Lisa Bennie Honours Thesis Reduced

7/31/2019 Lisa Bennie Honours Thesis Reduced

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A large atom number

magneto-optical trap for BEC

production

Author:Lisa Bennie

Supervisor:Dr. Lincoln Turner

A thesis submitted for the degree of

Bachelor of Science Advanced with Honoursin the School of Physics ofMonash University

November 5, 2010


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Abstract

An integral part of a Bose-Einstein condensate (BEC) machine is the magneto-

optical trap (MOT), where a cloud of atoms can be trapped and cooled by aspatially modulated force from atom-photon momentum exchange. From thiscloud of atoms, a BEC can be formed by evaporative cooling in a conservativetrap. This thesis presents three components required for BEC production:variable beam expanders for wide, intensity balanced laser trapping beams;an ultra-high vacuum system; and a high resolution objective lens for BECimaging. Six beam expanders for magneto-optical trapping were constructed,in which the magnification of the beam can be continuously varied between4.7 5.3, without losing collimation, by adjusting the position of a single lens.This allows the peak intensity to be varied by up to 24 % with no loss intotal power. The construction of an ultra-high vacuum system is discussed,

which reached a pressure of 2 108 Torr. Having obtained this pressure, itis expected that after bakeout the pressure will reach 1012 Torr. Lastly thedesign of a high resolution objective lens is presented, which has a resolutionof 0.9 m with a diffraction limited field of view of 340 m. Unlike commercialmicroscope objectives, this design corrects for spherical aberration caused bythe glass cell wall, which allows the objective lens to be located outside ofthe vacuum system. The objective lens contains four elements made of SF11glass, and is insensitive to element spacing variations of < 0.1mm.


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Acknowledgements

This thesis would not exist without the help, support and contribution of many

people. I would like to thank:

Lincoln Turner for his unfailing energy and enthusiasm, Russell Anderson for all of his assistance throughout the year The mechanical workshop, especially Stephen Downing, for their hard

work in constructing various items for the project. Without Stephen,the vacuum system would have literally fallen apart!

Nino Benci and the electronic workshop,

Everyone in the Monash BEC group for providing a supportive teamenvironment, Phil, for his love and encouragement, and finally all the Honours students for making this such a fun year!


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Contents

1 Introduction 11.1 Trapping in the MOT . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Optical molasses . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 The MOT - a damped harmonic oscillator . . . . . . . . 5

2 Variable beam expanders for magneto-optical trapping 9

2.1 The MOT trapping beams . . . . . . . . . . . . . . . . . . . . . 9

2.2 Beam expansion methods . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 A single collimation lens . . . . . . . . . . . . . . . . . . 10

2.2.2 Keplerian and Galilean beam expanders . . . . . . . . . 11

2.2.3 Galilean variable beam expanders . . . . . . . . . . . . . 12

2.3 Design of the MOT beam expanders . . . . . . . . . . . . . . . 142.3.1 Theoretical design . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Simulations in OSLO . . . . . . . . . . . . . . . . . . . . 16

2.4 Construction of the prototype . . . . . . . . . . . . . . . . . . . 19

2.4.1 Difficulties in obtaining straight beam propagation . . . 20

2.4.2 Difficulties in fibre collimation . . . . . . . . . . . . . . 20

2.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Construction of the MOT beam expanders . . . . . . . . . . . . 22

3 The ultra-high vacuum system 25

3.1 The preliminary vacuum system . . . . . . . . . . . . . . . . . 25

3.1.1 Vacuum pressure before bakeout . . . . . . . . . . . . . 263.1.2 Vacuum pressure after bakeout . . . . . . . . . . . . . . 27

3.2 Design of the UHV system . . . . . . . . . . . . . . . . . . . . . 28

3.3 Construction of the UHV system . . . . . . . . . . . . . . . . . 29

3.3.1 Evacuating the system . . . . . . . . . . . . . . . . . . . 31

3.3.2 Incorporating the MOT beam expanders . . . . . . . . . 33

4 A high resolution BEC imaging system 35

4.1 Imaging system requirements . . . . . . . . . . . . . . . . . . . 36

4.1.1 Spherical aberration . . . . . . . . . . . . . . . . . . . . 36

4.1.2 Lens design software . . . . . . . . . . . . . . . . . . . . 38

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4.2 An imaging system using catalogue lenses . . . . . . . . . . . . 394.2.1 Initial results . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 A custom objective lens . . . . . . . . . . . . . . . . . . . . . . 41

4.3.1 The final design . . . . . . . . . . . . . . . . . . . . . . 44

5 Conclusion 49

A A custom lens holder for the beam expanders 51

Bibliography 54


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Chapter1Introduction

The study of Bose-Einstein condensation is currently an active field of physics.The 2010 International Conference on Atomic Physics (ICAP) demonstratedthat there is intensifying interest in this area, both experimental and theoreti-cal. This thesis examines the construction of part of a machine used to createBose-Einstein condensates, known as the magneto-optical trap.

A Bose-Einstein condensate (BEC) is a state of matter first predicted in1924 by Nath Bose and Albert Einstein [1, 2][3, p.1]. Seven decades later,technology and cold atom physics advanced to the stage where the first BECwas created. This was achieved in 1995 by Carl Wieman, Eric Cornell andWolfgang Ketterle [4]. If a dilute gas of weakly interacting bosons in an

external trapping potential is cooled to several hundred nanokelvin, the bosonsbegin to occupy the lowest energy level of the trap potential. Bosons, unlikefermions, can multiply occupy the same energy level. Once one boson occupiesthe lowest energy level, it becomes energetically favourable for more bosons tooccupy the same level. In this manner, when a critical temperature is reacheda cascade occurs where the majority of atoms in the trap condense into theone energy level, forming a BEC. This process is shown in Figure 1.1(a).

Once a BEC has formed in the ground state of the trapping potential, itcan then be raised to an excited state of the potential. An example of a BECin an excited state is shown in Figure 1.1(b), where quantised vortices areobserved to form a regular lattice pattern [6]. The goal of the Spinor BEC

Laboratory at Monash University is to perform magnetic resonance imaging(MRI) on BECs to observe topological defects in the trapped condensate.

To form a BEC, a cloud of cold atoms must first be held in a trap, fromwhere the temperature can be lowered further. This is achieved using amagneto-optical trap (MOT). The MOT was developed by Steven Chu et al.in 1987 [7], and consists of three pairs of orthogonal, counterpropagating laserbeams and a magnetic field gradient increasing in strength from the centreof the trap outwards. Many MOTs have been made which trap 108 atoms,which results in a BEC of 105 atoms [8, 9]. Larger atom number MOTs of1010 atoms have been made, from which a BEC of 107 atoms can form [10, 11].Our aim was to build a MOT which can trap 1010 atoms with a high loading

1


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1.1. Trapping in the MOT 3

(a) (b)

Figure 1.2: (a) Schematic of a typical MOT, showing six counter-propagating, or-thogonal, circularly polarised laser beams, with two coils in an anti-Helmholtz con-figuration producing a magnetic field [12, p.192]. This magnetic field is zero at thecentre and linearly increases in strength with radial distance. (b) Contour lines fromtwo anti-Helmholtz coils [13]. Here Z is the distance along the axis of the two coils,the vertical direction shown in (a), and R is the radial distance from the z-axis. Atthe centre of the trap, the magnitude of the magnetic field increases outwards linearly

and has a steeper gradient along the z-axis. This steep gradient is usually orientedvertically, so that the greater trapping force counteracts the force of gravity.

this occured when the atom was travelling towards the photon, then the atomwould be slowed by this momentum exchange. The frequency of the photondetermines whether this interaction occurs in the correct direction for slowing.If the photon frequency corresponds to an atomic transition, then the atomwill absorb the photon. This is known as the resonance frequency. Due to theDoppler effect, the photon frequency will be different in the reference frameof the atom, depending on the direction and velocity the atom is travelling

with respect to the photon. If the photon frequency is tuned to the atomicresonance only when the atom is moving towards the photon, then the atomwill be slowed.

Now consider a laser beam incident on an atom. The rate of atom-photoninteraction will increase with the beam intensity, and will increase as thephotons approach the resonance frequency in the atomic rest frame. The netforce on an atom due to the laser beam is

F = hks0 /2

1 + s0 + [2( + D)/]2, (1.1)

where h is the reduced Planck constant, s0 = I/Is is the ratio of the laser


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4 Chapter 1. Introduction

Figure 1.3: Profile of the force in an optical molasses as a function of atom velocity[15, p.88]. For small velocities, the force is linear, as shown by equation 1.4, however

for larger velocities the slowing force decreases. When we include a magnetic field,the force on the atoms will have a similar profile, but will be spatially dependentrather than velocity dependent [15, 157].

intensity I to the saturation intensity Is of the atomic transition, is thelinewidth of the atomic transition, is the laser detuning from resonance andD is the Doppler shift of the laser in the atomic reference frame [15, 16].The laser detuning = l a is the difference between the laser angularfrequency l and the atomic resonance angular frequency a. The Dopplershift is D =

k

v where k is the wavenumber of the laser and v is the

velocity of the atom in the laboratory frame.

We see from this equation that if two identical counterpropagating laserbeams were directed onto a stationary atom, then the net force on that atomwould be zero, since the wavevectors k have opposite direction. However, thenet force is non-zero for the case where the lasers are tuned slightly belowresonance, and the atom is moving slowly along the axis of the beam. Asthe atom moves in one direction, the oncoming laser beam is blue-shifted(in the atomic rest frame) towards resonance, while the other laser beam isred-shifted. This means that the atom will absorb more photons from theoncoming laser than from the co-propagating laser, and as such a greaterforce will be exerted by the oncoming laser, resulting in a non-zero dampingforce, or optical molasses force, FOM. If the lasers were tuned above resonancerather than below, then the atom would be accelerated instead of slowed, whichwould not be the desired effect in an optical molasses.

Equation (1.1) can be extended to include the two directions of the beams,to give

F = hk s0 /21 + s0 + [2( |D|)/]2 , (1.2)

such that the optical molasses force is the sum of the forces in both directions;

FOM = F+ + F. (1.3)


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Expanding the denominator of (1.2) as a Taylor series, we then obtain

FOM=

8hk2s0v

(1 + s0 + (2/)2)2

v, (1.4)

where we have assumed small velocities [15, p.88]. This is a velocity dependentforce which is linear for small velocities (Figure 1.3). While this analysis onlyconsidered a one-dimensional case, it can be extended into three dimensionsusing six counter-propagating orthogonal laser beams instead of two.

1.1.2 The MOT - a damped harmonic oscillator

While atoms collect within the optical molasses, they are not trapped; atomsentering the molasses will eventually diffuse out. A magnetic field can modu-

late the atom-photon interaction such that the force becomes spatially depen-dent, creating a restoring force which keeps the atoms trapped in the opticalmolasses. The introduction of a magnetic field gradient transforms the opticalmolasses into a MOT.

When an atom is exposed to a magnetic field, the atomic energy levelsare shifted. This is known as the Zeeman effect [17]. Figure 1.4(a) showsthe hyperfine structure of the ground and first excited state in 87Rb. Eachhyperfine energy level, with associated spin F quantum number, can containthe spin projections mF = F , . . . , 1, 0, 1, . . . F . When a magnetic field isapplied to the atom, the degeneracy in these mF states breaks, and the energylevels become shifted, as shown in Figure 1.4(b). For the |F, mF groundstates |2, 2 , |2, 2 and excited states |3, 3 , |3, 3 this shift is linear withincreasing magnetic field strength, and the relative shift of these energy levels isdepicted in Figure 1.5(a). The transitions |2, 2 |3, 3 and |2, 2 |3, 3are called cyclic transitions; the atom can repeatedly excite and decay betweenthese energy levels.

As discussed before, the magnitude of the magnetic field in the MOTincreases linearly with distance from the MOT centre. Figures 1.4 and 1.5show that for the |2, 2 , |2, 2 ground and |3, 3 , |3, 3 excited states thetransition energy (and hence the detuning) changes linearly with increasingmagnetic field strength. We see therefore that the detuning will change linearlywith increasing distance from the MOT centre, but with opposing gradients.

Now consider the effect of circular polarisation of the counter-propagatinglaser beams, such that one beam is right-hand circularly polarised, +, and theother is left-hand circularly polarised, . Here again we are only consideringthe one-dimensional case, but as before this can easily be extended to three-dimensions. An atom in the |2, 2 ground state can be excited to the |3, 3state by the beam. If the light is red-detuned for an atom at rest in thecentre of the trap, then the tuning of this transition will approach resonanceas the atom moves towards the beam, and away from resonance as it movesin the opposite direction. The same effect is observed for an atom in the |2, 2state interacting with the + beam. This relationship is shown in Figure1.5. We see from this diagram that the probability of a transition occurring,


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(a) (b)

Figure 1.4: (a) The ground and first excited state of 87Rb, where each level hasthen been magnified to show the hyperfine structure [18]. The trapping beams in theMOT are tuned to (i) the cyclic transition, while a repump laser (ii) causes atoms toenter the cyclic transition. (b) The splitting of the hyperfine structure with increasingmagnetic field strength [19].

and hence the net force on the atom, increases as the atom moves out fromthe centre of the trap, and is always directed towards the centre. Once thedetuning reverses sign, the probability decreases. The restoring force on theatom therefore has a similar form to the force due to the optical molasses(Figure 1.3), except this force acts in position space rather than momentumspace. Notice that as an atom crosses the centre of the trap, it will no longerbe in the necessary state to be trapped by the oncoming laser. This doesnot result in a problem, because the atom will be optically pumped into thetrapping transition after several photons have been absorbed.

The total force on the atoms due to both the optical molasses and themagnetic field is

F = v r, (1.5)where was defined in equation (1.4), r is the distance from the trap centreand

=A

hk, (1.6)

where is the magnetic moment of the atomic transition and A is the mag-netic field gradient. We see from this equation why, in a first order approx-imation, the MOT can be regarded as a damped harmonic oscillator, with


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(a) (b)

Figure 1.5: (a) The energy levels of 87Rb used for trapping in the MOT, with

increasing magnetic field strength. The scale is arbitrary, and the diagram indicatesthe relative difference in splitting between the energy levels. Using this diagram, andthe knowledge that the magnetic field strength is linear with distance in the MOT,the change in detuning with distance can be found, and is shown in (b), again on anarbitrary scale. The background shading indicates the magnitude of the trapping forceon the atoms, with darker shading indicating stronger force. This force is directedtowards the centre of the trap.

as the damping coefficient and as the spring constant.To construct a MOT, wide trapping beams and an ultra-high vacuum

system are required. When the MOT is constructed, and BECs are condensedfrom the cloud of trapped atoms, a method of imaging the BEC is also needed.In chapter 2, we present the Galilean variable beam expanders used to formthe wide, intensity balanced MOT trapping beams. The ultra-high vacuumsystem was constructed, discussed in chapter 3, and finally the design of ahigh resolution BEC imaging system is presented in chapter 4.


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Chapter2Variable beam expanders for

magneto-optical trapping

2.1 The MOT trapping beams

As discussed in chapter 1, the MOT requires six counter-propagating orthogo-nal laser beams to form a region of optical molasses. A magnetic field gradientspatially modulates the atom-photon interaction such that a restoring forcetraps atoms at the centre of the MOT. To form a large atom number MOT,the trapping beams need to be:

well collimated over the length of the MOT setup,

circularly polarised in the correct relative orientation with respect to themagnetic field, to ensure that the trapping force is directed towards thecentre of the trap (see Figures 1.2(a) and 1.5(b)),

predominantly containing light red-detuned by 1 2 from the coolingtransition, with a small mixture (10%) of repumping light to returnatoms to the cooling transition, defined in Figure 1.4(a),

equal in intensity for each pair of beams, to ensure a balanced opticalmolasses,

with diameters as large as the geometric constraints of the MOT systemallow.

This last point is particularly important if the MOT is required to trap a largenumber of atoms. As the width of the beams increase, the interaction time be-tween atom and trap increases, allowing the trapping of initially faster atoms.Recent measurements on large atom number MOTs [11] clearly illustrate thiseffect (Figure 2.1).

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10 Chapter 2. Variable beam expanders for magneto-optical trapping

Figure 2.1: The number of atoms N trapped in a MOT increases with beam diameter

. The straight line shows the relation N 3.3

, which agrees with the data for upto 1010 atoms [11].

2.2 Beam expansion methods

While all BEC laboratories need to form wide, collimated MOT trappingbeams, there are several methods available to achieve this. If optical fibre isused, the beam diverging from the fibre is typically collimated using a singlelens with a long focal length, while free-space optics use a fixed Keplerian orGalilean beam expander. Using these methods, the intensity of the trapping

beams are adjusted by varying the beam power using a half-wave plate and apolarising beam splitter cube for every MOT beam.

We use optical fibre to transfer light to other areas of the lab. In particular,the light for the MOT beams is coupled into a single fibre, which then split intosix fibres with a fixed splitting ratio. To our knowledge, no other MOTs use ahome-made fibre-splitting system to produce the trapping beams [20]. Usingthis method, it is not simple to independently adjust the power in each beam.Instead we use Galilean variable beam expanders to adjust the intensities ofthe MOT beams by varying the width of each beam.

2.2.1 A single collimation lensA single collimation lens after a fibre is by far the simplest method for obtain-ing a wide MOT beam. The divergence of a Gaussian beam can be describedby the equation

w(z) = w0

1 +

z

w20

21/2, (2.1)

where w is the 1/e2 beam radius1, w0 is the beam waist (radius of the beamat the fibre exit) and z is the propagation distance from the beam waist (see

1All subsequent radii are given as the 1/e2 radii.


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2.2. Beam expansion methods 11

Figure 2.2: A Gaussian beam will diverge from a waist of 1/e2 radius w0 to a radiusof w(z) after a propagation distance of z, as described by equation (2.1).

Figure 2.2)[21, p.595].

If a convex lens is placed on the optical axis one focal length from thebeam waist, or the end of the fibre, then the lens will form a collimated beam.

In our case we require a beam diameter of 30 mm, as will be discussed insection 2.3.1. For light with wavelength 780 nm exiting a fibre with a modefield diameter of 5.3 m [Thorlabs P3-780-PM-FC-2], the required focal lengthto obtain a collimated beam with a 30 mm diameter is 160 mm. Not only doesthis method result in a long collimation length, it means that the width of thebeam cannot be adjusted without losing collimation, unless the lens is changedto one with a different focal length and re-positioned with respect to the beamwaist accordingly. This is not a simple operation.

To get a circularly polarised beam, a quarter-wave plate needs to be placedin a region where the beam is collimated, otherwise differences in optical pathlengths across the beam would result in a non-uniform elliptical polarisation.

We are using polarisation maintaining (PM) fibres which ensure that the out-put is linearly polarised, so we only need to use a quarter-wave plate to convertto circular polarisation. In this case, the quarter-wave plate would be locatedafter the collimating lens, and would have to be greater than 30 mm in diam-eter. A quarter-wave plate of this size becomes expensive for example, CVIMelles Griot sell a multiple-order quarter-wave plate only 30 mm in diameterfor $730 [22].

So although using a single lens to produce a wide collimated beam is avery simple design, the size of the beam cannot be adjusted, the distance offree-space propagation takes up unnecessary room in the experimental set-up,and including a quarter-wave plate makes the design expensive.

2.2.2 Keplerian and Galilean beam expanders

An improvement to collimating a beam with a single lens is to use either aKeplerian or a Galilean beam expander. A Keplerian beam expander consistsof two convex lenses, while a Galilean beam expander uses one convex andone concave lens [23, p.230]. Both types can take a small collimated beamand increase the beam diameter, but notably the Keplerian expander bringsthe beam to a focus between the two lenses first, while the Galilean expandermagnifies the beam over a shorter distance (Figure 2.3). In either case, a


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(a) Keplerian beam expander (b) Galilean beam expander

Figure 2.3: A standard (a) Keplerian beam expander and (b) Galilean beam ex-pander. In the Keplerian system, the focus allows for spatial filtering, but can cause

air breakdown in high power cases. The total length of the system is shorter in aGalilean design.

collimated beam is initally needed, so if the optical system did not alreadyproduce a collimated beam, an additional collimating lens would need to beplaced before the beam expander. This lens could have a much shorter focallength than required if no beam expander was used, so the total length of thesystem would be shorter than when using a single collimation lens.

Using a Keplerian expander has the advantage that the focus allows easyspatial filtering of the beam, while the Galilean expander produces a widebeam over a smaller propagation distance than either the Keplerian expanderor a single collimation lens. We chose to use a Galilean beam expander bothfor the shorter propagation distance and because air turbulence at the focusof a Keplerian expander can produce scintillation of the beam. However, wemodified the design to allow a variable output beam width which maintainscollimation. This is called a Galilean variable beam expander.

2.2.3 Galilean variable beam expanders

In a Galilean variable beam expander, the rear concave lens in the standardGalilean design is replaced by one convex and one concave lens (see Figure

2.4(a)). The concave lens has a higher power than the convex lens, meaningthat when these two lenses are close together they form an effective concavelens. Then the system behaves in the same way as a standard Galilean beamexpander. If the central concave lens is moved forward to the last convex lens,then the system is effectively reversed, and the output beam becomes smallerthan the input beam.

If only the central lens moved, the output beam would not remain colli-mated. To maintain collimation, the rear lens must be moved at the sametime, in a manner shown in Figure 2.4. When both lenses are moved withrespect to the output lens, the beam width can be continuously changed [24].

When both convex lenses have the same focal length, then the system is


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2.2. Beam expansion methods 13

(a) (b)

Figure 2.4: Schematics of a (a) symmetric and (b) non-symmetric Galilean variablebeam expander. If both rear lenses are moved as shown, the beam magnification canbe continuously changed while maintaining collimation. If the input and output lenshave equal focal lengths, then the system is centred at a magnification of one. If theinput lens has a shorter focal length than the output lens then the system is centredat a magnification greater than one.

a symmetric variable beam expander. When the total length of the system isat a maximum, then the output beam has the same size as the input beam,and the system is said to be centred at a magnification of one. Such a systemis shown in Figure 2.4(a). If the rear lens has a shorter focal length thanthe output lens, then the system is non-symmetric. When the system is at amaximum length, the output beam will be wider than the input beam, andthe system is then centred at a magnification greater than one. An exampleof this is shown in Figure 2.4(b).

In practice, it is preferable to keep the number of variable positions ordegrees of freedom to a minimum. Ideally, only one lens should need to beadjusted to change the beam width with a minimum of collimation loss. Toachieve this, we designed a lens system that produced the desired beam mag-nification when the rear lens is located at the maximum distance from thefixed front lens. In this position, in order to produce a small change in thebeam magnification the rear lens would only need to move position slightly tokeep the beam collimated. But in our design, the rear lens is kept fixed forsimplicity, and only the central lens is adjustable. When the magnificationis changed by moving the single lens by a small distance (defined in 2.3.2),the beam size does not change visibly over a distance much greater than the


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Figure 2.5: A diagram of the optical components of the Galilean variable beamexpanders (not to scale).

length of the MOT, and so it is considered to remain well collimated. Thedetails of the design of our MOT beam expanders are discussed in the next

section.

2.3 Design of the MOT beam expanders

2.3.1 Theoretical design

To design the MOT beam expanders, first the required magnification wascalculated. A diagram of the optical elements in the beam expanders is shownin Figure 2.5. We wished to use small quarter-wave plates with a diameter of11 mm, so the initial beam size was restricted to less than this diameter. The

final beam size needed to be less than the smallest dimension of the glass cell,being 30 mm. Using equation (2.1), a lens with (coincidentally) a focal lengthof 30 mm was used to initially collimate the fibre output. A beam expanderwith a magnification of five then gives a trapping beam 28 mm in diameter,as wide as the geometrical constraints of the glass cell allowed.

Next a set of three lenses were found to achieve the required magnification.The focal lengths f1, f2 and f3 of the three lenses in a Gaussian variable beamexpander are related to the magnification M by the set of equations

d1 = f1 + f2 +f1f2f3M

, (2.2)

and d2 = f2 + f3 + f2f3Mf1, (2.3)

where d1, d2 are lens spacings and M = r1/r2 where r1, r2 are beam radiias shown in Figure 2.6 [25]. However, these two equations do not specifywhether the design is symmetric and centred about a magnification of one, ornon-symmetric and centred about a magnification of five as required. Whenthe design is centred about the desired magnification, the total length of thesystem is maximised as the magnification is adjusted. This sets the conditionthat

(d1 + d2)

M= 0, (2.4)


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2.3. Design of the MOT beam expanders 15

Figure 2.6: Diagram showing the lens focal lengths f, separations d and beam radiir of a Galilean variable beam expander. These quantities are related by equations(2.2) and (2.3).

which, when applied to equations (2.2) and (2.3) yields the relation

f1 = M f3. (2.5)

Notice that when M = 1, the focal length of the output lens is the same asthat of the input lens, as is indeed the case for a symmetric variable beamexpander system. For the system to instead be centred on a magnification offive, the focal length of the output lens must be five times longer than theinput lens. This equation does not contain f2 the focal length of the concavelens determines the overall length and distances between the lenses. To ourknowledge, this simple relation between the focal lengths of the first and thirdlens in a Galilean variable beam expander has not been documented elsewhere.

Using equations (2.2) and (2.3) and the condition (2.5), it was possible tothen find a combination of standard lenses to produce a magnification of fivewith reasonable distances between the lenses. The lenses in the final designhave focal lengths of

f1 = 150 mm, f2 = 9 mm, and f3 = 30mm,

which were available as catalogue items [Thorlabs LA1417-B, LD2568-B andLA1027-B]. Equations (2.2) and (2.3) give corresponding separation distancesof

d1 = 132 mm, and d2 = 12mm.

The distance between the lenses was the main limiting factor in the designprocess. Often a combination of lenses would be found that would result in ei-ther d2 being negative, or so small that once the lens thickness was accountedfor there was little or no space left between the lenses for movement. Ifd2 wasincreased, then d1 would be so large that it offered little improvement in prop-agation distance over using a single collimation lens. These problems resultedin a magnification of 6 being rejected, as a satisfactory combination could notbe found. For example, with a magnification of 6, three possible lenses weref1 = 150 mm, f2 = 10 mm and f3 = 25 mm, however the distances betweenthese lenses were d1 = 130mm and d2 = 5 mm. The distance d2 was far toosmall to be achieved, since the distance from the optical centre to the front of


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Figure 2.7: A graph of the total length (d1 + d2) of our Galilean variable beamexpanders as a function of magnification, using lenses with focal lengths f1 = 150 mm,f2 = 9mm and f3 = 30 mm. With this lens combination, the total length ismaximised at a magnification of five. This means that for small variations around amagnification of five, the total length can remain constant, and hence the rear lenscan be fixed with minimal loss to collimation.

a 1 Thorlabs f = 25mm lens is 7.7 mm. This would require the two lensesto be overlapping, a slight physical impossibility!

It was confirmed that the design was centred at a magnification of 5 bygraphing the total length of the system as a function of magnification, for thechosen values of f1, f2 and f3. Figure 2.7 confirms that the total length of thesystem is maximised at a magnification of five, and that there is little changein the total length at this position for small variations in magnification.

While finding the desired combination of lenses, certain relations wereobserved between the focal lengths of the lenses and the distances betweenthem. If M and f2 were fixed, and f1 and f3 satisfied the relation (2.5), thenit was observed that

f1 = d2 = Md1. (2.6)

If instead f2 was the only focal length allowed to vary, then it was observedthat

f2 =d2

2=

d12

. (2.7)

With this knowledge, the best combination of lenses was found efficiently.

2.3.2 Simulations in OSLO

The next stage in the design process was to confirm than lenses with focallengths of f1 = 150mm, f2 = 9mm and f3 = 30 mm, separated by thedistances d1 = 132mm and d2 = 12 mm, would in fact give a collimatedoutput beam with a magnification of five. This was done by simulating thelens system using the ray-tracing program OSLO. The previous analysis usedthin-lens approximations, and so did not take into account any aberrationsintroduced by a real lens, or the effects of different aperture sizes. All thesedifferent factors can be included in simulations performed using OSLO.


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2.3. Design of the MOT beam expanders 17

In OSLO, a surface is specified by its radius of curvature, material, apertureradius and thickness to the next surface along the optic axis. This means thatto correctly simulate the beam expander system the distances between the

surfaces of the lenses (s1 and s2) need to be known - this is in contrast tod1 and d2, which are the distances between the optical centres of the lenses.For example, the beam expanders use a plano-convex f = 30 mm lens. Thedistance from the focal point of this lens to the optical centre is 30 mm, whilethe distance from the focal point to the centre of the curved surface is 27 .1mm.Calculating the surface separations of the lenses yielded s1 = 125.7mm ands2 = 7.6mm.

Using these separations, the simulated beam expander did not give satis-factory results. The output beam was not collimated but rather convergingto a focus 4 m from the system, and the beam was magnified at the surfaceof the final lens by a factor of only 3.3 (Figure 2.8(a)). Interestingly, these

results were better than if the rear lens was reversed so that the flat side facedthe concave lens. In that case, when the new separation was calculated, theoutput beam converged to a focus 3.8 m away with a magnification of only 2.3(Figure 2.8(b)). This is contrary to the convention of placing the convex sideof the lens facing the collimated beam, and the flat side of the lens facing thefocused beam, to minimise the angle between the rays and the surface.

These results are not surprising when we consider that the equations usedto find d1 and d2 assume thin, ideal lenses, and do not take into account thethickness of the lenses used. With this knowledge it seemed likely that theseparations of these lenses could be adjusted, without changing curvatures, sothat a collimated beam with a magnification of five could be achieved. This

was done manually in OSLO until the best result was found. When the lensesare separated by s1 = 118.2mm and s2 = 2.2 mm, the beam is magnified by afactor of 4.97 and is well collimated, coming to a focus 148 m away from thesystem (Figure 2.9(a)). The specifications of this design are shown in Table2.1. Again, if the rear lens is reversed, the performance is diminished, with afocal length of only 8 m and a magnification of 3.1 (Figure 2.9(b)).

Having found the ideal combination of focal lengths and lens separations,the effect of varying the magnification was simulated. The bi-concave lens

is mounted on a translation mount [Thorlabs SM1Z] with a 1.5 mm range ofmovement. By keeping the total length of the system constant, and movingthe bi-concave lens forward and backwards by 0.75 mm, the changing charac-teristics of the lens system was calculated in OSLO. These results are shown inFigure 2.10. When the central lens was moved 0.75 mm backwards, the beamremained fairly collimated, with a focal length of 18 m, and the magnificationincreased to 5.3. When the lens was moved forward by 0.75 mm the beam hada focal length of 16 m, and the magnification decreased to 4.7. In both cases,after the change in magnification the beam size did not change visibly acrossa distance of 4 m. As this is much greater than the distance across the MOT( 0.3 m), the beam can be considered to remain well collimated. This total


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(a) (b)

Figure 2.8: OSLO raytrace of the three chosen lenses with optical centres separatedby the distances calculated using equations (2.2) and (2.3). We found that with theseseparations, the system did not produce a collimated beam, and the magnification ofthe beam near the output lens was less than five. The first lens is plano-convex andwas simulated in both possible orientations. We found that (a) placing the convexside towards the concave lens yielded a higher magnification and longer focal lengththan when (b) the plane side was placed towards the concave lens. These simulations

caused us to alter the lens separations to achieve better results.

(a) (b)

Figure 2.9: The results of manually adjusting the lens separations in OSLO to givea magnification as close to five as possible, with the best collimation. In Figure (a),the system has a very long focal length of 148 m, and as such is well collimated overthe distance of a MOT trapping beam, and gives a magnification of 4.97. In Figure(b), the orientation of the plano-convex lens was reverse, and we again found thatthe quality of the system was diminished, having a focal length of only 8 m and amagnification of 3.1.

Surface Curvature Radius (mm) Thickness (mm) Diameter (mm) Glass

1 8.6 24.4 BK72 -15.4 2.2 24.4 Air3 -14.4 2 9 BK7

4 14.4 118.2 9 Air

5 7.3 50.8 BK76 -77.3 50.8 Air

Table 2.1: The surface data entered into OSLO giving the best beam expanderdesign shown in Figure 2.9(a). These are from left to right Thorlabs lenses LA1027-B, LD2568-B and LA1417-B. BK7 is the most common type of glass for optical lenses,with a refractive index of 1.51.


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2.4. Construction of the prototype 19

(a) backwards (b) forwards

Figure 2.10: Simulating the magnification range available in the final design. InFigure (b) the central lens was moved forwards by 0.75 mm and the focal length de-creased to 18 m while the magnification increased to 5.3. In Figure (a) the central lenswas moved 0.75 mm backwards, resulting in a magnification of 4.7 and a focal lengthof 16 m. This gives a total magnification range of 0.6 while maintaining satisfactorycollimation.

range in magnification corresponds to a change in intensity of13% to +11%.The fibre splitters used to form the six MOT beams do not split the powerof the input light source perfectly evenly; there can be a variation in powerbetween two MOT beams of up to 1%. The range of intensity available to usthrough the use of the variable beam expander is sufficient to counteract thisinbalance of the beam powers, such that we can form an intensity-balancedoptical molasses.

2.4 Construction of the prototype

A prototype was made to both test the design and to determine the equipmentneeded to make six beam expanders in total. We chose to construct the beamexpanders using cage system components, which in theory ensures that alloptical components are always positioned on axis, and removes some of thechallenges in aligning optics. As will be discussed in 2.4.1, this was not asstraight forward as anticipated, and some difficulties in Thorlabs componentswere overcome before achieving an axial beam. The polarisation of the beamwas not essential for this prototype, so the quarter-wave plate was not includeduntil the final design.

Initially, the beam exiting the fibre was collimated by placing a f =15.3 mm asphere roughly 15 mm away from the end of the fibre, and then the

lens position was carefully adjusted until the resulting beam was collimated.To find this position, a CCD camera beam profiler [Thorlabs BC106-VIS] wasused to measure the 1/e2 beam waist both close and far from the system. Itwas found that 13 cm away from the lens the beam had a width of 2.4mm,and 3.8 m away from the lens the beam had diverged slightly to a width of2.9 mm. Since it is the nature of Gaussian optics to diverge slightly over largedistances, and that the collimation of the beam was extremely sensitive tosmall changes in the position of the collimating lens, it was decided that thisbeam was sufficiently collimated. Later in the construction process, furtherdifficulties with this initial collimation were identified, and will be discussedin 2.4.2.


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Once the initial collimation of the beam was achieved, the positions of thethree beam expander lenses were adjusted to achieve a magnified, collimatedoutput beam. It was during this process that a number of challenges began

to appear in the construction of this prototype.

2.4.1 Difficulties in obtaining straight beam propagation

The first problem encountered was that the output beam did not appear tohave a Gaussian profile, but rather exhibited a definite bullseye pattern, whichwas initially thought to be caused by the beam diffracting off one of the lensapertures. Further investigation showed that the beam was being deflectedoff-axis by the bi-concave lens in the z-translation mount by roughly 15.

We discovered that this was due to two separate problems. The first wasthat the lens was not being held securely in the mount, so that the centre of thelens was no longer on axis, but rather had moved by approximately 0.2mm.This slight change in position would not normally have a large effect, howeversince the lens has such a short focal length, and hence a very small radiusof curvature, the effect was very pronounced. This problem was surprising inthat we were using the Thorlabs recommended mount for that particular lens.However, on closer inspection there was a small difference in radii between thering holder the lens was glued into and the lens mount in which the lens washeld. To remove this problem, a new lens mount was designed and constructedto hold the lens in the z-translation mount correctly. The mechanical drawingof this new lens mount is shown in Appendix A.

The second problem was due to the z-translation mount itself. Althoughthe lens was now being held centrally by the new lens mount, the translationmount was holding this new mount off-axis as well. A photograph of thetranslation mount is shown in Appendix A. To fix this problem, the translationmount was partially disassembled and reassembled. This released tension onthe screws responsible for positioning the lens, so that the lens was finally heldstraight and on-axis.

2.4.2 Difficulties in fibre collimation

Once the problems with the translation mount were solved, the intensity profile

of the output beam was observed to be improved, but still not ideal. Wediscovered that the single asphere used to collimate the fibre was the source ofthe remaining error in the beam profile; the profile was Gaussian after exitingthe fibre, but degraded after the asphere. This was unexpected, as an identicalasphere was used to couple the light into the fibre without difficulty. However,one of the advantages of using optical fibre is that the fibre only transmits thelowest order Gaussian mode of the beam. This means that even though theasphere used to couple light into the fibre introduced higher-order error to thebeam, this error is not transmitted along the fibre and the resulting outputwas the perfect Gaussian observed.

To try to solve this problem, a longer focal length asphere and a microscope


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2.4. Construction of the prototype 21

(a) (b)

Figure 2.11: Comparison of the Gaussian quality of fibre output collimated using(a) an asphere with a focal length of 15.3 mm and (b) a Thorlabs fibre collimatorcontaining an air spaced doublet. We observed that the asphere introduced error intothe beam, which can be seen as faint rings in the image, while the fibre collimator

successfully produced a Gaussian profile.

objective lens were both used to collimate the fibre output, but no significantimprovement was observed. Instead, we decided to purchase fibre collimators[F810APC-780], consisting of an air-spaced doublet lens positioned at a fixeddistance from the fibre connector, which are able to collimate the fibre outputto a beam 7.5 mm in diameter. A comparison of the beam quality using theasphere and fibre collimator is shown in Figure 2.11. Unfortunately, onlyone fibre collimator arrived before the conclusion of this project. As such,the prototype was completed using the original asphere despite the irregular

profile it introduced.

2.4.3 Results

The prototype was constructed as shown in Figure 2.12(a) and measurementswere made to determine the magnification. We found that the distance be-tween the input and central lens set the size of the beam immediately exitingthe system, while the relative position of these two lenses to the output lensdetermined the collimation.

The magnified beam was too wide to use the beam profiler to measure thebeam width, so a knife-edge measurement was performed. Instead of using

a knife-edge, a piece of thin cardboard was deemed sufficient for the accuracyrequired. To measure the width of the beam, the beam was gradually blockedby the cardboard edge, and the decreasing intensity of the beam was recordedby focussing the beam onto a power meter. The intensity measured as afunction of the edge position for a Gaussian beam is given by

I(x) =P

2

1 erf

2(x x0)

(2.8)

where P is the total power of the beam, x0 is the beam centre and is the 1/e2

width of the beam. This equation is the result of differentiating a Gaussian


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(a) (b)

Figure 2.12: (a) The beam expander prototype containing the (i) fibre connector,(ii) initial collimating lens with f = 30 mm, (iii) the beam expander input lens (f =30 mm), (iv) variable lens (f = 9 mm), held in a translation mount and (v) theoutput lens (f = 150 mm). (b) The data from a knife-edge measurement near theoutput lens, from which the magnification was calculated to be 5.3 0.1.

intensity profile. A model was constructed using this formula, and fitted tothe recorded data using the Mathematica routine NonLinearModelFit to findthe beam width.

In order to adjust the beam expander to produce a magnification of five,several knife-edge measurements were performed and analysed, and the lenspositions adjusted accordingly. A graph of the knife-edge data and the lineof best fit is shown in Figure 2.12(b), for the beam immediately exiting theprototype, showing the final result of a magnification of 5.3

0.1. Collimation

was confirmed by viewing the beam through a infra-red viewer. Having foundthat the beam expander could produce a magnification which was close to fiveand easily adjustable, measurements of the magnification range were reservedfor the final beam expanders.

2.5 Construction of the MOT beam expanders

For the MOT trapping beams, six Galilean variable beam expanders wereneeded in total. As mentioned earlier, we planned to use Thorlabs fibre col-

limators to initially collimate the optical fibre output, but shipping delaysmeant that they could not be used at this stage. Instead, we used six spheri-cal lenses already available four with a focal length of 30 mm and two witha focal length of 25 mm. The last two were chosen as we did not have sixmatching lenses, and a MOT can still be formed if one pair of trapping beamsis slightly smaller than the others.

Only the relative intensities in each beam pair need to be balanced, so atranslation mount was used in only three of the six beam expanders. Quarter-wave plates were included on a rotation mount to circularly polarise the light,and two mirrors were used in each beam expander which served to bend thedesign into a periscope arrangement. Not only did this mean that the beam


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2.5. Construction of the MOT beam expanders 23

expanders fit neatly around the glass cell with minimum space, but this alsoprovided full directional tilt adjustment to the output beams to make precisealignment possible. A close-up of one beam expander is shown in Figure

2.13(a), and the collection of all six beam expanders is shown in Figure 2.13(b).Duplicating a system by six can be time consuming. For the sake of effi-

ciency, one of the six beam expanders was carefully adjusted while performingknife-edge measurements until the output beam was magnified and collimatedas required. The size of this output beam was viewed on a card through aninfra-red viewer, and the apparent size was marked. This card was then usedas a reference to adjust the remaining five beam expanders.

Measurements of the first beam expander revealed that the beam wasmagnified by 5.2 0.2, and translation of the central lens gave a total changein magnification of 0.8 0.3. This agrees within uncertainty to the expectedmagnification range of 4.7 to 5.3. These uncertainties were calculated from

the confidence interval of the model fitted to the data. Any degradation ofcollimation during this movement was visually imperceptible over a distanceof four metres.

We have constructed six Galilean variable beam expanders centred on amagnification of five to produce 28 mm diameter trapping beams for the MOT.This magnification can be continuously varied from 4.7 to 5.3 by translatingthe single central lens along the optic axis, corresponding to a change in in-tensity of 24%, which is more than adequate to compensate for the slightlyunbalanced power splitting in each of the fibre splitters.

In chapter 3, we discuss the construction of the ultra-high vacuum system.Once this was completed, the six MOT beam expanders were arranged around

the glass cell, and a photograph of the final arrangement is shown in Figure3.8.


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(a)

(b)

Figure 2.13: (a) One of the six Galilean variable beam expanders constructed forthe MOT, with the (i) fibre connector, (ii) f = 30 mm initial collimating lens, (iii)

mirrors providing full tilt adjustment of the output beam, (iv) quarter-wave plate ona rotation mount, (v) f = 30 mm lens, (vi) f = 9 mm lens held in a custom brassholder, mounted in a (vii) translation mount to vary the beam magnification, and(viii) the output f = 150 mm lens. (b) The MOT beam expander collection. Thelower two beam expanders have been built such that the input and output beams areorthogonal, to fit the beam expander above and below the glass cell.


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Chapter3The ultra-high vacuum system

An ultra-high vacuum system is an essential component of the MOT. Disre-garding the high reactivity of rubidium with oxygen and water vapour, at at-mospheric pressures the collisional forces between air molecules and the rubid-ium gas would completely overwhelm the comparatively weak trapping forcesof the MOT. Even at high vacuum (103 to 109 Torr, where atmosphericpressure is 760 Torr), collisions between the trapped atoms and remaining airmolecules are the main limiting factor in the size of the MOT formed. For thisreason we require an ultra-high vacuum system at a pressure of 1012 Torr toform a large atom number BEC. As a comparison, this pressure is an order ofmagnitude lower than that found on the surface of the Moon.

This chapter will first discuss a small vacuum system constructed to testthe quality of our pumps and determine the procedures required to achieveultra-high vacuum. The construction of the ultra-high vacuum system willthen be presented, and finally we will show how the MOT beam expanderswere incorporated into the system.

3.1 The preliminary vacuum system

Before constructing the complete vacuum system, a smaller system was madeto check that the turbo pumps we planned to use were free from machine oil

contamination. Being second hand pumps, this was a necessary precautionto take. This preliminary vacuum system also served as a test of bakeout,which will be discussed later, and dictated the cleaning procedure required toachieve pressures of 1012 Torr.

The preliminary vacuum system consisted of a 4.5 ConFlat tee fitting,with the end (or flange) of the base of the T blanked off. A turbo pump[Pfieffer TMU065] was connected to one end of the tee, and a residual gasanalyser (RGA) to the other. This arrangement is shown in Figure 3.1(a). Theturbo pump, or turbomolecular pump, can reduce the pressure in a vacuumsystem to 1010 Torr and uses a series of tilted blades to strike atoms towardsthe exhaust. An interior view of a turbo pump is shown in Figure 3.1(b). The

25


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26 Chapter 3. The ultra-high vacuum system

(a) (b)

Figure 3.1: (a) The preliminary vacuum system, with (i) a roughing pump and(ii) turbomolecular pump connected to (iii) a tee fitting with (iv) a blanked-off flangeand (v) an RGA. (b) An internal view of a turbomolecular pump, showing the slantedblades which force molecules down towards the exhaust [26].

exhaust is connected to a roughing pump [Varian IDP3A01], which can achievepressures of 103 Torr and does not use oil, as many roughing pumps do, whichlowers the risk of vacuum system contamination. The RGA [Stanford ResearchSystems RGA100] consists of a heated filament and anode which ionises theresidual gas in the system and identifies the partial pressures of the different

gases and contaminants present.To achieve a high vacuum seal, all open ends of the vacuum componentswe used end in a ConFlat flange. To seal two flanges together, a copper gasket(or ring) is placed between the flanges into the grooves provided. When theflanges are then bolted together, a knife-edge circling each flange cuts intothe copper gasket, forming a metal seal. When bolting components together,the bolts must be gradually tightened in a star formation (see Figure 3.2) toensure that uniform pressure is applied. If the bolts are tightened in a circularpattern, then as the copper gasket is indented the resulting slight bulge willtravel around the system to meet the first tightened bolt. This will not resultin a vacuum seal and can damage the knife-edges of the flanges. All bolts used

were silver plated, to prevent steel-on-steel seizing after bakeout.

3.1.1 Vacuum pressure before bakeout

Once the vacuum system was constructed, the roughing pump and turbo pumpwere turned on to begin pumping down. After 20 hours, the pressure reached1.6 108 Torr, given by the total pressure measurement from the RGA. Agraph of the falling pressure over this time is shown in Figure 3.3(a), andFigure 3.3(b) shows the mass spectrum from the RGA. The remaining gas inthe chamber was dominated by water, but interestingly peaks characteristic ofmachine oil were also observed [28]. At this point, we were unable to determine


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3.1. The preliminary vacuum system 27

(a) (b) (c)

Figure 3.2: Recommended star formations for evenly tightening bolts on ConFlatflanges with (a) 6, (b) 8 and (a) 16 holes [27].

0 1 2 3 4 50

2. 107

4. 107

6. 107

8. 107

1. 106

1.2 106

Time hours

PressureTorr

(a) (b)

Figure 3.3: (a) Data recorded using the RGA of the pressure inside the vacuumsystem after the turbo pump was turned on. The pressure drop was exponential with

time, as expected, however there appear to be two distinct regions where the pressuredropped at different rates. This was perhaps caused by the turbo pump working ata different efficiency once the pressure fell below 2 107 Torr. (b) An analysis ofthe composition of the residual gas in the vacuum system before bakeout, when thepressure reached 1.6 108 Torr. Before bakeout the pressure was dominated by (i)hydrogen and (ii) water, with peaks at (iii) masses 39, 41, 43, 55 and 57 which indicatethe presence of machine oil [28]

if the machine oil was from the turbo pump, or from the walls of the tee fitting,and so we decided to bake the vacuum system to further reduce the pressureand hopefully remove the contamination.

3.1.2 Vacuum pressure after bakeout

To perform a vacuum system bakeout, heating tape made of wire surroundedby fibreglass was wrapped around the tee fitting. Thermocouples (type K)were placed around the surface to measure the temperature at different areas.This is shown in Figure 3.4(a). The tee was wrapped in several layers ofaluminium foil, creating a self-contained oven. The tee surfaces were heatedto 120C for 24 hours to remove the layer of water trapped on the inside surfaceof the vacuum system. During bakeout, the pressure increased by an order ofmagnitude before falling again. 24 hours after the vacuum system cooled to


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(a) (b)

Figure 3.4: (a) To bake out the vacuum system to remove the remaining waterand machine oil, the tee was wrapped in heater tape and thermocouples were placedaround the system to measure temperature. The tee was then wrapped in aluminiumfoil and heated to 120C. (b) The results from the RGA after bakeout, when thesystem reached 1.1 109 Torr. The scale on this graph is an order of magnitude lessthan that in Figure 3.3(b), and shows that while there is a small amount of hydrogen(2 u) and water (18 u) remaining, the machine oil was successfully removed.

room temperature, the pressure reached a minimum value of 1 .1 109 Torr,an order of magnitude lower than before bakeout. The mass spectrum fromthe RGA after bakeout is shown in Figure 3.4(b), where we can see thatthe oil peaks have disappeared. During bakeout the turbo pump was onlyheated to 30C, as it has a maximum operating temperature of 90C. Thistemperature would not have been sufficient to remove oil contamination, andso we concluded that the oil had been on the surface of the tee fitting, despitethe fact that it had come straight from the manufacturer. As a result, werealised that every component of the ultra-high vacuum system would need tobe thoroughly cleaned before assembly.

3.2 Design of the UHV system

A drawing of the ultra-high vacuum (UHV) system is shown in Figure 3.5,and was designed by two other members of the Spinor BEC laboratory, AlexWood and Russell Anderson. The UHV system includes:

the glass cell, which will contain the atoms trapped in the MOT andeventually the BEC,


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3.3. Construction of the UHV system 29

a turbo pump, which can be isolated from the UHV system using aright-angle all-metal valve [Huntington MV-250-T],

an ion pump [Gamma Vacuum 75S], to reduce the pressure from 109

Torrto 1012 Torr,

an RGA to detect residual gas components, a hot cathode, dual filament, nude ion gauge, which can measure the

total pressure of the system to a higher degree of accuracy than theRGA,

a getter [Alvatec AS-3-Rb-10-V vapour source], a temporary atom sourcecontaining 10 g of rubidium, which will be used until the rubidium ovenand atomic beam system is completed,

and a gate valve which allows later extensions to be made to the vacuumsystem, without needed to expose the baked system to atmosphere.

The UHV system will eventually be joined to the vacuum system housingthe rubidium oven, which forms a collimated beam of rubidium which is slowedby the Zeeman slower before entering the MOT [19, 29]. As this system is notyet completed, we will temporarily be using a getter as our atom source. Thegate valve on the opposite side to the atom source is currently shut, and allowsanother chamber to later be added to the vacuum system. This chamber iscommonly known as a science chamber, and a BEC can be moved into itto perform additional experiments that may not have been planned into this

existing system design.

3.3 Construction of the UHV system

The preliminary vacuum test revealed that components that arrived sealedfrom the manufacturer contained trace amounts of machine oil. Although forthe very small system the majority of this contamination was successfully re-moved through bakeout, to achieve pressures of 1012 Torr the vacuum systemmust be as clean as possible prior to baking out. The quantum optics group atCaltech University have compiled a very informative document on ultra-high

vacuum procedures [27]. Although there are many different opinions on theamount of cleaning required for UHV, we chose a procedure which used onlyacetone, methanol and deionised water.

First, all areas of the component that would be exposed to vacuum werewiped down using a low-lint tissue (Kimwipe) soaked in technical grade acetoneto remove oil and other contaminants. Then the majority of the acetoneresidue was removed using deionised water, and finally methanol (LichroSolvspectroscopic grade) was applied using a Kimwipe to remove the last of theacetone. Particular attention was paid to the knife-edge on each of the flanges,as dirt can collect easily in the grooves in that region. Unfortunately, knife-edges are sharp, and if this process is not carefully done even the low lint wipes


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(a)

(b)

Figure 3.5: (a) A drawing of the vacuum system constructed for the BEC machine.On the left is the rubidium oven and atomic beam source [29], with the ultra-highvacuum system discussed in this chapter to the right, separated by the Zeeman slowercoils. (b) A close-up of the UHV system.


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can leave lint behind. If this occurred, a lens tissue soaked in methanol wasused to remove all traces of lint. For the smaller components, an ultrasonicbath was used instead of Kimwipes, in which vibrations were coupled through

water to a beaker of methanol to thoroughly clean all contaminants from thepart. Most components however were too large to fit into the bath, or wouldhave used several litres of methanol, so cleaning with Kimwipes was sufficientin most cases.

As each component was cleaned, they were wrapped in aluminium foiland stored until required. To assemble the components a copper gasket wasplaced between each pair of flanges, which were then bolted together in thestar formation described earlier. Where possible, a torque wrench was usedto ensure than an even torque of 20Nm was applied to each bolt on thelarger 6 or 4.5 flanges, and 15 Nm on the smaller 2.75 flanges. To later

join the UHV and atomic beam systems, we decided that the rubidium beam

would be positioned 200 mm above the table. To hold the UHV system suchthat the centre of the glass cell was at this exact height, several high-loadbearing support structures were designed and constructed. A photograph ofthe completed UHV system is shown in Figure 3.6.

3.3.1 Evacuating the system

Once the UHV system was assembled, the roughing and turbo pumps wereswitched on. The ion pump was not activated at this stage, as the maximumpressure it can operate at is 104 Torr. The pressure reached 103 Torr after

only a few minutes and ceased to fall. This was unexpected, as the turbo pumpshould have been capable of producing a much lower pressure, as discussedearlier. It was concluded that there was a large leak present in the systempreventing the pressure from falling.

To identify the position of the leak, helium was slowly sprayed around allof the flanges. With the RGA running, we watched in real time for a peak toappear at a mass of four - when this occurred, we could tell that helium hadentered the system through a leak and reached the RGA. Using this method,a significant leak was found both near the getter and the RGA itself. Oncethe bolts on these flanges were tightened further, the leaks disappeared andthe pressure began to fall. Once the pressure dropped below 104 Torr the

ion pump was turned on, however the pressure reached 106

Torr and againceased to fall after pumping overnight.

After another slow helium leak test was performed on all the joins withno result, we deduced that there was a leak in the right angle valve nearthe turbo pump. This was located at the furthest point from the RGA, andthe geometry of the system meant that the helium was removed from thesystem by the turbo pump instead of reaching the RGA. To test this theory,we closed the right angle valve, isolating the turbo pump from the vacuumsystem, and after only five minutes the pressure fell to 107 Torr, and thenreached 108 Torr overnight.

After pumping down the system for two weeks, the pressure reached 2


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Figure 3.6: Two views of the UHV system, showing (i) the getter, (ii) the glass cell,(iii) the ion gauge, (iv) the right-angle valve, (v) the turbo pump, (vi) the ion pumpand (vii) the RGA.


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Figure 3.7: The mass spectrum of the vacuum system at 2 108 Torr, indicatingthat the system is dominated by (i) hydrogen (2 u) and (ii) water (18 u), with tracesof (iii) nitrogen (28 u), (iv) argon (40 u) and (v) carbon dioxide [30, p.494].

108 Torr and continued to slowly drop. For such a large system with a highsurface area, this is an expected pressure to reach before bakeout. Figure3.7 shows mass spectrum at this pressure. We can see that the pressure isdominated by hydrogen and water, and no oil peaks were detected. Thisindicated that our cleaning regime had been appropriate. The next stage willbe to bake out the system, after which we expect to reach ultra-high vacuumat 1012 Torr.

3.3.2 Incorporating the MOT beam expanders

While a pressure of 108 Torr is not low enough to form a large BEC, itis sufficient to form a MOT. So having reached this pressure we began toconstruct the MOT around the glass cell rather than baking the system tolower the pressure this will be done at a later stage. Figure 3.8 shows theMOT beam expanders and anti-Helmholtz coils in place around the glass cell.

Thorlabs optomechanics posts and post holders were used to position theMOT beam expanders around the glass cell. Four of the six beams are directedhorizontally through the cell, while the other two beams pass through thecell vertically. A small breadboard suspended above the glass cell was usedto mount one beam expander firmly above the glass cell. The other beam

expanders are held in such a way as to give full coarse adjust of the beamposition and angle. Once these are positioned, fine adjust of the beam directionis given by the mirror tilt controls.

To begin trapping rubidium atoms, we now need a laser light source withstable power, frequency and polarisation. This was the work of a separatehonours project [20], and has almost been completed, so we expect to have acompleted MOT in the near future.


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Figure 3.8: Two different views of the UHV system surrounded by the six MOTbeam expanders presented in chapter 2. One beam expander was held above the glasscell using a suspended breadboard, and (i) the temporary coils were attached to theupper and lower beam expander, with the glass cell at the centre.


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Chapter4

A high resolution BEC imaging

system

When the BEC machine is completed we will need to have a method of imagingthe condensate. Essentially all data collected during a BEC experiment is doneso by imaging the condensate in different ways. Imaging also provides a gooddiagnostic tool of the BEC machine performance; we can check that differentexperiments produce a BEC of the expected size or shape, which can revealmisalignment of any laser beams. Structures within a BEC such as vorticescan be imaged if a BEC is released from a trap. As it falls, the mean-field

repulsion of the atoms causes the BEC to expand. Being able to image vorticesin the expanded condensate will be used to confirm results found using themagnetic resonance imaging technique we plan to develop. This chapter willdiscuss the design of a high resolution imaging system for this purpose, whichwill be constructed at a later date.

We have designed a high resolution imaging system to perform absorptionimaging of the condensate. In absorption imaging, light passes through thecondensate and into the objective lens, after which it is collected by a CCDcamera [9]. Some of the light will be absorbed or scattered by the condensateand will not reach the detector, and this change in intensity reveals internal

structure in the BEC. Our imaging system uses an infinity-corrected micro-scope objective lens and an achromat which acts as an imaging lens to forma magnified image of the condensate. This chapter presents the design of along working distance objective lens which is capable of correcting aberrationscreated by the glass cell, allowing the objective lens to be located outside thevacuum. The working distance is the distance from the focus to the first opti-cal element, which in our case is the glass cell wall. A diagram of the imagingsystem is shown in Figure 4.1. The achromat will in reality have a much longerfocal length than is suggested by this diagram, of roughly 200 mm, chosen toachieve an appropriate magnification determined by the pixel resolution of thecamera.

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36 Chapter 4. A high resolution BEC imaging system

4.1 Imaging system requirements

To image structures within a trapped BEC a resolution of 1

2 m is required.

Once the BEC has been released from the trap, we will need an imaging systemthat is diffraction-limited to at least 50 m off-axis [31, p.101]. This is knownas the diffraction-limited field of view (FOV).

To achieve these requirements, a lens system with a high numerical aper-ture is required. The numerical aperture (NA) of a lens is a dimensionlessquantity given by

NA = n sin , (4.1)

where n is the refractive index of the medium between the object and the lens(usually air or vacuum) and is the angle of the widest scattered ray of lightfrom the object captured by the objective lens (see Figure 4.1). The numerical

aperture is related to the resolution R of the lens system by the equation

R =1.22

2 NA(4.2)

where 1.22 is Rayleighs constant and is the wavelength of light used (in ourcase, 780 nm) [23, p.548]. For example, to achieve a resolution of 1 m, thenumerical aperture of the system must be at least 0.48. In air or vacuum, thiscorresponds to a widest angle of scattered light of approximately 30.

One of the main factors limiting the numerical aperture of a system isthe physical lens size a larger diameter lens can capture a wider angle ofscattered light than a smaller lens. We will be imaging the top of the BEC,

so that when the BEC is released it does not move out of the field of view.This means that our imaging system will need to use lenses with a maximumdiameter of 50.8 mm, to fit within the magnetic field coils at the top of theglass cell (see Figure 4.1). The BEC will be located at the centre of the cell,15 mm from the inside surface of the glass wall. However, we want to be ableto image the BEC after it has fallen by up to 2 mm, requiring the objectivelens to have a working distance of 17 mm.

4.1.1 Spherical aberration

A high numerical aperture is not a sufficient requirement for a high resolution

imaging system. Equation (4.2) assumes that the system is diffraction limited.In other words, aberrations in the system must not be the limiting factor tothe imaging performance. As we will be using a single wavelength of light, wedid not need to consider chromatic aberrations, however aberrations such ascoma, astigmatism and spherical aberrations did need to be minimised. Themethods used to do this will be discussed in 4.1.2.

Spherical aberration occurs when light passing through different regionsof a lens is focussed at different points along the optic axis (Figure 4.2(a)).This effect is most pronounced in spherical lenses (lenses with constant radii ofcurvatures), and can be reduced with aspherical lenses although in some casesnot entirely, as we observed in the construction of the MOT beam expanders


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4.1. Imaging system requirements 37

Figure 4.1: A schematic of the imaging system in position near the glass cell (notto scale). The angle of scattered light collected by the objective lens defines thenumerical aperture and resolution of the system (equations (4.1) and (4.2)). An

achromat will be used to focus the light collimated by the objective lens to form animage.


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(a) (b)

Figure 4.2: (a) An exaggerated example of a lens causing positive spherical aber-ration. Rays passing though near the edge of the lens are brought to a closer focusthan rays passing through near the lens centre. (b) A microscope objective lens witha numerical aperture of 0.9 [23, p.590]. The light is first bent outwards by the rearelement, then gradually bent inwards by a series of different lenses. This process min-

imises aberrations in the system, and is characteristic of microscope objective lenses.In our case, we require a system in which the focal point is much further from thelast lens, in order to position our imaging system outside of the glass cell.

(see 2.4.2). Spherical aberration can occur in two directions; either the lightis focussed in a spread before the expected focal point, called positive sphericalaberration, or it is focussed in a spread after the expected focal point, callednegative spherical aberration [32, p.226]. If a lens produces positive sphericalaberration, then it can be corrected by using a second lens which introducesequal, negative spherical aberration [33, p.73]. This is the approach we usedto minimise spherical aberration in our imaging system. An example of ahigh numerical aperture objective lens which minimises spherical aberrationis shown in Figure 4.2(b).

The 5 mm silica wall of the glass cell causes a significant amount of spher-ical aberration which we needed to correct for using our objective lens. Thiscorrection requirement was the reason we cannot use a commercial long work-ing distance microscope objective.

4.1.2 Lens design software

The objective lens was designed using OSLO, which was previously used to

simulate the MOT beam expanders discussed in 2.3.2. An essential aspect ofthe design process of this objective lens was the optimisation routine whichcould be performed on the surface curvatures and lens thicknesses to achieve adiffraction limited result. This used a damped least squares method to adjustsystem variables until an error function, which included terms such as comaor spherical aberration, reached a local minimum value. This approach wasdependent on the initial configuration of the lens system one design maybe optimised to give a much smaller value of the error function than another.The optimisation algorithms are generally unable to find the global minimumdue to the massive size of the parameter space. Ultimately trial and error, andan educated intuition of the lens system behaviour, was required to achieve a


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4.2. An imaging system using catalogue lenses 39

Author NA Elements FOV (m) Glass Availability

(a) W. Alt [34] 0.29 4 1000 BK7 Thorlabs

(b) T. Ottenstein [35] 0.4 3 400 SF11 Custom,

Zeiss(c) D. Weiss [36] 0.55 ? 140 ? Custom

(d) S. Johnstone [37] 0.49 5 10 BK7& SF11

4 Thorlabs,1 Custom

(e) L. Bennie 0.52 4 340 SF11 Custom

Table 4.1: A comparison of the performance of our objective lens to other systemsdesigned for high resolution imaging of BECs or cold atoms. The final design of oursystem (e) has a numerical aperture of 0.52 with a field of view of 340 m, and willbe discussed in more detail in 4.3.1. It is similar in performance to entry (b), whichhas a similar field of view, but a lower NA of 0.4.

design which was both diffraction limited and manufacturable.

Using this software, it was logical to define the width of a collimated in-put beam, which would then be focussed by the array of lenses, rather thandefining rays emanating from a point source at a certain angle. For this rea-son, although our imaging system will be used to collect light scattered fromthe BEC, in the following simulations we will consider the reverse situation collimated light entering the system and being focused at the point where theBEC will be located.

The performance of the lens system was measured by the wavefront errorboth on- and off-axis. The wavefront error is the difference between the shape

of the wavefront produced by the lens system at the exit pupil of the lens (inour case the glass cell) and a perfectly spherical wavefront, which would beproduced if the system exhibited zero aberrations. If the average, or root-mean-squared (RMS), wavefront error is less than 0.07 , then the system isconsidered diffraction limited [33, p.191].

4.2 An imaging system using catalogue lenses

A comparison of other imaging systems used for imaging BECs reveals thata compromise tends to be reached between a high NA, a wide field of view

and affordability of the design. Table 4.1 lists the performances of four otherobjective lenses compared to our design.

The lens system we used as a basis for our design is shown in Figure 4.3(a),and is the first entry in Table 4.1. This system used four 25.4 mm diameterlenses to give a diffraction limited field of view of 1 mm, which is much widerthan we require, however it only had a numerical aperture of 0.29, whereas werequired at least 0.48. After replicating this design in OSLO, using the dataprovided in reference [34], the system was scaled to use 50.8 mm lenses. The


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(a) (b)

Figure 4.3: (a) The lens design listed in entry (a) of Table 4.1 which we initially usedas a basis for our design [34]. It used 25.4 mm Thorlabs catalogue lenses to achievean NA of 0.29 and a diffraction limited field of fiew of 1 mm. (b) A lens systemwe designed using Thorlabs catalogue lenses [from left to right: LC1315, LA1384(twice), LE1076]. It has a NA of 0.4, but could not be increased to greater than0.48 as required. As the analysis in Figure 4.4 shows, this design was not diffraction

limited on axis, and so it was rejected.

curvature of the last surface before the glass cell was set as a solved variablesuch that the numerical aperture of the system was set to 0 .4. This decreasedthe focal length of the system, and the glass cell was positioned such that thefocus was 17 mm from the inner cell surface. Interestingly, we observed thatthe distance between the glass cell and the lens closest to the focus had noeffect on the performance of the system. This meant that in the final designthe BEC can be imaged from two positions; first when held in a trap 15 mmfrom the glass cell, and then after it has fallen over a distance of up to 2 mm.

Introducing these changes to the system meant that the system was nolonger diffraction limited. The optimisation routine was then performed tominimise the error in the system. We found that the best way to do this wasto incrementally allow curvatures, and then thicknesses, to vary optimising thesystem between each new introduction. If all lens curvatures and thicknesseswere allowed to vary from the start, then although the error in the systemwould become very small, the result would not be manufacturable with acombination of overlapping lenses, distances of several metres between lenses,and lens thicknesses either greater than 10 cm or less than 0.5 mm. This couldhave been controlled by setting the range over which these quantities couldvary, however this feature was not discovered until later, as discussed in 4.3.

4.2.1 Initial results

When a design was found which had distances of no more than 20 cm betweenlenses, and no less than 1 mm, the lenses were set to Thorlabs catalogue lensesone by one. When one lens appeared to be a close match to a cataloguelens, then this lens was fixed as a catalogue lens, and the system was againoptimised. This process was repeated until all the lenses were catalogue lenses.This took several attempts, with trial and error, choosing a different cataloguelens for the first lens fixed, and fixing lenses in different orders. Frequently themeniscus lens (closest to the glass cell) was the first lens to be fixed, as there


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4.3. A custom objective lens 41

was a very limited choice of catalogue meniscus lenses. Once all the lenseswere specified as catalogue lenses, the air spaces were fixed one at a time,again performing optimisation each time, until finally there were no variables

left in the system.If we had been able to produce a satisfactory design using catalogue lenses,

then the imaging system would have been very inexpensive (compared to man-ufacturing custom lenses) and could have been tested in the laboratory in arelatively short amount of time. Unfortunately, we did not manage to achievethis. The best result obtained using Thorlabs catalogue lenses is shown inFigure 4.3(b). As the simulations of spot size and wavefront error shows, thesystem is not diffraction limited (Figure 4.4). This means that even thoughthe numerical aperture is 0.4, this objective lens could not be used to form animage with a resolution of 1.2 m as expected from equation (4.2).

In addition to this, we observed that with 50.8 mm diameter lenses made

of BK7 glass the numerical aperture could not be made higher than 0.4. Whenthe NA of the system was increased, the rays would begin to pass outside ofthe 2nd or 3rd lens. The NA of the system also depended on the diameterof the input beam, so even though we could prevent the marginal rays frompassing outside of the lens, this also had the effect of decreasing the numericalaperture.

4.3 A custom objective lens

To achieve our requirements, we decided to increase the refractive index of theglass and no longer limited the design to catalogue lenses. This resulted inthe successful design discussed in this section. A lens made of a higher indexglass does not need to be as curved to have the same focal length. Conversely,a higher refractive index lens will have a shorter focal length than a lowerindex lens with identical curvature. Because of this, the higher index lens cancollimate a beam diverging at a wider angle. Hence the numerical aperture ofan objective lens can be increased by increasing the refractive index. Figure4.5 demonstrates that higher refractive index glass also reduces the wavefronterror of an optical system. This is because of the decrease in curvature of thesurfaces.

To achieve a higher numerical aperture, the BK7 glass in the original design[34], scaled to 50.8 mm lenses, was replaced with SF11 glass. This glass has arefractive index of 1.77 for 780 nm light, which is higher than BK7 glass withn = 1.51. While we could have chosen a glass with an even higher refractiveindex, which in theory could have given a higher NA, higher refractive indexglass can have undesirable qualities. Some types of glass have low chemicalresistance, which can result in a cloudy surface after prolonged exposure to air.Others can be darker, having a lower transmittance of light at the requiredwavelength, or some types of glass expand or contract with temperature witha higher sensitivity [33, p.108]. BK7 and SF11 glass are both considered to bestable types of glass [33, p.363].


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(a)

(b)

Figure 4.4: Performance simulations of the lens system shown in Figure 4.3(b),which used Thorlabs catalogue lenses. (a) Ray tracing results showing that the light(green dots) did not fall within the diffraction limited Airy disc (black circle), therebyshowing that the system was not diffraction limited. This was simulated on-axis

(bottom row), 120 m off-axis (middle row) and 170 m off-axis (top row), from adistance along the optic axis

Lisa Bennie Honours Thesis Reduced

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