Deflecting Metastable Helium Atoms with Resonant Light

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Deflecting Metastable Helium Atoms

with Resonant Light

Bachelor Thesis

Bachelor of Physics and Astronomy (Joint Degree)

Cornelis Anthonius Jozef Keijzer

Studentnumber VU: 2529124

Studentnumber UvA: 10438041

VU University Amsterdam

LaserLab, Department of Physics and Astronomy

Daily Supervisor: Y. van der Werf

Supervisor: Dr. W. Vassen

Examinator: Prof. Dr. Piet Mulders

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Abstract

Photons do not have a mass, but they do carry momentum. In this thesis, a research on the deflection of metastable helium atoms using resonant 1083 nm laser light is described. By scattering photons, the metastable helium atoms gain the momentum of the absorbed photons in the direction of illumination. This results in a deflection of the atom beam.

The main goal of this research is to determine the relation between the deflection of the atom beams and the scattering force of the resonant laser light. This is done by varying parameters like the atom beam width, the direction of illumination and the light beam width. For every set of measurements, the deflection is measured for different light intensities and interaction times. The results are quantitatively analyzed and compared to each other. The results will be discussed by means of the theory of the scattering force, showing the limits of the theory and the effects of the assumptions made to test it.

It is shown that the measured deflections agree with the expected relations based on the theory and the corresponding assumptions. As for the results that deviate from the expectations, clear explanations are given that result in points of interest for follow-up projects.

One of the main difficulties of the research is found in the alignment of the light beam. An important suggestion for follow-up projects is to find a method to consequently align the light beam optimally in the same way.

Additionally, it is shown that the absolute deflection of the atom beam should not depend on the beam width of the atom beam or on the direction of illumination.

Finally, by increasing the beam width of the light beam and the interaction time of the laser light with the atom beam, it is shown that the measured deflections increase significantly and even exceed the theoretical approximation of the maximum value for the deflection. This results in a discussion of the approximations and assumptions made for this research, leading to a better understanding of the theory of the scattering force.

Hoewel fotonen geen massa hebben, dragen ze wel een impuls met zich mee. In dit verslag wordt een onderzoek beschreven naar de afbuiging van een bundel metastabiele heliumatomen met resonant 1083 nm laser licht. Door fotonen te absorberen en weer uit te zenden krijgen de metastabiele heliumatomen een snelheid mee in de richting van verlichting. Dit resulteert in een afbuiging van de atoombundel.

Het doel van dit project is het bepalen van de relatie tussen de mate van afbuiging van de atomen en de verstrooiingskracht van het resonante laser licht. Dit wordt gedaan door verschillende eigen-schappen van de lichtbundel te veranderen, zoals de breedte van de bundel, de intensiteit van het licht en de interactietijd van het licht met de atomen.

De resultaten worden kwantitatief geanalyseerd en met elkaar vergeleken. Daarnaast zal worden uitgelegd waarom bepaalde resultaten afwijken van de verwachtingen. Dit resulteert in een discussie over de theorie, waarin de beperkingen van de theorie en de gemaakte aannames ter discussie worden gesteld. Deze discussie leidt tot nieuwe idee¨en en suggesties voor vervolgonderzoek.

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Introduction

Even though light does not have a mass, the transfer of momentum from light to matter was already demonstrated back in 1933, when Frisch measured a small deflection of a beam of sodium atoms after illuminating it with resonant light [1].

Since then, the field of atomic physics has made great progress, and with it the science behind atom manipulation. The development of tunable, high intensity lasers has made it possible to deflect, decelerate and even trap atoms using a collimated light source [2].

In this thesis, a research on the deflection of metastable helium atoms with resonant light is described. Metastable helium atoms are in the 23S1 state and are denoted as He* atoms.

Metastable helium atoms are well suited for this research because of their strong cooling transition at 1083 nm. Fibers that emit light of this wavelength are well represented in the LaserLab. Like other kinds of atoms including Rubidium and Cesium, He* atoms can gain momentum by absorbing and emitting photons of a resonant frequency. What makes He* atoms more suitable for this research is their small mass. The momentum gained by absorbing one resonant photon is enough to give a He* atom a measurable velocity of about 0.1 m s−1. This in contrast to 0.006 m s−1 and 0.003 m s−1 for Rubidium and Cesium, respectively.

Light of a resonant frequency can wield a force on atoms and thereby change their momentum in the direction of illumination. This principle has resulted in the development of the Magneto-Optical Trap (MOT) [3]. Using magnetic fields and resonant light, a MOT traps a cloud of atoms and cools it down to temperatures in the order of 1 mK or even lower.

The achievement of a cloud of ultracold (≤ 1 mK) atoms using a Magneto-Optical Trap is an important step towards Bose-Einstein Condensation [4]. Using Bose-Einstein condensation allows for ultrahigh precision measurements to be performed, making it possible to determine different (quantum) properties of atoms and nucleons, like the proton radius [5].

An additional advantage of metastable helium atoms is their simplicity. Apart from atomic hy-drogen, helium is the simplest atom in the periodic table. The interactions between the electrons and nucleons in helium atoms are well understood. This makes metastable helium suitable for the ultrahigh precision measurements mentioned earlier.

Other applications of metastable helium include the use in researches on nanolithography [6]. This thesis is focused on deflecting a beam of He* atoms using resonant 1083 nm laser light. The main goal of this research is to test the theoretical relation between the deflection of a He* atom beam and the scattering force wielded on the atoms by the laser light. This is done by varying parameters like the atom beam width, the direction of illumination, the light beam width and the light intensity.

In addition, problems and factors that can influence the measurements and their accuracy are discussed in order to fully understand the results and contribute to further research.

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Chapter 2

Theory

2.1 Energy Levels and Resonant Light

To be able to understand the theory of the scattering force, it is necessary to get a clear view on the relevant energy levels of atomic helium.

Helium atoms in a gas typically reside in the 11S0 ground state. By means of a voltage discharge,

the excitation from the 11S0 ground state to the 23S1 metastable state is induced. This state is

called metastable because the He* atoms remain in this state for about 8000 s [7]. An important property of He* atoms is their internal energy of 20 eV.

Metastable helium atoms can absorb photons of certain resonant frequencies, with equivalent resonant wavelengths. One of the resonant wavelengths for He* atoms is λ = 1083 nm. Metastable helium atoms are highly sensitive to 1083 nm light. The absorption of a 1083 nm photon induces an excitation from the 23_S

1 metastable state to the 23P2 state. The lifetime of this 23P2 state is

short, typically about 98 ns. After this time photon emission in a random direction occurs. after the photon emission the atom falls back to the 23_S

1 metastable state, making this transition a

closed system. A schematic representation of the energy levels of atomic helium is shown in figure 2.1.

Figure 2.1: Schematic representation of some of the energy levels of helium, including the

relevant energy transitions [8].

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As long as the metastable helium atoms are illuminated by resonant light, the process of absorbing and emitting photons will repeat itself. One process of absorbing and subsequently emitting a photon is called a scattering event.

2.2 Photon momentum

Photons do not have a mass, but they do carry momentum. The momentum of a photon is given by p = ~k = h 2π 2π λ = h λ (2.1)

where h = 6.62607 · 10−34 m2 _{kg s}−1 _{is Planck’s constant and λ is the wavelength of the light.}

Right after a He* atom absorbs a 1083 nm photon, the momentum of the photon is transferred to the He* atom. Conservation of momentum dictates that the He* atom will gain the exact mo-mentum of the photon, both in magnitude and direction. After absorbing a photon, spontaneous emission occurs, giving the atom a second kick in a random direction. When the scattering events are repeated several times over a very large number of He* atoms, the net effect of the spontaneous emission will approximately average out to zero.

Helium atoms have a mass of 6.65 · 10−27kg. Knowing the mass of a helium atom and the average amount of momentum gained in the direction of illumination after one scattering event, the recoil velocity vrec can be calculated. The recoil velocity is the velocity gained by a He* atom after one

scattering event. The recoil velocity is equal to vrec = p mHe = h λmHe ≈ 0.092 m s−1 (2.2)

which means that the He* atoms gain a velocity of 9.2 cm s−1_{in the direction of illumination after}

each scattering event.

2.3 The scattering force

After determining the effect of one scattering event on the He* atoms, it is necessary to find out how many scattering events the He* atoms will process. Consequently, the next step is to determine at what rate the scattering events are processed. The scattering rate is given by

Rscatt = Γ 2 I /Isat 1 + I /Isat+ 4δ2/Γ2 (2.3) The scattering force is equal to the rate at which the light delivers momentum to the He* atoms times the photon momentum [9]. In other words, the scattering force is given by the momentum of the incident photons times the scattering rate. In equation form,

Fscatt = ~k Γ 2 I /Isat 1 + I /Isat+ 4δ2/Γ2 (2.4)

where I is the intensity of the light interacting with the He* atoms and Isat = 0.16 mW cm−2is the

saturation intensity [9], which is a measure for the minimum light intensity needed to continuously induce scattering events.

The quantity δ = (ω − ω0) + kv is called the frequency detuning from resonance [9]. To minimize

this detuning, monochromatic 1083 nm light with a frequency ω equal to the atomic resonance frequency ω0is used in this research. The remaining expression δ = kv is equal to the Doppler shift

of the light. The He* atoms gain a total scattering velocity vscatt = n · vrec in the direction of the

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Because of this velocity, the light becomes red-shifted in the reference frame of the He* atoms. Therefore the expression of the frequency detuning from resonance can be written as δ = kvscatt.

The natural linewidth of the 23_S

1 → 23P2 transition for metastable helium is given by Γ =

2π · 1.6 MHz [8]. Γ gives a rough estimate on how many scattering events the He* atoms can process before they become insensitive to the laser light due to the Doppler shift. This happens when δ > Γ.

We can now make an estimate on the maximum scattering velocity the He* atoms can gain before becoming insensitive to the laser light. This velocity, denoted vmax, is given by

vmax =

Γ k =

Γλ

2π (2.5)

where vmaxk is the value of Γ for which Γ = δ. The maximum scattering velocity is estimated to

be vmax ≈ 1.73 m s−1.

With the typical interaction times between the He* atoms and the resonant light in this research this maximum scattering velocity should not be exceeded. This means that for the measurements performed in this research δ ≤ Γ. For this reason the estimation can be made that the final term in equation (2.4) is negligible, so 4δ2_/Γ2 _{≈ 0. This approximation results in a simplified expression}

for the scattering force, which is now given by Fscatt= ~k Γ 2 I /Isat 1 + I /Isat (2.6) With this expression for the scattering force it is possible to estimate the total amount of momentum absorbed by the He* atoms. Since force is equal to the derivative of momentum over time, the amount of momentum absorbed by the He* atoms can be determined by integrating the scattering force over the interaction time, which is the time in which the He* atoms interact with the resonant light.

2.4 Beam width and light intensity

The interaction time is determined by the velocity of the He* atom beam perpendicular to the light beam and the interaction length. The velocity of the He* atom beam perpendicular to the light beam is 1200 m s−1 at the interaction region [8]. The interaction length L is the path length in which the He* atoms are illuminated by laser light of intensity I ≥ Isat. The interaction length

is calculated by means of the laser beam width w and the intensity I of the light.

The light beam is coming out of a fiber. Light from a fiber always has a Gaussian beam profile. The intensity I (x ) of a Gaussian light beam decreases as a function of the distance x from the center of the beam according to

I (x ) = Ipeak· exp(−2x2/w2) =

2Pmax

πw2 · exp(−2x

2_/w2₎ _(2.7)

where Pmax is the total power of the light beam and w denotes the beam width. It naturally

follows from equation (2.7) that the beam width w is defined as the distance from the center of the beam at which the intensity of the light has decreased by a factor e−2. Equivalently, at a distance w from the center of the beam, the intensity is approximately 13.5% of the peak intensity Ipeak of

the light beam.

For a given beam width w and total light power Pmax, one can determine the interaction length L

of the He* atoms with the resonant light, assuming that the He* atoms only interact with the light for intensities I ≥ Isat. To find the value of L, one needs to fill in I (x ) = Isat = 0.16 mW/cm2 in

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of the beam within which the light intensity is higher than or equal to the saturation intensity Isat.

The total interaction length is given by L = 2x , which results in

L = s 2w2_{· ln} 2Pmax Isatπw2 (2.8)

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Chapter 3

Setup

Figure 3.1 shows a schematic representation of the complete setup and its components. This figure also shows the coordinate system which is used to visualize the setup and the directions of the atom beam and the light beam.

A gas of 11_S

0 ground state helium atoms from a helium gas cylinder (1) enters the setup via the

source (3). A schematic representation of the source is shown in figure 3.2. The atoms go through a Tantalum needle, which is subject to a 2.5 kV discharge. Because of the discharge the helium atoms are excited from the 11S0ground state to the 23S1metastable state. Liquid nitrogen is used

to cool down the source.

The metastable helium atoms leave the source in the positive x-direction through a 250 µm diameter nozzle.

Figure 3.1: Schematic representation of the setup and its components (view from above).

The coordinate system with the relevant directions is also shown. A set of turbo pumps

is used to keep the setup in a vacuum.

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To collimate the atom beam, a slit (5) is placed 8.1 ± 0.5 cm from the nozzle. This slit has a width of 160 µm or 40 µm, depending on the measurement. After passing through the slit, the He* atoms cross the 1083 nm light beam. The wave vector of the light is directed in the y-direction, perpendicular to the He* atom beam. The center of the light beam is situated about 3 cm from the slit.

Because the slits used in this research have a height of about 3 mm, a collimating hole with a diameter of 4.00 ± 0.05 mm is placed 48.5 ± 0.7 cm from the nozzle. The hole is placed to make sure no light or other particles can reflect from the edges of the Zeeman slower and reach the end of the setup where a Faraday Cup detector (7) is placed. The Faraday Cup detector is discussed in section 3.2.

The hole has no effect on the horizontal collimation of the atom beam. After passing through the hole, the He* atoms enter the Zeeman Slower, which is a circular tube with an inner diameter of 3.5 cm.

3.1 Helium source and He* beam

The helium gas entering the source consists of 11_S

0 ground state atoms. The ground state atoms

go through a Tantalum needle. To induce the excitation from the 11_S

0 ground state to the 23S1

metastable state, a voltage discharge of 2.5 kV is applied to the Tantalum needle.

The metastable helium atoms leave the source through the 250 µm diameter nozzle. The atom beam entering the nozzle consists of a number of He* atoms in the order of 1014per steradian per second [10]. The helium source including the Tantalum needle and the nozzle is shown in figure 3.2.

Figure 3.2: Schematic representation of the helium source [8].

The plasma of He* atoms created by the discharge reaches high temperatures, which can deform the source and make the setup unusable. Using liquid nitrogen, the source is cooled down to a temperature of 77 K . The low source temperature also makes the transition from the ground state to the metastable state more efficient, creating more He* atoms. An additional effect of this low temperature is that the average velocity of the metastable helium atoms leaving the source is halved to about 1200 m s−1 [8].

3.2 Faraday Cup detector

A Faraday Cup detector is a detector that uses the charge or the internal energy of particles to create a signal. It is typically used to detect electrons or other charged particles.

The Faraday Cup detector consists of an aluminum cylindrical case with an 8 mm opening in the front. A 160 µm slit is placed directly behind the opening of the detector. An electrically

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isolated stainless steel plate is located about 2 cm behind the opening of the detector. A schematic representation of the Faraday Cup detector is shown in figure 3.3.

Figure 3.3: Schematic representation of the Faraday Cup Detector [8]. In this research, a

160 µm slit is placed directly behind the opening to increase the resolution of the signal.

The He* atoms enter the Faraday Cup through the 160 µm slit. This slit is placed directly behind the 8 mm opening to increase the resolution of the signal. The typical deflections measured in this research are in the order of 1 mm which is smaller than the original opening.

After entering the detector, the atoms hit the stainless steel plate. The metastable helium atoms created by the discharge have an internal energy of 20 eV. This is high enough to induce surface charge ionization on the stainless steel Faraday cup plate, because stainless steel has an ionization potential of 4.4 eV [8].

When a He* atom hits the stainless steel plate, the atom loses its internal energy and induces surface charge ionization of the plate. This results in the release of an electron from the plate. There is a potential difference of +300 V between the outer aluminum case of the detector and the plate. Because of this potential difference, the released electrons are attracted by the outer case of the Faraday Cup. In order to neutralize the resulting positive charge of the plate, a flow of electrons is induced from earth to the Faraday Cup plate. this flow of electrons results in a current which is measured by an external readout.

The current induced by the He* atom beam hitting the Faraday Cup plate is in the order of 1 pA, which is too low for a typical voltmeter to measure. For this reason a current amplifier is used which creates an output voltage increased by a factor 109V/A. To clarify this, an induced current of 1 pA results in an output signal of 109 pV = 1 mV. The induced current is a measure for the amount of He* atoms that hit the Faraday Cup plate.

The distance between the first slit (position 5 in figure 3.1) and the Faraday Cup plate is measured to be 272 ± 2 cm. The expected width of the image of the He* beam on the Faraday Cup can be de-termined by the combination of holes and slits. By using the 160 µm slit, the He* atom beam will be 14±1 mm wide at the Faraday Cup. When using the 40 µm slit, this image will be 11±1 mm wide. The Faraday Cup detector can be moved horizontally in the y-direction along the full width of the Zeeman Slower tube. The displacement of the detector is monitored with a translation stage attached to the detector.

3.3 Laser

The light source used in this research is a fiber which emits a round Gaussian beam of 1083 nm light. In order to make sure that the frequency of the light remains resonant with the 23_S

1 → 23P2

transition, the laser is actively locked to the resonant frequency by means of a helium cell. This is necessary because the frequency of the unlocked laser has a tendency to drift due to environmental factors like the temperature inside the laboratory.

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The total power of the light coming out of the fiber is about 2.3 mW. The light first passes a lens which collimates the light into a parallel beam. In order to make sure that no back reflections will enter the fiber and disrupt the laser, an optical isolator is used. An isolator blocks the light from one direction, while allowing the light from the other direction to pass [11]. This way it is made sure that the laser is protected from reflected light entering the fiber. The downside of the isolator is that about 25% of the power of the light is lost. The remaining power is about 1.7 mW. To make sure that the intensity of the light at the interaction region with the He* atom beam is high enough to induce scattering events, it is essential that further power loss is limited to a minimum. For this reason optical mirrors are used that are particularly coated for the reflection of 1083 nm light. The mirrors have an average reflectance of Rave > 0.99 for angles of incidence of

0 − 45 degrees. Power measurements behind the set of five mirrors showed a remaining light power of about 1.6 mW. The final two mirrors are used to make sure that the light beam arrives at the He* atom beam horizontally and directly perpendicular to the direction of the atom beam.

3.4 Controlling the light intensity

One of the main goals of this research is to measure the deflection of the He* atoms at different light intensities, in order to test the theoretical relation between the light intensity and the scat-tering force. A convenient way to vary the intensity is by means of a combination of a half-wave plate and a Polarizing Beam Splitter (PBS).

The coordinate system of a half-wave plate consists of a fast axis and a slow axis. When linearly polarized light passes a half-wave plate, its polarization direction is rotated [12]. The wave vector of an incoming wave has a certain direction of polarization. This direction of polarization can be divided in the two components of the coordinate system. One component in the direction of the fast axis of the half-wave plate (the fast component), and one component in the direction of the slow axis of the half-wave plate (the slow component). This way every wave vector can be drawn in a coordinate system with a fast axis and a slow axis. When passing the half-wave plate, the wave vector is mirrored with respect to the fast axis of the coordinate system. This happens because the slow component of the wave vector progresses slightly slower inside the half-wave plate than the fast component, shifting the phase of the wave by 180 degrees. Therefore, by turning the half-wave plate one is turning the direction of polarization of the wave. This is demonstrated graphically by figure 3.4.

After passing the half-wave plate, the light goes through the PBS. This optical component allows light of certain polarization directions to pass, while the light waves of other polarization directions are deflected sideways.

In short, by turning the half-wave plate, the half-wave plate/PBS combination allows part of the light to pass, while the rest is deflected. This way, by turning the half-wave plate, the light intensity can easily be changed.

The half-wave plate can be turned continuously over 360 degrees. Figure 3.5 shows the measured relation between the angle of rotation and the share of the incoming light allowed to pass through the combination of half-wave plate and PBS. The half-wave plate was turned in steps of 10 degrees. At every step, the output power was noted.

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Figure 3.4: Graphic representation showing the mirroring of the wave vector with respect

to the optical axis. The wave vector can be divided into a component in the direction of

the slow axis, and a component in the direction of the fast axis. Inside the half-wave plate,

the slow component of the wave vector progresses slightly slower than the fast component,

mirroring the wave vector. For this reason, a half-wave plate is also called a retarder. This

image is a modified version of the image found in the book by Hecht [12].

Figure 3.5: Graph showing the measurement of the relation between the degree of rotation

of the half-wave plate on the x-axis and the light power let through by the PBS on the

y-axis. A sinusoidal fit through the data points results in a minimum output power of

0.059 ± 0.016 mW and a maximum output power of 1.306 ± 0.016 mW.

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3.5 Vacuum pumps

The entire setup is hermetically closed off from the outside and pumped to a vacuum by a set of turbo pumps. During the measurements, when the helium source is running, the pressure inside the source chamber is kept at 5.1 · 10−5mbar. The pressure inside the rest of the setup is kept at 10−7 mbar.

It is desired that the He* atoms are not disturbed by other particles present in the setup. The atom beam must be as clean as possible. When a He* atom interacts with a neutral particle in the setup, the He* atom can lose its internal energy and fall back to the ground state. A helium atom in the ground state can no longer be deflected by resonant light and can no longer be detected by the Faraday Cup detector.

This research is dependent on metastable helium atoms. By applying a discharge on the Tantalum needle in the source, a plasma of excited metastable helium is created. To maintain this excitation, the pressure in the source chamber needs to be at 5.1 · 10−5 mbar level or lower. When the pressure in the source chamber is too high, other particles may pollute the metastable helium plasma, which can cause it to extinguish.

The presence of particles like oxygen and water vapor in the setup can create oxidation of the Tantalum needle or other parts of the setup. This can decrease the efficiency of the discharge and subsequently decrease the quality of the measurements.

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Chapter 4

Methods

4.1 Knife-edge measurements

The average number of scatterings the He* atoms will process is dependent on the interaction time of the atoms with the resonant light. It is therefore necessary to characterize the light beam, i.e. determine the beam width of the beam at the interaction region where the light and the He* atoms interact with each other.

To determine the beam width, a knife edge measurement is performed at different locations before and after the interaction region. The positions of measurement are 33.3 cm and 14.4 cm before the interaction region and 14.4 cm and 33.3 cm behind the interaction region.

The reason that knife-edge measurements are performed both before and after the interaction re-gion with the He* atom beam is because it is not only necessary to know the beam width w , but also whether the light beam is divergent or convergent. Divergent and convergent beams will result in curvature of the wavefront, which can influence the number of scattering events processed by the He* atoms, and therefore the deflection.

It is desirable that the light beam is perfectly collimated, resulting in an equal beam width at every location of measurement. In practice, the light beam will be slightly convergent or divergent. In this case the light forms a curved wavefront. In case of a curved wavefront, the number of scattering events that the He* atoms can process before they become insensitive to the light changes. For convergent beams, the maximum number of scattering processes increases because the cur-vature of the light beam moves along with the direction of movement of the He* atom beam. Therefore the effect of the Doppler shift of the light is eliminated to a certain extent. The di-rection of illumination remains perpendicular to the didi-rection of the He* atom beam for a longer period of time and the maximum scattering velocity gained by the He* atoms before they become insensitive to the laser light increases. For divergent beams, the effect is opposite and the maxi-mum number of scattering processes decreases.

A knife-edge measurement is performed by means of a sharp razor blade that is placed upon a translation stage. The translation stage is placed perpendicular to the direction of the light beam. The output power of the light beam is measured by a sensor connected to a power meter that measures the amount of incident 1083 nm light.

The razor blade is first placed in such a way that it blocks the light beam completely. No laser light reaches the sensor in this way. By means of the translation stage the razor blade is then pulled back in steps of typically 100 to 500 µm, unblocking an increasing part of the light beam in the process. At every step the output power of the laser beam is noted. This results in a graph showing the relation between the knife-edge displacement and the output power detected by the sensor.

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a knife-edge measurement is the integral over the Gaussian light beam. The integral of a Gaussian function is the error function, which is defined as

erf(x ) =√2 π

Z x

0

e−t2dt . (4.1)

To find the beam width w , a fit containing the error function from equation (4.1) is used on the knife-edge measurement. The fit is given by

P (x ) = P0+

1

2Pmax· erf( √

2(x − x0)/w ) (4.2)

In this function, P0 is the minimum power collected by the sensor, which is coming from

back-ground light, and Pmax is the maximum power collected by the sensor. As for the quantities within

the brackets of the error function, x is the position of the knife edge and x0denotes the peak

posi-tion of the Gaussian light beam. The beam width w is defined in secposi-tion 2.4 as the distance from the center of the beam where the intensity of the light has decreased to a factor e−2 of the peak intensity.

An example of a knife-edge measurement graph is shown in figure 4.1. This measurement was performed 33.3 cm from the interaction region with the He* atoms on a round Gaussian light beam. The measurement resulted in a beam width of 2.13 ± 0.03 mm

Figure 4.1: Knife-edge measurement performed on a round Gaussian light beam 33.3 cm

before the interaction region with the He* atom beam. The fit function shown in equation

(4.2) resulted in a beam width of w = 2.13 ± 0.03 mm.

4.2 Atom beam profiles

The metastable helium atoms move through the Zeeman slower until they reach the end of the setup where the Faraday Cup detector is placed. The Faraday Cup detector can be moved horizontally

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perpendicular to the He* beam. The Faraday Cup detector is moved sideways in small steps of 1 mm. The position of the cup can be read directly from a translation stage connected to the detector.

At every position, the output signal is given by an external voltmeter. This voltmeter gives a signal in the form of a voltage, which is created by a current amplifier that multiplies the current induced in the Faraday Cup detector by a factor 109 _{V/A. The voltage readout typically shows}

an output in the order of a few mV. As discussed in section 3.2, the output voltage is a measure for the number of metastable helium atoms collected by the Faraday Cup detector.

The step-by-step voltage output measurement along the He* atom beam results in a Gaussian-like atom beam profile. An example of a measured atom beam profile is shown in figure 4.2. For this profile, the 160 µm slit was placed in the atom beam (position 5 in figure 3.1). For this profile, no light was used to deflect the atoms.

Figure 4.2: Atom beam profile without deflection. The 160 µm was placed in front of the

atom beam, and no light was used to deflect the atoms. For this beam profile the Gaussian

fit from equation (4.3) was used that resulted in a peak position of 32.207 ± 0.059 mm.

The goal of this research is to explore the relations between the deflection of the He* atom beam, the intensity of the light and the beam width of the light beam. In order to do this, different parameters were varied in the measurements.

A total of four sets of measurements are explored. For the first three sets of measurements a round Gaussian light beam is used. Two of these sets are focused on the deflection in the positive y-direction; the first set uses the 160 µm slit, and second set uses the 40 µm slit. Using the 40 µm slit should result in a narrower, more collimated atom beam containing less He* atoms. This is done to see whether a narrower atom beam results in a more accurate determination of the deflection. The third set of measurements uses the 160 µm slit just like the first set, but explores the deflection in the negative y-direction to see whether this shows notable differences with the first two sets of measurements. In order to do this, two extra 1083 nm coated optical mirrors were used to redirect

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the light beam around the setup. This measurement is performed to see whether the results of the measurement in the negative y-direction are consistent with the results found for the measure-ment in the positive y-direction. It is a test to see whether the setup is consistent in both directions. The final set of measurements again explores the deflection in the positive y-direction, like the first two sets of measurements. The 160 µm slit is again placed in front of the atom beam, but the interaction time between the atoms and the laser light is increased. This is done by means of a setup of two cylindrical lenses which form a telescope that creates a wider collimated light beam. A wider light beam should result in a greater interaction length, more scattering events and therefore a greater deflection of the He* atom beam. This set of measurements is also used to see whether the theoretical maximum value of the deflection of 4 mm discussed in the next section can be met or even exceeded.

The final parameter that is varied in all sets of measurements is the intensity I of the light. In order to test the relation between the light intensity and the scattering force, every measurement set consists of measurements with different light intensities. The light intensity is varied by means of the half-wave plate and PBS discussed in section 3.4.

The results collected with the four sets of measurements will be compared quantitatively relative to each other. By means of these comparisons, the factors that cause the results to differ from the expecations based on the theory are discussed.

4.3 Determining the deflection

The four sets of measurements result in a number of atom beam profiles. For every profile in each set, a different light intensity is used. According to the theory of the scattering force, this re-sults in different amounts of momentum absorbed by the He* atom beam and therefore a different scattering velocity in the y-direction. Because of this scattering velocity, the measured He* atom beam profile will be shifted relative to the atom beam profile measured without the use of the light beam. This deflection of the atom eam profiles is what we want to determine.

The beam profiles are all analyzed in the same way. A Gaussian fit function of the form

V (x ) = V0+ A · exp(−0.5(x − xc)/w ) (4.3)

is used to determine the peak position of the Gaussian atom beam profile. In this function V (x ) is the output voltage of the Faraday Cup detector dependent on the detector position x , V0is the

minimum output voltage derived from a background offset, and A is an empirical fit parameter. The quantity w denotes the beam width of the atom beam with the same definition as the one for a light beam, and xc is the peak position of the atom beam. The shift in this peak position xc

relative to the peak position of the undeflected atom beam profile is defined as the deflection.

4.4 Testing the (F,I)-relation

An important part of this research is to test whether the measured values for the deflection behave according to the theory. This can be done by testing the relation between the light intensity and the scattering force. The equation for the scattering force used to test the (F , I )-relation is

Fscatt= ~k Γ 2 I /Isat 1 + I /Isat (4.4) which was also shown in equation (2.6). Every quantity in this equation has a fixed value except for the light intensity I . The intensity continuously changes over the interaction length of the light beam, which makes it mathematically very difficult to determine the scattering force. To solve this issue it is necessary to find a good approximation of the average intensity over the interaction

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length of the beam.

An approximation of the average intensity of the light beam over the interaction length is Iave=

Ipeak

2 =

Pmax

πw2 . (4.5)

where Pmax is the total light power of the light beam interacting with the He* atoms and w is the

beam width of the laser beam. This approximation says that the average intensity is half the peak intensity.

Important to note is that the average intensity for the final set of measurements is slightly different. The beam width of the light beam used for the fourth set of measurements is increased by a set of cylindrical lenses. Due to the cylindrical lenses this is no longer a round Gaussian light beam, but has an elliptic shape, and equation (4.5) no longer holds. In this case the average intensity is given by

Iave=

Pmax

πw1w2

(4.6) where w1 is the horizontal beam width of the elliptic laser beam. The vertical beam width w2 is

the same as the beam width w in the first two sets of measurements, because the beam is only broadened horizontally.

Now we can find the value of the total scattering force for each light intensity. The next step is to determine the total momentum transferred to the He* atoms. To do this, we need to know the interaction time of the light beam with the He* atoms. The interaction time is given by

τ = L

v (4.7)

where L is the interaction length given by equation (2.8) and v the average velocity of the He* atoms which is 1200 m s−1.

The total momentum absorbed by the He* atoms can be calculated by integrating the scattering force over the interaction time. However, using the approximations and assumptions mentioned earlier, the absorbed momentum can be calculated by simply multiplying the scattering force by the interaction time.

The final step in finding the total scattering velocity gained by the He* atoms is dividing the absorbed momentum by the mass of a helium atom. In conclusion, the scattering velocity gained by the metastable helium atoms is given by

vsc= ~k mHe Γ 2 Iave/Isat 1 + Iave/Isat L v (4.8)

The scattering velocity gained by the He* atoms determines the deflection of the He* atom beam. The distance between the interaction region and the Faraday Cup plate is measured to be 269 ± 2 cm. Taking the velocity of the He* atom beam to be 1200 m s−1, the time of flight for the He* atoms from the interaction region to the Faraday Cup plate is 2.24 ± 0.02 ms. The deflection is given by the scattering velocity multiplied by the time of flight of the He* atoms.

In section 2.3 it was shown that the maximum scattering velocity that the He* atoms can gain before becoming insensitive to the 1083 nm light is vmax ≈ 1.73 m s−1. Taking the time of flight

into account this comes down to a theoretical maximum deflection of about 4 mm for a perfectly collimated light beam.

With the assumption that the intensity of the light is constant over the full interaction length, it follows that the deflection is directly proportional to the scattering force. Because of this, the (F , I )-relation can be tested by determining the relation between the light intensity and the deflec-tion. Figure 4.3 shows the ideal relation between the scattering force and the light intensity. The

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graph shows equation (4.4) with the normalized intensity I /Isat on the x-axis and the scattering

force Fscatt on the y-axis.

In this research the deflection is measured for normalized intensities I /Isat as shown in figure 4.3.

The expectation is that the sets of measurements show a similar relation between the deflection and the normalized light intensity as the relation shown in figure 4.3.

Figure 4.3: graphic representation of the relation between the scattering force and the

light intensity interacting with the He* atoms according to the theory. The graph shows

the normalized intensity on the x-axis and the scattering force on the y-axis. Since the

deflection is assumed to be directly proportional to the scattering force, it is expected that

the measured relations between the deflection and the normalized light intensity will have

the same form.

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Chapter 5

Results and Discussion

For each measurement in the four sets of measurements discussed in section 4.2, the theoretically expected deflection is calculated by means of the beam width w and intensity I of the 1083 nm light beam and the interaction length L of the light with the He* atom beam.

The calculated values for the deflection result in a plot showing the theoretically expected relation between the light intensity I and the deflection of the He* atom beam.

The measured deflections are determined by means of the measured beam profiles and the Gaussian fit analysis discussed in section 4.3. The measured values for the deflection in their turn result in a plot showing the relation between the light intensity I and the measured deflection of the He* atom beam.

For each set of measurement the results are discussed and when necessary it is explained why the results differ from the expectations based on the theory.

The three main parameters that are varied are the slit width (and with it the number of atoms), the direction of illumination, and the interaction time. To see whether the variations of the parameters have the expected effect on the deflection of the atom beams, the sets of measurements are compared to each other and quantitatively analyzed.

5.1 160 µm slit measurement in the positive y-direction

For the first set of measurements, the 160 µm slit was placed in front of the He* atom beam. The deflection is measured in the positive y-direction, which we will call deflection to the left.

For this set of measurements, a round Gaussian light beam was used. The four knife-edge mea-surements performed on this light beam resulted in a beam width of w = 2.36 ± 0.02 mm at the interaction region with the He* atom beam. The light beam is divergent with an angle of curva-ture of about 0.7 mrad. The effect of a divergent light beam is that the the He* atom beam will sooner become insensitive to the laser light than it would with a perfectly collimated light beam, which will result in lower values for the deflection. However, an angle of curvature of 0.7 mrad is negligible and will not significantly affect the measured deflection.

The deflection was measured for different total light powers Pmax, resulting in different light

in-tensities and interaction lengths. This is done for every set of measurements.

The deflections were determined by measuring the shift in the peak positions of the measured He* atom beam profiles. The peak positions were determined with the Gaussian fit function from section 4.3. Two of the He* atom beam profiles are shown in figure 5.1.

The red plot shows the He* atom beam profile deflected by light with a power of 1.60 ± 0.01 mW light. The black plot shows the He* atom beam profile that is not deflected. The deflections are determined with respect to the peak position of the undeflected beam profile. The shift of the red plot with respect to the black plot was measured to be 0.859 ± 0.095 mm. This is significantly lower than the theoretical value of 6.17 ± 0.17 mm.

The deflections measured for each light power, and therefore different normalized intensities I /Isat,

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Figure 5.1: Graph showing two measured He* atom beam profiles using using the 160 µm

slit. The black plot shows the Gaussian fit on the He* atom beam profile that was not

deflected by light. The peak position was determined to be 32.207 ± 0.059 mm. The

red plot shows the Gaussian fit on the He* atom beam deflected by light with a power of

1.60±0.01 mW. The peak position of this profile was determined to be 31.348±0.074 mm.

deflections for each normalized intensity I /Isat. The red plot shows the measured deflections for

several normalized intensities.

The measured values for the deflection shown in figure 5.2 seem to show a linear relation with the normalized intensity I /Isat. The values for the measured deflection are also significantly lower

than the theoretically calculated values. The explanation for this can be found in the alignment of the light beam. It is possible that the He* atom beam was not illuminated by the center of the light beam, but by the region under (or above) the center of the light beam where the intensities are significantly lower.

Looking at figure 5.2, the maximum measured deflection is 0.859 ±0.095 mm. This is in accordance with a misalignment of approximately 4.3 mm. Since the alignment is performed by eye and the He* atom beam can not be seen, this misalignment is a plausible reason for the lower measured deflection.

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Figure 5.2: Graph showing the theoretically calculated deflections and the measured

de-flections for several normalized light intensities I /I

sat

. The measured deflection is

signif-icantly lower than the theoretical deflection. The measured deflections seem to show a

linear relation with the intensity.

5.2 The effect of a narrower slit

In this section, the comparison will be made between the measurements with the 160 µm slit and the 40 µm. For both sets of measurements, the deflection is to the left, in the positive y-direction. Since a greater part of the He* atom beam will be blocked due to the narrower slit, it is expected that the output voltage will be lower. Based on the dimensions of the slits and holes in the setup, it was discussed in section 3.2 that the image of the He* atom beam on the Faraday Cup detector with the 40 µm in front should be about 3 mm smaller than the image with the 160 µm slit. What we want to find out with this comparison is whether the use of a narrower atom beam results in more accurate results.

Figure 5.3 shows two deflected He* atom beam profiles. Both atom beams were deflected by the same light beam with a beam width of 2.36 ± 0.02 mm. The red plot shows an atom beam with the 160 µm slit in front. The black plot shows an atom beam with the 40 µm slit in front. Both are deflected in the positive y-direction by light with a total power of Pmax = 0.20±0.01 mW.

The expected deflection for this light power and beam width is 4.122 ± 0.294 mm. The measured deflection is measured to be 0.081 ± 0.075 mm for the atom beam profile with the 160 µm slit in front. With the 40 µm slit in front, the measured deflection is 1.439 ± 0.129 mm.

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Figure 5.3: Graph showing He* atom beam profiles measured with the 40 µm slit (black

plot) and the 160 µm slit (red plot) placed in front of the atom beam. The peak position

of the black beam profile is determined to be 30.53 ± 0.12 mm. The peak position of the

red beam profile is determined to be 32.13 ± 0.05 mm. The beam widths of the black and

red plot beam profiles are 3.80 ± 0.08 mm and 4.05 ± 0.20 mm, respectively.

The output voltage is significantly lower for the measurement with the 40 µm slit in front of the atom beam. The peak voltage is 1.80 ± 0.05 mV against a peak voltage of 3.75 ± 0.05 mV with the 160 µm slit in front of the atom beam. This is in accordance with the expectation.

The beam widths are found to be nearly equal and within each others error flags. This is not in accordance with the expectation. A possible explanation can be found in the deflection of the He* atom beams.

As shown in section 5.1, the He* atom beam with the 160 µm slit in front was hardly deflected due to misalignment of the laser light. The He* atom beam with the 40 µm slit in front was also deflected less than expected according to the theory, but it was clearly deflected.

To explain why the beam width of the beam profile with the 40 µm slit in front is this broad, we need to look at the deflection of the He* atoms individually. Not every He* atom is deflected in the same way. Some atoms are deflected more than others, because the number of scattering events processed by the individual He* atoms is not large enough to average out the effect of the photon emission in random directions. This can result in a wider Gaussian distribution of the deflection, and therefore a wider beam width.

The error bars are equal for every output voltage, because the uncertainty in the current readout was 0.05 mV for every output voltage. Because the output voltages are lower for the beam profile with the 40 µm slit in front, the relative error is greater for this set of measurements. The mea-surements are therefore not more accurate for a narrower atom beam.

The deflections measured for each normalized intensity I /Isatfor both the 160 µm slit measurement

set and the 40 µm slit measurement set are shown in figure 5.4. In this graph, the black line shows the theoretically calculated deflections for each normalized intensity. The blue and red plot show

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the measured deflections with the 160 µm slit and the 40 µm slit in front of the He* atom beam, respectively.

Figure 5.4: Graph showing the theoretically calculated deflection (black line) and the

measured deflections with the 160 µm slit and the 40 µm slit in front of the He* atom

beam (blue and red plot, respectively) for each normalized light intensity I /I

sat

. The

red plot and black line have more or less the same shape, indicating that the measured

relation between the deflection and the light intensity is in accordance with the theory.

The measured deflections are significantly lower than the theoretical values.

The red plot containing the measured deflections with the 40 µm slit in front of the He* atom beam has the same shape as the black line showing the theoretical deflections. This indicates a relation between the deflection and the light intensity that is in accordance with the theoretical relation. The saturation effect at higher intensities is clearly visible.

However, the measured deflections are again significantly lower. An explanation for this can be found in equation (2.4). In order to be able to calculate the momentum absorbed by the He* atoms, an important assumption was made. This assumption stated that for this research, the term 4δ2_/Γ2_{≈ 0.}

However, the measured deflections approach the theoretical maximum deflection of 4 mm calcu-lated in section 4.4, implying that for these measurements δ ≈ Γ. Taking this factor into account, the scattering force would be significantly lower because in that case 4δ2_/Γ2_{≈ 4. This would}

re-sult in a lower value for the scattering force and therefore lower theoretical values for the deflection. Another possibility is that the approximation for the value of Iave given in section 4.4 is too

high. This approximation says that the average intensity is half the peak intensity over the entire interaction length. However, the interaction length increases with the intensity. Therefore it is probable that the estimation of Iave is too high at higher intensities, because the light power is

divided over an increasing interaction length. This results in theoretical deflection values that are too high.

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5.3 Deflection in different directions

This section focuses on the comparison between illumination in the positive y-direction and illumi-nation in the negative y-direction, or left deflection versus right deflection. To be able to illuminate the He* atom beam in the negative y-direction, the round Gaussian beam of light used for the previously shown measurements is redirected around the setup and enters the setup from the other side.

The beam width at the intersection with the atom beam is determined to be 2.99 ± 0.01 mm for the redirected light beam, again showing an angle of curvature of about 0.7 mrad. For this set of measurements, the 160 µm slit was again placed in front of the He* atom beam.

Figure 5.5 shows two He* atom beams deflected by a round Gaussian light beam with a total power of 1.00 ± 0.01 mW. The 160 µm slit was placed in front of the atom beam. The red plot shows the He* atom beam deflected in the positive y-direction (to the left). The black plot shows the He* atom beam deflected by the redirected light in the negative y-direction (to the right). For this graph, the x-axis was adjusted so that the value of zero represents the peak position of the undeflected atom beams. This way the deflection can be seen more clearly.

Since all parameters for these two measurements are equal except for the beam width of the light beam, which is slightly narrower for the red plot, and the direction of deflection, it is expected that the plots look similar and show similar deflections. The reason for this is that the deflection should not be dependent on the direction of illumination.

Figure 5.5: Graph showing two measured He* atom beam profiles. The black plot shows

the beam profile deflected to the right. The red beam profile is deflected to the left. Both

profiles show a Gaussian fit. The peak position of the black plot is determined to be

2.191 ± 0.058 mm. The peak position of the red plot is −0.677 ± 0.072 mm.

In section 5.1 it is discussed that the alignment of the light beam was not optimal for these measurements, which resulted in small values for the deflections. Therefore it is logical that the measured deflections for the two profiles in figure 5.5 are not equal. The deflection measured with

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the black plot is 2.191 ± 0.080 mm to the right, and the deflection measured with the red plot is 0.677 ± 0.083 mm to the left.

Another interesting feature of figure 5.5 is that both the maximum output voltage and the width of the atom beam of the black plot are greater than for the red plot. The beam width of the black plot is determined to be 3.90 ± 0.12 mm, whereas the beam width of the red plot is 3.52 ± 0.09 mm. Since the deflection of the black atom beam profile is greater, it is logical that the profile is also wider. The reason for this is that not every He* atom is deflected in the same way, which increases the width of the Gaussian distribution. This is explained in section 5.2. However, a wider profile should show a lower peak value, because the total number of atoms measured in both profiles should be equal. The area under the graphs should therefore be equal, which is not the case. The reason for this can be found in the alignment of the slit with respect to the He* atom beam. The slit was aligned all over again between the different sets of measurements. It is therefore probable that the number of atoms going through the slit was greater for this set of measurements in comparison with the previous ones, resulting in a greater area under the plot.

It is important to note that the number of atoms going through the slit and reaching the Faraday Cup detector should not have any effect on the deflection of the atom beam. The deflection is only dependent on the light beam and its characteristics.

Because of this, it should not matter whether the 160 µm slit or the 40 µm is placed in front of the atom beam. The measured deflections should be similar for the same light beam. The same should hold for the deflection measured in the opposite direction. If all other parameters are the same, the deflection should be the same as well, only in the opposite direction.

Figure 5.6 shows the comparison between the measured deflections in the positive y-direction with the 40 µm slit placed in front of the atom beam and the measured deflections in the negative y-direction with the 160 µm slit in front of the atom beam. For both cases, the theoretical deflection values are shown as well. This is done because the light beam width is slightly greater for the light beam that is redirected around the setup.

Because the deflection should not depend on the slit width, and because the deflection curve from section 5.1 does not show the expected relation between deflection and light intensity, the choice has been made to compare the deflection to the right with the deflection in the to the left using the 40 µm slit instead of the 160 µm slit.

In the figure, the black and red lines show the theoretical deflection values for the deflection in the negative y-direction and the positive y-direction, respectively. The black and red plots show the measured deflections for the deflection in the negative y-direction and the positive y-direction, respectively.

The measured deflection curves look similar and clearly show the saturation effect at higher inten-sities. Although the theoretical values for the deflection in the negative y-direction are higher, the measured deflections are slightly lower. This is probably due to a misalignment of the light beam. In figure 5.6 the effect of an important flaw of the theory becomes clear. The theory does not take into account that the He* atoms become insensitive to the laser light after scattering a certain number of photons, due to the Doppler shift of the light. It was discussed in section 4.4 that the theoretical maximum deflection possible for a perfectly collimated light beam is about 4 mm. Taking this into account, the theoretical deflection plots should not exceed the maximum deflection and should be flattening out earlier.

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Figure 5.6: Graph showing the theoretically calculated deflections (black and red lines)

and the measured deflections (black and red plots) for both deflection in the negative

y-direction and deflection in the positive y-direction, respectively.

5.4 The effect of a longer interaction length

In order to reach the maximum deflection discussed in the previous section, a telescope was cre-ated using two cylindrical lenses. Using cylindrical lenses the interaction length is increased, and with it the interaction time of the laser light with the He* atoms, without losing too much in-tensity. The knife edge measurements resulted in a light beam width at the interaction zone of w = 5.59 ± 0.08 mm, with an angle of divergence of about 1.6 mrad, which is negligible. Like in the first two sets of measurements, the light is again directed in the positive y-direction, resulting in de-flection to the left. For this measurement the 160 µm slit was placed in front of the He* atom beam. Figure 5.7 shows two of the atom beam profiles for this set of measurements. The black plot shows the profile of the undeflected He* atom beam. This undeflected plot is similar to the undeflected plot in figure 5.1. This is logical, since the only difference between the two sets of measurements is the beam width of the light beam. The red plot shows the profile of the atom beam deflected by a 1.30 ± 0.01 mW light beam.

For this set of measurements the deflection was again determined with the Gaussian fit discussed in section 4.3. However, the plots are no longer nicely Gaussian. The reason for this is that the He* atoms are so far deflected that part of the atom beam no longer reaches the Faraday Cup. These atoms hit the edge of the Zeeman slower, lose their internal energy and no longer induce a signal.

The peak position of the deflected He* atom beam, which is the red plot in figure 5.7, is determined to be 27.561 ± 0.209 mm according to the Gaussian fit. Looking at figure 5.7 this is a reasonable approximation, since the fit goes through most of the error flags of the measured datapoints. Figure 5.8 shows the theoretical values and measured values for the deflection for different

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nor-Figure 5.7: Graph showing an undeflected atom beam profile and a beam profile deflected

by an elliptic Gaussian light beam with a power of 1.30 ± 0.01 mW and a beam width of

5.59 ± 0.08 mm. The measured deflection with the Gaussian fit is 4.550 ± 0.214 mm.

malized light intensities. Important in this figure are the black line and the black plot, which are the theoretical deflection values and the measured deflection values for the long interaction length, respectively.

The deflections measured with the Gaussian fit are significantly lower than the theoretically cal-culated deflections. The main reason for this is that the theory does not take into account that the He* atoms become insensitive to the laser light after scattering a certain number of photons. However, the measured deflections do exceed the theoretical maximum deflection of about 4 mm discussed in section 4.4.

The maximum deflection of 4 mm was calculated for a perfectly collimated light beam. The elliptic light beam used for this set of measurements was even slightly divergent, which means that this cannot be the reason for the higher measured deflections.

An explanation for the higher measured values of the deflection can be found in the shape of the beam profiles of the deflected He* atom beams. Because the profiles are not nicely Gaussian, the Gaussian fit is not as accurate as desired. It is possible that the actual peak positions result in lower deflection values. However, looking at the measured atom beam profile in figure 5.7, the deflection seems to be slightly underestimated. If anything, the deflection could be even greater. It is important to understand that the parameters Isatand δ are measures on how sensitive the He*

atoms are to the laser light. At light intensities under Isat, the He* atoms still process scattering

events. Similarly, when the Doppler shift of the laser light with respect to the He* atoms exceeds the boundary of Γ = δ discussed in section 2.3, the He* atoms will still scatter photons, just not as frequently as before. It is plausible that the He* atom beam processed enough scattering events to exceed the theoretical maximum deflection of 4 mm. Because of the longer interaction time, the probability that the He* atoms process scattering events above the boundary of Γ = δ increases.

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Figure 5.8: Graph showing the deflection values for each normalized light intensity I /I

sat

for the widened elliptic Gaussian light beam with a beam width of 5.59 ± 0.08 mm. The

theoretical values of the deflections for this long beam width are shown by the black line.

The measured deflections for the same light beam are shown by the black plot. Also shown

are the measured deflections in the +y direction for the short beam width of 2.36±0.02 µm

discussed in section 5.1, as well as the measured deflections in the -y direction for the short

beam width of 2.99 ± 0.08 µm discussed in section 5.3. For all measurements shown in

this graph, the 160 µm slit was placed in front of the atom beam.

5.5 Additional points of discussion

In section 3.1 it is said that the He* atom beam leaving the source contains a number of He* atoms in the order of 1014 _{per steradian per second. But is this in accordance with the measured output}

voltages?

The slit placed in the Faraday Cup detector has an area of 0.48 mm2_{and is located about 2800 mm}

from the source. Therefore the slit spans 0.48/28002 _{= 6 · 10}−8 _sr.

This comes down to 6 · 10−8 · 1014 _{= 6 · 10}6_{He* atoms per second hitting the Faraday Cup plate.}

The plate has an efficiency of about 70% [13] so the number of He* atoms inducing a signal is about 4.2 · 106 per second.

Since the charge of a single electron is 1.602 · 10−19 C, the current induced by the He* atoms is 6.7 · 10−13 A.

The output signal is created by a current amplifier that multiplies the induced current by 109_V/A,

resulting in an output voltage of 0.67 mV.

The actual output voltages are measured to be between 0.8 mV and 5.0 mV. This is in the same order of magnitude as the output voltage calculated above. Therefore it is very plausible that the measured signals are in accordance with the estimation of 1014_{He* atoms per steradian per second}

leaving the source.

By using the interaction length L which is calculated with I (x ) = Isat we automatically make

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In practice the scattering force is greater than zero for every value of I > 0. However, without this assumption it would be very hard and too time-consuming for this project to calculate the theoretical values for the deflection. For follow-up projects it would be very interesting to make more accurate approximations.

To measure the displacement of the Faraday Cup, a translation stage was used that was accurate up to 0.1 mm. Since the measured deflections are in the order of a few millimeters, this translation stage is not an ideal measuring device. A suggestion for follow-up research is to install a translation stage with which the displacement of the Faraday Cup can be determined up to 10 µm or even lower. For a long time we were not able to measure Gaussian He* atom beams. The shape of the beam profiles was not stable, and showed patterns we are still not able to explain.

To make sure that the He* atom beams were not polluted by electrons or other charged particles which could potentially disrupt the signal, a magnet was placed on the Zeeman slower. The magnet was strong enough to filter all charged particles out of the atom beam. It was immediately clear that the use of the magnet did not change the signal. Therefore the idea that charged particles interrupted the signal was eliminated.

After a couple of weeks in which the setup was checked for flaws and possible causes for the strange signals, a solution was found. The only part of the setup that was not tested yet was the Faraday Cup detector. The hypothesis was that the problem had to be originating from the detector. In an attempt to find a solution, the Faraday Cup was tilted downward in an angle of about 10 degrees. With the tilted Faraday Cup the atom beam profiles show a clear Gaussian form with which the deflection can be measured. A comparison between the signal measured with the tilted Faraday Cup and the signal measured with the horizontally placed Faraday Cup is shown in figure 5.9.

Figure 5.9: Graph showing a typical plot measured when the Faraday Cup was placed

horizontally, so not tilted, in the setup. The red plot shows a typical atom beam measured

with the tilted Faraday Cup detector.

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Cup detector downward, a greater part of the He* atom beam reached the Faraday Cup plate, creating a nice Gaussian signal.

To test this, the Faraday Cup detector was removed from the setup and the slit placed in front of the plate was moved downward. The detector was placed horizontally back in the setup. Again we measured strange signals like the black plot in figure 5.9. After tilting the detector, nice Gaussian atom beam profiles were measured again.

The decision was made to do the measurements with the tilted Faraday Cup detector. It is still unclear what the cause of the strange signals is, and why the atom beam profiles do look nicely Gaussian when the Faraday Cup detector is tilted. It is possible that the back electrodes of the Faraday Cup detector are reached by the He* atoms when the detector is tilted. Whether this has anything to do with the output signal is a question for follow-up research.

Some of the measurements in this project were subject to misalignment of the laser light. To be able to make better comparisons between the different sets of measurements, follow-up research should focus on a method to align the laser light consistently in the same way.

The beam profiles of deflected He* atom beams have a greater beam width than the beam profile of the undeflected atom beam. A greater beam width also results in a lower peak output voltage. The reason for this is that not all He* atoms are deflected in the same way. The deflection of the particles results in a wider Gaussian distribution. Because the total number of He* atoms collected by the Faraday Cup detector must remain approximately equal - atoms cannot disappear - the output peak of a wider beam profile is lower.

Some of the measurement on the deflection in the negative y-direction, discussed in section 5.3, did not show these features. The peak output voltage of some of the deflected atom beam profiles was higher than the peak output voltage of the undeflected profile. A clear explanation for this result has not yet been found. A suggestion for follow-up research is to repeat the measurements with a properly aligned light beam, and also measure the deflection in the negative y-direction for a light beam with a greater beam width and by using the 40 µm slit. The results of these measurements could lead to a better understanding of this problem.

Finally, every measurement showed a deflection that was lower than the theoretically expected value. It has been discussed that the main reason for this is that the theory does not take into account that the sensitivity of the He* atoms for the resonant light decreases for light intensities I < Isat and for values for the frequency detuning from resonance of δ > Γ. For this reason

approximations and assumptions were made. However, as shown in the results section, these approximations and assumptions were not sufficient.

It can be very interesting for follow-up projects to create extensive models, calculations and sim-ulations in order to accurately determine the theoretical values for the deflection.

Deflecting Metastable Helium Atoms with Resonant Light