The next generation of telescopes will be able to directly measure the light curves of exoplanets. This thesis demonstrates the potential of using light curve analysis to study exoplanetary ring systems, with the goal of retrieving their albedo map and optical depth. This can improve our understanding of the composition, formation, and evolution of such systems. We construct a mathematical model that describes how light interacts with ring particles through absorption, transmission, and scattering. Applying this model to data of Saturn’s rings from high-resolution images by NASA's Cassini space probe, albedo maps for the rings are constructed and the optical depth is found to be mostly consistent with existing literature. We then derive an analytic formula for the light curve of an exoplanetary ring system. This formula is converted into a numerical linear transformation that links the albedo map of a ring system to its light curve. Using the Moore-Penrose pseudoinverse and singular value decomposition this transformation is inverted, enabling the retrieval of the albedo map from the light curve. The method is tested using synthetic data of model ring systems while varying the axial tilt, observer direction, and signal-to-noise ratio from artificial Gaussian white noise. The results show that the method is generally effective at recovering the albedo map and optical depth for signal-to-noise ratios down to 10³.

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.

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.

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.

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.

This thesis develops a method to map the surface of an exoplanet. The problem of exoplanet mapping sounds easy to solve. Take a few photographs of an exoplanet and sew them together to create a map of the surface. The telescope required to do this is far beyond our technological capabilities. However, our current telescopes could allow us to measure the light reflected from the star on the surface of the star as a point source. This thesis constructs the surface map of an exoplanet using only a point source of reflected light as information. This method of planet mapping is called spin-orbit tomography, introduced by Cowan and Agol [2008] and more in depth by Fujii and Kawahara [2012]. Spin Orbit tomography is a method to construct a surface map from the reflected light curve, the total intensity of the light from the star, reflected on the surface of the planet, directed towards an observer.

In the near future, next-generation telescopes will be able to observe Earth-like exoplanets illuminated by their parent stars for long periods of time. As the distance between exoplanet and observer is enormous, exoplanets will only make up a pixel on our image. However, the observed intensity of this pixel will fluctuate over time as the exoplanet rotates about its axis and orbits around its parent star. These fluctuations in the reflected light contain information about the planet’s surface. Previous researchers have developed a retrieval method known as spin-orbit tomography which uses these intensity fluctuations to construct a surface map of the exoplanet assuming fully diffuse reflection as a model for the reflection of starlight. In this thesis, we aim to build on this method by introducing two new reflection models, namely Fresnel and Lommel-Seeliger reflection, and by using this composite reflection model for the starlight we attempt to reconstruct the surfaces of exoplanets based on their observed light curve. We will derive an analytical expression for the observed intensity of the light reflected off an exoplanet. Next, we will simulate intensity observations along the orbit of an exoplanet with a pseudo-randomly generated surface containing terrain types found on Earth.
As we assign a measure of reflectance for each reflection model, we will construct three distinct reflectance maps for one planet, nd simulate the observed light curve by linearly transforming these maps. Afterward, we attempt to retrieve the reflectance maps from the observed signal by inverting the transformation.
We show that the retrieval of all three maps is successful if the light curve is observed without noise, even when the number of samples is low. If artificial shot noise is imposed on our signal, we are still able to retrieve glossy reflectance maps for Lambertian and Lommel-Seeliger reflection by truncating the transformation matrix. Lastly, we show that even if the original light signal originates from an exoplanet that only reflects according to the Lambertian model, then the composite reflection model still retrieves the Lambertian map accurately. On the other hand, if the original light curve is simulated by the composite reflection model, then the Lambertian method does not retrieve an accurate surface map.

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.

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.

The Atmoscope is the idea to use the atmosphere of a planet as a telescopic lens. It uses the refractive properties of an atmosphere to converge light rays to a focal line. It has been hypothesized that using Earth’s atmosphere could, under favorable circumstances, yield an amplification of 55; 000 for a detector that has a surface area of 1m2. However, the precise effects of the oblateness of Earth in combination with the effects of physical effects such as turbulence inside the atmosphere have not yet been researched. Here we show that incorporating the eccentricity of Earth diminishes the amplification greatly. We found using a ray tracing model for gradient-index media that the amplification for ellipsoidal planets has the following relation with the detector or pixel size: A / D􀀀0:66. For spherical planets this relation is A / D􀀀1. Furthermore, we implemented velocity diffusion to simulate physical effects inside the atmosphere, which surprisingly did not affect the amplification significantly. We anticipate this result to be a starting point for more sophisticated ray tracing models. For example, a full map for the refraction in the atmosphere could be used to test the effect of a non-homogeneous atmosphere. Much research still needs to be done, before it can be decided definitively whether the Atmoscope could be a useful telescope.

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.

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.

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.

Mankind has always adapted materials to fullfil a need. Recent developments in nano-science allows us to construct objects with varying permittivity and permeability. One possible structure is an invisibility cloak device. This device is put around an object. Any plane wave that enters the cloak should go around this object. The wave exits the cloak as a plane wave, rendering the object on the inside invisible. Since making such a device is quite dicult, it is useful to simulate it first. A relatively new programming language is Julia [1], [2]. Julia has a high performance and this makes it an interesting language to use for these simulations. In this work, the material properties of such a device was calculated. A simulating tactic was developed. This tactic was a hybrid version between the Method of Moments (MoM) and the Finite Elements Method (FEM). Both these methods and the hybrid method were implemented in BEAST.jl [3]
and CompScienceMeshes [4] and were tested using simulations in EMwavesBEP.jl[5]. While the FEM and MoM gave great results, the hybrid algorithm had only a good result for the electric fields. The magnetic elds failed to show the wanted results. The previous determined cloak was simulated using the hybrid algorithm. The cloaking property was clearly visible and the wave
exited as a plane wave. A far field should be calculated to verify this further.

Ultra-wideband submillimeter observations are crucial to study the process of star and galaxy formation and for characterizing cosmic dust in the interstellar medium. The Deep Spectroscopic High-redshift Mapper 2.0 (DESHIMA 2.0) will use an integrated superconducting spectrometer chip with a 220-440 GHz band coverage and a spectral resolution of f/df = 500, enabling submillimeter observations with an unprecedented instantaneous band coverage.

However, the octave bandwidth of DESHIMA 2.0 poses a challenge: the atmospheric transmission is highly nonlinear in the broad frequency window of DESHIMA 2.0, complicating the removal of atmosphere noise from the signal.

In this thesis, I present the Time-dependent End-to-end Model for Post-process Optimization (TiEMPO). TiEMPO provides realistic time-dependent simulations of high-redshift galaxy observations. It consists of the following components:
Galaxy model. A galaxy is modeled using a two-component modified blackbody spectrum as a template. The model outputs the flux density, which is converted to power spectral density using the frequency-dependent effective aperture area of the telescope.
Atmosphere model. TiEMPO makes use of atmosphere model ARIS, which models a spatially and dynamically varying atmosphere and outputs Extra Path Length. TiEMPO converts this to precipitable water vapor using a relation that was found with the Smith-Weintraub value of the Extra Path Length and the ideal gas law.
Telescope beam. TiEMPO can be adapted to use any arbitrary beam shape for the near-field telescope beam. The far-field beam is modeled using the effective aperture area. Finally, the output of TiEMPO is given at multiple positions, enabling simulations of sky chopping and nodding in two directions.
Radiation transfer. A static model of the sensitivity of DESHIMA, determining the attenuation and the emission of the atmosphere and transmitting the signal through each component of the telescope and instrument.
Spectrometer chip. TiEMPO can adopt any filter transmission of the channels inside a spectrometer chip. In this work, they are approximated with Lorentzian curves. The photon and recombination noise are modeled with the NEP and the noise distribution is approximated with a normal distribution. The noise is incorporated with an integration over the filter response, treating photon-bunching over the wide bandwidth of DESHIMA 2.0 accurately.
Conversion to sky temperature.Finally, the power measured in the chip is related to the sky temperature with an interpolation made with a skydip simulation in the radiation transfer model.

We compare the first TiEMPO simulations to observation data by comparing the time signal, power spectral density and noise equivalent flux density. Apart from a small offset in the power spectral density, the simulation data closely resembles the observation data. TiEMPO allows us to test algorithms for atmosphere removal and galaxy detection, to study the effect of different weather conditions and to evaluate the performance of different observing techniques. TiEMPO is modular, making it usable for the original DESHIMA instrument and its successor DESHIMA 2.0. The use of TiEMPO can be extended to other spectrometers besides DESHIMA 2.0, like a grating spectrometer, and other telescopes, such as the promising 50m-aperture AtLAST/LST telescope.

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.

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 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.

DESHIMA is a superconducting on-chip spectrometer using MKIDs in the sub-mm wavelength regime (240-720 GHz, 1:3 bandwidth). This thesis presents (1) an analysis of the quasi-optical system used for DESHIMA on the ASTE telescope in Chile (2) design of the room-temperature optics of this design and (3) a preliminary investigation into an ultra-wideband leaky-lens antenna suitable for multi-pixel, constant aperture efficiency operation to be used as a feed to high f-number (>2) reflectors. The designed and analyzed optics show good performance and tolerance for the fall 2017 campaign of DESHIMA-on-ASTE. The wideband investigation leads to constant, >70% optical efficiency over the 1:3 bandwidth for a single pixel design (f-num=3.7) or >55% optical efficiency with multiple pixels spaced at 2*\lambda*f-number (f-num=5). Future work must be done to optimize the radiation efficiency of the antenna over the whole bandwidth, which is now the largest obstacle to frequency-dispersion free aperture efficiency.