We have proposed and demonstrated a technique for the measurement of the wavefront of a diode laser beam with a large dynamic range. Our technique is a modified version of Hartmann and Shack-Hartmann wavefront sensor. The modified version is capable of providing a large dynamic range (180 degrees). The wavefront measurement exhibits a precision of ( 0.02 degrees), subject to a standard deviation governed by the diffraction limit (~£f/d). Using the physical measurement of the wavefront, we are able to reconstruct the electric fields of a diode laser beam at any location, including the far-field and near-field. The reconstructed electric fields were computed form the data of the intensity and the phase distribution by means of Fourier transform. The information about the electric field can be very useful in the design of microlens for the efficient coupling of light source into optical components. The results indicate that the wavefront sensor with large dynamic range can provide a reliable method for measuring the wavefront distributions of diode lasers with large divergence angles. However, the numerical near-field intensity is 150% deviated from the measured near-field intensity because of the inherent inaccuracy in the wavefront measurement. In this study, we have measured the near field intensity distribution directly with an objective and a CCD camera. We found that the distribution of the mode field was symmetric at a distance of 8£gm from a diode laser and the mode field diameter was 4.75£gm. Using the phase retrieval algorithms, the radii of the near-field wavefront in the vertical axis and the horizontal axis were 8£gm and 41£gm, respectively. Through the geometrical optics, the optimum curvatures of elliptic-cone-shaped lensed fiber for efficient coupling in the vertical axis and the horizontal axis were 4£gm and 20.5£gm individually. Once we know the optimum curvatures of elliptic-cone- shaped lensed fiber, we can fabricate it using grinding and fusing.

This thesis is concerned with the development of a practical digital image processing system for recording and subsequent reconstruction of the magnitude and phase of an optical wave-front arriving from a coherently illuminated object disturbance. Since the wave-fronts of concern are coherent, the magnitude and phase of such waves are generally independent functions in the sense that the knowledge of one is not sufficient to uniquely deduce the other. To uniquely reconstruct and characterize optical disturbances both the magnitude and phase are required. In general, all recording media respond only to light intensity and no difficulty is encountered in recording the intensity and therefore the magnitude, because it is the square root of the intensity.

The angular resolution of an imaging system is sharply bounded by the diffraction limit, a fundamental property of electromagnetic radiation propagation. In order to increase resolution and see finer details of remote objects, the sizes of telescopes and cameras must be increased. As the size of the optics increase, practical problems and costs increase rapidly, making sparse aperture systems attractive for some cases. The method of Intensity Correlation Imaging (ICI) provides an alternative method of achieving high angular resolution that allows a system to be built with less stringent precision requirements, trading the mechanical complexity of a typical sparse aperture for increased computational requirements. Development of ICI has stagnated in the past due to the inadequacies of computational capabilities, but the continued development of computer technologies now allow us to approach the image reconstruction process in a new, more e ffctive manner. This thesis uses estimation methodology and the concept of transverse phase diversity to explore the modern bounds on the uses of ICI. Considering astronomical observations, the work moves beyond the traditional, single-parameter uses of ICI, and studies systems with many parameters and complex interactions. It is shown that ICI could allow significant new understanding of complex multi-star systems. Also considered are exoplanet and star-spot measurements; these are less promising due to noise considerations. Looking at the Earth imaging problem, we find significant challenges, particularly related to pointing requirements and the need for a large field-of-view. However, applying transverse phase diversity (TPD) measurements and a least-squares estimation methodology solves many of these problems and re-opens the possibility of applying ICI to the Earth-imaging problem. The thesis presents the TPD concept, demonstrates a sample design that takes advantage of the new development, and implements reconstruction techniques. While computational challenges remain, the concept is shown to be viable. Ultimately the work presented demonstrates that modern developments greatly enhance the potential of ICI. However, challenges remain, particularly those related to noise levels.

Optical images of astronomical objects viewed through ground-based telescopes are blurred by the atmosphere. The atmosphere is turbulent and as a consequence the density of air is not evenly distributed. This results in random, time-varying variations in refractive index. The wave-fronts passing through the atmosphere become aberrated, degrading the quality of the images. One solution is to include an adaptive optics system in the telescope. The system estimates the aberration of the wave-fronts and compensates the wave-front in real time using a corrector element, typically a deformable mirror. An important problem is how to estimate the aberrations optimally using only a small amount of light. This procedure is called wave-front sensing and is the subject of the research of this thesis. For turbulence with Kolmogorov statistics, the wave-front slope contains 87% of the energy of the aberrations. Hence, it is crucial to estimate the slope accurately. The displacement of an image is directly proportional to the wave-front slope and is used to estimate the slope. The conventional way of measuring the average slope of the wave-front in a Shack-Hartmann sensor is from the centroid of the image at the focal plane. It is demonstrated that using the centroid estimator produces an estimate with infinite variance. The Cramer-Rao lower bound (CRLB) is a theoretical lower bound for the variance of an unbiased estimator. The variance of the maximum-likelihood (ML) estimate for the displacement of a diffraction-limited image approaches the CRLB using a relatively small number of photons. The ML estimator is extended to the case where the image is randomly blurred by atmospheric turbulence. It is found that the variance of the error of the slope estimator can be improved significantly at low turbulence levels by using the ML estimator instead of the centroid. Curvature sensors use two defocused images to estimate the wave-front aberrations. It is shown using the CRLB that the focal plane is the optimal plane to measure the slope and the error using defocused images is quantified. The effect of using broadband light on the accuracy of the slope estimate is also investigated. When using laser guide stars, it is not possible to estimate the slope of the wave-front directly from the image because the beam is displaced on both the upward and downward journey. However, the displacement is a weak function of wavelength due to dispersion. In theory, the difference in wave-front slope as a function of wavelength is proportional to the absolute slope. Centering algorithms were implemented on experimental data taken at the Observatoire de Lyon to confirm this relationship. There is strong evidence pointing to a linear relationship between two pairs of differential tilt measurements, but not between the differential and the absolute tilt. However, the data appears to have been affected by a systematic experimental error and a new experiment is needed. Phase retrieval is a non-linear technique used to recover the phase in the Fourier domain using intensity measurements at the image plane and additional constraints. A method is described to solve the phase retrieval problem using linear iterations near the solution, which provides both analytical insight into phase retrieval and numerical results. The algorithm finds the maximum a posteriori estimate of the phase using prior information about the statistics of the noise and the phase and converges well in practice. When phase retrieval is performed on data from subdivided apertures, there is a loss of information regarding the relative piston terms of the subapertures and this error is quantified. It is found that there is a smaller wavefront error when estimating the phase from a full aperture than from a subdivided aperture. Using a combination of intensity measurements from a full and a subdivided aperture is shown to result in a small improvement at very high photon levels only. Curvature sensors measure the wave-front aberrations via a linear relationship between the curvature of the wave-front and the intensity difference between two defocused images. In practice, their performance is limited by their non-linear behaviour, which is characterised by solving simultaneously the irradiance transport equation and the accompanying wave-front transport equation. It is shown how the presence of non-linear geometric terms limits the accuracy of the sensor and how diffraction effects limit the spatial resolution. The effect of photon noise on the sensor is also quantified. A novel technique for deriving wave-front aberrations from two defocused intensity measurements is derived. The intensity defines a probability density function and the method is based on the evolution of the cumulative density function of the intensity. In one dimension, the problem is easily solved using histogram specification with a linear relationship between the wave-front slope and the difference in the abscissas of the histograms. This method is insensitive to scintillation. In two dimensions, the procedure requires the use of the Radon transform. Simulation results demonstrate that very good reconstructions can be attained down to 100 photons in each detector.

Coherent Diffractive Imaging (CDI) has become an increasingly popular frame work in which to solve the classic phase problem in imaging due to benefits in resolution and the facility of collecting data in this modality. In particular, there is considerable interest in using the short wavelength and high coherence of fourth generation x-ray sources with CDI techniques to phase non-crystalline, or nano-crystalline biomolecular samples. CDI provides an opportunity to determine the structure of proteins and other biological samples which are unable to be phased with the standard techniques of protein crystallography, typically due to lack of adequate crystalline samples. Methods of non-crystalline phase retrieval are legion, however many suffer from limitations in resolution or the inability to recover phase fields containing a pathological singularity. Wavefields containing phase singularities are common in optical fields. The creation of an x-ray wavefield containing a pathological phase singularity is demonstrated. In this thesis a form of CDI termed astigmatic diffraction is presented, that is able tophase uniquely this class of wavefield. This is achieved by illuminating the sample with beams containing known phase curvature. The theory of the method and simulations of its application to a nano-crystalline biomolecule are presented. The experimental recovery of the direction of the phase gradient of a sample illuminated with coherent x-rays produced by a synchrotron source is shown to verify this method.

As astronomers and astrophysicists seek to view ever-increasingly distant celestial objects, the desired angular resolution of telescopes is constantly being increased. Classical optics, however, has shown a proportional relationship between the size of an optical telescope and the possible angular resolution. Experience has also shown that prohibitive cost accompanies large optical systems. With these limitations on classical optical systems and with the drastic increase in computational power over the past decade, intensity correlation interferometry (ICI) has seen renewed interest since the 1950’s and 60’s when it was initially conceived by Hanbury Brown and Twiss. Intensity correlation interferometry has the advantage of less stringent equipment precision and less equipment cost when compared to most other forms of interferometry. ICI is thus attractive as a solution to the desire for high angular resolution imaging especially in space based imaging systems. Optical interferometry works by gathering information about the Fourier transform of the geometry of an optical source. An ICI system, however, can only detect the magnitude of the Fourier components. The phase of the Fourier components must be recovered through some computational means and typically some a priori knowledge of the optical source. This thesis gives the physics and mathematical basis of the intensity correlation interferometer. Since the ICI system cannot detect the phase of an optical source's Fourier transform, some known methods for recovering the phase information are discussed. The primary method of interest here is the error-reduction algorithm by Gerchberg-Saxton which was adapted by Fienup to phase retrieval. This algorithm works by using known qualities of the image as constraints; however, sometimes it can be difficult to know what these constraints are supposed to be. A method of adaptively discovering these constraints is presented, and its performance is evaluated in the presence of noise. Additionally, an algorithm is presented to adapt to the presence of noise in the Fourier modulus data. Finally, the effects of the initial condition of the error-reduction algorithm are shown and a method of mitigating its effect by averaging several independent solutions together is shown.

This thesis examines problems in electron diffraction and related areas of theoretical optics. It begins with a study of the phase of a quantum mechanical wave function and the behaviour of phase vortices and vortex cores. Several rules for vortex core evolution are given and simulated vortex trajectories are studied. These simulations show that in electron microscopy at atomic resolution and in other similar situations, vortices occur in the wave functions very frequently. This means any image processing methods which deal with the wave function phase must permit vortices to occur. In this context a number of methods of phase retrieval are compared and evaluated. The criteria of evaluation are the accuracy of the phase retrieval, its ability to cope with vortices, its numerical stability and its required computational resources. The best method is found to be an iterative algorithm similar in approach to the Gerchberg-Saxton method, but based on a through focal series of images. / Using this phase retrieval method as an essential tool, the thesis continues with a study of inverse problems in electron optics. The first problem considered is that of using a set of images taken to characterise the coherent aberrations present in a general imaging system. This problem occurs in many areas of optics and is studied here with a focus on transmission electron microscopy. A method of using software to simultaneously determine aberrations and subsequently remove them is presented and tested in simulation. This method is found to have a high level of accuracy in aberration determination. The second inverse problem studied in this thesis is the inversion problem in dynamical electron diffraction. This problem is solved for a periodic object, giving an accurate and unique solution for the projected potential in the multiple scattering case. An extension of this solution to objects which are non-periodic in the direction of the incident wave is investigated. Finally a model computation solving the general inversion problem for dynamical diffraction in an aberrated transmission electron microscope is performed, illustrating this and previous material and summing up the advances presented in this work.

Abstract This thesis is devoted to methods of analyzing optical spectra obtained from turbid liquids, i.e., liquids that are optically very thick and/or scatter light. Data for spectral analysis were obtained with a new, multifunction spectrophotometer developed for industrial liquid samples. One characteristic of the spectrophotometer is that spectral analysis methods can be implemented into the software. Here, the emphasis was on data inversion methods, particularly the Kramers-Kronig analysis and the maximum entropy method, which can be used to gain information on the wavelength-dependent complex refractive index of liquid samples. Relating to such characteristics as density and colour, the complex refractive index also helps to identify the species that form a liquid. The methods were applied to study the internal reflection of light from the prism-liquid interface of the probe and to analyze surface plasmon resonance spectra. This study provided new methods of investigating the optical properties of relatively difficult objects, like offset inks, and of assessing adhesion forces between ink and the substrate system. Another important part of the thesis was the exploration of spectral analysis methods to obtain optical properties of nanoparticles in a liquid matrix. Bounds for the optical properties of multi-component structures in a liquid were considered with the aid of Wiener bounds.

In this thesis we generalize the problem of phase retrieval of vector to that of multi-vector. The identification of the multi-vector is done up to some special classes of isometries in the space. We give some upper and lower estimates on the minimal number of multi-linear operators needed for the retrieval. The results are preliminary and far from sharp.

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