Showing papers in "Journal of the Optical Society of America in 1982"

Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry

Abstract: A fast-Fourier-transform method of topography and interferometry is proposed. By computer processing of a noncontour type of fringe pattern, automatic discrimination is achieved between elevation and depression of the object or wave-front form, which has not been possible by the fringe-contour-generation techniques. The method has advantages over moire topography and conventional fringe-contour interferometry in both accuracy and sensitivity. Unlike fringe-scanning techniques, the method is easy to apply because it uses no moving components.

Statistical ray optics.

Abstract: A statistical approach is taken toward the ray optics of optical media with complicated nonspherical and nonplanar surface shapes. As a general rule, the light in such a medium will tend to be randomized in direction and of 2n2(x) times greater intensity than the externally incident light, where n(x) is the local index of refraction. A specific method for doing optical calculations in statistical ray optics will be outlined. These optical enhancement effects can result in a new type of antireflection coating. In addition, these effects can improve the efficiency as well as reduce the cost of solar cells.

Diffraction analysis of dielectric surface-relief gratings

Abstract: Diffraction by a dielectric surface-relief grating is analyzed using rigorous coupled-wave theory. The analysis applies to arbitrary grating profiles, groove depths, angles of incidence, and wavelengths. Example results for a wide range of groove depths are presented for sinusoidal, square-wave, triangular, and sawtooth gratings. Diffraction efficiencies obtained from the present method of analysis are compared with previously published numerical results. To obtain large diffraction efficiencies (greater than 85%) for gratings with typical substrate permittivities, it is shown that the grating profile should possess even symmetry.

New theory of partial coherence in the space–frequency domain. Part I: spectra and cross spectra of steady-state sources

Abstract: It is shown that, under very general conditions, the cross-spectral density of a steady-state source of any state of coherence may be expressed in terms of certain new modes of oscillations, each of which represents a completely spatially coherent elementary excitation. Making use of this result, a statistical ensemble of strictly monochromatic oscillations, all of the same temporal frequency, is then introduced that yields the cross-spectral density as a correlation function in the space–frequency domain. From these results two new expressions for the Wiener–Khintchine spectrum of the source and also a new mode representation of the cross-correlation function of the source follow at once.

Anisoplanatism in adaptive optics

Abstract: We examine the consequences for an adaptive-optics system of the fact that the turbulence-induced wave-front distortion for two propagation paths with only slightly different propagation directions can be significantly different. We consider the implications of this fact for a compensated imaging system and for an adaptive-optics laser transmitter. Theory and numerical results are presented. The basic results are presented in terms of the average optical transfer function of a compensated imaging system and in terms of the average antenna gain of an adaptive-optics laser transmitter, each expressed as a function of the angular separation ϑ between the propagation path along which the reference signal arrives and the propagation path along which the adaptive-optics system is to provide performance. It is shown that for high spatial frequencies (for the compensated imaging system) and for large-aperture diameters (for the adaptive-laser optics transmitter), i.e., large compared with r0/λ and with r0, respectively, the magnitude of the anisoplanatism effect can be characterized by an isoplanatic patch angular size, which we denote by ϑ0. If the angular separation between the two propagation paths is ϑ, it is shown that the optical transfer function and the antenna gain are each reduced by a factor of exp[−(ϑ/ϑ0)5/3]. This simply expressed performance-reduction factor represents an asymptotic limit for high spatial frequencies and for large transmitter diameters. For lower spatial frequencies and smaller transmitter diameters the reduction factor is not so severe. Numerical results are presented to illustrate this.

Multicoated gratings: a differential formalism applicable in the entire optical region

Abstract: We present a new formalism for the diffraction of an electromagnetic plane wave by a multicoated grating. Its basic feature lies in the use of a coordinate system that maps all the interfaces onto parallel planes. Using Maxwell’s equations in this new system leads to a linear system of differential equations with constant coefficients whose solution is obtained through the calculation of the eigenvalues and eigenvectors of a matrix in each medium. Through classical criteria, our numerical results have been found generally to be accurate to within 1%. The serious numerical difficulties encountered by the previous differential formalism for highly conducting metallic gratings completely disappear, whatever the optical region. Furthermore, our computer code provides accurate results for metallic gratings covered by many modulated dielectric coatings or for highly modulated gratings. We give two kinds of applications. The first concerns the use of dielectric coatings on a modulated metallic substrate to minimize the absorption of energy. Conversely, the second describes the use of highly modulated metallic gratings to increase this absorption.

Asymmetric self-defocusing of an optical beam from the photorefractive effect

Abstract: A new kind of optical self-defocusing is described that in steady state is independent of optical beam power and is strongly asymmetric. The physical mechanism responsible is the photorefractive effect. We present a theory that explains the observed dependence of this self-defocusing on polarization, angle of incidence, beam size, and crystal orientation. Experimental results, using a single-domain crystal of BaTiO3, are presented that show excellent quantitative agreement with the theory. Possible device applications are discussed, including an optical diode and a low-power bistable device with permanent memory.

Finding the illuminant direction

Abstract: Shading is important for estimation of three-dimensional shape from the two-dimensional image, for instance, for distinguishing between the smooth occluding contour generated by the edge of a sphere and the sharp occluding contour generated by the edge of a disk. In order to use shading information to solve such problems, one must know the illuminant direction L. This is because variations in image intensity (shading) are caused by changes in surface orientation relative to the illuminant. Each illuminant direction L has a unique effect on the distribution of changes in image intensity dI, potentially permitting the estimation of L. However, because dI is a function of bothL and the surface curvature, L can be estimated from the image only by making an assumption about the imaged surface curvature. One assumption that is sufficient to disentangle L and surface curvature is that changes in surface orientation are isotropically distributed. This condition is true of images of convex objects bounded by a smooth occluding contour and is true on average over all scenes. Estimates made by using this assumption agree with estimates of illuminant direction given by human subjects for images of natural objects, even when both are objectively wrong. Further, there is a significant correlation between the variance of these estimates and the variance of the human subjects’ estimates.

Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields

Abstract: A recently formulated theory of partial coherence in the space-frequency domain is used to determine the mode structure of an important class of partially coherent sources and of the radiation fields generated by them. The effective number of modes is found to depend in a fundamental way on the ratio of the coherence length to the effective size of the source. The contribution of the effective modes to the far-field intensity is also analyzed.

Attenuation constant of a coherent field in a dense distribution of particles

Abstract: In a dense distribution of particles, the propagation characteristics of the coherent field are strongly affected by the pair-correlated distributions of scatterers. This paper presents an optical experimental study to show that, when the particle density is greater than about 0.1%, the attenuation constant departs markedly from the formula based on an uncorrected scatter assumption. It decreases sharply when ka < 1, whereas it shows a slight increase when ka ≫ 1. Experimental data are shown for the volume densities ranging from 10−3 to 40% and ka ranging from 0.529 to 82.793. Comparisons are given with some theoretical calculations.

Extended Jones matrix method

Abstract: An extended Jones matrix method is derived for treating the transmission of off-axis light through birefringent networks. Neglecting multiple reflections, this matrix algebra includes the effect of Fresnel reflection and refraction at the crystal-plate surfaces.

Irradiance moments: their propagation and use for unique retrieval of phase

Abstract: The functional dependence of irradiance moments with distance from the pupil plane is studied within the framework of Fresnel diffraction theory. The concept of analytic pupil function is introduced, and for such pupil functions it is shown that any finite-order irradiance moment exists, even in the presence of arbitrary continuous phase aberrations. The uniqueness of the relationship between pupil-plane phase and irradiance moments, when the moments are calculated over an orthogonal plane at a fixed point along the optical axis in image space, is obscure, and the relationship between phase and moments is generally nonlinear. However, by studying the behavior of irradiance moments throughout the neighborhood of a given axial point in image space, one may determine, for a large class of pupils, the pupil-plane phase uniquely (within an arbitrary additive constant), and only a linear problem need be solved for phase retrieval. In particular, unique phase retrieval may be accomplished by measuring moments in the neighborhood of either the pupil plane or the image plane. Examples of this technique are given.

First-order optics—a canonical operator representation: lossless systems

Abstract: The theory of canonical transforms is applied to establish the mathematical foundations of the operator algebra method, leading to useful relations between geometrical ray optics and the operator representation of wave optics. The transfer operator of a general first-order system is expressed in terms of the elements of the ray-transfer ABCD matrix. The theory is exemplified through the discussion of basic operator subgroups and special types of systems: Fourier, Fresnel, imaging, and afocal.

Strehl ratio for primary aberrations: some analytical results for circular and annular pupils.

Abstract: Imaging systems with circular and annular pupils aberrated by primary aberrations are considered. Both classical and balanced (Zernike) aberrations are discussed. Closed-form solutions are derived for the Strehl ratio, except in the case of coma, for which the integral form is used. Numerical results are obtained and compared with Marechal’s formula for small aberrations. It is shown that, as long as the Strehl ratio is greater than 0.6, the Marechal formula gives its value with an error of less than 10%. A discussion of the Rayleigh quarter-wave rule is given, and it is shown that it provides only a qualitative measure of aberration tolerance. Nonoptimally balanced aberrations are also considered, and it is shown that, unless the Strehl ratio is quite high, an optimally balanced aberration does not necessarily give a maximum Strehl ratio.

Reconstruction of the support of an object from the support of its autocorrelation

Abstract: The phase-retrieval problem consists of the reconstruction of an object from the modulus of its Fourier transform or, equivalently, from its autocorrelation. This paper describes a number of results relating to the reconstruction of the support of an object from the support of its autocorrelation. Methods for reconstructing the object’s support are given for objects whose support is convex and for certain objects consisting of collections of distinct points. The uniqueness of solutions is discussed. In addition, for the objects consisting of collections of points, a simple method is shown for completely reconstructing the object functions.

Development of scotopic sensitivity and the absorption spectrum of the human ocular media.

Abstract: Scotopic spectral sensitivity was measured for nine observers (aged 4.5 months to 66 years) from 400 to 650 nm (10-nm steps) by using a 42 degree naturally viewed stimulus. The dependent measure was the visually evoked cortical-potential amplitude that was phase locked to an 8-Hz flickering stimulus. Sensitivity was similar for all observers at middle and long wavelengths, but at short wavelengths there was a decrease in sensitivity with increasing age. The density of the preretinal ocular media was estimated by subtracting the log scotopic spectral-sensitivity function of each observer from the human rhodopsin-absorbance spectrum when the two sets of curves were pinned at long wavelengths. The density of the infant ocular media was lower than that for adults. To quantify the sequence of ocular-media development, scotopic sensitivity was determined for an additional 42 observers (aged 1 month to 70 years) at two spectral points: 553 nm, where the optic-media density is low, and 405-430 nm, where the density is high. From these data, optic-media density at 400 nm was calculated. Despite substantial individual differences within each age, a clear aging function emerged. Preretinal optic-media density increased monotonically from birth throughout adulthood. Thus optical density at 400 nm differs by about a factor of 22 between the average 1-month-old infant and the average 70-year-old adult.

Holography and the inverse source problem

Abstract: The inverse source problem for monochromatic sources Re[ρ(r, ω)e−iωt] to the scalar-wave equation is investigated. It is shown that a unique solution to the inverse source problem can be obtained by imposing the constraint that the solution minimize the source energy E = ∫d3r|ρ(r, ω)|2. For certain recording geometries the time derivative of the real image produced by a point-reference hologram is shown to be directly proportional to the time-reversed minimum energy source Re[ρ*ME(r, ω)e−iωt] in the short-wavelength limit.

Limits of spatial-frequency discrimination as evidence of neural interpolation

Abstract: We have studied the ability of observers to discriminate between suprathreshold vertical sinusoidal spatial-frequency gratings on the basis of spatial frequency. The results show that spatial-frequency discrimination is not a smooth function of spatial frequency but instead appears regularly segmented. Similar results were also obtained in an experiment in which observers discriminated between pairs of narrow vertical lines on the basis of their separation. Angular resolutions achieved for both discrimination tasks were less than the spacing between photoreceptors, requiring some type of neural interpolation. The similarity between the two sets of data indicates that discrimination between spatial-frequency gratings is probably based on the separation between two features exactly one cycle apart. We suggest that the segmentation reflects the existence of neural-image representations with discrete levels of spatial accuracy.

Fresnel scattering by a corrugated random surface with fractal slope

Abstract: Consideration of the density of rays emanating from an infinite, corrugated Gaussian surface with fractal slope reveals a hitherto unexplored shortwave scattering regime that is uncomplicated by the presence of caustics in the scattered wave field. The statistical and coherence properties of the ray-density fluctuations in this regime are calculated as a function of fractal dimension D, and it is shown that in the Brownian case (D = 1.5) the problem can be solved exactly. The properties of the intensity pattern in a coherent scattering configuration are also investigated. The contrast of the pattern is computed as a function of propagation distance, and the asymptotic behavior in the strong scattering limit is again found to be exactly solvable when D = 1.5. It is shown that the intensity fluctuations are K distributed in this case. The effects of a finite outer-scale size are evaluated and discussed.

Image-position error associated with a quadrant detector

Abstract: An exact expression is obtained that evaluates the error associated with angular-position measurement of an incoherently illuminated object when a quadrant detector is used in the image plane of an optical system. The accuracy of such a measurement is inversely proportional to the signal-to-noise voltage ratio associated with the four quadrants summed to act as a single detector. In the particular case of a circular object, the rms angular measurement error is given by the expression σθ = π[(3/16)2 + (n/8)2]1/2(λ/D)/SNRv, where n is the angular subtense of the object divided by the diffraction angle λ/D, SNRv is the signal-to-noise voltage ratio, λ is the wavelength of the light used, and D is the diameter of the limiting aperture in the optical system under consideration.

Coupled-mode theory of nonlinear propagation in multimode and single-mode fibers: envelope solitons and self-confinement

Abstract: A set of equations describing pulse propagation in multimode optical fibers in the presence of an intensity-dependent refractive index is derived by taking advantage of the coupled-mode theory usually employed for describing the influence of fiber imperfections on linear propagation. This approach takes into account in a natural way the role of the waveguide structure in terms of the propagation constants and the spatial configurations of the propagating modes and can be applied to the most general refractive-index distribution. The conditions under which soliton propagation and longitudinal self-confinement can be achieved are examined.

Lie algebraic theory of geometrical optics and optical aberrations

Abstract: A new method, employing Lie algebraic tools, is presented for characterizing optical systems and computing aberrations. It represents the action of each separate element of a compound optical system, including all departures from Gaussian optics, by a certain operator. These operators can then be concatenated, following well-defined rules, to obtain a resultant operator that characterizes the entire system. New insight into the origin and possible correction of aberrations is provided. With some effort, it should be possible to produce, by manual calculations, explicit formulas for the third-, fourth-, and fifth-order aberrations of a general optical system including systems without axial symmetry. With the aid of symbolic manipulation computer programs, it should be possible to compute routinely explicit formulas for aberrations of seventh, eighth, and ninth order, and probably beyond.

Theory of optical edge detection and imaging of thick layers

Abstract: The optical microscope measurement of small objects, 0.5 to 10 μm in diameter, is complicated by the apparent change in the dimension of the object with a change in the spatial coherence of the illumination. Coherent edge-detection methods have been developed for the measurement of line objects on integrated-circuit photo masks and wafers. A generalization is presented of the coherent threshold equation that permits the extension to any state of partial coherence of the illumination as well as extension to the measurement of nonplanar objects. In the latter case, a waveguide model is developed for imaging of lines patterned in thick layers and is compared with experimental data.

Infrared optical properties of thin H 2 O, NH 3 , and CO 2 cryofilms

Abstract: To identify and account for the effects from cryocontamination, the infrared spectral transmittances of cryofilms formed by CO2, H2O, and NH3 were measured. These 0.25–14-μm-thick films were cryopumped onto 20-, 80-, and in one case 50-K germanium substrates; the deposition pressures for the films deposited at 20 K and that for 80-K films were approximately 2 × 10−6 Torr. Transmittance spectra were obtained for the 500–3700-cm−1 range with a Fourier-transform spectrometer. Values of the optical properties (n, k) for the CO2, H2O, and NH3 cryofilms were derived from the experimental data using a thin-film-transmittance analytical model and the nonlinear least-squares method. Results from the least-squares method are compared with a Kramers–Kronig determination of the refractive index (n). The optical properties (n, k) of such cryofilms are essential for predicting the degradation of contaminated cryocooled optical surfaces.

Reappraisal of arrays of concentric annuli as superresolving filters.

Abstract: The superresolving filter design of Toraldo di Francia [ Atti. Fond. Giorgio Ronchi7, 366 ( 1952)], which consists of an array of concentric annuli of finite width, is studied. Such a filter compares favorably with other superresolving filter designs while being considerably more simple to fabricate. In particular, for given input energy, the intensity at the focal point is usually greater than for optimized filter designs. The manufacturing tolerances of such a filter are discussed.

Depolarization of electromagnetic waves scattered from slightly rough random surfaces: a study by means of the extinction theorem

Abstract: The scattering and depolarization of electromagnetic waves from perfectly conductive slightly rough random surfaces is studied using the small perturbation method through the Ewald–Oseen extinction theorem. This permits predictions in those cases in which the physical optics, or the Kirchhoff approximation, fails, namely, at grazing incidence and when the wavelength of the incident radiation is comparable with the correlation length of the random heights. In this way it is seen, for example, that, as with the Rayleigh–Fano method, the depolarization of the fields is obtained from the second order of the expansion and exists even in the backscattering and specular directions. Also, unlike the predictions of the Kirchhoff approximation, this depolarization depends on the surface shape. However, this approach yields in the specular direction the same result as the Kirchhoff approximation for those cases in which the latter is known to be valid, i.e., for large correlation lengths and non-grazing-incidence directions and establishes precise conditions under which the Kirchhoff approximation is retrieved in the backscattering direction.

Dependence of the focal shift on Fresnel number and f number

Abstract: The behavior along the axis of the intensity arising from the diffraction of a uniform, converging spherical wave at a circular aperture is studied on the basis of the theory of the boundary-diffraction wave. The results are used to determine the location of the principal intensity maximum and to elucidate the dependence of the focal shift both on the Fresnel number and on the f number of the focusing geometry. Analytic as well as numerical results are obtained. Comparison with microwave experiments of Farnell [ Can. J. Phys.36, 935 ( 1958)] is also made.

Reflectance and preparation of front-surface mirrors for use at various angles of incidence from the ultraviolet to the far infrared

Abstract: The increasing use of reflecting optics in ultraviolet (UV), visible, and infrared (IR) devices has stimulated a great amount of research on coatings for front-surface mirrors. Methods for measuring the reflectance of front-surface mirrors at various wavelengths and angles of incidence are discussed, and techniques for preparing reflecting films with maximum reflectance and durability are described. Data on the UV, visible, and IR reflectance of the most frequently used mirror coatings, Al, Ag, Au, and Rh, are presented. The application of single-layer and multilayer dielectric overcoatings for increasing their durability and normal-incidence reflectance are described, and the effect of these coatings on reflectance at higher angles of incidence is treated. It is shown that Al and Ag overcoated with rather thin layers of silicon oxides or Al2O3 (t = 1000–2000 A) have, in the IR from 8 to 12 μm, practically the same high reflectance as the unprotected metal at close to normal incidence but greatly decreased reflectance at angles larger than 40°. Since only the parallel component is responsible for the IR-reflectance decrease, such film combinations are suitable for producing highly efficient reflection polarizers for the IR, such as for CO2 10.6-μm laser radiation. The effect of water absorption in dielectric overcoatings on the mirror reflectance at 3 μm is discussed.

Effective number of degrees of freedom of partially coherent sources

Abstract: Orthogonal expansion of a partially coherent source with a given cross-spectral density W is used to define an effective number N of degrees of freedom and an effective number N of uncorrelated random variables characterizing the source. Relations N⩽N≡Tr W/λ0⩽(Tr W/∥W∥)2=Ve/Vce are established and discussed. Here Tr W, ‖W‖, and λ0 are, respectively, the trace, the norm, and the largest eigenvalue of W used as the kernel of a homogeneous Fredholm equation; Ve is an effective volume of the source; and Vce is its effective coherence volume. The main results are illustrated by a Gaussian Schell-model source.