The Application of Frequency Response Techniques in Optics
01 May 1962-Vol. 79, Iss: 5, pp 889-919
TL;DR: In this article, a resume of the treatment of image formation from the standpoint of the theory of passive linear systems is given, it being shown that image formation for an incoherent object satisfies the basic postulates of superposition and stationarity.
Abstract: A resume of the treatment of image formation from the standpoint of the theory of passive linear systems is given, it being shown that image formation for an incoherent object satisfies the basic postulates of superposition and stationarity. It then follows that the spatial frequency response of an optical system will be given by the Fourier transform of its impulse response, this latter being simply the distribution of intensity in the image of a narrow self-luminous line. There follows an account of work done in the author's image-assessment group at Imperial College. This includes the diffraction theory of optical frequency response, aberration tolerance theory and the numerical evaluation of frequency response and diffraction integrals together with examples of the response curves for particular cases. A number of methods for the measurement of frequency response are described, and the theory of these methods and results showing the comparison of theoretical and measured response curves are discussed. The final section describes methods for the measurement of the Fourier spectra of photographic images, and their application to the study of the influence of the detector properties on recorded images.
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TL;DR: The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object.
Abstract: The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object. There results a four-fold integral involving these functions, and the complex conjugate functions of the latter two. This integral is evaluated in terms of the Fourier transforms of the coherence function, the diffraction distribution function and its complex conjugate. In fact, these transforms are respectively the distribution of intensity in an 'effective source', and the complex transmission of the optical system-they are the data initially known and are generally of simple form. A generalized 'transmission factor' is found which reduces to the known results in the simple cases of perfect coherence and complete incoherence. The procedure may be varied in a manner more suited to non-periodic objects. The theory is applied to study inter alia the influence of the method of illumination on the images of simple periodic structures and of an isolated line.
566 citations
TL;DR: The optical MTF of the authors' subjects’ eyes is computed and it is found that the role of aberrations in degrading the MTF may be greater than generally believed.
Abstract: We have designed an aberroscope that differs from Tscherning’s classical instrument in that it makes use of an artificial astigmatism rather than an artificial myopia to defocus the image of a point source of light. A subject views the source through a ±5 D crossed cylinder lens with axes at 45° to the principal axes of an intercalated grid and sees a shadow image of the grid. The distortions of this grid image are quantitatively related to the wave aberration of the eye. Using this device we have obtained drawings for more than 50 subjects. These drawings of the grid pattern have been analyzed by means of a two-dimensional polynomial curve Fitting technique that computes Taylor polynomial terms to the fourth order. From the Taylor coefficients it is possible to reconstruct the wave aberration surface. Examination of the Taylor terms so obtained shows that the monochromatic aberrations of the eye are dominated by third-order Taylor terms within the range of physiological pupil sizes, and that spherical aberration frequently appears predominantly about one axis only, a condition that we have termed “cylindrical” aberration. We have computed the optical MTF of our subjects’ eyes and find that the role of aberrations in degrading the MTF may be greater than generally believed.
406 citations
TL;DR: The subjective crossedcylinder aberroscope method of Howland and Howland as discussed by the authors has been modified by the addition of a beam splitter and a camera to permit direct photographic recording of the distorted retinal image of the aberronscope grid.
Abstract: The subjective crossed-cylinder aberroscope method of Howland and Howland [J. Opt. Soc. Am. 67, 1508 (1977)] has been modified by the addition of a beam splitter and a camera to permit direct photographic recording of the distorted retinal image of the aberroscope grid. The ocular aberration can then be deduced from direct measurements of the grid distortion. Preliminary results on 11 subjects confirm earlier findings that comalike, third-order aberrations are more important than spherical or other fourth-order aberrations in degrading the retinal image and for the average subject, the diffraction-limited pupil size is approximately 3 mm. This new objective method for measuring wave aberration yields significantly less variance in population estimates of the coefficients of the wave-aberration polynomial than that of the previous subjective method.
238 citations
TL;DR: This article examined the contribution of optical and photoreceptor properties as well as receptor pooling to eccentricity-dependent variations in spatial vision by comparing the performance of ideal observers with that of human observers.
Abstract: We examined the contribution of optical and photoreceptor properties as well as receptor pooling to eccentricity-dependent variations in spatial vision by comparing the performance of ideal observers with that of human observers. We measured contrast sensitivity functions in human observers and calculated such functions in ideal observers for retinal eccentricities of 0-40 deg. Comparisons of human and ideal performance in a variety of tasks reveal that many aspects of the variation in spatial vision with eccentricity can be understood from an analysis of the discrimination information available at the retinal ganglion cells.
204 citations
TL;DR: In this article, the authors measured the contrast sensitivity of older subjects through natural pupils and compared the results with those from a group of younger subjects using a crossed-cylinder aberroscope and calculated modulation transfer functions (MTFs) and root-mean-squared (RMS) wave-front aberrations for fixed pupil diameters of 4 mm and 6 mm.
Abstract: We measured the contrast sensitivity (CS) of a group of older subjects through natural pupils and compared the results with those from a group of younger subjects. We also measured each subject’s monochromatic ocular wave-front aberrations using a crossed-cylinder aberroscope and calculated their modulation transfer functions (MTF’s) and root-mean-squared (RMS) wave-front aberrations for fixed pupil diameters of 4 mm and 6 mm and for a natural pupil diameter. The CS at a natural pupil diameter and the MTF computed for a fixed pupil diameter were found to be significantly poorer for the older group than for the younger group. However, the older group showed very similar MTF’s and significantly smaller RMS wave-front aberrations compared with the younger group at their natural pupil diameters, owing to the effects of age-related miosis. These results suggest that although monochromatic ocular wave-front aberrations for a given pupil size increase with age, the reduction in CS with age is not due to this increase.
151 citations
References
More filters
Journal Article•
TL;DR: The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object.
Abstract: The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object. There results a four-fold integral involving these functions, and the complex conjugate functions of the latter two. This integral is evaluated in terms of the Fourier transforms of the coherence function, the diffraction distribution function and its complex conjugate. In fact, these transforms are respectively the distribution of intensity in an 'effective source', and the complex transmission of the optical system-they are the data initially known and are generally of simple form. A generalized 'transmission factor' is found which reduces to the known results in the simple cases of perfect coherence and complete incoherence. The procedure may be varied in a manner more suited to non-periodic objects. The theory is applied to study inter alia the influence of the method of illumination on the images of simple periodic structures and of an isolated line.
566 citations
TL;DR: The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object.
Abstract: The theory of image formation is formulated in terms of the coherence function in the object plane, the diffraction distribution function of the image-forming system and a function describing the structure of the object. There results a four-fold integral involving these functions, and the complex conjugate functions of the latter two. This integral is evaluated in terms of the Fourier transforms of the coherence function, the diffraction distribution function and its complex conjugate. In fact, these transforms are respectively the distribution of intensity in an ‘effective source’, and the complex transmission of the optical system— they are the data initially known and are generally of simple form. A generalized ‘transmission factor’ is found which reduces to the known results in the simple cases of perfect coherence and complete incoherence. The procedure may be varied in a manner more suited to non-periodic objects. The theory is applied to study inter alia the influence of the method of illumination on the images of simple periodic structures and of an isolated line.
550 citations
TL;DR: In this paper, the response of a defocused aberration-free optical system to line-frequencies in the object is studied analytically, and curves are given showing the response as a function of line-frequency for a range of values of defect of focus.
Abstract: The response of a defocused aberration-free optical system to line-frequencies in the object is studied analytically. Curves are given showing the response as a function of line-frequency for a range of values of defect of focus. A comparison is made with the results to be expected from geometrical optics. A tolerance for defect of focus is obtained, which accords well with published experimental results. Both circular and rectangular apertures are considered.
489 citations
TL;DR: In this paper, it was shown that the diameter of the area of coherence on a plane illuminated by a source of angular radius is given by d = d = 0 √ √ n √ 16 λ n N ε, where N is the refractive index of the intervening medium.
Abstract: It is shown that a 'phase-coherence factor' may be defined in a manner which leads, without recourse to explicit statistical analysis, to the theorems established by van Cittert (1934) and Zernike (1938) for analogous factors. An invalid approximation made in their calculations of the phase-coherence factor for a plane illuminated directly by a source is corrected. The new treatment is applied to the theory of Young's experiment, the stellar interferometer, and illumination in the microscope. The phase-coherence factor defined here enables a general theory of the formation of optical images to be formulated. Further, it is shown that the diameter of the area of coherence on a plane illuminated by a source of angular radius $\alpha $ is given by d = $\frac{0\cdot 16\lambda}{N\,\sin \,\alpha}$, where N is the refractive index of the intervening medium.
252 citations