Electromagnetic Properties of the Proof of Principle Booster Setup for the MADMAX Experiment

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(1)T ECHNISCHE U NIVERSITÄT M ÜNCHEN. M ASTER T HESIS IN P HYSICS. Electromagnetic Properties of the Proof of Principle Booster Setup for the MADMAX Experiment. Author: Jacob E GGE. Supervisor: Béla M AJOROVITS. Technische Universität München Department of Physics Max Planck Institut für Physik. October 1, 2018.

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(3) iii. TECHNISCHE UNIVERSITÄT MÜNCHEN. Abstract Electromagnetic Properties of the Proof of Principle Booster Setup for the MADMAX Experiment by Jacob E GGE. The MADMAX experiment is a novel approach to search for axion dark matter in the range from 40 µeV to 400 µeV. A booster consisting of dielectric disks amplifies the tenuous axion signal coming from axion-photon conversion in a strong magnetic field to detectable levels. A Proof of Principle Booster Setup was built to study the electromagnetic properties and systematic effects of a small scale prototype booster. It is shown that the dielectric disks can be positioned with µm precision and thermal effects on disk positions are quantified. Unwanted reflections influence the electromagnetic response. A model to simulate the effect of unwanted reflections was developed and tested. It satisfactorily predicts electromagnetic responses for up to four disks at different distances. It was shown that the booster could be adjusted to a desired electromagnetic response with acceptable deviations of the boost factor for up to five disks. Short comings of the model and tilts of the disks were identified. These become limiting factors for adjusting more disks and introduce systematic effects that have to be investigated in the future..

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(5) v. Contents Abstract 1. Introduction. 1. 1.1. Motivation for Axions . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.1.1. The Strong CP Problem . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.1.2. Dark Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2. 1.1.3. Axion Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 1.1.4. Current Limits and Experiments . . . . . . . . . . . . . . . . . .. 4. Magnetized Disk and Mirror Experiment . . . . . . . . . . . . . . . . .. 4. 1.2.1. Working Principle . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 1.2.2. Proof of Principle Booster Setup . . . . . . . . . . . . . . . . . .. 8. 1.2. 2. 3. iii. Theory. 9. 2.1. Booster Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 2.1.1. Transfer Matrix Formalism . . . . . . . . . . . . . . . . . . . . .. 9. 2.1.2. Simulating the Booster with Electrical Circuits . . . . . . . . . . 12. Experiment and Methods 3.1. 13. Description of Proof of Principle Booster Setup . . . . . . . . . . . . . . 13 3.1.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. 3.1.2. Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Vector Network Analyzer . . . . . . . . . . . . . . . . . . . . . . 15 Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Parabolic Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Disk Adjuster Mounts . . . . . . . . . . . . . . . . . . . . . . . . 18 Carts and Rails . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Plane Mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Temperature Sensors . . . . . . . . . . . . . . . . . . . . . . . . . 20. 3.2. Description of Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. 3.2.2. Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Port and Power Meter . . . . . . . . . . . . . . . . . . . . . . . . 20 Power Splitter and S-parameter File . . . . . . . . . . . . . . . . 21 Air Gaps, Disks and Mirror . . . . . . . . . . . . . . . . . . . . . 22.

(6) vi 3.3. 3.4. 3.5. 4. Data Acquisition and Processing . . . . . . . . . . . . . . . . . . . . . . 24 3.3.1. Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 24. 3.3.2. Frequency and Time Gate . . . . . . . . . . . . . . . . . . . . . . 24. 3.3.3. Antenna Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 25. 3.3.4. Measuring Unwanted Reflections . . . . . . . . . . . . . . . . . 27. Fitting Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.4.1. Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 30. 3.4.2. Nelder-Mead Algorithm . . . . . . . . . . . . . . . . . . . . . . . 31. Systematics/Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.5.1. Thermal Response and Stability . . . . . . . . . . . . . . . . . . 31. 3.5.2. Positioning and Hysteresis . . . . . . . . . . . . . . . . . . . . . 32. 3.5.3. Unwanted Reflections . . . . . . . . . . . . . . . . . . . . . . . . 32. 3.5.4. Reproducing EM properties . . . . . . . . . . . . . . . . . . . . . 32. Results 4.1. 33. Thermal Response and Stability . . . . . . . . . . . . . . . . . . . . . . . 33 4.1.1. Thermal Response . . . . . . . . . . . . . . . . . . . . . . . . . . 33. 4.1.2. Short Term Stability . . . . . . . . . . . . . . . . . . . . . . . . . 35. 4.2. Motor Precision and Hysteresis . . . . . . . . . . . . . . . . . . . . . . . 37. 4.3. Unwanted Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3.1. Identification and Suppression . . . . . . . . . . . . . . . . . . . 41. 4.3.2. Measuring Response . . . . . . . . . . . . . . . . . . . . . . . . . 42. 4.3.3. Verification of the Model with Distance Tests . . . . . . . . . . . 46 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Reproducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Distance Test for up to four Sapphire Disks . . . . . . . . . . . . 49 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56. 4.4. 4.3.4. Static Test of Model . . . . . . . . . . . . . . . . . . . . . . . . . . 58. 4.3.5. Effect on spacing and BF . . . . . . . . . . . . . . . . . . . . . . . 59. 4.3.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62. Reproducing Electromagnetic Properties . . . . . . . . . . . . . . . . . . 64 4.4.1. Standard Case for 1 to 5 Disks . . . . . . . . . . . . . . . . . . . 64 Spacings Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 64 Power Boost Factor Uncertainty . . . . . . . . . . . . . . . . . . 69 Model Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . 71. 5. 4.4.2. Going to Higher Disk Number . . . . . . . . . . . . . . . . . . . 77. 4.4.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78. Conclusion. 79.

(7) 1. Chapter 1. Introduction 1.1 1.1.1. Motivation for Axions The Strong CP Problem. The Axion is a hypothetical particle originally conceived to solve the Strong CP problem, the inconvenient fact that the strong interaction appears not to violate chargeparity (CP) symmetry although naively it should. The Quantum chromodynamics (QCD) Lagrangian contains a term. LQCD ∝ −θ. αs a µν G G̃a 8π µν. (1.1) µν. a and its dual G̃ , the strong coupling with the gluon field strength tensor Gµν a. constant αs and the angle θ [1]. Note that in the introduction natural units are used, e.g. h̄ = c = 1. This term is in principle CP violating as long as θ is non-zero [1]. One example where such CP violation in the strong sector would manifest itself is the electric dipole moment (EDM) of the neutron [2]. However, the neutron EDM must be extremely small - if it exists at all - as current experimental limits put it at below 3.0 × 10−26 e cm [3]. This in turn limits θ to |θ | < 10−10 [2]. This is deemed unnatural as θ could have any value between −π and π but turns out to be consistent with zero. A popular approach to solve this problem is the Peccei-Quinn theory which provides a mechanism that drives θ towards zero without any fine-tuning. By introducing a new global U (1) PQ symmetry the static CP-violating angle θ is promoted to a dynamical CP-conserving field θ ( x ). The Peccei-Quinn-symmetry (PQ-symmetry) U (1) PQ is spontaneously broken at an energy scale f a . Excitations of θ ( x ) represent a new particle - the axion [4]. Additionally the vacuum energy density VQCD of QCD depends on θ and is minimized for θ ( x ) = 0 [5]. Thus the field θ ( x ) feels an effective potential and naturally becomes zero. This potential gives the axion an effective mass. It depends on f a and can be expressed as [6]: m a ≈ 6 eV. 106 GeV fa. (1.2).

(8) 2. Chapter 1. Introduction Originally f a was thought to be of the order of f EW , the electroweak energy scale. [4]. With Eq. (1.2) the axion mass would be O(10 keV) which was quickly excluded by experiment. Other models put f a f EW . These are called "invisible axion" models which comes apparent when looking at the axion coupling to photons. The axion can couple to two photons via kinetic mixing with η0 and π0 resulting in the effective Lagrangian [6]:. L aγγ =. a α Caγ E · B 2π fa. (1.3). with fine-structure constant α, axion field a and electric and magnetic fields E and B. Caγ is a model dependent parameter of order unity. A large f a means a small, "invisible" coupling. The two most important "invisible axion" models are the Kim, Shifman, Vainshtein, Zahkarov (KSVZ) [7, 8] model and the Dine, Fischler, Srednicki, Zhitnisky (DFSZ) [9, 10] model. For the scope of this thesis it suffices to point out that Caγ = −0.97 for KSVZ and Caγ = 0.36 for DFSZ. Defining a new coupling constant gaγγ that includes Caγ and f a cleans up Eq. (1.3) gaγγ =. Caγ α 2π f a. L aγγ = gaγγ aE · B. (1.4). (1.5). The coupling to photons can lead to conversion of axions into photons or vice versa in the presence of a static magnetic field, the so called Primakoff-effect [11]. Interestingly the coupling constant gaγγ again depends on f a so that the axion mass is directly proportional to its coupling strength. In addition to the axion described above, the so called QCD axion, one can arrive at axion-like particles (ALP) by loosening the relation between mass and coupling strength. ALPs can have masses and coupling strengths independent from each other but do not necessarily solve the strong CP problem. See [12] for a review.. 1.1.2. Dark Matter. Dark matter is a hypothetical type of matter that is thought to make up 85 percent of all matter in the universe [13]. Its existence is implied by a number of astrophysical and cosmological observations. One of the first hints was given by observations of galaxy rotation curves which, assuming Newtonian dynamics, cannot be explained by just the ordinary matter contained in galaxies as the mass of the visible matter like stars is much too low to keep galaxies together spinning at the observed velocities [14]. Other effects like gravitational lensing also imply that galaxies are much more massive than visible matter alone could explain [15]. Finally, precise measurements of the Cosmic Microwave Background are best fitted by models containing a significant amount of dark matter [13]. So far it is unknown what dark matter consists of other than that it must have mass and must be weakly-interacting. Nevertheless.

(9) 1.1. Motivation for Axions. 3. it is generally accepted by the scientific community. Alternative ideas most notably modify gravity at different scales (MOND, TeVeS) and are successful in solving some problems like galaxy rotation curves but struggle at others, like mass distributions observed for the famous bullet cluster collision of two galaxy clusters [16]. As a standard in cosmology, the ΛCDM model, which includes a cosmological constant Λ as dark energy and cold (as in non-relativistic) dark matter (CDM), puts the total dark matter density at Ωc h2 ≈ 0.12 coming from a fit to the cosmic microwave background [13]. Axions are bosons that have zero electric charge and interact very weakly. They are therefore a suitable candidate for dark matter.. 1.1.3. Axion Cosmology. One has to consider three major events in the early universe to understand the dynamics of axion dark matter. Inflation, the PQ symmetry breaking and the QCD phase transition. Inflation is a postulated epoch of exponential expansion of space in the very early universe. It is well supported as an explanation for the isotropy and flatness of our universe, the homogeneity of the Cosmic Microwave Background and the absence of magnetic monopoles [17]. QCD phase transition describes the transition from a deconfined quark-gluon plasma at higher temperatures to the familiar hadronic phase of confined quarks and gluons at lower temperatures [18]. Before the QCD phase transition, at energy scales above f QCD ≈ 150 MeV, strong interactions are pertubative and VQCD is extremely suppressed [1]. Now with f a f QCD this means that when axions first arose they felt no effective potential, are massless and any value of θ between −π and π is possible [19]. Consequently, after the PQ phase transition causally disconnected regions of spacetime can have a different initial misalignment angle θi . There are now two different scenarios depending on the order of PQ phase transition and inflation. In Scenario A inflation happens after the PQ phase transition. A small patch of homogeneous θi is expanded beyond the size of our current observable universe. The initial misalignment angle θi is unique for our universe and determines the axion mass. This way - assuming a suitable value for θi - axions could be the entirety of cold dark matter with any value of f a ∈ ( M p , 109 GeV) [20]. The axion mass can range from neV to a few µeV [21]. In Scenario B inflation happened before the PQ phase transition. Our observable universe then consists of many patches with random θi . One has to perform a statistical average of these patches to obtain an axion mass. Furthermore, one now has to consider topological defects coming from boundaries between different θi that can significantly contribute to the axion mass. Latest theoretical predictions put the axion mass at 25 − 500µeV in this scenario [22]..

(10) 4. Chapter 1. Introduction. 1.1.4. Current Limits and Experiments. An overview of current limits can be seen in Fig. (1.1). So far a lot of the expected parameter space for scenario A and B remains untouched. Excluding parameter space for the QCD axion requires to reach down to the "QCD axion line" as mass and coupling strength are proportional to each other (see Eq. (1.2) and (1.4)). Axions with m a > 30 meV are ruled out mostly by astrophysical observations [23]. As axions interact weakly, they would serve as a mechanism for energy loss in stars, white dwarfs and supernovae. "Heavy" axions would cool these astrophysical objects by an observable amount and are thus excluded. Dedicated experiments search for lower mass axions. A selected few are shortly presented here. The CERN Axion Solar Telescope (CAST) is a helioscope looking at the sun to detect possible axions that are created by photon-to-axion conversion in the stellar plasma. It uses a 9 T re-purposed LHC test magnet to convert these solar axions back to X-ray photons that can then be detected. It has excluded gaγγ > 0.66 × 10−10 GeV−1 for m a < 20 meV [24]. Another type of axion experiments are haloscopes. They rely on the conversion of axions in the local dark matter halo of the Milky Way to photons. The Axion Dark Matter Experiment (ADMX) uses a strong magnetic field in a cavity to resonantly convert axions to photons. They have been the first to exclude a part of the KSVZ parameter space (1.86 < m a < 3.36 µeV) [25] and recently even reached sensitivity to DFSZ couplings in a small mass range (2.66 < m a < 2.81 µeV) [26]. The HAYSTAC (Haloscope At Yale Sensitive To Axion Cold dark matter) experiment is another cavity based experiment that has excluded gaγγ & 2 × 10−14 GeV−1 (about a factor 2.7 above the benchmark KSVZ model) for 23.15 < m a < 24.0 µeV [27]. The axion community is still a growing field and listing all present and future experiments is beyond the scope of this thesis. A review can be found in [28]. Suffice to say that these are indeed exciting times for axion physics.. 1.2. Magnetized Disk and Mirror Experiment. The Magnetized Disk and Mirror Experiment (MADMAX) is a new proposed experiment to search for axions. It will probe the mass range from 40 µeV to 400 µeV while being sensitive to powers as low as 10−23 W, reaching DFSZ axion sensitivity. This mass range has not been covered well by other experiments. This section is based on [29] in which a detailed description of MADMAX and its working principle can be found. Axions can be converted to photons via the Primakoff-effect with corresponding frequency. A higher axion mass is equivalent to a higher photon frequency. The line width of the converted photons is ∼ 10−6 [21] times smaller than their frequency as.

(11) 1.2. Magnetized Disk and Mirror Experiment. 5. Log10 ωALP [Hz] 5. 10. 15. 20. 25. Υ(1S) ASP. PVLAS. LEP 2/3γ LHC. am. Be Du. -5. p. m. LSW [GeV- 1 ]. SN1987a DAMA. CAST. FERMI HESS. - 10. Global Sun Telescopes. SN γ- burst. xion. UF/RBF ADMX. Log10 - 15. HB. BBN CMB MADMAX. EBL ion ax. SZ. DF. VZ. ys. KS. Ra. ax io. n. X-. - 10. -5. 0. 5. 10. Log10 mALP [eV] Figure 1.1: Current axion and axion-like landscape as di-photon coupling constant gaγγ over axion mass m a . The currently untouched parameter space is white. Higher axion masses are excluded by astrophysical and cosmological constrains. The benchmark models KSVZ and DFSZ are plotted as solid lines. The prospected parameter range of MADMAX is shown as the dotted line. Adapted from [21]..

(12) 6. Chapter 1. Introduction. cold dark matter axions are relatively slow (v a ∼ 10−3 ). Cavity based experiments struggle with the fact that in order to go to higher frequencies one needs to build smaller cavities to still operate the cavity at its lowest resonant mode with a high quality factor. However, the power output of cavities scales with the volume, limiting cavities to mass ranges below ∼ 20 µeV. MADMAX uses a new approach to enhance axion to photon conversion. By placing dielectric disks into a strong magnetic field the axion field induces electromagnetic emissions from the disks that can resonate and constructively interfere.. 1.2.1. Working Principle. The slow cold dark matter axions have a large de Broglie wavelength (O(10 m) for m a = 100 µeV). Additionally axions must have a high number density to still produce the expected dark matter mass density with such a small axion mass. Due to these two properties it is sensible to treat the axion field as a classical field and add the axion-photon coupling to the classical electromagnetic Lagrangian. One then obtains modified Maxwell equations with an additional source term in the AmpereMaxwell law in a current-free dielectric medium with dielectric constant e:. ∇ × B − e Ė = gaγγ B ȧ. (1.6). Providing a static homogeneous external magnetic field Be , the axion field a induces an electric field: Ea = −. gaγγ Be a(t) e. (1.7). Its field strength depends on the dielectric constant e. Taking the limit v a → 0 yields a spatially static axion field: a(t) = a0 cos(m a t). (1.8). It oscillates with an angular frequency of m a . Defining E0 = gaγγ Be a0. (1.9). one gets: Ea = −. E0 cos(m a t) e. (1.10). If one now has two different dielectric media 1 and 2, there will be a discontinuity of E a at the boundary between the media. However, this is not permitted by the (modified) Maxwell equations. The overall electric and magnetic field components parallel to the dielectric boundary E|| and H|| must be continuous. Demanding E1,|| = E2,|| and H1,|| = H2,|| for the electromagnetic fields in medium 1 and 2 yields an emission perpendicular to the boundary to compensate the discontinuity of E a ..

(13) 1.2. Magnetized Disk and Mirror Experiment. 7. The amplitudes of this emission into both media with refractive index n are: E1 = ( E2a − E1a ) γ. e2 n 1 e1 n 2 + e2 n 1. E2 = −( E2a − E1a ) γ. e1 n 2 e1 n 2 + e2 n 1. H1,2 = −( E2a − E1a ) γ. (1.11). (1.12). e1 e2 e1 n 2 + e2 n 1. (1.13). This emission corresponds to a propagating electromagnetic wave which can in principle be detected. Maximizing the discontinuity by having one medium be a perfect mirror (e1 = ∞) and the other vacuum (e2 = 1), one gets E1 = H1,2 = 0 γ. γ. γ. and E2 = E0 where E0 = | E0 |. The power per area of a single mirror axion-induced emission is: E2 P0 W = 0 = 2.2 × 10−27 2 A 2 m. . B2 10T. . 2 Caγ. (1.14). Because this is very hard to detect with current technology, a power boost is needed. The idea of MADMAX is to use not just a mirror but many dielectric disks as well. Each dielectric disk again emits according to Eq. (1.11). As the de Broglie wavelength of the axion, λ a ∼ 10 m, is much bigger than the entire setup, each emission of each disk is coherent and can constructively interfere with one another. Additionally, emissions can be reflected back and forth between the disks acting as a resonator. Having a mirror at one end ensures that all emissions eventually end up propagating toward the receiver at the other end where they deposit a measurable power. The exact power boost is frequency-dependent and very sensitive to the arrangement of the disks. The amplitude boost factor is defined to be: β(ν) =. Eout (ν) E0. (1.15). The total emitted power is then: P = β2 (ν) P0. (1.16). β2 (ν) is called the power boost factor. First developments on a receiver system demonstrated the ability to detect P = 1.2 × 10−22 W over an integration time of 28 h [30]. This means that a power boost factor of β2 (ν) ≥ 104 is needed. To achieve that, the MADMAX collaboration plans to have 80 dielectric disks made of a highepsilon material (e ≈ 24 for lanthanum aluminate) with an area of ∼ 1 m2 . Motors are used to precisely control the disk arrangement inside a cryogenic vessel. A 10 T magnet surrounds the vessel. The receiver system will be cryogenic as well..

(14) 8. Chapter 1. Introduction. Figure 1.2: Boostfactor, reflectivity and group delay of 20 dielectric disks. The measurable reflectivity and group delay are related to the boost factor which cannot be measured directly. Taken from [29].. 1.2.2. Proof of Principle Booster Setup. A direct measurement of β(ν) is impossible as it would require an actual axion signal detection to do so. It is, however, extremely important to precisely know β(ν) in order to assure the mass (frequency) range in which the experiment is sensitive and to either exclude or confirm the mass and coupling strength of the axion or axion like particles. Fortunately one can measure related quantities like reflectivity. Fig. 1.2 shows the boost factor, reflectivity and group delay of 20 dielectric disks. One can then simulate β(ν) and match the reflectivity of the setup to that of the simulation. To understand the correspondence between reflectivity and boost factor, several small scale prototypes have been built, the latest and biggest being the 20 disk Proof of Principle Booster Setup. It can accommodate up to 20 disks with a diameter of 20 mm made of sapphire, a material with a high relative permittivity (e = 9.3 perpendicular, e = 11.5 parallel to crystal axis). It lacks a magnet as the main goal is to study the electromagnetic response and not to convert axions into photons. A vector network analyzer (VNA) and antenna are used to input microwaves and measure the response. Motors move the disks with µm precision to replicate the simulated electromagnetic response of the system. A more detailed description of the setup and its components can be found in the Experiment and Methods section. This Master thesis presents major measurements performed with the Proof of Principle Booster Setup contributing in particular in understanding effects of unwanted reflections and other systematics..

(15) 9. Chapter 2. Theory 2.1 2.1.1. Booster Physics Transfer Matrix Formalism. A Transfer Matrix Formalism is used to analyze electromagnetic propagation through layered media. A modified formalism that includes axion-induced emissions has been developed and a more detailed explanation can be found in [29]. It is used to calculate the electromagnetic response of a disk and mirror system as well as its boost factor. The boost factor β(ν) is due to the coherent sum of emissions by each disk as well as resonances between coherently emitted waves. Each dielectric region r with dielectric constant er contains a superposition of the axion induced electric field (assuming a homogeneous magnetic field) Era = −. E0 = − Ar E0 , er. (2.1). and the left and right propagating electromagnetic waves. with nr =. √. ErR = Rr e+iωnr ∆x. and. ErL = Lr e−iωnr ∆x. and. er Rr e+iωnr ∆x nr er HrR = Lr e−iωnr ∆x nr HrR =. (2.2) (2.3). er as we assume non-permeable materials with µr = 1.. ∆ x = x − xr measures the position x relative to the left boundary of every region r (see the schematic in Fig. 2.1). Demanding continuity between each dielectric boundary relates the electric and magnetic fields of region r to that of region r + 1 which reads as:. − E0 Ar + Rr eiδr + Lr e−iδr = − E0 Ar+1 + Rr+1 + Lr+1. (2.4a). er iδr er + 1 Rr e − Lr e−iδr = ( R r +1 − L r +1 ) nr n r +1. (2.4b).

(16) 10. Chapter 2. Theory with optical thickness δr = ωnr ( xr+1 − xr ). These continuity conditions can be. expressed in a transfer matrix formalism:   1  = Gr Pr   + E0 Sr    1 Lr L r +1. (2.5).   1  n r +1 + n r n r +1 − n r  Gr = 2nr+1 nr+1 − nr nr+1 + nr. (2.6a). . R r +1. . . Rr. . with.  Pr = . . e+iδi. 0. 0. e−iδr. . Sr =. (2.6b).  . A r +1 − A r  1 0  2 0 1. (2.6c). Gr is the regular transfer matrix between region r and r + 1. Pr describes the phase acquired by propagating through region r. Sr contains the axion-induces electromagnetic waves produced at each boundary between r and r + 1. Multiple layers are then described by iterating Eq. (2.5) from region 0 to region m:   1 R R    m  = T0m  0  + E0 ∑ Tm s Ss −1 1 L0 Lm s =1. (2.7). Tba = Ga−1 Pa−1 Ga−2 Pa−2 · · · Gb+1 Pb+1 Gb Pb. (2.8). . . . . m. with for b < a. Taa = 1 and P0 = 1. To further compactify one defines m. ∑ Tms Ss−1. (2.9).   1   = T   + E0 M   Lm L0 1. (2.10). T = T0m. and. M=. s =1. . Rm. . . R0. . The first part of Eq. (2.10) is the usual transfer matrix for in and outgoing electromagnetic waves while the second part is the sum of all axion-induced emissions. The quantities of interest for this master thesis are the reflectivity of a disk-andmirror system as well as its boost factor. Region 0 is to be a perfect mirror thus. |n0 | = ∞. The incoming wave Lm is that of the vector network analyzer which in practice is much larger than any axion-induced contributions (besides there is no magnetic field in the Proof of Principle Booster Setup anyway) so E0 = 0 is a valid.

(17) 2.1. Booster Physics. 11. Figure 2.1: Each dielectric region r = 0, 1, ..., m contains a right and left going wave, Rr and Lr . At each boundary between regions the continuity requirement demands Eq. 2.4. The transfer matrix formalism solves this iteratively and yields the outgoing amplitudes Rm and L0 . Setting e = ∞ in region 0 realizes the mirror closing off one end.. assumption. The amplitude reflection coefficient is defined as the ratio between outgoing (Rm ) and incoming (Lm ) electromagnetic wave amplitude which by solving Eq. (2.10) is:. R=. T[1, 2] Rm | R0 =0 = Lm T[2, 2]. (2.11). The square brackets denote the matrix indices. It is a frequency dependent complex quantity. The phase of R is particularly interesting as it is largely independent of attenuation and allows for much more precise measurements. A convenient way to represent the phase Φ(ω ) of R is group delay because it does not depend on the absolute distance between antenna and booster: τg = −. dΦ(ω ) dω. (2.12). For calculating the boost factor we assume no incoming electromagnetic waves, thus R0 = Lm = 0. The boost amplitude is the amplitude of outgoing electromagnetic wave, completely sourced by axion-induced emissions, divided by the amplitude of the emission by a single mirror E0 : B=. Rm = M[1, 1] + M[1, 2] E0. (2.13). B still contains phase information which mostly is of no interest. The amplitude and power boost factor are then: β(ν) = | B|. (2.14). β2 ( ν ) = | B |2. (2.15).

(18) 12. Chapter 2. Theory. 2.1.2. Simulating the Booster with Electrical Circuits. Scattering parameters (S-parameters) are used to describe n-port networks. Each port i has an incident wave amplitude ai and reflected wave amplitude bi . S-parameters describe the relation between incident and reflected amplitudes:       b1 S11 . . . S1n a      1 ..   ..   ..   .. .. .= . . . · .        bn Sn1 . . . Snn an. (2.16). The disks and mirror setup represents a 1-port network as a perfect mirror does not allow transmission. The scattering parameter S11 is then simply the reflection coefficient R. S11 = R =. b1 a1. (2.17). Wave impedance is the ratio of the transverse components of the electric and magnetic field. For transverse electric-magnetic (TEM) plane waves in a perfect dielectric medium with relative permittivity e and relative permeability µ the wave impedance Z is: r Z=. µ0 µ e0 e. (2.18). A lossless transmission line with capacitance C and inductance L has the characteristic impedance: r Zc =. L C. (2.19). Propagation of current I and voltage V are described by the Telegrapher’s equation which takes the form of a wave equation: ∂2 V 1 ∂2 V − =0 ∂t2 LC ∂x2. (2.20a). ∂2 I 1 ∂2 I − =0 ∂t2 LC ∂x2. (2.20b). This is equivalent to the electromagnetic wave equation in one dimension with the electric and magnetic field being replaced by voltage and current and permittivity and permeability replaced by capacitance and inductance. The electromagnetic response of the booster in one dimension can therefore be modeled by an electrical circuit. Solving the transfer matrix formalism (Eq. (2.10)) for m dielectric regions breaks down to solving 2m linear boundary conditions. By using transmission lines to simulate the dielectric regions these boundary conditions can efficiently be solved by integrated circuit simulation software like QUCS (Quite Universal Circuit Simulation) [31] or ADS (Advanced Design System) [32]..

(19) 13. Chapter 3. Experiment and Methods 3.1 3.1.1. Description of Proof of Principle Booster Setup Overview. The Proof of Principle Booster Setup is a small scale prototype to test the electromagnetic response of up to 20 dielectric disks and mirror with 20 cm diameter. To this end a Vector Network Analyzer (VNA) measures the reflectivity of the booster by generating a signal between 10 and 30 GHz which is sent out by a horn antenna. A parabolic mirror redirects and collimates the beam into the booster. The idea is to have the beam resemble a plane wave as much as possible. The booster consists of movable disks, a fixed plane mirror and the surrounding mechanics. An aluminium frame holds ten rails with 20 carts on which the dielectric sapphire disks are moved by high-precision motors. The frame has a length of 800 mm and a width and height of 500 mm. It sits on an optical table with aircushioned legs to reduce vibrations. The fixed plane copper mirror is placed midway into the frame with the disks in front of the plane mirror. The plane mirror and sapphire disks reflect the incoming beam, which is then refocused by the parabolic mirror onto the antenna which collects the signal to be analyzed by the VNA. By comparing the outgoing and incoming signals at different frequencies the VNA is able to measure the amplitude and phase of the frequency-dependent reflectivity. The antenna and surroundings can also reflect part of the returning signal back into the booster before reaching the VNA. These so called unwanted reflections have a significant effect on the reflectivity which will be studied later on. See Fig. 3.1 for a schematic and Fig. 3.2. for an image of the setup. Each component will be introduced and characterized separately in the following subsections.. Figure 3.1: Schematic of the setup..

(20) 14. Chapter 3. Experiment and Methods. (a) Whole Setup. (b) Disks and mirror Figure 3.2: (a) Overview of the whole setup. The VNA emits a signal towards the antenna. The antenna is in the focus of the parabolic mirror so that the beam is collimated into the booster. The booster is inside the aluminium frame and holds up to 20 disks and a plane copper mirror. In this case the disks are removed to better see the copper mirror. The booster reflects the collimated beam which is then refocused onto the antenna and analyzed by the VNA. (b) Side view of the sapphire disks and copper mirror. The disks can be moved on rails to desired positions by high-precision motors..

(21) 3.1. Description of Proof of Principle Booster Setup. 3.1.2. 15. Components. Vector Network Analyzer A VNA can measure the S-Parameters in both amplitude and phase of a device under test as a function of frequency by sweeping over a set frequency range. For a detailed description of the workings of a VNA see [33]. The model used is the MS4647B from Anritsu [34]. Its specifications are listed in Tab. 3.1. It has two ports, however, all measurements presented in this master thesis are reflectivity measurements using just one port. The reflectivity measurement is sampled at 6.67 MHz intervals resulting in 3001 data points per measurement. The port of the VNA is connected to the antenna by a microwave coaxial cable. An adapter to connect the cable to the antenna is needed. The VNA needs to be calibrated in order to remove the effects of the coaxial cable and adapter. An automatic calibration can be done with the Anritsu 36585V AutoCal module [35]. Through a series of measurements it determines the S-Parameters of the elements connected to the module so that they can be calibrated out. The adapters to connect to the AutoCal module and the antenna are slightly different in length which can be accounted for with the de-embedding tool of the VNA. frequency range [GHz] max. frequency resolution [Hz] power resolution [dB] phase stability [deg /◦C]. 0.01-70 1 0.01 < 0.5. Table 3.1: VNA specifications. Antenna The antenna is a pyramidal horn antenna from A-Info. The manufacturer’s specifications are listed in Tab. 3.2 [36]. Horn antennas are known for their broad bandwidth, moderate directivity (see Fig. 3.3 for an antenna pattern) and low voltage standing wave ratio (VSWR). By gradually increasing the size of the metal waveguide its wave impedance is continuously matched to that of free space (about 377 Ohms). This is important as any sudden change in wave impedance causes reflections and decreases the efficiency of the antenna. A perfectly matched antenna would need to be infinitely long which is obviously not the case. The specified VSWR result in about 1.2% of the incoming power being reflected. This is high enough to cause significant effects as can be seen later on..

(22) 16. Chapter 3. Experiment and Methods. length [mm] aperture [mm × mm] frequency range [GHz] VSWR forward gain [dB] 3dB Beamwidth [deg] polarization. 271 105 × 90 18-26.5 1.25 25 10 linear. Table 3.2: Antenna specifications. Figure 3.3: Antenna pattern at 20 GHz. The main lobe in the center and its suppressed side lobes can be seen. Taken from [36]. Parabolic Mirror In the ray approximation a point source in the focus of a parabolic mirror results in an outgoing collimated beam, that means an outgoing plane wave. Since the antenna is not an ideal point source and the ray approximation does not hold for microwaves in the dimensions involved, the outgoing beam is not a perfect plane wave. It is rather an imperfect Gaussian beam. However, as later results show, it is close enough to plane waves that the electromagnetic responses as predicted by the 1D model can be produced. The parabolic mirror is therefore a crucial part of the setup. In the proof of principle setup two parabolic mirrors were used so far. They are both off-axis parabolic mirrors with an effective focal length long enough to accommodate the antenna in the focal point. The effective focal point is twice the parent focal point of the parabola. The first one was a commercially available product. The reflective surface is plated in gold to increase reflectivity. It was later realized that the beam width is bigger than the small parabolic mirror. A new bigger version was then custom made by the MPP workshop. It is made from aluminium and has no plating. This new mirror increased signal strength by about 50 % as it captures more of the beam.. diameter [mm] effective focal length [mm] off-set angle [deg]. gold plated. aluminium. 101.6 152.4 90. 250 152.4 90. Table 3.3: Parabolic mirror specifications.

(23) 3.1. Description of Proof of Principle Booster Setup. 17. Figure 3.4: Aluminium parabolic mirror with antenna.. Disks The heart of the setup are the sapphire disks. The specifications are given in Tab. 3.4. Sapphire (Al2 O3 ) single crystals are chosen for their high dielectric constant and its commercial availability. Its dielectric constant depends on the orientation of the crystal lattice. The dielectric constant perpendicular to the plane of the disks is e = 9.3. The disk diameter must be big enough to warrant a 1D description of the booster, that means the Gaussian beam should in principle have negligible intensity at the edge of the disk. The variation in thickness was determined by measuring the disks with an ultrasonic device. Any big variation in thickness would be a deviation from the 1D description.. diameter [mm] thickness [mm] thickness variation [µm] e (at 25 ◦C, 10 GHz) tan(δ). 200 1 ±10 9.3 ∼ 10−4. Table 3.4: Disk specifications. Figure 3.5: Sapphire disk with holding structure.

(24) 18. Chapter 3. Experiment and Methods. Disk Adjuster Mounts The disks are installed on kinematic platform mounts that sit on carts. This allows to adjust the angular orientation of the disks. The mounts are the KM100B/M from Thorlabs [37]. Two knobs control tip and tilt to a resolution ∼ 0.01◦ . Springs hold the movable platform in place. They had to be adjusted to carry the weight of the disks by stretching them with washers and thus increasing the spring force. Tilting the disks has a huge influence on the electromagnetic response of the booster and a precise adjustment is needed. Carts and Rails The ten rails allow the disks to move on carts driven by a motor. Each rail holds two carts and motors. The two motors are oriented opposite to each other. They can be moved along the rails and fixed at a desired position on the rail. The rail model is the MN 42-790-G1-V1 Minirail from Schneeberger [38]. The dimensions and running accuracy are in Tab. 3.5. The good accuracy is needed to precisely move the disks to desired positions. To have the rails as parallel as possible after mounting, their orientation was measured with a mechanical measurement arm and adjusted. The slopes relative to the reference rail O vary between 70 µm m−1 and 700 µm m−1 [39]. The carts are the MNN 42 from Schneeberger. They slide along the rails with ball bearings as a contact. length [mm] width [mm] running accuracy [µm m−1 ]. 790 42 <6. Table 3.5: Rail specifications. Figure 3.6: Front view of the booster with annotated rails. Taken from [39]..

(25) 3.1. Description of Proof of Principle Booster Setup. 19. Motors The motors have to position the disks with micrometer precision. The used model is the MP-20 DC HLS (succeeded by the L-220 with same specifications) from Physical Instruments [40]. Specifications are listed in Tab. 3.6. They are mounted on separate sledges that can be fixed at desired positions on the rails. Their axle is connected to the carts with a bolt and nut. They are linear DC motors that have micrometer reproducibility in both directions. This again is needed to precisely position the disks. They are controlled by the C-885 master controller module that consists of 10 C-863 controller modules that can each control two motors. step size [nm] range [mm] bidirectional repeatability [µm]. 100 77 ±1. Table 3.6: Motor specifications. Figure 3.7: Motor installed on rail.. Plane Mirror The plane mirror is essentially a thick copper plate. See Tab. 3.7 for specifications. Like for the disks the diameter has to be large enough to accommodate the beam. Copper has a high reflectivity in the microwave range and is therefore a suitable mirror material. A mirror that is too thin will experience loss through the skin effect and might bend under mechanical stresses. It is mounted on a holding structure with adjustable angle. Its position was adjusted to be parallel to the frame front face using a mechanical measurement arm when the booster was assembled the first time. diameter [mm] thickness [mm]. 200 9. Table 3.7: Plane mirror specifications.

(26) 20. Chapter 3. Experiment and Methods. Temperature Sensors Changes in temperature cause the setup to expand or contract and change the positions of the disks. PT100 temperature sensors are used to monitor room temperature at various locations. Currently three sensors are in operation: One taped at the frame of the setup, another taped at the back of the plane mirror and the third suspended above the disks directly exposed to air in order to have independent measurements at key positions possibly subject to thermal expansions.. 3.2 3.2.1. Description of Model Overview. Modelling the electromagnetic response of the Proof of Principle Booster Setup is paramount to reproduce a desired electromagnetic response with the physical setup. It is the only way to infer the boost factor of the physical setup since it cannot be measured but only calculated. The model for simulating S-parameters and boost factor of the Proof of Principle Booster Setup is realized with an integrated circuit simulator. The first software used was the open-source Quite Universal Circuit Simulator (QUCS). Later the switch was made to Advanced Design Systems (ADS) from Keysight Technologies. ADS is in many regards the commercial counterpart of QUCS and offers more components (like power splitters) to work with. These components can be placed and connected with each other in a graphical user interface. To model a disk and mirror system, components that replicate the desired physical properties of disk, mirror and air are selected. Two types of simulation are done: A S-parameter simulation that simulates the reflectivity of the booster as a 1-port network and an AC simulation that uses voltage sources to mimic the axion-induced emission of disks and mirror to calculate a boost factor. The S-parameter simulation is equivalent to the reflectivity measurement of the VNA. An overview of the model can be seen in Fig. 3.8. In the following each component is introduced separately from right to left.. 3.2.2. Components. Port and Power Meter On the right end a port and a power meter element are placed. The port is the reference point for the S-parameter simulation. It is essentially the equivalent of the VNA in the model. The power meter element measures the power output during AC simulation. The power output is converted to a power boost factor by dividing the total power by that of a single mirror axion emission P0 (see Eq. (1.14)). S-parameters and power boost factor are written to a touchstone and csv file respectively..

(27) 3.2. Description of Model. 21. Figure 3.8: Model implemented in ADS. The port and power meter on the right measure Sparameters and power output respectively. The power splitter opens an alternative path for the signal so it can be reflected back into the system with a reflectivity defined by a S-parameter file that corresponds to unwanted reflections in setup. The mirror, air gaps and disks on the left constitute the booster and are all simulated by their own components. For simplicity, only two instead of 20 disks are shown.. Power Splitter and S-parameter File As can be seen later on, antenna and surrounding structures reflect part of the returning beam back into the booster. To simulate this in the model, a power splitter is placed to the left of port and power meter. It splits the signal coming back from the disk and mirror system. One part of the signal returns to the port and power meter as usual while the other propagates to a reflective element defined by a S-parameter file. The S-parameter file plays the role of the unwanted reflections back into the system from the antenna and possible other elements that have a frequency-dependent reflectivity. This has an important effect on the measured group delay. The idea is to measure the S-parameters of the unwanted reflective elements and give them to the model so it can simulate the effects. Details on how the S-parameters are obtained are in given section 3.3.4. The power splitter controls how much of the main signal is diverted towards the the S-parameter file element. It has three ports (see Fig. 3.8): Number 1 towards the booster, number 2 towards the port and number 3 towards the S-parameter file.. P P P S11 = S22 = S33 =0. (3.1a). P P S12 = S21. (3.1b). P P S13 = S31. (3.1c). P 2 P 2 (S12 ) + (S13 ) =1. (3.1d). The last line of Eq. 3.1 is to ensure passivity of the power splitter, e.g. no loss P and S P are set directly to the desired value. Usually so that it splits or gain. S13 12.

(28) 22. Chapter 3. Experiment and Methods. symmetrically as the S-parameters of the file element are scaled separately and control how strong the unwanted reflections are. When one does not want to model the P is set to zero and S P = 1. unwanted reflections S13 12. The air gap Air Refl. with length lr controls the position of the power splitter. The air gap Air Comp. compensates changes made to all other air gaps so that the total length of the simulation stays constant at a fixed ls . Air Gaps, Disks and Mirror On the left a series of air gap and disk elements are placed with a mirror element on the very left. They make up the booster. Mirror, disks and the air gaps are all simulated by using transmission lines with their respective dielectric constant. See Fig. 3.9 for schematics. The physical transmission line element allows setting its characteristic impedance in term of relative permittivity and permeability. It has a length l and a dielectric loss tangent tan(δ). The reference impedance of the simulation is Z0 = 120π Ω, which is a convenient approximation of the impedance of free space (Z0 = 119.916 9832 π Ω to be exact). The S-parameters of a transmission line with characteristic impedance Z and length l, connected at both ends to an impedance Z0 , can be calculated using the transfer matrix formalism with three dielectric regions with refractive indices n0 = n2 = 1 and n1 =. Z0 Z. so that T of (2.9) is T = G1 P0 G2. (3.2). and the resulting S-parameters are:. T T S11 = S22 =. r · (1 − p2 ) T[1, 2] = T[2, 2] 1 − r 2 · p2. (3.3a). T T S12 = S21 =. p · (1 − r 2 ) 1 = T[2, 2] 1 − r 2 · p2. (3.3b). with Z − Z0 Z + Z0 nl p = exp −iω c r=. (3.4a) (3.4b). One can rewrite the impedance Z of Eq. (2.18) as: Z0 Z= √ e with. r Z0 =. µ0 ≈ 120π Ω e0. (3.5). (3.6).

(29) 3.2. Description of Model. 23. The mirror is set to a very high e = 1020 thus practically Z = 0, leading to T = −1. This is equivalent to the procedure in the transfer matrix forRsMirror = S11. malism when |n0 | = ∞ for the leftmost dielectric region. Air gaps have e = 1 and the disks e = 9.3. The discontinuity of impedance between air gaps and disks not only reflects incoming electromagnetic waves but also causes the axion induced electric field emissions (see Eq. (1.11)). These can be modelled by voltage sources since voltages are equivalent to electric fields in the 1D model as explained in section 2.1.2. This is why mirror and disks have an additional voltage source next to the transmission line. Disks have them on both sides, the mirror only on one. They output a voltage Ua = ±. p. P0 Z0. 1 −1 e. (3.7). with P0 being the axion-induced power emission of a single mirror as of (1.14). It is set to P0 = 10−27 but since the boost factor is the fraction between total power emitted and P0 its value is irrelevant. The voltage Ua assumes that the disk element is connected to an air element with e = 1. The sign depends on the orientation of the discontinuity of impedance. That means that the voltage source on the right of a transmission line (+) has the opposite sign to the voltage source on the left (-). Disk elements can be disabled by setting e = 1. This leads to no axion induced T = 0 as of Eq. (3.3a). They then emission in Eq. (3.7) and no reflections since S11. behave just like an air gap element.. (a) Disk Element. (b) Mirror Element. (c) Air Gap Element. Figure 3.9: Disk, mirror and air gap element used in ADS. They are all modelled by a transmission line with corresponding length and dielectric constant. Disk and mirror have additional voltage sources to model the axion-induced emission..

(30) 24. Chapter 3. Experiment and Methods. 3.3. Data Acquisition and Processing. Reflectivity measurements from both model and setup are processed before further use. The raw signal in frequency space and its Fourier transformed counterpart in time domain can be seen in Fig. 3.10. They are filtered in frequency and time domain in order to remove unwanted elements in the signal. Frequency gating mainly removes the region below 15 GHz where the waveguide cutoff of the antenna reflects all of the outgoing signal right back to the VNA. Time gating removes early unwanted reflections arriving before the signal enters the booster system and late unwanted reflections caused by the antenna and surroundings. Gating in both domains requires transforming between them with a Fourier transform. To remove the transmission behaviour of the antenna an antenna calibration is done as well. At last, late unwanted reflections are measured in order to simulate them in the model. Each step will be described in detail in the following subsections.. 3.3.1. Fourier Transform. To transform between frequency and time domain a Fourier transform is used:. R(t) = F −1 { R(ω )} =. 1 2π. R(ω ) = F { R(t)} =. Z ∞ −∞ Z ∞ −∞. R(ω ) eiωt dω. (3.8). R(t) e−iωt dt. (3.9). Since the data is discrete, a fast Fourier transform (FFT) is used. The input data is a complex quantity in frequency domain and starts at 10 GHz. It is extended to 0 GHz by zero-padding and then converted to a symmetric doublesided spectrum by mirroring the complex conjugate of the single-sided spectrum onto the negative frequencies. This gives a hermitian function whose Fourier transform is always real.. 3.3.2. Frequency and Time Gate. Because the signal is transformed back and forth between frequency domain and time domain, a filter function that minimizes ringing artifacts is needed. The smooth filter function used is the Kaiser-Bessel window.. wa (t) =.  h √ i   I0 α 1−(t/τ )2. if |t| ≤ τ.  0. if |t| > τ. I0 [α]. (3.10). I0 is the zeroth Bessel function, τ the window length and α determines the shape. To filter the frequency domain the filter function is multiplied with the frequency domain reflectivity. R f iltered (ω ) = wa (ω ) · Run f iltered (ω ). (3.11).

(31) 3.3. Data Acquisition and Processing. 25. (a) Frequency domain. (b) Time domain. Figure 3.10: Raw frequency and time domain spectrum of just the mirror. A Fourier transform is used to transform between the two domains. In frequency domain one can see that the entire signal below 15 GHz is reflected by the waveguide cutoff. The time domain is smoothed and its absolute value is plotted. The main mirror peak is the dominant peak. The waveguide cutoff is the peak before the mirror peak. The peaks after the mirror peak are unwanted antenna reflections of ever increasing order.. To filter the time domain, the reflectivity is first Fourier transformed into the time domain, then multiplied by the filter function (with time t as argument) and finally Fourier transformed back into frequency domain. o n R f iltered (ω ) = F (wa (t) · F −1 Run f iltered (ω ). 3.3.3. (3.12). Antenna Calibration. The measurement from the VNA is a product of antenna transmission behavior, booster response and unwanted reflections. The antenna has a transmission coefficient that changes the signal when it passes through the antenna. The function that describes the change from input to output is called a transfer function. It can be measured once and then be applied to all future measurements with the antenna calibration. It is important to remove the antenna’s influence since the model assumes a perfect antenna and would otherwise deviate from uncalibrated measurements. Note that in the following the dependence on ω is implicit for reflectivity R and S-parameters S. With no disks present, the measured reflectivity of just the mirror with no back reflections is: 2iklm R0m = ηm S12,a Rm S21,a , mirror e. (3.13). with the antenna transmission coefficients S12,a = S21,a and reflectivity coefficient 2iklm is the phase factor accrued for the distance between Rm mirror of the mirror alone. e. antenna and mirror lm with wave number k =. ω c.. Some overall efficiency ηm is. assumed to account for losses between mirror and antenna. In the model, a perfect.

(32) 26. Chapter 3. Experiment and Methods. mirror and antenna are assumed (Rsmirror = −1 and S12,a = S21,a = 1) leading to: R0s = ηs (−1)e2ikls. (3.14). The modelled distance ls is not necessarily the same as in the measurement. Again some overall efficiency ηs can be applied to the model as well. By dividing reflectivity of the measurement by that of the model, one approximately gets the transfer function of the antenna: Hantenna =. R0m 2 2ik(lm −ls ) = S12,a Rm mirror e sR0s. (3.15). It also includes contributions of the physical mirror which must be close to perfect for the antenna calibration to work. The linear phase difference between measurement and model is irrelevant for magnitude and group delay. Scaling the reflectivity of the model by a factor s = ηm /ηs makes sure that overall efficiency of measurement and model are the same. Before installing any disks, an antenna calibration is done. In order to measure just the antenna transfer function folded with the mirror reflectivity, one has to cut out early and late reflections by applying a time gate on the total reflectivity Rm tot that just encompasses the main mirror peak (Eq. 3.12). The scaling factor s is found by comparing the mean reflectivity averaged over frequency of measurement and model. The transfer function is saved in a touchstone file and later applied to any measurement to remove the antenna transmission behavior. The calibrated measurement is then: Rm,calib =. Rm,raw Hantenna. (3.16). Dividing the measured mirror peak by e2ik(lm −ls ) effectively replaces phase factor of the measurement by that of the model: R0m,calib =. R0m Hantenna. = −ηm e2ikls. (3.17). The whole process chain for a measurement is: Rm,proc = F. . Rm,raw (ω ) ( w a ( t ) · F −1 w a ( ω ) Hantenna. (3.18). The model lacks the antenna calibration so that n o Rs,proc = F (wa (t) · F −1 {wa (ω ) Rs,raw (ω )}. (3.19). The final processed signal in frequency and time domain can be seen in Fig. 3.11..

(33) 3.3. Data Acquisition and Processing. (a) Frequency domain. 27. (b) Time domain. Figure 3.11: Frequency and time domain of processed signal. The time domain signal is smoothed and its absolute value is plotted. The frequency gate is 15 to 30 GHz and the time gate 3 to 14 ns, cutting out the waveguide cutoff reflection and including the first order of unwanted reflections. The antenna calibration ensures an overall flat magnitude in frequency domain and a well shaped mirror peak in time domain. The unwanted reflections cause the ripples on top the frequency domain magnitude.. 3.3.4. Measuring Unwanted Reflections. The reflectivity of the whole setup is a product of the reflectivity of the disk and mirror system and the reflectivity of the unwanted reflections. Reflections that happen before the signal enters the booster are ignored, as these can always be filtered out by a time gate. However, reflections that come after the signal leaves the disk and mirror system potentially overlap with the response of the disk and mirror system. Here one can divide the total reflectivity into different orders of unwanted reflections. The zeroth order is just the measured reflectivity of the disk and mirror system: The signal enters the booster, gets reflected and transmitted into the VNA to be measured. The first order is the part that is not transmitted into the VNA but instead is reflected with R R to enter the booster a second time before being transmitted into the VNA. Each time a small portion of the signal is not transmitted but reflected, creating the next higher order of unwanted reflections. To measure R R a reflectivity measurement Rm with just the mirror present is made. The zeroth order is then R0m described in Eq. 3.13. The distance between mirror and reflecting element is lm,r . Ignoring losses the first order is then: R1m = R0m R R e2iklm,r Rm mirror. (3.20). Since R0m and R1m are sufficiently disjoint in time domain, one can apply a time domain gate on Rm that just encompasses the reflection peak of first order to obtain R1m . An antenna calibration is made so that: R1m,calib = R0m,calib R R e2iklm,r Rm mirror. (3.21).

(34) 28. Chapter 3. Experiment and Methods. (a) Time domain with short gate. (b) Group delay with short gate. (c) Time domain with long gate reflections. (d) Group Delay with long gate. Figure 3.12: Group delay difference between including and excluding the first order antenna reflection with a time gate. Excluding the unwanted reflections yields a nicely shaped group delay while including causes ripples. As can clearly be seen, a model without unwanted reflections has no chance of fitting the group delay with ripples..

(35) 3.4. Fitting Procedure. 29. The obtained S-parameters of the unwanted reflection R1m have to be scaled in amplitude and shifted in time to fit into the model as the model has different overall efficiency and total distance than the measurement. This is achieved by scaling R1m with a generic α(k ) which depends on the wave number k and therefore on frequency. The important condition for α(k ) is that the relative amplitude and phase Φ between zeroth and first order reflection are the same in the model and setup in order to achieve a matching group delay: R0m,calib R1m,calib. =. R0s R1s. Φ( R0m,calib ) − Φ( R1m,calib ) = Φ( R0s ) − Φ( R1s ). (3.22). (3.23). One can express R1s simply by multiplying all S-parameters of elements in the model and scale it with α(k ): P 2 m R1s = α(k ) R0s e2ik(ls −lr ) (S13 ) R1. (3.24). P is the transmission coefficient of the power splitter towards the reflection S13. S-parameter file. Since it is passed back and forth, its transmission coefficient is squared. e2ik(ls −lr ) is the phase accrued by propagating the distance between mirror and S-parameter file which is the total length ls minus the length of the air gap Air Refl. lr . By inserting R1s from Eq. 3.24 and the calibrated R0m,calib from Eq. 3.17 into the conditions Eq. 3.22 and Eq. 3.23 one can solve for the scale factor α(k ) that satisfies the conditions:. |α(k)| =. 1 P )2 η (S13 m. Φ(α(k )) = −2ik (2ls − lr ). (3.25) (3.26). Writing α(k ) = |α|eikctc with time shift tc , the reflection S-parameters have to be scaled by |α| and shifted by tc = − 2c (2ls − lr ).. 3.4. Fitting Procedure. The boost factor can only be simulated, not measured. A correlated, measurable quantity like reflectivity is used to match the setup with the model. If the setup and model have the same reflectivity, one can infer the physical boost factor of the setup from the model. One can either fit the measurement to the model by adjusting the disk spacings in the setup or fit the model to the measurement by adjusting the disk spacings in the model. First a reference model or measurement is made. An algorithm then tries many different disk arrangements in either measurement or model until it finds the one that best fits the reference. To compare the reflectivity from measurement and model, they undergo the processing chain described in.

(36) 30. Chapter 3. Experiment and Methods. (a) Time domain. (b) Group delay with unwanted reflections. Figure 3.13: Implementation of first order unwanted reflections. The first order unwanted reflections are selected in time domain with a gate. They are transformed back into frequency domain and given to the S-parameter file in ADS. The resulting group delay of just a mirror and first order unwanted reflection is on the right.. section 3.3. The group delay τg of both is calculated. The chi-squared residual between measured and modellled group delay is the function that is optimized by the algorithm. It is: f max. χ2 =. ∑. . τgm (i ) − τgs (i ). 2. (3.27). i = f min. f min and f max are the minimum and maximum frequency of the group delay. Two algorithms were implemented to minimize χ2 .. 3.4.1. Genetic Algorithm. The first algorithm implemented is a simple genetic algorithm [41]. It randomly varies the disk spacings ds until a better one is found. It uses the current best disk spacing as a starting point for new variations. The procedure is: • generate start disk spacing ds by adding a random offset from a uniform dither with range ±v to an initial disk spacings. • perform a measurement/simulation and calculate χ2 . It is set as the current best. • repeat following steps until the dither range v is smaller than threshold: – generate a new random trial disk spacing dt from start disk spacing ds – perform a measurement/simulation and calculate χ2 – If χ2 is better than current best, set dt as the new start disk spacing ds . χ2 is set as the current best. If not keep old start disk spacing. – If no better χ2 can be found after a certain amount of trials N, the dither range is halved..

(37) 3.5. Systematics/Problems. 31. • dither range is sufficiently small and the best disk spacings are returned. 3.4.2. Nelder-Mead Algorithm. The Nelder-Mead algorithm is a popular algorithm for multidimensional unconstrained optimization. A detailed description can be found in [42]. The Nelder-Mead algorithm is simplex based. A simplex in Rn is defined as the convex hull of n + 1 vertices x0 , . . . , xn ∈ Rn . In two dimension it is simply a triangle. The algorithm evaluates the function to be optimized at these vertices and then transforms the simplex to find better values until a termination criteria is met. In a nutshell (based on [43]): • Construct the initial working simplex S. • Repeat the following steps until the termination criteria is met: – calculate χ2 at all vertices – if the termination criteria is not satisfied by the output, transform the working simplex. • Return the best vertex of the current simplex S and the associated function value. The termination criteria in this case is the variation between vertices. If it is sufficiently small, the algorithm stops. The key step is transforming the simplex. For the scope of this thesis the details are omitted. It involves replacing the worst vertex by a better point by using reflection, expansion or contraction with respect to the best side or shrinking the simplex towards the best vertex.. 3.5. Systematics/Problems. One goal of the Proof of Principle Setup is to understand and learn as much as possible about systematic effects that influence the ability to reproduce a desired electromagnetic response. The measurements and their results are presented in chapter 4. In this section each investigated effect is shortly introduced as well as the type of measurements made to understand it.. 3.5.1. Thermal Response and Stability. The whole setup is subjected to external effects like temperature, vibrations and air currents that effect disk positions. By measuring the reflectivity of a static disk over a long period, these effects can be quantified. Measuring temperatures in parallel allows to estimate the change in disk position per change in temperature. The short and long term variations in reflectivity determine the stability of the setup..

(38) 32. 3.5.2. Chapter 3. Experiment and Methods. Positioning and Hysteresis. The ability to position the disks with good accuracy and reproducibility is key to create a working booster. The achieved precision is investigated by repeatedly ordering a disk to the same position and measuring the reflectivity. A hysteresis between positions approached by either forward or backward motion is quantified and corrected for.. 3.5.3. Unwanted Reflections. As mentioned before, the beam propagating back to the antenna is partly reflected back into the booster. These unwanted reflections cause significant bias if not properly modelled. Identifying the sources of reflections is the first step to understanding them. Their behaviour at different distances is studied. The model including the unwanted reflections is tested at different distances and with different number of disks. This also gives an estimate on the curvature of the rails. Short comings of the model point to still unadressed effects like diffraction and tilts.. 3.5.4. Reproducing EM properties. The main goal of the Proof of Principle Setup is to reproduce the desired electromagnetic response. In the end, the actual disk positions and the effects influencing them are of little interest as long as the response of the setup matches the expectations. Adjusting the setup to the model as well as the other way around quantifies how well one can reproduce the desired electromagnetic response and hence it can be tested whether the required accuracy can be achieved..

(39) 33. Chapter 4. Results 4.1. Thermal Response and Stability. The setup is subject to thermal expansion and contraction and so is the disk position. To quantify the change in disk position, a single disk with mirror was installed and left untouched for a long period of time during which simultaneous reflectivity and temperature measurements were done repeatably. Two methods were used to quantify the thermal response: The first simply took the group delay peak frequency by interpolating the processed group delay with a cubic spline. This allows to make a measurement every few seconds. The group delay peak frequency corresponds to the distance between disk and mirror. This method also allows to quantify short term stability. The second method fits the disk spacing and total distance with the model to the measurement to directly see the influence of temperature on the fitting procedure.. 4.1.1. Thermal Response. In the group delay peak method a single disk was installed at a distance of d ∼ 5 cm to the mirror. At this distance the group delay consists of many higher order harmonic peaks which are sharper than the first order harmonic, hence allow for more precise determination of disk distance. The reflectivity measurement was processed with a frequency gate of 15 − 30 GHz and a time gate of 4 − 7.5 ns, filtering out unwanted reflections. The group delay was interpolated with a cubic spline to find the group delay frequency of the seventh harmonic peak. Every 10th reflectivity measurement a temperature measurement was made. Temperature sensors were attached at various locations. The VNA was ordered to make a measurement as often as possible which lead to uneven time differences between measurements. The measurement was interpolated to even time differences between time stamps. Furthermore, the temperature data was averaged between the different sensors and interpolated to the same time stamps. In Fig. 4.1a the group delay peak frequency and temperature is plotted against time. To find the temperature response, the group delay peak frequency and corresponding distance between disk and mirror was plotted against temperature (Fig. 4.1b). Since.

(40) 34. Chapter 4. Results. Peak Frequency Temperature 20. Temperature [°C] fmax [MHz]. fmax [MHz]. 8. 19. 6. 25. 8. 20. 6. 15. 4. 18. 4. 10. 2. 17. 2. 5. 0 2. 16 0. 5. 10. 15. 20. Time [h]. 25. 30. 2. 35. 400. 300. 300. 100. dm [nm] Counts. 100 0 100 200. 100. 300 0. 10. 20. Time[h]. 17. 18. 19. Temperature [°C]. 5. 20. 30. 200. d0m [nm] 100. 100. 200. 300. 400. 0.006. 200. 50. 50. 16. (b) Temperature dependence. 150. 0. 0. 0. (a) Thermal response. fmax [kHz]. Data Linear Fit. 10. dm [ m]. 10. 0.005 0.004 0.003 0.002 0.001 0.000. (c) Fluctuations. 100. 50. 0. 50. fmax [kHz]. 100. 150. (d) Fluctuations. Figure 4.1: Temperature dependence and stability of a static one disk setup with the group delay peak method. (a) The group delay peak frequency and temperature over time. The peak frequency follows the recorded temperature closely. (b) Group delay peak frequency and corresponding disk position over temperature. The linear dependence is clearly seen and quantifies the change in disk position per temperature change. (c) The group delay peak frequency of (a) is subtracted by a smoothed group delay peak frequency to show short term fluctuation in frequency and corresponding disk position over time. (d) The fluctuations are collapsed into a histogram to quantify the standard deviation of the fluctuations.. the seventh harmonic peak was measured, the corresponding distance is: dm =. 7 7 c λ= 2 2 f max. (4.1). A linear fit gives the frequency and distance change per temperature change:. ∆ f max ( T ) = (2.478 ± 0.001) MHz K−1 ∆T ∆dm,1 ( T ) = (−5.874 ± 0.003) µm K−1 ∆T The uncertainty comes from the least squares covariance matrix. However, repeated measurements of thermal response yielded varying results over the course of about half a year. It seems to be sensitive to changing setup configurations. In the fitting method one can directly observe the effect of temperature changes.

(41) 4.1. Thermal Response and Stability. 35. on the fitting procedure. Each reflectivity measurement of a static disk was fitted by the model. This also allowed to increase the the time gate to 3 − 14 ns, encompassing the first order unwanted reflections. However, since each fit takes several minutes, only long term changes can be resolved. Over a period of roughly 30 h the spacing between disk and mirror and the total distance between mirror and antenna was fitted. The disk was positioned at a spacing of 8.25 mm. The air conditioning was deliberately turned off to maximize the experienced temperature range. Three temperature sensors measured the temperature for each reflectivity measurement. The disk spacing and temperature over time can be seen in Fig. 4.2a. The temperature was averaged between the three sensors. Again, the disk spacing was plotted over temperature and fitted with a linear fit in Fig. 4.2b. The change in disk spacing per change in temperature is: ∆dm,2 ( T ) = (−7.45 ± 0.04) µm K−1 ∆T The total distance is subtracted by its mean to get the total distance residual. It is plotted against temperature in Fig. 4.2c. Clearly it does not follow the temperature linearly. Overall it changes by just a few µm. The total distance mainly controls the ripples in the group delay caused by unwanted reflections. The jump at around 29 ◦C could mean that the best fit total distance caused the ripples to skip one period relative to the measurement. See section 4.3.1 for more information. The change of disk position by thermal expansion is similar for both methods used. They differ by about a 1.5 µm K−1 . Repeated measurements yielded comparable results. In total the change in disk position per Kelvin is estimated to be the mean of both methods presented here with a conservative error estimate of ±2 µm: ∆d Final ( T ) = (−7 ± 2) µm K−1 ∆T. (4.2). A higher temperature results in a smaller disk spacing. Many parts in the setup can change the disk spacing by expanding and contracting. The motor axle for example is made from stainless steel and has a length of 77 mm. Stainless steel has an fractional expansion coefficient between 11 − 17 × 10−6 K−1 [44]. Thus the motor axle is expected to change between 0.8 − 1.3 µm per Kelvin. Consequently, the observed change must be a sum of different expanding parts.. 4.1.2. Short Term Stability. Another quantity of interest is the width of the peak frequency without long term thermal response which is a measure of the short term stability of the setup. To find the statistical variation in group delay peak frequency of the data obtained by the group delay peak method, it was smoothed with a Gaussian filter (scipy.ndimage.filters.gaussian) with a standard deviation of 10 minutes and then subtracted from the unsmoothed data, effectively removing fluctuations with time ˜ can be scales longer than 10 minutes. The remaining short term fluctuations ∆ f max.

(42) 36. Chapter 4. Results. Disk Separation Temperature. 10. 30. Temperature [°C]. dm [ m]. 5. 29. 0 5. 28. 10. 27. 15. 26. 20 25. 25 0. 5. 10. 15. Time [h]. 20. 25. 30. (a) Disk spacing and temperature. Data Linear Fit. dm [ m]. 20 10 0 10 25. 26. 27. 28. 29. Temperature [°C]. 30. (b) Temperature dependence. ls [ m]. 4 2 0 2 25. 26. 27. 28. 29. Temperature [°C]. 30. (c) Total distance and temperature Figure 4.2: Temperature dependence by fitting the model to the setup. (a) The fitted disk spacing and recorded temperature over time. Again, the disk spacing follows the temperature closely. (b) Fitted disk spacing plotted against temperature. The linear dependence is described by a linear fit. (c) Fitted total distance plotted against temperature. It does not follow the temperature and varies by just a few µm..