Laser Lock for Laser Cooling and Trapping of Potassium-40

University of Amsterdam

Bachelor thesis physics and astronomy

Laser Lock for Laser Cooling and Trapping

of Potassium-40

Author:

Maarten Mooij 10365451

Supervisor: Prof. Dr. Florian Schreck Daily supervisors: Wouter Meinster and Dr. Benjamin Pasquiou

Abstract

Laser cooling of atomic gases requires a laser with a small linewidth. However, dis-ruptions due to for example temperature fluctuations or external sound can increase the linewidth of the laser too much. An active feedback system is needed to keep the light close to the desired frequency. In this thesis such a laser lock is described. The laser lock makes use of Doppler-free polarization spectroscopy, which means a probe and a pump beam are used to receive a clear absorption signal on the probe beam. Furthermore, an external magnetic field splits the energies of the σ+ and the σ− transitions. Therefore,

the polarization components of the probe beam are transmitted for different laser frequen-cies. Splitting and subtracting the probe beam by polarization gives a dispersive error signal. A proportional-integral-derivative controller (PID controller) is used to generate a control signal from the dispersive signal. This control signal acts on control elements in the laser source and changes the laser’s frequency in such a way that the error signal is pushed towards zero, thereby locking the laser to the desired frequency. While locked, the linewidth of the laser is about 2.5 MHz, which is well within the linewidth of 6 MHz needed for potassium. Furthermore, the PID controller reduces the noise by 6 dB for frequencies up to 10 kHz. Further work can reduce the linewidth by attenuating the noise even more and also attenuating frequencies higher than 10 kHz.

Populair wetenschappelijke samenvatting

Al meer dan honderd jaar wordt er veel onderzoek gedaan aan atomen. Atomen bestaan uit een kern waar omheen elektronen vliegen. Het blijkt dat deze elektronen niet willekeurig rond de kern draaien, maar in specifieke banen bewegen. Een beetje zoals de planeten in het zonnestelsel. Elektronen willen het liefst zo dicht mogelijk bij de kern in het midden zitten en daarom zullen ze altijd proberen in de baan het dichtste bij het midden te komen. Om ze verder weg van de kern te krijgen moeten ze extra energie krijgen, net zoals op aarde je wat extra energie nodig hebt om omhoog te gaan. Het blijkt dat de energie van de elektronen voor deze banen erg precies ligt. Elektronen met niet precies de goede energie, kunnen niet in die bepaalde baan zitten. En omdat de banen zo precies zijn, zijn de verschillen in energie tussen de banen ook erg precies.

Een manier om een elektron in een andere baan te brengen is met licht. Licht bestaat namelijk uit energiepakketjes, fotonen. Zo kan een foton zijn energie afgeven aan een elektron waar-door het elektron naar een hogere baan gaat. Het foton is zo opgenomen in het atoom. Maar dat elektron wil graag terug naar die lagere baan en het duurt dan ook niet lang voordat dit gebeurt. Op dat moment verliest het elektron wat energie en dat wordt weer omgezet in een foton. Dat foton vliegt vervolgens een willekeurige kant op.

had namelijk een beweging en die beweging geeft die door aan het atoom. Als vervolgens weer een foton wordt uitgezonden, duwt het atoom zichzelf de andere kant op. De beweging van het atoom voor en na het absorberen van het foton is daardoor anders. Gebeurt dit voor heel veel fotonen achter elkaar, dan zal de beweging in de richting van waar de fotonen komen steeds worden geremd, terwijl de beweging achteraf middelt tot netto geen verplaatsing. Op deze manier kan de beweging van atomen gestuurd worden met licht.

Omdat de banen van de elektronen zo precies liggen, reageren de atomen maar voor hele spec-ifieke frequenties van licht. Een standaard lampje gebruiken om een elektron in een andere baan te leiden, heeft daarom weinig zin, want van de vele frequenties die dat lampje uitzendt, worden er maar een paar opgenomen. Wat veel beter werkt is een laser. Die zendt veel licht

van ´e´en frequentie uit. Als die frequentie precies de juiste is voor een bepaald atoom, zijn er

meteen een heleboel fotonen die de beweging van atomen kunnen sturen. Het is dan wel erg belangrijk dat het licht altijd die specifieke frequentie behoudt, want een klein beetje andere frequentie betekent dat een atoom het licht van de laser helemaal niet meer ziet. De frequentie moet zo precies zijn, alsof je een tafel van twee meter op tien nanometer precies moet meten. Het lastige is dat de frequentie van een laser helemaal niet zo stabiel is. Een beetje geluid of luchtstroom is al genoeg om de frequentie ver van de gewenste te krijgen. De laser moet daarom de hele tijd gecorrigeerd worden voor alle kleine veranderingen. De laser moet als het ware gelocked worden op een specifieke frequentie.

Om een laser goed stabiel te houden moeten een systeem gebruikt worden wat voor kleine verschuivingen van de frequentie een duidelijke verandering in signaal geeft. Dan is meteen duidelijk dat de frequentie verschuift en kan ervoor gecorrigeerd worden. Ook moet de fre-quentie waarop de laser stabiel gehouden wordt precies degene zijn die we nodig hebben om atomen te besturen.

Een hele slimme truc zorgt hier precies voor. Namelijk, je tapt een klein beetje licht van de grote laserstraal af en laat dat licht door een wolkje van precies dezelfde atomen gaan als die je wilt besturen. Na dit wolkje meet je de intensiteit van het licht en kan je dus meteen zien of het licht wordt opgenomen of niet. Voor alle frequenties zal veel licht worden gemeten, behalve voor de frequentie die wordt opgenomen door de atomen. Door een systeem te maken dat altijd probeert de laser naar het dipje in het signaal te sturen, is de laser op precies de goede frequentie gelocked.

Tijdens dit bachelorproject is een lock gemaakt die de laser altijd binnen 2.5 MHz van de gewenste frequentie locked. Hierbij is gebruik gemaakt van kalium. Voor kalium mag de frequentie maxiamaal 6 MHz verschuiven. Daardoor is er door deze laser-lock onderzoek aan atomen mogelijk.

Contents

1 Introduction 4

1.1 Quantum gases . . . 4

1.2 Magneto-optical trap . . . 4

2 Theory 9 2.1 Doppler-free polarization spectroscopy . . . 9

2.2 Dispersive error signal . . . 12

2.2.1 Magnetic field . . . 12

2.2.2 Circular polarization of the pump beam . . . 14

2.3 Using a PID controller to generate a correction signal . . . 14

3 Experimental work 15 3.1 Laser alignment . . . 15 3.1.1 Laser . . . 15 3.1.2 AOM . . . 17 3.1.3 Spectroscopy . . . 19 3.2 Electronics . . . 20

3.2.1 Slow feedback loop . . . 21

3.2.2 Fast feedback loop . . . 21

3.3 Fabry-P´erot cavity . . . 21

4 Discussion 23 4.1 Short time stability . . . 23

4.2 Long term drift . . . 23

4.3 Attenuation of the noise by the slow and the fast feedback loop . . . 25

4.4 Attenuation of acoustic- and electronic-added noise by the fast feedback loop 28 5 Conclusion 30 6 Acknowledgments 30 A Electronics 32 A.1 Subtractor . . . 32

1

Introduction

1.1 Quantum gases

Since the creation of the first ultracold quantum gas in 1995 [1], the interest in ultracold quantum gases has increased. These gases are interesting because of the quantum behaviour that becomes visible when atoms are extremely cold and close to each other.

There are mainly two sorts of quantum gases, one of bosons and one of fermions. Initially research focused on making a Bose Einstein condensate (BEC) of ultracold atoms. This state of matter is a cloud of bosons that are all in the energy ground state, the lowest possible energy state a particle can be in, see Fig. 1. Because of the low energy, the cloud is said to be very cold, theoretically zero Kelvin, in practice typically 100 nK [2]. Since all atoms are in the same state, they do not behave as individual particles, but act as a group together. This unnatural and extreme circumstance discloses many interesting physical phenomena.

After the first BEC was made 1995, also a Fermi sea became of great interest. However, a Fermi sea has an additional difficulty. Fermions, unlike bosons, cannot occupy the same quantum state. Therefore, if one fermion is in the ground state, the lowest possible state for another fermion is the state adjacent to the ground state. Yet another fermion can then occupy only a state of ground state plus two times a little energy and so on, see Fig. 1. In December 2003, Regal et al. were the first to produce a Fermi sea of ultracold atoms [3].

1.2 Magneto-optical trap

One way of creating a gas of ultracold atoms is by using a magneto-optical trap (MOT), which consists of a quadrupole magnetic field and laser light sent towards the quadrupole field center from six orthogonal directions.

Bosons Fermions

ygr

en

E

EF

Figure 1: Simplified representation of the lowest possible energy configuration for some atoms. Left: Bosons can occupy the same state as other bosons and therefore all bosons will be in the ground state. Right: Fermions cannot occupy the same state, therefore only one atom is in the ground state, the next one in the state directly above and so on.

(a) (b)

(c)

Figure 2: Schematic drawing of the mechanism of laser cooling. (a) An atom moves towards a laser beam. (b) The atom absorbs a photon by exciting an electron to a higher energy level. The momentum of the photon is passed on to the atom and therefore the atom is slowed down. (c) After some time the electron will fall back to the lower energy state under spontaneous emission of a photon. The emission direction of the photon, and thereby the momentum kick received by the atom from that emission, is arbitrary. By absorbing and emitting many photons, on average the movement of the atom towards the laser light will be slowed down.

Laser cooling Lasers can be used to slow down the movements of atoms. The laser cooling

makes use of the property of atoms to absorb light and emit it. When an atom encounters a light beam, it can absorb a photon, see Fig. 2. The momentum of the photon is then transferred to the atom. With spontaneous emission the atom will emit light in an arbitrary direction and will therefore lose momentum in that random direction. Consequently, when an atom absorbs and emits lots of photons, the momentum of the atom will slow down opposite to the direction of the laser beam.

Energy levels All atoms have specific frequencies of light they can absorb, given by the

energy levels of the atom. Only photons with energies close to the energy of those transitions can be absorbed. See Fig. 14 for relevant energy levels of potassium. The linewidth of the energy transition is mostly quite narrow, for potassium it is typically 6 MHz [4].

Doppler shift The energies of transitions given in Fig. 14 are for atoms with no velocity.

When atoms do have some motion, the energies of the photons must be different. Atoms moving towards the light, absorb light with a slightly lower energy than motionless atoms would absorb. Atoms moving away from the light absorb higher frequencies, see Fig. 3. This ”Doppler shift” is described by

Figure 3: Schematic drawing of the Doppler effect for absorbing light. Atoms moving towards the light absorb light with lower frequencies than the resonance frequency (red arrow). Atoms with no velocity in the direction of the light absorb light with no frequency shift. For atoms moving away from the light, the frequency has to be higher.

ν1∼ ν − v,

ν2∼ ν + v,

(1)

where ν is the frequency of the light that will be absorbed by motionless atoms and ν1 and

ν2 are the frequencies for which the light will be absorbed by atoms moving towards the light

and moving way from it respectively.

Optical molasses When six orthogonal beams with a frequency ν1 are used, an optical

molasses is produced. Because of the small shift in frequency, the light mainly encounters atoms moving towards the light, while motionless atoms or atoms moving away from the light are not effected. With six orthogonal beams, the motions of the atoms are slowed down for all directions and therefore the atoms are cooled down.

MOT Six orthogonal beams cool the atoms, but do not accumulate them in a cloud together.

To create a cloud atoms from outside the center of the cloud must be pushed towards the center. To create such a force, a quadrupole magnetic field, which is a magnetic field that increases from a center towards the outside, is used. A magnetic field induces a Zeeman

shift of the mf-states of the atom. Without a magnetic field these states are degenerated

with the same energy as the energy of the associated F-state. A magnetic field lifts the

degeneracy, which means the energy of transitions between the mf-states change. For a

quadrupole magnetic field, the shift in energy increases for atoms further away from the center. Exploiting the polarization of the laser beams, it can be assured that atoms are pushed back to the center, see figure 4.

To control the movements of the atoms by exciting atoms with laser light, the laser has to be stable at the correct frequency. A free running laser is not stable enough and therefore a control system is needed, a so-called ’laser lock’.

Laser Laser





-Magnetfield B

x





-



-



-F=0

F=1

m=-1 m=0 m=+1

B<0

B=0

B>0

Force

Force= 0

Force





Energy

Zeeman

Figure 4: Simplified drawing of the magneto-optical trap. The magnetic field split the mf

-states of the hyperfine structure of the atoms differently on opposite sides of the center. LHCP

light and RHCP light push the atoms back to the center driving the σ+ transition of atoms

on the left side of the center and the σ−transition on the right side of the center respectively.

The atoms in the middle are not effected because the energy of both laser beams is too low. The figure is taken from [5]

In tensit y Frequency (a) In tensit y Frequency (b)

Figure 5: Two shapes of an error signals. (a) For a peak the change in intensity of the error signal is similar for lower or higher frequencies while (b) for a dispersive error signal this change is different.

Locking a laser The laser is locked onto a specific atomic transition using Doppler-free

po-larization spectroscopy. The popo-larization technique is used to create a dispersive error signal rather than a peak shaped error signal at the resonance frequency, see figure 5. A dispersive error signal has the advantage over a peak because the intensity of the signal is different for both sides of the resonance frequency. In that way it is simple to detect if the frequency is too high or too low.

This thesis will discuss how such a laser lock can be produced. Firstly, the creation of a dispersive error signal will be described theoretically. This section will be followed by the description of the experimental setup used to achieve a stabilized laser. Next we discuss the precision of the laser lock along with suggestions for improvements. The conclusion sum-maries the characteristics of the laser lock. The appendices describe the used elements in more detail.

2

Theory

2.1 Doppler-free polarization spectroscopy

When a laser beam is sent through a gas without saturating the gas, absorption dips will be measured for frequencies belonging to energy transitions of the electron configuration of the atom. Therefore, by sweeping the laser frequency slowly, the transmissions can be traced over frequency, see Fig. 6. However, only for a gas with no thermal energy (zero Kelvin) the absorption will be limited to a narrow range of frequencies. Atoms at room temperature are constantly moving and therefore light with other frequencies will be absorbed as well due to the Doppler shift, see equation 1 and Fig. 3. This results in a very broad range of frequencies absorbed by the gas, see Fig. 6b.

One way to avoid this broad absorption dip is using a combination of two beams [6], see Fig. 7a. One beam with a low intensity, the probe beam, and another beam with higher intensity, the pump beam. They overlap and are aligned in opposite direction. Due to the low intensity of the probe beam, it does not saturate the gas and thus all frequencies around an optical transition will be absorbed. The pump beam has an intensity such that it satu-rates the gas completely, which is called hole burning. Because the pump beam and probe beam have opposite direction, they encounter different atoms, see Fig. 8a and 8b. Only for frequencies that can be absorbed by atoms with no velocity in the directions of the beams, the atoms will absorb both the pump and probe beam, see Fig. 8c and 8d. Because the pump beam saturates the gas completely, the probe beam cannot be absorbed, and therefore for this particular frequency the probe beam can pass. With the same sweeping laser used for figure 6b, the same Doppler shape will occur as before, but with peaks for frequencies where the pump and the probe approach the same atoms, see Fig. 7.

In the case of39K three peaks appear, see Fig. 7b. These peaks are the peaks of transitions

from the two different F-states 2_S

1/2, F0 = 1 and 2S1/2, F0 = 2 to the 2P3/2 state, see Fig.

14. The third peak is the crossover of the two states. This crossover optically pumps atoms of a certain velocity class e.g. from the F’=1 state into the F’=2 state, while the probe beam is resonant with F’=2 state atoms of that velocity class (or the inverse for the atoms with opposite velocity). The overall effect is that the probe laser is absorbed by more atoms, resulting in a dip of the spectroscopy signal. The peak with the highest frequency belongs to

for the transition from the F’=1 state to the2P3/2 state, because this is the highest energy

transition, while the peak with the lowest frequency belongs to the F’=2 to2_P

3/2 state. The

Photo

diode K. spectroscopycell LASER

z (a) - 800 - 600 - 400 - 200 0 200 Frequency (MHz) 0.0 0.5 1.0 1.5 2.0 2.5 Intensity (V ) Doppler spectroscopy (b)

Figure 6: The intensity of the laser after passing a cell with a natural potassium vapor.

The laser is scanned over a range of frequencies around the D2 line of39K. (a) The optical

setup used for recording the spectroscopy. (b) Spectroscopy using a single beam propagating through the gas. Due to the Doppler effect not only the resonance frequency is absorbed, but also many other frequencies.

HWP3 K. spectroscopy cell Pump beam Pr obe beam Photo diode LASER z (a) 0.0 -800 -600 -400 -200 0 200 Frequency (MHz) 0.5 1.0 1.5 2.0 2.5

Intensity (V )Doppler free saturation absorption spectroscopy

F=1 F=2

Crossover

(b)

Figure 7: Doppler-free satiation absorption spectroscopy. The intensity of the laser beams is measured similar to Fig. 6. (a) A counter propagating pump beam with a high intensity is added to the optical setup, which induces hole burning. (b) The probe beam has a low intensity and can therefore not pass the cell. Only for frequencies the pump beam saturates the gas the probe beam is not absorbed and therefore peaks arise.

-4 -2 2 4 vz 0.05 0.10 0.15 0.20 Number of atoms

Pump beam Probe beam

(a) + V ν₁ ν₁ absorbde by atoms with - V absorbed by atoms with + V - V

PUMP BEAM PROBE BEAM

(b) -4 -2 2 4 vz 0.05 0.10 0.15 0.20 Number of atoms Pump beam Probe beam (c)

PUMP BEAM PROBE BEAM

ν ν

saturation

absorption by other atomsnot absorbed

Vhorizontal= 0

(d)

Figure 8: Schematic drawing of hole burning (left) and of light passing the potassium cell (right). (a) and (b) For frequencies off resonance the probe beam and pump beam address different atoms. Therefore the probe beam will be entirely absorbed. (c) and (d) For frequen-cies on resonance pump and probe beam are absorbed by the same atoms. The pump beam saturates the atoms entirely and therefore the probe beam can pass the cell with (almost) no absorption.



F = 0

F’ = 1



m

= +1

m

= -1 m

= 0

Figure 9: Optical transitions between an F=0 ground state and an F’=1 excited state within

the mf-states of the hyperfine structure of the atom. Circular polarization drives the σ+ and

the σ−transitions, changing the angular momentum projection mf by one. When a magnetic

field is added, the mf-states split to different energies. See [7] for a complete scheme of

relevant transitions in 39_K

2.2 Dispersive error signal

To create a dispersive signal, a combination of a magnetic field and polarisation is used.

2.2.1 Magnetic field

The polarization of the probe beam is linear, which can be seen as a combination of right hand circular polarization (RHCP) and left hand circular polarization (LHCP). Since the magnetic field is applied parallel to the propagation direction of the light, those RHCP and

LHCP drive the σ+ and the σ− transitions within the mf-states of an atom, see Fig. 9 for

a simplified scheme. When a magnetic field is applied, the mf-states will split to different

energies. Therefore the RHCP and LHCP have a different resonance frequency compared to the original resonance frequency. A quarter wave plate (QWP) converts the left- and right-hand circular polarizations to orthogonal polarizations. Using a polarizing beam splitting cube to split the horizontal and the two polarization components are sent onto two photodiodes to record their intensities. Because the resonance peaks of the two photodiode signals are different the subtraction of one of the other gives a dispersive signal, see Fig. 10b. For

a high magnetic field the Zeeman shift within the mf states is so big that the transition

from 2S1/2, F0 = 1 to 2P3/2 does not split only into two resonance frequencies but far more.

Therefore, small peaks can be seen within the error signal, see Fig. 10c. Such a signal is not useful anymore.

QWP1 QWP2 HWP K. spectroscopy cell

LASER

z

B-field

Error signal

Intensity (V )Error signal with optimum magnetic field

(b) -200 0 200 400 Frequency (MHz) -0.6 -0.4 -0.2 0.2 0.4 0.6

Intensity (V )Error signal in high magnetic field

(c)

Figure 10: Doppler-free polarization spectroscopy. The error signal is measured in a similar way as in Fig. 6 and 7, but with a extended setup (a) A magnetic field inside the spectroscopy

cell splits in energy of the mf-states. A QWP inside the path converts the left- and- right

hand circular polarizations of the probe beam to orthogonal polarization. The different po-larizations are split by a cube, measured by two photodiodes and subtracted electronically, resulting in an error signal. (b) The error signal with the steepest possible slope around zero.

(c) For a too high magnetic field the mf-states split too much, resulting in more small peaks

2.2.2 Circular polarization of the pump beam

To obtain a good dispersion error signal more than one technique can be used. Here a second technique is described that is used in the setup. This technique is based on using a circular pump beam, which means light of only LHCP or RHCP. This field effects the gas in two ways, i.e. the birefringence of the gas as well as the dichroism. Birefringence of a gas is the effect that the refractive index for light going through that gas is different for different polarizations. Dichroism means that the absorption coefficient of the gas is frequency dependent. In [8] and [9] the calculation is done for the resulting intensity of the subtraction of the beams depending on the frequency. The result is a dispersive signal in the intensity.

The technique of splitting the peaks due to the magnetic field assumed that the pump beam also is a superposition of two directions of circular polarized light, saturating the gas for both

directions. When instead the pump beam is circularly polarized, only for one (σ+ or σ−)

transition hole burning by the pump beam happens sufficiently, which means for the probe beam the other transition is much more absorb. Therefore the intensity of the two beams are different after the cell. Rotating the QWP before the cube corrects for this difference in intensity. Finally, with the correct tuning of the components the two techniques results in the dispersive signal of Fig. 10b.

2.3 Using a PID controller to generate a correction signal

The laser is locked to the frequency for which the steep slope of the dispersive error signal crosses the zero intensity line, see Fig. 10b. The frequency of the laser can be tuned by changing the current going to the laser, or by changing the voltage to the piezoelectric crystal inside the laser. When the frequency of the laser shifts, these two signals have to change in such a way that the frequency is corrected. To generate a correction signal a proportional-integral-derivative controller is used. This compares the incoming signal (here the subtraction) with a reference value, generally zero voltage. But instead of only looking what the difference is (proportional), it also measures the change in difference (derivative) and the difference over a certain time (integral). The advantage of the derivative part is, that it can react faster for because it reacts on a fast changing error signal.

For a slowly changing signal, the signal is difficult to distinguish for some time. By integrating this small derivation it becomes already after a relative short time a clear signal. Therefore a combination of integrate (slow) function and a fast derivative part is a good way of locking a laser.

Laser

_{Saturated absorption}

spectroscopy

Two photodiodes

PID

Piezo

Subtractor

I MOD

Figure 11: Schematic drawing of the feedback loops used to lock the laser.

3

Experimental work

The laser is sent through a spectroscopy cell. Due to this spectroscopy a frequency dependent signals are measured by two photodiodes. A subtractor subtracts one photodiode signal of the other, which results in a dispersive signal. A PID controller is used to generate correction signals by which the frequency of the laser is adjusted, see Fig. 11. In the following we describe the setup of the laser lock in more detail.

3.1 Laser alignment

3.1.1 Laser

We use a diode laser with an external diffracting grating, controlled by the MLD-1000 modu-lar laser driver system of Sacher Lasertechnik Group. The broadband grating splits the light in different directions, depending on the frequency. The zero order leaves the laser and is used in the alignment. The minus first order of a specific frequency depending on the grating angle is exactly sent back to the laser, see Fig. 13. This retro-reflected light will stimulate emission in the laser diode, thereby selecting the frequency at which the laser lase. With a mirror at the back side of the diode the light goes back to the grating, forming the laser cavity. Almost all of the minus first order light will again be reflected into the diode, and by repeating this process many times, in the end all the light is of one frequency, the one of the minus first order. The grating is designed in such a way that a small part of the light can leave the laser as the zero order and can be used in the optical setup. On the backside of the grating, a piezo is placed to control the angle of the grating. Because the frequency of the minus first order depends on the angle of the grating the frequency of the laser can be controlled by the piezo.

FABRY-PÉROT CAVITY QWP1 QWP2 HWP3 NEUTRAL DENSITY FILTER, 0.4 OD LONGPASS FILTER, >700nm SHORTPASS FILTER, < 950nm AOM PHOTO DIODES K. SPECTROSCOPY CELL HWP2 CYLINDRICAL LENSES OPTICAL ISOLATOR HWP1 LASER f = 200mm f = 200mm POLARIZING BEAM SPLITTING CUBE BEAM SPLITTER MIRROR REMAINDER OF OPTICAL SETUP PUMP BEAM PROBE BEAM ZERO-ORDER FIRST ORDER

z

B-field

Figure 12: Schematic drawing of the optical setup used for the laser lock. A diode laser with an external grating provides the light. Only a fraction of the light is used for the laser lock. To create an error signal, Doppler-free polarization spectroscopy is used, for which a spectroscopy cell, wave plates and two photodiodes are used. An AOM is used to correct the

light for the different energies of the transition of39K and40K. A Fabry-P´erot Cavity is used

to check the laser for single-mode operation.

first order

Gr

ating

Diode

Figure 13: Schematic drawing of the laser. The laser consists of a diode, grating and a mirror. The minus first order is the laser beam reflected inside the laser cavity, the zero order is used in the laser setup.

light coming from the laser has an elliptical transverse mode profile due to the geometry of the diode chip and to obtain a circular profile, a cylindrical lens telescope is used.

Subsequently a half wave plate (HWP) is places in front of a polarizing beam splitting cube (in short cube). The HWP determines the ratio between the vertical and horizontal polari-sation. The cube splits the light by polarisation into two beams. Therefore the HWP can be used to tune the intensities of the beams. The cube is used to split the light in a part needed for the laser lock and the remainder of the beam is used for cooling and trapping. The laser lock only needs small intensities of light and therefore we use a beam with an intensity of 2 mW, while the intensity of the beam of the remainder of the optical setup is 55.9 mW.

3.1.2 AOM

The MOT consists of 40K and therefore the laser has to be exactly at the correct frequency

for this isotope. However, the abundance of40_{K in our natural sample is only 0.0117(1) %. In}

contrast, 39K has an abundance of 93.2581(44) %, resulting in a much stronger spectroscopy

signal. Therefore 39K is used to lock the laser. In order to have the laser source lasing at

the40K frequency, the laser light must be shifted by the isotope shift between39K and 40K

before the lock.

For cooling and trapping a frequency close to the transition of the 2S1/2, F0 = 9/2 state to

2_P

3/2, F0 = 11/2 state of40K is used. The associated frequency is 766.701 nm+651.1 MHz, see

Fig. 14. In the spectroscopy cell the laser light addresses the transition from the2_S

1/2, F0 = 1

state to the2P3/2 state of39K with accompanying frequency is 766.701 nm + 288.6 MHz. The

difference in frequency is therefore 362.5 MHz, see figure 15.

In the remainder of the optical setup some acousto-optic modulators (AOMs) are used, which shift the frequency of the light. Those AOMs shift the light by 80 Mhz. Therefore the master

laser must be locked 80 Mhz below the transition of the 2S1/2, F0 = 9/2 to 2P3/2, F0 = 11/2

state. The difference between the master laser and the transition from the 2S1/2, F0 = 1 to

2_P

3/2state of39K, used in the lock, is therefore 282.5 Mhz. Therefore an AOM inside the laser

path of the lock must be added, which lowers the frequency of the master laser by 282.5 Mhz.

An AOM consists of a piece of glass through which a sound wave propagates, generated by a piezo-electric transducer. A fraction of a light beam sent through this glass and adopts the sound wave inside the glass, which results in a shift in frequency of that fraction of the light. Depending on the orientation of the light beam with respect to the glass the frequency of is lower or higher than the initial laser beam. The signal for the piezo-electric transducer is generated by a voltage controlled oscillator (VCO) and amplified in two stages. The power

2_P 3/2 2_S 1/2 2_P 1/2 (1285.8) 40

_K

_K

_K

Figure 14: Optical transitions of the D1 and D2-lines of39K, 40K and 41K. For the lock the

transition from 2S1/2, F0 = 1 to 2P3/2 of39K is used, while for the MOT the transition from

the 2S1/2, F0 = 9/2 state to 2P3/2, F0 = 11/2 state of 40K is used. The schematic is directly

FREQUENCY (MHZ) MASTER LASER LASER LOCK 39_K 2_S 1/2,F’=1 2P3/2 MOT 40_K 2_S 1/2,F’=9/2 2P1/2,F’=11/2 +288.6 MHZ 362.5 MHZ +80 MHZ AOMSINREMAINDER OFOPTICALSETUP -282.5 MHZ

AOM INLASERLOCK

766.701NM +571.1 MHZ +651.1 MHZ

39_K 2_S

1/2 2P3/2

Figure 15: Schematic representation of the different frequencies needed for the setup. These frequencies are the frequency needed of the laser lock and the MOT and the frequency of the master laser.

analogue converter (DAC) connected to a control computer. The control voltage determines the frequency of the VCO. The efficiency of the AOM depends strongly on the frequency. Since no standard AOM with highest efficiency at 282.5 Mhz is available, we used an AOM with a resonance frequency of 150 MHz. We removed some loops of the two impedance match-ing coils inside the AOM, in order to lift the highest efficiency to 282 MHz.

To increase the diffraction efficiency of the AOM, we concentrate the laser beam to a narrow beam inside the AOM. For this narrow beam, two lenses, one in front and one behind the AOM, operate jointly to concentrate and collimate the laser beam. The lenses we use are LA1708-B lenses from Thorlabs, with a focal length of 199.3 mm.

The AOM splits the light into two beams, one with the shifted frequency (the first order) and the other consisting of the light that isn’t affected (the zero order). The first order beam has a small angle compared to the zero order beam and therefore the beams can be easily separated. The intensities are 1.14 mW and 0.76 mW for the first order and zero order re-spectively, corresponding to an efficiency of 60 %. In the following, we will discuss first the path of the first order beam, used for the spectroscopy, and afterwards the path of the zero order beam, used for the cavity.

3.1.3 Spectroscopy

Now we are describing the spectroscopy. This part of the laser lock consists of the cell with

vapor, the density of the gas is increased by heating the cell to roughly 65oC [2].1 A coil around the cell induces a magnetic field in the cell of approximately 5 Gauss parallel to the laser beams. This magnetic field induces a Zeeman shift by which the dispersion signal is generated, see Sec. 2.2.1.

The probe and the pump beam are sent through the cell, aligned on top of each other and propagating in opposite directions. The intensities are 83 µW and 379 µW respectively. The pump and probe beams are derived from the light entering the spectroscopy setup by a cube, using a HWP in front of the cube to manage the proportion of the intensity of the beams. To establish a circular polarised pump beam, a QWP is placed in the path of the beam. Because the intensity of the probe beam is relatively low, potential noise of the cube becomes significant. To reduce this noise, we used a probe beam split off by the cube with an intensity almost twice as high as 83 µW, which is attenuated by 40 % before the cell. For this attenuation we used a neutral density filter from Thorlabs of 0.4 OD.

After the cell, the probe beam first is sent through a QWP, which transforms the left- and right hand circular polarization to orthogonal polarization, followed by a cube, which split the probe beam by the polarization components into two beams. These beams have an intensity of about 7 µW and are equal within a small frequency interval of the laser beam around the optical transitions. The intensities of the beams are measured by two photodiodes. To remove background light, which would add noise to the spectroscopy signal, two filters are placed in front of the photodiodes. Those filters remove all wavelengths below 700 nm and above 950 nm.

3.2 Electronics

The signal of the photodiodes is subtracted by a subtractor. This subtractor consists of an instrumental amplifier (INA114) along with some other components, see Appx. A.1 for a detailed schematic figure of the subtractor. The signal generated by the subtractor is the error signal for the PID. Because the subtractor is enclosed in the PID box, the signal is directly soldered to the SIGNAL IN of the PID, see schematic Fig. 24 in Appx. A.2.

For correcting the laser frequency, only the integrating and differentiating parts of the PID are used. These branches of the PID are connected to the FAST OUT and the SLOW OUT respectively. The proportional part is not used, but it has an output to the MONITOR OUT. The reason for using two of them is that the laser can only be controlled by the current to the diode of the laser and by controlling the voltage to the piezo.

For the optimum correction, the PID must generate a correction signal for noise of a broad range of frequencies with an optimum intensity. For a low gain, the frequency of the laser

can drift of too much before corrected sufficiently, while for a correction signal with too high a gain, the laser can be over corrected and therefore the feedback loop can start to oscillate.

3.2.1 Slow feedback loop

The main lock is performed by the slow branch of the PID. The SLOW output is connected to the PIEZO IN of the laser piezo driver and thereby with the piezo of the laser. Between the SLOW out and the PIEZO IN, an attenuator of Mini Circuits with HAT 1 (1 dB attenuation) is placed, followed by a switch and a low pass filter The attenuator is placed to attenuate a small electronic noise as well as to avoid resonance loops. The switch can flip between a frequency generator for finding the error signal of Fig. 10b and the output of the SLOW OUT for locking the laser on a specified frequency. The low pass filter is placed to avoid feedback loop oscillations. Without the low pass filter, even for a low gain, a resonance of mainly 5.7 kHz occurs. A low pass filter with a cut-off frequency of 1.6 kHz solves this problem and enables the use of a much stronger gain, which results in a better locked laser.

3.2.2 Fast feedback loop

For the fast feedback loop the FAST OUT is connected to I MOD, the input for current modulation of the laser current driver. Between these two a transformer of Mini Circuits is placed with best transfer frequencies between 0.2 and 500 MHz. The transformer is needed to overcome the difference in offset between the different grounds. Without the transformer the laser jumps to another current in case of a gain too small for effective corrections. Also a 10 dB attenuator is placed between the FAST OUT and the transformer, to avoid over correction with relatively small gain.

With all this components the laser can be stably locked at the correct frequency. The exact properties will be covered in the discussion section.

3.3 Fabry-P´erot cavity

All of the above deals with locking a laser to one frequency. But even when the lock is working, a laser can emit several frequencies simultaneously. This behaviour is called multi-mode operation. For cooling and trapping, the laser has to operate at exactly one frequency, which is called single-mode operation. To check for unwanted multi-mode operation a

Fabry-P´erot cavity is used. While the first order beam of the AOM is needed for spectroscopy, the

zero order is not yet used for anything particular. Therefore we use this beam for the cavity. A cavity consists of two mirrors opposite each other. On one side the beam enters the cavity,

-100 -50 0 50 100 150 Frequency (MHz) 0.2 0.4 0.6 0.8 1.0 1.2

Intensity (V ) Fabry-Pérot cavity

Figure 16: The frequency spectrum of the Fabry-P´erot cavity. When standing waves occur,

the intensity measured increases. By changing the cavity length, the frequency for which standing waves occur is scanned. Therefore the number of frequency components at which the laser emits can be measured. Because the peak in the figure is smooth, the laser is of one frequency (single-mode operation).

frequency is in resonance with the cavity. The resonance condition is met if the length of the cavity is equal to a multiple of half the wavelength of the light. On resonance, some light will leave the cavity through the second mirror and can be detected on a photodiode.

To avoid tedious searching for standing waves, one of the mirrors is oscillating with a frequency controlled by a frequency generator. Thereby the distance between the mirrors changes and consequently the frequency for standing waves changes as well. For a laser with exactly one frequency, only one peak should be measured. For multi-mode operation, multiple frequencies will be seen. From Fig. 16 it follows that the laser is in single-mode operation.

4

Discussion

The lock described in the section above typically stabilizes the laser for at least a day. In this section the characteristics of the lock will be described. This includes measurements of the short time stability as well as long term stability. Furthermore, the effect of the slow and the fast feedback loops will be characterized.

4.1 Short time stability

When the laser is locked, some noise in the error signal can be seen. The noise comes from imperfect corrections of the PID as well as internal noise in the electronics and optics. This leads to a wider linewidth of the laser. To measure the linewidth of the laser while it is locked, the noise on the error signal is measured with an oscilloscope, see Fig. 18. The amplitude of the oscilloscope is given as a voltage. To calculate the difference in frequency, the following relation is used, ∆fshift= ∆fF0_{=1 to F}0₌₂ ∆tF0_{=1 to F}0₌₂ ∆terror signal ∆Verror signal ∆Vshift, (2)

where ∆fF0_{=1 to F}0₌₂ = 461.7 MHz, see Fig. 14. ∆t_F0_{=1 to F}0₌₂, ∆t_{error signal} and ∆V_{error signal}

can be measured from Fig. 17 and 10b. The exact values of ∆tF0_{=1 to F}0₌₂ and ∆t_{error signal}

on the oscilloscope depend on the amplitude of the frequency generator and can change with

time. Also ∆Verror signal depends on various components of the setup and can therefore easily

change. Thus, all these values are always measured together with ∆Vshift. From Fig. 18 it

follows that the linewidth of the laser is typically 2.5 MHz. This linewidth is well within the required 6 MHz mentioned in Sec. 1.2.

4.2 Long term drift

Short time fluctuations effect the frequency of the laser, and when the error signal is recorded for a range of frequencies, these fluctuations are visible by horizontal motion of the entire error signal of figure 17. The laser lock corrects for those fluctuations and in this way the frequency of the laser is held stable.

However, the dispersion can also move vertically. In this case the dispersion is no longer symmetric around zero, which means that the frequency belonging to the zero intensity is changed. Because the laser is locked on this zero intensity, the laser is locked on the wrong frequency. Besides, the lock is also less stable because the laser can escape relatively easy at one side.

Vertical movements can be caused by a changing magnetic field in the potassium cell. The magnetic field influences the two polarised beams differently and therefore the intensity of

-500 0 500 Frequency (MHz) -1.0 -0.5 0.5 Intensity (V ) Error signal F=1 F=2

Figure 17: The error signal measured using the complete optical setup. Using this error signal and Eq. 2, the relation between the intensity as a voltage and the frequency drift of the laser can be fixed. - 0.04 - 0.02 0.00 0.02 0.04 - 1.5 - 1.0 - 0.5 0.0 0.5 1.0 Time (ms) Frequency (MHz )

Noise of the laser

Figure 18: The noise on the laser measured with an oscilloscope. The intensity of the signal is converted to frequencies using equation 2. For this graph fifteen measurements are shown on top of each other, showing the maximum displacement of the laser for different times.

the error signal changes with a changing magnetic field. Also, changing temperature in the cell could affect the error signal, due to the absorption that is temperature depend. These fluctuations change the error signal slowly but can increase within a day to an error signal not functional for locking. The precise effect of these components will not be characterised here. But it would be good to investigate this drift in further research.

To characterize the shift in frequency due to the long term drift, the shift of the middle of the dispersive error signal of Fig. 17 is measured over some time. Using equation 2 the shift in frequency is calculated, which leads to a shift of 1.2 MHz after one night. To maintain the error signal centred around zero, the magnetic field can be changed, as well as the angles of HWP2 and QWP2, see Fig. 12.

4.3 Attenuation of the noise by the slow and the fast feedback loop

The effect of the slow and fast feedback loops on the background noise can be measured. This attenuation is performed by measuring the noise in dB with high and minimum gain and subtracting these results. In Fig. 19 and 20 the attenuation in dB per frequency is shown for the slow and the fast feedback loop respectively. Here a positive value means that the noise is smaller for high gain. Because this attenuation can only be measured with a locked laser, the minimum gain of the slow branch of the PID is not zero. For the fast branch the minimum gain is zero.

To achieve the clear shape in the Fig. 19 and Fig. 20, firstly the intensity is measured and averaged two times for each gain. Secondly, the ’movingaverage’ function of Mathematica is used, which averages the value for each point, with a number of points around that point. For Fig. 19 this number is 50, which means that all points in the figure are an average of 50 data points around that point. At the borders the first and last 25 points of the data are not included in the figure. For Fig. 20 number of points for which is averaged is 45. Due to this function, extreme values are faded out by the average and hence a clear signal appears from the data.

From Fig. 19 it follows that the slow feedback loop mostly works for low frequencies. This means up to 700 Hz the slow feedback loop is significant, whereas from 1.7 kHz the slow feed-back loop does not correct powerfully. Meanwhile, the fast feedfeed-back loop starts working from roughly 1.5 kHz and decreases around 9.5 kHz, with some small peaks after that, see Fig. 20. Considering that inside the slow feedback loop a low pass filter with cut off frequency of 1.6 kHz is placed, it is not surprising that the correction of the slow feedback loop decreases around this frequency. The small peaks around 3.1 kHz are therefore not completely clear.

0 1000 2000 3000 4000 5000 0 2 4 6 Frequency (Hz) Correction (dB )

Frequency dependent correction bythe slow feedback loop

Figure 19: Noise reduction of the slow feedback loop for different frequencies. This reduction is measured with an oscilloscope using the Fourier transform math function. Subtracting the signal with low gain (almost de-locked) from the signal with the highest gain (strongest locked), the attenuation by the slow feedback loop is measured. To reduce the noise on this subtraction, all data points are the mean value of the nearest 50 points. These mean values are calculated using the ’movingaverage’ function of Mathematica. From this graph follows that the slow feedback loop reduces the noise with 6 dB up to 700 Hz sufficient, and after 1700 Hz no significant correction is measured.

loop works perfectly fine at low frequencies and the fast feedback loop is not designed for low frequencies, it is not surprising that the fast feedback loop starts only working from the point at which the slow feedback loop becomes less effective. It is not clear why the fast feedback loop stops working at around 10 kHz. Possibly, this upper limit of attenuation is caused by the fact that the fast feedback loop is tuned towards relatively small frequencies through which it lost its effectiveness for higher frequencies.

Thus, the combination of the slow and the fast feedback loop decreases the noise by 6 dB for frequencies between zero and 10 kHz.

0 5000 10 000 15 000 20 000 25 000 0 2 4 6 8 Frequency (Hz) Correction (dB )

Frequency dependent correction bythe fast feedback loop

Figure 20: Noise reduction of the fast feedback loop for different frequencies. This reduction is measured analogue to Fig. 19. For measuring the difference, the gain of the fast branch of the PID was zero or maximal. For this figure, the mean value of the nearest 45 points is used for the ’movingaverage’ function of Mathematica. From here a clear attenuation of 6 dB between 1 kHz and 10 kHz can be seen, for higher or lower frequencies, no significant attenuation is measured.

4.4 Attenuation of acoustic- and electronic-added noise by the fast feed-back loop

Another way of testing the system is by adding noise of some particular frequency. Only the fast feedback loop is tested, because the slow branch of the PID cannot easily switch between low and high gain. The attenuation is tested for two different sources of noise, acoustic and electric.

The acoustic noise is made by placing a loudspeaker in front of the setup. Driven by a fre-quency generator software for frequencies between 500 Hz and 20 kHz the attenuation between no gain and maximum gain of the fast is measured. In figure 21 the attenuation is plotted. It turns out that the attenuation is about 4 dB for frequencies between 1 kHz to 10 kHz. For higher frequencies no attenuation is observed.

The electric noise is made by a frequency generator. The amplitude of the noise is 1.00 V peak-to-peak and frequencies between 1 kHz and 1.5 MHz are measured. Above 17 kHz no attenuation is measured. Below 17 kHz an attenuation of 3 dB is measured, see Fig. 22. For both acoustic and electric attenuation measurements the absolute intensity with and without using the fast feedback loop are very unstable. The uncertainty therefore should be taken up to 3 dB. Therefore, the results in Fig. 21 and Fig. 22 can only be taken for the general result: the fast feedback loop works up to 10 kHz for about 4 to 6 dB.

Further work could improve the lock to be more precise as well as better characterized. To reduce the linewidth of the laser, the gain of the slow and the fast feedback loops should be increased, whereby more frequencies of the noise should be attenuated by the PID. A lock without the low pass filter in the slow feedback loop is desired, as well as a transformer with a lower limit frequency of 10 kHz instead of 500 kHz. Influence on long term drift should be characterised elaborately before clear improvements can be carried out.

-1 0 1 2 3 4 5 6 7 8 500 5000 )B d( noi ta un ett A Frequency (Hz)

Attenuation by fast feedback loop of external-added

acoustic noise

Figure 21: Acoustic-added noise attenuation of the fast feedback loop. The acoustic noise is generated by a frequency generator connected to a loudspeaker. For each frequency the intensity is measured without and with using the fast feedback loop and subtracted one of the other, analogue to Fig. 19 and 20. Because the exact values of the intensity were not very stable, the precision of the attenuation is within ±3 dB. Therefore the precise effect per frequency is not reliable. However, the general effect strengthen the results before, that is, fast feedback loop attenuates up to 10 kHz for this noise about 4 dB.

-1 0 1 2 3 4 5 6 7 1 10 )B d( noi ta un ett A Frequency (kHz)

Attenuation by fast feedback loop of external-added

electronical noise

Figure 22: Electric-added noise attenuation of the fast feedback loop. This figure is analogue to Fig. 21.

5

Conclusion

The idea of the project, making a decent laser lock to keep the frequency of the laser stable

around some frequency, is accomplished. The lock keeps the laser stable for a sufficient

amount of time. Studies of the noise show that the linewidth is about 2.5 MHz, which is within the linewidth needed for potassium. Measuring the attenuation by the slow and fast feedback loops shows an attenuation of the noise of about 6 dB for frequencies up to 10 kHz and no attenuation for higher frequencies. When noise is added to the system, the same effect can be seen. Therefore the locked laser useful for cooling and trapping atoms. Further work could be improving the linewidth of the lock as well as characterizing and reducing the long term shift.

6

Acknowledgments

During my three-month bachelor project I worked together with several people. Therefore I would like to thank them for their help and cooperation. Firstly I would like to thank Wouter (Meinster) for facilitating and supervising this project alongside his own master project. In addition he gave me a proper introduction to experimental research in atomic physics. Also, I want to thank Florian (Schreck), for giving me the opportunity to do my bachelor project in his research group. Furthermore, I would like to thank Benjamin (Pasquiou), for helping me understanding the more theoretical part of the project, as well as helping me to solve difficult problems in the lab. Finally, I would like to thank Alex (Bayerle) for his help with electronics. For this I also want to thank the electronic workshop.

References

[1] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell. Observation of Bose-Einstein condensation in a dilute atomic vapor. Science, 269(5221): 198–201, 1995.

[2] Tobias Gerard Tiecke. Feshbach resonances in ultracold mixtures of the fermionic quantum gases 6Li and 40K. 2009.

[3] C. A. Regal, M. Greiner, and D. S. Jin. Observation of resonance condensation of fermionic atom pairs. Phys. Rev. Lett, 92:040403, Jan 2004.

[4] T.G. Tiecke. Properties of potassium. 2010.

[5] Alois Mair Martin Horbanski, Jan Krieger. F20: Magneto-optische falle, presentation. FP-Seminar, June 2005.

[6] C. Wieman and T. W. H¨ansch. Doppler-free laser polarization spectroscopy. Phys. Rev.

Lett., 36:1170, May 1976.

[7] H.J. Metcalf and P. van der Straten. Laser Cooling and Trapping,. Graduate Texts in Contemporary Physics. Springer New York, 1999. ISBN 9780387987286.

[8] CP Pearman, CS Adams, SG Cox, PF Griffin, DA Smith, and IG Hughes. Polarization spectroscopy of a closed atomic transition: applications to laser frequency locking. Journal of Physics B: Atomic, Molecular and Optical Physics, 35(24):5141, 2002.

[9] Yutaka Yoshikawa, Takeshi Umeki, Takuro Mukae, Yoshio Torii, and Takahiro Kuga. Frequency stabilization of a laser diode with use of light-induced birefringence in an atomic vapor. Appl. Opt., 42(33):6645–6649, Nov 2003.

2 3 6 7 4 Photo diode 1 Photo diode 2 +15V PID -15V PID Subtraction to PID N.C. N.C. 5 GND PID GND PID 0.1µF 0.1µF INA114 GND PID

Figure 23: Schematic drawing of the subtractor, used to create the error signal out of the two signals of the photodiodes.

A

Electronics

A.1 Subtractor

To generate an error signal from the two signals of the photodiodes, we use an instrumentation amplifier, the INA114 of Burr-Brown. The subtractor has three connections, two of the photodiodes and one of the output signal to the PID. The grounds of the photodiodes are not connected to the subtractor, to avoid ground loop effects by unnecessary ground connections. Also to avoid ground loops, the subtractor is connected to the ground of the PID and uses the power supply of the PID, and therefore no external power supply is needed. Because the subtractor is connected to the PID, it is integrated into the PID housing. The output of the subtractor is directly soldered to the PID IN-connections, see Appx. A.2.

A.2 PID schematics

The PID has four connections, which are from top to bottom the INPUT, FAST OUT, MON OUT and SLOW OUT. The MON OUT is the proportional signal, not used for the lock. The input is connected to the subtractor by wires soldered inside the PID and not by the BNC-connection of the front plane.

The PID is powered by a power supply with +18 V, −18 V and ground. The connection for the power is on the backside. The power for the subtractor is connected to the output of the +15 V and −15 V voltage regulators. Inside the PID two different types of switches are used, that is SIl12-IA85-7604K for the slow off/on switch and SIL05-IA72-71D for the fast off/on switch. Both are from MEDER elements. While testing the lock, the SIL05 switch of the slow feedback loop broke. Because the voltage on the switch is 12 V in stead of 5, we replaced

1 1 2 2 3 3 4 4 D D C C B B A A A 2 _A4 _A6 _A8 _A10 _A12 _A14 _A16 _A18 _A20 _A22 _A24 _A26 _A28 _A30 _A32 C2 C4 C6 C8 _C10 _C12 _C14 _C16 _C18 _C20 _C22 _C24 _C26 _C28 _C30 _C32 P1 2x16 +18 V -18 V GN D GN D +18 V -18 V IN 2 1 OU T 3 GN D U 2 _MC79M15CT 100u F C2 Elko GN D -18 V 33u F C5 Elko GN D -15 V GN D IN 1 2 OU T 3 GN D U 1 _MC7815AC T 100u F C1 Elko GN D +18 V GN D 33u F C4 Elko GN D +15 V IN 1 2 OU T 3 GN D U 3 _MC7805AC T IN 2 1 OU T 3 GN D U 4 _MC79M05CT 10u F C6 Elko 10u F C7 Elko GN D GN D +5 V -5 V GN D GN D IN 1 2 OU T 3 GN D U 5 _MC78M12A CT +18 V GN D 10u F C3 Elko GN D +12 V P2 Signal IN GN D 1 2 3 P6 Jum pe r2 8 5 3 2 6 7 4 1 U 6 _AD805 5 56 R2 1 Res1 GN D 1k R2 Res1 1k R3 Res1 +5 V -5 V 1 2 3 P7 Jum pe r2 8 5 3 2 6 7 4 1 U 7 _AD805 5 56 R4 ' Res1 100p F C24 ' K erk o 1k R5 ' Res1 GN D -5 V +5 V 3k3 R6 ' Res1 330p F C25 ' K erk o 100k R7 ' Res1 GN D 1n F C26 ' K erk o 330 R8 ' Res1 1k R9 ' Res1 GN D +5 V -5 V 220n F C8 Tanta l 220n F C9 Tanta l GN D 220n F C1 0 Tanta l 220n F C1 1 Tanta l 100n F C1 6 Tanta l 100n F C1 7 Tanta l 1n F C1 2 K erk o 1n F C1 3 K erk o 1n F C1 4 K erk o 1n F C1 5 K erk o +15 V -15 V P3 FAST Ou t GN D GN D 10k R1 1 Res1 10k R1 2 Res1 1 2 3 P8 Jum pe r2 10k R13 ' Res1 GN D R14 10k ' Res1 1u F C27 ' K erk o GN D 10k R1 5 Res1 GN D 10k R1 6 Res1 10k R17 ' Res1 10k R18 ' Res1 P4 PIEZO Out GN D 8 5 3 2 6 7 4 1 U1 1 AD805 5 GN D 1k5 R1 9 Res1 1k5 R2 0 Res1 +5 V P5 Mon. Out GN D -5 V +15 V +15 V -15 V -15 V 1 2 3 S2 PIEZ O I nt 1 2 3 S3 PIEZ O Sw eep 1 2 3 S1 FAST I nt +12 V +12 V +12 V GN D GN D GN D 1 3 2 _{R1 FAST 1k} Am p. 1 3 2 Value: 10k R1 0 PI EZ O A m p. 1 2 3 4 Re 1 FAST I nt 1 2 3 4 Re 2 PI EZ O I nt 1 1 2 3 4 Re 3 PI EZ O Sw eep GN D GN D GN D Int 1 Int 2 Int 3 Int 1 Int 2 Int 3 220n F C2 9 Tanta l 220n F C3 0 Tanta l 1n F C3 1 K erk o 1n F C3 2 K erk o steckba r steckba r steckba r steckba r steckba r steckba r steckba r steckba r steckba r steckba r steckba r steckba r steckba r steckba r 2 3 1 A 4 11 U8 A AD713 K N 5 6 7 B 4 11 U8 B AD713 K N 10 9 8 C 4 11 U8 C AD713 K N 12 13 14 D 4 11 U8 D AD713 K N -15 V +15 V 1 2 P9 Sweep ou t 10k R2 3 Res1 -15 V +15 V 2 3 1 A 4 11 U9 A AD713 K N 5 6 7 B 4 11 U9 B AD713 K N 10k R25 ' Res1 22k R24 ' Res1 GN D +15 V -15 V -15 V +15 V GN D 100n F C28 ' K erk o steckba r 100k R26 ' Res1 steckba r steckba r steckba r GN D 100n F C3 3 Tanta l 100n F C3 4 Tanta l +15 V -15 V 10 9 8 C 4 11 U9 C AD713 K N GN D 12 13 14 D 4 11 U9 D AD713 K N 50 R27 * Res1 GN D O PT IONA L 1 2 P10 Signal 2 In GN D 10k R28 * Res1 _{TIO OP} NA L -15 V +15 V GN D 10k R27 ' Res1 10k R28 ' Res1 steckba r steckba r 1 2 P11 LED GN D 1k R2 9 Res1 REMOVED +15V INA of SUB TR AC TOR -15V INA of SUB TR AC TOR SUB TR AC TOR OUT SUB TR AC TOR GND SIL12 SIL05 REMOVED

Figure 24: Schematic drawing of the PID controller. This diagram is taken from the wiki of the group of Florian Schreck. Some components are changed from the original PID, see for more details the schematics. Furthermore, the subtractor is integrated into the PID housing, using the same power supply and the connection between them is directly soldered inside the