VIII Evaluation, Design and Extrapolation Methods for Optical Signals, Based on Use of the Prolate Functions

Publisher Summary This chapter discusses applications, specifically dealing with the performance characteristics of optical systems. The linear prolate functions are a set of band limited functions, which like the trigonometric functions, are orthogonal and complete over a finite interval. However, unlike the trig functions, they are also complete and orthogonal over the infinite interval. Fourier transform of a linear prolate function is proportional to the same prolate function. The mathematical properties of the circular prolates, which make them readily applicable to the analysis of optical imagery, are summarized in the chapter. The optical applications arise from the modern use of the prolates as a convenient set of one-dimensional, orthogonal functions. These applications have been on the general subject of optical systems analysis. Performance characteristics of the laser have been established, along with the ultimate ability of lens and lensless systems to form high quality images. There is one physical phenomenon that unifies these applications—diffraction at a finite aperture—and it can be said that this phenomenon is optimally analyzed by use of the prolate functions.

[1]  D. Auston,et al.  Transverse mode locking , 1968 .

[2]  G. D. Boyd,et al.  Generalized confocal resonator theory , 1962 .

[3]  J. C. Heurtley.  Hyperspheroidal Functions-Optical Resonators with Circular Mirrors , 1964 .

[4]  J. Gordon,et al.  Confocal multimode resonator for millimeter through optical wavelength masers , 1961 .

[5]  D. Slepian.  Analytic Solution of Two Apodization Problems , 1965 .

[6]  B. Frieden.  Image Evaluation by Use of the Sampling Theorem , 1966 .

[7]  C. Niven.  V. On the conduction of heat in ellipsoids of revolution , 1880, Philosophical Transactions of the Royal Society of London.

[8]  R. Lytle,et al.  Prolate spheroidal antennas in isotropic plasma media , 1969 .

[9]  Guy Lansraux,et al.  MAXIMUM OF THE FACTOR OF ENCIRCLED ENERGY , 1961 .

[10]  G. D. Francia.  Degrees of Freedom of Image , 1969 .

[11]  G. Buck,et al.  Resolution limitations of a finite aperture , 1967 .

[12]  B. Roy Frieden,et al.  On Arbitrarily Perfect Imagery with a Finite Aperture , 1969 .

[13]  C. K. Rushforth,et al.  Restoration, Resolution, and Noise , 1968 .

[14]  H. Wolter.  V On Basic Analogies and Principal Differences Between Optical and Electronic Information , 1961 .

[15]  R. Barakat.  Application of the Sampling Theorem to Optical Diffraction Theory , 1964 .

[16]  B. Roy Frieden,et al.  Band-Unlimited Reconstruction of Optical Objects and Spectra* , 1967 .

[17]  D. Slepian.  Prolate spheroidal wave functions, Fourier analysis and uncertainty — IV: Extensions to many dimensions; generalized prolate spheroidal functions , 1964 .

[18]  D. Slepian.  Some Asymptotic Expansions for Prolate Spheroidal Wave Functions , 1965 .

[19]  von F. Zernike.  Beugungstheorie des schneidenver-fahrens und seiner verbesserten form, der phasenkontrastmethode , 1934 .

[20]  William Streifer,et al.  Optical Resonator Modes—Rectangular Reflectors of Spherical Curvature* , 1965 .

[21]  J. L. Harris,et al.  Diffraction and Resolving Power , 1964 .

[22]  Itoh Yoshinobu.  Evaluation of Aberrations Using the Generalized Prolate Spheroidal Wavefunctions , 1970 .

[23]  Richard Barakat,et al.  Solution of the Luneberg Apodization Problems , 1962 .

[24]  C. W. Barnes.  Object Restoration in a Diffraction-Limited Imaging System , 1966 .

[25]  B. Frieden.  The extrapolating pupil, image synthesis, and some thought applications. , 1970, Applied optics.

[26]  C. Helstrom.  Image Restoration by the Method of Least Squares , 1967 .

[27]  D. Slepian,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — II , 1961 .

[28]  C. Rino.  Bandlimited Image Restoration by Linear Mean-Square Estimation , 1969 .