Ptychography Coherent Diffractive Imaging Systems for
Extreme Ultraviolet and X-ray SourcesbyCharles Schrepferman Bevis
B.A., University of Colorado at Boulder, 2011
M.S., University of Colorado at Boulder, 2017

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
2020
Committee Members:
Margaret Murnane, Chair
Henry Kapteyn
Rafael Piestun
Sadegh Yazdi
Mathias Weberii
Bevis, Charles Schrepferman (Ph.D., Physics)
Ptychography Coherent Diffractive Imaging Systems for Extreme Ultraviolet and X-ray Sources
Thesis directed by Prof. Margaret Murnane

Ptychography coherent diffractive imaging is a rapidly developing method for microscopic
imaging with coherent X-ray and extreme ultraviolet sources. The technique, which shifts therole of image formation from a physical optic to a computational algorithm, provides a route
toward photon efficient imaging systems with diffraction-limited resolution. When combined withhigh-energy coherent light sources, a microscope capable of probing the nanoworld with exquisite
elemental and chemical sensitivity is realized. However, these systems come with concessions suchas the large data volume required to form an image, long data acquisition times, and relatively
complex image reconstruction methods.
This thesis focuses on the development and extension of ptychography coherent diffractive
imaging to higher throughput modalities. This is demonstrated with the formulation of the multiple
beam ptychography method that uses several beams to simultaneously probe different parts of asample. This is first shown with beams of different wavelengths and later with different polarization
states. The technique is then extended for beams with identical wavelengths and polarizations bycontrolling the aliasing of the measured signal.
Finally, an X-ray spectromicroscopy study of a highly heterogeneous meteoric grain is discussed. Ptychography coherent diffractive imaging is combined with scanning transmission X-ray
microscopy and X-ray absorption spectroscopy to reveal the meteorite’s mineralogical properties
through chemical, elemental, and textural identification with a resolution more than an order ofmagnitude lower than previous similar studies.

Dedication

For my mom. I love ya mumzy, and I miss you every day.

iv

Acknowledgements

My time at JILA and CU has given me the opportunity to collaborate with some of the brightest,
kindest, and genuine people I’ve met in my life. None of which would be possible without my advisors
Profs. Margaret Murnane and Henry Kapteyn. They gave me the support and advise to shape my scientificdevelopment over the past five years and the open and encouraging environment of their research grouphas been a wonderful place to develop my skills as a researcher and scientist. I have to thank the KM
group, past and present, for all of the support, interesting insight, and spontaneous coffee breaks over theyears. I’d like to thank the KM Imaging Team in no particular order: Elisabeth Shanblatt, Dennis Gardner,
Daniel Adams, Giulia Mancini, Michael Tanksalvala, Robert Karl, Yuka Esashi, Bin Wang, Nicholas Jenkins,
Peter Johnson, Christina Porter, Guan Gui, Zhe “Joe” Zhang, Chen-Ting Liao, Emma-Elizabeth CatingSubramanian, Michael Gerrity, and Michael Hemmer. This thesis would not have happened without theircollective mentoring, collaboration, laughs, hard work, and late nights in the depths of B1B19. There are
too many to thank, but I want to mention Joshua Knobloch, Travis Frazer, Begoña Abad, Christian Gentry,
Nathan Brooks, and Quynh Nguyen for the interesting conversations, collaborations, and friendships. Thanks
to my collaborators Mike Lo, Arjun Rana, and John Miao at UCLA for their fruitful collaborations at the
COSMIC beamline. The STROBE center has given me access to so many different opportunities I wouldn’t
have looked for and really strengthened my belief in the necessity and power of interdisciplinary collaboration.
Thanks to Lauren Mason, Ellen Keister, Sarah Schreiner, and Tess Eidem for making them happen. Thankyou to the NSF for their support through the Graduate Research Fellowship Program. Finally, thank you
to my friends and family for the love and support. This PhD. thing was hard, and I couldn’t have done it
without all the encouragement to see it through to the end.

v

Contents

Chapter
1 Introduction
1.1

1.2

1

Introduction to Imaging with High Energy Light . . . . . . . . . . . . . . . . . . . .

2

1.1.1

Development of Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

1.1.2

X-ray and EUV Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.1.3

X-ray and EUV Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

The Physics of Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.2.1

The Wave Equation of Light . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

1.2.2

The Helmholtz Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.3

Angular Spectrum Solution to the Helmholtz Equation . . . . . . . . . . . . . 11

1.2.4

The Fresnel Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.5

The Fraunhofer Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.6

The Fresnel Number and the Validity of Propagation Approximations . . . . 17

1.2.7

The Born Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2.8

The Ewald Sphere, Far-field Diffraction, and the Born Approximation . . . . 20

1.3

The Refractive Index and Contrast Generation . . . . . . . . . . . . . . . . . . . . . 22

1.4

Relevant Concepts in Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4.1

Amplitude and Phase Information . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4.2

Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.4.3

Coherent vs. Incoherent Scattering . . . . . . . . . . . . . . . . . . . . . . . . 26

vi
2 Coherent Radiation Sources

28

2.1

Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2High Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.1

The Three-Step Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.2

Phase Matching. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Ptychography Coherent Diffractive Imaging

37

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2

The Phase Problem

3.3

Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4

3.5

3.6

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.1

Transverse Coherence Length . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.2

Longitudinal Coherence Length . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.1

Some implications of a theorem due to Shannon . . . . . . . . . . . . . . . . 43

3.4.2

Sampling and Experimental Geometry . . . . . . . . . . . . . . . . . . . . . . 44

Iterative Phase Retrieval in CDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5.1

Constraint Sets and Generalized Projections . . . . . . . . . . . . . . . . . . . 45

3.5.2

Phase Retrieval Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Ptychography CDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.6.2

The Ptychography Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.6.3

The Ptychography Phase Retrieval Algorithm . . . . . . . . . . . . . . . . . . 54

4 Spatial, Spectral, and Polarization Multiplexing in Ptychography

58

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2

Multiple Mode Ptychography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3

Spatial Separation of Mutually Incoherent Probes . . . . . . . . . . . . . . . . . . . . 60
4.3.1

Spectral Multiplexing with Spatially Separate Beams . . . . . . . . . . . . . . 61

vii

4.4

4.3.2

Polarization Multiplexing with Spatially Separate Beams . . . . . . . . . . . 64

4.3.3

Temporal Multiplexing with Spatially Separate Beams . . . . . . . . . . . . . 67

Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5 Multiple Beam Ptychography with Mutually Coherent Beams

70

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3

5.4

5.2.1

Autocorrelation Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.2

Alias Cloaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.1

Autocorrelation Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3.2

Alias Cloaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.3.3

Image Quality Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 Spectroscopic X-ray Imaging of the Allende Meteorite

89

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.2.1

Allende Meteorite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2.2

Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.2.3

X-ray Imaging Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3

STXM Data Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.4

Ptychography Data Analysis

6.5

Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

7 Conclusions and Future Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

100

7.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

7.2

Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

viii
Bibliography

103

ix

FiguresFigure1.1

Robert Hooke’s Compound Microscope . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

The Electromagnetic Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Focusing Techniques for X-ray Light . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.4

Different CDI Experimental Configurations . . . . . . . . . . . . . . . . . . . . . . .

8

1.5

Coordinate System of a Scattering Event . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.6

Geometry of a Scattering Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.7

Incoherent vs. Coherent Scatter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1

The Layout of the Advanced Light Source . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2

Types of Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3

The Three-Step Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4

Phase Matching in a Gas-filled Waveguide . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1

Spatial Coherence Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2

Longitudinal Coherence Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3

Generalized Projection Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4

The Fourier Amplitude Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5

The Error Reduction Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6

Schematic of a Ptychography Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1

Spatially Separated, Spectrally Multiplexed Ptychography Experiment Setup . . . . 61

x
4.2

Object Reconstructions from Spatially Separated, Spectrally Multiplexed Ptychography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3

Chromatic Aberration of Spectrally Separate Probes . . . . . . . . . . . . . . . . . . 63

4.4

Spatially Separated, Polarization Multiplexed Ptychography Experiment Setup . . . 64

4.5

Measured and Reconstructed Spatially Separated, Orthogonally Polarized Probes . . 65

4.6

Orthogonally Polarized Reconstructions with Spatially Separate Beams . . . . . . . 66

4.7

Temporal Multiplexing Multiple Beam Ptychography Setup . . . . . . . . . . . . . . 69

5.1

The Experimental Schematic for Multiple Beam Ptychography . . . . . . . . . . . . 71

5.2

The Process of Autocorrelation Filtering . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.3

The Effect of Aliasing on Autocorrelation Filtering . . . . . . . . . . . . . . . . . . . 77

5.4

Experimental Results for Autocorrelation Filtering . . . . . . . . . . . . . . . . . . . 80

5.5

Experimental Results for Aliased Autocorrelation Filtering . . . . . . . . . . . . . . 82

5.6

Experimental Results for Aliased Autocorrelation Filtering . . . . . . . . . . . . . . 83

5.7

Laser Interferometry of Rabbit Cell Sample . . . . . . . . . . . . . . . . . . . . . . . 85

5.8

Comparison of Power Spectral Density Between Multiple Beam and Single Beam
Ptychography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.9

Comparison of Single and Multiple Beam Ptychography Resolution . . . . . . . . . . 88

6.1

X-ray Imaging Scheme at COSMIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2

Fe STXM Spectromicroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.3

STXM Absorption Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.4

Ptychography Amplitude Reconstructions Near Al, Fe, Mg, and Ni Absorption Edges 96

6.5

X-ray Ptychography Absorption Contrast . . . . . . . . . . . . . . . . . . . . . . . . 96

6.6

Spatial Resolution of Ptychography Reconstructions . . . . . . . . . . . . . . . . . . 97

6.7

Scattering Quotient Maps of Ptychography Reconstructions . . . . . . . . . . . . . . 98

Chapter 1

Introduction

The past decade has seen tremendous progress in the field of nanoscale imaging. Whileelectron and atomic force microscopy continue to provide unprecedented resolution on the atomic
scale, the short wavelengths and high brightness of coherent X-ray and extreme ultraviolet lightsources make them an ideal probe for the nanoworld. With increased penetration depths andless stringent sample preparation requirements, X-ray and EUV imaging offers exquisite elemental
and chemical contrast while probing a wide range of systems with length scales from of one toone hundred nanometers. Recent advances in coherent diffractive imaging (CDI) have enabled
diffraction-limit resolution microscopy systems free of aberrations due to costly and lossy X-ray
optics. This thesis focuses on the development of new imaging modalities for CDI systems, inparticular the throughput and performance of ptychography CDI is investigated and improved.
This thesis is organized as follow. The remainder of chapter 6.1 provides a background andreview of important concepts in EUV and X-ray CDI. Chapter 2 details the coherent, high energy
light sources used in this thesis. Chapter 3 provides an introduction to the concepts and algorithmsof CDI, ending with a discussion of ptychography CDI. In chapter 4, multiple mode ptychography
is discussed, along with an extension to multiple beam ptychography. Chapter 5 offers an extension
to multiple beam ptychography that permits large field of view imaging with identical probes. In
chapter 6 work done on a newly commissioned X-ray microscopy beam line at the Advanced Light
Source is discussed, where X-ray microscopy is used to infer the mineral composition of an Allendemeteorite fragment with unprecedented resolution. This thesis concludes with chapter 7 with a2summary of ongoing and future work.

1.1

Introduction to Imaging with High Energy Light

1.1.1

Development of Microscopy

When thinking of a microscope, an image of a compound microscope commonly appears in
one’s head. In this configuration, the objective is used to magnify an image of the original object
and the eyepiece is used by the observer to view the magnified image. Though the principles ofmagnification were known to the ancient Greeks, it would take more than sixteen hundred years
before these ideas were organized into useful imaging systems such as the telescope and microscope[1]1 .
It was Robert Hooke who published the first scientific treatise on the subject of microscopy
in 1665, Micrographia. Here, using both a single lens and compound microscope, Hooke describeda study of the microscopic world, including several revolutionary scientific findings such as thefirst observation of a cell and microorganisms [3]. Hooke also offered detailed descriptions andillustrations of his microscopes. In addition, Hooke was the first scientist to suggest that engineeringand controlling the light source was as crucial for the quality production of images. Figure 1.1
depicts Hooke’s suggested light source paired with his compound microscope.
The desire to view ever smaller systems spurred interest and research into the theory of optics
and light-matter interactions. Among the advancements to both the design and understanding of
the microscope, came an exploration into the fundamental limits of the performance of such an
imaging system.
An important advancement came in 1873 when Ernst Abbe formulated the diffraction limit oflight, which placed a lower limit on the resolving power of a microscope at one half the illuminating
wavelength [4]. Abbe’s resolution criteria states that objects smaller than ∼200 nm cannot be
resolved using conventional visible light-based imaging system.
1
Contrary to popular belief, Galileo Galilei did not invent the telescope, but rather he created his first design
based on a description he heard regarding a similar system built by a Dutch spectacle-maker [2].

3Figure 1.1: Robert Hooke’s Compound Microscope. A detailed drawing of Robert Hooke’s
compound microscope with an artificial light source. This design allowed for ideal viewing conditions of the specimen regardless of ambient light. The drawing is from [3].

The drive to beat the diffraction limit has spawned a variety of super-resolution imaging
techniques including evanescent near-field imaging [5], well controlled fluorophores in fluorescent
microscopy [6, 7], and mulitphoton excitation techniques [8]. A brute-force method to lowering theresolution limit is to simply scale the wavelength of the illumination by a desired factor – mucheasier said than done. The imaging system with the highest resolving power to date is the electron
microscope, which uses this philosophy.
These microscopes use electrons as an illumination source with de Broglie wavelengths λ =h/p, where h is Plank’s constant and p is the electron’s momentum [9]. The wavelength of theseelectrons are typically picometers and theoretically allows for picometer-scale imaging. In practice,
this is usually reduced to Ångström levels due to aberrations in the electron imaging system [10].

1.1.2

X-ray and EUV Light

An important discovery in the field of light science came in 1893, when Wilhelm Róntgen
discovered X-ray light. Róntgen noted that when accelerated electrons struck a thick target, they
subsequently emitted rays that were invisible to the eye, but with a fluorescent screen, shadowgrams

4of objects placed in their path could be recorded [11]. These “X-rays”, as Róntgen called them,were high energy electromagnetic radiation produced by bremsstrahlung caused by the decelerationof electrons hitting a thick target [12].
The medical applications of these rays was almost immediately realized [13], with the application to material science and crystallography following in the next half-century [14–16]. The
next revolution in X-ray light came in the late 1940’s when the synchrotron light source was created. These sources use a finer control of the electron’s acceleration to produce bright, high energy
photons that can be made coherent through filtering [17].
In 1987, a third revolution in X-ray generation began with the discover and development ofhigh harmonic generation light sources [18]. In the last 30 years, high harmonic generation has
matured into an almost turn-key solution for a laboratory scale, high energy, coherent light source.
This technique uses an extremely nonlinear process to up convert light of some driving frequency
to a high order harmonic and is capable of producing bright, coherent flux at extreme ultraviolet(EUV) and X-ray wavelengths when properly phase matched [19–22].

Figure 1.2: The Electromagnetic Spectrum. The spectrum ranges from the infrared (IR),visible, ultraviolet, extreme ultraviolet (EUV), soft X-ray, and hard X-ray. Adapted from [23].

Like all electromagnetic radiation, EUV and X-ray light is characterized by its wavelength(λ), or equivalently its wavenumber (k = 2π/λ), frequency (f = c/λ), or photon energy (~ω, with
ω = 2πf ). The exact feature that is used to describe is light depends on the area of study andwhich feature is most directly related to the sample under investigation. For example, in imaging,5
wavelength is often used as it provides an approximate reference for the physical length scale ofthe sample or performance of the system. For spectroscopic studies, it is more common to referto the photon energy, where a particular energy structure of a sample may be of interest. EUV
light contains light with wavelength from about 100 nm - 10 nm, with soft X-rays following withwavelengths from approximately 10 nm - 1 Å. The distinction between “hard” and “soft” X-rays
is made to denote wavelengths that can and cannot penetrate air [14].
According to Abbe’s resolution criteria, EUV and soft X-ray light should be able to natively
study nanoscale systems if used as the source of a microscopy system. In addition, this region of thespectrum contains K and L absorption features of many elements, giving elemental and chemical
contrast not present at other spectral regions [23].

1.1.3

X-ray and EUV Microscopy

This field of study is the marriage of scientific advancements outlined in the previous two
sections. The shorter wavelength offers a lower resolution limit in these microscopes. Furthermore,
the long penetration depth and lower energy of these microscopes is suitable for a wider arrayof sample sizes and requires less stringent sample preparation methods compared to electron microscopy. In order to exploit these properties for high resolution imaging, a method to manipulate
the light source is needed. Unfortunately, conventional lenses cannot be fabricated for EUV and
X-ray light. Luckily, there are refractive, reflective, and diffractive solutions to focus and form
images with high energy light.
Refractive optics such as the compound refractive lens (Fig. 1.3(a)) have direct parallels
to refractive lenses for visible light. Instead of a single lens, the compound lens consists of many
discrete lenses which all impart a small deflection to the incident beam [25, 26]. These optics
have proven effective for focusing high energy X-rays that are not strongly absorbed, but are stillchallenging to use with EUV and soft X-ray light.
A Fresnel zone plate (Fig. 1.3(b)) is a popular choice in X-ray microscopy and acts as adiffraction-based analogue to a refractive lens. These elements work by carefully designing the6

Figure 1.3: Focusing Techniques for X-ray Light. (a) A compound lens consists of many small,
discrete lenses in series, each providing a small deflection to the incident light. (b) The Fresnel zone
plate is a diffractive optical element that focuses light by carefully controlling the interference of thetransmitted beam. (c) Kirkpatrick-Baez mirrors are two curved mirrors set at 90◦ relative to eachother. Light is focused in each direction by bouncing of each mirror. They are used in a glancing
incidence geometry to maintain high efficiency. (d) The Laue lens is the one dimensional analogue
to the Fresnel zone plate. This vertically symmetric geometry allows for higher manufacturing
tolerances and therefore tighter focusing. Adapted from [24].

structure of the optic such that transmitted X-rays constructively interfere at only one location,effectively focusing the beam [27]. This is done through patterning alternating regions of transparent and opaque concentric rings, which selectively block light that would destructively interfere at
the focal point. Though easy to align and compact, their resolution is limited by the fabrication
quality of the outer-most zone. This presents a difficult manufacturing problem for high resolution
imaging systems, where resolutions less than 30 nm are needed. Nonetheless, high quality zone
plates can be used for X-ray microscopy systems [28].
The Laue lens can be thought of as a one dimensional zone plate (Fig. 1.3(d)) and is
the diffractive analogue of a cylindrical lens [29]. The Laue lens was created to circumvent the
fabrication challenges associated with zone plate. By moving to a vertically symmetric geometry,the lens can be made via sputtering methods, allowing for finer layer thickness and therefore a7tighter focus [30]. Two Laue lenses can be used in succession to create a uniform, tight focal spot.
Reflective optical elements, such as the Kirpatrick-Baez mirrors (Fig 1.3(c)) are among theoldest solutions for manipulating X-ray light. The system consists of two parabolic or ellipticalmirrors oriented 90◦ with respect to each other. X-rays are directed onto the mirrors at a glancing
incidence to maintain high reflectivity, and independently focused in each direction. This configuration has been used for imaging, but resolution is limited due to aberrations [31].
These elements or a combination there of are used to build different types of X-ray microscopy
systems. For example, a zone plate can be used in place of a lens to build a conventional imaging
system for X-ray sources, so long as the source is coherent. This type of microscope forms a
direct image on a detector. On the other hand, some of these optics are designed to only focusing
light. These optics are still useful for imaging, when operating in a point-scanning modality.
Scanning transmission X-ray microscopy (STXM) is one such method where a beam is tightly
focused and scanned across the sample, building up an image point-by-point [32, 33]. Thoughthese X-ray microscope designs are appropriate for many applications, they are inherently limitedby the fabrication quality of these components.
An imaging method that avoids this reliance on optic fabrication quality is coherent diffractive
imaging (CDI). This technique transfers the task of image formation from a physical optic to acomputational algorithm that solves for what a sample must look like given a measurement of its
scatter pattern and additional knowledge about the experimental setup. Because there are no optics
between the sample and detector, it is capable a achieving diffraction-limited resolution as well as
being the most photon-efficient type of EUV or X-ray imaging [34]. Furthermore, the techniqueprovides both phase and amplitude contrast, allowing for a wide range of sample to be imaged.
CDI has matured into a well-developed technology in EUV and X-ray imaging over the last twenty
years, taking on many specialized configurations to probe different aspects of a sample, see Fig 1.4.
Work is this thesis is focused on expanding these modalities to higher throughput, higher resolution
configurations.

8

Figure 1.4: Different CDI Experimental Configurations. (a) Plane-wave CDI: A coherent
plane wave illuminates a sample and the far-field diffraction pattern is measured. (b) Bragg CDI:
the diffraction pattern is acquired at the Bragg angle from a nanocrystal. (c) Ptychography CDI: alocalized probe is scanned over an extended sample and a series of diffraction patterns are collectedfrom overlapping positions. (d) Fresnel CDI: a sample is positioned slightly out of focus of acoherent probe and the Fresnel diffraction pattern is measured by a detector. (e) Reflection CDI:
a coherent probe is reflected off the surface of a sample and the reflected diffraction is measured ona detector. Modified from [35].

1.2

The Physics of Diffraction
To understand the mechanics of CDI, it is important to understand the behavior of light as itpropagates through an imaging system. In this case, the imaging system is simply the interaction ofan illuminating probe with a sample and propagation of that light through free-space to a detector.
This section briefly reviews the theory of of wave propagation and diffraction.
The complete theory of diffraction physics requires an understanding of electromagnetism,
quantum mechanics, and field theory. However, for the work discussed in this thesis and the fieldof CDI, a purely classical treatment based on the wave theory of light will suffice. Emphasis isplaced on topics stemming from Fourier optics, a subfield of optics on which much of the physicalmathematical modeling of CDI is based. The derivations presented here are largely adapted from[12, 36, 37]. The eager student is directed to these sources for deeper understanding and a moredetailed (and perhaps coherent) treatment.

9
1.2.1

The Wave Equation of LightAs with any good derivation in classical electromagnetism, this one starts with Maxwell’sequations. This collection of four relationship between the properties of the electric and magnetic
field describe the complete theory of classical electromagnetism. Since light is electromagnetic radiation, insight into its behavior is gleaned by observing and manipulating these relations. Maxwell’sfour equations are:Gauss’s Law: ∇ · D = ρ(1.1)

Gauss’s Law for Magnetism: ∇ · B = 0

(1.2)

∂B
∂t
∂D
Ampère’s Law: ∇ × H = J +∂tFaraday’s Law: ∇ × E = −

(1.3)
(1.4)

where ∇ is the three dimensional gradient operator, D is the electric displacement, E is the electricfield, H is the magnetizing field, B is the magnetic field, ρ is the free charge, and J is the freecurrent. Here, the bold notation E denotes a vector quantity with components [Ex , Ey , Ez ].
It is now assumed that the electric and magnetic fields are propagating inside a dielectric
medium – more specifically, a linear, isotropic, homogeneous dielectric medium – and that there
is no free charge (ρ = J = 0) in the region of interest. In linear dielectric materials, a linear
dependence exists that relates D to E and B to H.
D = E(1.5)

B = µH(1.6)

where  is the permittivity and µ is the permeability of the material.
With the foundations in place, the wave equation is derived by first taking the curl of Eqn.
1.3 (Faraday’s Law).
∇ × ∇ × E = −∇ ×

∂B
∂t

(1.7)

10
Recalling the vector identity ∇ × ∇ × Q = ∇ (∇ · Q) − ∇2 Q, Eqn. 1.7 is rewritten as∇ (∇ · E) − ∇2 E = −∇ ×

∂B
.
∂t

(1.8)

Using Eqn. 1.1 (Gauss’s Law) and rearranging the time derivative on curl operators on the righthand side gives∇2 E =∂ (∇ × B)
.
∂t

(1.9)

Finally, an application of Eqn.: 1.4 (Ampère’s Law) rewritten in terms of E and B using Eqns.
1.5 and 1.6 gives the relation∇2 E = µ

∂2E
.
∂t2

(1.10)

Drawing from outside knowledge, a few strategic definitions are introduced. The refractive index
of the medium is defined as
r
n=

µ
,
µ0 0

(1.11)

with µ0 , 0 the vacuum permeability and permittivity, respectively. Furthermore, the speed of lightin vacuum is defined asc= √1= 299 792 458 m/s.
µ0 0(1.12)

Using these definitions, the wave equation for electromagnetism is written in the familiar form
∇2 E =

n2 ∂ 2 E,
c2 ∂t2(1.13)

where c/n is interpreted as the speed of wave propagation within the dielectric medium.
An nearly identical derivation for the wave equation for magnetism is carried out by taking
the curl of Eqn. 1.4 and following similar steps. Work done in this thesis follows from application,
approximations, and manipulations of Eqn. 1.13.

1.2.2

The Helmholtz Equation

Eqn. 1.13 is simplified by the application of a separable ansatz. In this case, the assertionis made that the wave equation is separable in space r = [x, y, z] and time (t). Beginning with a11monochromatic solution, it is assumed the solution can be written as
E(r, t) = E(r)e−i(ωt+φ) ,(1.14)

where ω is the frequency of the wave, and φ is some time independent, constant phase offset.
Dropping this into 1.13 gives∇2 E(r)e−i(ωt+φ) = −n2 ω 2E(r)e−i(ωt+φ) .
c2(1.15)

Canceling the common factor gives the familiar formulation of the Helmholtz equation∇2 E(r) + n2 k 2 E(r) = 0,

(1.16)

with the wave number k ≡ ω/c. Furthermore, the periodicity of the phase element in Eqn. ??
implies a temporal oscillation period of T = 2π/ω and it follows that the spatial distance of one
oscillation defines a wavelength λ = 2π/k.
Given the symmetry and separability of Eqn. 1.16, this vector equation is separable into
distinct scalar equations over each component of E(r), providing a relation that describes the spatial
evolution of a monochromatic stationary field through a homogeneous medium. For a polychromaticwave, a solution is obtained by spectrally decomposing into monochromatic components
1
E(r, t) = √2πZ ∞E ω (r)e−iωt dω

(1.17)

0where E ω (r) is the spatial component of pure wave with a single frequency ω.

1.2.3

Angular Spectrum Solution to the Helmholtz Equation

The standard and most sought after solution in the field of diffraction is the calculation of afield at some arbitrary point is space, given a the field distribution at some other known location.
This is a colloquial definition of a “propagation” problem. Typically, this problem is formulated asa boundary-value problem for Eqn. 1.16. The problem is usually set up by assuming that sources ofradiation exist in some lower half space z < 0 and the spatial distribution of some monochromatic

12
field ϕω (r) is known at z = 0. Given this, find the spatial distribution ϕω (r) anywhere in the upper
half space z > 0.
A practical first approach is to search for potential solutions on planes with z = δz > 0,
where any solution is a propagated field from the known distribution at the boundary z = 0. In
this formulation, the coordinate system is selected such that the direction of propagation is along
the z-axis, ẑ, often aligned with the direction of the optical axis.
The solution to this problem is usually found through the Kirchhoff or Rayleigh-Sommerfeld
diffraction integrals. This treatment is formulated to produce the the field ϕω (r p0 ) at a point p0
in the upper half space, z > 0, given the field on a surface S – and its normal derivative in thecase of the Kirchhoff formulation – enclosing p0 . With an application of Green’s Theorem and an
appropriate choice of Green’s function, these integrals are produce a relation between the field atplanes with z > 0 to the field at the boundary plane, z = 0. A different approach is consideredhere where a solution is found by considering the lateral Fourier decomposition of the wave.
Before beginning this derivation, it is convenient to examine a very general solution to theHelmholtz equation, a time-independent plane wave
p(r) = eik·r ,(1.18)

with k = 2π
λ [fx , fy , fz ]. The components fα are spatial frequencies, a normalized version of the
wave number commonly used in imaging science with f = k/2π = 1/λ. The components of the
wave vector k define the direction of propagation, with the components interrelated through
fz =

1
λ

q
1 − (fx )2 − (fy )2 .

(1.19)

With these relations, in a fixed plane at z = 0, a complex exponential exp[i2π(fx x + fy y)] isinterpreted as a plane wave who’s propagation direction in completely defined using Eqn.: 1.19.
Consider now the typical propagation problem, where a solution to Eqn. 1.16, ψ(x, y; 0), is
given at the plane z = 0. From this, the field at some other plane ψ(x, y; z1 ) is sought. The field
ψ(x, y; z) can be spectrally decomposed in the lateral dimensions x = [x, y], where the field and13decomposition are related via a Fourier transform1
ψ(x, y; z) =2πZψ̃(fx , fy ; z)ei2π(fx x+fy y) dfx dfy ,(1.20)

where ψ̃(fx , fy ; z) is the two dimensional Fourier transform of ψ(x, y; z) over the lateral dimensions(x, y). A notational convention is also introduced here. When considering the field in a fixed (x, y)plane, z is constant and is separated by a semicolon from other variables.
Inserting Eqn. 1.20 into Eqn. 1.16 gives the expression
12π

Z∂z2 ψ̃(fx , fy ; z) +



2πn
λ

2




1 − λ2 (fx2 + fy2 ) ψ̃(fx , fy ; z)dfx dfy = 0.

(1.21)For this integral to be zero for all functions, the integrand must vanish. So the above expression isequivalent to the differential equation∂z2 ψ̃(fx , fy ; z) = −



2πn
λ

2




1 − λ2 (fx2 + fy2 ) ψ̃(fx , fy ; z).

(1.22)The simple solution to Eqn. 1.22 is
i2πnψ̃(fx , fy ; z) = ψ̃(fx , fy ; 0)e λ√1−λ2 (fx2 +fy2 )z.

(1.23)

There are two different cases of this solution to consider. If λ2 (fx2 + fy2 ) > 1 the argument under
the square root becomes complex and the exponential term in Eqn. 1.23 becomes real and decays
to zero within a distance of a few wavelengths. This solution is known as an evanescent wave andmust only be considered for propagation on the length scale of the wavelength. For λ2 (fx2 + fy2 ) < 1
the exponential oscillates and the effect of propagation over some z distance is understood as achange of relative phases of the components of the lateral spectral decomposition. Since each plane
wave component propagates at a different angle, each travels a different distance between the twoplanes, which results in relative phase delays. Each component is a plane wave propagating in aunique angular direction defined by k so the lateral decomposition in Eqn. 1.20 is often referred toas the angular spectrum. Inserting Eqn. 1.23 into Eqn. 1.20 gives
1
ψ(x, y; z) =
2π

Z

i2πn

ψ̃(fx , fy ; 0)e λ√1−λ2 (fx2 +fy2 )z i2π(fx x+fy y)edfx dfy ,(1.24)

14
which provides the field ψ(x, y; z) given the angular spectrum of the field in some other planeψ̃(fx , fy ; 0). It should be noted that Eqn. 1.24 describes a relative relation. Meaning once the field
is know in some plane z0 , it can be calculated in any other plane z1 by finding the relative distance∆z = z1 − z0 . For practical and implementation purposes, this relation is often written to relate afield at two different z-planes using Fourier transforms−1
ψ(x, y; z1 ) = F⊥{F⊥ {ψ(x, y; z0 )} H(fx , fy ; ∆z)} ,(1.25)−1with F⊥ and F⊥
representing the forward and inverse Fourier transform over the lateral dimensions(x, y) and (fx , fy ), respectively, and wherei2πnH(fx , fy ; ∆z) = e λ√1−λ2 (fx2 +fy2 )∆z.

(1.26)The quantity H is referred to as the “free space transfer function” or “free space propagator”, but
is equally valid in homogeneous media with constant n.

1.2.4

The Fresnel Approximation

While Eqn. 1.25 provides an exact solution to the propagation of a general monochromaticelectric field between two parallel planes, there are situations where approximations can be leveraged
to simplify the relation. Because of the wavelengths and experimental geometries commonly usedin CDI systems, the Fresnel – or sometimes the paraxial or small-angle – approximation is often
valid. In this regime, it is assumed that the lateral extent of the field at plane z = z0 and thediffraction pattern in plane z = z1 are small compared to ∆z = z1 − z0 . Or in terms of spatial
frequencies, (fx2 + fy2 )  1/λ2 .
In this situation, the propagating plane waves in the angular spectrum are restricted to thosethat move at small angles relative to the optical axis. If this condition is met, then a binomial
approximation can be made in the exponential in Eqn. 1.26q

1 − λ2 (fx2 + fy2 ) ≈ 1 −
λ2 2
fx + fy2 .
2(1.27)

15
Substituting the expansion into the free space propagator then gives
H(fx , fy ; ∆z) ≈ e≈ei2πn∆zλ2
1− λ2 (fx2 +fy2 )i2πn∆zλe−iπnλ∆z (fx +fy ) .

2

2

It is seen above that within the Fresnel approximation, the spatial phase dispersion is approximatedas a quadratic phase dispersion. The propagated field under the Fresnel approximation is thenψ(x, y; z1 ) = ei2πn∆z
λ

n
o
2
2
−1
F⊥
F⊥ {ψ(x, y; z0 )} e−iπnλ∆z (fx +fy ) .

(1.28)

Fresnel propagation can be reformulated into an expression that only requires a single Fouriertransform. To see this, the convolution theorem is applied to Eqn. 1.28
n
n
n
ooo
i2πn∆z22−1−1e−iπnλ∆z (fx +fy ).
F⊥ {ψ(x, y; z0 )} F⊥ F⊥
ψ(x, y; z1 ) = e λ F⊥

(1.29)= ψ(x, y; z0 ) ⊗ D(x, y; ∆z)),with the Fresnel propagation operator
D(x, y; ∆z)) = ei2πn∆z
λ

o
n
2
2
−1
F⊥
e−iπnλ∆z (fx +fy ) .

(1.30)

Considering the limiting form of the Fresnel integrals gives an explicit form of D(x, y; ∆z) [38]:D(x, y; ∆z) =i2πn∆z −iπn221e λ e λ∆z (x +y ) .
iλ∆z(1.31)

Looking now at the explicit form of the convolution integral in Eqn. 1.29i2πn∆ze λψ(x, y; z1 ) =
iλ∆z

ZZ

−iπn

2

2

ψ(ξ, η; z0 )e λ∆z ((x−ξ) +(y−η) ) dξdη.

(1.32)

Expanding the last exponential inside the integrand and partially factoring it outside the integral
gives
i2πn∆ziπne λ22ψ(x, y; z1 ) =e λ∆z (x +y )
iλ∆z

ZZ h

i −i2πniπnψ(ξ, η; z0 )e λ∆z (ξ 2 + η 2 ) e λ∆z (xξ+yη) dξdη.

(1.33)Aside from multiplicative factors, the above expression is recognized as a Fourier transform of the
field at z = z0 multiplied by a quadratic phase factor. Written in terms of the Fourier transform
operatori2πn∆z
n
o
iπn
iπn
e λ
2
2
ψ(x, y; z1 ) =e λ∆z (x +y ) F ψ(ξ, η; z0 )e λ∆z (ξ 2 + η 2 ) ,iλ∆z(1.34)

16yxwhere the Fourier transform has conjugate variables (ξ, η) and ( λ∆z, λ∆z). For the work detailed

in this thesis, Eqn. 1.34 is used for the propagation of light. Though Eqn. 1.28 and 1.34 are analytically equivalent, they present different advantages when implemented numerically. Specifically,
Eqn. 1.28 is better suited for small propagation distances ∆z, while Eqn. 1.34 is better suitedfor larger distances. The reason for this is that small ∆z in Eqn. 1.34 results in a steep phasecurvature in the parabolic phase term, which becomes difficult to adequately sample while being
computationally tractable. And similarly for Eqn. 1.28 and large ∆z.

1.2.5

The Fraunhofer Approximation

A final useful approximation when considering wave propagation is the Fraunhofer – or farfield – approximation. In this limit the field propagation expression has yet another simplification,
reducing propagation to a single Fourier transform. When the propagation distance ∆z becomesvery large relative to the aperture size∆z π(ξ 2 + η 2 )max,
λ(1.35)

then the quadratic phase can be ignored and Eqn. 1.34 reduces to
i2π∆z

iπ
e λ
2
2
ψ(x, y; z1 ) =e λ∆z (x +y ) F {ψ(ξ, η; z0 )} .
iλ∆z(1.36)Up to multiplicative phase factors, the far-field diffraction pattern is simply the Fouriertransform of a field evaluated at frequencies
fx = x/λ∆z(1.37)
fy = y/λ∆z.
When numerically implementing this propagation, care must be taken to correctly setup and spatial
and frequency grids in each plane. Fortunately the resolution and maximum extend of the grids are
related through the usual Fourier conjugate relation such that if the grid at z0 is known, then thegrids at z1 are easily computed. Defining the grid spacing dξ in plane z0 and maximal extent ξmax ,then the frequency element resolution and grid extent is simply df = 1/ξmax and fmax = 1/dξ,
respectively. Thus, a straightforward procedure to convert from ξ → f → x.

17
1.2.6

The Fresnel Number and the Validity of Propagation Approximations

There are several different ways to treat wave propagation depending on system geometry.
The question remains how to determine which regime is appropriate for a given situation. One
convenient method is the dimensionless quantity Fresnel Number
FN =

a2
,
λ∆z

(1.38)

where a is a characteristic size such as aperture diameter in the initial field at z = z0 . With thiscriteria, three different regimes are defined:
FN  1: Fraunhofer Regime
FN ∼ 1: Fresnel Regime(1.39)FN  1: Spectrum of Plane Waves Propagation.
With this as a broad guide, the value of FN can be used as a general guide to determine which
method is appropriate for propagation.

1.2.7

The Born Approximation

The theory of diffraction is formulated here using the language of scattering physics. A
monochromatic light field scattered by an object is considered, and the scattering event is assumed
to be elastic – meaning the frequency of the incident light ω is assumed to be unchanged during
the scattering process. This derivation is largely derived from [37, 39]. To begin, the Helmholtz
equation is written with explicit spatial dependence with n → n(r),∇2 ψ(r) + n2 (r)k 2 ψ(r) = 0.

(1.40)

This is rewritten as an inhomogeneous equation∇2 ψ(r) + k 2 ψ(r) = −k 2 (n2 (r) − 1)ψ(r).

(1.41)The scattering potential f (r) is now introduced and is defined as f (r) = k 2 (n2 (r) − 1). An
important assumption of this method is that there is a single scattering event caused by a single

18
scattering potential. An ansatz for the solution is made where the field at all locations is written
as the sum of the incident wave ψi (r) and the scattered wave ψs (r),ψ(r) = ψi (r) + ψs (r).

(1.42)

 2

∇ + k 2 ψs (r) = −f (r)ψ(r),(1.43)

Inserting this into Eqn. 1.41 yields

where the fact that ψi (r) satisfies the homogeneous Helmholtz equation is used.
Solving this is relation requires the use of Green’s functions and Green’s first theorem [12].
A Green’s function, denoted as G(r, r 0 ), is an impulse response from a point source located at
location r 0 observed at location r. For the system considered here, the Green’s function needed isFigure 1.5: Coordinate System of a Scattering Event. An incident plane wave eiki ·r scattersfrom an object located at position r 0 from origin O. The response is observed at position r. Adaptedfrom [39].

the well-known out-going free space impulse response for the Helmholtz equation, a spherical wave
0G(r, r 0 ) =eik|r−r |.
|r − r 0 |(1.44)

With this, a solution to Eqn. 1.43 is found such that the scattered wave is written asZψs (r) =0f (r 0 )eik|r−r |ψ(r)d3 r 0 .
|r − r 0 |(1.45)

19
This expression is starting to look tractable, however the wave inside the integral is the total wave
and thus depends on knowledge of the scattered wave on the left hand side. To move forward, a few
assumption and approximation must be made. First, it is assumed that the scattering potential isweak such that ψs (r)  ψi (r). Second, it is assumed that the incident wave field is a simple plane
wave and can be written ψi (r) = eik0 ·r , with k0 the incident wave vector. Now, the total wave field
can be replaced with just the incident fieldZψs (r) =0

eik|r−r | ik0 ·r0 3 0
f (r )ed r.
|r − r 0 |0(1.46)

Finally, the observation point is taken to be sufficiently far away that the spatial extent of thescatterer can be neglected, or |r|  |r 0 |. Then the approximation |r − r 0 | ≈ r − r̂ · r 0 holds with r̂
the unit vector in the r direction. With this, the scattered wave vector k = k0 r̂ is defined and achange of variables into momentum transfer space is performed with momentum transfer
q = k − k0 .

(1.47)The scatter field is thenZψs (r) =0

eik0 r−k·r i(k·r0 −q·r0 ) 3 0
f (r )ed r,
r
0(1.48)

where the less stringent approximation |r − r 0 | ≈ r is made in the denominator of the integrand.
Continuing to simplify0

eik0 r−ik·r i(k·r0 −q·r0 ) 3 0
ψs (r) = f (r )
e
d r
r
Z
eik0 r0(1.49)=f (r 0 )eiq·r d3 r 0
reik0 r ˜=
f (q),
r
where it is seen that the scattered wave ψs (r) is equal to the three dimensional Fourier transform ofZ0

the scattering potential f (r). The implication being that if light scattered from an object is observedor recorded as a diffraction pattern, then the object’s spatial distribution can be recovered by asimple Fourier transform.

20
1.2.8

The Ewald Sphere, Far-field Diffraction, and the Born Approximation

With the approximations made in section 1.2.7, the momentum transfer was introduced asa convenient coordinate space to formulate the scattered wave generated by a given scattering
potential. Because only elastic scattering is considered, |k| = |k0 | = 2π/λ, then Eqn. 1.47 implies
that all possible q are confined to a two dimensional spherical shell embedded in a three dimensional
space, the so-called Ewald sphere.
The Ewald sphere has radius 2π/λ and the incident and scattered wave vectors k0 and k
pointing out from the center of the sphere. Typically, a coordinate system is chosen such that theincident wave vector is simply k0 = kẑ. If a detector is placed a distance z from the scatterer suchthat z is much larger than the sample dimensions, a relation between the detector plane coordinates[x, y] and momentum transfer coordinates [qx , qy , qz ] is derived through similar triangles, shown in
Fig. 1.6. Notably, the transverse components of q and k are equivalent:
qx = k x = k pxx2 + y 2 + z 2yqy = k y = k p
2
x + y2 + z2
qz = k

(1.50)!

zp−1 .
x2 + y 2 + z 2

Fig. 1.6 shows that there is a relation between the accessible parts of the Ewald sphere andthe experiment geometry through the angle θ. The larger the angle θ, the more information of theFourier distribution of the sample is captured on the detector. The information captured at larger
angles corresponds to higher spatial frequency information about the sample, so the achievable
resolution of a scattering experiment is directly linked to θ. This is parameterized through thenumerical aperture (NA) of an imaging system and is simply defined by the detector size andexperiment geometry for a lensless imaging system. For an experiment with detector size 2D andobject-to-detector distance z, the NA is defined by
NA = sin θ = √D
D2 + z 2.

(1.51)

21

Figure 1.6: Geometry of a Scattering Experiment. (a) An incident plane wave with wave
vector k0 = kẑ scatters with an element and produces scattering vector k that makes angle θ withthe optical axis. (b) The experimental geometry viewed in frequency space. Note that the scatteringexperiment is completely described by the momentum transfer vector q, which is restricted to values
that lie on the Ewald sphere surface (red arc). (c) If a detector (green dotted line) with spatial
extent D is placed a distance z that is much larger than the extent of the scatterer, then thedetector coordinates [x, y] and zẑ form a triangle that is mathematically similar to the one definedby the components of scattering vector [kx , ky ] and kz . The experiment geometry defines the NAof the system and along with the wavelength λ sets the spatial resolution limit of the experiment.
At small angle, or large Ewald sphere radius, qz ≈ 0 and q is confined to a plane. Figure adaptedfrom [40].

In the small angle approximation, the Ewald sphere is approximated as a plane and the22coordinate relations simply toxzyqy ≈ kz

qx ≈ k

(1.52)

qz ≈ 0.
If the experiment is configured to have a high NA, the curvature of the Ewald sphere cannot
be ignored and the scattered wave should no longer be described using only two dimensions. The
curvature will result in a distortion in the higher spatial frequencies due to the projection of theEwald sphere on to a two dimensional flat plane. The situation is further complicated if a system isconfigured with a non-normal incidence sample-beam geometry, where effects of conical diffraction
nonsymmetrically distort the recorded diffraction pattern. This poses problems for the practicalimplementation of wave propagation methods that often rely on the fast Fourier transform (FFT)
algorithm, which requires that the input signal is sampled on a linear grid. Fortunately, thisnonlinear signal can be interpolated back to a linear grid spacing using a procedure called tilted
plane correction (TPC) [41–43].

1.3

The Refractive Index and Contrast Generation
The first interpretation of the refractive index (n) that is often learned is that it is the ratioof the speed of a wave a it travels through a vacuum (c) to its speed inside a material (v), withn = c/v. While true, this definition glosses over some of the more subtle aspects of the index,
most importantly that its value is dependent on the wavelength of light λ. When considering EUV
and X-ray wavelengths, the value of the refractive index is often very close to 1. In this regime,
it is common to write the refractive index as a continuous function of both space r = [x, y, z] andwavelength λ [23]:n(r, λ) = 1 − δ(r, λ) − iβ(r, λ).

(1.53)

23
The real and imaginary parts of the index describe the dispersive and absorptive components of alight-matter interaction and are given byna (r) re λ2
f1 (λ)
2π
na (r) re λ2β=f2 (λ) ,2π
δ=(1.54)(1.55)

with na the spatially dependent atomic number density and re is the classical electron radius. f1
and f2 are called the atomic scattering factors and are measures of the scattering power of individual
atoms as a function of wavelength. They are semi-empirical quantities that can be calculated andfound in look-up tables for energies far from absorption features [44]. Close to absorption edges,
the value of f1 and f2 depending strongly on local conditions of the atoms in a material so it isusually more accurate to rely on measured values instead of calculated predictions [45].
The monochromatic wave ψ(r) exiting an object with refractive index distribution n(r) may
be parameterized using only two dimension with r = [x, y, z] = [x, z] asψ(x) = O(x)P (x),(1.56)

where O(x) is the complex object transmission function and P (r) is the wave incident on the sample
propagating along the z-axis. This formulation is known as the the projection approximation and
it is valid if the incident wave is sufficiently constant over the thickness of the object along thedirection of propagation. For an in depth study on the validity of the approximation, see [46–48].
Under this assumption, the complex object transmission function can be writtenO(x) = A(x)eiφ(x) .

(1.57)and contains information directly related to the absorption (imaginary) and phase shifting (real)parts of the complex refractive index through
2πA(x) = e− λRTβ(x,z)dz(1.58)

δ(x, z)dz,(1.59)

0and
φ(x) =

2π
λ

Z T
0

24
with T the object thickness along the axis of propagation. In CDI systems, this exit wave is
propagated to a detector where a diffraction pattern is recorded. If the initial exit wave is retrieved,
then a map of the spatially varying refractive index is obtained and material information about the
sample is recovered.

1.4

Relevant Concepts in Microscopy
In this section some important concepts in imaging and microscopy are reviewed that are

relevant to the work done in this thesis.

1.4.1

Amplitude and Phase Information

As detailed in section 1.3, light can be treated as complex field with
ψ (x) = A (x) eiφ(x) ,(1.60)

where the amplitude A (x) and phase φ (x) are both real. When this field is directly measured ona detector, only the intensity I is observed with
I = |ψ (x) |2 = A (x)2 .

(1.61)

If the field possesses A (x) = 1, then the wave appears uniform when detected, no matter the
structure of the phase. Fortunately, as a field propagates, amplitude and phase content begin
to mix, meaning that a measurement of the intensity of the field in a different plane could giveinformation about its phase structure elsewhere, the simplest example of this being interference
fringes.
The amplitude of a wave field scattered from a sample often contains information aboutsample absorption while the phase carries information about structure. Together, they paint a
complete picture about a sample’s elemental and chemical composition as well as thickness anddensity.
There are a variety of microscopy technique that are developed specifically to measure phase
information. Zernike phase contrast microscopy uses wave plates to convert phase difference pro-25
duced by scattering from a sample into amplitude contrast in the formed image [49]. Holography
uses an interferometric technique to simultaneously record both amplitude and phase information
about a sample by interfering a scattered wave with a known reference beam [36]. Chapter 3 details
how coherent diffractive imaging can be used to recover both phase and amplitude of a sample.

1.4.2

Resolution

Resolution is often the first characteristic thought about when investigating an imaging system. Ernst Abbe arrived at his resolution criteria by applying concepts of Fourier analysis to the
problem of image formation. Abbe considered an object to be composed of a series of sinusoidal
components with varying frequency and angle, and an image is formed by the interference between
the diffraction from these different components [4].
When illuminated, the sinusoids produce diffraction orders propagating at different angles
proportional to their frequency. Abbe contended that the a sinusoid component was detectable if
the first order diffraction was collected by the imaging system. Abbe formulated this analysis intoa resolution criteria that depends on the geometry of an imaging system
δr =

λ
2NA

(1.62)

where δr is the smallest resolvable feature in an image, λ is the illuminating wavelength, and NA
is the numerical aperture of a system. The NA is define by the maximum angle subsumed by thesystem aperture and the refractive index NA = n sin θ. Imaging system that reach this performanceare said to have “diffraction-limited” resolution.
The Rayleigh and Sparrow criteria have also been used to quantify resolution, differing from
Abbe’s definition through a numeric factor added to right hand side of Eqn. 1.62, (1.22 in the caseof Rayleigh, and 0.94 for Sparrow) [50]. Spatial frequency analysis of an imaging system allows forfurther definitions using the frequency response of the system [36].
Typically, the resolution of a microscope is thought of as being system dependent. Meaningthat images produced with a system will have some resolution and that resolution is dependent on

26
the configuration of the microscope. For diffractive imaging systems, such as the ones discussed inthis thesis, this is not entirely accurate. Though the resolution limit is very much set by the systemgeometry (i.e. Eqn. 1.62) the actual resolution of an image depends on the scattering power of asample. If the sample cannot efficiently scatter to the maximum acceptance angle of the NA, thenthe resolution is limited by the sample, not the system.

1.4.3

Coherent vs. Incoherent Scattering

The particular character of coherent scattering is important for coherent diffractive imaging.
To illustrate why, consider an inhomogeneous scattering medium, for this example: a collection of
nanoparticles, illuminated by a beam of light, with its diffraction pattern measured in the far-field.
There are three important length scales in this problem, the illumination wavelength λ, the average
distance between nanoparticles d, and the size of the illuminated area a [40].

Figure 1.7: Incoherent vs. Coherent Scatter. Diffraction from a collection of nanoparticles withinter particle distance d and illumination width a. (a) Under incoherent illumination, a continuous
diffraction ring is observed. (b) With coherent illumination, a speckled diffraction pattern uniqueto the particular particle configuration is produced. Figure reproduced from [40].

If the illumination is incoherent (Fig. 1.7(a)), then the recorded diffraction pattern exhibits
a maximum intensity at angle ∼ λ/d. This results from the diffraction pattern representing an
average over the many different wavefronts incident on the sample and can equivalently be thought

27
of as an average over many different particle ensembles. Varying a will not change the diffraction
pattern, only the area over which the incoherent wavefronts are averaged.
When coherent illumination is used (Fig. 1.7(b)), a similar dependence is observed, but finerfeatures with angular width ∼ λ/a appear – the speckle. These arise from interference between
wavefronts scattered from different particles. If a single particle position changes, all interferences
are affected, and the speckle pattern changes.
With this picture, it is understood that diffraction from coherent illumination reflects aparticular realization of the system, whereas incoherent illumination provides an average over an
ensemble of slightly different system configurations. It follows that from the coherent scatterpattern, an object’s unique structure may be recovered. Details of how to perform this inversion
are discussed in chapter 3.

Chapter 2

Coherent Radiation Sources

Coherent sources of X-ray and EUV light fall generally into two categories: facility scale
sources and tabletop sources. Facility scale sources such as free electron lasers (FEL) and synchrotron sources leverage relativistically accelerated electrons to generate high brightness and highenergy radiation that can be configured to emit over a very broad spectral bandwidth [51]. Throughfiltering, these sources can be made spatially coherent. However, as a user facility these sources areoften very large – sometimes the size of several buildings – and have limited availability for users,
as well as a high cost of operation and maintenance. These sources are best for studies requiring ahight flux and high energy light source.
Tabletop sources, on the other hand, tend be far cheaper and readily installable in researchand industrial settings. This gives greater access and allows for rapid prototyping and greater
customization for the specific needs of an experiment. Tabletop sources such as X-ray lasers andand high harmonic generation systems tend to have lower flux and spectral bandwidth, but research
into these sources is rapidly closing the gaps with their facility-scale counterparts. Work in thisthesis focused on HHG and synchrotron light sources, a more detailed discussion of these sources
is presented here.

2.1

Synchrotron Radiation
The synchrotron light source grew out of early efforts from high-energy physicists to constructparticle accelerators to study high-energy and nuclear physical phenomena [52, 53]. As accelera-

29
tor technology continued to develop and the understanding and benefits of coherent light sources
became apparent, synchrotron particle accelerators were developed to specifically act as a highbrilliance electromagnetic radiation sources [54]. A diagram of the Advanced Light Source (ALS)
at the Lawrence Berkeley National Lab – where work in this thesis was carried out – is depicted inFig. 2.1, showing a schematic layout of the components of a synchrotron radiation source.
A synchrotron generates radiation by accelerating electrons to extremely high, relativistic
speeds through a curved trajectory. An electron gun produces electrons that travel into a linear
accelerator that accelerates the electrons using a series of radio frequency cavities. The electron
beam (current) is inserted into a booster ring which increases the electron energy using a series ofbending magnets similar to the larger storage ring.

Figure 2.1: The Layout of the Advanced Light Source. A diagram of the Advanced Light
Source, a third generation synchrotron light source. A series of three accelerators (linear accelerator,
booster ring, and storage ring) accelerate electrons to 1.9 GeV. Synchrotron radiation is produced
in bending magnets and undulators. Beam lines tap into the radiation and direct the light intoexperiment end stations. Image from [55].

Once the electrons are accelerated, they are inserted into the storage ring, which maintains a30high current and electron energy using bending magnets to confine the electrons’ path. Beam lines
are setup around the storage ring to tap into the radiation produced by the accelerating electrons.
Synchrotron sources are not continuous, rather electrons bunches are created at the nodes of theaccelerating radio-frequency field. The brilliance and spectrum of the synchrotron is altered andenhanced through the placement of insertion devices in the storage ring [23].

Figure 2.2: Types of Synchrotron Radiation. Synchrotron radiation is produced with three
types of devices. Bending magnets, used to confine the electron trajectory to a closed path, arethe earliest device and produce radiation in with a broad spectrum. Wiggler and Undulators are
periodic magnetic arrays that cause deviations in the electron path. These deviation result inincreased acceleration and therefore radiation flux. These devices differ primarily in the strength oftheir magnetic field, which can be set to produce a bright but highly divergent beam with a broad
spectrum (wiggler), or a more collimated and narrow band emission (undulator). Reproduced from[23].

An insertion device is a periodic magnetic structure that causes the electron path to transversely oscillate. Wigglers and undulators are similar insertion devices, that deviate only in thestrength and period of their magnetic structure. Wigglers use a much higher magnetic field thatcauses a significant deviation in the electron beam, generating more synchrotron radiation at theexpense of poor beam divergence. Undulators use a weaker magnetic field to produce smaller oscil-

31
lations in the electron beam. These smaller oscillations allow the radiation bursts from successive
oscillations to interference, producing a much more narrow-band radiation spectrum with a smaller
divergence [56]. A schematic of the different types of synchrotron radiation is shown in Fig. 2.2.

2.2

High Harmonic Generation
High harmonic generation (HHG) is an extremely nonlinear process where an intense, ultra-

fast burst of coherent light of some fundamental frequency is upconverted to a higher frequency
harmonic [18, 57]. The polarization and coherence properties of the input light is preserved in thesimplest geometry while the pulse duration is actually reduced. However, it is possible to tune
the coherence and polarization properties using selection rules or molecular targets [58–60]. The
technique provides a compact and flexible way to produce coherent light in the VUV to hard X-ray
regimes when properly phase matched [19–22, 61, 62] and is driving a revolution in uncovering newunderstanding of interactions in quantum materials [61, 63–66], nanoscale transport [67, 68], andcoherent imaging [69–71].

2.2.1

The Three-Step Model

The three-step model is an elegant, simplified solution to the problem of HHG [72–74]. The
model is semi-classical, ignoring many details of the fully quantum mechanical description of theprocess. Nonetheless, the model provides an intuition into the HHG process and accurately predicts
a number of important features.
The three-step model is formulated by considering the interaction of a single atom with anultrafast laser pulse. The three-step model process, depicted in Fig. 2.4, is as follows:(1) An electron is ionized by the extremely strong electric field of the laser pulse. The ionization
is a result of the suppression of the Coulomb field at the peak of the intense field of thepulse.
(2) The electron is accelerated away from the parent atom, driven by the upcycle of the laser’s

32electric field. On the down cycle, the electron’s acceleration reverses and it is driven backtoward the parent ion, gaining momentum and kinetic energy.
(3) The electron arrives back at the parent ion where there is a finite probability of recombination that produces a photon releasing the energy accumulated by the free electron.

Figure 2.3: The Three-Step Model. Step 0: The ground state before the atom interacts withlaser field. Step 1: The strong laser field causes an electron to tunnel ionize into the continuum.
Step 2: The free electron is accelerated by the external field, gaining energy. Step 3: The freeelectron is driven back to its parent ion where it recombines, releasing its excess energy as a highenergy photon. Adapted from [75].

The energy of the emitted photon depends on both the ionization potential of the parent atom
as well as the amount of kinetic energy gain while in the continuum. This energy is understoodby considering the kinematic equations of motion of a free electron driven by an oscillating electricfield. Depending on which part of the external field cycle the electron is ionized at, the particle will
have differing amounts of kinetic energy when it recombines with the parent ion. By considering
all the different cycle points where ionization can occur, the maximum kinetic energy is found to
be Ekmax = 3.17Up . Where Up is the ponderomotive energy, the energy the electron experiences inan oscillating external field – sometimes called the cycle-averaged “quiver” energy – is given by [72]
Up =

e2
λ2 I
,
c0 me 4π 2 c2 4

(2.1)

33
where e is the electron charge, me is the electron mass, λ is the driving laser wavelength, and I is
the driving laser peak intensity. The maximum photon energy, or “cutoff energy”, is given by thesum of the ionization potential of the parent atom Ip and the maximum kinetic energy [72, 74]hωcutof f ≈ Ip + 3.17U p ∝ Iλ2 .

(2.2)

Eqn. 2.2 reveals an interesting feature of HHG: longer wavelength driving lasers can produce
higher energy photons during HHG. This can be understood by again considering the kinematics
of a free electron in an oscillating external field. The longer the wavelength of the driving laser, thelonger the electron is accelerated. This higher energy comes at the expense of HHG flux, as manyof the fundamental energy photons are required to create one high energy photon.
With the three-step model giving an intuitive understand of the HHG process, it is seenthat with a sufficiently strong laser field, HHG is a fairly robust method for producing EUV light.
Indeed, HHG has been demonstrated with many different media, including solid [76] and plasma[77] targets, as well as a diluted gas [78] and gas-filled waveguides [79]. However, only the single
emitter picture has been considered. To get bright, high energy EUV light, a coherent summationof many single emitters is needed.

2.2.2

Phase Matching

With the intuitive understanding of the HHG process through the three-step model, attention
must now be placed on how to generate bright, coherent beams of EUV light. In order to buildup
the the emission from many single emitters, their emission must constructively interfere at themacroscopic level. This process is called phase matching and refers to matching the dispersion of
the HHG EUV light with the driving laser field within the generating medium. Many different
media and techniques have been investigated for phase matching, the discussion here is restricted
to phase matching in a gas-filled capillary waveguide [19, 80], which was the HHG method used forthe work in this thesis. Due to the large difference in the EUV and fundamental field wavelengths,
there is an inherent dispersion mismatch that must be compensated. This is done by adjusting the34properties of the generating medium by varying experimental parameters [19, 81, 82].
The phase matching condition is a statement that the phase velocity vp = w/k of the fundamental field must be matched with the EUV harmonics. For the q th harmonic order and driving
laser with wave number kL , the phase matching condition is∆k = kq − qkL = 0.

(2.3)

Inside a gas-filled capillary waveguide, this dispersion mismatch has three primary contributions and is written as [19, 80]:
µ 2 λL2πP (1 − η)∆k ≈ q 11 2 − q(∆δ + n2 ) + qP ηNa re λL .
{z
}
|
λL
| 4πa
{z } |{z} free electrons
geometric

(2.4)

atoms

Here q is the harmonic order, µ11 is the mode factor – a geometric constant from the waveguide
coupling [83] – that is the first zero of a Bessel function of the first kind, λL is the fundamental
laser wavelength, a is the radius of the waveguide, P is the pressure inside the waveguide, η is theionization fraction, re is the classical electron radius, Na is the number density of atoms/atm, ∆δ isthe difference in the indices of refraction of the gas for the two wavelengths of light, and n2 = ñ2 IL
is the nonlinear index of refraction at λL .
The first term is a geometric dispersion term arising from the propagation of the fundamental
wavelength through the waveguide. This term results in phase shifts in the laser as a function of
propagation distance, and affect the relative phases of the emitted harmonics. The second term
describes the phase of the neutral atoms, including contributions from both dispersion and thenonlinear refractive index. The third term is the dispersion from electrons as a result of theatom’s ionization by the intense laser pulse. Note that both the neutral atom and the free electron
dispersion terms depend on the pressure inside the capillary P , while the geometric and free electron
terms provide positive dispersion and the neutral atom term is negative. This indicates that thepressure inside the waveguide is a convenient experimental knob to tweak to achieve phase matching.
Using the phase matching condition ∆k = 0 with Eqn. 2.4, the phase matching pressure is

35

Figure 2.4: Phase Matching in a Gas-filled Waveguide. (a) HHG in a gas-filled capillary
waveguide. An intense, ultrafast IR pulse is focused into a waveguide triggering the three-stepmodel and harmonic generation. When properly phase matched, bright, coherent EUV light isemitted collinearly with the driving laser, but with a different divergence. (b) A diagram of properly
phase matched HHG, showing the coherent addition of EUV photon emission. Adapted from [75].

found
P ≈

µ211 λ2L1.
4πa2 2π(1 − η)(∆δ + n2 ) − ηNa re λ2L(2.5)

From Eqn. 2.5, there are some values of the ionization fraction which results in a nonphysicalpressure for phase matching. This occurs when the denominator goes to zero. Solving for thispoint gives the critical ionization level [79]ηc =1
Na re λ2L
1 − 2π(∆δ+n
2)

,

(2.6)

which sets an upper limit on the the intensity of the driving laser light that can be used for HHG,
as a higher intensity leads to a larger η. Recalling Eqn. 2.2, it is seen that the cutoff energy of
the HHG process cannot be driven higher by simply increasing the intensity of the laser field, as

36
the ionization will prevent phase matching above ηc . Therefore, to obtain higher energy photons
through HHG, longer wavelength high intensity laser systems are needed to drive the process.

Chapter 3

Ptychography Coherent Diffractive Imaging

3.1

Introduction
Coherent diffractive imaging (CDI) is a broad assortment of imaging techniques that seek

to produce a real space image of a specimen using only measurements from the specimen’s scatterpattern. These techniques use phase retrieval, a method that transfers the task of image formationfrom physical optics – i.e. lenses – to a computational algorithm that solves for the phase of an object
given its measured diffraction intensity and additional a priori information about experimental
constraints.
Without the need for image forming optics, CDI has exploded in popularity in the X-ray
and EUV microscopy communities where quality, high power image forming optics are difficult
to fabricate. Since its first demonstration using coherent X-rays in 1999 [84, 85], CDI has foundwide applications across varied fields such as material science [86–88], semiconductor science [89,90], biological science [91, 92], and many more [35, 93]. With the added benefits of diffractionlimited resolution and phase-contrast sensitivity, CDI is an instrumental tool driving the nanoscale
microscopy revolution of the last decade. This chapter discusses the formulation of the phaseretrieval problem and its solution as through iterative optimization.

3.2

The Phase Problem
The phase of light is not directly measurable. In so many words, that is the phase problem.

Indeed, when light is collected on any detector – the eye, photographic paper, or charge-coupled38
devices (CCD) – only the intensity of the wave is measured, leaving the phase information to beobserved through interferometric techniques. More formally, given a light wave with complex scalaramplitude ψ = Aeiφ with A and φ real, then an intensity-based detector will measure
I = |ψ|2 = A2 ,(3.1)

and the phase information is lost on detection. This is unfortunate as according to Eqns. 1.36 and
1.34, the wave at the detector plane and directly behind the object – the exit surface wave (ESW) –are related through simple Fourier transform relationships. If the phase of the measured diffraction
is recovered, then together with the measured intensity, the ESW is found through a simple inverse
Fourier transform. This task is known as phase retrieval.
Mathematically, half of the information carried in the diffracted wave is lost during theintensity measurement. To account for this, additional constraints must be enforced on the system
to uniquely determine the object. Usually, this is done with known or previously measured aspectsof the system geometry or a priori information about the sample. The phase retrieval problem canthen be thought of as an optimization to find an ESW that both produces the measures intensity
and satisfies the additional constraints.

3.3

Coherence
As the name suggests, CDI requires a certain degree of spatial and temporal coherence.

Coherence is a measure of how well correlated a wave is with itself in both space and time. Moreformally, a coherent wave has a fixed phase relationship between space and time such that if the
wave is measured at spatial point x1 and temporal point t, then at some different place x2 anddifferent time t + τ the wave can be perfectly predicted. For a wave ψ(x, t), the mutual coherencefunction is defined to be the correlation between between the wave at points x1 and x2 and time
separation τ .
Γ(x1 , x2 , τ ) = hψ(x1 , t)ψ ∗ (x2 , t + τ )iT ,

(3.2)

39
with h·iT represents the ensemble average over time period T . Typically this function is normalized
to give the complex coherence factor
Γ(x1 , x2 , τ )Γ(x1 , x2 , τ )γ(x1 , x2 , τ ) = p=pΓ(x1 , x1 , 0)Γ(x2 , x2 , 0)I(x1 )I(x2 )(3.3)

where I(x) is the intensity of the field at point x. The modulus of 3.3 gives the ratio of coherent
intensity to total intensity and is a measure of the coherence properties of the wave field, with|γ| = 1 being fully coherent and |γ| = 0 being fully incoherent [37]. In practice, it is often useful
to find the distinct length and time scales over which the degree of coherence is high for a given
source, these will set physical limits on the CDI experiment geometry and attainable resolution.

3.3.1

Transverse Coherence Length

Transverse coherence, also referred to as spatial coherence, is the distance |x1 − x2 | for which
γ falls to 0. In a CDI experiment, the probing illumination area must be fully spatially coherent
in order for the object and diffraction pattern to be related through a Fourier transform.
A good measure for spatial coherence is interference fringe visibility in the diffraction froman object illuminated by a light source. The simplest form being a Young’s double slit experiment,
which is used to construct a relation between experimental geometry and transverse coherence.
Consider the Young’s double slit experiment shown in Fig. 3.1. In this setup, a source of widtha is placed a distance R before a screen with two slits separated by distance d. The diffraction
from the two slits is viewed on a screen placed a distance L away. The transverse coherence length
can be quantified as the slit separation d for which the visibility of the fringes in the diffraction
goes to zero. This means that the minima of the diffraction pattern produced by illuminating the
double slit with a point source on the optical axis coincides with the maxima of the diffraction
produced by the double slits illuminated by a point on the edge of the extended source [40]. In
the small angle approximation, the angle of the mth maximum for on axis illumination is given asθmax = mλ/d and the mth minimum occurs at θmin = (m + 1/2)λ/d for illumination on the opticalaxis[94]. When the point source is shifted a distance a/2 off the optical axis, the location of the

40

Figure 3.1: Spatial Coherence Measurement. A simple geometry to measure transverse coherence length of an extended light source. The interference fringes produced from a illumination on
the optical axis (solid black lines) and on the edge of the source of size a (red dashed lines). The slit
distance d for which the fringes are exactly out of phase corresponds to the transverse coherence
length of the system L. Reproduced from [40].

extrema are shifted by an angle ∆θ = a/(2R). Equating the maxima of the on-axis illumination
with the minima of the off-axis source gives the relation λ/(2d) = a/(2R). Solving for d then gives
the transverse coherence length L = d = λR/a. In practice, an extended source may be asymmetricso a transverse coherence length in the horizontal and vertical direction with source dimension ahby av are given by
Lh = λR/ah ,Lv = λR/av .

(3.4)

In terms of a CDI system, the slit separation used in the example above can be thought of as
the width of the object – or illumination in the case of ptychography– D. The interference fringes
considered are thought to arise from two point source emitter on either side of the object. With
this context, it can be seen that the transverse coherence length of the source must be greater thanD to ensure the visibility of the interference effects that the imaging technique relies on [95].

41
3.3.2

Longitudinal Coherence Length

The longitudinal coherence length, or temporal coherence, is a measure of the coherence ofa wave along its direction of propagation. More formally, it is the self-correlation of the wave at a
fixed point, but at two different times. The coherence time Tc is the time over which the phase of
the wave can be predicted given its measurement at some earlier point. This time can be converted
to a longitudinal coherence length through a factor of the speed of light, Ll = cTc . To derive a
relation for Ll , consider two waves, one with wavelength λ and the other with wavelength λ + ∆λ.
These waves depart from the same point in space and time and propagate. Ll is defined as the
distance the waves propagate until they are exactly out of phase, see Fig. 3.2 [40].

Figure 3.2: Longitudinal Coherence Measurement. Two waves with wavelengths λ and λ+∆λoriginate from the same point in space and time. After propagating the longitudinal coherence
length Ll , the waves are exactly π out of phase. Reproduced from [40].

After this distance, the first wave has made N oscillations, and the second wave must havemade N − 1/2 oscillations. If the waves are in antiphase, then N λ = (N − 1/2)(λ + ∆λ). Solving
for N and recalling that Ll = N λ, the coherence length is found to be
Ll =

1 λ2
.
2 ∆λ

Note that the exact pre-factor is dependent on the spectral density of the source [96].

(3.5)

42
The longitudinal coherence has an interesting effect on CDI system: it limits the attainable
resolution. For waves scattered at an angle θ from either end of an object with width D, the path
length difference between the two waves is given by ∆dpath = D sin θ. In order to have coherent
interference Ll > ∆dpath . Recalling the relations from section 1.2.8 gives
λ2q> D sin θ ≈ D .
∆λ
k

(3.6)

If the largest detectable momentum transfer is considered, qmax , a relationship between source
fractional bandwidth and real space resolution δx can be found using δx = 2π/qmax
δx
∆λ
<
.
λ
D

(3.7)

Unless the above relation is satisfied, there will not be coherent interference at the edge of themeasured diffraction pattern.
Similarly, when using a pulsed light source, the temporal coherence is limited to the pulse
duration τp . The path length distance has an associated time delay ∆tpath = ∆dpath /c. If ∆tpath >τp , then even if the pulse is coherent throughout its duration, the pulse does not exist long enoughto get the required interference and the diffraction pattern will be blurred. Following a similar
process to the one outlined above, the maximum resolution (i.e. highest angle scatter) that can becollected for a given object size D and pulse duration is δx = λD/τp c.

3.4

Sampling
The use of pixelated detectors and digital imaging processing for CDI data acquisition andimage reconstruction means that continuous functions and mathematical operations must be adequately represented through discrete arrays. As such, care must be taken when selecting thesampling interval such that the original continuous function is accurately represented by discrete
points on a finite grid. The condition for this is the Shannon-Nyquist sampling theorem, whichstates that in order to reproduced a continuous waveform from discrete samples, it must be sampled
at an interval twice that of the highest frequency present in the waveform [97].

43
3.4.1

Some implications of a theorem due to Shannon

When taking the discrete Fourier decomposition of a discretized function with N samples
and sample spacing δx, the sampling rate in reciprocal frequency space δf is given through therelationδx δf =

1
.
N

(3.8)

With the transform centered on the gird, the largest frequency fmax contained in the decompositionis thenfmax =

N
1
δf =
.
2
2δx

(3.9)

This means that if a wave form contains frequencies such that f > fmax , then that informationis aliased down to lower frequencies where it adds additional power to lower frequency Fouriercomponents. This is a consequence of the periodic boundary conditions assumed by the discrete
Fourier transform. In order to properly sample a function, it must be bandwidth-limited such thatany content with f > fmax is zero.
The Nyquist frequency fN y is defined as the cut-off frequency for a function’s bandwidthwith fN y 6 fmax . Using Eqn. 3.9, a minimum sampling rate is found to be twice the Nyquist
frequency δx 6 1/(2fN y ) [98].
In Sayre’s seminal works on CDI, the idea is proposed that the phase problem could be
solved through adequate sampling of an object’s diffraction pattern. In his brief article “Someimplications of a theorem due to Shannon”, Sayre considers a function o(x) which is only nonzero
over a finite size a in real space [99]. To determine the required sampling rate to uniquely determine
the Fourier transform of o(x), Sayre considered a the real space analogue of twice the Nyquist
frequency. Accordingly, the Fourier transform õ(f ) must be sampled at a rate δf = 1/a. However,in a diffraction experiment, |õ(f )|2 is measured which corresponds to the Fourier transform offunction’s autocorrelation o(x) ? o(x). The width of a function’s autocorrelation is twice that of
the function itself, so the measured diffraction must be sampled at δf > 1/2a to uniquely record
the intensity. Initially, this idea was proposed as a method for crystallography, but did not receive44much attention as the diffraction patterns are composed of distinct peaks with spacings δf = 1/a.
However, the idea was later expanded to noncrystalline samples, with the characteristic dimensiona reinterpreted as the localized sample or illumination width [100].

3.4.2

Sampling and Experimental Geometry

Consider now a diffraction pattern recorded with a pixelated detector of size Nx × Ny withpixel area px × py positioned a distance z from the object scattering the light. Using Eqn. 1.52, itis seen that the frequency sampling interval is determined by the experimental geometry with
px
,
λz
py
δfy =
.
λz

δfx =

(3.10)

Through the discrete Fourier transform, a relation between the object sampling interval andthe detector plane frequency content is established. Using Eqn. 3.8 the object sampling rates are
δx =

λz
,
Ny py

λz
.
δy =
Ny py

(3.11)

It is now assumed that the object is represented by a function o(x, y) with finite extent Dx × Dy .
Then to sample the diffraction pattern intensity without aliasing
1
,
2Dx
1
δfy 6
.
2Dx

δfx 6

(3.12)

Equating Eqn. 3.10 and 3.12 gives the dimensionless quantity
σx,y ≡λz> 2.
px,y Dx,y(3.13)σ is referred to as the oversampling and the condition that it be greater than two is actually justthe Nyquist sampling condition for the autocorrelation of the object. The condition that σ > 2
is generally required for CDI experiments [84, 101–103]. Because the diffraction pattern is themeasurement of interest, not the intensity of the diffraction pattern, it is common to refer to a45diffraction pattern as “oversampled” in an experiment. Recalling the diffraction-limited resolutionof an imaging system is given by λ/2NA with NA inversely proportional to z, it is clear that there
is a trade off between resolving power and oversampling when designing a CDI system.
An important implication of the sampling requirements in CDI is a restriction on the experimental geometry. For example, taking σ = 2 and rearranging Eqn. 3.13, a maximum extent on
the localized object or probe illumination is found in terms of the other experimental parameters
Dx,y 6

3.5

λz
.
2px,y

(3.14)

Iterative Phase Retrieval in CDI
Over the past 70 years, the initial ideas of phase retrieval have been adapted and expanded

into a formulation as a non-convex optimization problem [104] . The algorithms in modern phaseretrieval problems typically take the form of iterative projections (or reflections) onto constraint
sets in reciprocal spaces where some knowledge about the object is known and enforced. This
process continues iteratively until come convergence criteria is met – i.e. some residual error metricis below a preset threshold or the algorithm has run for a pre-defined number of iterations. A
brief introduction and overview of these algorithms is given here. For a more in-depth analysis thereader is referred to [105, 106].

3.5.1

Constraint Sets and Generalized Projections

When formulating a phase retrieval algorithm, it is useful to understand the spaces and sets
typically involved. Consider an image with N complex-valued pixels. This image can be represented
as a complex vector ν in an N -dimensional Euclidean space VN . The Fourier transform of this vectorν̃ is also contained with VN as the dimensionality of ν is preserved under the Fourier transform.
The Fourier transform acts as a rotation within VN with the magnitude of ν preserved due toParseval’s Theorem [98].
Within this space, constraint sets in both real and reciprocal space are expressed as theset of all points contained in VN where the image satisfies the corresponding constraint. In CDI,

46
one of these constraint sets CA is known as the amplitude constraint or Fourier constraint andconsists of the set of all images whose Fourier magnitude is equivalent to the one measured during
the experiment. The real space constraint can vary depending on the algorithm in use, examples
include the non-negativety of real space image pixel value [107], constraints on the image histogram[104], or most commonly the support or isolation constraint [100]. The support constraint CS issimply a statement that the object is band-limited and thus has been Nyquist sampled in reciprocal
space. Formally, the support constraint is a statement what all pixel values are zero outside somedefined support region D.
Within this formalism, the phase retrieval problem is cast as finding the image vector ν̂ thatis an element of the union between CA and CS . The solution is found by starting with a random
guess and iteratively projecting between the constraint sets, see Fig. 3.3. For the problem to be
uniquely solvable, the constraint sets must intersect at only one point. In practice this is rarely thecase due to noisy measurements and other experimental uncertainties.
The generalized projections are mappings from a general image vector ν on to a vector
contained in one of the constraint sets ς such that |ν − ς| is minimized [108]. Practically, thissimply means that the vector ν is mapped to the closest point in the constrain set C when aprojection is applied. The projection operation for the support constraint is applied to the real
space representation of ν and is given byΠS {νx } =νx , x ∈ D(3.15)0, else.
Practically, this projection amounts to setting the values of ν to zero outside some defined support
D.
The projection onto the amplitude constraint set CA is best performed in Fourier space.
Given the measured Fourier intensity values I, the amplitude projection is given by ΠA {ν} =47

Figure 3.3: Generalized Projection Algorithm. A representation of a generalized projection
phase retrieval algorithm. An image vector ν0 is initialized an some point in VN and iterative
projections on to the Fourier amplitude constraint set CA and the finite support constraint set CS
are performed until convergence. It can be seen that the convergence behavior depends on where
the algorithm is initialized.

F −1 {PA {F {ν}}} with
PA {ν̃} =



√ ν̃


 I |ν̃|, ν 6= 0√ I , else.

(3.16)

This operation replaces the Fourier amplitude of the vector ν with that of the measured values√
I

while preserving the updated phase values of ν̃. Issues with the convergence of these algorithms
largely stem from this projection operation, which can be multivalued when ν̃ = 0. This is seen
when the projection is view in the complex plane in Fig. 3.4. The measured amplitude defines acircle in this plane and projections for nonzero ν are defined by picking the closest point on thiscircle to ν. However if ν = 0, the all points on the circle are equally distant and the projection ismultivalued. The nonuniqueness of this projection is a result of the nonconvexity of the amplitude
constraint and makes analyzing the convergence behavior of these algorithms difficult to predict[105, 106]. This is illustrated in Fig. 3.3 where the convergence of the algorithm is seen to depend

48
on the initialization.

√
Figure 3.4: The Fourier Amplitude Projection. The measured Fourier amplitudes I define
a circle in the complex plane. The projection operation on the vector ν projects it to the closest
point on the circle, while preserving its phase defined by angle α. If ν = 0 then the projection
operation is not well defined. Adapted from [106].

3.5.2

Phase Retrieval Algorithms

The first algorithm for diffractive imaging was created by Gerchberg and Saxon (GS) [109]and actually predates the early formative conceptual work on CDI and sampling [100, 110]. Still,
the GS algorithm provides the general structure that nearly all phase retrieval algorithms still
follow, which involves the iterative projection between constraints in reciprocal spaces. The mostcommon algorithms for phase retrieval only differ in their choice of the real space update function
using the generalized projection. It is useful to understand the workings of these algorithms inbuilding toward more complicated phase retrieval schemes used in this thesis.
The general flow of a single diffraction pattern phase retrieval algorithm is shown in Fig. 3.5
and requires the following steps:(1) Initialize an guess for the two dimensional image reconstruction g0 (x).
(2) Propagate gk (x) the detector plane via a Fourier transform to get g̃k (f ).

49

Figure 3.5: The Error Reduction Algorithm. The phase of an exit-surface wave is recovered by
iteratively enforcing constraints in the object plane (real space) and detector plane (Fourier space).
After a random guess is initialized, the wave is numerically propagated to the detector plane where
the amplitude of the wave’s Fourier transform is forced to match that of the measured data. The
updated wave is then propagated back to the object plane where the finite support constraint isenforced, meaning the wave is isolated and of finite extent. This process is repeated until thealgorithm converges to a solution. Note that if the support constraint is replaced with some moregeneral real space update condition, then this diagram generalizes to many different phase retrieval
algorithms.

0(3) Apply the amplitude constraint to find g̃k (f ) = PA {g̃k (f )}.
0(4) Propagate back to the object plane to obtain gk (x).
(5) Update the object guess using the algorithm’s object update rule to calculate gk+1 (x).
(6) Repeat (2)-(5) until a suitable solution is found.
The progress of the algorithm is usually monitored using some type of error metric. The

50
exact metric can vary, but typical choices include a Fourier errorPEF =|g̃k (f )| −
P√
I

√

I
,

(3.17)

where the relative error between the Fourier amplitude of the k th object guess is compared against
the measured value. As the algorithm progresses this error metric should, overall, decrease in value.
Another useful metric to track is the convergence of the algorithm
EC =X|gk (x)| − |gk+1 (x)| .

(3.18)

This metric tracks the relative change in the reconstruction between iteration and gives insight into
whether the algorithm has converged or stagnated.
The GS algorithm uses a direct intensity measurement in both the object (real space) anddetector (Fourier space) planes. Meaning an amplitude projection is performed in both real andFourier space. The real space intensity measurement is often difficult to obtain in X-ray and EUV
microscopy, but is quite simple in electron microscopy and relatively easy in visible imaging. As
such, the GS algorithm has found widespread use for phase-contrast imaging and is still widelyused today. It is most interesting here as a precursor for more common phase retrieval algorithms
used in CDI.
The error-reduction (ER) algorithm was designed by Fienup, who generalized the GS algorithm by including real space constraints like non-negativity and finite support [107, 111]. Whilethese constraints are implicitly enforced with the real space intensity measurement in GS, removing
it was the pivotal step that allowed an object to be reconstructed from only a measurement of its
0

scatter pattern. The object update in ER takes the form gk+1 (x) = ΠS {gk (x)}. This algorithm iseasy to implement, but is known to stagnate in local minima during optimization.
One of the most widely used CDI algorithms is Fienup’s hybrid input-output (HIO). In his
tests, it was determined this algorithm had the best convergence behavior [107]. The object update

51
rule in HIO is given as
gk+1 (x) =




0

g (x) , x ∈ D
k(3.19)0
gk (x) − βgk (x) , else,
where β is a relation parameter with value [0.5, 1] with β > 0.9 a common choice [107, 112].
Since the development of these algorithms, improvements have been made to help theirconvergence and overcome limitations. One such method is the iterative refinement of the support
region – colloquially know as the shrinkwrap algorithm – that was shown to improve convergence
speed [112]. Relaxing the requirement for an isolated object to satisfy the finite support constraint
were implemented by tightly focusing the illuminating light [113, 114] or masking the illumination
close to the sample [115, 116], at the expense of a required known curvature in the illumination and areduction in flux. Further research into the nature of the algorithms brought improved convergence
performance and reconstruction quality with the introduction of more complex operations such as
generalized reflections and proximal mappings [117–119].

3.6

Ptychography CDI
The ER and HIO algorithms all require a single diffraction pattern measurement, while

GS requires two. While these offer some advantage in terms of required data and relatively low
complexity in implementation, it comes at the cost of robustness. These algorithms are prone
to stagnation when searching for the optimal solution and there are known ambiguities that are
inherent to the algorithms themselves [110, 120]. Many of these issues are overcome if multiple
diffraction pattern measurements are used to reconstruct a single image. This scanning modality
for CDI is known as ptychography and it has quickly grown to be one of the most wide spread
imaging methods in X-ray and EUV microscopy. For an in depth review of the theory and historyof the technique, interested readers are referred to [121, 122].

52
3.6.1

Background

Like single diffraction pattern CDI, ptychography also has its roots in crystallography. The
idea was first proposed by Hoppe in 1969, where is was thought that if the Bragg peaks fromcrystalline diffraction are made to interfere, then information about their relative phases can belearned [123]. However, the Bragg peaks are highly localized in the standard implementation of
crystallography so instead of using the usual plane wave illumination, Hoppe suggested that a finite
coherent illumination be used instead. By the convolution theorem, the resulting Bragg peaks inthe far-field diffraction are convolved with the Fourier transform of the finite illumination. By
choosing an appropriate size for the illumination, the interference of neighboring peaks is observed.
By recording a second diffraction pattern with the illumination slightly shifted, a unique solution
for the phases is recovered. The name “ptychography” was later introduced from the Greek word
“ptycho” meaning “to fold” [124–126]. In German, the word for “convolution”, “faltung”, is thesame as “to fold”. Ptychography is thus translated as “convolutional imaging”.
Initially, the idea received little attention and only a few proof-of-principle studies were
performed [127]. The next important milestone was the creation of the Wigner distribution deconvolution method which was the first ptychographic method to reconstruct non-crystalline samples[128]. The technique was successfully implemented with an optical [129] and electron microscope[130], and later using a scanning transmission X-ray microscope [131, 132]. Despite this success,the reconstructed resolution was limited by the relative translation between adjacent positions. So
scanning a large field of view with high resolution quickly becomes intractable.
Ptychography CDI began its rapid rise is popularity with the advent of the first ptychographic
phase retrieval algorithms [133, 134]. These algorithms are structurally very similar to HIO withseveral recorded diffraction patterns, however a new and very powerful real space constraint was
introduced: the overlap constraint. This novel constraint essentially states that regions of an object
illuminated by several different, but overlapping, illumination scan positions must be consistent.
Meaning that the object update at different illumination positions are no longer independent. Each53
object update benefits from the refinement provided by adjacent position updates, leading to faster
algorithm convergence and an increased robustness to noise. Performance was further increased byreformulating the object update function via a gradient-descent optimization routine [135]. Thisalgorithm now known as the “Ptychographic Iterative Engine” (PIE), and its extended form (ePIE)
is one of the most popular implementations of ptychography CDI.
Initially, these algorithms required that the illumination function be known a priori, but thislimitation was subsequently lifted when it was realized that the data redundancy of a ptychographic
dataset allowed for the blind deconvolution of the sample transmission function from the illumination [136–138]. This was a key extension that allowed ptychography to become a real alternative
to other X-ray microscopy techniques as precise illumination characterization is difficult at thesewavelengths.
Since its inception, several improvements have been introduced to the basic ptychographyalgorithm. Similar to single diffraction pattern CDI, the introduction of advanced optimization routines have seen increased performance and convergence [139–143]. Fermat spiral scanning patterns
are now standard to combat the known “raster grid pathology” [137, 144]. Methods for refining thescan grid positions during image reconstruction alleviate some of the strict requirements of systemmechanical stability [145, 146]. Continuous scanning modes allow for multiple scan position to be
multiplexed into a single diffraction measurement, reducing data volume [147–151]. The illumination of a ptychography microscope can be carefully engineered to allow for the entire dataset
to be captured in a single exposure, though at the cost of a reduced spatial resolution [152, 153].
The coherence requirements of ptychography have also been relaxed to allow for hyperspectral andmultimode imaging, this will be discussed in detail in chapter 4.
By exploiting the high amount of data redundancy in the ptychography dataset, low frequency
information lost when recording data with a beam block can be recovered [154]. Similarly, theresolution of a ptychographic reconstruction can be enhanced by computationally increasing the NAof the recorded diffraction [155]. Even the strict oversampling requirement, a necessary condition
in all other forms of CDI, can be relaxed in ptychography [156, 157].

54
Ptychography is used to study three dimensional systems by combining the technique with
tomography [91, 158–160] and by incorporating multiple scattering effects into the reconstruction
algorithm [48, 90, 161–163]. Recently these two method have been combined for improved reconstruction quality [164, 165]. Finally, ptychographic CDI has been implemented in a reflection modefor high-resolution surface microscopy [41–43, 69, 166].

3.6.2

The Ptychography Dataset

The dataset for ptychography consists of N diffraction patterns In collected by scanning afinite illumination function, P (x) – the “probe” function – across an object transmission function
O (x), previously discussed in section 1.3. Diffraction is recorded at scan positions xn = [xn , yn ].
The diffraction patterns are given as
In = |F {O (x) P (x − xn )}|2 ,(3.20)

where Fraunhofer propagation has been used, though Fresnel or another suitable propagation
method can be substituted.
It is important that the scan positions xn are chosen such that adjacent scan position havea high degree of overlap in the illuminated area of the sample, typically between 60% - 70%. Thisoverlap introduces redundancy into the dataset and makes the reconstruction process more robust[136]. In the literature, the exit surface wave ψn (x) = O (x) P (x − xn ) is often introduced andrepresents the wave field directly behind the object. The diffraction patterns In and scan grid xn
are used as inputs to the reconstruction algorithm.

3.6.3

The Ptychography Phase Retrieval Algorithm

Similar to the single diffraction pattern phase retrieval algorithms detailed in section 3.5.2,
ptychography phase retrieval algorithms alternate between real and reciprocal space where constraints are enforced on an iteratively updated object and probe guess. The reciprocal spaceconstraint is identical to Eqn. 3.16 and it is enforced with each recorded diffraction pattern in-

55

Figure 3.6: Schematic of a Ptychography Algorithm. The ptychography algorithm is summarized with a data set containing four diffraction patterns collected with a circular scan overfour overlapping positions. The algorithm iterates of the different scan positions in the outer loop
(purple arrows). At the nth position, the exit surface wave ψn (x) = O (x) P (x − xn ) is calculated.
An updated guess of the exit wave is calculated after applying the Fourier amplitude constraint
by propagating ψn (x) to the detector (red and blue arrows). From this, an updated guess at theobject and probe is made. The algorithm then moves to position n + 1 and repeats the process.
However, the overlap between adjacent position means that the next position’s update benefits
from the previous positions. Figure is adapted from [167].

dividually. The object space update is less obvious. There are numerous algorithms available for
ptychography, which mainly differ in their object update function and whether the different scan
positions are processed serially or in parallel [136, 137, 139].
Work in this thesis is mainly based on the ePIE algorithm, which is a serialized gradientdescent based optimization method. Serialized versions of ptychography are observed to have fasterconvergence behavior, at the cost of being more susceptible to stagnation in local minima. This

56
stagnation is largely due to the serial nature of the algorithm; a parallel implementation is trying
to globally optimize the entire image simultaneously, while serial versions optimize sub regions of
the image sequentially. The serial ePIE algorithm is summarized here following [136], but many of
its properties are generalizable to other ptychography reconstruction algorithms.
The set of measured intensities In , scan positions xn , and object and probe function initializations O(0) (x) and P (0) (x) are required as inputs to the reconstruction algorithm. The updated
object O(j+1) (x) and probe P (j+1) (x) are then calculated with the following steps:(j)(1) For position n, calculate the ESW ψn (x) = O(j) (x)P (j) (x − xn ).
n
o
(j)
(j)
(2) Propagate the ESW to the detector to find ψ̃n (f ) = F ψn (x) .
(j)(3) Apply the amplitude constraint by replacing the Fourier modulus |ψ̃n (f ) | with the mea√
sured intensity amplitude In .
()
p ψ̃n(j) (f )0 (j)−1ψn (x) = F
In (j).
(3.21)|ψ̃n (f ) |
n 0
o
0 (j)
(j)
(4) Propagate the updated ESW back to the object plane to calculate ψn (x) = F −1 ψ̃n (f ) .
(5) Use the updated ESW to find O(j+1) (x) and P (j+1) (x) with update conditionsO(j)∗ (x + xn )  0 (j)
(j)
ψ
(x)
−
ψ
(x)
,
2
O(j) (x + xn )P (j)∗ (x − xn )  0 (j)(j)
O(j+1) (x) = O(j) (x) + α
ψ
(x)
−
ψ
(x)
.
2
P (j) (x − xn )P (j+1) (x) = P (j) (x) + α(3.22)(6) Move to scan position xn+1 and repeat steps (1)-(5). The shifted probes at P (j) (x − xn )and P (j) (x − xn+1 ) should overlap and constrain the regions of the object present in both
illumination positions.
(7) Continue iterating through all N positions to complete one full update iteration for theentire scanned object area.
The parameters α and β in Eqn. 3.22 are relaxation parameters similar to β in the HIO
algorithm. Typically their values are set in the range [0.9, 1] and control the step size in the

57
gradient optimization. To avoid the possibility of dividing by zero, the denominator in Eqn. 3.22
is converted to the maximum value of the object or probe function. An error metric is usually
defined to monitor the algorithm progress, with a common choice being the RMS error between themeasured diffraction amplitude and the algorithm’s current guess for the diffraction amplitude.

Chapter 4

Spatial, Spectral, and Polarization Multiplexing in Ptychography

4.1

Introduction
Since its initial rediscovery, the scope of ptychography has been extended to simultaneously

reconstruct several different incoherent object and/or probe modes with no increase in the dataacquisition volume [168, 169]. An incoherent superposition of diffraction from multiple modes isrecorded on a detector during a ptychography scan and computationally separated during reconstruction. This technique, known as multiple mode ptychography or ptychographic multiplexing, isreviewed below. Later, it is shown that by introducing spatial separation between mutually incoherent modes, the field of view of ptychography reconstructions can be expanded without sacrificing
spatial resolution. Results discussed here are previously published in [170].

4.2

Multiple Mode Ptychography
Multiple mode ptychography was first derived to account for the complication of decoher-

ence during the imaging process [169]. The high coherence requirements for CDI imposes strict
constraints on the experimental setup, such as negligible vibrations of mechanical components, astationary sample or sufficiently short ultrafast pulses from the source, and a highly monochromatic
source. These competing requirements often lead to a reduction in the usable coherent flux in agiven system. Thus, relaxing the coherence needs of a CDI algorithm directly leads to an easingof experimental design. Later, the idea of multiple mode ptychography was expanded to allow forspectrally sensitive objects and polychromatic illuminating probes [168].

59
Regardless of the particular source of decoherence the algorithm is attempting to compensate,
the basic formulation of multiple mode ptychography remains the same. Namely, in a system withsome type of decoherence – source stability, mechanical vibrations, or source polychromaticity –the measured diffraction pattern is treated as an incoherent sum of several mutually incoherent
modes withIj (f ) = |F {ψ1 (x − xj )} + F {ψ2 (x − xj )} + . . . + F {ψN (x − xj )}|2 ,(4.1)

where Ij (f ) is the j th measured intensity scatter pattern, ψn is the exit-surface wave corresponding
to the nth incoherent mode, N is the total number of incoherent modes modeled in the system. The
multiple mode ptychography algorithm leverages differences in the diffraction produced from eachmode – from either a mode-specific object response or probe profile, or differing scattering angles
due to various wavelengths in the probe – to tease apart the contributions of each mode to thesum total recorded on the detector. An error metric E is defined that takes into account multiple
incoherent modes and is minimized using a gradient descent optimization with
"
!
#2
X
X
E=
ψ̃j,n (f ) − |Ij (f )| ,
j

(4.2)

n

where ψ̃j,n (f ) = F {ψn (x − xj )}. This is similar to the error metric used for ePIE [136], butwith an additional summation over all the incoherent modes. The object and probe update rules
for multiple mode ptychography are similarly modified to take into account the presences of themultiple incoherent modes. The real space update condition for the k + 1 iteration of the probe
and object at the j th scan position and nth incoherent mode are given by
Onk ∗ (x − xj )  k 0
k
ψ
(x)
−
ψ
(x)
n,j
n,j
2
|Onk (x − xj )|P k ∗ (x + xj )  k 0
k+1
kkOn,j+1(x) = On,j(x) + β n
ψ
(x)
−
ψ
(x)
,
n,j
n,j
2
|Pnk (x + xj )|k+1
kPn,j+1(x) = Pn,j(x) + α(4.3)

0 (x) is the j th Fourier amplitude updated exit-surface wave of the nth mode. The Fourier
where ψn,j

amplitude update rule is also modified to take into account the incoherent sum of mode intensities

60
measured during an experiment. The modulus update rule is now given by




p




I
(f
)
j
k
k0
−1
r
ψn,j (x) = Fψ̃n,j (f ) .


2


 P ψ̃ k (f )

n

(4.4)

n,j

Eqns. 4.3 and 4.4 are applied sequentially as the scan positions and multiple modes are looped
through, forming one complete iteration of the multiple mode ptychography algorithm. Like single
mode ptychography, this algorithm can be formulated to run with either a parallel or serial update
condition and similarly takes on the various advantages and setbacks associated with each. The
multiple mode ptychography imaging modality has been successfully used to perform hyperspectral
imaging with EUV light [171], to relax the monochromaticity requirement with both optical and
EUV light [172, 173], measure the unique phase structure of vortex beams [174], characterize
the coherence of X-ray beams generated from synchrotron sources [175], and investigate dynamic
systems [176], among other applications [177].

4.3

Spatial Separation of Mutually Incoherent Probes
In the original formulations of multiple mode ptychography, the mutually incoherent modes

were assumed to spatially overlap with one another on the sample. However, this spatial overlap
is not required or imposed by the ptychography algorithm. With this in mind, multiple mode
ptychography can be employed to image separate locations of a sample simultaneously, as long as
the multiple illuminating probes are mutually incoherent with each other, i.e. non-interfering at
the detector plane.
Imaging in this modality allows for a larger field of view in the reconstructed image with noloss in spatial resolution. The mutual incoherence of the probes can be obtained in a number ofways. The following sections detail proof-of-principle experiments where the incoherence is achieved
using probes of different wavelengths (sec. 4.3.1), with orthogonal polarization states (sec. 4.3.2),
and an ongoing experiment where the probes are delayed temporally (sec. 4.3.3).

61
4.3.1

Spectral Multiplexing with Spatially Separate Beams

Figure 4.1: Spatially Separated, Spectrally Multiplexed Ptychography Experiment
Setup. A schematic for the visible ptychography microscope that uses two illuminating probesof different wavelengths that are spatially separate. The different colored probes are combinedwith a beamsplitter and then spatially offset using a pair of diffraction gratings. The additional
diffraction orders are blocked by a pinhole. The two unblocked beams propagate parallel to eachother and are imaged onto a 1951 USAF test pattern sample. The diffracted light is focused onto a
detector where the overlapping, non-interfering diffraction patterns are collected. Schematic is not
to scale. Reproduced from [170].

In this initial proof-of-principle, a transmission-mode, visible ptychography microscope with
two spatially and spectrally separate beams was constructed, as shown in Fig. 4.1. Two monochromatic, fiber-coupled laser diodes [Blue Sky Research, FTEC2 440-20, λ = 420 nm and FTEC2
658-60, λ = 658 nm] are combined using a beam-splitting cube [Thorlabs, BS013] causing bothbeams to propagate collinearly. The beams then pass through a pair of diffraction gratings [Rainbow Symphony, 01604, 500 lines/mm] which gives the beams a horizontal offset while keeping them
parallel with each other. The additional diffraction orders are blocked by a 300 µm pinhole after
the second diffraction grating. The beams are imaged onto the sample through an imaging systemwith a demagnification of M = 0.1. The sample used in this experiment is a negative 1951 USAF
resolution test target [Thorlabs, R3L3S1N]. The diffracted light subsequently passed through a62Fourier transform lens with f = 2 cm, which focused the light on a CMOS detector [Mightex,
SCN-B013-U].
Diffraction patterns were collected in 10 x 10 semi-random rectilinear scan grid, with approximately 60% overlap between adjacent positions. Multiple exposures were captured at each scan
position with exposure times ranging from 0.05 ms - 750 ms and frames were combined to extend
the dynamic range of the detector. The composite diffraction patterns were shifted to the center ofthe numeric grid using a center of mass of the two DC peaks of the constituent diffraction patterns,
which were separated due to chromatic aberration in the transform lens. Using the multiple mode
ptychography algorithm described in [168], and two objects corresponding to different areas of thesample were reconstructed for each probe wavelength, shown in Fig. 4.2.

Figure 4.2: Object Reconstructions from Spatially Separated, Spectrally Multiplexed
Ptychography. Reconstructed objects for the red (a) and blue (a) probe beams in the spectrally
multiplexed ptychography experiment. Scale bar is 20 µm and the scale is equivalent in (a) and(b). (c) Relative locations of the imaged areas on a USAF Resolution Test Target. The twoareas are separated by 175 µm, which is the same as the measured separation between the probes.
Reproduced from [170].

63
To verify the accuracy of the image reconstructions, the two illuminating beams were directly
imaged by translating the detector to the sample plane. The reconstructed objects in Fig. 4.2
correspond to areas on the sample separated by 175 ± 5 µm. This agrees with the directly-imaged
probes on the detector, having a measured separation of 179 ± 5 µm. A further verification was
performed by computationally propagating the reconstructed probes for each wavelength to theimage plane of the imaging lens in the experiment. The resulting beams are show in Fig. 4.3.
The blue probe had to be propagated further in order to reach the imaging plane. The difference
between the two image planes of the different colored probes was numerically determined to be
550 ± 100 µm. This agrees with the calculated value of 420 ± 4 µm for chromatic aberration of theimaging lens for the two wavelengths used in the experiment.

Figure 4.3: Chromatic Aberration of Spectrally Separate Probes. The amplitudes of theblue (a) and red (b) beam recovered from the multiple mode ptychography reconstruction. The
propagation of the blue (c) and red(d) probes along the optical axis of the experiment. The white
dotted lines indicate the location of the reconstructed probes, while the dashed white lines indicate
the actual focal plane of the different colored probes. A transverse profile of the probe beams inthe blue (e) and red (f) image plane, obtained from computation propagation of the reconstructedprobes. The difference between the image planes of the different colored probes is consistent withchromatic aberration expected from the imaging lens in the system. The scale bar in (a) is 20 µm
and is shared with (b), (e), and (f). The scan bar in (c) is 1 mm and is shared with (d). Reproduced
from [170].

64
4.3.2

Polarization Multiplexing with Spatially Separate Beams

In the previous section, separate wavelengths were used to so that the there was no interference between the beams on the detector and a multiple mode ptychography algorithm could be
used. However, this is not the only way to remove this interference; any mutually incoherent modes
will suffice [169]. In this section, ptychographic multiplexing is demonstrated using orthogonallypolarized probe modes. In this setup, depicted in Fig. 4.4, a single red (λ = 658 nm) was spatiallysplit into two orthogonal polarization states using a beta barium borate (BBO) crystal [Eksma
Optics, 9830; 2.7 mm thickness, θ = 23.4, φ = 90] before illuminating a two spatially separated
portions of a negative USAF resolution test target [Thorlabs, R3L3S1N]. The scattered light wascollected by a lens (f = 2 cm) placed 2 cm after the sample, which focused the light onto a CMOS
detector.

Figure 4.4: Spatially Separated, Polarization Multiplexed Ptychography Experiment
Setup. A schematic depiction of the visible, transmission-mode ptychography experiment using
two spatially separated probes with orthogonal polarization states. The beam from a single red(λ = 658 nm) laser diode is split into two orthogonally polarized beams using a birefringent crystal.
The beams are then incident on a USAF resolution test target and the scattered light is collectedby a Fourier transform lens that forms a diffraction pattern on a pixelated detector. Reproduced
from [170].

By directly imaging the two beams at the sample plane, the separation between the two beams

65
was measured to be 216 ± 20 µm. The directly imaged beams are shown in Fig. 4.5. Without the
spatial separation between the beams, the sample would produce identical diffraction patterns forboth polarizations. In the case where a sample does not have a polarization-dependent opticalresponse, the spatial separation between the beams allows the orthogonally polarized modes to be
deconvolved without any ambiguity.

Figure 4.5: Measured and Reconstructed Spatially Separated, Orthogonally Polarized
Probes. (a) A Direct image of the orthogonally polarized, spatially separated probes used insample plane of the experiment. The separation between the beams is 216 ± 20 µm. (b-c) The
reconstruction of the êk polarized beam in both amplitude and phase. (d-e) The reconstruction of
the ê⊥ probe in amplitude and phase. The scale bar is 20 µm in (a) and the scale is equivalent in(a-e). Reproduced from [170].

A 64 position ptychography scan was taken with both beams incident on the sample. Data
were collected and pre-processed using an approach identical to the one described in sec. 4.3.1. The
successful reconstruction is shown in Fig. 4.6. The two reconstructed areas show that the parallel
polarized probe scanned a region of group 6 and the perpendicular polarized probe scanned group 7
on the USAF resolution test pattern. Locating these areas on the sample shows that the separation
between the two regions is 170 ± 20 µm, which is consistent with the separation measured in a

66
direct imaged of the beams at the sample plane.

Figure 4.6: Orthogonally Polarized Reconstructions with Spatially Separate Beams. The
left column shows the amplitude reconstructions for the parallel (a) and perpendicular (b) polarized
beams. (c) The location of the relative position of the scanned region on the USAF resolution test
target. The two areas are separated by 170 ± 20 µm, which corresponds to the measured separation
between the two probes. The scale bare in (a) is 20 µm and is shared with (b). Reproduced from[170].

In Fig. 4.5(b), group 7, subgroup 5 of the USAF resolution test target is resolved, indicating
an upper bound for the microscope resolution of 2.46 µm, which is equivalent to the Abbe resolution
limit for this microscope system. This indicates that using multiple beams for ptychography has noadverse effect on the achievable spatial resolution of an multiplexed ptychographic imaging system.

67
4.3.3

Temporal Multiplexing with Spatially Separate Beams

When performing ptychography with a pulsed source, it is possible to temporally multiplex
information from the interaction of the sample with several pulses into a single detector exposurethat is much longer than the individual pulse length or repetition rate of the system. Because
consecutive pulses in the pulse train are temporally isolated, they will never arrive at the detector
at the same time, and thus cannot interference. If the detector is allowed to expose such that it
will integrate over pulses, then the recorded diffraction pattern can be treated as an incoherent
sum of the signals produced from each pulse. Hence, the model of ptychographic multiplexing canbe applied to this system where each time delayed pulse is treated as a single mutually incoherent
mode.
Configuring an ptychography imaging system to make use of temporal multiplexing unlocks
the potential for studying spatially resolved system dynamics without increasing the data volume
or acquisition time. Alternatively, if a sample of interest is static, then temporal multiplexing
can be used to perform multiple mode ptychography with probes that are otherwise identical (i.e.
spectrally, polarization, etc.).
Modern research continues to push the development of coherent high energy systems thatuse geometries with more than one beam [178, 179]. Combining these novel sources with spatiallyseparated multiple mode ptychography, the benefits of imaging with higher energy sources - highresolution and chemical sensitivity - is possible with higher throughput. Leveraging the naturally
pulsed nature of an HHG EUV source, a spatially separated, temporally multiplexed ptychography
microscope has been constructed using two time-delayed EUV probes.
In this setup, a bright, phase-matched, fully spatially coherent HHG EUV beam centered at
λ = 56 nm was generated using a custom built HHG source. Harmonics were generated by frequency
doubling a pulse train with central wavelength of 790 nm, pulse duration of 45 fs, pulse energy of
8 mJ, and repetition rate of 1 kHz, and focusing the resulting 395 nm, 2 mJ pulses into a 150 µm
diameter hollow-core waveguide filled with Ar at a backing pressure of 84 Torr. The HHG process

68
up converts the wavelength of the driving source into a spectrum of higher energy light, centered at
the 7th harmonic (22 eV). After the HHG process, the residual light at the fundamental frequency
is absorbed using two 200 µm thick Al filters.
The EUV light is then directed in an imaging chamber where it reflects off of an off-axis
toroidal mirror with a focal length of 27 cm and it begins to focus. As the light propagates, thebeam is split using the edge of a D-mirror. The divided beams are directed on to two flat mirrorsthat redirect the beams on to two pinhole masks of diameters 30 µm and 50 µm to clean up the
spatial mode and easily identify the different beams during reconstruction. The shaped beams are
incident on the sample, which is placed approximately 1 cm after the pinhole mask. The scattered
light then propagates 4.6 cm to a pixelated detector where the scatter from both beams is recorded
simultaneously. A schematic of the imaging chamber is shown in Fig. 4.7.
The path length traversed by the two different beams differs by 0.3 mm, corresponding toa temporal delay of 10 ps between the two pulses. The pulse duration of the EUV light – limitedby the driving laser pulse duration – is only tens of femtoseconds in length, so the pulses are well
separated in time at the detector. This allows temporal multiplexing in this setup.

4.4

Discussion and Conclusion
A novel method to increase the throughput of a ptychographic imaging system was demon-

strated. By spatially separating probes in a multiplexed ptychography scan, multiple areas of asample are imaged simultaneously. This increase in throughput does not require a reduction innumerical aperture of either individual probe and therefore preserves the high spatial resolution
offered by CDI and is applicable to any set of non-interfering probes. This has been explicitly
demonstrated using two probes of different wavelengths and two orthogonally polarized probes ofthe same wavelength. A further demonstration using temporally offset coherent pulses is currently
underway.
This technique is appealing for use with coherent EUV and X-ray sources. In particular,
coherent EUV sources generated with HHG are a natural choice as the HHG process produces a

69

Figure 4.7: Temporal Multiplexing Multiple Beam Ptychography Setup. A schematic of
the EUV multiple beam ptychography experiment with two temporally offset probes. A single
EUV beam (λ = 56.4 nm) is generated via HHG and then focused by an off-axis toroidal mirror
and split by partially reflecting off the edge of a D-mirror. The split beams propagate to two flat
mirrors that redirect the beams onto a sample, with a controllable spatial separation. The scattered
light from both beams is collected by pixelated detector simultaneously. The path length difference
traversed by the different legs of the experiment is 0.3 mm, corresponding to a relative delay of 10
ps, allowing for temporal multiplexing in this setup.

discrete comb of spectrally distinct harmonics, which combine to form extremely sort pulses. Witha few additional optics, existing EUV ptychographic imaging systems can be modified to spatiallyseparate these harmonics, or temporally delay two or more pulses, allowing for spectral or temporal
multiplexing, respectively.
With the need for simultaneous high resolution and large field of view imaging to fully
investigate the relationship between microscopic order and macroscopic function, spatially separate
multiple mode ptychography provides a possible path forward. Ideas developed during the proof-ofprinciple experiments described here are already implemented at the Spring-8 synchrotron source[180].

Chapter 5

Multiple Beam Ptychography with Mutually Coherent Beams

5.1

Introduction
This chapter addresses an extension to the mutually incoherent multiple beam ptychographyalgorithm discussion during chapter 4. A significant accomplishment of ptychography CDI is its
shift of the Nyquist sampleing criteria that previously limited sample size in diffractive imaging
systems [95]. It does this by shifting this constraint from the object under investigation to thebeam used to probe the system and operating in an area-by-area scanning modality. This allows
for an image field-of-view (FOV) that grows with the size of a scan, or number of scan positions.
This comes at the cost of increased scan time and data acquisition required to form an image. It
is this limited throughput that remains an obstacle for large field-of-view ptychographic imaging.
This chapter presents a diffraction-limited, large FOV ptychographic imaging technique thatuses several mutually coherent, spatially separated illuminating probes to scan a sample. This increases the throughput of the microscope by a factor corresponding to the number of probes without
increasing data acquisition time or volume. Using an optical transmission mode ptychography CDI
microscope, two related but distinct methods are demonstrated. These experiments are carried out
on the setup depicted in Fig. 5.1 with various pinhole sizes and separations in the pinhole array.
Finally, an analysis of the image fidelity of the reconstructed objects using both multi-beam andsingle-beam ptychography is presented. These techniques are previously published in [181, 182].

71

Figure 5.1: The Experimental Schematic for Multiple Beam Ptychography. A visible
laser diode (λ = 656nm) is spatially filtered and collimated before it is sent onto a pinhole array.
The exact specifications of the pinhole array vary depending on the technique, but it functions togenerate multiple mutually coherent probes that are then imaged onto the sample via the imaging
lens. A Fourier transform lens is used to propagate the exit wave to the far-field and produce a
diffraction pattern that is then recorded by the detector.

5.2

Methods
In ptychography CDI, the illuminating probe is subject to an oversampling constraint [84,183] that restricts its transverse size. Satisfying the oversampling constraint allows the resulting
diffraction intensities to be Nyquist sampled by the detector. When a second illuminating probe
is added to the system, interference terms between the two probes are introduced at the detector.
Traditional ptychography algorithms interpret these two individual probes as a single illumination
function, which greatly restricts the allowable size and separation of the probes on the sample, as
the entire probe system must satisfy the oversampling condition.
It is possible to suppress this interference and effectively convert the measured signal to an
incoherent sum of the diffraction from all the probes. This suppression can be achieved through72
digital filtering of the data after acquisition, which we refer to as autocorrelation filtering in section
5.2.1. In a complementary approach, the experimental parameters can be tuned to naturally remove
the interference from the measured signal, which we refer to as alias cloaking in section 5.2.2.

5.2.1

Autocorrelation Filtering

Consider a transmission ptychography experiment with two identical, mutually coherent
beams with wavelength λ and finite support incident on an extended sample and separated bya center-to-center distance ∆x. Working in the projection approximation, the exit surface wave
just after the sample is given by the sum of the contributions from each probe functions multipliedby the object transmission function, i.e.: ψ (x) = O (x) P1 (x) + O (x) P2 (x − ∆x), where ψ (x)is the exit surface wave, O (x) is the object transmission function, and Pn (x) is the nth probe
function. During a scan, a detector positioned in the far-field a distance z away from the sample,
will record the intensity of the diffracted wave as
I (f ) = |F {ψ (x)}|2 ,(5.1)

where F {} is the 2D Fourier transform operator. Using the form of ψ (x) given above, the Fouriertransform is computed with the convention G̃ (f ) ≡ F{G (x)}:F {ψ (x)} = F {O (x) P1 (x) + O (x) P2 (x − ∆x)}= Õ (f ) ⊗ P̃1 (f ) + Õ (f ) ⊗ P̃2 (f ) e2πi(∆x·f ) ,(5.2)

where ⊗ is the 2D convolution operator. Combining Eqn. 5.1 and Eqn. 5.2 gives the expression
for the detector intensity as22
I (f ) = Õ (f ) ⊗ P̃1 (f ) + Õ (f ) ⊗ P̃2 (f ) e2πi(∆x·f )hi+ 2R Õ (f ) ⊗ P̃1 (f ) × Õ (f ) ⊗ P̃2 (f ) e2πi(∆x·f ) ,(5.3)

where ⊗ is a 2D convolution and R [ ] indicates the real part of a complex quantity. Comparing
Eqn. 5.3 and Eqn. 4.1, we notice that the only difference between the expressions is the third termin Eqn. 5.3. Thus, if this term is removed, the resulting intensity distribution is identically that of
one produced by two mutually incoherent propagating beams.

73
Now, in a procedure similar to [184], a Fourier transform of the intensity given in Eqn. 5.3
is computed to isolate the final term in the expression. This gives the autocorrelation of the exit
surface wave22Õ (f ) ⊗ P̃1 (f ) + Õ (f ) ⊗ P̃2 (f ) e2πi(∆x·f )h
i o+ 2R Õ (f ) ⊗ P̃1 (f ) × Õ (f ) ⊗ P̃2 (f ) e2πi(∆x·f ),A (η) =Fn(5.4)

where A is the autocorrelation with variable η. Using the Fourier shift theorem and the convolution
theorem, Eqn. 5.4 is written as

  ∗
∗
A (η) = O (η) P 1 (η) ⊗ O (η) P 1 (η)

  ∗
∗
+ O (η) P 2 (η − ∆x) ⊗ O (η) P 2 (η + ∆x)

  ∗
∗
+ O (η) P 1 (η) ⊗ O (η) P 2 (η + ∆x)

  ∗
∗
+ O (η) P 2 (η − ∆x) ⊗ O (η) P 1 (η)

(5.5)

n
o
where G (η) = F G̃ (f ) . The first two terms in Eqn. 5.5 contain information pertaining to
the autocorrelation of each term of the exit surface wave ψ (x) and are referred to as the “DC”
terms as they are centered at η = 0. The last two terms are cross-terms and can be though of as
the cross-correlation of the different contributions of ψ (x) from the two probes. These terms are
centered at η = ±∆x and are referred to as the “AC” terms. Because the probe functions P1 (x)and P2 (x) have finite support, each term in Eqn. 5.5 is also of finite extent, with a width equal to
the sum of the widths of the contributing probe functions for each term. This results in distinct
autocorrelation and cross-correlation peaks when data is viewed in this space.
Taking only the DC terms from Eqn. 5.5 and performing an inverse Fourier transform gives
the filtered intensity:
n


  ∗
∗
O (η) P 1 (η) ⊗ O (η) P 1 (η)
o
  ∗
∗
+ O (η) P 2 (η − ∆x) ⊗ O (η) P 2 (η + ∆x) ,IA (f ) =F −1(5.6)

where IA (f ) is the autocorrelation filtered intensity. This expression further simplifies to22
IA (f ) = Õ (f ) ⊗ P̃1 (f ) + Õ (f ) ⊗ P̃2 (f ) e2πi(∆x·f )
,

(5.7)

74
which is now equivalent to an intensity measurement produced by two mutually incoherent, noninterfering beams. The mathematical procedure described by Eqn. 5.1 - 5.7 is depicted in Fig.
5.2.

Figure 5.2: The Process of Autocorrelation Filtering. (a) Two coherent probes (green
squares) are incident on an object. (b) The resulting field propagates to a detector located in
the far field, recording a diffraction pattern that contains high frequency interference from the interaction of the exit surface wave of each probe. (c) Fourier transforming the signal intensity brings
it into autocorrelation space, where the (d) incoherent sum of the exit surface waves (DC peak)can be filtered from the interference terms (AC peaks), provided the system satisfies Eqn 5.9. (e)Inverse transforming the filtered signal yields an incoherent sum of the diffraction from each probe
which can then be used for image reconstruction. Reproduced from [181].

Diffraction intensity measurements treated with this filtering method can now be used directlywith any of the existing multiple mode ptychography reconstruction algorithms to retrieve images

75
pertaining to the regions of the sample scanned by each probe. This filtering procedure is referred
to as “autocorrelation filtering”.
As long as the AC and DC peaks do not overlap in autocorrelation space, filtering is possible.
This is a condition on both the size of peaks and the separation between them. First, consider
that that the AC peaks must be sufficiently offset from the DC peaks such that there is no overlap
between them. This means the magnitude of the separation between the probes, |∆x|, must belarger than the full width of the DC peak. The autocorrelation of a function with finite support istwice the width of the function itself. Defining the characteristic probe width in the object plane,D, we have the condition|∆x| ≥ 2D.

(5.8)

where ∆x is the probe separation. This constraint alone is not sufficient as it does not consider
the size of the AC peaks themselves. Namely, in order for the AC and DC peaks to be separated,the extent of the probes must be sufficiently small. The spatial extent of autocorrelation space isdetermined by experimental parameters and geometry and is defined as X = λzp where λ is the
wavelength of the illumination, z is the propagation distance between the detector and the sample
plane, and p is the detector pixel pitch. The ratio of this extent to the beam diameter is an alternate
way to express the oversampling ratio previously described in Eqn. ??, σ = X/D. The AC and DC
peaks must all fit within the autocorrelation space without overlap, all of which have width 2D.
The optimal way to ensure there is no overlap while maximizing the size of the illumination is torequire that σ = 4. This is in contrast to most CDI techniques which require the looser constraintthat σ = 2 to properly sample the diffraction amplitude [183, 185].
Furthermore, as the separation of the beams is increased, the interference is pushed to ahigher spatial frequency. Referring to Eqn. 5.5, this has the effect of moving the AC peaks in
autocorrelation space. Eventually this frequency will exceed the sampling rate of the detector (i.e.
the detector pixel size) and the interference signal will alias. This can cause the AC and DC peaks
to overlap once again, preventing the isolation of the DC terms. However, if the separation is

76
further increased, the interference will alias back to an acceptable frequency and autocorrelation
filtering is again permitted. The effects of aliasing gives rise to repeated regions values of |∆x|where autocorrelation filtering is permitted. These regions are shown in Fig. 5.3.
The full condition on the beam separation must include the effect of aliasing on the recorded
diffraction intensity. Taking this into account, the condition is best expressed with the dimensionlessinequality2 ≤ mod

∆x
,σ
D


≤σ−2

(5.9)

where mod ( ) is the modulus operator.

5.2.2

Alias Cloaking

Referring to Fig. 5.3, it is observed that there is a decrease in the intensity of the interference
fringes as the probe separation is increased. This effect is explored through a study of a normalized
error which compares the diffraction intensity pattern produced by a two beam system that isallowed to interfere with an equivalent system that does not include interference. This error isdefined assErms =1 X[SI (∆L) − SN (∆L)]2 ,M m(5.10)

where Erms is the root-mean-squared error, M is the total number of elements within a signal, SI
is the signal with interference fringes, SN is the equivalent signal without interference, and ∆L is
a normalized probe separation such that ∆L = ∆x/D.
As the probe separation increases, the overall trend of Erms is to decrease. This is dueto the aliasing of inter-beam interference which suppresses the fringe intensity. Once the beam
separation increases past a critical point, the period of the interference is less than the extent of adetector pixel. In this scenario, the interference is measured as a single value corresponding to theaverage of the fringe within the pixel. As ∆L tends towards infinity, this average value becomes
more uniform across the detector, eventually becoming a constant background added to the signal
measured by each pixel. As the interference is averaged away, the diffraction pattern approaches

77

Figure 5.3: The Effect of Aliasing on Autocorrelation Filtering. The difference between
an intensity pattern produced by mutually incoherent (non-interfering) and mutually coherent
probes is plotted as a function of normalized probe separation ∆L = ∆x/D. The striped, green
regions correspond to probe separations where autocorrelation filtering is permitted. Grey regions
indication those where filtering is forbidden, noting regions where the AC and DC peaks overlap in
autocorrelation space. Note that there are many repeating regions where filtering is permitted, thisis due to the effect of aliasing on the interference frequency as the probe separation increases. There
are regions where the difference falls to zero. These occur at values of the probe separation with thespatial frequency of the interference precisely matches the sampling rate of the detector. In thesecases, the interference is measured as a uniform background and is removed from the measured dueto the experimental configuration. These points are where alias cloaking is permitted. The small
inserts depict the layout of autocorrelation space for various vlaues of ∆L . Reproduced from [181].

the non-interfering case, resulting in the overall trend in the normalized error.
As ∆L is increased, there are points in Fig. 5.3 where the error naturally drops to zero.
These points occur when the detector pixel size is equal to an integer number of interference fringe
periods. When this occurs, there is no difference in the measured average fringe value between
pixels across the detector and the inference appears as a uniform background. Here, the aliasing
has naturally removed or “cloaked” the interference and the diffraction appears identical to that

78
produced by an incoherent beam system. This process is called alias cloaking. Data collected in thisconfiguration is immediately ready for reconstruction with a multi-mode ptychography algorithm
without additional filtering.
The critical beam separations where alias cloaking occurs are completely determined by
the experimental geometry. Referring to Eqn. 5.3 and recalling the geometric relation between
frequency and detector coordinates, f = x/λz [36], the interference on the detector occurs withsome mean frequency fI = ∆x/λz. The diffracted wave is sampled on the detector with a frequency
fs = 1/p. The alias cloaking condition is satisfied when fI is an integer multiple of fs . Settingthese quantities equal and solving for the beam separation gives the critical probe separation
∆xc =

N λz
,
p

(5.11)

where N is an integer.
Due to the finite size of the detector pixels, the visibility of the interference at frequencies inthe vicinity of fI are also suppressed. As ∆L becomes larger, the slope of the error surrounding
the minima decreases, meaning the system becomes more resilient to misalignment as the beam
separation is increased. Fig. 5.3 can be further extended to show that at large values of ∆L, theerror introduced by the interference is naturally suppressed at all values of ∆x and restrictions on
the system parameters to obtain alias cloaking are greatly relaxed.

5.3

Experimental Results
The schematic of the visible-wavelength transmission-mode ptychographic microscope is shownin Fig. 5.1. Although the two methods described above require a slightly different apparatus, theyshare common elements to produce and shape the beam, as described here. A fiber-coupled laser
diode [Blue Sky Research FTEC2 658-60, = 656 nm] propagates through a spatial filter [Newport,
900: with a 5 m pinhole 900PH-5 and M-10X objective] to clean up the laser mode. The filtered
beam is then collimated by a lens with focal length f = 5 cm. At this point the setup must be
optimized for the different techniques and samples, as described for each experiment below.

79
5.3.1

Autocorrelation Filtering

This section demonstrates the feasibility of configuring a ptychographic CDI microscope to
use the autocorrelation filtering method described in section 5.2.1. Using the setup described inFig. 5.1, the beam is transmitted through a pinhole mask consisting of three pinholes in an L-shapeformation, fabricated with anodized aluminum foil [Thorlabs, BKF12] in which three pinholes were
punched with a pushpin [Staples, 32014]. The pinholes ranged in diameter from 90 - 175 µm, withcenter-to-center separations of approximately 0.75 mm. The pinhole array is imaged onto a sample
using an imaging lens with focal length f = 10 cm; the imaging system has magnification M = 1.
The light is scattered though a Negative United States Air Force (NUSAF) resolution test chart[ThorLabs, R3L3S1N] and is captured by a Fourier transform lens with focal length f = 2 cm,
which focuses the scattered light onto a CMOS detector [Mightex, SCN-B013-U]. A schematic of
the experiment is shown in Fig. 5.4(a). Given these system parameters, the oversampling of eachindividual beam ranged from 10–15, with the parameter ∆L falling between 7.8 - 11.8, satisfying
Eqn. 5.9 without any aliasing in the signal.
Data were recorded at 121 scan positions in an 11x11 rectilinear grid with random offsets
of 20% to avoid periodic artifacts [158]. At each position, multiple exposures were captured with
exposure times ranging from 0.05 ms to 750 ms. The resulting images were combined, increasing
the dynamic range of the detector from 68 dB to 100 dB [186]. Complex valued images for theobject and probe associated with each of the three beams were reconstructed with a modified
version of the multimode ptychography algorithm described in [168]. The reconstructed objects
and probes are shown in Fig. 5.4. The probes and objects are displayed in complex amplitude withamplitude mapped as brightness and phase as hue. The three reconstructed regions correspond to
three different areas of the USAF test pattern.
To further study the capabilities of autocorrelation filtering, the microscope was adjusted
such that the frequency of interference between the beams was aliased on the detector. Thisaliasing property allowed for a larger beam separation. This experiment is schematically depicted

80

Figure 5.4: Experimental Results for Autocorrelation Filtering. (a) The experimental
setup used to test the autocorrelation filtering technique, the small inset displays an example
of autocorrelation space for the pinhole array used during this experiment. (b-c) The complex
amplitude of the reconstructed probes recovered from the multimode ptychographic reconstruction
after autocorrelation filtering, where brightness is amplitude and hue represents phase. (e-f) The
complex amplitude of the reconstructed image of the NUSAF test chart sample. (h) A diagram ofgroups 4-7 of a USAF resolution test chart. The dashed lines show the relative positions of eachreconstructed area in (e-f). The recovered probes and objects are paired by row. The scale bar of
200 µm and color wheel shown in (b) are common for (c-g). Reproduced from [181].

in Fig. 5.5(a). A new pinhole mask was fabricated, consisting of two circular pinholes, each with81
a diameter equal to 0.4 mm, separated by a center-to-center distance of 3.7 mm. Due to thelarge probe separations in this experiment, the transform lens was replaced with an aspherical lens[Thorlabs AL2520-B] of equivalent parameters to reduce the spherical aberrations on the recorded
diffraction data.
The imaging system was removed from the previous experiment in favor of placing the pinhole
mask in close proximity to the sample to simplify the experimental setup. The sample consists of atransverse slice of nymphaea of aqustio stem mounted on a microscope slide and preserved in cedar
wood oil [Amscope PS25]. With these parameters, the system had an individual beam oversamplingof σ = 6.3 and normalized probe separation ∆L = 9.3. These parameters place this experimental
geometry in the second allowed filtering region in Fig. 5.3, meaning the interference has aliased to
to a lower frequency.
A 64-position ptychography scan was recorded arranged in an 8x8 rectilinear grid, with 20%
random offsets between adjacent scan positions. Data were processed identically to the previous
experiment, where autocorrelation filtering was applied to remove the interference terms and reconstructed using a multimode ptychography algorithm [168]. The reconstructed amplitudes of eachobject and probe are shown in Fig. 5.5(b-c).

5.3.2

Alias Cloaking

Alias cloaking is demonstrated experimentally to validate the method outlined in section
5.2.2. The setup for this experiment uses the laser system depicted in Fig. 5.1, with the pinhole
array and imaging system shown in Fig 5.6(c). The pinhole array consists of three circular apertures
arranged in a linear pattern, each with a diameter of 1.26 mm and a center-to-center separation of
2.50 mm. These system parameters give an individual beam oversampling of σ ≈ 2 and satisfies
the critical beam separation given in Eqn. 5.11 with N = 1. Referring the Fig. 5.3, note that thesystem is sitting in the first null with ∆L = σ. Due to the large peripheral error for surrounding
values of ∆L, this configuration is the least ideal experimental setup to perform alias cloaking as
it is most sensitive to misalignment.

82

Figure 5.5: Experimental Results for Aliased Autocorrelation Filtering. (a) The experimental schematic for aliased autocorrelation filtering, the small inset displays an example ofautocorrelation space for the pinhole array used during this experiment. (b-c) The reconstructedcomplex amplitudes of the nymphaea of aqustio sample and the illuminating probes. The scalebars represent 200 µm in the object field-of-view and probe inset field-of-view respectively. The
scale bars and color scheme are equivalent for both b and c. In these images brightness indicatesthe amplitude and hue corresponds to phase. Reproduced from [181].

A ptychography scan was recorded using a test sample consisting of a thin transverse slice ofa rabbit testicle mounted on a microscope slide and preserved in cedar wood oil [AmScope, PS25].

83

Figure 5.6: Experimental Results for Aliased Autocorrelation Filtering. (a) Large FOV
phase reconstruction of a cross-sectional slice of a rabbit testicle obtained using alias cloaking
ptychography. (a) is composed of reconstructed objects from three identical beams simultaneously
incident on the sample, each imaging one third of the specimen. (b) An enhanced region shownby the black square outline in (a). (c) Experimental apparatus used to obtain the reconstructions
shown in (a, b), where the small inset displays an example of autocorrelation space for the pinhole
array used during this experiment. Notice the cross-correlation peaks are naturally suppressed.
Scale bars are 1 mm (a) and 200 um (b). Reproduced from [181].

The scan consisted of 784 scan positions arranged in a 28 x 28 square grid with a step size of
1/6 the beam radius, modulated with a 20% random offset. The large number of scan positions,coupled with the scan position step size and beam diameter, lead to overlapping fields of viewin the reconstructed objects. Because the diffraction-limited resolution of each reconstruction is
independent of the collective probe system, each object retains a high spatial resolution.
The three reconstructed images were manually registered. Once registered, the images were84
averaged together using a cosine smoothing image stitching algorithm. This algorithm weights
image pixels more heavily the further they are from their respective image edge. This stitchingalgorithm is used to combat the fall off in reconstruction quality near the edge of a scan area. Thisis due to the decreased level of overlap near the periphery of a ptychography scan region. The
single large field-of-view, high-resolution image is shown in Fig. 5.6(a). The total reconstructedarea is 21.5 mm2 . Fig. 5.6(b) shows an enlarged region given by the black square outline in Figure
5.6(a).
In the original reconstruction, a quadratic phase was present in all three reconstructed objects.
The presence of this phase indicates a slight bow in the sample. This curvature was verified byplacing the sample in an interferometer [Zygo GPI XP]. Phase data measured in the interferometeris show in Fig. 5.7 along with a cubic fit. The fit demonstrates a second-order phase across the
sample. For viewing purposes, this phase was removed from Fig. 5.6(a, b).

5.3.3

Image Quality Analysis

Previous work on multimode ptychography algorithms show that multiplexing data has nodetectable effect on reconstructed image resolution of the complex object transmission function[168]. To investigate the effect of alias cloaking on the system resolution, reconstructions of equivalent fields of view of the sample obtained using alias cloaking and single beam ptychography are
analyzed. The two reconstructions are show in Fig. 5.8 (c, d). A short glance shows that thereconstructions look nearly identical, except for a low frequency oscillation present in the multibeam reconstruction. This modulation is due to a persistent back reflection, present in all thediffraction patterns taken during the scan. Thus modulation is not an artifact due to multiple
beam techniques.
To compare the two reconstructions, the phase retrieval transfer function is calculated [69,187, 188]. The function is defined as* P
PRTF (fr ) =nhψnrec (xj )iip
I meas (xj ) j

+
j

,
ϕ

(5.12)

85

Figure 5.7: Laser Interferometry of Rabbit Cell Sample. Phase data measured with a laserinterferometer on the rabbit cell sample used for alias cloaking experiments. Data are displayedwith a cubic fit. The fit displays a quadric phase present in the sample. This phase indicates acurvature in the sample.

where ψnrec (xj ) is the j th reconstructed diffraction pattern generated by the nth probe, I meas (xj )is the intensity measurement corresponding to the j th scan position, h im corresponds to an averageover parameter m, with i indicating an average over independent reconstructions, n is the individual
reconstructed probe, j is the scan positions, and ϕ is the azimuthal angle of the diffraction pattern.
This calculation used 100 independent ptychography reconstructions produced with 500 iterations
each. The phase retrieval transfer function for the single and multiple beam reconstructions areshown in Fig. 5.9.
An estimate for the resolution is found by locating the spatial frequency corresponding to thepoint where the phase retrieval transfer function falls below 0.5 [188]. The location of this point

86
provides an estimated spatial resolution of 2.85 µm for the multiple beam case and 2.70 µm for the
single beam case. The Abbe diffraction-limited resolution of the microscope is calculated at 2.48µm.
The two imaging modalities are further compared by examining their azimuthally averaged
power spectral densities. The power spectral density used here is a measure of the azimuthally
averaged power as a function of spatial frequency, defined as
DE
PSD (fr ) = |F {I (x)}|2 ,ϕ(5.13)

where PSD (fr ) is the azimuthally averaged power spectral density and I (x) is the image of interest.
The power spectral densities and the multiple and single beam reconstruction used in this analysis
are shown in Fig. 5.8 (a, c, d). The two spectra are nearly identical, with the average ratio between
them being 0.98.
It is expected that the maximum number of allowable beams for use in multiple beam ptychography is limited by the finite dynamic range of the detector, as is the case for multiplex holography[189]. The exact relation between reconstruction quality, number of beams, and detector dynamic
range is currently unknown, but provides an interesting subject for further analysis of multiple
beam ptychographic imaging systems.

5.4

Discussion and Conclusion
The throughput of a ptychographic imaging system is a key factor that limits the applicabilityof this powerful imaging technique. The diffraction-limited, high-throughput imaging with inherent
phase contrast obtained with CDI is of great interest in the field of bio-imaging and material characterization, where large scale properties are inherently linked to the behavior and characterizationof local order, interactions, and phenomena. In this work, a proof-of-principle demonstration of two
novel techniques allowing for the use of multiple coherent illuminating probes to simultaneously
image multiple areas of a sample was provided. These techniques provide a method for removing
the interference between the coherent probes, allowing the data to be processed using a multiple

87

Figure 5.8: Comparison of Power Spectral Density in Between Multiple Beam andSingle Beam Ptychography. (a) The azimuthally averaged power spectral density of two phasemap reconstructions. One obtained while using an alias cloaking experimental geometry (c) and
one obtained using conventional single beam ptychography (d). The field-of-view and scales areequivalent in (c-d). (b) displays the ratio of the power spectral density shown in (a) a functionof spatial frequency. The mean value of this ratio is 0.98. The similarity of the PSD for thetwo different imaging modalities indicates that the two imaging techniques produce similar results.
Reproduced from [181].

mode ptychography algorithm.
Multiple beam ptychography provides large field-of-view, high-throughput imaging while minimizing the required data acquisition. This imaging technique is implemented by controlling the88
1
Single BeamMulti Beam

0.9

0.8

PRTF

0.7

0.6

0.5

0.4

0.3
0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

fr [µm -1 ]Figure 5.9: Comparison of Single and Multiple Beam Ptychography Resolution. The
phase retrieval transfer function of the single and multiple beam ptychography reconstructions ofequivalent fields of view, shown as a function of spatial frequency. A value of 0.5 is used as acutoff value to determine the resolution of a reconstruction. In this analysis, the multiple beamreconstruction has a slightly higher cutoff frequency, indicating a higher resolution than the single
beam image.

relative placement of the illuminating probes and can be applied in either data post-processing
through autocorrelation filtering, or experimentally through alias cloaking. High fidelity reconstructions are obtained when samples were simultaneously illuminated by either two or three coherent probe beams. This technique could be applied in the X-ray and EUV spectral ranges, wherelarge field-of-view images with nanometer spatial resolution provide key insight into the relationshipbetween microscopic order and macroscopic material properties.

Chapter 6

Spectroscopic X-ray Imaging of the Allende Meteorite

6.1

Introduction
The last decade of intense focus and research has seen ptychography mature in a well estab-lished technique for nanoscale microscopy [190–193]. Applications of X-ray ptychography include
two and three dimensional investigations of biological systems [194, 195] and integrated circuit
structure [196, 197], where a lateral resolution of tens of nanometers is regularly obtainable. Recent developments have extended soft X-ray imaging toward a diffraction limited lateral resolutionof 5 nm [198, 199].
Scanning transmission X-ray microscopy (STXM) is a valuable tool when spatially resolved,
spectrally sensitive information is required. This technique involves tightly focusing an X-ray beamand raster scanning a sample, where a bucket detector records the localized X-ray absorption of
the illuminated areas. In this configuration, an image is built up pixel by pixel, with the resulting
image resolution limited by the focal spot size of the X-ray beam. Combining STXM with soft
X-ray absorption spectroscopy, affords a system that can map the chemical-specific composition ofoxygen, nitrogen, carbon, and transition metals like iron, manganese, and nickel in for biologicaland material systems [200].
In this chapter, STXM and X-ray ptychography spectromicroscopy are used to investigate
an Allende meteorite grain. Complex X-ray transmission images of a meteorite grain are obtainedwith spatial resolution of 15 nm, an improvement of more than an order of magnitude over previous studies [201]. The high resolution images and spectral information are used to infer mineral

90composition of the grain and potential formation processes of the meteorite. Results and methods
presented here are part of a correlative microscopy study of the Allende meteorite with electron
and X-ray imaging, previously published in [202].

6.2

Methods

6.2.1

Allende Meteorite

The Allende meteorite is a CV3 carbonaceous chondrite that fell in Chihuahua, Mexico on
February 8, 1969. Breaking down during its entry into Earth’s atmosphere, fragments of themeteorite were spread over an area approximately 8 km by 50 km, one of the largest strewnfields
known on Earth. Hundreds of meteorite fragments were collected ranging in size from approximately
1 g to 110 kg; overall 2 - 3 tons of meteoric stone was collected, making it one of the most abundantsources of meteoric samples for study [203]. Due to the timing of its arrival and abundance,
Allende was well studied as many laboratory facilities had been established to receive samples fromthe Apollo program which was expected to return with the first samples of the lunar surface in justa few months. Consequently, the Allende meteorite is colloquially referred to as “the best-studied
meteorite in history” [204].
The meteorite is a heterogeneous aggregation of calcium and aluminum rich inclusions andmillimeter-sized chondrules, set in a fine-grained matrix of micro- to sub-micrometer sized oxides,
suflides, silicates, and metals. The chondrules are small stony spherules which formed as molten
droplets in space before being accreted into a larger body and are among the oldest matter observed on Earth, predating the Earth itself by 30 million years. These chondrules contain valuable
information about conditions during the early formation of the solar system [205]. The chemical
diversity and nanometer sized petrographical features make the meteorite an interesting sample for
X-ray spectromicroscopy.

91
6.2.2

Sample Preparation

In this study, unsorted Allende meteorite fragments were added to an agate mortar andimmersed in ethanol, then pounded by an agate pestle into sub-micrometer grains. The resulting
suspension was sonicated for 2 minutes to further disperse the particles. The solution was thendrop cast using a pipette onto a carbon film 200 mesh Cu transmission electron microscope (TEM)
grid. Samples were then air dried for 24 hours before data acquisition.

6.2.3

X-ray Imaging Data Acquisition

Ptychography and STXM data were recorded at the COherent Scattering and MICroscopy(COSMIC) beamline – beamline 7.0.1 at the Advanced Light Source (ALS) at Lawrence Berkeley
National Lab (LBNL). COSMIC has a tunable, coherent soft X-ray source, suppling monochromatic
light with energy ranging from 250 eV to 2500 eV, spanning the carbon and sulfur K-edges. X-rays
were focused using a Fresnel zone plate with outer zone width of 45 nm, giving a total coherent
flux of 109 photons/s at the sample position. The beamline endstation, known as Nanosurveyor
2, incorporates the sample area hardware of an FEI CM200 series TEM to leverage the existing
design of a TEM for a high stability sample manipulation [198]. The design uses a standard FEI
CompuStage sample manipulator where the sample is secured with a Hummingbird 3 mm half-gridtip. This common sample mounting scheme allows for easy transfer of sample between X-ray andelectron microscope for correlative study. The design preserves much of the function of a standard
TEM, meaning any sample manipulator for the CM200 series CompuStage can be used in the
Nanosurveyor 2. This allows for tomographic, cryo-biased, and other in-situ experiments to be
performed in this setup.
Once the sample was mounted and inserted into the sample chamber, it was pumped down to1 x 10-6 torr. Diffracted X-ray light was recorded with a novel fast charge-coupled device developed
at LBNL. The device was configured withe a 50 frames/s frame rate, 15 bit dynamic range, a 12 bit
analogue-to-digital converter, and a 512 x 512 pixel chip with 26 µm pixels. During the experiment,

92
the on-chip binning was set to 2. All diffraction data were collected without a beamstop. The X-ray
imaging system is schematically shown in Fig. 6.1.

Figure 6.1: X-ray Imaging Scheme at COSMIC. Allende meteorite grains deposited on a TEM
grid was mounted in the COSMIC soft X-ray beamline for ptychographic and spectromicroscopic
studies. The beamline incorporates a FEI CM200 CompuStage for sample mounting and manipulation. This scheme allows for easy sample transfer for correlative electron microscopy studies.
Ptychography and STXM datasets are collected in a transmission mode, where X-ray light ranging
from 707 - 1551 eV was used to probe the sample. High energy resolution spectroscopic image
stacks were captured using STXM at the Fe L3 edge (707 eV), Ni L2/3 edge (865/848 eV), Mg K
edge (1302 eV), and Al K edge (1551 eV). Ptychography data were collected 5 eV before and on
resonance of the absorption edges. Modified from [202]

STXM data were recorded by raster scanning a region of the sample in an 80 x 80 square scan
grid with a 40 µm step size and a 10 ms dwell time per point. The resulting image had a 3.2 µm
by 3.2 µm field of view. X-ray absorption spectra can be collected over a specified spectral range
at a single location, along a line with a specified direction, or throughout the entire image fieldof view. Line scans were initially performed to calibrate the desired photo energies corresponding
to an absorption edge for a specific element, then image stacks were recorded with the energies

93
varying across the absorption feature. STXM absorption spectroscopy were recorded at the Fe L3edge (707 eV), Ni L2/3 edge (865/848 eV), Mg K edge (1302 eV), and Al K edge (1551 eV). Theenergy resolution varied within the scan, beginning with a 1 eV step size -15 eV before the edge,
moving to 0.25 eV with ±5 eV of the absorption resonance, then back to a 1 eV step size until theenergy of the X-rays was 15 eV above the edge. STXM energy stack would take between 30 to 60
minutes to record depending on the absorption edge width.
Ptychography data were collected using a double exposure mode. Dwell times of 15 ms and
150 ms were used to enhance the dynamic range of the recorded diffraction patterns. Square scangrids of 80 x 80 or 100 x 100 positions with a step size of 30 nm cover a field of view of 3.2 - 5.8µm2 . With these scan parameters and experimental geometry, the reconstructed pixel size was 8
nm/pixel. Each ptychography dataset consisted of 6,400 - 10,000 diffraction patterns of 256 x 256
pixels. Ptychography scans were collected both 5 eV before and on resonance a the Fe L3 edge, Ni
L2/3 edge, Mg K edge, and Al K edge. These dataset provide a differential measurement, at eachabsorption feature, allowing the element to be easily isolated in the sample.

6.3

STXM Data Analysis
STXM absorption spectroscopy was used to provide local chemical information in the me-teorite grain. A STXM image stack recorded at different X-ray energies was processed using the
MANTiS software to calculate and isolate the composition of the meteorite [206]. The spectral
range of a STXM energy stack was chosen to span different atomic resonances, then converted to
optical density by normalizing to a fully transmitting regions in the acquired field of view. Hot
pixel were replaced by the average of the six neighboring pixels and the background was subtracted.
After this, the image stack was aligned iteratively using center of mass and common line alignment
methods. Spectra in the 3D image stack were decomposed using principle component analysis(PCA) and k -means clustering to group clusters with similar spectra. Last, singular value decomposition was used to produce chemical maps and corresponding absorption spectra. The recorded
spectrum and corresponding STXM images for the Fe L3 edge are shown in Fig. 6.2.

94

Figure 6.2: Fe STXM Spectromicroscopy. (a) The Fe X-ray absorption spectrum generated
from STXM energy scans across the Fe L3 edge. (b-i) The corresponding STXM images at various
point along the spectrum (dashed lines in (a)). A drastic change in the absorption throughout themeteorite grain is seen in the on resonance image (c) as Fe is distributed throughout the grain.
Adapted from [202].

PCA generated spectra for the four major elements in the grain are shown in Fig. 6.3. The Al
K edge spectrum agreed well with previously published spectrum of Al2 O3 [207] and the Mg K edge
spectrum was consistent with those of magnesium silicates [208]. Similarly, the extracted Ni L2/3
edge is similar to those of synthetic nickel sulfide and pure nickel [209, 210]. Two unique spectra
for Fe were found from PCA (Fig. 6.3 (b)). These spectra closely match previously publishedreference spectra of iron sulfide and iron silicate, corresponding to different oxidation states of Fe[211]. The recovered iron silicate spectra is shown in Fig. 6.3 (b), solid purple line and iron sulfide
is displayed in Fig 6.3 (b), dashed yellow line. The two major absorption peaks in the Fe L-edge
are labeled L3a and L3b .

6.4

Ptychography Data Analysis
Ptychography images of the Allende meteorite were initially reconstructed at COSMIC using

the distributed GPU-based ptychographic solver SHARP provided by the ALS [212]. The highspeed data transfer and collective computing power granted by the ALS GPU-cluster allowed fornear-real time feedback during data collection, but came at the expense of carefully pre-processing

95

Figure 6.3: STXM Absorption Spectra. Absorption spectra generated from STXM energyscans across absorption edges of (a) Al, (b) Fe, (c) Mg, and (d) Ni. Each spectra reveals the uniquespectral fingerprint left by each element present in the meteorite grain. The relative peak intensities
between the Fe L3a and L3b suggest that Fe2+ is the predominant chemical species of Fe in thesample. Adapted from [202].

data to remove artifacts and tuning algorithm parameters to maximize reconstruction quality. Eachptychography dataset was reprocessed to account for beamline intensity fluctuations and to remove
background noise on the detector. Images were then reconstructed using a parallel ptychographyalgorithm with an update condition derived from the hybrid projection-reflection algorithm [213].
Reconstructions of each dataset ran for 5,000 iterations, with the probe update turned on after
iteration 5 and continuing for the remaining iterations.
High fidelity, complex images were reconstructed on and below the absorption resonance for
Al, Fe, Mg, and Ni; the four main elements composing the meteorite grain. Fig. 6.4 shows thereconstructed amplitude images of the four elements on (a-d) and pre- (e-h) edge. By dividing theon-edge and pre-edge absorption images, the spatial dispersion of each element is revealed with both
high contrast and spatial resolution, shown in Fig. 6.5. The spatial resolution of the reconstructed
images was estimated using a 10 - 90 knife edge test. The test is shown in Fig. 6.6, giving a lateral
resolution of approximately 20 nm. The resolution varies across the different images as the image
resolution is energy dependent.
The quantitative information available from ptychography CDI comes in the form of thereconstructed amplitude and phase of the complex transmission or reflection function of a sample.
In general, this function is dependent on not only material properties, such as the complex refractive

96

Figure 6.4: Ptychography Amplitude Reconstructions Near Al, Fe, Mg, and Ni Absorption Edges. Amplitude reconstructions from ptychography data obtained across the absorption
edges of the four main elements in the meteorite grain. (a-d) Pre-edge images at energies 1,545.5
eV, 700 eV, 1,295 eV, 842 eV, respectively. (e-h) On-edge images at energies: 1,551 eV, 707 eV,
1,302 eV, 848 eV. Reproduced from [202].

Figure 6.5: X-ray Ptychography Absorption Contrast. The four major elements composing
the Allende meteorite grain are localized with high contrast and spatial resolution by dividing
the pre-edge and on-edge amplitude images at the Al, Fe, Mg, and Ni edges shown in 6.4. The
red arrows in (a-c) indicate the locations of Al melt pockets in the meteorite grain. This highcontrast viewing method highlights the presence of Fe in these pockets as well as the absence of
Mg. Reproduced from [202].

index n, but also sample material thickness and density. To eliminate this sample ambiguity, thescattering quotient of the complex transmission function is examined [214–216]. The scatteringP

N (x)β (E)(x;E)|i
Pi i
quotient is defined as fq ≡ ln|T
φ(x;E) =Ni (x)δi (E) . Here, |T (x; E)| and φ (x; E) are the complexi97

Figure 6.6: Spatial Resolution of Ptychography Reconstructions. Knife-edge resolution testof ptychographic absorption reconstruction at the Fe L3 edge. Spatial resolution is quantified at
20nm using the 10-90 criterion. Adapted from [202].

transmission function amplitude (0 6 |T (x; E)| 6 1) and phase retardation (φ (x; E) 6 0) produced
via ptychographic reconstruction at photon energy E and spatial location x, Ni is the atomic
number density of the ith element in the sample, and δ and β are the real and imaginary parts ofthe complex refractive index, n = 1 − δ + iβ.
Referring to the relation between the complex transmission function and the complex refractive index in section 1.3, the scattering quotient is independent of effects due to sample thickness
variation [216, 217]. This feature makes the scattering quotient uniquely well suited for studying
heterogeneous materials where it is difficult to tease apart changes due to thickness variations fromthose due to a changing material composition. In the case of two dimensional images, the corresponding pixel values of the scattering quotient maps are averages along projections through thesample. Viewed in this way, the scattering quotient offers a novel contrast mechanism sensitive
only to sample composition changes [218, 219].
Scattering quotient maps for the on and pre-edge ptychography reconstructions were calculated for Al, Fe, Mg, and Ni. Fig 6.7 shows this alternate contrast mechanism to visualize sample

98
composition independent of variations in sample thickness.

Figure 6.7: Scattering Quotient Maps of Ptychography Reconstructions. Novel contrast
is gained by calculating the scattering quotient map from the meteorite grain ptychography reconstructions. (a-d) Scattering quotient images calculated from on-edge reconstruction. (e-h) Quotient
maps calculated from pre-edge data. Reproduced from [202].

6.5

Discussion and Conclusion
Specific, local chemical and elemental composition information is extracted from the X-ray

images. From this information, specific mineral phases of the meteorite grain can be inferred with20 nm spatial resolution. X-ray absorption spectra extracted from the meteorite grain for Al,
Mg, and Ni are consistent with with previous measurements of their respective mineral phasesin the literature. Two dominant spectra were recovered from the Fe X-ray absorption analysis,corresponding to iron sulfide and iron silicate. In the case of iron silicates, the relative intensityof the Fe L3a edge spectra are directly related to the redox state of Fe in the meteorite grain.
Specifically, the relative intensity of the L3a peak decreases linearly relative to the L3b peak as thePratio of Fe3+ / Fe decreases [220]. Since L3a has a higher peak intensity than L3b , it is concluded
that the the majority of Fe in iron silicate is in the reduced form Fe2+ . This narrows the potential
mineral candidates present in the grain.
X-ray ptychography images provide high resolution textural information that provide insight

99
to the processes acting on the Allende parent body during and after accretion. In particular, shock
veins indicate that the meteorite, at some point, experienced impact-induced heating, melting,and cracking [221]. The observation of nanoscopic Fe and Al veins in the larger silicate matrix
provides strong evidence of melts from impact shocks or heat-induced metamorphism [222, 223].
The absence of Mg from these veins (Fig. 6.5(c)) indicates that no melting of phases including Mg
occurred.
Information extracted about the Allende meteorite agrees well with previous studies using
lower resolution techniques [224, 225]. However, the finer details offered by these advanced X-ray
microscopy techniques allow for the study of fine-grained matrix material and the observation ofmelt channels unobservable at coarser resolutions. In particular, the Allende meteorite is classified
as shock stage 1, or unshocked, but its parent body must have undergone impact events during
accretion and later to eject meteorites. The nanoscale melt channel observed here may provide
evidence of these impacts and future studies can constrain the timing and pressures of these events(i.e. pre-accretion vs. post-accretion).
High resolution X-ray ptychography and STXM absorption microscopy were used to study apiece of the Allende CV3 carbonaceous chondrite meteorite. The multidimensional data provides
local chemical and textural information on the heterogeneous composition of the meteorite grain.
Work presented here is part of a larger correlative microscopy project where additional insight isgained by imaging the same meteorite grain with using TEM. Further information is derived from
ptychography data using a newly-developed semi-quantitative analysis method to convert scattering
quotient maps to two-element ratio maps given a fixed amount of a third element [202, 226].

Chapter 7

Conclusions and Future Work

7.1

Summary
In the last decade, coherent diffractive imaging (chapter 3) has grown into a mature, general-purpose imaging method of coherent EUV and X-ray light sources (chapter 2). In particular, therobust reconstruction methods of ptychography CDI leverages the redundancy in many overlapping
diffraction measurements to produce a high quality, complex transmittance or reflectance map of
a sample. This thesis described work to expand the capabilities and improve the performance ofcoherent diffractive imaging techniques for both tabletop EUV and facility-scale X-ray sources.
By using multiple, illuminating probes (chapter 4), the throughput of a ptychography CDI
system is dramatically increased. Furthermore, using spatially separated beams with orthogonalmodes allows for spectral, and polarization resolved imaging, with all wavelengths and polarizations resolved in a single scan. Additionally, temporal multiplexing will enable the simultaneous
measurement of multiple frames of a nanoscale dynamic system.
The ideas of multiple beam ptychography can be expanded to make use of mutually coherent
probes (chapter 5). In this modality, aliasing is strategically used to remove the need for mutually
incoherent probes for multiple beam ptychography. Thus easing the experimental requirements forhigh throughput, large field of view ptychography CDI.
A high resolution ptychography CDI microscope was recently commissioned at the Advanced
Light Source (chapter 6). This microscope combines decades of engineering from electron microscopy with a high brightness synchrotron light source to create a ptychography CDI system ca-

101pable of spectromicroscopy with nanometer resolution. This microscope is used to spectroscopicallyimage a granule from the Allende meteorite with unprecedented resolution. The multidimensional
data provides local chemical and textural information of the heterogeneous meteorite sample.

7.2

Future WorkWork presented in this thesis represents first steps toward efficient, versatile ptychography

CDI microscopy. Techniques and principles outline in this work can be readily combined withexisting methods to build more flexible imaging systems suitable for a wide range of samples andlength scales.
Ongoing research in algorithm development stands to improve the performance of multiple
beam ptychography. Recently, an algorithm was proposed that will enable CDI on the attosecond
time scale [227]. This algorithm provides a leap in the performance of multiple mode ptychographyalgorithms, which directly translates to the performance of the multiple beam ptychography technique. An improved performance allows for a larger number of simultaneous probes, increasing thethroughput during data acquisition. Furthermore, development of HHG and X-ray sources resulting
in more than one beam is ongoing which is immediately useful for a multiple beam ptychography
imaging system [178, 179].
As the stability and performance of short wavelength coherent source continues, the performance of CDI systems will advanced with them. In particular, the resolution of these imaging
systems is increased by using shorter wavelength illumination or increasing the NA. For certain
spectromicroscopy applications, imaging with a particular wavelength to observe an absorption
feature is necessary. Thus, if smaller features want to be studied, the NA of the microscope must
be increased. This can present significant engineering challenges as the CDI systems become increasingly and necessarily compact. Nevertheless, the highest wavelength-to-resolution CDI system
was recently demonstrated using as 13nm HHG source [71], where sub-wavelength resolution wasreported by increasing the NA of an EUV ptychography CDI microscope. The combination of highNA imaging with CDI techniques has the potential to provide tabletop EUV coherent microscopes

102
capable of resolving nanometer-scale features with unprecedented temporal resolution.
Finally, a challenging but exciting frontier in nanoscale imaging comes in the form of correlative, real-time CDI. Many natural systems are comprised of heterogeneous structures with ahierarchical organization. Despite advancements in imaging technology, there is no one techniquecapable of characterizing a sample so that the macroscopic properties are fully understood in termsof microscopic structure and function. Instead, a hybrid imaging approach that incorporates EUV
and X-ray, electron, and visible imaging modalities can leverage the advantages of each system in
a complimentary manner. For example, visible microscopy is used to scan a large field of viewat lower resolution to determine areas of interest for higher resolution electron or X-ray imaging.
This process is further streamlined if a system capable of these three imaging modalities share acommon sample holder that can quickly switch between the desired imaging modality without theneed of a sample transfer.
Still, system throughput of X-ray and EUV imaging is largely limited by camera readout
rates, where a ptychography scan consisting of less than one minute of exposure time can take thirty
minutes or more to collect because of the detector readout. Fortunately, coming advancements in
X-ray and EUV CMOS technology coupled with advancements in distributed GPU reconstruction
algorithms [142, 228, 229] pave the way toward video-rate full-field nano-imaging.

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