Stuart Lee Jackson- Holographic Interferometry on the ZaP Flow Z-Pinch

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Holographic Interferometry

on the ZaP Flow Z-Pinch

Stuart Lee Jackson

A thesis submitted in partial fulfillment of

the requirements for the degree of

Master of Science in Aeronautics and Astronautics

University of Washington

2003

Program Authorized to Offer Degree: Aeronautics & Astronautics




Graduate School

This is to certify that I have examined this copy of a master’s thesis by

Stuart Lee Jackson

and have found that it is complete and satisfactory in all respects,

and that any and all revisions required by the final

examining committee have been made.

Committee Members:

Uri Shumlak

Brian A. Nelson

Date:



In presenting this thesis in partial fulfillment of the requirements for a Master’s

degree at the University of Washington, I agree that the Library shall make its copies

freely available for inspection. I further agree that extensive copying of this thesis is

allowable only for scholarly purposes, consistent with “fair use” as prescribed in the

U.S. Copyright Law. Any other reproduction for any purpose or by any means shall

not be allowed without my written permission.

Signature

Date




Abstract

Holographic Interferometry

on the ZaP Flow Z-Pinch

by Stuart Lee Jackson

Chair of Supervisory Committee:

Associate Professor Uri ShumlakDepartment of Aeronautics & Astronautics

The ZaP Flow Z-Pinch Experiment is a basic plasma physics experiment designed to

investigate the effects of a sheared axial velocity profile on the stability of a Z-pinch

plasma. A holographic interferometer is used to determine the density profile at the

midplane of the Z-pinch. Chord integrated density information is recorded during

a plasma pulse using the expanded beam of a pulsed ruby laser and holographic

techniques. The chord integrated measurement from this holographic interferogram

is inverted using an Abel inversion to determine the radial electron number density

profile of the plasma. Holographic interferograms are made at different times on

successive plasma pulses to study the evolution of the density profile with respect

to time. During the quiescent period of the Z-pinch, the density profiles show an

approximately radially symmetric Z-pinch with an electron number density of about

1 × 1017 cm−3 and a radius of 1 to 1.5 cm.



TABLE OF CONTENTS

List of Figures iii

List of Tables vii

Chapter 1: Introduction 1

Chapter 2: Overview of the ZaP Flow Z-Pinch Experiment 3

Chapter 3: Theory and Practice of Holographic Interferometry 8

3.1 Overview of Holographic Interferometry . . . . . . . . . . . . . . . . . 8

3.2 Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Holographic Interferometry . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4 Holographic Reconstruction . . . . . . . . . . . . . . . . . . . . . . . 13

Chapter 4: Pulsed Ruby Laser Operation 16

4.1 Characteristics and Setup . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2 Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Chapter 5: Holographic Methods on the ZaP Flow Z-Pinch 29

5.1 Recording the Interferogram . . . . . . . . . . . . . . . . . . . . . . . 29

5.2 Developing the Interferogram . . . . . . . . . . . . . . . . . . . . . . 33

5.3 Reconstructing the Interferogram . . . . . . . . . . . . . . . . . . . . 36

i



Chapter 6: Density Profile Determination 38

6.1 Fringe Shift Measurement . . . . . . . . . . . . . . . . . . . . . . . . 386.2 Chord-Integrated Electron Density Determination . . . . . . . . . . . 46

6.3 Inversion Method for Determining the Radial Density Profile . . . . . 46

Chapter 7: Error Analysis 57

7.1 Path Length Errors due to Refraction . . . . . . . . . . . . . . . . . . 57

7.2 Errors due to Incorrect Center Location . . . . . . . . . . . . . . . . 59

7.3 Random Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter 8: Results of Density Profile Investigation 65

8.1 Results Obtained with the Double-Pass Holographic Interferometer . 65

8.2 Results Obtained with the Single-Pass Holographic Interferometer . . 70

Chapter 9: Summary and Conclusions 82

Bibliography 85

ii



LIST OF FIGURES

1.1 m = 0 “sausage” and m = 1 “kink” instabilities in a Z-pinch. . . . . . 2

2.1 ZaP Flow Z-Pinch experimental apparatus. . . . . . . . . . . . . . . . 4

2.2 Formation of a ZaP Flow Z-Pinch. . . . . . . . . . . . . . . . . . . . . 5

2.3 Plasma current and normalized m=1 mode at the Z-pinch midplane,

with the quiescent period indicated (Pulse 204018). . . . . . . . . . . 6

2.4 Photos of Z-pinch emission showing quiescent period and growing in-

stability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.1 Interference of wavefronts of scene and reference beams forms a diffrac-

tion pattern at the holographic plate. . . . . . . . . . . . . . . . . . . 10

3.2 Holographic interferograms of a candle flame . . . . . . . . . . . . . . 12

3.3 Phase shift measurement based on fringe shift in a finite-fringe holo-

graphic interferogram . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.1 Laser rail assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2 Cooler and power supply . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 A fiber optic cable is mounted on the laser rail behind the rear reflector

and used to monitor the laser pulse. . . . . . . . . . . . . . . . . . . . 20

4.4 Electrical schematic of the photodiode circuit used to monitor the laser

pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.5 Output of monitor circuit for a representative laser pulse. . . . . . . . 21

iii



4.6 Electrical schematic of the remote control panel used to control the

pulsed ruby laser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.7 Electrical schematic of the receiver box used to trigger the laser head

power supply and the Pockels cell shutter electronics. . . . . . . . . . 23

4.8 Flashlamp and ruby rod. . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.9 Before it is triggered, the Q-switch prevents laser light from reaching

the rear reflector, spoiling the gain of the laser cavity. . . . . . . . . . 26

4.10 When triggered, the Q-switch allows light from the laser head to reach

the rear reflector, completing the laser cavity and resulting in the emis-sion of a high-intensity laser pulse. . . . . . . . . . . . . . . . . . . . 27

5.1 Optical arrangement of the double-pass holographic interferometer. . 30

5.2 ZaP Flow Z-Pinch experimental apparatus with an inner electrode

nosecone and a conical hole in the end wall. . . . . . . . . . . . . . . 32

5.3 Optical arrangement of the single-pass holographic interferometer. . . 33

5.4 Process used to develop holographic plates. . . . . . . . . . . . . . . . 34

5.5 Optical arrangement used to reconstruct and photograph the holo-

graphic interferograms. . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.1 Holographic interferograms with and without plasma present. . . . . 39

6.2 Conversion of the noisy periodic fringe pattern in a raw holographic

interferogram to a square fringe pattern (Pulse 21029011). . . . . . . 40

6.3 Locations of peaks and troughs are indicated with “x’s” along a row of

the interferogram, plotted as a dashed line. The same row of the raw

interferogram is plotted as a solid line. . . . . . . . . . . . . . . . . . 41

6.4 Bright and dark fringes plotted over the raw holographic interferogram. 43

iv



6.5 Fringe shift measurement. . . . . . . . . . . . . . . . . . . . . . . . . 45

6.6 Chord-integrated electron density (Pulse 31029011). . . . . . . . . . . 476.7 Shell model used to obtain the radial electron density profile from the

chord-integrated density using a discrete Abel inversion method. . . . 48

6.8 Graphical depiction of each coefficient Aki as half the normalized path

length through a cylindrical shell of plasma. . . . . . . . . . . . . . . 51

6.9 Radial electron density profile (Pulse 31029011). . . . . . . . . . . . . 53

6.10 Performance of the inversion method when used on six simulated den-

sity profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.1 Effect of incorrect center location on the performance of the inversion

method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7.2 Performance of the inversion method when used on the r squared sim-

ulated density profile. Error bars are included for every other point to

indicate the effect of random errors on the radial density profile. . . . 63

7.3 Radial electron density profile with error bars plotted for every other

point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8.1 Holographic interferograms made using the double-pass holographic

interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

8.2 Chord-integrated density determined from the fringe shift in the holo-

graphic interferograms of Fig. 8.1. . . . . . . . . . . . . . . . . . . . . 68

8.3 Radial electron density profiles resulting from inversion of the chord-

integrated electron density data of Fig. 8.2. . . . . . . . . . . . . . . . 69

8.4 Normalized m = 1 mode, plasma current, and holography laser monitor

associated with the holographic interferograms of Fig. 8.1. . . . . . . 71

v



8.5 Radial electron density profiles for hydrogen, 50% methane/50% hy-

drogen, and helium Z-pinches (Pulses 20910027, 21029011 and 30204007). 728.6 A holographic interferogram made using the single-pass holographic

interferometer (Pulse 30204010). . . . . . . . . . . . . . . . . . . . . . 73

8.7 The holographic interferogram of Fig. 8.6 with its bright fringes high-

lighted and numbered. . . . . . . . . . . . . . . . . . . . . . . . . . . 74

8.8 Chord-integrated electron density obtained from fringes 5 through 9 in

Fig. 8.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

8.9 Radial electron density profiles for helium Z-pinches during formationof the Z-pinch (Pulses 30205010, 30205014, and 30205018). . . . . . . 77

8.10 Normalized m = 1 mode, plasma current and chord-integrated electron

number density for the helium Z-pinches used to study the density

profile during formation (Fig. 8.9). The holography laser monitor shows

when each interferogram was made. . . . . . . . . . . . . . . . . . . . 78

8.11 Radial electron density profiles for helium Z-pinches during and after

the quiescent period (Pulses 30204007, 30204019, 30204021). . . . . . 808.12 Normalized m = 1 mode, plasma current and chord-integrated electron

number density for the helium Z-pinches used to study the density pro-

file during and after the quiescent period (Fig. 8.11). The holography

laser monitor shows when each interferogram was made. . . . . . . . 81

vi



LIST OF TABLES

2.1 ZaP Flow Z-Pinch typical operating parameters. . . . . . . . . . . . . 7

4.1 Pulsed ruby laser characteristics . . . . . . . . . . . . . . . . . . . . . 17

5.1 Holographic film and plate characteristics. . . . . . . . . . . . . . . . 34

vii



ACKNOWLEDGMENTS

The author wishes to express sincere appreciation to Ed Crawford, Uri Shumlak,

Brian Nelson, Ray Golingo and the “ZaP Team” for their guidance and assistance in

this endeavor and to his family and friends for their understanding and support.

viii



1

Chapter 1

INTRODUCTION

One of the earliest and most basic magnetic confinement concepts studied by fusion

scientists is the Z-pinch. A Z-pinch is a column of plasma with an axial current flowing

through it that creates an azimuthal magnetic field. The magnetic pressure from the

azimuthal field confines and compresses the column, creating a hot, dense plasma.

Typically, a Z-pinch is made by connecting a wire of tungsten or frozen deuterium

between two electrodes. When a large potential is applied across the electrodes, the

wire vaporizes and then ionizes, forming the Z-pinch [4].

Unfortunately, a static Z-pinch formed in this manner is susceptible to two types of

magnetohydrodynamic (MHD) instabilities. The m = 0 “sausage” mode occurs when

the plasma column begins to become thinner at any point along its length. Magnetic

pressure builds at this point, causing the column to become thinner still, until finally

it breaks and the plasma current is disrupted. The m = 1 “kink” instability occurs

when the plasma column begins to kink or bend. Magnetic pressure builds inside the

bend, pushing it farther out until the column is broken and the current is lost. Figure

1.1 shows the m = 0 “sausage” and m = 1 “kink” instabilities in a Z-pinch. These

MHD instabilities usually destroy the Z-pinch within tens of nanoseconds, limiting

its usefulness as a fusion reactor. Z-pinches are commonly used in experiments where

the production of a large amount of x-ray radiation is desired [4].

Several methods have been proposed to reduce the destructive effects of MHD



2

(a) (b)

Figure 1.1: m = 0 “sausage” and m = 1 “kink” instabilities in a Z-pinch. (a) m = 0“sausage” mode. (b) m = 1 “kink” mode.

instabilities on the Z-pinch. One such method is sheared-flow stabilization. Numerical

simulations show that a linear velocity shear of vz/a > 0.1kV A inhibits the growth

of the m = 1 mode [14]. Similar behavior has been shown for the m = 0 mode.

This result is supported by sheared-flow stabilization experiments conducted on the

ZaP Flow Z-Pinch. These experiments have produced Z-pinch plasmas that exhibit

characteristics of stability for 700 times the theoretical instability growth time [12, 13].

The purpose of this thesis is to investigate the density profile of a sheared-flow

Z-pinch plasma using holographic interferometry.



3

Chapter 2

OVERVIEW OF THE ZAP FLOW Z-PINCH

EXPERIMENT

The ZaP Flow Z-Pinch experiment at the University of Washington is a plasma

physics experiment designed to investigate the behavior of a sheared-flow Z-pinch.

The experimental apparatus, shown in Fig. 2.1, is composed of two coaxial, cylin-

drical electrodes enclosed in a vacuum chamber [3]. The outer electrode is a 150 cm

long, 20.5 cm inner diameter hollow copper cylinder with a copper end wall. The

inner electrode is a 100 cm long, 10 cm outer diameter hollow copper cylinder that

ends 50 cm before the outer electrode. The area between the electrodes before the

inner electrode ends is called the “acceleration region.” The space from the tip of the

inner electrode to the outer electrode end wall is referred to as the “assembly region.”

Hydrogen gas is puffed in the acceleration region between the two electrodes at a

point about halfway down the length of the inner electrode. A potential difference of

1 to 10 kV is applied across the two electrodes, causing the gas to ionize. Current is

carried between the two electrodes by the plasma, causing a magnetic field to form

behind the plasma. The radial current and the azimuthal magnetic field interact to

create a Lorentz force that accelerates the plasma axially down the electrodes towards

the assembly region.

One end of the forming Z-pinch attaches at the tip of the inner electrode as the

other end moves down the outer electrode and attaches to the end wall. The Lorentz

force is now directed radially inward towards the center of the plasma, confining and



4

Figure 2.1: ZaP Flow Z-Pinch experimental apparatus.

compressing the Z-pinch. This formation process results in a flowing Z-pinch with

axial velocities of 10 cm/µs [12, 13]. Figure 2.2 shows ZaP Flow Z-Pinch formation.

Formation of the flowing Z-pinch is followed by a 10-20 µs long quiescent period,

characterized by low magnetic mode activity. Figure 2.3 is a plot of the plasma current

and the normalized m=1 mode at the Z-pinch midplane. The quiescent period, defined

as where the magnitude of the normalized m=1 mode is less than 0.2, begins at 20

µs and ends at 40 µs for the plasma pulse shown. Photos of plasma emission also

indicate the existence of a quiescent Z-pinch plasma during this time. Figure 2.4 is a

series of images taken at the midplane of the Z-pinch at the times indicated during

the plasma pulse of Fig. 2.3. These images show a quiescent Z-pinch plasma and the

growth of an apparent instability towards the end of the quiescent period. Table 2.1

lists typical operating parameters for the ZaP Flow Z-Pinch.



5

(a) (b)

(c) (d)

Figure 2.2: Formation of a ZaP Flow Z-Pinch. (a) Hydrogen gas is puffed betweenthe inner and outer electrodes. (b) The gas is ionized and accelerated by Lorentzforces. (c) Current flows between the two electrodes as the Z-pinch begins to form.(d) Flowing Z-pinch is formed.



6

Figure 2.3: Plasma current and normalized m=1 mode at the Z-pinch midplane, withthe quiescent period indicated (Pulse 204018).

Figure 2.4: Photos of Z-pinch emission showing quiescent period and growing insta-bility. Photos were taken through a 500-600 nm bandpass filter (Pulse 204018).



7

Table 2.1: ZaP Flow Z-Pinch typical operating parameters.

parameter symbol value

ion velocity vion 10 cm/µs

Alfven speed vA 12 cm/µs

quiescent period length τ q 15 µs

total temperature T e + T i 150 eV

electron number density ne 1017 cm−3

edge magnetic field Bedge 1.8 T

Z-pinch radius a 1 cm

Z-pinch length L 50 cm

peak current I p max 250 kA

peak power P p max 1.5 GW

capacitor charge voltage V mb 9 kV



8

Chapter 3

THEORY AND PRACTICE OF HOLOGRAPHIC

INTERFEROMETRY

3.1 Overview of Holographic Interferometry

Holographic interferometry is a measurement technique that has been used by engi-

neers and scientists since the 1960’s. Applications include recording the vibration or

deformation of opaque objects or measuring the density of sufficiently dense, trans-

parent fluids or plasmas. When applied to plasmas, holographic interferometry works

based on the same principles as traditional Mach-Zehnder interferometry, but pro-

vides a two-dimensional “picture” of the chord-integrated plasma density, instead of

recording the density information along only one chord.

3.2 Holography

A hologram is a record of the interaction of two beams of light, in the form of a

microscopic pattern of interference fringes [11]. Usually one of the beams has bounced

off of or passed through some object of interest and the other is a reference beam used

to record the state of the first. The difference between a photograph and a hologram is

that a photograph contains only information about the intensity of the incident light,

while a hologram retains information about the incident wavefront’s intensity and

phase. When properly illuminated, a hologram can be used to reproduce the original

wavefront from the object. This reconstructed wavefront produces an image that



9

can be viewed with full parallax—when the viewpoint is changed, a different portion

of the reconstructed wavefront is intercepted by the viewer and the image appearsthree-dimensional. A photograph appears flat because the intercepted wavefront is

the same at different viewpoints. It is important that the recorded hologram and the

image not be contaminated by information from the light source. Laser light is most

often used because its coherence and monochromicity mean that it consists of planar

wavefronts that contain no information.

Leith-Upatnieks off-axis holography is generally used when making plasma density

measurements using holographic interferometry [9]. To make the holographic expo-sure, the laser beam is expanded to the desired diameter and split into two parts,

the reference beam and the object or scene beam. The reference beam does not pass

through the plasma; instead it bounces off several mirrors and then is incident upon

the holographic plate. The scene beam travels the same path length as the reference

beam, but passes through the plasma before hitting the plate.

Interaction between the laser light and the plasma electrons causes a phase shift in

the scene beam with respect to the reference beam that is proportional to the integralof the refractive index along the path of the laser light:

∆Φ =2π

λ

(n − n0) dl, (3.1)

where ∆Φ is the phase shift in radians, λ = is the wavelength of the laser light, n is

the refractive index of the plasma, and n0 is the refractive index of the background

medium [16]. The background medium is usually air or vacuum, so that n0 is ap-

proximately equal to one. For a plasma where the magnetic field is negligible, the

refractive index is related to the electron number density by

n =

1− ne

nc

1/2

, (3.2)

where ne is the plasma electron number density and nc is the plasma cutoff density



10

Scene Beam

R e f e r e nc e Be am

Holographic Plate

BrightFringes

DarkFringes

Figure 3.1: Interference of wavefronts of scene and reference beams forms a diffractionpattern at the holographic plate.

above which the laser light will not propagate. Equations 3.1 and 3.2 are combined

to yield the phase shift in terms of the integral of the electron number density along

the path of the laser light,

∆Φ = −2π

λ

1 −ne

nc1/2

− 1

dl. (3.3)

This phase difference manifests itself as a microscopic diffraction pattern on the

holographic plate where the two beams come together at an angle. This diffraction

pattern consists of a series of bright and dark fringes produced by constructive and

destructive interference between the wavefronts, as shown in Fig. 3.1. The width of

the fringes is affected by the phase shift in the scene beam. In this manner, density

information is recorded on the holographic plate in two dimensions. After developing,

the hologram can be illuminated again by the reference beam to reconstruct the

wavefront of the scene beam that passed through the plasma [16].

In reality, a single holographic exposure records phase differences due to windows

and path length differences along with the phase shift due to the plasma density,



11

making the density information difficult or impossible to recover. The techniques of

holographic interferometry are used to eliminate these unwanted effects and isolatethe phase shift due to the plasma density.

3.3 Holographic Interferometry

In holographic interferometry, two holographic exposures are made. The first is made

before the plasma pulse, with no plasma present in the vacuum chamber, to serve as

a baseline for the second. The second exposure is made during the plasma pulse, with

plasma present in the chamber. Both exposures are made on the same holographic

plate. It is the interference pattern due to the reconstructed wavefronts of these two

holograms that forms the holographic interferogram.

Density information could be recorded using only the second exposure, but the

system used has two advantages over a single exposure. The diffraction pattern made

during the first exposure takes into account phase shifts due to imperfect windows,

path length differences, and other experimental irregularities. The second exposure

has these same irregularities, plus the phase shift due to the plasma, so that the only

difference recorded between the two exposures is due to the presence of the plasma

[6]. The path length difference must still be less than the coherence length of the

laser to reliably record a fringe pattern.

The second advantage is the introduction of reference fringes. Figure 3.2 shows

a photograph of a candle flame and a series of interferograms of that candle flame.

Without reference fringes, the interferogram is a series of horizontal fringes of varying

widths, as shown in Fig. 3.2(b). The spacing between successive fringes indicates that

there is a difference in refractive index between the two locations, but not the sign

of the refractive index gradient. It is impossible to determine from the interferogram

alone whether the refractive index increases or decreases between adjacent fringes



12

(a) (b)

(c) (d)

Figure 3.2: Holographic interferograms of a candle flame. (a) Photograph of thecandle flame used to make the holographic interferograms in Figs. (b)–(d). (b) Infinitefringe holographic interferogram. (c) Finite fringe holographic interferogram. Axis of mirror tilt is vertical. (d) Finite fringe holographic interferogram. Axis of mirror tiltis horizontal.



13

[16]. This type of interferogram is called an infinite-fringe interferogram because the

width of the fringes depends on the refractive index gradient.In a finite-fringe interferogram, fixed-width reference fringes are introduced to

eliminate the sign ambiguity. This is accomplished by tilting a mirror in the reference

beam slightly between exposures, producing vertical or horizontal wedge fringes, as

shown in Fig. 3.2(c) and Fig. 3.2(d). The displacement of the wedge from the straight

part of each fringe indicates the phase shift of the laser light due to the presence of

the candle flame. The width of the fringes can be adjusted as desired by increasing

or decreasing the mirror tilt, and the axis of the tilt should be parallel to the densitygradient of interest [6].

In the case of a plasma interferogram, the phase shift is due to the electrons in the

plasma. As shown in Fig. 3.3, a displacement of half a fringe width corresponds to a

phase shift of π, while a whole fringe width corresponds a phase shift of 2π. “Fringe

width” refers to the distance between consecutive bright or dark fringes. The phase

shift is often expressed in terms of the fringe order, the displacement of the fringe

normalized by the fringe width. The fringe order, f , is related to the phase shift by

f =∆Φ

2π, (3.4)

so a fringe order of one corresponds to displacement of one fringe width and a phase

shift of 2π.

3.4 Holographic Reconstruction

After the holographic interferogram is exposed and developed, it can be illuminated

again by the reference beam in order to reconstruct the original scene and reference

wavefronts. In practice, a HeNe laser is usually used to mimic the original reference

beam for convenience. The reconstructed scene beam wavefronts of the two holo-

graphic exposures produce an interference pattern that can be recorded on ordinary



14

Figure 3.3: Phase shift measurement based on fringe shift in a finite-fringe holographicinterferogram



15

photographic film. As described in Sec. 3.3, the phase shift is determined by measur-

ing the shift of the fringes in the interference pattern. The relationship in Eq. 3.3 isthen used to find the chord-integrated electron density based on this phase shift.



16

Chapter 4

PULSED RUBY LASER OPERATION

4.1 Characteristics and Setup

The laser used to make the interferograms is a pulsed ruby laser with a pulse width

of approximately 50 ns and a wavelength of 6943 A. Characteristics of the ruby laser

are listed in Table 4.1.

Figure 4.1 shows components of the laser rail assembly and Fig. 4.2 shows the

power supply. A fiber optic cable is mounted as shown in Fig. 4.3 on the laser rail

behind the rear reflector and linked to a photodiode circuit to monitor the laser pulse.

With the double-pass system described in Sec. 5.1.1, the fiber optic cable is mounted

in the same manner behind the frosted back of the first mirror in the reference beam.

Figure 4.4 is a schematic of the photodiode circuit used to monitor the laser pulse.

Figure 4.5 shows the output of the monitor circuit for a representative laser pulse,

showing a full width at half max (FWHM) of roughly 50 ns. The response time of

the detector is unknown, and it is possible that the laser pulse width is less than 50

ns.

4.2 Operation

The laser is controlled by the operator outside the laboratory using a remote control

panel. A schematic is shown in Fig. 4.6. When the charge button is pressed, it lights

up and the 5 kV laser head power supply is charged to the voltage set by the dial on



17

Table 4.1: Pulsed ruby laser characteristics

characteristic type/value

laser head Korad K-1 Ruby Laser Head

power supply Korad K-1 Power Supply (5 kV)

cooler Korad KWC Laser Cooler

Q-switch Korad K-QS2 Pockels Cell

Q-Switch Assembly

shutter Korad K-QS2 Pockels Cell

Shutter Electronics

wavelength 6943 A

specified pulse width < 15 ns

operating pulse width 50 ns

specified pulse energy 1.1 J

operating pulse energy 600 mJ

polarization horizontal



18

Q-switchlaser pointer

for alignment

rear reflector iris

laser head

etalon

beam expander

Figure 4.1: Laser rail assembly

the front of the power supply. The voltage used for most holograms is 3.6 kV, slightly

above the measured lasing threshold of 3.5 kV. It should be noted that the voltage

indicator gauge on the front of the laser power supply is not correctly calibrated, so

that it reads 4.2 kV when the power supply voltage is actually 3.67 kV, as measured

by a multimeter. When the power supply reaches the set voltage, the charge light

turns off and the fire button lights. The laser can be fired manually using the fire

button if the “fire” cable is connected from the back of the remote control panel

to the back of the power supply and the delay dial on the front of the laser is set

to “internal.” However, the laser is usually fired through the ZaP data acquisition

system.

Two channels of a Data Design Corporation DG11 Digital Delay Generator module

are set 920.2 µs apart, with the second channel set 0.88 µs before the desired laser firing

time. When the operator triggers the experiment, the first DG11 channel triggers a



19

Figure 4.2: Cooler and power supply



20

Q-switchrear reflector

and mount

fiber optic cable

and mountindex card

(protects alignment

laser pointer)

alignment laser

pointer mount

Figure 4.3: A fiber optic cable is mounted on the laser rail behind the rear reflectorand used to monitor the laser pulse.



21

Figure 4.4: Electrical schematic of the photodiode circuit used to monitor the laserpulse.

Figure 4.5: Output of monitor circuit for a representative laser pulse.



22

F i g ur e 4 . 6 : E l e c t r i c a l s c h e m

a t i c of t h e r e m o t e c o n t r o l p a n

e l u s e d t o c o n t r o l t h e p u l s e d r u b y l a s e r .



23

Figure 4.7: Electrical schematic of the receiver box used to trigger the laser headpower supply and the Pockels cell shutter electronics.

light emitting diode (LED) that sends a light pulse through a fiber optic cable to a

receiver box in the lab, visible in Fig. 4.2. A photodiode in the receiver box detects

the light pulse, and the receiver box sends a 45 V trigger signal to the laser power

supply. A schematic of the receiver box is shown in Fig. 4.7. The Pockels cell shutter

electronics module is triggered by the second DG11 channel and a second receiver box

in a similar manner. The 920.2 µs delay time was set based on the internal delay time

used during initial testing, although other delay times were not investigated. The 0.88

µs delay time between the second trigger and the actual laser pulse was set using an

ICCD spectrometer. To determine this delay time, the spectrometer was set to record

data at the desired laser firing time over a wavelength range that included the ruby

laser wavelength. The trigger time of the second DG11 module was stepped earlier

in small increments from the desired laser firing time until laser light was observed in

the ICCD spectrum.

Inside the housing of the K-1 laser head shown in Fig. 4.1 is a 3/8 in diameter

synthetic ruby rod, surrounded by a spiral flashlamp. The ruby rod and flashlamp



24

Figure 4.8: Flashlamp and ruby rod.

are shown in Fig. 4.8. When the laser power supply is triggered, the energy in its

capacitor bank is discharged through the flashlamp by means of a transformer. The

xenon gas in the flashlamp becomes hot and begins to radiate. Green and blue light

from the flashlamp is absorbed by the ruby rod, exciting the chromium ions in the

ruby and causing a population inversion. This population inversion causes the rod to

have a positive gain at the ruby laser wavelength, 6943 A. The rear reflector, laser

head, and etalon shown in Fig. 4.1 form the laser cavity. Light from the laser head

bounces between the rear reflector and the etalon, passing multiple times through the

ruby rod. Because the total gain from multiple passes through the laser rod is greater

than one, the amount of power in the laser cavity grows until the lasing threshold is



25

reached and a laser pulse is emitted.

A Q-switch is included in the laser cavity to spoil the cavity gain and prevent

the lasing threshold from being reached. Until it is triggered, it prevents light from

reaching the rear reflector and being reflected back to the laser rod. This allows energy

to accumulate in the ruby rod over time. When the Q-switch is finally triggered,

completing the laser cavity, the cavity gain is much higher than it otherwise would

be because of the increased energy in the laser rod. A high-intensity laser pulse is

emitted, draining the energy in the laser rod. Without the Q-switch, a series of low-

intensity pulses would be emitted as the cavity repeatedly reached the lasing threshold

[7].

Figure 4.9 shows the Q-switch assembly with its cover removed, mounted on the

laser rail between the laser head and the rear reflector. It consists of a Pockels

crystal and a polarizer stack. Before the Q-switch is triggered, horizontally polarized

light from the ruby rod is incident upon the Pockels crystal and passes through

with its polarization direction unchanged. The polarizer stack, which passes only

vertically polarized light, prevents the horizontally polarized light from reaching the

rear reflector.

When the Q-switch is triggered, a bias voltage is applied to the Pockels crystal

by the Pockels cell shutter electronics. The phase of the electric field vector of the

horizontally polarized light is retarded 90 deg by the energized crystal, changing it

to vertically polarized light, as shown in Fig. 4.10. The polarizer stack passes the

vertically polarized light, allowing it to reflect off of the rear reflector and pass back

through the polarizer. The polarization direction of the light is once again transformed

as it passess through the Pockels crystal, and the resulting horizontally polarized light

returns to the ruby rod [8].

In addition to the Q-switch, an adjustable iris is also included in the laser cavity,



26

F i g ur e 4 . 9 : B e f or e i t i s t r i g g e r e d , t

h e Q- s w i t c h pr e v e n t s l a s e r l i g h t f r o m

r e a c h i n g t h e r e ar r e fl e

c t or , s p o i l i n g t h e

g a i n

of t h e l a s e r c a v i t y.



27

F i g u r e 4 . 1

0 : W h

e n t r i g g e r e d ,

t h e Q - s w i t c h a l l o

w s l i g h t f r o m t h e l a s e r h e a d t o

r e a c h t h e r e a r r e fl e c t o r , c o m p

l e t i n g

t h e l a s e r c a v i t y a n d r e s u l t i n g i n t h e e m i s s i o n o f a h i g h - i n t e n s i t y l a s e r p u l s e .



28

between the Q-switch and the laser head. The iris is used to clean up the edges of

the laser beam and to eliminate transverse modes. It can also be used to control thebeam diameter. Use of the iris reduces the output energy of the laser from 720 mJ

to 460 mJ when the laser power supply is charged to 3.5 kV. For most holograms,

an iris diameter of 4 mm and a power supply charge of 3.6 kV was used, yielding an

output energy of 600 mJ.

The etalon serves as one end of the laser cavity, reflecting light back to the laser

head while improving the monochromicity of the laser. Light emitted from the ruby

rod has a certain width in frequency space about the frequency associated with theideal wavelength. The laser cavity, however, acts to select certain resonant frequencies

from among those emitted by the ruby rod. These resonant frequencies have an

integral number of half-wavelengths across the laser cavity. The etalon, a small piece

of glass with two parallel faces, resonates at frequencies more widely spaced than

those of the laser cavity, and acts to further reduce the frequency spread of the laser.

The combined effects of the ruby rod, laser cavity, and etalon produce laser light that

is extremely monochromatic, and the laser pulse is said to be single-mode [10].The final component on the output end of the laser rail is a beam expander, used

to expand the diameter of the beam from 4 mm to roughly 5 cm. On the far end of

the rail, a laser pointer is mounted behind the rear reflector and the fiber optic cable

described in Sec. 4.1. This laser pointer, with an output less than 5 mW, is used for

beam alignment. The fiber optic cable and the index card must be removed during

alignment. During ruby laser operation, however, either the fiber or the card must

be placed in front of the laser pointer, as shown in Fig. 4.3, to prevent it from being

damaged by the ruby beam.



29

Chapter 5

HOLOGRAPHIC METHODS ON THE ZAP FLOW

Z-PINCH

5.1 Recording the Interferogram

Holographic interferograms are made using two different experimental arrangements.

In both cases, the vacuum tank window that serves as the holography port is located

at the center of the 50 cm long assembly region. It is aligned with a 5 cm diameter

hole in the outer electrode that allows the ruby laser pulse to pass through the plasma

and out an identical hole and port on the opposite side of the chamber. The location

of the port allows the experimenter to make density measurements at the midplane

of the Z-pinch.

5.1.1 Double-Pass Recording

A double-pass holographic interferometer is used to measure the density of plasmas

made with the experimental configuration shown in Fig. 2.1. The optical arrangement

is shown in Fig. 5.1.

The laser pulse is expanded to a 5 cm diameter beam that is split by a 50%

beamsplitter into the scene and reference beams. The scene beam passes through

the plasma, hits a mirror on the opposite side of the vacuum tank, and then passes

through the plasma a second time before hitting the holographic film or plate. This

double-pass system doubles the laser’s path length through the plasma, resulting in



30

ZaP Vacuum Vessel

Holographic Plate

Ruby Laser

Z-pinch Plasma

Scene BeamReference Beam

6943A

NotchFilter

Tilting

Mirror

Figure 5.1: Optical arrangement of the double-pass holographic interferometer.



31

a larger and more easily-measured fringe shift.

Several mirrors are used along the path of the reference beam to increase its path

length to equal that of the scene beam. The last mirror in the reference path is the

tilting mirror used to make the vertical wedge reference fringes in the interferogram.

Its axis of tilt is out of the page in Fig. 5.1. The mirror is tilted by passing a current

through a wire attached to one side of the back of its mount. The other side of the

back of the mount is attached at two points, defining the axis of tilt. When the wire

is heated by the current, it expands, slightly tilting the mirror. The current can be

adjusted to control the amount of tilt and, as a result, the width of the fringes in the

interferogram. In practice, a current of about 0.32 A was used.

It is important to mention that after its second pass through the vacuum tank, the

scene beam is split a second time by the beamsplitter. Half of the beam’s intensity is

transmitted by the beamsplitter and continues on its way to the holographic film, and

the other half is reflected towards the laser. To prevent the beam from reflecting back

into the laser cavity, the mirror on the opposite side of the vacuum tank is aligned

so that the laser beam hits the beamsplitter in a slightly different place on its second

pass.

This intentional misalignment introduces only a small difference in the path of the

scene beam on its first and second passes through the plasma. It introduces a more

noticeable difference in the paths of the beam through the beamsplitter, causing part

of the beam to be cut off. The area of interferograms made with the misaligned scene

beam is therefore smaller than it would be if the beam could pass through the center

of the beamsplitter on both passes. For this reason, an effort was made to minimize

the misalignment of the beam. A better solution would have been to use a larger

beamsplitter, because on at least one occasion the scene beam was not intentionally

misaligned enough, and the ruby laser rod was damaged.



32

Figure 5.2: ZaP Flow Z-Pinch experimental apparatus with an inner electrodenosecone and a conical hole in the end wall.

5.1.2 Single-Pass Recording

A single-pass holographic interferometer is used with the experimental configuration

shown in Fig. 5.2. A nosecone is added at the end of the inner electrode and the solid

end wall of Fig. 2.1 is replaced with an end wall with a conical hole in its center.

Nine gas puff valves are used to inject gas into the area between the electrodes,

instead of three. The single-pass optical arrangement is shown in Fig. 5.3. After the

beam expander, the laser beam is shot upward and then toward the vacuum tank

before it hits the beamsplitter. The scene beam travels down to a mirror next to the

holography port and then passes through the machine. On the far side of the vacuum

tank, a 305 mm focal length lens is used to image the plasma onto the holographic

film or plate, with a 1:2 image-to-object magnification ratio. The reference beam

passes over the vacuum tank and two mirrors are used to direct it through a 305 mm

focal length lens and onto the holographic film or plate. The tilting mirror is tilted

about a vertical axis in Fig. 5.3. The purpose of the lenses is two-fold. First, in the

scene beam, imaging the plasma onto the film or plate improves the quality of the



33

Figure 5.3: Optical arrangement of the single-pass holographic interferometer.

interferogram and reduces the effect of dirty or imperfect vacuum tank windows [6].

Second, in both beams, the 1:2 magnification decreases the size of the interferogram.

This demagnification allows the interferogram to fit on the narrower holographic film

and increases the power per unit area that hits the film, which is less sensitive than

the plates to the ruby laser wavelength. A 35 mm camera body with a shutter is usedwith the 35 mm holographic film, simplifying the data collection process and allowing

interferograms to be made at a faster rate than with plates.

5.2 Developing the Interferogram

After the interferogram is recorded, the holographic plate or film is developed using

darkroom techniques. AGFA-Gavaert 8e75 or 10e75 plates are used, the specifications

of which are shown in Table 5.1 [6, 15]. They are made of clear glass covered on one

side by a thin emulsion of minute silver halide crystals in gelatin. Tiny patches of silver

form on the crystals when they are exposed to light. The steps in the development

process are shown in Fig. 5.4. The plates are soaked for five minutes in the dark



34

Table 5.1: Holographic film and plate characteristics.

manufacturer AGFA-Gevaert Slavich

type Holotest 10E75 & 8E75 PFG-01

exposure 40–300× 10−7 J/cm2 10−2 J/cm2

(at λ = 6943 A)

resolution 2800–3000 lines/mm 3000 lines/mm

plate/film 4x5 in plate (other sizes

exist)

specially-ordered 35 mm

film

notes production ceased in late

1990’s

plates and other sizes

available

Developing Holographic PlatesIn the Dark:

Lights on, if desired:

4. Rinse five minutes with running water in a tray in the sink.5. Bleach with Chromium Intensifier until exposed portion of plate is almost transparent6. Rinse 10 minutes with running water in a tray in the sink7. Pour used bleach into a bottle and label for disposal. DO NOT DUMP IN SINK.

15 minutes in

Developer D-19

Solution

210 seconds in KodakIndicator Stop Bath

Solution (X-Ray

Indicator)

35 minutes inKodak Rapid

Fixer Solution

Figure 5.4: Process used to develop holographic plates. The process used for film isidentical, except the film is soaked in the developer for ten minutes.



35

in Kodak developer D-19 to complete the conversion of the exposed silver halide

crystals to metallic silver. Ten seconds in an x-ray indicator stop bath solution endsthis conversion process. Five minutes in a rapid fixer solution removes the unexposed

silver halide crystals [2]. After fixing, the lights are turned on and the plates are rinsed

in running water for five minutes. This process produces an amplitude interferogram

that can be reconstructed and viewed as described in Sec. 5.3. The intensity of

ruby laser light absorbed by the plate is not important to the density analysis, so

the diffraction efficiency of the plate can be improved by bleaching it, allowing more

light to pass through during reconstruction. A 10:1 or stronger dilution of potassiumdichromate with water is used to bleach the plate, transforming the metallic silver

in the plate to transparent silver salt. This procedure transforms the amplitude

interferogram into a phase interferogram. The phase information is preserved in the

silver salt, whose index of refraction differs from that of the gelatin covering the rest

of the plate [16]. The plate is rinsed for ten minutes in running water and allowed to

air dry. The bleaching process can be repeated as needed as the plates darken over

time due to exposure to light.Specially-ordered 35 mm PFG-01 holographic film is also used, since production

of AGFA plates ceased in the late 1990’s. Specifications are shown in Table 5.1.

Processing of the film is the same, except that it requires ten minutes in the developer

solution instead of five. A film can is used during development so that all of the

processing steps can be done under normal lighting conditions. Despite its lower

sensitivity, film is more convenient and cheaper than plates.

Whether a plate or film is used, the holographic recording medium must be able

to resolve the spatial frequency of the fringe pattern, given by

f f r =sin θ

λ, (5.1)

where θ is the angle between the scene and reference beams and λ is the wavelength



36

of the laser light [16]. For both the double-pass and the single-pass holographic

interferometer, θ is 15 deg and λ is 6.943 × 10−4

mm, so the fringe pattern in eachhologram has a spatial frequency of 372 lines/mm. This provides a course resolution

limit. From Table 5.1, the resolution of the plates and film used is much finer than

the spatial frequency of the fringe pattern, a requirement for high-quality holograms

and interferograms.

5.3 Reconstructing the Interferogram

The developed and bleached holographic plate looks very much like a clear pane of

glass with a slightly darkened spot where it was struck by the laser pulse. To view

and photograph the interferogram, the plate must be illuminated by the same type

of light used during exposure (see Sec. 3.4)—parallel wavefront, 6943 A laser light.

For convenience, a helium-neon (HeNe) laser with a wavelength close to the ruby

laser wavelength is used to mimic the original reference beam. When this beam

strikes the holographic plate, it is diffracted by the two holograms on the plate to

create a wavefront that reflects the phase differences between the two exposures, one

with plasma and one without. The interference pattern created by this wavefront is

recorded on photographic film.

The optical arrangement used to reconstruct and photograph the holographic in-

terferograms is shown in Fig. 5.5. A microscope objective and a lens function as

a beam expander to expand the reconstruction beam to the roughly 2.5 cm diam-

eter required to illuminate the interferogram. An aperture is included in the beam

expander to select the cleanest part of the beam. The expanded beam then passes

through a 305 mm focal length lens, identical to the lens used in making the inter-

ferogram. This lens is removed when reconstructing interferograms made with the

double-pass system of Sec. 5.1.1, which does not include the lens. The reconstruction



37

Figure 5.5: Optical arrangement used to reconstruct and photograph the holographicinterferograms.

beam hits the holographic plate, or holographic film mounted on a glass plate, at

approximately the same angle as the original reference beam, roughly 15 deg. This

angle is chosen so that the reconstructed object beam propagates perpendicular to the

plate. It is slightly different than the angle of the original reference beam due to the

difference in wavelength. The reconstruction beam is diffracted by the holographic

plate to reconstruct the recorded fringe pattern, and this reconstructed object beampasses through a series of lenses and into the camera body. Inside the camera body,

the reconstructed object beam passes through a shutter, which is used to adjust the

exposure time of the photographic film according to the diffraction efficiency of the

interferogram. An adjustable iris is used to clean up the interferogram by reducing

stray light and removing non-parallel rays from the beam. The reconstructed object

beam is then incident upon the film at the back of the camera. The resulting fringe

pattern is recorded on the black and white film, either Polariod Polapan 667, ISO3200 or Polaroid Polapan 665, positive/negative, ISO 80 film.



38

Chapter 6

DENSITY PROFILE DETERMINATION

6.1 Fringe Shift Measurement

Once the holographic interferogram has been made, developed, reconstructed, and

photographed, the photograph of the interferogram is scanned and stored in a grayscale

Tagged Image Format (TIF) file. A resolution of 300 pixels per inch is typically used,

but a higher resolution could be used to reduce pixel effects. Figure 6.1(a) is a photo-

graph of a finite fringe holographic interferogram (see Sec. 3.3) where no plasma was

present during either of the two holographic exposures. The dark horizontal lines in

the interferogram are from reference wires attached to the port on the far side of the

vacuum tank from the laser. They are half a centimeter above and below the center

of the vacuum tank, respectively, and serve as a reference for determining the size

and location of the Z-pinch. The straight reference fringes are a product of tilting the

mirror slightly between exposures, and indicate zero plasma density.

Figure 6.1(b) is a photograph of a holographic interferogram where the second

holographic exposure was made during a ZaP plasma pulse. The presence of the

plasma has caused a measurable shift of the fringes in the interferogram, resulting in

the characteristic wedge shape. The amount by which the curved part of each fringe

is shifted with respect to the straight part is an indication of the relative phase shift

of the ruby laser light due to the plasma density. Several Interactive Data Language

(IDL) programs are used to measure the fringe shift in the interferogram and, from

that, to determine the phase shift.



39

(a) (b)

Figure 6.1: Holographic interferograms with and without plasma present. (a) No

plasma present. (b) Z-pinch plasma (Pulse 21029011).

Image editing software such as Microsoft Photo Editor or Adobe Photoshop is used

to edit the scanned interferogram. The image is cropped, and all white space outside

the area of the interferogram is filled or removed. If necessary, the interferogram is

rotated so that the reference wires are horizontal.

The computer program analyze interferogram.pro, written in the IDL program-

ming language, is used to isolate and store the fringes in the hologram as IDL vari-

ables. Other relevant quantities, such as the locations of the reference wires and the

number of bright and dark fringes in the hologram, are also stored.

When analyze interferogram.pro is compiled in an IDL session and executed, the

user is prompted to select the TIF image file of the interferogram. When selected, the

image file is opened, its contents are read into an array, and this raw hologram array

is cropped in IDL to a rectangular box that encloses the fringe pattern. The raw

hologram array is smoothed twice, using three and five-pixel-square windows. The

noisy periodic fringe pattern is converted to a square fringe pattern by maximizing

or minimizing the brightness of points in the fringes, as shown in Figure 6.2. Points

whose brightness is at or above a certain fraction of the brightness of the brightest



40

(a) (b)

(c) (d)

Figure 6.2: Conversion of the noisy periodic fringe pattern in a raw holographicinterferogram to a square fringe pattern (Pulse 21029011). (a) Raw holographic in-terferogram. (b) Holographic interferogram converted to square fringe pattern. (c) Arow of the raw holographic interferogram, showing the noisy periodic fringe pattern.

(d) The same row of the converted holographic interferogram, showing the squarefringe pattern.



41

Figure 6.3: Locations of peaks and troughs are indicated with “x’s” along a row of the interferogram, plotted as a dashed line. The same row of the raw interferogramis plotted as a solid line.

point are maximized, and the rest are minimized. This fraction is typically greater

(0.95) in the center than in the rest of the interferogram (0.75) because the diffraction

efficiency of the interferogram and brightness of the reconstruction beam are greatest

there. This conversion process reduces the effects of high-frequency noise and irreg-

ularities in the interferogram, and allows a simple peak-finding algorithm to be used

in the row-by-row analysis to follow. The method of converting the noisy periodic

fringe pattern to a square fringe pattern could be improved by using a variable bright-

ness fraction. A brightness fraction that adjusts to maintain a constant fringe width

along the length of a fringe might further reduce the effects of irregularities in the

interferogram.

A row-by-row analysis of the interferogram is performed to locate the center of

each square bright or dark fringe. The left and right edges of bright fringes are placed

where the difference in brightness between consecutive pixels changes from positive



42

to zero and from zero to negative. The peaks in each row are chosen as the pixels

halfway between these two points for each bright fringe. The magnitude of the peaksis set to a fixed value. The left and right edges of dark fringes are placed where

the difference in brightness between consecutive pixels changes from negative to zero

and from zero to positive. The troughs are chosen as the pixels halfway between

these pixels, and their magnitude is set to a second fixed value. Figure 6.3 shows a

row of the interferogram, plotted as a dashed line, with “x’s” indicating the peaks

and troughs. The corresponding row of the raw interferogram is also plotted as a

solid line in the figure. The peaks and troughs in each row are collected along withthe closest peaks and troughs in the other rows and assigned to fringes. Peaks and

troughs that are more than a certain fraction (usually one-fifth) of a fringe-width

from the corresponding valid peaks and troughs in the previous row are thrown out

as erroneous outliers. This results in a set of data points corresponding to each bright

or dark fringe. Each fringe is smoothed using a seventeen-pixel-wide window to reduce

the effects of localized irregularities. The bright fringes formed from the peaks in each

row are plotted over the raw interferogram in Fig. 6.4(a). Figure 6.4(b) shows thedark fringes formed from the troughs. To facilitate measurement of the fringe shift, a

straight line is fit to each fringe, as shown in Fig. 6.4. This straight line connects the

mean of the ten farthest left points in the lower half of the fringe to the mean of the

ten farthest left points in the upper half of the fringe. The bright and dark fringes,

straight lines, and other relevant quantities are stored in a data file as IDL variables.

A second IDL program, invert interferogram.pro, is used to measure the shift

of a selected fringe, calculate the chord-integrated electron density based on that

shift, and invert the chord-integrated density to determine the radial electron density

profile. The first few lines of the program contain parameters that can be edited

by the user, such as whether to invert a peak or a trough, which peak or trough to



43

(a)

(b)

Figure 6.4: Bright and dark fringes plotted over the raw holographic interferogram.(a) Bright fringes formed from the peaks in each row. (b) Dark fringes formed fromthe troughs in each row.



44

invert, how many inversions should be attempted in iterating for the center, and how

far to shift the top and bottom of the straight line. When invert interferogram.prois compiled and run in an IDL window, the user is prompted to open the data file

produced by analyze interferogram.pro, restoring the bright and dark fringes and other

related variables. Figure 6.5(a) shows a selected fringe and the associated straight

line plotted on top of the raw holographic interferogram. The top or bottom of the

straight line can be shifted left or right by the user, to force the pixel shift to be

zero at the edges. Figure 6.5(b) shows a selected fringe and the shifted straight

line. The resolution used to scan the image (300 pixels per inch) is apparent fromthe fringe. The dotted lines in Figs. 6.5(a)-6.5(b) mark the centroid of the fringe

and the points where the fringe and the straight line intersect. Points outside the

intersection of the fringe and the straight line are discarded, reducing the effects of

aberrations at the edge of the holographic interferogram. The shift of the selected

fringe from the corresponding straight line is measured in pixels and smoothed twice,

with nineteen and twenty-nine-pixel-wide windows. The double-pass interferograms

were smoothed only once with a seven-pixel-wide window. The number of pointsused in each smoothing window was determined empirically so that irregularities

are reduced, but the overall characteristics of the fringe shift are maintained. The

smoothed pixel shift is plotted in Fig. 6.5(c). The dotted line represents the centroid

of the pixel shift. The pixel shift is converted to the fringe order (see Sec. 3.3)

by dividing the shift in pixels by the average fringe width in pixels. This average

fringe width is the mean of the distances between the straight lines corresponding to

consecutive bright or dark fringes. The fringe order is plotted in Fig. 6.5(d) versus the

impact parameter, in centimeters. The impact parameter of the fringes is converted

from pixels to centimeters using the 1 cm distance between the reference wires as a

conversion factor. Negative values of the impact parameter are assigned to points



45

(a) (b)

(c) (d)

Figure 6.5: Fringe shift measurement. (a) A selected bright fringe, associated straightline, and raw holographic interferogram. Dotted lines mark the centroid and edgesof the fringe shift. (b) A selected bright fringe and shifted straight line. Dotted linesmark the centroid and edges of the fringe shift. (c) Smoothed pixel shift. A dotted

line marks the centroid of the pixel shift. (d) Fringe order vs. impact parameter incentimeters. A negative value of the impact parameter is assigned to points belowthe centroid of the fringe order.



46

below the centroid of the fringe shift.

6.2 Chord-Integrated Electron Density Determination

The chord-integrated electron number density is calculated in invert interferogram.pro

from the fringe order using the relation

f =∆φ

2π=

1

2λnc

nedl, (6.1)

where f is the fringe order, ∆φ is the phase shift, ne is the electron density, nc is the

plasma cutoff density and λ is the ruby laser wavelength. This equation is obtained

from Eq. 3.3 by assuming the refractive index (Eq. 3.2) is nearly one and can be

approximated by

n ≈ 1 − 1

2

ne

nc

. (6.2)

Equation 6.1 simplifies to

N e =

nedl = 3.212 × 1017f [cm−2], (6.3)

where N e is the chord-integrated electron number density in cm−2 [5]. Figure 6.6

shows the chord-integrated electron number density versus the impact parameter.

The impact parameter equals zero at the centroid of the chord-integrated density.

6.3 Inversion Method for Determining the Radial Density Profile

6.3.1 Abel Inversion

The chord-integrated electron number density is inverted in invert interferogram.pro

using a discrete Abel inversion method to yield the radial electron density profile.The plasma is assumed to be radially symmetric, so that the chord-integrated

number density at a given impact parameter is given by the Abel transform equation

N e (y) = 2

∞

y

ne (r) rdr

(y2 − r2)1/2, (6.4)



47

Figure 6.6: Chord-integrated electron density (Pulse 31029011).



48

y

r

ne0

ne1

neI-1

neI-2

nei+1

nei-1

nei

r I-1

r I

r I-2

r i

r i+1

r i-1

r 2

r 1

∆r

Figure 6.7: Shell model used to obtain the radial electron density profile from thechord-integrated density using a discrete Abel inversion method.

where y is the impact parameter of the chord and r is the distance of a point on

the chord from the center of the Z-pinch. The “2” in this expression is a “4” when

the double-pass system of Sec. 5.1.1 is used instead of the single-pass system. To

discretize the integral at each of I pixels across the fringe, the plasma is modeled as I

concentric, cylindrical shells, as shown in Fig. 6.7. nei, the number density across shell

i, is assumed to be constant from ri to ri+1. The chord-integrated electron number

density at the impact parameter corresponding to radius i depends on the density of

and path length through shell i and all the shells outside it:

N ei = 2

I −1k=i

nek

rk+1

rk

rdr

(r2 − r2i )1/2

, (6.5)



49

where nek is the electron number density throughout each shell k, from k = i to

k = I − 1. The integral in Eq. 6.5 is simplified as follows, rk+1

rk

rdr

(r2 − r2i )1/2

=

r2 − r2i1/2

rk+1

rk

=

r2k+1 − r2i1/2 − r2k − r2i

1/2. (6.6)

Substituting ri = i∆r, where ∆r is the shell width, into Eq. 6.6, we get rk+1

rk

rdr

(r2

−r2i )

1/2= ∆r

(k + 1)2 − i2

1/2 −

k2 − i2

1/2

= ∆rAki, (6.7)

where

Aki =

(k + 1)2 − i21/2 − k2 − i2

1/2

. (6.8)

The coefficients Aki are used to rewrite the discrete Abel transform (Eq. 6.5) as

N ei = 2∆rI −1k=i

nekAki. (6.9)

Eq. 6.9 can be solved for nei to arrive at the discrete Abel inversion formula,

I −1k=i

nekAki =1

2∆rN ei

Aiinei +

I −1k=i+1

nekAki =1

2∆rN ei

nei =1

Aii

1

2∆rN ei −

I −1k=i+1

nekAki

. (6.10)

The model has been set up so that the density is zero at the outside edge of the

last shell, neI = 0, and so that the density at the center is

ne0 =1

A00

1

2∆rN e0 −

I −1k=1

Ak0nek

. (6.11)



50

Substituting for Ak0 = {k + 1 − k} = 1, the density at the center becomes

ne0 = 12∆r

N e0 −I −1k=1

nek. (6.12)

As shown in Fig. 6.8, the coefficients Aki are half the path lengths through each

cylindrical shell, normalized by ∆r, the width of each shell. In the above case where

the coefficients are Ak0 = 1, the path length through each shell is simply 2∆r .

In effect, using the Abel inversion method is equivalent to the following process.

The electron density of the outermost shell (shell I) is calculated by dividing the

chord-integrated density at the first chord by the path length through the shell atthat impact parameter:

ne(I −1) =1

A(I −1)(I −1)

1

2∆rN e(I −1). (6.13)

The electron density of the second outermost shell (shell I-1) is calculated in a similar

manner using the chord-integrated density at the second chord, except there are two

shells involved. The contribution of the outermost shell can be calculated, since the

electron density and the path length through the outermost shell are known. This

contribution is then subtracted from the chord-integrated density to yield the chord-

integrated density due to the second outermost shell alone. This value is divided by

the path length through the second outermost shell to yield the electron density of

that shell:

ne(I −2) =1

A(I −2)(I −2)

1

2∆rN e(I −2) −A(I −1)(I −2)ne(I −1). (6.14)

This process is repeated, working inward, until the innermost shell is reached and

the electron density at every shell has been determined. Each interferogram contains

enough information to produce two radial density profiles by this method; one is based

on the lower part of the interferogram, and the other is based on the upper part.

In practice, it is more efficient to use IDL’s matrix inversion function than a

user-defined back-substitution method to solve the system of equations described by



51

r/∆r

AI-1 I-1

AI-2 I-2

AI-1 I-2

AI-1 I-3

A I-1 6

AI-1 5

AI-1 4

AI-1 3

AI-1 2

AI-1 1

I-1 0 A

AI-2 I-3

AI-2 0

AI-2 1

AI-2 2

AI-2 4

AI-2 3

A

AI-2 5

I-2 6

A0 0

A A A

A A A

A A

A

1 0 2 0 3 0

1 1 2 1 3 1

2 2 3 2

3 3

A A

A A

A A

A A

A A

A

4 0 5 0

4 1 5 1

4 2 5 2

4 3 5 3

5 44 4

5 5

0 r /∆r I

Figure 6.8: Graphical depiction of each coefficient Aki as half the normalized pathlength through a cylindrical shell of plasma.



52

Eq. 6.10. The IDL function is an optimized, executable Gaussian elimination method.

Equation 6.9 is expressed in matrix form as

Ne = 2∆rAne (6.15)

and solved for ne, yielding the Abel inversion equation in matrix form:

ne =1

2∆rA−1Ne. (6.16)

The matrix A has as its elements the coefficients Aki, and the vectors Ne and ne con-

tain the chord-integrated density and the electron density in each shell, respectively.

The matrix form of the Abel inversion equation, Eq. 6.16, is the form used in the

procedure invert interferogram.pro.

As discussed in Sec. 7.2, the location of the center of the shell model affects the

results of the inversion method. For this reason, the Abel inversion method is run

several times in invert interferogram.pro, with the shell model centered about a dif-

ferent point each time. The range of these points is controlled by the value of the

variable center range. By default, they surround the centroid of the chord-integrated

density profile, but the user is prompted to change their location if desired. The

two radial density profiles that result at each center are compared using the differ-

ence in the means of the electron densities in their ten innermost shells. The center

where this difference is smallest yields the best shell model, and the corresponding

radial density profiles are stored as structures in the data file that is the output of

invert interferogram.pro.

Figure 6.9 shows the radial electron density profile that results when the chord-

integrated density of Fig. 6.6 is inverted using this method. A dotted line divides the

radial density profiles based on the lower (r < 0) and upper parts of the interferogram.



53

Figure 6.9: Radial electron density profile (Pulse 31029011).



54

6.3.2 Verification of the Inversion Method

The performance of the inversion method was verified using six simulated density

profiles, shown in Fig. 6.10. The equations used to create the profiles in Fig. 6.10(a)-

6.10(f) are

Fig. 6.10(a), constant profile, ne ∝ constant, (6.17)

Fig. 6.10(b), hollow profile, ne ∝

r3 for 0 < r ≤ 0.5 ,

(1 − r)3 for 0.5 ≤ r ≤ 1 ,

(6.18)

Fig. 6.10(c), linear profile, ne ∝ (1 − r)∆r

, (6.19)

Fig. 6.10(d), plateau profile, ne ∝

constant for 0 < r ≤ 0.5 ,

(1−r)∆r

for 0.5 ≤ r ≤ 1 ,

(6.20)

Fig. 6.10(e), r squared profile, ne ∝ (1 − r)2, (6.21)

Fig. 6.10(f), square root of r profile, ne ∝√

1 − r, (6.22)

where ne is the simulated electron number density, r is the radial coordinate, and ∆r

is the distance between consecutive points in the simulated profile. Each simulated

electron density profile was numerically integrated to produce the chord-integrated

density shown. This chord-integrated density was inverted using the Abel inversion

method to produce the inverted electron density profile shown.

The inverted profiles closely match the corresponding simulated profiles, with

some exceptions. The disturbances near the edges of the inverted constant profile

in Fig. 6.10(a) are caused by one of the assumptions in the Abel inversion: that

the density is zero outside the plasma column. This effectively introduces a large

discontinuity at the edge, where the density plummets from a finite value to zero.

The sharp peaks in Figs. 6.10(b), 6.10(c), 6.10(e) and 6.10(f) are not accurately

reflected in the inverted profiles because of the nineteen and twenty-nine-pixel-wide



55

( a )

( b )

( c )

( d )

( e )

( f )

F i g u r e 6 . 1

0 : P e r f o r m a n c e o f t h e i n v e r s i o n m e t h o d w h e n u s e d o n s i x s i m u l a t e d d e n s i t y p r o fi l e s . T h e i n v e r t e d

p r o fi l e r e s u l t s w h e n t h e i n v e r s i o n m e t h o d i s u s

e d t o i n v e r t t h e c h o r d - i n t e g r a t

e d d e n s i t y m a d e f r o m t h e s i m u l a t e d

p r o fi l e .

( a ) C o n s t a n t p r o fi l e .

( b ) H o l l o w p r o fi l e .

( c ) L i n e a r p r o fi l e .

( d ) P l a t e a u p r o fi l e .

( e ) r s q u a r e d p r o fi

l e .

( f )

S q u a r e r o o t o f r

p r o fi l e .



56

double smooth performed prior to the inversion, as described in Sec. 6.1. The double

smooth tends to artificially flatten the peaks, leading to lower values of the invertedelectron density in those regions.



57

Chapter 7

ERROR ANALYSIS

Possible sources of error in the radial electron density profile include path length

errors due to refraction by the plasma, errors in the placement of the center of the

shell model, and random errors in measurement of the fringe shift that are propagatedthrough the inversion method.

7.1 Path Length Errors due to Refraction

A simplified model of the plasma is used to estimate the importance of path length

errors due to refraction by the plasma. If the plasma is modeled as a block with

refractive index varying in the y direction alone, a ray of the scene beam incident in

the x direction will be bent in the y direction as it passes through the plasma. This

bending will introduce a path length error which is negligible (less than λ/10) if

(n)2 L3

n0λ< 0.3, (7.1)

where n is the gradient of the refractive index with respect to y, L is the path length

of the ray through the plasma, n0 is the refractive index at y = 0, and λ is the

wavelength of the laser light [16].

If we assume a density profile that varies as y2, the refractive index is given by

n = n0

1 − y

a

2, (7.2)

where a is the width of the block of plasma in the y direction. The gradient of the



58

refractive index with respect to y is

n = −2n0a

1 − y

a

. (7.3)

Substituting Eq. 7.3 into Eq. 7.1, we get for the neglection criterion

4n20

a2

1 − y

a

2L3 < 0.3. (7.4)

Since the electron density is known, the refractive index at y = 0 can be found

from

n2 = 1− ne

nc, (7.5)

where ne is the electron density and nc is the critical electron density, above which

the laser light will not propagate [5]. This critical density is

nc = ω2m0/e2. (7.6)

For the laser light in vacuum,

ω = 2πc/λ =2π (2.998× 108 m/s)

(6943× 10−10 m)= 2.713 × 1015 rad/s. (7.7)

Substituting into Eq. 7.6,

nc =(2.713 × 1015 rad/s)2 (9.109 × 10−31 kg)(8.854 × 10−12 F/m)

(1.602 × 10−19 C)2

= 2.313× 1027 m−3. (7.8)

From Eq. 7.5, with ne = 2 × 1023 m−3, the refractive index at y = 0 is

n0 =

1 − 2 × 1023 m−3

2.313 × 1027 m−3= 0.999957. (7.9)

Assuming a = 0.01 m and L = 0.02 m, Eq. 7.4 can be evaluated at y = 0.005 m,

now that n0 is known:4n2

0

a2

1 − y

a

2L3 =

4 (0.999957)2

(0.01 m)2

1 − 0.005 m

0.01 m

2

(0.02 m)3

= 0.08

< 0.3. (7.10)



59

The criterion established in Eq. 7.1 is satisfied, so path length errors due to refraction

by the plasma can be safely neglected.

7.2 Errors due to Incorrect Center Location

The location of the center of the shell model affects the results of the inversion method.

As mentioned in 6.3, the inversion is performed with a range of center locations, and

the most symmetric model is chosen. Figure 7.1 illustrates the importance of choosing

the correct center. The first two figures show the results of inverting the r squared

and hollow simulated density profiles with a center location ten pixels (0.068 cm)

away from the actual center location. The resulting inverted profiles are asymmetric,

and inversion of the hollow profile results in negative values near the center. The

third figure shows the radial density profiles that result when the chord-integrated

density from Fig. 6.6 is inverted with its center location ten pixels (0.085 cm) from

that selected by invert interferogram.pro. It can be compared to Fig. 6.9, which uses

the center chosen by invert interferogram.pro.

7.3 Random Errors

Random errors in the fringe shift measurement and chord-integrated density are prop-

agated through the inversion method and result in errors in the radial density profile.

These errors are estimated using the formalism outlined in [1]. If x is the weighted

sum or difference of u and v,

x = au± bv, (7.11)

the uncertainty, σx, in x is

σx =

∂x

∂u

2

σ2u +

∂x

∂u

2

σ2v ± 2

∂x

∂u

∂x

∂u

σ2

uv

1/2

, (7.12)



60

( a )

( b )

( c )

F i g ur e 7 . 1 : E ff e c t of i n c or r e c t c e n t

e r l o c a t i o n o n t h e p e r f or m a n c e of t h e i n v e r s i o n m e t h o d .

( a )

r s q u ar e d pr o fi l e

w i t h

a c e n t e r l o c a t i o n t e n p i x e l s a w

a yf r o m

t h e a c t u a l c e n t e r l o c

a t i o n.

( b ) H o l l o w

pr o fi l e w i t h

a c e n t e r l o c a t i o n

t e n p

i x e l s a w a yf r o m

t h e a c t u a l c e n

t e r l o c a t i o n.

( c ) R a d i a l e l e c t r o

n d e n s i t y pr o fi l e ( P u l s e 2 1 0 2 9 0 1 1 ) w i t h a c e n t e r

l o c a t

i o n t e n p i x e l s a w a yf r o m

t h a t

s e l e c t e d b y i n v e r t i n t e r f e r o g r a

m. p r o.



61

where

σ2u = variance of u, σu = uncertainty in u,

σ2v = variance of v, σv = uncertainty in v and

σ2uv = covariance of u and v.

Typically, the standard deviations of u and v are used for the uncertainties. If u and

v are completely uncorrelated, the covariance is σ2uv = σuσv. The partial derivatives

of x with respect to u and v are the constants

∂x

∂u= a and (7.13)

∂x

∂v= b, (7.14)

so the uncertainty is

σx =

a2σ2u + b2σ2

v ± 2abσ2uv

1/2. (7.15)

Equation 7.15 can be used to find the standard deviation of x, a measure of the error

in our estimate of x. This result can then be added to or subtracted from each value

of x to yield error bars that are plotted about the values of x.

In the case of the radial electron density profile, the density at shell i, nei, described

by Eq. 6.10, is the weighted sum of the chord-integrated density at the corresponding

impact parameter, N ei, and the densities nek of shells i + 1 to I −1. This relationship

reduces to the weighted sum of the chord-integrated densities from shell i outward,

since each term nek is itself a weighted sum of the chord-integrated densities from

shell k outward. Keeping the nek terms in the equation for simplicity, the uncertainty

in the electron density at shell i is given by

σnei=

1

A2ii

1

2∆r

2

σ2N ei

−I −1

k=i+1

A2kiσ2

nek

1/2

, (7.16)



62

where σN ei is the uncertainty in the chord-integrated density at impact parameter

yi = ri and σnek is the uncertainty in the electron density at each shell outsideshell i. The covariance term in Eq. 7.16 is zero because the values of nek are not

independent of each other. The uncertainties of the nek terms, σnek, are calculated

at each shell, based ultimately on the uncertainty in the chord-integrated density

at each value of the impact parameter. The uncertainty in the chord-integrated

density at each impact parameter, σN ei, is calculated by taking the the difference

between the raw chord-integrated density values and the smoothed chord-integrated

density values. In practice, the matrix form of Eq. 7.16 is used in the IDL procedureinvert interferogram.pro:

σne =

1

2∆r

2 A−1

2σ2Ne

1/2

, (7.17)

where σne and σNeare vectors containing the variance in the electron density and the

variance in the chord-integrated density, respectively, and each element of the matrix

(A−1)2

is the square of the corresponding element in the matrix A−1 from Eq. 6.16.

Figure 7.2 shows the r squared simulated density profile of Fig. 6.10(e), except

that random error with standard deviation 1×1015 cm−2 has been added to the chord-

integrated density. The resulting inverted profile is shown with error bars calculated

using Eq. 7.16. The simulated density profile lies within the error bars, indicating that

the random error in the chord-integrated density profile has been correctly estimated

and propagated through the inversion method.

Figure 7.3 shows the radial density profile of Fig. 6.9 with error bars plotted for

every other point. The effects of smoothing data stored as pixels in the interferogram

image are visible in the error bars in Fig. 7.3, which grow from small to relatively

large to small again every few data points. In addition, the error bars show that the

effects of random errors on the radial electron density profile are localized.



63

Figure 7.2: Performance of the inversion method when used on the r squared sim-ulated density profile. Error bars are included for every other point to indicate theeffect of random errors on the radial density profile.



64

Figure 7.3: Radial electron density profile with error bars plotted for every otherpoint. The dotted line at r = 0 separates the radial electron density profile basedon the lower part of the interferogram from that based on the upper part (Pulse

21029011).



65

Chapter 8

RESULTS OF DENSITY PROFILE INVESTIGATION

Density profiles of the flowing Z-pinch plasmas produced by the ZaP Flow Z-pinch

were investigated using the methods described in the preceding chapters. Plasmas

composed of hydrogen, methane, and helium were studied, and all measurements weremade at the midplane of the Z-pinch. An attempt was made not only to determine

the radial density profile of the plasma, but to study its evolution with respect to

time.

8.1 Results Obtained with the Double-Pass Holographic Interferometer

An initial investigation into the radial density profile of the Z-pinch plasma was per-

formed using the double-pass holographic interferometer of Sec. 5.1.1 and the exper-

imental configuration of Fig. 2.1. Hydrogen plasmas were used for this investigation.

Two of the resulting holographic interferograms are shown in Fig. 8.1.

The chord-integrated density determined from the fringe shift in the interferograms

is shown in Fig. 8.2. A dotted line marks the center of the Z-pinch, found using the

method of Sec. 6.3 and set to impact parameter zero.

The chord-integrated density shown in Fig. 8.2(a) is determined from the fringe

shift of the fourth dark fringe from the left in Fig. 8.1(a). The large shift yields a

chord-integrated density that is peaked at approximately 5.5× 1017 cm−2. A Z-pinch

radius of slightly less than 1 cm is also apparent from the right half of Fig. 8.2(a)

(r > 0). Since the lower edge of the fringe shift is not visible in Fig. 8.1(a), the



66

(a) (b)

Figure 8.1: Holographic interferograms made using the double-pass holographic in-terferometer. (a) Pulse 10705009. (b) Pulse 10515011.



67

chord-integrated density information in the left-hand half of Fig. 8.2(a) is missing.

This leads to inaccuracies in the radial density profile.

The chord-integrated density of Fig. 8.2(b) is determined from the more modest

fringe shift of the sixth dark fringe from the left in Fig. 8.1(b). Its maximum value is

approximately 2.5 × 1017 cm−2 and the Z-pinch radius is just over 1 cm. The chord-

integrated density is slightly hollow in the center, leading to a more dramatically

hollow radial density profile.

The radial electron density profile shown in Fig. 8.3(a) is obtained by inverting the

chord-integrated electron density data of 8.2(a) and corresponds to the interferogram

of Fig. 8.1(a). It is highly peaked at approximately 3 × 1017 cm−3 at the center of

the Z-pinch. For the double-pass interferograms of this section, the chord-integrated

density was smoothed only once with a seven-pixel-wide window, as mentioned in

Sec. 6.1. The radial electron density profile of Fig. 8.3(a) is therefore somewhat

jagged. The left half of the figure is not as reliable as the right half because the

chord-integrated density is truncated, as mentioned above. Its unusual shape is due

to this truncation, and not to the plasma density gradient. The innacuracy is due to

a systematic error in the fringe shift measurement and is not reflected in the error

bars, which indicate the effects of random errors.

The radial electron density profile of Fig. 8.3(b) comes from the slightly hollow

chord-integrated density data of Fig. 8.2(b) and the interferogram of Fig. 8.1(b). It

is noticeably more hollow, with a maximum value of approximately 1 × 1017 cm−3

at about r = 0.3 or 0.5 cm. It is no surprise that the radial density profile for this

pulse is hollow, since the chord-integrated density decreases near its center as the

path length through the plasma increases.

Figure 8.4 shows the plasma current and normalized m = 1 mode for the plasma

pulses associated with the holographic interferograms of Fig. 8.1. The output of



68

( a )

( b )

F i g ur e 8 .2 :

C h or d - i n t e gr a t e d

d e n

s i t y

d e t e r m i n e d

f r o m

t h e f r i n

g e s h i f t i n

t h e h o l o gr a

p h i c i n t e r f e r o

gr a m s of

F i g. 8 . 1 .

( a ) P u l s e 1 0 7 0 5 0 0 9 .

( b ) P

u l s e 1 0 5 1 5 0 1 1 .



69

( a )

( b )

F i g u r e 8 . 3 : R a d i a l e l e c t r o n d e n s i t y p r o fi l e s r e s u l t i n g f r o m i n v e r s i o n o f t h e c h

o r d - i n t e g r a t e d e l e c t r o n d e n s i t y d a t a

o f F i g .

8 . 2 .

( a ) P u l s e 1 0 7 0 5 0 0 9 .

( b ) P u l s e 1 0 5 1 5 0 1 1 .



70

the circuit used to monitor the ruby laser pulse is also shown in the figure, and its

peak occurs at the time the interferogram was made. Figure 8.4(a) shows that theinterferogram of Fig. 8.1(a) was made at 22 µs, before the peak of the plasma current

and during the first half of the quiescent period, which begins at 21.5 µs and ends

at 24 µs. The interferogram of Fig. 8.1(b) was made at 27 µs, just after the peak of

the plasma current and during the latter half of the quiescent period, which begins

at 21.5 µs and ends at 28 µs.

Based on the radial density profiles obtained from these interferograms, the radial

electron number density of the Z-pinch is highly peaked at the beginning of thequiescent period and develops a hollow profile later in the quiescent period as the Z-

pinch widens. The contrast in the radial electron density profiles of Pulses 10705009

and 10515011 led to a more detailed investigation of the evolution of the radial electron

density profile using the single-pass holographic interferometer. The switch to a single-

pass system was made following damage to the laser rod that occured while making

measurements with the double-pass holographic interferometer (see Sec. 5.1.1).

8.2 Results Obtained with the Single-Pass Holographic Interferometer

The single-pass holographic interferometer of Sec. 5.1.2 was used to investigate the

time evolution of the density profile of Z-pinches made using the experimental con-

figuration shown in Fig. 5.2. Plasmas composed of hydrogen, a 50% methane/50%

hydrogen mixture, and helium were studied. As shown in Fig. 8.5, a Z-pinch formed

with helium has a higher density than a methane/hydrogen or pure hydrogen Z-pinch.

This leads to a maximum fringe order of about one fringe for a helium Z-pinch and a

higher signal-to-noise ratio in the chord-integrated density. As a result, features of the

chord-integrated density and the radial electron density profile are more discernible.

Figure 8.6 is an example of an interferogram made during a helium plasma pulse



71

quiescentperiod

(a)

quiescent

period

(b)

Figure 8.4: Normalized m = 1 mode, plasma current, and holography laser monitorassociated with the holographic interferograms of Fig. 8.1. (a) Pulse 10705009. (b)Pulse 10515011.



72

Figure 8.5: Radial electron density profiles for hydrogen, 50% methane/50% hydrogen,and helium Z-pinches (Pulses 20910027, 21029011 and 30204007).



73

Figure 8.6: A holographic interferogram made using the single-pass holographic in-terferometer (Pulse 30204010).

using the single-pass holographic interferometer. A comparision of Fig. 8.6 with

Fig. 8.1 shows the interferograms made with the single-pass holographic interferometer

are much higher quality than the double-pass interferograms. This higher quality

is the result of improvements made to the optical configuration of the single-pass

holographic interferometer. Many of the mirrors used were designed to be used with

a 45 deg angle of incidence. This requirement was adhered to in the optical setup of the

single-pass system, but it was not rigorously observed in the double-pass holographic

interferometer, leading to lower-quality interferograms. The addition of imaging lenses

to the scene and reference beam paths of the single-pass holographic interferometer

also contributed to the quality improvement.

The magnitude and shape of the fringe shift in Fig. 8.6 varies from the left side of

the interferogram to the right side, indicating that the density of the Z-pinch varies

axially as well as radially. Figure 8.7 shows the same interferogram with its bright

fringes and the associated straight lines labeled and highlighted. The chord-integrated



74

Figure 8.7: The holographic interferogram of Fig. 8.6 with its bright fringes high-lighted and numbered.

density from fringes 5-9 is shown in Fig. 8.8. In this case, a value of zero for the impact

parameter corresponds to the center of the vacuum chamber, y = 0. A change in the

shape and magnitude of the chord-integrated density are apparent from one side of

the interferogram to the other. The location of the plasma with respect to the center

of the vacuum chamber and how it varies across the interferogram is also evident from

the chord-integrated density.

The radial asymmetry of the chord-integrated density from this particular inter-

ferogram indicates that it cannot be reliably inverted using the Abel inversion method

described in Sec. 6.3.1. Even if it were symmetric, the decrease in the chord-integrated

density near the center of many of the fringes is great enough that it causes the Abel

inversion method to output negative density values near the center of the plasma col-

umn. This behavior indicates that the plasma would be better modeled using some

other model, such as two plasma filaments. In the majority of cases, however, the

radially symmetric, single-column model is the appropriate model.



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Figure 8.8: Chord-integrated electron density obtained from fringes 5 through 9 in

Fig. 8.7.



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The holographic interferometer was used on successive helium plasma pulses to

investigate the time evolution of the radial electron density profile. Figure 8.9 showsdensity profiles during formation of the Z-pinch. To within the resolution and view

of the holographic interferometer, the density profile is initially flat or zero as the

plasma arrives at the axial location where the holographic measurements were made

(Pulse 30205010). The density profile becomes more peaked during the early part of

the Z-pinch’s formation (Pulse 30205014). During the latter part of the formation

process, the density profile becomes relatively flat or zero, indicating that either the

electron density is below the resolution of the holographic interferometer or its edgesare out of view (Pulse 30205018).

The observations made during formation with the holographic interferometer are

in agreement with the two-chord He-Ne interferometer measurements shown along

with the normalized m = 1 mode, total plasma current and holography laser monitor

in Fig. 8.10. The normalized m = 1 mode and plasma current are plotted in the first

column for the helium plasma pulses used to study formation of the Z-pinch. Plotted

in the second column is the chord-integrated electron number density, as recorded

by the two-chord He-Ne interferometer at the midplane of a helium Z-pinch. Impact

parameters y = 0 cm and y = −1.5 cm are measured from the center of the vacuum

chamber. The holography laser monitor included in both columns shows when each

interferogram was made. The chord-integrated electron density is approximately the

same at y = 0 cm and y = −1.5 cm during arrival and late formation, when the

interferograms of Pulses 30205010 and 30205018 were made. It is higher at y =

0 cm during late formation, corresponding to the peaked profile obtained from the

interferogram of Pulse 30205014.

Figure 8.11 shows density profiles during and after the quiescent period of the Z-

pinch. A peaked profile was observed during the middle of the quiescent period (Pulse



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Figure 8.9: Radial electron density profiles for helium Z-pinches during formation of the Z-pinch (Pulses 30205010, 30205014, and 30205018).



78

(a) (b)

(c) (d)

(e) (f)

Figure 8.10: Normalized m = 1 mode, plasma current and chord-integrated electron

number density for the helium Z-pinches used to study the density profile duringformation (Fig. 8.9). The holography laser monitor shows when each interferogramwas made. (a)–(b) Pulse 30205010. (c)–(d) Pulse 30205014. (e)–(f) Pulse 30205018.



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30204007). The density profile becomes less peaked late in the quiescent period (Pulse

30204019) and flattens following the quiescent period (Pulse 30204021). Figure 8.12shows the normalized m = 1 mode, plasma current, holography laser monitor, and

chord-integrated electron number density from the two-chord He-Ne interferometer

for the helium plasma pulses used to study the density profile during and after the

quiescent period. The measurements made with the He-Ne interferometer once again

agree with the density profiles obtained using the holographic interferometer. The

chord-integrated electron density measured at y = 0 cm is much higher than that

measured at y = −1.5 cm during the middle of the quiescent period, when the in-terferogram of Pulse 30204007 was made. Later in the quiescent period, when the

interferogram of Pulse 30204019 was made, the value of the chord-integrated density

at y = 0 drops, leading to a less-peaked density profile. The interferogram of Pulse

30204021 was made after the quiescent period, when the chord-integrated density at

both y = 0 cm and y = −1.5 cm is low. The decrease in the peak of the density

profile late in the quiescent period is typically sharper for helium than for hydrogen

or hydrogen/methane.



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Figure 8.11: Radial electron density profiles for helium Z-pinches during and afterthe quiescent period (Pulses 30204007, 30204019, 30204021).



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(a) (b)

(c) (d)

(e) (f)

Figure 8.12: Normalized m = 1 mode, plasma current and chord-integrated electron

number density for the helium Z-pinches used to study the density profile during andafter the quiescent period (Fig. 8.11). The holography laser monitor shows when eachinterferogram was made. (a)–(b) Pulse 30204007. (c)–(d) Pulse 30204019. (e)–(f)Pulse 30204021.



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Chapter 9

SUMMARY AND CONCLUSIONS

Holographic interferometry is used on the ZaP Flow Z-Pinch Experiment to inves-

tigate the radial electron density profile of a sheared-flow Z-pinch plasma. During a

plasma pulse, a single-time measurement is made using a pulsed ruby laser and holo-

graphic techniques, yielding a two-dimensional map of the chord-integrated electron

density. This chord-integrated measurement is inverted using an Abel inversion to

yield the radial electron density profile of the Z-pinch. Holographic interferograms

are made at different times on successive plasma pulses to study the evolution of the

density profile with respect to time.

Several working gases were used to make the various Z-pinches studied. In gen-

eral, it was found that helium Z-pinches have a higher electron density than hydro-

gen/methane or hydrogen Z-pinches and therefore provide a better signal-to-noise

ratio for making density measurements.

Some of the holographic interferograms were made using a double-pass holographic

interferometer, and some were made using a single-pass holographic interferometer.

Higher-quality interferograms were produced using the single-pass interferometer, due

to an improved optical configuration. A better signal-to-noise ratio was obtained

using the double-pass interferometer, because of the increased fringe shift that results

from the longer path through the plasma. For this reason, it is advisable to use

the double-pass holographic interferometer when making density measurements of

hydrogen or hydrogen/methane plasmas. Imaging optics should be used to reduce



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diffracted and stray light in the double-pass holographic interferometer, and optical

components such as max-R mirrors should be carefully positioned. It is also advisableto take measures to prevent the returning scene beam in the double-pass holographic

interferometer from reflecting from the beamsplitter and back into the laser cavity.

These precautions are necessary to protect the ruby laser rod from damage.

The data collection rate was increased by switching to specially-ordered 35 mm

Slavich PFG-01 holographic film instead of holographic plates. Because the film is less

sensitive at the ruby laser wavelength than AGFA plates, lenses are used to increase

the power per unit area of the laser light that exposes the film. The process used

to develop and reconstruct the film is nearly identical to the procedure used for the

plates.

Measuring the fringe shift of often hazy interferograms is the most difficult part

of obtaining the radial density profile. This task is performed by the IDL procedure

analyze interferogram.pro, which was continuously modified over the course of the

investigation. The improvement in clarity that occurred with the addition of lenses

into the single-pass beam paths made fringe shift measurement easier.

The Abel inversion method reliably inverts both synthetic test profiles and ex-

perimental data. Each fringe of the interferogram yields two radial electron density

profiles—one from the upper part of the interferogram, and one from the lower part.

Iteration of the method is used to find the central point that yields the most sym-

metric radial electron density profile. This central point is selected as the center of

the Z-pinch, and the corresponding profile is selected as the radial electron density of

the Z-pinch.

In general, the radial electron density profiles show a Z-pinch with an electron

number density of around 1 × 1017 cm−3 and a radius of 1 to 1.5 cm. The Z-pinch

is radially symmetric, to the level of accuracy reflected in the error bars. Both the



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radial density profile and the chord-integrated density often vary from one fringe to

the next across an interferogram, indicating an axial density gradient.The time evolution of a helium Z-pinch is shown in Chapter 8, and the results ob-

tained using the holographic interferometer are corroborated by measurements made

using a He-Ne interferometer. Both interferometers show that the electron density

or the location of the plasma fluctuates during formation of the Z-pinch. A peaked

radial electron density profile is observed on-axis during the middle of the quiescent

period. The density profile becomes less peaked later in the quiescent period. After

the end of the quiescent period, the dense plasma disappears.



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applications. Addison-Wesley, 1978.

[11] G. Saxby. Practical holography . Prentice Hall, 2nd edition, 1994.

[12] U. Shumlak, R. P. Golingo, B. A. Nelson, and D. J. Den Hartog. Evidence of

stabilization in the Z-pinch. Physical Review Letters, 87(20):205005–1–205005–4,2001.

[13] U. Shumlak, R. P. Golingo, B. A. Nelson, S. L. Jackson, E. A. Crawford, and

D. J. Den Hartog. Sheared flow stabilization experiments in the ZaP flow Z

pinch. Physics of Plasmas, 10(5):1683–1690, 2003.

[14] U. Shumlak and C. W. Hartman. Sheared flow stabilization of the m=1 kink

mode in Z pinches. Physical Review Letters, 75(18):3285–3288, 1995.

[15] UAB Geola. Emulsions for holography: technical product specifications and sales

information brochure, 2001.

Stuart Lee Jackson- Holographic Interferometry on the ZaP Flow Z-Pinch

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