Closing the Gap to the Diﬀraction Limit: Near
Wavelength Limited Tabletop Soft X-Ray
Coherent Diﬀractive Imaging
byRichard Lunt Sandberg
B.A., Brigham Young University, 2004

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
2009This thesis entitled:
Closing the Gap to the Diﬀraction Limit: Near Wavelength Limited Tabletop SoftX-Ray Coherent Diﬀractive Imaging
written by Richard Lunt Sandberg
has been approved for the Department of Physics

Prof. Henry C. Kapteyn

Prof. Margaret M. Murnane

Date

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

iii
Sandberg, Richard Lunt (Ph.D., Physics)
Closing the Gap to the Diﬀraction Limit: Near Wavelength Limited Tabletop Soft XRay Coherent Diﬀractive Imaging
Thesis directed by Prof. Margaret M. Murnane and Prof. Henry C. Kapteyn

Light microscopy has greatly advanced our understanding of nature. The achievable resolution, however, is limited by optical wavelengths to around 200 nm. Usingnovel imaging and labeling technologies, resolutions beyond the diﬀraction limit can beachieved for specialized specimens using techniques such as near-ﬁeld scanning optical
microscopy, stimulated emission depletion microscopy and structured illumination microscopy [1–3]. This dissertation presents a versatile soft x-ray diﬀraction microscopewith 50 nm resolution using tabletop coherent soft x-ray sources. This work represents
the ﬁrst high resolution demonstrations of coherent diﬀractive or lensless imaging using
tabletop extreme ultraviolet and soft x-ray sources [4, 5].
This dissertation also presents the ﬁrst use of ﬁeld curvature correction in x-ray
coherent imaging which allows high numerical aperture imaging and near-diﬀractionlimited resolution of 1.5λ. The relevant theory behind high harmonic generation, theprimary tabletop source used in this work, will be discussed as well as the theory behind coherent diﬀractive imaging. Additionally, the ﬁrst demonstration of tabletop soft
x-ray Fourier Transform holography is shown with important applications to shorter
wavelength imaging with high harmonic generation with limited ﬂux. A tabletop soft
x-ray diﬀraction microscope should ﬁnd broad applications in biology, nanoscience, andmaterials science due to its simple optical design, high resolution, large depth of ﬁeld,
3D imaging capability, scalability to shorter wavelengths, and ultrafast temporal resolution.

Dedication“Trust in the Lord with all thine heart; and lean not unto thine own understanding.
In all thy ways acknowledge him, and he will direct thy paths.”’ Proverbs 3:5-6

To my best friend and wife Deannec Copyright by Richard L. Sandberg 2008

All Rights Reserved

v

Acknowledgements

First of all, I feel very lucky to have worked with my advisors, Margaret Murnaneand Henry Kapteyn. Thanks for your generosity, perspective, support, encouragement,
and advice. I was extremely blessed to have had great coworkers, life-coaches, andfriends in so many who were instrumental in this work, especially Ariel Paul, Daisy
Raymondson, and Bill Schlotter. From UCLA, John Miao, Changyong Song, Kevin
Raines and others for image reconstructions and being excellent teachers, friends, andcollaborators. Many thanks to all in KM Group: Chan La-o-vorakiat, Matt Seaberg,
Ethan Townsend, Luis Miaja-Avila, Mark Siemens, Amy Lytle, Paul Arpin, Ra’anan
Tobey, Dave Gadiosi, Oren Cohen, Steﬀen Hädrich, Jim Holtsnider, Scott Hoch, Nick
Wagner, Rachael and Matt Tearle, Kim Reid, Sterling Backus and KM Labs and somany others that I don’t have room to list. Thanks to the EUV ERC: at CSU Przemek
Wachlak, Mario Marconi, Carmen Menoni, Jorge Rocca, and Sheila Davis; at CXRO
Anne Sakdinawat, David Attwood, Yanwei Liu, and Farhad Salmassi. In OSEP: Ricki
Hadow, Dana Anderson, Matt Squiers, Barb Tennis, Dan Dessau at CU and Dave
Pappas at NIST and groups, NSF and OSA (for $). In JILA/CU: Hans Greene, Dave
Alchenberger, all Instrument, Electronics, and other staﬀ and the summer Ultimate
lunch crew. Thanks to former undergrad friends and mentors at BYU and Bruno La
Fontaine and group at AMD. I’m sure I’m forgetting so many, thanks to you too.
Finally, thanks to family, friends, especially my parents, and to Deanne, our beautiful
kids Karadyn, Savannah, and Nathan, and to God for everything.

viContents

Chapter
1 Introduction to Tabletop Coherent Diﬀractive X-ray Imaging

1

1.1

Imaging at the nanoscale. . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Comparison of diﬀerent microscopies . . . . . . . . . . . . . . . . . . . .

3

1.2.1

Optical Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2.2

Electron Microscopy . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.2.3

X-ray Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . .

9Overview of work to be discussed . . . . . . . . . . . . . . . . . . . . . .

11

1.3

2 High Harmonic Generation as a Bright, Coherent Source of EUV and SXR Radiation

13

2.1

Introduction to high harmonic generation . . . . . . . . . . . . . . . . .

13

2.2

Ultrafast laser technology . . . . . . . . . . . . . . . . . . . . . . . . . .

14

2.3

Theory of High Harmonic Generation. . . . . . . . . . . . . . . . . . .

16

2.3.1

Discovery and three step model . . . . . . . . . . . . . . . . . . .

18

2.3.2

Path integral interpretation of HHG . . . . . . . . . . . . . . . .

25

2.3.3

Phase matching . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

Coherence and mode quality . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.4.1

Deﬁnitions of coherence . . . . . . . . . . . . . . . . . . . . . . .

31

2.4.2

Coherence properties of HHG through phase matching in a hollow

2.4

waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

vii
2.5

Comparison to other tabletop SXR sources . . . . . . . . . . . . . . . .

41

2.6

Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

3 Theory of Coherent Diﬀractive Imaging

44

3.1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.2

Diﬀraction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.3

Sayre and oversampling . . . . . . . . . . . . . . . . . . . . . . . . . . .

47

3.4

Phase Retrieval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.5

Implementing XCDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

3.5.1

Experimental developments of XCDI . . . . . . . . . . . . . . . .

54

3.5.2

Other CDI techniques . . . . . . . . . . . . . . . . . . . . . . . .

55

System requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

3.6.1

Geometrical considerations . . . . . . . . . . . . . . . . . . . . .

58

3.6.2

Data processing and resolution requirements . . . . . . . . . . .

61

3.6.3

Scalability and limits to resolution set by radiation damage . . .

63

Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

3.6

3.7

4 First Tabletop Demonstration of Soft X-ray CDI
4.1

4.2

4.3

67

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

4.1.1

Phase matching and coherence optimization . . . . . . . . . . . .

69

4.1.2

Beam blocks and alignement . . . . . . . . . . . . . . . . . . . .

71

First XCDI Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72

4.2.1

Reﬂection mode results with 150 µm apertured Quantifoil . . . .

74

4.2.2

Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

First High Resolution Tabletop XCDI . . . . . . . . . . . . . . . . . . .

76

4.3.1

Results with “J” slit sample . . . . . . . . . . . . . . . . . . . . .

76

4.3.2

Phase retrieval reconstruction using GHIO

. . . . . . . . . . . .

79

4.3.3

Results with 15 µm apertured Quantifoil . . . . . . . . . . . . . .

80

viii
4.4

4.5

Pushing the resolution: High Numerical Aperture XCDI . . . . . . . . .

83

4.4.1

Experimental setup for high NA XCDI . . . . . . . . . . . . . . .

83

4.4.2

High NA results with 46.9 nm SXR laser. . . . . . . . . . . . .

87

4.4.3

Curvature correction for high NA diﬀraction . . . . . . . . . . .

87

4.4.4

High NA results with 29 nm HHG . . . . . . . . . . . . . . . . .

91

Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

5 Pushing the Limits: Tabletop Fourier Transform Holography

96

5.1

Toward shorter wavelength tabletop XCDI . . . . . . . . . . . . . . . . .

96

5.2

Theory of Fourier Transform Holography . . . . . . . . . . . . . . . . . .

97

5.2.1

Mathematical representation of FT holography . . . . . . . . . .

99

5.2.2

Multiple reference FT holography . . . . . . . . . . . . . . . . . . 100

5.3

5.4

First Demonstration of Tabletop X-ray FT Holography . . . . . . . . . . 101
5.3.1

Experimental setup for tabletop SXR FT holography . . . . . . . 101

5.3.2

Results of 29 nm tabletop FT holography . . . . . . . . . . . . . 103

5.3.3

High resolution FT holography with short exposures . . . . . . . 105

5.3.4

Eﬀects of high NA on FT holography . . . . . . . . . . . . . . . 107

5.3.5

Hybrid holography phase retrieval approach . . . . . . . . . . . . 107

Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Outlook: Future developments and applications
6.1

6.2

113

Future directions: a practical tabletop water window microscope . . . . 113
6.1.1

Recent developments at 13 nm . . . . . . . . . . . . . . . . . . . 114

6.1.2

Pathways to usable water window . . . . . . . . . . . . . . . . . 116

Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Bibliography

118

ix

FiguresFigure1.1

Description of three popular optical techniques to beat the diﬀraction
limit using ﬂuorescent proteins as markers from [1]. A) Description ofconfocal microscopy where a pinhole in the focal plane of the objective
microscope reduces out of focus light from ﬂuorescence (green wave) of
a green ﬂuorescent protein (GFP) illuminated by the blue wave. B) Description of the 4Pi technique whereby two counter propagating beams(red waves) interfere and produce a sub-diﬀraction limited spot. C) Description of STED, where a probe laser (blue wave) is used to excite alarge spot of many GFP molecules. A second laser (orange wave) and asingular optics is used to produce a phase vortex and donut mode thatquenches ﬂuorescence from all GFP except what is in the center of thebeam. Thus ﬂuorescence from only a very small spot is gathered. . . . .

1.2

5Electromagnetic spectrum with emphasis on the extreme ultraviolet (EUV)and soft x-ray imaging (SXR) portion. EUV/SXR microscopy has beenshown to be useful because of the numerous absorption edges that exist
in this region [6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

x
2.1

Schematic of a typical chirped-pulsed-ampliﬁer (CPA) system with cryocooling. The output of a titanium-sapphire oscilator is stretched in a
grating based stretcher and then ampliﬁed in a multipass ampliﬁer. Finally, the ampliﬁed pulse is telescoped to a larger diameter and thencompressed with a grating based compressor. This speciﬁc design is a
KMLabs DragonT M design [7, 8]. . . . . . . . . . . . . . . . . . . . . . .

2.2

15

Typical HHG spectrum of Ar gas as seen in Li et al. [9] Notice the steadydecrease in intensity in the ﬁrst few harmonics, followed by the plateau
region and eventually the cutoﬀ near the 33rd harmonic. The plateau
and cutoﬀ are typical for HHG spectra. Li et als experiment focused1064 nm light from a 10 Hz, 36 ps pulsed Nd:YAG laser into a 15 Torrgas jet. Also shown are various ionization energies for argon. . . . . . .

2.3

19

Pictoral representation of the three step model [10,11] of how the electron
is ionized and accelerated by the fundamental intense, femtosecond laser
beam and then is recombined with the atom, producing EUV and soft
x-ray light as shown in Kapteyn, Murnane, and Christov [12]. The cycle
shown represents one half cycle of the driving electric ﬁeld. The fringes
in the electron wave packet show its evolution throughout the trajectory.

2.4

20

Diﬀerent photoionization processes of extreme nonlinear optics dependent
upon the driving laser intensity: a) multiphoton absorption ionization,b) tunneling ionization, and c) barrier suppression ionization. The redline corresponds to the Coulomb potential suppressed by the laser ﬁeld.
Adapted from reference [13]. . . . . . . . . . . . . . . . . . . . . . . . . .

21

xi
2.5

One dimensional electron trajectories of freed electrons after tunnelingfor various initial laser ﬁeld phases. An initial phase of zero corresponds
to the electron being released at the peak of the driving laser ﬁeld. Noticethat for initial phases greater than 90 degrees, the electron never returns
to the parent ion (x = 0). These trajectories are given by the classicalequations of motion (Equations 2.7 and 2.8) [14]. . . . . . . . . . . . . .

2.6

23

Return kinetic energy of the electron in units of ponderomotive energy(Up ) as a function of the phase of the driving laser ﬁeld (initial phase). A
phase of zero degrees corresponds to the electron tunneling at the peakof the laser electric ﬁeld. The maximum value of return kinetic energy is
given by 17.2 degrees and gives a kinetic energy of 3.17Up [14]. . . . . .

2.7

25

Comparison of calculated HHG spectrum using a generalized form of theLewenstein model and solving the time dependent Schrödinger equation.
The Lewenstein model slightly overestimates the dipole strength due tothe neglect of the lateral spread of the electron wavefunction as it propagates, thus decreasing the probability of recombination [13]. . . . . . .

2.8

26

As the EUV signal from the harmonics and the laser signal propagate,
they will go in and out of phase. The length over which the two remainin phase is called the coherence length, Lc . If phase matching is achieved,
the signal will increase quadratically until it is limited by reabsorption of
the neutral gas [12]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.9

29

Schematic showing high harmonic generation in a hollow fused silica
waveguide. When the pressure of the gas inside the waveguide is tuned,the phase velocities between harmonics and driving laser are matchedand the harmonic ﬂux builds up coherently. The harmonic beam thenhas very small divergence (<1 mrad).

. . . . . . . . . . . . . . . . . . .

30

xii
2.10 Illustration of the two types of coherence: a) spatial coherence is well deﬁned phase travsversely to the wave propagation and can be quantiﬁed by
the transverse coherence length, ξt (from Attwood [6]), temporal coherence is well deﬁned phase along the direction of wave propagation and canbe quantiﬁed by a longitudinal coherence length, ξl (from Schlotter [15]).

32

2.11 Schematic of a Young’s double pinhole (wavefront interferometer) used
to test spatial coherence from [16]. . . . . . . . . . . . . . . . . . . . . .

36

2.12 Measurement of the coherence of phase matched high harmonic generation from a hollow wave guide reproduced from Bartels et al [17]. Measurements were made where the beam diameter was about 1 mm with two
pinholes (about 40 µm diameter each) at diﬀerent separations as shown.
Notice the very high fringe visibility out to almost 80% width of the beam. 37
2.13 Illustration of the phase properties and temporal coherence of a diverging
Gaussian beam (from Attwood [6]). . . . . . . . . . . . . . . . . . . . . .

38

2.14 (Top) Measured optimized 29 nm mode (log scale) on the CCD after two
multilayer mirror reﬂections. (Bottom) Gaussian ﬁt to horizontal line outof HHG mode showing excellent ﬁt with R2 = 0.9986. . . . . . . . . . .
3.1

Schematic showing the relevant parameters and variables for setting up
diﬀraction theory (from Peatross and Ware [18]). . . . . . . . . . . . . .

3.2

40

45

An illustration of the no-density region generated by oversampling thediﬀraction pattern. Here we surround the object with a no-density region
where there are no scatters. . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

49

Illustration of hybrid input-output (HIO) algorithm from Fienup et al[19]. In this dissertation a genetic algorithm version of HIO was used
known as guided HIO or GHIO [20]. . . . . . . . . . . . . . . . . . . . .

52

xiii
3.4

Illustration of novel coherent x-ray imaging techniques: a. Ptychography[21]; b. Keyhole Diﬀractive Imaging or curved wave imaging [22]. . . . .

57

3.5

Setup of relevant parameters in oversampled coherent diﬀraction imaging. 59

3.6

Study of radiation dose versus image resolution as a study of radiation
damage with electron and x-ray microscopy (from Marchesini et al [23]).

4.1

65

Experimental setup for lensless imaging using coherent high harmonic
beams at a wavelength of 29 nm. A single harmonic order is selected
and focused using a pair of normal-incidence multilayer mirrors. The
sample stage is positioned near the focus, where it scatters the soft-x-ray
beam onto a CCD. Inset, measured logarithmically scaled soft-x-ray beam
proﬁle that is a near-perfect gaussian T EM00 over 4 orders of magnitude(see section 2.4.2) [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Large (>1 mm) and medium (∼ 0.5 mm) beamblocks used to increase
dynamic range of the diﬀraction patterns [24]. . . . . . . . . . . . . . . .

4.3

71

Reﬂection geometry used with previous vacuum chamber for imaging ofRon the ﬁrst Mo/Si mirror. . . . . . . . . . . . . . . . . . . .
Quantifoil

4.4

69

73

R
“Multi-A” carbon foil [25]. The holey carbon
SEM image of Quantifoil

ﬁlm is about 20 nm thick and rests on a 50 µm picth copper grid with
about 10 µm wide gride lines. . . . . . . . . . . . . . . . . . . . . . . . .
4.5

73

A portion of the high spatial frequency data for the reﬂection geometry ona linear scale (top), the reconstructed image showing some basic features
but poor resolution of about 2 µm (bottom) [24]. . . . . . . . . . . . . .

4.6

75

(a) Optical image of J-slit. (b) Oversampled diﬀraction pattern from thissample where the diﬀraction at the edge of the image corresponds to a
momentum transfer of 0.018 nm−1 . (c) Magnitude of the reconstructed
lensless image. [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

xiv
4.7

Logarithmic raw diﬀraction patterns from the “J” slit sample: (top) 1
minute exposure with no beam block, (left) 10 minute exposure with a
200 µm beam block, (right) 120 minute exposure with a 3 mm beam
block. The ﬁgures represent the full area of the camera and have notbeen cropped. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

78

(a) SEM image of a masked carbon TEM ﬁlm; (b) oversampled softx-ray diﬀraction pattern where the diﬀraction at the edge of the image
corresponds to a momentum transfer of 0.059 nm−1 ; and (c) magnitude of
the reconstructed lensless image. The correspondence in size, number andposition of holes, and aspect ratios between the EUV image and the SEM
image can be seen. One pixel in the reconstructed image corresponds to
107 nm. The inset in (c) show a line-scan taken along the direction notedwith the green solid line, demonstrating a spatial resolution of 214 nm [4]. 81

4.9

Scanning Electron Microscope images of 40 nm thick holey carbon ﬁlm
placed approximately 56 µm above a 15 µm aperture: (a) SEM stage at
0 degrees tilt, (b) SEM stage at -1.8 degrees tilt, (c) SEM stage at +1.8
degrees tilt, (d) SEM image at +8 degrees tilt. . . . . . . . . . . . . . .

82

4.10 Setup of the tabletop soft x-ray diﬀraction microscope. A coherent soft
x-ray beam is produced by high harmonic generation or by spatially ﬁltering the emission from a capillary discharge laser in an Ar-ﬁlled hollow
waveguide. The beam is gently focused by a multilayer mirror pair onto
the sample, and the diﬀraction pattern is collected on an x-ray CCD.
A moveable beam block allows for brighter portions of the diﬀraction
pattern to be blocked to avoid saturating the CCD when acquiring thehighest angle diﬀracted light [5]. . . . . . . . . . . . . . . . . . . . . . .

84

xv
4.11 Lensless imaging using coherent soft x-ray laser beams at 47 nm. (a)Reﬂection image of the waving stick ﬁgure sample (scale bar = 1 micron)using an SEM. (b) Coherent soft x-ray diﬀraction pattern (in transmission) after curvature correction (maximum momentum transfer of 0.166
nm−1 at the edge of the CCD). (c) Reconstructed image with curvaturecorrection. (d) Line-out of the image along the legs (shown in the inset),
verifying a resolution of 71 nm [5]. . . . . . . . . . . . . . . . . . . . . .

88

4.12 Curvature correction in high numerical aperture lensless imaging. (a), (d)Coherent soft x-ray diﬀraction pattern at 47 nm wavelength (in transmission) before and after curvature correction (maximum momentum transfer of 0.166 nm−1 at the edge of the CCD). (b), (e) Magniﬁed diﬀraction
pattern at high diﬀraction angles showing the radial blurring near theupper right edge in (b), which is absent in (e). (c), (f) Reconstructed
image of waving stick ﬁgure without and with curvature correction. The
lower width and increased sharpness of the curvature-corrected stick-girlimage can be seen in the lineouts shown in the insets [5]. . . . . . . . . .

90

4.13 Lensless imaging using coherent high harmonic beams at 29 nm. (a)Reﬂection image of the sample (scale bar = 1 micron) using an SEM. (b)Coherent soft x-ray diﬀraction pattern (in transmission) after curvaturecorrection (maximum momentum transfer of 0.134 nm−1 at the edge of
the CCD). (c) Reconstructed image (d) lineout of the image along thelegs (shown in the inset), demonstrating a resolution of 94 nm [5]. . . .

92

xvi
4.14 A previous demonstration of lensless imaging using coherent high harmonic beams at 29 nm. (a) Reﬂection image of the sample (scale bar =1 micron) using an SEM. (b) Coherent soft x-ray diﬀraction pattern (intransmission) after curvature correction (maximum momentum transfer
of up to 0.134 nm−1 at the edge of the CCD). (c) Reconstructed image (d)lineout of the image along the legs (shown in the inset), demonstrating a
resolution of 92.5 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1

94

Schematic of Fourier relationship and Fourier Transform holography in
1 dimension (from Schlotter [15]). Column one shows diﬀerent transmission functions (g(x)). Column two shows |G|2 , the magnitude squaredof the Fourier transform (far ﬁeld diﬀracted intensity) of g(x) in column

one. Column three shows the autocorrelation F |G|2 of the diﬀracted
wave. Row (a) shows the transmission, diﬀraction, and autocorrelationof a single slit. Row (b) shows the same information, except for a double
slit. Notice now that in the autocorrelation there is a self-correlation
peak (center) and cross-correlations peaks between the two slits (sidepeaks). Row (c) shows a sample (trapezoid) and delta-function-like reference point in transmission, diﬀraction, and autocorrelation. Now the
autocorrelation cross-correlation peaks produce an image of the trapezoid
sampel and its complex conjugate. This can be seen expecially in Row(d) where the trapezoid sample is complex. . . . . . . . . . . . . . . . .

98

xvii
5.2

Schematic of multiple reference Fourier transform holography with atabletop soft x-ray source. A coherent soft x-ray beam with λ = 29
nm from a tabletop high harmonic source illuminates a test pattern surrounded by ﬁve reference holes. The scattered soft x-ray light interferesand is recorded on a CCD camera in the far ﬁeld as a hologram (logscale). Due to the high scattering angles recorded in the hologram, curvature correction is necessary. The spatial autocorrelation of the object
can be retrieved by taking the squared magnitude of the Fourier transform of the hologram. A single sub-image from the hologram is shown
demonstrating 89 nm resolution (Photo credit Greg Kuebler) [26]. . . . 102

5.3

Logarithmic mode of a 29 nm Gaussian HHG beam from a 5 cm waveguidewith about 0.7 nW of power in a bandwidth of about 250. . . . . . . . . 102

5.4(a) Curvature corrected logarithmic hologram from composite 4800 s of
exposure (8 separate 600 s exposures with maximum momentum transfer
of Qmax = ±0.211 nm−1 ). The inset shows the detail of the center ofthe diﬀraction pattern. (b) Magnitude squared of the autocorrelation
reconstruction of the sample - the variations in sub-image intensity are
most likely due to variations in reference hole transmitted intensity. (c) A
single sub-image reconstruction with a sample line cut (blue dotted line).
(d) Plot of the line-out demonstrating 89 nm resolution with a signal tonoise of 73 [26]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

xviii5.5(a) Hologram recorded from a 30 s exposure (maximum momentum transfer of Qmax = ±0.045 nm−1 ). (b) Sub-image from the magnitude squaredof the autocorrelation for a 30 s exposure showing a lineout (blue dashedline). (c) Lineout of sub-image showing 125 nm resolution and a signal
to noise ratio of 12.7. (d) Average of 20 lineout resolution measurements(4 per sub-image) per time step as a function of exposure time with error
bars, showing the standard deviation of the resolution. Each diﬀraction
pattern array was 2048 x 2048 pixels. The array was set to zero outsideof the region where the signal from the diﬀraction pattern dropped below
the noise. This reduced the noise in the reconstructed autocorrelation [26].106

5.6(a) Log of the non-curvature corrected hologram; (b) sub-image from autocorrelation without curvature correction showing image distortion; (c)log of the curvature corrected hologram; (d) sub-image from autocorrelation with curvature correction showing improved image quality. . . . . 108

5.7(a) Scanning electron microscope image of the ﬁve-reference-hole sample.
(b) Log of the curvature corrected hologram used in the iterative phaseretrieval (maximum momentum transfer of Qmax = ±0.211 nm−1 ). (c)Reconstructed image of the sample using the GHIO phase retrieval algorithm to reﬁne the resolution to 53 nm, as demonstrated in the lineout(blue dotted line) shown in (d) [26]. . . . . . . . . . . . . . . . . . . . . 111

6.1

Current and future region where HHG may be a useful tabletop sourceof x-rays (adapted from Attwood [6] courtesy of M. Murnane). . . . . . 114

xix
6.2

Preliminary results of 13 nm Fourier transform holography: (a) SEM
image of FT holography sample with seven reference holes (b) 13.5 nm
hologram from 40 minute exposure (log scale) (c) autocorrelation reconstruction of the sample (d) line cut through one sub-image showing 184
nm resolution [27]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Chapter 1

Introduction to Tabletop Coherent Diﬀractive X-ray Imaging

This thesis will present the ﬁrst results of a high resolution (<100 nm) diﬀractive
x-ray microscope using a tabletop, coherent x-ray source. Originally developed at large,
3rd generation synchrotron sources, this technique for nano-scale imaging is elegant
when applied to more limited ﬂux, tabletop x-ray sources because of its stability, robustness, and ability to produce high resolution images without the need of diﬃcult tomanufacture and lossy imaging optics. Additionally, as will be shown, this technique isapplicable in high numerical aperture schemes meaning that the resolution is ultimately
limited not by the point spread function of the imaging optic, but by source brightness,
wavelength, and radiation damage of the sample [23]. The source used for the majorityof the work presented in this thesis is phase matched high harmonic generation of an
intense femtosecond laser. This source is very attractive because of its complete spatial
coherence and high temporal coherence, compactness, stability, and ability to produce
sub-femtosecond pulses of x-rays (< 10−15 s). However, ﬁrst this chapter will review
the basic principles of microscopy as well as compare several microscopy modalities inorder to provide a framework against which to present the results.

1.1

Imaging at the nanoscale
Microscopy is a critical enabling tool for understanding the nano-world. Since

Hooke ﬁrst applied the microscope to viewing the cellulose structure in cork [28], mi-2croscopy has had an especially important impact on biology. Standard visible lightmicroscopes can image living cells with a resolution as low as 200 nm [29], especiallywith the use of ﬂuorescent markers. However, this resolution is fundamentally limitedby the wavelength of visible/near-UV light. Increasing the resolution in imaging has
driven technology and science in areas as disparate as biotechnology, semiconductor
fabrication, materials science, and defense.
Traditionally, the resolution of an imaging system is described by the so called
Rayleigh Criterion [6, 29]. This limit is given by
r=

0.61λ
NA

(1.1)

where r is the transverse or spatial resolution, λ is the illuminating wavelength, N A
is the numerical aperture of the imaging system, and the prefactor (0.61) comes from
Rayleigh’s criterion for resolving two points imaged with system with circular apertures.
Abbe put forth a similar resolution criterion in his seminal work on microscope design,where the prefactor is 0.5 [1,30,31]. To further increase resolution, two major strategies
have been taken: increasing the numerical aperture and decreasing the wavelength.
Increasing the numerical aperture of a microscope is a common practice, andhigh quality, high numerical aperture lenses have found numerous applications. The
numerical aperture (N A) of a lens is given by
N A = n sin θ(1.2)

where n is the index of the refraction of the material between the sample and objective
lens and θ is the half angle of the light collected by the objective lens [32]. High quality,
short focal length objective lenses are available for optical microscopes with a practicalcollection angle of up to about 72◦ (N A = sin(72◦ ) ∼= 0.95). With the developmentof immersion microscopy–where the air gap between the sample and objective lensis replaced with a higher index of refraction material–numerical apertures of 1.4 are3practical with high quality immersion oil (n = 1.51) [29, 33, 34]. Immersion optics
have found numerous applications in biological imaging and semiconductor lithography
where they have allowed sub-wavelength information to be extracted (as in microscopy)
or imprinted (as in lithography) [1, 2, 35]. However, even with these techniques theresolution of optical microscopes is limited to about 200 nm [36]. In the next section
we will discuss diﬀerent types of imaging techniques that work to beat this opticalresolution barrier and compare their advantages and disadvantages.

1.2

Comparison of diﬀerent microscopies
This section will compare and contrast diﬀerent imaging techniques that strive topush imaging from the microscopic regime to the nanoscopic regime. The discussion will
focus on far-ﬁeld techniques that can be used to extract three dimensional informationof thick samples as opposed to near ﬁeld or scanning probe techniques. These near ﬁeld
and scanning probe techniques, such as atomic force microscopy or near ﬁeld optical
microscopy can achieve very high resolution, even atomic resolution in some cases, but
are limited to studies of the surfaces of samples [37]1.2.1

Optical Microscopy

Optical microscopies have been the work horse of microscopy for centuries. Dueto their compact and relatively inexpensive nature, optical microscopes are ubiquitousin elementary school class rooms and biotechnology ﬁrms alike. However, until thelast decade the resolving power of optical microscopes has been limited by a diﬀraction
limited spot of the optical wavelength (∼ λ/2) [1, 31].
One of the main advances in optical biological microscopies has been the useof green ﬂuorescent proteins (GFP) as markers or labels that can be introduced into
biological systems. This fact was recognized by the awarding of the 2008 Nobel prize in
Chemistry to Osamu Shimomura, Martin Chalﬁe, and Roger Y. Tsien for “the discovery

4and development of the green ﬂuorescent protein” [38]. These proteins typically absorb
light in the blue region of the spectrum and ﬂuoresce at longer wavelengths in thegreen portion. Several techniques utilize these markers to increase or isolate the signal
from these proteins through the means of spectral or spatial ﬁltering. Several of thesetechniques are summarized in Figure 1.1 from the excellent review by Stefan Hell entitled“Far-ﬁeld optical nanoscopy” [1]. The ﬁrst pane of Figure 1.1 demonstrates a very
common optical technique known as confocal microscopy. Confocal microscopy places
a pinhole in the focal plane of the objective lens and thus rejects out of focus light fromthe sample. Additionally, confocal microscopy can be used in a scanning modality wherea single, diﬀraction limited spot can be scanned around a sample. Thus, it is possible
to illuminate a single GFP molecule and locate it to ∼ 200 nm. While an increase inimage quality ensues, there is no true enhancement of resolution unless the pinhole size
is reduced which sacriﬁces image intensity and acquisition speed. The next techniquedescribed is 4Pi, where two counter propagating beams illuminate a sample and set up
a standing wave [1, 39]. A three dimensional interference pattern is produced wherea sub-wavelength spot exists at the focus of the system. Thus in a similar fashion as
confocal microscopy, a single small volume of the sample can be interrogated. However,
4Pi has the drawback of having side lobes of the interference pattern next to the focal
spot that must be accounted for.
A similar technique to 4Pi that holds great promise is structured illumination (SI)microscopy as demonstrated by Gustafsson [40]. In SI, pupil ﬁll ﬁltering or apodization
is used to produce a sinusoidally varying illumination pattern where the period of thesinusoidal illumination can be as small as half the wavelength of light. Multiple imagesof a sample illuminated with such a pattern are taken while the illumination pattern isrotated and shifted by λ/2. These images, which are Moiré patterns, are then processedin a computer and an enhancement of a factor of 2 in resolution to about 100 nm
can be achieved. A nonlinear version of this techniques that involves saturation of

5

Figure 1.1: Description of three popular optical techniques to beat the diﬀraction limit
using ﬂuorescent proteins as markers from [1]. A) Description of confocal microscopy
where a pinhole in the focal plane of the objective microscope reduces out of focus
light from ﬂuorescence (green wave) of a green ﬂuorescent protein (GFP) illuminatedby the blue wave. B) Description of the 4Pi technique whereby two counter propagating
beams (red waves) interfere and produce a sub-diﬀraction limited spot. C) Descriptionof STED, where a probe laser (blue wave) is used to excite a large spot of many GFP
molecules. A second laser (orange wave) and a singular optics is used to produce aphase vortex and donut mode that quenches ﬂuorescence from all GFP except what isin the center of the beam. Thus ﬂuorescence from only a very small spot is gathered.

6
the GFP has also been demonstrated with ∼ 50 nm resolution [3]. One advantage ofthis technique is that it is a full-ﬁeld technique as opposed to a scanning techniquesuch as confocal scanning microscopy. The drawback to this technique is that it uses
a computer reconstruction algorithm, that data collection can take a long time, andthat in the case of the nonlinear technique high intensities are needed that can damage
biological samples.
One ﬁnal set of optical imaging techniques that perhaps has the most promise is a
set that uses the ability of the GFP to be placed in a so called dark state where the GFP
will not ﬂuoresce [1]. This can be achieved by a variety of means, but most modalities
use saturation of the protein as means to increase resolution by essentially turning oﬀ
proteins you do not want to image. One of the most popular techniques in this category
is stimulated emission depletion microscopy (STED). STED is depicted in panel (C)of Figure 1.1. As shown in the ﬁgure, this technique uses two illuminating beams.
The ﬁrst beam illuminats the GFP and excites it. The second beam passes through asingular optic which produces a phase vortex donut beam which has a minimum at theaxial center. This beam saturates all the GFP illuminated by the ﬁrst beam, except
in the central null. In this way, ﬂuorescence from a sub-diﬀraction limited spot canbe collected. Many variations on technique have been introduced both in scanningand full ﬁeld modalities and have produced images with resolution down to some 10’sof nanometers when combined with time resolved readout and computer algorithms
for ﬁtting of point spread functions [1]. One ﬁnal imaging technique that is worth
mentioning is multiphoton microscopy where the second or third harmonic of an intense
ultrafast laser pulse is detected. This technique utilizes the fact that the second order or
third order nonlinear cross section of materials is smaller than the ﬁrst order absorption
cross section. However typically this means using red or near IR pulses which moves adiﬀraction limited spot size more towards 400 nm rather than 200 nm.
These modern optical techniques have truly broken the 200 nm barrier, but they7
have some drawbacks. First, the optical systems have become much more complicatedand expensive requiring $100,000 to upwards of $1 million dollars for systems that can
produce three dimensional images with 50 nm resolution or less. Much of this expense
comes from high quality, closed loop piezoelectric stages that can provide nanometerstep resolution and nanometer stability. Additionally, mechanical stability of less thanthe optical resolution is required for these techniques. Finally, all of these techniques
except for multi photon microscopy require labeling with GFP. While this technique has
become a gold standard in biological imaging, it has not found as many implementations
in materials science imaging.

1.2.2

Electron Microscopy
After optical microscopy, the next most commonly used microscopic technique isthe electron microscope. Since Ernst Ruska built the ﬁrst electron microscope in 1931,
it has found countless applications in biology, physical science, and material science andengineering. Due to the large impact of his invention, Ruska was awarded the 1986
Nobel prize in Physics [38]. The electron microscope uses the fact that high energy
electrons have an extremely short wavelength. The quantum mechanical description of
the electron’s de Broglie wavelength is given by
λ = h/p = √h
2m0 eU(1.3)

where h is Planck’s constant, m0 is the electron’s mass, e the electrons charge, and U theaccelerating electric potential. Most electron microscopes have accelerating potentials
between 10kV and some few 100kV, making the electrons velocity relativistic. Hencethe relativistic wavelength of the electron wavelength will be1h
λ = h/p = √2m0 eU 1 + eU

(1.4)

2m0 c2where c is the speed of light [41]. Thus, an electron that is accelerated across a 30
kV potential will have a wavelength of 0.012 nm (0.12 Å). Thus we see from equation

8
1.1 that even with modest numerical aperture electron microscopes atomic resolution is
feasible and it has been demonstrated [41].
Electron microscopes can be classiﬁed into two basic classiﬁcations: scanning electron microscopes (SEM) and transmission electron microscopes (TEM) [41]. Scanningelectron microscopes typically raster scan a focused electron spot over a sample and collect the secondary electrons released or the backscattered electrons. Therefore, SEMs
are very surface sensitive and often require labeling or coating with materials that will
increase the contrast of the electron intensity image. Transmission electron microscopes
also typically scan a focused spot of electrons over the sample, but instead detects thetransmitted electron intensity. Both SEM and TEM can be used for elemental speciﬁcstudies by spectrally resolving the energy of the transmitted or secondary electrons andcomparing these spectra to known binding energies. Thus, electron microscopes havebecome a very powerful tool for nanoscale imagine with elemental speciﬁcity.
While electron microscopes have found broad application, there are several drawbacks that should be discussed here. First, the free mean path of high energy electronsin air is very short requiring electron microscopes to work in vacuum [1, 41, 42]. Thisadded level of complexity may require many steps of sample preparation such as freeze
drying, cryo-ﬁxing, or chemical ﬁxing, especially in biological or wet materials. Second,
electrons are charged particles and interact very strongly with matter, giving them avery short mean free path of only a few hundreds of nanometers (<500 nm according
to Spence [41]). Thus when trying to image thick samples such as a whole cell (fewµm diameter) or when trying to produce three dimensional images, electron microscopy
often requires sectioning of chemically or cryo-ﬁxed samples possibly introducing structural damage or artifacts. Additionally, the high energies of the electrons used meansthat they are ionizing radiation and cause sample damage [23, 41]. Finally, when hightemporal resolution is required of dynamical processes, electron microscopy is limited in
the shortness of the pulse durations it can produce to about 100 fs because of Coulombic

9repulsion of the charged particles. In addition to electron microscopes, a gamut of otherparticle microscopes have come on line recently such as focused ion beam microscopes
and neutral atom (typically He) microscopes. These microscopes are similar enough toelectron microscopes in the function and application that they will not be discussed,
but references [43, 44] provides more information on these devices.

1.2.3

X-ray Microscopy

X-ray microscopy provides an alternative approach to electron microscopy wherehigh resolution imaging of thick samples is required. By virtue of its short wavelength,
x-ray microscopy has made it possible to obtain nanometer-scale resolution images ofwhole unstained cells, magnetic permalloy wires, internal structures in nanocrystals,
and magnetic domains in thin ﬁlm samples [45–48]. Extreme ultraviolet and soft x-ray(EUV/SXR) radiation is particularly useful because it can penetrate relatively thicksamples in comparison to electron microscopy, thus opening up the possibility for threedimensional tomography of nano-objects. This region covers the electromagnetic spectrum from about 1 nm to 50 nm wavelength, or alternatively, from about 30 eV to 8 keVof photon energy as shown in Figure 1.2 from Attwood [6]. A very useful relationshipbetween the photon energy, E in electron-volts (eV), and wavelength, λ in nm, is givenbyE[eV ] =1240[nm ∗ eV ]
.
λ[nm]

(1.5)

Furthermore, as is shown in Figure 1.2, numerous core-level absorption edges and widely
varying elemental absorption cross sections provide excellent inherent image contrast in
the soft x-ray region of the spectrum. Two such examples are biological imaging around
400 eV or magnetic contrast imaging through x-ray circular dichroism in magnetic
nanostructures around 800 eV [6, 45, 48]. Since EUV and SXR radiation is so strongly
absorbed these wavelengths do not propagate readily through air, thus requiring a vacuum environment. However, at higher photon energies (5-10 keV) the mean free path

10

Figure 1.2: Electromagnetic spectrum with emphasis on the extreme ultraviolet (EUV)and soft x-ray imaging (SXR) portion. EUV/SXR microscopy has been shown to be
useful because of the numerous absorption edges that exist in this region [6].

of x-rays in air is suﬃciently long such that samples can be studied at atmosphere [6].
At those photon energies the dominant imaging process is phase contrast rather thanabsorption, so high contrast imaging becomes diﬃcult. Additionally, radiation damage
is a factor since EUV, SXR, and x-ray radiation are ionizing sources. These eﬀects will
be discussed later in chapter 3.
Two challenges have limited the broader application of soft x-ray microscopy:
the diﬃcult fabrication of x-ray imaging optics and the limited number of suitable xray sources available worldwide. Since soft x-rays are strongly absorbed by matter,
diﬀractive or reﬂective optics (rather than refractive optics) must be used. The highest
spatial resolution x-ray microscope demonstrated to date (∼ 10 nm) used zone-platecondenser and focusing lenses with feature sizes as small as 15 nm [49]. These optics arethus challenging to fabricate and limited in availability. Moreover, the bright, tunable
x-ray sources required for soft x-ray microscopy have generally only been available at
a limited number of large-scale synchrotron facilities. Additional comparisons between
zone plate imaging verses coherent imaging are discussed thoroughly in Ariel Paul’s
dissertation, chapter 5 [24].
However, recently, signiﬁcant progress has been made that promises to open

11
up soft x-ray microscopy for more widespread use. First, new diﬀractive microscopy
techniques using coherent x-ray beams have made possible nanometer-scale resolution
imaging by essentially replacing the imaging optics in a microscope with a computer
reconstruction algorithm [20, 22, 47, 48, 50–53]. Second, advances in coherent soft x-ray
generation using tabletop ultrafast lasers have made it possible to implement practical
tabletop soft x-ray sources with suﬃcient coherent ﬂux for soft x-ray imaging [17,54–58].

1.3

Overview of work to be discussed
This dissertation will present recent work demonstrating coherent diﬀractive mi-

croscopy with an oversampling iterative phase retrieval algorithm (also called lenslessimaging) using coherent, tabletop, soft x-ray sources. Chapter 2 will present the theory behind high harmonic generation (HHG) as a tabletop coherent soft x-ray source.
Chapter 3 will present the important theory behind diﬀraction, sampling, and lensless or
coherent x-ray imaging. Chapter 4 will discuss the ﬁrst demonstration of high resolution(<100 nm) tabletop coherent x-ray imaging. The experimental setup and initial lower
resolution results will be discussed as well as improvements necessary for higher resolution imaging. Image resolutions of 94 nm were obtained using 29 nm light produced byhigh-order harmonic upconversion of an ultrafast laser, while resolutions of 71 nm wereobtained using a 46.9 nm soft x-ray laser [4, 5]. Much of this early work was done with
Ariel Paul and additional details, especially on the experimental setup, can be foundin his dissertation [24]. Next, another technique, Fourier Transform holography, will bedemonstrated in chapter 5 with near 50 nm resolution using the same coherent tabletop
x-ray source. This holographic technique has important applications for lower ﬂux imaging with shorter wavelength sources. Finally, chapter 6 will present pathways towards
shorter wavelengths and higher resolution as well as discussing important applicationsof a tabletop coherent x-ray microscope.
These techniques show great promise for further rapid progress. High harmonic

12
generation, for example, is attractive as a soft x-ray light source because it generates
fully coherent beams with pulse durations of <10 fs using an extreme nonlinear opticalupconversion technique. Moreover, the ultrafast laser systems required to implement
a high harmonic light source are widespread and readily available. Currently, highharmonic beams with suﬃcient ﬂux for diﬀractive or zone plate microscopy can begenerated at wavelengths longer than 13 nm, where the nonlinear upconversion process
can be phase matched [17, 55]. However, this source can also produce reduced ﬂux athigher photon energies such as in the biologically important water window region [59,60]
and even as high as 1 keV [61]. It has also been shown that high harmonic yields atthese higher photon energies can be enhanced several orders of magnitude through theapplication of quasi-phase matching techniques [62–65] or through phase matching withlonger wavelength lasers [66].

Chapter 2

High Harmonic Generation as a Bright, Coherent Source of EUV andSXR Radiation

2.1

Introduction to high harmonic generation
The development of a new generation of ultrafast lasers during the 1990’s has

had a signiﬁcant impact on science and technology. These lasers have made accessible the regimes of extremely short time scales (femtosecond to attosecond dynamics),
high bandwidth (hundreds of gigahertz), and very high ﬁeld intensities (greater than apetawatt/cm2 ) [13, 67]. The high ﬁeld intensities achievable with ultrafast lasers havemade possible entirely new types of nonlinear optical processes, and this in turn has
spawned new techniques and technologies. One of these techniques, high harmonic generation (HHG), converts optical radiation at photon energies of ∼ 1.5 eV to extreme
ultraviolet (EUV) or soft x-ray (SXR) radiation with photon energies up to several keV(∼ 1 nm wavelength) [12, 61].
This chapter presents an introduction to high harmonic generation (HHG) from
intense ultrafast lasers as a tabletop source of coherent x-rays. A brief history of thedevelopment of HHG is reviewed, and the semiclassical three-step model that describes
this highly nonlinear optical eﬀect is discussed. Various numerical models are considered that describe the quantum mechanical aspects of HHG. Finally, it will be shownhow eﬃcient HHG can be obtained through phase matching techniques and how phasematching in hollow waveguides produces highly coherent soft x-ray beams.

14

2.2

Ultrafast laser technology
As we will see in the following discussion, a necessary forerunner of tabletop highharmonic generation is the ability to produce very strong short laser pulses. One way
to achieve this is through very intense optical laser pulses. Soon after the laser was ﬁrst
demponstrated in 1960 by Maiman, it was realized that this new device was a very useful
tool for creating high peak ﬁelds [68]. Within two years the ﬁeld of nonlinear optics wasborn when Franken and colleagues demonstrated optical harmonics by tightly focusing alaser into a quartz crystal [69]. Laser technology advanced rapidly in the next 40 years,
and with that advancement came ever increasing peak intensity. However, until shortpulsed, broadband lasers were developed the ionization of atoms with laser ﬁelds was
not attainable. Speciﬁcally, the development of ultrafast lasers was needed that produce
pulses with durations on the order of picoseconds (ps or 10−12 s) or femtoseconds (fsor 10−15 s). Here we brieﬂy review the developments that are pertinent to HHG asoutlined in Backus et al [67].
One of the early pioneering methods developed in ultrafast laser science wasmodelocking a laser. When a laser is modelocked, it ceases to emit a steady beam ofcoherent radiation, but instead emits a train of short pulses. Throughout the 1980’s,
modelocked Neodymium-glass (Nd:glass) lasers or synchronously pumped and modelocked dye lasers were common. These lasers were very unstable and would often dropout of the modelocked state. Also, these laser systems did not have enough bandwidth
to produce very short duration pulses, thus limiting the duration of available pulsed
lasers to the 100 fs to 1 ps range. Even though temperamental at times, these lasers
were capable of high average power (10s of Watts). However, since the pulses were still
relatively long (0.1-100 ps), peak laser intensities necessary for HHG (approximately
1013 W/cm2 ) were limited to a few groups of scientists with state-of-the-art lasers. In
the late 1980’s and early 1990’s, many developments in ultrafast lasers brought great

15

Figure 2.1: Schematic of a typical chirped-pulsed-ampliﬁer (CPA) system with cryocooling. The output of a titanium-sapphire oscilator is stretched in a grating based
stretcher and then ampliﬁed in a multipass ampliﬁer. Finally, the ampliﬁed pulse is
telescoped to a larger diameter and then compressed with a grating based compressor.
This speciﬁc design is a KMLabs DragonT M design [7, 8].

progress in HHG [67]. Older style lasers such as the Nd:glass were improved and newlasers were introduced such as the titanium-doped sapphire laser (Ti:sapph). The modelocked Ti:sapph laser was introduced by Sibbet and his group in 1990 [70]. Ti:sapph isa very desirable laser material because it has a very broadband gain from approximately
700 nm to 1100 nm, meaning that ideally all of these wavelengths can be ampliﬁed in
the crystal simultaneously [67]. Additionally, new laser ampliﬁer systems were developed such as chirped-pulse ampliﬁcation or CPA. CPA uses two components known asa stretcher and compressor that spread out the pulse in time before ampliﬁcation andthen recompress it after ampliﬁcation. This is necessary because the large pulse energies
of approximately 1 mJ can easily damage the optics in the ampliﬁer if the pulses are too
short. After compression, the pulse duration can be as short as 5 fs with 1 mJ of energy
at a repetition rate of several thousand Hz [67] These newer femtosecond lasers enabledresearchers to obtain peak intensities that were previously unattainable with tabletop
sources. In a 100 µm focus of a standard CPA ultrafast laser beam, peak intensities of1018 W/cm2 are easily achievable [67]. The peak electric ﬁeld strength of these laser
pulses can be many times higher than 109 V/cm, much higher than the binding ﬁeld of16valence electrons of atoms [13].

2.3

Theory of High Harmonic Generation
High-order harmonic generation (HHG) of ultrashort-pulse lasers has emergedin the past decade as an attractive table-top source of EUV/SXR light with uniqueproperties: HHG light is both coherent and can be used to generate light pulses evenshorter than 100 as (attosecond is 10−18 s) in duration [12, 71–73]. High harmonic
generation occurs when intense ultrashort laser pulses interact with atoms. Noble gases
are typically used as targets. Noble gases are self restoring after being ionized (unlike
crystals or solids) and have the highest ionization potential of all atoms. Their highionization potentials allow the noble gas atoms’ electrons to remain bound at higher
laser intensities. Other compact sources of EUV radiation that are currently being used
and further developed include laser produced plasmas and soft x-ray lasers [74, 75]. A
good review of these sources is given in Attwood chapters 6-7 [6].
When light is incident on a material, the electric ﬁeld polarizes the atoms or
molecules of the material. This polarization in turn aﬀects the propagation of thelight in a way that is described by Maxwell’s equations [76]. The induced polarization
reradiates, both at the same frequency as the incident light, and in some cases, at highharmonics. For low intensities (I  1011 W/cm2 ), the induced polarization is linearly
dependent on the electric ﬁeld, E, and is given by
P = 0 χE(2.1)

where 0 is the permittivity of free space and χ is the electric susceptibility of thematerial [13]. If the polarization is oscillating, then the oscillating dipoles will reradiate
light at the frequency of their oscillation as described by electromagnetic theory. As theincident intensity of the light is increased, the induced polarization reaches a nonlinear

17
regime where the induced polarization can be described by
P = 0 χE + a1 E 2 + a2 E 3 + ...

(2.2)

where the ith -order term in this expansion is P = ai E i+1 and the nonlinear coeﬃcientai is given byai = 0 χiijk(2.3)

χiijk where is the ith -order electric susceptibility of the material. Equations 2.2 and 2.3
describe a perturbative expansion of the polarization that is valid as long as the higher
order terms decrease monotonically. For a sinusoidal driving ﬁeld E(t) = E0 cos(ωt), theinduced polarization includes higher order terms with frequencies of 2ω, 3ω, etc. These
constitute second, third, and higher order harmonic generation. In these processes,
multiple fundamental laser photons are absorbed and reemitted with the combinedenergy qh̄ω in a single harmonic photon.
As mentioned earlier, if the intensity of the peak driving laser is increased evenfurther to approximately 1013 to 1015 W/cm2 , the laser electric ﬁeld will begin to approach the same strength as the Coulomb ﬁeld binding the outermost electron of an
atom [9]. In this range, the perturbation approach described above will not accurately
describe the induced polarization because the electron will ionize, and the laser target
is converted into a plasma. At these intensities, equation 2.2 does not converge, so
traditional nonlinear optical theory does not suﬃce. This nonperturbative regime is
known as the strong-ﬁeld regime. The physics of multiphoton absorption and emission
processes such as second harmonic generation and two photon absorbtion have corresponding higher-order processes in the strong ﬁeld regime–HHG and above thresholdionization (ATI) [13] The later is a multiphoton ionization process in which more fundamental ﬁeld photons are absorbed by an electron than are needed to exceed the binding
Coulomb potential. In both HHG and ATI, the energy spectrum of either the photons or
the photoelectrons produced in the process has a structure consisting of discrete peaks

18that approximately correspond to multiples of the fundamental laser frequency.

2.3.1

Discovery and three step model

The transition from the perturbative regime to the strong ﬁeld regime can be seenin the photoelectron and photon energy spectrum in ATI and HHG. In the perturbative
regime, the intensities of successive peaks decrease monotonically so that the expansion
in equation 2.2 converges. However, as the driving laser intensity is increased, more
peaks appear, and all peaks increase in strength until a plateau of peaks with nearly
equal intensity becomes apparent. These plateaus are characteristic of HHG spectra
and were ﬁrst observed in ultrafast laser studies of noble gases in the late 1980’s [9, 77].
Figure 2.2 shows a typical HHG spectrum from Argon [77]. Notice the monotonic
decrease of the ﬁrst three peaks to the plateau region (harmonic order 7-29) followedby an abrupt cutoﬀ (about the 33rd harmonic). It is also interesting to note that HHG
only produces odd harmonics. This is due to symmetry considerations in the nonlinear
optical process – the generation of even harmonics requires a medium without inversion
symmetry, and of course an atomic gas cannot have any such asymmetry.
Over the last several years, HHG has progressed from a novel nonlinear process
to a robust table-top source of high energy photons. A key part of this progress wasa development of the theoretical understanding of the process. The next section ofthis paper is dedicated to reviewing the pertinent theoretical models that have aided inunderstanding HHG.
In the early 1990’s, the eﬀorts of several groups, most notably Krause, Schafer,and Kulander [10] and Corkum [11], produced a semiclassical description of HHG known
colloquially as the three step model. Several adaptations of this model have been
demonstrated [13], but all have the same underlying features. These features are shown
schematically in Figure 2.3. The three step model describes how the electron is ﬁrst
ionized by the laser ﬁeld, then the electron is accelerated by the electric ﬁeld of the19

Figure 2.2: Typical HHG spectrum of Ar gas as seen in Li et al. [9] Notice the steadydecrease in intensity in the ﬁrst few harmonics, followed by the plateau region andeventually the cutoﬀ near the 33rd harmonic. The plateau and cutoﬀ are typical for
HHG spectra. Li et als experiment focused 1064 nm light from a 10 Hz, 36 ps pulsed
Nd:YAG laser into a 15 Torr gas jet. Also shown are various ionization energies forargon.

20

Figure 2.3: Pictoral representation of the three step model [10,11] of how the electron is
ionized and accelerated by the fundamental intense, femtosecond laser beam and thenis recombined with the atom, producing EUV and soft x-ray light as shown in Kapteyn,
Murnane, and Christov [12]. The cycle shown represents one half cycle of the driving
electric ﬁeld. The fringes in the electron wave packet show its evolution throughout the
trajectory.

laser, and ﬁnally it recombines with the ion producing a high energy photon. The highest photon energy is accurately described by the cutoﬀ law ﬁrst suggested by Krause,
Schafer, and Kulander and further explained by Corkum. This energy is given by
Emax = h̄ωmax = Ip + 3.17Up(2.4)

where h̄ is Planck’s constant divided by 2π, Emax is the highest photon frequency
produced by the HHG process, Ip is the ionization potential of the most weakly boundelectron, and Up is the ponderomotive energy of the electron in the oscillating electricﬁeld. The ponderomotive energy, which is the average kinetic energy of an electron ina sinusoidal electric ﬁeld, is given byUp =e2 E 2= 9.33 × 1014 eV ∗ Iλ2
4me ω02(2.5)

where e is the electron charge, E is the magnitude of the maximum electric ﬁeld of thelaser, m is the mass of the electron, ω0 is the laser frequency, I is the maximum laser
intensity, and λ is the laser wavelength. Thus, we see that the maximum photon energy(or shortest wavelength) is propotional to the peak laser intensity I and to the squared
driving wavelength, λ2 . The discussion that follows will brieﬂy review the three stepmodel and cutoﬀ law as they apply to HHG.

21

Figure 2.4: Diﬀerent photoionization processes of extreme nonlinear optics dependent
upon the driving laser intensity: a) multiphoton absorption ionization, b) tunneling ionization, and c) barrier suppression ionization. The red line corresponds to the Coulomb
potential suppressed by the laser ﬁeld. Adapted from reference [13].

2.3.1.1

Step 1: Ionization

The ﬁrst step in the model occurs when the electric ﬁeld of the driving laser is
suﬃcient to ionize the target atom. The process behind the ionization diﬀers depending
upon the peak laser intensity, as shown in Figure 2.4. The ﬁrst type of ionization described earlier is multiphoton ionization, including ATI. In this process, several driving
laser photons, typically with wavelengths of 0.8 − 1.06 µm, are absorbed simultaneously, thus giving the electrons enough kinetic energy to escape the binding Coulomb
potential. Multiphoton ionization occurs at peak intensities below 1014 W/cm2 for thesewavelengths. As the laser intensity increases, the Coulomb potential becomes more andmore distorted until only a narrow barrier prevents the electron from being released.
At this point, the electrons tunnel through the distorted barrier as shown in Figure 2.4.
Finally, if the laser ﬁeld strength is large enough, the Coulomb barrier will be suﬃciently
suppressed such that the electron can escape over it. This process is known as barrier
suppression ionization (BSI) and occurs at peak intensities above 1015 W/cm2 where
the peak electric ﬁeld of the laser exceeds the atomic binding electric ﬁeld of about 1010
V/m. However, BSI usually only occurs for the shortest of laser pulses (∼ 5 fs) because
longer length pulses will cause the electron to tunnel out before the intensity reaches thelevel needed for BSI. The Keldysh parameter is the common yardstick used to determine
which of these various ionization processes occurs [78]. The Keldysh parameter, γ, is22given by
γ=

ω



2mIp
eE

(2.6)

where e is the electron charge, E is the magnitude of the peak electric ﬁeld of the laser,m is the mass of the electron, ω is the laser frequency, and Ip is the ionization potentialof the valence electron. This parameter can be understood as the laser frequency times
the approximate tunneling time. When γ < 1, the electron most likely undergoes
barrier suppression ionization. If γ > 1, then the electron most likely is ionized throughtunneling. Accordingly, the nonperturbative nonlinear optics regime can be classiﬁed as
the regime where γ  1 [13]. A more accurate way of determining the ponderomotiveenergy to be used in the cutoﬀ energy estimates (Equation 2.4) is to use the laser
intensity at which ionization occurs based on the tunneling ionization rates [14]. These
rates are determined from a quantum mechanical calculation by Ammosov, Delone, andKrainov and are known as the ADK rates [79]. From these rates we can calculate thenumber of atoms that have been ionized and thus determine how many electron-ion
pairs can produce EUV light through HHG after being accelerated and recombining.

2.3.1.2

Step 2: Electron Acceleration

After ionization, the next step in the three step model is the propagation of thefree electron in the driving laser ﬁeld. The assumption that the electron is free after
ionization is justiﬁed because outside of the atomic potential, the laser ﬁeld is muchgreater than the Coulomb potential. Corkum [11] explains that classically the electron
moves according to the equations of motion given byx = x0 cos ωt + v0 t + x0

(2.7)

vx = v0 sin ωt + v0

(2.8)

and

23

Figure 2.5: One dimensional electron trajectories of freed electrons after tunneling forvarious initial laser ﬁeld phases. An initial phase of zero corresponds to the electron
being released at the peak of the driving laser ﬁeld. Notice that for initial phases greaterthan 90 degrees, the electron never returns to the parent ion (x = 0). These trajectories
are given by the classical equations of motion (Equations 2.7 and 2.8) [14].

where ω is the laser frequency, x0 is the initial displacement of the electron from theion, and v0 is the initial velocity of the electron after tunneling. Depending upon whenthe electron is freed from the atomic potential by tunneling, these equations might be
modiﬁed by a phase shift due to the relative phase of the driving laser. The initial
phase of the released electron with respect to the driving laser ﬁeld will determine thevalue of the initial velocity and the phase of the oscillating portions of equations 2.7 and2.8. These equations, in turn, will determine the velocity of the electron upon returning
to the vicinity of the parent ion. This return velocity will determine the value of the
kinetic energy which the electron will be able to give up in addition to the ionization
potential when the electron recombines with the parent ion. This energy is released asa high harmonic photon.

2.3.1.3

Step 3: Recombination

The ﬁnal step of the three step model is recombination of the propagating freeelectron with the parent ion. The electron has gained kinetic energy while being accelerated by the driving laser ﬁeld. As the ﬁeld direction reverses, the electron is accelerated

24
toward the parent ion. For linearly polarized light, the electron will return to the vicinity of the ion and can recombine, giving up its energy as a high energy photon. Thus,
the energy of the photon is given by the ionization potential of the atom plus the kinetic
energy of the electron represented by the ponderomotive energy term in Equation 2.4.
As shown in equations 2.7 and 2.8, the maximum velocity and kinetic energy of the
returning electron will depend upon the phase of the driving laser ﬁeld at the time theelectron escapes. Figure 2.5 shows the one dimensional trajectories of the free electrons
according to equation 2.7 for several initial laser ﬁeld phases [14]. An initial phase ofzero corresponds to the electron tunneling at the peak of the driving laser ﬁeld. Noticethat for initial phases greater than 90 degrees the electron never returns to the ion. The
kinetic energy of the electron is dependant on its velocity when it recombines, which is
given by the slope of the trajectory as it crosses zero. The greatest slope corresponds toan initial phase of 0.3 radians or 17.2 degrees. These are the trajectories that produce
the highest harmonics near the cutoﬀ region. For electrons that are released before orafter the time corresponding to a phase of 0.3 rad, their return kinetic energy will beless as shown in Figure 2.6 [14]. As can be seen, there is a degeneracy in the return
kinetic energy for electrons not released at the peak of the pulse where two diﬀerent
electron trajectories contribute photons of the same energy. Often these trajectories
are refered to as long trajectories (which spend more time in the continuum) and theshort trajectory (less time in the continuum). The harmonics made up of photons fromthese trajectories make up the plateau region of the spectrum. Thus, the 3.17 lawof the three step model predicts that the maximum photon energy of the harmonics
produced will be the sum of the ionization potential and the maximum return kinetic
energy. This model uses classical arguments to describe the electron’s propagationand the maximum recombination energy with quantum mechanical tunneling. Thisprediction has been veriﬁed by comparison to experimental results [9, 12, 13, 77]. While
the three step model is useful for an intuitive understanding of HHG and for estimating

25

Figure 2.6: Return kinetic energy of the electron in units of ponderomotive energy (Up )as a function of the phase of the driving laser ﬁeld (initial phase). A phase of zero
degrees corresponds to the electron tunneling at the peak of the laser electric ﬁeld. The
maximum value of return kinetic energy is given by 17.2 degrees and gives a kinetic
energy of 3.17Up [14].

the highest photon energies possible, the three step model cannot produce the harmonicpeaks of the spectrum. To obtain a more complete understanding of HHG, quantum
mechanics must be invoked.

2.3.2

Path integral interpretation of HHG

After the success of the semiclassical three step model, researchers sought to understand the quamtum mechanical aspects of HHG. Nearly all early approaches werebased on Krause, Schafer, and Kulander’s initial approach [10]. This involved numerically solving the time dependent Schrödinger equation (TDSE) for the electron
wavefunction in order to calculate the dipole acceleration. The square of the Fouriertransform of the dipole acceleration is proportional to the single atom power radiation.
The dipole acceleration is given bya (t) =d2 ψ |r| ψ
dt2

(2.9)

26

Figure 2.7: Comparison of calculated HHG spectrum using a generalized form of theLewenstein model and solving the time dependent Schrödinger equation. The Lewenstein model slightly overestimates the dipole strength due to the neglect of the lateral
spread of the electron wavefunction as it propagates, thus decreasing the probability ofrecombination [13].

where |ψ is the electron eigenfunction of the TDSE. The single atom radiated power isgiven byP (ω) =

e2 |a(ω)|26π0 c3(2.10)

where e is the electron charge, a(ω) is the Fourier transform of the dipole acceleration,
0 is the permittivity of free space, and c is the vacuum speed of light. Most methods involve using a single active electron approximation with various approximationsof the Coulomb potential [13]. Since the oscillating electric ﬁeld is periodic, the HHG
process is also periodic, thus leading to discrete energy peaks in the spectrum. This approach, while eﬀective in reproducing experimental results, does not give us an intuitive
understanding of the quantum mechanics of the HHG process.
Another quantum mechanical method for calculating a HHG spectrum that ismore intuitive is that presented by Lewenstein et al [80]. The spectra produced by
the TDSE method and a modiﬁed Lewenstein method are compared in Figure 2.7. The
important aspects of the Lewenstein model can be summarized in the following equation

27
for the induced dipole moment x(t): tdt

x(t) = i
0







d3 pd∗ (p − A(t ))E(t )d(p − A(t ))e−iS(p,t,t ) + c.c.

(2.11)

where t is the time of electron ionization, t is the time at which the electron recombineswith the atom, p is the momentum of the electron, A(t) is the laser vector potential,
E(t) is the driving electric ﬁeld of the laser, d(p−A(t )) are the dipole transition matrix
elements, and S(p, t, t ) is the classical action integral given byS(p, t, t ) =

 t
t

dt





[p − A(t )]2
+ Ip
2


(2.12)

with Ip as the ionization potential. As stated by Lewenstein et al. [80], this formulation
has a very satisfying interpretation as a product of probabilities. The E(t )d(p − A(t ))term gives the probability of an electron transitioning from a bound state to a continuum
state with momentum p at time t . The phase of the propagating electron from timet to time t is represented with the exponential including the classical action integral.
Notice that the Coulomb potential is ignored in the classical action due to the fact thatthe laser ﬁeld dominates once the electron is ionized. Finally, the d∗ (p − A(t )) is the
dipole transition probability back into the ground state at time t. These probabilities
are summed over all paths and all ionization times by the integrals. Therefore, thedipole acceleration that will be Fourier transformed and used in equation 2.10 to givethe single atom power spectrum is obtained by taking the second time derivative ofx(t) in equation 2.11. As seen from Figure 2.7, the Lewenstein model gives excellent
agreement with the TDSE method except for the overestimation of the dipole strength.
This overestimation is a result of the neglect of the fact that the electron wavepacket
will spread transversely as it propagates, thus reducing the probability of recombination.
Several adaptations of the Lewenstein model include the wavepacket spreading as wellas other considerations. For details on these adaptations, see Brabec and Krausz [13].

28
2.3.3

Phase matching

One of the factors limiting the wider application of HHG as a light source is thelimited amount of ﬂux. The highest conversion eﬃciencies reported to date are on theorder of 10−5 to 10−6 [6,12,13]. These eﬃciencies were obtained though matching of thephase velocity of the fundamental laser ﬁelds and the produced harmonic signal. The
phase velocity of the light pulses is aﬀected by the diﬀerent components of the gas target.
Neutral atoms in the gas target slow down the optical phase velocity; however, the eﬀectof the plasma is just the opposite. The optical and near-IR phase velocities in the laser
pulses speed up in plasmas. When the phases of the laser light and produced harmonics
accumulate a phase diﬀerence of π, the newly produced SXR radiation destructively
interferes with the SXR radiation created from previous cycles. Thus the coherence
length over which the HHG process constructively adds is given by
Lc =

π
∆k

,

∆k = kq − qk0(2.13)

where kq is the wavevector of the q th order harmonic and ∆k is the phase mismatch
between the fundamental and harmonic ﬁelds. However, if phase matching is achieved(∆k = 0), the produced harmonic signal increases quadratically with interaction length
until the absorption of the harmonics by the neutral gas limits the signal. This is demonstrated in Figure 2.8. Early attempts at phase matching involved adjusting whether thelaser focused before or after a gas jet of a noble gas [81]. This allowed researchers tobalance the phase mismatch of the neutral gas on the laser ﬁeld and the Gouy phaseshift due to the Gaussian beam focusing. However, the interaction length was limited,
so eﬃcient EUV production was not achieved. Later, phase matching by pressure tuning
in hollow gas-ﬁlled ﬁbers was demonstrated [14, 82, 83]. This technique, by tuning thepressure of the gas in the waveguide, allows the dispersion of the neutral atoms to bebalanced with the anomolous dispersion of the plasma. Additionally, this allowed phasematching over longer interaction lengths [12]. Additionally, the SXR radiation from

29

Figure 2.8: As the EUV signal from the harmonics and the laser signal propagate,
they will go in and out of phase. The length over which the two remain in phase iscalled the coherence length, Lc . If phase matching is achieved, the signal will increase
quadratically until it is limited by reabsorption of the neutral gas [12].

30these ﬁbers exhibits high spatial coherence [17], eﬀectively making HHG a coherent,
tunable, femtosecond SXR source. This technique has been shown to produce HHG
eﬃciently (in the 10−5 range) for lower photon energies at and below about 50 eV. For
higher photon energies, the laser intensity must be increased, which raises the ionization and introduces negative net dispersion of the produced harmonics. This dispersion
increases the phase mismatch and prevents eﬃcient harmonic generation above about
80 eV [12]. Above this energy, quasi-phase matching (QPM) techniques are needed. In
QPM, the HHG process is eﬀectively shut oﬀ periodically after each coherence length.
This allows the EUV and fundamental signals to get back into phase and prevents thedestructive interference. Periodic poling in nonlinear crystals achieves the same result
for lower order harmonic generation [12]. Preliminary QPM has been demonstrated inﬁbers with periodically modulated inner diameters [84, 85], with counter-propagating
beams [62,86,87], and with periodic gas jets [64]. While promising as a technique to enhance shorter wavelength HHG, bright soft x-ray radiation has yet to be demonstrated
with quasi-phase matching.

2.4

Coherence and mode quality
This chapter has thus far reviewed the important background and theory of highharmonic generation, but has yet to describe its coherence properties that make it usefulas a tabletop source for coherent imaging. This section will describe what coherence

Figure 2.9: Schematic showing high harmonic generation in a hollow fused silica waveguide. When the pressure of the gas inside the waveguide is tuned, the phase velocities
between harmonics and driving laser are matched and the harmonic ﬂux builds up
coherently. The harmonic beam then has very small divergence (<1 mrad).

31
means at short wavelengths and why high harmonic light from hollow waveguides exhibits a high level of spatial and temporal coherence.

2.4.1

Deﬁnitions of coherence

Coherence is the property of electromagnetic radiation to have a well deﬁned
phase relationship either as a function of time or space. Typically, this property canbe divided in two directions across an electromagnetic wave [6]. Spatial coherence isthe property of an electromagnetic wave to have a well deﬁned phase at two points
separated in a direction perpendicular to the propagation of the wave and is closely
related to mode quality. Temporal coherence is the property of an electromagnetic
wave to have a well deﬁned phase at two points that are separated only in time, or
equivalently, two points separated in a direction parallel to propagation. As the topic ofcoherence at short wavelengths is covered extensively in chapter 8 of Attwood [6] and insection 2.3 of Schlotter [15], the discussion here will be limited to only what is relevant
for coherent imaging with tabletop soft x-ray sources.
These two types of coherence are illustrated in Figure 2.10 [6, 15]. Figure 2.10
illustrates the idea of spatial coherence in part (a) and in part (b) it illustrates the ideaof temporal coherence. A source with full spatial coherence is a perfect plane wave andtherefore all points perpendicular to the direction of propagation will have a well deﬁned
phase relationship, namely they will all have the same phase. Since perfect plane waves
do not exist, no source exhibits perfect spatial coherence, however a spherical wave far
from a point source approaches an ideal plane wave. Therefore, spatial coherence canoften be described in terms of source size as is shown in Figure 2.10 (a) (reproduced
from Attwood chapter 8.1 [6]). A smaller source will produce a more spherical wave
front. As a source increases in size, the phase of the radiated electromagnetic radiationfrom the individual radiators in the extended source (atoms, electrons, molecules, etc.)
can become uncorrelated, thus reducing the coherence of the average radiated wave.

32

a)

ξ t = z∆θ
ξt =

zλ
2πd

b)

Figure 2.10: Illustration of the two types of coherence: a) spatial coherence is well deﬁned phase travsversely to the wave propagation and can be quantiﬁed by the transverse
coherence length, ξt (from Attwood [6]), temporal coherence is well deﬁned phase along
the direction of wave propagation and can be quantiﬁed by a longitudinal coherence
length, ξl (from Schlotter [15]).

33
The relavant quantity used in describing spatial coherence is the transverse coherence
length, ξt as given byξt = z∆θ =zλ
2πd(2.14)

where ∆θ is the divergence angle of the source, z is the distance from the source (assuming the small angle approximation), λ is the wavelength, and d is the source diameter.
This equation as derived by Attwood from the Heisenberg uncertainty principle diﬀers
from Schlotter’s deﬁnition by a factor of 1/2π [6, 15].
There are multiple ways by which this coherence length can be measured; however,
the most commonly used technique is with a wave front interferometer or Young’s
double slit experiment. This experiment introduces two small pinholes in an otherwise
opaque mask some distance z away from a source with divergence angle ∆θ and theinterfererence from these two pinholes is measured on a distant screen. Once the pinholes
are separated by a greater distance than ξt , the phase relationship between the two
pinholes ceases to be correlated and the interference fringes on the distant screen willlose contrast [6].
While spatial coherence is a measure of the beam quality of a source, temporal coherence is a measure of the temporal bandwidth of the source [6,15]. Temporal coherence
describes the phase relationship between two points located parallel or longitudinallyalong the direction of propagation as is shown in Figure 2.10 (b) from Schlotter [15].
Temporal coherence can be quantiﬁed by the longitudinal coherence length, ξl , given by
ξl =

λ2
∆λ

(2.15)

where λ is the central wavelength of the source and ∆λ is the full width half max(FWHM) bandwidth of the source [15]. Again, there is some ambiguity in this deﬁnition
as Attwood has an additional factor of 1/2 in the deﬁnition of the longitudinal coherence
length [6]. The longitudinal coherence length can be measured with a spectrometer,
such as a grating, or by a Mach-Zender or Michelson interferometer [18]. In the case of

34
an interferometer, the longitudinal coherence length takes on an especially signiﬁcantdescription. As additional path length is added to one arm of the interferometer, fringes
will appear on the output of the interferometer until the the path length diﬀerence
exceeds the longitudinal coherence length, ξl , at which point the phases of the diﬀerent
wavelengths of light in the source cease to be well correlated. Thus, the deﬁnition of
longitudinal coherence length given in equation 2.15 is intuitive as the ratio λ/∆λ canbe interpreted as the number of wavelengths of path diﬀerence over which a source hasstrong phase coherence longitudinally. This distance can also be thought of in terms ofa coherence time, τc , as shown in Figure 2.10 [15].
2.4.2

Coherence properties of HHG through phase matching in a hollow
waveguide

Now that we have deﬁned what coherence in x-ray sources means, we can explore why phase matched HHG in hollow waveguides exhibits such a high degree of
spatial coherence. As mentioned earlier in section 2.3.1, HHG occurs when an electron
rescatters with its parent ion under the inﬂuence of an intense, ultrafast laser driving
ﬁeld. As described by Bartels and colleagues, since the laser ﬁeld driving the nonlinear
up-conversion is coherent and the HHG process is deterministic classically, it would
be expected that the harmonic radiation produced would also be coherent [17]. However, this was not the case until Rundquist and colleaques demonstrated pressure tuned
phase matching in hollow waveguides [17, 82]. Before phase matching in hollow waveguides, several groups had studied the coherence properties of high harmonic generationin gas jets [88, 89]. These groups measured limited spatial and temporal coherence ofthe harmonics due to several macroscopic eﬀects. First of all, since HHG occurs whenintense, ultrafast laser pulses ionize a gas target, the index of refraction of the produced
plasma forms a negative lens and rapidly varies in time. This rapidly changing index
of refraction prevents the coherent buildup of ﬂux and destroys the mode quality of the

35
harmonic generation process [17]. Furthermore, as was mentioned in section 2.3.1.3,
multiple electron trajectories can contribute to each harmonic and these diﬀerent trajectories tend to have diﬀerent divergence angles [88, 90]. The longer trajectories havea larger divergence angle and the shorter trajectories a smaller divergence angle. Thus,
this eﬀect of multiple trajectories gives the non-phase matched harmonic wavefront a
multimodal behavior [17].
Phase matching of the harmonic up-conversion process in a hollow waveguide
overcomes these challenges and produces high coherence. Previously it had been shownthat when phase matching in a gas jet a single electron quantum trajectory could essentially be selected [81]. This greatly improves the temporal coherence of the source
and also the mode quality. Phase matching in a hollow waveguide also selects a narrow
range of electron trajectories improving the mode quality and temporal coherence of thesource [14,17,82]. Additionally, since the driving laser propagates in a well deﬁned modeof the waveguide, plasma defocusing and index eﬀects are reduced [17]. Thus, the harmonic ﬁeld from a phase matched hollow waveguide source does take on the coherence
eﬀects of the conﬁned laser mode as was shown by Bartels and colleagues [16, 17].
Bartels and colleagues measured the coherence properties of the phase matchedhigh harmonic source from hollow waveguides using a Young’s two pinhole setup as
shown in Figure 2.11 and as described earlier in section 2.4.1. They showed that theharmonics had a very low divergence angle (<1 mrad) and essentially full spatial coherence as demonstrated in ﬁgure 2.12 [17]. In their experiment, two pinholes, eachwith a diameter of about 40 µm, and with varying distances between them, were placed
about 3 m from the exit of a hollow waveguide where the beam was about 1 mm indiameter. The interference of the transmitted light was recorded on a charged-coupleddevice (CCD) camera about 1 m after the pinholes. The interference in Figure 2.12
shows essentially unity fringe visibility out to about 80% of the diameter of the beam.
This suggests essentially full spatial coherence accross the beam diameter. The fact that

36

Figure 2.11: Schematic of a Young’s double pinhole (wavefront interferometer) used totest spatial coherence from [16].

the ﬁnge visibility does wash out at larger separations (400-800 µm) can be attributed
to the fact that the phase matched source in argon still produced a comb of about ﬁve
harmonics centered at about the 19th harmonic (about 30 eV). Let us now analyze
the measurement from Bartels et al in light of transverse and longitudinal coherence
length [17]. In terms of longitudinal coherence length, the authors state that the phasematched beam produces a spectrum with about 4-5 harmonics (17th to 23rd harmonics). This corresponds to a bandwidth, ∆λ, of about 12 nm at a central wavelength,
λ, of about 40 nm. This ratio, λ/∆λ, is about 3.5, suggesting that after 3.5 wavelengths diﬀerence in path length (∆r from Figure 2.11) between the two pinholes theinterference pattern will wash out. This path diﬀerence then exceeds the longitudinalcoherence length of ξl = 3.5λ = 140 nm. This would correspond to about three and ahalf fringes from the central maximum, which agrees well with the number of fringes
shown in Figure 2.12. Therefore, any reduction in fringe visibility is most likely due
to limited longitudinal coherence length (temporal coherence) not transverse coherence
length (spatial coherence).

37

Figure 2.12: Measurement of the coherence of phase matched high harmonic generationfrom a hollow wave guide reproduced from Bartels et al [17]. Measurements were madewhere the beam diameter was about 1 mm with two pinholes (about 40 µm diametereach) at diﬀerent separations as shown. Notice the very high fringe visibility out toalmost 80% width of the beam.

38

Figure 2.13: Illustration of the phase properties and temporal coherence of a diverging
Gaussian beam (from Attwood [6]).

2.4.2.1

Gaussian mode

If we calculate the transverse coherence length according to equation 2.14 witha 40 µm source size with a central wavelength of 40 nm as given by reference [17] anda z distance of 2.85 m, then we get a coherence length of ∼ 500 µm (corresponding toa radius). This amount is equal to the divergence angle by the relationship θ = r0 /z,
suggesting that the beam is completely coherent spatially. In fact, Attwood describes
how for a Gaussian beam, the transverse coherence length is the same as the far ﬁeld
divergence of the beam. This conclusion is logical since a Gaussian beam is the lowest
order and simpliest mode in which electromagnetic radiation will naturally propagate
as a beam [18]. The far ﬁeld divergence (and hence transverse coherence length) for a
Gaussian beam is shown in Figure 2.13 (reproduced from Attwood [6]). This fact can√be shown mathematically as follows [6]. The radius at which the intensity is 1/ e thevalue of the peak intensity of a Guassiam beam, r(z), is given as a function of distance,z, from the smallest radius, r0 , (known as the waist) in

λz 2
) .
r(z) = r0 1 + (4πr02(2.16)

39
If we calculate the far ﬁeld divergence angle, ∆θ, (deﬁned as θ in Figure 2.13) far fromthe waist position (z0 ) we obtain∆θ(z  z0 ) =r(z) ∼ λ
.
=
z
4πr0

(2.17)

Finally, if we multiply this value by the distance from the waist (which can be considered√a source in this discussion) in order to calculate the far ﬁeld radius (1/ e), we obtainr(z) = z∆θ =zλ
2πd(2.18)

where d is the source diameter (2r0 ). This result is exactly the same result as thetransverse coherence length in equation 2.14. Thus, a Gaussian beam exhibits totaltransverse (or spatial) coherence.
Because of the foregoing discussion, it might not come as a suprise that a phasematched high harmonic beam from a hollow waveguide will produce a Gaussian beam.
The far ﬁeld beam of the 27th harmonic (29 nm) is shown in Figure 2.4.2.1. This beam
was measured on a CCD camera after bouncing oﬀ of two multilayer mirrors in a setupthat will be described in chapter 4. These mirrors act as a monochromator, eﬀectively
selecting a single harmonic. Figure 2.4.2.1 (top) shows the logarithmic Gaussian highharmonic beam which covers almost the full dynamic range of the camera (∼ 65000) and(bottom) a Gaussing ﬁt to the verticle lineout of the Gaussian beam showing very good
agreement. The true measure of how well a beam matches to a TEM00 Gaussian modeis given by measuring a beam’s M 2 (M-squared) value which compares the divergenceof the beam with the ideal divergence of a Gaussian beam. Measuring the M 2 of ourphase matched harmonic beam is a future project that needs to be completed.

2.4.2.2

Temporal coherence of phase matched high harmonic generation

As mentioned earlier, phase matching of the high harmonic process can eﬀectively
select eﬀectively a single electron trajectory. The harmonic resulting from this single

40

Figure 2.14: (Top) Measured optimized 29 nm mode (log scale) on the CCD after two
multilayer mirror reﬂections. (Bottom) Gaussian ﬁt to horizontal line out of HHG mode
showing excellent ﬁt with R2 = 0.9986.

41
trajectory would be expected to have a very narrow bandwidth and long longitudinalcoherence length. However, as we discussed earlier in this section, Bartel’s measurementof coherence suggests a λ/∆λ of 3.5. This, however, is due to the fact that theirphase matched source was optimized for brightest ﬂux resulting in 4-5 harmonics being
phase matched. The individual bandwith of each harmonic may be much narrower
than 3.5. Single harmonics can be selected by use of a grating or multilayer mirror
monochromator. Such a measurement of a single harmonic produced from a gas jetreports a λ/∆λ of 71 [91]. However, this source is not fully phase matched, thus we might
expect a broader bandwidth. Bartels et al, in a separate work involving optimizationof harmonic generation with pulse shaping measure spectrometer resolution limited
bandwidth of the 27th harmonic (29 nm or 42 eV) to be 0.25 eV [92]. This corresponds
to a λ/∆λ bandwith of about 170. Subsequent observations in work contributing tothis dissertation from diﬀraction patterns and spectrometers have measured λ/∆λ ≥250 from the 27th harmonic after two bounces from multilayer mirrors which each havea bandpass of about 3.3 eV (or 2.5 nm). Thus, phase matched high harmonic generation
can produce truly laser like EUV and SXR radiation with high temporal and spatial
coherence. In the next chapter it will be shown why this high coherence is necessary forhigh resolution coherent imaging.

2.5

Comparison to other tabletop SXR sources
Before moving onto the next chapter, we should brieﬂy overview other tabletop

sources of EUV/SXR radiation. We will start with the most incoherent sources andmove to progressively more coherent sources. Laser produced plasmas have been shown
to be a very bright source of EUV to hard x-ray radiation [6, 74, 93]. These sources
operate by focusing a high power (∼ 100 mJ/pulse, nanosecond pulse duration) laser
into liquid or solid targets producing hot, dense plasmas [6]. These highly ionizedplasma species can then produce atomic line radiation on top of a broad continuum of42
Bremsstrahlung radiation. Thus these sources can have very high temporal coherence
but low spatial coherence as the radiation is emitted into 4π steradians (sr). One
particularly nice example of zone plate imaging with a laser produced plasma comesfrom Hans Hertz group in Sweden where a 3 ns Nd:YAG laser (100 mJ/pulse, 100 Hz)
is focused into a liquid nitrogen jet target. The hydrogen-like liquid nitrogen Kα plasmaline produces bright radiation at 2.478 nm which is in the biologically important water
window [94]. While this source is extremely bright (∼ 1014 photons/s) this is into 4π
steradians [94]. The ﬂux on the sample is ∼ 106 photons/(s × µm2 ) after a condensor
optic (either a multilayer mirror or zone plate). This group demonstrates approximately
30 nm resolution in a ﬁve minute exposure, making practical nanometer scale imaging
in the water window. While useful for many applications, these spatially incoherent
plasma sources are not very tunable as they rely on atomic transitions. Additionally,
they have relatively long pulse durations (nanoseconds) making them unsuitable forultrafast imaging. More information on laser produced plasmas is available in Attwood,
chapter 6 [6].
Another attractive source of tabletop EUV/SXR radiation is SXR lasers [6, 95].
These sources are more coherent than laser produced plasmas as they have high gain on
highly ionized single atomic transition with spectral bandwidths of up to λ/∆λ ≥ 104
and lower divergence (≤ 10 mrad divergence) [95]. Although they are lasers, their spatial
coherence is lower than high harmonic generation sources due to the fact that the lasing
action builds up from noise. Thus the extended sources are an ensemble of uncorrelated
radiators. These lasers can broadly be classiﬁed into two groups: electrical discharge
pumping of gas ﬁlled capillaries and laser pumped solid targets [6]. The electricallydischarged lasers can show essentially full spatial coherence in a gain saturated longcapillary [96], high power (∼ 1 mW at 1-5 Hz repitition rate) in nanosecond pulses [75],and with both coherent imaging and zone plate imaging have shown wavelength limited
resolution (47 nm) [57, 97]. Laser pumped solid target lasers have demonstrated lasing

43
down to 10.9 nm with Ni-like plasmas of metals (Te, Sn, Cd, Ti), with modest powers(∼ 1 µW at 1-5 Hz) in picosecond pulses [98], and have demonstrated 38 nm resolution
at 13.9 nm wavelength using zone plates [54]. These sources are very attractive for
imaging, interferometry, and even nanopatterning due to their high ﬂux and narrow
bandwidth, but they are not tunable, have limited ps temporal resolution, and require
a lot of pump energy (GJ of power for electrically discharged lasers). The practicalityof scaling these sources to shorter wavelengths is diﬃcult as the pump power necessary
goes like 1/λ4 , thus scaling to single wavelength lasers is not practical on tabletop
systems [6]. A recent very exciting development with tabletop SXR lasers is seeding of
the SXR laser with a high harmonic source [58], thus giving the source the divergenceand mode properties of HHG with the spectral bandwidth of an SXR laser. While still
limited to wavelengths ∼ 10 nm, this source demonstrates unprecedented temporal andspatial coherence with ≤ 1 µW of power, making it an excellent source for tabletop
coherent imaging.

2.6

Chapter Summary
With the development of eﬃcient HHG, table-top, coherent, femtosecond EUV

sources are available. HHG has developed from ﬁrst being observed in the late 1980’s
into an eﬃcient EUV/SXR source through phase matching techniques. The development of the three step model and the use of numerical quantum calculations has
aided in explaining the HHG process and predicting how it can be made more eﬃcient. Now, EUV/SXR radiation from HHG is ﬁnding applications in spectroscopy,
imaging, and perhaps in the future lithographic techniques [6, 12, 13]. Additionally,
HHG has made possible the production of isolated sub-femtosecond or attosecond radiation pulses [71–73, 99]. Thus HHG presents a fruitful area for further research andtechnological development especially in ultrafast coherent imaging.

Chapter 3

Theory of Coherent Diﬀractive Imaging

3.1

Overview
As mentioned in chapter 1, an attractive alternative to traditional x-ray mi-

croscopy involving imaging optics is x-ray coherent diﬀractive imaging. This rapidly
developing technique would greatly beneﬁt from a coherent tabletop source of x-rays,
such as high harmonic generation. This chapter will describe the relevant backgroundand theory of x-ray coherent diﬀractive imaging (XCDI). Speciﬁcally, this chapter will
describe the oversampling phase retrieval algorithm, including the system requirements
and limits of this technique. Additionally, this chapter will describe some recently
developed techniques of CDI that will make this technique much more applicable.

3.2

Diﬀraction Theory
This section will review aspects of scalar diﬀraction theory important to under-

standing CDI. We will seek to describe the Fourier relationship between the transmitted electromagnetic wave immediately following an illuminated sample (known as anexit surface wave) with the scattered light (diﬀraction pattern) on a distant screen.
Diﬀraction is a very common subject of electromagnetic radiation and multiple excellent sources are available on the subject; speciﬁcally, Goodman is a seminal work on
Fourier optics, diﬀraction, and diﬀractive imaging [100]. However, for nostalgic reasons,
we will follow the discussion from Peatross and Ware [18]. For brevity, we will begin

45

Figure 3.1: Schematic showing the relevant parameters and variables for setting up
diﬀraction theory (from Peatross and Ware [18]).

with the Fresnel-Kirchhoﬀ diﬀraction formula 3.1. This formula describes the scatteredelectromagnetic radiation on a distant screen from a sample that is illuminated with acoherent bean as shown in Figure 3.1 [18]. We have made only a few assumptions tothis point. Speciﬁcally, these are that the aperture or sample size, D, and the distancefrom the sample to screen, d, are many times larger than the illumination wavelength:D, d  λ. Additionally, we have assumed that the measurements made on the distant
screen are in the paraxial approximation, thus the measurement screen is in the forward
direction so that the scattering angles are small (see Peatross and Ware chapter 10 [18]).
The diﬀracted electric ﬁeld distribution, E(x, y, z = d), on the distant screen is givenby the Fresnel-Kirchhoﬀ formula2

2−ieikd e 2d (x +y )λdikE(x, y, z = d) =

 

2

2





E(x , y  , 0)e 2d (x +y ) e− d (xx +yy ) dx dy  (3.1)
ik

ik

where k = 2π/λ, E(x , y  , 0) is the coherent illuminating wave, and the integration isover the aperture or sample. The coordinate system is given by coordinates x, y, z withpositive z in the direction of beam propagation. The aperture or sample is located at
z = 0 which we will refer to as the sample plane and the distant screen (or later in thischapter, the CCD) is located at z = d, with d  λ, which we will call the detector
plane. It should be noted that the two phase factors in front of the integrals are global
phase factors that do not aﬀect the intensity. Additionally, we note the quadratic phase46
ik(x2 + y 2 ), in the exponential in the integrand that describes how the pattern
factor, 2devolves as it propagates in z. Equation 3.1 gives the observed diﬀraction pattern in thedetector plane and is said to be in the near ﬁeld.
All of the measured diﬀraction patterns in this dissertation were measured in
the so called far ﬁeld of the aperture. The far ﬁeld diﬀraction intensity is given by
the Fraunhofer diﬀraction formula. The Fraunhofer diﬀraction pattern is a simpliﬁed
version of the Fresnel diﬀraction pattern, and is described as follows by Peatross andWare [18]. The Fresnel diﬀraction pattern is valid everywhere after the aperture aslong as the previously described assumptions are satisﬁed. As the scattered light propagates farther away from the aperture it evolves according to the quadratic phase factor.
However, at some point, the diﬀraction stops evolving and the transverse electric ﬁeld
distribution stays the same except for growing larger proportionally to the distance fromthe aperture. At the point that the diﬀraction pattern stops evolving, it is said to bein the far ﬁeld. The far ﬁeld distance is given by
d

ik
k 2
2
2
(x + y 2 ) so that e 2d (x +y ) ≈ 1
2

(3.2)

where x2 + y 2 = D2 is the squared aperture or sample diameter, D. Equation 3.2 canbe simpliﬁed to be expressed only in terms of the aperture or sample to measurement
screen distance, d, the wavelength, λ, and the sample diameter, D, as follows:
d

πD 2
.
λ

(3.3)

When this condition is satisﬁed, the diﬀraction pattern electric ﬁeld is given by the
Fraunhofer diﬀraction formula2

2−ieikd e 2d (x +y )E(x, y, z = d) =
λd
ik

 





E(x , y  , 0)e− d (xx +yy ) dx dy  .
ik

(3.4)

Several aspects of equation 3.4 are worth noting. First, we note that the Fraunhofer diﬀraction formula is essentially a two dimensional Fourier transform of the exit

47
surface wave at the sample plane except for the global phase factors in front of the integrals. Thus, the ‘shadow’ of the sample and the far ﬁeld diﬀraction pattern represent
a Fourier conjugate relationship. The coordinates in the far ﬁeld can be thought of as
spatial frequencies with
fx =

x
y
and fy =
zλ
zλ

(3.5)

where x, y are the coordinates or characteristic distances in the sample plane [100]. This
Fourier relationship is one of the founding principles of XCDI and is ubiquitous in thereconstruction process, usually in the form of the fast Fourier transforms (FFT).

3.3

Sayre and oversampling
Now that we understand the fundamental Fourier relationship between the sample

exit surface wave and the far ﬁeld diﬀraction pattern, we can discuss the next fundamental building block of XCDI: oversampling of far ﬁeld diﬀraction patterns. This idea
was presented as a natural extension of x-ray crystallography by David Sayre in 1952
in his seminal work “Some implications of a theorem due to Shannon” [101]. X-ray
crystallography has been the primary tool for determining the atomic scale structure ofmany materials, including many important biomolecules such as DNA [102, 103].
However, many important biological molecules and also whole cells cannot becrystallized easily. Thus, traditional x-ray crystallography techniques cannot easily
determine their structure. In 1952, Sayre recognized that, even though a recorded
diﬀraction pattern contains no phase information (only the intensity is measured), if
the diﬀraction pattern is sampled at a suﬃcient rate the phase information can beextracted [101, 102, 104]. This process can be thought of as solving a set of nonlinearly
coupled equations that describe the Fourier relationship between the exit surface wave ofa sample and the far ﬁeld diﬀraction pattern. Therefore, if we can determine the phaseof the electromagnetic wave in the far ﬁeld, the surface exit wave of the sample can be

48
determined. If the sample diﬀraction pattern is sampled at the spatial Nyquist frequency(fN = D/zλ where D is the sample diameter), then the number of unknowns (phases)in the set of coupled nonlinear equations exceeds the number of knowns (amplitudes).
However, if the diﬀraction pattern is sampled at a higher rate than the spatial Nyquist
frequency then the set of equations can be solved. This technique is called oversampling.
The oversampling ratio describes the amount of oversampling. In the sample
plane, the oversampling ratio can be described asσ=electron density region + no density region
electron density region(3.6)

where σ ≥ 2 [101]. The terminology of electron density comes from x-ray crystallography
because the x-rays interact almost exclusively with the atomic electrons (valence to coredepending on the energies) through processes of scattering and absorption [6]. Thus, therecovered exit surface wave or sample shadow is related to the sample electron density
through Babinet’s principle [18]. In the detector plane, the oversampling ratio canbe thought of as the number of sampling points in the detection plane per diﬀraction
speckle. In a linear sense for a two dimensional sample, the linear oversampling is given
1by O = σ 2 where
O=

zλ
1
=
pfD
Dp

(3.7)

where fD is the spatial frequency of the largest sample distance (diameter) D given byequation 3.5, p is the CCD or detector pixel width, and z is the sample to detector
distance [105]. Practically speaking, the speckle size on the CCD that corresponds tothe spatial frequency fD is the smallest speckle in the diﬀraction pattern and is givenby the ‘spatial period’ 1/fD [100].
Now we will present a very insightful empirical derivation of the oversampling
theorem from Miao and Sayre [102] which follows closely Ariel Paul’s description inchapter 5 of his dissertation [24]. Figure 3.2 also presents the Fourier relationship ofoversampling between the sample and detector planes. Typically the diﬀraction pattern

49

Figure 3.2: An illustration of the no-density region generated by oversampling thediﬀraction pattern. Here we surround the object with a no-density region where there
are no scatters.

is measured on a CCD or other array detector with N × N sampling points. Because
the number of pixels in the reconstructed sample cannot exceed the number of detected
points, the sample cannot be sampled at any higher rate than N × N . This is the socalled ‘conservation of pixels’ rule.
The following discussion will present how sampling the far ﬁeld diﬀraction patternis synonymous to surrounding the sample with a no density region. If the electron
density of an object is assumed to be ρ(x), then the Fourier transform of this object isgiven byF (k) = F [ρ(x)] = ∞−∞ρ(x) exp[2πik · x]dx

(3.8)

where x is the three-dimensional spatial coordinate in real space and k is the threedimensional frequency coordinate in reciprocal space. If F (k) is sampled at the Nyquist
frequency of the object (fN = f D = N/D), then we can replace the continuous Fouriertransform is replaced by the discrete Fourier transform
N −1

ρ(x) exp[2πik · x/N ],F (k) =
x=0

k = 0, . . . , N − 1.

(3.9)

50
where x and k have become discrete and range from 0 to N − 1 in each dimension.
However, if F (k) is sampled at twice the Nyquist frequency, thenN −1ρ(x) exp[2πik · x/(2N )],F (k) =

k = 0, . . . , 2N − 1.

(3.10)

x=0

Let us introduce a new function g(x) such that⎧⎪⎪⎪
⎨ρ(x) 0 ≤ x ≤ N − 1g(x) =⎪⎪⎪⎩0N ≤ x ≤ 2N − 1(3.11)

If g(x) is substituted into Equation 3.10, then2N −1

g(x) exp[2πik · x/(2N )],F (k) =

k = 0, . . . , 2N − 1.

(3.12)

x=0

Equation 3.12 represents the relation between the density function g(x) and its fastFourier-transform pattern sampled at the Nyquist frequency of g(x). Thus, oversampling the discrete Fourier transform of the object’s density function by sampling moreﬁnely than the object’s Nyquist frequency generates a no-density region surrounding
the electron density of the object as illustrated in Figure 3.2 [102]. Clearly, by the same
logic, increasing the degree of oversampling increases the size of this no-density region.
Thus, increasing the oversampling rate is increasing the ratio of knowns to unknowns inthe set of nonlinear equations from earlier by surrounding the sample with known zero
values. This no density region is known as the support and plays a crucial role in thephase retrieval process as we will see next [106].

3.4

Phase Retrieval
Even though Sayre suggested that an image of a sample could be retrieved fromthe diﬀraction pattern alone, it took several years to develop practical analytical tools
for performing the phase retrieval [102,104]. Twenty years later, Gerchberg and Saxton
proposed the ﬁrst phase retrieval algorithm based on Sayre’s idea [107]. A few years

51
later Fienup et al improved upon Gerchberg and Saxton’s work with two phase retrieval
algorithms based on a ﬁnite support and a positivity constraint based on the fact that all
actual physical samples will have to have a positive amplitude of the complex electron
density [106].
Fienup’s algorithms, error reduction and input-output, demonstrated that via
phase retrieval a complex object can be reconstructed from the magnitude of the Fouriertransform (diﬀraction pattern) alone. Various other algorithms have been developed anddemonstrated and two excellent reviews compare and contrast these techniques [19,108].
Most of these algorithms are similar to Fienup’s hybrid input-output (HIO) algorithm
in their use of a ﬁnite support although it should be noted that Marchesini et al’s
‘shrinkwrap’ algorithm has proven to be very robust and fast due to the use of a dynamic support [52]. This technique allows the support to periodically update based on
an intensity threshold, thus “shrink-wrapping” the sample electron density region. For
all the images presented in this dissertation, a genetic algorithm version of Fienup’s
original hybrid input-output (GHIO) algorithm was used. This technique was developed by our collaborators at University of California at Los Angeles in Professor John
Miao’s group [20, 47, 109]. The basic HIO algorithm is described in Figure 3.3. The
basic aspects of the HIO iterative phase retrieval algorithm are [19, 106]:

1. The square root of the measured diﬀraction pattern intensity (the amplitude)
is seeded with an initial random phase.
2. Each reconstruction iterates back and forth between real and reciprocal spacewith FFT (or inverse FFT) based on the Fourier relationship between the sample planeand detector plane. In the sample plane (sometimes called real space), two constraints
are applied the support constraint and the positivity constraint. For the support constraint, a support is deﬁned as a rectangular area containing the sample, with its size
estimated from the linear oversampling ratio as in equation 3.7. The sample density

52

Figure 3.3: Illustration of hybrid input-output (HIO) algorithm from Fienup et al [19].
In this dissertation a genetic algorithm version of HIO was used known as guided HIOor GHIO [20].

53
outside the support and the negative real or imaginary part of the electron density inside
the support are slowly pushed to zero (positivity constraint).
3. Next, after applying constraints in the sample plane, the FFT of the sample
plane is performed in order to return to the detector plane (or reciprocal space). In thesample plane, the diﬀerence between the new magnitude of the FFT and the original
measured diﬀraction pattern amplitude are compared in order to compute an error.
Then, the amplitude of the FFT is replaced with the original measured diﬀraction
pattern amplitude (square root of the diﬀracted intensity) but the phase is allowed tocontinue onto the next iteration.
4. This iterative process is continued by applying the support and positivity constraint in the sample plane and the amplitude constraint in the detector plane until
the phase converges and a faithful high resolution image of the sample plane is reconstructed. This can be achieved by setting a threshold on the error at which point thealgorithm is terminated.

Additional techniques can be applied such as running multiple reconstructions
on diﬀerent random phase starts in a genetic algorithm (known as GHIO) or by using
a certain number of HIO iterations followed by a number of error reduction iterations[20,108,109]. The GHIO technique will be described in the next chapter in more detail.
These phase retrieval techniques can be thought of as the spatial version of the ultrafast
pulse measurement technique knows as Frequency Resolved Optical Gating (FROG)[110]. Additionally, it has been shown that these algorithms are very unique in theirsolutions. It has been shown that multiple solutions are “pathologically rare” andthat “in general the 2-D phase retrieval problem will have a unique solution if one
exists” [111]. Thus, iterative phase retrieval techniques have proven to be an eﬀective
coherent imaging method for oversampled far ﬁeld diﬀraction patterns. Eﬀectively an
imaging optic is replaced by the iterative phase retrieval algorithm.

54

3.5

Implementing XCDI

3.5.1

Experimental developments of XCDIEven though by the mid-1980’s the theoretical and computational framework for
XCDI existed, it took some ﬁfteen years more until the ﬁrst demonstration. There areseveral reasons as to why this delay occurred. First, as personal computers became
cheaper and more powerful, they began to appear in ever more laboratories. The phaseretrieval techniques discussed in the previous section, when applied to high resolution
imaging, are very demanding computationally. Reconstructions can take days on asingle processor because often it requires 100,000’s of iterations, with each iteration
involving two FFT’s on array sizes up to several mega pixels. Second, the introductionof third generation synchrotrons with insertion devices such as wigglers and undulators
greatly increased the amount of bright, coherent x-ray ﬂux [6].
The ﬁrst experimental demonstration of x-ray coherent diﬀractive imaging (orlensless imaging) was performed at Brookhaven National Lab at the National Synchrotron Light Source in the groups of David Sayre and Janos Kirz by John Miao andcolleagues [50]. Since this time upwards of 40 groups across the world have pushed thetechnology and found numerous applications [47]. Perhaps one of the most attractive
potential applications is the use of XCDI to image single biomolecules with a single,
very intense femtosecond burst of x-rays from a x-ray free electron laser (FEL or fourth
generation synchrotron) [51, 112].
Before proceeding, we should note several of the advantages and disadvantagesof CDI in light of the phase retrieval algorithms. Several aspects of XCDI make it
an extremely elegant and appealing technique. Because lensless imaging only requires
illumination with a plane wave, ∼ mm accuracy in sample placement is suﬃcient. No
multi-step focusing process is necessary, as is the case for imaging systems that require
sample placement and stability at the micron level. With imaging optics, the magniﬁca-

55tion is a purely geometric function of the object to image distance ratio, often requiring
the detector to be a meter or more from the imaging optic. In contrast, for tabletop
XCDI, the entire imaging apparatus ﬁts in 0.5 m x 1.5 m. It should also be noted thatthe same experimental setup will work at any wavelength at which narrowband coherent
light can be produced.
Additionally, as noted previously, CDI with phase retrieval does not require any
imaging optics, which at x-ray wavelengths are lossy, diﬃcult to manufacture, and impose a limited numerical aperture and ﬁnite point spread function. Thus, XCDI has thepotential to be truly limited only by the wavelength of illuminating light, opening up opportunities for further development and potentially sub-wavelength or super-resolution
techniques in analogy to optical wavelengths (see chapter 1). However, some of thedisadvantages are the fact that for traditional XCDI image information is only available
after signiﬁcant computer processing and the need for fully coherent sources. Furthermore, for traditional XCDI, only isolated objects can be imaged because of the needfor a ﬁnite support [19]. We will see in a coming section how this constraint has been
overcome with curved beam techniques.

3.5.2

Other CDI techniques

Recently there has been a rush of activity to develop XCDI techniques that do not
require a compact support. Of particular interest has been the development of curved
beam XCDI techniques that greatly increase the stability and speed of the phase retrieval
algorithms and which will allow imaging of extended samples [21,22]. These techniques
use the illumination proﬁle as a support instead of demanding that the sample be ﬁnite.
This adds a level of complexity by requiring a measurement and reconstruction of thebeam using traditional phase retrieval algorithms. This can be achieved by removing
the sample altogether. Additionally, these curved beam techniques are a step towards“structured illumination” XCDI techniques that will open various enhancements such

56
as extended ﬁeld of view or increased resolution with the same exposure times.
The ﬁrst technique, Keyhole Diﬀractive Imaging illuminates the sample with acurved wavefront by placing the sample in the Fresnel regime after the focus of an x-ray
lens as shown in Figure 3.4 (b) [22]. Illuminating the sample with a curved wavefront
allows for faster and more stable reconstruction of the image via phase retrieval andthe ability to zoom in on regions of interest. This technique can be combined with ascanning coherent imaging technique, ptychography, in order to image extended samples
eﬃciently without prior knowledge of the illumination support in Figure 3.4 (a) [21].
For this ptychographic technique, an additional sample plane constraint is added byrequiring that the individual exposure regions overlap.
These techniques have faster converging reconstruction algorithms based on the
additional phase curvature information on the illuminating beam. They also allow
imaging of extended samples; however, they have some disadvantages worth noting.
First, they require an optic that can produce a tight focus meaning that zone plates orhigh NA mirrors are again introduced. Additionally, they require mechanical stabilityof the beam-sample-detector line below the resolution desired. This is challenging if thegoal is to achieve 10 nm resolution. This is a complication that is avoided in traditional
XCDI when the sample is overﬁlled with the illuminating beam. In this case, vibrations
are allowed up to the limit of a detector pixel size (∼ 10 µm instead of 10 nm) greatly
relaxing this restriction.
In addition to coherent imaging with iterative phase retrieval algorithms, holography has also proven to be capable of high resolution imaging with x-rays. The simplest
technique for x-rays is known as Gabor or in-line holography in which a sample is simply illuminated with a coherent beam and the interference between the diﬀracted lightand illuminating beam is recorded in the far ﬁeld [113, 114]. The phase informationis encoded in the interference between these two beams, thus an image can be reconstructed with a single Fourier transform instead of an iterative phase retrieval algorithm.

57

a)

b)

Figure 3.4: Illustration of novel coherent x-ray imaging techniques: a. Ptychography[21]; b. Keyhole Diﬀractive Imaging or curved wave imaging [22].

58
The ﬁrst x-ray coherent imaging demonstrated with tabletop SXR sources was with a
HHG source using Gabor holography by Bartels et al in the same work in which they
demonstrated the spatial coherence of HHG [17]. These intial demonstrations had limited resolution of a few µms. Gabor holography has two notable disadvantages. First,
high resolution Gabor holography requires a tightly focused beam in order to achieve
good interference at high spatial frequencies. Second, there is a phase ambiguity in thereconstructed image leading to the ‘twin image’ problem which can cause ringing anddistortions in the image [115].

3.6

System requirements
When implementing XCDI experimentally, it is necessary to keep several require-ments in mind. This section presents a list of necessary relationships or ‘governing
equations.’ Another treatment of this material can be found in Paul, chapter 5 [24] andin reference [105]. Figure 3.5 shows the relevant setup and parameters for the following
discussion.

3.6.1

Geometrical considerations

The two most crucial relationships to consider for CDI are satisfying the requirements of Sayre’s oversampling theorem and the phase retrieval algorithms. The
oversampling ratio was given earlier in equation 3.7, but is so fundamental to CDI thatis is repeated here: O = zλ/pD where z, is the sample to detector distance, λ is the
wavelength, p is the detector pixel size, and D is the sample size or diameter. Anothercrucial relation to remember is that the detector needs to be in the far ﬁeld, so werestate the far ﬁeld distance from equation 3.3: z  πD2 /λ.
The next aspect necessary for CDI is the coherence of the source. As we discussed
in section 2.4.1, in order to encode phase information in diﬀraction patterns through highcontrast interference fringes a source must exhibit high spatial and temporal coherence.

59

Figure 3.5: Setup of relevant parameters in oversampled coherent diﬀraction imaging.

60
As the coherence of the source degrades, these fringes begin to wash out (in the case ofdegraded spatial coherence) or spread into each other (in the case of degraded temporal
coherence). Miao et al present empirical relations for how coherent a source needs tobe for high resolution CDI [105]. The temporal coherence requirement for CDI is
OD
λ
≥
∆λ
r

(3.13)

where as deﬁned before (and shown in Figure 3.5) λ is the central wavelength of thesource and ∆λ the bandwidth, O is the oversampling ratio, D is the sample size (diameter), and r is the highest resolution obtainable in the image reconstruction. Thus, we
see that the spectral bandwidth places a limit on the image resolution by
r≥

OD
.
λ/∆λ

(3.14)

This relation shows that more oversampling has the eﬀect of lessening the spectral
bandwidth requirement for lensless imaging.
In terms of longitudinal coherence length, ξl , the temporal coherence limit is
ξl =

ODλ
λ2
≥
.
2∆λ
2r

(3.15)

Ariel Paul gives a very intuitive and satisfying interpretation of this requirement interms of path length diﬀerence (see chapter 5 in [24]) by using the Bragg diﬀraction
condition, λ = r sin(θ) (θ is the scattering angle), to replace λ/r with sin(θ) as follows
ξl ≥OD sin(θ).
2(3.16)Paul interprets equation 3.16 in terms of path length diﬀerence. Essentially what thisrequirement states is that the longitudinal coherence length, ξl , must be longer thanthe path diﬀerence between any two points that are separated by O/2 times larger than
the sample size D. Thus, for the minimum oversampling necessary for CDI, this path
length diﬀerence would arise from points on the opposite sides of the sample.

61
For spatial coherence, the requirement on transverse coherence length, ξt is moreeasily understood and is simply
ξt ≥ OD(3.17)

so that the transverse coherence length is larger than the total area illuminated OD[105]. As pointed out by Miao et al, the higher the oversampling, the ﬁner the detailsof the diﬀraction pattern that must be faithfully sampled. Thus, higher oversamplingallows a source to be less coherent temporally but at the expense that the source must
become more coherent spatially.
Several other experimental aspects of XCDI are worth mentioning brieﬂy but arecovered in more detail in Ariel Paul’s thesis [24]. These include light tightness, stability,
and illumination. Since image resolution is achieved by sampling diﬀraction patterns
at high angles where diﬀraction eﬃciency is low, the signal to noise ratio of thesehigh spatial frequencies is very important. Thus the enclosure of the detector must be
shielded from any stray light and the detector performance must be maximized for low
signal detection and read out. Furthermore, the pointing and intensity stability of thesource is very important so that a faithful diﬀraction pattern is recorded. Additionally,
the sample should be illuminated with close to a uniform intensity which usually requires
somewhat of a large radius of curvature on the beam and ﬂat phase front. Increasing
the beam size with respect to the sample size also tends to improve stability.

3.6.2

Data processing and resolution requirements

When setting up a CDI experiment, several factors are important to consider. The
geometrical issues discussed previously such as oversampling, far ﬁeld distance, coherence, etc., will allow an experimental setup that should produce high quality diﬀraction
patterns. However, this alone will not guarantee a successful experiment. As in any
imaging experiment, two parameters that are very important to consider are resolution
and ﬁeld of view (here we focus only on two dimensional imaging, but if three dimen-

62
sional imaging is considered then also depth of focus is an important parameter). With
CDI, the resolution and ﬁeld of view are determined by the Fourier relationship of thescattered light. Typically, this scattering process is quantiﬁed by the scattering angle orphoton momentum transfer that is imparted to the photons. Figure 3.5 also describes
the relevant parameters in this discussion. The momentum transfer, Q, is given byQ=

4π sin(θ).
λ(3.18)

where sin(θ) = y/z where y is the transverse distance from the no deﬂection, zero
momentum transfer (Q = 0) point. Thus the maximum momentum transfer measurable,
Qmax , corresponds to y = L/2 = N p/2 with L being the width of the detector array, N
the number of pixels, and p the detector pixel width. The minimum momentum transfer
measurable, Qmin , is set by the transverse distance y = p.
Due to the Fourier relationship between the sample plane and the detector plane,
the minimum measurable increment in one will set the largest distance in the otherand vice a versa. This can be demonstrated by the fact that the detector pixel size
determines the reconstructed image ﬁeld of view, LF OV by
LF OV = 2

2π
λ
.
=
Qmin
sin(θmin )(3.19)

If we make the small angle approximation so that sin(θmin ) ≈ θmin ≈ p/z, which is
generally true for far ﬁeld detection and appropriate oversampling, then we obtain
LF OV =zλ= 1/fmin .
p(3.20)

which we can also equate to the smallest spatial frequency fmin . Equation 3.20 can also
be used to calculate the reconstructed image pixel size, pim , through the principle ofconservation of number of pixels between the detector and image. In this discussion,
we will focus on linear pixels (as along one edge of the detector), N . The reconstructed
image pixel size is a very important consideration when setting up an experiment, as
it must be smaller than any detail of a sample that must be resolved. Therefore, the63absolute limit of an image resolution theoretically, r, with CDI is set by this value, pim .
Practically at least two, and preferably many image pixels must be used to resolve thisfeature so that r ≥ 2pim . The image pixel size is given by
pim =

zλ
LF OV
=
.
N
pN

(3.21)

Alternatively, one can calculate pim = 2π/2Qmax analogously to equation 3.19, where
there will be a small discrepancy due to the small angle approximation. Furthermore,from the viewpoint of oversampling (LF OV = OD, see 3.5), the image pixel size can becalculated as follows
pim =

zλD
zλ
OD
=
=
N
pN D
pN

(3.22)

which is exactly what we calculated in equation 3.21.
Finally, the practical limit on reconstructed image resolution that can be expected
is set by the maximum detectable momentum transfer, Qmax . The exact value of Qmax
is diﬃcult to determine, and good signal to noise considerations are important. Bill
Schlotter’s thesis has a good review of signal to noise in chapter two [15] and in his
article in reference [116]. An estimate of highest image resolution (half pitch) possible
rdif f that is supported by diﬀraction will be
rdif f =

2π
λ
=
Qmax
2 sin(θmax )(3.23)

where Qmax is the maximum detected momentum transfer (with good signal to noise),corresponding to the scattering angle subtended by the angle θmax . We note that thisis essentially the same as equation 1.1, except with the prefactor of 1/2 as set by theAbbe resolution criterion. This equation and the ability to calculate the image pixel
size, pim , are very useful to remember when analyzing data and determining resolution.
3.6.3

Scalability and limits to resolution set by radiation damage

Earlier, this thesis discussed the possibility that XCDI was capable of producing
wavelength limited resolution. This achievement would be extraordinary, especially64
since hard x-ray energies like those envisioned at future generation FELs might approach
1 Å or less, making atomic scale resolution possible. While the basic technique of
CDI is capable of such resolutions, unfortunately the samples are not unless they are
specially prepared. One of the main reasons for this is that both electron microscopy
and x-ray microscopy use ionizing radiation, meaning that the energies that atoms or
molecules receive when absorbing these particles are higher than at least the binding
energies of the atom’s valence electrons and often of the core electrons too. Other
challenges are resolution dependence on wavelength/photon energy and ﬂux due toscattering eﬃciency.
The ﬁrst limit on the scalability of XCDI to atomic resolution is radiation damage. This topic has been studied by many groups along with comparisons to electron
microscopy doses [23, 112, 117–119]. Miao points out that it has been widely acceptedthat at atomic scale resolution a radiation dose of 2 × 107 Gy will cause destruction ofprotein crystals in protein crystallography (1 Gray = 1 J/kg of absorbed energy) [47].
Marchesini et al present a very instructive study of radiation damage and its limits on
resolution as shown in Figure 3.6. For illumination with about 1 keV of photon energy,
the resolution limit before radiation damage destroys the feature to be resolved is about
5-8 nm in a frozen biological sample [23]. Thus, the radiation damage limited resolutionof XCDI with long exposure times is a few nanometers. However, if a suﬃcient doseof x-rays can be delivered in a few fs pulse (such as from a FEL) it has been shown
theoretically that sub-nm resolution may be possible [112, 120]. Neutze et al present
a study of such a keV FEL pulse and show that radiation damage will manifest itself
in about 10 fs [112]. This pulse duration is 1-2 orders of magnitude shorter than whatis expected to be available in the ﬁrst generation of FELs such as the Stanford Linear
Coherent Light Source (LCLS).
Finally, it is important, especially for tabletop sources with limited ﬂux, to pointout how resolution depends on x-ray dose. Both Shen et al and Marchesini et al describe

65

Figure 3.6: Study of radiation dose versus image resolution as a study of radiation
damage with electron and x-ray microscopy (from Marchesini et al [23]).

66
how in general resolution scales with photon ﬂux [23,119]. This dependence arises fromthe dependence of scattering eﬃciency as a function of spatial frequency or angle, among
other parameters. The maximum resolution, r, as a function of incident ﬂux, I0 , is given
by
r∝

⎧
⎪
⎪
⎪
⎨I −1/40

F or 2D imaging⎪⎪⎪⎩I0−1/3

F or 3D imaging.

(3.24)

Here we have assumed that the wavelength is ﬁxed. Additionally, Shen et al and Marchesini et al point out that as photon energies increase from 1-10 keV the absorption andscattering cross sections have a 1/λ2 dependence. Therefore as wavelengths are decreased in this regime, more ﬂux will be needed to achieve the same spatial resolution.
Therefore, most XCDI schemes that are focused on achieving the highest resolution arefocusing on the 1 keV range. For shorter wavelengths around EUV and SXR wavelengths
this dependence on wavelength will not be the same as there are multiple absorption
edges in this region. Therefore, in the EUV/SXR region, resolution will scale more liker ∝ λ assuming that the scattered ﬂux (not incident ﬂux) is held constant, a diﬃcult
assumption to achieve experimentally due to the numerous absorption edges.

3.7

Chapter Summary
In this chapter we have discussed the important theory involved in x-ray coher-

ent diﬀraction imaging. We reviewed relevant diﬀraction theory in order to understandhow diﬀraction patterns form a two dimensional Fourier transform of the sample surface exit wave. Next, this chapter reviewed the idea of oversampling and how thisidea, in combination with phase retrieval algorithms, provides a powerful technique for
x-ray coherent imaging. We reviewed relevant geometric relations important for successfully implementing XCDI. Finally, we discussed the limitations on the resolution
attainable.

Chapter 4First Tabletop Demonstration of Soft X-ray CDI

As was discussed in chapter 3, given the need for coherent illumination, mostsmall-scale EUV/SXR sources are not suitable for lensless imaging. Thus, to-date,lensless imaging has been the sole domain of large x-ray facilities such as synchrotrons
or free-electron lasers, where the bright beams can be made coherent by spatial ﬁltering.
Recently, the ﬁrst lensless imaging using a soft x-ray, free electron laser facility at 32 nm
was demonstrated. In that work, the high per-pulse energy of the FEL allowed single
shot diﬀraction data to be collected [51]. However, because the sample was destroyed in
the process, increasing sample statistics or time-delay measurements on a single sample
become diﬃcult. Also, low spatial frequency information about the sample was missing
due to a hole in the detector which allows the very intense FEL pulse to pass throughwithout damaging the detector. Since this ﬁrst demonstration, several other lenslessimaging demonstrations have been published, including time resolved images of pumpprobe experiments [121–124].
As was shown in chapter 2, high harmonic generation in a gas-ﬁlled waveguide
generates spatially coherent EUV beams and is ideally suited for lensless imaging [17,59, 125]. This light source has already been used for Gabor holography with resolution<10 µm [17]. Here, we present the ﬁrst experimental demonstration of high resolutiontabletop CDI using a tabletop source of coherent soft-x-rays. Much of this chapter will
come from results published in two papers by Sandberg et al with invaluable assistance in

68
the image reconstruction from members of Prof. Janwei Miao’s Coherent Imaging Groupat UCLA, primarily with the help of Dr. Changyong Song [4,5]. The ﬁrst of these papers
was coauthored with Dr. Ariel Paul, and many speciﬁc details of the development andimplementation of the experimental setup as well as alignment procedures are discussed
in chapter 6 of his dissertation [24].
Section 4.1 discusses the compact XCDI microscope setup. Section 4.2 describes
the ﬁrst CDI results obtained in reﬂection mode imaging with a few µm resolution as
well as necessary changes that needed to occur to achieve high resolution. The ﬁrst highresolution, ∼ 200 nm, images of a thick sample are presented in section 4.3 [4]. Finally,
we were able to push the resolution down to 1.5 times the illumination wavelength bygoing to a high numerical aperture regime and by using a 47 nm tabletop x-ray laser.
These results will be presented in section 4.4 [5]. These results thus demonstrate thatlensless coherent diﬀractive imaging can be successfully implemented using tabletop
soft x-ray light sources, with broad potential application in nanoimaging and biologicalimaging.

4.1

Experimental Setup
The setup used for most of the results discussed in this dissertation are shownin Figure 4.1. In our experiment, 1.3 mJ, 25 fs pulses from a titanium doped-sapphirelaser ampliﬁer system (KMLabs DragonT M [8]) are focused with a 50 cm focal length
lens into a gas-ﬁlled, hollow fused-silica waveguide at intensities of 5 × 1014 W/cm2 .
While the waveguide is evacuated, the lens to waveguide distance is varied for maximum
power coupling and mode quality. A typical distance from lens to waveguide entrance
is typically ∼ 60 cm. Phase matching of the conversion process is achieved by pressuretuning the gas as described in chapter 2. In this regime, bright emission over a combof odd-order harmonics from 25 - 31 is obtained at wavelengths near 30 nm [14, 82, 83].
The hollow waveguide is a 150 µm inner-diameter, 10 cm long fused silica capillary ﬁlled

69

Figure 4.1: Experimental setup for lensless imaging using coherent high harmonic beams
at a wavelength of 29 nm. A single harmonic order is selected and focused using a pairof normal-incidence multilayer mirrors. The sample stage is positioned near the focus,
where it scatters the soft-x-ray beam onto a CCD. Inset, measured logarithmicallyscaled soft-x-ray beam proﬁle that is a near-perfect gaussian T EM00 over 4 orders ofmagnitude (see section 2.4.2) [4].

with argon gas at ∼ 70 Torr pressure.

4.1.1

Phase matching and coherence optimization

When optimally coupled, an SXR beam is generated with a radius of ∼ 25 µm(FWHM) and a divergence of ∆θ < 1 mrad, giving a transverse coherence length ofξt equal to the full beam width and a ﬂux of ∼ 1011 photons/second. In the initial
work described in this chapter, a 10 cm long waveguide was used to insure full spatial
coherence of the 29 nm source. However, in later studies it was found that 5 cm longwaveguides could be used which increased the ﬂux by about an order of magnitude
while still maintaining the mode quality and spatial coherence. More details on the
production, characteristics, and geometry of the v-groove waveguide and in addition
other experimental details are available in chapter 6 of Dr. Ariel Paul’s dissertation [24].
Two 200 nm thick aluminum ﬁlters (Lebow Company, Inc.) are used to eliminate
the fundamental laser light (one is held in a custom light-tight ﬁxture). A pair ofnarrowband, molybdemum/silicon multilayer (40 layer pairs, one ﬂat, one with 50 cm
ROC) mirrors centered at ∼ 29 nm acts as both a monochromator and a condenser,
to gently focus the beam onto the sample with a beam waist to about 1.5-2 times the

70sample diameter, D. The position of these mirrors is adjusted so as to optimally focus
the beam on the sample by using a Gaussian beam propagation model (BeamView).
Typical distances are around 1 m from the ﬁber output to the ﬁrst mirror, about 10 cm
between the mirrors, and about 20-30 cm between the second mirror and the sample.
The narrowband mirrors each have a reﬂectivity of 25% in a 2.5 nm bandpass, toeﬀectively select a single harmonic order. After reﬂecting from these mirrors, the soft
x-ray beam has a narrow bandwidth of λ/∆λ > 250, giving a longitudinal coherenceξl = λ/∆λ > 7.3 µm. The sample is held in an x-y stage controlled by high-precision
closed-loop dc motors (Physik Instrumente L.P.). The diﬀraction of the SXR light fromthe sample is recorded on a large-area x-ray CCD camera (Andor) with a 2048 x 2048
array of 13.5 µm pixels [126].
The bandwidth of the source is substantially smaller than that set by the mirrors,
thus the narrow-band structure of the selected harmonic indicates a very high level ofphase matching and the selection of essentially a very narrow band of electron trajectories. While the extent of how narrow this bandwidth can be made still needs to bestudied, there are several logical reasons why we could expect it to be this narrow. As
was mentioned earlier in section 2.4.2, Bartels et al demonstrated a very narrow bandwith (λ/∆λ = 170) by optimizing the HHG with shaped laser pulses [92]. We suggestthat a similar process is occuring here. When the HHG is optimized prior to insertinga sample into the beam line, the SXR 29 nm mode is viewed on the CCD camera after
it has bounced oﬀ the two multilayer mirrors. Thus we are already viewing essentiallya single harmonic (27th ). To optimize the ﬂux, all parameters possible are adjusted
to maximize the ﬂux and mode quality. These inlcude driving laser chirp and pulse
timing, laser focus into the waveguide, an iris that controls the diameter of the beam
before the focusing mirror, the waveguide coupling and pointing, and the gas pressure
inside the waveguide. Thus, it is feasible that by adjusting all of these parameters while
paying particular attention to the mode, we select essentially a single narrow band of

71

Figure 4.2: Large (>1 mm) and medium (∼ 0.5 mm) beamblocks used to increase
dynamic range of the diﬀraction patterns [24].

trajectories that produce a single harmonic, phase matched, Gaussian beam.

4.1.2

Beam blocks and alignement

Beam blocks of varying size are placed in the center of the diﬀraction pattern
to block the intense diﬀraction coming from low spatial frequencies. A large-diameterbeam block (>1 mm) allows us to acquire long exposure images to record the highest
spatial frequencies diﬀracted from the sample, while a small beam block (<200 µm
diameter) is used to record low spatial frequencies. These diﬀraction patterns are thenstitched together, essentially extending the dynamic range of the camera from 4 to 12
orders of magnitude. These beam blocks are supported by a 12.5 µm wire tethered to a2-inch diameter mounting ring, held in a kinematic x-y lens translator which allows ﬁne
control of the beam block position. For additional information about the beam blocks,
see Paul chapter 6 [24].
As was mentioned in chapter 3, a high quality, spatially coherent, uniform soft-xray mode is needed for lensless imaging to ensure the sharpest diﬀraction data. Figure
4.1 (also Figure 2.4.2.1) shows the mode proﬁle of the 29 nm SXR beam, which ﬁts to72
a near-perfect Gaussian to the full dynamic range of the camera (∼ 104 ). The sample
is carefully placed to optimize the illumination, diﬀraction quality, and oversampling
ratio. A ﬂat SXR intensity proﬁle, with a variation less than 10 percent, is required.
Overﬁlling the sample leads to a ﬂatter intensity and also tends to ensure long-term
illumination stability, mode quality, and a large radius of curvature of the soft-x-ray
beam compared to the aperture size.
The focal spot size, numerical aperture, and oversampling can be adjusted bymoving the mirrors and sample stage in z (along the beam axis). The sample also
must be placed far enough from the detector to guarantee a far ﬁeld diﬀraction pattern,
given by z ≥ D2 /λ, where z is the sample to CCD distance, D is the sample diameter,
λ is the wavelength. Finally, the distance from the sample to the CCD is chosen to
give an appropriate linear oversampling ratio (>5) that allows for easily reconstructablediﬀraction patterns with high resolution [105]. As described in section 3.6, the linear
oversampling ratio relates the smallest diﬀraction pattern speckles to CCD pixels, andis given by O = zλ/pD, where p is the pixel size of the CCD camera. For the two images
described here (‘J’ slit and 15 µm Quantifoil), the linear oversampling ratios were ∼10. In these reconstructed images, each image pixel (not to be confused with a CCD
pixel size p), corresponds to a size pim given by pim = OD/N , where N is the numberof pixels.

4.2

First XCDI Results
While the previous section described the experimental setup used for the majorityof the work, this section will brieﬂy discuss an experimental setup and results where
the sample was held in reﬂection mode. This work was never published but has some
importance, as two of the applications that will be pursued with tabletop XCDI will
be defect inspection of EUV lithography substrates and metrology of EUV lithography
optics, both of which will need to be done in reﬂection mode.

73

Figure 4.3: Reﬂection geometry used with previous vacuum chamber for imaging ofRon the ﬁrst Mo/Si mirror.
Quantifoil

R
Figure 4.4: SEM image of Quantifoil
“Multi-A” carbon foil [25]. The holey carbon
ﬁlm is about 20 nm thick and rests on a 50 µm picth copper grid with about 10 µm
wide gride lines.

74
Prior to the current “EUV Lensless Imaging Testbed” (ELITE) that was desinged
and manufactured by Ariel Paul and Hans Greene (see Paul, chapter 6 [24], we useda smaller vacuum chamber that was large enough for only two Mo/Si mirrors. This
setup is shown in Figure 4.3. The reﬂection geometry setup used the same laser, highharmonic cell, metal ﬁlters, and Mo/Si optics as described in the previous section. Thus,
we expect the ﬂux and coherence properties of the light to be the same.

4.2.1

Reﬂection mode results with 150 µm apertured Quantifoil

R“Multi-A” carbon foil [25]. The
The sample we used here was a Quantifoil

sample consists of a commercially available transmission electron microscopy (TEM)
grid that was aﬃxed to a 150 µm pinhole on a thin metal shim. The TEM grid consistsof thin 10 µm wide copper grids spaced by about 50 µm (approximately 40 µm x 40
µm open area) with a holey carbon ﬁlm on top that is approximately 20 nm thick as
is shown in Figure 4.4. The pinhole and TEM grid were then aﬃxed to the ﬁrst (ﬂat)
Mo/Si multilayer mirror with the TEM holey ﬁlm immediately aﬃxed to the mirror. In
order to increase the opacity of the TEM ﬁlm, ∼ 100 nm of gold were sputter coatedonto the ﬁlm before it was aﬃxed to the mirror.
The results of the reﬂection mode imaging are shown in Figure 4.5 with about
1 hour of exposure time. The top pane of Figure 4.5 shows the high spatial frequency
data with about 1 hour of exposure with an oversampling ratio of O = 7.7. A large
beam block (>1 mm diamter) blocks the more intense spatial frequencies and a vacuum
tube connecting the chamber and CCD apertures some high spatial frequency data.
Additional exposures without the beam block (not shown here) were also taken in order
to ﬁll in the missing low spatial frequency data. Figure 4.5 (bottom pane) shows thereconstruction using the GHIO algorithm (to be described later). The basic structureof the Quantifoil can be seen (refer to Figure 4.4), however the grid spacing to pitch
ratio of the copper grid seems to be smaller here (perhaps 1/4 or 1/3) compared to75

Figure 4.5: A portion of the high spatial frequency data for the reﬂection geometry
on a linear scale (top), the reconstructed image showing some basic features but poor
resolution of about 2 µm (bottom) [24].

76
the sample (1/5). This is probably due to a limiting of the open aperture caused by
the sample being held on a mirror at slightly less than normal incidence (∼ 8 degrees).
The resolution here is on par with one of the larger holes in the ﬁlm, so about 2 µm as
determined from the highest momentum transfer, Qmax .
4.2.2

Improvements

While performing this ﬁrst demonstration in reﬂection mode, Ariel Paul and I
realized that in order to achieve ∼ 100 nm resolution, we needed to get the sample muchcloser to the camera. This initiated a new design that led to the current ELITE chamber.
Additionally, a number of stability improvements were initiated such as ﬂoating theoptical table in order to reduce vibrations and enclosing the laser beam path in boxes
to reduce air currents. These improvements as well as the eﬀect they had on the data
are described in Paul chapter 6 [24].

4.3First High Resolution Tabletop XCDI
After the new ELITE chamber was ﬁnished, the experiment was reconﬁgured into

the setup described earlier in section 4.1. With the new setup, we were ready to try
a simple transmission sample for demonstration purposes. These results are describedin the paper that Ariel Paul and I coauthored with others in Physical Review Letters,
and much of the text in this section comes from that paper [4] (see also Paul chapter
6 [24]). We used two objects for these experiments: a rectangular “J” aperture with alength of 80 µm, and the same 20 nm thick carbon Quantifoil with holes of various sizes
described in section 4.2, placed over a 15 µm diameter aperture in a thin metal ﬁlm.

4.3.1

Results with “J” slit sample

For the “J” object, the linear oversampling ratio was 9, with a sample to CCD
distance z = 33 cm and a sample size D = 80 µm, resulting in an image resolution of

77

Figure 4.6: (a) Optical image of J-slit. (b) Oversampled diﬀraction pattern from thissample where the diﬀraction at the edge of the image corresponds to a momentum
transfer of 0.018 nm−1 . (c) Magnitude of the reconstructed lensless image. [4]∼ 1 µm. For the carbon TEM ﬁlm, the linear oversampling ratio was 13, with z = 9
cm and D = 15 µm. Figure 4.6 (b) shows the coherent diﬀraction pattern from the “J”
slit. Three diﬀerent diﬀraction patterns, shown in Figure 4.7, were stitched together
to increase the dynamic range, using a 1 minute exposure with no beam block, a 10
minute exposure with a ∼ 200 µm beam block, and a 120 minute exposure with a ∼3 mm beam block. Figure 4.6(c) shows the ﬁnal image of the ‘J’ pattern, with eachpixel corresponding to 347 nm. The reconstructed image is consistent with the optical
microscope image shown in Figure 4.6 (a). However, electron density noise can be seenin this image, arising from long-term instability of the SXR source and the ﬁnite spectral
bandwidth of λ/∆λ ∼ 250 at 29 nm combined with the large path diﬀerences in thediﬀraction pattern that arise with this large sample.
To reduce noise in the intensity of the ﬁnal diﬀraction patterns, we applied an inverse Fourier transform to the measured intensity to obtain the autocorrelation functionof the sample. Since the linear oversampling ratio is  2, the autocorrelation function
is surrounded by a large region that should have no signal. However, because of camera
noise that signal is not exactly zero. We therefore applied a low-pass ﬁlter to force thisregion to be zero. We then numerically integrated the diﬀraction intensity by binning

78

Figure 4.7: Logarithmic raw diﬀraction patterns from the “J” slit sample: (top) 1
minute exposure with no beam block, (left) 10 minute exposure with a 200 µm beam
block, (right) 120 minute exposure with a 3 mm beam block. The ﬁgures represent thefull area of the camera and have not been cropped.

79
3 × 3 pixels into 1 pixel and applying a deconvolution to remove the artifacts in thediﬀraction pattern due to intensity integration [20]. This step signiﬁcantly enhanced
the signal-to-noise ratio of the coherent diﬀraction pattern. The analyzed diﬀraction
pattern has a linear oversampling ratio ∼ 1/3 smaller, and an array size of 680 × 680
pixels. This also eases the requirement on the temporal coherence by a factor of 3.

4.3.2

Phase retrieval reconstruction using GHIO

Phase retrieval of the coherent diﬀraction pattern is carried out using the guided
hybrid-input-output (GHIO) algorithm (see section 3.4 or references for more details [20,47, 109]). This algorithm starts with 16 independent reconstructions of the diﬀraction
pattern, where random initial phases are used as the input. Each reconstruction iterates
back and forth between real and reciprocal space. In real space, a support is deﬁned
as a rectangular area containing the sample, with its size estimated from the linear
oversampling ratio. The sample density outside the support and the negative real or
imaginary part of the electron density inside the support are slowly pushed to zero.
In reciprocal space, the magnitude of the Fourier transform (i.e. square root of thediﬀraction intensity) remains unchanged, and the phase is updated with each iteration.
After 2000 iterations, 16 images are reconstructed, which is deﬁned as the 0th generation. An error value (magnitude of retrieved amplitude minus measured amplitude
divided by measured amplitude) is calculated for each image based on the diﬀerence between the measured and calculated magnitude of Fourier transform. The image with thesmallest R value is selected as the seed. The inputs for the 1st generation are obtainedby multiplying the seed with each of the 16 images and taking the square root of the
product. We repeat the procedure for the next generation, and after the 8th generation,
the 16 reconstructed images became consistent. From the reconstructed images, we deﬁne a tight support that represents the true envelope of the object. Using this tightsupport, we use another GHIO run to obtain the ﬁnal reconstructed image. Because

80
the diﬀraction pattern is non-centro-symmetric, the electron density of the sample iscomplex, which in principle makes the phase retrieval more diﬃcult than for real objects. By using a tight support with GHIO, and imposing a positivity constraint on
both the real and imaginary parts, we have shown that complex objects can be reliably
reconstructed from oversampled diﬀraction patterns.

4.3.3

Results with 15 µm apertured Quantifoil

A lensless image of the carbon ﬁlm, apertured to 15 µm, is shown in Figure
4.8. The same beam blocks were used as with the ‘J’ sample, and exposure times werecomparable except for the large beam block data which was about 2 hours. Here wewill only present the patched together ﬁnal diﬀraction pattern.Figure 4.8(a) is an SEM
picture of the sample, while Figure 4.8(b) shows the diﬀraction pattern. The noncentral-symmetry of the diﬀraction pattern indicates that the sample has absorption
and that the sample density is complex. Figure 4.8(c) shows the magnitude of thedensity for the lensless image, which agrees well with the SEM image. The slightdisagreement between the alignment of the holes in the SEM and lensless images is due
to parallax. The carbon ﬁlm and mounting aperture are separated by approximately 56µm - thus even a few degrees of tilt in the SEM stage slightly alters the parallax andthe exact alignment of the smaller holes in the SEM image with respect to the large
aperture. A line scan of the reconstructed image indicates that the current tabletop
lensless microscope has a resolution of 214 nm.
The parallax of the sample was discovered when one of the reviewers pointed outthat the hole patterns in the carbon ﬁlm between Figure 4.8 (a) and c did not agree.
We checked experimentally that the diﬀerences between the SEM and x-ray images canbe wholly explained by parallax resulting from the diﬀerent viewing angles of the SEM
images. As stated above, the plane of the Quantifoil ﬁlm and the aperture are separated
by some distance (56 m), as is made clear in the diﬀerent SEM images shown in Figure

81

Figure 4.8: (a) SEM image of a masked carbon TEM ﬁlm; (b) oversampled soft-xray diﬀraction pattern where the diﬀraction at the edge of the image corresponds toa momentum transfer of 0.059 nm−1 ; and (c) magnitude of the reconstructed lenslessimage. The correspondence in size, number and position of holes, and aspect ratios
between the EUV image and the SEM image can be seen. One pixel in the reconstructed
image corresponds to 107 nm. The inset in (c) show a line-scan taken along the direction
noted with the green solid line, demonstrating a spatial resolution of 214 nm [4].

82
4.9. By tilting the SEM stage by only a few degrees, we see that the number of holesthat appear above the large 15 m aperture change. This ﬁnding adds strength to theargument that XCDI has a long depth of focus. Note that by measuring the shift in theholes, we can estimate the separation of the two ﬁlms. With a tilt of 8 degrees, the holes
in the ﬁlm shift by 8 µm with respect to the aperture, giving us a separation of the twoplanes (aperture and quantifol) of about 56 µm. Since this distance is more than threetimes the size of the 15 µm laser-ablated aperture, there would be a signiﬁcant amountof parallax. Thus we see that Figure 4.8 (c) and Figure 4.9 (b) agree well.
While these initial results were impressive, the resolutions are still low; comparable to a conventional optical microscope (see chapter 1). Three approaches are possiblefor improving the resolution of the microscope. First, higher spatial frequencies could
be captured by going to a higher numerical aperture. This could be accomplished either
by using a larger detector or moving the sample closer to the CCD. Second, the eﬀective spectral bandwidth of the source could be improved by narrowing the individual
harmonic peak or by using a tabletop SXR laser. Third, shorter wavelength HHG light
could be used. The ﬁrst two of these suggestions will be presented in the next section.

Figure 4.9: Scanning Electron Microscope images of 40 nm thick holey carbon ﬁlm
placed approximately 56 µm above a 15 µm aperture: (a) SEM stage at 0 degrees tilt,(b) SEM stage at -1.8 degrees tilt, (c) SEM stage at +1.8 degrees tilt, (d) SEM image
at +8 degrees tilt.

83

4.4

Pushing the resolution: High Numerical Aperture XCDIThis section will describe work conducted to push XCDI to the sub-100 nm regime.

The work continues with 29 nm HHG generation, but in a higher numerical aperture
regime. Additionally, it will describe the work conducted on a tabletop SXR capillarydischarge laser operating at 46.9 nm [75]. Both of these demonstrations will use highnumerical aperture (NA) and achieve sub-100 nm resolution. Due to the high NA, acurvature correction scheme was developed to account for the high angle diﬀraction
and allows near-diﬀraction-limited resolution of 1.5 λ. The SXR laser work was donewith Przemek Wachulak and collaborators at Colorado State University in Fort Collins,
Colorado in the groups of Mario Marconi, Jorge Rocca, and Carmen Menoni in theElectrical and Computer Engineering Department. The results of this section were
published in the Proceedings of the National Academy of Sciences of the USA and muchof the text in this section comes from that publication [5].

4.4.1

Experimental setup for high NA XCDIThe high NA experimental setup is shown in Figure 4.10. The same chamberas described earlier in section 4.1 was used. Coherent beams from either a soft xray laser or a high harmonic source are focused onto a sample using the same simple
two-mirror relay arrangement. Light scattered from the object is then detected using a
CCD camera positioned a few centimeters from the sample (typically 1.7 cm). Thin ﬁlm
ﬁlters are used to block unwanted longer wavelength radiation such as plasma emission
from the laser. In the case of the soft x-ray laser, a single 100 nm thick aluminumﬁlter was used to block plasma emission at longer wavelengths. For the high harmonic
beams, two 200 nm thick aluminum ﬁlters blocked the driving laser light at 780 nm.
Two multilayer mirrors, one ﬂat and one with a 50 cm radius of curvature, were used
to gently focus the soft x-ray beams onto the sample. Diﬀerent bilayer thickness and84

Figure 4.10: Setup of the tabletop soft x-ray diﬀraction microscope. A coherent soft xray beam is produced by high harmonic generation or by spatially ﬁltering the emission
from a capillary discharge laser in an Ar-ﬁlled hollow waveguide. The beam is gently
focused by a multilayer mirror pair onto the sample, and the diﬀraction pattern iscollected on an x-ray CCD. A moveable beam block allows for brighter portions of
the diﬀraction pattern to be blocked to avoid saturating the CCD when acquiring thehighest angle diﬀracted light [5].

85
materials (Sc/Si for 46.9nm or Mo/Si for 29 nm) were used to optimize reﬂection forthe diﬀerent wavelengths. The soft x-ray laser mirrors are 40% reﬂective at 46.9 nm,
while the mirrors used to focus the high harmonic beam are 28% reﬂective at 29 nm.
The soft x-ray laser source is based on fast electric-discharge excitation in a thin
capillary that creates a highly-ionized neon-like argon plasma lasing at λ = 46.9 nm.
It produces pulses of energy ∼ 0.2 mJ with ∼ 4.5 mrad divergence at a repetition rateof 1 Hz [75]. Lasing takes place on the 3s1 P1 to 3p1 S0 transition of neon-like argon
by exciting argon gas in an alumina capillary (3.2 mm diameter, 27 cm long) with acurrent pulse of amplitude ≈ 24 kA with a 10% to 90% rise time of ≈ 40 ns [75,127,128].
This laser occupies a footprint of 0.4 x 1 m2 . The fast current pulse is produced bydischarging a cylindrical capacitor through a spark-gap switch connected in series withthe capillary load. The current pulse rapidly compresses the plasma column to achievea dense and hot plasma ﬁlament where a population inversion is created by electron
impact excitation of the laser upper level and rapid radiative relaxation of the laser
lower level [6]. The capacitor employs a water dielectric which provides high energy
density and also cools the capillary. A continuous ﬂow of Ar maintains a pressure of
490 mTorr in the capillary channel.
This laser is based on high-gain ampliﬁcation of spontaneous emission from thelaser transition and although it does not use a resonator, it nevertheless generates abeam with high spatial and temporal coherence. The longitudinal coherence length isdetermined by the Doppler-broadened linewidth of the laser transition at λ/∆λ ∼ 104 ,corresponding to a longitudinal coherence length of ξl ∼ 500 µm. The degree of spatial
coherence increases with the length of the plasma column, and approaches full coherence
for 36 cm long capillaries [96]. In this experiment, the laser output was spatially ﬁltered
to ensure good spatial coherence using a 1.5 mm diameter pinhole placed approximately
1.5 m from the output of the laser, where the beam is ≈ 1.5 cm in diameter. We estimate(using the van Cittert-Zernike theorem) that the beam has at least 85% fringe visibility

86
at the edges of the pinhole [6].
For both soft x-ray sources, the sample size was 7 µm. For the soft x-ray laser at
46.9 nm, the sample to CCD distance was 17 mm, giving an oversampling ratio of 8.5.
For the high harmonic source at 29 nm, the sample to CCD distance was 42 mm, giving
an oversampling ratio of 13. The samples used for both sources were 7 µm tall “waving
stick girl” ﬁgures etched into a 100 nm thick silicon nitride window with electron-beamlithography by Anne Sakdinawat of the Center for X-ray Optics and Lawrence Berkeley
National Laboratory. Since the silicon nitride is ∼ 4% transmissive to 29 nm photons,
200 nm of gold was electroplated on the sample used with the high harmonic beam.

4.4.1.1

Phase retrieval of high NA data

Three separate exposures were used with similar beam blocks as described in
section 4.1.2 and section 4.3.1 for both data sets, however for the high NA HHG data
multiple shorter exposures were used. Similar data patching and processing steps weretaken as was the case with the intial HHG XCDI data.All data sets were cropped
down to an array of good signal to noise ratio data. For the SXR laser this was 1980
pixels square. For the HHG data included in the publication it was 2028 pixels. Afterbackground subtraction and ﬁltering, the diﬀraction intensity was binned by numericallyintegrating 3x3 pixels into 1 pixel and then deconvolved with a sinc function [20].
Due to the strong absorption of soft x-rays by the specimen, the diﬀraction patterns in this section are non-centro-symmetric, resulting in the complex electron density. Phase retrieval of the complex electron density was carried out by using the guided
hybrid-input-output (GHIO) algorithm (see sections 3.4 and 4.3.2) [20, 105]. This work
used 20 initial reconstructions. After 3000 iterations, 20 images were reconstructed,
which was deﬁned as the 0th generation. An R-value was calculated for each image
based on the diﬀerence between the measured and calculated magnitude of the Fouriertransform. A seed image with the smallest R-value was selected for the reconstruction.

87
By multiplying the seed with each of the 20 images and taking the square root of the
product, a new set of 20 images was obtained, which was used as the initial inputs forthe next generation. We repeated the procedure for the next generation, and after the
7th generation, the 20 reconstructed images became consistent. The ﬁnal reconstruction
was chosen from the average of the best 3 images with the smallest R-values.

4.4.2

High NA results with 46.9 nm SXR laser

The coherent diﬀraction pattern from the ﬁrst stick girl sample when illuminatedby the soft x-ray laser for an exposure time of 300 shots (5 minutes at 1 Hz) is shown in
Figure 4.11 (a). The ﬁnal summed and processed diﬀraction pattern was then curvaturecorrected (explained in the next section) and is shown in Figure 4.11 (b). This diﬀraction
has good signal to noise out to a 38 degree angle, or a NA = 0.62. This numericalaperture represented at the time the highest numerical aperture x-ray imaging at any
wavelength.
Figure 4.11 (c) shows the ﬁnal reconstruction of the stick ﬁgure. A lineout of thereconstructed image is shown in Figure 4.11 (c), indicating a resolution of 71 nm. This
resolution, at 1.5 λ, compares very well with other lensless imaging experiments doneat large scale facilities. Furthermore, our calculations have indicated that the resolution
could reach the diﬀraction limit. The current resolution of 71 nm is limited by thesize of the CCD detector, because the diﬀraction pattern in Figure 4.11 (b) was clearlycut oﬀ by the CCD edges. Using a larger CCD detector and the curvature-correction
procedure, it should be possible to achieve diﬀraction-limited resolution using a tabletop
soft x-ray diﬀraction microscope.

4.4.3

Curvature correction for high NA diﬀraction

The experimental setup was conﬁgured to enhance the spatial resolution by placing the detector as close to the sample as possible, which increases the numerical aper-

88

Figure 4.11: Lensless imaging using coherent soft x-ray laser beams at 47 nm. (a) Reﬂection image of the waving stick ﬁgure sample (scale bar = 1 micron) using an SEM.
(b) Coherent soft x-ray diﬀraction pattern (in transmission) after curvature correction(maximum momentum transfer of 0.166 nm−1 at the edge of the CCD). (c) Reconstructed image with curvature correction. (d) Line-out of the image along the legs
(shown in the inset), verifying a resolution of 71 nm [5].

89
ture of the microscope (up to NA = 0.62 here). In this mode of operation, the approximation of sin(θ) ≈ θ is no longer valid, and therefore the diﬀraction intensity on theplanar CCD is signiﬁcantly distorted at higher angles, as can be seen in Figures 4.12(a) and (b). To remove the distortion, we interpolated the diﬀraction pattern from aplane onto a spherical surface. Figures 4.12 (d) and (e) show the curvature-corrected
diﬀraction pattern, where the radial stretching and skew of the speckles near the edgeof the diﬀraction pattern (corresponding to high diﬀracted angles) are clearly removed.
Figures 4.12 (c) and (f) show the magnitude of the reconstructed image before andafter curvature-correction, indicating that curvature-correction indeed signiﬁcantly improves the quality of the reconstruction. The lower width and increased sharpness ofthe curvature-corrected stick-girl image can be seen in the lineouts shown in the inset
of Figure 4.12 (c) and (f). This correction of mapping the diﬀraction pattern onto acurved surface is really only a ﬁrst approximation correction. A forthcoming publicationthat is an extension of this work with a more detailed curvature correction will soon besubmitted with Kevin Raines and colleagues from John Miao’s group at the Universityof California at Los Angeles.

90

Figure 4.12: Curvature correction in high numerical aperture lensless imaging. (a), (d)Coherent soft x-ray diﬀraction pattern at 47 nm wavelength (in transmission) before
and after curvature correction (maximum momentum transfer of 0.166 nm−1 at the edgeof the CCD). (b), (e) Magniﬁed diﬀraction pattern at high diﬀraction angles showing
the radial blurring near the upper right edge in (b), which is absent in (e). (c), (f)Reconstructed image of waving stick ﬁgure without and with curvature correction. The
lower width and increased sharpness of the curvature-corrected stick-girl image can beseen in the lineouts shown in the insets [5].

91
4.4.4

High NA results with 29 nm HHG

After the success of the high NA XCDI on the tabletop SXR laser, we repeated theexperiment with a similar sample on the 29 nm HHG source. As mentioned before, thissample was coated with 200 nm of gold to prevent transmission of the 29 nm photons
through the 100 nm Si3 N4 substrate. Also, as was mentioned in section 4.1, for thiswork a shorter 5 cm waveguide was used which increased the 29 nm ﬂux by almost
an order of magnitude (to around 108 photons/s or 0.7 nW on the sample) while still
maintaining a fully Gaussian mode and full spatial coherence. Two data sets are shown
here that both have maximum scattering angles of about 18 degrees corresponding toa NA = 0.31. The ﬁrst data set shown here actually came after the second, because
only the ﬁrst data set was published due to its higher image quality [5]. However bothdata sets demonstrate sub-100 nm resolution, a factor of two improvement over the
Quantifoil data [4] and previously reported HHG zone plate microscopes [55].
Figure 4.13 summarizes the ﬁrst data set of high resolution imaging results using
a high harmonic source. A reﬂection image of a gold coated stick girl sample using an
SEM is shown in Figure 4.13 (a), while Figure 4.13 (b) shows the curvature correctedhigh dynamic range soft x-ray diﬀraction pattern in transmission (NA = 0.31). Figure
4.13 (c) shows the reconstruction of the 7 µm stick ﬁgure, with a resolution of 94 nm
veriﬁed in the lineout of Figure 4.13 (d). The reconstructed image has excellent ﬁdelity
with the SEM image, with the exception of missing the right hand. This is due to theﬁnite resolution of the reconstruction, which is greater than the hand feature size, andpossible incomplete etching of the substrate in this region.
Compared with Figure 4.11 (c), Figure 4.13 (c) shows slightly more internal density ﬂuctuations, which is likely due to the lower temporal coherence (higher spectral
bandwidth) of the high harmonic source compared with the soft x-ray laser. On theother hand, the region outside the sample boundary in Figure 4.13 (c) is much cleaner,

92

a

b

c

d

Figure 4.13: Lensless imaging using coherent high harmonic beams at 29 nm. (a)Reﬂection image of the sample (scale bar = 1 micron) using an SEM. (b) Coherent
soft x-ray diﬀraction pattern (in transmission) after curvature correction (maximum
momentum transfer of 0.134 nm−1 at the edge of the CCD). (c) Reconstructed image(d) lineout of the image along the legs (shown in the inset), demonstrating a resolutionof 94 nm [5].

93
which is attributed to the higher spatial coherence of the high harmonic source. In
addition, the higher spatial coherence of the high harmonic source allows all the output ﬂux to be used, while the higher spectral bandwidth can support sub-10 fs pulse
duration for dynamic imaging experiments. Furthermore, the bandwidth of the highharmonic source can be engineered through a variety of coherent control techniques,
which will make it possible to precisely determine the time-space resolution tradeoﬀ forsuch a source. In the case of the soft x-ray laser, an increase of the capillary length
can simultaneously make the output essentially fully spatially coherent and signiﬁcantly
increase the pulse energy, making single shot exposure possible. These two sources arecomplementary and have actually been combined in a recent demonstration of HHG
ampliﬁcation in a tabletop SXR laser [58]. This source will prove very useful for single
shot XCDI as it exhibits full spatial and temporal coherence.
Figure 4.14 shows the older high NA results from 29 nm HHG. This very interesting sample was sputter coated with a small DC sputtering chamber at JILA after
already having been etched through using standard electron beam lithography. The
sample in Figure 4.13 was coated with gold before etching, thus it has much cleaner
edges. This result had signiﬁcantly more noise in the diﬀraction pattern and reconstruction probably due to instability in the diﬀraction after an hour of exposure time for thelarge beam block data. All previous results in this chapter, besides Figure 4.13, weretaken in the same way. A single hour long exposure on a CCD allows for signiﬁcantaccumulations of thermal and cosmic ray noise. However, the data in Figure 4.13 wasrecorded through 6 separate 20 minute exposures for 2 hours of exposure. This method
allowed the individual exposures to be compared in order to make removal of cosmic
rays easier. Additionally, this allowed for monitoring of the diﬀraction signal between
the individual exposures. If the signal started to drift, the coupling into the front of
the waveguide could be adjusted without aﬀecting the harmonic beam pointing, thus
allowing us to bring back the HHG signal. This greatly increased the stability of the94

Figure 4.14: A previous demonstration of lensless imaging using coherent high harmonic
beams at 29 nm. (a) Reﬂection image of the sample (scale bar = 1 micron) using
an SEM. (b) Coherent soft x-ray diﬀraction pattern (in transmission) after curvaturecorrection (maximum momentum transfer of up to 0.134 nm−1 at the edge of the CCD).
(c) Reconstructed image (d) lineout of the image along the legs (shown in the inset),demonstrating a resolution of 92.5 nm.

95
signal and quality of the diﬀraction data.

4.5

Chapter Summary
This chapter presents the ﬁrst high resolution XCDI results with tabletop coher-

ent x-ray sources. Both reﬂection and transmission imaging schemes were presented.
Approximately 200 nm resolution was achieved in a thick (∼ 56 µm) sample using a 29
nm HHG source [4]. Higher resolutions (sub-100 nm) were shown using high numericalaperture geometries with two sources: a 46.9 nm capillary electrical discharge laser andthe same 29 nm HHG source [5]. This high NA scheme, with NA up to 0.62, was aﬁrst for x-ray imaging and made possible near wavelength limited resolution (1.5 λ).
As we scale these techniques to shorter wavelengths it may be possible to obtain sub-10
nm resolution with 3 nm water window light. However, at these shorter wavelengths
tabletop sources will have lower ﬂux levels. The next chapter will discuss a method of
increasing image signal to noise with very low exposure levels.

Chapter 5

Pushing the Limits: Tabletop Fourier Transform Holography

The previous chapter demonstrated that tabletop coherent SXR sources can enable near wavelength limited XCDI [4,5]. However, the wavelengths used (29 and 47 nm)
are not useful for probing thicker samples because they are very strongly absorbed [6].
Therefore, shorter wavelengths are desired to truly enable the usefulness of tabletop
XCDI. As these shorter wavelength sources become available, they will initially have
limited ﬂux. Multi-reference holographies have demonstrated that good signal to noise
images can be produced with relatively low ﬂux [15, 116]. This chapter demonstrates
one such holography, multi-reference Fourier transform holography on a tabletop coherent SXR source. These results came as part of an ongoing collaboration with Bill
Schlotter and have been accepted for publication in Optics Letters [26]. Much of thetext in this chapter has come out of drafts for that paper. Bill Schlotter’s dissertation
is an excellent source for the theory, application, and experimental considerations of FT
holography [15].

5.1

Toward shorter wavelength tabletop XCDI
As was mentioned in section 3.5.2, another diﬀractive imaging technique that iscomplementary to CDI with iterative phase retrieval is Fourier transform holography(FT holography). Recent work using light from synchrotron sources and the FLASH freeelectron laser demonstrated that the image quality of FT holography can be improved

97with short exposure times by employing multiple or parallel references beams [116,123, 129, 130]. Thus FT holography is an attractive technique for tabletop soft x-ray
microscopy where sources have limited ﬂux.
In this work, we present the ﬁrst implementation of multiple reference Fouriertransform holography on a tabletop coherent x-ray source. We achieve 89 nm image
resolution using 29 nm light, showing for the ﬁrst time that multiple reference Fouriertransform holography is viable for imaging with high numerical apertures (as high as
NA >0.5). Then, using oversampling and an iterative algorithm to reﬁne the Fouriertransform holography images, we improve our image resolutions to <53 nm, demonstrating the highest numerical aperture x-ray holography to date. Finally, we show that
Fourier transform holography can be used eﬀectively with exposure times as short as
30 s for an image resolution of 125 nm. This represents a signiﬁcant improvement overprior demonstrations of lensless imaging using high harmonic sources - where exposure
times of more than an hour are required for comparable image quality [4, 5]. Because
the ultrafast laser system used is available in many laboratories worldwide, this work
can ﬁnd application in many areas of science and technology.

5.2

Theory of Fourier Transform Holography
In FT holography, an unknown sample object and a known reference structure are

illuminated with a single coherent beam. Interference between the scattered light fromreference and object are detected in the far ﬁeld as a Fourier transform hologram [113,114,131–134]. This is demonstrated schematically in 1D in Figure 5.1. The hologram isrecorded as a diﬀraction pattern on an area detector-typically a charged-coupled device(CCD) camera. An image of the sample is then easily reconstructed by computing themagnitude of the two dimensional Fourier transformation of the detected hologram.
To obtain the highest possible image resolution, the reference beam should diverge
very strongly (corresponding to a small point spread function), thus requiring a small

98

Figure 5.1: Schematic of Fourier relationship and Fourier Transform holography in 1
dimension (from Schlotter [15]). Column one shows diﬀerent transmission functions(g(x)). Column two shows |G|2 , the magnitude squared of the Fourier transform (farﬁeld diﬀracted
 intensity) of g(x) in column one. Column three shows the autocorre
lation F |G|2 of the diﬀracted wave. Row (a) shows the transmission, diﬀraction,
and autocorrelation of a single slit. Row (b) shows the same information, except fora double slit. Notice now that in the autocorrelation there is a self-correlation peak(center) and cross-correlations peaks between the two slits (side peaks). Row (c) showsa sample (trapezoid) and delta-function-like reference point in transmission, diﬀraction,
and autocorrelation. Now the autocorrelation cross-correlation peaks produce an imageof the trapezoid sampel and its complex conjugate. This can be seen expecially in Row(d) where the trapezoid sample is complex.

99
reference source size located in the same plane as the sample. The size of the referencehole ultimately puts the limit on the image resolution obtainable in FT holography.
Schlotter sets this limit at about 70% of the reference hole diameter in section 2.2[15]. Several methods have been used to produce a small reference beam at soft xray wavelengths, such as tightly focusing a beam with a Fresnel zone plate lens [135],
introducing a small scattering particle [136, 137], or drilling a small reference hole on
the same slide that holds the sample [15, 48, 116, 138].

5.2.1

Mathematical representation of FT holography

To further understand the relevant theory of FT holography, we present a simpliﬁed mathematical treatment of it as presented in Bill Schlotter’s dissertation [15]. If
an opaque mask consisting of an object with transmission to (x, y) and a reference holewith transmission tr (x, y) are simultaneously illuminated with a coherent light sourcewith amplitude Eo , then the transmitted wave immediately behind the mask will be asum
E(x, y) = Eo to (x, y) + Eo tr (x, y) = r(x, y) + o(x, y)(5.1)

where we have made the substitution o(x, y) = Eo to (x, y) and r(x, y) = Eo tr (x, y).
In the far ﬁeld, the diﬀracted wave or hologram will take on the form of the Fouriertransformation of Eh (X, Y ) as follows
Eh (X, Y ) = F [(x, y)] = F [r(x, y)] + F [o(x, y)](5.2)

Eh = R + O.

(5.3)

Here the capital letters R and O represent the Fourier transform of r and o respectively
and X, Y are the far ﬁeld coordinates. Now, when we measure the hologram Eh in thefar ﬁeld, we only detect the intensity, Ih as+ OO∗ + OR∗ + RO∗ = |R|2 + |O|2 + OR∗ + RO∗
Ih = |R + O|2 = RR∗   
AC

XC

AC

XC

(5.4)

100
where ∗ represent complex conjugation, AC are the diﬀraction from each aperture alone
and XC represents the interference between the two apertures.
Now, in order to retrieve the image imformation, we perform the spatial autocorrelation of the hologram by Fourier transformation
F [Ih ] = F [|R|2 ] + F [|O|2 ] + F [OR∗ ] + F [RO∗ ](5.5)+ o o + o r 
+ r o
Erec = r r 

(5.6)

AC

where

XC

represnts the complex correlation, r and o are the original transmitted ﬁelds ofthe reference and object respectively and Erec is the recovered ﬁeld of the spatial autocorrelation. We see that the recovered ﬁeld contains the AC self correlation terms whichappear at the center (DC) of the autocorrelations and that these will be surrounded by
the XC cross correlation terms. These XC terms will represent a cross-correlation of
the reference hole with the object and its complex conjugate. If the reference is verysmall compared to the object, then the cross correlation is a high resolution image.

5.2.2

Multiple reference FT holography

To increase the signal-to-noise of images produced using FT holography, multiple reference beams can be used [15, 116]. In multiple reference FT holography, eachreference beam produces a separate image in the autocorrelation reconstruction. These
images can then be averaged to improve the signal to noise and image quality [15, 116].
Schlotter et al demonstrated that a high resolution image could be obtained with only
2500 detected photons with this technique [15]. Some care needs to be taken when
surounding a sample with multiple references in order to assure that the individual reconstructed sub-images do not overlap. The treatment of this issues is dealt with in
Schlotter section 4.2 [15]. This technique could be useful for obtaining images with low
ﬂux HHG sources at shorter wavelengths.

101

5.3

First Demonstration of Tabletop X-ray FT Holography
This section summarizes the experimental setup and results of tabletop FT holog-

raphy from a coherent 29 nm HHG source. To our knowledge, this is the ﬁrst demonstration of x-ray FT holography on a tabletop source with sub-100 nm resolution.

5.3.1

Experimental setup for tabletop SXR FT holography

The schematic of the experimental setup is shown in Figure 5.2. The driving
laser, harmonic setup, and illumination optics are all the same as described earlier insection 4.1 and in [4, 5]. However, here, as in the high numerical aperture work, a 5 cm
ﬁber was used in stead of a 10 cm ﬁber for increased 29 nm ﬂux (∼ 108 photons/secondor about 0.7 nW onto the sample). The mode quality and coherence properties are stillpreserved with this waveguide. Figure 5.3 shows the mode of the 29 nm beam from a
5 cm waveguide on a logarithmic scale with a power of about 0.7 nW in a bandwidthof about λ/∆λ = 250 at 29 nm. The beam saturated the CCD camera in less than 100
ms.
The test object used for this work is a nanostructured pattern inspired by the1951 USAF test target, surrounded by ﬁve fully transmissive reference holes of ∼ 125±6nm diameter, as seen in Figure 5.2. The substrate for the sample is a 100 nm thick
low stress, non-stoichiometric silicon nitride membrane. An x-ray opaque 400 nm Au
ﬁlm was deposited on the membrane by DC magnetron sputtering. The test object
was milled into the 400 nm of gold using a focused ion beam, leaving only the silicon
nitride membrane. The silicon-nitride membrane transmits ∼ 20% of the incident 29
nm light, yielding a total photon ﬂux through the test object roughly matched to thetransmission through each one of the reference holes. This ensures good contrast in theinterference fringes of the hologram, and thus of the reconstructed image. Schlotter 3.4
discusses the speciﬁcs of the sample preparation technique with FIB milling [15].

102

Figure 5.2: Schematic of multiple reference Fourier transform holography with a tabletop
soft x-ray source. A coherent soft x-ray beam with λ = 29 nm from a tabletop highharmonic source illuminates a test pattern surrounded by ﬁve reference holes. The
scattered soft x-ray light interferes and is recorded on a CCD camera in the far ﬁeld
as a hologram (log scale). Due to the high scattering angles recorded in the hologram,curvature correction is necessary. The spatial autocorrelation of the object can be
retrieved by taking the squared magnitude of the Fourier transform of the hologram. A
single sub-image from the hologram is shown demonstrating 89 nm resolution (Photo
credit Greg Kuebler) [26].

6
5
4
3
2
1
104
Figure 5.3: Logarithmic mode of a 29 nm Gaussian HHG beam from a 5 cm waveguidewith about 0.7 nW of power in a bandwidth of about 250.

103
5.3.2

Results of 29 nm tabletop FT holography

Figure 5.4 summarizes our experimental results using FT holography alone. Figure 5.4 (a) shows a logarithmic, curvature corrected hologram of the sample shown in
Figure 5.2. There is only 17 mm between the sample and the center of the detector(momentum transfer of Q = 0). Nevertheless signal can be detected above the noise
more than 700 pixels away from the center (Qmax = ±0.211 nm−1 ). Such a large numerical aperture (NA=0.5) does not comply with the paraxial approximation because
the wavefront originating at the sample will be signiﬁcantly distorted when recorded
on a ﬂat detector. Correction of the curvature is accomplished by mapping the ﬂathologram onto a spherical surface as described in section 4.4.3. This technique corrects
for distortions such as pixel skew and stretching at higher angles [5].
The hologram in Figure 5.4 (a) is a summation of 8 separate 600 second exposures that were each individually background subtracted. Additionally, ﬁxed patternnoise and cosmic rays were removed for each exposure. After the individual exposures
are summed, the hologram is curvature corrected. Because the curvature correction
introduces a region of zeros at high Q, a Gaussian ﬁlter is applied to smooth the edgeof the data. This last step prevents ringing after Fourier transformation of the hologram that would otherwise arise due to the presence of a hard edge in the data. The
magnitude squared of the correlation reconstruction is shown in Figure 5.4 (b) and results from the magnitude of the spatial Fourier transformation of the hologram. At thecenter of the autocorrelation is the self-correlation of the sample. Surrounding theseself-correlation terms are ﬁve object-to-reference cross correlations and their complex
conjugates. Surrounding these cross-correlation sub-images are the reference-reference
cross correlations. The diﬀerence in the intensities of the sub-images is caused by ﬂuctuations in the ﬂux transmitted through the reference holes due to variations in the sizeof the holes and non-uniform illumination of the sample.

104

Figure 5.4: (a) Curvature corrected logarithmic hologram from composite 4800 s of
exposure (8 separate 600 s exposures with maximum momentum transfer of Qmax =±0.211 nm−1 ). The inset shows the detail of the center of the diﬀraction pattern. (b)Magnitude squared of the autocorrelation reconstruction of the sample - the variations
in sub-image intensity are most likely due to variations in reference hole transmitted
intensity. (c) A single sub-image reconstruction with a sample line cut (blue dottedline). (d) Plot of the line-out demonstrating 89 nm resolution with a signal to noise of
73 [26].

105
One of the ﬁve sub-images from the autocorrelation shown in Figure 5.4 (b) is
enlarged in Figure 5.4 (c). The signal-to-noise ratio in this image is measured by taking
the mean of the autocorrelation signal magnitude in the red solid box and dividing it by
the standard deviation of the pixels in the dashed red box which represents a zero signal
background region. The signal-to-noise of this image is 73. Figure 5.4 (d) shows a lineout
along the dashed blue line shown in Figure 5.4 (c), demonstrating a resolution of 89 nm.
This resolution is more than four times higher than the highest resolution holographic
image previously recorded using a tabletop source [56], and is only three times theilluminating wavelength of 29 nm. This result once again demonstrates that diﬀractive
imaging can be used at very high numerical aperture with tabletop sources to achieve
very high spatial resolution [5]. Given the very high signal-to-noise demonstrated in
Figure 5.4, the exposure time can be reduced substantially. The fundamental resolution
limit for this geometry where the object is convolved with a circular reference is ∼ 70%of the diameter of the 125 nm reference holes, or ∼ 88 nm. In our current experiment,a resolution limited by the size of the reference holes, rather than the hologram signalto-noise ratio, is obtained after only about 120 seconds of exposure.

5.3.3

High resolution FT holography with short exposures

In microscopy, a sample is often searched at lower resolution to identify a particular region of interest. Fourier transform holography is inherently a fast and simple
imaging technique and the use of multiple references has myriad attributes including
improved image redundancy [15,116]. This is particularly advantageous for rapidly surveying a sample. We demonstrate this by showing that useful image resolution can beachieved after only 30 s of exposure. Figure 5.5 (a) shows the hologram recorded witha 30 s exposure, containing approximately 15,000 detected (29 nm) photons. Figure
5.5 (b) shows one sub-image, where the signal-to-noise was calculated in the same way
as for Figure 5.5 (c), and where a lineout was taken along the blue dashed line. The

106

Figure 5.5: (a) Hologram recorded from a 30 s exposure (maximum momentum transfer
of Qmax = ±0.045 nm−1 ). (b) Sub-image from the magnitude squared of the autocorrelation for a 30 s exposure showing a lineout (blue dashed line). (c) Lineout of
sub-image showing 125 nm resolution and a signal to noise ratio of 12.7. (d) Averageof 20 lineout resolution measurements (4 per sub-image) per time step as a function of
exposure time with error bars, showing the standard deviation of the resolution. Eachdiﬀraction pattern array was 2048 x 2048 pixels. The array was set to zero outside of
the region where the signal from the diﬀraction pattern dropped below the noise. This
reduced the noise in the reconstructed autocorrelation [26].

107
signal-to-noise ratio is 7, and the lineout resolution is 125 nm, as shown in Figure 5.5 (c).
We note also that the current setup has a low ratio of detection photons to generated
EUV light ( 1%), due to the mirrors and ﬁlters in the beam path. Substantial reductions in exposure time are possible using a higher throughput setup, perhaps enablingnear real-time diﬀractive imaging. Figure 5.5 (d) shows how the resolution scales as afunction of exposure time for FT holography by taking the average of 20 lineouts (4 per
sub-image) for exposure times of 30 s, 120 s, 600 s, 1800 s, 4800 s. Figure 5.5 (d) showsthat the maximum image resolution achievable with the 125 ± 6 nm reference holes of
89 nm was attained after approximately 120 s of exposure.

5.3.4

Eﬀects of high NA on FT holography

The importance of curvature correction for high numerical aperture Fourier transform holography is illustrated in Figure 5.6. Figure 5.6 (a) shows the non-curvature corrected composite 4800 s exposure hologram. A single autocorrelation sub-image fromthe non-curvature corrected data is shown in Figure 5.6 (b). These are compared to thecurvature corrected hologram and corresponding autocorrelation sub-image in Figure
5.6 (c) and Figure 5.6 (d). The curvature correction eliminates the apparent aberrationand thus substantially increases the image sharpness and clarity.

5.3.5

Hybrid holography phase retrieval approach

In microscopy, often a particular region of interest is isolated for a detailed higher
resolution image. It is possible to signiﬁcantly enhance the resolution of Fourier transform holographic images by applying an iterative algorithm to the oversampled hologram [22, 123, 129, 130, 138, 139]. The only input necessary for this hybrid approachis the previously recorded hologram. In real space the resolution of the holographic
image reconstruction is limited by the extent of the reference. But for the hologram,in reciprocal space this restriction on resolution equates to the angular extent of the

108

Figure 5.6: (a) Log of the non-curvature corrected hologram; (b) sub-image from autocorrelation without curvature correction showing image distortion; (c) log of the curvature corrected hologram; (d) sub-image from autocorrelation with curvature correction
showing improved image quality.

109
reference wave, which is simply the Airy disk resulting from diﬀraction by the referenceholes (see Figure 5.2). If light scattered from the unknown object extends beyond thereference Airy disk of the hologram, then higher spatial frequency information about
the unknown object is recorded in the hologram - although it cannot be accessed using Fourier transform reconstruction alone. However, this information can be recovered
using iterative phase retrieval algorithms on the oversampled diﬀraction pattern or hologram [22,48,123,129,130,138,139]. This allows further improvement of the reconstructed
image resolution. As stated earlier, over a cumulative 4800 s exposure, measurable signal in the diﬀracted light was observed out to a 29◦ diﬀracted angle (momentum transfer
of 0.211 nm−1 ), corresponding to a potential image resolution of 30 nm. The measured
diﬀraction pattern was preprocessed ﬁrst by removing cosmic X-rays and subtracting
the background. Following summation of the separate 600 s exposures the cumulative
diﬀraction pattern was curvature-corrected. To maximize the signal-to-noise ratio ofthe diﬀraction pattern while maintaining the oversampling, the data were binned into
3 X 3 pixel blocks. Next, the entire pattern was deconvolved with a sinc kernel in real
space to compensate for the ﬁnite pixel size of the CCD detector [20, 47, 109].
Following this preprocessing, the iterative algorithm is applied to enhance theresolution. First an initial rectangular loose support is estimated from the oversampling
ratio. With this loose support, 16 independent reconstructions are computed with theguided hybrid input and output algorithm (GHIO) [20,47,104,109]. Each reconstruction
starts with a random phase set as the initial input and is iterated back and forth between
real and reciprocal space. In real space the electron density outside the support andnegative density inside the support are slowly pushed to zero. In reciprocal space, themeasured Fourier modulus (i.e. the square root of the diﬀraction intensity) is updated
in each iteration. After 2000 iterations, 16 images are reconstructed, deﬁned as the
0th generation. An R-factor is calculated for each image, i.e. the diﬀerence between
the calculated Fourier transform modulus from the image and the measured diﬀraction

110pattern modulus. A seed for the next generation is chosen by selecting the image withthe lowest R-factor. By multiplying the seed with each of the 16 images and taking
the square root of the product, we obtain a new set of 16 images, which is used as theinitial input for the next generation. This process is repeated through 9 generations.
After the ﬁnal generation, the best seeds are averaged and a new tighter support wasdetermined. With this new tight support, the diﬀraction pattern is again submitted to
the GHIO reconstruction process, this time yielding the ﬁnal image with a substantially
improved resolution of 53 nm, as illustrated in Figure 5.7.
Figure 5.7 (a) shows a scanning electron microscope image of the FT holography
sample. Figure 5.7 (b) shows the preprocessed, curvature corrected hologram from acomposite 4800 s exposure that was used in the iterative phase retrieval. Figure 5.7 (c)shows the ﬁnal reconstruction of the sample, demonstrating 53 nm resolution as shownby the lineout along the dashed blue line (shown in Figure 5.7 (d)).
This holography/oversampling phase retrieval hybrid technique has several notable advantages. First, while image recovery using phase retrieval takes tens of hoursof processor time, reconstruction using the FTH method takes only seconds. Thus theholographic/oversampling phase retrieval hybrid technique allows quick feedback on low
resolution information, with the possibility of further improving the resolution to thelimit set by wavelength, numerical aperture, and signal-to-noise via phase retrieval.
Also, as pointed out previously, the reference holes in Fourier transform holography
break the symmetry of the phase retrieval problem, allowing for higher image quality
and quicker convergence [123, 130]. This technique also eases the requirement for smallreference features that are diﬃcult to manufacture.

5.4

Conclusion
In conclusion, we have introduced a practical diﬀractive soft x-ray microscopy

system capable of fast feedback, high resolution imaging (near 50 nm) using Fourier

111

Figure 5.7: (a) Scanning electron microscope image of the ﬁve-reference-hole sample.
(b) Log of the curvature corrected hologram used in the iterative phase retrieval (maximum momentum transfer of Qmax = ±0.211 nm−1 ). (c) Reconstructed image of thesample using the GHIO phase retrieval algorithm to reﬁne the resolution to 53 nm, as
demonstrated in the lineout (blue dotted line) shown in (d) [26].

112transform holography. All of the data presented are from a tabletop soft x-ray sourceand demonstrate that high numerical aperture (NA >0.5) coherent diﬀractive imaging
is practical. By combining oversampling iterative phase retrieval with Fourier transform
holography, we achieve near wavelength-limited 53 nm resolution using 29 nm light, thus
demonstrating the highest numerical aperture x-ray holography to date. Moreover, goodresolution images are possible with exposure times of only 30 s. This work represents
a signiﬁcant step forward over past work, and uses ultrafast laser systems that areavailable in many laboratories worldwide.
In the future, new phase matching schemes promise to increase the high harmonicﬂux at short wavelengths down to ∼ 1 nm, for applications in ultrafast biological andmaterials imaging on the nanometer scale as well as a wide range of applications inchemical and materials spectroscopies. Because FT holography is photon eﬃcient, it
is an ideal imaging scheme to use with high harmonic sources, particularly when combined with phase retrieval to achieve the highest spatial resolution. Finally, since thesediﬀractive imaging schemes can exploit the ultrafast pulse duration capability of highharmonic generation, time resolved coherent imaging will be possible in the future withsimultaneously high spatial and temporal resolution.

Chapter 6

Outlook: Future developments and applications

This dissertation has demonstrated that tabletop x-ray coherent diﬀractive imaging is possible at high resolution (∼ 50 nm). It has demonstrated that high harmonic
generation (HHG) is a compact, bright, and extremely coherent source of EUV/SXR radiation. Additionally, tabletop soft x-ray lasers, although not as coherent as HHG, canbe spatially ﬁltered to provide very bright coherent sources. Both of these sources wereused in the ﬁrst demonstrations of tabletop x-ray coherent diﬀractive imaging (XCDI)or lensless imaging. This section will review the future developments that will need totake place to truly make this microscope useful and some of the suggested applications
for such a technique.

6.1

Future directions: a practical tabletop water window microscope
While these demonstrations are compelling, a practical tabletop coherent x-ray

microscope will need to access shorter wavelengths (higher photon energies) in the 1-6
nm range (200-1000 eV). These wavelengths are not absorbed as strongly as the 30-50
nm range thus making imaging of thicker samples or even three dimensional tomography
possible. We have already discussed in section 2.5 that SXR lasers have a punishingpump power scaling of 1/λ4 , thus accessing these short wavelengths on tabletop SXR
laser sources will be diﬃcult [6]. Therefore, it is hoped that HHG will scale to usable

114

Figure 6.1: Current and future region where HHG may be a useful tabletop source of
x-rays (adapted from Attwood [6] courtesy of M. Murnane).

wavelengths in the single nm wavelength range. Figure 6.1 shows the region in which
HHG produces ﬂux levels that make scientiﬁc or technological applications possible
currently (∼ 1 µW), and where HHG has potential for useful ﬂux levels through thedevelopment of higher repetition rate lasers and better nonlinear optics. This section
will discuss some preliminary work conducted at repeating these results at the 13.5 nm
wavelength under the direction of fellow graduate student Daisy Raymondson. Then it
will also discuss pathways towards useful water window imaging with HHG.

6.1.1

Recent developments at 13 nm

Section 2.3 described how HHG is a very tunable source of coherent SXR radiation. Our group previously demonstrated that helium can be phase matched to produce
narrow divergence 13.5 nm x-rays [125], however imaging at this wavelength has notbeen demonstrated by our group. HHG from a helium gas jet has been used in a zone
plate imaging conﬁguration to achieve 200 nm resolution elsewhere, but imaging with
13.5 nm at sub-100 nm resolutions with HHG has yet to be achieved [55]. One potential reason is that available photon ﬂux at 13.5 nm from helium is about 20 times lessthan 29 nm ﬂux from argon due to the re-absorption of harmonic radiation in helium.
This added absorption comes from, among other factors, the fact that helium is phase115
matched at about 400 torr of pressure compared to phase matched argon at about 50
torr of pressure. The 13.5 nm wavelength has extremely important technological implications because of the development of extreme ultraviolet lithography (EUVL) for
the semiconductor industry at that wavelength [6, 35]. Thus, if high resolution imaging(<50 nm) at 13.5 nm can be demonstrated on a high repetition rate tabletop source, it
could have important applications in EUVL mask and mirror blank defect inspection.
Daisy Raymondson has demonstrated preliminary image resolutions down to 70
nm resolutions (see Daisy A. Raymondson Dissertation and forthcoming articles). She
uses the same laser setup described in chapter 4, but focuses the fs laser pulses into a 100µm diameter waveguide and phase matches in helium at a pressure of about 400 torr.
Some preliminary data using FT holography with ∼ 200 nm reference holes is shownin ﬁgure 6.2. This result demonstrates approximately 184 nm resolution in 40 minutesof exposure. Additionally, Raymondson is working on higher repetition rate lasers (15kHz instead of 3 kHz) that could increase the available 13.5 nm ﬂux by a factor of 5 ormore. With improved nonlinear optics, higher repetition rate lasers, and higher throughput optical/harmonic discrimination schemes, 13.5 nm imaging with HHG may be able
to achieve the same resolution-to-wavelength ratio as described in chapters 4 and 5,
making sub-30 nm resolutions practical.

Figure 6.2: Preliminary results of 13 nm Fourier transform holography: (a) SEM imageof FT holography sample with seven reference holes (b) 13.5 nm hologram from 40
minute exposure (log scale) (c) autocorrelation reconstruction of the sample (d) line cut
through one sub-image showing 184 nm resolution [27].

116
6.1.2

Pathways to usable water window

Another potentially very large application of tabletop XCDI is biological imaging
in the water window (between 284 eV and 543 eV) [6]. It has been said that one of
the reasons x-ray water window imaging has not found broader applications in biology
is that biologists do not want to have to take their laboratories to synchrotrons forimaging. Therefore, a tabletop source of coherent water window radiation source could
achieve much broader application in the biological community. However, as discussed in
section 2.3, HHG cannot be phase matched through traditional methods at these photon
energies with driving laser wavelengths of 800 nm due to the high levels of ionization [87,90]. Quasi-phase matching has been demonstrated with HHG up to photon energies ofabout 200 eV (∼ 6 nm) but not yet into the water window [62,85,87,140]. However, Tenio
Popmintchev and colleagues in our group have recently demonstrated phase matchedwater window generation with mid-IR driving wavelengths (1.4 µm) [66]. This is possible
due to the lower ionization levels that occur when using mid-IR wavelengths. Currently
eﬀorts are underway to measure the coherence and demonstrate coherent imaging withthis source. Therefore, biological tabletop water window XCDI looks very possible in
the next two to three years.

6.2

Summary and outlook
This section has shown that the tabletop XCDI demonstrations from this disser-

tation have many potential applications to biology and materials science and technology.
Water window biological imaging and EUVL metrology are only two of these that should
be realized in coming years. With the development of new phase matching and quasiphase matching techniques, better lasers, and better management of the harmonic lightthere is real possibility to have bright, coherent SXR radiation up to 1 keV (∼ 1 nm).
Additionally, XCDI with HHG may open a whole new ﬁeld of tabletop ultrafast x-ray

117
imaging with femtosecond to attosecond resolution. Already eﬀorts are underway in ourgroup to repeat Ra’anan Tobey and Mark Siemens’ work of studying the photoacoustic
properties of thin ﬁlms in imaging mode [141]. Additional time resolved studies will
begin soon on magnetic materials. This may open the door to femtosecond dynamic
studies of magnetic thin ﬁlms with photon energies ∼ 800 eV and at resolutions approaching <10 nm [48, 142]. Thus, many interesting applications are already waiting
for tabletop XCDI with HHG as soon as the SXR ﬂux becomes available.

Bibliography[1] S. W. Hell, “Far-ﬁeld optical nanoscopy,” Science, vol. 316, no. 5828, pp. 1153–
1158, 2007.
[2] S. M. Hurtley and L. Helmuth, “The future looks bright,” Science, vol. 300, no.
5616, p. 75, 2003.
[3] M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: Wide-ﬁeld
ﬂuorescence imaging with theoretically unlimited resolution,” Proceedings of the
National Academy of Sciences of the United States of America, vol. 102, no. 37,
pp. 13 081–13 086, 2005.
[4] R. L. Sandberg, A. Paul, D. A. Raymondson, S. Hadrich, D. M. Gaudiosi, J. Holtsnider, R. I. Tobey, O. Cohen, M. M. Murnane, H. C. Kapteyn, C. Song, J. Miao,
Y. Liu, and F. Salmassi, “Lensless diﬀractive imaging using tabletop coherent
high-harmonic soft-x-ray beams,” Phys Rev Lett, vol. 99, no. 9, p. 098103.
[5] R. L. Sandberg, C. Song, P. W. Wachulak, D. A. Raymondson, A. Paul, B. Amirbekian, E. Lee, A. E. Sakdinawat, O. V. C. La, M. C. Marconi, C. S. Menoni,
M. M. Murnane, J. J. Rocca, H. C. Kapteyn, and J. Miao, “High numerical aperture tabletop soft x-ray diﬀraction microscopy with 70-nm resolution,” Proc Natl
Acad Sci U S A, vol. 105, no. 1, pp. 24–7.
[6] D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation: Principles andApplications. Cambridge University Press, 2007.
[7] D. Müller, S. Backus, K. Read, M. Murnane, and H. Kapteyn, “Very high-peakpower lasers cryogenic cooling multiplies ti: sapphire output,” Laser Focus World,
vol. 41, no. 10, p. 65, 2005.
R[Online]. Available: http://www.kmlabs.com[8] “Kmlabs.”[9] X. F. Li, A. LHuillier, M. Ferray, L. A. Lompr, and G. Mainfray, “Multipleharmonic generation in rare gases at high laser intensity,” Physical Review A,
vol. 39, no. 11, p. 5751.
[10] J. L. Krause, K. J. Schafer, and K. C. Kulander, “High-order harmonic generation from atoms and ions in the high intensity regime,” Physical Review Letters,
vol. 68, no. 24, p. 3535.

119[11] P. B. Corkum, “Plasma perspective on strong ﬁeld multiphoton ionization,”
Physical Review Letters, vol. 71, no. 13, p. 1994.
[12] H. C. Kapteyn, M. M. Murnane, and I. P. Christov, “Extreme nonlinear optics:
coherent x rays from lasers,” Phys. Today, vol. 58, no. 3, pp. 39–46, 2005.
[13] T. Brabec and F. Krausz, “Intense few-cycle laser ﬁelds: Frontiers of nonlinear
optics,” Reviews of Modern Physics, vol. 72, no. 2, pp. 545–591, 2000.
[14] A. R. Rundquist, “Phase-matched generation of coherent, ultrafast x-rays using
high harmonics,” Ph.D. dissertation, Washington State University, 1998.
[15] W. F. Schlotter, “Lensless fourier transform holography with soft x-rays,” PhD
Dissertation, Stanford University, 2007.
[16] R. A. Bartels, A. Paul, M. M. Murnane, H. C. Kapteyn, S. Backus, Y. Liu,and D. T. Attwood, “Absolute determination of the wavelength and spectrum ofan extreme-ultraviolet beam by a young’s double-slit measurement,” Opt. Lett.,
vol. 27, no. 9, pp. 707–709, 2002.
[17] R. A. Bartels, A. Paul, H. Green, H. C. Kapteyn, M. M. Murnane, S. Backus, I. P.
Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Generation of spatially coherent
light at extreme ultraviolet wavelengths,” Science, vol. 297, no. 5580, pp. 376–8.
[18] J. Peatross and M. Ware, “Physics
http://optics.byu.edu/textbook.aspx.

of

light

and

optics,”2008,[19] J. R. Fienup, “Reconstruction of an object from the modulus of its fourier transform,” Opt. Lett, vol. 3, no. 1, pp. 27–29, 1978.
[20] C. Song, D. Ramunno-Johnson, Y. Nishino, Y. Kohmura, T. Ishikawa, C. C. Chen,
T. K. Lee, and J. Miao, “Phase retrieval from exactly oversampled diﬀraction
intensity through deconvolution,” Physical Review B, vol. 75, no. 1, p. 12102,
2007.
[21] P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiﬀer, “Highresolution scanning x-ray diﬀraction microscopy,” Science, vol. 321, no. 5887, pp.
379–382, 2008.
[22] B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer,
M. de Jonge, and I. McNulty, “Keyhole coherent diﬀractive imaging,” Nat Phys,
vol. 4, no. 5, pp. 394–398.
[23] S. Marchesini, H. Chapman, S. Hau-Riege, R. London, A. Szoke, H. He, M. Howells, H. Padmore, R. Rosen, and J. Spence, “Coherent x-ray diﬀractive imaging:
applications and limitations,” Optics Express, vol. 11, no. 19, pp. 2344–2353, 2003.
[24] A. J. Paul, “Coherent euv light from high-order harmonic generation:
Enhancement and applications to lensless diﬀractive imaging,” PhD Dissertation, University of Colorado at Boulder, 2007. [Online]. Available:
http://jilawww.colorado.edu/pubs/thesis/paul ariel/paul thesis.pdf

120
Rcarbon ﬁlms.” [Online]. Available: http://www.quanitfoil.com[25] “Quantifoilholey[26] R. L. Sandberg, D. A. Raymondson, C. La-o vorakiat, A. Paul, K. J. Raines,
J. Miao, M. M. Murnane, H. C. Kapteyn, and W. F. Schlotter, “Tabletop soft
x-ray fourier transform holography with 50 nm resolution,” Optics Letters, 2009(in press).
[27] R. L. Sandberg, D. A. Raymondson, W. F. Schlotter, K. J. Raines, C. La-o vorakiat, A. Paul, M. M. Murnane, H. C. Kapteyn, and J. Miao, “Near diﬀraction
limited coherent diﬀractive imaging with tabletop soft x-ray sources,” Journal of
Physics: Conference Series (Proceedings of the Conference on X-ray Microscopy),
2008 (in review).
[28] R. Hooke, Micrographia. Courier Dover Publications, 2003 (Original year 1665).
[29] R. J. Oldﬁeld, Light Microscopy: An Illustrated Guide. CV Mosby, 1994.
[30] E. Abbe, “Beiträge zur theorie des mikroskops und der mikroskopischen
wahrnehmung,” SPIE milestone series, vol. 178, pp. 12–24, 2004 (Original 1873).
[31] E. Abbe, Gesammelte Abhandlungen. Jena: G. Fischer, 1904.
[32] M. Spencer, Fundamentals of Light Microscopy.
1982.

Cambridge University Press,[33] M. Parry-Hill and M. W. Davidson, “Microscope objectives:
Immersion oil and refractive index,” 2008 11 22 2008. [Online]. Available:
http://www.microscopyu.com/[34] J. J. Cargille, “Immersion oil and the microscope,” 2008 11 22 1985.
[35] H. J. Levinson, Principles of Lithography. SPIE Press, 2005.
[36] M. W. Davidson, “Microscope objectives: Immersion oil and refractive index,”
2008 11 24 2008. [Online]. Available: http://www.microscopyu.com/[37] W. A. Hofer, A. S. Foster, and A. L. Shluger, “Theories of scanning probe microscopes at the atomic scale,” Reviews of Modern Physics, vol. 75, no. 4, p.
1287.
[38] “Nobel prize website.” [Online]. Available: http://nobelprize.org/nobel prizes[39] B. Bailey, D. L. Farkas, D. L. Taylor, and F. Lanni, “Enhancement of axial resolution in ﬂuorescence microscopy by standing-wave excitation,” Nature, vol. 366,
no. 6450, pp. 44–48.
[40] M. G. L. Gustafsson, “Surpassing the lateral resolution limit by a factor of two
using structured illumination microscopy,” Journal of Microscopy, vol. 198, no. 2,
pp. 82–87, 2000.
[41] J. C. H. Spence, Experimental high-resolution electron microscopy. Oxford University Press, 1988.

121[42] W. Meyer-Ilse, D. Hamamoto, A. Nair, S. A. Lelievre, G. Denbeaux, L. Johnson,
A. L. Pearson, D. Yager, M. A. Legros, and C. A. Larabell, “High resolution
protein localization using soft x-ray microscopy,” Journal of Microscopy, vol. 201,
no. 3, pp. 395–403, 2001.
[43] S. B. McLane, E. W. Mller, and O. Nishikawa, “Field ion microscopy with an
external image intensiﬁer,” Review of Scientiﬁc Instruments, vol. 35, p. 1297,
2004.
[44] L. A. Giannuzzi and F. A. Stevie, “A review of focused ion beam milling techniques
for tem specimen preparation,” Micron, vol. 30, no. 3, pp. 197–204, 1999.
[45] C. A. Larabell and M. A. Le Gros, “X-ray tomography generates 3-d reconstructions of the yeast, saccharomyces cerevisiae, at 60-nm resolution,” Mol Biol Cell,
vol. 15, no. 3, pp. 957–62.
[46] G. Meier, M. Bolte, R. Eiselt, B. Krger, D. Kim, and P. Fischer, “Direct imaging
of stochastic domain-wall motion driven by nanosecond current pulses,” PhysicalReview Letters, vol. 98, 2007.
[47] J. Miao, T. Ishikawa, T. Earnest, and Q. Shen, “Extending the methodology of
x-ray crystallography to allow structure determination of non-crystalline materials, whole cells and single macromolecular complexes,” Annu. Rev. Phys. Chem.,
vol. 59, p. 387, 2008.
[48] S. Eisebitt, J. Lüning, W. F. Schlotter, M. Lörgen, O. Hellwig, W. Eberhardt,
and J. Stöhr, “Lensless imaging of magnetic nanostructures by x-ray spectroholography,” Nature(London), vol. 432, no. 7019, pp. 885–888, 2004.
[49] W. Chao, B. D. Harteneck, J. A. Liddle, E. H. Anderson, and D. T. Attwood,“Soft x-ray microscopy at a spatial resolution better than 15 nm,” Nature, vol.
435, no. 7046, pp. 1210–1213, 2005.
[50] J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of
x-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature, vol. 400, p. 342, 1999.
[51] H. N. Chapman, A. Barty, M. J. Bogan, S. Boutet, M. Frank, S. P. Hau-Riege,
S. Marchesini, B. W. Woods, S. Bajt, W. H. Benner, R. A. London, E. Plonjes, M. Kuhlmann, R. Treusch, S. Dusterer, T. Tschentscher, J. R. Schneider,
E. Spiller, T. Moller, C. Bostedt, M. Hoener, D. A. Shapiro, K. O. Hodgson,
D. van der Spoel, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M. Seibert,
F. R. N. C. Maia, R. W. Lee, A. Szoke, N. Timneanu, and J. Hajdu, “Femtosecond
diﬀractive imaging with a soft-x-ray free-electron laser,” Nat Phys, vol. 2, no. 12,
pp. 839–843.
[52] S. Marchesini, H. He, H. N. Chapman, S. P. Hau-Riege, A. Noy, M. R. Howells,
U. Weierstall, and J. C. H. Spence, “X-ray image reconstruction from a diﬀraction
pattern alone,” Physical Review B, vol. 68, no. 14, p. 140101, 2003.

122[53] D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz,
E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray
diﬀraction microscopy,” Proceedings of the National Academy of Sciences of theUnited States of America, vol. 102, no. 43, pp. 15 343–15 346, 2005.
[54] G. Vaschenko, C. Brewer, F. Brizuela, Y. Wang, M. A. Larotonda, B. M. Luther,
M. C. Marconi, J. J. Rocca, C. S. Menoni, and E. H. Anderson, “Sub-38nm
resolution tabletop microscopy with 13nm wavelength laser light,” Optics Letters,
vol. 31, no. 9, pp. 1214–1216, 2006.
[55] M. Wieland, C. Spielmann, U. Kleineberg, T. Westerwalbesloh, U. Heinzmann,
and T. Wilhein, “Toward time-resolved soft x-ray microscopy using pulsed fs-highharmonic radiation,” Ultramicroscopy, vol. 102, no. 2, pp. 93–100.
[56] P. W. Wachulak, R. A. Bartels, M. C. Marconi, C. S. Menoni, J. J. Rocca, Y. Lu,
and B. Parkinson, “Sub 400 nm spatial resolution extreme ultraviolet holography
with a table top laser,” Optics Express, vol. 14, no. 21, pp. 9636–9642, 2006.
[57] P. W. Wachulak, M. C. Marconi, R. A. Bartels, C. S. Menoni, and J. J. Rocca,“Soft x-ray laser holography with wavelength resolution,” J. Opt. Soc. Am. B,
vol. 25, no. 11, pp. 1811–1814.
[58] Y. Wang, E. Granados, F. Pedaci, D. Alessi, B. Luther, M. Berrill, and J. J.
Rocca, “Phase-coherent, injection-seeded, table-top soft-x-ray lasers at 18.9 nm
and 13.9 nm,” Nature Photonics, vol. 2, no. 2, pp. 94–98.
[59] C. Spielmann, N. H. Burnett, S. Sartania, R. Koppitsch, M. Schnurer, C. Kan,
M. Lenzner, P. Wobrauschek, and F. Krausz, “Generation of coherent x-rays in
the water window using 5-femtosecond laser pulses,” Science, vol. 278, no. 5338,
pp. 661–664, 1997.
[60] Z. Chang, A. Rundquist, H. Wang, M. M. Murnane, and H. C. Kapteyn, “Generation of coherent soft x rays at 2.7 nm using high harmonics,” Physical Review
Letters, vol. 79, no. 16, pp. 2967–2970, 1997.
[61] J. Seres, E. Seres, A. J. Verhoef, G. Tempea, C. Streli, P. Wobrauschek,
V. Yakovlev, A. Scrinzi, C. Spielmann, and F. Krausz, “Laser technology: Sourceof coherent kiloelectronvolt x-rays,” Nature, vol. 433, no. 7026, pp. 596–596.
[62] X. Zhang, A. L. Lytle, T. Popmintchev, X. Zhou, H. C. Kapteyn, M. M. Murnane, and O. Cohen, “Quasi-phase-matching and quantum-path control of highharmonic generation using counterpropagating light,” Nature Physics, vol. 3, pp.
270–275, 2007.
[63] M. Zepf, B. Dromey, M. Landreman, P. Foster, and S. M. Hooker, “Bright quasiphase-matched soft-x-ray harmonic radiation from argon ions,” Physical Review
Letters, vol. 99, no. 14.
[64] J. Seres, V. S. Yakovlev, E. Seres, C. Streli, P. Wobrauschek, C. Spielmann,and F. Krausz, “Coherent superposition of laser-driven soft-x-ray harmonics fromsuccessive sources,” Nat Phys, vol. 3, no. 12, pp. 878–883.

123[65] A. L. Lytle, X. Zhang, P. Arpin, O. Cohen, M. M. Murnane, and H. C. Kapteyn,“Quasi-phase matching of high-order harmonic generation at high photon energies
using counterpropagating pulses,” Optics Letters, vol. 34, p. 174, 2008.
[66] T. Popmintchev, M.-C. Chen, O. Cohen, M. E. Grisham, J. J. Rocca, M. M.
Murnane, and H. C. Kapteyn, “Extended phase matching of high harmonics driven
by mid-infrared light,” Opt. Lett., vol. 33, no. 18, pp. 2128–2130, 2008.
[67] S. Backus, C. G. Durfee Iii, M. M. Murnane, and H. C. Kapteyn, “High power
ultrafast lasers,” Review of Scientiﬁc Instruments, vol. 69, p. 1207, 1998.
[68] T. H. Maiman, “Stimulated optical radiation in ruby,” Nature, vol. 187, no. 4736,
pp. 493–494, 1960.
[69] P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of opticalharmonics,” Physical Review Letters, vol. 7, no. 4, pp. 118–119, 1961.
[70] D. E. Spence, P. N. Kean, and W. Sibbett, Optics Letters, vol. 16, p. 42, 1991.
[71] I. P. Christov, M. M. Murnane, and H. C. Kapteyn, “High-harmonic generationof attosecond pulses in the single-cycle regime,” Physical Review Letters, vol. 78,
no. 7, pp. 1251–1254, 1997.
[72] M. Hentschel, R. Kienberger, C. Spielmann, G. A. Reider, N. Milosevic, T. Brabec,
P. Corkum, U. Heinzmann, M. Drescher, and F. Krausz, “Attosecond metrology,”
Nature, vol. 414, no. 6863, pp. 509–513.
[73] E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz,
and U. Kleineberg, “Single-cycle nonlinear optics,” Science, vol. 320, no. 5883, pp.
1614–1617, 2008.
[74] M. M. Murnane, H. C. Kapteyn, M. D. Rosen, and R. W. Falcone, “Ultrafast
X-ray Pulses from Laser-Produced Plasmas,” Science, vol. 251, no. 4993, pp. 531–
536.
[75] J. J. Rocca, V. Shlyaptsev, F. G. Tomasel, O. D. Cortzar, D. Hartshorn, andJ. L. A. Chilla, “Demonstration of a discharge pumped table-top soft-x-ray laser,”
Physical Review Letters, vol. 73, no. 16, p. 2192.
[76] F. Zernike and J. E. Midwinter, Applied nonlinear optics. Wiley New York, 1973.
[77] A. McPherson, G. Gibson, H. Jara, U. Johann, T. S. Luk, I. A. McIntyre, K. Boyer,
and C. K. Rhodes, “Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases,” J. Opt. Soc. Am. B, vol. 4, no. 4, pp. 595–601, 1987.
[78] L. V. Keldysh, “Ionization in the ﬁeld of a strong em wave,” Sov. Phys. JETP,
vol. 20, p. 13071314, 1965.
[79] M. Ammosov, N. Delone, and p. . Krainov, V.P. 64 (1986), “Tunnel ionization ofcomplex atoms and of atomic ions in an alternating electromagnetic ﬁeld,” Soviet
Phys. JETP, vol. 64, no. 6, pp. 1191–1194, 1986.

124[80] M. Lewenstein, P. Balcou, M. Y. Ivanov, A. LHuillier, and P. B. Corkum, “Theoryof high-harmonic generation by low-frequency laser ﬁelds,” Physical Review A,
vol. 49, no. 3, p. 2117.
[81] P. Balcou and A. LHuillier, “Phase-matching eﬀects in strong-ﬁeld harmonic generation,” Physical Review A, vol. 47, no. 2, p. 1447.
[82] A. Rundquist, C. G. Durfee Iii, Z. Chang, C. Herne, S. Backus, M. M. Murnane,and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science,
vol. 280, no. 5368, p. 1412, 1998.
[83] C. G. Durfee III, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and
H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,”Physical Review Letters, vol. 83, no. 11, pp. 2187–2190, 1999.
[84] I. Christov, H. Kapteyn, and M. Murnane, “Quasi-phase matching of highharmonics and attosecond pulses in modulated waveguides,” Opt. Express, vol. 7,
no. 11, pp. 362–367, 2000.
[85] A. Paul, R. A. Bartels, R. Tobey, H. Green, S. Weiman, I. P. Christov, M. M.
Murnane, H. C. Kapteyn, and S. Backus, “Quasi-phase-matched generation ofcoherent extreme-ultraviolet light,” Nature, vol. 421, no. 6918, pp. 51–54, 2003.
[86] J. Peatross, S. Voronov, and I. Prokopovich, “Selective zoning of high harmonic
emission using counter-propagating light,” Optics Express, vol. 1, no. 5, pp. 114–
125, 1997.
[87] A. L. Lytle, X. Zhang, R. L. Sandberg, O. Cohen, H. C. Kapteyn, and M. M.
Murnane, “Quasi-phase matching and characterization of high-order harmonic
generation in hollow waveguides using counterpropagating light,” Optics Express,
vol. 16, no. 9, pp. 6544–6566, 2008.
[88] P. Salières, A. L’Huillier, and M. Lewenstein, “Coherence control of high-orderharmonics,” Physical Review Letters, vol. 74, no. 19, p. 3776.
[89] T. Ditmire, E. T. Gumbrell, R. A. Smith, J. W. G. Tisch, D. D. Meyerhofer,and M. H. R. Hutchinson, “Spatial coherence measurement of soft x-ray radiation
produced by high order harmonic generation,” Physical Review Letters, vol. 77,
no. 23, p. 4756.
[90] A. L. Lytle, “Phase matching and coherence of high-order harmonic generation in hollow waveguides,”PhD Dissertation,
UniversityofColoradoatBoulder,
2008.
[Online].
Available:
http://jilawww.colorado.edu/pubs/thesis/lytle amy/lytle thesis.pdf[91] M. Wieland, R. Frueke, T. Wilhein, C. Spielmann, M. Pohl, and U. Kleineberg,“Submicron extreme ultraviolet imaging using high-harmonic radiation,” Applied
Physics Letters, vol. 81, p. 2520, 2002.
[92] R. Bartels, S. Backus, E. Zeek, L. Misoguti, G. Vdovin, I. P. Christov, M. M.
Murnane, and H. C. Kapteyn, “Shaped-pulse optimization of coherent emissionof high-harmonic soft x-rays,” Nature, vol. 406, no. 6792, pp. 164–166.

125[93] T. Wilhein, S. Rehbein, D. Hambach, M. Berglund, L. Rymell, and H. M. Hertz,“A slit grating spectrograph for quantitative soft x-ray spectroscopy,” Review of
Scientiﬁc Instruments, vol. 70, p. 1694, 1999.
[94] P. A. C. Takman, H. Stollberg, G. A. Johansson, A. Holmberg, M. Lindblom, andH. M. Hertz, “High-resolution compact x-ray microscopy,” Journal of Microscopy,
vol. 226, no. 2, pp. 175–181.
[95] J. J. Rocca, “Table-top soft x-ray lasers,” Review of Scientiﬁc Instruments, vol. 70,
no. 10, pp. 3799–3827.
[96] Y. Liu, M. Seminario, F. G. Tomasel, C. Chang, J. J. Rocca, and D. T. Attwood,“Achievement of essentially full spatial coherence in a high-average-power soft-xray laser,” Physical Review A, vol. 63, no. 3, p. 033802.
[97] C. A. Brewer, F. Brizuela, P. Wachulak, D. H. Martz, W. Chao, E. H. Anderson, D. T. Attwood, A. V. Vinogradov, I. A. Artyukov, and A. G. Ponomareko,“Single-shot extreme ultraviolet laser imaging of nanostructures with wavelength
resolution,” Optics Letters, vol. 33, no. 5, pp. 518–520, 2008.
[98] Y. Wang, M. A. Larotonda, B. M. Luther, D. Alessi, M. Berrill, V. N. Shlyaptsev,
and J. J. Rocca, “Demonstration of high-repetition-rate tabletop soft-x-ray lasers
with saturated output at wavelengths down to 13.9 nm and gain down to 10.9
nm,” Physical Review A (Atomic, Molecular, and Optical Physics), vol. 72, no. 5,
p. 053807.
[99] K. J. Schafer and K. C. Kulander, “High harmonic generation from ultrafast pumplasers,” Physical Review Letters, vol. 78, no. 4, p. 638.
[100] J. W. Goodman, Introduction to Fourier Optics, Second Edition.
Graw Hill, 1996.

Boston: Mc-[101] D. Sayre, “Some implications of a theorem due to shannon,”
Crystallographica, vol. 5, no. 6, pp. 843–843, 1952.

Acta[102] J. Miao and D. Sayre, “On possible extensions of x-ray crystallography throughdiﬀraction-pattern oversampling,” Acta Crystallographica Section A, vol. 56,
no. 6, pp. 596–605, 2000.
[103] J. D. Watson and F. H. C. Crick, “Molecular structure of nucleic acids: A structure
for deoxyribose nucleic acid,” Nature, vol. 171, no. 4356, pp. 737–738.
[104] J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of
the fourier transforms of nonperiodic objects,” Journal of the Optical Society ofAmerica A, vol. 15, no. 6, pp. 1662–1669, 1998.
[105] J. Miao, T. Ishikawa, E. H. Anderson, and K. O. Hodgson, “Phase retrieval ofdiﬀraction patterns from noncrystalline samples using the oversampling method,”
Physical Review B, vol. 67, no. 17, p. 174104.

126[106] J. R. Fienup, “Reconstruction of a complex-valued object from the modulus ofits fourier transform using a support constraint,” Optical Society of America,
Journal, A: Optics and Image Science, vol. 4, pp. 118–123, 1987.
[107] R. W. Gerchberg and W. O. Saxton, “Practical algorithm for determination ofphase from image and diﬀraction plane pictures,” Optik, vol. 35, no. 2, pp. 237–&.
[108] S. Marchesini, “Invited article: A uniﬁed evaluation of iterative projection algorithms for phase retrieval,” Review of Scientiﬁc Instruments, vol. 78, no. 1, p.
011301.
[109] C. C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization
technique to noncrystalline x-ray diﬀraction microscopy: Guided hybrid inputoutput method,” Physical Review B, vol. 76, no. 6, p. 64113, 2007.
[110] R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort
Laser Pulses. Kluwer Academic Publishers, 2002.
[111] R. Barakat and G. Newsam, “Necessary conditions for a unique solution to twodimensional phase recovery,” Journal of Mathematical Physics, vol. 25, no. 11,
pp. 3190–3193, 1984.
[112] R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, “Potential forbiomolecular imaging with femtosecond x-ray pulses,” Nature, vol. 406, no. 6797,
pp. 752–757.
[113] D. Gabor, “A new microscopic principle,” Nature, vol. 161, no. 4098, pp. 777–778,
1948.
[114] D. Gabor, “Holography, 1948-1971,” Science, vol. 177, no. 4046, pp. 299–313,
1972.
[115] J. B. Tiller, A. Barty, D. Paganin, and K. A. Nugent, “The holographic twin image
problem: a deterministic phase solution,” Optics Communications, vol. 183, no.
1-4, pp. 7–14.
[116] W. F. Schlotter, R. Rick, K. Chen, A. Scherz, J. Stöhr, J. Lüning, S. Eisebitt,
C. Gunther, W. Eberhardt, O. Hellwig, and I. McNulty, “Multiple reference fouriertransform holography with soft x rays,” Applied Physics Letters, vol. 89, no. 16,
pp. 163 112–3.
[117] D. Sayre, J. Kirz, R. Feder, D. M. Kim, and E. Spiller, “Transmission microscropyof unmodiﬁed biological materials: comparative radiation dosages with electrons
and ultrasoft x-ray photons,” Ultramicroscopy, vol. 2, no. 4, pp. 337–49, 1977.
[118] R. Henderson, “The potential and limitations of neutrons, electrons and x-rays foratomic resolution microscopy of unstained biological molecules,” Q Rev Biophys,vol. 28, no. 2, pp. 171–93, 1995.
[119] Q. Shen, I. Bazarov, and P. Thibault, “Diﬀractive imaging of nonperiodic materials with future coherent x-ray sources,” Journal of Synchrotron Radiation, vol. 11,
no. 5, pp. 432–438, 2004.

127[120] J. Miao, K. O. Hodgson, and D. Sayre, “An approach to three-dimensional structures of biomolecules by using single-molecule diﬀraction images,” Proceedings of
the National Academy of Sciences of the United States of America, vol. 98, no. 12,
pp. 6641–6645, 2001.
[121] H. N. Chapman, S. P. Hau-Riege, M. J. Bogan, S. Bajt, A. Barty, S. Boutet,
S. Marchesini, M. Frank, B. W. Woods, W. H. Benner, R. A. London, U. Rohner,
A. Szoke, E. Spiller, T. Moller, C. Bostedt, D. A. Shapiro, M. Kuhlmann,
R. Treusch, E. Plonjes, F. Burmeister, M. Bergh, C. Caleman, G. Huldt, M. M.
Seibert, and J. Hajdu, “Femtosecond time-delay x-ray holography,” Nature, vol.
448, no. 7154, pp. 676–679.
[122] M. J. Bogan, W. H. Benner, S. Boutet, U. Rohner, M. Frank, A. Barty, M. M.
Seibert, F. Maia, S. Marchesini, S. Bajt, B. Woods, V. Riot, S. P. Hau-Riege,
M. Svenda, E. Marklund, E. Spiller, J. Hajdu, and H. N. Chapman, “Single
particle x-ray diﬀractive imaging,” Nano Lett., vol. 8, no. 1, pp. 310–316, 2008.
[123] S. Marchesini, A. E. Sakdinawat, M. J. Bogan, A. Barty, H. N. Chapman,
M. Frank, S. P. Hau-Riege, A. Sz, C. Cui, and D. A. Shapiro, “Massively parallel
x-ray holography,” Nature Photonics, vol. 2, no. 9, p. 560, 2008.
[124] A. Barty, M. J. Bogan, S. Hau-Riege, S. Marchesini, K. Sokolowski-Tinten, N. Stojanovic, H. Ehrke, A. Cavalleri, D. Stefan, and M. Frank, “Ultrafast single-shot
diﬀraction imaging of nanoscale dynamics,” Nature Photonics, vol. 2, no. 7, p.
415, 2008.
[125] X. Zhang, A. R. Libertun, A. Paul, E. Gagnon, S. Backus, I. P. Christov, M. M.
Murnane, H. C. Kapteyn, R. A. Bartels, and Y. Liu, “Highly coherent light at 13
nm generated by use of quasi-phase-matched high-harmonic generation,” Optics
Letters, vol. 29, no. 12, pp. 1357–1359, 2004.
[126] G. Beutier, G. van der Laan, A. Marty, and F. Livet, “Back-illuminated ccd forcoherent soft x-ray imaging,” The European Physical Journal Applied Physics,
vol. 42, no. 2, pp. 161–167, 2008.
[127] B. R. Benware, C. D. Macchietto, C. H. Moreno, and J. J. Rocca, “Demonstrationof a high average power tabletop soft x-ray laser,” Physical Review Letters, vol. 81,
no. 26, p. 5804.
[128] C. D. Macchietto, B. R. Benware, and J. J. Rocca, “Generation of millijoule-levelsoft-x-ray laser pulses at a 4-hz repetition rate in a highly saturated tabletop
capillary discharge ampliﬁer,” Opt. Lett., vol. 24, no. 16, pp. 1115–1117.
[129] S. Marchesini, S. Boutet, A. E. Sakdinawat, M. J. Bogan, S. Bajt, A. Barty, H. N.
Chapman, M. Frank, S. P. Hau-Riege, and A. Szoke, “Ultrafast, ultrabright, x-ray
holography using a uniformly-redundant array,” Arxiv preprint arXiv:0801.4969,
2008.
[130] L. M. Stadler, C. Gutt, T. Autenrieth, O. Leupold, S. Rehbein, Y. Chushkin, and
G. Grbel, “Hard x ray holographic diﬀraction imaging,” Physical Review Letters,
vol. 100, no. 24, p. 245503, 2008.

128[131] G. W. Stroke and D. G. Falconer, “Attainment of high resolutions in wavefrontreconstruction imaging,” Physics Letters, vol. 13, no. 4, pp. 306–309, 1964.
[132] J. T. Winthrop and C. R. Worthington, “X-ray microscopy by successive fouriertransformation,” Physics Letters, vol. 15, no. 2, pp. 124–126, 1965.
[133] G. W. Stroke, “Lensless fourier-transform method for optical holography,” Applied
Physics Letters, vol. 6, no. 10, pp. 201–203, 1965.
[134] G. W. Stroke and D. Gabor, An introduction to coherent optics and holography.
Academic Press, New York, 1966.
[135] I. McNulty, J. Kirz, C. Jacobsen, E. H. Anderson, M. R. Howells, and D. P.
Kern, “High-resolution imaging by fourier-transform x-ray holography,” Science,
vol. 256, no. 5059, pp. 1009–1012.
[136] H. He, U. Weierstall, J. C. H. Spence, M. Howells, H. A. Padmore, S. Marchesini,and H. N. Chapman, “Use of extended and prepared reference objects in experimental fourier transform x-ray holography,” Applied Physics Letters, vol. 85,
no. 13, pp. 2454–2456, 2004.
[137] H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau-Riege, C. Cui,
M. R. Howells, R. Rosen, H. He, and J. C. H. Spence, “High-resolution ab initio
three-dimensional x-ray diﬀraction microscopy,” Journal of the Optical Society ofAmerica A, vol. 23, no. 5, pp. 1179–1200, 2006.
[138] S. Eisebitt, M. Lörgen, W. Eberhardt, J. Lüning, S. Andrews, and J. Stöhr,“Scalable approach for lensless imaging at x-ray wavelengths,” Applied Physics
Letters, vol. 84, p. 3373, 2004.
[139] H. He, S. Marchesini, M. Howells, U. Weierstall, H. Chapman, S. Hau-Riege,
A. Noy, and J. C. H. Spence, “Inversion of x-ray diﬀuse scattering to images using
prepared objects,” Physical Review B, vol. 67, no. 17, p. 174114, 2003.
[140] A. L. Lytle, X. Zhang, J. Peatross, M. M. Murnane, H. C. Kapteyn, and O. Cohen,“Probe of high-order harmonic generation in a hollow waveguide geometry using
counterpropagating light,” Physical Review Letters, vol. 98, no. 12, p. 123904,
2007.
[141] R. I. Tobey, M. E. Siemens, O. Cohen, M. M. Murnane, H. C. Kapteyn, and K. A.
Nelson, “Ultrafast extreme ultraviolet holography: dynamic monitoring of surface
deformation,” Opt. Lett., vol. 32, no. 3, pp. 286–288, 2007.
[142] S. Eisebitt, M. Lörgen, W. Eberhardt, J. Lüning, and J. Stöhr, “Lensless xray imaging of magnetic materials: basic considerations,” Applied Physics A:
Materials Science & Processing, vol. 80, no. 5, pp. 921–927, 2005.