.2 Theory ofa parabolic x-ray lens

4.2.1 Geometry

The parabolic equation of the lens surface is x = y

²

2R , x < R

2 , (4.1)

19

(24)

2-8 cm 150−500

µm

∼∼

Figure 4.1: Schematic of an ideal parabolic compound refractive lens with approx- imate dimensions. For clarity only 10 out of a few hundred holes are shown.

as plotted in Figure 4.2. A circle is also shown for comparison. The parabola is cut at x = R/2 to make the lens twice as compact as the circular one. The cut is arbitrary and has no inﬂuence on the performance

¹

.

0 50 100 150 200

0 20 40 60 80 100

x (µ m)

y (µ m)

R=100µm

void

material

Figure 4.2: Illustration of the lens surface and a circular shape for comparison.

The parabolic shape is twice as compact as the circular. The parabola is cut at y=R.

4.2.2 Geometrical optics

Gain for a monochromatic beam

The main parameter to be estimated is the gain, i.e. the ratio of the ﬂux at a slit placed in the focal plane with and without the lens present in the beam. This can be calculated within the framework of geometrical optics as long as the dimensions of the focal spot is larger than the diﬀraction limit. We know that the focal length

1See however Section 7.2 for a note on the impact from surface roughness as a function of the number of holes.

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4.2. THEORY OF A PARABOLIC X-RAY LENS 21

f of a single thin lens is determined by the diﬀerence, δ, between the index of refraction in the lens (air) and that of the surrounding medium, and the radius of curvature R of the lens:

f = R

2δ . (4.2)

An array of N closely spaced lenses gives a device with total focal length F = f

N = R

2δN . (4.3)

We make the assumption that the rays undergo a negligible lateral displacement in passing through the system (we will check this assumption later with ray-tracing).

The rectangular source is located a distance s

1

from the thin lens. The slit through which the x-rays must pass is located a distance s

2

beyond the lens. The beam is focused at s

2

+ ∆.

h

d/2

s₁ s₂

It follows that

1 s

1

+ 1

s

2

+ ∆ = 1

F , (4.4)

and d/2

∆ = h

s

2

+ ∆ . (4.5)

The maximal angle a ray can make with the horizontal and still strike the slit is θ = h

s

1

= d/2 s

1

s

2

1 , (4.6)

where

=

1 s

1

+ 1 s

2

− 1 F

. (4.7)

The absolute value makes the relation valid even if the focus lies in front of the slit. However, h must not be greater than the radius of curvature of the lens, R, in which case the ray would miss the lens entirely. In the absence of the lens, the fraction of the x-rays emitted by the source that would strike the slit would be (we omit the normalization factor of 1/2π, since it will fall away)

I

0

= d s

1

+ s

2

. (4.8)

With the lens present, but with no absorption of the x-rays, this would be increased to

I

lens

= 2θ. (4.9)

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To include absorption we incorporate a factor for the attenuation of the beam passing through the material. The parabolic equation of the proﬁle is

x(y) = 1

2R y

²

. (4.10)

A ray that has lateral displacement y traverses a length of absorbing material equal to

x

tot

(y) = x(y) · 2N = N y

²

R , (4.11)

and is attenuated by a factor

exp(− N y

²

Rl ), (4.12)

where l is the attenuation length in the lens material. Thus the root-mean-square (rms) spread of the Gaussian beam is (using Eq. 4.3)

σ

abs

=

r

Rl 2N = √

F δl. (4.13)

Including absorption, the ﬂux falling on the slit is given by an integral over the angle α of the ray from the source:

I

_lens^abs

=

Z min(θ,R/s1)

min(θ,R/s1)

exp

−Ns

²1

α

²

Rl

dα. (4.14)

If θ < R/s

1

,

I

_lens^abs

=

r

πRl N

1 s

1

erf θs

1

r

N Rl

!

= √ 2πσ

abs

1 s

1

erf

1 2 √

2s

2

d σ

abs

, (4.15)

and if R/s

1

< θ

I

_lens^abs

= √ 2πσ

abs

1 s

1

erf

√ R 2σ

abs

. (4.16)

Here we have used the error function erf(z) = 2

√ π

Z z

0

exp(−x

²

)dx. (4.17)

For perfect focusing, → 0 and thus θ → ∞. Using Eqs. 4.8 and 4.16 we obtain the gain

G(R) = I

lens^abs

/I

0

= √

2π s

1

+ s

2

s

1

σ

abs

d erf

√ R 2σ

abs

. (4.18)

The maximal gain occurs when R → ∞ G

max

= √

2π s

1

+ s

2

s

1

σ

abs

d . (4.19)

This is clearly an unphysical limit. However, the error-function approaches

unity quickly. It is also worth noting that the length of the lens L = RN = R

²

/2δF ,

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4.2. THEORY OF A PARABOLIC X-RAY LENS 23

0 20 40 60 80 100 120 140 160

0 1 2 3 4 5 6

R / µm

Gain (solid) and length (dashed)

Length:Be Length:PMMA Length:Al Gain:Be Gain:PMMA Gain:Al

Figure 4.3: Gain (solid) and length in cm (dashed) of the lens as a function of the aperture of the lens. Source=100 µm, slit=50 µm, E=20 keV, source-to-slit distance=100 cm.

and therefore grows quadratically with the radius for a ﬁxed focal length. The gain and the length of the lens are shown for lenses made of beryllium, PMMA (polymethyl methacrylate, also referred to as Plexiglas) and aluminum in a typical geometry in Figure 4.3. Reasonable values of R would be 150, 100 and 40 µm for Be, PMMA and Al, respectively. Parameters are shown in Table 4.1.

Material σ

abs

(µm) R (µm) N L (cm) Gain

Be 68 150 396 5.9 5.0

PMMA 47 100 3353.4 3.4

Al 18 40 66 0.26 1.3

Table 4.1: Parameters for parabolic refractive x-ray lenses made of Be, PMMA and Al. Source=100 µm, slit=50 µm, E=20 keV, source-to-slit distance=100 cm.

Gain as a function of energyfor ﬁxed parameters

To calculate the gain when we are out of focus we have to generalize Eq. 4.15. An exercise in geometry for a point source a distance y above the horizontal axis yields the result

G(y) =

r

π 2

s

1

+ s

2

s

1

σ

abs

d [ζ(y) + η(y)] , (4.20)

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0 5 10 15 20 25 30 35 40 0

1 2 3 4 5 6

E / keV

Gain

Be PMMA Al

Figure 4.4: Gain as a function of energy for a lens optimized for 20 keV.

Source=100 µm, slit=50 µm, E=20 keV, source-to-slit distance=100 cm.

where

ζ(y) = erf

√ d/2 2σ

abs

s

2

− y

√ 2σ

abs

s

1

, (4.21)

and

η(y) = erf

√ d/2 2σ

abs

s

2

+ y

√ 2σ

abs

s

1

. (4.22)

The gain for a ﬁnite source size is thus calculated by integrating this over the source distribution. Using this expression, gain as a function of energy for the case above is plotted in Figure 4.4. The slope to the left of the peak is steeper since both over-focusing and increased absorption work against us. Towards higher energies the deterioration from under-focusing is mitigated by reduced absorption.

We notice that the curve for Al is somewhat peculiar, since the latter eﬀect is stronger then the former. This is not a general observation, however, and it should be noted that the shape of the curve is a strong function of the geometry.

Gain as a function of the material properties

From Figure 4.3 and Table 4.1 it is obvious that aluminum is a bad choice of lens material. Let us look somewhat more carefully at this issue. We saw in Eq. 4.19 that the intensity gain is proportional to the eﬀective lens aperture, given by σ

abs

=

√ F δl, which gives us the functional dependence on the pure material properties δ (refractive index) and l (attenuation length). This quantity is maximized by choosing a material with as low an atomic number as possible. Fig. 4.5shows √

δl

as a function of energy for Be, PMMA and Al, respectively.

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4.2. THEORY OF A PARABOLIC X-RAY LENS 25

0 10 20 30 40 50 60

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8x 10⁻⁴

E / keV (δ⋅ l)1/2 /m1/2

Be PMMA Al

Figure 4.5: Combination of material properties proportional to gain as a function of x-ray energy. Curves are shown for Be, PMMA and Al, respectively.

As shown in Chapter 3 we have l = 1

µ ∝ Z

ρσ , (4.23)

and we also know that δ ∝ ρE

²

. Note that the product δl is independent of the material density. The optimum energy for a given Z can be calculated from (using Eq. 3.18)

d

dE (δ · l) = 0 ⇒ E

opt

= 2.78·Z

^1.07

keV. (4.24) Characteristic for this energy is that the cross-section for photo-absorption is twice the cross-section for Compton-scattering. For Be, PMMA and Al, the optimal energies are 12 keV, 19 keV and 43 keV, respectively.

It is also interesting to keep the energy ﬁxed and study the dependence on the atomic number Z. We can look at two extreme cases.

1. Low energy, Compton scattering can be neglected:

µ ∝ Z

^3.2

⇒ G ∝ √

δ · l ∝ Z

^1.6

(4.25)

2. High energy, photo-absorption can be neglected:

µ indep. on Z ⇒ G ∝ √

δ · l indep. on Z (4.26)

We see this Z-dependence in Fig. 4.6. For 5keV the gain decreases strongly

with Z, whereas for 60 keV the curve is almost ﬂat. Thus, at low energies the

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4 6 8 10 12 14 0

0.2 0.4 0.6 0.8 1 1.2x 10⁻⁴

Z (δ⋅ l)1/2 [m1/2 ]

5 keV 30 keV 60 keV

Figure 4.6: Combination of material properties proportional to gain as a function of the atomic number. Curves are shown for 5, 30 and 60 keV, respectively

choice of lens material is critical, but for higher energies it is less important. For energies ranging from a few keV to about 20 keV, beryllium shows signiﬁcantly better performance than PMMA. For energies between 20 and 40 keV, PMMA seems like a good choice. For even higher energies aluminum is feasible, perhaps even preferable, since the length of the lens can be kept down due to the higher refraction in aluminum.

Of course, this analysis is just one aspect of the choice of lens material. In the broad perspective other factors, such as availability of manufacturing techniques, heat tolerance and material and fabrication cost, must be taken into account.

4.3 Ray-tracing

4.3.1 Validity ofassumptions and approximations

In the geometrical optics treatise we made some assumptions and approximations, which are not obviously valid. For example, can each individual lens really be treated as a thin lens? A focal length of a “thick” lens can be written (Ref. [24], section 6.1)

1 f = (n

l

− 1)

1 R

1

− 1 R

2

+ (n

l

− 1)d n

l

R

1

R

2

, (4.27)

where the last term is the correction to the thin lens equation (also known as the

lensmaker’s formula). Assume −R

1

= R

2

= R = 100 µm. The thickness of the

lens at the optical axis, d, is equal to R in our case. The relative correction will

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4.3. RAY-TRACING 27

thus be (using n

l

− 1 = −δ)

−δd/n

l

R

²

−2/R = δd

2n

l

R = δ

2 + O(δ

²

), (4.28)

which shows that the thin lens approximation is correct to a very high accuracy.

Perhaps more worrying is the assumption that the N individual lenses are “closely spaced”, or more precise, that the rays undergo a negligible lateral displacement when passing through the compound lens. To a ﬁrst approximation this should be true if L F . However, this condition is not always valid and it takes some more investigation to trust the approximation.

4.3.2 Ray-tracing algorithm

Writing a ray-tracing program for this type of lens is a straight-forward task. Each individual lens is treated as a unit with transfer function F, and tracing through the lens is a simple iteration of this function. The geometry is described by the following parameters: C = radius of curvature, y

a

= half the aperture, and , which describes the lenticular shape. The program is general and can handle circular, elliptical, parabolic and hyperbolic shapes. To calculate the attenuation, the path length in the lens material has to be summed-up for each ray, i.e. the thick lines in the ﬁgure.

(1 −

²

)x

²

− 2Cx + y

²

= 0

= 0 : circle 0 < < 1 : ellipsoid

= 1 : parabola

> 1 : hyperbola

(ok, y

out

, α

out

, ξ) = F(y

in

, α

in

, δ)

y_in

y_out

α

_out

α

in

y_a

0 x

y

-y_a

Calculating the intersection of the ray and the lens surface amounts to solving

0 y

in

+ k

cos α

in

sin α

in

=

x(y) y

. (4.29)

Some algebraic manipulation yields a second-degree equation in k k

²

1 C −

²

C cos

²

α

in

+ 2k

y

in

C sin α

in

− cos α

in

+ y

²_in

C = 0. (4.30) The ﬁrst point of intersection can then be calculated as

x

1

= k cos α

in

y

1

= y

in

+ k sin α

in

, (4.31)

(32)

where k is the path traveled within the material. At the point of intersection, the tangential to the surface has to be calculated and the law of refraction evaluated.

Finally a new angle α

1

is derived.

dy

dx

=

^C_y

− (1 −

²

)

^x_y

β = α

in

+

^π₂

− tan

¹

dy dx

γ = sin

¹

[(1 − δ) sin β]

α

1

= α

in

+ γ − β

ⁱⁿ

α

₁

α

(x ,y )

1 1

β γ

The next steps are similar and we do not go into the details. After each iteration a Boolean variable ok tells whether the ray is still alive or went out of bounds, in which case the trace is terminated. To calculate the flux profile after the lens from a finite source, we let rays start from n equidistant points on the source and from each point m uniformly distributed rays hit the aperture of the lens. For good results, 100 x 100 rays are typically needed.

The algorithm was implemented in a C-program and the output stored in ASCII-files. The data file was then read by a Matlab-program to calculate gain and beam profiles.

4.3.3 Ray-tracing results

We start by checking the straight-line approximation. Fig. 4.7 shows the path of 25-keV x-rays through a compound lens in PMMA made up of 600 lenses with R = 100 µm. The rays originate from a point-source located on the optical axis 60 cm from the lens. The focal length of the lens is 19.4 cm at 25keV. The ray entering the ﬁrst surface at y = 20 µm has a lateral deviation of only 1.4 µm, whereas the one entering at y = 90 µm deviates 6.5 µm. Thus, the straight-line approximation does not seem completely justiﬁed. We will make a small error when calculating the absorption in the lens. However, this does not imply imperfect focusing.

An initial test of the resolution and possible aberrations is to send in a parallel beam. The width of the focal spot (or, rather line in our case) is a good indication of the quality of focusing. For a perfect lens and neglecting diﬀraction this should of course be zero within numerical errors. The result for the parabolic lens is shown in Fig. 4.8. Focusing seems perfect, although the focal plane is slightly shifted compared to the theory. The circular lens, however, shows strong spherical aberration and poor resolution. This may not be disastrous in a “ﬂux-collection”

application, but certainly bad in an imaging conﬁguration.

As a ﬁnal ray-tracing test, we check the validity of Eq. 4.20. The result for

parabolic and circular voids is shown to the left in Fig. 4.9 compared to theory. A

lens made of PMMA optimized for 20 keV is projecting a 50 µm source on a 10 µm

slit 1 meter away. It comprises 540 holes with 100 µm radius of curvature. For the

parabolic case there is an almost perfect consistency with theory. The maximum

gain is about 10% lower for the circular lens, owing to the aberration and increased

absorption.

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4.4. MANUFACTURING AND LENS PARAMETERS 29

0 200 400 600

14 16 18 20 22

y (µm)

Hole number

0 200 400 600

84 86 88 90 92

Hole number

y (µm)

Figure 4.7: Two rays entering at diﬀerent distance to the optical axis traced through the 600 voids with R = 100 µm. Rays enter at y = 20 µm and y = 90 µm, respectively, from a point-source on the optical axis 60 cm from the ﬁrst lens surface.

E = 25keV; material is PMMA; F = 19.4 cm.

Also shown to the right in Fig. 4.9 is the beam profile in the focal plane. We see that the parabolic lens images the source close to perfectly to the 10 µm slit. In the case of the circular lens, the image of the source is slightly smeared out by the finite point-spread function of the lens. This should be a few µm wide, estimating from the figure. This spread is obviously more significant when focusing to smaller dimensions.

4.3.4 Conclusions

The ray-tracing results indicate that the theoretical frame-work developed in Sec- tion 4.2 gives accurate results. The assumptions and approximations seem justiﬁed, at least from a pragmatic perspective. Furthermore, the importance of avoiding spherical aberration by using parabolic lenses has been demonstrated.

4.4 Manufacturing and lens parameters

The key to facilitate manufacturing of the parabolic voids is to make the lens in two halves. In this way the established technology of high-precision diamond-cutting could be applied. Polymethyl methacrylate (PMMA) with an average atomic num- ber of approximately 6 and good mechanical properties was chosen as lens material.

See Fig. 4.10 for a drawing of the lens half. It was decided to make 600 grooves with R = 100 µm, i.e. a 6 cm long lens. This gives a focal length at 25keV

F = R

2δN = 100 µm

2 · 4.29 · 10

⁷

· 600 = 19 cm.

Six lens halves were diamond-turned by Reed Precision Microstructures, Santa

Rosa CA, U.S.A. The lenses were attached to aluminum plates that were joined

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−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

−100

−50 0 50 100

µm

focal plane Theoretical

Real focal plane P a r a b o l i c h o l e s

−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25

−100

−50 0 50 100

µm

C i r c u l a r h o l e s

Figure 4.8: Ray-tracing for a parallel incoming beam of 25-keV x-rays. 600 holes in PMMA with R=100 µm gives a theoretical focal length of 19.4 cm. Top: Parabolic proﬁle, 6 cm long lens. Focusing is perfect within numerical errors. The focal plane is slightly shifted (≈ 1 cm) from the expectation of a single N times stronger lens.

Bottom: Circular holes, 12 cm long lens. We see the strong spherical aberration,

resulting in a poor resolution and low gain.

(35)

4.4. MANUFACTURING AND LENS PARAMETERS 31

15 20 25

0 2 4 6 8 10 12

E [keV]

Gain

R.T. para.

R.T. circ.

Theory

−150 −10 −5 0 5 10 15

5 10 15

µm

Gain

R.T. circ.

R.T. para.

Figure 4.9: Comparison between ray-tracing and theory. Material is PMMA; lens is optimzed for 20 keV; 50 µm source; 10 µm slit 1 meter away; 540 holes with R = 100 µm. Left: Gain as a function of energy. Right: Beam proﬁle in the focal plane at 20 keV.

with micrometer screws and adjusted under a microscope. Figure 4.11 shows the assembly.

60 mm, 600 grooves 100 µm

100 µm

5 mm 15 mm

Figure 4.10: Drawing of one half of the parabolic refractive lens. Only 20 of the 600 grooves are shown and the depth dimension is scaled down for clarity.

To evaluate the profile and surface finish, one of the lenses was examined with a scanning transmission electron microscope, see image in Fig. 4.12. The surface had to be vapor-deposited with a thin gold layer to be conductive. The crests were damaged at a few positions, but the effect should be too small to influence the performance of the lens.

To examine the proﬁle, we also used a scanning confocal microscope with sub-

micron resolution. Images were taken over the whole region of the lens, which

(36)

Figure 4.11: Lens halves with aluminum support. The two aluminum plates were joined and adjusted with micrometer-screws under a microscope.

revealed no systematic deviation of the shape. There was also no visible diﬀerence between the six copies, except one that was damaged and discarded. Three images are shown in Figure 4.13. The proﬁle is slightly asymmetric. While the right slopes of the teeth look perfect, there is a small deviation from the desired shape near the bottom on the opposite side, likely caused by the radius of the tip of the diamond tool. The impact from this should be small since the contribution from this part of the lens is small due to high absorption far from the optical axis.

4.5 Experimental evaluation

4.5.1 Experimental setup

An experimental setup was designed to evaluate the lens and measure gain and beam proﬁles. The source was an ordinary x-ray tube equipped with a tungsten anode and the voltage was variable between 10 and 60 kV. The real source size was 0.4 x 12 mm

²

, but by adjusting the angle of the optical axis to the horizontal, the apparent vertical source size could be varied. For these measurements this was ﬁxed to 50 µm.

We used a cadmium-zinc-telluride detector (Amptek XR-100T-CZT) with close to 100% detection eﬃciency over the energy region of interest and 1.3 keV FWHM energy resolution at 60 keV when thermo-electrically cooled to −25

^o

C. After am- pliﬁcation the signal was put into to a multi-channel analyzer.

The x-ray lens and the slit in front of the detector were both placed on vertical

translation stages with 5 µm precision. The slit width was measured to 30 µm with

a microscope. The translation stages were put on an arm that could be rotated to

point back to the source for coarse adjustment. Fine adjustment was subsequently

made with the translation stages. A schematic of the setup is shown in Figure 4.14.

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4.5. EXPERIMENTAL EVALUATION 33

Figure 4.12: Scanning electron microscope image of the PMMA parabolic lens surface.

Figure 4.13: Parabolic lens proﬁle measured with a confocal microscope. The proﬁle is slightly asymmetric and the radius of the tool tip can be seen at the bottom.

Rotation stage Translation

stage 1

Translation stage 2 Lens

Slit Source

α=7

Detector

ο 330 mm

640 mm

A multi-prism lens for hard X-Rays