EUV Microscopy with a Tabletop High Harmonic
Generation Source: Generalizing Coherent Diffractive
Imaging to Extended Samples in Transmission, Reflection,and Hyperspectral Modalities
byBosheng Zhang
B.S., University of Science and Technology of China, 2007

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2015This thesis entitled:
EUV Microscopy with a Tabletop High Harmonic Generation Source: Generalizing Coherent
Diffractive Imaging to Extended Samples in Transmission, Reflection, and Hyperspectral
Modalities
written by Bosheng Zhang
has been approved for the Department of Physics

Prof. Margaret M. Murnane

Prof. Henry C. Kapteyn

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

iii
Zhang, Bosheng (Ph.D., Physics)
EUV Microscopy with a Tabletop High Harmonic Generation Source: Generalizing Coherent
Diffractive Imaging to Extended Samples in Transmission, Reflection, and Hyperspectral Modalities
Thesis directed by Profs. Margaret M. Murnane and Henry C. Kapteyn

Imaging at the nanoscale is of great interest for applications in materials science, nanoscienceand biology. The microscopy method developed in this thesis combines a tabletop coherent EUV/Xray source based on high harmonic generation, and an image-forming method based on coherent
diffractive imaging. This microscopy method offers truly diffraction-limited resolution; however,previous work has been limited to thin, isolated samples in transmission mode. This thesis work
extends this tool for imaging non-isolated samples, and for working in reflection mode to image
surface features of thick samples. The quantitative phase information of the reflection image enables
surface profilometry capability with sub-nanometer precision. The microscope developed in thiswork is also demonstrated to have hyperspectral capability with simultaneous multi-wavelength
illumination, without the need for wavelength scanning or energy-resolved detectors. In the future,
by taking advantage of the short-pulse nature of the high harmonic illumination, this microscope will
be able to image nanoscale ultrafast dynamics with 10 femtosecond temporal resolution, opening
the door for imaging at the space-time limits.

Dedication

To my mother Guozhen Wang, my father Shihuan Zhang, and my wife Fengmei Liu.

v

Acknowledgements

First of all, I would like to thank my advisors, Margaret Murnane and Henry Kapteyn for theirvision, advice, support, and encouragement. They have provided me access to various resources,
opportunities for skill development, connections with experts in both academia and industry, andvaluable experiences such as patent application.
I feel fortunate to have Dan Adams and Matt Seaberg as my colleagues throughout the
majority of my PhD career. Much of the work in this thesis would not have been possible without
their many ideas, suggestions, and direct contribution in experiments. I also owe them a lot for theirwork on the lasers. I have benefited from Dan’s strong leadership, and excellence in both scienceand project management. I thank him for pushing and helping me to achieve and for teaching
me various things including UI design. He demonstrates to me a spirit of getting the job done nomatter what it takes, which is truly touching and stimulating. Matt has been a partner that iscreative, forward-thinking, and knowledgable with great ability in both experiments and theories,
and he is a go-to guy when I want ideas, suggestions, or a fruitful and inspiring discussion. Great
challenges have been less daunting with him on the team. Dennis has made great contributions topart of this thesis work. He became an invaluable member of the team with his high work ethic,
and versatile skills in both hardware and software. Although Liz joined the group not as long ago,she has already become well-known for a lot of things, especially fabrication of a sample that hasbrought us a series of successful results presented in this thesis work. I am hopeful that ultrafast
dynamic imaging, a holy grail that has driven me and other previous members, is now possible
with the project in the good hands of those two and other younger members. Days and nights withvi
the imaging team when we made the breakthroughs will always be a memorable experience that I
am proud of.
Tenio was my mentor when I started in the KM group. I want to thank him for teaching me
HHG both before and after I joined the imaging team. I also want to thank Ming-Chang, Xibin,
and Isabell for teaching me various things before I started working on imaging. Thanks to all KM
group members that I have worked with for help and company. Thanks to Justin, Weilun, Eric,and Farhad for collaboration. I have to thank the instrument, electronics, and computing staff who
have helped and taught me, and who have made JILA a wonderful place to work at: Dave, Todd,
James, Hans, Kim, Ariel, Blaine, Tracy, J.R..
Thanks to friends: Chenjie, Wei, Cong, Jing, Hongcheng, Bo, Linqiang, Xia, Yan, Hao,
Qing, Long, Xiaoshi, Jianfeng, Honghua and many others that I don’t have room to list, for help,
company, and fun time.
Thanks to my parents and my wife for their love, encouragement and support.

viiContents

Chapter
1 Introduction

1

1.1

Imaging probes and image-forming methods . . . . . . . . . . . . . . . . . . . . . . .

2

1.2

Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62 Imaging Probe: High Harmonic Generation Producing Coherent EUV/X-ray

7

2.1

Coherent EUV/X-ray imaging contrast mechanism . . . . . . . . . . . . . . . . . . .

7

2.2

Theory of high harmonic generation . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2.1

Microscopic three-step model . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.2.2

Macroscopic phase matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3

Properties of HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Image-forming Method: Coherent Diffractive Imaging
3.1

3.2

14

Review of CDI developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.1

Single-diffraction CDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.2

Ptychography CDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Diffraction theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.1

History of diffraction theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.2

Wave equations from Maxwell’s equations . . . . . . . . . . . . . . . . . . . . 20

3.2.3

2D Scalar Diffraction Theory with Green’s function approach: RayleighSommerfeld diffraction integral . . . . . . . . . . . . . . . . . . . . . . . . . . 22

viii

3.3

3.2.4

Approximations of Rayleigh-Sommerfeld diffraction integral . . . . . . . . . . 27

3.2.5

2D Scalar diffraction with angular spectrum approach . . . . . . . . . . . . . 28

3.2.6

2D vectorial diffraction

3.2.7

Non-normal incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.8

3D diffraction with volume integral . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2.9

Summary of 2D diffraction formulas . . . . . . . . . . . . . . . . . . . . . . . 31

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Iterative algorithms for diffraction phase retrieval . . . . . . . . . . . . . . . . . . . . 32
3.3.1

Projection onto the modulus constraint set on the detector plane . . . . . . . 35

3.3.2

Projection onto the support constraint set on the sample plane in singlediffraction CDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.3

Projection onto the “overlap” constraint set on the sample plane in ptychography CDI

3.3.4
3.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Combining two projections for a full iteration cycle . . . . . . . . . . . . . . . 37

Numerical implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Keyhole CDI for Extended Samples in Transmission

44

4.1

Apertured illumination CDI: first attempts for imaging extended samples . . . . . . 45

4.2

Keyhole CDI for extended samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3

Keyhole CDI for semi-transparent samples . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 First Demonstration of General Reflection CDI
5.1

5.2

60

Initial attempts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.1.1

Apertured illumination CDI in reflection . . . . . . . . . . . . . . . . . . . . . 60

5.1.2

Imaging periodic samples in reflection . . . . . . . . . . . . . . . . . . . . . . 61

First demonstration of a general purpose ptychography reflection microscope . . . . 63
5.2.1

Sample fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2.2

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

ix

5.3

5.2.3

Image reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2.4

Oxide layer measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2.5

Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2.6

Comparison between CDI reconstruction and SEM and AFM images . . . . . 75

5.2.7

High harmonic beam characterization through ptychography . . . . . . . . . 75

5.2.8

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Keyhole CDI in reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 Surface Nano-imaging with High Lateral and Axial Resolution Using Tabletop EUV Ptychography

81

6.1

Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2

Image reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.3

Lateral resolution characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.4

Axial resolution characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.5

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

7 Hyperspectral Imaging with Harmonic Combs and Ptychography

93

7.1

Decomposing incoherent superposition with ptychography . . . . . . . . . . . . . . . 93

7.2

Experiment and results

7.3

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

8 Future

98

Bibliography

100

x

TablesTable
3.1

Key milestones in the history of diffraction theory . . . . . . . . . . . . . . . . . . . 21

3.2

Summary of different algorithms for two successive projections . . . . . . . . . . . . 38

xi

FiguresFigure1.1Classification of imaging techniques based on their two key components: probes(illumination sources) and image-forming methods . . . . . . . . . . . . . . . . . . .

3

2.1

A comparison between an X-ray tube and a HHG source that has a waveguide geometry 11

2.2

A representative HHG spectrum in the EUV range . . . . . . . . . . . . . . . . . . . 13

3.1

Illustration of the coherent diffractive imaging technique . . . . . . . . . . . . . . . . 15

3.2

Diffraction geometries for homogenous volume and inhomogeneous volume . . . . . . 24

3.3

Diffraction geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.4

Applicability of different diffraction formulas for numerical calculations . . . . . . . . 33

3.5

Illustration of the iterative projections used for phase retrieval in CDI . . . . . . . . 34

3.6

Geometric representation of combined projections onto two constraint sets that are
represented by two lines intersecting . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.7

Illustration of the mapping from spatial frequencies to the detector grid for threedifferent situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1

The apertured illumination CDI scheme isolates an extended transparent sample byimaging a aperture onto the sample plane . . . . . . . . . . . . . . . . . . . . . . . . 46

4.2

Tabletop EUV keyhole CDI of a sample with an opaque background . . . . . . . . . 49

4.3

Tabletop EUV keyhole CDI of a sample with a semi-transparent background . . . . 54

4.4

Image reconstruction results for a sample that has a semi-transparent background . 55

xii
4.5

Height comparison between AFM and keyhole CDI . . . . . . . . . . . . . . . . . . . 57

5.1

Mapping the diffraction from a tilted sample to a diffraction pattern linear in frequency space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2

Visible laser apertured illumination and tilted sample correction by reconstructing
a 1951 USAF resolution target in a reflection mode geometry . . . . . . . . . . . . . 62

5.3

EUV microscope image in reflection mode of a 2D array of nickel nano-pillars . . . . 64

5.4

Experimental setup for reflection mode Fresnel ptychography . . . . . . . . . . . . . 68

5.5

Diffraction data and ptychographic reconstruction . . . . . . . . . . . . . . . . . . . 69

5.6

Height profile comparison between CDI and AFM

5.7

A visual comparison between the reconstructed CDI amplitude and phase with images obtained using SEM and AFM

5.8. . . . . . . . . . . . . . . . . . . 74

. . . . . . . . . . . . . . . . . . . . . . . . . . . 76

A comparison of separate reconstructions of the HHG illumination beam, using thebeam as the object in one case and as the probe in the second case . . . . . . . . . . 78

5.9

Comparison between the illumination reconstructed as a ptychographic probe andpropagated to the detector, and the unscattered illumination measured directly on
the detector (raw data) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.10 Reflection keyhole CDI reconstruction with the probe reconstructed from ptychography 80
6.1High NA reflection EUV ptychography experiment . . . . . . . . . . . . . . . . . . . 83

6.2

Images reconstructed from ptychography and comparison with SEM and AFM . . . 86

6.3

Characterization of the lateral resolution . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.4

Height calculated from the reconstructed phase . . . . . . . . . . . . . . . . . . . . . 91

7.1

Setup and results of hyperspectral imaging with four harmonic orders using ptychographical information multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Chapter 1

Introduction

In December of 1959, Richard Feynman gave a talk titled “There’s Plenty of Room at the
Bottom” [1] at an annual meeting of the American Physical Society at Caltech. In this famous
lecture, Feynman laid the conceptual foundations for the field now called nanotechnology when
he imagined a day when things could be miniaturized – when huge amount of information could
be encoded onto increasingly small spaces, and when machinery could be made considerably morecompact. Feynman’s vision has been proven by the development of many scientific and technologicalfields, such as materials science, and electronics: machines and devices have been made on everdecreasing length scales to make them faster, cheaper and more efficient or to exploit the peculiar
quantum mechanical properties. For these, and many other fields such as biology, microscopy is
the basic, and perhaps the most import, tool to investigate and classify small structures on thenanometer scale. It has become one of the few methodologies applied to nearly every field of scienceand technology in use today.
This chapter will discuss two key components of every microscopy (or more generally imaging)technique: imaging probes (or illumination sources), and image-forming methods. This discussion
provides a context to introduce the unique microscopy technique developed in this thesis, onethat combines coherent extreme ultraviolet (EUV) from high harmonic generation (HHG) as theimaging probe, and coherent diffractive imaging (CDI) as the image forming method. Following
the discussion on microscopy, this chapter will provide an overview of this thesis.

2

1.1

Imaging probes and image-forming methods
The first microscope is usually credited to Zacharias Jansen in Middleburg, Holland, aroundthe year 1595. Since then, various techniques have been developed in the field of of microscopy, or
more generally imaging. Imaging techniques can be analyzed and classified based on their two key
components: the type of probe used to interact with the sample to be imaged, and the method
used to form an image, as shown in Fig. 1.1.
The first key component of every imaging technique is the probe it uses. Essentially, imaging
is about mapping of the interaction between the selected probe and the object to be imaged, andthe type of probe used directly determines its contrast mechanism. Furthermore, the probe type
can put an inherent limit of the resolution by its wavelength. Far field imaging systems have adiffraction-limited resolution given by (using the Rayleigh Criterion):
r=

0.61λ
.
NA

(1.1)

In addition, the choice of probe determines how thick a sample can be imaged by its penetration
depth.
In principle, to image a sample, any type of wave or particles can be used as the probe. As
shown in Fig. 1.1, the following probes have been used for imaging:(1) Light or photons. This probe type has imaging contrast mainly based on elastic scatteringof the photon by the sample, or some excited processes such as fluorescence. It can further
classified based on the wavelength of the illuminating photons, such as visible light, EUV,
X-ray, etc.
Visible light microscopy (or light microscopy, optical microscopy) is the oldest and probably
the most familiar one to us – after all, human beings’ eyeball imaging system relies on thisprobe type. It is used extensively in microelectronics, biotechnology, pharmaceutic research,mineralogy and microbiology. Its resolution is limited to ≈ 200 nm due to the relatively
long wavelength.

3VisibleIncoherent: LPP DPP, X-ray tubes
Light

EUV/X-ray

Synchrotron/XFEL
Coherent

EUV laser
High harmonic generation

Probe

Other lightElectrons
Ions
Local tips

Imaging

Ultrasound wave
Radio wave
Other probes
Point-by-poing scanning
Conventional refractive lens
Lens
Image-forming

Fresnel zone plate
Electromagnetic lens

Full field+

HHG CDI
Microscopy. High temporal resolution. 3D capability. elemental specificity. non-contact. High resolution. fast, full field. Amplitude & Phase contrast

ShadowgraphLensless

Pinhole camera
Holography (some)
Coherent diffractive imaging

Figure 1.1: Classification of imaging techniques based on their two key components: probes (illumination sources) and image-forming methods. The microscopy in this thesis work combines highharmonic generation as the probe and coherent diffractive imaging as the image-forming method,and have combined advantages from these components.

4
A fluorescence microscope uses fluorescence and phosphorescence instead of the conventional transmitted or reflected light for imaging. Super-resolved fluorescence microscopy
overcomes the diffraction limit in optical microscopy and brings it into the nanodimesion,
opening up important applications in biology and life sciences.
Shorter wavelength probes, such as EUV or X-ray, can yield higher resolution. Several
image forming methods can be applied to this probe type, including scanning transmission
X-ray microscopy (STXM), transmission X-ray microscopy(TXM) [2] and CDI [3]. X-ray
microscopy can image with elemental and chemical contrast. It also has three-dimensional
imaging capability due to the long penetration depth of X-rays.
(2) Electrons. Due to the electron’s short wavelength, electron microscopes have a much higher
resolution than optical microscopes. There are several types of electron microscopes: scanning electron microscope (SEM), transmission electron microscope (TEM), scanning transmission electron microscope (STEM), etc. In SEM, interaction of the electrons with thespecimen result in several possible products, including low-energy secondary electrons andhigh-energy backscattered electrons, light emission (cathodoluminescence) or X-ray emission, all of which provide signals carrying information about the properties of the specimen
surface, such as its topography and composition. TEM can image with a really high resolution of below 0.5 Å [4] but is limited to thin samples (≈ 100 nm or less).
(3) Ions. A representative example is scanning helium ion microscope. Ions have even shorter
wavelengths, and do not suffer from a large excitation volume, so an ion microscope canhave even higher resolution than SEM. It usually has a better material contrast too.
(4) Local tips such as a mechanical tip used in atomic force microscopy (AFM), a wellestablished technique for surface profilometry. Here “local tips” refer to the probes used inscanning probe microscopy (SPM). They represent a special category of imaging probes, including a large variety of probes that that interact with the object locally through different
types of interaction, such as atomic force, magnetic force, etc.

5(5) Other probes include ultrasound wave used in sonography for imaging body structures,radio waves together with magnetic fields used in magnetic resonance imaging (MRI), etc.
The second key component of every imaging technique is the image-forming method it uses.
As shown in Fig. 1.1, both point-by-point scanning and full field imaging techniques exist. In pointby-point scanning techniques, either the beam, such as in SEM, or the sample, such as in STXM,
AFM or other SPM, can be raster scanned to form a two-dimensional image. Full field imaging
techniques include those with an image forming optics, such as a lens, and those without (termed“lensless” here). The type of lens includes conventional refractive lenses used in optical microscopy,
Fresnel zone plates used in TXM, electromagnetic lenses used in electron or ion microscopy; Inthese imaging techniques, the lens design is the core issue for improving resolution and decreasing
aberration.
A simple example of lensless, full field imaging techniques is the shadowgraph imaging used in
X-ray radiography. A pinhole camera is also a lensless imaging technique. Two import techniquesin this category include some types of holographic imaging (such as in-line holography invented by
Gabor [5]) and CDI [6]. While it is possible for these two techniques to employ certain optics or
lenses to condense the illumination onto the sample, the image-forming process itself does not relyon the lenses. CDI uses computer algorithms to replace lenses to form an image, as will be seen in
Chapter 3.
The microscopy developed in this thesis combines coherent EUV/X-ray generated from HHG
as the imaging probe and CDI as the image forming method. As shown in Fig. 1.1, advantagesof EUV/X-ray from HHG as a probe include high temporal resolution of 10 fs due to its shortpulse duration, 3D capability, elemental specificity because many elements have their resonancefrequencies in this wavelength range, non-contact working mode, and higher resolution offeredby shorter wavelengths compared to conventional optical microscopy. CDI as an image-forming
method have advantages, such as truly diffraction limited resolution without aberration from imageforming optics or limitation from the NA of the optics, inherently much faster than point-by-point

6
scanning methods because it is a full field technique, and it provides quantitative amplitude andphase information at the same time, enabling comprehensive and definitive characterization of thesample. Combining HHG and CDI would lead to a microscope that combines the advantages fromboth sides.

1.2

Overview of this thesis
HHG CDI microscopy was first demonstrated in 2007, by Richard Sandberg, Ariel Paul etal. from our group [7]. In 2011, my colleagues Matt Seaberg, Dan Adams et al. demonstrated
a record, 22 nm resolution imaging with a 13 nm HHG source [8]. Both of these results arefor isolated samples in transmission mode with monochromatic light. This thesis will focus on
further developments of this microscopy method, including generalizing it to extended samples intransmission, reflection, and hyperspectral modalities.
The two key components of this microscopy, coherent EUV/X-ray from HHG as the probe,and coherent diffractive imaging as the image-forming methods, are described in Chapter 2 andChapter 3. The following chapters will discuss the results with this microscopy method. CDI
was conventionally limited to small, isolated samples, and has very limited success in reflection
geometry which is important for imaging surfaces, and structures on thick substrates. Chapter 4
will discuss the first demonstration of generalizing it to non-isolated samples with a HHG source.
Chapter 5 will discuss the first demonstration of a general-purpose reflection CDI microscope basedon ptychography. Chapter 6 will discuss results on further improvement of reflection ptychography
CDI. They enable high lateral and axial resolution surface imaging, with excellent imaging fidelity
compared with well-established methods such as SEM and AFM. Chapter 7 will discuss resultson hyperspectral imaging with multi-colored HHG combs. Instead of monochromatic illumination,
hyperspectral ptychography imaging uses multiple colors to illuminate the sample. The incoherent
sum of intensities from different colors are decomposed by the ptychography algorithm, enabling
spectromicroscopy with separate responses of the sample at each illuminating color. Finally, in
Chapter 8, ideas for future work with this microscope are discussed.

Chapter 2

Imaging Probe: High Harmonic Generation Producing Coherent EUV/X-ray

This chapter describes the first of the two key components for our unique microscopy: coherent EUV/X-ray from high harmonic generation used as the imaging probe. First, the constrast
mechanism is discussed for this probe. Then the theory of the high harmonic generation is described. Finally, relevant properties of this imaging probe are discussed.

2.1

Coherent EUV/X-ray imaging contrast mechanism
The response of materials to electromagnetic radiation can be characterized by their index ofrefraction n(ω), where ω is the frequency of the radiation. For the spectral range of EUV or X-ray,n(ω) is typically written as [9]:n(ω) = 1 − δ + iβ

(2.1)

where
2

re λ 0δ = na2π
f1 (ω)(2.2)

r e λ2 0β = na2π
f2 (ω)

where na is the average density of atoms, re is the classical electron radius which is about 2.8 ×
10−15 m, λ is the wavelength of the radiation. Here f10 (ω) and f20 (ω) refer to the real and theimaginary parts of the complex atomic scattering factor:
f 0 (ω) =Xsgs ω 2.
ω 2 − ωs2 + iγω

(2.3)

where γ measures the phenomenological damping force, ωs is the resonance frequency associatedwith electron energy level s, gs is the so-called “oscillator strength”, which in the simple semi-8classical model is the number of electrons. We see that n(ω) depends on the material’s composition
and the density in the sample.
For a transmission geometry, if the sample is thin enough to satisfy the projection approximation [10], the exit surface wave Eout for a coherent incident wave Ein can be computed by amultiplication:
Eout (x0 , y 0 ) = Ein (x0 , y 0 ) · t(x0 , y 0 )

(2.4)

where
t(x0 , y 0 ) = e−k0Rβ(x0 ,y 0 ,z 0 ) dz 0 −ik0eRδ(x0 ,y 0 ,z 0 ) dz 0

(2.5)

Here the phase of Eout (x0 , y 0 ) is relative to the phase corresponding to vacuum of the same thickR

0

0

0

0

ness. The sample would show an amplitude contrast of e−k0 β(x ,y ,z ) dz ], and a phase contrast ofR−k0 δ(x0 , y 0 , z 0 ) dz 0 in its two-dimensional exit wave. Spatial variation of the sample’s composition, density, or thickness if the complex wave Eout , or the complex transmission function t(x0 , y 0 )is to be imaged, which is true in coherent diffractive imaging.
For a reflection geometry, Eout can be written as:
Eout (x0 , y 0 ) = Ein (x0 , y 0 ) · r(x0 , y 0 )

(2.6)

where
r(x0 , y 0 ) = rm (x0 , y 0 ) · exp(i−4πh(x0 , y 0 ) cos θi
)
λ

(2.7)

or
 

−4πh(x0 , y 0 ) cos θi0 0r(x , y ) = |rm (x , y )| · exp i(φ rm (x , y ) +
.
λ
0

0

0

0

(2.8)

Here rm (x0 , y 0 ) is the complex reflection efficient form the material of the reflecting surface itself,h(x0 , y 0 ) is the surface height distribution, and θi is the angle of incidence. rm is determined byn(ω) through Fresnel equations. So coherent imaging of Eout (x0 , y 0 ) or r(x0 , y 0 ) will show contrast
from both the composition and the height profile.

9

2.2

Theory of high harmonic generation
After seeing the contrast mechanism of imaging with coherent EUV/X-ray light, let us see

how to generate them. Four approaches exist for generating coherent radiation at these short wavelengths: synchrotrons, free electron lasers, EUV/soft X-ray lasers and high harmonic generation(HHG). Compared with synchrotrons and free electron lasers, HHG is a very compact source thatcan be put on a tabletop. EUV/Soft X-ray lasers have pulse durations of ≈ picosecond, while HHG
has a much shorter pulse duration, on the order of 10 fs or less, allowing for much higher temporal
resolution.
HHG, as an extreme nonlinear process that upconverts the driving laser frequency to its highorders, was first observed in late 1980’s [11, 12]. Typically it is driven by ultrafast lasers composedof an oscillator that generates femtosecond pulses with low pulse energy, and an amplifier that
increases the pulse energy to the mJ level. HHG can be understood on two levels: the microscopic,
single atom level, which explains the emission of high energy photons, and the macroscopic level
where phase-matching is considered.

2.2.1

Microscopic three-step model

On a microscopic, single atom level, the emission of high-energy photons in HHG can be
explained with a semi-classical, three-step model [13] that is quite analogous to what happens in
X-ray tubes. The photos and schematics of these two sources are compared in Fig. 2.1. In an X-ray
tube, as illustrated by Fig. 2.1(b), electrons in the cathode are first ionized, then accelerated in thestrong electric filed provided by the high voltage between the cathode and the anode, and finallycollide with the anode and emitting the high kinetic energy they have gained from the external
electric field. The three steps (ionization, acceleration, and collision) are roughly what happen in
HHG on a single atom level. A HHG source in a waveguide geometry is illustrated in Fig. 2.1(d).
Intense femtosecond laser pulses with a linear polarization (recent work shows circular polarization
can also work [14]) are focused into a waveguide that is filled with a certain type of noble gases,

10such as argon or helium. The electrons of the noble gas atoms are first ionized by the strong fieldof the laser. The the free electrons are accelerated by the electric field of the laser pulse, andas the electric field oscillates and changes direction, the electrons will deaccellerate, turn back to
the parent ions, and finally collide and recombine with the ions. At recombination, the energy isreleased in the form of high-energy photons, typically in the EUV or X-ray range. Depending on
at the phase of the electric field at the moment of ionization, the calculated energy released can beas high as:hνmax ≈ Ip + 3.17Up(2.9)

where Ip is the ionization potential of the atoms, and Up is the so-called ponderomotive potentialof the laser field, defined asUp =e2 E 24me ω02(2.10)

where e is the electron charge, E is the linearly polarized electric field amplitude, ω0 is the laser
carrier frequency and me is the electron mass.

2.2.2

Macroscopic phase matching

Macroscopically, HHG requires phase matching to be efficient, similar to other nonlinear
optics processes, such as second harmonic generation, sum and difference frequency generation,
parametric amplification and oscillation. Phase matching for HHG means emissions from atoms at
different positions along the propagation axis are in phase, so the beam adds up coherently. The
phase mismatch ∆k is defined as:∆k = qk1 − kq

(2.11)

where q is the harmonic order, k1 is the wave number for the driving fundamental frequency, andkq is the wave number for qth harmonic. ∆k has contributions from three parts: the neutral atoms,
the plasma due to ionization, and the waveguide geometry(in contrast to free space that is phasemismatch free). It can be written as [15, 16]:∆k ≈ qu211 λ1
2π− qP (1 − η) (∆δ + n2 ) + qP ηNa re λ0
4πa2
λ1(2.12)

11ab

X-rayscdFemtosecond
laser
Waveguide filled with
noble gas

HHG EUV/X-rays
Coherent
buildupelectron

Figure 2.1: A comparison between an X-ray tube and a HHG source that has a waveguide geometry.
A photo and a schematic are shown in (a) and (b) for an X-ray tube, and in (c) and (d) for a HHG
source.

12
where we assumed the light is coupled into EH11 mode described by the Bessel J0 function, andu11 is the root for the first zero, λ1 is the fundamental (first order) wavelength, a is the waveguide
radius, P is the pressure in atmospheres, ∆δ is the difference in index of refraction of the neutral
gas at the fundamental and the qth order harmonic wavelengths, η is the ionization percentage, Na
is the number density of the gas at one atmosphere, and re is the classical electron radius. We haveused the approximation the harmonic order q  1, and at EUV/X-ray wavelengths, the index ofrefraction is close to 1. The two main parameters used in experiments to achieve phase matching
are the pressure of the noble gas, and the pulse energy which affects the ionization percentage η.
The specific parameters can be found in later chapters on experimental results.

2.3

Properties of HHG
EUV or X-rays produced by HHG are spatially coherent, in contrast to incoherent X-ray

radiation from X-ray tubes. It is driven by a laser which is spatial coherent, and the propagationin the waveguide can purify the spatial modes of the coupled laser light. With Young’s double slit
measurement, this source exhibits spatial coherence across nearly the whole beam extent in the
EUV [17] and X-ray [18] wavelengths.
A typical HHG spectrum is shown in Fig. 2.2 [19]. Usually, several harmonics with odd orders
are phase matched simultaneously. As estimated from the spectrometer measurement represented
by the dashed line in Fig. 2.2, for each harmonic, λ/∆λ ≈ 100. Usually only one harmonic isselected through multilayer mirrors. Imperfect monochromaticity causes blurring of diffraction
fringes at high diffraction angles, thus a decrease of spatial resolution in diffractive imaging. A
discussion with a specific experimental situation will be presented in Chapter 4.

13

Figure 2.2: A representative HHG spectrum in the EUV range obtained from a spectrometer(dashed curve) and from a double-pinhole measurement. Figure adapted from [19].

Chapter 3

Image-forming Method: Coherent Diffractive Imaging

Coherent diffractive imaging (CDI) [3, 6] is the image forming method of the microscopy
developed in this thesis work. Also called “lensless imaging”, CDI is a computational imaging
technique that replaces image forming optic(s) with a computational algorithm. As illustrated inFig. 3.1, in CDI, a coherent beam of wavelike particles, such as photons or electrons, is incident
on the object to be imaged, and the diffraction magnitude is measured on a pixel-array detector.
With the measured diffraction magnitude, and additional known constraints about the sample,
iterative algorithms are applied to retrieve the phase of the diffraction that is missing during themeasurement of the magnitude. With both the magnitude and the phase of the diffraction, and themathematical relationship of the complex diffraction field from the object, the image is calculated,with both amplitude and phase contrast.
This chapter will first provide a historical review of the CDI techniques including both singlediffraction CDI and ptychography CDI [20]. The next two sections will discuss two theoreticalaspects of CDI: the diffraction theory, and iterative algorithms for diffraction phase retrieval. The
section on the diffraction theory aims to establish a systematic framework, which covers different
diffraction formulas, the approximations they use, and their ranges of applicability. This section
aims to make it clear which diffraction formula to use with given experimental conditions. The
section on iterative algorithms will explain how to retrieve phase for the measured diffraction
modulus using iterative algorithms.
After the theories behind CDI, this chapter will discuss the numerical implementation of

15

Detector
Coherent
beam

SampleDiffraction
pattern

Reconstruction

Image
Figure 3.1: Illustration of the coherent diffractive imaging technique. A spatially-coherent beam
illuminates the sample, and the diffracted intensity is measured on a pixel-array detector. Then
iterative algorithms are applied to retrieve the phase that cannot be physically measured for thediffracted wave, and reconstruct an image of the sample.

16
these theories in CDI.

3.1

Review of CDI developments
Here the CDI techniques divided into two categories: single-diffraction CDI and ptychography

CDI. To reconstruct a 2D image of the sample, only one diffraction pattern is needed in singlediffraction CDI, while multiple diffractions measured at different scanning positions are needed in
ptychography CDI.

3.1.1

Single-diffraction CDI

In 1895, Röntgen discovered X-ray and in 1912 Laue discovered X-ray diffraction from crystals. Afterwards, the field of X-ray crystallography developed rapidly, and became an important
tool for identifying the atomic and molecular structures of a crystal. Since many materials can form
crystals — such as salts, metals, minerals, semiconductors, as well as various inorganic, organicand biological molecules — X-ray crystallography has been fundamental in many scientific fields.
It is used to reveal the structure and function of biological molecules, including vitamins, drugs,
proteins, and nucleic acids. For example, in 1953, Watson and Crick suggested the double-helix
model of DNA structure using this tool.
X-ray crystallography was the first experimental discipline to face up to the consequencesof lost phase: when the diffraction was measured, only the amplitude (intensity, or flux) can bephysically measured but not the phase. Had both the amplitude and the phase of the complex wave
been known, the wave can then be directly propagated back from the detector plane to the sample
position to get an image. Indeed, this type of inverse problem — the reconstruction of an object frommeasurements of its scattered intensity — has occupied physicists for over a century, and arises infields as varied as optics, astronomy, medical tomographic imaging, holography, electron microscopy,
and particle scattering, besides X-ray crystallography. Fortunately for crystalline structures, there
is usually large quantities of stereochemical a priori information making phase retrieval much more
tractable than for generalized, non-crystalline structures.

17
Extending X-ray crystallography to a general diffraction imaging technique started in 1952,
when Sayre [21] observed that Bragg diffraction undersamples the diffraction intensity pattern relative to Shannon’s theorem, and noted out that there may be enough information in the diffraction
itself to uniquely solve the diffracting object, if only one could measure midway between Bragg
peaks. In 1980, he pointed out [22] that a single isolated non-peoriodic object permits such ahigher density sampling scheme. He envisaged extending X-ray crystallography to general imaging of non-periodic samples using diffraction — the method known as coherent diffractive imaging(CDI) now. However, Sayre’s early work does not prove that a properly sampled far-field diffraction
pattern would necessarily yield sufficient information to determine an image of the object, primarily
because it does not deal with issues of the independence of the additional measurements. In 1978,
Fienup, working independently of Sayre, argued that [23] it should be possible to reconstruct anobject form its Fourier modulus, and proposed some iterative algorithms (input-output approach)that built on the Gerchberg-Saxton algorithm developed in 1972. In 1982, Bates published a paper on the existence of a unique solution for the phase recovery problem [24]. He argued thatthe diffraction from an isolated object would certainly lead to a unique solution apart from sometrivial ambiguities (lateral translation, complex conjugate, spatial inversion, and absolute phase).
In early 1980’s, Fienup’s further development of iterative algorithms with feedback [25] produces
a remarably sucessful optimization method for phase retrieval. The iterations can be viewed as
Bregman projections in Hilbert space, and this insight has allowed theoreticians to analyze andimprove on the baisc Fienup algorithm [26].
On the experimental side, the successful demonstration of X-ray CDI did not happen until
1999, by Miao et al. on a synchrotron source [6]. There was great effort from my group to move CDI
from the huge light sources to our unique, tabletop HHG source. In 2007, Richard Sandberg, Ariel
Paul et al. from our group first demonstrated it with a tabletop, high harmonic EUV source [7].
In 2011, my colleagues Matt Seaberg, Dan Adams et al. demonstrated a record, 22 nm resolution
imag with a 13 nm HHG source [8].
Conventional CDI does have a serious limitation: it is applicable to samples that are small

18
and isolated. In 2008, Abbey et al. demonstrated the keyhole CDI method, which successfully
removed the isolation limitation. Two factors contribute to its success: first, instead of requiring
the sample to be isolated, it uses isolated illumination by putting an aperture in the beam. The
beam maintained a sharp edge and a finite support when it propagated to the sample plane. So
although the sample is not isolated, the exit surface wave is, making the isolation constraint in thesample plane still valid. Second, instead of illuminating the sample with plane wave, they useda beam with a phase curvature. This phase curvature provides a low-resolution in-line hologram
image of the sample, and also contributes to eliminate ambiguities and facilitates convergenceof the iterative algorithm. I, together with Matt, Dan et al., demonstrated keyhole CDI with
HHG [27], with results shown in Chapter 4. Dennis Gardner, I and other members of the groupalso demonstrated apertured illumination CDI, which is a more demanding yet more straightforward
approach, is based on the same idea of isolating the illumination instead of isolating the sample.
To date, single-diffractin (non-ptychographical) CDI has been used to extract the structureand dynamics of a variety of objects, including biological samples [28, 29], nanocyrstals [30, 31],
strain fields inside a nanocrystal [32] and integrated circuits [33].

3.1.2

Ptychography CDI

Instead of retrieving phase from a single-diffraction pattern, ptychography CDI (or just ptychography) [20] scans the sample relative to the beam, with overlap in between adjacent scans. The
overlap , instead of the isolation constraint in single-diffraction CDI, provides redundant information that allows the phase problem to be solved.
Ptychography CDI has gone through a quite different history than conventional CDI. Between about 1968 and 1973, Hoppe [34–38] conceived the concept of ptychography to solve thephase recovery problem in crystallography. The concept uses convolution theorem, and in German,“convolution” shares the same meaning with “folding”, for which the Greek word is “ptycho”; hence
the name “ptychography”. In 1989 and 1992, Bates and Rodenburg published two papers on the
Wigner distribution deconvolution method used to recover the phase for general, non-crystalline

19
structures, and in 1996, Chapman demonstrated this method using an X-ray source [39]. In 2004,
Rodenburg and Faulkner introduced and demonstrated a faster and more efficient iterative method
for ptychographic phase retrieval [40, 41], later known as ptychographical iterative engine (PIE).
This method was demonstrated with X-ray in 2007 by Rodenburg et al. In 2008, Thibault et al.
discovered that not only the object, but also the illumination beam can be solved with ptychography [42]. This has pushed the capability of the ptychography CDI to a new level. A detailed
account of the history of ptychography can be found in Ref. [43].
In 2014, we demonstrated for the first time generalized CDI in a reflection geometry with HHG[44], as shown in Chapter 5. Afterwards, together with Dennis Gardner and Dan, I demonstrated
reflection CDI with high lateral and axial resolution, and unprecedented quality. The images
from our microscope compare favorably with those from well-established microscopy methods, such
as SEM and AFM. It has a unique, powerful contrast mechanism, making it a promising tool forsurface metrology. This work is covered in Chapter 6. Another exciting result involves hyperspectral
imaging: the sample is illuminated with multiple colors at the same time and the incoherent sum is
detected; then a computational method is used to decouple the incoherent sum, and separate thereponse of the sample at different wavelengths. This spectromicroscopy method exploits the powerof computational imaging, and has no need for an energy-resolved detector, and serial scanning of
wavelengths. This work is covered in Chapter 7.

3.2

Diffraction theory
After the review of historical developments in the previous section, this section focuses on thefirst theoretical aspect of CDI – diffraction theory. I will first review the history of diffraction theory.
Then I will provide a short derivation of the key diffraction formulas for different experimental
conditions. Most of these formulas can be found in textbooks covering the topic of diffraction [45,46],
while some are new results; all of them are put in a coherent, systematic framework. After thissection, it should be clear to the readers which diffraction formula to use for the specific CDI
experimental condition they are faced with.

20
3.2.1

History of diffraction theory

The origin of diffraction theory can be dated back to 1660, when Italian scientist Francesco
Maria Grimaldi coined the word “diffraction” and was the first to record accurate observationsof the phenomenon. Since then, the understanding of diffraction has deepened along with thedevelopment of theories on the nature of the light. Table 3.1 showcases a short summary of key
milestones, extracted from Refs. [45, 47].

3.2.2

Wave equations from Maxwell’s equations

We start from Maxwell’s equations (Eq. 6.6 in Ref. [46], with the same notations) whichgovern electromagnetic phenomena in the classical (non-quantum) regime:
~ ·D
~ =ρ
∇
~ ·B
~ =0
∇
(3.1)
~ ×E
~ = −∂t B~∇~ ×H~ = J~ + ∂t D~∇
, where~ = µ0 H
~ +M
~,
B

(3.2)

~ = 0 E~ + P~
D~ and scalar potential Φ which relate to the fields E~ and B~ in
In terms of vector potential A
the following way:
~ =∇
~ ×A
~
B
(3.3)
~ = −∇Φ − ∂t A~
E
and which we choose to satisfy the Lorenz condition:~ ·A~ + 1 ∂t Φ = 0
∇
c2

(3.4)

, then the wave equations are:(∇2 − c12 ∂tt )Φ = − ρ0~ = −µ0 J~(∇2 − c12 ∂tt )A

(3.5)

21

Table 3.1: Key milestones in the history of diffraction theory.
Year
1660
1678

1704
1801
1818

1860
1882
1896

1939-1947

Event
Grimaldi coined the word “diffraction” and recorded accurate observation of
the diffraction phenomenon for the first time.
Huygens proposed the principle named after him: “Every point on a wave-front
may be considered a source of secondary spherical wavelets which spread out inthe forward direction at the speed of light. The new wave-front is the tangential
surface to all of these secondary wavelets.”
Newton published Opticks, in which he proposed the particle theory of light.
Young performed the double slit experiment, strengthened the wave theory oflight.
Fresnel published his famous memoir on the Diffraction of Light, calculated the
distribution of light in diffraction patterns with excellent accuracy based on
work by Hyugens and Young.
Maxwell identified light as electromagnetic wave, and in the following two years
published an early form of equations named after him.
Kirchhoff formulated the so-called Huygens-Fresnel principle.
Sommerfeld modified Kirchhoff’s formula using Green’s function, resulting inRayleigh-Sommerfeld diffraction theory, a rigorous solution for two-dimensional
scalar diffraction problem.
Stratton, Chu, Schelkunoff and Smythe [48–50] formulated vectorial diffraction
theory.

22
For radiation generation problems, we consider situations where there are free sources ofcharge and current; while for radiation diffraction or propagation problems, we consider a volume
inside which there are no free sources of charge and current: ρ = 0, J~ = 0. Then in terms ofpotentials, the wave equations become:(∇2 − c12 ∂tt )Φ = 0

(3.6)

~=0
(∇2 − c12 ∂tt )A~ the wave equation is:
In terms of elctric field E,~ = ∇(∇~ · E)~ + µ0 ∂tt P~ + ∂t ∇~ × µ0 M~[∇2 − 0 µ0 ∂tt ]E(3.7)~ the wave equation is:
In terms of electric displacement field D,~ − 0 µ0 ∂tt D~ = −∇~ × (∇~ × P~ ) + 0 ∂t ∇~ × µ0 M
~
∇2 D

(3.8)

The above three wave equations form the bases for all the further treatment of the diffraction
regime in this section. We will use Eq. 3.6 for two-dimensional (2D) vectorial diffraction theory,and Eq. 3.7 for 2D scalar diffraction theory. Here “two-dimensional” means diffraction is calculated
against a two-dimensional distribution of fields, usually on the “exit surface”. We will use Eq. 3.8
for three-dimensional (3D) diffraction theory; here “three-dimensional” means the diffraction is
calculated against the three-dimensional distribution of certain properties of the scatterers.

3.2.3

2D Scalar Diffraction Theory with Green’s function approach: RayleighSommerfeld diffraction integral

Scalar diffraction theory ignores the vector nature of the electromagnetic fields. To be specific,~ and H~ and between different components of E
~ [48,
boundary conditions lead to coupling between E51]. This treatment simplifies the problem, and turns out to provide accurate enough result when
the feature dimension of the scattering object is much larger than the wavelength. Mathematically,~ = 0 in Eq. 3.7. We also consider the situation where thewe set the magnetic response M

23
medium is isotropic and linear dielectric with χ(~r) being the electric susceptibility, then we can~ Eq. 3.7 thus becomes
write P~ = 0 χ(~r)E.
~ = −2∇(E~ · ∇ ln n) + χ(~r)0 µ0 ∂tt E~[∇2 − 0 µ0 ∂tt ]E(3.9)~ = E(~~ r, t) as the combination of its monochromatic components:
We write E~ ≡ E(~~ r, t) =EZ~ r, ω)e−iωtdω E(~(3.10)and plug it in the wave equation above, then we have:~ ω = f~s[∇2 + k02 ]E(3.11)~ ω (~r) ≡ E(~~ r, ω) and fs denotes the source term:
where E~ ω − ∇(E~ ω · ∇ ln(1 + χ(~r))).
f~s (~r) = −χ(~r)k02 E(3.12)We are going to use Green’s function approach to solve the wave equation Eq. 3.11, but
before doing that, we should keep in mind that Green’s theorem is applicable for functions thatare continuous over a complete bounding surface. Again, the boundary conditions at the sharpedges on the scattering object, which necessitate the consideration of the vectorial nature of theelectromagnetic field, can also make the field discontinuous at these places.
Consider a Green’s function, which satisfies [∇2 + k 2 ]G(~r, ~r0 ) = δ(~r − ~r0 ), with V denoting a
volume containing the position ~r in consideration, we can write:~ ω (~r) =E~ ω (~r0 )(∇2 + k 2 )G(~r, ~r0 )d~r0 ER~ ω (~r0 )(∇2 + k 2 )G(~r, ~r0 ) − G(~r, ~r0 ) (∇2 + k 2 )E~ ω (~r0 ) − f~s (~r0 )= V d~r0 E
RR~ ω (~r0 )] +~ ω (~r0 )∇0 2 G(~r, ~r0 ) − G(~r, ~r0 )∇0 2 E= V d~r0 [E
rGf~s
V d~
R

~ ω (~r0 )δ(~r, ~r0 ) =
r0 E
V d~

R

V

(3.13)

Applying Green’s theorem to the above euqation leads to~ ω (~r) =
E

I

0

0

~ ω (~r0 ) − E~ ω (~r0 )∂n0 G(~r, ~r0 )) +d~r (G(~r, ~r )∂n0 E

SZ

dr~0 Gf~s(3.14)

V

where S is the surface enclosing the chosen volume V , ∂n0 is the directional derivative along ~n, thenormal vector of surface S pointing to inside V .

24

Figure 3.2: Diffraction geometries for homogenous volume and inhomogeneous volume. Region I
contains the scattering sources. Region II is the diffraction region. Figure adapted from [46].

25
There are two possible diffraction geometries as shown in Fig. 3.2: homogeneous and inhomogeneous geometries. First we consider the homogeneous case where the volume under consideration
contains no scattering sources (f~s = 0), then Eq. 3.11 becomes the homogeneous Helmholtz equation:[∇2 + k 2 ]ψ = 0(3.15)~ and the solution Eq. 3.14 becomes:
where we use ψ to denote a scalar component of E,Z
ψ(r) =

d~r0 (G(~r, ~r0 )∂n0 ψ(~r0 ) − ψ(~r0 )∂n0 G(~r, ~r0 ))(3.16)

S

We choose the Green’s function to be Dirichlet Green’s function:
G = GD ≡ GRS− = −1 eikR eikR̃
(
−
)
4π R
R̃

˜ 0 |,~r
˜ 0 is the image of the ~r0 relative to S1 surface. It satisfies:
where R = |~r − ~r0 |, R̃ = |~r −~rGRS− 0=0~r ∈S1

(3.17)(3.18)

∂n GRS− ~r0 ∈S = 2∂n GK S
1
1
where
GK (~r, ~r0 ) = −1 eikR
4π R(3.19)

is the Green’s function used in Kirchhoff diffraction integral.
If we choose S2 to be part of an infinitely large sphere, then with Sommerfeld radiation
condition limr0 →+∞ r0 (∂n0 + ik)ψ(~r0 ) = 0, the surface integral has contribution only from S1 , but
not from S2 . We then get the first Rayleigh-Sommerfeld solution, also called Rayleigh-Sommerfeld
diffraction integral :Z
ψRS− (~r) =S1

d~r0 ψ(~r0 )1
1 eikR z(1 −)iλ R R
ikR(3.20)

where λ is the wavelength; ~r0 is a source position; ~r is the position where we calculate the diffraction.
For practical use, we usually choose S1 to be a flat surface immediately after an object that isbasically a two-dimensional object, the so-called exit surface. The geometry is illustrated in Fig.
3.3.

26

R
y

r

r'
x

z

S1
Figure 3.3: Diffraction geometry.

27
3.2.4

Approximations of Rayleigh-Sommerfeld diffraction integral

While Rayleigh-Sommerfeld diffraction integral shown in Eq. 3.20 is the rigorous solution
for the scalar diffraction problem, it is difficult for numerical calculation. In this section, I will
discuss two situations in which it can be approximated into formulas that are easy for numerical
implementation. Both situations meet the following two conditions:r  r0(3.21)rλwhere r0 = |~r0 |; r = |~r|.
In the first situation, there is an additional paraxial condition:rp
x2 + y 2
4 λ

.
z
z

(3.22)

Then Rayleigh-Sommerfeld diffraction integral Eq. 3.20 can be approximated by Fresnel diffraction
integral :
k 1
k 1
1
2
2 1
02
02
ψ(x, y) ≈ eikz ei 2 z (x +y ) F[ei 2 z (x +y ) ψ(x0 , y 0 )|S1 ] (fx = 1 x ,fy = 1 y ) .
λ z
λ z
i
λz

(3.23)

Here F denotes Fourier transform, and it is evaluated at a spacial frequency (fx , fy ) depending on
the coordinate of the position under consideration (x, y).
In the second situation, the additional condition is the far field condition:r  zR ≡πr02
λ(3.24)

Here I call zR the Rayleigh range of the sample, similar to the definition of “Rayleigh range” for a
Gaussian beam with waist radius r0 . Then Rayleigh-Sommerfeld diffraction integral Eq. 3.20 canbe approximated by what is termed far field diffraction integral in this thesis:
ψ(x, y) ≈eikr 1 z()F[ψ|S1 ] (fx = 1 x ,fy = 1 y ) .
λ r
λ r
i λ r2(3.25)

Surprisingly, this situation is not treated in most textbooks, but it is more useful for the experiments
covered in this thesis work than the frequently mentioned “Fraunhofer diffraction formula” in theliterature. The latter requires both far field condition and the paraxial condition to be true, so it
is limited to low NA situation.

28
3.2.5

2D Scalar diffraction with angular spectrum approach

Besides Green’s function approach, there is another approach to solve the homogeneous
Helmholtz equation Eq. 3.15: the so-called angular spectrum approach [45]. While Green’s function
approach works in the real space, the angular spectrum approach works in the Fourier space. We
decompose the field at a certain plane with constant z into its plane wave components by using twoRR
dimensional Fourier transform: ψ(x, y, z) = F −1 {A(fx , fy , z), x, y} =dfx dfy ei2π(fx x+fy y) A(fx , fy , z).
We plug this in Eq. 3.15, and for the angular spectrum A(fx , fy , z) we get:(∂z2 + fz2 )A(fx , fy , z) = 0(3.26)

where
s
fz =

1
1
1 − ( 1 fx )2 − ( 1 fy )2 .
λ

(3.27)

λThe solution is:A(fx , fy , z) = A(fx , fy , 0)H(fx , fy )(3.28)H(fx , fy ) = ei2πfz z .
q
For sub-wavelength spatial frequencies, or fx2 + fy2 > λ1 , fz is imaginary, and the wave
experiences exponential decaying rather than increase of the phase when propagating along z andis called evanescent wave.

3.2.6

2D vectorial diffraction~ and H,~ as well as their
Boundary conditions on the sample introduces coupling between E
various scalar components. Scalar diffraction theory ignores this coupling and entails some degreeof error. This error is significant when feature size is on the same order of the wavelength. Anotherscenario in which we should probably use vectorial diffraction theory is when polarization matters,such as a reflection geometry with non-normal incidence.
For vectorial diffraction solution, we start from wave equations Eq. 3.6, and with the aid of

29a Green’s function subject to Neumann boundary condition:
~=
A

Z
ds((−
S1

1 eikR~)~n · ∇A)
2π R

(3.29)~ the above equation is equivalent to:
In terms of B,~ = 1 ∇×
B
2π

Z
ds
S1

eikR
~
~n × B
R(3.30)~ and E,~ we have (for z > 0, E
~ =E
~ diff in Ref. [46] Eq. 10.101):
And based on symmetry between B~ = 1 ∇×
E
2π

Z
ds
S1

eikR
~
~n × E
R(3.31)

Under the same conditions we derive the far field diffraction integral Eq. 3.25, the above
vectorial diffraction integral can be approximated by the following far-field vectorial diffraction
formula:
~ =e
E

3.2.7

ikr

1
F[E|S1 ](~n × ˆi ) × r̂i λr

(3.32)

Non-normal incidence

Momentum transfer plays an important role in the evaluation of diffraction. In the framework
constructed in this section, it is a natural result of Fourier shift theorem. For non-normal incidence,the exit surface (S1 : z 0 = 0) wave has a linear phase shift across the plane, which we factor out inthe following way:00

ψ(x0 , y 0 , z 0 = 0) = ei2π(f0,x x +f0,y y ψ0 (x0 , y 0 )(3.33)

Among the several two-dimensional scalar diffraction integrals that can be approximated using
Fourier transforms, we choose the far field diffraction integral Eq. 3.25 for an example. We get:
ψ(x, y) ≈eikr 1 z
00( 2 )F[ei2π(f0,x x +f0,y y ψ0 (x0 , y 0 )] (fx = 1 x ,fy = 1 y ) .
λ r
λ r
i λr

(3.34)

eikr 1 z()F[ψ0 (x0 , y 0 )] (fx = 1 x −f0,x ,fy = 1 y −f0,y ) .
λ r
λ r
i λ r2(3.35)

With Fourier shift theorem:
ψ(x, y) ≈We see that the Fourier transform is performed at the “momentum transfer” frequencies, in otherwords, the final spatial frequency subtracting the initial spatial frequency.

30
3.2.8

3D diffraction with volume integral

This thesis mainly works on 2D imaging; yet, for completeness and comparison with 2D, the3D diffraction is also discussed here. If the volume we choose contains the scatters, as shown inthe second panel in Fig. 3.2, then we would work with the inhomogeneous wave equation Eq. 3.8
with the right-hand side unequal to 0.
~ ω , for D~ ω:
Similar to the way we get Eq. 3.14 for E~ ω (~r) =DI~ ω (~r0 ) − D~ ω (~r0 )∂ 0 G(~r, ~r0 )) +d~r0 (G(~r, ~r0 )∂n0 D
nZ

dr~0 Gf~sD(3.36)VS

where~ × (∇~ × P~ ) − iω0 µ0 ∇~ ×M~
f~sD = −∇(3.37)

With S chosen to be infinitely large sphere, using Sommerfeld radiation condition, the surface
integral would vanish. We choose the Green’s function G = Gk as defined in Eq. 3.19, then under
the conditions:r  r0(3.38)r  zR
we have:ikr~ s (~r) = 1 e k 2 F[f~sD1 ]| ~ 1
D
f = λ r̂4π r

(3.39)

1~
fsD1 = ~n × P~ × ~n − ~n × M
c(3.40)

where~ =D~0 + D~ s , where D0 is the incident wave, and Ds is the scattered save.
The total field D
We then use Born approximation:~s  D~0D(3.41)

which means the scattering wave is much weaker than the indicident wave. For linear and isotropic
medium, weak scattering means χ  1 and χB ≡ µ − 1  0. Then the source term can beapproximated by:
~ ×D
~ 0)
~ 0 × ~n − 1 ~n × (χB c 1 ∇fsD1 ≈ ~n × χDcik

(3.42)

31
Equation 3.39 with the source term approximated by Eq. 3.42 provides the solution ofdiffraction under Born approximation with a 3D integral.
For 3D CDI, there are two approaches. In the first approach, 2D diffraction slices are assembled into a 3D diffraction volume, then images are reconstructed with phase retrieval algorithms
with diffraction formulas similar to the one given above (assembling before reconstructing). Thisapproach is used in Ref. [52] for example. For the X-ray wavelength, the index of refraction is close
to 1; and if the sample is relatively thin as is true in this work, the Born approximation conditionis satisfied, while for other cases, it can fail. Then it is favorable to use the second approach: First
2D projection images are reconstructed at different orientation angles; then using the computed
tomography approach to obtain the 3D structure (reconstructing before assembling). The second
approach is used in several works on ptychographic X-ray computed tomography [53, 54].

3.2.9

Summary of 2D diffraction formulas

Here I give a summary of the 4 key results and their conditions for 2D diffraction problems.
They are Fresnel diffraction integral, far field scalar diffraction integral, angular spectrum diffraction
formula, and far field vectorial diffraction integral, respectively: 1 eikz ei k2 z1 (x2 +y2 ) 1 F[ei k2 z1 (x02 +y02 ) E|
~ S ],


1
i
λz





ikr


~ S ]
 e i λrz 2 F[E|1~
E(~r) =


q


1
1
1
−1
2
2

~
F
exp[i2π λ z 1 − ( 1 fx ) − ( 1 fy ) ] · F[E|S1 ]



λ
λ





 eikr 1 F[E| ](~n × ˆ ) × r̂

i

λr

S1

i

√
Fresenl, r  max[r0 , λ];x2 +y 2

z

q
4

λ
z
02

Far field scalar, r  max[r0 , λ]; z  πrλAngular spectrum,02

Far field vectorial, r  max[r0 , λ]; z  πrλ(3.43)For numerical calculations, there might be more constraints on their applicability range. For
example, although the angular spectrum diffraction formula is a rigorous across the whole space,difficulty is met when it is applied to far field. This is because the transfer function has fast
oscillation in the phase and cannot be well sampled in this situation.

32
For Fresnel diffraction integral, it is convenient if the diffraction plane has a field of view atleast that of the sample plane; in other words: λz/∆xSmp > DSmp . Soz > zF ≡

∆xSmp
DSmp
λ

(3.44)

The applicability of different diffraction formulas is shown in Fig. 3.4.

3.3

Iterative algorithms for diffraction phase retrieval
This section discusses the second part of the theories behind CDI: the iterative algorithms

that retrieve the phase of diffraction patterns. The basic idea is illustrated in Fig. 3.5. Each fulliteration cycle, or updating cycle, is composed of two successive single projections: the projection
onto the detector plane constraint set, or the modulus constraint set, Pm ; and the projection onto
the sample domain constraint set Ps .
Both single-diffraction CDI and ptychography CDI makes use of the same detector-plane
constraint: the modulus constraint, meaning that the modulus of the diffracted field has to be equal
to what is measured by the detector. They differ in the sample plane constraints: single-diffraction
CDI makes use of finite support constraint while ptychography CDI uses “overlap constraint”.
In this section, each of these three constraints and its corresponding projection is investigatedseparately first. Then algorithms that assemble two successive single projections into a full iteration
cycle are presented.
For ease of discussion, the relationship between the sample plane and the detector plane canbe expressed as:
U =Tu
where
• u is a quantity related to the sample to be reconstructed;• U is a quantity related to the detected diffraction intensity (or counts);• T is a transform, or a propagator that calculates U from u.

(3.45)

33

Figure 3.4: Applicability of different diffraction formulas for numerical calculations. Angular spectrum formula performs well in the near field; Fresnel diffraction integral performs well after zF for
low numerical aperture; Far field diffraction integral, both scalar and vectorial, should be preferred
for the far field region for any numerical aperture.

34

a

c

y

Random phase

Randominitial guess

Solutionxb

Detector plane
ConstraintU'u'

Solution

U

T

T-1

u

Detector plane
Constraint

Sample plane
Constraint

Sample plane
Constraint
Randominitial guess

Figure 3.5: Illustration of the iterative phase retrieval algorithms used in CDI. (a) A simple example
for solving two linear equations. A random staring point, after successive projections to one andthen the other line, finally will arrive at the intersection which is the solution subject to the twoconstraint sets represented by the two lines. (b) In CDI, a random guess is first projected to thedetector plane constraint set, then to the sample plane constraint set to finish a full updating cycle.
After many iterations, the solution is found at the intersection of the two constraint sets. (c) A
simple specific implementation of the approach of iterative projections in CDI. The phase is notmeasured on the detector, so a random phase distribution is assigned to the measured modulus toform EDetector as a staring point. The complex wave is then propagated to the sample plane using
a transform T based on diffraction theory. Then the sample plane constraint is applied. The new
sample field ESample is then propagated to the detector plane with T . The phase of the updated
EDetector is kept while its modulus is replaced by the measured one, to form a new wave on thedetector. This process is repeated until the algorithm converges.

35
3.3.1

Projection onto the modulus constraint set on the detector plane

For Pm , assuming u is the exit surface wave (here we focus on two-dimensional diffraction)
to be reconstructed, projection to the modulus constraint set simply requires the modulus of U to
be replaced with the measured modulus, |U |m . This projection, or updating rule, can be expressedas:
Pm u = T −1 [|U |m exp{iφ(T [u])}](3.46)

Basically, we propagate u from the sample plane to the detector plane to get U , keep its phase
while replacing its magnitude with the measured values, and then propagate back to finish thissingle projection.

3.3.2

Projection onto the support constraint set on the sample plane in singlediffraction CDI

For Ps in single-diffraction CDI, it is the projection to the support constraint set: the exit
surface wave u has to have a finite support S, outside which u = 0. The realization of this projection
Ps,Support is very straightforward: we simply set u outside S to be zero:Ps u = S · u(3.47)

where S is the mask for the support; it is 1 inside the support and 0 outside the support. There
are several ways to obtain S. It can be from an image of the sample from a lower resolution,
optical microscope, although this method requires a separate measurement. It can be from the
autocorrelation, which is at most twice the object size. Marchesini [55] devised an automatically
updating method, the “shrink-wrap” method that works really well. In this method, the support iscreated by thresholding the amplitude |u| at a certain level (4%, for example) followed by convoving
with a Gaussian function.
In practice, the finite support constraint alone does not work well enough. Indeed, it issimply not a powerful enough constraint. Combinations with other sample-plane constraints basedon some a priori information greatly enhances the convergence of the iterative reconstruction. One

36
typical example of this extra sample-plane constraint is the positivity constraint, or more generally,
the phase constraint: the phase of the sample or the exit-surface wave is within a certain range. In
Ref. [27], I provided a unified approach that combines the projections to the conventional amplitude
constraint and the phase constraint in the sample plane. Besides the amplitude support, one cancreate an additional “phase support” SPhase . For pixels that have their phase within the known
range, SPhase = 1, and otherwise it is 0. We use the new combined support, which is the productof S = SAmplitude and SPhase , instead of only SAmplitude when doing projection Ps as shown in Eq.
Ps . For objects for which phase variation is much less than 2π, or any essentially binary object,
this approach can be readily applied. In our keyhole CDI results shown in Chapter 4, we used thisapproach in our reconstruction.

3.3.3

Projection onto the “overlap” constraint set on the sample plane in ptychography CDI

Ps in ptychography CDI is the projection to the “overlap” constraint set: for any two diffractions at two scan positions that have overlap, the retrieved phases have to yield reconstructedsample images that are the same in the overlapped region.
For the transmission geometry, specifically for j th scanning position Rj , Eq. 2.4 can be
written as:~ j)
Et,j (~r0 ) = Ei (~r0 ) · t(~r0 − R(3.48)∗ 0~ j ) = Ei (~r ) Et,j (~r0 )t(~r0 − R
|Ei (~r0 )|2(3.49)

orwhere Ei∗ means the complex conjugate of Ei .
There can be two types of ptychographical projection derived from the above simple equation:
the parallel projections algorithm [42], and the extended ptychographical iterative engine (ePIE)[56]. Both the sample and the incident beam can be reconstructed by ptychography algorithms.
(1) In parallel projections algorithm, t is updated using data from all the positions by a single

37
step:

P

~ j )Et,j (~~j)
Ei∗ (~
r0 +R
r0 +R
0

t(~r ) = j P
0 ~2
r +Rj )|
j |Ei (~


Ei (~r0 ) =P(3.50)·~ j )]r0 ) t∗ (~r0 −R
j [Et,j (~P
0 −R~ j )|2|t(~rj(2) In ePIE, t and Ei are updated at one scanning position after another, in a serial version.
∗ r0 )

i (~tn+1 (~r0 − Rj ) = tn (~r0 − Rj ) + α maxE|E[Et,j,n+1 (~r0 ) − Et,j,n (~r0 )]r0 )|2i (~(3.51)

∗ (~
0 −R
~

t
r
)
j
Ei,n+1 (~r0 ) = Ei,n (~r0 ) + β
[Et,j,n+1 (~r0 ) − Et,j,n (~r0 )]~ j )|2
max |t(~r0 −Rwhere α and β are relaxation parameters; subscripts n and n + 1 means the nth and n + 1th
iteration.
In this section, Et plays the same role as the previously used symbol u, both meaning theexit-surface wave. With the updated Ei and t provided above, their multiplication will provide theupdated Et or u, completing a full single projection Ps,Ptychography . It is not explicitely given here,
as it is more convenient to use the above updating formulas Eq. 3.50 or Eq. 3.51.
The above discussion in the transmission geometry can be easily extended to the reflection
geometry by replace t with r.

3.3.4

Combining two projections for a full iteration cycle

Given projections onto a single constraint set, either on the detector plane, or on the sample
plane, different algorithms exist to combine the two successive consecutive projections to form
a full iteration of reconstruction (or updating): error reduction (ER) [25], hybrid input-output(HIO) [25], relaxed averaged alternating reflections (RAAR) [57], and difference map [26]. The
combined projections for these different algorithms are presented in Table 3.3.4. A good review
and comparison of these different algorithms can be found in Ref. [58] by Marchesini. In thispaper, Marchesini also gave an intuitive explanation of how these projections work, by providing
a geometric representations, with two straight lines representing the two constraint sets, and theirintersection representing the desired solution. This geometric representation is shown in Fig. 3.6

38
Table 3.2: Summary of different algorithms for two successive projections. β is a relaxation parameter. [P1 , P2 ] can be either [Pm , Ps ] or [Ps , Pm ].
Algorithm
Error reduction
Hybrid input-output
Relaxed averaged alternating reflections
Difference map

Combined projection
P1 P2((1 + β)Ps − β)Pm + 1 − Ps1
2 β((2Ps − 1)(2Pm − 1) + 1) + (1 − β)Pm
1 + {P1 [(β + 1)P2 − 1] − P2 [(β − 1)P1 + 1]}

In this thesis work, for single-diffraction CDI, the RAAR algorithm is used most frequently;
while for ptychography CDI, ePIE projection Eq. 3.51 combined with the error reduction algorithm
is chosen for image reconstruction.

3.4

Numerical implementations
To numerically implement the phase retrieval algorithms described in the previous section,

it is desirable to find u, |U |m (magnitude of U from the measurement), and T that satisfies:|T u| = |U |m(3.52)

where(1) the whole equation U = T u is equivalent to one of the diffraction formulas given in Eq.
3.43.
(2) U is some quantity that can be derived from the counts on the detector;(3) u is some quantity that relates to the sample;(4) T is a transform that can be realized by the Fast Fourier Transform (FFT) algorithm whichis computationally efficient.
To achieve this, it is first necessary to establish the relationship between the detector counts~ since the diffraction formulas in Eq.
distribution N with the electric field complex amplitude E,
3.43 are written in terms of electric field.

39

Figure 3.6: Geometric representation of combined projections onto two constraint sets that are
represented by two lines intersecting. (a) Error reduction algorithm. (b) HIO algorithm. (c) Error
reduction algorithm with a reflector rather than a projector. (d) Difference map algorithm. The
symbol ρ here plays the same roles as u in the text. Figure adapted from Marchesini [58].

40

N (~r) = η∆tp2 n̂ ·

1 ~
S(~r)
hν

(3.53)

where N (~r) means the count on a detector pixel located at position ~r, η is the efficiency for thedetector pixel to convert a photon to a count, ∆t is the exposure time, p is the width of the pixel
which is assumed to be square, n̂ is the normal vector of the detector, ν is the frequency of the~ is the time-averaged Poynting vector of the beam. Sincebeam, and S~ r) = 1 |E(~r)|2 k̂
(3.54)
S(~
2Z0
p
where Z0 is the impedence of free space µ0 /0 ≈ 377 Ω; k̂ is normalized wave vector. Thenss11
N (~r)|E(~r)| =(3.55)
1 1 2
η hνpn̂CCD · k̂ ∆t
2Z
0

Now that the relationship between the directly measured counts N and the electric field
amplitude |E| is established, we can combine it with the diffraction formulas shown in Eq. 3.43.
Here we choose the far field scalar diffraction formula for discussion, and the extension to otherformulas is straightforward.

|EDet (~r)| =

r

1
1
1
p2
η hν
2Z0

q

N (~
r)
1
n̂CCD · k̂ ∆t(3.56)= [ 1i eikr ] λrz 2 F[ESmp ] fx ≡ 1 x −f0x ,fy ≡ 1 y −f0y
λ zds

λ zds

It is then useful to examine the relationship between Fourier transform and the FFT algorithms used in numerical calculations. In MATLAB, for example, it is evaluated with the function“fft2” for the two-dimensional case. It is useful to see the exact relationship between Fourier transform and what FFT provides. We consider a two-dimensional complex function g(x, y), sampled
on a grid with uniform distance in both x and y directions:xjx = x1 + (jx − 1)∆x, jx = 1, 2, . . . Nx
yjy = y1 + (jy − 1)∆y,(3.57)

jy = 1, 2, . . . Ny

Then, the Fourier transform can be calculated with “fft2” using:F[g(x, y)] = ∆x∆y[fftshift fft2 ifftshift g(x, y)](3.58)

41
where “fftshift” is the function shifting the zero frequency to the center of the grid, and “ifftshift”
is inverse function. The corresponding spatial frequency grid is:fx,jfx = fx,1 + (jfx − 1)∆fx , jfx = 1, 2, . . . Nx ;(3.59)

fy,jfy = fy,1 + (jfy − 1)∆fy , jfx = 1, 2, . . . Nx .
N

Here ∆fx = Nx1∆x , ∆fy = Ny1∆y , fx,1 = −floor[ N2x ], fy,1 = −floor[ 2y ]. For convenience, we define
FT2 ≡ fftshift fft2 ifftshift .

(3.60)

F[g(x, y)] = ∆x∆yFT2 [g(x, y)](3.61)

Then Eq. 3.58 becomes:

Using the derived relationship shown in Eq. 3.61, Eq. 3.56 can be rewritten as:s
1 N (~r)
λr2
FT2 [cNFT ESmp ] fx = 1 x −f0x ,fy = 1 y −f0y =(3.62)λ r
λ r
zn̂ · k̂ ∆tq
1 1 2
where the constant cNFT = ∆xSmp ∆ySmp η hν
2Z0 p So for far field scalar diffraction, u, Um , andT in search are:(1) u = cNFT ESmp (proportional to ESW ESmp ) immediately the sample plane.
(2) Um = λrz

2

q

r)
1 N (~; After this calculation from N (~r), it should be then remapped from an̂ · k̂ ∆tunequally spaced spatial frequency grid fx = λ1 xr − f0x , fy = λ1 yr − f0y to a chosen uniformly
spaced spatial frequency grid which is required by FFT. This procedure is described inRef. [59]. More recently, I together with others have filed a provisional application for apatent on a faster and potentially more accurate version [60]. The increased speed offeredby this version is important for fast diffractive imaging, especially real-time diffractive
imaging.
Usually some pre-processing steps for the raw measured diffraction patterns need to happen
beforehand, such as removing the dark frame background, removing hot pixels due tocosmic rays with median filtering, and combining short(unsaturated) and long(saturated)
exposures for high dynamic range (HDR).

42(3) T = FT2 .
We will see in Chapter 4 that u and T can be slightly modified so that the amplitude orphase support constraint can be directly imposed.
The remapping used in obtaining Um can be easier to understand with the help of Ewald
sphere analysis. Figure 3.7 illustrates the way that spatial frequencies mapped onto the detector gridfor three different situations: normal incidence with low numerical aperture, normal incidence withhigh numerical aperture, non-normal incidence with high numerical aperture. We see that alongwith the increasing of the numerical aperture (or scattering angle), the mapping from frequency tothe detector grid is not linear anymore. The diffraction pattern needs to be remapped to a linear
grid to be compatible with the FFT algorithm.

43

a

r+
y

x

Sample+zSample

b

r-

-

g

h

i

j

Detector

r+

c
y

x

d
+

z

Sample-rDetector
r+

e
x

+

y
z

Samplef-r-

Detector

Figure 3.7: Illustration of the mapping from spatial frequencies to the detector grid for threedifferent situations. (a) Normal incidence with small angle scattering. (b) Simulated diffraction
pattern for a square mask function. The mapping from spatial frequency q± to the detector gridis symmetric and linear. (c) and (d) are for normal incidence but with high angle scattering. The
mapping from q± to the detector grid is nonlinear but still symmetric. (e) and (f) are for nonnormal incidence, high angle scattering. The mapping is nonlinear, and asymmetric. The calculated
nonlinear frequency grid is shown in (g), and the chosen equally spaced spatial frequency grid isshown in (h). The measured diffraction pattern (i) leads to a corrected one shown in (j), after
remapped from grid (g) to (h), to be compatible with the FFT algorithm.

Chapter 4

Keyhole CDI for Extended Samples in Transmission

Single-diffraction CDI requires the sample to be on a finite support, or isolated, which presents
a severe limitation to this imaging method. Several techniques have been developed to overcome
this limitation, including ptychography CDI [61], keyhole CDI [62], and apertured illumination
CDI [59]. As we have seen previously, in ptychography CDI, adapted from an electron beamtechnique and later demonstrated using a synchrotron light source [61], a beam is scanned acrossan object. The additional information provided by the overlap between adjacent scans is then used
to reconstruct one large field-of-view image. Keyhole CDI, first demonstrated using light from asynchrotron source, uses a FZP focusing optic with smaller diameter than the incident beam toconfine the illumination on the sample [62]. Finally, apertured illumination CDI projects an imageof an aperture onto the sample plane to ensure isolated illumination. This technique requires highquality focusing optics and also a precise determination of the imaging plane of the aperture, andhas been demonstrated using a He-Ne laser [59].
In this chapter, initial attempts for imaging extended samples with apertured illumination
CDI are described. While we achieved success with visible light, we were not able to implement
this approach with HHG sources, as this technique is more demanding than keyhole CDI. The
next following sections will focus on our implementation of the keyhole CDI technique with a HHG
source. It enabled the first general tabletop EUV coherent microscope that can image extended,
non-isolated, non-periodic, objects. By implementing keyhole coherent diffractive imaging withcurved mirrors and a tabletop high harmonic source, we achieve improved efficiency of the imaging

45
system as well as more uniform illumination at the sample, when compared with using Fresnel zone
plates. Moreover, we show that the unscattered light from a semi-transparent sample can be used asa holographic reference wave, allowing quantitative information about the thickness of the sample
to be extracted from the retrieved image. Finally, we show that excellent tabletop image fidelity
is achieved by comparing the retrieved images with scanning electron and atomic force microscopy
images, and show superior capabilities in some cases.

4.1

Apertured illumination CDI: first attempts for imaging extended samples
As discussed before, conventional CDI relies on the samples to satisfy the “isolation con-straint”, or “support constraint”, i.e., features exists in a small, isolated region, outside which, theelectric field is zero. Instead of requiring the sample to be isolated, we transform the beam so it
is isolated in a finite area, and ideally has a relatively sharp edge to form a top-hat shape. Our
first attempt to implement this idea is to directly image an aperture to the sample plane, which weterm “apertured illumination CDI” (AICDI).
In order to implement and test the AICDI technique and data processing algorithms, we firstdeveloped a proof-of-concept system using a polarized 632.8 nm HeNe laser. A schematic diagramof the setup is shown in Fig. 4.1(a). First we spatially filter and collimate the beam to overfill a
300 µm wide circular aperture. The aperture is imaged to the sample plane using a one-to-one 4f
imaging system. A positive lens placed directly after the sample sends the scattered light into the
Fourier plane at the CMOS detector (Mightex Systems MCE-B013, 5.2 µm pixel size). In general
the positive lens after the sample is unnecessary, however, the detectors used in this experimentwere small enough that a demagnification of the far field was required in order to use a wavelength
as large as 633nm. The aperture size is selected to satisfy the oversampling criterion, where thedistance between the lens and the detector is 11.6 mm, corresponding to an NA of 0.22 for withour 5.3 mm diameter detector.
We illuminated a bundle of suspended, 26 µm diameter copper wires with an image of theaperture (Fig. 4.1(a) inset). An example of a scatter pattern obtained is shown in Fig. 4.1(b). The

46

Figure 4.1: The apertured illumination CDI scheme isolates an extended transparent sample byimaging a aperture onto the sample plane. (a) A schematic of the setup, including a traditional
bright-field microscope image of the sample. (b) The scatter pattern recorded by the detector in the
Fourier plane scaled by the fourth root. (c) The exit surface wave reconstructed from the scatterpattern. The circle outlines the area illuminated by the aperture. (d) A reconstruction when theillumination is subtracted out. (e) Overlay of the images from many scan positions. The circles
represent the outline of the aperture illumination at different scan positions. Figure reproduced
from Gardner et al. [59].

47
scatter pattern we obtain is proportional to the modulus of the Fourier transform of the illuminatedportion of the sample. We use the RAAR algorithm as outlined in Ref. [57] (including the correct
form of non-negativity, a fast ramp in the feedback parameter and a modified amplitude constraint)with “shrinkwrap” [55] to recover the phase. With the recovered phase, we are able to reconstruct
the exit surface wave (Fig. 4.1(c)). This technique constitutes a bright-field imaging microscope. If
instead we first record the scattered light with the sample removed (leaving the rest of the system
unchanged) and reconstruct the electric field at the image plane of the aperture, then during thereconstruction process we can subtract the complex amplitude of the illumination at the detector
plane per iteration. Subtracting out the illumination results in the wires appearing bright insteadof dark (Fig. 4.1(d)); this constitutes a dark-field microscope. By scanning the sample in a plane
perpendicular to the optical axis, we are able to reconstruct different areas of the extended sample
independently. With overlap between the scan positions, we can register adjacent reconstructions
to build up a large-area, high-resolution image. Figure 4.1(e) shows a number of reconstructionsof this sample, which were overlaid in post-processing. Colored circles indicate the area that was
illuminated by the imaged aperture for each individual reconstruction.

4.2

Keyhole CDI for extended samples
In keyhole CDI, an aperture is inserted into the beam to isolate the illumination, similar toapertured-illumination CDI. However, it does not require the aperture to be exactly imaged onto
the sample plane. Instead, the sample is put wherever there is significant phase curvature, whichhelps for the robustness of reconstruction [62, 63]. This techniqure requires the phase of the beam
to be characterized carefully.
First, we demonstrated keyhole CDI with a tabletop HHG EUV source using a sample with
an opaque background. A schematic diagram of the setup in the imaging chamber is shown in Fig.
4.2(a). We focused a 25 fs pulse duration, 1 mJ pulse energy, 3 kHz repetition-rate, Ti:sapphirelaser beam with wavelength centered around 780 nm, into a 150 µm diameter hollow waveguidefilled with Ar gas to generate phase-matched, spatially coherent, harmonics at wavelengths near 29

48
nm. After the waveguide, the HHG beam and CDI microscope were in medium vacuum (≈ 10−6torr) to avoid absorption of the EUV light from air. We used a pair of silicon mirrors oriented
at near the Brewster’s angle for the 780 nm light to effectively absorb the unconverted 780 nm
light while reflecting the EUV beam. The 780 nm light was further filtered out by two 200 nm
thick aluminum filters. The harmonics of only odd orders [18] are separated in energy by about
3 eV (twice of the fundamental 780 nm photon energy), and only one harmonic (27th order) was
selected and focused on the sample with a flat and a curved EUV multilayer mirror (radius ofcurvature 25 cm). The EUV mirrors have a full width at half maximum (FWHM) bandwidth of 2.1
eV centered around 43.2 eV (corresponding to a wavelength of λ = 28.7 nm). Typically, a single
EUV harmonic at this photon energy has a FWHM bandwidth of about 0.4 eV (a representative
spectrum of EUV harmonics can be found in [19]), corresponding to a longitudinal coherence lengthof Lc = λ2 /∆λ = 3.1 µm. This means that for a field of view (FOV) D at the sample, the maximum
scattering angle θmax , limited by finite Lc , is determined by tan θmax = 2Lc /D (See [64]. Noticehere we use a different definition of Lc , and do not use the approximation of tan θ ≈ θ when θ issmall). For all the experiments performed in this paper, D ≤ 25 µm. For D = 25 µm, the NA isrequired to be ≤ 0.3, which is greater than the experimental NA used in this section (0.23) as wellas that in next section (0.20).
The EUV mirror pair has generally higher throughput than the FZP used in conventional
keyhole CDI, while also inducing a curved wavefront that is very beneficial for the reconstruction
algorithm [65]. The non-normal incidence of the HHG beam on the EUV curved mirror producessome astigmatism in the beam. A 50 µm diameter pinhole was placed about 2 mm before thehorizontal focus (x-focus) to introduce a sharp edge on the HHG beam and enforce the isolation
requirement on the illumination, rather than the sample. We measured the positions of the x-focus
and y-focus with ±50 µm accuracy by use of the pinhole as a knife-edge scanner. The separationof x-focus and y-focus was measured to be ≈ 0.55 mm. The sample was composed of a 100 nm
thick gold layer (which has a negligible transmission of 5 × 10−5 for 28.7 nm light), deposited on athin Si3 N4 membrane. Features were etched into the gold layer, as shown in the dark areas in Fig.

49

(a)

CCD

sampleflat
EUV mirror

(b)

II

pinholeI

curved
EUV mirror

(d)

(e)

(f)

1

0.5

I

5 μm -1

I

10 μm

II

5 μm -1II

10 μm

Amplitude [a. u.]

(c)

10 μm

0

Figure 4.2: Tabletop EUV keyhole CDI of a sample with an opaque background. (a) Setup in
the CDI vacuum chamber. The EUV mirror with a 25 cm radius of curvature focuses the HHG
beam, and puts a curved wavefront on the sample. A pinhole placed before the x- and y-fociintroduces a sharp edge onto the beam. (b) Image of the sample using a high magnification optical
microscope. Two regions of interest, I and II, are circled, with corresponding measured diffraction
patterns (cropped and centered) shown in (c) and (e), and their corresponding reconstructions of
the electric field amplitude (normalized to unity at maximum with arbitrary units) shown in (d)and (f). The color map as shown on the right of (f) is shared by (d) and (f). Figure reproduced
from Zhang et al. [27].

50
4.2(b). Keyhole CDI enables the imaging of any region of interest on such an extended sample;
here we selected region I and region II as indicated in Fig. 4.2(b). The imaging FOV can be
adjusted by placing the sample at different distances from the focus positions, which corresponds
to different HHG beam spot sizes. For the measured diffraction shown in Figs. 4.2(c) (for regionI) and 4.2(e) (for region II), the sample was placed 1.3 mm and 0.9 mm downstream of the circleof least confusion (the midpoint of x- and y- foci) respectively, resulting in a FOV of D ≈ 25 µm
and D ≈ 18 µm. The diffraction patterns were recorded on an X-ray CCD (Andor iKon-L, 2048
× 2048 pixel array, 13.5 × 13.5 µm2 pixel size), as shown in Figs. 4.2(c) and 4.2(e), with a total
exposure time of 30 minutes for each. The CCD was positioned at a distance of 44.6 mm from thecircle of least confusion.
For experiments performed in this section, NA ≈ 0.23, and the calculated D/NA is 104 µm
for Fig. 4.2(c) and 75 µm for Fig. 4.2(e), both of which are much greater than the thickness of thesample (≈ 150 nm), so Eq. (2.4) is valid. The electric field at (x, y) position on the detector planeis given in the paraxial Fresnel approximation by
2πei λ z i π (x2 +y2 )Et,Det (x, y) =
e λz
iλz

ZZ ∞

2π

0

0

π

02

02

e−i λz (xx +yy ) ei λz (x +y ) Et,Smp (x0 , y 0 ) dx0 dy 0 ,

(4.1)

−∞

where z is the distance between the sample and the detector [45]. In the case of a binary sample
with an opaque background level as used in this experiment, if we let tc be the constant transmission
value for the feature area, then t(x0 , y 0 ) can be written as t(x0 , y 0 ) = |t(x0 , y 0 )| exp(iφtc ), where φAdenotes the phase of a complex quantity A. Then Eq. (4.1) is equivalent to
π

02

02

iφEi,Smp (x0 ,y 0 )U = F[ei λz (x +y ) eu(x0 , y 0 )],

(4.2)

1
π
2
2
−1
where U is defined as U = Et,Det (x, y){ iλz
exp[i 2π
λ z + i λz (x + y ) + iφtc ]} , having an amplitude

proportional to the measured |Et,Det |, F is the Fourier transform, and u(x0 , y 0 ) = |Ei,Smp (x0 , y 0 )t(x0 , y 0 )|is the quantity to be reconstructed. This equation provides the required transform that relates thedetector plane to the sample plane. The precise characterization of the incident beam Ei,Smp is
possible through techniques such as ptychography CDI [42]; in this paper, we use an approximation:

51
we consider only the quadratic phase (including astigmatism) while ignoring higher order phases,
thus Ei,Det = |Ei,Det | exp[i πλ (x2 /zdfx + y 2 /zdfy )], where |Ei,Det | is the measured amplitude of thebeam (with the sample out of the beam path) on the detector, zdfx is the distance between thedetector and the x-focus, and zdfy is the distance between the detector and the y-focus. We thenback-propagated Ei,Det to the sample plane to obtain Ei,Smp .
Reconstruction of u was conducted using a modified RAAR algorithm [66]. We started froman initial guess of random phase for the diffracted wave, and each iteration was composed of thefollowing three steps:(1) Calculate u using the inverse transform for Eq. (4.2):πu = exp[−i λz(x02 + y 02 ) − iφEi,Smp (x0 , y 0 )]F −1 U ;(2) Apply the support constraint provided by the finite illumination, and a constraint of thephase of u to be within [−π/4, π/4] rad, which is equivalent to the non-negativity constraint[26];(3) Calculate U using the transform defined in Eq. (4.2) and apply the modulus constraint.
The algorithm iterates until a suitably low error is achieved or more importantly, the derivativeof the error reaches a constant value; usually below 100 iterations. After the algorithm converged,
100 iterations were averaged together to produce the object domain reconstructions, as shown in
Figs. 4.2(d) and (f) for region I and II respectively. Based on Abbe Theory [67], the theoreticalresolution achieved in this first demonstration is 0.82λ/NA = 102 nm, which can easily be improvedin the future by use of a shorter wavelength illumination or by increasing the NA.
The original implementation of keyhole CDI [62] made use of a FZP with a smaller diameter
than the incident beam to constrain the extent of the illumination on the sample. A central
beam-stop is required in this case, in order to prevent any unfocused light from illuminating thesample. This results in an annular beam. In the implementation of keyhole CDI discussed here, the
FZP is replaced by a curved EUV multilayer mirror, which is generally more efficient. A detailed

52
discussion of the efficiency of EUV zone plates can be found in [68]. An ideal (without fabrication
imperfections, with no substrate that introduces extra absorption) FZP of alternately opaque andtransmissive zones has a theoretical efficiency (diffracted into +1 order) of 1/π 2 ≈ 10%; while for
an ideal phase reversal zone plate, the theoretical efficiency is 4/π 2 ≈ 41%. Due to absorptionof materials for the EUV wavelength, it is not possible to build a genuine phase reversal zone
plate with transparent phase-shifting zones. Based on the results and with materials that “seemsuitable” from [68], for 28.7 nm as used in this paper, an ideal FZP made of aluminum, assuming
no oxidation in addition, has a calculated efficiency of 28%; while for 13 nm (the wavelength usedin EUV lithography), an ideal FZP made of beryllium has a calculated efficiency of 24%. The
efficiency will be even less for implementation in keyhole CDI due to the use of the central beamstop. In comparison, our 28.7nm EUV mirror has a measured efficiency of 47%, and the 13 nm
EUV mirror has a measured efficiency of 66%. In addition to the improved efficiency, EUV-mirrorbased keyhole CDI does not require a central beam-stop; therefore the sample is more uniformly
illuminated. To our knowledge, these results represent the first demonstration of a general andefficient tabletop coherent EUV microscope that can image extended (i.e. non-isolated) samples,
as well as the first demonstration of keyhole CDI using a tabletop EUV source.

4.3

Keyhole CDI for semi-transparent samples
Next we performed EUV-mirror-based keyhole CDI on a sample with a semi-transparent

background (Fig. 4.3). The HHG source and EUV mirrors were similar to those used in theprevious measurements but with some improvements, such as the use of a higher laser repetitionrate of 5 kHz, and a larger, 200-µm-diameter waveguide. These improvements resulted in an
enhancement of the HHG flux from ≈ 109 to ≈ 1010 photons per second in a single harmonic at theexit of the waveguide. For the sample used in this section, which has less scattering efficiency thanthe previous sample, these improvements decrease the required exposure times for this experimentfrom about 2 hours to 14 minutes.
The second sample consists of 30 nm chromium deposited on a 45-nm-thick Si3 N4 membrane.

53
The patterned features, etched in the Cr/Si3 N4 sample by use of focused ion beam, are shown in
the inset SEM image of Fig. 4.3(a). Unlike the first sample, this sample was ≈ 8.5% transparent.
As a result, the diffraction pattern contains a large amount of un-scattered light.
As discussed above, the non-normal incidence of the EUV beam on the curved mirror introduced astigmatism, with a separation between horizontal and vertical foci of 1.5 mm in thiscase. To implement keyhole CDI, a 200-µm-diameter pinhole aperture was placed in the beam 16
mm upstream of the circle of least confusion. Due to the transparent nature of this sample, thescattered light from the aperture was of similar amplitude to that from the sample and had to be
removed. To accomplish this, a second 50 µm diameter pinhole aperture was placed between thefirst aperture and the sample to spatially filter most of the unwanted scattered light from the hardedge of the first pinhole (see the sketch of the experimental setup in Fig. 4.3(a)). This second
aperture was placed 1.4 mm upstream of the circle of least confusion. As shown in Figs. 4.3(b)and 4.3(c), the second aperture removed the majority of the unwanted light scattered from the firstaperture. The sample as positioned at the circle of least confusion, where the illumination spot
size was 8 µm in diameter. The detector was placed 5.71 cm away from the sample. The measured
diffraction patterns of the sample and the beam are shown in Fig. 4.3(d) and its inset, respectively,
corresponding to an NA = 0.20, leading to a theoretical resolution of 0.82λ/NA = 118 nm.
In this experiment, the thickness of the sample(≈ 75 nm) is much less than D/NA = 40
µm, so Eq. (2.4) is again valid. We write the total complex transmission function as: t(x0 , y 0 ) =t0 (x0 , y 0 ) + ∆t(x0 , y 0 ), where t0 (x0 , y 0 ) is the transmission coefficient of the background unpatterned
Cr/Si3 N4 layers of the sample (≈ 0.29), and ∆t(x0 , y 0 ) is the modification to the transmissionof the sample due to the etched features. This reconstruction approach is similar to that used
0

0

,y )
π
in Fresnel CDI [63]. We write U = F{exp[i λz(x02 + y 02 )]Ei,Smp (1 + ∆t(xt0 (x0 ,y 0 ) )} where U has amagnitude proportional to the measured magnitude of the electric field at the detector, and ∆t/t0
is the quantity to be reconstructed in the iterative algorithm. We first calculated the incident field
Ei,Smp using the same approach explained in [69], with the sagittal and tangential slices throughthe propagated beam shown in Figs. 4.4(a) and 4.4(b). The quantity ∆t/t0 is non-zero only in the

54

(a)

CCD
flat
EUV mirror

(b)

(c)

pinhole 2(d)samplepinhole 1
curved
EUV mirror

1μm

5 μm -1

Figure 4.3: Tabletop EUV keyhole CDI of a sample with a semi-transparent background. (a)Schematic of the setup. A second pinhole is inserted into the beam to remove scatter light form thefirst pinhole. The inset shows an SEM image of the sample, composed of a 30 nm Cr film deposited
on top of a 45-nm-thick Si3 N4 membrane. (b) and (c) A zoomed view of the beam on the CCD
before and after inserting the second pinhole. (d) Diffraction pattern (cropped and centered) fromthe sample shown to the 1/4 power. The inset shows the diffraction pattern of the beam when thesample is removed. Figure reproduced from Zhang et al. [27].

55

(a)

Sagittal (c)Top(e) Amplitude

1

0.5

(b)

Tangential (d)Amplitude [a. u.]

1μm

0Bottom

(f)

Phase
2

73 mm

Phase [rad]1

0

Figure 4.4: Image reconstruction results for a sample that has a semi-transparent background
Image reconstruction results for a sample that has a semi-transparent background. (a),(b)Sagittal and tangential slices through the focusing EUV beam. The dashed lines show thepositions of the two foci. (c) SEM image of the top side of the sample (geometrically scaled toaccount for a 52◦ tilt of the sample plane). (d) SEM image of the bottom side of the sample.
Only the darkest parts on the sample are completely etched through. (e), (f) Reconstructed
0 0
amplitude (normalized to unity at maximum with arbitrary units) | ∆tt0 | and phase φt (x , y ) of thesample. (d) (e) (f) have the same scale bar as (c). Figure reproduced from Zhang et al. [27].

56
feature areas, allowing us to use the shrink-wrap dynamic support constraint [55]. An additional
constraint on the phase of ∆t/t0 to be within [φ0 − ∆φ/2, φ0 + ∆φ/2], where φ0 = 2.0 rad and∆φ = π/2 rad were determined empirically, was found to significantly speed up the convergenceof the iterative reconstruction. The magnitude of t0 was determined from the ratio of the beam
intensity on the CCD with the sample’s Cr/Si3 N4 layers in the beam and that of the sample outof the way. Starting with an initial guess of random phase on the detector, we typically used
20 to 100 iterations of the RAAR algorithm [57] followed by 10 iterations of the error reduction
algorithm [25] to retrieve the sample diffraction phase on the detector. We averaged over 10
0 0
independent reconstructions, and the amplitude (normalized) | ∆tt0 | and phase φt (x , y ) (the phaseof the background is chosen as the zero-phase reference) are shown in Figs. 4.4(e) and 4.4(f). They
are in very good agreement with the SEM images shown in Figs. 4.4(c) (top) and 4.4(d) (bottom).
Moreover, it was also possible to reconstruct the image of the small 50-nm-diameter hole, seen in
the topside SEM image Fig. 4.4(c).
The ESW can be written as Et = Ei (t0 + ∆t). We see that a non-zero transmission factor t0
produces a reference wave while ∆t produces an object wave for the in-line holography geometry[5, 62]. Thus, for the sample used in this experiment, the diffraction pattern in Fig. 4.3(d) is in
actuality an in-line hologram. However, since there is only a small area on the CCD where thereference wave has significant intensity to produce interference with the scattered wave from thesample, the image from a conventional in-line holographic reconstruction [5, 70] would not have as
high resolution as CDI. Moreover, in-line holographic reconstructions also suffer from twin-image
artifacts [71]. Using the known reference wave, CDI can be used to extract phase information
about the object. Furthermore, with this extracted phase information, and the value(s) of index ofrefraction for the material, we can obtain thickness or depth information.
A reference topography measurement of the sample is shown in Fig. 4.5(a) by use of a Digital
Instruments Dimension 3100 atomic force microscope (AFM), with a probe tip size of 3 nm anda scan step size of 16 nm. For comparison, we then calculate the depth map from the keyhole
CDI reconstruction. For the sample used in this experiment, if we define the top plane (completely

57(b) CDI

Depth
[nm]

(c)

0

-20

-20
-40
-60

0
Depth [nm](a) AFM
1μm

-40

AFM

-60
-80

CDI

-100

(d)

AFM

Cr
Si3N4

(e)

CDI

1μm

75 nm
Figure 4.5: Height comparison between AFM and keyhole CDI. (a), (b) Depth maps of the sample
using AFM and keyhole CDI respectively. (c) Comparison of lineouts along the dashed line in (b)with associated error bars. (d) and (e) 3D profiles of the sample based on the depth values in (a)and (b). The top 30 nm Cr layer and bottom 45 nm Si3 N4 layer are shown in different colors.
Figure reproduced from Zhang et al. [27].

58
unetched) as the zero depth level, and write the index of refraction for EUV as n = 1 − δ + iβ, thenthe phase of t as a function of the depth d (≤ 0) can be written as− 2π δ · d
for − hCr ≤ d ≤ 0,
λ Cr
φt (d) =
− 2π [δ · (−h ) + δ· (d + h )] for d ≤ −h ,
λ

Cr

Cr

Si3 N4

Cr

(4.3)

Cr

where hCr is the thickness of the top Cr layer. The deposited Cr film has a density of 6.55
g/cm3 (8.9% less than the bulk Cr density 7.19 g/cm3 ), determined from an X-ray reflectivity
measurement. By use of the known δCr and δSi3 N4 values at the illuminating wavelength [72], as
well as hCr = 30 nm, we calculated the depth map of the sample from the reconstructed phase φtwith Eq. (4.3). To determine the uncertainty of our quantitative analysis, we considered the major
error sources. First, the ±50 µm uncertainty in determining the positions of the two astigmatic
foci results in an error in the calculated phase of the incident beam and thus the reconstructedsample phase and thickness. Second, the uncertainty in the wavelength (28.7 ± 0.7 nm due to the
bandwidth of the multilayer mirrors) causes uncertainty in the values of index of refraction, leading
to error in the thickness calculation. Third, the imperfect repeatability of reconstructions leads tosmall fluctuations in the calculations. We scanned the focus positions and the wavelength in theiruncertainty range, and for each parameter set, we performed 10 independent reconstructions; fromall reconstructions, we found the maximum and minimum depth at each position, with the meanvalues shown in Fig. 4.5(b). An AFM is a surface measurement device, meaning that those portionsof the sample that are completely etched through will result in an artificially deep measurement.
We find, using -75 nm as the lower threshold, that the AFM image gives the same shape in thecompletely etched regions as the SEM image of the bottom side of the sample, as shown in Fig.
4.4(d). These images confirm that the 50 nm wide hole and the edges of the larger features are notfully etched through. Fig. 4.5(c) compares the depth profiles along the dashed lines in Fig. 4.5(b)between keyhole CDI and AFM. The error bars plotted in Fig. 4.5(c) indicate the maximum andminimum depth values found within the range of uncertainty. We see good quantitative agreement
between the AFM and tabletop EUV keyhole CDI images. The depth of the 50 nm wide hole is59
20 ± 5 nm as measured by AFM, and 28 ± 9 nm from keyhole CDI. The AFM uncertainty of ±5
nm is determined from a two-dimensional grating calibration standard with a 180 nm step height,
and is given for a 95% confidence level.
Keyhole CDI illuminated by EUV high harmonics has several advantages in comparison with
AFM. First, the depth profile from the entire FOV can be measured simultaneously so that point-bypoint scanning is unnecessary. Thus, there is the potential for much higher data acquisition speeds,
limited only by the illumination flux. Second, AFM images can be influenced by nonlinearity,hysteresis, creep of the piezoelectric material, and cross-talk between the x, y, and z axes. In
practice, software enhancement and filtering are used to improve AFM image quality, but this postprocessing can also flatten out real topographical features. Keyhole CDI has no such problems.
Third, keyhole CDI allows a long working distance and no contact with the sample, thus avoiding
potential sample damage. A limitation of our current technique is that it is only applicable torelatively thin samples; the sample should be thin enough so that the light can penetrate, andmuch thinner than D/NA as mentioned before. However, this limitation can be overcome in the
future by extending tabletop keyhole CDI to reflection mode [59].

4.4Conclusion
Using a new approach to keyhole coherent diffractive imaging, we have demonstrated a table-

top EUV microscope that can image extended, non-isolated, aperiodic samples for the first time.
We achieve increased efficiency of the imaging system and a more uniform illumination at the sample when compared with previously reported methods based on Fresnel zone plates. Quantitative
depth information about the object can also be retrieved, in very good agreement with AFM measurements and with significant added benefits such as non-contact, non-destructive measurement
capabilities. In the future, when combined with advances in bright HHG sources with < 1 nm
wavelength [18], this approach can be used to image nanoscale dynamics, including ultrafast spin,heat, strain and current flow [73–75] with combined few femtosecond time resolution and sub-10
nm spatial resolution, in thick samples, with elemental and chemical sensitivity.

Chapter 5First Demonstration of General Reflection CDIThis chapter describes the work on extending the CDI technique form transmission geometry
to reflection geometry. Although it seems straightforward for this extension, the truth is that only
very limited successes had been achieved for reflection CDI. Previous work was either limited tohighly reflective EUV lithography masks in a normal incidence geometry [76], restricted to low
numerical aperture through the use of a transmissive mask [77], or restricted to isolated objects[78, 79].

5.1

Initial attempts

5.1.1

Apertured illumination CDI in reflection

Thw sample isolation requrement can be harder for reflection geometry: most of the materials
do not have zero reflectivity, although they can have close to zero transmissivity as long as they
are thick enough. So techniques that allow for imaging non-isolated objects are needed. We first
tried the apertured illumination CDI approach.
We modified the transmission mode setup of apertured-illumination CDI so that the sample
is at an angle α = 30 degrees (Fig. 5.2(a)). The same detector and Fourier transform lens wereused as in the transmission setup, but were repositioned such that they were aligned along the
specular reflection from the sample (Fig. 5.2(a)). Thus the NA was kept at 0.22 and the resolution
at 1.4 µm. The sample used was a positive 1951 USAF Resolution Target. Figure 5.1(a) shows ascatter pattern form the vertical bars of group 5 (element 1) of the resolution target. The black61

Figure 5.1: Mapping the diffraction from a tilted sample to a diffraction pattern linear in frequency
space. (a) The raw data from light scattered by the sample at an angle of 30 degrees. (b) The data
mapped onto a linear grid in frequency space. The dotted black lines are added to highlight thecurvature seen in the tilted sample diffraction shown in (a). Both images have been scaled by the
fourth root. Figure reproduced from Gardner et al. [59].

dashed lines are overlaid to illustrate the curvature in the diffraction resulting from a tilted sample.
In Fig. 5.1(b) we show the scatter pattern after mapping the diffraction onto a grid that is linear
in spatial frequency. After interpolation, the scatter pattern is proportional to the modulus ofFourier transform of the sample. Using the same iterative phase retrieval algorithm as mentioned
in the previous section, we are able to reconstruct any arbitrary position of the target. These
reconstructions are overlaid and shown in Fig. 5.2(b). An objective based bright-field microscopy
image is also shown in Fig. 5.2(c) for comparison.

5.1.2

Imaging periodic samples in reflection

For a periodic sample, if an averaged image of a unit cell, instead of an image of every unit
cell in the illumination area, is the goal, then the isolation requirement would mean for each unit
cell, the feature area is less than 50%. So here we used a periodic sample to avoid the sample
isolation requirement for initial attempts. This allows us to decouple the difficulties of satisfying
isolation requirement and applying CDI to a reflection geometry.
The sample was a two-dimensional array of identical square, nickel nano-pillars, each ∼ 2µm
in width and 20nm high, patterned on a sapphire substrate. Rather than using AICDI, a slightly
simpler geometry was used where the beam was loosely focused directly onto the object, with aspot size of approximately 25 µm, so that many pillars were illuminated (Fig. 5.3(a)). The incident

62

Figure 5.2: Visible laser apertured illumination and tilted sample correction by reconstructing a1951 USAF resolution target in a reflection mode geometry. (a) A schematic of the setup. Note thata negative USAF pattern is shown, but a positive USAF was used in the experiment. (b) Several
reconstructions with different scan positions are overlaid to show the AICDI reconstruction. (c) A
traditional bright-field microscopy image of the sample. Figure reproduced from Gardner et al. [59].

63
angle of the HHG beam on the sample was 45 deg, and as a result the scatter pattern in Fig. 5.3(b)displays a high degree of asymmetry, making this specimen a good demonstration of the need fortilted plane correction. Figures 5.3(b) and 5.3(c) show uncorrected and corrected scatter patterns
respectively. The 27.6 mm square detector with 13.5 µm pixels (Andor iKon) was placed 4.5 cm
past the object, resulting in a NA of 0.29. An integration time of 20 minutes was required in orderto obtain the diffraction pattern in Fig. 5.3(b). The missing center in the diffraction pattern is theresult of a beam stop used to prevent saturation of the bright zero order peak. The beam stop wasplaced as close to the detector as possible (≈ 2mm) in order to minimize edge diffraction effects.
This specimen can be thought of as a convolution between a Dirac comb function with a single
nickel nano-pillar of 1/4 duty cycle. With this in mind, and using the convolution theorem, we canconsider the Fourier transform of this sample to be the product of the individual Fourier transformsof the Dirac comb and a single (averaged) nano pillar. This means that in the diffraction plane, aDirac comb samples a sinc function, which is the Fourier transform of a single pillar. Using this idea
we can increase the signal-to-noise ratio, after applying the tilted plane correction, by extracting thepeak values (spaced by the period of the Dirac comb) of the scatter pattern and placing them on anew, coarser grid, shown in Fig. 5.3(d). This new grid was used in the averaged pillar reconstruction
shown in Fig. 5.3(e), producing an image with ∼ 100 nm theoretical resolution. The reconstructions
were carried out in the same manner and with the same algorithm as in the case of the 632.8 nm
illumination. Clearly this method of resampling the data onto a separate grid by extracting thepeaks in the diffraction plane is only applicable for arrays of identical objects. However, provided
a contrast mechanism exists, the same apertured illumination technique discussed above can be
implemented more generally for full field imaging of nanostructures in the EUV.

5.2

First demonstration of a general purpose ptychography reflection microscope
After initial attempts of reflection CDI with apertured illumination CDI technique and witha periodic sample described in the previous section, this section describes our work using pty-

64

Figure 5.3: EUV microscope image in reflection mode of a 2D array of nickel nano-pillars.(a) A
schematic of the setup. Uncorrected (b) and corrected (c) diffraction patterns. The data in (c)was resampled onto a coarser grid, shown in (d), containing only the peak intensity points. (e)Reconstructed image using the diffraction pattern in (d). The missing data shown in (b), (c) and(d) was left as a free parameter and is a result of a beam stop used to prevent saturation of thebright zero order peak on the camera. Figure reproduced from Gardner et al. [59].

65
chography CDI with a non-periodic sample for reflection CDI. We demonstrate the most generalreflection-mode coherent diffractive imaging to date using any light source, by combining the ePIE
algorithm [56] with curved wavefront illumination [80]. This allows extended (non-isolated) objects
to be imaged at any angle, which will enable tomographic imaging of surfaces. This work also
represents the first non-isolated-object, high fidelity, tabletop coherent reflection imaging, which
expands the scope of applications for CDI significantly. This work demonstrates a powerful newcapability that can impact a very broad range of science and technology. First, our approach removes restrictions on the numerical aperture, sample, or angle, so that general extended objects
can be imaged in reflection mode at any angle of incidence. Second, illumination of the sample
with a strongly curved wavefront removes the need for a zero-order beam-stop by reducing the
dynamic range of the diffraction patterns. The curved illumination also allows the size of the beam
to vary according to the sample size, alleviating the need for a large number of scan positions. This
also results in fewer necessary scan positions when imaging a large field of view. Third, reflection
ptychography produces surface images containing quantitative amplitude and phase information
about the sample that are in excellent agreement with AFM and SEM images, and also removes
all negative effects of non-uniform illumination of the sample or imperfect knowledge of the sample
position as it is scanned [81]. The result is a general and extensible imaging technique that can
provide a comprehensive and definitive characterization of how light at any wavelength scattersfrom an object, with resolution limited only by the wavelength and the numerical aperture of thesystem. This complete amplitude and phase characterization thus is fully capable of pushing full
field optical imaging to its fundamental limit. Finally, because we use a tabletop high harmonic
generation (HHG) 30 nm source [18], in the future it will be possible to image energy, charge andspin transport with nm spatial and fs temporal resolution on nanostructured surfaces or buriedinterfaces, which is a grand challenge in nanoscience and nanotechnology [74, 82].

66
5.2.1

Sample fabrication

The sample used in the experiment was fabricated on a super-polished silicon wafer. The
wafer was rinsed with acetone, isopropanol, and methanol, and baked on a hotplate for 20 minutes
at 250◦ C. It was then spin-coated with Microchem 2% PMMA in anisole, molecular weight 950
at 4000 r.p.m. for 45 seconds. Afterwards it was baked at 180◦ C for 90 seconds. Electron beamlithography was performed using a FEI Nova NanoSEM 640, using Nanometer Pattern Generation
System (NPGS) software and patterns. The resist was then developed by immersion in a 1:3
solution of methyl-isobutyl-ketone:isopropanol for 30 seconds. Approximately 30 nm of titanium
was evaporated onto the surface using a CVC SC3000 3-boat thermal evaporator. The lift-off step
was accomplished in acetone using a sonicator.

5.2.2

Experiment

The experimental geometry for reflection mode Fresnel ptychography is shown in Fig. 5.4. A
Ti:sapphire laser beam with wavelength ≈785 nm (1.5 mJ pulse energy, 22 fs pulse duration, 5 kHz
repetition rate) is coupled into a 5 cm-long, 200 µm inner diameter, hollow waveguide filled with60 torr of argon. Bright harmonics of the fundamental laser are produced near a center wavelengthof 29 nm (27th harmonic) since the high harmonic generation process is well phase-matched [19],ensuring strong coherent signal growth and high spatial coherence. The residual fundamental laser
light, which is collinear with the high harmonic beam, is filtered out using a combination of two
silicon mirrors (placed near Brewster’s angle for 785 nm light) and two 200 nm-thick aluminum
filters. The EUV beam is then sent through an adjustable ≈1 mm aperture, placed ≈1 m upstreamof the sample, to remove any stray light outside the beam radius. A pair of Mg/SiC multilayer
mirrors then select the 27th harmonic of the Ti:sapphire laser at 29.5 nm. The first mirror is flat,
while the second mirror has a radius of curvature of 10 cm. This mirror pair focuses the HHG
beam onto the sample at an angle of incidence of 45◦ . The focus position is 300 µm downstream ofthe sample, so that the HHG beam wavefront at the sample plane has significant curvature. The

67
angle of incidence on the curved mirror is approximately 2◦ , which introduces small amounts ofastigmatism and coma onto the HHG beam.
The sample consisted of ≈30 nm-thick titanium patterned on a silicon substrate using e-beamlithography. A scanning electron microscope (SEM) image of this object is shown in Fig. 5.5b. The
scattered light from the object is measured using an EUV-sensitive CCD detector (Andor iKon,
2048×2048, 13.5 µm square pixels), placed 67 mm from the object, and oriented so that the detector
surface was normal to the specular reflection of the beam. The sample was positioned 300 µm before
the circle of least confusion along the beam axis, so that the beam diameter incident on the sample
was approximately 10 µm. Diffraction patterns were measured at each position of 10 adjacent 3 × 3
grids, with 2.5 µm step size between positions. The positions were randomized by up to 1 µm inorder to prevent periodic artifacts from occurring in the ptychographic reconstruction [83].

5.2.3

Image reconstruction

The process for obtaining the reconstruction was as follows:(1) Tilted plane correction was applied to each of the 90 diffraction patterns in the full dataset.
(2) The standard ePIE algorithm [56] was applied to the corrected data, with subpixel scan
position precision handled as in Maiden et al. [84]. A starting guess for the probe wascalculated using knowledge of the sample-to-focus distance (300 µm). The object starting
guess was set to unity and the probe guess was normalized to contain the same energy as
the average diffraction pattern in the dataset. The algorithm was allowed to update theprobe guess in parallel with the object guess at each sub-iteration. The algorithm was
run in this way for 20 full ptychographic iterations, at which point the probe guess had
made much more progress towards convergence than the object guess. The object guess
was reinitialized to unity, and the algorithm was restarted using the new probe guess, andallowed to run for 100 iterations, long enough for both the object and probe to converge tostable solutions.

68

Figure 5.4: Experimental setup for reflection mode Fresnel ptychography. The EUV beam propagates through an adjustable ≈1 mm aperture, and a single harmonic is selected using a pair ofmultilayer mirrors centered at 29.5 nm, and focused onto the sample. The scattered light is collected
on a CCD detector placed directly after the sample. The inset shows a height profile reconstructed
through ptychography. Figure reproduced from Seaberg et al. [44].

69

Figure 5.5: Diffraction data and ptychographic reconstruction. (a) Representative diffraction pattern, scaled to the 14 power, taken from the 90-scan dataset. (b) SEM image of the Ti patterned Si
sample. Note that the large defect circled in the SEM image resulted from contamination after theptychography measurement. (c) Reconstructed amplitude (thresholded at 5%) of the HHG beam.
The inset shows the reconstructed phase (displayed modulo-2π). (d) Ptychographic reconstruction of the object shown in (b). The reconstruction is plotted as the complex amplitude, wherebrightness represents reflected amplitude and hue represents the phase of the reconstruction. Notethat the majority of defects seen in the SEM image of the Ti nanostructures are reproduced in theptychographic reconstruction. The scale bar in (b) is shared among (b)-(d). Figure reproduced
from Seaberg et al. [44].

70(3) The object guess was re-initialized as described in step 2, and the probe guess was set tothat found at the end of step 2. The subpixel position correction method [81] was applied
to the ePIE algorithm, and the overlap constraint was applied with subpixel shifts of theprobe [84]. The position correction feedback parameter β was started at a value of 50,
and automated as in Zhang et al. [81]. The probe guess was not allowed to update during
this step. Again, the algorithm was run for 100 iterations, until the position corrections
converged to < 0.1 pixel.
(4) Finally, using the probe found in step 2 and the corrected scan positions found in step 3,
and with the object guess reinitialized to unity, the algorithm was run for 200 iterations toachieve the final reconstruction.
Each full iteration (cycling through all 90 diffraction patterns) took approximately 30 seconds
on a personal computer, leading to a total reconstruction time of 3.5 hours.

5.2.4

Oxide layer measurement

The thickness of the oxide layer on the Si wafer used for sample fabrication was determinedthrough ellipsometry (Gaertner Scientific L117F300). The measurements were made at 70◦ angle
of incidence using a He-Ne laser at 632.8 nm. The ellipsometric angles ψ and ∆ were determined
to be 10.20(0.04)◦ and 171.00(0.07)◦ , respectively. The angles are defined asrp= tan ψ e−i∆
rs

(5.1)

where rp and rs are the complex reflectivity coefficients for p- and s-polarized light, respectively.
The thickness of the oxide layer was calculated by numerically solving the argument of Eq. (5.1),
assuming an index of refraction for the oxide layer of 1.474(0.003) and an index of refraction for thesilicon substrate of 3.89(0.02) + i 0.011(0.009) at 632.8 nm wavelength. The result, was an oxide
layer thickness of 3.0(0.1) nm.

71
5.2.5

Results and discussion

Due to the non-normal angle-of-incidence on the sample, the patterns must be remapped
onto a grid that is linear in spatial frequencies of the sample plane, in order to use fast Fouriertransforms (FFTs) in the data analysis. We used tilted plane correction to accomplish this [59].
An example of a corrected diffraction pattern is shown in Fig. 5.5a. The diffraction patterns were
cropped such that the effective numerical aperture was 0.1, enabling a half-pitch resolution of 150
nm. The image was reconstructed using the ePIE, along with the sub-pixel position determination
method [81, 84]. A starting guess for the probe was calculated based on the estimated distanceof the sample from the focus. The reconstructed complex amplitude of the object is shown in
Fig. 5.5d. During the course of the reconstruction, the algorithm was used to further solve forthe complex amplitude of the probe as well, resulting in the illumination shown in Fig. 5.5c. The
reconstructed probe is completely consistent with a measurement of the unscattered beam at thedetector. The high fidelity of the CDI reconstruction is evident by the fact that the majority ofsmall defects visible in the SEM image of the Ti patterns (Fig. 5.5b) are also clearly visible in the
CDI reconstruction (Fig. 5.5d). Note that the large defect circled in the SEM image in Fig. 5.5b
was the result of sample contamination after the ptychography measurement.
Ptychography solves for the complex amplitudes of both the object and the probe (or incidentbeam) simultaneously [56, 83]. As a result, reliable quantitative information about the object canbe obtained from the reconstruction, since the effect of the probe on the diffraction patterns isessentially divided out. Quantitative surface relief information can be obtained from the phase of
the reconstructed object as well. The titanium was patterned at a thickness of approximately 30
nm. The round trip path difference of the reflected light is −2h cos θ, where h is the height abovea reference (such as the substrate) and θ is the angle of incidence. At 45◦ angle of incidence for afeature thickness of 30 nm, the round trip path length difference between the silicon substrate andthe patterned titanium features is 42.4 nm. At 29.5 nm wavelength, this corresponds to between
1 and 2 wavelengths path length difference. Additionally, the phase change upon reflection can be72highly variable for absorbing materials. In the case of this sample, both silicon and titanium havenative oxide layers which must be taken into account when calculating this phase change. Thus,
some prior knowledge is required in order to retrieve the absolute height of the features. The method
for calculating the phase change upon reflection from a thin-film system with complex indices ofrefraction is described in Born and Wolf [67]. The indices of refraction at 29.5 nm wavelength
necessary for this calculation were obtained from the Center for X-Ray Optics (CXRO) [72]. The
thickness of the silicon oxide layer was measured using ellipsometry to be 3.0(0.1) nm, and thethickness of the titanium oxide layer was assumed to be 2.9(0.2) nm based on the literature [85,86].
For the SiO2 /Si region, the phase change δSi was calculated to be −1.22(.03) radians and thetheoretical reflectivity was calculated to be 0.33%. For the TiO2 /Ti patterns, the phase change
δTi was calculated to be −1.92(0.09) radians and the theoretical reflectivity was calculated to be
10.9%. The object reconstruction shows a ratio of ≈17 between the reflectivity of the titanium
and the silicon surfaces based on a histogram of the reconstructed amplitude, in reasonably good
agreement with the calculated values, which assumed no surface roughness.
A flattening method was applied to the reconstructed phase of the silicon substrate, similar
to that used in atomic force microscopy, due to some residual phase curvature reconstructed on
the flat substrate. The peak-to-valley height variation of the subtracted surface fit was < 4 nm
over the full 35 × 40µm2 field of view. After flattening, the reconstruction shows an average of 4.26
radians of phase difference between the titanium and silicon surfaces, corresponding to a 46.2(0.7)
nm path length difference (when 2π is added and after taking the phase changes upon reflection into
account). This corresponds to a 32.7(0.5) nm average thickness of the titanium patterns. A height
map of the sample could then be produced by assuming that 2π should be added to any part of
the reconstruction that exhibited an amplitude above 25% of the maximum (based on the relativereflectivities of titanium and silicon, as discussed above). Additionally, the reflection phases δSi andδTi were subtracted from the Si and Ti regions using the same criteria. The result of this analysis isdisplayed in Fig. 5.6a, and represents a significant improvement in image quality compared with alltabletop coherent reflective imaging to date. After the ptychography measurements were taken, an

73
independent height map of the sample was obtained using a Digital Instruments Dimension 3100
AFM. The resulting AFM height map is shown in Fig. 5.6b, after applying the same flattening
method as that used for the CDI reconstruction. The AFM measurement shows an average height
for the titanium features of 32.7 nm, which agrees exactly with the ptychography result withinerror bars.
Many small pieces of debris are visible in the AFM image shown in Fig. 5.6b, with heights
above that of the patterned titanium. None of the EUV work was done in a cleanroom environment.
The reason these are not visible in the CDI height map (Fig. 5.6a) is that the 3D information
relies on the phase difference of light reflecting from the substrate versus the features (at 45◦ )and not on the absolute height difference. While the debris locations are still evident in the CDI
reconstruction (Fig. 5.5d), the modulo 2π ambiguity of the phase information combined with thevery short wavelength prevents us from extracting the absolute height information of all features.
However, a tomographic or multi-wavelength approach would enable full 3D reconstructions of allfeatures on a surface [87].
Finally, we note that previously it was believed that full knowledge of the probe was necessary
when using Fresnel (curved wavefront) ptychography for phase retrieval [80]. However, we find thatfor ptychographic grids of 3 × 3 and larger with sufficient overlap between adjacent probe positions(60-70% area overlap [56]), the algorithm converges to a consistent result for the probe provided
that the phase curvature of the starting guess differs by no more than 50% of the actual phasecurvature. Even this condition is relaxed entirely in the case of isolated objects. To demonstrate
this, we performed a separate ptychographic retrieval of the probe by scanning a 5 µm diameter
pinhole across the beam near the focus. The probe that is retrieved using this method can bepropagated to the sample plane for comparison to the probe found in the course of the sample
reconstruction. We found very good agreement between the two probe reconstructions, independentof the accuracy of the starting guess for the probe.

74

Figure 5.6: Height profile comparison between CDI and AFM. (a) 3D profile of the object based on
ptychographic reconstruction. (b) 3D profile of the object based on an AFM measurement. Any
features taller than 40 nm were thresholded to 40 nm for the 3D rendering. (c) Histograms of theheight profiles shown in (a) and (b). The histograms were used to calculate the average feature
thickness of 32.7 nm based on the both the CDI and AFM measurements. The scale axis shown in(a) is shared by both (a) and (b). Note that the large debris spot on the right of the AFM image
was introduced after the CDI image was taken.Figure reproduced from Seaberg et al. [44].

75
5.2.6

Comparison between CDI reconstruction and SEM and AFM images

There are a number of defects visible in the sample image reconstructed through ptychographic coherent diffractive imaging (CDI) which are also visible in scanning electron microscope(SEM) and atomic force microscope (AFM) images. A visual comparison between the three techniques is shown in Fig. 5.7. Of the 7 defects pointed out in the figure, only defects 1-5 are visible
in all of the images. The 6th and 7th defects are only visible in the CDI phase image and the AFM
image. This is a demonstration of the fact that CDI has both amplitude contrast (analogous toSEM) and phase/height contrast (analogous to AFM).

5.2.7

High harmonic beam characterization through ptychography

To ensure that our recovery algorithm was correctly retrieving the probe illumination, we firstcharacterized the extreme ultraviolet (EUV), high harmonic generation (HHG) beam by scanninga 5 µm diameter pinhole across the beam near its focus and reconstructed the illumination using
ptychography. In this case, the pinhole can be thought of as the probe, while the beam is aneffective object. The scan consisted of a 6 x 6 grid with 1 µm step size between adjacent scan
positions. The reconstructed beam is shown in Fig. 5.8a.
The reconstructed beam was propagated to the sample position (200 µm upstream of thepinhole probe location) and calculated on the tilted plane (at 45◦ ) using tilted plane correction,
shown in Fig. 5.8b. Immediately after this ptychography scan, the pinhole probe was removed andthe sample was translated such that the beam illuminated one of the star patterns on the sample(with reconstruction shown in Fig. 5.5d). We performed a 3 x 3 ptychographic scan across thestar feature, with 2.5 µm step size. In this case, a probe starting guess consisting of a Gaussian
amplitude profile with random phase sufficed to consistently retrieve the probe amplitude shownin Fig. 5.8c. As can be seen by comparison of Figs. 5.8b and c, the two beam characterization
methods show very good agreement between both the phase and the amplitude. It should be notedthat the HHG beam drifted slightly inside the adjustable aperture during the course of the two

76

Figure 5.7: A visual comparison between the reconstructed CDI amplitude and phase with images
obtained using SEM and AFM. (a) Reconstructed CDI amplitude image of the sample. (b) Phase of
the reconstructed image. (c) SEM image of the sample. (d) AFM image of the sample. In the above
images, 7 defects have been pointed out (located above and to the right of each number). Defects
1-5 are visible in all of the images, whereas defects 6 and 7 are only visible in the reconstructed
phase and in the AFM image. The circled defect in (c) and (d) was a result of contamination after
the CDI measurements were taken.Figure reproduced from Seaberg et al. [44].

77scans, resulting in slightly different beam structure during the two measurements.
As a further consistency check, the probe reconstruction (shown in Fig. 5.5c) was propagatedto the detector, and the tilted plane correction was undone in order to examine the result in thereal coordinates of the detector. The result of these steps is shown in Fig. 5.9a. A comparison wasmade with a direct measurement of the unscattered beam by translating the sample to a featurelessregion of the silicon substrate, shown in Fig. 5.9b. As can be seen in Figs. 5.9a and b, while it isevident that, as in the above sample plane comparison, some beam drift occurred during the courseof the ptychographic scan, the reconstructed probe is entirely consistent with the high harmonic
beam used to illuminate the sample.

5.2.8

Conclusions

We have demonstrated the first general, tabletop, full field reflection mode CDI microscope,capable of imaging extended nanosurfaces at arbitrary angles in a non-contact, non-destructive
manner. This technique is directly scalable to shorter wavelengths and higher spatial and temporal
resolution, as well as tomographic imaging of surfaces. By combining reflection-mode CDI with
HHG sources in the keV photon energy region, it will be possible to capture nanoscale surface
dynamics with femtosecond temporal and nanometer spatial resolution.

5.3

Keyhole CDI in reflection
We had tried reflection keyhole CDI, even before reflection ptychography actually. The greatchallenge for we had met is that our knowledge of the probe beam is not accurate enough withprevious methods. In transmission keyhole CDI discussed in Chapter 4, we obtain the probe
beam by measuring foci positions, and assuming a spherical phase centered at these foci when it
propagates to the detector. This method proves to be good enough with the two transmission
results shown in Chapter 4. Accurate knowledge of the probe beam was not needed in those two
results probably because: (a) for the first result, the sample is a binary object, allowing us to put
on a strong amplitude support and phase constraint; (b) for the second result, the features are

78

Figure 5.8: A comparison of separate reconstructions of the HHG illumination beam, using thebeam as the object in one case and as the probe in the second case. (a) Reconstruction of the HHG
beam near the focus using a 5 µm diameter pinhole probe. The main image displays the amplitude
and the inset displays the phase. The scale bar has width 2 µm. (b) The result of propagating
the reconstructed beam from (a) to the tilted sample plane. Again, the main image shows theamplitude and the inset shows the phase. The scale bar has width 5 µm. (c) The amplitude (mainimage) and phase (inset) of the reconstructed probe based on a 3 x 3 ptychographic scan acrossthe one of the features on the titanium sample discussed in the text. The scale bar is shared with(b). Note that the beam amplitudes in (b) and (c) are displayed in the tilted sample coordinates,
resulting in elongation in the horizontal direction. Figure reproduced from Seaberg et al. [44].

Figure 5.9: Comparison between the illumination reconstructed as a ptychographic probe andpropagated to the detector, and the unscattered illumination measured directly on the detector(raw data). (a) The probe reconstruction from Fig. 5.5c, propagated to the detector plane. (b)The HHG beam measured directly on the detector by translating the sample to a featureless regionof the silicon substrate. The scale bar in (a) has width 1 mm and is shared by (a) and (b). Figure
reproduced from Seaberg et al. [44].

79
located within in an area that is much smaller than the beam size, thus the beam can regarded
as uniform anyway. For the sample used in reflection ptychography in the last section, these two
conditions are not met, and very accurate knowledge is necessary for the success of the keyhole
CDI approach.
Thanks to ptychography, which reconstructs not only the object, but also the probe beam,with high accuracy, this challenge is overcome. With the beam reconstructed from the ptychography, we achieved the first reflection-mode keyhole CDI result on a tabletop in conjunction with thefirst general full-field demonstration of reflection ptychography CDI.
To reconstruct the image using keyhole CDI, tilted plane correction was first performed
on one of the diffraction patterns, and the reconstruction was then achieved using the method
described in [27]. First, a random phase was generated at the detector, and then the electric field
was propagated back to the sample plane. The total reflectivity r was separated into two parts: aconstant baseline reflectivity r0 (the reflectivity of the substrate) and the difference in reflectivities
between the features and substrate, ∆r. The magnitude of r0 can be found either by trial-and-error,
or from the ptychographic reconstruction. Next, the product of the beam and r0 is subtracted fromthe total exit surface wave (ESW). This quantity is then multiplied by the conjugate of the probe
phase, which gives the exit surface wave of just the object. We used the shrink-wrap approach [55]to the field at the sample plane to apply the support constraint. The reconstructed sample is shownin Fig. 5.10b.
While the ptychography approach typically offers higher quality images than the keyhole CDI
approach thanks to information redundancy in the multiple scans with overlap, it is impossible,
difficult, or not desirable to perform ptychography scanning for certain situations. For example,
in the “diffraction before destruction” regime, a single pulse is enough to damage the sample,
not allowing the original sample to be scanned and illuminated with successive pulses. In thesesituations, it is preferable to use the keyhole CDI approach. One can simply use ptychography CDI
as a beam characterization step, while the goal of sample reconstruction is achieved using keyhole
CDI.

80

a

Reconstructed Probeb

keyhole CDI Reconstruc�on

π
10 µm

0
-π

Figure 5.10: Reflection keyhole CDI reconstruction with the probe reconstructed from ptychography. (a) Reconstructed probe from ptychography. Adapted from Seaberg et al. [44]. (b) Reconstructed sample using only one diffraction pattern using the keyhole CDI approach. Scale bar, 10µm, shared by (a) and (b).

Chapter 6

Surface Nano-imaging with High Lateral and Axial Resolution Using Tabletop
EUV Ptychography

Building on the work described in the previous chapter, the work in this chapter is on surface nano-imaging with high lateral and axial resolution, and unprecedented image quality. The
images from our tabletop EUV ptychography compare favorably with SEM and AFM, which are
well-established techniques for imaging surfaces with nanometer resolution. This is achieved by
performing reflection ptychography with a HHG source at a high numerical aperture (around 0.4).
It works in a parallel, area-by-area scanning mode, providing powerful quantitative contrast, capable of capturing composition and topographical variation. Short wavelength illumination and highangle scatter leads to a very high, 40 nm by 80 nm (half-pitch), lateral resolution while quantitative
phase information enables surface profilometry with ultra-high, 6-angstrom, axial resolution.

6.1

Experiment
To drive the HHG process, we focus ultrashort pulses from a titanium doped sapphire am-

plifier (central wavelength around 780 nm, pulse energy of 1.4 mJ, repetition rate of 5 kHz, 22 fs)
into a 150 µm diameter hollow waveguide filled with argon (36 torr backing pressure) generatingphase-matched, spatially coherent harmonics at wavelengths near 29 nm. After the waveguide, theHHG beam and CDI microscope were in medium vacuum (≈ 10−6 torr) to avoid absorption of the
EUV light. We use a pair of silicon rejecter mirrors set near Brewster’s angle for 780 nm (|r| ≈ 0.36,
29 nm) and two 200 nm thick aluminum filters to completely extinguish the fundamental light. As

82shown in Fig. 6.1a , two 45◦ angle-of-incidence (AOI) flat multilayer mirrors select the 27th harmonic (29 nm) which is then focused near the sample by an ellipsoidal nickel-coated mirror set at
5◦ grazing incidence. The beam illuminates the sample at 50.57◦ AOI, and has a slight negative
phase curvature due to the position of the sample relative to the focus. In this work the sample
was titanium shapes patterned with E-beam lithography on a silicon substrate (SEM image shownin Fig. 6.1b). During the period after the sample was fabricated, incredibly fine height variations
caused by contamination formed over the surface of the sample. As a demonstration of the quantitative contrast mechanism in CDI we measure the height of these variations with unprecedented
accuracy in a non-destructive, non-contact manner and compare them to AFM and SEM for verification. An EUV sensitive CCD (Andor iKon-L, 2048×2048 pixel array, 13.5×13.5 µm2 pixel size)
is positioned 3.17 cm away from the sample and perpendicular to the specular reflection, leading
to a high nominal numerical aperture of about 0.4. The sample is scanned with a step size of
about 3 µm in both x- and y-directions, on a 18×11 rectangular grid (random offsets within 20%of the step size). With each exposure taking 0.1 seconds and 3 accumulated exposures for each of
the 198 diffraction patters, the total exposure time was less than 1 minute for the whole 65 µm
x 40 µm field of view, corresponding to 44 µm2 per second exposure speed. To account for the
non-normal incidence on the sample and the high angle scattering, tilted-plane correction [59, 79]
must be performed to obtain the Fourier transform of the reflected exit surface wave on a linear gridof spatial frequencies. We employ an advanced, proprietary version of the correction for each of the
198 diffraction patterns. A representative measurement is shown in Fig. 6.1c, and the corrected
amplitude is shown in Fig. 6.1d. The simple effect of tilting the sample away from normal incidence
results in an increased geometric field of view with decreased resolution. As shown in Fig. 6.1d, theactual NA in x-direction is significantly reduced, compared with y-direction, resulting in decreased
resolution in x-direction. The dashed rectangle in Fig. 6.1d shows the actual spatial frequency
window we used for the reconstruction corresponding to a pixel size of 76 nm in x- and 38 nm in
y-direction, sampled at 512 × 1024 with the same sampling frequency in both directions.

83

Figure 6.1: High NA reflection EUV ptychography experiment. (a) Schematic of the tabletop EUV
microscope. (b) SEM of the sample (scale bar 10 µm). (c) Representative diffraction pattern. (d)Diffraction pattern after tilted plane correction.

84

6.2

Image reconstruction
To obtain the initial guess of the probe (beam), we measured the far-field beam reflected froma flat, substrate area. We assume a quadratic phase with x- and y-radius taken as the distancefrom the measured x- and y-focus to the detector [27]. We performed tilted plane correction on the
beam field guess and then back-propagate to the sample plane to produce the initial probe guess.
We use a constant value for the initial object guess. We use the ePIE [56] algorithm, and update
only the object in the first two iterations. In the following 30 iterations, we update both the object
and the probe. For the next 50 iterations, we update the object with position refinement [81].
For the next 600 iterations, we update the object, the probe and use position refinement. We
finish the reconstruction procedure with 150 iterations of object update only. Our ePIE algorithm
is implemented in C++, with OpenCV library support, on a NVIDIA Tesla K40c GPU. Each
iteration takes at most 4.6 second (with probe, object update and position refinement on) for all
the 198 diffraction patterns, each of which has a grid size of 512 by 1024. Main features become
clear at less than 32 iterations (2.5 minutes), and the whole 832 iterations take less than one hour.
The reconstructed object has complex values proportional to |r| exp[φr +i2π/λ · (−2h cos θi )],where r is the complex Fresnel reflection coefficient of the sample with amplitude |r| and phase
φr , λ is the illumination wavelength, h is the height distribution of the sample, and θi is the angle
of incidence. Due to the asymmetry in pixel x- and y-dimension, we interpolated the image to
up-sample in x-direction by a factor of 2. The reconstructed amplitude and phase are shown in
Fig. 6.2a and b, and for comparison Fig. 6.2c shows an SEM (model: FEI Nova NanoSEM 630)
measurement. We find excellent agreement with the SEM in both the ptychography amplitude andphase images; while ptychography captures details of the slight surface variation that the SEM
does not. In the highlighted region shown as a black rectangle in Fig. 6.2c, we compare images
from 4 different measurements: ptychography amplitude, ptychography phase, SEM (with higher
resolution and smaller field of view than Fig. 6.2 c) and AFM, in Fig. 6.2d-g respectively. We seethat all measurements agree quite well except very faint details, such as slight height variation of

85
the surface due to debris, circled in white and red (dashed), which are clearly discernable in theptychographic amplitude and phase but are nearly invisible to the SEM. The features circled in red(dashed), are completely unrecognizable in the SEM image while easily visible in the ptychographic
phase image. It is evident that reflection mode CDI provides a powerful technique for surface
profiling, enabling clear visualization of composition and topography with high contrast. The
amplitude and phase of the reconstructed beam are shown in Fig. 6.2h and i. The beam had a 1/e2
diameter of ≈ 8 µm in x- and 6 µm in the y-direction. The majority of height variations within thesample area are mostly within 30 nm (except for sparse debris), much smaller than D/NA (D isthe beam diameter), justifying modeling the exit surface wave as a multiplication of incident wave
and the object response [42].

6.3

Lateral resolution characterization
We present three methods to characterize the lateral resolution: the phase retrieval transfer

function (PRTF) [28,52,88], the modulation transfer function (MTF) (that is, the Fourier transformof the point-spread-function (PSF)), and the knife-edge assessment (10%-90% edge width).
The PRTF is the counterpart of MTF in coherent diffractive imaging. In conventional imaging, the MTF describes the response of an imaging system to various spatial frequencies. In
coherent diffractive imaging the image forming hardware is replaced by a phase retrieval algorithmand correspondingly, the PRTF plays a similar role to the MTF by quantifying the repeatability
and accuracy of the retrieved phase as a function of spatial frequency. The PRTF evaluates the
combined effect of the signal-to-noise-ratio (SNR) of the diffracted wave and systematic errors that
corrupt the convergence of the retrieval algorithm. Non-normal incidence causes a significant asymmetry of the measured range of spatial frequencies in the horizontal versus the vertical directions
so we evaluate the PRTF in these two directions separately; PRTF is calculated as:
PRTF(f1D ) =rec. (f , f )i |i i
hh|hEi,j
x y i j 1D
meas.
hh|Ej
(fx , fy )|ij i1D(6.1)

where |Ejmeas. (fx , fy )| is the magnitude of the measured electric field on the detector corresponding

86

Figure 6.2: Images reconstructed from ptychography and comparison with SEM and AFM. (a)Ptychographic amplitude reconstruction. Scale bar: 10 µm, shared with (a-c). (b) Ptychographic
phase reconstruction. (c) SEM for comparison. The area enclosed in the black rectangle is zoomed
in and compared with the ptychography amplitude (d), ptychography phase (e), higher resolution
SEM (f), and AFM (g) with 80 nm sampling step. Scale bar in (g), 2 µm, shared by (d-g). Areas
highlighted in circles are compared in detail for the 3 techniques. (h) and (i) are the amplitude andphase of the reconstructed, incident high harmonic beam. Scale bar, 10 µm. The red and white
rectangular regions are used later in the text.

87
rec. (f , f ) is the reconstructed electricto spatial frequency (fx , fy ) and the j th scan position, Ei,jx yfield at the ith iteration, corresponding to spatial frequency (fx , fy ) and the j th scan position,
h ii and h ij are averages over iterations and scan positions respectively, h i1D means averageover one direction to transform a 2-dimensional signal to a 1-dimensional signal. We averageover all different fy values when evaluating PRTF(|fx |), and over all different fx values whenevaluating PRTF(|fy |). We started with the 832nd iteration, which we considered the convergedreconstruction. We averaged 100 iterations during which time only the object was updated. The
calculation result is shown in Fig. 6.3 a. We observe that the PRTF values only slightly decrease
with increasing spatial frequency in both horizontal and vertical directions and are still above 0.98
at the maximum spatial frequencies. This large PRTF value is a manifestation of good SNR acrossthe whole frequency range and accurate measurement of parameters. We conclude that the PRTF
approach supports a spatial resolution of 73.2 nm in horizontal direction and 36.6 nm in verticaldirection but is not sufficient to fully characterize the lateral resolution.
In the second method, we evaluated the modulation transfer function of our CDI microscope.
In the region of interest shown in Fig. 6.2 c (red dashed rectangle), we took a corresponding AFM
measurement with a 10 nm probe diameter, and 40 nm sampling step (model: Digital Instruments
Dimension 3100). Fig. 6.3c shows the power spectral density (PSD) of the two images from theptychography phase measurement and AFM measurement plotted together (ptychographic PSD infine dashing). By taking a square root of the ratio of the ptychographic phase PSD and the AFM
PSD, we obtain a relative MTF: the MTF of the ptychography phase imaging divided the MTFof the AFM image. In the absence of a much higher resolution AFM image, we are limited toclaiming a relative MTF rather than absolute. However, if we reasonably assume the AFM image
has a MTF value above 50% at the highest spatial frequency (39.1 nm, half pitch), the fact that therelative MTF is above 22% in both x- and y- directions at the maximum spatial frequency (clearly
above 11%; which is the Rayleigh resolution contrast level) allows us to conclude our resolution is73.2 nm in horizontally and at least 39.1 nm vertically. Finally, we characterize the resolution of
our microscope by measuring the 10% to 90% width of two edges. Compared with AFM, where

88

Figure 6.3: Characterization of the lateral resolution. (a) Phase retrieval transfer function. (b) Relative modulation transfer function of the ptychographic phase and AFM (grey dashed (horizontal)
line: 11% contrast level corresponding to Rayleigh resolution). (c) Power spectral density. Both
AFM and CDI PSDs are shown; lines with fine dashing are CDI. (d) shows the edges where wemeasure the 10% to 90% edge width of the phase reconstruction. (e) Profile measured nominallyin the x-direction. (f) Profile measured nominally in the y-direction. Fits to the error function
compliment, and are also shown in (e-f) as solid lines (blue) (both R2 values ≈ 0.99).

89
the image contrast comes solely from height differences, ptychography derives its contrast fromthe reflected complex exit surface wave. We choose two profiles nominally in the x-direction andnominally in the y-direction (shown as white solid lines in Fig. 6.3 d). The feature edges along these
profiles are clean, sharp, and near the center of the ptychographic scan area avoiding insufficient
overlap that can cause decreased reconstruction quality at the egdes. The phase along these two
profiles are shown in Fig. 6.3e andf respectively. We see that in x-direction, the 10%-90% width
occurs across ≈ 2 pixels (73.2 nm), and the y-direction rises in only 1 pixel (36.6 nm). The tworising edges agree well with the error function compliment (ERFC) fit that have 10%-90% widthsof 73.2 nm and 20.9 nm (smaller than 36.6 nm) respectively. (The R2 value in either ERFC fitswas ≈ 0.99). The resolution characterized via three different approaches all support the conclusionthat we have spatial resolution of 73.2 nm in horizontal direction and 36.6 nm in vertical direction,corresponding to the highest measured spatial frequencies.

6.4

Axial resolution characterization
In reflection CDI, we can calculate the height from the reconstructed phase in a similar way

to holography. The reconstructed phase is the sum of the geometric phase that is proportional
to height, 2π/λ · (−2h cos θi ), and the phase of the complex reflection coefficient φr . Figure 6.4ashows the reconstructed phase in a selected region, and Fig. 6.4b shows the calculated height. Dueto the lack of an absolute reference in CDI only relative phase values are accessible so when thereflecting material is monolithic, the retrieved phase is purely geometric and φr cancels exactly.
When considering features that have different composition, φr is also different and must be taken
into account. In the white rectangular area shown in Fig. 6.2c, the region between the stars has
the same composition, a contamination layer above the Si substrate, thus we consider only thereflection from this top layer and neglect the contribution from the silicon substrate since it has amuch weaker reflection (less than 10 % determined from Fig. 6.2 a). Visually, the ptychographic
height measurement (Fig. 6.4d) shows good qualitative agreement with the AFM (Fig. 6.4e). To
characterize the axial resolution of our microscope we examine surface variations highlighted in the

90
white rectangular region, shown in Fig. 6.2c. It is important to note that we do not expect thelateral ptychographic resolution to be comparable with the AFM resolution in this area: because
wide-angle diffraction from weakly scattering features has poor SNR, the combination of strong
signal from the titanium features and detector saturation leads to a loss of fidelity in this region.
However, this does not affect the reconstructed height when the ratio of the feature width to thewidth of the PSF is much greater than unity, which is true for these surface features. On the other
hand, this condition is not met for the very fine debris apparent in the AFM image. In order to moreaccurately compare the height variations measured by either AFM or ptychography we apply two
filters. First, we simply remove points outside the wide, trench features using a histogram filter inboth the AFM and ptychographic images, shown in Fig. 6.4g. Second, we convolve a Gaussian PSF(σ = 70 nm) with the AFM measurement. The width of the Gaussian PSF was chosen to minimize
the overall error between the AFM and ptychographic reconstruction. The results of either thehistogram-filtering or PSF-filtering are shown in Fig. 6.4f and g; these are compared against theptychographic height map, Fig. 6.4d. The three panels shown in Fig. 6.4c are a histogram of theheight difference between the ptychographic reconstruction and the AFM image for different filters.
The top-most histogram and corresponding Gaussian fit with reported width, σ = 4.3 Å compares
the unfiltered AFM (Fig. 6.4e) directly to the ptychographic reconstruction. The center histogramand corresponding Gaussian fit with reported width, σ = 3.7 Å compares the histogram-filtered,ptychographic height map to the histogram-filtered AFM (Fig. 6.4f). The bottom-most histogramand corresponding Gaussian fit with reported width, σ = 3.2 Å compares the ptychographic height
map to the Gaussian-PSF filtered AFM image (Fig. 6.4g). Two representative profiles are shownin Fig. 6.4c corresponding the white dashed line in Fig. 6.4e. The top-most panel is a directcomparison between the unfiltered AFM and ptychographic height reconstruction. The bottommost panel compares the PSF-filtered AFM with the ptychographic reconstruction. The threecomparisons (Fig. 6.4c) are in good relative agreement however; the fact that both filtered versionof the AFM image show lower relative difference results from the omission of spurious debris.

91

Figure 6.4: Height calculated from the reconstructed phase. (a) Three-dimensional rendering of
surface features imaged by ptychographic CDI. Scale bar, 10 µm. (b) Histograms of the height
differences measured by ptychography and AFM. The top panel includes no filtering (comparing(d) to (e) directly). The middle panel uses a histogram filter to remove values outside the trenchfeatures (comparing (d) to (f)). The bottom panel used a Gaussian-PSF to smooth fine debris in
the AFM (comparing (d) to (g)). Scale bar, 2 µm shared with (d-g). (c) Comparative profiles
taken along the dashed (white) line in (e). The top panel compares (d) to (e). The bottom panel
compares (d) to (g). After smoothing, the 2-σ (95% confidence interval) width, in height differenceis ≈ 6 Å.

92

6.5

Discussion
We have demonstrated surface imaging with unprecedented fidelity comparing favorably withwell-established techniques such as SEM and AFM. We achieve a lateral resolution of 40 nm vertically by 80 nm horizontally, and a sub-nanometer axial precision. EUV reflection provides a
powerful imaging contrast mechanism; it has composition sensitivity unlike AFM, and improved
contrast compared to SEM. Instead of serial, point-by-point scanning, EUV ptychography employs
parallel, area-by-area scanning for imaging, significantly decreasing the time for scanning, making
source flux the only practical limit for high volume imaging. The increase of the imaging speed
makes this tool attractive for real applications involving large-area imaging, such as semiconductor
inspection. In contrast to AFM that works in a contact or tapping mode, this microscopy provides a long working distance: in this work, only limited by sample-detector distance (31.66 mm).
Ptychography CDI does not require the sample to be conductive as SEM does. The resolution
reported in this work can be improved with shorter wavelengths, although for most materials, thiswill come at the cost of decreased reflectivity. At smaller grazing incidence angle, by taking an
additional data set with the sample rotated at 90 degrees, we can ensure high resolution in both
directions.

Chapter 7

Hyperspectral Imaging with Harmonic Combs and Ptychography

While previous chapters present the work on removing obvious roadblocks on the way ofmaking HHG CDI a general microscopy tool, this chapter discusses a powerful capability of ptychography CDI that is unexpected from the early beginning – the capability to decompose incoherent superposition of different modes, due to information redundancy provided by the overlap inthe scanning.
This chapter will first introduce the principles of decomposing incoherent superposition withptychography. Then preliminary results of our ongoing work are presented on decomposing incoherent superposition from different wavelengths in a multi-order HHG source. These results
represent a proof-of-principle demonstration of hyperspectral imaging using a HHG source and theptychography method.

7.1

Decomposing incoherent superposition with ptychography
We have seen in the previous chapters that information redundancy provided by the overlap

in the scanning have brought several unique capabilities to this technique: the reconstructionof the sample is more robust; not only the sample but also the illuminating beam itself can bereconstructed; some measurement errors in the experiment, such as position errors in the scanningstages, can be automatically corrected. Recently, another advantages have been added to the list:
there can be enough redundant information in ptychography to decompose incoherent superpositionon the detector. In 2013, Thibault and Menzel published a paper titled “Reconstructing state

94
mixtures from diffraction measurements” [89]. In this paper, they demonstrated experimentallythe diffractive imaging of a sample with X-ray that has only partial spatial coherence. They also
successfully decomposed simultaneously the five dominant spatial modes in the X-ray beam. Thisis a conceptually surprising result: as the name indicates, CDI relies on high spatial coherenceof the illumination for imaging; but for ptychography, partially coherent illumination source canalso work. There is enough information provided by the overlap in the scanning to decompose theincoherent superposition into its individual coherent components.
Several sources of decoherence in scattering experiments are identified in Ref. [89], such as
spatial and temporal incoherence of the probing radiation, vibration of the sample, etc. Other
sources include superposition of different polarizations. The case of temporal incoherence is very
interesting: if a multi-colored illumination is used, can ptychography diffractive imaging separate
the responses of the sample to different colors simultaneously? A positive answer to this question
was given by Batey, et al. [87]. In their experiments, wavelengths of 410 nm, 530 nm, 632 nm
from three laser diodes illuminated the sample at the same time. They successfully recovered
the sample response for each of these three colors with ptychography. They called this technique“ptychographical information multiplexing” (PIM).
It is apparent that the modulus constraint used in multi-color diffractive imaging should be
different from that used in monochromatic CDI as shown in Eq. 3.46. The projection to this newmodulus constraint can be written as [87]:|U |mP̃m,MC Uj = qPUj
2
j |Uj |

(7.1)

where ˜ indicates the projection happens in the Fourier space instead of the real space, subscript“MC” indicates multicolor, Uj is the field on the detector plane for the jth color, |U |m is themeasured modulus of the field. This formula works for other sources of incoherence too, in whichcase j means the jth mode, or state.

95

7.2

Experiment and results
In our work, we combined the PIM technique with our EUV HHG source. The experimental

setup is shown in Fig. 7.1(a). The HHG process is driven using a Ti:sapphire amplifier system(KMLabs Dragon, 785 nm, 5 kHz, 1.5 mJ, 22 fs). The laser is coupled into a 150 µm inner diameterhollow-core waveguide, which is filled with argon at 60 torr to produce several phase-matchedharmonics near 29 nm. In contrast to previous experiments, this experiment does not not use
any EUV multilayer mirror to select one harmonic; instead all the four phase-matched harmonics
are sent onto the sample. A low-resolution spectrum measured by a two-dimensional grating on
the same plane as the sample is shown in Fig. 7.1(b). For the multiple-wavelength hyperspectral
measurement, the harmonics are refocused using an elliptical mirror at 5◦ glancing incidence. The
sample (titanium features patterned on silicon) was placed at the beam focus at 50◦ angle ofincidence, where all the harmonic beams have diameter ≈10 µm. The incoherent superposition of
the diffraction intensity from all four colors was measured by an EUV-sensitive CCD detector placed
55 mm away from the sample. The detector was positioned normal to the specular reflection of thebeam from the sample. The hyperspectral ptychographical data set consisted of scatter patterns
collected at 150 different scan positions, with a step size of 1 µm between positions.
We use the PIM technique to reconstruct the sample’s response to each of the four harmonicorders, with the amplitudes shown in Fig. 7.1 (c-f) and phases shown in Fig. 7.1(g-j) for fourcolors respectively. These reconstructions seem to have degraded quality compared with single
color reconstruction. The probable reason is that while the signal-to-noise ratio for the sum of all
four colors is about the same as for the single-color situation, after divided, or decomposed into
four colors, each color has much lower SNR leading to lower image quality. This is especially true
for the fourth color. As seen from the spectral weights in Fig. 7.1b, the fourth color is much weaker
than other three colors; and as shown in the reconstructions, the sample is not recognizable for the
fourth color.

96

Sampleb

CCD
Elliptical
Mirrorλ1

λ2

1Spectrum [a.u.]

a

0.8
0.6

λ1

λ3

0.4

λ4

0.2
0

25

30

35
λ [nm]

40

λ3

λ2

λ4

c

d

e

f

g

h

i

j

Figure 7.1: Setup and results of hyperspectral imaging with four harmonic orders using ptychographical information multiplexing. (a) Experimental setup. (b) Measured spectrum with a low
resolution 2D grating sample. Simultaneous reconstructions for the four wavelengths are shown in(c-f) for the amplitude and (g-j) for the phase. Scale bar: 10 µm, shared by (c-j).

97

7.3

Conclusion
Here the source in our experiment has a λ/∆λ of ≈ 27%. Before the invention of the techniqueused in this chapter, it is extremely difficult to use such a broadband light source to perform
diffractive imaging. It represents a great step forward to reconstruct anything with a broadband
source, but even more is achieved here: ptychography allows for individual reconstruction of thecolors that illuminate the sample simultaneously, which represents a simple yet powerful approachfor hyperspectral imaging, or spectro-microscopy.
More colors bring more information and less ambiguity in imaging, a capability that colorvision brings to humankind. For the specific experiment in this chapter, with phase reconstructed
for more than one color, the calculated height has less ambiguity resulting from the 2π phasewrapping problem [87]. In the future, this technique promises to enable spectroscopic EUV/X-ray
imaging by taking advantage of the various element-specific absorption edges at EUV wavelength.
For example, when combined with this technique, the water window wavelengths generated from
HHG [16] can be used to obtain elemental contrast in biological specimens.

Chapter 8

FutureWork presented in the previous chapters opens up various applications for this HHG CDI
microscope. At the top of the list is ultrafast dynamic imaging that makes use of the ultrashort,
femtosecond pulse duration of the HHG source. Investigating ultrafast dynamics at the nanometer scales can find applications in the study of ultrafast heat transport and surface acoustic wave
dynamics at the nanoscale [73, 74, 90], ultrafast magnetic dynamics [75, 82, 91, 92], fracture dynamics, shock formation, ablation, plasma formation under extreme conditions. Two methods exist
to exploit the ultrashort pulse duration of a HHG source. The first is the pump-probe method,which relies on repeatable excitation of the sample. This method with a HHG source has been
applied to study of dynamics of ultrafast heat transport and surface acoustic wave , and ultrafast
magnetics [75,82,91,92]. These studies can be extended to an imaging mode to investigate the local
behavior. When radiation-induced damage and intrinsic sample movement prevent the repeatable
excitation of the sample, the single-shot method is desirable. Proof-of-principle experiments havebeen demonstrated on both FEL [93, 94] and HHG [95] sources. Single-shot experiments typicallyrequires high photon counts from a single pulse, but the photon count requirement can be dramatically lowered in some cases with advanced algorithms such as the expectation maximization
algorithm [96]. It seems possible to apply these algorithms to a current HHG source to perform
single-shot experiments.
Any improvement on the HHG probe, or the CDI image-forming method, can bring improvements to the microscopy that combine these two elements. On the HHG probe side, it has been

99
demonstrated that this source can generate water-window photons (between the carbon and oxygen
K-shell absorption edges at 284-540 eV) [16]. This spectral range is of particular importance forbiological imaging as in this range water appears transparent while nitrogen and other biologicalelements are absorbing. In 2012, Popmintchev, Chen et al. further extended the HHG photon
energy to above 1.6 keV [18]. Sub-nanometer resolution is promised if this new high photon energy
HHG source can be applied to CDI, which would face two challenges: limited flux at this highenergy, and the supercontinua, ultra-broadband spectrum: it has a continuous spectrum ranging
from 200 eV all the way to 1.6 keV. Either it needs to be monochromatized, or certain imaging
algorithm, such as the PIM algorithm shown in Chapter 7 that deals with broadband illuminationneeds to be applied to the diffraction data. A third interesting development in the HHG source isthe generation of circularly polarized HHG [14]. Combined with CDI, it would enable imaging ofmagnetic nanodomains based X-ray magnetic circular dichroism [97]. On the CDI image-forming
side, Chapter 7 provides a proof-of-principle demonstration of hyperspectral imaging, where thedifference in contrasts for each color comes from the phase relative to the surface profile. One possible extension of hyperspectral imaging would be to image the distribution of different elements
inside the spectrum.
Looking further ahead, the microscopy that combines HHG and CDI in this thesis is poised toimage with nanometer spatial resolution and femtosecond temporal resolution, studying the world
at the ultrasmall space limit and the ultrafast time limit.

Bibliography[1] Richard P Feynman. There’s plenty of room at the bottom. Engineering and science, 23(5):22–
36, 1960.
[2] Anne Sakdinawat and David Attwood. Nanoscale X-ray imaging. Nature photonics, 4(December), 2010.
[3] Henry N. Chapman and Keith A. Nugent. Coherent lensless X-ray imaging. Nature Photonics,
4(12):833–839, December 2010.
[4] Rolf Erni, Marta Rossell, Christian Kisielowski, and Ulrich Dahmen. Atomic-Resolution Imaging with a Sub-50-pm Electron Probe. Physical Review Letters, 102(9):096101, March 2009.
[5] D. Gabor. A new microscopic principle. Nature, 161(4098):777–778, May 1948.
[6] Jianwei Miao, Pambos Charalambous, Janos Kirz, and D Sayre. Extending the methodology of
X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens. Nature,
400(July):342–344, 1999.
[7] Richard L Sandberg, Ariel Paul, Daisy A Raymondson, Steffen Hädrich, David M Gaudiosi,
Jim Holtsnider, Raanan I Tobey, Oren Cohen, Margaret M Murnane, Henry C Kapteyn,
Changyong Song, Jianwei Miao, Yanwei Liu, Farhad Salmassi, and Steffen Ha. Lensless Diffractive Imaging Using Tabletop Coherent High-Harmonic Soft-X-Ray Beams. Physical Review
Letters, 99(9):098103, August 2007.
[8] Matthew D Seaberg, Daniel E Adams, Ethan L Townsend, Daisy A Raymondson, William F
Schlotter, Yanwei Liu, Carmen S Menoni, Lu Rong, Chien-Chun Chen, Jianwei Miao, Henry C
Kapteyn, and Margaret M Murnane. Ultrahigh 22 nm resolution coherent diffractive imaging
using a desktop 13 nm high harmonic source. Optics Express, 19(23):22470–9, November 2011.
[9] D Attwood. Soft X-Rays and Extreme Ultraviolet Radiation: Principles and Applications.
Cambridge University Press, 2007.
[10] D Paganin. Coherent X-Ray Optics. Oxford Series on Synchrotron Radiation. OUP Oxford,
2006.
[11] J Bokor, P. H. Bucksbaum, and R. R. Freeman. Generation of 355-nm coherent radiation.
Optics Letters, 8(4):217, April 1983.

101[12] A. McPherson, G Gibson, H Jara, U Johann, T S Luk, I. A. McIntyre, K Boyer, and C K
Rhodes. Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases.
Journal of the Optical Society of America B, 4(4):595, April 1987.
[13] P. Corkum. Plasma perspective on strong field multiphoton ionization. Physical Review
Letters, 71(13):1994–1997, September 1993.
[14] Ofer Kfir, Patrik Grychtol, Emrah Turgut, Ronny Knut, Dmitriy Zusin, Dimitar Popmintchev,
Tenio Popmintchev, Hans Nembach, Justin M. Shaw, Avner Fleischer, Henry Kapteyn, Margaret Murnane, and Oren Cohen. Generation of bright phase-matched circularly-polarized
extreme ultraviolet high harmonics. Nature Photonics, 9(February):99–105, December 2014.
[15] Charles G Durfee Iii, Andy R Rundquist, Sterling Backus, Catherine Herne, Margaret M
Murnane, Henry C Kapteyn, and Charles Durfee. Phase Matching of High-Order Harmonics
in Hollow Waveguides. Physical Review Letters, 83(11):2187–2190, September 1999.
[16] M.-C. Chen, P. Arpin, T. Popmintchev, M. Gerrity, B. Zhang, M. Seaberg, D. Popmintchev,
M. M. Murnane, and H. C. Kapteyn. Bright, Coherent, Ultrafast Soft X-Ray Harmonics Spanning the Water Window from a Tabletop Light Source. Physical Review Letters,
105(17):173901, October 2010.
[17] Randy A Bartels, Ariel Paul, Hans Green, Henry C Kapteyn, Margaret M Murnane, Sterling
Backus, Ivan P Christov, Yanwei Liu, David Attwood, and Chris Jacobsen. Generation of
spatially coherent light at extreme ultraviolet wavelengths. Science, 297(5580):376–378, 2002.
[18] Tenio Popmintchev, Ming-Chang Chen, Dimitar Popmintchev, Paul Arpin, Susannah Brown,
Skirmantas Alisauskas, Giedrius Andriukaitis, Tadas Balciunas, Oliver D Mücke, Audrius
Pugzlys, Andrius Baltuska, Bonggu Shim, Samuel E Schrauth, Alexander Gaeta, Carlos
Hernández-Garcı́a, Luis Plaja, Andreas Becker, Agnieszka Jaron-Becker, Margaret M Murnane, Henry C Kapteyn, Skirmantas Ališauskas, Tadas Balčiunas, and Andrius Baltuška.
Bright coherent ultrahigh harmonics in the keV x-ray regime from mid-infrared femtosecond
lasers. Science (New York, N.Y.), 336(6086):1287–91, June 2012.
[19] R. A. Bartels, A. Paul, M. M. Murnane, H. C. Kapteyn, S. Backus, Y. Liu, and D. T. Attwood.
Absolute determination of the wavelength and spectrum of an extreme-ultraviolet beam by aYoung’s double-slit measurement. Optics Letters, 27(9):707–709, May 2002.
[20] JM Rodenburg. Ptychography and Related Diffractive Imaging Methods. Advances in Imaging
and Electron Physics, 5670(07):87–184, 2008.
[21] D Sayre. Some implications of a theorem due to Shannon. Acta Crystallographica, page 843,
1952.
[22] D Sayre. Imaging Processes and Coherence in Physics. Springer, Berlin, 1980.
[23] J R Fienup. Reconstruction of an object from the modulus of its Fourier transform. Optics
Letters, 3(1):27, July 1978.
[24] R H T Bates. Fourier phase problems are uniquely solvable in mute than one dimension. I:
Underlying theory. Optik, 62(3):247–262, 1982.

102[25] J R Fienup. Phase retrieval algorithms: a comparison. Applied optics, 21(15):2758–69, August
1982.
[26] Veit Elser. Phase retrieval by iterated projections. Journal of the Optical Society of America.
A, Optics, image science, and vision, 20(1):40–55, January 2003.
[27] Bosheng Zhang, Matthew D. Seaberg, Daniel E. Adams, Dennis F. Gardner, Elisabeth R.
Shanblatt, Justin M. Shaw, Weilun Chao, Eric M. Gullikson, Farhad Salmassi, Henry C.
Kapteyn, and Margaret M. Murnane. Full field tabletop EUV coherent diffractive imaging ina transmission geometry. Optics Express, 21(19):21970, September 2013.
[28] David Shapiro, Pierre Thibault, Tobias Beetz, Veit Elser, Malcolm Howells, Chris Jacobsen,
Janos Kirz, Enju Lima, Huijie Miao, Aaron M Neiman, and David Sayre. Biological imaging
by soft x-ray diffraction microscopy. Proceedings of the National Academy of Sciences of theUnited States of America, 102(43):15343–15346, October 2005.
[29] Jianwei Miao, Tetsuya Ishikawa, Qun Shen, and Thomas Earnest. Extending X-ray crystallography to allow the imaging of noncrystalline materials, cells, and single protein complexes.
Annual review of physical chemistry, 59:387–410, January 2008.
[30] I. Robinson, I. Vartanyants, G. Williams, M. Pfeifer, and J. Pitney. Reconstruction of the
Shapes of Gold Nanocrystals Using Coherent X-Ray Diffraction. Physical Review Letters,
87(19):195505, October 2001.
[31] G. Williams, M. Pfeifer, I. Vartanyants, and I. Robinson. Three-Dimensional Imaging ofMicrostructure in Au Nanocrystals. Physical Review Letters, 90(17):175501, April 2003.
[32] Mark A Pfeifer, Garth J Williams, Ivan A Vartanyants, Ross Harder, and Ian K Robinson.
Three-dimensional mapping of a deformation field inside a nanocrystal. Nature, 442(7098):63–
6, July 2006.
[33] Brian Abbey, Garth J. Williams, Mark A. Pfeifer, Jesse N. Clark, Corey T. Putkunz, Angela
Torrance, Ian McNulty, T. M. Levin, Andrew G. Peele, and Keith A Nugent. Quantitative
coherent diffractive imaging of an integrated circuit at a spatial resolution of 20 nm. Applied
Physics Letters, 93(21):214101, 2008.
[34] W Hoppe. Diffraction in inhomogeneous primary wave fields. 1. Principle of phase determination from electron diffraction interference. Acta. Crystallogr. A, 25:495–501, 1969.
[35] W Hoppe and G Strube. Diffraction in inhomogeneous primary wave fields. 2. optical experiments for phasedetermination of lattice interferences. Acta Crystallographica Section A
Foundations of Crystallography, 25:502–507, 1969.
[36] W Hoppe. Diffraction in inhomogeneous primary wave fields. 3. amplitude and phase determination for nonperiodic objects. Acta Crystallogr. A, 25:508–515, 1969.
[37] Reiner Hegerl and W Hoppe. Dynamische theorie der kristallstrukturanalyse durch elektronenbeugung im inhomogenen prim{ä}rstrahlwellenfeld. Berichte der Bunsengesellschaft f{ü}rphysikalische Chemie, 74(11):1148–1154, 1970.
[38] R Hegerl and W Hoppe. Phase evaluation in generalized diffraction (ptychography). Proc.
Fifth Eur. Cong. Electron Microscopy, pages 628–629, 1972.

103[39] Henry N. Chapman. Phase-retrieval X-ray microscopy by Wigner-distribution deconvolution.
Ultramicroscopy, 66(3-4):153–172, December 1996.
[40] H. Faulkner and J. Rodenburg. Movable Aperture Lensless Transmission Microscopy: A Novel
Phase Retrieval Algorithm. Physical Review Letters, 93(2):023903, July 2004.
[41] J. M. Rodenburg and H. M. L. Faulkner. A phase retrieval algorithm for shifting illumination.
Applied Physics Letters, 85(20):4795, 2004.
[42] Pierre Thibault, Martin Dierolf, Andreas Menzel, Oliver Bunk, Christian David, and Franz
Pfeiffer. High-resolution scanning x-ray diffraction microscopy. Science (New York, N.Y.),
321(5887):379–82, July 2008.
[43] J. M. Rodenburg. Ptychography: early history and 3D scattering effects. In Davide Bleiner,
editor, Proc. of SPIE, volume 8678, page 867809, December 2012.
[44] Matthew D. Seaberg, Bosheng Zhang, Dennis F. Gardner, Elisabeth R. Shanblatt, Margaret M.
Murnane, Henry C. Kapteyn, and Daniel E. Adams. Tabletop nanometer extreme ultraviolet
imaging in an extended reflection mode using coherent Fresnel ptychography. Optica, 1(1):39,
July 2014.
[45] Joseph W Goodman. Introduction to Fourier Optics. McGraw-Hill physical and quantum
electronics series. McGraw-Hill, 2nd edition, 1996.
[46] J D Jackson. Classical Electrodynamics. Wiley, 1998.
[47] S Celozzi, R Araneo, and G Lovat. Electromagnetic Shielding. Wiley Series in Microwave andOptical Engineering. Wiley, 2008.
[48] J. Stratton and L. Chu. Diffraction Theory of Electromagnetic Waves. Physical Review,
56(1):99–107, July 1939.
[49] S. Schelkunoff. On Diffraction and Radiation of Electromagnetic Waves. Physical Review,
56(4):308–316, August 1939.
[50] WR Smythe. The Double Current Sheet in Diffraction. Physical Review, 72(11):1066–1070,
December 1947.
[51] S. A. Schelkunoff. Some Equivalence Theorems of Electromagnetics and Their Application to
Radiation Problems. Bell System Technical Journal, 15(1):92–112, January 1936.
[52] Henry N Chapman, Anton Barty, Stefano Marchesini, Aleksandr Noy, Stefan P Hau-Riege,
Congwu Cui, Malcolm R Howells, Rachel Rosen, Haifeng He, John C H Spence, Uwe Weierstall,
Tobias Beetz, Chris Jacobsen, and David Shapiro. High-resolution ab initio three-dimensional
x-ray diffraction microscopy. Journal of the Optical Society of America. A, Optics, image
science, and vision, 23(5):1179–200, May 2006.
[53] Martin Dierolf, Andreas Menzel, Pierre Thibault, Philipp Schneider, Cameron M Kewish,
Roger Wepf, Oliver Bunk, and Franz Pfeiffer. Ptychographic X-ray computed tomography at
the nanoscale. Nature, 467(7314):436–9, September 2010.

104[54] M Holler, a Diaz, M Guizar-Sicairos, P Karvinen, Elina Färm, Emma Härkönen, Mikko Ritala,a Menzel, J Raabe, and O Bunk. X-ray ptychographic computed tomography at 16 nm isotropic
3D resolution. Scientific reports, 4:3857, January 2014.
[55] S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells, U. Weierstall,and J. C. H. Spence. X-ray image reconstruction from a diffraction pattern alone. PhysicalReview B, 68(14):140101, October 2003.
[56] Andrew M Maiden and John M Rodenburg. An improved ptychographical phase retrieval
algorithm for diffractive imaging. Ultramicroscopy, 109(10):1256–62, September 2009.
[57] D Russell Luke. Relaxed averaged alternating reflections for diffraction imaging. Inverse
Problems, 21(1):37–50, February 2005.
[58] S Marchesini. Invited article: a unified evaluation of iterative projection algorithms for phaseretrieval. The Review of scientific instruments, 78(1):011301, January 2007.
[59] Dennis F Gardner, Bosheng Zhang, Matthew D Seaberg, Leigh S Martin, Daniel E Adams,
Farhad Salmassi, Eric Gullikson, Henry Kapteyn, and Margaret Murnane. High numericalaperture reflection mode coherent diffraction microscopy using off-axis apertured illumination.
Optics Express, 20(17):19050–9, August 2012.
[60] Bosheng Zhang, Matthew D. Seaberg, Daniel E. Adams, Henry C. Kapteyn, and Margaret M.
Murnane. Coherent diffractive imaging with arbitrary angle of incidence. Patent US 62/043,132(Provisional), 2014.
[61] J. Rodenburg, A. Hurst, A. Cullis, B. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs,
and I. Johnson. Hard-X-Ray Lensless Imaging of Extended Objects. Physical Review Letters,
98(3):034801, January 2007.
[62] Brian Abbey, Keith A. Nugent, Garth J. Williams, Jesse N. Clark, Andrew G. Peele, Mark A.
Pfeifer, Martin de Jonge, and Ian McNulty. Keyhole coherent diffractive imaging. Nature
Physics, 4(5):394–398, March 2008.
[63] G. Williams, H. Quiney, B. Dhal, C. Tran, K. Nugent, a. Peele, D. Paterson, and M. de Jonge.
Fresnel Coherent Diffractive Imaging. Physical Review Letters, 97(2):025506, July 2006.
[64] J C H Spence, U Weierstall, and M Howells. Coherence and sampling requirements for diffractive imaging. Ultramicroscopy, 101(2-4):149–52, November 2004.
[65] H M Quiney, K A Nugent, and A G Peele. Iterative image reconstruction algorithms using
wave-front intensity and phase variation. Optics letters, 30(13):1638–40, July 2005.
[66] Daniel E Adams, Leigh S Martin, Matthew D Seaberg, Dennis F Gardner, Henry C Kapteyn,
and Margaret M Murnane. A generalization for optimized phase retrieval algorithms. Optics
Express, 20(22):24778–90, October 2012.
[67] Max Born and Emil Wolf. Principles of Optics. null. Cambridge University Press, 7th edition,
1999.
[68] Janos Kirz. Phase zone plates for x rays and the extreme uv. Journal of the Optical Societyof America, 64(3):301, March 1974.

105[69] H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent. Diffractive imaging ofhighly focused X-ray fields. Nature Physics, 2(2):101–104, January 2006.
[70] H J Kreuzer, M J Jericho, I A Meinertzhagen, and Wenbo Xu. Digital in-line holography withphotons and electrons. Journal of Physics: Condensed Matter, 13(47):10729–10741, November
2001.
[71] Gabriel Koren, François Polack, and Denis Joyeux. Iterative algorithms for twin-image elimination in in-line holography using finite-support constraints. Journal of the Optical Society ofAmerica A, 10(3):423–433, March 1993.
[72] B.L. Henke, E.M. Gullikson, and J.C. Davis. X-Ray Interactions: Photoabsorption, Scattering,Transmission, and Reflection at E = 50-30,000 eV, Z = 1-92. Atomic Data and Nuclear Data
Tables, 54(2):181–342, July 1993.
[73] Mark E Siemens, Qing Li, Ronggui Yang, Keith A Nelson, Erik H Anderson, Margaret M
Murnane, and Henry C Kapteyn. Quasi-ballistic thermal transport from nanoscale interfaces
observed using ultrafast coherent soft X-ray beams. Nature materials, 9(1):26–30, January
2010.
[74] Damiano Nardi, Marco Travagliati, Mark E Siemens, Qing Li, Margaret M Murnane, Henry C
Kapteyn, Gabriele Ferrini, Fulvio Parmigiani, and Francesco Banfi. Probing thermomechanics
at the nanoscale: impulsively excited pseudosurface acoustic waves in hypersonic phononic
crystals. Nano letters, 11(10):4126–33, October 2011.
[75] Stefan Mathias, Chan La-O-Vorakiat, Patrik Grychtol, Patrick Granitzka, Emrah Turgut,
Justin M Shaw, Roman Adam, Hans T Nembach, Mark E Siemens, Steffen Eich, Claus M
Schneider, Thomas J Silva, Martin Aeschlimann, Margaret M Murnane, and Henry C Kapteyn.
Probing the timescale of the exchange interaction in a ferromagnetic alloy. Proceedings of the
National Academy of Sciences of the United States of America, 109(13):4792–7, March 2012.
[76] Tetsuo Harada, Masato Nakasuji, Yutaka Nagata, Takeo Watanabe, and Hiroo Kinoshita.
Phase Imaging of Extreme-Ultraviolet Mask Using Coherent Extreme-Ultraviolet Scatterometry Microscope. Japanese Journal of Applied Physics, 52(6S):06GB02, June 2013.
[77] S Roy, D Parks, K A Seu, R Su, J J Turner, W Chao, E H Anderson, S Cabrini, and S D
Kevan. Lensless X-ray imaging in reflection geometry. Nature Photonics, 5(April):243–245,
2011.
[78] Michael Zürch, Christian Kern, and Christian Spielmann. XUV coherent diffraction imaging
in reflection geometry with low numerical aperture. Optics express, 4(May):118–123, 2013.
[79] Tao Sun, Zhang Jiang, Joseph Strzalka, Leonidas Ocola, and Jin Wang. Three-dimensional
coherent X-ray surface scattering imaging near total external reflection. Nature Photonics,
6(September):586–590, 2012.
[80] D. J. Vine, G. J. Williams, B. Abbey, M. A. Pfeifer, J. N. Clark, M. D. de Jonge, I. McNulty,
A. G. Peele, and K. A. Nugent. Ptychographic Fresnel coherent diffractive imaging. PhysicalReview A, 80(6):063823, December 2009.

106[81] Fucai Zhang, Isaac Peterson, Joan Vila-Comamala, Ana Diaz, Felisa Berenguer, Richard Bean,
Bo Chen, Andreas Menzel, Ian K Robinson, and John M Rodenburg. Translation position
determination in ptychographic coherent diffraction imaging. Optics express, 21(11):13592–
606, June 2013.
[82] Stefan Mathias, Chan La-o vorakiat, Justin M. Shaw, Emrah Turgut, Patrik Grychtol, Roman
Adam, Dennis Rudolf, Hans T. Nembach, Thomas J. Silva, Martin Aeschlimann, Claus M.
Schneider, Henry C. Kapteyn, and Margaret M. Murnane. Ultrafast element-specific magnetization dynamics of complex magnetic materials on a table-top. Journal of Electron
Spectroscopy and Related Phenomena, 189(2013):164–170, August 2013.
[83] Pierre Thibault, Martin Dierolf, Oliver Bunk, Andreas Menzel, and Franz Pfeiffer. Probe
retrieval in ptychographic coherent diffractive imaging. Ultramicroscopy, 109(4):338–43, March
2009.
[84] Andrew M Maiden, Martin J Humphry, Fucai Zhang, and John M Rodenburg. Superresolution
imaging via ptychography. Journal of the Optical Society of America. A, Optics, image science,and vision, 28(4):604–12, April 2011.
[85] Toshiaki Ohtsuka, M Masuda, and N Sato. Ellipsometric study of anodic oxide films
on titanium in hydrochloric acid, sulfuric acid, and phosphate solution. Journal of The
Electrochemical Society, 132(4):787, 1985.
[86] Maria Advincula, Xiaowu Fan, Jack Lemons, and Rigoberto Advincula. Surface modificationof surface sol-gel derived titanium oxide films by self-assembled monolayers (SAMs) and nonspecific protein adsorption studies. Colloids and surfaces. B, Biointerfaces, 42(1):29–43, April
2005.
[87] Darren J Batey, Daniel Claus, and John M Rodenburg. Information multiplexing in ptychography. Ultramicroscopy, 138:13–21, March 2014.
[88] R N Wilke, M Priebe, M Bartels, K Giewekemeyer, A Diaz, P Karvinen, and T Salditt. Hard
X-ray imaging of bacterial cells: nano-diffraction and ptychographic reconstruction. Optics
express, 20(17):19232–54, August 2012.
[89] Pierre Thibault and Andreas Menzel. Reconstructing state mixtures from diffraction measurements. Nature, 494(7435):68–71, February 2013.
[90] Qing Li, Kathleen Hoogeboom-Pot, Damiano Nardi, Margaret M. Murnane, Henry C.
Kapteyn, Mark E. Siemens, Erik H. Anderson, Olav Hellwig, Elizabeth Dobisz, Bruce Gurney,
Ronggui Yang, and Keith a. Nelson. Generation and control of ultrashort-wavelength twodimensional surface acoustic waves at nanoscale interfaces. Physical Review B, 85(19):195431,
May 2012.
[91] Chan La-O-Vorakiat, Mark Siemens, Margaret M Murnane, Henry C Kapteyn, Patrik Grychtol, Roman Adam, Claus M Schneider, Justin M Shaw, Hans Nembach, and T J Silva. Ultrafast
Demagnetization Dynamics at the M Edges of Magnetic Elements Observed Using a Tabletop
High-Harmonic Soft X-Ray Source. Physical Review Letters, 103(25):257402, December 2009.
[92] Emrah Turgut, Chan La-o vorakiat, Justin M. Shaw, Patrik Grychtol, Hans T. Nembach,Dennis Rudolf, Roman Adam, Martin Aeschlimann, Claus M. Schneider, Thomas J. Silva,

107
Margaret M. Murnane, Henry C. Kapteyn, and Stefan Mathias. Controlling the Competition
between Optically Induced Ultrafast Spin-Flip Scattering and Spin Transport in Magnetic
Multilayers. Physical Review Letters, 110(19):197201, May 2013.
[93] Henry N. Chapman, Anton Barty, Michael J. Bogan, Sébastien Boutet, Matthias Frank,
Stefan P. Hau-Riege, Stefano Marchesini, Bruce W. Woods, Saša Bajt, W. Henry Benner,
Richard A. London, Elke Plönjes, Marion Kuhlmann, Rolf Treusch, Stefan Düsterer, Thomas
Tschentscher, Jochen R. Schneider, Eberhard Spiller, Thomas Möller, Christoph Bostedt,
Matthias Hoener, David a. Shapiro, Keith O. Hodgson, David van der Spoel, Florian Burmeister, Magnus Bergh, Carl Caleman, Gösta Huldt, M. Marvin Seibert, Filipe R. N. C. Maia,
Richard W. Lee, Abraham Szöke, Nicusor Timneanu, and Janos Hajdu. Femtosecond diffractive imaging with a soft-X-ray free-electron laser. Nature Physics, 2(12):839–843, November
2006.
[94] Rui Xu, Huaidong Jiang, Changyong Song, Jose a Rodriguez, Zhifeng Huang, Chien-Chun
Chen, Daewoong Nam, Jaehyun Park, Marcus Gallagher-Jones, Sangsoo Kim, Sunam Kim,
Akihiro Suzuki, Yuki Takayama, Tomotaka Oroguchi, Yukio Takahashi, Jiadong Fan, Yunfei
Zou, Takaki Hatsui, Yuichi Inubushi, Takashi Kameshima, Koji Yonekura, Kensuke Tono,
Tadashi Togashi, Takahiro Sato, Masaki Yamamoto, Masayoshi Nakasako, Makina Yabashi,
Tetsuya Ishikawa, and Jianwei Miao. Single-shot three-dimensional structure determinationof nanocrystals with femtosecond X-ray free-electron laser pulses. Nature communications,
5(May):4061, January 2014.
[95] a. Ravasio, D. Gauthier, F. Maia, M. Billon, J-P. Caumes, D. Garzella, M. Géléoc, O. Gobert,
J-F. Hergott, a M. Pena, H. Perez, B. Carré, E. Bourhis, J. Gierak, a. Madouri, D. Mailly,
B. Schiedt, M. Fajardo, J. Gautier, P. Zeitoun, P. Bucksbaum, J. Hajdu, and H. Merdji.
Single-Shot Diffractive Imaging with a Table-Top Femtosecond Soft X-Ray Laser-Harmonics
Source. Physical Review Letters, 103(2):028104, July 2009.
[96] Hugh T Philipp, Kartik Ayyer, Mark W Tate, Veit Elser, and Sol M Gruner. Solving structurewith sparse, randomly-oriented x-ray data. Optics express, 20(12):13129–37, June 2012.
[97] S Eisebitt, J Lüning, W F Schlotter, M Lörgen, O Hellwig, W Eberhardt, and J Stöhr. Lenslessimaging of magnetic nanostructures by X-ray spectro-holography. Nature, 432(7019):885–8,
December 2004.