Showing papers in "Journal of the Optical Society of America in 1966"

Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and Very Short Exposures

Abstract: A theoretical foundation is developed for relating the statistics of wave distortion to optical resolution. The average resolution of very-long- and very-short-exposure images is studied in terms of the phase- and log-amplitude-structure functions, whose sum we call the “wave-structure function.” Those results which are comparable are in agreement with the findings of Hufnagel and Stanley who studied the average modulation transfer function of long-exposure images. It is found that the average short-exposure resolution can be significantly better than the average long-exposure resolution, but only if the wave distortion does not include substantial intensity variation.

Spatial and Temporal Contrast-Sensitivity Functions of the Visual System

Abstract: T HE dependence of the form of the spatial contrast-sensitivity function for a square-wave test grating upon the duration of exposure of the target has been investigated by Schober and Hilz. 1 Kelly 2 has pointed out an analogous dependence of the form of the temporal contrast (modulation) sensitivity function upon the angular extent of the test target. The reciprocal nature of these spatio-temporal interactions can be particularly clearly ap­ preciated if the threshold contrast is determined for a grating target whose luminance perpendicular to the bars is given by where m is the contrast, v the spatial frequency, and ƒ the temporal frequency of the target. FIG. 2. Temporal contrast-sensitivity (reciprocal of threshold contrast) functions for different spatial frequencies. The points are the means of four measurements and the curves (two with dashed low-frequency sections) differ only in their positions along the contrast-sensitivity scale, O 0.5 cycle per degree, ● 4, ∆ 16, ▲ 22 cycles per degree. FIG. 1. Spatial contrast-sensitivity (reciprocal of threshold contrast) functions for different temporal frequencies. The points are the means of four measurements and the curves (one with a dashed low-frequency section) differ only in their positions along the contrast-sensitivity scale O 1 cycle per second, ● 6, ∆ 16, ▲ 22 cycles per second. Such a pattern was set up as a display on a cathode-ray oscil­ loscope and Figs. 1 and 2 show the results of threshold-contrast measurements made by the author (a well-corrected myope). Viewing was binocular at a distance of 2 m. The grating pattern subtended 2.5°×2.5° in the center of a 10°× 10° screen illuminated to the same mean luminance of 20 cd/m 2. The general similarity of the two sets of contrast-sensitivity functions is immediately evident but two features are particularly remarkable. First, the form of the fall-off in sensitivity at high

Analysis of response patterns of LGN cells.

Abstract: A detailed analysis was made of the response characteristics of single cells in the lateral geniculate nucleus of the macaque monkey. The goal was to understand how these cells contribute to the processing of visual information. Data were analyzed from a representative sample of 147 cells, whose responses to equal-energy spectra (presented as diffuse flashes of monochromatic light) were recorded at three radiance levels. On the basis of their responses, the cells were divided into two general classes: (a) spectrally nonopponent cells which respond to all wavelengths with either an increase or decrease in firing rate, (b) spectrally opponent cells (about two-thirds of the sample) which respond with an increase in firing rate to some parts of the spectrum and a decrease to other parts. Four types of opponent cells were found: (i) red excitatory and green inhibitory (+R−G), (ii) green excitatory and red inhibitory (+G−R), (iii) yellow excitatory and blue inhibitory (+Y−B), (iv) blue excitatory and yellow inhibitory (+B−Y). Comparisons with psychophysical data indicated that nonopponent cells transmit brightness information; opponent cells, however, carry information about color, the hue of a light being determined by the relative responses of the four types. The saturation of spectral lights appears to be related to the differences in responses of opponent and non-opponent cells.

Optical Systems with Resolving Powers Exceeding the Classical Limit. II

Abstract: The fundamental invariant of an optical system is the number N of degrees of freedom of the message it can transmit. The spatial bandwidth of the system can be increased over the classical limit by reducing one of the other constituent factors of N. As examples of this invariance theorem N=const. established in Part I of this series [ J. Opt. Soc. Am.56, 1463 ( 1966)], we discuss (a) a system whose spatial-bandwidth increase is achieved by a proportional reduction of its temporal bandwidth, and (b) the airborne synthetic-aperture, terrain-mapping radar, whose spatial resolution comes from exploitation of the temporal degrees of freedom of the received signal. The increase of the spatial bandwidth beyond the classical limit is, however, limited by the appearance of evanescent waves.The number of degrees of freedom of the object wave field stored in a hologram is discussed. The storage capacity of the photographic plate, which is proportional to its size times its spatial cutoff frequency, is fully exploited only by single-sideband Fraunhofer but not by single-sideband Fresnel holograms.

Frequency Doubling in Visual Responses

Abstract: This paper reports an unexpected visual phenomenon. When a wide, photopic stimulus field is sinusoidally modulated in both space and time, over a certain frequency range the apparent spatial frequency of the stimulus is doubled. In its original form, the (deLange) flicker-fusion model which has been accepted by the author and others cannot account for this result. But it can be explained by assuming that there is a second (low-pass) filtering operation which follows the nonlinear (brightness) response of the visual system, rather than preceding it. If this hypothesis is correct, then the frequency-doubling effect is the result of neural mechanisms which are more central than the locus of flicker fusion.

Diffraction of a Plane Wave at a Sinusoidally Stratified Dielectric Grating

Abstract: The problem of diffraction at a sinusoidally stratified dielectric grating is treated. The analysis of the reconstruction process from a hologram formed in a “thick” photographic emulsion leads to this problem, but it may also occur in other areas. A rigorous solution of this problem is presented for both polarizations of the electric wave. The solution is evaluated numerically for values of the different parameters that are typical for holograms. It is shown that the amplitude of the diffracted light has a maximum if the light is incident at the Bragg angle, a feature already observed experimentally. Results are given for different values of the period as well as the thickness of the grating and for both polarizations.

Correction of Asymmetric Interferograms Obtained in Fourier Spectroscopy

Abstract: A method is given of correcting the asymmetry of an interferogram, caused by either defects in the interferometer or off-center sampling. The method makes use of the phase error, which can be derived from the central portion of the interferogram or from a separate calibration. The technique avoids the disadvantages of the double-sided transform, normally used to derive the spectrum from such interferograms.

Limiting Resolution Looking Down Through the Atmosphere

Abstract: The turbulence of the atmosphere places an upper limit on the quality of an image of ground objects obtained by long-exposure photography from high altitudes in the atmosphere or in space. (By making the imaging optics good enough, the film resolution fine enough, and the platform stable enough, this limit could be approached but not exceeded.) A useful quantity for indicating the magnitude of this limit is the integral of the MTF associated with the turbulence. Treating the integral as a two-dimensional bandwidth, one-half the inverse of its square root can be associated with a resolution length, or angle, in the same manner that an electrical engineer associates a rise time with one-half the inverse of the bandwidth of an R–C filter. Based on published data for typical strength of atmospheric turbulence, the integral of the MTF was calculated as a function of altitude and the corresponding resolution computed. This resolution is shown to correspond to a length of about 4.6 cm on the ground. It is shown that as an observer goes deeper into space, this limiting ground resolution remains constant, but the diameter of the optics needed to approach the limit goes up. Graphs of achievable ground resolution at any altitude and of the diameter of the optics needed to approach this limit are presented.

Near-Infrared Planetary Spectra by Fourier Spectroscopy. I. Instruments and Results

Abstract: A near-infrared, two-beam interferometer has been built for astronomical observations by Fourier transform spectroscopy. Various improvements, especially a highly accurate interferometrically controlled stepping drive, have resulted in the production of laboratory spectra with 0.1-cm−1 resolution and unusually clean instrumental line shape, and spectra of Venus and Mars with about 1-cm−1 resolution.

Spatial Resolution by the Human Visual System. The Effect of Mean Retinal Illuminance

Abstract: This paper reports experimental results showing the effect of mean retinal illuminance on the modulation transfer function (MTF). The line spread function of the human visual system was computed. The test pattern in which the luminance varied sinusoidally in the horizontal direction was generated on the face of an oscilloscope by modulating the intensity of the beam.The behavior of the system changes with the mean retinal illuminance. At the highest mean retinal illuminance (1000 td) the MTF shows two peaks which gradually disappear at lower mean retinal illuminances. The line spread function of the entire visual system also exhibits a change in shape dependent on the mean retinal illuminance. In particular, what are deemed inhibitory influences become less prominent at lower mean retinal illuminance. From the line spread function of the optical part up to the retina, the line spread function of the physiological part of the visual system is also evaluated.A common asymptote appears at high frequencies when the characteristics are plotted in terms of absolute deviation from the mean versus spatial frequency for threshold perception. The asymptote provides an upper bound for the behavior of visual acuity at any retinal illuminance.

Eye-Movement Responses to Step and Pulse-Step Stimuli

Abstract: A spot of light is presented to an observer who tracks its movement visually, doing so as quickly and accurately as possible. The positions of the eye are continuously recorded so that direction and magnitude of eye movements as a function of time can be assessed. Without warning, the target spot steps from its resting position, moving 6° horizontally to one side, followed after a time W by a 12° step in the opposite direction. The result is a pulse-step pattern of target motion with the time interval W msec defining the pulse duration. The directions of the pulse and step are always opposite but otherwise are unpredictable. Trials consisting of pulses followed by steps are intermixed randomly with a larger number of trials consisting of 6° steps alone. The experiments demonstrate that the visual system is sometimes able to cancel an eye-movement response to a pulse, on the basis of information contained in the subsequent step, to which it responds instead. As the step is delayed by progressively longer pulses, the probability increases that a response to the pulse will occur. If a response does occur in the direction of the step, it begins about 325 msec after the beginning of the step. This latency is independent of pulse time W and is about 40 msec longer than the latency of responses to steps presented alone. It is concluded that the visual system utilizes this 40 msec to operate upon a latent response to a pulse, and thereby to cancel its overt manifestation (eye movement) before initiating a response to the second, incompatible stimulus.

Polarization, Directional Distribution, and Off-Specular Peak Phenomena in Light Reflected from Roughened Surfaces

Abstract: Directional distribution of light reflected from roughened surfaces with measurements made of plane-polarized components of reflected light as well as of mixed radiation

Dispersion of Nitrogen

Abstract: The refractivity of nitrogen gas has been measured, with high relative precision, at vacuum wavelengths from 4679 A to 20 586 A. The absolute value of refractivity for 5462 A, reduced to one atmosphere at 0°C., is 2.991×10−4. A dispersion formula fitting the data well at 15°C. is 108 (n−1)= 6497.378+3 073 864.9/(144-σ2), where σ is wave number in reciprocal microns. This is compared to Svensson’s dispersion formula for air. Alternative dispersion formulas are discussed. Quantities related to dispersion are deduced: mean absorption frequency, effective total oscillator strength., dielectric constant, Verdet constant, and Rayleigh scattering coefficient.

Lightness change of grays induced by change in reflectance of gray background.

Abstract: A relatively small gray sample looks lighter (darker) when it is placed on darker (lighter) background. This phenomenon is an example of the well-known contrast effect. The purpose of this paper is to make a quantitative determination of the effect, and to derive a formula for it.For any given gray sample and gray surround on the left side, the observer was instructed to choose, for the different gray surround on the right, a different gray sample appearing as light as that on the left. It was found that sample lightness changes rapidly with reflectance when sample reflectance is close to that of the background. This effect was named the “crispening effect.”Several models (von Kries coefficient law, Hurvich–Jameson induction) were tried, but none of them reproduced the experimentally discovered crispening effect. A fairly successful empirical formula was developed by adding a term for the crispening effect to the formula for the induction theory.

Image Evaluation and Restoration

Abstract: The evaluation of an image must depend upon the purpose for which the image was obtained and the manner in which the image is to be examined. Where the goal is extraction of information and where the image is to be processed prior to viewing, the information content of the image is the only true evaluation criterion. Under these conditions, the improvement achieved by processing can be evaluated by comparing the ability of the human observer to extract information from the image before and after processing. The extent to which the processing approaches the optimum can be evaluated by determining the fraction of the total information content of the image which can be visually extracted after processing. The basic mathematical concepts of image processing are indicated, relating the input point spread function (p. s. f. of the unprocessed image), the processing point spread function (p. s. f. which defines the processing operation), and the output point spread function (p. s. f. of the processed image), and their Fourier domain equivalents. Examples are shown of images which have been processed and the details of the processing operations are described.

Metal Photocathodes as Secondary Standards for Absolute Intensity Measurements in the Vacuum Ultraviolet

Abstract: Metal photocathodes as secondary standards for absolute intensity measurements in vacuum UV

Theory of Interferometric Analysis of Laser Phase Noise

Abstract: The line width of a well-stabilized laser operating far above threshold is determined by random fluctuations of the phase. This paper discusses several types of experiments which can give information about the details of this phase random process. In order to study the laser phase noise experimentally the laser signal (containing phase noise only) must be passed through some type of interferometer which will convert the phase noise to intensity noise. The various properties of this derived intensity noise which may then be determined are its probability density, first and second moments, autocorrelation function, and spectrum. These measurable quantities depend on two factors; the first and more fundamental is the joint probability distribution for the change in phase in a given time. The second factor is the manner of operation of the interferometer in changing phase to intensity noise. We discuss both two-beam and multiple-beam interferometers and derive theoretical expressions for the above-mentioned properties of the output intensity fluctuations. It is interesting that although in both cases the output intensity fluctuations are nongaussian random processes, it is nevertheless possible to derive a number of useful theoretical results.

Effects of Coherence on Imaging Systems

Abstract: The effects of imaging with coherent light are investigated and related to the parameters of an imaging system. Theoretical results are presented to describe the imaging parameters and the coherent field effects. The experimental results show the coherent field effects of edge ringing and shifting, speckling and field-of-view limitation. Incoherent results are presented in some cases for comparison.

Far-Infrared Spectrum of Liquid Water*

Abstract: The infrared spectra of H2O and D2O in the liquid state at ambient temperature (30°C) have been remapped in the spectral region between 10 and 330 μ. The major features observed were extremely intense absorption bands with maxima at 685 and 505 cm−1 in H2O and D2O, respectively. These major bands are overlapped at the low-frequency ends by much less intense bands producing transmittance minima near 193 and 187 cm−1, respectively. No evidence was obtained for the series of narrow bands recently reported by Stanevich and Yaroslavskii. Extinction coefficients have been determined for the range 170–50 cm−1 and are compared with recent data; present data on linear absorption coefficients for H2O in the range 1500–1100 cm−1 are in fair agreement with the results of previous workers. The influence of temperature variations on the frequencies of infrared bands has been studied for all bands in the region between 4000 and 32 cm−1. Theoretical interpretation of the results is discussed briefly.

Interference Microscope with Total Wavefront Reconstruction

Abstract: An interference microscope is described, constructed in 1951–56, in which three-dimensional objects can be reconstructed, correctly incorporating amplitudes and phases, from two photographs simultaneously taken on one plate. These photographs are “holograms,” that is to say, records of the interference of the image-carrying wave, split in two, with a coherent background wave, also split in two. A phase difference of a quarter wave is produced, for the two otherwise identical photographs, between the image-carrying waves and their respective coherent backgrounds. The two photographs are in sine–cosine or “quadrature” relation; between them they contain the full optical information. If they are illuminated in such a way that there is a difference of a quarter wave in the phases at two corresponding points, and if the two beams are united, the original image-carrying wave is restored correctly in amplitude and in phase. An essential part of the instrument is a “quadrature prism”; a beam splitter with a three-layer sandwich, which establishes the quadrature relation in the taking of the holograms.The microscope has the advantage that it need not be focused, as it gives a three-dimensional reconstruction. Moreover, photographs can be taken with 1/10 or even 1/100 of the light required for exposure going through the object; the rest of the energy is supplied by the background beam, which is 10–100 times stronger, and which goes around the object.Many details of the instrument are now out of date owing to the invention of the laser. Alternative methods which have now become possible are discussed for realizing the principle of total reconstruction.

Wide-Angle Michelson Interferometer for Measuring Doppler Line Widths*

Abstract: The field-compensation principle, which has been applied to interferometric spectroscopy independently by P. Connes and by L. Mertz, allows the useful solid angle accepted by an interferometer to be increased by an amount that can be very large. This paper is concerned with a particular application of this principle using the Michelson interferometer. Although the technique is difficult to utilize where a wide range of path differences is required, the interferometer takes an extremely simple form when constructed for a narrow range of path difference about a fixed central path difference. While such an instrument has a limited use in spectroscopy, there is one type of measurement which it is admirably suited to perform: the determination of the width of a single isolated atomic line whose analytical shape is known. A description is given of the theory and construction of a wide-angle Michelson interferometer now being used for the measurement of Doppler temperatures from the width of the 5577 A atomic oxygen line in the nightglow and aurora. This line is known to be accurately gaussian in shape, and is well-isolated from other lines, making it an ideal subject for this instrument.

Detection of Differences in Real Distributions

Abstract: If two coherently illuminated distributions are spatially separated at the front focal plane of a spherical lens and the resulting diffraction pattern is square-law recorded in the back focal plane, the Fourier transform of the record contains comparative data about the two input distributions. The transform of the record presents data in a reference frame that simplifies the problem of identifying and locating differences between the two input distributions. The process of transforming, detecting, and retransforming is a method of convolving the two input distributions. If the inputs are identical, the autocorrelation function is derived. If only real functions are evaluated, the autocorrelation function is symmetrical about the correlation peak. An asymmetry indicates a difference between the inputs, i.e., a cross-correlation function. The value of the technique is that it reduces the procedure for comparing spatial distributions to a search for asymmetry about an easily defined point. The same process may be used to identify spatial distributions by heterodyning a reference distribution with the field of view to be searched. The cross correlation function derived contains a sharp correlation peak for each target within the field of view.

Photographic Recording of Spatially Modulated Coherent Light

Abstract: Photographic recording of spatially modulated coherent light, such as is encountered in wavefront reconstruction or coherent optical data processing, is described in terms of the amplitude transmittance (Ta) vs exposure (E) curve of the film. It is proposed that the Ta−E curve is more appropriate for this application than the D−log E curve. The nonlinear effects of recording are analyzed in this context using a nonlinear “circuit” model. A general expression for the resulting amplitude transmittance is derived by use of a Fourier-transform technique. A specific nonlinearity, the error-function limiter, is analyzed in detail to illustrate the effects of the film nonlinearity.

Measurement of the Radiative Lifetimes of the First Excited States of Na, K, Rb, and Cs by Means of the Phase-Shift Method*

Abstract: The lifetimes of the first excited states of the elements Na, K, Rb, and Cs were measured using the phase-shift method described by Brewer et al. [ Rev. Sci. Instr.33, 1450 ( 1962)]. The data were taken over a wide range of vapor densities to allow the effects of radiation entrapment and of scattered exciting light to be taken into account. The following values were obtained: Na 3p2P 16.1±0.3 nsec, K 4p2P 27.8±0.5 nsec, Rb 5p2P3/2 27.0±0.5 nsec, and Cs 6p2P3/2 30.5±0.7 nsec.

New Method of Making Fresnel Transforms with Incoherent Light

Abstract: A new method for recording Fresnel transformations of two- and three-dimensional scenes illuminated by spatially incoherent light is described. The technique is based on the properties of the triangular interferometer and the afocal optical system. Experimental results with one- and two-point objects have verified the basic principles of the method.

Optical Scattering Cross Sections for Polydispersions of Dielectric Spheres

Abstract: Optical scattering cross sections for polydispersions of dielectric spheres, showing dependence on ratio of third to second moment of shape and size distribution function

Equivalent Layers in Multilayer Filters

Abstract: Extending the work of Epstein, the theoretical basis and the properties of equivalent layers in multilayer configurations are presented. The results are applied to the design of practical long-, short-, and bandpass filters.

Object Restoration in a Diffraction-Limited Imaging System

Abstract: This paper presents a formal solution to the problem of object restoration in a one-dimensional, diffraction-limited imaging system. It is found that if the illumination in the object space is confined to a finite region, then the imaging equation can be solved for the object in terms of the image. The solution can be expressed as a series expansion on the eigenfunctions of the imaging operator.