## Airy beam for three-dimensional optical imaging

##### Abstract

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Airy beam is a self-sustained light beam that propagates in free space along a parabolic trajectory with a constant lateral acceleration. It is a diffraction-free beam with an electric field pattern described by the Airy function containing a main lobe and side lobes. Unlike Gaussian beam that expands after the focus, a practical finite-energy Airy beam keeps its pattern without spreading over a relatively long distance during its propagation. This unique feature makes the Airy beam a great potential of designing optical systems for three-dimensional (3D) imaging. In this work, we develop a new optical 3D imaging methodology using Airy beam. The proposed imaging method has an advantage of increased depth of field (DOF) that is important in biological optical imaging. In the first step, we perform a fundamental research on specific parameters that can affect the properties of the Airy beams. The finite-energy Airy beams are generated by optical Fourier transform using a lens with the incoming plane wave that is truncated by an aperture. Our simulation and experiments showed that by changing the aperture size and the focal length of the lens, the Airy beam pattern is modified. In addition, the DOF, the beam size and the main lobe energy are changed. As a result, Airy beam can be designed for specific imaging applications. Next, we consider the phase in the cubic phase pattern for the phase modulation to generate the Airy beam. This problem is related to the wavelength dependent phase modulation since the phase in cubic phase pattern is designed for a specific wavelength whereas a broadband light source is normally used in optical coherence tomography (OCT). A liquid crystal display (LCD) panel from a projector is used as the SLM. The cubic phase patterns with different gray values displayed on the computer screen provide different phase modulations in the SLM. The experimental and simulated results show that the phase modulation affects the beam shape of the Airy beam. If the maximum phase modulation is larger than 1.7pi, the Airy beam can keep its pattern and the unmodulated Gaussian beam can be neglected. Further, we design a 3D imaging system using the phase-space method based on the Wigner distribution function (WDF). We theoretically build the WDF for the Airy beam and show that the WDF of the Airy beam is shifted in the phase-space as the transverse position of incoming Airy beam changes, and the WDF is tilted as changing the axial position (along the propagation direction). A larger truncating factor can reduce the energy contribution in WDF for the side lobes and modify the shape of the main lobe close to a straight line. In this case, the WDF of the Airy beam is similar to that of the Gaussian beam. However, the DOF of the Airy beam is greatly improved. We perform experimental measurements of WDF using the basic idea of measuring both the space and spatial frequency information in a 4-f system. Our experimental results match the simulation and validate the 3D imaging using Airy beam in the phase space.

##### Degree

Ph. D.