Coherent EUV Light from High-Order Harmonic
Generation: Enhancement and Applications to
Lensless Diffractive Imaging
byAriel J. Paul
B.A., University of Pennsylvania, 1999

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
2007

This thesis entitled:
Coherent EUV Light from High-Order Harmonic Generation: Enhancement andApplications to Lensless Diffractive Imaging
written by Ariel J. Paul
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.

iiiPaul, Ariel J. (Ph.D., Physics)
Coherent EUV Light from High-Order Harmonic Generation: Enhancement and Applications to Lensless Diffractive Imaging
Thesis directed by Profs. Henry C. Kapteyn and Margaret M. Murnane

The first half of this thesis presents the first demonstration of quasi-phase matching in the coherent high-order harmonic conversion of ultrafast laser pulses into the
EUV region of the spectrum. To achieve this quasi-phase matching, a novel method of
fabricating hollow waveguides with a modulated inner diameter was developed. This
technique lead to significant enhancements of EUV flux at wavelengths shorter thanwere previously accessible by known phase-matching techniques. In the second half of
this thesis, the first tabletop demonstration of lensless diffractive imaging with EUV
light is presented using HHG in a gas-filled hollow waveguide to provide coherent illumination. This tabletop microscope shows a spatial resolution of ∼ 200 nm and a large
depth of field. Furthermore, the technique is easily scalable to shorter wavelengths ofinterest to biological imaging.

Dedication

To my grandmother Josepha Ligonv

Acknowledgements

Foremost, my family, especially my parents, Philip & Genia Paul, for the knowledge and support to pursue a successful graduate career. Profs. Margaret Murnaneand Henry Kapteyn for my unique research path. Pre-CU: Jeff Klein (amazing mentor), Kane Bros., Dave Weitz, UPPSMS, Darin Rhodes, Brad Baker, Jen Close, NOAA
coworkers. Early: Charles Bailey, Tim Black, Joe Britton, Kevin Holman, Dustin
Hoover, Chris Kelso, Jason Schmidt, James Walker. TA: amazing students, Jerry Leigh,Mike Dubson, John Cumalat, Tom DeGrand. Early work: Randy Bartels, Sterling
Backus, Emily Gibson. Instrument Shop: Dave Alchenburger, Todd Asnicar, Leslie
Czia, Hans Green (awesome friend/teacher), Kim Hagen, Blaine Horner, Andy Hytjan,
Tracy Keep, Alan Pattee, Lee Thornhill, Seth Weiman. Support staff: Sam Jarvis,
Pam Leland, Rachel Tearle, Lisa Roos, the supply office, electronics shop. Lab: Luis
Miaja-Avilla, Oren Cohen, Etienne Gagnon, David Gaudiosi, Erez Gershgoren, Mike
Grisham, Steffen Haedrich, Jim Holtsnider, Nathan Lemke, Amy Lytle, Daisy Raymondson, Richard Sandberg, Guido Sartoff, Arvinder Sandhu, Ra’anan Tobey, Adrienne
Van Allen, Nick Wagner, Andrea Wuest, all K/M group past & present. Jilans: Ashley Carter, Manuel Castellanos, Konrad Lehnert, Brandon Smith, Oliver Monti, Mark
Notcutt. Boulder High & tutoring students. Other Friends: Mike Alessi, Kathe Baker,
Jessica Bedwell (lv/sp), Adrienne Bentley, Corey Dickerson, Craig Emrich, Aaron Fortner, Halston Hoverstein, Amy Johnson, Heidi Metzler, Mark Morlino, Dione Rossiter,
Michael Urban. Companions: Dixie, Macie, Dani, George, Zoey, Lilly.

viContents

Chapter
1 Introduction

1

2 High-Order Harmonic Generation and Hollow Gas-Filled Waveguides

4

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2

Basic Principles of High Harmonic Generation . . . . . . . . . . . . . . .

5

2.2.1

Extreme Nonlinear Optics . . . . . . . . . . . . . . . . . . . . . .

5

2.2.2

3-Step Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6Phase-Matched Frequency Upconversion in Hollow Waveguides . . . . .

7

2.3.1

Phase Matching in Nonlinear Optical Frequency Conversion . . .

7

2.3.2

Phase Matching of HHG in Gas-Filled Hollow Waveguides . . . .

10

2.3

2.4

3-Section Hollow Gas-Filled Waveguide Design with Outer-Capillary Fixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Quasi-Phase Matching in Modulated Hollow Waveguides

13
17

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

3.2

Quasi-Phase Matching for Nonlinear Optical Frequency Conversion . . .

17

3.3

Motivations for a Modulated Waveguide . . . . . . . . . . . . . . . . . .

21

3.4

Using Modulated Waveguides for High-Order Harmonic Generation . . .

24

3.4.1

The First Demonstration of Quasi-Phase Matching with HighOrder Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

vii
3.4.2
3.5

Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

Producing Modulated Waveguides . . . . . . . . . . . . . . . . . . . . .

32

3.5.1

49

Limitations of Modulated Waveguides and Perspective . . . . . .

4 Hollow Gas-Filled Waveguides

50

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.2

Hollow Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

4.2.1

Modes and Mode Beating . . . . . . . . . . . . . . . . . . . . . .

51

4.2.2

Attenuation and Bending Losses . . . . . . . . . . . . . . . . . .

52

4.2.3

Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

Modifications to the Hollow Waveguide Design . . . . . . . . . . . . . .

56

4.3.1

Why Change the Design? . . . . . . . . . . . . . . . . . . . . . .

56

4.3.2

Initial Hollow Waveguide Improvement Attempts . . . . . . . . .

57

4.3.3

Single-Piece Inner Capillary Hollow Waveguides

58

4.3.4

V-groove Fixture for Single-Piece Hollow Waveguides

4.3.5

Simple Set-up for Comparing Transmission of Hollow Waveguides

4.3

4.4

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

62with a Spatially-Filtered HeNe Laser . . . . . . . . . . . . . . . .

67

Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

5 Coherent Imaging in the Extreme Ultraviolet

77

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

5.2

Coherent vs. Incoherent Illumination and Imaging . . . . . . . . . . . .

79

5.3

Spatial and Temporal Coherence . . . . . . . . . . . . . . . . . . . . . .

82

5.4

Imaging in the Extreme Ultraviolet . . . . . . . . . . . . . . . . . . . . .

83

5.5

Lensless Diffractive Imaging . . . . . . . . . . . . . . . . . . . . . . . . .

87

5.5.1

The Phase Problem . . . . . . . . . . . . . . . . . . . . . . . . .

88

5.5.2

The Oversampling Method . . . . . . . . . . . . . . . . . . . . .

89

5.5.3

General Description of Algorithm . . . . . . . . . . . . . . . . . .

92

viii
5.5.4

Experimental Requirements . . . . . . . . . . . . . . . . . . . . .

6 Table-Top Lensless Microscopy
6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2

Lensless Diffractive Imaging and HHG in Hollow Waveguides: A Natural
Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1

6.3

6.4

6.5

6.6

98
98

99

Coherence of High-Order Harmonic Emission Generated Using a
Hollow-Waveguide Geometry . . . . . . . . . . . . . . . . . . . .

6.2.2

93

99

Efficient Use of Photons . . . . . . . . . . . . . . . . . . . . . . . 100

Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3.1

Illumination by the EUV Source . . . . . . . . . . . . . . . . . . 102

6.3.2

Geometry, Microscope Vacuum Chamber, and Detector . . . . . 106

6.3.3

Beam Blocks for Increasing Dynamic Range . . . . . . . . . . . . 109

Imaging Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.4.1

Initial Attempts and Improvements . . . . . . . . . . . . . . . . . 114

6.4.2

J-Slit Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6.4.3

200 nm Resolution with QUANTIFOIL R Sample . . . . . . . . 122

Light-Tightness with Thin Metal Filters . . . . . . . . . . . . . . . . . . 126
6.5.1

Thin Metal Filters for Separating Pump and EUV . . . . . . . . 126

6.5.2

NW-40 Baffle Design . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.5.3

Magnetically-Coupled Vacuum-Compatible Filter Wheel . . . . . 131

Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.6.1

Improving Resolution . . . . . . . . . . . . . . . . . . . . . . . . 133

7 Conclusion

135

Bibliography

137

ix

FiguresFigure2.1

Phase-matched signal growth . . . . . . . . . . . . . . . . . . . . . . . .

11

2.2

A to-scale 3-d rendering of one end of the 3-section design. . . . . . .

15

2.3

A picture of an outer capillary for the 3-section design . . . . . . . . . .

16

2.4

A schematic diagram of an outer capillary for the 3-section design . . .

16

3.1

An idealized QPM geometry . . . . . . . . . . . . . . . . . . . . . . . . .

19

3.2

Normalized pressure for phase matching vs. normalized ionization fraction 23

3.3

A schematic of the partially modulated waveguide in the 3-section configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

Experimentally measured HHG spectra for unmodulated and modulated
waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

26

29

Experimentally measured HHG spectra (log scale) from He for three different periodicities of the modulated waveguides. . . . . . . . . . . . .

31

3.6

A short section of capillary with a 1 mm modulation period . . . . . . .

39

3.7

The second-generation glass-blowing set-up . . . . . . . . . . . . . . . .

40

3.8

The current third-generation glass-blowing set-up . . . . . . . . . . . . .

42

3.9

The third-generation custom glass-blowing lathe getting a tune up . . .

44

3.10 A short section of capillary with a 0.25 mm modulation period . . . . .

47

3.11 A hole cut in the capillary with the CO2 laser showing how misalignmentof the beam pointing can easily be diagnosed. . . . . . . . . . . . . . . .

48

x
4.1

A simulation by Amy Lytle of mode beating in a hollow waveguide . . .

53

4.2

To-scale 3-d rendering of the slotted innner capillary . . . . . . . . . . .

59

4.3

Microscope image of an abrasively cut slot . . . . . . . . . . . . . . . . .

60

4.4

Stereoscope image of a laser poked hole . . . . . . . . . . . . . . . . . .

61

4.5

A 3-d exploded rendering of the v-groove fixture . . . . . . . . . . . . .

64

4.6

An assembled v-groove fixture . . . . . . . . . . . . . . . . . . . . . . . .

64

4.7

A 3-dimensional rendering of the custom silicone gasket . . . . . . . . .

65

4.8

A schematic of the set-up for testing waveguide transmission . . . . . .

69

4.9

A measurement of transmitted power for 10 cm waveguides comparing
those resting in a v-groove with those held in an outer-capillary fixture .

73

4.10 A measurement of transmitted power for 10 cm waveguides resting in a
v-groove without holes versus those with 2 laser-poked holes . . . . . . .

73

4.11 An example of the comparison of the output modes for three different
waveguides placed in a v-groove (top) and the corresponding waveguides
held in an outer capillary fixture (bottom) . . . . . . . . . . . . . . . . .

74

5.1

SEM micrograph of a zone plate . . . . . . . . . . . . . . . . . . . . . .

85

5.2

An illustration of the no-density region generated by oversampling thediffraction pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.3

A schematic representation of the iterative phase retrieval algorithm . .

94

6.1

The mode of the 29 nm HHG light near its focus, plotted on a log scale.

105

6.2

A schematic of the current transmission imaging geometry . . . . . . . . 107

6.3

3-d rendering of the beam block ring . . . . . . . . . . . . . . . . . . . . 111

6.4

A large and medium beam block suspended within the ring holder . . . 111

6.5

The x-y kinematic lens mount used to steer the beam blocks into position 113

6.6

SEM micrograph of the QUANTIFOIL R MultiA carbon film . . . . . . 116

xi
6.7

Examples of data and a reconstruction from the initial lensless imaging
attempt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.8

Data taken before the laser was isolated from air currents (top) and after(bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.9

The diffraction data, J-slit sample, and reconstructed image . . . . . . . 123

6.10 (a): SEM image of the sample over its 15 µm aperture , (b): the stitcheddiffraction data on a logarithmic scale, (c): the reconstructed image, (d):a line out of the reconstructed image taken across the small blue bar in (c)124
6.11 Schematic diagram of the NW-40 Baffle . . . . . . . . . . . . . . . . . . 130
6.12 3-dimensional renderings of an exploded and assembled vacuum-compatible
filter wheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Chapter 1

Introduction

My graduate career has been somewhat unique in the types of problems to which
I have devoted much of my time and effort. The problems on which I have thrived
the most have required hands-on mechanical answers that were motivated and solved
through an understanding of the underlying physics. Additionally, whereas students
are generally most concerned with experimental difficulties that affect their individual
experiments, I took great pleasure in developing elegant solutions to several experimental issues faced regularly in our research. Of course, the somewhat unique nature of
the Kapteyn/Murnane group, with many students making use of similar technologies
and sharing equipment, allowed me these opportunities, and meant that such solutions
quickly benefited many experiments. My advisors allowed me a tremendous amount ofautonomy and freedom to pursue several of these types of projects, and the results havebeen some of the more rewarding moments of my research.
Most of the results presented in this thesis appear (or will appear) in the scientific literature. So, I have made a special effort to include not only experimental results,
but also technical details that will help a student continue the work to which I havecontributed. In particular, I have included detailed sections describing the process ofmaking modulated waveguides, the construction and benefits of a v-groove fixturingsystem for hollow waveguides, the development of light-tight fixtures for holding EUV
filters, as well as alignment and data acquisition details for our lensless diffractive imag-

2
ing experiments. Whenever appropriate, I have used a ‘conversational’ approach. Thisapproach allows me to interweave some of my personality with the scientific discussion,as well as giving the reader information on the scientific process I used to obtain the
result, in addition to the result itself.
In Chapter 2, I present a basic introduction to high harmonic generation (HHG)
and discuss the relevance of phase matching to nonlinear optical frequency conversion.
The low-order process of second harmonic generation is used to motivate the fundamental theory of phase matching, and then the method of pressured-tuned phase matchingof HHG in a gas-filled hollow-waveguide is explored. This chapter concludes with a description of the original 3-section hollow waveguide fixture used by our group to phasematch HHG.
Chapter 3 describes the world’s first demonstration of quasi-phase matching(QPM) in the realm of extreme nonlinear optics using periodically-modulated hollow
gas-filled waveguides [1]. In a similar fashion to my approach towards discussing traditional phase matching, I begin with a treatment of QPM in the context of second
harmonic generation. The implications of QPM are extended to the process of HHG,
and I explain the motivations for attempting to fabricate waveguides with a periodicallymodulated inner diameter. After discussing the demonstration of QPM in such waveguides, I present an in-depth account of the glass-blowing process used to fabricate themodulated waveguides, as well as of the manner in which this process evolved.
Since I exerted a significant portion of my research efforts to improving the useof hollow waveguides, Chapter 4 is devoted to a discussion of these waveguides. The
basic issues of coupling and losses for a hollow waveguide are treated, after which I delve
into the details of modifying the 3-section design and the development of a specialized
v-groove fixturing system. A simple optical set-up for accurately comparing the transmission characteristics for different hollow waveguide arrangements is also shown. My
work on hollow waveguides, which improved the consistency and modes of the coherent

3extreme ultraviolet light (EUV) we produce through HHG, ties together my early workwith the coherent imaging work presented after this chapter.
Chapter 5 introduces the topic of coherent imaging in the extreme ultraviolet(EUV) region of the spectrum and details some of the unique aspects of imaging at
this wavelength range as compared to the visible end of the spectrum. I briefly review the idea of coherence and its relevance to imaging, after which I present some of
the common methods of imaging in the EUV. The background of the relatively new
method of lensless diffractive imaging, which we chose to attempt with our EUV source,
is detailed. This imaging method replaces traditional optics with a computerized algorithm that retrieves phase information from a measured diffraction pattern subject toconstraints. Although we do not perform the phase retrieval ourselves, understanding
the theory of this process is highly relevant to grasping the experimental requirements
for successful lensless imaging. To conclude the chapter, I enumerate and clarify thesevarious experimental requirements.
The experimental details and results from the world’s first demonstration of lensless diffractive imaging with a tabletop high harmonic source are presented in Chapter
6. The suitability of this technique for our unique EUV source is made clear, andits advantages are explored. I also describe in detail the experimental geometry andits components, as well as the improvements that were made to produce success of theimaging technique. The imaging results definitively show that we have built and demonstrated a tabletop lensless diffractive microscope with resolutions in the range of 200 nm
using coherent ∼ 30 nm illumination. In addition, I describe some of the novel devices
used to ensure a good signal-to-noise ratio in our data, including a magnetically-coupledvacuum compatible filter wheel. Finally, I discuss the scalability of our microscope toshorter wavelengths and its promising future.

Chapter 2

High-Order Harmonic Generation and Hollow Gas-Filled Waveguides

2.1

Introduction
In July of 1960, Theodore Maiman demonstrated the first laser [2]. This lightsource could be focused to higher intensities in the laboratory than any other lightsources previously available, and opened the door to the new and remarkable field of
nonlinear optics. After the birth of the laser, the demonstration of nonlinear opticalphenomena revolutionized the science of optics, bringing new understanding to theinteractions of light and matter.
More recently, high-power ultrafast laser amplifiers have been developed that produce high energy pulses with durations in the femtosecond (fs) regime [3]. These systems
rely on some version of the chirped pulse amplification (CPA) scheme to avoid nonlinear effects and optical damage during the amplification process; the pulse is stretchedin time before amplification and then compressed afterwards using gratings or othermeans [4, 5]. The phenomenally short pulse widths and high powers of such lasers not
only allow time-resolved pump-probe experiments, but have also allowed researchers topush the limits of high harmonic generation (HHG) [6]. HHG represents a most extreme
realm of nonlinear optics in which 10’s or even 100’s of photons from a visible laser are
combined together in a frequency upconversion process that can extend into the x-ray
region of the spectrum. Much of my experimental work has revolved around increasing
the efficiency of this process, and using the coherent short-wavelength light that results.

5

2.2

Basic Principles of High Harmonic Generation

2.2.1

Extreme Nonlinear Optics

The very high brightness of lasers allowed researchers to obtain large optical fields
that were not previously accessible. Before the advent of the laser, optical systems weretraditionally analyzed according to a theory in which a linear relationship between theapplied electric field E and a material’s response to this field, the polarization P, wasassumed. This relationship is given by
P = ǫ0 χE(2.1)

where χ is the susceptibility tensor. However, when tightly focused, a typical laser beam
can produce electric field strengths in excess of 1010 V/m. These field strengths are on
the order of atomic Coulomb fields, and possess the ability to drive a material’s response
past the linear regime. In many cases, we can understand the nonlinear response using
a perturbative approach in which we describe the relationship between P and E with apower series:
Pi ω = Pi 0 +

X
j

X

j,l,m

χij Ej ω +Xj,lχijl ∇l Ej ω +χijlm Ej ω1 El ω2 Em ω3 +X

X

χijl Ej ω1 El ω2 +j,lχijl Ej ω1 Bl ω2 + ....

(2.2)

j,l

This approach explains numerous phenomena, including sum- and difference-frequency
mixing, the Pockels effect, the Raman effect, and the Kerr effect, to name a few of themore frequently used nonlinear processes [7]. However, for some extreme nonlinear processes, such as high harmonic generation (HHG), even the perturbative approach fails.
HHG represents a most extreme version of nonlinear optics. In lower-order harmonic
processes, the optical field perturbs the atom. In contrast, the laser field strengths required to drive HHG are so enormous that the atom effectively perturbs the field, andthe perturbative approach can no longer be relied upon to yield accurate predictions.

6
2.2.2

3-Step Model

Although the most accurate description of HHG involves numerical integrationsof the nonlinear Schrödinger equation, a quasi-classical three-step model exists that
accurately predicts the general features of HHG and provides a physical picture of theinteractions involved [8, 9]. In the three-step model, the laser field ionizes a valence
electron, which proceeds to follow a trajectory determined by the laser field until it
recombines with its parent ion. Essentially, in this model, the powerful deceleration of
the electron during recombination produces bremsstrahlung radiation.
In the first step, the intense laser field suppresses the Coulomb potential of theatom, allowing the electron to ionize by absorbing multiple photons, or by tunnelingthrough the Coulomb barrier. For very strong fields, tunneling becomes the more probable ionization mechanism. In addition, the intensity dependent rate of ionization may
be accurately predicted by incorporating a quantum mechanical calculation known as
the ADK rate (named for its inventors Ammosov, Krainov, and Delone) [10]. Once theelectron ionizes, the second step involves its evolution in the laser field. Since the laser
field dominates the Coulomb potential, we consider the electron as ‘free’ during thisstep. As the optical field oscillates, it first propels the electron away from the ion, andthen, when the field reverses, the field accelerates the electron back toward the ion. We
see that ionization or recombination can occur on every half-cycle of the driving field.
In addition, we note that only linearly polarized fields lead to electron trajectories that
return the electron to the ion [11]. During the second step, the electron gains a timeaveraged kinetic energy, associated with being subjected to a sinusoidal field, known as
the ponderomotive energy. The ponderomotive energy Up is given byUp =e2 E 24me ω02(2.3)

where e the electron charge, me is the electron mass, E is the electric field of the driving
laser and ω0 is its frequency. For a given laser frequency, Up is directly proportional

7
to the square of the electric field. Therefore, Up scales linearly with the driving laser’s
intensity. Finally, in the third step, the electron recombines with its parent ion, and in
so doing, releases its kinetic energy and the ionization potential of the atom as a highharmonic photon. Considering this third step, an accurate prediction of the highest
achievable photon energy can be made and is given by this simple cutoff rule(h̄ω)max = Ip + 3.2Up(2.4)

where Ip is the ionization potential of the gas species being used, and the factor of
3.2 relates to the optimum phase for the ionization of the electron with respect to theoscillating electric field [8,9]. From the single atom point of view, the cutoff rule suggeststhat to increase the highest obtainable photon energies, we should use atoms with highionization potentials, like the noble gases, and ionize them with the most intense laser
pulses available. Of course, maximizing the flux of high energy photons is a much more
complicated matter, that requires navigating the effects of many radiating atoms underdynamic conditions.

2.3

Phase-Matched Frequency Upconversion in Hollow Waveguides

2.3.1

Phase Matching in Nonlinear Optical Frequency Conversion

In the summer of 1961, P.A. Franken et. al. at the University of Michigan realized
the possibility of exploiting the extraordinarily intense electric fields of focused laser
beams to produce optical harmonics [12]. They reported the first nonlinear optical signal
in the form of second harmonic generation (SHG). The generation of optical harmonics
represents one of the most important applications of nonlinear optics, and has lead tothe development of many novel coherent light sources. However, to efficiently produce
optical harmonics, the process must be phase matched: i.e. the phase relationshipbetween the fundamental and signal light must be maintained throughout the nonlinear
medium.

8
In nonlinear optical frequency conversion, this concept of phase matching remains
one of the most fundamental challenges. Looking at the issues associated with phasematching in the context of SHG provides a useful introduction. In the SHG process, twophotons of a fundamental driving field are combined in a nonlinear crystal to produce asingle photon that possesses twice the frequency of the fundamental light. From a moreclassical electromagnetic perspective, the fundamental wave with frequency ω induces
a polarization wave at frequency 2ω via an interaction with the materials second-ordernonlinear susceptibility. Driven by the fundamental wave, this polarization wave travels
with an identical phase velocity that depends on n(ω). However, the second-harmonic
wave radiated by the polarization wave propagates at a speed determined by n(2ω) [7].
In general, in SHG, the fundamental and second harmonic waves will travel at
differing phase velocities. Thus, the two waves will possess a time evolving phase relationship. The relative phase difference between the two waves determines the directionof power flow between the waves [13]. So, maintaining the proper phase relationshipproves essential for efficient SHG. In the case of SHG, the process of ensuring zero
phase slip between the fundamental and harmonic fields is referred to as phase matching. When phase matched, the generated second harmonic at any point in the medium
always adds in phase with that generated earlier in the medium. When the conversion
process is not phase matched, the coherence length equals the distance in a crystal overwhich the phase slip between the fundamental and second harmonic equals 180◦ [7].
Therefore, unless an SHG process is phase-matched, the coherence length is the longest
length over which a macroscopic output signal can be generated.
Phase matching is not automatic in most media. In general, the fundamental
beam and the second harmonic beams, with angular frequencies ω and 2ω respectively,
travel at differing speeds as they propagate through a material. This effect arises fromclassical dispersion; the phase velocity is given by vω = c/nω , and nω is frequency
dependent. In terms of k-vectors, where kω = ω/vω = ωnω /c , for second harmonic

9
generation we have
ω + ω = 2ω

(2.5)

which is a statement of the conservation of energy, since the energy of a photon is given
by E = h̄ω. Furthermore, we have∆k = k2ω − (kω + kω ) =2ω(n2ω − nω )
c

(2.6)

which is a statement of the conservation of momentum, since the momentum of a photon
is given by p = h̄k. Since nω 6= n2ω due to dispersion, clearly, ∆k 6= 0 in general.
Furthermore, if we assume a lossless nonlinear medium, the Poynting vector S is given
by
(2ω)2
S2ω =
2nω nω n2ω



µ
ǫ0

3
2

2 2

d L Sω Sωsin(∆kL/2) 2(∆kL/2)(2.7)

where L is the distance traveled in the material, and d is known as the nonlinear opticalcoefficient and describes the strength of the nonlinearity. This equation shows that whenno phase matching occurs, the energy in the second harmonic varies as sinc2 (∆kL/2),
and it will be maximum at a distance
L = Lc =

π
∆k

(2.8)

known as the coherence length. After the second harmonic wave propagates over a
coherence length, energy begins to return to the fundamental wave [7].
Employing some actual values to calculate a typical coherence length reveals an
immediate obstacle to efficient SHG. Suppose we wish to frequency double a wavelengthof 1 µm to a wavelength of 500 nm in the visible region of the spectrum using a KDP
crystal as the nonlinear medium. For SHG,
Lc =λ4(n2ω − nω )(2.9)

So, in this case, nω =1.50873, n2ω =1.529833, and the coherence length is a meager
11.8 µm [12]. Most nonlinear materials exhibit similar coherence lengths. So, the10second harmonic signal only grows for a small number of optical cycles, leading to
extremely low conversion efficiencies. Clearly, from the definition of Lc given in Equation
2.8, decreasing ∆k increases the coherence length. Moreover, from Equation 2.7 for
the Poynting vector, as ∆k approaches zero, the second harmonic signal will growproportionally to L2 [7]. The general features of phase-matched versus unphase-matchedsignal growth are illustrated in Figure 2.1.

2.3.2

Phase Matching of HHG in Gas-Filled Hollow Waveguides

Unfortunately, the nature of HHG presents several significant obstacles to phasematching. Perhaps the most daunting, is that the process takes place in a gas. As
such, the nonlinear medium is optically isotropic, and does not afford the possibilityof traditional phase-matching techniques based on differing source and signal polarizations encountering separate indices of refraction. Additionally, the most straightforwardmanner of producing high harmonics involves focusing high intensity laser light into apulsed gas jet. A pulsed gas jet minimizes the gas load on a vacuum system, and thedriving laser merely needs to be focused with sufficient intensity into the jet to produce
harmonics. However, when traversing a focus, a laser beam undergoes a π phase shift,
known as the Guoy phase shift [14]. This intrinsic phase shift destroys the possibility ofphase matching in the gas jet method over extended propagation distances (over shortdistances, the Guoy phase actually can be used to compensate for other sources of phasemismatch).
Still, with the innovation of a hollow-waveguide geometry, phase-matched HHG
was realized in the late 1990’s [15]. The hollow waveguide approach to phase matchingrelies on several important features of the waveguide. First, when laser light is coupled
into the guide, it does not go through a focus and travels instead as a plane wave. Second,
the light experiences a frequency dependent dispersion while propagating through the
waveguide. Finally, since the waveguide confines the gas in which the HHG takes place,

11

Figure 2.1: Phase-matched signal growth (blue) compared to unphase-matched signal(green) for several coherence lengths

12
the pressure of the gas may be controlled, allowing the strength of the dispersion causedby the gas to be tailored. The k-vector for the fundamental light propagating througha hollow waveguide in the lowest loss mode can be written ask≈2π 2πP (1 − η)δ(λ)µ2 λ++ (1 − η)n2 I − P ηNatm re λ − 11 2λλ
4πa(2.10)

where λ is the wavelength in the guide, Natm is the number density of neutral atoms at
atmospheric pressure, δ(λ) is a function corresponding to the dispersive characteristicsof the gas species being used, P is the gas pressure, η is the ionized fraction, re is theclassical electron radius, µ11 is the first root of a Bessel function of zeroth kind J0 corresponding to the lowest loss mode, and a is the radius of the waveguide [16]. Factoringout the pressure term from Equation 2.10 and neglecting the small contribution of thenonlinear index of refraction gives
k≈



µ2 λ
2π
2π(1 − η)δ(λ)+P− ηNatm re λ − 11 2λλ
4πa(2.11)In the case of the EUV wavelengths of interest, the waveguide has a negligible effect on
the phase velocity, which is then slightly greater than c for the harmonics at ionizationlevels where the pressure dependent term remains positive. For phase matching, we
need∆k = kq − qk0(2.12)

where q is the harmonic order so that kq is the k-vector of the qth harmonic and k0 is the
k-vector of the fundamental driving laser. Clearly, kq and k0 for light traveling in the
waveguide will be given by Equation 2.11 but with λ replaced by λq or λ0 respectively.
So, for the HHG process taking place in a waveguide, and using the knowledge that for
high-order harmonics qλq = λ0 and that qλ0 ≫ λq ,∆k ≈qλ0 µ211
4πa2
where,+ P
2π(1 − η)[∆(δ)]ηNatm re (qλ0 − λq ) −λq∆(δ) = δ(λ0 ) − δ(λq )(2.13)

13
Looking at Equation 2.13, the pressure dependent term will have an opposite sign than
the waveguide term for low levels of ionization. Thus, by tuning the gas pressure in the
waveguide for a given ionization fraction, we can achieve the phase-matched condition
for a wide range of experimental parameters [16].
One of the main limitations of the pressure-tuned phase-matching technique in ahollow waveguide is the level of ionization at which the technique still works. Clearly,
if the gas is fully ionized, the overall sign of the pressure dependent term in Equation
2.13 will be the same as the waveguide term, and pressured-tuned phase matching is
no longer possible. So, the pressure dependent term can be solved for the ionization
fraction at which the overall term value equals zero. We refer to this ionization as thecritical ionization and label it ηcr . Discarding the λq term, we find that−1Natm re λ20
ηcr = 1 +2π(∆δ)(2.14)

Typical values of ηcr limit pressure-tuned phase matching to ionization levels < 5%,and in turn, to photon energies of ∼ 100 eV [11]. Methods of quasi-phase matching toovercome this limitation are discussed in the following chapter.

2.4

3-Section Hollow Gas-Filled Waveguide Design with OuterCapillary Fixture
When I first began my research, the 3-section hollow waveguide design using anouter-capillary fixture was well established in our group. The HHG process must takeplace in a tenuous gas. However, the EUV light produced must travel through vacuum
so as not to be absorbed. The most popular method of overcoming this experimental
dilemma involves using a pulsed gas jet. The gas jet allows a high pressure region toexist within the vacuum for a short time without putting a continuous high gas load on
the pumping apparatus. However, as discussed earlier in the beginning of Section 2.3.2,
a gas jet does not allow phase matching over a long interaction region.

14
The 3-section design solves the basic experimental dilemma of confining a low
pressure gas to a waveguide in an otherwise evacuated environment via differential
pumping, and maintains a constant and tunable gas pressure within the central section[15]. The design consists of three pieces of fused silica capillary held in place by vacuum
compatible epoxy in a larger outer capillary that serves as an alignment and gas handling
fixture. Fused silica is used as the waveguide material as it provides excellent properties
in terms of resistance to thermal and laser damage, and does so in a cost-effective
manner. The three smaller capillary pieces are referred to as the inner capillary, andhave a typical outside diameter of 1.20 mm. The outer capillary has a 0.250” outer
diameter and an inside diameter that only slightly exceeds the outside diameter of theinner capillary (typically by ∼ 10 µm). The inner capillary can be obtained in several
inside diameters, but in the experimental work discussed in this thesis, all inner capillary
pieces have an inside diameter of 150 µm. In general, the length of the waveguide isreferred to by the length of the central inner capillary, since the HHG process occurs
within this region. The length of the waveguide generally varies between 2.5 cm and10 cm, depending on the gas being used, since the short absorption lengths of some
noble gases in the EUV limit the useful interaction length supplied by the waveguide.
The central inner capillary has a further piece of capillary at each end, referred to as
an end piece. These end pieces are typically 5 mm long, and are placed within theouter capillary such that they maintain a 0.5 mm gap from the ends of the central
inner capillary. Figure 2.2 displays a to-scale 3-d rendering of one end of the 3-section
hollow waveguide design and shows the relative placement and geometry of the innerand outer capillary components. The 3-section hollow capillary assembly is designed tobe pumped from each end of the outer capillary fixture while gas flows into the gaps
separating the central inner capillary and its end pieces. Since the end pieces are small
apertures with low conductance, a differential pressure can be maintained on each sideof them, allowing the central inner capillary to achieve a constant gas pressure while

15

Figure 2.2: A to-scale 3-d rendering of one end of the 3-section design

the outer capillary ends are held at vacuum pressures. The gaps between the central
inner capillary and its end correspond to radially drilled holes in the outer capillary thatprovide the passageways for positive gas flow. The outer capillary is made such that its
ends are ∼ 3.5 cm from these gaps and a standard 14 ” Ultra-Torr R fitting can be seatedover each end and leave room for a 14 ” Ultra-Torr R tee to form a seal over the gas input
holes. The other two radial holes straddling the gas input holes at each end of the outercapillary serve as gluing ports. A picture of an outer capillary for holding a 10 cm inner
capillary is shown in Figure 2.3, and a schematic with the relevant measurements isshown in Figure 2.4 . Using a thinned down q-tip end, epoxy is pushed into these holes
to cement the inner capillary pieces into position. Note that Figure 2.2 and Figure 2.4
show that the axial hole centered on the end piece goes slightly deeper than the inner
bore; the idea being that epoxy can be forced all the way around the end piece ensuringthat the gas flow only occurs through the small conductance of the end piece inner bore.

16

Figure 2.3: A picture of an outer capillary for the 3-section design

Figure 2.4: A schematic diagram of an outer capillary for the 3-section designChapter 3

Quasi-Phase Matching in Modulated Hollow Waveguides

3.1

Introduction
The extension of frequency conversion techniques leads to novel light sources andconsequently new scientific tools. One of the main themes of HHG research is extending
the tunability of this source, and the flux it produces, especially as the process is pushed
towards shorter wavelengths. As discussed in Chapter 2.3.2, phase matching greatly
increases the efficiency of this source, but pressure tuned phase matching faces serious
limitations as the ionization of the gas in the waveguide exceeds a critical ionizationlevel. However, we have demonstrated techniques for quasi-phase matching (QPM) of
HHG that overcome this limitation. To make a clear intuitive picture of the QPM
process, I first explore a theoretical approach to QPM for SHG, and then extend thispicture to a basic approach to QPM for HHG in a hollow waveguide. Afterwards, themotivations for using structural changes to a waveguide to achieve QPM are discussed
along with the successful implementation of this idea.

3.2

Quasi-Phase Matching for Nonlinear Optical Frequency Conversion
Under conditions not conducive to traditional phase matching, or where tailoring

the nonlinear response is desirable, the method of quasi-phase matching (QPM) may
be used. In the QPM scheme, a structural periodicity integrated into the nonlinear

18
medium corrects the phase relationship between the fundamental and signal light atregular intervals.
Since the pioneering work of Franken, nonlinear optical frequency conversion has
grown into an immensely powerful scientific tool, as evidenced by the prolific use ofsuch equipment as frequency doubled pump lasers and optical parametric oscillators.
The most popular methods of achieving frequency conversion rely on birefringent phasematching. For all crystals not exhibiting the cubic class of symmetry, the index ofrefraction becomes a function of wavelength, propagation direction, and polarization.
Such optically anisotropic crystals are referred to as birefringent, since, for a given direction of propagation, a light beam can interact with two different indices of refraction,
depending on the beam’s polarization [17]. This special property allows the possibility of perfect phase matching for SHG. If the fundamental and second harmonic lightpropagate collinearly in a birefringent crystal, but with mutually orthogonal polarizations, a condition can often be found wherein both beams experience the same index
of refraction, and so, propagate at identical phase velocities. Ironically, the idea for
QPM, devised independently by Armstrong et. al and Franken and Ward, predates theuse of birefringent phase matching [18, 19]. However, the realization of QPM did notoccur until the technologies associated with accurately manufacturing structured materials came of age. As demonstrated by nonlinear materials such as periodically poled
LiNB03 (PPLN) that have been used for quasi-phase matched SHG, QPM continues togain importance and utility [20].
In the QPM scheme, the relative phase mismatch between the driving and signal
fields is corrected at regular intervals by the use of a periodic modification integrated
into the nonlinear medium. Modifications useful for QPM alter the nonlinearity of thematerial, either by changing its strength, reversing its sign, or eliminating the nonlinearity altogether [13]. Figure 3.1 provides a useful introduction to this idea. This figure
represents an alternating stack of crystals used for the QPM of an arbitrary SHG pro-19
cess. Each slab has a thickness equal to the coherence length, and we assume that theindex of refraction as a function of wavelength remains the same throughout the stack.
However, every other slab lacks a nonlinear response. In this idealized arrangement,
the SHG signal builds in the first slab. Then, just before energy begins to flow backto the fundamental field, the pump and signal waves enter the second slab. Energy
transfer ceases at this point since no nonlinear response is present. However, the phaserelationship between the pump and signal waves continues to evolve in the second slab.
When the waves are back in phase, they encounter the third slab. This slab has anonlinear response, so energy proceeds once again to flow to the harmonic wave. Moreover, since the correct phase relationship exists, the signal produced in the third slab
adds constructively with the signal produced in the first slab, and so on [7]. A more

Figure 3.1: An idealized QPM geometry wherein each slab of crystal is a coherence
length thick and every other slab of crystal lacks a nonlinear response

mathematically rigorous analysis of QPM SHG that closely follows the logic of Fejer
et. al. shows some important insights into the QPM process [13]. Working under theassumptions of low conversion efficiency, weak focusing, long-pulse interactions, and alossless medium, the slowly varying amplitude equation that describes the development

20of the second-harmonic field is given by
dE2ω
dz

′

= Γd(z)e−i∆k z

Γ ≡

(3.1)

iωEω2
n2ω c

where E2ω is the amplitude of the second-harmonic field, z is the distance along thedirection of travel, d(z) is the spatially dependent nonlinear coefficient for SHG, and∆k′ is the wave vector mismatch. To find the strength of the second harmonic field after
traversing a material of length L, equation 3.1 is integrated with respect to z, giving
E2ω = ΓZ L′

d(z)e−i∆k z dz

(3.2)

0

Considering this integral as a Fourier transform leads to a transparent interpretation of
the effect of QPM. Letg(z) ≡d(z)def fwhere − 1 ≤ g(z) ≤ 1(3.3)

then the Fourier transform of g(z) isZ′
1 L
g(z)e−i∆k z dzG(∆k ) =
L 0
′

(3.4)

Furthermore, assuming g(z) is a function periodic in z, with period Λ expressed byg(z) =∞
X

Gm eiKm z

where Km =

−∞

2πm
Λ

(3.5)

For all Km far from the value ∆k′ , the integrand will be a rapidly oscillating function
averaging to zero. However, if Km is near ∆k′ , the integral is given byE2ω ≈ ie−∆k·L/2 ΓdQ Lsinc(−∆k · L/2)

(3.6)

where dQ = def f Gm and ∆k ≡ k2ω − 2kω − Km . This approach yields an especiallysimple interpretation of QPM; i.e. the structure may be viewed as adding an effective
wave vector to the phase-matching conditions.

21

3.3

Motivations for a Modulated Waveguide
As discussed in Chapter 2, HHG performed in a hollow waveguide lends itself

to phase matching through pressure tuning of the gas filling the waveguide. Still, thismethod encounters limitations when attempting to reach higher energy harmonics. The
driving laser generates higher EUV photon energies at high intensities that correspondto higher levels of ionization. At these high levels of ionization, the phase velocity of thedriving laser beam exceeds the limit of pressure tuned phase matching. As the ionization
approaches a certain level, which we call the critical ionization, the effect of neutral
atoms grows smaller while the plasma dispersion increases. Thus, an asymptoticallyincreasing pressure is required for phase-matching. Above the critical ionization level,
phase-matching cannot be achieved in the hollow waveguide geometry. However, toovercome this obstacle, the concepts of QPM may be applied to HHG.
To induce QPM of the highest accessible harmonics, Christov et. al. proposedusing a hollow waveguide with a periodic modulation of the inner diameter [21,22]. The
primary assumption behind this idea is that the laser mode adiabatically follows the
waveguide. Under this assumption, the laser mode expands or shrinks to fill the periodically varying cross-sectional area of the waveguide. Therefore, the waveguide regularly
modulates the intensity of the driving field. The phase and the cutoff of the HHG
emission depends on the laser intensity with extreme sensitivity, so the modulationsin the waveguide effectively change the nonlinear interaction, allowing QPM to occur.
Calculations by Christov suggested that significant enhancements at higher harmonicorders were possible with the correct modulation conditions [21].
Extending the formalism presented in the preceding section to the realm of HHG
reveals that introducing a QPM k-vector, ∆k in a waveguide under the influence of a
QPM mechanism will be Equation 2.13, but with an extra term arising from QPM, and22having the formqλ0 µ211
2πq(1 − η)[∆(δ)]∆k ≈− Km + P ηNatm re (qλ0 − λq ) −
4πa2
λq


(3.7)

[23]. First-order QPM corresponds to modulating the waveguide (and harmonic emission) every coherence length and we will assume here that m = 1. Also, a critical
periodicity, Λcr , can be defined where the QPM k-vector is equal to the k-vector of anevacuated unmodulated waveguide. This critical periodicity and its ratio to the actual
periodicity (Λ)are given byΛcr =8π 2 a2qλ0 µ211

β=

Λcr
Λ

(3.8)

where the dimensionless quantity β has been introduced to help characterize the generaleffect of the modulations.
To obtain a general illustration that summarizes the modifications to the phasematching pressure due to the presence of a modulated waveguide, we introduce several
dimensionless quantities. For a given set of experimental conditions, an optimum pressure for phase matching can exist. Calling this pressure Popt , we define P0 as the valueof Popt when η = 0 in the absence of any QPM k-vector. Solving Equation 2.13 for P0
we find thatP0 =λ0 λq µ211
8π 2 a2 [∆(δ)](3.9)

Now, the ratio of Popt to P0 can be thought of as a normalized and dimensionlesspressure, and the ratio of η to ηcr (defined in Equation 2.14) gives a normalized anddimensionless ionization level. Figure 3.2 shows a plot of Popt /P0 versus η/ηcr for several
different ranges of β. The dotted line is where Popt /P0 = 1, and curves falling below the
dotted line are not physical. For the blue curve, β = 0, and the trend of pressure-tuned
phase matching in an unmodulated waveguide appears; i.e. an asymptotically increasingpressure is needed to achieve phase matching as the ionization level approaches thecritical ionization. For the green curve, β < 1, and the waveguide modulations actually

23

Figure 3.2: A graph plotting normalized pressure for phase matching versus normalized
ionization fraction (courtesy Randy Bartels)

24serve to decrease the range of ionization for which phase matching can be obtained. The
most interesting curve is plotted in red, and it shows the situation for β > 1. This curve
suggests that the useful range for which QPM operates is actually above the critical
ionization! Thus, since higher photon energy harmonics are created at higher levels of
ionization, adding a modulation to the waveguide that introduces a sufficiently large
k-vector to Equation 3.7 should allow us to find a pressure at which we can quasi-phasematch the HHG process for harmonic orders that were not accessible for the pressuredtuned phase-matching scheme in a unmodulated waveguide. In fact, as is shown in thefollowing section, this QPM scheme is successful.

3.4

Using Modulated Waveguides for High-Order Harmonic Generation

3.4.1

The First Demonstration of Quasi-Phase Matching with HighOrder Harmonics

Although developing fabrication techniques to produce modulated waveguides
took considerable effort (described in Section 3.5), the use of modulated waveguides forhigh-order harmonic generation immediately generated a number of impressive results.
The description of the first experimental results from these waveguides that follows
relies heavily on a paper I wrote with Randy Bartels, “Quasi-phase-matched generationof coherent extreme-ultraviolet light” Nature, vol. 421, pp. 51-54, 2003 [1].
As mentioned previously, pressure-tuned phase matching in unmodulated hollow
waveguides is limited to relatively low EUV photon energies. HHG can produce photons
in this high ionization regime with energies of several hundred eV up to > 1 keV [24–26].
However, efficient phase-matched HHG had only been demonstrated at photon energies
of between 50−100 eV prior to the work presented here [15]. This limitation prevents theuse of this source for applications such as in vitro imaging of small cellular structures,

25
which requires EUV light in the water window (∼ 300 eV) region of the spectrum.
Coherent sources in the 13-nm wavelength of interest to EUV lithography would also
benefit from increased efficiency.
This flux limitation can be partially overcome by applying QPM techniques to
HHG. Using a hollow waveguide with a periodically modulated diameter, the energy
range over which we efficiently generate high harmonics can be increased significantly.
In the experiment, 25-fs-duration pulses from a high-repetition rate (25 kHz, 1mJ per
pulse) Ti:sapphire laser system operating at 760 nm were focused into 150µm innerdiameter hollow waveguides filled with various noble gases [27]. The spectrum of the highharmonic radiation from these waveguides was recorded using a glancing incidence EUV
spectrometer. The inner diameter of some of the waveguides was periodically modulated
to alter the laser intensity traveling inside the waveguide. As represented in Figure 3.3,
the modulations were approximately sinusoidal, with a period of 10.5 mm, and a radial
depth of the order of 10µm, corresponding to a 13% modulation of the waveguide inner
radius. The periodic structure modulates the generation of high harmonics and appears
to restrict HHG to regions with favorable constructive interference. This work showsthat sophisticated concepts of nonlinear optical photonics and engineered structures canbe applied even to the extreme nonlinear optics of HHG. The highly nonlinear natureof the HHG process, though complicated, allows for new control mechanisms that do
not exist in conventional nonlinear optics.
To interpret our results, we use previous theoretical predictions of the potential
utility of QPM frequency conversion in the EUV [21, 28]. HHG is extremely sensitive
to intensity; thus, a periodic modulation in the intensity of the driving laser will modulate the HHG emission. Generation of the highest harmonic orders is turned off when
the waveguide bulge increases the mode diameter in the waveguide, thus preventing
back-conversion of the EUV light. Therefore HHG will be quasi-phase matched when
the period of the modulation matches the period of the phase mismatch between the

26

Figure 3.3: A schematic of the partially modulated waveguide in the 3-section configuration

pump and signal (that is, twice the coherence length). We also note that two otherexperimental works used QPM in low-order harmonic generation in gases [29,30]. However, these approaches are not applicable to the EUV. Other proposed schemes, suchas suppressing HHG with interfering beams, were first demonstrated with gas jets inthe EUV [31]. More recently, work in our own group has shown remarkable enhancements using counter-propagating light in unmodulated hollow waveguides to quasi-phasematch the HHG process [32, 33].
An intuitive picture of our experimental results can be obtained in the perturbative approach. The signal corresponding to the qth harmonic (Eqω (L)) after propagation
through a generation medium of length L is given by:Eqω (L) ≈Z L
0Eωq (z)d(z)ei∆kz dz(3.10)

where Eω (z) is the laser field, d(z) is the effective nonlinear coefficient for HHG, and∆k = qkω − kqω is the net phase mismatch between the fundamental and harmonic field.
In the absence of phase matching or quasi-phase matching, the rapid ei∆kz phase term
will cause the integral in Equation 3.10 to average to zero. This integral will be non-zero
if either ∆k = 0, corresponding to traditional phase matching in the visible or EUV, or27
if d(z) ≈ cos(∆KM z); corresponding to traditional QPM in the visible. For the latter,
the sign on the nonlinear coefficient d(z) is modulated periodically using periodicallypoled materials [13, 34–36]. The modulation period Λ (where ∆KM = 2π/Λ) is chosen
to be equal to the coherence length of the signal, that is, the distance over which thefundamental and harmonic fields undergo a phase slip of π, to ensure that the harmonic
signal from different regions in the interaction length always adds in phase.
In this work, a new type of QPM can be implemented in the EUV where thesignal wave itself is modulated periodically, that is,Eωq ≈ cos(∆KM z)(3.11)and d is independent of z. The periodic QPM structure can modulate the generationof high harmonic orders in several ways. Harmonics near cut-off that require higher
laser intensity for generation can be turned on and off by the modulation. However,other effects (such as the periodic evolution of the phase of the driving pulse) can
also contribute to a periodic modulation of the harmonic generation process. In fact,
virtually any periodic change in the character of the harmonic emission can allow QPM
to operate, particularly in a very-high-order nonlinear process where very small phasechanges can dramatically change the output [37].
The significance of the quasi-phase matching discussed here is that it permits
phase matching of HHG at higher ionization levels and hence higher photon energies
than previously possible. This can be made apparent by calculating the coherence
length that results from ionization of the gas. The plasma-induced change in index ofrefraction corresponds to a phase mismatch in Equation 3.10 of∆kplasma ≈qne e2 λ4πme ǫ0 c2(3.12)

where λ is the driving laser wavelength, e is the charge of the electron, me is themass of the electron, ǫ0 is the vacuum permittivity, and ne is the electron density.

28
For fully ionized argon at a pressure of 1 torr, and for q = 29, ne = 3.3x1016 cm−3 andπ∆k ≈ 23cm−1 . Therefore, the coherence length, Lc , given by Lc = ∆k
, is 1.4 mm. Thus,

very substantial levels of ionization can be compensated for using the QPM techniquedemonstrated here, with experimentally realizable modulation periods in the millimeter
range. In our experiments, because we could modulate at most a 2.5-cm-long sectionof capillary, we used higher pressures (30 torr). This higher pressure implies that the
1 mm modulation period compensates approximately for an additional 4% fractional
ionization at harmonic order 29, and can be compared with a unmodulated waveguidewhere phase matching occurs at ionization levels of 24%. Thus, QPM more than doubles
the allowable level of ionization in this case. For He, the effect of the QPM is greater,
because the fractional ionization at which high harmonics are efficiently generated ismuch lower (<0.6% for q = 61, for example). Therefore, a small change in allowableionization results in a large change in the energy of the highest phase-matched harmonicorder.
Figure 3.4 shows the HHG spectrum from He, Ne and Ar gas for unmodulated(blue) and modulated (red) waveguides. These spectra were taken in the phase matchedregime of the unmodulated waveguide. For all cases, the flux measured from the modulated waveguides is significantly greater (by factors of 25) than that measured from
unmodulated waveguides. In this case, the modulated section was 1 cm in length, with a
1 mm periodicity, placed near the end of the waveguide. The measured flux corresponds
approximately to 1 nJ per harmonic per pulse for Ar, and 20 pJ per harmonic per pulse
for He at repetition rates of 2 kHz. Most significantly, the modulated waveguide increases the brightness of higher harmonic orders by at least two orders of magnitude. In
the case of He, for the unmodulated waveguide the comb of harmonics spans an energy
range from 60 to 80 eV, with a spectral peak at 68 eV (Figure 3.4a, blue trace). For themodulated waveguide, the spectral peak shifts by 27 eV, from 68 to 95 eV (Figure 3.4a,red), while the QPM harmonic comb also spans a broader energy range from 63 to 112

29

Figure 3.4: Experimentally measured HHG spectra for unmodulated (blue) and modulated (red) waveguides. Data are shown for He gas (a), Ne gas (b) and Ar gas (c), atpressures of 150 torr, 47 torr and 45 torr, respectively. From [1]30
eV. Note that the wavelength region over which we achieve QPM easily includes the 94eV (13 nm wavelength) region of interest for the next generation of EUV lithographies.
In the case of Ne, use of a modulated waveguide extends the observed high-harmonic
emission from 70 to 90 eV, while significantly increasing the flux (Figure 3.4b). In thecase of Ar, the HHG spectrum from a unmodulated waveguide consists of a comb ofphase-matched harmonics, peaked around 37 eV (Figure 3.4c, blue). For the modulated
waveguide (Figure 3.4c, red), the HHG output spectral peak shifts to 47 eV, while theflux increases significantly.
We also investigated HHG from longer (2.5 cm) modulated waveguides with different modulation periods. Figure 3.5 shows the experimentally measured HHG spectrafrom He for three different periodicities of the modulated waveguides. All spectra weretaken through two 200 nm zirconium filters, each of which transmits EUV in this range
with ∼ 50% efficiency, while reflecting or absorbing visible light [38]. The gas pressure
was 111 torr, and the laser intensity was about 5x1014 W/cm2 . The different curves
were taken at different positions of the grating in the EUV spectrometer, because of
the finite spectral region that can be captured simultaneously on the CCD. Therefore,
the cut-off at low energy is artificial, limited by the spectral window. The high-energy
cut-off is limited by the available laser intensity. The spectra were also normalized tohighlight the trend of increasing harmonic energy with decreases in the modulation period. As the modulation period is reduced from 1 mm, to 0.75 mm, to 0.5 mm, thehigh-energy cut-off increases from 112 to 175 eV. This cut-off was limited by the available laser intensity, and it increased in later experiments (see Section 3.4.2) when longer
modulated sections, shorter laser pulses, and higher laser intensities were implemented.
Finally, by decreasing the modulation period from 1 to 0.5 mm (Figure 3.5),
the amount of ionization that can be compensated increases, allowing for even higher
harmonic orders to be phase matched. Furthermore, by increasing the length of themodulated waveguide and reducing the pressure, very high levels of ionization might

31

Figure 3.5: Experimentally measured HHG spectra (log scale) from He for three different periodicities of the modulated waveguides, each 2.5 cm in length. Blue, 1 mm
periodicity; red, 0.75 mm periodicity; green, 0.5 mm periodicity. From [1]32
be compensated for, leading to the generation of very-high-order harmonics, perhaps
even from fully ionized gasses. This approach also leads to a significant increase in
output flux, because the transparency of the gas is increasing with increasing photon
energy over the range we investigate, while the effective nonlinearity is not changing
rapidly. For He and Ne, for example, the absorption length is increasing with increasing
harmonic photon energies. Therefore, we observed both higher harmonic orders andincreased overall flux in the data of Figure 3.4.

3.4.2

Further Results

After the initial results presented in the preceding section, I performed further
explorations of modulated waveguides with Emily Gibson, who was at the time a seniorgraduate student in our group. A thorough treatment of the results and interpretationsof those experiments already exists in her thesis entitled “Quasi-Phase Matching of SoftX-Ray Light from High-Order Harmonic Generation Using Waveguide Structures” [39].
The reader is referred to this reference and the corresponding scientific papers for a
detailed account [40, 41].

3.5

Producing Modulated Waveguides
When I first arrived in the Kapteyn/Murnane group, I was assigned to work with

Dr. Sterling Backus. Ster, as he is known, is one of the world’s experts on high power
amplifiers for ultrashort laser pulses, and at that time was the senior leader of the labs.
Although quite busy with his own duties and research, he was an excellent mentor, andrecognized that he could suggest tasks to me that would require independent work andcreativity. Still, being a young graduate student, I required guidance, and I wanted
to make sure that at times when Ster was unavailable I could focus my efforts on a‘back burner’ type of project. I had noticed that an interesting device for modifying
the central inner capillary of the 3-section hollow waveguide consisting of a custom

33
miniature glass-blowing lathe built by Seth Weiman in the JILA Instrument Shop had
been sitting idle in our lab together with a compact CO2 laser operating at a 10 µm
wavelength as a heat source.
The basic idea was to implement simple glass-blowing techniques to achieve a
periodic modulation of the inner diameter of the inner capillary. The major differences
between a glass-blowing lathe and a standard lathe are that a glass-blowing lathe has
two headstocks whose motion is made synchronous through mechanical couplings, andone headstock generally accommodates a freely rotating feedthrough for gas pressure
control. Glass-blowing lathes find their main application in scientific glass blowing,
making glassware from custom test tubes and vessels, to specialized vacuum cells andflow controls. The primary advantages of such a lathe are that it allows cylindricallysymmetric application of heat from a focused heat source, as well as a much higher
degree of accuracy than manual rotation of the workpiece. However, if a cylindricalpiece of glass is held in such a lathe and the headstocks do not rotate synchronously, atorque will be placed on the piece about the axis of rotation. As the glass is heated andbegins to yield, it will twist about the soft point. Similarly, radial stresses will deformthe glass with the application of heat, so a glass-blowing lathe must be accurately aligned
and run true to avoid unwanted kinks in the workpiece. For a tube of glass placed in theglass-blowing lathe, the internal gas pressure must be controlled to create the desired
effect; positive pressures will create a bulge, while negative pressures will constrict the
tube. When heated in a glass-blowing lathe at slow rotation speeds for a given diameter,where the centripetal acceleration of the glass is less than the acceleration due to gravity,
glass tubes tend to sag radially inward. Controlling the gas pressure compensates forthis sag, increases it, or in the case presented here, can be used to bulge the innerdiameter radially outward.
As mentioned in Chapter 2.4, the capillary that forms the 3-section waveguide ismade of high purity fused silica. High purity fused silica glass is the noncrystalline or34
amorphous equivalent of quartz. It possesses some unique characteristics that make it
useful for high intensity laser waveguides. However, these characteristics must also be
understood in order to yield the desired results from the glass-working processes. As
compared to Pyrex glass, fused silica has some remarkable properties; a much higher
softening point of nearly 1700◦ C, and nearly an order of magnitude smaller coefficientof thermal expansion. Whereas normal pyrex can be worked with a propane and oxygen
flame, fused silica requires the heat of a hydrogen and oxygen flame. Although quitetransparent in the visible part of the spectrum, fused silica is completely opaque forwavelengths above 7 µm for any reasonable thickness (i.e., several wavelengths) [42].
Therefore, a CO2 laser operating in the far IR with sufficient power constitutes anideal heat source for precision glass-blowing of fused silica. Amazingly, due to its low
coefficient of thermal expansion, a piece of fused silica remains remarkably robust underthermal stress, as it can be heated white hot and then quenched in cold water without
cracking. The high transmittance of fused silica at visible wavelengths, coupled with its
resilience to thermal stress, combine to give it a high damage threshold for the ultrafast
amplified pulses used in HHG. However, this material also possesses a low thermal
conductivity, much like regular glass, that must be carefully taken into account in theglass-blowing processes.
When approaching the question of how to perform accurate glass-blowing on ourdelicate fused silica waveguides, I wanted to settle on the minimum number of adjustableparameters. Heat deposition, the area over which this heat acts, net force on the inner
capillary wall, and the time this force acts while the glass is soft are the heart of the
problem. However, rotation speed, convective cooling, thermal conductivity, coolingtime, incident laser power, laser focal spot size, exposure time, and gas pressure all affectthe final result of the glass-blowing operation. Although some literature exists describing
the process of tapering capillaries with a CO2 laser, it would appear that the type ofglass-blowing I performed was novel to some extent [43,44]. With such a large parameter35
space, and mainly trial and error as a guide, I decided to settle on a fixed rotation
speed, internal gas pressure, output laser power, and focal spot size. Therefore, laser
exposure time became the main adjustable parameter. For a complicated set of thermaltransport conditions, I reasoned that an exposure time could be found that would create
the desired modulation depth. The laser power was chosen using an optical pyrometerviewing a heated piece of capillary to verify that I could reach and maintain the softening
temperature; I settled on around 6 W as a starting point. For pressurizing gas, I chose
dry nitrogen, as I was concerned with maintaining the cleanliness of the capillary’s innerbore. After some back-of-the-envelope calculations, I chose 50 psi as a working pressure,believing that once the fused silica had reached its softening point a pressure above 10
psi would produce a several micron change in the inner diameter within a couple ofseconds. The original set-up focused the CO2 laser with a single 10 cm focal length
Zn-Se lens to an approximately 0.75 mm spot diameter. Nominally, considering heattransport and the Gaussian intensity distribution of a focused beam, a focal spot 75%
the width of desired modulation period produces the correct, approximately sinusoidal,
modulation.
When initially testing the apparatus, I quickly found that a high rotation speed
produced the best results, which compelled me to begin the first in a series of modifications of this set-up. When I took on the capillary modulation project, the set-up was
almost entirely hand operated and mostly untested. The rotary motor was coupled to
the lathe drive shaft with an o-ring, and the drive shaft was coupled to the headstocks
with miniature chains. To accurately create a periodic effect, the original glass-blowing
lathe was mounted on a long-throw linear translation stage and driven by a micrometerscrew. Moreover, the laser power and exposure time were also manually adjusted. The
CO2 laser pulses at a repetition of 5 kHz, and its power is adjusted by changing thepulse duration of the individual pulses from between 5 and 195 µs, with a maximum
power of around 10 W; the percentage of the 200 µs pulse window used is referred to36
as the duty cycle of this laser. The very first modulated waveguides made on this setup
required me to fire the laser, time the exposure, time the cooling period during which
I would translate the lathe to the next location, and then refire the laser. I settled on
3000 rpm as the rotation speed, which can be calibrated with a strobe light or a precision rpm indicator. The reasons for the success of a high rotation speed were twofold.
First, the capillary must be convectively cooled in between laser exposures, or else theresidual heat will cause a steadily increasing modulation depth. To this end, I added
a fan to the set-up, but the high rotation speeds aided tremendously. Furthermore, toobtain consistent results, the heating of the capillary wall must be quite even. The laser
is only interacting with slightly less than half the capillary perimeter at any given time,
so ideally the time it takes for the heat to transfer to the inner wall will be on the orderof half the rotation period, the rotation period being 0.02 s for 3000 rpm in this case.
Although the thermal conductivity, denoted K, of fused silica varies with temperature,
it will begin with a value of ∼ 2 Wm−1 C −1 [45]. As a rough estimate, the time it takes
for a given quantity of heat to transfer through a rectangular solid is given byt=

QL
K(∆T )A(3.13)

where Q is the heat, L is the thickness of material, T is the temperature, and A isthe area [42]. Considering a patch of fused silica exposed to 6 W for 0.01 s, 0.5 mm
thick, with a temperature gradient of 1500◦ C, 0.75 mm on a side, the time for heattransfer will be about 0.02 s. Of course, this time can be increased by convective cooling,but it suggests the need for a high rotation speed. Miniature drive chains were not well
suited to this application due to large vibration, nor was the original rotary gas couplingfor pressurizing the capillaries. So, I refitted the lathe with miniature timing belts, alow-friction rotary coupling, and special clamps and mounts for reducing vibration.
The initial sets of modulated waveguides that I produced had been made withlong exposure times at moderate laser powers. This ‘slow roasting’ approach was prob-

37
lematic in terms of laser power drift and the deformation of the outside diameter of thecapillary. The CO2 laser power is quite sensitive to its operation temperature, and thetemperature of the building cooler water changes subtly over the day. More importantly,
as the inside diameter of the capillary expands and the capillary wall moves outward,
the outside diameter also increases. Since the 3-section outer capillary fixture relies
on a tight fit between the inner and outer capillary components, I was forced to handpolish the modulated inner capillary to the correct outside diameter. The polishing was
an extraordinarily delicate task that required a great deal of patience and a pragmatic
acceptance that attrition of capillaries would inevitably occur during the process. To
accomplish this task, I retrofitted a commercial glassblowing lathe with precision miniature chucks. With the capillary held in these chucks and the lathe rotating at less than
100 rpm, I would hold a diamond file gingerly to the underside of the capillary and slide
it horizontally back and forth until the outside diameter had been ground down enoughto fit into the outer capillary fixture. During this procedure enough pressure had to be
applied for effective cutting action of the file, but too much pressure would easily snap
the capillary. I finished by smoothing the outside with a ruby stone and cleaning off
any residual glass dust.
Considering that the original paper suggested using a modulated waveguide calling for a change to the the inner diameter on the order of 1%, I originally focused myefforts on devising an accurate active measurement of the modulation depth. I considered capacitive methods, differential absorbtion of gas, interference, and imaging.
However, at the time, the most straightforward approach turned out to be inspecting
capillaries subject to various laser exposure times, and iterating the exposure time until
the desired modulation appeared as judged by eye. Eventually, Amy Lytle perfecteda nondestructive measurement technique, whereby the modulated capillaries were immersed in index matching fluid and photographed with a CCD camera mounted to ahigh-quality microscope. From the CCD image, and knowing the correspondence of38
CCD pixel size to actual measurements at a given magnification, the difference in innerdiameter between modulated and unmodulated sections of capillary can be judged towithin ∼ 1 µm. In part, I originally decided to use the rather unsophisticated microscope inspection technique because I received a challenge from my advisors that went
something to the effect of “You know that ‘back burner’ project of yours, well we havea grant review coming up...think you could make it work in two weeks?” At this point,
I had improved the set-up and had a general feel for the modulation process, as well as
the polishing technique necessary to incorporate the capillary into the 3-section design.
Surprisingly, one of the first fully functional modulated capillaries I produced and incorporated successfully into a 3-section fixture generated notably unique spectra whenused for HHG. The section of modulated capillary shown in Figure 3.6 is an example
of one of the first successful modulated capillaries, made by the hand operated lathe
described earlier. The results obtained from these initial efforts are described in thepreceding section 3.4.1.
Clearly, as the utility of making modulated waveguides became apparent, more
precision, flexibility, and automation was necessary in the glass-blowing apparatus.
With the help of Randy Bartels, we added computer control to the laser exposure
settings and the cooling time. The original apparatus had a limited travel range of 1
cm, so the lathe was reattached to a larger translation stage and precision realigned. In
addition, I fitted a stepper motor and lead screw to the apparatus such that the lathe
position could be automatically and accurately driven. This second-generation apparatus is shown in Figure 3.7. This set-up included two lenses, a 25 cm focal length formaking the heating laser beam approximately the diameter of the capillary at its focus,
and a 5 cm ZnSe cylindrical lens for creating a tight line focus along the axial directionof the capillary so that the period of modulations could be made as small as 0.5 mm.
I should note that the optics were aligned such that the capillary would intersect theheating laser beam at the focus for the smallest periodicities and before the focus for39

Figure 3.6: A short section of capillary with a 1 mm modulation period, note thedeformation of the capillary’s outside diameter

40

Figure 3.7: The second-generation glass-blowing set-up

41
longer periodicities; the idea being that the laser beam characteristics would be lessaffected if dust particles could fall through the focus during heating. The miniaturetiming belt drive configuration can also be seen in Figure 3.7, along with the motor
mount and rotary coupling clamps.
Progressively, we were interested in extending the utility of the modulated waveguides, and wanted a glass-blowing system that would be flexible in terms of both thelength of capillary that could be modulated and the periodicity of modulation. The eventual solution came in the form of the third-generation glass-blowing apparatus shownin Figure 3.8. The desire to modulate sections of capillary over 10 cm in length mademounting the entire lathe on a large translation stage impractical. Henry Kapteyn suggested that we build a new custom lathe, and mount the entire CO2 laser on a moveable
rail, with completely in-line optics. For this project I collaborated with Lee Thornhillof the JILA Instrument Shop, and he designed and built the first version of this thirdgeneration set-up. Along with members of the shop, I have subsequently modified manyof its components to optimize the design.
Since a 10 µm wavelength is not visible to the naked eye, nor to standard IR
viewers, precise alignment of this far-IR laser beam becomes challenging, but a linear
array of optics highly simplifies matters. The optics train consists of three cylindrical
ZnSe lenses. For clarity, I will refer to the direction parallel to the capillary axis as thehorizontal dimension, and the direction perpendicular to this as the vertical dimension.
The first lens has a divergent -5 cm focal length and expands the beam in the horizontal
dimension of the capillary to increase the f-number on the third lens, which has a 5
cm focal length. This geometry can focus the beam as tightly as 75 µm in width, asmeasured in the horizontal direction. I should note that despite the tighter focus available in this optics arrangement, the functional limit of this apparatus remains around
0.2 mm periodicity, most likely due to the difficulty of confining heat conduction when
the periodicity becomes considerably shorter than the wall thickness of the capillary.

42

Figure 3.8: The current third-generation glass-blowing set-up

43
The second lens controls the beam size in the vertical dimension. The divergent lensis kept fixed, but the other two lenses can move along an optics rail to change thebeam conditions depending on the operation being performed. For various modulation
periodicities, the third lens can be moved such that the capillary intersects the beamat different points from the focus. During modulation operations, the second lens is set
such that the beam size in the vertical direction matches the capillary diameter. Duringhole poking applications, as will be discussed later in this section and in Section 4.3.3,
the second lens is set such that it generates a round focal spot.
In addition to the laser and optics being made into a mobile monolithic subassembly, the apparatus shown in Figure 3.8 incorporates several other major upgrades. The
laser in the current set-up is a Synrad series J48-1W with an optional closed-loop powerstabilizing head that provides a feedback loop for establishing an output power that fluctuates less than 1% during each exposure. The CO2 laser does not provide an absolute
power control, but for a given duty cycle, the power will remain quite steady for eachexposure as long as the temperature of the laser remains constant. An IR temperature
sensor is mounted to the laser body, and I adjust the cooling water flow such that a
constant temperature of around 14◦ C is maintained during operation. Furthermore,
the timing is accurately controlled with a digital delay generator (Quantum Composers
Model 9512 pulse generator).
Great care has also been taken in the alignment of the lathe. The metal rods
on which the lathe components sit are precision ground solid rods of hardened stainlesssteel that have been selected for straightness. Furthermore, the lathe is mounted on ahardened plate of ground tool steel. The lathe is carefully aligned using dial indicatorsand precision pin gauges, as shown in Figure 3.9, such that the components have amutual run out of less than 0.0005”. With the help of Mark Siemens, a set of gas jetsfor convective cooling were attached to the optics mounting rail, so that a rush of gas
could flow over the heat affected zone after each laser exposure. Since they are attached44

Figure 3.9: The third-generation custom glass-blowing lathe getting a tune up

to the optics rail, these jets move along with the laser, and their flow is controlled by
a computer automated solenoid valve, allowing precise optimization and coordinationof the cooling period. For additional convective cooling, I mounted a fan on a steadyrest that tracks the motion of the laser. Unlike the original lathe, both headstocks
are pressure sealed. Before the current set-up, I was forced to flame polish one endof the capillary shut so that the inner bore could be effectively pressurized, but thisdelicate operation is no longer needed. Additionally, the stepper and driver motors
in the third-generation are more powerful and precise (5107.68 steps/mm, HT23-400
motor driven by Si3540 stepper motor driver, both from Applied Motion Products).
In striking contrast to its humble beginnings, the current apparatus is fully automatedonce the capillary has been loaded into place.
Serendipitously, while endeavoring to increase the efficiency of the modulatingprocess, I found that with the correct combination of laser intensity and exposure time,
I could eliminate the polishing of the capillaries. Initially, the laser exposure times
were on the order of 40 s with nearly 2 minutes of cooling time between exposures.
Modulating a 2.5 cm section of capillary at a periodicity of 0.25 mm translated to 100
modulations, and a highly time consuming task. The active cooling jets significantly
reduced the cooling time down to 1 or 2 seconds. However, a quick calculation suggeststhat the time needed to heat the capillary should be more on the order of 1 s. The simple

45
relation Q = mc∆T relates the heat Q that changes a piece of material with mass mand specific heat c by a temperature ∆T . Considering a length of solid cylindrical pieceof fused silica h long with a radius R and a density ρ being heated by a laser with power
P, the time t to change by ∆T is given byt=P
πR2 hρc(∆T )(3.14)

Although the specific heat of fused silica varies somewhat with temperature, a valueof ∼ 1000 J kg−1 C −1 will suffice for this estimate [45]. The density of fused silicakg
is known to be 2200 m
3 [42]. Assuming the cylinder is 1 mm in length with a radiusof 0.60 mm and is being heated with a 6 W laser with 100% absorption, the timenecessary for a change of 1500◦ C will be 1.6 seconds. Convective cooling will certainly
lengthen this estimate, but still, the glass-blowing operation time appeared as though
it could be easily shortened by an order of magnitude. While trying to decrease thelaser exposure time with an increased laser power, I noticed an interesting phenomenon.
At a ‘sweet spot’ of intensity, the CO2 laser beam could both heat the capillary andsimultaneously ablate its outer surface. Thus, when operating in this regime, polishing
the outer surface of the capillary became an unnecessary step, as the original outside
diameter was preserved at the unheated locations, and made slightly smaller at theheated locations. Unfortunately, finding this regime can be tricky, since too muchintensity will lead to rapid ablation, and actually less heating of the capillary. If too
much ablation occurs too rapidly, the heat deposited by the laser is carried away by
evaporative cooling: small molten fused silica debris flies off the capillary or is vaporized
by the laser.
So a careful balance of laser power and exposure time must be found for successful
modulation of the capillary. Currently, the apparatus has a ‘test’ mode wherein the
exposure time can be varied for each laser exposure. Once I identify the power andfocusing parameters to bulge the inner diameter of the capillary, while also ablating

46
its outer surface, I make several test pieces to identify the optimal exposure time.
Generally, I use a chirped exposure, and increase the exposure time in discrete steps
every four exposures. When inspected under a microscope, these test pieces quickly
reveal the optimal exposure time for a given set of modulation parameters. The resultof the many improvements to the glass-blowing apparatus and procedure can be seenin Figure 3.10. This section of modulated capillary exhibits extremely regular andconsistent modulations, with only a subtle change to the outside diameter.
Alignment of the apparatus described here is particularly challenging because of
the invisibility of 10 µm light. If the laser is not directed along the capillary’s radius,
the outer wall will be ablated before the inner wall softens; therefore proper alignmentis crucial. In fact, considering the travel of the beam from the output of the CO2 laser
to the point where it heats the capillary (around 40 cm), an angular deviation fromthe aligned laser path of 1x10−4 radians, or about 20 arc seconds, is enough to cause
improper function. Fortunately, fused silica glows a very bright white when heated with
the CO2 laser, so this glow, along with the manner in which the laser cuts the capillary,can be used for alignment and for diagnosing beam parameters. For rough alignment, I
use a hand held piece of capillary to identify the laser beam position in much the same
way we normally use a plain white index card. To adjust the focusing optics, I turn thelaser power down till only a faint glowing spot appears on a stationary capillary held in
the lathe. Then the focusing optics can be adjusted to maximize this glow, indicating
the tightest focal spot on the surface of the capillary. Since the Rayleigh length zR given
by
zR =

πω0 2
λ

(3.15)

where ω0 is the focal spot radius and λ is the laser wavelength, is on the order of several
mm, the focusing optics can be accurately adjusted by this maximum glow technique.
Once the position giving the tightest focus has been found, I note the positions of the47

Figure 3.10: A short section of capillary made in the third-generation glass-blowing setup with a 0.25 mm modulation period. This picture was taken without index matching
fluid, and the periodic bright spots are merely an artifact of the microscope illuminationand lensing effects of altered capillary

48
focusing optics and reference other configurations to these positions. For fine alignmentof the beam pointing, I allow the laser to cut a hole in a stationary capillary and judge
the hole’s orientation relative to the radius of the capillary as shown in Figure 3.11. In
addition, when the focusing is set to give a line focus, features cut into a stationary
capillary allow me to judge the alignment of the line focus relative to the capillary as
well as the astigmatism of the focused beam.

Figure 3.11: A hole cut in the capillary with the CO2 laser showing how misalignmentof the beam pointing can easily be diagnosed.

Although the current apparatus functions well, two main improvements could bemade to make it more of a ‘turn key’ system. First, the CO2 laser should be given a
dedicated closed-loop water chiller. The absolute power of this laser is quite sensitive
to its operating temperature and requires the exposure time to be readjusted even in
the case of a drift as small as 1◦ C in the water temperature. Second, the ideal lathe
design would incorporate commercially available precision collets. Currently, the weaklink in the alignment of the lathe is the specially made collets used to hold the capillary
and form a pressure seal to the headstocks. The current collets are made from small
pieces of hard plastic, but are difficult to machine to the tolerances needed, and areeasily damaged with improper handling. This second issue may require a redesign ofthe headstocks of the lathe and perhaps an alternative pressure seal. However, thebenefits to long term dependable operation would be worth the effort.

49
3.5.1

Limitations of Modulated Waveguides and Perspective

Of course, solving challenges of precise technical glass blowing at the 100’s ofµm scale posed an interesting and rewarding task in of itself. However, I currently
believe the main utility of this work was to demonstrate the feasibility of modulating
the driving light field to achieve QPM in the HHG process. Although more could be done
to improve the consistency and efficiency of producing modulated waveguides, they are
an inherently tricky structure to create. Moreover, discovering the correct conditions
to optimize QPM with this structure involves a slow feedback loop. A much more
attractive solution is to employ alternative methods that allow adaptive real-time controlof the driving laser field. One such approach, using weak counterpropagating light inan unmodulated hollow waveguide and recently employed by our group, has shown
dramatic enhancements of flux at short wavelengths and holds great promise for future
results [33]. Perhaps even more interestingly, this technique has proved to be a real-timeprobe of the HHG process within a waveguide, allowing us insight into the dynamics
and optimization of QPM with the hollow waveguide [32]. Another technique using the
beating of different driving laser modes within a hollow waveguide has also recently been
proposed and claimed dramatic improvements of flux at short wavelengths [46,47]. The
addition of tailored pulse shapes and wavefronts, as well as more accurate models anddiagnostic feedback tools will likely lead to even further improvements and extensions of
HHG. Indeed, the ultimate solution to optimizing flux at difficult to reach wavelengths
most likely will involve a combination of these approaches and tools.

Chapter 4

Hollow Gas-Filled Waveguides

4.1

Introduction
Over the past century, the technologies associated with transmitting high fre-quency electromagnetic radiation have rapidly grown in importance, particularly withthe telecommunications revolution. One of the most important technologies, especiallyin recent years, has been the application of waveguide structures to the optical domainof the electromagnetic spectrum in the form of optical fibers for long distance communication. Waveguides come in many geometries, and are made from many materials,
both metallic and dielectric [48]. However, for the purposes of guiding high intensity
laser pulses for various nonlinear applications, hollow dielectric waveguides have become a popular choice [16, 49]. My personal experiences modifying hollow waveguides
and incorporating them into the outer capillary fixtures lead me to question some ofour established designs and assembly practices. Eventually, this questioning led me to
a complete redesign of our fixturing system that incorporated my knowledge of physics,
mechanical design, and glassblowing.

4.2

Hollow Waveguides
Since hollow waveguides are one of the essential elements in phase-matched pro-duction of high-order harmonics, I believe the basic parameters that can affect their performance should be explored. The results briefly summarized here come from two papers

51that have long influenced the use and understanding of hollow waveguides. The first paper, by E. A. J Marcatili and R. A. Schmeltzer, is ”Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell System Technical Journal,
vol. 43, pp. 1783-1809, (1964) [50]. This paper is one of those rare gems for which the
PDF cannot yet be found on the internet, and one must actually go to the library andmake a photocopy of it. However, the paper gives the definitive treatment of the modes
and loss mechanisms present in the waveguides we use. The second paper, by Richard
L. Abrams, ”Coupling Losses in Hollow Waveguide Laser Resonators,” IEEE Journalof Quantum Electronics, vol. 8, pp. 838-843, (1972), describes the effect of different
focusing conditions on the amount of power coupled into a hollow dielectric waveguidefor the lowest loss propagation mode [51]. Since the HHG process depends so sensitively
on laser intensity, the features of the mode coupled into the hollow waveguide, as wellas the nature of this mode’s propagation down the guide, affect the spectrum, mode,and flux of the upconverted EUV light.

4.2.1

Modes and Mode Beating

Marcatili and Schmeltzer describe the electromagnetic modes that propagate in ahollow dielectric waveguide. They perform their analysis assuming that the waveguide’s
inner bore radius is much larger than the free space wavelength of the field. Under
this condition, the waveguide can be considered to guide the mode through a series ofglancing incidence reflections. They also work under the assumption of infinite wall
thickness (interestingly, they mention this condition is the same as frosting the externalsurface of a glass waveguide). Finally, they only work with low loss modes where thepropagation constant in the waveguide is nearly that of free space. For the fused silica
waveguides of interest, the lowest loss modes are the hybrid EHnm modes. Althoughthese modes have a longitudinal component to the electric field of the propagating wave,
its magnitude is small compared to the transverse fields, so the modes are essentially52transverse. Their electric field profile as a function of the radial direction is given
approximately byE(r) = Jn−1

µ

nm r

a



(4.1)

where Jn−1 is a Bessel function of the (n − 1) kind, µnm is the mth root of the equation
Jn−1 (µnm ) = 0, r is the radial coordinate, and a is the radius of the waveguide’s innerbore [50].
In most of the calculations for phase matching and other aspects of high harmonic
generation, the lowest loss EH11 mode is assumed. However, a significant amount of
power can also be coupled into some of the higher-order modes, and these modes canhave a significant effect on the high harmonic process. In particular, these higher ordermodes have distinct phase and group velocities, and can beat against the fundamental
mode. An example simulation of this situation is shown in Figure 4.1. In general, it
would be ideal to avoid mode beating during the HHG process, as it will suppress theintensity of our driving laser at various points along the guide. However, several authors
have suggested that under the right conditions, mode beating could be used as a QPM
scheme [46, 52].

4.2.2

Attenuation and Bending Losses

Marcatili and Schmeltzer also quantify the attenuation along the waveguide. They
give the attenuation constant αnm as
αnm =

µ

nm

2π

2 λ2

"

1 (ν 2 + 1)Re
a3
2 (ν 2 − 1) 12

#

(4.2)

where λ is the wavelength of the radiation traveling down the hollow core of the waveguide, and ν is the ratio of the indices of refraction of the guide material and the gas
filling the guide. For our case, fused silica has an index of refraction of 1.45, and thecore of the guide will have an index of refraction very near to 1 [42]. For a coupledwavelength of 800 nm in a 150 µm inner diameter fused silica waveguide, α11 works

53

Figure 4.1: A simulation by Amy Lytle of mode beating between equal contributionsof the EH11 and EH12 modes within a hollow waveguide. Note that the beating has an
approximately 2 cm period. A similar period is experimentally observed in the form of
periodic variation of plasma emission intensity along a hollow waveguide in which gas
is being ionized by a driving laser used for HHG

54
out to be 0.33 m−1 . So for our typical 10 cm waveguides, we should have roughly 3%
attenuation for the lowest loss EH11 mode.
Furthermore, the loss due to bending of the waveguide can be given in terms of
the radius of curvature which doubles the straight guide attenuation, labeled R0 . For
the EH1m modes
2
R0 = √
3



2π
µ1m

2


 12
1
a3
3 (ν 2 − 1)
1+ 2 +
cos 2θ0λ2µ1m 4 (ν 2 + 1)

(4.3)

where θ0 corresponds to the orientation of the field polarization with respect to thecurvature of the waveguide. For θ0 = 0 the field at the waveguide’s center is in theplane of curvature and the attenuation is maximum. Similarly, the attenuation will bea minimum when the polarization is perpendicular to the plane of curvature [50]. So,for bending losses, we should notice several key points: the higher order modes are lesssensitive to bending loss, the bending losses decrease dramatically for decreasing innerdiameter, and the bending loss is sensitive to the polarization of the coupled mode. For
example, for an EH11 mode at 800 nm wavelength in the maximum attenuation polarization, and with a 150 µm inner diameter fused silica waveguide, the bend radius thebend radius that results in a bending loss equal to the intrinsic EH11 loss is 6.2 m. Thus,
waveguides should be held as straight as possible to minimize bending. Interestingly
though, if loss of lower order modes becomes desirable, inducing a bend may prove to
be an interesting way of tuning the loss.

4.2.3

Coupling

Coupling the lowest loss EH11 mode efficiently into a hollow waveguide requires
a combination of the correct focal spot size and a mechanical system with the precisionand sufficient degrees of freedom to align the guide. Abrams calculated that optimum
coupling of a TEM00 free space mode of the laser to the EH11 mode of the guide is given

55byω0 = 0.6435a(4.4)

where ω0 is the radius of the free space beam at its focus, and a is the radius of the
waveguide. In theory, when this coupling condition is met, 98% of the energy fromthe TEM00 mode will couple into the EH11 mode of the waveguide [51]. As the focal
spot size deviates from this condition, more energy will go into the lossier modes of the
waveguide. Ideally, the driving laser should have a collimated free space beam, and thefocusing lens focal length is chosen to meet the condition of Equation 4.5. This way,
the lens can be translated parallel to the propagation direction of the laser and change
the focal spot position without changing its size. For a collimated driving laser beamof diameter Dcol , and wavelength λ0 , the lens focal length f will be given by,
f = 0.6435

πaDcol2λ0(4.5)

In general, we find that the most successful method for coupling into a hollow
waveguide is to position the waveguide to the laser beam, not the laser beam to the
waveguide. Usually this alignment is accomplished by noting the path of the laser beam
before introducing the waveguide, and then, after inserting the waveguide, adjusting the
waveguide alignment until the beam path is unaltered by the waveguide and emerges
with a clean spatial mode. Next, the coupling can be optimized using a power measurement and iterative adjustment of focal spot position and waveguide alignment (moving
the focal spot invariably steers the laser beam slightly). This approach requires thatthe waveguide be held in such a way as to allow it 4 degrees of freedom, and that the
coupling optic can be translated parallel to the beam propagation direction. Although
the waveguide could be placed on a single stage that included horizontal and verticaladjustment, along with azimuthal and planar rotation, we use a much simpler systemof placing a vertical and horizontal adjustment at each end of the waveguide. In general, adjustments at the front of the waveguide (since they are the most critical to the56coupling) are used to maximize the power, while the rear adjustments determine thepointing and mode quality of the laser beam emerging from the waveguide. In addition,
once the alignment has been optimized, if the rear adjustment is left undisturbed, small
corrections to the front of the waveguide to correct for laser pointing drift will not appreciably alter the beam pointing emerging from the waveguide for most experiments.
Finally, for the 150 µm inner diameter waveguides we use, the positioning must be done
to better than 10 µm for optimum results.

4.3

Modifications to the Hollow Waveguide Design

4.3.1

Why Change the Design?

During my work with modulated waveguides, I became highly interested in ways inwhich we could improve the consistency and quality of our 3-section hollow waveguides.
In terms of consistency, the coupling efficiency of laser light and the output mode qualityof the original 3-section design varies substantially. Considering that one of our main
goals remains enhancing the flux of EUV, especially at shorter wavelengths, I wanted tostart with an optimized unmodulated capillary and be sure that we were able to quantify
enhancements in a fair and repeatable manner. Making and assembling the componentsof the original design in a precise, consistent, and repeatable manner requires a great
deal of practice and patience. Outer capillaries are easily broken during the drilling
process, and the sanding and handling of individually cleaved inner capillary end pieces
must be accomplished with a set of forceps and a steady gentle hand.
The final assembly of the optical-quality inner sections into the outer-capillary
fixture presents its own set of difficulties. Sliding the inner capillary pieces into thetight fitting outer capillary can easily break or chip the inner components, while an
overzealous application of epoxy can foul the assembly. Of course, maintaining theposition of the inner components while jabbing epoxy into the gluing holes can be57a tricky endeavor as well. Furthermore, the outer capillaries are fragile and relatively
expensive. Once the inner pieces are glued into place, the outer capillary is not reusable.
Therefore, the assembly’s effective lifetime depends either on how long the inner capillary
survives laser damage or the amount of time before the assembly is stressed and cracked
during alignment. Unfortunately, waveguides seem to preferentially expire at moments
when they are most needed, so the experimenter must have a backup, or be left waiting
many hours for glue to dry.

4.3.2

Initial Hollow Waveguide Improvement Attempts

I began this improvement project by upgrading the manufacture and interchangeability of the outer-capillary fixture. Originally, separate members of the group would
produce the radial holes in the fixture using a laborious glass drilling process. To makethe quality and consistency of the outer capillaries better, as well as to increase the efficiency of production, I had a local technical glass company ultrasonically mill the holes
for us. In this process, a milling tool cuts the three holes at each end of the capillary
simultaneously, keeping a precise relationship between the positions of the holes relative
to the capillary ends. Next, I turned my attention to the inner-capillary pieces. Originally, separate members of the group would cleave their own end pieces, sand them,and then glue them in place, leaving substantial variability in the size and placementof the end pieces. I developed a procedure to produce and place the inner capillary end
pieces. I waxed many pieces of inner capillary together, and cut them simultaneously
on a wet saw while using a carefully-measured stop to maintain a length of 5 mm forall batches. Now that the outer capillaries were all made to the same dimensions, thecentral inner capillary could be centered between the gas feedthrough holes. After thecentral inner capillary was placed, I used a specially made depth gauge to place the end
pieces so that they maintained a 0.5 mm gap from the ends of the central piece.
Although these various improvements greatly facilitated the production of the

58
3-section waveguides, the coupling efficiency and output modes remained inconsistentand sometimes unsatisfactory. Endeavoring to solve these problems lead me to novel
ways of producing the 3-section inner capillary, and eventually a complete redesign of
3-section hollow waveguide assembly.

4.3.3

Single-Piece Inner Capillary Hollow Waveguides

Having the inner capillary split into three separate sections allows for simple gas
feedthroughs and differential pumping, but we grew concerned that this arrangement
was leading to misalignment and unnecessary coupling losses. The gaps between the end
pieces and the central piece clearly cause coupling loss, as evidenced by the scatteringof laser light. In addition, since the outer capillary’s inner diameter is not a perfect fit
to the inner capillary, we were concernced that the end sections might be slightly tilted
so that all three capillary pieces did not share a common axis. Nick Wagner cleverly
realized that we could make a 3-section waveguide from a single piece of inner capillaryby partially slicing through the capillary wall with a wet saw. He initially attempted thisprocedure by hand; first, by marking the inner capillary with a sharpie at the positions
to be slotted, and then holding the capillary gently to the wet saw blade. Althoughsome capillary attrition inevitably occurred, the basic idea was a success. A single-pieceinner capillary greatly simplifies the production and assembly of hollow waveguides. To
facilitate this manufacturing technique, I quickly designed and machined jigs for holding
multiple inner capillaries so that they could be slotted and trimmed simultaneously using
a surface grinder. This set-up represented another major step forward in improving thehollow waveguide assembly process. The number of major components had now been
reduced to 2, and the length of the inner capillary, as well as the size and depth of theslots, could be precisely set with the surface grinder.
A 3-dimensional rendering of one end of the slotted single piece inner capillary1
of an inch wide and made with a diamondis shown in Figure 4.2. The slot is 32

59

Figure 4.2: To-scale 3-d rendering of the slotted innner capillary

wet saw. Although slotting the inner capillary allows for a single-piece implementation, the mechanical process of cutting the slot can make fine chips in the capillary, as
seen in the microscope image of Figure 4.3. The mechanically-made slots did represent a step forward, but the resulting chipping and roughness produced less-than-ideal
laser transmission. The output mode, which refers to the mode structure of the laser
beam emerging from the output of the capillary, for these slotted capillaries was betteroverall than with past designs. However, the intense fs amplified laser was producing
white light at the face of the slots, suggesting unwanted nonlinear interactions withthe capillary material. After considering several ideas for mechanically polishing theslots, my advisor, Prof. Henry Kapteyn, made the critical suggestion that perhaps instead of abrasively cutting the gas feedthrough in the inner capillary, I might be able
to efficiently produce a smooth hole by using the CO2 laser set-up I had developed formodulating waveguides. When tightly focused at near full power, the CO2 laser ablates
fused silica as well as delivering heat, allowing it to ‘drill’ a smooth hole as shown in
Figure 4.4. One of the key elements of the existing CO2 laser glass-working set-up is its

60

Figure 4.3: Microscope image of an abrasively cut slot

ability to pressurize the capillary with inert nitrogen gas. As the focused laser reaches
the inner wall of the capillary tube and the viscosity of the fused silica drops sufficiently
due to heating, the pressure forces the remaining wall radially outwards. Therefore, nomolten ablated fused silica debris gets lodged on the inner wall of the capillary. I foundthat a careful balance of focal spot size, laser power, gas pressure, and exposure time
produces beautiful radial holes in the capillary. The focal spot size determines the sizeof the hole, and a ∼ 100 µm diameter spot works well. The laser power must be highenough to ablate material, and the optimum power corresponds to a range where it also
fire polishes the hole; typically 6 W (65% duty cycle for the CO2 laser). The capillaryinternal pressure was set to 15 psi. Although the gas pressure is not exceedingly critical,
a gas pressure that is too high pushes the internal wall outwards too soon, leading toa constriction rather than a subtly tapered hole. As for laser exposure time, too shortof an exposure time obviously does not allow the laser to penetrate the capillary wall,
while too long of an exposure time causes damage to the other side of the inner wall. An
exposure time of around 2 s generally works well with the above parameters, making theprocess quick and efficient. When the capillary is heated by the laser, it glows a bright

61

Figure 4.4: Stereoscope image of a laser poked hole

white. However, when the laser penetrates to the inner wall, the pressurized gas flows
from the hole. The outward flowing gas cools the capillary, which causes the glowing todim. This change provides a quick feedback for rough adjustment of the exposure time,
but I make the final adjustments by inspecting the holes under a microscope.
As an important side note, I also investigated improved methods for cleaning
the inner capillaries. Since the inner capillary ends must be sanded to obtain the bestmode and simplify the laser coupling process, small fused silica particles inevitably
enter the inner bore. These particles, especially larger ones, are quite pernicious anddifficult to remove. Moreover, they can cause massive transmission loss, improper gas
flow, and mode corruption. In the past, the generally-accepted method was to spraymethanol into the capillary and use an air jet to blow it out over and over again until
the capillary had been cleaned. This method was laborious, and often failed to remove
lodged particles. The next step was to take a piece of fine wire slightly smaller than the
inner bore and use it to push out any unwanted debris. This method also proved neither
an easy nor enjoyable task, requiring a steady hand (putting fine wire into a 150 µm
diameter hole is considerably more difficult than threading a needle). Discussions with

62
Hans Greene of the JILA instrument shop lead me to try using an ultrasonic cleaner.
This method turned out to be a tremendously effective, reliable, and simple solution to
the cleaning dilemma. The capillaries are placed in the ultrasonic cleaner for several
minutes in an upright position so that debris that is shaken loose falls out of the bottom
end, then rinsed with filtered water and blown out. After cleaning in this manner, theinner capillaries show no signs of debris, either when inspected under a microscope, orby coupling a HeNe laser into the capillary and looking for scattering.

4.3.4

V-groove Fixture for Single-Piece Hollow Waveguides

Improvements to the outer and inner capillary components of the hollow waveguide assembly greatly improved their utility and uniformity. However, the inconsistency
in transmission and output mode (though ameliorated) continued. The need for a revolutionary design became apparent when Arvinder Sandhu, a post-doc in our group, washaving great difficulty getting good coupling or decent modes from a 1 m long hollow
waveguide that he wished to use for self-phase-modulation and pulse compression. In
trying to determine the nature of the problem, we decided to test the straightness of theouter holding capillary. For a cylindrical object, a simple first test is to roll the object
back and forth on a surface plate. While rolling the object, one looks for any gaps
between the object and the surface plate as well as watching the ends of the object for
any run out. Interestingly enough, although the outer capillary is extremely straight,
Arvinder and I noticed that the inner bore of the capillary appeared not to maintain aconstant position with respect to the axis of the tube.
The outer capillary material is known as precision bore tubing, and because of its
dimensions and the aspect ratio of its inner and outer diameters, it is not manufacturedby the traditional methods of draw towers. Precision bore tubing is made by melting apiece of thick-walled glass tubing and pulling it over a precision mandrel made of highly
polished solid metal. Thus, the surface quality of the metal rod determines the surface

63
quality of the inner bore of the capillary. Moreover, when the thermal expansion of
the metal has been taken into account, the inner diameter of the capillary is extremely
accurate to tolerances better than 0.001”. The outer surface of precision bore tubing
is ground and polished, leaving it very straight and round. Unfortunately, the processof making precision bore tubing does not in general provide reliable concentricity ofthe outer and inner diameters of the tube. Thus, the inner diameter can ”snake”
through the tube. For our applications this bending constitutes a highly undesirable
feature. It distorts the inner capillary tube through which the laser is coupled and musttravel, leading to unwanted attenuation and mode corruption. This discovery, thoughsuggesting a clear path to fixing the nagging inconsistencies of the 3-section design, wassomewhat disheartening since the problem of bending induced by the outer capillary
fixture seems so glaring in retrospect.
A complete redesign of the 3-stage hollow waveguide assembly presented a majormechanical and experimental challenge, particularly in devising appropriate gas seals.
Kinematically speaking, a v-groove beneath and a line contact above represents theideal geometry for holding a long cylindrical object in a straight and unstressed manner. Ironically, the original work demonstrating pulse compression in hollow waveguides
also emphasized the use of a very straight v-groove [49]. The design presented here wascreated with the invaluable help and expertise of Tracy Keep in the JILA instrument
shop. My goals were simple; create a robust, modular, and flexible fixture that guaranteed the straightness of the inner capillary (from now on to be referred to simply as
the capillary). Prototyping and perfecting this design consumed many months, but it
continues to make a lasting contribution to our research as well as allowing exciting
possibilities for HHG.
A 3-dimensional exploded rendering of the v-groove fixture is displayed in Figure
4.5 and a picture of an assembled fixture appears in Figure 4.6. This assembly completely eliminates the outer precision bore tubing fixture. The assembly consists of a

64

Figure 4.5: A 3-d exploded rendering of the v-groove fixture

Figure 4.6: An assembled v-groove fixture

65
v-grooved bar with an o-ring holder and end piece on each end, blocks for feeding gas
into the capillary, and hold-down bars for securing a clear plastic cover for the capillary.
The v-groove bar is a single piece of aluminum machined into a half I-beam configuration to reduce weight but maintain stiffness. The v-groove itself is machined into thebar as the final machining operation to prevent warping or bowing of the groove.
The components attached to the v-groove bar form gas and vacuum seals to thecapillary, as well as providing proper seating in the groove. The o-ring holders haveseats for an 002 size o-ring that forms the vacuum seal on the ends of the capillary.
Silicone o-rings must be used to withstand the heat dissipated into the capillary fromthe coupled laser. The o-rings are compressed between the end pieces and the o-ringholder so that they seat onto the capillary’s outside wall and form a tight seal that does
not require vacuum grease. Additionally, this design allows an inspection window to
be placed in the end piece that aids with coupling the driving laser into the capillary.
However, the ends of the capillary must be beveled to prevent cuts in the sealing o-ring
during assembly. A custom-made silicone gasket forms the gas seal to the laser-drilledholes on top of the capillary. At first, I tried using a standard o-ring. Unfortunately, tosqueeze an o-ring onto the cylindrical surface of the capillary required too much load,
resulting in an unreliable seal as well as deformation and occasional fracturing of thecapillary. A 3-dimensional rendering of the gasket is shown in Figure 4.7. The large

Figure 4.7: A 3-dimensional rendering of the custom silicone gasket

66
surface area and conforming shape of the gasket allow it to create a reliable gas seal
with minimal stress on the capillary. I make this gasket from silicone tubing that has an
inner diameter that closely matches the outer diameter of the capillary. Using a razor
blade, I slice a short length of tubing axially in half, and then use a piece of 17 gauge
hypodermic tubing sharpened at one end as a gasket punch to cut the gas feedthroughhole (a thick piece of paper such as an old punch card forms an excellent backing during
the hole punching operation). Using a spare gas block as a fixture, I align the gas
feedthrough hole in the gasket with its counterpart on the bottom of the gas block andtrim the half-cylinder gasket to length with a razor blade. The gas block and gasket
together hold the capillary firmly in place at the ends of the fixture, but the capillarymust be held in tight contact with the v-groove along its entire length to avoid bowing.
I use a sheet of clear plastic cut from a 12 ” diameter piece of Tygon R tubing to holdthe capillary in place. Much like the silicone gasket, I slice the Tygon R tubing axiallyand then cut it to size, placing it on the capillary such that the Tygon R sheet curves
away from the v-groove bar and must be pulled down over the capillary using the twohold-down bars. The end pieces each hold a 14 ” thin walled stainless steel tube that
forms the vacuum connection to Ultra-Torr R fittings.
The v-groove fixture possesses several distinct advantages over the outer capillary
fixture, particularly in the configurability of its components. Not being made of afragile material, the v-groove fixture is quite robust. Since the older outer capillary
design employed Ultra-Torr R tee fittings to create the gas seal, large portions of thecapillary were not visible, especially the capillary entrance. Being able to view thefront of the capillary greatly simplifies the coupling and alignment procedure, while
being able to view the length of the capillary allows one to clearly see the plasma
created by the driving laser. Maximizing the light emission from this plasma often
allows a rough optimization of the EUV flux, and noting the color of the plasma aids introubleshooting contamination of the gas system. Furthermore, since the stainless tubes

67
on each end are thin walled, they have a much better conductance for pumping than theouter capillary. Residual gas at the capillary entrance can cause undesirable ionization
losses and plasma-induced defocusing of the laser beam that leads to decreased coupling
efficiency and mode corruption. The design of the v-groove fixture is also extremely
flexible. We have made v-groove bars as long as 1 m and as short as 0.5 cm. The
shortest bars have a single gas feedthrough, but I have also made a 10 cm fixture with
5 gas feedthroughs. Although the silicone gaskets must occasionally be replaced whenchanging the capillary, and the o-rings are disposed of with each capillary replacement,
all other components of the v-groove fixture are completely reusable, saving long termcost. Moreover, a laser-damaged capillary can easily be replaced within 10 minutes
yielding consistent results results consistent with the previous capillary, saving valuable
time, and providing peace of mind.

4.3.5

Simple Set-up for Comparing Transmission of Hollow Waveguides
with a Spatially-Filtered HeNe Laser

As I began to work more with hollow waveguides and improve their performance,
I became interested in making a careful and repeatable measurement of their throughput under near-perfect coupling conditions. Being able to compare waveguides in thismanner allows us to make determinations of waveguide quality between various manufacturers, understand better the effect of different fixturing designs, evaluate the effectof waveguide modifications, and ideally understand and tune the mode structure within
the waveguide. My goal was to make a reliable measurement within a 1%-2% uncertainty. However, after realizing that precise power measurement at low powers can be atricky endeavor, I consulted with Nobel Laureate Jan Hall who has tremendous expertisein perfecting such measurements. The set-up described here reflects several extremely
insightful suggestions by Jan. Specifically passive optical isolation, a reference arm formonitoring power fluctuations, and tips for the use of sensitive photodiodes. The set-up

68
is shown schematically in Figure 4.8. Having a stable light source in terms of powerand high quality mode represents the first step in making a reliable transmission measurement. The HeNe laser gives a polarized output of around 4 mW and is held in acustom-machined Delrin R block for stability. Jan mentioned that HeNe lasers are particulary susceptible to power fluctuations from back reflections that return to the laser
cavity and incite erratic feedback and mode hopping. So, the first several optics andtheir alignment was suggested by Jan to minimize this undesirable effect. The HeNelaser beam immediately goes through a linear polarizer (P) set at the output beampolarization, and then a quarter-wave plate (WP). Together these two optical elements
form a passive optical isolator. As the beam passes through the quarter-wave plate
it becomes circularly polarized. Any back reflections after this point must once again
traverse the quarter-wave plate, and in turn regain linear polarization. However, theeffect of going through the quarter-wave plate twice will cause the back reflected light tobe rotated by 90◦ to the original polarization state, and thus, it is rejected by the initialpolarizer. To further protect against back reflections, the polarizer and quarter-wave
plate are set at an angle deviating from the normal to the beam path such that the backreflected light misses the HeNe laser output coupler. Still, for best results, the HeNelaser should be given 30 minutes or so of warmup time to allow for thermal equilibrium.
The rest of the set-up ensures a high quality mode and proper coupling conditions,as well as providing a reference arm to monitor power fluctuations and establish atransmission measurement without moving the waveguide or detector. After the opticaloscillator, the beam passes through a spatial filter (SF, Newport model 900 Spatial
Filter) to ensure a near-perfect TEM00 Gaussian mode. Two lenses (L1 and L2) mounted
on linear translation stages serve as a telescope to collimate the beam to an adjustablebeam diameter, depending on the desired coupling conditions for the waveguide. The iris
immediately following this telescope (I1) eliminates the residual high order modes leftfrom spatial filtering. The next two mirrors (M1,M2) allow alignment of the reference

69

Figure 4.8: A schematic of the set-up for testing waveguide transmission. P: polarizer,
WP: quarter-wave plate, S.F.: spatial filter, L1-L3: 7.5 cm, 10 cm, and 50 cm focal
length plano-convex AR coated lenses respectivley, I1-I5: irises, M1-M6: silver mirrors,
PBS: polarizing beam splitter cube.

70arm. Since the light is circularly polarized after the quarter-wave plate following the
HeNe laser, it is evenly split when it encounters the 50/50 polarizing beam splitter cube(PBS). This cube itself and a mirror (M3) allow alignment of the waveguide testingarm. Although different lenses can be used for focusing into the waveguides, the set-up
is currently optimized for 150 µm inner diameter waveguides using a 50 cm focal length
lens (L3) mounted on a linear translation stage. The beam size and mode quality at
the focus of the 50 cm lens are diagnosed using a CCD camera. Since the beam going
into the focusing lens (L3) is collimated, the lens can be adjusted to optimize coupling
without changing the focal spot characteristics. To ensure the alignment of both thetesting and reference arm, a collared iris can be placed in mounts at four positions along
the beam paths (I2-I5). The waveguide to be tested sits in a v-groove held on a stage
that provides vertical and horizontal positioning perpendicular to the beam propagation
direction, as well as tilt and rotation adjustment. The front end of the waveguide sitsreferenced to the front end of the v-groove for repeatable placement. To properly alignthe waveguide, a 45◦ turning mirror (M4) can be translated into the beam path where it
redirects the laser light to the bottom of a shelf above the set-up. This mirror is placedin the beam before the waveguide is put into place, and the position of the beam iscarefully marked. A poorly-coupled waveguide will steer the incident HeNe laser beamfrom its original path as well as exhibit high order output modes. So, once some light canbe seen coming through the waveguide, careful adjustments are made to the waveguide
position and orientation so that the beam returns to its original position with a goodmode. Two final mirrors (M5, M6), guide the beams from the testing and reference
arm onto the power detector (Newport 818-UV measured with Newport 1825-C power
meter).
With the set-up properly aligned, making a power measurement becomes straightforward. A large area photodiode acts as the power detector. The photodiode was tested
to verify that the detection characteristics across its surface were homogeneous, and it

71
is positioned such that the reference and test arm beams impinge on its surface withapproximately equal sizes. The photodiode is very sensitive, and will give varying measurements if moved or tilted with respect to the beam. Therefore, a mount firmly locks
the photodiode in place, and it is not disturbed during the measurement. To increase theaccuracy of the measurement, I custom made a filter holder for the photodiode and fitted it with a HeNe laser notch filter. This filter rejects all light with wavelength outside
the range of 633±2 nm. Since the photodiode is optimized for low power measurements, even reflections of room light from a hand or shirt can alter the measurement
without this filter in place. Furthermore, following a suggestion from Jan, the filter
holder is shiny aluminum to reduce IR background on the photodiode. Without the
waveguide in place, a ratio can easily be established between the power throughput of
the measurement arm and the reference arm that naturally takes into account all losses.
Since only a relative measurement is desired, with the waveguide in place and aligned,
the arms can be alternately blocked and power measurements taken. Double-checking
the reference arm ensures that the power has not fluctuated during the measurement
and allows a transmission measurement without moving the waveguide or the detector.
Finally, a CCD camera can be inserted at the waveguide output to record the output
mode structure and quality.
This transmission measurement apparatus had been for the most part a side
project, but I was lucky enough to have an extremely talented and meticulous visiting
student, Steffen Haedrich, complete some waveguide characterization experiments forme. Steffen came from the University of Jena in Germany, home of optics legends Carl
Zeiss and Ernst Abbe, so he was well acquainted with precise optical measurements.
He perfected the mode quality of the coupled beam, as well as carefully setting its size
to the optimum of 0.64 times the waveguide diameter for introducing the lowest loss
mode [51]. I first requested that he compare the transmission of waveguides cut fromthe capillary stock of 2 different suppliers, as we had long been curious if either man-

72
ufacturer produced better quality products. He quickly determined that no significant
statistical difference appeared between the two manufacturers. So, I had him quantify the difference in transmission between several waveguide configurations. Figure 4.9
shows a comparison of 10 cm waveguides (meaning 11 cm total length, since we usually
have 5 mm end sections) resting in the v-groove to those held in an outer-capillary
fixture. Waveguides 1-8 had no laser-poked holes. They were first measured resting in
the v-groove and then remeasured held in different outer-capillary fixtures. This data
exhibits a nearly 10% lower average transmission for the outer capillary fixture, as wellas illustrating the inconsistency inherent in that design, as indicated by the standard
deviation. Figure 4.10 compares 2 sets of 10 cm waveguides resting in a v-groove, those
without any modifications and those with a laser-poked hole at each end. This datashows that only a small loss of transmission occurs with the laser-poked hole modification. To further understand the waveguide laser coupling issues, Steffen captured
images of the output mode from the waveguides placed in various configurations with a
CCD camera. A very relevant example for work presented later in this thesis on imaging
with coherent EUV light, shows the mode corruption induced with the old outer capillary fixture. Figure 4.11 shows the output modes of three different waveguides when
resting in a v-groove (top) and the output modes of the same waveguides placed inthree different outer capillary fixtures (bottom). Clearly, this data emphasizes the needfor a consistent and reliable fixture for maintaining EUV mode quality. Hopefully thissetup will be used for future transmission measurements and mode comparisons as new
waveguide configurations come into use. It should also be helpful if tuning the coupling
conditions and waveguide configurations to a specific set of waveguide modes arouses
more interest, as has been suggested as a QPM scheme [47].

73

Figure 4.9: A measurement of transmitted power for 10 cm waveguides comparing thoseresting in a v-groove with those held in an outer-capillary fixture

Figure 4.10: A measurement of transmitted power for 10 cm waveguides resting in a
v-groove without holes versus those with 2 laser-poked holes

74

Figure 4.11: An example of the comparison of the output modes for three different
waveguides placed in a v-groove (top) and the corresponding waveguides held in anouter capillary fixture (bottom)

75

4.4

Future Directions
Several interesting possibilities exist for future work using the v-groove fixture de-sign with 3-section laser drilled capillaries. Perhaps most straightforwardly, this fixture
will allows us to perform HHG in very long waveguides, which may be useful for highflux applications requiring low pressure gasses with low absorption at the wavelengthof interest. Another exciting possibility will be a 2-stage waveguide arrangement, inwhich a different gas is used in each stage. Whereas the traditional arrangement forcompressing amplified fs pulses calls for a self-phase modulation (SPM) cell followedby carefully chosen and aligned chirped mirrors, previous work in our group has shownthat under certain conditions, a short hollow waveguide filled with Ar can efficiently
self-compress these pulses to shorter time durations [53, 54]. Ideally, the first section of
the 2-stage waveguide would be filled with Ar to compress the driving laser pulse, andthe second section would be filled with Ne or He, which with their large ionization potentials can produce the highest energy harmonics. Such an arrangement promises thepossibility of a high quality mode of a very short pulse duration (∼ 10 fs) with extremely
straightforward coupling into the second stage. We attempted a 2-stage waveguide in
the past, but with disruptions due to the gas inlets, the outer capillary fixture design
had an unacceptably poor transmission and output mode. I have already designed andbuilt such a 2-stage waveguide with the v-groove fixture, and it exhibits good transmission characteristics despite having 5 laser poked holes in an 11 cm overall length.
However, it has not yet been tested for its original purpose of HHG with self-compressed
pulses. Ironically, although I designed the v-groove fixture with the straightness of the
waveguide as the main consideration, a well-defined and tunable bend may actually beof interest. Recent experiments suggests that mode-beating within the waveguide canbe used as a QPM scheme [46,52]. In this work, the authors rely on coupling conditions
to launch high order modes. High order modes are more lossy than low order modes in

76a hollow waveguide. However, high order modes are actually less susceptible to bending
losses than lower order modes. Therefore, a tunable bend could be used to adjust theratio of high and low order modes, and optimize the mode beating QPM. As for thecapillary itself, a tapered end section might form a useful modification leading to longer
capillary lifetimes, and should also be useful for high gas pressures where larger gas flows
and bigger conductances are beneficial. I believe such an end section could be easily andreliably made in much the same way the laser drilled holes are formed in the capillary.
Although such a task would require delicate and repeatable alignment and placementof the capillary, I believe it is well within the limits of laboratory precision.

Chapter 5

Coherent Imaging in the Extreme Ultraviolet

5.1

Introduction
The development of novel light sources,such as the monumental invention of thelight bulb and the x-ray tube, has spurred major advancements in our ability to image
the natural world in both the macroscopic and microscopic realms. Recently, coherent
laser light sources have played an important role in extending the utility of the microscope. Coherent light sources make possible spectacular new imaging techniques suchas laser scanning confocal microscopy, holography, and laser tweezers. The coherent
EUV light generated using HHG performed in a hollow waveguide is a relatively uniquelight source as it exhibits near-perfect coherence at a range of short wavelengths in the
EUV [55]. Considering the uniqueness of this source, and its wavelength of operation,
we wished to demonstrate a suitable imaging technique.
Although established microscopy techniques are already exceedingly capable atimaging on micron and sub-micron length scales, high-resolution imaging (i.e. nanometerscale) of ‘thick’ samples remains an outstanding problem in imaging. Using innovative
imaging and labeling techniques, far-field light microscopes can image living cells witha resolution as high as 200 nm [56]. Of course, this resolution is fundamentally limitedby the wavelength of the visible/near-UV light. Using the much shorter wavelengths ofmoderate-energy electrons to surpass the diffraction limit of optical microscopy, electron
microscopy has demonstrated resolution at the atomic level [57]. However, many bio-

78
logical specimens, as well as samples of interest for materials science, are too thick forelectron microscopy. Inelastic scattering of electrons changes their wavelength, leading
to image blur. In addition, the phase contrast information of the electrons is decreasedby plural elastic scattering, but this information forms the primary image contrast mechanism for electron microscopes. These effects are especially prevalent in ‘thick’ samples,and limit electron microscopes to imaging thin samples, typically < 500 nm [58]. Thus,
a resolution gap exists between optical microscopes and electron microscopes; optical
microscopes can view thick specimens with an excellent depth of field but limited resolution, whereas electron microscopes have excellent resolution but are not suitable forthick samples. Especially in the realm of biology, this resolution gap obscures some of
the most important and exciting mysteries of life and its development; examples include
internal cell features, morphology and architecture of groups of chromosomes, as wellas groups of cells during embryonic development. Understanding of such enigmatic biological structures will benefit greatly from new microscopy techniques that bridge thisresolution gap [59].
The use of short wavelength light in the EUV or soft-x-ray (SXR) regions of
the spectrum, represents one of the most promising alternative approaches for highresolution imaging of thick samples [60]. EUV/SXR light can be used for nondestructive imaging applications requiring high resolution in thick samples [61]. Furthermore,
numerous core-shell absorption edges and widely varying elemental absorption cross sections provide excellent inherent image contrast. For instance, with these unique characteristics, light at 800 eV has been used for imaging magnetic nanostructures [62, 63].
However, some of the most exciting possibilities exist in the realm of biology. Between
the K-shell absorption edges of carbon at ∼290 eV and oxygen at ∼530 eV, nature has
provided a ‘water window’ where organic materials are up to 10 times more absorbing
than water [58]. Using synchrotron sources, imaging in this ‘water window’ region of thespectrum has already allowed imaging of several micron thick samples with high intrin-

79
sic contrast, and 3-dimensional tomography of a cell at impressive resolutions (∼ 60 nm)
has also been demonstrated [61, 64]. The long-term goal of the work I am presenting inthis thesis is to enable these exciting capabilities using a tabletop-size microscope setup.

5.2

Coherent vs. Incoherent Illumination and Imaging
Human vision, i.e. the imaging of light transmitted through or reflected fromobjects, forms the basis of our perception of reality. Not surprisingly, the qualities ofthe light illuminating an object, such as its frequency, polarization, and intensity, candramatically change our perception of the object. Along with these qualities, we mustalso consider whether the illumination is coherent or incoherent. For instance, when apainted wall is illuminated by the sun, it looks quite different than the speckle pattern weobserve when the same surface is illuminated by a visible laser beam. In simple terms,
if we can merely add the square of the amplitudes of the light waves, the illumination
is incoherent, but if we must first add the amplitudes of the waves and then square
the sum, the illumination is coherent [65]. For instance, if I take two separate mirrors
and use them to reflect light from an incoherent source like the sun onto a screen, theillumination will be brighter where the two beams overlap. However, if I shine lightfrom two coherent sources, such as two identical lasers, onto a screen, the brightness
where the two beams overlap will depend on the phase relationship between the two
waves in the overlapped region.
Of course, the coherence of illumination in most experiments is neither perfectly
coherent nor perfectly incoherent, but instead lies somewhere between these two extremes. The following discussion of partial coherence, briefly summarizes a treatment
by Fowles [17]. Strictly speaking, the time averaged light flux at a given point where
two monochromatic fields E1 and E2 meet is given by the irradianceI = hE · E∗ i = h|E1 |2 + |E2 |2 + 2Re(E1 · E∗2 )i

(5.1)

80
where the sharp brackets imply the time average. Making the assumption that thisaverage remains constant regardless of the definition of the time origin, and furthermore,that the fields possess identical polarizations, so we may ignore their vectorial nature,
Equation 5.1 becomes
I = I1 + I2 + 2RehE1 · E2∗ i
I1 = h|E1 |2 i

(5.2)

I2 = h|E2 |2 i

In a generalized interference experiment, the fields E1 and E2 come from a common
source but take differing optical paths to some point of overlap. In general the fields
take unequal amounts of time to traverse these divergent paths, and a phase difference
arises between the fields at the overlap point. Calling the time to reach the overlap
point t for E1 and t + τ for E2 , the interference term in Equation 5.2 is written as
2ReΓ12 (τ )(5.3)

Γ12 (τ ) = hE1 (t) · E2∗ (t + τ )i

(5.4)

where

This function, Γ12 (τ ) is known as the mutual coherence function, and it measures, asa function of separation in space and time, the extent to which we can predict theelectric field at one point if we know its value at some other point [60]. However, forconvenience, this function is often used in a normalized form known as the degree ofpartial coherence, and given by
γ12 (τ ) =Γ12 (τ )1(I1 I2 ) 2(5.5)

Using the degree of partial coherence the irradiance is expressed by
1

I = I1 + I2 + 2(I1 I2 ) 2 Reγ12 (τ )(5.6)

In general, γ12 (τ ) is a complex periodic function of τ , and an interference pattern
will result when |γ12 (τ )| =6 0. The types of coherence can be summarized in terms of81|γ12 (τ )| =6 0 as follows:|γ12 | = 1(complete coherence)

0 < |γ12 | < 1(partial coherence)|γ12 | = 0(5.7)(complete incoherence)

For an interference pattern, the contrast of fringes varies between a maximum
and minimum value given by Imax and Imin where1

1

Imax = I1 + I2 + 2(I1 I2 ) 2 |γ12 | Imin = I1 + I2 − 2(I1 I2 ) 2 |γ12 |(5.8)

and the fringe visibility V is defined asV =Imax − Imin
Imax + Imin

(5.9)

V = |γ12 |(5.10)

and for I1 = I2 , V will be given by

In essence, the fringe visibility in an interference experiment reveals the level of coherence of a source [17]. In fact, a measure of fringe visibility in a double pinhole
interference experiment has been used previously by our group to study and quantify
the level of coherence of our HHG source [55, 66].
The type of illumination chosen, along with the imaging method used, the qualityof the instrument, and the skill and experience of the observer, all mix together in the
final process of image interpretation. In general, the question of whether coherent orincoherent illumination is better has no definitive answer unless applied to a specific setof imaging parameters. For instance, for a conventional optical microscope in the brightfield imaging mode, highly coherent illumination tends to be associated with several
undesirable effects. These effects include a ‘ringing’ in the image from sharp edges, agranular speckle background coming from randomly phased scatters in the illuminating

82
light diffuser, and a high sensitivity to imperfections such as dust or scratches along theoptical train of the instrument [67].
In contrast, any imaging technique that uses phase information as a contrast
mechanism requires at least some degree of coherence. In optical microscopy, sometechniques requiring coherence include phase contrast, differential interference contrast,
and polarization contrast microscopy. These techniques can detect phase changes induced by small variations in a specimen’s index of refraction. Even the incandescentfilament lamps used in optical microscopes exhibit partial coherence, and the degree of
coherence can be adjusted by selecting portions of the source light with a condenser
aperture. In this case, the coherence originates from strong interactions that occur between neighboring atoms in microscopic domains during photon emission. These effects
give rise to localized synchronous emission of discrete bundles of light over the filament’s
surface [68]. On the other hand, holographic imaging techniques require a very highdegree of coherence. In classic holography, the light from a coherent source is split into
two beams, one illuminating a sample and the other serving as a reference. After theilluminating beam interacts with the a specimen, it interferes with the reference beamand the resulting interference pattern is recorded. This interference pattern essentially
encodes the phase information carried by the illuminating wave as intensity modulations [17]. The relatively new technique of lensless diffractive imaging, which is theimaging method we have chosen to mate with our HHG source, also necessarily requires
coherent illumination and will be discussed in more detail in Section 5.5.

5.3

Spatial and Temporal Coherence
The discussion in the preceding section concerning coherence dealt with fairly

idealized sources. Light sources in practice have a finite source size, a finite spectral
bandwidth, and radiate with some restricted angular extent. Therefore, it is often
more appropriate in practical terms to determine the region over which the light from a83realistic source can be considered coherent, in directions both perpendicular and parallelto the light’s propagation direction [60].
The coherence parallel to the direction of propagation is referred to as the temporal coherence, and a temporal coherence length LT coh can be defined as
LT coh =λ22∆λ(5.11)

where λ is the wavelength of the light and ∆λ is its FWHM bandwidth. This temporal
coherence length equates to the length of propagation that causes two waves that are
separated in wavelength by exactly the bandwidth to slip 180◦ out of phase [60]. So,
in a diffraction experiment, path length differences from the diffracting object to theobservation plane should be kept under this length to maintain a well-defined diffraction
pattern.
Similarly, the coherence transverse to the direction of propagation is referred toas the spatial coherence, and a spatial coherence length LScoh can be defined as
LScoh =zλ2πD

(5.12)

where z is the distance from the source, and D is the source’s diameter [60]. LScoh gives
the spatial extent over which two points in a transverse slice of the beam can still beconsidered to have a well correlated phase relationship. So, in an experiment to record
diffraction from an object, care should be taken to keep LScoh of the illuminating lightlarger than the transverse dimension of the sample object.

5.4

Imaging in the Extreme Ultraviolet
Producing images in the EUV involves a set of challenging parameters not preva-

lent with visible light sources. The glass lens is an ancient scientific device, with its
origins tracing back more than a thousand years ago. In terms of imaging, this elegant optical element forms the backbone of even the most advanced modern optical

84
microscopes. Unfortunately, for imaging at EUV wavelengths, the venerable glass lenscannot be used. A glass lens relies on the phenomenon of refraction for its ability tofocus light and produce images. However, refractive imaging does not exist at EUV
wavelengths due to the nature of the index of refraction for this region of the spectrum.
The following explanation is a brief summary of a discussion given by Attwood in [60].
At EUV wavelengths the index of refraction varies only slightly from unity andis expressed asn(ω) = 1 − δ + iβ

(5.13)To understand the δ and β terms in Equation 5.13, consider a plane wave
E(r, t) = E0 e−i(ωt−k·r)(5.14)

propagating in material with an initial amplitude E0 and having the magnitude of k ask=ω
(1 − δ + iβ)c(5.15)

Then, for the propagation direction that satisfies k · r = kr−(2πβ/λ)r
E(r, t) = E0 e−i(ωt−r/c) e|−i(2πδ/λ)r{z } e| {z }|{z}(5.16)

the first exponential term in Equation 5.16 represents the phase shift that would haveresulted from propagation in vacuum, the second exponential describes the phase shift
induced by the material, and the third exponential represents decay of wave strength due
to absorption. The values of δ and β in the EUV are such that no significant refraction
can occur in material before the incident wave is completely absorbed. Therefore, optics
in the EUV must rely on reflection or interference effects to focus, image, and manipulate
the light field. [60].
Fresnel zone plate lenses, usually just called zone plates, are the most common
diffractive optical elements used in high resolution imaging at EUV and shorter wavelengths. These diffractive optics are basically chirped transmission diffraction gratings

85
in a circular geometry. An SEM micrograph of a zone plate is shown in Figure 5.1.
The central region of the zone plate is opaque to the wavelength of interest. Going

Figure 5.1: A zone plate as seen by a scanning electron microscope, picture by Eric
Anderson, LBNL. From [60]radially outward from this central portion, concentric zones alternate between beingtransmissive and opaque while also getting finer in spacing. So, as a beam of lightshines on this structure, it has a small angle of diffraction near the inner most zones,
and a large angle of diffraction at the outermost zones. With the proper arrangementof zones, the diffraction through the zone plate leads to constructive interference at awell-defined point. This situation equates to defining a focal point [60]. In his book,“Soft X-Rays and Extreme Ultraviolet Radiation: Principles and Applications,” David
Attwood presents a thorough treatment of zone plates, and the reader is referred tothis reference for a detailed overview. However, I wish to summarize a few of his key
points for the purpose of comparison. First, the resolution attainable with a zone plateis on the order of the size of its smallest outer zone, labeled ∆r. Thus, the manufacturing process must produce dimensional tolerances that are much smaller than ∆r. To
achieve this demanding specification, advanced electron beam lithography and etching

86
techniques are employed [69]. Additionally, the focal length f, depth of focus DOF, andspectral bandwidth requirement of a zone plate lens are given as follows:f=4N (∆r)2
λ

DOF = ±

2(∆r)2λλ≥N∆λ(5.17)

where N is the number of zones, and λ is the illuminating wavelength [60]. As withconventional lenses, focal lengths of zone plate lenses must be quite short to obtain
good magnification. Spatial resolutions as high as 15 nm have been achieved with the
XM-1 microscope at the Advanced Light Source in Berkeley using a state-of-the-art
zone plate with ∆r = 15 nm, 500 zones, and a 1.5 nm illuminating wavelength [69].
Using this zone plate as an example, we see that for f = 300 µm, the depth of focus is
λ
300 nm, and ∆λmust be better than 500. So at high magnifications, zone plates havea relatively short working distance, a small depth of focus, and the potential for large
chromatic abberation. Additionally, the design of a zone plate, which blocks much of thelight it receives, makes them typically around 10% efficient in the first-order diffracted
light. Despite these challenges, resolutions down to 38 nm have been demonstratedusing tabletop EUV lasers that exhibit superbly narrow spectral bandwidth [70, 71].
Unfortunately, the range of these lasers does not extend into the ‘water window’ regionof the spectrum that tabletop HHG sources have already reached, albeit with limited
flux. Currently, however, high harmonic sources have only demonstrated resolutions of∼ 200 nm with zone plate imaging using 13 nm light [72].
Holography represents another possible imaging method in the EUV, and was
first demonstrated with x-rays in 1987 using Gabor holography [73]. Gabor holography
is a very straightforward holographic technique in which the illuminating light wave
impinges on a weakly scattering object, after which the undisturbed portion of theilluminating wave becomes its own reference wave and interferes with the scattered wave
in the far field [74, 75]. In an on-axis Gabor holography experiment, this situation canbe accomplished by using a small scattering specimen in comparison to the illuminating

87
beam size. The need to surround a specimen by a large highly transmissive region limits
the usefulness of Gabor holography, particularly in the EUV region of the spectrum [67].
Natural extensions of Gabor holography, such as Fourier transform holography haveyielded sub-100 nm resolutions using spatially filtered coherent x-rays at synchrotron
facilities [76].
Lensless diffractive imaging is a relatively new coherent imaging technique thatis an extension of holography [77–80]. It requires spatially coherent beams and replaces
imaging elements in the optical system with a computerized phase retrieval algorithm.
By obviating the need for physical imaging optics, this technique is well-suited to the
EUV, and was first demonstrated in 1999 using spatially-filtered x-ray light from asynchrotron source [77]. In principle, lensless diffractive imaging is free of aberrations,has a very large depth of focus, and can achieve diffraction limited resolution. Using
this technique at a synchrotron, a freeze-dried yeast cell has been imaged to better than
30 nm resolution [81]. In other exciting work, 3-dimensional reconstructions of objects
have been achieved to better than 50 nm resolution [82] Very recently in 2006, the firstlensless diffractive microscopy using a soft-x-ray free-electron laser at 32 nm wavelength
was demonstrated, with a claimed resolution of ∼ 60 nm [83].

5.5

Lensless Diffractive Imaging
In the late 1800’s the German physicist and influential microscope designer Ernst

Abbe introduced the notion that Fourier analysis could be applied to the theory ofimaging. In an analogy to the manner in which an arbitrary bounded waveform with
only finite discontinuities can be thought of as being decomposed to a series of elementary functions consisting of sines and cosines with different frequencies, Abbe suggested
that an object to be imaged with coherent illumination could be thought of as being
composed of a series of gratings with varying periodicities, positions, and orientations.
So, the total diffraction from an irregular object can be realized as the sum of the

88diffraction contributions from its various component gratings combined with the correct background component. However, this idea is strictly not the same as suggestingthat an object can be considered as the overlay of many varied gratings. When using a lens to image the object, the diffraction contributions interfere at any point at
which overlap between them occurs, but only near the image plane does this interference
produce a meaningful reproduction of the object [65]. Amazingly, lensless diffractive
imaging, as the name implies, performs this feat without the use of a physical lens.

5.5.1

The Phase Problem

Under the conditions of coherent plane wave illumination and validity of the Born
approximation, the light wave diffracted from an object forms the Fourier transform of
the object [84]. Considering this idea from the Abbe point of view suggests that thephase information carried by the diffracted wave contains the critical components to
reproducing a faithful image of the object. Writing the Fourier transform F of anarbitrary finite object described by the function g(x, y) as
F {g(x, y)} = G(kX , kY )(5.18)

where the transformation is from the space domain to the frequency domain, and kX andkY are coordinates in the frequency domain, the two dimensional Fourier shift theorem
states that
F {g(x − a, y − b)} = G(kX , kY )exp[−i2π(kX a + kY b](5.19)

Or, stated in words, translating a function in the space domain corresponds to creating
a linear phase shift in the frequency domain [67]. Thinking about the diffracted wave as
coming from the contributions of many gratings, the positions of these conceptualized
gratings will relate to the shape of the diffracting object. Thus, the phase information
contained in the diffracted wave must be preserved or retrieved to allow for meaningful
imaging. A lens preserves this phase information, but a simple energy detector (such as

89photographic film or a CCD array) cannot.
A traveling electromagnetic wave carries both amplitude and phase information.
However, detectors of electromagnetic waves in optics only record the intensity of the
waves given by |E|2 , leaving no possibility for direct detection of the phase information.
This phase problem is pervasive in areas of imaging and diffraction. Thus, solving thisphase problem is an active topic in areas as diverse as wave front sensing, astronomy,
crystallography, and diffractive imaging [85]. In a general form, the phase problem
involves determining an object f (x) solely from intensity measurements of its Fouriertransform F (u). F (u) can be expressed as
F (u) = |F (u)| exp[iψ(u)] = F [f (x)]Z ∞=f (x) exp[2πiu · x]dx

(5.20)−∞

where x is an M-dimensional spatial coordinate, and u is an M-dimensional spatial
frequency coordinate . Thus, the Fourier transform F (u) can be written as an amplitude
and a phase. With only intensity measurements of F (u), to fully recover f (x), one must
determine the phase function given by ψ(u) [85].

5.5.2

The Oversampling Method

The problem of retrieving phase information has been well studied in the fieldof x-ray crystallography. In x-ray crystallography the diffraction from many identicalunit cells leads to constructive interference and very strong Bragg peaks. However,
the oversampling technique can handle the situation that arises in diffraction from
noncrystalline objects where detectable intensity appears between Bragg peaks, or where
no Bragg peaks occur, and the diffraction pattern is continuous [84]. Sampling such adiffraction pattern at a frequency finer than that given in the situation of Bragg peaks
is referred to as oversampling [86]. The idea that oversampling the diffraction patternmight be able to reveal the phase information was suggested by David Sayre in 1952 [87].

90
However, to make it possible to resolve features in an oversampled diffraction patternwith good fidelity, the diffraction pattern must have a high dynamic range and be
produced with illumination that is both spatially and temporally coherent [86].
In 1998, Miao et al. gave the currently accepted explanation for the oversampling
method [88, 89]. They suggested that each intensity point in the diffraction pattern canbe viewed as a nonlinear equation. Solving the phase problem reduces to finding the
unknown ‘electron density’ from this set of equations (since the oversampling method has
its roots in the field of crystallography, the object of interest to be reconstructed when
the phase problem is solved is the sample’s electron density). However, when samplingis done at the Bragg peak frequency, twice as many unknowns exist as independentequations [84]. Miao calls the oversampling degree in real space σ, and defines it asσ=electron density region + no-density region
electron density region(5.21)

Furthermore, he calls the linear oversampling ratio for two dimensions in reciprocal
space O, and defines it as1

O = σ2(5.22)[86]. For σ larger than 2, the number of independent equations exceeds the number of
unknowns, and the phase information is embedded within the diffraction pattern [88].
Unfortunately, just performing oversampling cannot directly produce phase information. The phase itself must be retrieved using an iterative algorithm described in thefollowing section. In addition, several experimental parameters must also be optimized
concerning the sample, its placement relative to the detector, and the illumination.
In their paper entitled “On possible extensions of X-ray crystallography throughdiffraction-pattern oversampling,”Acta Crystallographica, vol. 56, pp. 596-605, (2000),
John Miao and David Sayre present a very straightforward derivation of the oversampling theorem [84]. For an N −by−N sized sample in real space, the Nyquist frequency inreciprocal space will be 1/N . This theorem suggests that sampling (in reciprocal space)

91
the Fourier transform of an object at a frequency finer than its Nyquist frequency corresponds to generating a no-density region surrounding the electron density of the object(in real space). This theorem forms one of the primary experimental considerations
for applying the oversampling method, as it means that a transmissive object must be
surrounded by an opaque mask whose size scales with the degree of oversampling. The
following derivation is a near verbatim version of the one in the aforementioned paper.
If the electron density of an object is assumed to be ρ(x), then the Fourier transformof this object is given by
F (k) = F [ρ(x)] =

Z ∞

−∞

ρ(x) exp[2πik · x]dx

(5.23)

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, thenF (k) =

N
−1
X
x=0

ρ(x) exp[2πik · x/N ],k = 0, . . . , N − 1.

(5.24)

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, thenF (k) =

N
−1
X
x=0

ρ(x) exp[2πik · x/(2N )],k = 0, . . . , 2N − 1.

Let us introduce a new function g(x) such thatρ(x) 0 ≤ x ≤ N − 1g(x) =
0N ≤ x ≤ 2N − 1(5.25)(5.26)

If g(x) is substituted into Equation 5.25, thenF (k) =

2N
−1
X
x=0

g(x) exp[2πik · x/(2N )],k = 0, . . . , 2N − 1.

(5.27)

Equation 5.27 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 more

92
finely than the object’s Nyquist freqency generates a no-density region surrounding theelectron density of the object as illustrated in Figure 5.2 [84]. Clearly, by the same logic,
increasing the degree of oversampling increases the size of this no-density region.

Figure 5.2: An illustration of the no-density region generated by oversampling thediffraction pattern

5.5.3

General Description of Algorithm

Our collaborator John Miao and his group at UCLA perform iterative phaseretrieval on our diffraction data to yield a reconstructed image of our sample. The
algorithm used to perform this phase retrieval is based on an approach suggested by
Fienup [90]. Furthermore, the algorithm incorporates constraints concerning the electron density. For illumination with hard x-rays with energy far from any absorption
edges, the electron density is positive and real. For lower energy illumination (EUV),
the electron density is complex. The real part of this density is usually positive, andonly becomes negative near absorption edges. The imaginary part, representing absorption by the sample, is positive definite [84]. The current algorithm, known as the guided
hybrid input output (GHIO) algorithm, was developed by John Miao and consists of

93
the following four basic steps in each iteration [86]:1. The square root of the measured diffraction intensities is the magnitude of the Fouriertransform of the sample with some experimental noise. This magnitude of the Fouriertransform is combined with a phase set to give a function with both amplitude andphase for each pixel. For the first iteration, a random phase set is used. For subsequent
iterations, the best phase set available is used.
2. An inverse fast Fourier transform is applied to the outcome of the first step. This
inverse transform provides a new electron density function.
3. The electron density function from the second step is now subject to the constraints
that it must be positive inside the support and zero outside the support. To enforce
the positivity constraint,the negative electron density inside the support is pushed close
to zero, and the positive electron density inside the support is retained. To enforce theconstraint outside the support, the electron density in this region is also pushed close
to zero. In this manner, a new electron density is defined.
4. To calculate a new set of phases, a fast Fourier transform is applied to the electron
density created in step three. The phase of the central pixel is set to zero, and this newphase set is employed in the following iteration [88].
The algorithm is shown schematically in Figure 5.3 [84].

5.5.4

Experimental Requirements

To obtain a given resolution with a high quality reconstructed image we must
consider the sample, the placement of the sample with respect to the detector, qualitiesof the illuminating light, and the nature of the collected diffraction data. The sample
should be illuminated with a plane wave and must be placed far enough from the detector
to guarantee a far field diffraction pattern. In practical terms, plane wave illuminationmeans that the sample aperture dimensions should be much smaller than the radiusof curvature of the wavefront impinging on the aperture (we currently use a factor of

94

Figure 5.3: A schematic representation of the iterative phase retrieval algorithm

95∼ 100). For a far-field diffraction pattern, the Fraunhofer criterion must be satisfied,
which for plane wave illumination is given by
z≫

D2
λ

(5.28)

where in this case, z is the sample to CCD distance, D is the sample diameter, andλ is the illuminating wavelength. Essentially, the condition of Equation 5.28 ensuresthat the diffraction pattern is also in a plane wave condition and its curvature can beneglected [17]. From a practical point of view, z larger than the right hand side ofEquation 5.28 by a factor of 3 is often sufficient, but a factor of 10 or more is muchpreferred.
The distance from the sample to the CCD is chosen to give an appropriate linear
oversampling ratio (> 5) that allows for easily reconstructable diffraction patterns withhigh resolution [86]. The linear oversampling ratio relates the smallest diffraction pattern speckles to CCD pixels. The linear oversampling ratio O in terms of experimental
quantities is given by
O=

zλ
Dp

(5.29)

where z is the sample to CCD distance, λ is the wavelength, D is the sample diameter,and p is the pixel size of the CCD camera. In the reconstructed image, each image pixel(not to be confused with a CCD pixel), corresponds to a size r given by
r=

OD
zλ
=
N
pN

(5.30)

where N is the linear number of pixels in an N-by-N CCD array. This image pixel size issuggestively labeled as r since it sets the ultimate resolution for any given geometry; witha pixelated image the sharpest discernable feature is at minimum one pixel wide. Since
their is a one-to-one mapping of pixels in reciprocal space to pixels in real space (thereconstructed image), this condition can be thought of as coming from a ‘conservation’of pixels. So, the larger we make O, the more pixels in the reconstructed image we must

96
allow to become part of the no-density region. So for a sample 1000 nm across, if O = 1
and we have 1000 pixels across in the reconstructed image, each pixel corresponds to 1
nm. However, if O = 2 the sample in the reconstructed image will only be 500 pixels
across and be surrounded by a no-density region, so each reconstructed image pixel willcorrespond to 2 nm [84]. Of course, having the pixel size in the reconstructed image
correspond to a particular dimension does not guarantee that we can achieve such aresolution. Clearly, we would like to have a detector with a large number of pixels,
and the smallest pixel size possible. Several other important considerations affect theresolution. To begin, we must actually capture the diffraction information corresponding
to features of a given size. Viewing features of the desired resolution as being diffraction
gratings with a slit spacing d, we recall the familiar formula for first order constructive
interferenced sin(θ) = λ(5.31)

where we also note that d sin(θ) is the path length difference between waves emanatingfrom two adjacent slits and meeting at a common point. So, for a desired resolution d,
we must ensure that the geometric conditions are such that the CCD camera captures
this diffraction angle. However, we must also take care that the oversampling ratio in
this geometry is sufficiently large for obtaining a high fidelity reconstructed image, andsimultaneously, sufficiently small such that the pixel size in the reconstructed image
allows resolving of features possessing the desired resolution. In general, for increasing
resolution, these constraints require placing the sample closer to the detector to capture
high diffraction angles, and shrinking the sample aperture to maintain the oversampling
ratio and far-field condition. Additionally, the spectral bandwidth of the illuminating
beam will affect the quality of the diffraction pattern and the reconstructed image.
Thinking from the point of view of a simple diffraction grating, clearly if the lighthitting such a grating is not monochromatic, the diffraction speckles will blur since

97
slightly different wavelengths will diffract at slightly different angles. Miao gives the
spectral bandwidth requirement for lensless imaging as
λ
OD
≥
∆λ
d

(5.32)[86]. To clarify how the desired resolution comes into Equation 5.32, I think it isuseful to put this condition into terms of temporal coherence. Multiplying each side ofEquation 5.32 by λ gives
λ2
λ≥ OD · = OD sin(θ)∆λ
d(5.33)

which now substituting into Equation 5.11 we can write
LT coh ≥OD
sin(θ)
2(5.34)

So the spectral bandwidth condition can be interpreted as suggesting that the temporal
coherence length of the illuminating light should be larger than the path differences
arising in the diffraction from illuminating an area larger than the sample aperture by
O/2.

Chapter 6

Table-Top Lensless Microscopy

6.1

Introduction
Perhaps the most important factor in the success of the optical microscope as atool for science has been its accessibility. The elegant compactness of this device allows
scientists around the world to not only explore the wonders of the microscopic world,
but it also gives them the freedom to invent and perfect new imaging and sample preparation techniques. The motivation for the work presented in this chapter is to forge
a pathway towards a practical tabletop EUV microscope that takes advantage of theunique properties of our EUV source, and that can be scaled to use at shorter wavelengths. Here I describe the world’s first demonstration of lensless diffractive imaging
in the EUV using a table-top light source, and I discuss the possibilities of extending
this work to shorter wavelengths and higher resolutions.
Much of the description of the imaging results presented here is an extension ofa paper I authored with Richard Sandberg that will be published in Physics Review
Letters. Furthermore, all of the reconstructed images presented here were produced by
Changyong Song of John Miao’s Coherent Imaging Group at UCLA.

99

6.2

Lensless Diffractive Imaging and HHG in Hollow Waveguides:
A Natural Fit

6.2.1

Coherence of High-Order Harmonic Emission Generated Using a
Hollow-Waveguide Geometry

Coherence of the illuminating light source forms one of the primary requirements
for successful lensless imaging. Since this imaging technique relies on collecting diffraction from the specimen, the visibility of the diffraction pattern, and ultimately the
resolution of the reconstruction, is directly related to the coherence of the illumination. Considering this requirement, most small-scale EUV and x-ray sources, as wellas large-scale sources such as synchrotrons or free-electron lasers (FEL’s) lack intrinsic
coherence. Therefore, for coherent imaging techniques, the light must be sent througha monochromator, as well as made spatially coherent via spatial filtering through microscopic pinholes, causing a large loss of flux. Although the bright large-scale sources
can handle this loss of flux, most small-scale sources are left without sufficient lightfor imaging in any practical manner. Thus, before the work presented here, lenslessimaging experiments in the EUV and x-ray have only been conducted at synchrotrons
or FEL’s.
In some of my earliest days of graduate school research, I made several of theexperimental pieces that helped Randy Bartels and Sterling Backus demonstrate thatHHG performed in a gas-filled hollow waveguide generates essentially fully spatiallycoherent EUV beams at wavelengths of ∼ 30 nm. The spatial coherence was measured
via the visibility of fringes produced from a double pinhole experiment. To demonstrate
the coherent imaging potential of this light source, it was also employed for Gaborholography with moderate resolutions of < 10 µm [55]. Later work in our group, using
the same double pinhole technique, showed that HHG performed in the hollow waveguide
geometry also gives spatial coherence at 13 nm, suggesting further that the trend is

100
intimately linked to phase matching of the HHG process and will continue to evenshorter wavelengths [66, 91].

6.2.2

Efficient Use of Photons

Considering the unique nature of our high-harmonic source, the appropriate experimental approach is to investigate imaging techniques that take advantage of thissource’s intrinsic spatial coherence, and efficiently use its photons. Originally, we had
considered performing imaging using zone plates, as we have collaborators at Berkeley
that are world experts in the manufacture and use of these diffractive optics. However,traditional zone plate imaging would require us to destroy some of the spatial coherence
to avoid ‘ringing’ artifacts in our images. Moreover, the low efficiency of zone plates
at our operating wavelengths would severely limit our useable flux. Scanning transmission microscopy is another possibility that would actually benefit from our source, as
its coherence should allow us to focus to near diffraction-limited spot sizes. However,this technique would again require zone plates and discard large amounts of flux. Since
it involves rastering over many points in a sample, scanning transmission microscopy
would also require extremely accurate positioning mechanisms, as well as methods fornormalizing flux at each position. A purely reflective optics geometry using ultra-highquality multilayer mirrors could in principle be used to create high quality images andpreserve flux [60]. However, such systems generally require incoherent illumination,and the optics arrangements are not usually conducive to a compact geometry. On theother hand, Gabor holography would take full advantage of our coherence, and has been
demonstrated with tabletop sources at submicron resolution [92]. However, for on-axis
Gabor holography, only a small fraction of the beam interacts with the sample. Since
the signal coming from the diffracted wave is small, this technique is not well suited to
our flux levels.
Oren Cohen, a post-doc in our group, originally suggested the possibility of trying

101lensless diffractive imaging. Oren and Ra’anan Tobey performed some initial diffraction
experiments that bolstered the feasibility of his suggestion, and afterwards, Richard andI pursued its implementation. Several aspects of lensless imaging make it an extremely
elegant and appealing microscopy technique for our high-harmonic EUV source. The
source, while bright, certainly does not compete with the overall flux available at asynchrotron or FEL. In the setup discussed here, the longest integration time is around
120 minutes. Therefore, ease of sample setup remains an important concern, as wellas the long-term stability of the source. As long as the conditions outlined in Section
5.5.4 are met: including proper illumination, the appropriate sample size and sample to
CCD distance, the technique should, in principle, work. Of course, the sample to CCD
distance sets the linear oversampling ratio and ultimately determines the highest angleof diffraction that can be captured. Yet, placement of the sample is non-critical; ∼mm placement accuracy is completely acceptable in our imaging geometry. In addition,no multi-step focusing process is necessary, in contrast to the case of physical imaging
optics that require sample placement and stability at the micron level. Furthermore,when performing imaging with a single optic, the magnification is a purely geometricfunction of the object to image distance ratio, often requiring the detector to be ameter or more from the imaging optic for high-resolution work. In contrast, for tabletop lensless imaging, the entire imaging apparatus fits in 0.5 m x 1.5 m, and large
magnifications are achieved with only centimeters separating the sample and detector.
I should also note that the same experimental setup will work at any shorter wavelengths
at which bright coherent light can be produced, reflected, and made narrowband withreasonable efficiency.

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6.3

Experimental Set-up

6.3.1

Illumination by the EUV Source

A 10 cm hollow waveguide is used to ensure full spatial coherence of the illuminating EUV light. The pump laser is typically run at a 3 kHz repetition rate, with 4.2 W of
average power, a 780 nm center wavelength, and 25 fs pulses. The 150 µm inner diameter hollow waveguide is held in the v-groove geometry described earlier and filled with∼ 65 torr of Ar. It should be noted that the best flux for the 27th harmonic producedin this hollow waveguide filled with Ar is observed at a phase-matching pressure of ∼35
torr. However, the EUV mode generated at this lower pressure of Ar is less Gaussian
and appears to have a less narrow spectral bandwidth than the EUV produced at 65
torr. Consequently, though the flux is greater, it fails to produce crisp diffraction patterns. Preliminary tests of spectral bandwidth of our source with an EUV spectrometer
λof around 200 for data taken at 65 torr and 120 at 35
at the two pressures suggest a ∆λtorr. These estimates of the spectral bandwidth appear consistent when compared withthe radial extent of clearly defined speckles seen in the diffraction data at the highest
recorded diffraction angles (where smearing due to lack of monochromaticity will bemost evident).
When optimally coupled, an EUV beam is generated with a beam waist radius of
about 25 µm and approximately 1 milliradian divergence, with ∼ 5 harmonics centerednear the 30 nm wavelength. Two 200 nm thick aluminum filters are used to extinguish
the fundamental laser light. The second filter encountered by the laser light is held ina special light-tight filter wheel described later in this chapter. A pair of narrowband
Mo/Si multilayer mirrors acts as both a monochromator and a condenser. The narrowband mirrors each have a peak reflectivity of about 25% centered 29 nm in a 2.5 nm
bandpass, making it possible to effectively select a single harmonic order, in this case,the 27th harmonic of the fundamental laser light. Having two mirrors narrows the band-

103
pass, and this bandpass can also lead to a slight narrowing of the spectrum of a single
harmonic. The first mirror is flat, while the second has a 50 cm radius of curvature to
gently focus the illuminating beam down to a beam diameter of a few hundred microns
at the sample position.
To appropriately calculate the EUV beam size illuminating the sample, simple
Gaussian beam propagation methods provide a useful and quick solution. However,care must be taken to include the initial radius of curvature of the wavefront exiting
the waveguide in the input parameters of the calculation. A 1 milliradian divergencecorresponds to an initial radius of curvature of about 2 cm. If the beam waist radius at
the output of the waveguide is taken to be 25 µm, and its radius of curvature assumed
to be 2 cm, the calculations give an estimate of the illuminating beam size along its
optical path with sufficient accuracy. We confirmed the validity of these assumptions
by changing the focusing geometry several times, taking careful measurements of thebeam path, and recording the beam size on the CCD camera.
The position of the sample relative to the condenser mirrors and detector is chosen
to optimize the illumination, the diffraction quality, and the oversampling ratio. We
wish to use high EUV flux on the sample, while still maintaining a fairly flat intensity
profile. To maintain high flux on a small sample, we focus the illuminating beam witha curved multilayer mirror. To make the illumination more uniform, the focused beam
size overfills the sample. Assuming a perfect Gaussian beam, overfilling the sample bya factor of ∼ 4 and using the most intense portion of the beam leads to an intensity
distribution with a variation ∼ 10%. Overfilling the sample not only leads to a more
uniform illumination, but also tends to ensure long-term illumination stability, mode
quality, and a large radius of curvature of the EUV beam compared to the aperture size.
Though overfilling the sample seems wasteful, conventional imaging would also require
uniform illumination for the best results.

104
6.3.1.1

Mode Quality

A high quality and uniform EUV mode is needed for lensless imaging, to ensure
the best spatial coherence and the best resolved diffraction data. Figure 6.1 shows themode of the 29 nm beam after reflecting from the two multilayer mirrors. Strikingly,
the mode profile fits to a near perfect Gaussian. This excellent beam spot quality
results from both the development of consistent alignment techniques, and the major
improvements in waveguide design described in Chapter 4.
The data plotted in Figure 6.1 represents a total of 5.2x107 counts on the CCD
detector in a 2 second exposure. Knowing the number of counts and the efficiency of the
optical elements lends itself to a flux estimate. To begin, the number of observed counts
must be converted to a number of detected photons. As quoted by the manufacturer,
the number of counts Nc seen per pixel at a gain setting of g for each incident photonof energy E is given by
Nc =

E
g3.65

(6.1)

For the readout time used in this measurement, the camera had a gain setting of 2.
In addition, only one harmonic order was detected, corresponding to 45 eV (29 nm).
Therefore, the camera detected 4.2x106 photons per second. Next, the throughput of
the optical system must be combined with the efficiency of the camera to determine
the ratio of photons output by the waveguide to those detected by the camera. For
all further quoted transmission, reflection, and efficiency in this estimate an energy of
45 eV will be assumed. After the waveguide, the EUV beam encounters two 200 nm
aluminum filters, a set of two multilayer EUV mirrors, and finally, the detector. 400
nm of aluminum is about 40% transmissive at the energy of interest [38] However, purealuminum oxidizes, and typically there will be about 10 nm of aluminum oxide per
filter [93]. This 20 nm of aluminum oxide has a transmission of about 45%. Each mirror
has a reflectivity of around 25%, and likely a 10 nm SiO2 layer on each mirror (meaning

105

Figure 6.1: The mode of the 29 nm HHG light near its focus, plotted on a log scale.

106effectively 40 nm total, since the beam must pass through this layer twice on eachmirror bounce). 40 nm of SiO2 has a transmission of 25%, giving a total throughputof the mirrors of 1.6%. Finally, the camera’s manufacturer claims a quantum efficiency
of approximately 33%, giving an overall throughput of the system as 9.3x10−4 . This
throughput amounts to an estimate of about 4.5x109 photons per second or about 32 nW
in a single 45 eV harmonic. However, the actual number may be an order of magnitude
or so higher, as the CCD chip also surely had oxide layers. Moreover, after having
antifreeze from a separate experiment condense on its surface repeatedly, the chip had
been manually cleaned with methanol many times before this measurement, leaving afilm of contamination that is difficult to quantify.

6.3.2

Geometry, Microscope Vacuum Chamber, and Detector

A schematic of the current imaging geometry is shown in Figure 6.2. Following
the hollow waveguide and aluminum filters, the EUV light proceeds into the vacuum
chamber where it reflect from the two multilayer mirrors in a z-fold configuration and is
focused onto the sample. The two multilayer mirrors are preceded by a mirror held ina motorized flipper mount that can be inserted into the beam path. This mirror can beused to monitor the coupling of the driving laser light into the hollow waveguide, andalso protects the sample from focused laser light when the aluminum filters are removed
from the beam path. The first multilayer mirror is flat, and the second has a 50 cm r.o.c;both are mounted in compact motorized mounts so that their pointing may be finely
adjusted under vacuum. To reduce astigmatism of the illuminating beam, the multilayer
mirrors are positioned to bring the angle of incidence of the illuminating beam as close
to the normal as possible. The sample is held on a 2-axis stage driven by closed loop
DC motors that have a minimum incremental motion of ∼ 30 nm. This stage provides
for precise alignment of the sample aperture and illuminating beam transverse to thebeam propagation direction. A special mount for holding beam blocks precedes the

107

Figure 6.2: A schematic of the current transmission imaging geometry (top) and thecustom microscope vacuum chamber (bottom)

108
CCD detector, and allows fine positioning of the beam blocks relative to the diffraction
data.
I created the initial design of the current microscope vacuum chamber, however,Hans Green performed the actual machining of the chamber and made some essential
modifications to my original conception. The chamber has an interior dimension of8.5”x22”, and a 6” depth. The chamber walls are made from a solid piece of aluminum,and they seal via an o-ring to the 1.25” thick base. The chamber base includes anintegrated breadboard of 14 −20 holes on a 1 inch square grid, allowing flexibility andeasy of alignment. Furthermore, the entrance port for the EUV illumination is vertically
centered at 4” above the optic table height (the standard beam height in our optics
experiments), and horizontally positioned such that it corresponds to a line of holes on
the internal breadboard. The camera port is similarly designed, but with a 1” horizontal
offset from the illumination port. A set of tapped holes encircles the camera port on
the inside of the chamber to accommodate the beam block holder. To provide for quick
pumping of the system, a turbo pump mates directly to the side of the chamber withno constrictions to limit its conductance. The mass and design of the chamber makeit quite stable and we experience no sensitivity to vibrational noise induced by theturbo pump. Since the experiment requires frequent venting to change beam blocks(described in Section 6.3.3), the chamber lid is a single piece that relies solely on theforce of atmosphere to keep it sealed and can be removed immediately upon venting. The
chamber also includes 8 NW-40 ports for electronic feedthroughs, vacuum diagnostics,
and to add flexibility to the design. Finally, a window with a light-tight removable
cover has been included so that the coupling of the driving laser beam into the hollow
waveguide may be monitored while the system is under vacuum.
The detector is a large-area x-ray CCD camera (Andor DO436). The camera
bolts directly to the chamber and is sealed with a large o-ring in such a manner thatits clear aperture is completely free of obstructions and a sample can be placed within

109
a minimum range of ∼ 2 cm from the CCD chip. The camera has a square array of
2048x2048 pixels each measuring 13.5 µm on a side. As a 16 bit device, it can read frombetween 0 and 65,356 counts per pixel. However, from a practical standpoint, this range
varies from about 50 to 65,356 counts per pixel. The lower end of this range does notreach quite to zero due to intrinsic thermal noise and electrical noise during readout ofthe pixels. To reduce the ‘dark’ counts due to thermally excited electrons, the CCD chipis cooled to a temperature of −60◦ C. The readout of this camera can be made highly
real-time by binning the pixels together in multiples of 2 up to 32. This binning feature
becomes especially useful for sensitive feedback during alignment and flux optimization
procedures. The camera also possesses the option of four readout times at 1, 2, 16, and32 µs per pixel. Although the shortest readout time is quite useful for prompt feedback,the longest readout time corresponds to a significant reduction in background electricalnoise in the image, and is therefore used for all final data sets.

6.3.3

Beam Blocks for Increasing Dynamic Range

When we go outside on a sunny day, the stars are still present but are not visible to the naked eye. In this case, the bright sunlight scattered in the atmosphere
overwhelms the feeble light of the stars. Much like the human eye, a CCD camera
has a limited dynamic range. If exposed to a light source past the point of saturation,
the electrons in a CCD pixel begin to bleed into adjacent pixels. As pixels become
over-saturated, information is lost and eventually the entire image is washed out. Unfortunately, the standard dynamic range of an x-ray CCD camera cannot capture boththe strongly diffracted low spatial frequency information and the weakly diffracted highspatial frequency information from a sample simultaneously without over-saturating.
Just as a solar eclipse allows the stars to become visible during the daytime,
beam blocks allow us to capture the weak large angle diffraction that corresponds tohigh spatial frequency information. These beam blocks vary in size and are placed in the

110central high intensity region of the diffraction pattern. The larger-diameter beam blocks
are used while acquiring long exposure images to record the high spatial frequencies
diffracted from the sample, and the smaller beam blocks are used during the relatively
shorter exposures to record diffraction emanating from low spatial frequencies. The
beam blocks are positioned close to the CCD chip surface (within ∼ 3 cm) to minimize
any diffraction occurring around the edges of the beam block. Moreover, the blocks
must be suspended in the beam path in a manner so that they block a minimum of the
CCD area, other than the central spot.
Creating an effective beam block holder presented an enjoyable challenge. I soughta suitable wire with which to suspend the beam blocks. Tungsten thin wire is a mostremarkable material. It has the ductility to be drawn into extremely fine diameters,
yet possesses enormous tensile strength. The tungsten wire used to support the beam
blocks has a diameter of 12.5 µm (nearly 5 times thinner than the average human hair).
I designed the holder ring to allow easy replacement of the suspension wires, as well as
have them cross at the exact center of the ring. A 3-dimensional rendering of the beam
block holder ring is shown in Figure 6.3, and two different size beam blocks suspended
on the holder ring are pictured in 6.4. Four identical dowel pins are pressed into thering and are offset from a diametric line by their radius so that a wire stretched between
them will be positioned in the center of the ring. The ring also includes a 0.010” lipacross which the wires are stretched to tension them, and the securing screws are placedsuch that their clockwise tightening also tensions the wire.
Ideally, the beam blocks are circles with smooth and well-defined edges to takeadvantage of symmetry in the diffraction pattern and aid in their proper placement.
The larger beam blocks are punched out of brass shim stock with a close tolerance
blanking punch, and then superglued to the intersection of the suspension wires. In
the case of the smallest beam blocks, I view the wires under a stereoscope and apply
a small amount of 5-minute epoxy to the crossing point until I form an appropriately

111

Figure 6.3: 3-d rendering of the beam block ring

Figure 6.4: A large and medium beam block suspended within the ring holder

112
sized epoxy ball. To gauge the size of the epoxy beam block when viewed with thestereoscope, I use a wire with a diameter equal to the desired diameter of the beam
block for reference. In this manner, I can reliably produce beam blocks from the tens
to hundreds of µm in diameter.
We record diffraction patterns of the sample with different size beam blocks inplace, and subsequently stitch the patterns together. This procedure essentially extending the dynamic range of the camera from 3 to almost 10 orders of magnitude. However,for this procedure to remain most effective, the fidelity of the diffraction pattern andits position on the CCD camera must be carefully maintained. As such, the mounting
ring for the beam blocks sits in a 2” kinematic x-y lens translator (Standa 5ZYP-2)
that I retrofitted with two piezo stepper motors (New Focus tiny Picomotor actuator
8351) to allow fine control of the beam block position relative to the diffraction pattern.
This retrofitted translator is shown in Figure 6.5. In addition, the beam block sizes
must be carefully chosen so that enough well defined features of the diffraction patternoverlap between the various data sets to allow proper stitching and scaling of the data.
Without the ability to accurately position the various size beam blocks, the diffraction
pattern would need to be steered onto the detector, compromising its consistency andcomplicating the image-stitching procedure. Moreover, the moveable beam block saves
enormous amounts of time. Originally, we used stationary beam blocks, that we could
position nearly, but never exactly, to the center of the camera. Walking the diffraction pattern onto the beam block involves a laborious procedure of first adjusting theillumination beam pointing and then carefully adjusting the sample to compensate. Especially for the higher spatial frequency data, where weak diffraction makes feedback
times painfully long (and yes, even 5 seconds for feedback is a long time in the wee
hours!), the moveable beam block considerably shortens the overall image acquisition
process.
As the resolution of the lensless microscope is pushed to shorter length scales,

113

Figure 6.5: The x-y kinematic lens mount used to steer the beam blocks into position

114
and the acquisition times decrease through increased flux, improvements to the beam
block configuration will be necessary. Most importantly, the ideal realization would
include beam blocks that could be interchanged within the vacuum chamber with noneed to vent the system and manually replace them. With a larger vacuum chamber,
one possibility would be to have a fixture containing multiple beam blocks. Such a
fixture could be translated to select the appropriate beam block, and then the fixture
could be finely positioned to steer the chosen beam block with respect to the diffraction
pattern. Although the wires suspending the beam blocks are quite fine, a small amountof information is lost with their presence. A slightly more fanciful, though feasible,
solution would be to magnetically levitate spherical beam blocks and have a mechanism
for exchanging different sized spheres [94].

6.4

Imaging Results

6.4.1

Initial Attempts and Improvements

When we began developing a lensless microscope, we performed our initial attempts in a reflection geometry in which I glued the sample to a mirror and masked all
but a small portion with a pinhole aperture. Before Oren’s suggestion to try lenslessdiffractive imaging, I was compelled to begin work on a microscope using zone plates. So,
I developed several components that would turn out useful for the lensless version. I had
already machined a small rectangular vacuum chamber from a solid block of aluminum
to form a stable base for the microscope. Furthermore, with a condenser applicationin mind, we had previously made narrowband multilayer mirrors for operation around
29 nm, and we had also purchased compact piezo driven optic mounts for steering thebeam (New Focus 8887). For motion control of the sample, I modified a two-axis microtranslation stage (Standa 7T228T-9S35) so that it could hold a 1” diameter mirror anditself be held in a 1” optic mount. I also retrofitted this stage with the piezo stepper

115
motors that would later be used on the motorized beam block positioner.
Unfortunately, the dimensions and design of the vacuum chamber I had machined,
along with the available positioners, were not conducive to the straightforward transmission geometries that had been demonstrated at synchrotron facilities. The geometry
we used was quite simply a z-fold of the EUV beam using two mirrors. The first mirror
was flat and held the sample. This mirror was held by the two-axis micro translation
stage, which was in turn held by a motorized optic mount. So, the first mirror had
4 degrees of freedom; it could be tipped and tilted to steer the EUV light hitting thesample, and it could also be moved horizontally and vertically to align the sample. Although this arrangement functioned, it was quite unwieldy, and the lever arms involved
made the beam steering particularly sensitive to adjustment. The second mirror had a
1 m r.o.c. so that the diffraction data could be collected and imaged onto the camera.
Using this focusing optic was far from ideal. One of the advantages of lensless imaging
is that in principle it is aberration free, but by introducing an optic downstream fromthe diffraction of the sample we introduced aberration and distortion of the wavefront.
Furthermore, since the vacuum chamber had been designed for a different application,some of the clear aperture of the camera was blocked by the vacuum connections to thechamber, limiting the ultimate resolution of this geometry.
Despite these limitations, we decided to push forward to gain knowledge andinsight while planning for a more optimized microscope design. We wished to begin
by imaging an interesting test object that would provide a distinct diffraction pattern
and that would be readily recognizable if the reconstruction efforts were successful.
My advisor, Henry Kapteyn, suggested we use an electron microscope test grid. In
particular, he pointed out a pattern made by Quantifoil Micro Tools GmbH that had
the qualities which we were seeking. This pattern is shown in Figure 6.6, and consistsof a ∼10 nm thick perforated plastic film stretched over a fine copper mesh and coatedwith ∼10 nm of carbon. The copper grid bars are nominally 10 µm wide, and the clear

116

Figure 6.6: SEM micrograph of the QUANTIFOIL R MultiA carbon film, from [95]aperture of each grid cell is 50x50 µm. To increase the opacity of the grid we sputter
coated ∼ 100 nm of gold onto the surface. To mount this specimen to the mirror, I
found that a stereoscope was an invaluable tool. Due to its large working distance,
depth of field, and depth perception, a stereoscope allows one to view small objects andmanipulate them with success. Konrad Lehnert has been kind enough throughout thelensless imaging work to let me use his stereoscope and micromanipulators to prepare
samples. While viewing the mirror under the stereoscope, I placed the grid on the
mirror, and used a micromanipulator to hold the grid in place while I applied small
dots of glue to its edges. Next I placed a pinhole aperture over the grid and followed
the same procedure to affix it.
Some of the original data from the reflection geometry is shown in Figure 6.7.
Unfortunately, the original sample for this data was destroyed soon after the diffraction
data had been collected, which taught us the valuable lesson of recording an optical or
SEM picture of the sample before collecting any diffraction data. The diffraction data
shown in Figure 6.7 represents only about half of the camera pixels, as the full data
set includes a large blacked out region coming from the narrow vacuum connection to
the camera. The sample was apertured with a 150 µm pinhole, so that portions of

117

Figure 6.7: An portion of the high spatial frequency data taken for the reflection geometry on a linear scale (top), the reconstruction attempt showing some basic features
but poor resolution (bottom)

118
4 grid cells were visible. The diffraction data shows reasonably well defined speckles,
partially due to the amplification of signal by several identical grid cells. In addition, theEUV beam was many times larger than the sample aperture, so small changes in beam
pointing had little effect on the data. Basic features of the grid geometry are visible
in the reconstruction, but clearly the resolution and image quality are poor. Althoughpromising, the initial attempts showed relatively serious limitations in terms of efficientuse of flux, resolution, ease of alignment, and overall flexibility of the imaging system.
To fix some of these issues, we built a larger vacuum chamber specifically designed
to the task of lensless imaging in a transmission geometry as described above. Thisgeometry removes any optics between the path of the diffracted light and the detector.
Furthermore, since it allows tight focusing onto a small sample, the efficient use offlux has been markedly increased. The transmission imaging geometry allows for astraightforward and highly effective alignment technique. To begin, the driving laser is
set to run in a CW mode where no ultrafast seed pulse is introduced into the amplifying
crystal, so that the only light comes from amplified spontaneous emission (ASE). The
ASE provides rough alignment of the capillary and the steering mirrors within thechamber. When kept at low power, ASE is much safer to align with and will notdamage the optics or sample. Next, we pump the system down, and optimize the EUV
flux and mode with the pulsed laser at full power through a combination of adjusting
capillary alignment, coupling conditions, and gas pressure within the capillary. After
this point, care is taken not to disturb the pointing of the capillary, in particular thex-y adjustments at its output. The EUV mode is then centered on the camera using
the internal chamber steering optics, and a baseline is established for exposure times.
Next, we vent the system, put the sample in place, and set the driving laser to ASE
at a power such that the laser focus is just visible to the naked eye when reflecting off
the sample aperture. A bright light is placed behind the sample so the naked eye seesa small point of white light emanating from the sample aperture. Using an inspection

119
mirror, the overlap between the pump laser and the sample aperture can be viewed. The
EUV beam path has been set to be centered on the CCD camera, so the sample itself
is adjusted horizontally and vertically to match the illuminating light. Once we find areasonable overlap, a white card is placed between the sample and the camera. With theroom darkened, we view the light passing through the sample aperture with an IR viewer
and maximize this light’s intensity with fine movements of the sample translation stages.
Since the v-groove fixture establishes an extremely straight waveguide, the pointing of
the driving laser light coupled through the waveguide remains extraordinarily collinear
to the produced EUV light. Therefore, when the system is again pumped down, only
minor tweaks to the sample alignment are necessary to maximize the EUV diffraction.
Isolating the fundamental laser beam from air currents proved to be one of themost significant improvements for acquiring stable and crisp diffraction patterns. One of
the unique characteristics of an ultrashort laser pulse is its relatively broad bandwidth.
In terms of wavelength, our amplified ultrashort laser produces pulses with around
80 nm FWHM bandwidth. Considering classical dispersion, the various wavelengths
forming the ultrashort pulse will experience different indices of refraction through agiven material, and correspondingly, different phase delays. Stable material dispersions
can be compensated with the gratings forming the heart of the well-known chirpedpulse amplification scheme [4]. However, air currents produce randomly fluctuating gas
densities and so correspond to randomly fluctuating indices of refraction that lead tohighly undesirable degradation of pulse stability. For the long acquisition times necessary for high resolution imaging, even slowly varying air currents cause a disturbance.
In particular, air currents around the components of the amplifier at which the beamis spectrally dispersed lead to variations in the pulse’s pointing, spectrum, and beam
shape at the output of the amplifier [96]. These variations affect laser coupling into
the hollow waveguide as well as the extreme nonlinear interaction of the laser with thegas in the hollow waveguide. Ultimately then, air currents disturbing the fundamental

120
amplified laser vary the flux, pointing, and spectrum of the generated EUV light. Afterseeing substantial improvements in laser beam stability with the simple addition of a
cardboard tube around the portion of the beam being focused into the hollow waveguide, I decided the time had come to properly seal the laser system and path fromair currents. With the assistance of Nathan Lemke I proceeded to make a series ofacrylic enclosures for the optics of the front end pump lasers, as well as the steering
optics on the output of the amplifier leading to the hollow waveguide. All the enclosures
are assembled in an ingenious manner suggested by Etienne Gagnon, whereby the sidepieces are held together with heavy duty thick double-sided sticky tape. Enclosuresconstructed in this manner not only form a quality air seal, but also allow for quick
and easy reconfiguration. The enclosures are connected with a series of acrylic tubes
and rubber gaskets, and the removable tops sit on foam weather stripping to further
protect against stray air currents. As shown in Figure 6.8, the difference in diffraction
pattern quality is astounding between data taken before and after the implementationof the enclosures. The clear radial smearing in the first picture of Figure 6.8 suggeststhat the spectral bandwidth of the EUV light was drifting during the acquisition most
likely from variations in the fundamental laser pulse shape, power, or duration.

6.4.2

J-Slit Sample

Once we moved into the transmission geometry, we eventually decided to use aless ambiguous and more binary sample. To increase the ease of reconstruction, and geta feel for optimizing the transmission geometry, we wanted to try a rudimentary sample
that was strictly either opaque or transmissive. We happened to have a small aperturein the shape of a ”J” that had been laser machined into a piece of stainless steel shimstock. This ”J-slit” turned out to be an excellent diagnostic sample.
The J-slit is about 80 µm tall (D), and was placed at a distance z = 33 cm
from CCD. Referring to Equations 5.29 and 5.30, these values work out to a linear

121

Figure 6.8: Data taken before the laser was isolated from air currents (top) and after(bottom)

122
oversampling ratio of 9, and an image pixel size of 0.3 µm. Pictures of the stitcheddiffraction pattern, an optical microscope image of the J-slit, and the reconstructed
image are shown in Figure 6.9. Three different diffraction patterns were stitched together
in order to form the high dynamic range data shown. The lowest spatial frequencies
were captured with no beam block in about 1 minute. Next, a small beam block, ∼200 µm in diameter, was used to get slightly higher spatial frequencies in about 10
minutes. Finally, a larger beam block, ∼ 3 mm in diameter, was used to capture thehighest spatial frequencies in about 120 minutes. The main purpose of this sample
was to troubleshoot our system. Therefore, we were not concerned about optimizing
the ultimate resolution, but rather, with convincing ourselves that we could obtain areasonable reconstruction that gave a faithful reproduction of the sample. Clearly, thisreconstruction shows the J-slit, and even preserves the aspect ratio of its components.

6.4.3

200 nm Resolution with QUANTIFOIL R Sample

To push the resolution of the system we returned to the QUANTIFOIL R sample,
but this time with a 15 µm aperture. The sample was again gold coated, but with only
about 20-30 nm of Au which should have a transmission between 10% and 3% for ourilluminating wavelength. The sample to CCD distance was z = 10 cm, giving a linear
oversampling ratio of about 14.5. Figure 6.10 displays the sample, diffraction data,
reconstructed image, and a line-out of the reconstruction to demonstrate resolution.
The picture in Figure 6.10a is a SEM micrograph of the sample taken here with the
instrument in the JILA Keck Lab. This picture reveals that the sample has torn and apiece has folded back upon itself. I should note that the foil itself is sitting ∼ 50 µm
above the aperture (Richard Sandberg estimated this distance by tilting the sample in
the SEM and performing straightforward trigonometric calculations from the change in
relative location of features). Moreover, the subtle difference in the observed aperture
position between the SEM picture and the reconstructed image shown in Figure 6.10c is123

Figure 6.9: The diffraction data, J-slit sample, and reconstructed image

124

Figure 6.10: (a): SEM image of the sample over its 15 µm aperture , (b): the stitcheddiffraction data on a logarithmic scale, (c): the reconstructed image, (d): a line out of
the reconstructed image taken across the small blue bar in (c)125merely from parallax. Figure 6.10b shows the coherent EUV diffraction pattern, and thenon-centro-symmetry of the diffraction pattern indicates that the sample has absorption
and that the sample density is complex. Finally, the line scan of the reconstructed image
in 6.10d indicates that the current tabletop lensless microscope has a resolution of 214
nm.
By looking at the fidelity of the diffraction pattern speckles near the edge of
the CCD camera (high scattering angle) we estimated a lower bound on our spectral
bandwidth of ∼ 200, and this estimate corresponds well to preliminary measurements
taken using an EUV spectrometer. However, for the claimed resolution the spectral
bandwidth requirement would appear to me much higher, more on the order of 1000.
Fortunately, since the experimental linear oversampling ratio was much higher than 2,
our collaborators were able to bin the data together, and reduce the linear oversampling
ratio by a factor of 3. This reduction simultaneously eases the spectral bandwidth
requirement, and the claimed resolution appears reasonable.
Several small white blotches appear on the foil in the reconstructed image that
do not appear in the SEM image. Clearly, the feature in the lower left hand corner of
the aperture appears in both images, but I think it is interesting to speculate on thenature of the other discrepant features. The mysterious features could be artifacts of
the reconstruction. However, we note that where the torn piece has folded back on itself(near the blue line) the contrast in the reconstructed image consequently changes. It
would appear we did not coat enough gold to make the foil completely opaque, which is
not surprising considering that we merely timed our sputtering process and assumed acertain deposition rate. Moreover, quoting the manufacturer’s website,“In the final stageof the production process of QUANTIFOIL R holey film, carbon is evaporated onto aplastic holey film, and the plastic is dissolved. The grids are washed with chloroform,
ethylacetat and aceton. However, not all plastic can be removed. Therefore, the final
thickness of the foil is about 20 nm, although only 10 nm of carbon is evaporated onto

126
the plastic. [95]” So, the discrepant features may in fact be blobs of plastic not evenly
washed away in the manufacture of the foil, or perhaps specks of dust. Unfortunately,
the SEM does not have the depth of field to focus through the backside of the apertureand image the plastic side of the film. We also do not wish to detach this delicate
sample. However, we should look at some other grids to at least confirm this possibility.
One final quality of the reconstructed image worthy of note is its depth of field.
Considering that the foil sits many microns above the aperture, but the aperture outline
can be seen sharply in the reconstructed image, the foil and aperture together can bethought of as a ‘thick’ sample. However, in the SEM image the aperture is quite blurrywhen the foil is in focus.

6.5Light-Tightness with Thin Metal Filters

6.5.1

Thin Metal Filters for Separating Pump and EUV

Using some typical values, it becomes apparent that massive extinction ratios
are required to maintain good signal to noise ratios on the CCD detector. The typicalparameters for the ultrafast amplified laser driving the harmonic generation process are
a 4.25 W average power running at 3 kHz repetition rate with a center wavelength ofaround 780 nm. So, the laser produces about 1.4 mJ per pulse, equating to approximately 8.8x1015 photons per pulse at 1.59 eV per photon. Or equivalently, 2.6x1019
photons per second.
The highest spatial frequency diffraction information of the data sets will appear
at the highest diffraction angles and tends to have the weakest diffraction efficiency.
Therefore, long integration times are necessary to capture this information, typically
2 hours. During such an exposure, the dark count background will be on the orderof 100 counts, while the weakest diffraction speckles may only be 50 counts above thisbackground. Considering the quantum efficiency of the CCD chip, and the gain of

127
the readout electronics, an approximately one-to-one correspondence exists between
impinging visible photons and the associated counts registered by the CCD camera. 50
visible photons on each pixel will be enough to wash out the highest spatial frequency
information. A less than 50 count noise background from visible photons across the2048x2048 pixel camera in a 2 hour exposure requires an extinction ratio of 1 part in
1015 !
To achieve this feat, thin metal foils are used to filter the 780 nm driving laser
light from the generated EUV light. The idea of using thin metal foils as EUV filters
was first explored in the 1960’s as part of the space program [97]. The basic techniques
for making unbacked metal foils in the several hundred nanometer thickness range were
initially developed to be used for EUV spectroscopy experiments [98] [99]. Presently,
these foils are produced commercially and can be purchased in a variety of materials,
thicknesses, geometries, and backings. The filters in this work are unbacked 200 nm
thick aluminum mounted on a metal annulus. Thin foils of aluminum are particularly
well suited for removing amplified ultrafast laser light from EUV at 29 nm. At 780 nm,
Al has an extinction coefficient of 8.597 [42]. Accordingly, after going through 400 nm
of Al, the intensity of 780 nm light should drop by a factor 8.7x10−25 . Yet, 400 nm of
Al has a transmission of around 40% for 29 nm EUV light. In addition, aluminum isquite resistant to thermal damage when placed in an intense ultrashort laser beam due
to its high reflectance in the near IR and its high thermal conductivity. A single 400 nm
Aluminum foil appears to be an ideal solution to the filtering needs of the experiment.
Unfortunately, major issues complicate the use of thin metal foils for filtering. The
foils are extraordinarily delicate since they must be thin enough to efficiently transmit
the EUV. Foils can be made more mechanically stable by using a supporting grid structure, but this added support cannot be used when mode quality of the EUV beam must
be preserved. Pinholes pose another difficulty, as they are a nearly unavoidable propertyof thin metal films and compromise the light rejecting properties of the film [100]. More-128
over, the high absorption of EUV in atmosphere requires experiments to be conductedunder vacuum, so the films must survive the viscous forces applied during pumping andventing of the experiment.
Clearly, the proper choice of filter geometry and an inventive holding arrangement
becomes necessary with the enormous extinction ratios needed for successful lenslessimaging with HHG. The issues caused by pinholes can generally be avoided by using
two thin filters, rather than a single filter of the necessary thickness. Pinholes form
in random locations, so the chances of two pinholes in separate filters lining up aresmall. If the filters are placed sufficiently close to one another (usually several cm),
unwanted light transmitted through the first filter’s pinholes will not have a chance todiverge enough to be transmitted through the second filter. To ensure light-tightness,stray visible light cannot be allowed to find a reflected path around the fixture used
to hold the filter. Unfortunately, a perfectly light-tight fixture is seemingly also gastight, so during pump-out and venting large viscous flow forces threaten to destroy
the filter unless pressures on both its faces are well balanced. Two specialized fixtures
were developed in association with the imaging work presented here to overcome thechallenges intrinsic to holding EUV filters in a light-tight manner: a baffle, and an
in-vacuum filter wheel.

6.5.2

NW-40 Baffle Design

The need for an inventive ‘baffle’ design for holding EUV filters was first suggested
to me by Randy Bartels, and the design described here was prototyped by myself withinvaluable guidance from Todd Asnicar of the JILA instrument shop. By a baffle, I mean
an arrangement whereby gas but not light can efficiently bypass the filter. The generaldesign philosophy of this sort of baffle involves a convoluted, yet high conductance,
pumping path around the filters that guarantees that any stray light attempting tocircumvent the filters will undergo many diffuse reflections from an absorptive material.

129
The baffle holds two EUV filters and consists of an insert, a housing, and threaded
retaining rings for securely holding the filters. Figure 6.11 shows a 3-d rendering of
the baffle parts and their relative positions when assembled. The central portion of
the housing forms an NW-40 centering ring holder, so that the baffle may be inserted
between two standard NW-40 flanges. As can be seen in Figure 6.11, the insert andthe housing each have four large pumping holes. Each pair of 180◦ opposed pumping
holes in the insert is rotationally offset by 90◦ from the corresponding pair of holes in
the housing, and the two pairs of holes in each piece are rotationally offset by 90◦ fromeach other. The insert is cemented with epoxy into the housing, and the insert ends
form a seating surface for the filters. The filters are held in place by coarsely threaded
retaining rings which mate with the housing (the coarse threading of the retaining rings
avoids stripping of the threads).
All pieces of the baffle are made from aluminum and then anodized black toincrease absorption. Furthermore, the inner and outer surfaces of the insert, as well as
the inner surface of the housing are bead blasted before anodizing to increase scatteringof incident light. Stainless steel wool is placed between the insert and housing to further
increase diffuse scattering of any stray light while maintaining high conductance. For
light to go around the filters it must enter one of the housing pump holes, reflect aroundto the corresponding insert pump hole, reflect off the inner insert walls to the second setof insert pump holes, and then reflect around to the second set of housing pump holes
while avoiding scattering from the steel wool in the process.
Overall, this design effectively eliminates stray visible light while allowing filters
to survive viscous forces during pumping. However, several aspects of the design limit
its utility. The large surface area of anodized aluminum (which is quite porous andabsorbs water vapor when at atmosphere) increases pump out times as well as limiting
the ultimate pressure of the vacuum system in which the baffle is placed. Since the filters
are always in the pumping path of the vacuum, they remain susceptible to mechanical130

Figure 6.11: Schematic diagram of the NW-40 Baffle

131damage. In addition, if either filter becomes compromised, the entire apparatus must
be removed from the vacuum system.

6.5.3

Magnetically-Coupled Vacuum-Compatible Filter Wheel

The idea for the unique and novel filter wheel described here was initially suggested by my advisor Henry Kapteyn. The execution of the design was a joint effort
between myself and Tracy Keep of the JILA instrument shop. The original motivation for this apparatus stemmed from the desire to study and optimize HHG at various
wavelength regions within the EUV spectrum. To efficiently access these different wavelengths, several different filter materials must be used to optimize EUV transmission.
Furthermore, filters with sharp transmission ”edges” in the wavelength regime of interest
act as calibration standards for EUV spectroscopy. The ideal experimental setup allows
filter sets to be interchanged and calibration filters to be inserted without dismantling
the vacuum system.
Figure 6.12 shows a 3-d rendering of the filter wheel in both an exploded anda fully assembled configuration. The key design elements incorporated in the wheel
include a magnetic rotational coupling, a light tight seal, NW-40 flange connections, a“safe” position, an inspection window, and a low profile in the EUV beam propagation
direction. A brass ring seated on the exterior of the wheel housing holds six rareearth magnets that couple to the internal filter wheel, allowing the filter wheel to beturned without the use of a dynamic vacuum sealing o-ring. To further improve vacuum
performance, the interior of these wheels contains no anodized material, although theoutside is anodized for laser safety and durability concerns. Detents in the wheel housing
ensure proper positioning of the internal wheel, while the inspection window provides
a visual check of both wheel position and filter integrity. The wheels hold three filters
each and have a large high conductance aperture that serves as the “safe” position
during evacuation and venting of the vacuum system. When the experiment is left idle

132

Figure 6.12: 3-dimensional renderings of an exploded and assembled vacuum-compatible
filter wheel

133
under vacuum, the wheel is placed in this safe position to avoid filter damage due tounintentional venting (power outages, mechanical failures, operator error). The low
profile of the design also accommodates the close stacking of several wheels, providing
the ability to have many filter combinations available in a compact geometry. The
wheel itself bears against an internal o-ring that forms a light-tight seal when a filter
is rotated in place in front of it. During exposures times exceeding 2 hours using oneof these devices, no detectable visible light ‘bleed-through’ has occurred. I should notethat nearly every EUV experiment in the Kapteyn/Murnane group currently employs
one or more of these filter wheels.

6.6

Future Directions

6.6.1

Improving Resolution

Three approaches are possible for further improving the ultimate resolution of
the microscope. First, if a larger detector were used, higher spatial frequencies could be
captured while maintaining the oversampling ratio. Equivalently, a CCD camera with asmaller pixel size could be used, and the sample to CCD distance reduced. Second, theeffective spectral bandwidth of the source could be improved using either narrower band
mirrors or by narrowing the individual harmonic peaks. Third, shorter wavelength HHG
light could be used. Techniques for quasi phase matching, and selective enhancementof a single harmonic order, have recently been demonstrated that allow for the efficient
generation of shorter wavelength harmonics. These techniques also accomplish spectral
selectivity that reduces the need for filtering, and thus can increase the throughputof the condenser optics [33, 37]. These improvements are quite feasible, and should
increase the ultimate resolution to 10’s of nm. As the laser repetition rates are increased
from 3 kHz to tens of kHz, the EUV flux will be simultaneously increased, and image
acquisition time will be dramatically reduced from hours to minutes. Furthermore, as134computing power increases reconstruction times will subsequently shrink, making thelensless technique more accesible.
Given a bright coherent source in the ‘water window’ and appropriate multilayer
mirrors, the set up described above could immediately begin producing reconstructible
data. The multilayer mirror technology presently exists, with the possibility of narrowband mirrors with over 30% reflectivity [101]. Of course, smaller samples, with a more
tightly focused illuminating beam, would need to be used to maintain the far field condition and the proper oversampling ratio. The extraordinary time resolution affordedby high harmonics also leaves open the tantalizing possibility of tabletop time-resolved
imaging on femtosecond timescales. So, a tabletop EUV and even a soft x-ray lensless
microscope is likely to become increasingly practical and has wonderful possibilities foruse in metrology in support of next-generation lithographies, nanoscience, biologicalimaging and perhaps even dynamics studies.

Chapter 7

Conclusion

In conclusion, my thesis reports on two major achievements. The first, in the fieldof nonlinear optics, involved ground-breaking work that extended traditional nonlinear
optics techniques for quasi-phase matching to the extreme realm of HHG. This work notonly extended the range over which efficient HHG could take place, but demonstrated
the feasibility of modulating the driving light field as a QPM mechanism. Since mywork done with modulated waveguides, several other techniques based on similar ideas
have shown impressive enhancement factors [33, 46]. The second achievement is theconstruction and demonstration of the world’s first tabletop lensless diffractive microscope using coherent EUV light from high harmonics as an illumination source. This
microscope showed a resolution of ∼ 200 nm, and already competes with the resolution
available from zone-plate microscopy with HHG sources.
Additionally, this thesis describes several original scientific devices that are bothnovel and elegant. To the best of my knowledge, the custom glass-blowing apparatus
is a one-of-a-kind device. Furthermore, the vacuum-compatible filter wheel and thev-groove fixture are examples of enabling technologies, without which, the achievablequality of our lensless microscope would be severely compromised.
As amplified ultrafast lasers are pushed to higher repetition rates, and the efficiency of HHG sources continues to improve at shorter wavelengths, using techniques
directly influenced by my work, the practicality of a tabletop EUV lensless microscope136with sub-100 nm resolution for the biological and material sciences will increase dramatically. Such a microscope promises to have broad applications, and should fueladvancements in technology, fundamental understanding of the microscopic world, andperhaps even unlock some of the mysteries of how life develops.

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