James E. Harvey

Fourier treatment of near‐field scalar diffraction theory

A general treatment of scaler diffraction theory is presented and some interesting concepts are discussed which yield new insight into the phenomena of diffraction throughout the whole space in which it occurs. The direct application of Fourier transform theory to the diffraction process results in two equivalent descriptions of the diffracted wave field: one describes the wave field as a superposition of plane‐wave components and corresponds to the transfer function approach in linear systems theory, and the other describes the wave field as a superposition of hemispherical‐wave components (or Huygen wavelets) and corresponds to the impulse response approach in linear systems theory. The convolution of the initial disturbance with the impulse response results in the well‐known Rayleigh‐Sommerfeld formula for near‐field diffraction. This formula is then rewritten in the form of the Fourier transform integral of a generalized pupil function which includes phase variations in the diffracting aperture. Any d...

LIGHT-SCATTERING CHARACTERISTICS OF OPTICAL SURFACES

A scalar theory of surface scattering phenomena has been formulated by utilizing the same Fourier techniques that have proven so successful in the area of image formation. An analytical expression has been obtained for a surface transfer function which relates the surface micro-roughness to the scattered distribution of radiation from that surface. The existence of such a transfer function implies a shift-invariant scattering function which does not change shape with the angle of the incident beam. This is a rather significant development which has profound implications regarding the quantity of data required to completely characterize the scattering properties of a surface. This theory also provides a straight-forward solution to the inverse scattering problem (i.e., determining surface characteristics from scattered light measurements) and results in a simple method of predicting the wave length dependence of the scattered light distribution. Both theoretical and experimental results will be presented along with a discussion of the capabilities and limitations of this treatment of surface scatter phenomena.

The spot of Arago: New relevance for an old phenomenon

The ‘‘spot of Arago’’ has been a controversial topic since its inception in 1818 when Poisson predicted its existence in an attempt to discredit Fresnel’s wave theory of light. Arago performed the experiment and found the surprising prediction was true, thus putting Fresnel’s theory on a firm technical foundation. In recent years, the spot of Arago, which exists as a bright spot at the center of the geometrical shadow of a circular obstruction, has caused substantial grief in various high‐energy laser applications and has come to be considered more of a nuisance than a curiosity. This paper suggests that the size and shape of the spot of Arago is characteristic of the wave‐front aberrations of the incident beam and can therefore be used to advantage as a beam sample for wave‐front analysis of annular beams. The implementation of this wave‐front sampling scheme would eliminate the requirement for a special beam‐sampling optical component and thus reduce to a minimum the deleterious effects upon the beam fr...

Transfer Function Characterization Of Grazing Incidence Optical Systems

By using Fourier techniques and linear systems theory we have derived an analytic expression for a generalized transfer function for grazing incidence optical systems operating at ultraviolet and x-ray wave-lengths that includes the effects of optical fabrication errors over the entire range of relevant spatial frequencies. The Fourier transform of this transfer function yields the image distribution (or point spread function) from which encircled energy characteristics or other image quality criteria can be obtained. This transfer function characterization of grazing incidence optical systems allows parametric trade studies and sensitivity analyses to be performed as well as the derivation of fabrication tolerances necessary to satisfy a given image quality requirement.

Surface Scatter Phenomena: A Linear, Shift-Invariant Process

Empirical experimental scattering data from conventional optical surfaces is shown to exhibit shift-invariant behavior with respect to incident angle when plotted in direction cosine space. This implies the existence of a surface transfer function that completely characterizes the scattering properties of the surface, and permits the application of linear systems theory and Fourier techniques in modeling the scattering effects of optical surfaces. A theoretical basis for this behavior is illustrated by showing that scalar diffraction phenomena (conical diffraction from gratings) is shift-invariant with respect to incident angle only in direction cosine space, and surface roughness can be considered to be composed of a superposition of sinusoidal phase gratings. The fact that many optical surfaces of interest deviate from this shift-invariant behavior does not invalidate the usefulness of the linear systems formalism. The ideal behavior of a shift-invariant scattering process can still be used for making engineering calculations and retained as the reference from which scattering from real surfaces is compared. This is completely analogous to the universally accepted transfer function characterization of imaging systems in spite of the fact that few real imaging systems are isoplanatic (no field-dependent aberrations).

A Parametric Study Of Various Synthetic Aperture Telescope Configurations For Coherent Imaging Applications

Synthetic aperture telescopes fall into two generic types; those consisting of a segmented primary mirror (perhaps substantially thinned or diluted) with a common secondary mirror, and those made up of an array of independent telescopes. Advantages and disadvantages of each type will be discussed. The diffraction-limited optical performance of several subaperture configurations (both redundant and non-redundant) will be presented and compared in terms of point spread function (PSF) characteristics and encircled energy plots. In practice, this ideal optical performance is degraded by various design, manufacturing, and operational errors. On-axis optical performance degradation due to rms phase errors, rms pointing errors, and rms focus errors between the subapertures making up the synthesized aperture will then be discussed. Coherent imaging with phased arrays of independent afocal telescopes essentially break down into two operations; 1) a pupil mapping operation in which the width-to-separation of the reduced beams entering the beam combining telescope must replicate that of the collecting subapertures, and 2) a Fourier Transform operation which forms the coherent image from these combined beams or subapertures. A quantitative analysis of the off-axis optical performance degradation due to pupil mapping errors will be presented, as will the field dependent effects of residual design aberrations of the independent telescopes. These inherent limitations will be discussed with respect to the field-of-view requirements of various applications of scenarios.

Diffraction Effects in Grazing Incidence X-Ray Telescopes

There is increasing interest worldwide in the use of tightly nested grazing incidence imaging mirrors for high-throughput x-ray telescopes. Diffraction effects of x-ray optical systems are often (justifiably) ignored due to the small wavelength of the x-ray radiation. However, the extremely large obscuration ratio inherent to grazing incidence optical systems produces profound degradation of the diffraction image over that produced by a moderately obscured aperture of the same diameter. Although many of the intended applications are moderate-resolution spectroscopic instruments, there is always a desire for high-resolution imaging as well. In this paper we show that diffraction effects can dominate other potential error sources at the low-energy (long-wavelength) end of the intended operating spectral range of some existing or planned x-ray telescopes. Parametric performance predictions are presented and compared with x-ray astronomy performance goals.

Optical performance of synthetic aperture telescope configurations

The optical performance of synthetic aperture arrays of independent telescopes, used for coherent imaging or beam transmission, is degraded by various design, manufacturing, and operational errors. Diffraction-limited performance will be presented in terms of point spread function (PSF) profiles and encircled energy plots. On-axis optical performance degradation due to RMS phase errors, RMS pointing errors, and RMS focus errors will then be presented as will off-axis optical performance degradation due to pupil mapping errors and field curvature of the individual telescopes.

Design and performance of ranging telescopes: monolithic versus synthetic aperture

For the special case of monolithic ranging telescopes, a family of characteristic design curves for various system performance requirements is discussed. The required secondary mirror displacement and the tolerance on this position, as well as the corresponding depth of field for any desired range, can be readily obtained from this normalized family of curves. The hyperfocal distance (closest range within depth of field when infinite range is also in focus) is also displayed on this set of curves and can be used to determine when it is necessary to activate ranging. The degradation in system performance is then plotted versus range, and the closest effective range is determined. This range is a strong function of telescope diameter and is crucial to certain ranging telescope applications. This degradation in system performance is then deter-mined for various properly phased synthetic-aperture systems (multiple-mirror telescopes) and compared to the monolithic telescope of equivalent aperture. For certain applications these results provide a strong motivation for going to synthetic-aperture telescopes based upon optical performance alone.

Systems engineering analysis of image quality

A linear systems approach (multiplying MTFs or convolving PSFs) to performing a complete systems engineering analysis of image quality is described. This includes not only the traditional diffraction analysis and the evaluation of image degradation from residual design errors: but also includes image degradation due to scattering effects from residual optical fabrication errors, assembly and alignment errors, and all other potential error sources appearing in a detailed error budget tree. The effects of mosaic detector arrays upon systems performance and the optimum system design will also be discussed. This analysis allows optical fabrication tolerances to be determined during the design phase of a program, frequently resulting in substantial cost and schedule savings. Inaccuracies in the linear systems assumption will be presented for several different applications.