Frequency stabilisation of a 1054 nm laser with a frequency comb laser

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Frequency stabilisation of a 1054 nm

laser with a frequency comb laser

Marjolein Gelauff

10409904

Report Bachelor Project Physics and Astronomy

Size 15 EC; Conducted between 18-04-2017 and 26-07-2017

Date of submission: 22-06-2018

Supervisor: Prof. dr. K.S.E. Eikema

Daily supervisors: L.S. Dreissen & C. Roth

The Ultrafast Laser Physics and Precision Metrology Group

Second assessor: Dr. Stefan Witte

Faculty of Science

Natuur- en Sterrenkunde

Universiteit van Amsterdam

LaserLab

Department of Physics and Astronomy

Vrije Universiteit Amsterdam

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Abstract

Precision spectroscopy in electronic (normal) and muonic hydrogen can be used to measure the size of the proton. However, the results differ by 4% (5σ), which is now called the Proton Radius Problem. To solve this puzzle an experiment at the VU is prepared to do precision spectroscopy on helium ions to measure the alpha particle radius. Part of this experiment aims to trap a helium ion in an ion trap together with a beryllium ion, so that experimental research can be done on the helium ion.

Ions can be captured in an ion trap with oscillating electrical fields and laser cooling techniques. However a laser beam part of the extreme ultraviolet (XUV) spectrum would be required to cool a helium ion, a laser with such a wavelength cannot be created. Since a beryllium ion can be cooled using a longer wavelength, the cooling laser beam will be created with a wavelength of 313 nm, using a 1054 nm laser and a 1542 nm laser. The effectiveness of the cooling process of the beryllium ion is dependent on the stability of the 1054 nm laser. This bachelor project contributes to the stabilisation of the 1054 nm laser to a frequency comb laser.

To do this, a mode containing the 1054 nm frequency was selected from the frequency comb beam. Locking and referencing the 1054 nm laser with the selected comb mode and the ultra-stable laser sets an upper limit to the accuracy and drift in the experiments with the 1054 nm laser. The highest possible accuracy of the 1054 nm laser can be found by first determining the difference in optical frequency between the laser and the reference laser beam, i.e. the beat note. Then by determining the frequency of the reference laser beam the 1054 nm laser frequency can be found and the two frequencies can be locked.

For this bachelor project a set-up was built to measure the beat note between a 1054 nm laser and an ultra-stable frequency comb laser. To create the reference laser beam, a mode containing the 1054 nm frequency was selected from the 1050 − 2000 nm spectrum of the frequency comb beam. The other wavelengths were blocked by 1064 nm mirrors, a transmission grating and a slit. With a fiber coupler the reference laser beam was connected to the same fiber as the 1054 nm laser beam. As a final result of the bachelor project fBN = 34.7765 MHz ± 1.7 kHz has been measured.

This result is the first step to stabilise the 1054 nm laser, which then can be used to better cool helium ions.

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Populair wetenschappelijke

samenvatting

Laserspectroscopie is sinds het begin van de laatste eeuw een fundamentele experimentele on-derzoeksmethode. Tijdens mijn bachelor project heb ik een bijdrage geleverd aan een experimenteel onderzoek dat ambieert een helium ion en een beryllium ion in een ionenval in te vangen om op het helium ion te experimenteren. Ionen kunnen ingevangen worden in een ionenval met behulp van oscillerende elektrische velden en laserkoeltechnieken. Voor het koelen van helium ionen zou er een laserstraal met een hele korte golflengte nodig zijn. Deze golflengte is uit een deel van het elektromagnetische spectrum waar geen laserstralen mee gemaakt kunnen worden. Daarom wordt in het LaserLab van de VU in Amsterdam een experiment uitgevoerd waarin een ander ion (beryllium), dat wel gekoeld kan worden, samen met een helium ion ingevangen zal worden. Het beryllium ion wordt gekoeld met een laser met een langere golflengte die wel als een laserstraal geproduceerd kan worden.

De onderzoeksgroep waar ik mijn project heb gedaan creëert de golflengte van de koellaser bij ongeveer 313 nm met behulp van een 1054 nm laser. Deze laserstraal zal gebruikt worden om het beryllium ion te koelen. Hoe preciezer de frequentie van deze laserstraal, hoe nauwkeuriger er gekoeld kan worden. Het behalen van de hoogste nauwkeurigheid kan door het verschil in frequentie tussen de laser en een andere zeer nauwkeurige laserstraal te bepalen. Dit verschil in frequentie wordt ook wel de beat-note genoemd. De beat-note moet zo sterk mogelijk zijn, een groter signaal is een betere meting en leidt tot een stabiele laser.

Bij het creëren van een beat-note wordt de verschilfrequentie tussen de twee laserstralen ge-vormd. Door het bepalen van de frequentie van de referentie laserstraal die een beat-note maakt met de 1054 nm laserfrequentie, zal de 1054 nm laserfrequentie bepaald kunnen worden en kunnen de twee frequenties gelockt worden. In dit project is deze beat-note gecreëerd. Om de referentie laserstraal te kunnen creëren is uit de laserbundel van een ultra-stabiele frequentiekamlaser met een spectrum van 1050 − 2000 nm enkel de 1054 nm geselecteerd. De rest van de golflengtes is geblok-keerd met het gebruik van 1064 nm spiegels, een transmissietralie en een spleet. Vervolgens is deze (1054 nm) referentie laserstraal door middel van een fiberkoppelaar (fiber coupler) in dezelfde fiber gevoegd als de 1054 nm laserstraal. Deze fiber is met een optische spectrometer gemeten waarmee de beat-note frequentie gevonden is. Deze beat-note kan gebruikt worden om de frequentie van de 1054 nm laser te stabiliseren. Met een stabiele laser kan in de toekomst nauwkeurig gekoeld worden in He+experimenten.

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Introduction

Experimental verification has been a validation process for many physical constants. Precision spectroscopy of atomic hydrogen can be used to find a value for the proton radius. The same can be done with muonic hydrogen where the electron is replaced with a muon, but it leads to a 4% smaller value (a 5σ effect), which is known as the "Proton Radius Puzzle" [1–5]. The use of spectroscopy with frequency comb lasers can lead to a larger precision in determining the energy structures of atoms [6, 7]. Testing with other ionic and muonic atoms, may lead to an even more definite and consistent solution for the proton size [1].

The group of prof. dr. K.S.E. Eikema is conducting experimental efforts on repeating the hydrogen experiment with a helium ion (He+) instead of hydrogen. This is done by combining a high power laser with high accuracy. It is pursued to combine the research with the results of muonic He+ (ηHe+) research to obtain an alpha particle radius [5]. In the experiment, He+ and ionised beryllium (Be+) will be trapped in one ion trap. Helium ions do not have a transition usable for cooling, but Be+does [5, 8]. The near 30 nm wavelength that would be needed for cooling He+, is part of the XUV-spectrum and such a wavelength cannot be created as a laser beam. However, the 313 nm of Be+ can be created as a laser beam [8]. Because of this, in the experiment Be+ as well as He+ are trapped in a linear Paul (ion) trap [5]. Both ions experience a harmonic potential with the use of electronic fields. In the ion trap the Be+transition provides efficient cooling, which is transferred to He+ through the Coulomb interaction [5]. Maintaining the required energy and temperature keep the ions in the minimum of the potential, and thus in the ion trap [5]. Because the He+ and Be+can be contained in an ion trap, further experimentation can be conducted with the He+.

To do high precision measurements an accurate laser is a requirement. A frequency comb can be used to increase the accuracy of lasers frequency [9]. The frequency comb is a pulsed laser with a range in the near infrared (NIR) spectrum. In this bachelor project a 1054 nm beam of frequency comb modes has been selected from the Ultra-Stable FC1500-250-ULN Frequency Comb and used to create a beat note. This beat note is the difference frequency of the selected frequency comb mode and the frequency of a 1054 nm (1kHz) laser from NKT Photonics, further referred to as the 1054 nm laser. The beat note frequency will make it possible to find and control the exact frequency of the 1054 nm laser [9]. This 1054 nm laser will be used in the Be+ cooling process during the experiment, which in turn is cooling He+.

During the experiment a near 313 nm beam is needed and created from the 1054 nm laser beam [5]. The purpose of this project is to stabilise a 1054 nm laser in order to create a stable 313 nm laser beam, which can be used for cooling Be+. To create this laser beam, the 1054 nm laser needs to be stable and the frequency ( f1054) needs to be found. A beat note has been created with the

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1054 nm laser and the 1054 nm frequency comb mode from the frequency comb, in order to find the beat note frequency ( fBN). After the frequency of the selected 1054 nm beam of the frequency

comb has been found, both frequencies can be used to calculate the frequency of the 1054 nm laser. To create the beat note, the frequency comb with a spectral range of 1050 − 2000 nm has been used. A beat note was created by selecting the mode nearest to 1054 nm in a beam and summing with the 1054 nm laser beam. This beat note has been measured with the spectrum analyser. Since the frequency lines of the modes of the frequency comb are spaced at 250 MHz, the beat note frequency ( fBN) always has a value between 0 − 125 MHz. Once fBN is found and f1054 is

calculated, the value can be used to stabilise the laser and provide stable laser cooling conditions. The 1054 nm laser can be stabilised to the 1Hz level with the use of locking it directly to the ultra-stable laser, through the selected frequency comb mode [1]. The beat note of the selected frequency comb mode with the 1054 nm laser beam, reveals the difference between their frequencies. In this project a compatible mode has been selected and used to create a beat note.

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Chapter 2

Methods

During this project a set-up has been built to select a near 1054 nm beam from the broadened frequency comb. The set-up is shown in Fig. 2.1. In the following paragraphs the components of the set-up are explained, following the beam path from the frequency comb to the measuring device. After the set-up was built a beat note was created between the selected frequency comb beam and the 1054 nm laser. The general principle is that the (lowest frequency) beat note between the comb and 1054 nm laser is made with as few modes of the comb laser as possible. In this manner the noise of the modes that do not contribute to the beat note is suppressed, leading to a better signal.

Figure 2.1: Schematic drawing of the set-up in which the frequency comb beam has been collimated and the mode containing the 1054 nm wavelength is selected. The set-up consists of: a half-wave plate (λ/2), lenses L1, .., L4 with f = −200 mm (L1) and f = 100 mm (L2, L3 and L4), a high-efficiency pulse compression transmission grating of the T-160-1060s series, a slit with a highly accurate mechanical adjustability, 1064 ± 10 nm mirrors M1, .., M8, the frequency comb and the fiber coupler with ratio 90:10.

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2.1 The 1054 nm laser and the frequency comb

The 1054 nm laser is a Koheras Adjustik Low Noise Single-Frequency laser from NKT Photonics and has a 1054 nm wavelength. Its frequency ( f1054) is to be found and stabilised. The frequency

comb frequency ( fFC) has not been calculated during this project. The frequency comb used is an

Ultra-stable FC1500-250-ULN (Ultra Low Noise) from Menlo Systems.

The range of the frequency comb is divided in modes, with equal linewidth 1Hz. The exact linewidth is limited by the resolution of the bandwidth analyser. The phase lock of the comb is correlated with the ultra-stable laser and its optical references. The ULN frequency comb will be used to stabilise the 1054 nm laser. The 1054 nm laser will be stabilised with an optical lock to one of the frequency comb modes and by a reference to the Cs (III) atomic clock from Symmetricom (referred to as atomic clock) [7].

The frequency comb spectrum is widened to a spectral range of 1050 − 2000 nm with Near-Infrared modulation (M-NIR). The modes containing the 1054 nm wavelength are to be isolated in a beam. The set-up of Fig. 2.1 filters the modes other than the closest to the 1054 nm wavelength. The frequency comb has mode spacing of 250 MHz. The frequency comb and the 1054 nm laser are joined by the use of the polarisation-maintaining fiber coupler, designed for 1064 nm wavelengths with a 90:10 ratio. The APD 310 Photodiode has been used to measure the beat note, during measurement other devices have been used on its location in Fig. 2.1. The beam path in Fig. 2.1 starts at the frequency comb and follows the red line to the measuring device.

2.2 Collimation of the frequency comb beam

This particular frequency comb is a recent addition to the LaserLab. The beam, which has a wavelength range from 1050 to 2000 nm, was less collimated than described in the factory manual. The beam has a Gaussian shape, which can be observed directly after the exit of the beam from the frequency comb. The beam shape can be collimated by placing a lens with focal length f = −200 mm in the beam path at 20.0 cm before the focal point. Because the set-up has been built in a limited space, the length of the beam path needs to be as short as possible.

The average output power of this type of frequency comb should be higher than 200 mW according to the factory manual. The observed optical output power (OOP) was 410 mW, with an uncertainty of 20 µW from the NOVA OPHIR power meter. First the power of the laser beam needs to be reduced, since high power could cause damage to the camera during the capture of the beam propagation and while placing a lens in the set-up. The power of the beam is reduced to 15 mW using two wedge prisms (wedges). One wedge is placed on the location of the first half-wave plate in Fig. 2.1, between the mirrors M1 and M2. The other is placed in the beam path between the mirror M2 and lens L1. By removing the wedges, the power is returned to its initial value of 410 mW.

To calculate the spot size of the beam (ω), the following equations have been used: zR= πω2 0 λ (2.1) ω(z) = ω0∗ r 1+ ( z zR )2 _(2.2)

In these equations,ω0 is the beam waist, λ the wavelength of the laser, z the distance from the

waist and zRthe Rayleigh length.

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Figure 2.2: Schematic drawing of a Gaussian beam, which illustrates the diverging propagation of a laser beam propagating in the z-direction. The radius of the beam (ω(z)) can be calculated (see Fig. 2.3a) and used to find the beam waist (ω0) at the focal point. The Rayleigh length (zR)

correlates the beam waist with the spot size.

be found in the appendix (Fig. A.1 and A.2). The Gaussian shape of the beam and its parameters are shown in Fig. 2.2. A Gaussian shaped beam first converges until its radius remains equal to the beam waist, after which the beam diverges again. The diameter of the beam has been measured at 28 different points between M3 (Fig. 2.1) and 22 cm after M3. To measure the spot size of the beam a webcam, type TRUST SC-012-259+9855, has been used. This webcam has a sensor of 4 mm × 3 mm, which is 640 × 480 pixels. The resolution of the webcam sensor is therefore 6.25 µm/pixel. An example of the measured beam size is shown in Fig. 2.3. The raw data is shown in 2.3a and the intensity at the middle of the beam is shown in 2.3b. The profile plot of 2.3b is a representation created with ImageJ and the data is used in the Mathematica code (Fig. A.1). The beam spot size is calculated for different z-values. The difference, between these spot sizes and the correlating z-values, shows that the focus (z0) is located between 401.2 mm and 406.5 mm. The

beam waist is therefore positioned at z= 40.39(26) cm and was equal to 0.61 mm. Placing lens L1, with focal length f = −200 mm at z = 20.0 cm, creates a collimated beam with the desired beam size of a few millimetres.

(a) _(b)

Figure 2.3: Intensity images measured at the beam path, at 145 mm after the exit point of the frequency comb. Fig. 2.3a is a capturing profile of the beam size. The highest intensity expresses in white pixels and zero intensity shows black. Fig. 2.3b is a profile plot of the beam shown in Fig. 2.3a where the intensity per pixel is presented.

2.3 Selecting 1054 nm modes

After the frequency comb beam is reflected by the 1064 nm mirrors, its wavelengths range from 1054 nm to 1074 nm. A transmission grating with a line density of 1600.0 lines/mm is used to select the near 1054 nm wavelengths. The grating transmits wavelengths in the range of 1060 ± 20 nm.

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The grating efficiency depends on the angle of incidence of the grating, which is 58.0 ± 1◦_{. The}

angle between the incident beam and transmitted beam (beam path angle) is 64.0◦_.

After transmission through the grating the different optical frequencies come out at different angles. Each colour propagates in the direction of its own angle. By placing a lens with focal length f = 100 mm in the beam path, a Fourier plane is created where the colours are focused and spatially separated. This is visualised in Fig. 2.4. After the lens, the beam is focused and the different wavelengths are propagated parallel to each other. A slit is used to select the 1054 nm wavelength beam. The slit blocks the wavelength other than the 1054 nm (b) and the wavelength larger than the 1054 nm (d). The 1054 nm wavelengths (c) remain from the beam before the grating (a) after the slit.

Figure 2.4: Representation of the wavelength separating process. In the set-up a 1600 lines/mm grating is used, suited for 1060 ± 10 nm. The lens has a focal point f = 100 mm and is places 10 cm after the grating, which is 10 cm before the slit. The frequency comb beam before the grating (a), the wavelengths< 1054 nm (b), 1054 nm wavelength (c) and wavelengths > 1054 nm (d) are schematic drawn.

The slit (VA100/M) is an adjustable mechanical slit which has a maximum slit width of 6.0 mm when fully opened and 0.0 mm when fully closed. Because the slit has such a small opening, the beam diverges after exiting the slit. The beam size is in the order of some hundreds of µm after exiting the slit. The beam size is increased to ω0 = 1.708(6) mm when it reaches lens L3, with

focal point f = 100 cm, 10 cm after the slit, as shown in Fig. 2.1. Lens L3 collimates the beam to a good size to ensure coupling of the beam into the fiber core.

The frequency comb has four diodes, for which the current has been tuned to obtain the highest optical output power (OOP), see appendix Fig. B.1. The OOP of the frequency comb was set to 406 ± 1 mW. The 1 mW variance is caused by the temperature dependency. The output power of 112 µW remained after selecting the mode with the 1054 nm wavelength and collimation of the beam. Decrease in power of the remaining beam is expected since all the modes with wavelengths other than near 1054 nm have been filtered out. The polarisation of the light also influences the intensity of this propagating beam after the fiber. Only one polarisation direction can propagate through the fiber. Because the beam before the fiber is polarized in the same direction as the fiber, the loss due to polarisation is minimised. A warm-up period for the laser beam of approximately an hour is required before measuring, to stabilize the power of the beam [10].

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2.4 Creating the beat note

After the beam is coupled into the fiber, which theoretically could retain 90% of the source power, a power of 49.2 µW is measured. This output power is approximately 44% of the incoming beam power. To minimise this loss, the polarisation state of the light needs to be further regulated. A half-wave plate can change the polarisation of the light by 90◦ _{without changing the intensity of}

the beam. Rotations of the half-wave plate by an angleα changes the polarisation with an angle 2α. In the set-up (Fig. 2.1), a half-wave plate was used before the grating, to polarise the beam for optimal transmission through the grating. The angle of the half-wave plate was determined by rotating the half-wave plate while measuring the power after the grating. The angle where the measured power was highest has been identified and used in this set-up. A second half-wave plate is used before coupling the beam into a fiber, such that the polarisation of the light aligns with the slow axis of the fiber. The light coming out of the fiber should be polarized in one direction.

The collimated beam has been guided into the fiber core with fine tuning of the mirrors M5 and M6 (Fig. 2.1). Two mirrors are needed to change the position and angle of the beam and therefore have full flexibility in aligning the beam into the fiber. A beat note can only be created with linearly polarized light, when both components have the same polarisation direction. After exiting the fiber coupler, the beam was focused on the high sensitivity detector unit APD310 using the 1064 nm mirrors M7 and M8 (Fig. 2.1).

The PM Fiber Coupler 1064 nm (fiber coupler) has been used such that it has two input fibers and one output fiber. The 1054 nm laser beam and the frequency comb beam are used as input for the fiber coupler. Because the ratio of the fiber coupler is 10 : 90, the combined beam has a power of 10% of the 1054 nm laser beam, and 90% of the frequency comb beam.

Creating a beat note between two beams is best done once the polarisation of both beams is in the same direction and when the wavelengths in the beams are the same. The spectral overlap is tuned with the slit and the selected wavelengths from the frequency comb are obtained with the use of the ANDO Optical Spectrum Analyzer. Fig. 2.5 shows the spectral overlap between the 1054 nm laser beam and the frequency comb beam around 1054.15 nm. This graph has been created using the slit to tune the selected wavelength. Fig. 2.5 shows the measured spectrum, which has its maximum at 1054.15 nm.

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Figure 2.5: Graph of the optical output spectrum measurements with the ANDO optical spectrum analyser. The intensity of the 1054 nm beam (black), the frequency comb beam (red) and the sum of beams (blue) are plotted. The plot was used to tune the wavelength selection of the frequency comb. The graph shows the result after tuning.

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Chapter 3

Results

The difference between the frequencies f1054 and fFC has been measured. This difference is called

the beat note. The analysis of the beat note has been conducted with a E4440A PSA Spectrum Analyzer. Fig. 3.1 shows a spectrum analyser trace of the measured beat note. The plot is fitted with the Gaussian fit of equation 3.1. The parameters of the fit are presented in Table 3.1.

y= y0+ A ∗ e−0.5∗((f −fBN)/ω)

2

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Relevant Fit Value and Error Parameters fBN 34.7765 MHz ± 1.7 kHz ω 0.14187 MHz ± 1.84 kHz FWHM 0.33408 MHz ± 4.33 kHz A 36.11 dB ± 0.39 dB y₀ −104.45 dB ± 0.11 dB

Table 3.1: Parameter results of the fit of the beat note, including the standard errors. The top of the Gaussian fit (maximum) is at the frequency fBN. The full width at half maximum (FWHM)

correlates according to equation 3.2 with the width (ω) and the amplitude (A) and the offset (y0)

of the fit, which are presented in decibels.

The frequency of the peak ( fBN= 34.7765 MHz) is the frequency difference between the selected

mode with the 1054 nm wavelength and the single mode of the fiber laser that needs to be locked. The FWHM is 0.33408 MHz. The width ω can now be calculated with equation 3.2. The width ω (the standard deviationσ) is 0.14 MHz.

FWHM=pln(4) × 2ω (3.2)

For the main experiment a beat note frequency of some tens of MHz is required. The difference frequency of the beat note will always have a value between 0 − 125 MHz. Fig. 3.1 shows the measurement from the spectrum analyzer. These data are fitted with a Gaussian fit. The fFC,

which will be known in the future, and the measured fBN can be used to calculate the f1054 with

the use of equation 3.3.

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Figure 3.1: The graph above shows the measured beat signal. The measurement points have been fitted with a Gaussian fit. The beat signal between the 1054 nm laser and the selected 1054.19 ± 0.07 nm frequency comb beam has been measured with the E4440A PSA Spectrum Analyzer. The Gaussian fit peaks at fBN = 34.7765(17) MHz. Other results of the Gaussian fit are

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Chapter 4

Discussion

In this project a set-up has been built for selecting the 1054 nm frequencies from an Er-fiber ultra-low-noise frequency comb (Fig. 2.1). This resulted in finding the frequency of the beat note ( fBN)

between the fiber laser that needed to be stabilised and the comb laser. Finding this frequency is the first step towards finding the frequency of the 1054 nm laser ( f1054) and locking this frequency.

The second step will be determining fFC, which has not been done during this bachelor project.

The values needed for calculating the fFC will be found in future research to complete equation

3.3. To calculate fFC, equation 4.1 will be used. In this equation, n is the mode number, f0 is the

carrier offset frequency and fr is the comb tooth spacing [7, 9]. The values of f0, fr and n will be

found in the next step.

fFC= fn= f0+ n × fr where f0< fr (4.1)

It is important to know the exact value of f1054 because in following experiments this frequency

needs to be well known and stabilised.

In these experiments, a 313 nm beam will be created and the Be+ and He+ will be trapped in an ion trap [8]. First the 1054 nm laser will be combined with a 1542 nm laser, and this is sent through a crystal creating a 626 nm beam. As a next step, this 626 nm beam will be sent into a doubling cavity resulting in the desired 313 nm beam [8, 9]. This 313 nm beam makes laser cooling of Be+and (indirectly) He+ possible [8].

The beat note frequency has been presented and was fBN= 34.776 MHz. This means that the

cooling laser can have a similar accuracy and stability, which would be more than sufficient for laser cooling Be+.

4.1 Conclusion

The goal of this project was to make a set-up to measure a beat note ( fBN) between a 1054 nm

laser and an ultra-stable frequency comb laser for future stabilisation of the cooling laser beam near 313 nm. In conclusion, the value of the beat note was fBN= 34.7765(17) MHz. The frequency

difference between the beams turned out to be small in comparison to the frequency of the two lasers. The fFC will be determined in future research to find f1054. Then it will be known exactly

with which precision the laser is locked. Although this project was small in comparison with the total effort to perform precision spectroscopy on the 1S−2S transition in He+, it contributed to an experiment which may result in validating the nuclear charge radius of He+.

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Acknowledgements

During this bachelor project selection of the 1054 nm frequency mode from the frequency comb beam has been done. It was my job to figure out how the selecting of this frequency from the comb could be achieved. This is a brief word of thanks for my supervisors in the lab PhD-students Laura Dreissen and Charlaine Roth. After loads of trial and error and even more questions to Laura and Charlaine, I found a way to use the limited space, the material present in the laboratory, and even with orders of new material a way to create a beat note and find its frequency value. Without their guidance it would have been difficult to create the whole set-up, built for selecting from the frequency comb, and the final creation of the beat note. In this I used most of my knowledge gained by the advice and explaining words of Laura and Charlaine. The lunch break discussions and random pop-up talks with the group and group leader Kjeld Eikema were as much of a con-tribution to the report as the articles used in the references were. A lot of insight has been gained during this project.

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Bibliography

[1] R. Altmann, L. Dreissen, E. Salumbides, W. Ubachs, and K. Eikema, “Deep-ultraviolet fre-quency metrology of h 2 for tests of molecular quantum theory,” Physical review letters, vol. 120, no. 4, p. 043204, 2018.

[2] R. Pohl, A. Antognini1, F. Nez, F. D. Amaro, F. Biraben, J. M. R. Cardoso, D. S. Covita, A. Dax, S. Dhawan, L. M. P. Fernandes, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C.-Y. Kao, P. Knowles, E.-O. Le Bigot, Y.-W. Liu, J. A. M. Lopes, L. Ludhova, C. M. B. Monteiro, F. Mulhauser, T. Nebel, P. Rabinowitz, J. M. F. d. Santos, L. A. Schaller, K. Schuhmann, C. Schwob, D. Taqqu, J. a. F. C. A. Veloso4, and F. Kottmann, “The size of the proton,” Nature, 2010.

[3] I. Sick, “Proton charge radius from electron scattering,” Atoms, vol. 6, no. 1, p. 2, 2017. [4] M. Ahmadi, B. X. R. Alves, C. Baker, W. Bertsche, E. Butler, A. Capra, C. Carruth, C. Cesar,

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[8] H. Ball, M. Lee, S. Gensemer, and M. Biercuk, “A high-power 626 nm diode laser system for beryllium ion trapping,” Review of Scientific Instruments, vol. 84, no. 6, p. 063107, 2013. [9] P. Del’Haye, O. Arcizet, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Frequency

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[10] A. C. Wilson, C. Ospelkaus, A. VanDevender, J. A. Mlynek, K. R. Brown, D. Leibfried, and D. J. Wineland, “A 750-mw, continuous-wave, solid-state laser source at 313 nm for cooling and manipulating trapped 9 be+ ions,” Applied Physics B: Lasers and Optics, vol. 105, no. 4, pp. 741–748, 2011.

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Appendix A

Mathematica code

This Mathematica code has been used to find the beam size radius.

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Appendix B

Frequency comb settings

Table B.1: Results of testing a significant amount of possible combinations of the settings of the diodes D1, D2, D3and D4in the frequency comb.

OOP (mW) D1(mA) D2(mA) D3(mA) D4(mA)

406(1) 525 950 950 950 404 500 950 950 950 390 400 950 950 950 373 300 950 950 950 254 200 950 950 950 340 150 950 950 950 400 525 850 950 950 394 525 750 950 950 384 525 650 950 950 376 525 550 950 950 367 525 450 950 950 357 525 350 950 950 347 525 250 950 950 400 525 950 850 950 392 525 950 750 950 384 525 950 650 950 376 525 950 550 950 366 525 950 450 950 357 525 950 350 950 347 525 950 250 950 400 525 950 950 850 392 525 950 950 750 384 525 950 950 650 376 525 950 950 550 367 525 950 950 450 357 525 950 950 350 347 525 950 950 250 321 350 950 950 950 305 250 950 950 950 280 150 950 950 950

(21)

OOP (mW) D1(mA) D2(mA) D3(mA) D4(mA) 271 100 950 950 950 262 100 850 950 950 251 100 750 950 950 241 100 650 950 950 225 100 500 950 950 214 100 400 950 950 202 100 300 950 950 191 100 200 950 950 173 100 200 800 950 166 100 200 700 950 147 100 200 600 950 134 100 200 500 950 121 100 200 400 950 107 100 200 300 950 92,4 100 200 200 950 77 100 200 100 950 69,2 100 200 100 200 36 100 200 100 100

Frequency stabilisation of a 1054 nm laser with a frequency comb laser