The modulation function in optics
O'Neill, Edward L
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It is the purpose of this paper to present a method for dealing with optical processes in terms of their capability for preserving object detail in the final image. In general, the main body of the dissertation is divided into three sections. Section I presents, in detailed fashion, a comprehensive background for the subsequent use of Fourier series and integrals. As a starting point, the Sturm-Liouville differential equation is taken and the orthogonality principle for eigenfunctions is developed from it. Applying the results to the one dimensional wave equation, the Fourier series representation is derived along with the expression for discrete eigenvalues. By extending the finite boundaries to infinity, the Fourier integral representation is presented along with the expression for continuous eigenvalues of periodic and transient functions respectively is then presented. For problems possessing rotational symmetry, the Fourier Bessel series is developed from the wave equation in polar coordinates. A demonstration of the orthogonality condition for Bessel functions over an infinite range is presented. The section closes with a brief description of the conditions for performing a Fourier transformation and the interpretation of a linear passive system. Section II deals exclusively with the application of linear network theory to several problems arising in diffraction theory. Emphasis is placed on the methods of evaluating and the interpretation of the modulation function as a contrast rendition versus spatial frequency curve for sine wave targets. Starting with the wave equation for one of the components of the electromagnetic field quantities, the resultant Kirchhoff-Huygens formulation of the diffraction problem is derived making use of the Green's function. With the usual feasible approximations made in the optical case, it is shown that the amplitude of the disturbance in the image plane is the two dimensional Fourier transform of the phase function over the aperture, a result said to have first been pointed out by Michelson. The phase function is such that for an aberrationless system, the actual wave front coincides with the reference sphere and hence the phase function is unity over the aperture and zero elsewhere. With the added relation that the modulation function and intensity distribution for a point source are Fourier transform pairs, it is then shown that the modulation function can be evaluated by means of a convolution integral of the phase function with its complex conjugate. This latter relationship was pointed out by Luneberg and made use of recently by Marechal. The various Fourier relationships are then demonstrated for the one dimensional problem of diffraction by a single slit. The interpretation of the modulation function in terms of contrast rendition and resolution limit is then emphasized for periodic targets. In the two dimensional problem of the circular aperture the geometrical scheme and Steel's method for computing the modulation function are demonstrated. However, since the modulation function is defined as the Fourier transform of the point image intensity distribution, a method for taking the Fourier-Bessel transform of the intensity distribution for a circular aperture is presented. The method used is that of integration with respect to a parameter, the results of which are in complete accord with the other methods of determining the modulation function. Finally the same method of integration is applied to the problem of the annulus and a family of modulation curves with the ratio of inner to outer radii as a parameter is plotted. In section III, the Fourier analysis of several simple scanning systems and an ideal defocused lens are presented. Included in the analysis is a discussion of the phenomena of spurious resolution for periodic targets. A general treatment of edge deterioration and image quality is then given for any symmetrical, degrading point image intensity distributions. Attention is then focused on the problem of synthesizing an optical system as opposed to analysis previously treated. Before such a procedure is outlined a general treatment of random functions is presented and the need for space averages of the functions involved is shown. This discussion leads into the concepts of correlation functions and power density spectra as Fourier transform pairs. In the light of the fact that most optical systems employ incoherent illumination, various restrictions and limitations on the design of an optimum linear filter in the Wiener-Hopf sense are presented. A brief description of specific synthesizing procedures actually in existence is then listed. Finally recommendations for future research in this field are outlined along with the summary of the dissertation proper.
This item was digitized by the Internet Archive. Thesis (Ph.D.)--Boston University
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