Showing papers in "Journal of the Optical Society of America in 1973"

Accurate two-dimensional eye tracker using first and fourth Purkinje images

Abstract: Although a number of devices are currently in use for monitoring eye position, none is both accurate and convenient to use. Methods based on the use of contact lenses can provide high accuracy but have obvious inconveniences. Other techniques—e.g., skin-mounted electrodes, or eyeglass-mounted photoelectric pickups—are relatively convenient, but eye position can be measured to an accuracy of no better than about 0.5° to 1°. A novel eye-tracking instrument has been developed that makes use of two Purkinje images. The instrument operates in the infrared, so that it does not interfere with normal vision; it requires no attachments to the eye; it has a sensitivity and accuracy of about 1 min of arc, and operates over a two-dimensional visual field of 10° to 20° in diameter. The basic principle of the instrument is described, and operating records are shown.

Image formation using self-imaging techniques*

Abstract: Two situations in which self-imaging techniques can be applied to advantage are presented: the pinhole-array camera and transmission through an optical fiber. The experimental procedure and results are presented for the case of a pinhole array illuminated with an extended incoherent object distribution. In the Fresnel-image planes, more images are formed than there are pinholes in the array, which is in contrast to the case of the pinhole-array camera. An optical fiber or thin film working in the kaleidoscope mode may form an image, provided that its length fulfills the self-imaging condition.

Optics in smoothly varying anisotropic planar structures: Application to liquid-crystal twist cells*

Abstract: Optical transmittance and reflectance of continuously varying anisotropic planar media, such as nematic liquid crystals in Schadt–Helfrich twist cells or cholesterics between parallel rubbed surfaces, have previously been computed with a 4 × 4 matrix method by considering the medium as broken up into many thin parallel layers and treating each as if it had homogeneous anisotropic optical parameters. A matrix multiplication was done for each layer, and unless each layer was much less than one wavelength thick, several more multiplications were done within each layer. Here we show how to do numerical computations with equal accuracy using much thicker layers. We use a truncated power series to approximate the variation of optical parameters through each layer. We also show two ways to obtain fast convergence of numerical computations with layers of homogeneous anisotropic material that are several wavelengths thick. We use the method to get a better understanding of the optical properties of twist cells, particularly for oblique rays. The possibility of measuring elastic constants by comparing measured with computed transmittance of twist cells is suggested.

Hermite–gaussian functions of complex argument as optical-beam eigenfunctions

Abstract: Optical-resonator modes and optical-beam-propagation problems have been conventionally analyzed using as the basis set the hermite–gaussian eigenfunctions ψn (x,z) consisting of a hermite polynomial of real argument Hn [√2x/w(z)] times the complex gaussian function exp [−jkx2/2q(z)], in which q(z) is a complex quantity. This note shows that an alternative and in some ways more-elegant set of eigensolutions to the same basic wave equation is a hermite-gaussian set ψˆn(x,z) of the form Hn[√cx]exp [−cx2], in which the hermite polynomial and the gaussian function now have the same complex argument √cx ≡ (jk/2q)1/2x. The conventional functions ψn are orthogonal in x in the usual fashion. The new eigenfunctions ψˆn, however, are not solutions of a hermitian operator in x and hence form a biorthogonal set with a conjugate set of functions ϕˆn(√cx). The new eigenfunctions ψˆn are not by themselves eigenfunctions of conventional spherical-mirror optical resonators, because the wave fronts of the ψˆn functions are not spherical for n > 1. However, they may still be useful as a basis set for other optical resonator and beam-propagation problems.

Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence

Abstract: A new technique (speckle interferometry) has been developed by Gezari, Labeyrie, and Stachnik, which allows the measurement of stellar diameters from a series of photographs obtained from large-aperture ground-based telescopes. The series of photographs is processed to obtain the Weiner spectrum of the photographic image, i.e., the ensemble-averaged modulus-squared Fourier transform obtained from the series of images. Gezari, Labeyrie, and Stachnik have measured stellar diameters as small as 0″.05, about 20 times better than is usually possible. In this paper, mathematical expressions are obtained for the Wiener spectrum of the image of a point source. As is well known, the Wiener spectrum of the image of an extended, incoherently radiating object, is expressible as a product of this point-source spectrum and the object spectrum. Calculations are performed using the Rytov approximation and assuming that the underlying atmospheric turbulence is describable by a Kolmogorov spectrum. Asymptotic closed-form expressions are obtained for angular frequencies much less than, and much greater than, the conventional seeing limit. In the latter case, the Wiener spectrum is found to be proportional to the optical transfer function.

Raman cross section of some simple gases

Abstract: The differential Raman cross sections of the main Raman-active vibrations have been measured in the gases N2, O2, H2, CO, NO, CO2, SO2, N2O, H2S, NH3, ND3, CH4, C2H6, and C6H6 using 488.0-nm laser light. The present results are compared with previous measurements made at other wavelengths. The Raman cross sections of the rotational lines in the diatomic gases were also measured, as were the vibrational-rotational lines of O2 and N2. Absolute measurement of the Raman cross sections were performed two ways: (i) by calibrating the Raman spectrometer, and (ii) by comparing the unknown against liquid benzene (for which the Raman cross section has been measured). Results of these measurements compare reasonably well with previous determinations for which corrections for the υ4 frequency dependence were made.

Heterochromatic additivity, foveal spectral sensitivity, and a new color model*

Abstract: In three separate experiments it is shown that (i) heterochromatic additivity failures for foveally viewed trichromatic lights can be predicted using a vector model derived from bichromatic additivity data, (ii) near-threshold bichromatic additivity failures are not qualitatively different from threshold-level failures, and (iii) foveal spectral sensitivities obtained by direct brightness matching and threshold methods are greater in the long- and short-wavelength ends of the spectrum than sensitivities obtained by flicker photometry. A new opponent-colors model that is appropriate for threshold-level color vision is expressed as a transformation of the CIE standard observer. The model allows the derivation of a light unit that correlates with signal detectability and predicts (a) confusion lines for deuteranopic and tritanopic vision, (b) spectral sensitivity as measured by flicker photometry (i.e., a sensitivity function much like the CIE Vλ function), (c) spectral sensitivity as measured by threshold (or direct-matching) techniques, (d) threshold-level heterochromatic additivity failures, (e) the apparent saturation of a threshold-level spectrum, (f) wavelength discrimination for a near-threshold spectrum, (g) loci of constant lightness-to-luminance ratios within the CIE chromaticity diagram, and (h) the essential quantitative differences between threshold and near-threshold heterochromatic additivity failures.

New approximation to the Voigt function with applications to spectral-line profile analysis

Abstract: An approximation to the Voigt function is described, which is valid over the entire domain of the independent variables that characterize it and which is accurate to the order of 0.0001 of the peak value of the function. Relations between the parameters of the function are also given. The approximation is used to develop a procedure for fitting observed lines with Voigt functions; the class of asymmetric lines that arise from the superposition of two Voigt functions is considered in some detail, and methods for extracting the Voigt parameters of the components from the observed contour of the envelope are given. The measurement of the width and shape of a single line with a Fabry–Perot interferometer is also discussed. All of the calculations described here can be handled by programmable calculators or small computers.

Brightness function: Effect of area and duration

Abstract: The dependence of perceived brightness on flash luminance and duration was determined for dark-adapted observers, with target size, retinal location, and wavelength varied parametrically. In the first series of experiments, observers made magnitude estimations of the brightness of flashes of varying luminance and duration. Perceived brightness varied as a power function of luminance, with simple fractional exponents: 1, (1/2), (1/3). The exponents for brightness depend upon both target size and flash duration. The second series of experiments determined how the Broca–Sulzer brightness enhancement shifts to briefer or longer durations with changes of luminance. Observers adjusted the duration of constant-luminance flashes to produce a maximally bright flash. The flash duration producing maximum brightness varied as a power function of luminance with simple fractional exponents: −(1/2) for point sources and −(1/3) for extended sources. The regularity of the exponents suggests that a simple mechanism underlies the encoding of brightness information in the dark-adapted state.

Calculation of diffraction efficiency in hologram gratings attenuated along the direction perpendicular to the grating vector

Abstract: A thick hologram grating with modulation that decreases along the direction perpendicular to the grating vector has been analyzed using coupled-wave theory developed by Kogelnik. General solutions of two predominant waves, undiffracted and first-order diffracted waves, are given for transmission and reflection types of hologram gratings. Calculated efficiency of diffraction for some special cases is compared with that for the uniform grating. The influence of the attenuation of grating modulation upon the validity of the theory when applied to the Bragg reflection is also discussed.

Contribution of the crystalline lens to the spherical aberration of the eye

Abstract: The influence of the corneal shape on the quality of the retinal image has been studied. We have distinguished between the role of the cornea and that of the crystalline lens; the wave surface behind the cornea was determined. First, the corneal topology was measured by photographic keratometry. Then, for the same subject, the spherical aberration of the eye was determined by the Foucault knife-edge test. The shapes of the anterior and posterior surfaces of the crystalline lens have been determined from experimental data.

Experimental determination of two-dimensional spatiotemporal power spectra of stellar light scintillation Evidence for a multilayer structure of the air turbulence in the upper troposphere

Abstract: Two-dimensional spatiotemporal power spectra of shadow patterns associated with stellar light scintillation has been determined experimentally by optical filtering of the spatial frequencies. The results show evidence for a multilayer structure of the air turbulence in the upper troposphere. Altitude and wind speed can be deduced for each layer. They are in good agreement with wind profiles from meteorological data.

Diffraction by thick, periodically stratified gratings with complex dielectric constant

Abstract: This paper extends Burckhardt’s solution for diffraction from a thick grating to include complex dielectric constants and nonsinusoidal stratifications. This allows any realistic periodic structure to be handled. Computed results are compared with coupled-wave theory, as described by Kogelnik, with emphasis on strongly absorbing gratings such as those made by photographing an interference pattern. Finally, some experimental holographic data are compared with computations that take into account the photographic nonlinearity between exposure and dielectric constant.

Smith–Purcell radiation from a point charge moving parallel to a reflection grating

Abstract: A rigorous solution is obtained for the problem of radiation from an electric point charge that moves, at a constant speed, parallel to an electrically perfectly conducting grating. The relevant vectorial electromagnetic problem is reduced to two two-dimensional scalar ones. A Green’s-function formulation of the two problems is employed. For both cases, an integral equation of the second kind for the remaining unknown function is derived. This integral equation is solved numerically by a method of moments. Some numerical results for the radiation from a moving point charge above a sinusoidal grating are presented; in particular, the power losses of the point charge have been estimated.

Power transfer between optical fibers

Abstract: Previous papers have treated power transfer between HE11, TE01, and TM01 modes propagating on identical cylindrical fibers. Here we extend the theory to include power transfer between modes of any order propagating on uniform circular fibers of different radii and dielectric constant. A simple analytical expression for the coupling coefficient is derived. The error in using the decoupled two-mode form of the coupled-mode equations is determined. Examples are given to illustrate the extension of the two-fiber results to arrays of fibers with different properties. All results are presented in a dimensionless form applicable to circularly cylindrical fibers of arbitrary physical parameters.

Spectrum and the energy levels of neutral argon, Ar I

Abstract: The spectrum of Ar i has been reobserved with grating spectrographs in large parts of the photographic range. The energy levels have been calculated taking into consideration interferometric and grating observations made since the publication of Atomic Energy Levels (1949) and by use of the dispersion formula for air adopted by the International Astronomical Union. Four ground-level combinations between 1066 and 876 A have been carefully measured on spectrograms taken with the 10.7-m vacuum spectrograph at the National Bureau of Standards; two in the second and two in the fourth order. They give for the lowest excited level, 3p5 (2P1(1/2))4s[1(1/2)]2 1s5 in Paschen’s notation), the value 93143.76 ± 0.05 cm−1 relative to the ground level zero. By use of this level, the 3p5nl level system as a whole has been accurately located relative to the ground level. From the levels, 34 lines in the vacuum ultraviolet between 1066 and 792 A have been calculated according to the combination principle. Between 894 and 806.8 A, these should be accurate to ± 0.0004 A or better, whereas for the 1066- and 1048-A lines and several lines below 806.8 A, the limits of error are somewhat larger. From interferometrically determined values of levels of types n s (n > 5) and n d (n > 3) accurate combinations with those of type 4p in the 10068–4768-A range have been calculated. The (3p5 2P1(1/2)) ionization energy, referred to the ground state of Ar ii, is 127109.80 ± 0.10 cm−1.

Determination of optical transfer function by inspection of frequency-domain plot

Abstract: In cases in which there are zeros in the optical transfer function, display of the absolute value of the Fourier transform of a blurred image may allow these zeros to be seen, if the noise level is low enough. This can be used to identify the optical transfer function, provided that it is one of the suitable common simple forms, or to determine certain parameters of the optical transfer function if its form is known. Examples of the use of this technique in the generation of restoring filters for image enhancement are presented.

Interferometric technique for recording and restoring images degraded by unknown aberrations

Abstract: A method for recording and restoring images that have been degraded by unknown aberrations is described. Several images are recorded with different masks placed in the pupil of the optical system. If these images are Fourier analyzed and certain phases are combined algebraically in the proper manner, the effects of the unknown aberrations can be removed, with an uncertainty concerning the absolute location of the object with respect to the imaging system. Experimental results are reported. The sensitivity of the method is analyzed, and the applicability of the technique in astronomical imaging is considered. The method shows considerable promise for high-resolution imaging of fine-grain structure on the solar surface.

Asymptotic theory of unstable resonator modes

Abstract: The integral equation that describes the mode structure of unstable resonators of rectangular aperture is solved by a new method that makes use of an asymptotic expansion of the integral for large Fresnel number. This expansion is carried out to terms of order 1/F, where F is the Fresnel number, and includes effects of diffraction at the edges of the feedback mirror, as well as a term corresponding to the geometric-optics approximation. Results obtained by this method are in good agreement with those obtained by gaussian integration, even at an effective Fresnel number of unity. The method has the advantage that it involves finding the roots of a polynomial, rather than the eigenvalues of a matrix, and therefore requires less computer time and storage. The results of the present work are at variance with a class of theories based on geometric optics, as regards the higher-loss modes even for very large Fresnel number, although the lowest-loss symmetric mode is correctly given by geometric optics in this limit.

Spectral losses of unclad vitreous silica and soda-lime-silicate fibers

Abstract: Spectral transmission losses of unclad optical fibers drawn from various types of commercial vitreous silica and of soda-lime-silicate glasses were measured in the wavelength range between 0.5 and 1.12 µm. The automated technique employed was both convenient to use and sensitive enough to measure losses of 6 dB/km, i.e., 75% loss per km, with an estimated accuracy of ±2 dB/km. Bulk-loss measurements performed with calorimetric and bridge-type techniques were in good agreement with the fiber-loss measurements.

Visual acuity as a function of exposure duration

Abstract: Changes of visual acuity with exposure durations shorter than the critical duration for detection can be attributed to simple light summation; however, changes of visual acuity with longer exposure durations must be otherwise accounted for. This paper shows changes of photopic acuity with prolonged exposure durations, and considers several possible underlying mechanisms. The acuity threshold was found to decrease with exposure durations up to 400 ms and possibly longer. Thus, pupillary and accommodative fluctuations were investigated, as mechanisms concerned, but were found not to have an effect on the phenomenon. A task-specific summation period was sought; however, no evidence for such was found. Also, similar results were found whether the presentation consisted of a single uniform exposure or two discrete exposures with some interval between.

Efficiency of optical-grating couplers

Abstract: The distributed coupling between one guided wave and a finite number M of freely propagating waves is considered as it applies to optical-grating couplers and prism couplers. Starting from a scattering-matrix representation, general amplitude and power relations among the various incident and outgoing waves are discussed. These relations are based on the reciprocity, symmetry, uniformity, or losslessness of the coupler. It is shown that the high efficiency (80%) of the prism coupler can also be obtained with a grating coupler, provided that the guided wave interacts with only one free wave (M = 1). Various methods of achieving this are proposed. Among them is a scheme that yields a wavelength-selective coupler. The optimization of couplers is discussed for a number of typical coupling situations. The effects of losses in couplers are also treated.

Guided normal modes of two parallel circular dielectric rods

Abstract: Some modes of two parallel, circular, lossless, dielectric rods are discussed for identical and nonidentical rods The electromagnetic boundary-value problem is solved by use of an expansion of the mode field in circular harmonics Examples of the distribution of flux density are given The beat wavelength for the beating of energy between the rods computed for identical rods is compared with approximate results of coupled-mode theory

Threshold and suprathreshold perceptual color differences.

Abstract: Observers matched a series of colors. They also matched a given neutral difference by producing, with a colorimeter, a color difference for one color attribute at a time (producing a saturation difference while keeping hue and lightness constant, producing a hue difference while keeping lightness and saturation constant, and producing a lightness difference while keeping hue and saturation constant). We found it possible for observers trained in color scaling to abstract the individual color attributes of hue, saturation, and lightness; that a unit suprathreshold perceptual color ellipsoid can be described about a given color in color space; that the precision of color-difference matching seems to be a function of the size of the perceptual color interval of the reference colors; and, finally, that there is a tendency for color-difference-matching ellipses to orient toward the nearest colorimeter primary.

Optical constants of ice in the infrared

Abstract: The normal-incidence spectral reflectance of ice at −7° C has been measured in the range 300–5000 cm−1. A Kramers–Kronig phase-shift analysis of the measured spectral reflectance has been employed to provide values of the real and imaginary parts of the refractive index. The resulting values of these optical constants are suitable for use in Mie-theory computations of scattering by ice particles in planetary atmospheres. The optical constants of ice at −7° C are compared in detail with those of liquid water at several temperatures and with those recently determined for ice at −170° C.

Causality calculations in the time domain: An efficient alternative to the Kramers–Kronig method*

Abstract: The consequences of causality and analyticity are commonly invoked in procedures for determining optical constants from reflectance data using the Kramers–Kronig relation. Here an entirely elementary argument is advanced, which exploits only the parity of Fourier transforms and the vanishing of the impulse response for negative times, and which avoids the concept of analyticity. This leads to a simply understood algorithm for such computations. The new procedure shows large gains of computational efficiency over the classical Kramers–Kronig approach. The method is applied first to model data and compared with exact results; it is then applied to real data and compared with the result obtained by the standard method. Excellent agreement is obtained in all cases.

Wave propagation in anisotropic thin-film optical waveguides

Abstract: Propagation of guided waves in layered crystalline media is investigated. Two-dimensional guided-mode dispersion curves are computed and classified for a three-layer uniaxial configuration with arbitrary optic-axis orientation in each layer. It is found that the dispersion curves can be either circular or elliptical and can also be discontinuous. Other interesting aspects of wave behavior include wave tilt of the evanescent fields and independent TE and TM mode-existence criteria.

Short-term average optical-beam spread in a turbulent medium

Abstract: On the basis of the extended Huygens–Fresnel principle, a general expression is derived for the short-term average optical-beam spread, as measured with respect to the instantaneous center of energy of the beam, of an initially coherent optical-beam wave propagating in a weakly inhomogeneous medium. The present analysis applies to the near field of the effective coherent transmitting aperture, where the beam wanders (dances) as a whole and does not break up into multiple patches or blobs. Central to the analysis is the short-term average mutual coherence function (MCF) of a spherical wave. This quantity is obtained from the corresponding long-term MCF by removing the random tilt of the wave front. Analytic expressions for the short-term beam spread are presented for the case of a Kolmogorov spectrum and the short-term average MCF derived by Fried. As expected, the short-term, turbulence-induced beam spread is always less than the corresponding long-term beam spread. Analytic and numerical results are given for the short-term average irradiance at focal range f, which is always greater than the corresponding long-term average irradiance.

Limitations of fringe-parameter estimation at low light levels*

Abstract: Fundamental limitations of estimating the amplitudes and phases of interference fringes at low light levels are determined by the finite number of photoevents registered in the measurement. By modeling the receiver as a spatial array of photon-counting detectors, results are obtained that permit specification of the minimum number of photoevents required for estimation of fringe parameters to a given accuracy. Both a discrete Fourier-transform estimator and an optimum joint maximum-likelihood estimator are considered. In addition, the Cramer–Rao statistical error bounds are derived, specifying the limiting performance of all unbiased estimators in terms of the collected light flux. The performance of the spatial sampling receiver is compared with that of an alternate technique for fringe-parameter estimation that uses a barred grid and temporal sampling of a moving fringe.