Effects of external parameters on the line shape of hyperfine lines in molecular iodine

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Rijksuniversiteit Groningen

Bachelor Thesis

Effects of external parameters on the line shape of hyperfine lines in molecular iodine

Author:

J.R. Maat

Student nr.: S2567423

Supervisors:

Dr. L. Willmann Prof. Dr. K.H.K.J. Jungmann

July 2016

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Abstract

This work focuses on high-precision spectroscopy on molecular iodine, which is executed at the Van Swinderen Institute in Groningen. This work also provides information as well as a number of improvements that have been made in the barium ion experiment. The barium ion experiment will attempt to measure Atomic Parity Violation in the future. In these experiments the properties of the laser sources determine the precision of the measurement.

These properties include the spatial intensity distribution, the polarization, the positioning stability to overlap the light with the well-localized ion in the trap and the absolute frequency of the light. Improvements that have been made include mode cleaning and measuring power broadening of one of the lasers used in the experiment. High-precision spectroscopy on molecular iodine provides an optical reference with uncertainties of 10^-12 for the barium ion experiment. This thesis discusses the implementation of a digital lock-in amplifier, TEM Messtechnik’s LaseLock, in the experimental set-up for frequency modulated spectroscopy, in order to compare it to an analog lock-in amplifier, the Scitec 420. The comparison includes line shapes and signal-to-noise ratios of spectra of hyperfine lines in molecular iodine. Similar results are obtained with both lock-in amplifiers, having signal-to-noise ratios up to 300, for acquisition times of 100 ms per data point. Further, the influence of a number of parameters of the LaseLock such as sampling rate, digital filter characteristics and modulation bandwidth were investigated. No significant difference between the analog and digital lock-in amplifiers has been found for the application in frequency modulation spectroscopy of iodine, and it can be concluded that the lock-in amplifier is not the limiting noise source in the set-up.

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1 Introduction

This thesis work has been executed in the framework of the single trapped Ba⁺ ion at the Van Swinderen Institute at the University of Groningen. In such experiments accurate understanding of the parameters of laser light, for example intensities, polarization and absolute frequencies, is indispensable.

This thesis starts with the description of the scientific context of the Ba⁺ experiment, which aims at a measurement of effects in atomic systems due to the weak interaction, which is known as Atomic Parity Violation (APV).

As part of this thesis an additional laser beam which is used to induce light shifts on the Ba⁺ energy levels has been implemented. One of the requirements is spatial overlap of a laser beam with a well localized trapped ion to better than 10 µm accuracy. In order to determine the local intensity of a focused laser beam, the spatial mode has to be controlled to a high precision, in particular since the spatial mode of a diode laser is rather complicated. The output of the diode laser has been transferred to a laser beam with Gaussian intensity profile. Further, an optical isolator was installed directly after one of the lasers. This reduces the amount of unwanted feedback to the laser cavity and ensures good stability of the frequency of the laser light. A measurement on power broadening of the 5d²D3/2 −6p²P1/2 transition in barium demonstrated the working of this additional laser system.

High-precision spectroscopy on molecular iodine provides an optical reference with uncertainties of 10⁻¹² and requires precise knowledge on the influence of experimental parameters. We studied the hyperfine structure of the R(25)(6-5) transition at a wave length of 649.87 nm. The effects of vapor pressure in the iodine cell on the transition are studied. The pressure of the iodine on which spectroscopy is done can be controlled, and influences features in the spectrum.

In the iodine spectroscopy set-up, a lock-in amplifier is used to filter the signal out of noise that covers it. A new, digital lock-in amplifier was installed, and measured results are compared to results measured by the current, analog lock-in amplifier. Here the bandwidth of the lock-in amplifiers, the achievable signal-to-noise ratio and the observed line shape have been investigated.

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2 ATOMIC PARITY VIOLATION 4

2 Atomic Parity Violation

Applying a parity transformation to a physical system reflects the three spatial coordinates of that system in the origin. When the system is invariant under parity transformations, it is said to be parity conserving. However, there are also physical systems that do not conserve parity. Those systems are said to be parity violating. Examples of systems that are parity violating include atoms. It is possible, in experiments, to measure the strength of parity violation in atoms. Experiments that aim at measuring such effects are known as experiments on Atomic Parity Violation. The barium ion experiment, executed at the Van Swinderen Institute, is an attempt to measure Atomic Parity Violation in barium ions,¹³⁸Ba⁺. This section aims at introducing the theory behind Atomic Parity Violation (APV). Measurements on Atomic Parity Violation may hint to new physics, which is a

reason to execute the so-called barium ion experiment at the Van Swinderen Institute.

2.1 The Standard Model

The Standard Model (SM) [1, 2, 3] is a theory that describes all experimental observations of fundamental particles and three of the four interactions of nature. It permits to predict results from measurements. Although the theory shows good agreement with experiments up to the energy range that is accessible for experiments, it is incomplete, and there is physics beyond the Standard Model [4].

One way to find evidence for physics beyond the Standard Model is to measure quantities with high precision, and compare them to values which can be derived in the Standard Model. Examples of these measurements include precise determination of the strength of Atomic Parity Violation effects and searches for permanent Electric Dipole Moments (EDMs). Atomic Parity Violation is discussed as an example in this chapter. More

information on permanent EDMs can be found in [5] and [6].

2.2 Parity

The Standard Model includes three discrete symmetries: parity symmetry (P), time reversal symmetry (T) and charge conjugation symmetry (C). By applying the parity operation, all spatial coordinates are reflected in the origin [4]. A mathematical representation is:

P ~ˆr = −~r (2.1)

P ψ(~ˆ r, t) = pψ(−~r, t) (2.2)

where ˆP is the parity operator, p its eigenvalue, ~r are the spatial coordinates, ψ is the wave function of the particle under consideration, and t is time. Since applying the parity operator twice should return the initial wave function (mathematically: ˆP²ψ(~r, t) = p ˆP ψ(−~r, t) = p²ψ(~r, t)), the two eigenvalues are p = ±1. Comparable considerations can be done concerning time reversal symmetry and charge conjugation symmetry, and are found in [4], but we will focus on parity symmetry here.

2.3 Parity Violation in experiments

In the first half of the twentieth century, the paradigm was that parity was conserved by the strong interaction, the electromagnetic interaction and the weak interaction. However, in 1950 E.M. Purcell and N.F. Ramsey suggested the idea of parity violation in permanent Electric Dipole Moments (EDMs) [7]. In 1956, Tsung-Dao Lee and Chin-Ning Yang

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had investigated whether or not there was experimental evidence available to decide on the question of parity conservation in the weak interaction, and suggested a number of experimental tests [8]. The first physical experiment that demonstrated parity violation, was executed by Chien-Shiung Wu in 1957 [9]. There is no evidence yet for possible parity violation in electromagnetic and strong interaction.

2.4 Electroweak unification and the Weinberg angle

Two of the fundamental forces which the Standard Model [1, 2, 3] describes are the electromagnetic and the weak interactions. The electromagnetic interaction is mediated by massless photons (γ), and thus has an infinite range, whereas the weak interactions are mediated by massive W^± and Z⁰ bosons. The fact that the carriers of the weak force are massive implies it has a finite range.

The unification of the weak and the electromagnetic interaction is described by the so-called electroweak interaction [10, 11]. Figure 1 shows the three terms arising in the electroweak interaction. The purely weak term is orders of magnitude smaller than the electromagnetic term, and is thus not accessible in experiments. It is possible, however, to investigate the weak interaction by measuring the interference term between the electromagnetic and the weak interaction.

Figure 1: Interference terms of the electromagnetic (EM) and weak interaction. The strength of the purely electromagnetic term is about six orders of magnitude larger than that of the interference term. Since the purely weak term is twelve orders of magnitude smaller than the purely electromagnetic term, it is not accessible in experiments. Figure taken from [10].

The electric charge e is the coupling constant of the electromagnetic interaction, and g_W is the coupling constant of the weak interaction. The Weinberg angle, also known as the weak mixing angle, is a parameter in the electroweak interaction, and is given by:

sin²θ_W = e²

g_W² (2.3)

The square of the sine of the Weinberg angle can be measured. Because of radiative corrections, its value depends on the energy scale at which one measures it (Figure 2).

2.5 Strength of Atomic Parity Violation effects

It can be estimated that the strength of Atomic Parity Violation effects in an atom is proportional to the cube of the atomic number Z of that atom [13], multiplied by a relativistic factor [14] (see Figure 3). As one can see from this graph, the significance of the relativistic factor increases for Z & 50. The effects that can be measured in radium are therefore significantly larger than those measured in barium. Problems concerning radium ions, however, include that radium is scarce, radioactive and that it has no stable isotopes.

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Figure 2: The value of the Weinberg angle depends on the energy scale. The light blue line displays the value expected by the Standard Model. The existence of an additional Z-boson, as for example the recently suggested dark Z-boson [12], will modify the expectation for sin²θW especially at low momentum transfer. In the top of the graph the anticipated uncertainties of planned and ongoing experiments are displayed. Figure taken from [10].

Figure 3: Scaling of the strength of atomic parity violation effects with respect to the atomic number Z of the nucleus on which the effects are measured. For larger Z the Atomic Parity Violation effects increase faster than Z³ due to relativistic contributions. Figure taken from [14].

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Since barium has an atomic structure similar to that of radium, it is most efficient to build a well-working set-up for measurement of Atomic Parity Violation in barium ions, before attempting measurements of Atomic Parity Violation in radium ions [10].

2.6 Light shifts to measure Atomic Parity Violation

The observable in a single ion Atomic Parity Violation experiment is a light induced shift of the atomic levels [15]. Good control of the light shift inducing laser field with respect to intensity, frequency and polarization is required.

By applying intense laser fields to ions, one can influence the ion [10]. In a two-level system, the atomic transition from the ground state |gi to the excited state |ei is driven by an on-resonant light field. On-resonant light has the same frequency as the transition it drives in an atom. After some time, the atom decays from the excited state to the ground state, and the process starts over again. This process of exchanging populations between two states is known as Rabi oscillation. When an off-resonant light field is applied to an atom, the energy levels in the atom are shifted. This shifting of energy levels is known as a light shift, or an AC Stark shift. For a two level system, the light shift is:

∆E = ±~Ω²

4δ (2.4)

where ∆E is the shift in energy levels, Ω is the Rabi frequency and δ is the detuning of the laser, given by δ = ω0− ω_L. ω0 and ωL are the frequencies of the atomic transition and the laser respectively. Figure 4 shows the light shift in a two-level system.

Figure 4: Applying an off-resonant light field to an atom, results in a light shift of the energy levels in that atom. Figure from [16].

By applying well chosen high-intensity laser fields to barium ions, it is possible to measure Atomic Parity Violation [10].

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3 THE BARIUM ION EXPERIMENT 8

3 The barium ion experiment

The barium ion experiment is an attempt to measure Atomic Parity Violation in singly charged barium ions,¹³⁸Ba⁺, with high precision [17]. This section will focus on the experimental set-up and improvements and measurements on this set-up. The first improvement is mode cleaning of the laser beam. In mode cleaning, the beam profile of a laser is made Gaussian to close approximation. Having a Gaussian beam is required in experiments, firstly since it can be focused to a small area, resulting in a high intensity, and secondly since one assumes laser beams to be Gaussian in calculations.

An optical isolator was installed in directly after the light shift laser to prevent unwanted feedback of laser light into the laser cavity. Unwanted feedback may be a source of laser power and frequency fluctuations, which need to be prevented in high-precision experiments.

Finally, measurements were done on power broadening of the 5d²D3/2 − 6p²P1/2

transition. Having information on power broadening is one of the requirements for a precise measurement on Atomic Parity Violation in barium ions.

3.1 Experimental set-up

This description of the experimental set-up of the barium ion experiment is limited to the relevant parts. A complete description can be found in [10].

Figure 5: The experimental set-up of the barium ion experiment. Ba⁺ ions are trapped in a hyperbolic Paul trap [10]. The ions are cooled, in order to bring them to the center of the trap, and to remove kinetic energy which would cause Doppler shifts on all transitions.

Trapped barium ions can be recorded from their fluorescence. A photo multiplier tube (PMT) measures the amount of photons emitted, and an electron multiplying CCD camera (EMCCD) records the spatial position from which photons originate. With the two photon

counters it is possible to measure spectra, and to know the position of ions in the trap.

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Trapping allows us to observe the system for a very long time, needed for precision experiments. To achieve cooling, the 6s²S1/2 −6p²P1/2 transition needs to be driven (see Figure 6). A Ti:Sapphire laser providse about 300 mW of light with a wave length of 987 nm. This light is frequency doubled in order to obtain light with a wave length of 493.5 nm. From the 6p²P1/2 level, ions can decay back to the 6s²S1/2 ground state, or to the metastable 5d²D3/2 state. In the latter case, dye laser is operated at a wave length of 650 nm in order to provide light for re-pumping.

More information on the trapping and cooling of barium ions in the barium ion experiment can be found in [10] and [6].

Figure 6: Energy levels in barium ions, including their lifetimes and the wave length required to drive transitions. This diagram was taken from [10].

3.2 Spatial mode of a typical diode laser

This section describes changes made to improve the quality of the a laser beam at 650 nm from a diode laser. The spatial mode of such a laser shows a complicated structure.

The beam profile shows a number of high-intensity regions in addition to a number of low-intensity regions (see Figure 7). Improvements can be made by making the laser’s beam profile more Gaussian. Beams with a Gaussian profile have the highest quality, and small deviations from a Gaussian beam profile can cause large deteriorations in beam quality [18]. Also, Gaussian beams can be focused on very small areas, creating high intensities. Finally, when describing the beam mathematically in calculations, one often assumes the beam has a Gaussian profile, so having a beam that approximates a Gaussian beam well is preferable.

Since a Gaussian beam is preferable, it is useful to mode clean the beam. Mode cleaning a beam will remove modes other than the fundamental Gaussian profile. Mode cleaning is done with a polarization maintaining single-mode optical fiber: the output of such a fiber is to very close approximation Gaussian. In order not to lose too much power in mode cleaning, it is important to make sure the focus of the beam is exactly at the core of the

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optical fiber, and that the angle at which the beam enters is such that power losses are minimum. A power meter is put at the end of the optical fiber and mirrors reflecting the beam from its source into the fiber are tuned such that transmitted power is maximum.

The input power of the optical fiber was 12.9 mW and the output was 4.7 mW, meaning that a coupling efficiency of 36 % has been achieved. A picture of the beam profile after mode cleaning the 650 nm laser light can be seen in Figure 7 as well. From this figure it can be seen that the beam quality has strongly increased, meaning that the mode cleaning of the laser beam has been successful.

Figure 7: Left (A): the beam profile of a diode laser at a wave length of 650 nm before mode cleaning. The profile is clearly non-Gaussian. Right (B): the beam profile of the 650 nm laser after exiting the single-mode optical fiber. This beam profile approximates a Gaussian beam well. Both beams have similar diameters.

3.3 Installation of an optical isolator

In addition to performing mode cleaning, an optical isolator was installed directly after the laser, in order to prevent unwanted feedback from light reflecting back from a mirror into the laser cavity. Unwanted feedback can be a cause of power and especially frequency fluctuations of the laser light, which is undesired in a precision measurement as the barium ion experiment. The optical isolator works as follows (see Figure 8) [19]. The front of the optical isolator contains a vertical polarizer, which ensures all light entering is polarized vertically. By the Faraday effect, the Faraday rotator inside the optical isolator changes the polarization of light that passes through it by 45°. At the end of the optical isolator, a second polarizer is placed at an angle θ of 45°. This polarizer is also known as the analyzer, and only allows light polarized at 45° with respect to the vertical to pass. Light reflected by any surface, causing unwanted feedback, can only enter when polarized at 45°with respect to the vertical. This reversly going light is rotated by another 45°, and is horizontally polarized. It cannot pass the vertical polarizer at the entrance of the optical isolator. This implies the optical isolator strongly suppresses unwanted feedback.

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Figure 8: Schematic drawing of an optical isolator. The amount of rotation Q is calculated by multiplying the Verdet constant ν of the optical material, the magnetic field strength H and the path length through the optical material L. A 45° rotation should be obtained.

Figure taken from [20].

The optical isolator was positioned directly in front of the mirror, to reduce the amount of unwanted feedback as much as possible. Approximately 70% of laser power was transmitted through the optical isolator. However, sufficient power remained, and power losses due to the installation of an optical isolator are of little concern here.

3.4 Power broadening by the light shift laser

The light from this laser was delivered to the experiment. In order to check the performance, the laser was scanned over the 5d²D3/2 −6p²P1/2 transition, see Figure 9. An increase in width of peaks in this spectrum due to an increase in laser power is known as power broadening. This section will investigate the amount of power broadening caused by increasing the light shift laser’s power and compare results to theory. Since the power broadening caused by the light shift laser is something that has to be taken into account in precision measurements, it is required to obtain knowledge on the power broadening measured in this experiment.

To determine the amount of power broadening, spectra were measured for different voltages applied to the light shift laser. The light shift laser power was calibrated. With the help of this calibration it is possible to find the width of peaks depending on the laser power. The width of peaks in these spectra, the so-called line width, was determined by estimating the full width at half maximum (FWHM) (see Figure 9).

Figure 10 shows measured line widths as a function of laser power. According to theory, at low laser intensities, line width depends on laser intensity as follows [21]:

∆ν = ∆ν₀ r

1 +I_l

Is (3.1)

where ∆ν is the line width, ∆ν0 the natural line width, and Iland Isare the laser intensity and the saturation intensity. Note that laser power and intensity scale linearly. From measured data, the natural line width of the 6p²P3/2 −6s²S1/2 transition appears to be

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Figure 9: The full width at half maximum (FWHM) of this peak is equal to the length of the red line in the center of the peak. The position of this red line is the average of the other two red lines, which display the peaks maximum and a constant PMT count rate that is measured when the laser is off resonance. The laser scans at 30(3) MHz per second, and the scan range is estimated to be 1.7(1) GHz. The FWHM of this peak was estimated to be 11.0(5) seconds, which corresponds to 330(30) MHz.

approximately 200 MHz. However, it was calculated before that the natural line width is approximately 20 MHz [10]. At high laser power, line width becomes saturated, meaning that there is no further increase in scattering rate on resonance. According to theory, the width should increase as pI/Is. The inconsistency between measured and calculated natural line width indicates that the system is not yet fully understood.

LS laser power [mW]

0 0.5 1 1.5 2 2.5 3

Peak FWHM [MHz]

50 100 150 200 250 300 350

Figure 10: Peak FWHM of the 5d²D3/2 −6p²P1/2 transition as a function of laser power.

Measured results do not agree with predictions from theory well.

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3.5 Conclusions

The installation of an additional light source for the Ba⁺ experiment was successful. A near Gaussian beam with known power is delivered to the trapped ion. Spectroscopy of the 5d²D3/2 −6p²P1/2 transition shows a good overlap of the trapped ion with the laser beam.

The set-up was exploited in an other bachelor thesis in the same period. In this bachelor thesis, the first light shift measurements are discussed [22].

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4 HIGH-PRECISION SPECTROSCOPY ON MOLECULAR IODINE 14

4 High-precision spectroscopy on molecular iodine

An other strict requirement on the laser sources for the Ba⁺ ion experiment concerns the absolute frequency stability of the laser frequencies. For some of the lasers a stability of 1 part in 10¹² or better is required. The performance of such a laser system and the technology to frequency stabilize lasers is investigated on the example of frequency modulation spectroscopy on molecular iodine,¹²⁷I2, in this thesis.

This chapter will introduce the high-precision spectroscopy on molecular iodine. Un- derstanding the method of frequency modulated spectroscopy, including phase-sensitive detection for signal recovery, is required to understand the measurements that were done with two different lock-in amplifiers.

Spectroscopy on molecular iodine is done in many applications for stabilization of laser frequencies [23]. The laser at a wave length of 650 nm in the barium experiment for driving the 5d²D3/2 −6p²P1/2 transition in barium ions, is operated close to hyperfine transitions in molecular iodine [24]. A diode laser is locked to one of these hyperfine transitions. The hyperfine transitions that are used in this experiment are those in the R(25)(6-5) transition in molecular iodine [24]. Locking a laser means stabilizing its frequency to the frequency of the hyperfine transition to which it is locked. Taking advantage of a transition in an atom or a molecule is known as creating an optical reference.

Since molecular iodine has many hyperfine lines, it can be used as an optical reference for transitions in many atoms other than barium, such as hydrogen, deuterium, positronium and muonium [25]. Other applications of optical references, amongst others using the hyperfine lines in molecular iodine, include metrology, space applications and navigation [26]. Frequency modulated spectroscopy, is also required to measure high sensitivities detecting extremely weak molecular overtones [27].

This chapter will provide information about frequency modulated spectroscopy and the role of the lock-in amplifier. It will consider the experimental set-up that is operated in this experiment for executing high-precision spectroscopy on molecular iodine. Finally, effects of one of the experimental parameters, namely the vapor pressure in the cell holding the iodine, will be discussed.

4.1 Frequency modulation spectroscopy

This section will explain how frequency modulation can be achieved in spectroscopy on molecular iodine. In spectroscopy on molecular iodine, we measure absorption of laser light by iodine molecules,¹²⁷I2. Figure 11 depicts an absorption peak.

Instead of scanning the laser over a molecular transition, it is also possible to modulate the laser frequency around a certain frequency ν0. Frequency modulation is indicated by the red sine wave in Figure 12. The frequency of the laser is now given by ν(t) = ν0+ ν1sin(ωt), where ν1 is the modulation depth and ω/2π is the modulation frequency. Since a frequency decrease from ν0 results in a reduced absorption, and an increase in frequency results in an increased absorption, the result of the applied frequency modulation is that the absorption is also modulated. This can be seen by the blue sine wave in Figure 12. The absorption is modulated at the same frequency ω/2π as the laser frequency, and can be given by A(t) = A₀+ A₁sin(ωt), in which A0 is the absorption that corresponds to laser frequency ν0 and A1 is the amplitude of the modulation of the absorption.

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Figure 11: This figure displays an absorption peak. Note that this peak is not necessarily Lorentzian, whereas actual absorption peaks are Lorentzian. This picture is only used as an illustration to explain how frequency modulation functions. Peaks like the one shown in this figure can be obtained by scanning the laser frequency over an atomic or molecular transition. The maximum absorption is achieved when the laser is at resonance with the transition. Figure taken from [28].

Figure 12: The absorption peak as shown in Figure 11, including the frequency modulation signals as well as the resulting modulated absorption signals.

When frequency modulation is done below the resonance frequency, the frequency modulation and the modulated absorption are in phase. However, when the laser is not frequency modulated at the left, but at the right side of the peak, there exists a phase shift of 180° between the frequency modulation ν1sin(ωt) and the modulation in absorption A⁰₁sin(ωt + π). This can be seen from the black sines in Figure 12. An increase in frequency causes a decrease in absorption, and a decrease in frequency causes an increase in absorption. In addition, the amplitude of the modulated absorption signal depends on the slope of the absorption peak. When one modulates the frequency on the part of the absorption peak that has a small slope the modulated absorption A1 is small. Frequency modulation around a frequency where the slope of the absorption peak is large, leads to

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a large modulated absorption signal A⁰₁ (Figure 12). Frequency modulation around ν0, a frequency at which the absorption peak has a small slope, results in an absorption signal with smaller amplitude than the same frequency modulation sine around frequency ν0⁰, a frequency where the absorption peak has a large slope. In other words, the amplitude A1

of the modulated absorption is to first approximation proportional to the derivate of the absorption peak.

4.2 Phase-sensitive detection

The modulated absorption signal can be demodulated by phase-sensitive detection, also known as lock-in amplification. The main achievement of lock-in detection is that it filters a small-amplitude signal from a seemingly overwhelming amount of noise [28]. This is achieved as follows.

The lock-in amplifier filters a signal from a high-intensity noise by averaging the product of a periodic input signal and a periodic reference signal in time. For the input signal we take the modulated absorption signal, and the frequency modulated signal is used as a reference signal. The output signal S(t) of the lock-in detector is thus given by:

S(t) = 1 T

Z T 0

A(t) sin(ωt + θ_mod)dt (4.1)

In which T is the averaging time for a measurement, and A(t) = A0+ A₁sin(ωt + θ_abs) is the modulated absorption signal and sin(ωt + θmod) the frequency modulation. θabs and θ_mod are the phases of the signals, and will be discussed further in section 5.4.3. With the help of a trigonometric identity, A(t) sin(ωt + θmod) can be rewritten as

A(t)ν(t) = A₀sin(ωt + θ_mod) + A₁sin(ωt + θ_abs) sin(ωt + θ_mod)

= A₀sin(ωt + θ_mod) + A₁

2 {cos(θ_abs− θ_mod) + cos(2ωt + θ_abs+ θ_mod)} (4.2) When this is averaged in time, the first term quickly vanishes. The second term is constant in time, and will therefore result in a DC output signal. The last term will vanish as well. This implies that the lock-in amplifier has a signal S(t) ∝ A1 as its output. The output of the lock-in amplifier is thus proportional to the derivative of the absorption peak. However, noise that enters the lock-in amplifier is unable to pass. This is explained as follows. Noise is found at all frequencies, and the power of noise at frequency ω is small. Multiplying any periodic input signal with a frequency Ω 6= ω with the modulation signal, and averaging the product in time, will give zero as a result. This implies that all signals with a frequency Ω 6= ω are not able to pass through the lock-in amplifier. In conclusion, the lock-in amplifier blocks noise at frequencies different from the modulation frequency, and allows a DC signal proportional to the derivative of the absorption peak to pass through. It provides a band-pass filter with a width of 1/2πT around the modulation frequency ω/2π.

The amplitude A1 of the modulated absorption signal is proportional to the slope of the absorption peak. The output signal S(t) of the lock-in amplifier is in turn proportional to the amplitude of the modulated absorption signal. This means that the output of the lock-in amplifier is proportional to the derivative of the absorption peak, as can be seen in Figure 13. In this figure it can be seen that the lock-in output is bipolar, or in other words, dispersive. The zero-crossing of the signal is at the resonance frequency.

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Figure 13: Absorption signal and the response of the lock-in detector. The derivative shows a steep decline around the zero-crossing. Figure adapted from [28].

In spectroscopy on molecular iodine, it is advantageous that the output of the lock-in detector is proportional to the derivative of the absorption signal. Firstly, the output presented by lock-in detection is often more accurate in determining the zero-crossing than the absorption signal itself. This is caused by the fact that the lock-in detector’s output signal is quickly decreasing around the zero-crossing, meaning that a small frequency offset implies a relatively large lock-in response. An equal frequency offset will cause a smaller difference in the absorption signal. This means frequency offsets are more easily determined by the derivative of the absorption signal than by the signal itself. Secondly, since the response of the lock-in detector is dispersive, it can be used for stabilizing the frequency modulated laser. The steep slope around the zero-crossing is a property of the lock-in response that can be used for correcting the laser when it drifts off. The error, or actually, the difference between the measured value and the value at the zero-crossing, is fed back to the laser, and thus used to stabilize the lasers frequency. This method of frequency stabilization is called top-of-fringe stabilization.

4.3 Doppler-free saturated absorption spectroscopy

To determine with high accuracy and precision the spectrum of atomic or molecular transitions, in this specific case the R(25)(6-5) rovibrational transition in molecular iodine, Doppler-free saturated absorption spectroscopy is done. Doppler-free spectroscopy is required to be able to visualize hyperfine transitions in molecular iodine. This is the case, because the Doppler effect [29] increases the line width of transitions in the spectrum so much that separate transitions are not recognizable anymore: spectral lines of various transitions overlap, and cannot be resolved. Doppler-free spectroscopy is a way to achieve sub-Doppler resolution, and thus to be able to resolve hyperfine lines. More extended considerations on this topic can be found in [29] and [30].

Figure 14 shows the experimental set-up that is worked with to do frequency modulated absorption spectroscopy.

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Figure 14: The experimental set-up of the spectroscopy done on molecular iodine in this experiment. The coming sections will focus on explaining the various parts of this set-up.

Doppler-free spectroscopy is done in the central part of the set-up that is rimmed by a gray box. In the top right part of the set-up, that is also rimmed by a gray box, frequency modulation is achieved. This figure was adapted from [31].

Doppler-free saturated absorption spectroscopy has been incorporated into the experimental set-up in the following manner [6, 31]. 30 mW of light with a wave length of 649.87 nm is emitted by a diode laser (HL6366DG/67DG) and passed through an optical isolator.

Effectively, an optical isolator employs the Faraday effect to prevent unwanted feedback to the diode laser [30].

With a beam splitter, two low-intensity beams are split off the laser beam. We name the two beams that have been split off from the main beam the probe and the reference beam, and we call the high-intensity remainder the pump beam. The reference and probe beam both have a power of approximately 0.5 mW, and a radius of 0.4 mm. Using some optical devices, the pump beam is send through the iodine vapor cell in a direction counter- propagating that of the probe and reference beam, which propagate through the vapor cell parallel to each other. The probe beam overlaps with the pump beam, the reference beam does not. Due to the Doppler effect, only those molecules that have velocities purely perpendicular to the direction of the light, can interact with both beams, i.e. absorb photons from both beams. If this is the case, the high-intensity pump beam will excite a large number of molecules, reducing the absorption of the probe beam. This implies that the overlapping probe beam will have a larger transmission than the non-overlapping reference beam.

Finally, a balanced photo-detector measures and amplifies the intensity difference

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between the reference beam and the probe beam. The signal measured by this photo- detector, displayed by the blue dashed line in Figure 14, is the modulated absorption signal, and is sent to two lock-in amplifiers, see the orange squares in Figure 14. Both lock-in amplifiers demodulate the signal, and remove noise from it. To do this, both lock-in amplifiers are fed with the frequency modulation signal, displayed by the red dashed lines, which they use as a reference signal. Note that results obtained in this chapter were measured with the analog Scitec 420 lock-in amplifier. This lock-in amplifier used an externally generated reference signal. This externally generated signal was also worked with to modulate the laser frequency. Chapter 5 will focus on results of measurements by the two lock-in amplifiers. The data measured by the two lock-in amplifiers is sent to a data acquisition system (DAQ) that records and stores data.

The laser is scanned over the frequency range of interest, to measure the all hyperfine lines in the absorption spectrum of the R(25)(6-5) transition. At the end of this chapter, more information will be given on the hyperfine lines in this rovibrational transition.

In the set-up, frequency modulation is obtained from a double passing acousto optic modulator (AOM) set-up (see Figure 14). After splitting the laser beam into a pump beam, a reference and a probe beam, the pump beam is send through a polarizing beam splitter (PBS). This PBS sends vertically polarized light to an AOM, which imposes a frequency modulation of ν1sin(ωt) onto the laser beam. In the process of frequency modulation, the beam passes a λ/4 wave plate twice. This means the polarization is rotated by 90°, and is now horizontal. The beam can then pass the PBS and travels to the iodine cell. More details of the set-up are found in [6].

It is important to note that there is some residual amplitude modulation present in the system, even though the pump beam is not amplitude modulated.

4.4 The 21 hyperfine lines in molecular iodine

The set-up permits the observation of the spectrum of hyperfine lines in the R(25)(6-5) transition in molecular iodine. This spectrum looks as shown below (see Figure 15). The lock-in signal indeed displays the derivative of the absorption signal of the hyperfine transitions. The large slope of the signal at resonance shows small line width, and a well-defined zero-crossing of the lock-in signal.

The a3 line is used to lock the light shift laser in the barium ion experiment, because it is very close to the 5d²S3/2 - 6p²P1/2 transition in barium. In addition, the spectrum is clear of lines for frequencies above the transition frequency of a3, which also makes it easier to lock the laser to the a3 line.

4.5 The fit function

In the analysis of the data obtained in this experiment, a fit function that uses line center, line width, modulation depth, modulation amplitude and amplitude modulation as parameters is used. This fit function is defined as follows [31]:

S(ν) = Z ^2π

ωm

0

X

i

V (ν₀+ ν₁sin(ωt)) sin(nωt + φ)dt (4.3) where V is a Voigt function (the convolution of a Lorentzian and a Gaussian) that describes the line shape, ω the modulation frequency, n the order of the derivative, ν1the modulation depth and φ the phase. One of the results of using this fit function is that 100 times smaller

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0 0.2 0.4 0.6 0.8

-2 -1 0 1 2

Lock-i n sig nal [V]

ƒ [GHz]

a

₁

a

₂

a

₃

a

₂₁

Figure 15: The spectrum of the hyperfine lines of the R(25)(6-5) transition in molecular iodine. Lines a1, a2, a3 and a21 have been noted for clarity. The frequency that is denoted on the horizontal axis is the frequency with respect to the center frequency of line a3. contributions to the spectrum can be resolved. The following figure shows the first three peaks, as fit by the fit function.

Figure 16: The fit function was fit through the data points measured when the laser was scanning over the frequency range of the first three hyperfine transition lines a1, a2 and a3. Parameters included in the fit are the offset signal and the slope of the background, amplitude modulation, and centers frequencies, the Lorentzian width and the amplitudes of the three lines. The fit has a χ²/ndf of 5.2.

4.6 Effects of increasing vapor pressure

One of the experimental parameters that affects the absorption spectrum of molecular iodine is the vapor pressure in the iodine cell. It influences the center frequency and the line width of lines [30]. The good signal-to-noise and the understanding of the line shape permits the extraction of pressure shift and broadening.

A cold finger on the side of the iodine cell is used to control this vapor pressure (see

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Figure 14). The vapor pressure in the cell depends on the temperature of the cold finger by the following empirical formula [32]:

log(P ) = −3512.830

T − 2.013 · log(T ) + 18.37971 (4.4) in which P is the vapor pressure of in the iodine cell in Pascals, and T is the temperature of the cold finger in Kelvin. Measurements of the vapor pressure were done by single pass absorption in the iodine cell, where the vapor pressure was increased between measurements.

The linear relation between the absorption and the vapor pressure permits the determination of the vapor pressure from the measured absorption. The measured data agree very well with the empirical formula 4.4, up to a temperature of approximately 30 °C. At this temperature the vapor pressure has reached approximately 82 Pa: all iodine in the cell is in the gas phase.

Figure 17: Measured data points (blue squares) lie close to the black line, which shows the empirical results given by Equation 4.4. At temperatures above 30 °C, vapor pressure becomes saturated at 82 Pa. Figure taken from [31].

4.6.1 Pressure shift

Increasing the vapor pressure changes the center frequency and the line width of transitions because of the increase in collision rate of the molecules in the cell [30, 29]. A change in center frequency, caused by an increase in vapor pressure is what is named a pressure shift. The pressure shift of three hyperfine lines in molecular iodine as a function of vapor pressure can be seen in Figure 18. Note that the uncertainty is caused mainly by the uncertainty of the vapor pressure in the iodine cell, which is approximately 1 Pa/K.

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Figure 18: A linear line (black line) was fitted through the extracted center frequencies (blue squares) to extrapolate the center frequency of the three hyperfine transitions at zero vapor pressure. The slope of the graphs shows the pressure shift in MHz per Pa of pressure.

Figure taken from [31].

By fitting a linear line, and extrapolating the data, the center frequency at zero vapor pressure could be determined for the three lines a1, a2 and a3. From the slope of the fit, the decrease in center frequency per Pascal of vapor pressure can be found. The extrapolated center frequencies for zero vapor pressure and pressure shifts of all three lines as found from these graphs can be found in Table 1. The center frequency of the hyperfine lines could be determined to the order of 10⁻³ of the line width.

Figure 19 shows that the relative shifts of two pairs of lines show similar behavior.

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From Figures 18 and 19 the hyperfine structure splitting can be determined, and it can be concluded that all three lines show approximately equal pressure shifts.

Figure 19: Relative shifts of the center frequencies of different pairs of lines (above: a2

and a1, below: a3 and a1). The relative shift between two lines is independent of vapor pressure. In addition, both graphs have the same shape, meaning that all three absorption lines are shifted from their center frequencies by approximately equal amounts at equal vapor pressures. The straight line gives the difference in center frequencies of the different lines. Figure taken from [31].

Table 1: Extrapolated center frequency at zero vapor pressure and pressure shift of lines a1, a2 and a3.

Line Center frequency (MHz) Pressure shift (MHz/Pa) a1 461312279.9330(5) -0.0105(3)

a2 461312315.1104(5) -0.0102(3) a3 461312346.9020(6) -0.0101(1)

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4.6.2 Pressure broadening

As the center frequency, the line width of a transition also depends on the vapor pressure in the iodine cell. An increase in line width due to an increase of vapor pressure is called pressure broadening. In Figure 20, one can see the relation between vapor pressure and the extracted Lorentzian width of the three hyperfine transitions in molecular iodine. The extracted parameters are summarized in Table 2.

Figure 20: Extrapolated line width of the three hyperfine transitions as a function of iodine vapor pressure. The slope of these graphs gives the pressure broadening in MHz per Pa of pressure. Figure taken from [31].

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Table 2: Line width extrapolated to zero vapor pressure and pressure broadening of lines a1, a2 and a3.

Line Line width (MHz) Pressure broadening (MHz/Pa)

a1 5.564(3) 0.059(9)

a2 5.101(3) 0.060(6)

a3 6.198(3) 0.060(7)

4.7 Conclusions

In this chapter has explained frequency modulated spectroscopy and the operation of the lock-in amplifier were described. The results of implementing the lock-in amplifier in a set-up for frequency modulated spectroscopy are demonstrated on the spectrum of hyperfine lines in the R(25)(6-5) transition in molecular iodine. The accuracy of the frequency is better than 10⁻¹¹. One of the hyperfine lines in this transition is used as an optical reference for the barium ion experiment.

The effects of the vapor pressure in the iodine cell on the line shape of transition lines in the spectrum were investigated. A spectrum measured at low vapor pressures, preferably below 10 Pa, provides the best signal-to-noise ratio for the determination of absolute frequencies due to minimized broadening and shift.

The center frequencies of the lines in molecular iodine are shifted by approximately -10 kHz/Pa. The center frequency of the lines can be determined with an accuracy of 10⁻¹² after correction of the pressure shift. In addition, the line width of the lines is increased by approximately 60 kHz/Pa. This means that the line width increases by 10⁻³ of the line width for each Pa of vapor pressure.

Measured results are consistent, as can be seen from the fact that the three hyperfine lines behave similarly. This means that molecular iodine is a reliable optical reference. An other conclusion that can be taken from the fact that measured results are consistent, is that the fit function is valid over a large range of experimental parameters, including vapor pressure. Results as obtained in this section are required to improve the understanding and the control of experimental parameters in the experiment. Increasing knowledge of the effects of experimental parameters is required to further improve the accuracy with which the spectrum can be measured.

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5 COMPARISON OF TWO LOCK-IN AMPLIFIERS 26

5 Comparison of two lock-in amplifiers

Lock-in amplifiers play an important role in the generation of signals from modulation spectroscopy (see Section 4.2). Phase-sensitive detection and lock-in amplification are different terms describing the same method of recovering a signal from an overwhelming amount of noise. In an attempt to improve the signal-to-noise ratio and the line shape of the iodine spectrum, a digital lock-in amplifier (TEM Messtechnik’s LaseLock [33]) was employed (see Figure 21a). This lock-in amplifier has a bandwidth of 300 kHz, whereas the Scitec 420 analog lock-in amplifier (see Figure 21b) has a bandwidth of 100 kHz.

Implementing a lock-in amplifier with a higher bandwidth might improve the precision at which measurements can be taken.

The digital lock-in amplifier has a number of features that the analog one does not possess: it provides a modulation signal, and characteristics of the low-pass filter through which the signal passes can be selected. In this chapter, the results of changing these parameters will be discussed.

(a) TEM Messtechnik LaseLock. (b) The Scitec 420.

Figure 21: The two lock-in amplifiers.

5.1 Noise

Noise can be defined as "an unwanted signal that interferes with the communication or measurement of another signal" [34]. Since noise reduces the quality of conducted measurements, it is important to qualify sources of noise, and to keep their influence as small as possible.

Sources of noise can be divided into two groups: intrinsic and external noise. It is impossible to remove intrinsic noise, because it appears due to inescapable physical processes in the electronics that are used in the experiment. Although effects of intrinsic noise cannot be avoided, understanding them can help to characterize the measured signal better. External noise is caused by sources such as power supplies, vibrations and frequency instabilities. External noise effects can be reduced, for example by improvement of shielding of the experiment from the outside environment.

The following sections will shortly describe the main sources of noise that together form the noise spectrum.

5.1.1 Intrinsic noise

There are multiple kinds of intrinsic noise. This section describes the ones that have the largest effect on the total noise spectrum: white noise and colored noise, of which pink noise plays the most important role in this experiment.

White noise is fully random noise which theoretically carries equal power at every equally broad frequency interval [34]. In other words, the power spectral density, which shows the amount of power a signal carries per Hertz, is constant over the complete frequency spectrum. Since this would imply that white noise carries infinite power, this

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definition of white noise does not hold for the full frequency spectrum. However, for a system with a given bandwidth, noise carrying equal power at all frequencies in the bandwidth is perceived as white noise by this system and by measurement apparatus in this system.

White noise carrying a constant energy over all frequencies is a theoretical concept.

This is why band-limited white noise is introduced. Band-limited white noise is defined as noise carrying a constant power over the frequency range from zero to the given bandwidth B, and is absent for frequencies above the bandwidth:

P (f ) =

(p0 if f ≤ B

0 if f > B (5.1)

Where P is the noise power at frequency f and p0 is the constant power that noise carries at all frequencies within the bandwidth.

In addition to white noise, there is colored noise. Colored noise can be defined as "all broadband noise that has a non-white spectrum" [34]. The most important colored noise in this experiment is pink noise, which is also known as flicker noise or 1/f noise. Pink noise is intrinsic noise in electronic systems for which the power spectral density is inversely proportional to the frequency f. In other words, the noise power drops by 1/f.

5.1.2 External noise

External noise plays an important role in the total noise spectrum. External noise is caused by the environment. Contrary to internal noise, external noise often appears as peaks at certain frequencies on the noise spectrum, which, not surprisingly, appear at the frequencies at which the sources of external noise operate.

5.2 Analog versus digital lock-in amplification

There are a number of differences between analog and digital lock-in amplifiers. First of all, the way in which signals are processed is not the same in both types. Analog lock-in amplifiers use electronic circuits to achieve phase-sensitive detection. In digital lock-in amplifiers, the signals are first digitized by an ADC with and then processed mathematically, using a digital signal processing (DSP) chip [35]. The number of bits and the exact sampling frequency of the ADC are not known. We will now discuss some disadvantages of digital lock-in amplification.

5.2.1 Discretization error and dynamic range

Before an analog input signal can be processed by a digital lock-in amplifier, or any digital apparatus, it needs to be digitized. This is done by an analog-to-digital converter (ADC).

More information on ADC’s can be found in [36]. The most important feature of ADC’s that will be discussed here is that they are the source of so-called discretization errors.

These errors occur, since a continuous analog signal needs to be converted to a discrete digital signal. The output signal of the ADC is not continuous: when increasing the analog input voltage, the ADC output increases in steps, see Figure 22. This implies that the input signal is represented by the output signal with an error.

An other limitation of lock-in amplifiers is their dynamic range. The dynamic range is the ratio between the largest and the smallest quantity that can be measured by an apparatus. When an input signal is smaller than the dynamic range, meaning that it is

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Figure 22: Output of a 3-bit ADC, depending on the input voltage. Increasing the number of bits will increase the number of steps that can be taken, and thereby it will decrease the size of the steps as well as the discretization error. Figure from [36].

smaller than the size of one step, it cannot be extracted by a digital apparatus. Therefore, there is a minimum signal amplitude required to perform lock-in detection with a digital lock-in amplifier. Since the number of bits of the ADC of the LaseLock is not known, the minimum signal amplitude required for the LaseLock is not known either.

5.3 Installing LaseLock in the experimental set-up

This section explains in detail how the digital lock-in amplifier was connected to the experimental set-up as described in section 4, and how measurements of both lock-in amplifiers are read by the data acquisition system.

5.3.1 Implementing the LaseLock in the experimental set-up

To compare the digital and analog lock-in amplifiers properly, it is required that both are implemented in the set-up of the experiment in parallel. This enables us to take measurements with both lock-in amplifiers simultaneously. When the lock-in amplifiers are connected in parallel, laser noise, such as amplitude variations and frequency fluctuations, contributes in the same way to the output of both lock-in amplifiers. The analog lock-in amplifier was already installed. The LaseLock, however, has been implemented in parallel to the analog lock-in amplifier.

The LaseLock can provide the modulation signal which is sent to the modulation of the laser frequency and the analog Scitec lock-in as a reference signal. The signal from the photo diode is duplicated by an SRS amplifier with two outputs and provided at the input of both lock-in amplifiers.

The modulation signal of the LaseLock is used as the control voltage to the VCO (see Figure 14). The parameters are an offset voltage of 6.8 V and an amplitude of 400 mV which result in a frequency of 73.33(1) MHz with a modulation depth of 2 MHz.

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Figure 23: The spectrum of three hyperfine lines in molecular iodine, as measured by the LaseLock. From scans like these, one can determine the signal-to-noise ratio, and find information about the line shape. The red box indicates in which part of the spectrum information can be found concerning noise. This will be discussed further on. Since the Scitec 420 produces scans as the one in the figure simultaneously, a comparison can be made between the two lock-in amplifiers. The laser scanned at a rate of 0.46(1) MHz/s.

This scatter plot was produced using raw data recorded every 100 ms.

The photo diode signal that is sent through the LaseLock, the signal passes through a low-pass filter after phase-sensitive detection. The type of low-pass filter can be set to a Bessel filter, a Butterworth filter or a 0.5, 1.0, 2.0 or 3.0 dB ripple Chebyshev filter. More information on filters can be found in [37]. The low-pass filter gain can be set in the range from 1 to 100, and the cut-off frequency of the filter can be chosen from 100 Hz to 850 kHz. The sampling divider of the filter, which determines the number of samples taken per second, is set to 2. This means that the LaseLock takes 1.25 million samples per second (MSps).

5.3.2 The data acquisition system

In order to compare the measured signals properly, it is required to record and store measurement results of both lock-in amplifiers. Recording and storing data is done by the data acquisition system (DAQ) [38]. Both the barium experiment and the iodine spectroscopy employ this data acquisition system. The DAQ works in the following manner.

Voltages put out by both lock-in amplifiers are converted to digital data at a frequency of 10 Hz. The data acquisition gives a data point every 100 ms, for two output channels of the LaseLock, and for three of the Scitec 420. In addition to the lock-in data, the control parameters for the frequency of the diode laser are recorded [38].

5.4 Comparing measurement results of both lock-in amplifiers

The spectrum of hyperfine lines in molecular iodine is measured by scanning the laser over the frequency range of these hyperfine transitions. These are the same transitions as analyzed in Chapter 4.

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The signal-to-noise ratio S/N is calculated by dividing the signal’s amplitude by the measured noise. The amplitude of the signal can be taken from graphs like the one in Figure 23, by taking the maximum and minimum data points measured for one of the peaks. The second peak in the spectrum is utilized to determine the signal amplitude, since the second peak is the one that has the maximum amplitude. In addition, regardless of whether the scan was backward or forward, the second line is always the middle line.

The noise is determined from the part of the spectrum where no lines are found. In Figure 23 this would be the part of the spectrum indicated by the red box. Data points in the box are projected on the vertical axis (see Figure 24). This results in a distribution of measured output values. These values represent the noise measured. We define the noise to be the standard deviation σ of the distribution. The signal-to-noise ratio is given by:

S

N = Smax− S_min

σ (5.2)

where S is the difference in maximum and minimum of the second line, and σ is the standard deviation of the noise distribution. The signal amplitude uncertainty is two times the noise, meaning that the uncertainty in signal-to-noise ratio is 2: ∆_N^S = 2. This is the case, since there is noise on the minimum and the maximum of the signal. Adding these noises quadratically leads to an uncertainty of √

2. To be on the safe side, for all measurements, we take 2 as the uncertainty. The error bars of the signal-to-noise ratios are smaller than the symbols representing data points, and are not visible.

LaseLock output [V]

−0.04 −0.02 0 0.02 0.04

counts

0 5 10 15 20 25

Figure 24: An example of a histogram that is employed to determine the amount of noise. This histogram contains 626 data points, taken with intervals of 100 ms. The red line shows a fit to the distribution, from which the standard deviation σ is determined.

In this example, σ = 0.01577 V. The signal from the measurement from which this noise distribution was taken, has an amplitude of 3.31(0.03) V. The signal-to-noise ratio measured by the LaseLock in this measurement is therefore 210(2). The χ²/ndf of the fit is 0.74.

The next section will show that both lock-in amplifiers measure the same noise under equal conditions, meaning that results of the two lock-in amplifiers can be compared without worrying about the possibility that they are affected differently by noise in the system.

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5.4.1 Noise

The first comparison is the noise for a fixed laser frequency at one of the transitions. Figure 25 shows the noise measured by the two lock-in amplifiers.

Time [s]

0 500 1000 1500 2000 2500

Scitec output [V]

−0.1

−0.05 0 0.05 0.1

(a) Noise measured by the Scitec 420. This scatter plot was produced using raw data recorded every 100 ms.

Time [s]

0 500 1000 1500 2000 2500

LaseLock output [V]

−0.1

−0.05 0 0.05 0.1

(b) Noise measured by the LaseLock. This scatter plot was produced using raw data recorded every 100 ms.

Figure 25: Noise measured by the two lock-in amplifiers.

Both figures look the same. The only visible difference is that the average amplitude of noise measured by the LaseLock is less than that of the Scitec 420. When plotting the output of one lock-in amplifier against the output of the other, a straight line appears. The relative gain of one lock-in amplifier with respect to the other is equal to the slope of this graph. The relative gain of the LaseLock with respect to the Scitec 420 was determined to be 0.69, meaning that it was necessary to multiply the output of the Scitec 420 by 0.69 in producing Figure 26 to correct for unequal gains of the two lock-in amplifiers. This figure shows the difference in noise measured by the LaseLock and the Scitec 420, for the same time scale as the previous two figures.

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Time [s]

0 500 1000 1500 2000 2500

LaseLock output - Scitec output [V]

−0.1

−0.05 0 0.05 0.1

Figure 26: Noise measured by the Scitec 420 subtracted from noise measured by the LaseLock. A straight line appears, signifying that the LaseLock and the Scitec 420 measure the same noise levels at all times. Moreover, it can be concluded that the lock-in amplifiers either do not add noise to the measured spectrum. All, or at least almost all measured noise is therefore created in the rest of the experimental set-up. This scatter plot was produced using raw data recorded every 100 ms.

The fact that the lock-in amplifiers do not add noise to the measurement improves the credibility of the comparison of the lock-in amplifiers. If the two would not measure the same noise, or create significant noise internally, they could not be prepared properly:

measured differences between the lock-in amplifiers could just as well be caused by noise, not by actual differences. In conclusion, the fact that both lock-in amplifiers measure the same noise levels allows us to compare results that were measured simultaneously.

5.4.2 Line shape

This section will focus on possible differences in line shapes measured by the two lock-in amplifiers. If there are differences in line shape, then one of the lock-in amplifiers processes the signal produced by the experimental set-up differently than the other, or the signals the lock-in amplifiers receive are not the same.

Differences in line shapes measured by the two lock-in amplifiers can be found by subtracting the spectrum measured by one from the spectrum measured by the other.

However, the outputs of the lock-in amplifiers do not necessarily have the same amplitude.

This requires us to find the relative gain of one of the lock-in amplifier with respect to the other. This is done by plotting the spectrum measured by one against the spectrum measured by the other, as can be seen in Figure 27.

Effects of external parameters on the line shape of hyperfine lines in molecular iodine

Rijksuniversiteit Groningen

Bachelor Thesis