The goal of this thesis is to investigate the viability of reflection-based biphoton Hong-Ou-Mandel Interferometry (HOMI) for measuring a sub-nanometer step size. We attempt to push this technology forward by designing for larger separation than has been done before in the literature, resulting in higher possible precision.
This is a Fisher-information-based estimation method. We show that the Quantum Cramér-Rao (QCR) bound can be saturated with our proposed measurement. We propose a mostly common-path interferometer design, where the two optical paths are distinct in polarization instead of spatial mode. This reduces the risk of creating accidental which-path information.
For the production of the photon superposition, we propose a novel biphoton source design specialized for large detuning between the two downconverted wavelengths, similar to the more common beamdisplacer entangled photon sources. The proposed photon source can be designed for type-0 and type-II SPDC.
We suggest a detection system based on a combination of visible and NIR single-photon detectors to handle larger detunings than is possible with a single type of detector. The best combination of detectors and type of SPDC was Si-SPAD (visible) and SNSPD (NIR) with weak downconversion focusing (ξ ≪ 1).
This experiment has an expected measurement time of 7.4 seconds for 0.1 nm precision. We conclude that biphoton HOMI is indeed feasible for high-precision metrology.

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.

In this report we will consider a method for minimising surfaces with given boundaries, which makes this problem convex and converges without pre-requisite knowledge of the topology of the minimal surface. We will translate the method, which was originally described in the language of differential forms to vector fields. We will also discuss some applications of minimal surfaces.

Imaging by inversion of acoustic or electromagnetic wave fields have applications in a wide variety of areas, such as non-destructive testing, biomedical applications, and geophysical explorations. Unfortunately, each modality suffers from its own application specific limitations, typically being difficulties in distinguishing different materials or tissues from each other in the case of acoustic wave fields and a low spatial resolution in the case of electromagnetic wave fields. To exploit the advantages of both imaging modalities, methods to combine them include image fusion, usage of spatial priors and application of joint or multi-physics inversion methods. The latter can be based on empirical relations between acoustic and electromagnetic medium properties or on structural similarity. In this work, two joint inversion algorithms based on structural similarity are presented. To account for the structural similarity the error-functional of standard Born inversion is extended with an additional penalty term. This additional term is either based on the L2-norm of the cross-gradient (CG), i.e. the cross product of the gradients of the acoustic and electromagnetic contrasts or on the L2-norm of the gradient difference (GD), i.e. the difference between the normalized gradients of both contrasts. To test the proposed methods, two synthetic models are considered; one with the gradients of the contrasts pointing in the same direction and one where the gradients point in opposite directions. Results show that the GD constraint significantly improves the resolution for the electromagnetic reconstruction compared to separate BI. The mean square errors (MSE) of the reconstructed profiles for the separate BI are 0.12 for the acoustic and 0.51 for the electromagnetic case, and for the joint GD inversion, 0.09 for the acoustic and 0.46 for the electromagnetic case. The joint GD inversion fails when using the model with the gradients of the contrasts pointing in opposite directions. The joint CG inversion does not enhance the reconstructed images, but shows similar performances for the different models. In conclusion, joint inversion based on structural constraints is shown to improve the electromagnetic resolution, especially using the GD constraint. Further research needs to be conducted to extend the functionality of the GD constraint to acoustic and electromagnetic contrasts with opposite contrast gradient directions.

In optical imaging, image quality is not only determined by the system itself but also by the media in which light traverse. Differences in the refraction index of the media encountered by a light wavefront produces phase aberrations which distort the image received from the original object. Currently, there are two main approaches for solving this problem: adaptive optics, which rely on deformable mirrors and wavefront sensors for correcting the phase aberration before it reaches the imaging sensor; and post-processing techniques, which try to estimate the object after receiving distorted images.

MFBD methods are a family of algorithms that are capable of reconstructing the object by fusing the information carried by a set of differently aberrated images. These techniques are widely used in current optical systems, allowing a notable increase in image quality in most situations. However, there are cases in which they are not applicable; for example, looking at a dynamic object (e.g., a bird flying) or looking through static aberrations (e.g., in microscopy applications); but certain modifications in the optical system can be used for solving this problem.

Actual optical devices have only one aperture, thus creating one full-size image on the imaging sensor but, by segmenting the pupil, several images can be retrieved at the same time with different aberrations (i.e., light follows a distinct path for each aperture). However, using a multi-aperture system implies that there is less imaging sensor area available for each aperture, thus obtaining images with less resolution. Nonetheless, MFBD algorithms can usually be extended in order to support SR, a technique that allows the increase of the object resolution by retrieving extra information from the displacement between images.

This thesis is focused on the development of a functional prototype of a multi-aperture optical system that can do real-time object reconstruction. As a MFBD technique is needed, the novel TIP algorithm (developed at Delft Center for Systems and Control) is selected. In order to achieve a fast and reliable reconstruction, the algorithm is: modified for increasing its robustness against noise, expanded in order to support SR and implemented efficiently in both CPU and GPU. Finally, the system is tested in a real environment, showing promising results.