In this work, we consider how to optimize an optical system, specifically one with diffractive optical elements (DOE). We start by describing optical theory called Fourier optics also known as wave optics. This type of optics is found by making assumptions from the Maxwell equations for magnetic and electrical fields. This leads us to the Rayleigh-Sommerfeld diffraction integral, which we need to propagate light. To optimize an optical system, we introduce the standard optimization methods used when gradients are available and also dive into data-driven methods. Two wellknown algorithms in each category: the Adam optimization method, which is an extension of normal gradient descent methods, and the UNet convolutional neural network. To make the optimization methods work with our physics simulation, we use an automatically differentiable implementation which gives the gradients for the optimization. Combining the two optimization methods with our optics engine, we optimize optical designs such that the resulting intensity on the sample plane resembles some target intensity. We are able to optimize systems with single and multiple DOE and for high and low resolution DOE designs. We find that more lenses makes the optimization better and increases the variability in the created projection. We also find that increasing the resolution severely slows down the optimization with the Adam method. Although, the optimization method Adam is well suited for this optimization task. It becomes computationally very expensive on high resolution due to the physics simulation at every optimization step. Some physical simulations require high resolution to make sure the simulation does not contain to much artefacts. We show that the data-driven approach has potential to solve this issue. We train a network that takes as input a target intensity and outputs the lens that produces that intensity. Combining these results, we conclude that modern optimization methods are well suited for optical system optimization and we find that there is a large untapped potential for data-driven methods in optics.

In this report a semi-analytic solution to the Laplacian magic window is proposed. The Laplacian magic window is a term recently introduced in 2017[2]. When a uniform wavefront hits a refractive surface, it creates an illumination distribution behind the surface. When the curvature of the surface is sufficiently small, it can be related linearly to the target illumination, and thus creates a ‘magic window’. The main idea of the semi-analytic solution is that a target illumination or a surface given in terms of Zernike polynomials can be solved analytically and expressed again in Zernike polynomials. The Zernike polynomials are set of complete and orthogonal polynomials that are already used in the field of optics to describe wavefronts of optical surfaces. The results of the semi-analytic solution agree qualitatively with the numeric results and work for complex and diverse inputs. The implementation is done in 2D, but the method is general enough so it can be extended to 3D.

There exist a lot of methods to find the electromagnetic field behind a lens, so called the forward lens problem. In contrast very few methods can do the inverse, namely finding a lens that would produce a given image or light distribution for a known source. Here a solution to the forward problem, the Fraunhofer approximation, is used to find an approximate solution of the inverse problem using a neural network. Given an input image the network would predict the lens that can produce this image. In general this lens is not the classic convex/concave shape but is a freeform lens. The predicted image produced by the predicted lens can be computed by the Fraunhofer approximation. To train the network this predicted image is compared to the desired image. This unsupervised training method is similar to that of Physics Informed Neural Networks (PINN), which is a recent approach to solve PDE’s. The phase contour of the lens is represented by a B-spline curve. The control points of this spline are the output of the network. In this way very few output variables can be specified by the network to achieve a smooth detailed lens shape. To give a proof of concept a one dimensional version of the Fraunhofer approximation is used. For this case the training already depends on many parameters. These are optimised for the lowest resulting loss. The Fraunhoffer approximation limits the images that can be created. If the network, however, is trained on images that can definitely be created, it is able to almost exactly predict a lens that delivers the desired image. The potential of the unsupervised training method in combination with a spline approximation should therefore be explored with other solutions of the forward problem. Namely a ray-tracing algorithm could be used, since this can create a wider variety of images. This would be computationally heavy compared to the Fraunhofer approximation.

This project aims to recreate intensity patterns using Fraunhofer diffraction as a means of simulation. These intensity patterns are created by phase shifting specific parts of an incoming field of light. These phase shifts are determined by a B-spline surface, which is in turn controlled by so-called control points. Only a handful of control points can describe a whole surface. The position of these control points is then determined using machine learning and specifically a technique inspired by ‘physics-informed neural networks’, which were introduced last year by Raissi et al. [1]. With this method, simple experiments which sought to recreate intensity patterns known to be in the solution space were carried out. These experiments showed some success, but suffered from the fact that they used too sensitive parameters in the input of the machine learning model, reducing the sophisticated method to a Monte Carlo search, or they used no input at all, which degraded the machine learning model to simple parameter optimization. Nevertheless, these experiments showed that this method has the potential to be used in more flexible optical setups, where multiple configurations can yield the same intensity pattern or where changing the parameters defining the setup do not induce enormous changes in the resulting intensity pattern. In addition, the proposed method relies upon Fraunhofer diffraction, which, when discretized for numerical computation, introduces aliasing issues when the incoming field changes too rapidly. This phenomenon was especially apparent when using point sources that create spherical wave fronts. A possible solution for this issue is to consider ray tracing techniques in future research.

Extremely large telescopes (ELTs) are expected to be one of the most promising astronomical observation instruments when it comes to observing exoplanets. During observations, however, Earth’s atmosphere introduces optical distortion, which needs to be corrected. For this reason, a wavefront sensor (WFS) will be developed that comprises a 100 ⇥ 100 array of kinetic inductance detectors (KID), superconductor-based photodetectors. The main advantage of KIDs in an array is their ability to be multiplexed on a single read out line by designing each KID to operate at a different frequency. The signal of an individual KID can essentially be described as a dip in the transmission signal of the readout signal. Upon the fabrication of these KID arrays, however, these dips may have shifted as so-called frequency scatter (or f-scatter), which severely disorders the readout signal. The f-scatter cannot be determined based solely on the readout signal as it contains no information about which KID each individual signal originates from. This paper manages to provide a solution in the form of an experimental method in which the KID array is scanned along the x- and y-axis to produce a spatial map. To realize this, an optical setup was designed and constructed, based on a single-lens system with a magnification of M = 1.22. By implementing a CCD camera, the KID array could be live imaged to align the setup accordingly. Thanks to numerous other alignment measures, the setup was able to scan the entire KID array along the x- and y-axis in 45 scans per axis. Based on the difference in transmission, a 45x45 spatial map was constructed, which was manually scaled down to 20x20. This 20x20 spatial map presented the location of KIDs on the fabricated detector array based on their measured operating frequency. By comparing this with the spatial map of the KID array design, the standard deviation in the fractional frequency deviation was determined, σ_δf/f = 5.92 · 10^-3. This result shows the experimental method is able to effectively unveil the severity of the f-scatter in the studied 20x20 KID array. Despite the potency of the presented method, the biggest giveaway of its flaws is a reduced pixel yield on the 20x20 spatial map from 400 to 374 pixels. The developed data analysis is able recognize multiple complications, which can be mitigated by optimizing readout parameters to improve the quality of the signal. Additionally, one-to-one imaging is far too restricting for advanced scanning techniques due to its fixed M. Appropriate suggestions for improving the lens systems would be implementing the Cooke triplet or a two-lens system that utilizes the principle of virtual imaging.

Janus particles are colloidal particles for which one half of the surface has different attributes than the other half. One property of a spherical dielectric particle with half of its surface covered by a layer of another dielectric or metal is that it has a non-uniform scattering pattern when exposed to light. However, the angle with which the light is shone on the particle has a large effect on the scattering pattern produced. Thus it is important that we are able to orient these Janus particles. The orientation can be controlled if we apply an electric field to the particle for example. The movement of colloidal particles with an electric field is widely studied and this field is called dielectrophoresis. For a Janus particle, the calculations for the force and torque become complicated. The movement and rotation of these particles have been studied, however, no analytic solution has been found. In this report, we derive a semi-analytic description of the force and torque due to an external electric field on a spherical Janus particle. For this, first the potential due to an external electric field is determined and then the force and torque are calculated with two methods: the dipole approximation and the Maxwell Stress Tensor method. In the dipole approximation, there is no force on the Janus particle. But, there is a torque on the particle in the dipole approximation. Due to this torque, the Janus particle will orient itself such that its cap points in the direction perpendicular to the applied field. For the torque calculated with the Maxwell Stress Tensor, we get a similar result as in the dipole approximation. On the other hand, according to the calculations with the stress tensor, there is a relatively small force on the particle.

Distributed feeding in photo-conducting antennas (PCAs) leads to simple optical designs, less prone to overheating, as well to terahertz (THz) antennas that can operate non-dispersively over wide bandwidths. However, the efficient analysis of pulsed PCAs was so far limited to architectures characterized by feeds small with respect to the THz wavelengths. In this thesis, an efficient and rigorous procedure for the time domain analysis of an infinitely long slot antenna printed on photo conductive material and excited by a distributed pulsed laser is presented. The procedure is electromagnetically rigorous, and relies on spectral representations of the fields in both the frequency and spatial domains.

A gas-cell-based calibration setup was designed to evaluate the wideband response of sub-millimeter spectrometers like DESHIMA (DEep Spectroscopic High-redshift MApper). The use of low pressure gas emission spectra allowed for accurate calibration of the absolute frequency response, and to test the detectability of faint emission spectra with long integration times. This is important to understand and evaluate systematic errors and noise profiles of sub-millimeter astronomical spectrometers before their telescope campaigns. The setup consisted of a low pressure (~mbar) gas at room temperature in a high vacuum (<10-3 mbar) chamber in front of a 77K N2 background. A double-winged rotating chopper was used for signal modulation of the on- and off-source paths to reduce the low-frequency noise profile. The setup has been able to successfully detect the emission spectra of nitrous oxide at 30 mbar and methanol at 1 mbar in the frequency range of 332 to 377 GHz with the prototype DESHIMA spectrometer. Our models showed that lower pressures should be detectable over similar averaging times. The standing spectrum showed to be too irregular for detecting spectral lines in a single measurement. A second measurement was required to subtract the standing features, which extended the total time required beyond the current system stability. Detailed analysis into optical resonances has shown the importance of anti-reflective (AR) coatings on the main optical interfaces to improve the detectability of the emission spectra. We adapted sub-wavelength pyramid gratings milled into TOPAS windows to reduce a standing wave in the output spectrum of the gas cell setup. Stability of the setup was shown for observation times of up to ~103 seconds before environmental noises became dominant. Extensive stability testing has shown the impact of key components in the setup. A two-stage post-processing algorithm was developed to successfully reduce instabilities in the data by removing linear drifts and by removing the common profile over simultaneous read-out data.

The beauty of optical birefringence lies in the fact that it provides an independent control of light over different polarization directions, which leads to many important applications in today's optical systems. This thesis describes the applications of existing naturally-occurring birefringent materials as well as the engineered formed-birefringent meta-materials via nano-fabrication. In particular the application of birefringent materials in optical trapping, waveguide engineering and imaging are discussed. The investigation methods include numerical simulation, nano-fabrication and experimental validation. @en

The development of optical metamaterials in recent years has enabled the design of novel optical devices with exciting properties and applications ranging across many fields, including in scientific instrumentation for space missions. This in turn has led to demand for computational methods which can produce efficient device designs. Traditional optical devices admit a closed-form solution for this inverse design problem. However, in the presence of strong multiple scattering, which is often the case when considering optical metamaterials, the inverse problem becomes ill-posed. As a result, many optimization and machine learning techniques have been applied towards discovering good solutions. In this MSc thesis project, several of the most promising of these techniques are applied to a specific problem, the discovery of silicon metamaterial lens designs for the CoPILOT high-altitude balloon project. Ultimately, a software tool capable of producing effective and admissible designs is produced and demonstrated. First, an overview of the CoPILOT design problem is presented. Next, relevant background material topics, including properties of metamaterials and computational methods for simulating them, are covered in some detail. After this, methods used to solve optical design problems in past literature are described and contrasted. Then, a comprehensive explanation of the method developed and used for this project, including important design considerations, is given. The best solutions found using this lens optimization method are shown and compared. Finally, fruitful areas of future work on this topic are listed.

Interference of light can be used to determine the concentration of a gas, called gas sensing. The absorption of the light by the gas molecules is measured based on the phase change of the light. In this report, a hollow core photonic crystal fiber is treated. The gas sample is inserted into the hollow core. Photonic crystal fibers possess the characteristic to have electromagnetic modes confined to the core with low attenuation. This means that there can be a high interference rate between the gas and light, which is desirable for gas sensing. First, less complicated optical fibers were studied. The analytical solution of the electric and magnetic field for the TE and TM modes of the simple fiber were derived and for the TE modes of the step-index fiber. These results were compared to simulation done with COMSOL Multiphysics. Photonic crystal fiber with circular core was then simulated with COMSOL Multiphysics, but the results did not match with the literature. Therefore another photonic crystal fiber was simulated with a star-shaped core. The simulation results found modes concentrated in the core, with low attenuation. It was attempted to get similar results with a three layer step-index fiber, by varying the imaginary refractive index of the middle layer. The attenuation of the three layer fiber was much higher than that of the photonic crystal fiber for all simulations. This indicates that the three layer step-index fiber does not support propagation modes that are confined to the core. Further research could be done by studying the effect of the radius of the middle layer and the real part of the refractive index.

Suppose we have a target radiance distribution, a light source, and a plane for receiving light. How do we design a reflector that can give a similar result as the target radiance distribution? The inverse reflector problem is of high interest for light designers and related industries, such as lamp manufacture, headlight and street light design, and interior designs. There have been some researches on this specific topic. Nevertheless, the existing algorithms are either not fully compatible with parallel acceleration or have a narrow application scope (can only handle the far-field problem). This thesis's goal is to design a method that can have a fast approximation of the inverse reflector problem and handle different application scenarios. The core of the proposed method in this thesis is a modified simulated annealing algorithm. I first give the definition of the inverse reflector problem and explain why I choose simulated annealing in the proposed method by analysing the related works and various optimisation algorithms. Then, I detail what issues the plain simulated annealing algorithm will have and why it cannot achieve satisfactory results. Moreover, to solve the issues and improve performance, I propose additional strategies, such as Phong tessellation, randomisation strategy, multi-level strategy, history-decision strategy and penalising strategy. I also discuss the system design based on the proposed method and how to accelerate the optimisation process in this system. I evaluate and validate the proposed method by running test cases in different scenarios and compare the results with other algorithms. The results show that the method, as a general optimisation algorithm, can achieve good results in different application scenarios. The compatibility, speed and accuracy are the main advantages. The result of the proposed method can also serve as the initial guess for a finer optimisation.

Differentiable ray-tracing is an exciting new development in computer graphics to approach all sorts of 3D scene design problems by obtaining gradients of renders produced by ray-tracing with respect to parameters that define the scene. These gradients can then be incorporated in a gradient-descent type optimization pipeline. One such type of design problem is caustic design with free-form lenses. This is the process of obtaining the geometry of lenses in an optical system such that the light that passes through this system forms a desired illumination distribution on a target screen. A typical application of this is distributing the light from streetlights or car headlights in a pleasing and efficient way over the street. The non-imaging optics literature offers many methods to design free-form lenses for caustic design, but differentiable ray-tracing is largely unknown in this field. Therefore this thesis proposes a method of optimizing free-form lenses for caustic design with differentiable ray-tracing. The optimization pipeline starts with a multi-layer perceptron neural network which outputs parameters that define one free-form lens side in the form of a B-spline surface. A specially implemented ray-tracer then produces a caustic render by tracing through this lens from either a plane wave or a (grid of) point source(s). Back-propagation and optimization takes place using automatic differentiation in PyTorch and the Adam optimizer. The results, which are verified with LightTools, show great promise for this technique, especially for the plane wave optimizations. The point source grid optimizations proved to be more challenging, but also here the optimization was able to achieve improvement. This shows that the proposed technique also has potential in positive etendue optimizations.

The concept of optical refrigeration dates back to 1929, when Pringsheim recognized that thermal energy associated with the translational degrees of freedom of isolated atoms could be reduced by the process of anti-Stokes fluorescence. Optical refrigeration of a solid was first experimentally demonstrated in 1995 with the Ytterbium-doped fluorozirconate glass by Epstein and his team and since then this invigorating field has garnered much scientific interest for development of an all optical refrigerator. The present works discusses the recent candidate materials including crystals, semiconductors, and ionically doped glasses. Cooling processes and necessary conditions for cooling are outlined, and general thermodynamic limitations are discussed. 10% wt. Ytterbium doped Yttrium Lithium Fluoride (Yb+3:YLF) is chosen as the candidate active material. The Carnot efficiency for laser and sun-light as a pump source is evaluated using a narrow-band approximation outlined by Stephen and his team. A quantum-mechanical cooling model based on Epstein and his team, is developed. In the proposed system, the candidate material is placed on a magnetically suspended platform inside a vacuum chamber and illuminated with laser light with the appropriate wavelength in the near infrared region. The dynamics of important cooling parameters are simulated and studied. The cooling effects due to radiative relaxation compete with the heating effects due to parasitic absorption and non-radiative relaxation but net cooling is observed confirming validity of light source and material parameter selection. In addition to laser, the conventional source of pump radiation, sun-light as a pump input to the quantum-mechanical model is simulated and the effects on the cooling power and efficiency are studied. To enhance the energy efficiency of the system, fluorescence recovery schemes using photovoltaics are built and studied. Suggestions for experimental realization are given. The developed model can be base for designing a practical optical refrigeration system for laser and sun-light based optical sources.

In this thesis we are interested in distinguishing patterns of mesoscale cloud patterns in the trades. Specifically, whether Sugar, Gravel, Fish and Flowers patterns can objectively be identified using physical quantities. For this purpose, we use cloud fraction data attained by the CORAL Ka-Band cloud radar at the Barbados Cloud Observatory during the boreal winter seasons of 2018, 2019 and 2020. These cloud fraction data represents the curves up until a height of 4 km for a given 6-hour interval of time. We do this to see if these clusters match up with the labels assigned to each cloud fraction curve obtained from a classification model used in Schulz (2021). Firstly, we map the cloud fraction curves onto points on a finite dimensional space using functional principal component analysis. We subsequently apply K-means, Gaussian Mixture Models and Mean Shift clustering onto the pre-processed dataset to identify any robust clusters. We have been able to attain robust Sugar-like clusters for K-means for 3 and 4 partitions and Mean Shift with bandwidth λ ≈ 585. This provides evidence that we are able to use cloud fraction data to distinguish Sugar. However, the same can not be said for Gravel, Fish and Flowers as we have not been able to identify them in our analysis. It is suggested for future research to do sensitivity analysis in the height interval of the cloud fraction data, that outliers are omitted and that the labeled data from Schulz (2021) are used instead of the mean and spread of the data pertaining to those labels.

In this thesis, we provide a method for reconstructing a planet's surface map from its reflected light curve. We are going to derive an equation for the reflective light-curve under the assumption that the surface map is characterized by four different surface types (ocean, vegetation, sand and snow), is stationary (no clouds), and that all reflection is diffuse (Lambertian). We will show that the transformation is a linear function of the surface map and we will work out the transformation for arbitrary observer inclination and axial tilt. Using this knowledge, we create mock light curve data of self-generated planets. Afterwards, the transformation is inverted using the Moore-Penrose pseudo-inverse and the mock data will be used to demonstrate the surface map recovery for edge-on and face-on observations of planets with different axial tilts. Furthermore, we also provide a method for recovering the planet's axial tilt from its reflected light curve. Even when a realistic amount of photon shot noise is added to the light curve, we are able to retrieve the planet's surface map and axial tilt fairly well, especially when the planet's tilt has larger axial tilt.

In this report, we will focus on simulating galaxy observations with the Deep Spectroscopic High-redshift Mapper (DESHIMA). To do so, we will discuss and evaluate two main parts needed to accurately perform such a simulation: Firstly, we will answer the question whether Time-dependent End-to-end Model for Post-process Optimization (TiEMPO), the modelling software used for DESHIMA observation simulations, is able to accurately simulate real life galaxy observations conditions. To do so, the simulation program is fed artificially created atmospheric data and its output is compared with sky brightness data of real measurements. More specifically, the time signal, power spectral density and noise equivalent flux density of both the simulation and the measurement data are derived and compared. This comparison showed, apart from a linear drift of the time signal data and a small offset of the power spectral density, good agreement between the simulation and the measurement. The second part of this thesis discusses whether we can detect an artificially created galaxy, using the already verified atmospheric model of TiEMPO. To do so, the output of the simulation is run through a series of algorithms that calculate the observation spectrum of the telescope, as if it were a real measurement. In addition, the application of different observation tactics and telescope parameters are tested and visualised. Most importantly, two observational position-switching (chopping) techniques are applied and compared: the dual point and ABBA chopping techniques. To test the effectiveness of the two chopping techniques, both will be used to simulate atmospheric filtration using stationary, i.e. without telescope movement, simulation and measurement data, which do not contain the (to be detected) galactic data. As there is no telescope movement, nor galactic data, the spectra should ideally fluctuate around zero. However, as we will see in this report, this is not obtained in all cases. After further analysis, two main types of offsets could be identified: the first one originating from the linear drift of the measurement's time signal data, whereas the second one is due to the spatial displacement of the chopping positions. The former can be corrected by applying the ABBA chopping technique rather than the dual chopping method, whereas the latter cannot with either of the two. Using the insights we acquire from running these simulations with observation conditions for DESHIMA, we are able to perform an actual galaxy observation simulation. The galactic data acquired from this observation simulation shows good agreement with the input values of the galaxy data of TiEMPO, assuring that TiEMPO can be used for galaxy observation simulations.

The aim of this research is to study a method to find exomoons and to find specific characteristics that point to the existence of exomoons. Exomoons are natural satellites orbiting an extrasolar (exo)planet in an extrasolar system. By looking at eclipses on exoplanets, we can find these characteristics. We consider the reflected light signals of exoplanets with an exomoon. The reflected light signal is the intensity of light the planet reflects from the host star towards the observer. We derive the reflected light signal in a planetary system with one exoplanet and in a system with an exoplanet and an exomoon. We made the assumptions that the exoplanets and exomoons have a homogeneous surface (albedo is 1) and move in circular orbits around a star with an inclination angle between the orbital planes. Along the orbit the planet and the moon have changing phases. The moon is always close to its planet, so the phases of the exomoon and the exoplanet are the same. For exoplanets, it is not possible to spatially separate a moon from its planet. One only sees the total signal of both bodies, the light originating from the star, reflected by the two bodies towards the observer. Maybe we can find an exomoon by examining eclipses. Eclipses occur when the exoplanet and exomoon are aligned with the star, so that the body closest to the star blocks the light towards the body farthest from the star. Finding eclipses is one of the few methods to discover exomoons. The systems are modeled with the assumption that any total eclipse happens every time r = R or −r = R, we see short dips in the reflected light signal. These dips are periodic and make the complete signal quasi-periodic. That is why we also calculate the Fourier transform of the reflected light signal. This quasi-periodicity causes the Fourier spectrum to have side bands that are repeated and are copies of themselves. In this research, the orbit of the exomoon is tilted to see the effect of inclination in the reflected light signal. The result is fewer eclipses. At most twice a year for a short amount of time an eclipse can occur. In the Fourier domain, this results in more peaks, but the pattern is still repeated. From this research, we cannot conclude whether or not an exomoon is present from measured data. What we learned, is that if an exomoon is present, short dips in the received light signal occur and this results in periodic side bands in the Fourier domain with respect to the system of one planet. The duration of an eclipse is very short, so the detection of the eclipse can easily be missed. The Fourier transform signal becomes stronger when a longer time has been measured.

Context. In the near-future, exoplanets can be observed directly through telescopes. Although the resolution of the planet's image will only be one pixel at first, the intensity of this pixel will change over time because of the orbit around its host star and its diurnal rotation. This intensity as a function of time is called the light curve of an exoplanet. The changes in the light curve as a result of annual and diurnal rotation can in turn be used to obtain information about the surface of the planet, this is called spin-orbit tomography. Aims. The aim of this study is to determine if an exoplanet's surface can be retrieved from its light curve for planet surfaces that can be described by Lambertian, Lommel-Seeliger or Fresnel reflection, or a combination of these. The variation in the light curve due to differences in the planet's surface will be used to find a map of its continents and oceans and to determine what surface types the planet is made of. Methods. This thesis starts by composing a near-equal area segmentation of a sphere to maximize the retrieval of information per pixel of the exoplanet's surface. Additionally, a method for generating artificial planets is described, such that the following method can be tested on light curves, since the current telescopes are not powerful enough to measure an exoplanet's light curve. A linear transformation from the surface to the light curve is constructed to obtain the light curve from the surface. Consequently, this transformation is inverted in order to obtain information about the surface from the light curve. This method is applied to exoplanets with a stationary surface, i.e. no clouds or changing ice caps and is consistent of the following surfaces: water, vegetation, sand and snow, each described by a different reflection model. Lastly, surface retrieval is tested from a light curve with a realistic amount of photon shot noise (SNR ≈ 14). Results. The composed near-equal area segmentation of a sphere is the Voronoi diagram of the Fibonacci lattice. It is a very appropriate near-equal area segmentation, because the maximum difference in facet area is 12% for 1001 points. Furthermore, the retrieval of an exoplanet's surface from its reflected light curve is close to perfect for exoplanets that are described by a combination of the three reflection models if the light curve does not contain noise and there are a sufficient number of data points. If the light curve does contain shot noise, parts of the surface that are described by the Lommel-Seeliger law, are not retrieved correctly. However, the general shape of the surface that is described by Lambertian or Fresnel reflection is still retrieved correctly. If the surface can be described by one single reflection model, the planet's features are retrieved correctly from a light curve with shot noise regardless of the reflection model. Conclusions. Spin-orbit tomography in the form of a linear transformation between the light curve and albedo map of an exoplanet is a very accurate method to retrieve the albedo map from a single observed pixel, even with a realistic amount of shot noise.

The Kuramoto model (KM) is a well known mathematical model of coupled oscillators that is frequently used to study synchronization phenomena. In this bachelor thesis we investigate the effects of noise on synchronization in Kuramoto-type networks. In the first part we follow the methods of Maggi and Paoluzzi [1], but include detailed in between steps, to obtain an analytical expression for the critical coupling strength, $k_c$, of the KM in the thermodynamic limit under the influence of time-correlated noise (i.e non-white noise). The coupling strength, $k$, is a parameter in the KM that essentially determines to what extent oscillators influence each other. When $k > k_c$ we start to see synchronization. Our numerical simulations agree with the results found in [1], in that the analytical expression for $k_c$ holds up for low values of correlation time, but quickly breaks down as correlation time increases. In the second part of this thesis we consider a Kuramoto-type adaptive dynamical network that is also investigated in Fialkowski et al. [2]. The dynamical phenomena that are observed in [2] are also present in our simulations. We explain these dynamical phenomena with the help of a variety of plots. Subsequently, the Kuramoto-type adaptive dynamical network is expanded to include white noise terms in the coupling dynamics. We find, through the use of simulation, that under the same conditions as in [2], synchronization is observed for significantly lower values of coupling strength. This result is explained qualitatively. Simulations also show, that for specific values of noise strength, the hysteric behaviour observed in [2] is not present. Other conclusions, like the degree to which the noise can reduce the coupling strength required for full synchronization, or beyond what value of noise strength full synchronization can no longer occur, are unable to be drawn. Additional simulation work and a further analytical work is recommended for an ensuing study.