Determination of the focal length of a lens generating aberrations

Abstract

In most laser applications it is necessary to focus, modify or shape the laser beam by using lenses and other optical elements. The most important characteristic of a lens is its focal length. The focal length of a lens gives a measure of how strongly the lens either converges or diverges light. When the lens is imperfect, it generates aberrations. This imperfection could be due to disfigured or imperfectly figured optics and misalignments of the lens in an optical system. It’s important to determine how the aberrations affect the focal length and hence the imaging properties of the lens and find ways of minimizing or eliminating these effects. This will improve the quality of the experiments and results achieved. Also in case of imperfect lenses e.g. those with cracked or deformed surfaces, cost of new purchases will be saved in that the imperfect lenses can still be used and the aberrations generated corrected. This thesis reports a theoretical and experimental investigation of the focal length of a lens generating aberrations. A theoretical model was formulated by considering the general case of a Gaussian laser beam passing through a lens with arbitrary aberrations and finite aperture. By considering the difference between the incoming and outgoing effective radii of curvature, an expression for the lens focal length as a function of the aberrations present was derived. In order to test the model, a lens with various aberrations was simulated with phase holograms in the laboratory. Primary aberrations with selected coefficients were programmed onto a phase Spatial Light Modulator whose Liquid Crystal Display (LCD) was active in reflective mode only. On striking the LCD, the incoming laser beam was reflected off it and into a two-dimensional Shack-Hartmann wavefront sensor. The sensor used binary optic lenslet arrays to directly measure the wavefront slope (phase gradient) of the laser beam. By integrating these measurements over the lens aperture, the wavefront or phase distribution was determined and the laser beam parameters presented in terms of Zernike polynomial coefficients. From these, Zernike primary aberrations affecting the field curvature immediately before and after the lens which in turn affects the accurate determination of its focal length was analysed from the graphs of lens power against aberrations and calculated using Mathematica software. The theoretical results show that out of all the primary aberrations used, only three, namely spherical, defocus and x-astigmatism aberrations have significant effect on the lens focal length. The presence of x- astigmatism creates two focal planes, one in the horizontal plane and the other in the vertical plane. This means that if x- astigmatism is non zero then the lens will have two focal planes. The experimental result validates the theoretical model satisfactorily.