High-accuracy absolute distance metrology

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High-accuracy absolute distance metrology

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 25 september 2006 om 17:30 uur

door

Bastiaan Lucas SWINKELS

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Dit proefschrift is goedgekeurd door de promotor: Prof. dr. ir. J.J.M. Braat.

Samenstelling promotiecommissie:

Rector Magnificus, voorzitter

Prof. dr. ir. J.J.M. Braat, Technische Universiteit Delft, promotor Dr. N. Bhattacharya, Technische Universiteit Delft

Prof. dr. ir. P.H.J. Schellekens, Technische Universiteit Eindhoven

Dr. Z. Sodnik, ESA-ESTEC, Noordwijk

Prof. dr. W. Ubachs, Vrije Universiteit, Amsterdam Prof. dr. ir. L.J. van Vliet, Technische Universiteit Delft Prof. dr. H.P. Urbach, Technische Universiteit Delft

Prof. dr. ir. A. Gisolf, Technische Universiteit Delft, reservelid

Dr. N. Bhattacharya heeft als begeleidster in belangrijke mate aan de totstandkoming van het proefschrift bijgedragen.

This work was supported by TNO Science and Industry, Delft. ISBN-10: 90-78314-04-4

ISBN-13: 978-90-78314-04-2

Cover: Beam-splitters in the mist, see also Fig. 4.2. Photo by Maarten van Turnhout. Copyright c_{2006 by B.L. Swinkels.}

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Summary

In the coming years, a number of space missions will be carried out that consist of multiple satellites that fly in close formation. Examples include missions for gravitational wave detection, X-ray telescopes and synthetic aperture telescopes. For most of these, a metrology system that measures and controls the formation with high accuracy is an essential component to enable the science measurement. The background of our research is the Darwin mission, which will be launched by ESA around 2020 and is aimed at detecting planets around nearby stars. It consists of up to 4 free-flying telescopes and a central satellite, which interferometrically combines the collected light to obtain an angular resolution that is higher than achievable with a single telescope. The interferometric detection poses very high demands on the satellite pointing and the stability of the mutual distances. To observe the ‘white-light’ interference fringes, it is necessary that the optical path length experienced by the starlight is equal along the different paths from telescopes to beam-combination to within a few wavelengths. This is only possible with a complex metrology system that monitors all the distances, angles and velocities in the system. The measurements made with the various systems will be used to control the optical path lengths by moving delay lines and by steering the satellites with milli- and micro-newton thrusters. Our research focuses on the possible implementation of the sub-system that should measure the absolute distance between two satellites with high accuracy. For Darwin, the required accuracy would be 70 micrometers over a distance of up to 250 meters.

To achieve these high-accuracy distance measurements, we have been studying a technique called frequency sweeping interferometry. This technique uses a single tunable laser, which is swept over a well known frequency difference. The total phase difference observed in the interferometer is then directly proportional to the optical path length. Contrary to several other techniques, such as double-wavelength interferometry, the method yields a truly absolute distance measurement without any ambiguities. What is unique to our approach is that we define the two ends of the frequency sweep by locking the laser to two resonances of a very stable, high finesse Fabry-P´erot cavity. The stabilization of the laser is performed using the Pound-Drever-Hall technique, so we can take advantage of the incredibly high stabilities that have been obtained in other fields like cold atom spectroscopy. Another key concept of our approach is the use of a digital signal processor to control the sweeping and locking of the laser. This allows us to sweep the laser, find a resonance and lock to that resonance, in a quick and reliable way.

Frequency sweeping interferometry normally suffers from one large problem, namely that any changes in the distance occurring during the sweep will cause enormous errors in the distance calculation. We developed a simple interpolation algorithm, which can correct these errors in case of smooth changes in distance, which would be the case in our intended application. This allows us to use only a single laser, which is very attractive for use in space. The interpolation scheme yields both the phase at a single wavelength and the phase difference as in double-wavelength interferometry. We can thus calculate both an absolute and a relative distance.

An experimental setup to demonstrate the feasibility of frequency sweeping interferometry was designed, built and characterized. This setup was used to measure optical path lengths

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up to 110 meters. Comparing the absolute and the relative measurements, we demonstrated a repeatability of 100 micrometers (1σ). We have also shown that scaling between incremental and absolute measurements agrees to a level of 1 · 10−4_{, which is to within the experimental}

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Samenvatting

In de komende jaren zullen een aantal ruimtemissies worden uitgevoerd waarbij een aantal sa-tellieten op korte afstand in formatie vliegen. Voorbeelden zijn missies voor het detecteren van zwaartekrachtgolven, r¨ontgentelescopen en missies die gebruik maken van optische apertuur syn-these. Deze missies zijn zouden niet mogelijk zijn zonder een metrologiesysteem, dat nauwkeurig de onderlinge posities van de satellieten meet en regelt. De achtergrond voor ons onderzoek is de Darwin missie, die rond 2020 gelanceerd zal worden door ESA en tot doel heeft om planeten te detecteren rond naburige sterren. Deze missie bestaat uit vier vrij vliegende telescopen en een centrale satelliet, waarop het licht van de telescopen interferometrisch wordt gecombineerd om zo een hoekresolutie te behalen die hoger is dan mogelijk zou zijn met een enkele telescoop. De interferometrische detectie stelt hoge eisen aan de stand van de satellieten en de stabiliteit van hun onderlinge afstand. Om de wit-licht fringes te kunnen observeren, is het noodzakelijk dat de optische weglengte die het sterrenlicht ervaart langs de verschillende paden van de telescopen tot aan de bundelcombinatie gelijk is tot op een paar golflengten. Dit is alleen mogelijk met een metrologiesysteem dat nauwkeurig alle afstanden, hoeken en snelheden meet. De metingen kunnen worden gebruikt om de optische weglengten te stabiliseren door delay-lines te verschui-ven of door de satellieten aan te sturen met milli- en micronewton stuurraketjes. Ons onderzoek richt zich op de mogelijke implementatie van een subsysteem dat de absolute afstand tussen twee satellieten met hoge nauwkeurigheid meet. Voor Darwin moeten afstanden tot 250 meter met een nauwkeurigheid van 70 micrometer worden gemeten.

Om dergelijke nauwkeurige afstandsmetingen te doen hebben we onderzoek gedaan naar een techniek genaamd ‘frequency sweeping interferometry’ (‘verstembare frequentie interferometrie’). Bij deze techniek wordt gebruik gemaakt van een verstembare laser die over een nauwkeurig be-kend frequentie-interval wordt gescand. Het totale faseverschil dat dan in de interferometer kan worden gemeten is evenredig met de optische weglengte in de interferometer. Anders dan bij andere technieken, zoals tweegolflengte-interferometrie, kan de afstand daadwerkelijk absoluut worden gemeten zonder enige dubbelzinnigheid in afstand. Uniek in onze opzet is dat we de eindpunten van het frequentie-interval defini¨eren door de laser te stabiliseren op twee resonan-ties van een zeer stabiele Fabry-P´erot cavity met een hoge finesse. Voor het stabiliseren van de laser gebruiken we de Pound-Drever-Hall techniek, die eerder in gebieden zoals koude atoom-spectroscopie is gebruikt om extreem hoge stabiliteit te bereiken. Een essentieel element van onze aanpak is het gebruik van een digitale signaalprocessor (DSP) voor het aansturen van het verstemmen en stabiliseren van de laser. Op deze manier kunnen we snel en betrouwbaar de laser verstemmen, een resonantie zoeken en daaraan stabiliseren.

Een belangrijk nadeel bij gangbare implementaties van frequency sweeping interferometry is dat elke verandering in de afstand tijdens het verstemmen van de laser leidt tot een grote fout in de berekening van de afstand. Wij hebben een eenvoudig interpolatiealgoritme ontwikkeld dat deze fouten kan corrigeren wanneer de afstand geleidelijk verandert, wat het geval is in onze beoogde toepassing. Hierdoor kunnen we volstaan met het gebruik van slechts ´e´en laser, wat erg aantrekkelijk is bij gebruik in de ruimte. Met het interpolatiealgoritme verkrijgen we zowel de fase die zou worden gemeten bij het gebruik van een enkele golflengte, als het faseverschil dat

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zou worden gemeten bij tweegolflengte-interferometrie. Hiermee kunnen we zowel een absolute als een relatieve afstand berekenen.

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Introduction

1.1 Background: exo-planets

One of the most intriguing questions is if we, as living creatures, are alone in this universe: Is there any life outside our own planet? The incredibly large number of stars that can be observed suggests that some of them will have planets just like our sun and that life must have evolved on some of these planets. The first direct evidence of a planet orbiting a star was found in 1995 [1]. Since then, almost 200 so called exo-planets have been found and this number is increasing steadily. So far, only large, Jupiter-like planets have been found, but this is probably caused by the bias towards detecting large planets. It is generally expected that a lot of stars will also have smaller planets similar to our earth. If such planets are found, an interesting question to ask is if they can sustain life. Assuming a form of life that is not extremely different from our own, this requires some limits on the temperature of the planet and its distance to the star. Planets that fulfill these criteria are said to be located in the habitable zone. Another requirement for life would probably be the presence of water, which could be confirmed by examining the optical spectrum of the planet for signs of water vapor. It should even be possible to find evidence for life by detecting the presence of oxygen and ozone, which can also be achieved using optical spectroscopy. These are gases that cannot exist without a continuous replenishment by a process like photosynthesis. We will now discuss the various methods that can be used to detect planets.

1.1.1 Indirect detection methods

The light sent by a planet will always be vastly overwhelmed by the light of the star around which it orbits. The easiest method for detecting an exo-planet is therefore to study the indirect effects that the planet induces to the star.

A planet and a star will rotate around their common center of mass. The star will thus appear to wobble around its average position. For heavy planets in close orbits, this effect is so large that the velocity component in the direction of earth can be measured due to a Doppler-shift in the optical spectrum of the star. Observing this over a long time will yield the period and the amplitude of the motion of the star, from which the properties of the planet can be deduced. This is called the radial velocity method and was used to detect the first known exo-planet [1]. This method has a large bias toward detecting heavy planets that rotate in a close orbit around the sun, the so-called ‘hot Jupiters’. It is also biased toward planetary systems that are seen almost edge-on.

Similar to the radial velocity method, the existence of a planet can also be inferred from the lateral motion of a star. This technique is called astrometry and will be used in a number of future space missions [2, 3]. The sensitivity of this method decreases with distance and can thus only be used for nearby stars.

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12 CHAPTER 1. INTRODUCTION In a final indirect detection technique, the intensity of a star is monitored with high accuracy over long periods of time. If a planet passes exactly in front of the star, a reduction of the intensity can be observed for a few hours. Such an event is called a transit and was first reported in [4]. This method works only for planetary systems that are seen exactly edge-on. Stars that are known to have planets according to the radial velocity method are thus good candidates for observing transits. This method is also biased toward detecting large planets in close orbits. The chance of detecting a transit is very rare. Several missions are planned to tackle this problem in a brute-force way: A single telescope observes the same position of the sky for several years and measures the intensity of a large number (105_{) of stars very accurately.}

Statistics will then ensure that some planets will be found [5, 6].

1.1.2 Direct detection methods

In direct detection methods, it is attempted to observe the features of the planet itself. This is much more difficult than the indirect methods described above, but they are important to learn more about the planet. One problem with directly observing a planet is the very small angular separation between a star and its planet. The minimum diameter of the telescope needed to separate the two is determined by the diffraction limit. Even for some close stars, this would require a diameter of the telescope of around 10 meters. A second difficulty is the large difference in brightness between the planet and the star. A planet will reflect the light of its parent star at visible wavelengths (peaking around 500 nm for the sun) and will also emit in the far infrared due to its own temperature (peaking at 10 µm for an earth-like planet). For this reason the spectrum of the planet is largely a copy of the spectrum of the star, but with an intensity that is a factor of 109 _{lower. The exception is a small bump in the spectrum in the far infra-red,}

caused by the planet’s thermal emission. Detecting the planet is thus ideally performed in the far infra-red, where the ratio is the smallest, but still a factor of 106. Even if the star and the planet are separated based on the diffraction limit, it is still possible that the signal of the planet is overwhelmed by one of the diffraction side-lobes of the image of the star. The challenge of observing a planet is often compared to detecting the light of a firefly buzzing at one meter next to a lighthouse, from a distance of 1000 kilometers. Nevertheless, some first results have been obtained using large telescopes [7]. The problem of the high contrast might be partially solved by a technique called coronography, in which the light of the star is blocked in an intermediate image in the telescope.

Because there are practical limits on the diameter of a single telescope, a solution for increas-ing the angular resolution is to use aperture synthesis. In this technique, the light of several smaller telescopes is brought to interference to yield a resolution that would correspond to that of a single telescope with a diameter as large as the separation of the smaller telescopes. Two different schemes can be used. In the first, called imaging, the phase and the amplitude of the interference fringes are measured for a large number of different baselines. This corresponds to sampling the image of the sky in the Fourier domain at spatial frequencies that are related to the various baselines. The use of long baselines will allow the measurement of high spatial frequencies. If sufficient measurements are obtained, the image at the sky can be reconstructed in the data-analysis [8]. In a second technique, called nulling, the light of the star is suppressed as much as possible, while the light of the planet is passed through. When interfering the light of two telescopes, the light in one arm can be shifted by a phase π. Light from the on-axis star will then interfere destructively. Light from the planet, which is slightly off-axis, will experience an extra path length difference equal to the baseline times the angle. If this path length difference corresponds to a phase of π, the light of the planet will interfere constructively [9]. In practice more complex schemes are used with more telescopes.

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1.2. DARWIN 13

Figure 1.1: Artist’s impression of the Darwin mission (ESA, 2002).

which the gravity of a heavy star acts as a lens to image a different star that is located far behind the lens. A first planet has been found in this way recently [10]. These events are very rare and only last a few hours, so a great amount of experimental luck is needed.

1.2 Darwin

As described above, direct observation of a planet is most easily done in the far infrared. Since the atmosphere of the earth partly blocks this light (due to the same gases that you actually want to observe at the exo-planet), it is better to do the observation in space. Other advantages of measuring in space are the lack of any turbulent atmosphere and the lack of thermal background radiation. For these reasons, the European Space Agency (ESA) is planning to launch a space mission to investigate exo-planets using aperture synthesis in the far infrared. The mission is called Darwin and will be launched around 2020. The main interferometric technique will be nulling, although imaging would also be possible. According to the original plans, Darwin consists of six free flying telescopes (the ‘collectors’), one beam-combiner spacecraft (the ‘hub’) and possibly a small communication satellite, see Fig. 1.1. These satellites will fly in close formation, with a maximum distance of 250 meters between a telescope and the hub. Each of the collector satellites contains a telescope that captures the light of the star and relays it to the central beam-combiner satellite. The diameter of the telescopes will be 1–3 meters [11]. More recent plans are to use only four telescopes and a single beam-combiner spacecraft [12]. The exact details of the mission are not yet known, since some of the required technologies are still in active development.

1.3 Metrology

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14 CHAPTER 1. INTRODUCTION physical observable quantity. The most important units for our purpose are the second, which is defined as 9192631770 cycles of the radiation that corresponds to the transition between two chosen levels of a Cesium atom and the meter, which is defined as 1/299792458 part of the distance traveled by light in one second in vacuum. In future, these definitions might change as more accurate definitions become available.

An important concept in metrology is traceability. This means that for every measurement that is performed there should be an unbroken chain of comparisons or calibrations against known standards, leading all the way back to the primary standards. These activities are the responsibility of the national standard bureaus such as NIST (United States) and the NMi (the Netherlands). Another key concept of metrology is that every measurement result is combined with an uncertainty, which gives an indication of how far the measurement could deviate from the true value [14].

More recently, the term metrology is also used to describe a measurement system that is part of a larger system and usually measures (the dimensions of) the larger system itself. Some modern telescopes, for example, are so big that moving them around would bend their structure too much. Similarly, tiny changes in temperature might cause the telescope to expand or contract. For an accurate measurement of a star, the length of the telescope itself should therefore also be measured. Such a metrology system can be used to correct the errors (e.g. by bringing the telescope back in focus) or allows for compensating the errors in the later calculations.

1.4 Darwin metrology

The main scientific measurement to be performed in the Darwin mission consists of detecting fringes using white-light interferometry in the far-infrared. Since the coherence length of the light is only a few tens of micrometers, all the optical path-lengths via the different telescopes to the central beam combination should be made equal to within a few wavelengths to observe the fringes. This will only be possible if the mutual positions of the different satellites are accurately known, which requires a complex metrology system. Not only the mutual distances between the satellites should be known, but also mutual speeds and attitudes. It is impossible to do these measurements with a single system, so the metrology system consists of several sub-systems [15, 16]. These will be switched on in several steps with increasing accuracy:

1. At the start of a measurement, the attitude of the various satellites should already be known with an accuracy of 10 arcseconds using conventional star-trackers. Similarly, the mutual positions of the satellites should be known with an accuracy of several centimeters using a local, GPS-like radio frequency system [17]. If all the satellites are located in a single plane, the accuracy for the out-of-plane position will be degraded to a few meters. Reducing this value to centimeter level will require a coarse lateral metrology system, which might be implemented as a divergent source on one satellite and a camera on the other. The position and attitude errors should now already be small enough for transferring optical beams between two satellites for the subsequent metrology systems.

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1.5. OTHER SPACE MISSIONS WITH DISTANCE METROLOGY 15 This will be implemented using a collimated beam at one satellite and a position sensitive detector at the other satellite. The lateral position should be known to high accuracy since it directly influences the coupling-efficiency for the transfer of starlight from the telescopes to the hub and thus the stability of the fringe contrast in the science measurement. The second reason is that a lateral movement in the direction of the star will have a direct influence on the path-length difference of the interferometer.

3. In this step, the absolute distance between a telescope and the hub should be measured with an accuracy of around 70 µm. This distance has a direct influence on the optical path length difference. This should be reduced to below a centimeter to allow the path length to be equalized using the delay line. The distance should, however, be measured with an accuracy much better than a centimeter, since scanning the whole delay line to find the white-light fringe might take too much time.

4. Next, the rate at which the optical path-length changes should be damped to allow the acquisition of the white-light fringe. This might require an incremental metrology system that can measure path-length changes with nanometer resolution.

5. In a final metrology step, which is closely related to the science measurement itself, the position of the white-light fringes is observed by scanning the delay line.

The measurements of the metrology system will be used to correct the positions of the satel-lites using micro- and milli-newton thrusters. The final equalization of the path lengths will be performed with a delay line with a stroke of 1 cm. Our work will focus on the possible implementation of the absolute distance metrology of step 3.

1.5 Other space missions with distance metrology

Apart from Darwin, there are a number of different space missions that critically depend on distance metrology. For some of these missions, distance metrology is a technology that enables a different measurement. For other missions, the distance measurement itself is the main scientific measurement. We will discuss a few of these missions.

1.5.1 Inter-satellite distance metrology

The first need for measuring distances in space was for the docking of different spacecraft, which happened as early as 1966. The first few dockings were performed manually by the Americans with the aid of radar, but soon after, the Russians performed the first automated docking of satellites using radar and cameras [18]. The distance measurements needed for docking are not extremely challenging, since only an accuracy of a few centimeters is required. Moreover, this accuracy is only needed at close range, with more relaxed requirements at larger distances.

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16 CHAPTER 1. INTRODUCTION Finally, there is a group of missions in which the measured distance itself is the main property of scientific interest. One example is the Grace mission, which tries to measure the gravitational field of the earth. It consists of two satellites flying 220 km apart. The distance between the two is measured with micrometer accuracy using radio frequency waves [20]. In a future follow-up mission, the accuracy could be improved using a stabilized laser [21]. A final, and most spectacular example is the LISA mission, in which three satellites fly with a mutual separation of five million kilometers to detect gravitational waves [22]. Averaged over time scales of a year, the astonishing strain sensitivity of around 10−23 should be reached!

1.5.2 Internal distance metrology

A final example of the use of distance metrology in space concerns measurements that need to be performed so accurately, that the instrument can no longer be considered to be rigid. This can be compensated by equipping the whole instrument with internal metrology sensors. The measurement can then be corrected for flexing of the spacecraft later in the data-analysis. Two missions employing this concept are SIM and GAIA, both of which try to perform micro-arcsecond astrometry for detecting exo-planets [2, 23].

1.6 Goal of our research

As described above, the Darwin mission requires a metrology system that can measure absolute distances with high accuracy. The intended accuracy is 70 µm at a distance of 250 meter. In our research, we will review the various techniques for performing distance measurements and we will try to actually build such a system. We will limit ourselves to experiments that can be performed in the laboratory. We will thus not try to build a system that is totally qualified for use in space. This would require a lot of resources and will have to be done in industry, for example by the sponsor of our research. Parallel to our research, ESA has already awarded an industrial study into various components of the metrology system [15]. No contract for the final implementation has been awarded yet.

1.7 Outline of this thesis

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Chapter 2

Theory

This chapter introduces the theory that is necessary to understand the various concepts that are used in our experiments. First, the basics of interferometry are explained, both for an ideal monochromatic source and for a more realistic case of a laser with a finite coherence length. Next a detailed discussion of the Fabry-P´erot cavity is given since it is the central component in our experimental setup. Several techniques for stabilizing lasers to such a cavity are discussed with a focus on the Pound-Drever-Hall technique and its dynamic response. The chapter concludes with a short overview of feedback-theory.

2.1 Basic interferometry

Consider a simple unbalanced Mach-Zehnder interferometer as in Fig. 2.1. For now, the source is assumed to be an ideal, monochromatic laser. Its electric field E as function of time t can be described as

E(t) = E0exp(i2πν0t), (2.1)

with E0 the amplitude and ν0 the optical frequency. In the interferometer, the light is split into

two branches using a beam-splitter. In the long branch the light is delayed with respect to the short branch by an amount τ . The two branches are combined on a second beam-splitter and the power of the combined beams is measured with a detector. The electric field at the detector caused by the two branches can be described as

E1(t) = k1E(t) = k1E0exp(i2πν0t)

E2(t) = k2E(t − τ) = k2E0exp(i2πν0(t − τ))

(2.2) with k1 and k2 two real constants that account for the power splitting and other losses at the

beam-splitters. Summing and squaring these two fields yields the power P at the detector P = |E1+ E2|2 = P0k21+ k22+ 2k1k2cos(φ0)≈ P0/2 [1 + cos(φ0)] , (2.3)

laser

detector

t

Figure 2.1: Schematic of a Mach-Zehnder interferometer where the light in the long branch is delayed by an amount τ with respect to the short branch.

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18 CHAPTER 2. THEORY

laser

oscillator

~

detector

t

AOM

Figure 2.2: Schematic of a Mach-Zehnder interferometer with an acousto-optic modulator (AOM) in one of the arms.

where the last approximation is valid for ideal ‘50–50’ beam-splitters and perfect alignment. It is assumed that the amplitude is dimensioned such, that the power P0 = |E0|2. The phase φ0 is

defined as φ0 = 2πν0τ = 2π ν0L c = 2π L λ. (2.4)

Here, L is the optical path length of the long branch, c the speed of light and λ the wavelength of the light in vacuum. Eq. 2.3 explains the light and dark fringes that appear when changing the path length in any interferometer.

2.1.1 Quadrature phase retrieval

In interferometry, the goal is usually to obtain a phase, based on the measurement of an optical power. This is not possible with a single power measurement, since Eq. (2.3) has no unique solution. A common solution is to somehow obtain a second interferometric measurement of the same phase shifted by π/2. This is called quadrature detection. The two measured values Px

and Py can be described by

Px= ox+ g cos(φ0)

Py= oy+ g cos(φ0− π/2) = oy+ g sin(φ0),

(2.5) with ox and oy two offsets and g a common gain. The phase φ0 can then be recovered by

φc= arctan4q Py− oy Px− ox = arctan4q sin(φ0) cos(φ0) = φ0 mod 2π, (2.6)

where φc is the calculated phase. Instead of the normal arctangent function, a four-quadrant

version arctan4q should be used, which yields a phase from −π to π instead of −π/2 to π/2.

The absolute value of the phase φ0 is thus lost, but it is possible to follow phase changes by

unwrapping φc over time to remove the 2π phase jumps.

2.1.2 Heterodyne detection

A different technique for retrieving the phase is called heterodyne detection. In this method, the optical frequency in one of the arms of the interferometer is shifted by the heterodyne frequency fh. This is usually achieved with an acousto-optic modulator (AOM), see Fig. 2.2. The electric

field in the second arm stays the same as Eq. (2.1), but the field of the first arm should now be written as

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2.2. LASER PHASE NOISE 19 Similar to Eq. (2.3), the optical power after beam combination is obtained as

P (t) = P0

k2₁+ k2₂+ 2k1k2cos(2πfht + φ0)

. (2.8)

The detected signal thus contains a DC-component and a component that oscillates at the het-erodyne frequency fh. The phase φ0can be obtained with an electrical phase meter by comparing

the (band-pass filtered) detected signal with the modulation signal cos(2πfht). The main reason

for using heterodyne detection is that the information is detected at a higher frequency, which has some advantages for the detection electronics. Similar to quadrature detection, the phase is recovered modulo 2π, so only phase differences can be measured.

2.2 Laser phase noise

In practice, a laser is not truly monochromatic, but appears as a peak with a finite line-width in the optical spectrum. This becomes especially apparent when doing interferometry over path length differences that are no longer small compared to the coherence length of the source. One way to model this effect is to describe the electric field of the source as

E(t) = E0exp(iϕ(t)), (2.9)

with ϕ(t) the time-varying optical phase. In a more general case the amplitude E0 would also

be considered to vary over time, but the effect of amplitude noise can usually be neglected compared to that of phase noise. For a quasi-monochromatic laser the electric field oscillates with a high frequency around an average frequency ν0. This part can be split off from the phase

to distinguish between the average frequency and a part ϕn(t) that contains the noise

ϕ(t) = 2πν0t + ϕn(t). (2.10)

The instantaneous frequency ν(t) can now be defined as the derivative of the phase ν(t) = 1 2π dϕ(t) dt = ν0+ 1 2π dϕn(t) dt = ν0+ νn(t), (2.11)

where νn is the time-varying part that contains the noise. The phase noise ϕn(t) and the

frequency noise νn(t) can be seen as two alternative ways to describe the same phenomena,

since the instantaneous frequency is uniquely defined by the phase.

Applying this new description of the electric field to Eq. (2.3) shows that the power at the detector is now time-varying

P (t) = P0

k2₁+ k2₂+ 2k1k2cos(φ(t))

, (2.12)

with the time-varying interferometric phase φ(t) given by

φ(t) = ϕ(t) − ϕ(t − τ) = 2πν0τ + ϕn(t) − ϕn(t − τ) = φ0+ ∆ϕn,τ(t). (2.13)

Here, φ0 = 2πν0τ is the nominal phase encountered in Eq. (2.4) and ∆ϕn,τ(t) is the phase error

that contains the noise, which is a function of time and the delay τ . The phase error can be derived from the instantaneous frequency

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20 CHAPTER 2. THEORY where the convolution has been defined as

f ∗ g (t) ≡ Z ∞

−∞f (τ )g(t − τ) dτ

(2.15) and the boxcara,b function as

boxcara,b(t) ≡

(

1 a < t < b

0 elsewhere. (2.16)

The total accumulated phase error can thus be expressed as the convolution of the instantaneous frequency with a rectangular window of width τ . A convolution in the time domain leads to a product in the frequency domain, so the Fourier transform of ∆ϕn,τ(t) can be described by

F {∆ϕn,τ} (f) = 2πF {νn} (f)·F {boxcar0,τ} (f) = 2πτF {νn} (f) exp(−iπfτ)

sin(πτ f )

πτ f , (2.17) where the Fourier transform has been defined as

F {g(t)}(f) ≡ Z ∞

−∞exp(−2πift)g(t) dt.

(2.18) Since νn(t) is generally a stochastic signal that cannot be integrated, it is more appropriate to

consider the power-spectral-densities S∆ϕn,τ and Sνn

S∆ϕn,τ = 4π 2_τ2 sin(πτ f ) πτ f 2 Sνn, (2.19)

where S∆ϕn,τ is defined in units rad

2_{/ Hz and S}

νn in Hz

2_{/ Hz. The phase is usually detected}

with a system with a limited bandwidth B, so the variance of the phase error becomes ∆ϕn,τ2= Z B 0 S∆ϕn,τdf ≈ 4π 2_τ2Z B 0 Sνndf. (2.20)

Usually, the detection bandwidth is much smaller than the inverse of the delay (B ≪ 1/τ), in which case the sinc-function of Eq. (2.19) can be approximated to be unity. [24, 25] The coherence time τccan now (somewhat arbitrarily) be defined as that delay, for which the variance

of the phase error is equal to two (∆ϕn,τc2

= 2). For many lasers, the frequency noise spectrum Sνn can be approximated as white noise with a level of C (in Hz

2_{/ Hz). Integrating}

Eq. (2.20) up to infinity (without using the approximation for the sinc-function and substituting R∞

0 sinc2(x) dx = π/2) then yields the coherence time

τc= 1

Cπ2 (2.21)

and finally the coherence length Lc

Lc= cτc=

c

Cπ2. (2.22)

2.3 Fabry-P´

erot cavity

if the multiply reflected waves interfere constructively. This is a resonant phenomenon which only occurs when the length of the cavity equals an integer number of half-wavelengths. The resonances can be made very narrow by increasing the reflectivity of the mirrors, which makes the Fabry-P´erot cavity a useful tool in spectroscopy. The Fabry-P´erot cavity is of interest for our measurement since it is the essential part of our laser stabilization scheme and it forms the stable reference for our length measurement.

2.3.1 Basics properties

Consider an optical cavity constructed of two mirrors with their semi-reflecting surfaces facing each other at a distance d. The refractive index of the medium in between the mirrors will be designated with n0, while the mirror substrates will have refractive indices n1 and n2. For the

moment it is assumed that no energy is absorbed at the mirror surfaces and that no additional phase shift takes place on reflection. Only plane waves that are perpendicularly incident on the (infinitely large) mirrors will be considered.

Waves traveling in the cavity will be transmitted through and reflected from the surfaces with amplitude transmission and reflection coefficients t and r. The transmission and reflection coefficients for waves originating from inside the mirror substrate will be designated with t and r. According to the Stokes theorem [27], which holds in the loss-less case, the coefficients are related as

¯ r = −r

r2+ t¯t = 1. (2.23)

Furthermore, the transmittance T and reflectance R are defined as T = t¯t

R = r2, (2.24)

which indicate which fraction of the incident power is transmitted or reflected from a surface. Combining Eqs. (2.23) and (2.24) yields

R + T = 1, (2.25)

which shows energy conservation at each surface.

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22 CHAPTER 2. THEORY set of equations is obtained

E1 = ¯t1E0+ r1E4 E2 = exp(−iψ/2)E1 E3 = r2E2 E4 = exp(−iψ/2)E3 Et= t2E2 Er= ¯r1E0+ t1E4, (2.26)

where ψ is the phase delay experienced by a wave traveling up and down the cavity, so

ψ = 4πνn0d/c. (2.27)

Solving Eq. (2.26) for the amplitude transmission and reflection ratios Gt and Gr yields

Gt= Et E0 = ¯t1t2exp(−iψ/2) 1 − r1r2exp(−iψ) = (1 − R) exp(−iψ/2) 1 − R exp(−iψ) (2.28) Gr = Er E0 = r¯1+ r2(t1¯t1− r1r¯1) exp(−iψ) 1 − r1r2exp(−iψ) = r(exp(−iψ) − 1) 1 − R exp(−iψ), (2.29) where we have used Eqs. (2.23) and (2.24) and assumed that both mirrors are equal. Squaring the amplitude ratios yields the power transmission and reflection ratios GT and GR

GT = |Gt|2 = (1 − R) 2

1 + R2_{− 2R cos (ψ)} (2.30)

GR= |Gr|2= 2R(1 − cos (ψ))

1 + R2_{− 2R cos (ψ)}. (2.31)

Plotting GT and GRshows the well known transmission and reflection graphs for a Fabry-P´erot

cavity, see Fig. 2.4. The transmission is maximum when ψ is an exact multiple of 2π, so, by using Eq. (2.27), the optical frequency of a resonance νres is found as

νres= p

c 2n0d

= p ∆νfsr, (2.32)

with p generally a large integer indicating the longitudinal mode number of the resonance. The difference between two successive resonance frequencies is called the free spectral range ∆νfsr

∆νfsr=

c 2n0d

. (2.33)

2.3.1.1 Approximation around resonance

The formulas above show a periodic behavior in ψ with period 2π and in ν with period ∆νfsr.

It is thus possible to replace ψ and ν by e_{ψ = ψ − p 2π and e}_{ν = ν − p ∆ν}fsr. The behavior of the

transmission around resonance can then be found by approximating the cosine in Eq. (2.30) to second order when_eψ_{≪ 1}

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2.3. FABRY-P ´EROT CAVITY 23 0 1 G T (−) ∆ν_FSR ∆ν_res −0.50 0 0.5 1 1.5 2 2.5 1 (ν − ν res)/∆νFSR (−) G R (−)

Figure 2.4: Transmission and reflection coefficients for a Fabry-P´erot cavity with a finesse of 20.

This is a Lorentzian function with a full width at half maximum ∆νres

∆νres=

∆νfsr(1 − R)

π√R . (2.35)

It is now possible to define the finesse F as the ratio between the free-spectral-range and the width of a resonance [27], which is commonly used to describe the quality of a Fabry-P´erot cavity F = _∆ν∆νfsr res = π √ R 1 − R. (2.36)

For use later in this chapter Gr, will also be approximated around resonance by expanding the

exponentials of Eq. (2.29) to first order, which is again valid for eψ close to zero e Gr≈ r((1 − i eψ) − 1) 1 − R(1 − i eψ) = −i eψr 1 − R + i eψR = −i(2πeν/∆νfsr)r 1 − R + i(2πeν/∆νfsr)R = − i2eν ∆νfsr πr 1 − R 1 1 + i2νre ∆νfsr πr 1−R ! = −_∆νi2eν fsrF 1 1 + i2eνr ∆νfsrF = −_∆νi2eν res 1 1 + i2eνr ∆νres ! ≈ −_∆νi2eν res (2.37)

The first approximation is valid for eν ≪ ∆νfsr, but the second approximation only for eν ≪ ∆νres.

See Fig. 2.5 for a complex plot of Gr. Far away from resonance, Gr is close to -1, but around

resonance it will quickly sweep along the circle. Exactly at resonance the function moves parallel to the imaginary axis.

2.3.1.2 Dynamic response

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24 CHAPTER 2. THEORY −1 0 1 −1 1 ν Re Im

Figure 2.5: Complex plot of Gr for a cavity with a finesse of 20. Values are calculated for

e

ν/∆νfsr ranging from -0.2 till 0.2 with a step size of 0.002. The curve is traversed clockwise

when the frequency increases, assuming that r > 1.

instantaneously the light already in the cavity will not disappear immediately since it can bounce many times from the highly reflecting mirrors. Over time the energy inside the cavity will slowly decay due to the tiny amount of energy that is either absorbed or transmitted by the mirror. At each round-trip through the cavity the light will reflect twice from a mirror, so after N bounces the amplitude is a factor (r2)N = RN times its original amplitude. An effective number of bounces Neff can be defined as the number of round-trips it takes for this factor to drop to a

value of 1/e (r2)Neff _{= R}Neff ≡ exp(−1). (2.38) Rewriting yields Neff = 1 1 − R ≈ F π, (2.39)

where the approximations are valid for highly reflecting mirrors. Since the optical path length of one round-trip through the cavity is 2n0d, a storage length Ls can be defined as

Ls= 2n0dNeff =

2n0dF

π (2.40)

and a storage time τs as

τs= Ls c = 2n0dF cπ = 1 π∆νres . (2.41)

See for example [28] for a further analysis of the dynamic effects of a Fabry-P´erot cavity. 2.3.1.3 Lossy mirrors

So far it was assumed that no energy is absorbed at the mirrors. For highly reflecting mirrors this can no longer be neglected. Eq. (2.25) should now be augmented with the absorptance A to uphold energy conservation

R + T + A = 1. (2.42)

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2.3. FABRY-P ´EROT CAVITY 25 should now be multiplied with an efficiency factor η, which can be shown to be

η = T T + A 2 . (2.43)

Note that the absorption coefficient A contains both the effects of absorption and scattering due to roughness of the mirror. Incident power that is not reflected or transmitted is thus either dissipated as heat in the mirror or scattered outside the cavity.

2.3.2 Gaussian optics

Beams of light with a finite cross-section no longer behave as plane waves, but are influenced by diffraction effects. Although different descriptions are possible, one description of great practical use applies to beams with a Gaussian intensity profile. For a paraxial beam traveling along the z-axis with its beam waist at the origin, the scalar field can be written as

E(x, y, z) ∝ _w(z)1 exp iθ(z) −x 2_{+ y}2 w2_(z) − ik(x2_{+ y}2₎ 2ρ(z) − ikz , (2.44)

with w(z) the beam radius as a function of z w(z) = w0

q

1 + z2_/z2

R, (2.45)

ρ(z) the radius of curvature of the wave-front as a function of z

ρ(z) = (z2+ z_R2)/z (2.46)

and θ(z) an extra phase shift as a function of z

θ(z) = arctan(z/zR). (2.47)

In these equations zR is the so called Raleigh distance which only depends on the minimum

beam waist radius w0 as

zR= πw20/λ. (2.48)

Constructing a Fabry-P´erot cavity out of flat mirrors would lead to high losses due to diffraction. If concave mirrors with the right amount of curvature are used this effect can be compensated. This leads to eigenmodes of the cavity that maintain a stable shape while traveling once up and down the cavity. One intuitive way of obtaining a stable solution is to start with a Gaussian beam with a given beam waist and to insert two mirrors that have a radius of curvature that coincides with the local curvature of the wavefront defined by Eq. (2.46). For a symmetric cavity with two equal mirrors with radius of curvature R that are separated by a distance d the Raleigh distance becomes

zR=

p

(R − d/2)d/2, (2.49)

which totally defines the cavity mode [30]. 2.3.2.1 Higher order transverse modes

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26 CHAPTER 2. THEORY where Hq and Hr are Hermite polynomials of order q and r. The functions ρ(z), θ(z) and w(z)

are the same as in Eq. (2.44). Since the transverse behavior is described by the product of a Gaussian and a Hermite function, these modes are known as Hermite-Gaussian modes.

Similar to the plane mirror cavity described in section 2.3.1, a curved mirror cavity will show resonant behavior for those frequencies, for which the phase of a complete round-trip through the cavity equals a (large) integer times 2π. This yields the resonant frequency νres

νres=

c 2n0d

(p + (q + r + 1) K) , (2.51)

with p the axial mode number, q and r the transverse mode numbers and K a constant defined as K = 2 πarctan d 2zR . (2.52)

Note that the resonance frequencies are no longer an exact multiple of the free-spectral-range ∆νfsr as in Eq. (2.32). Apart from this shift in resonance frequency, however, the resonant

behavior remains unchanged.

If K (0 ≤ K ≤ 1) becomes a rational number it is possible that a zero-th order transverse mode (q = 0, r = 0) becomes degenerate with a higher order transverse mode with a lower axial mode number [30]. This is not desirable when the cavity is used for laser stabilization. Low order degeneracy should be avoided by proper design of the cavity. Higher order degeneracy can be prevented by proper mode-matching.

2.4 Stabilizing lasers to Fabry-P´

erot cavities

Free running lasers have an optical frequency that is not perfectly stable over time. At longer timescales they are usually sensitive to changes in temperature of the environment, which in-fluences the length of the laser cavity and thus the central frequency. On shorter timescales the optical frequency will fluctuate because of mechanical vibrations of the cavity and due to physical limitations of the lasing process itself, causing a finite coherence length. For this reason, laser physicists have long tried to link the frequency of lasers to standards that have a higher stability than a free-running laser. These standards can be based on spectroscopic features [31], physical standards such as interferometers and more recently stable clocks [32] For this project we are interested in locking a laser to a resonance of a Fabry-P´erot cavity. This requires giving feedback to the laser based upon an error signal, which should change sign at the frequency of interest. Several schemes to generate such a signal will be discussed.

2.4.1 Side-of-fringe locking

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2.4. STABILIZING LASERS TO FABRY-P ´EROT CAVITIES 27

2.4.2 Low frequency modulation: lock-in detection

A common measurement technique method is to somehow impose a modulation frequency to a system and synchronously detect the response of the system at the modulation frequency. This is called lock-in detection and has the advantage that the electronic signal processing can be performed at higher frequencies, thereby avoiding sensitivity to some low-frequency noise sources. The technique can be used for generating an error signal from a resonance of a Fabry-P´erot by using an optical phase modulator. The electric field after the modulator becomes

E(t) = E0exp[i(2πνt + βmsin(2πfmt))], (2.53)

where fm is the modulation frequency and βm the modulation depth. Analog to Eq. (2.11) the

electric field thus has an instantaneous optical frequency ν(t) = 1

2π d

dt(2πνt + βmsin(2πfmt)) = ν + fmβmcos(2πfmt) (2.54) If the modulation frequency is slow enough for the field inside the cavity to adapt to the changes (implying that fm ≪ 1/τs≈ ∆νres), the transmitted power can be written as

P (t) = P0GT(ν(t)) = P0GT(ν + fmβmcos(2πfmt)) (2.55)

If the amplitude of the modulation of the instantaneous frequency is small enough (fmβm ≪

∆νres) this can be approximated to first order as

P (t) ≈ P0GT(ν) + P0fmβmcos(2πfmt)

d

dνGT(ν) (2.56)

By detecting this signal on a detector and detecting synchronously at frequency fm, an error

signal Veis obtained that is proportional to the first derivative of GT with respect to the optical

frequency Ve= κP0fmβm d dνGT(ν) = − 8κP0fmβm∆νres2 νe (∆ν2 res+ 4eν2)2 , (2.57)

where κ is the detector sensitivity in Volts per Watt. Equation 2.34 is used to approximate GT.

The maximum sensitivity σ close to resonance is σ = d deνVe(eν) e ν=0= − 8κP0fmβm ∆ν2 res (2.58) A lock-in amplifier usually operates by electrical mixing of the detected signal with the modulation signal followed by a low-pass filter to get rid of any signals oscillating at the modu-lation frequency, see Fig. 2.6. The cut-off frequency of the filter should be much lower than the modulation frequency, which in turn should be lower than the cavity line-width. Overall, the achievable bandwidth of this method is thus much lower than the cavity line-width. A trade-off thus has to be made between a high bandwidth of the error signal (implying a large line-width) or high frequency discrimination (implying a small line-width) [34].

The lock-in technique is also widely used for locking lasers to spectroscopic features. For this application, the detected signal is usually demodulated with the triple of the modulation frequency instead of the modulation frequency itself. If Eq. (2.56) is expanded into more terms it appears that the term oscillating at the triple modulation signal is proportional to the third derivative of GT(ν). This yields a clearer error signal that is not sensitive to constant slopes

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28 CHAPTER 2. THEORY mixer detector oscillator LP LF tunable laser Fabry-Pérot (a) 0.0 0.5 1.0 Transmission (−) (b) −5 0 5 −1 0 1 Frequency (MHz) Ve (−) (c)

Figure 2.6: Lock-in method. (a) Experimental scheme, as used in [34]. Note that only one input of the laser is used for applying both the modulation signal and the feedback signal. LP: low-pass filter, LF: loop filter. (b) Calculated transmission as a function of optical frequency for a cavity with a line-width of 1MHz. (c) Matching error signal. The locking point is indicated with a dot.

2.4.3 High frequency modulation: Pound-Drever-Hall stabilization

A different technique arises when the modulation frequency is increased to a value much larger than the cavity line-width. The technique was described first by Pound for microwave applica-tions [35]. Its first application for stabilizing lasers was reported by Drever and Hall [36] for use in gravitational wave detection. The method is now in widespread use for locking lasers and is commonly called Pound-Drever or Pound-Drever-Hall (PDH) stabilization. The method for gen-erating the error signal was described earlier for both atomic absorption lines and Fabry-P´erot cavities by Bjorklund, but was then only used for spectroscopy and not for laser stabilization [37]. Contrary to the lock-in method, the PDH method uses the light reflected from and not transmitted by the cavity, see Fig. 2.7.a.

2.4.3.1 Quasi-static case

Consider again a laser that is phase modulated according to Eq. (2.53), but now with a modula-tion frequency fm that is much higher than the cavity line-width (fm≫ ∆νres). In this case the

use of GR is not allowed, since the cavity has not enough time to adjust to the instantaneous

frequency caused by the modulation. The proper way to proceed is to expand the sine inside the exponent with the aid of Eq. (A.2). The electric field after the phase modulator can then be written as E (t) = E0exp(i2πνt) ∞ X k=−∞ Jk(βm) exp(i2πkfmt). (2.59)

The light after the modulator thus consists of a carrier at the optical frequency ν surrounded by sidebands at multiples of the modulation frequency fm. The reflected electrical field is found

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2.4. STABILIZING LASERS TO FABRY-P ´EROT CAVITIES 29

mixer detector

oscillator

LP

LF

tunable

laser

Fabry-Pérot

phase

modu-lator

(a) 0 0.2 0.4 0.6 Transmission (−) (b) −40 −20 0 20 40 −0.4 −0.2 0 0.2 0.4 Optical frequency (MHz) Ve (−) (c)

Figure 2.7: Pound-Drever-Hall method. (a) Experimental scheme. LP: low-pass filter, LF: loop filter. (b) Calculated transmission as a function of optical frequency for a cavity with the same properties as described in section 4.5. The large peak at the origin is the optical carrier. The two smaller peaks at ± 20 MHz are the modulation sidebands. (c) Matching error signal. The locking point is indicated with a dot.

is obtained by squaring the resulting electric field Pr(ν) = P0 ∞ X k=−∞ ∞ X k′_=−∞ Jk(βm)Jk′(β_m)G_r(ν + kf_m)G∗_r(ν + k′f_m) exp[i2π(k − k′)f_mt]. (2.60)

This expression is built up of cross-terms which are located at zero frequency or oscillate at multiples of fm. If the modulation index βm is kept much smaller than 1, all the terms of order

2 and higher can be neglected. Furthermore, since the signal is detected synchronously at the modulation frequency fm, only those terms for which k −k′ = ±1 will survive the demodulation.

Only two terms plus their complex conjugates remain Pr(ν) fm = P0[J0(βm)J−1(βm)Gr(ν)G∗r(ν − fm) exp(i2πfm)+ J1(βm)J0(βm)Gr(ν + fm)G∗r(ν) exp(i2πfm)+ complex conjugates]. (2.61)

With help of Eqs. (A.3) and (A.4) this can be written as Pr(ν) fm = 2P0J0(βm)J1(βm)[ Re{Gr(ν)G∗r(ν + fm) − Gr(ν − fm)G∗r(ν)} cos(2πfmt)+ Im{Gr(ν)G∗r(ν + fm) − Gr(ν − fm)G∗r(ν)} sin(2πfmt)]. (2.62) This signal is detected on a photodiode after which it is electrically multiplied with the original modulation signal and low-pass filtered. By changing the phase shift between the two signals before multiplying it is possible to obtain either the cosine or the sine term. Of these two, the sine term is the interesting one, since Gr moves along the imaginary axis when the frequency is

swept through a resonance, see Fig. 2.5. The demodulation then yields the error signal Ve

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30 CHAPTER 2. THEORY fm fd 1 2 3 4 7 8 5 6

Figure 2.8: Schematic of the power spectrum of a laser that has a frequency modulated distur-bance at frequency fd and that is frequency modulated at frequency fm. The numbers indicate

the 8 combinations of peaks that are separated by exactly fm± fd.

The resulting error signal has a very steep slope around the location of the resonance, see Fig. 2.7. The sensitivity around resonance can be determined by differentiating with respect to frequency.

σ = d dνVe(ν) ν=νres = 2κP0J0(βm)J1(βm) d dνIm {Gr(ν)G ∗ r(ν + fm) − Gr(ν − fm)G∗r(ν)} ν=νres = −4κP0J0(βm)J1(βm) d deν Im e Gr(eν) e ν=0 = 8κP0J0(βm)J1(βm)/∆νres (2.64) where we used that Gr(νres± fm) = eGr(±fm) − 1. The highest sensitivity is reached when

the term J0(βm)J1(βm) is maximized, which happens for βm ≈ 1 [38]. Contrary to the lock-in

method, the Pound-Drever-Hall method does not require that the resonance width should be larger than the modulation frequency. The sensitivity can thus be increased by choosing a cavity with a very small resonance width.

2.4.3.2 High frequency response of the Pound-Drever-Hall method

The simple analysis of the Pound-Drever-Hall method as described above only holds when both the amplitude and the frequency of the disturbances of the optical frequency are much smaller than the line-width of the cavity.1 It is often mentioned casually (e.g. [36, 39]) that the error signal has a low pass character with a cut-off frequency that equals the line-width of the cavity, but no detailed derivation was found in literature. Since the exact dynamic behavior is important for designing a feedback system based on the PDH error signal, a detailed derivation is carried out in this section.

So far, it was assumed that the frequency of the laser changes quasi-statically, so that the optical field inside the cavity can adjust itself to the incoming field. Especially for high finesse cavities this assumption no longer holds. To study the dynamic response of the PDH method, it is assumed that the disturbance can be modeled as a frequency modulation on top of a fixed optical frequency ν. This can be introduced in a way similar to how we described the effect of the phase modulator in Eq. (2.53):

E(t) = E0exp[i(2πνt + βdsin(2πfdt))] (2.65)

1

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2.5. CONTROL THEORY 31 with fd the frequency of the disturbance and βd the modulation depth of the disturbance. The

instantaneous frequency is thus

ν(t) = ν + fdβdcos(2πfdt) (2.66)

Similar to the case without a disturbance, the light is phase modulated at fm

E(t) = E0exp[i2π(νt + βdsin(2πfdt) + βmsin(2πfmt)]

= ∞ X k=−∞ ∞ X l=−∞ Jk(βm)Jl(βd) exp[i2π(ν0+ kfm+ lfd)t]. (2.67)

The disturbance thus appears as extra side-bands around all the side-bands caused by the modulation, see Fig. 2.8. To calculate the Pound-Drever-Hall error signal we have to proceed in a similar way as above. All the sidebands will reflect from the cavity with the appropriate value of Gr and the power on the detector Pr is again obtained by squaring the total reflected field.

The resulting expression contains a summation over 4 indices Pr(ν) = P0 ∞ X k=−∞ ∞ X l=−∞ ∞ X k′_=−∞ ∞ X l′_=−∞ Jk(βm)Jl(βd)Jk′(β_m)J_l′(β_d) Gr(ν + kfm+ lfd)G∗r(ν + k′fm+ l′fd) exp[i2π((k − k′)fm+ (l − l′)fd)t]. (2.68)

Fortunately we can make a large number of simplifications, which finally yields the complex transfer function GPDH(fd) GPDH(fd) = 8κP0J0(βm)J1(βm) ∆νres 1 1 + i2fdr ∆νres = σ 1 1 + ifd fc , (2.69)

where the cut-off frequency fc = ∆νres/2r ≈ ∆νres/2 and σ is the sensitivity calculated for the

quasi-static case in Eq. (2.64). The complete derivation is given in Appendix A.2. Note that in our calculations we obtain a bandwidth that is half the line-width of the cavity. This is half the value that is often mentioned casually in the literature, see e.g. [39].

2.4.3.3 Large amplitude disturbances

So far, it was assumed that the disturbances have a small amplitude, which means βd ≪ 1. If

this assumption can no longer be made, higher order terms of the Jacobi expansion have to be added. The analysis could be carried out analogously to section 2.4.3.2, but the complexity of the formulas increases dramatically. It is, however, not too difficult to imagine some effects that will take place. The calculated power will contain the same cross-terms as in Eq. (2.68), but now we have some extra terms that oscillate at frequencies fm± 2fd. After down-converting the

frequency, some new terms with frequency 2fd will survive, which means that the error signal

will no longer be linear. This will lead to some deterioration of the feedback performance for small non-linear effects and to a total breakdown of the feedback for larger non-linear effects.

2.5 Control theory

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32 CHAPTER 2. THEORY

G

_p

G

_c process controller nm np measurement

s

p sr sm se sc + + + + + _

G

_m

Figure 2.9: Basic blocks of a feedback system.

controlled by feedback is shown in Fig. 2.9. The process has a single control signal sc and a

single output signal sp. The output reacts to changes on the input with a transfer function Gp.

The process variable is measured by a measurement system or sensor with transfer function Gm

which produces the measured signal sm. An error signal se is then obtained by comparing the

measured value with a reference signal sr, which might be a fixed value or a certain function

of time. Finally, the loop is closed by a controller, which drives the process with the control signal sc based on the error signal. All transfer functions are assumed to be linear but complex

functions of frequency. Noise sources are modeled by introducing the process noise np and the

measurement measurement noise nminto the loop. Combining all the elements of Fig. 2.9 yields

the set of equations

sp = Gpsc+ np

sc = Gcse

se = sr− sm

sm = Gmsp+ nm.

(2.70)

Solving this for the process variable sp yields

sp = GpGc 1 + GpGcGm sr− GpGc 1 + GpGcGm nm+ 1 1 + GpGcGm np, (2.71)

where the product of the all the transfer functions is usually called the loop gain Gl

Gl= GpGcGm. (2.72)

Since the transfer function of the measurement instrument Gm is ideally unity (or just a scaling

value to convert between different units), the process variable can be made to closely follow the reference signal by increasing the loop gain to a large value (Gl ≫ 1). In this case the process

variable can be approximated by

sp≈ 1 Gm sr− 1 Gm nm+ 1 Gl np. (2.73)

Compared to the free running process (where sp = np), the feedback reduces the process noise

np by a factor as large as the loop gain, but introduces a new error source nm caused by the

measurement system. A low noise measurement system is thus essential. The noise sources np

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2.5. CONTROL THEORY 33 reference signal is set to zero, the power spectral density of the process variable Ssp(f ) in the

closed loop system becomes Ssp(f ) = 1 1 + Gl 2 Snp(f ) + GpGc 1 + Gl 2 Snm(f ), (2.74)

assuming that the two noise sources are statistically independent.

2.5.1 Stability

As indicated by Eq. (2.71) it is desirable to make the total loop gain GpGcGmas large as possible,

both to suppress the process noise np and to make the process follow the reference signal sr as

accurately as possible. Unfortunately, the loop gain cannot be increased indefinitely because of phase delays at higher frequencies, which exist in any physical system. At some high frequency the total phase delay will be 180 degrees, which changes the negative feedback into positive feedback and causes the system to oscillate. A stable and robust system can be obtained by designing the feedback according to guidelines that impose a number of margins in phase and amplitude at unity feedback. One popular method to obtain a stable feedback system is to use a proportional, integrating and differentiating (PID) controller. The proportional part determines the overall loop gain. The integrating part will cause an infinite gain at low frequencies, which will drive the static error to zero. The differentiating part is able to counteract some of the phase delays at high frequencies, which allows increasing the gain further than without this compensation. In the time domain the effect of the PID-controller can be described by

sc = P (se+ I

Z

sedt + D

d

dtse), (2.75)

where P is the proportional constant, I is the integral constant and D is the differentiating constant.2 In the frequency domain the amplitude of the controller scales inversely with fre-quency for f < I/2π and scales linear in frefre-quency for f > 1/(2πD). The optimal value of these parameters can be determined with an empirical procedure called Ziegler-Nichols tuning.

2

Different definitions can be found in literature. In our case P is defined in units of sc per units of se, I in

(34)

(35)

Chapter 3

Methods for measuring absolute

distances

Measuring distances is probably one of the earliest measurements that have been performed by humans. Accurate measurements probably started with the work of Michelson at the end of the 19th century. Modern high accuracy measurements are possible with the invention of the laser around 1960 and the advancement of high speed electronics in the last 20 years. The field is thus very wide. A good overview of some earlier work is given in [41].

This chapter will start with a review of the various methods that can be used to measure absolute distances. We will limit ourselves to the accurate optical methods used for measuring distances larger than a few meters. A description of our method is then given, with some justification of our specific choices. Finally, a detailed error analysis of our method is given.

We use the convention that L indicates optical path length and D a distance. The relation between the two is simply

L = 2nD, (3.1)

where n is the refractive index averaged over the total path length. The factor of two arises from the fact that the light has to travel the distance to the reflective object twice. For most terrestrial measurements, the uncertainty in the refractive index is one of the limiting factors. This is not an issue in our case, since our intended application is a measurement in vacuum, so we can assume that the refractive index n is exactly unity.

3.1 Methods for measuring absolute distances

3.1.1 Time-of-flight

Of the various ways to measure an absolute distance, the time-of-flight method is conceptually the easiest. A pulse is sent out to a reflective object and the time it takes for the pulse to travel back to the source is recorded. This technique is used in various forms in radar, sonar and optics. The unknown path length L is then simply obtained as the product of the round-trip time τ and the speed of the pulse v

L = vτ. (3.2)

For optical pulses, v is the speed of light c. The benefits of this method are its simplicity and the ability to operate at very large distances. The accuracy at large lengths is limited by the relative stability of the clock (which can be extremely high if an atomic clock is available) and the knowledge of the speed of the pulse, which is the limiting factor when measuring in air. The resolution is usually limited to around a millimeter, because of the finite resolution in measuring time differences.

High-accuracy absolute distance metrology