Fraunhofer Diffraction at Apertures in the Form of Regular Polygons. I.

In this first contribution the symmetry properties of the Fraunhofer diffraction phenomena at the apertures possessing a symmetry are discussed. These properties are used for the derivation of the wave function describing Fraunhofer diffraction at apertures of the form of regular polygons. Besides the general formula expressions are derived giving the wave function in the directions where the diffraction patterns show some particular properties.     

Multiplexed computer-generated holograms with irregular-shaped polygonal apertures and discrete phase levels

We propose a novel type of multiplexed computer-generated hologram (MCGH) with irregular-shaped polygonal apertures and discrete phase levels. Each elementary cell forming the new MCGH is divided into a central aperture and several peripheral apertures. The new MCGH allows us to exploit the huge space-bandwidth product provided by standard lithography technologies. With use of the Abbe transform, the Fraunhofer diffraction patterns from the polygonal apertures and, therefore, the layout coefficients can be computed with simple algebraic expressions. Several symmetries related to the polygonal apertures also facilitate the layout-coefficient computation. In the novel iterative subhologram design algorithm (ISDA), we consider all subholograms equally and apply the image-plane constraint to the total reconstructed image, which is the coherent addition of the subimages from the subholograms. We designed MCGHs with several billions of pixels per period, which cannot be achieved with the classical iterative Fourier transform algorithm, because of the prohibitive computational cost and memory limitation. MCGHs with irregular polygonal apertures and discrete phases, which were designed by the ISDA, reconstruct a desired image of large size with high diffraction efficiencies and low reconstruction errors.     

Far-field diffraction patterns of circular sectors and related apertures

In studies of scalar diffraction theory and experimental practice, the basic geometric shape of a circle is widely used as an aperture. Its Fraunhofer diffraction pattern has a simple mathematical expression easily determined by use of the Fourier-Bessel transform. However, it may require considerable mathematical effort to determine the far-field diffraction patterns of aperture shapes related to the circular geometry. From a computational point of view, the mathematical difficulties posed by other aperture geometries as well as more-general apertures with irregular shapes can be surpassed by means of optical setups or fast numerical algorithms. Unfortunately, no additional insight or information can be obtained from their exclusive application, as would be the case if mathematical formulas were available. The research presented here describes the far-field diffraction patterns of single-sector apertures as well as their extension to double symmetrical sectors and to sector wheels formed by interleaved transparent sectors of equal angular size; in each case, full or annular sectors are considered. The analytic solutions of their far-field amplitude distribution are given here in terms of a series of Bessel functions, some interesting properties are deduced from these solutions, and several examples are provided to illustrate graphically the results obtained from approximate numerical computations. Our results have been verified numerically with the fast-Fourier-transform algorithm and experimentally by means of a spherical wavefront-single-lens Fourier-transform architecture.     

Light scattering from an optically active sphere into a circular aperture

To show how apertures affect measurements of the circularly polarized components of light scattered to a detector, we develop two methods of averaging the V and I Stokes parameters over a circular aperture that collects light scattered from an optically active sphere. One method uses a two-dimensional numerical integration that is appropriate for small apertures, and the other gives analytical expressions for scattering into a solid angle of any size. We identify the aperture locations that, independent of aperture size, give an average V (and an effective degree of circular polarization) of zero for scattering from an optically inactive sphere and of nonzero for scattering from an optically active sphere.     

Iterative method for the design of a dual-phase-conjugation-mirror resonator with multiple apertures

Apertures have been used to select the low-order transverse modes in resonators. The additional diffraction losses result in a change in the transverse-mode structure, and the presence of apertures inside a resonator generally distorts the mode shape. The optimization of a multiple-aperture resonator demands an approach that differs from the conventional method in which the mode theory is used. We demonstrate an iterative design method to find optimal phase profiles for the reflector surfaces to build a resonator with multiple apertures to produce a lowest-order mode with much smaller diffraction loss and to satisfy the phase-conjugation conditions at the mirrors. The results are compared with conventional stable resonators, and we show that substantial improvement in round-trip loss and beam quality can also be obtained.     

Nanoscale ridge aperture as near-field transducer for heat-assisted magnetic recording

Near-field transducer based on nanoscale optical antenna has been shown to generate high transmission and strongly localized optical spots well below the diffraction limit. In this paper, nanoscale ridge aperture antenna is considered as near-field transducer for heat-assisted magnetic recording. The spot size and transmission efficiency produced by ridge aperture are numerically studied. We show that the ridge apertures in a bowtie or half-bowtie shape are capable of generating small optical spots as well as elongated optical spots with desired aspect ratios for magnetic recording. The transmission efficiency can be improved by adding grooves around the apertures.     

Diffraction effects on broadband radiation: formulation for computing total irradiance

I present a formulation for treating diffraction effects on total irradiance in the case of a Planck source; earlier work generally depended on calculating diffraction effects on spectral irradiance followed by summation over spectral components. The formulation is derived and demonstrated for Fraunhofer diffraction by circular apertures, rectangular apertures and slits, and Fresnel diffraction by circular apertures. The prospects for treating other sources and optical systems are also discussed.     

Fraunhofer Diffraction at Apertures in the Form of Regular Polygons. II

In this second contribution the Fraunhofer diffraction patterns from apertures of the form of an equilateral triangle, a square, a regular pentagon, a hexagon and an octagon are compared with the calculated maps of the intensity distribution. For each shape of the aperture the first few maxima and minima of the intensity are tabulated and the formulae for the wave function are given characterizing the diffraction in the directions of the slowest and the steepest decrease of intensity.     

Analytical expressions for the longitudinal evolution of nondiffracting pulses truncated by finite apertures

Starting from some general and plausible assumptions based on geometrical optics and on a common feature of the truncated Bessel beams, a heuristic derivation is presented of very simple analytical expressions capable of describing the longitudinal (on-axis) evolution of axially symmetric nondiffracting pulses truncated by finite apertures. The analytical formulation is applied to several situations involving subluminal, luminal, or superluminal localized pulses, and the results are compared with those obtained by numerical simulations of the Rayleigh-Sommerfeld diffraction integrals. The results are in excellent agreement. The present approach can be rather useful, because it yields, in general, closed-form expressions, avoiding the need for time-consuming numerical simulations, and also because such expressions provide a powerful tool for exploring several important properties of the truncated localized pulses, such as their depth of fields, the longitudinal pulse behavior, and the decaying rates.     

Guided light and diffraction model of human-eye photoreceptors

The photoreceptors of the living human eye are known to exhibit waveguide-characteristic features. This is evidenced by the Stiles-Crawford effect observed for light incident near the pupil rim, and by the directional component of light reflected off the retina in the related optical Stiles-Crawford effect. We describe a model for the coupling of light to/from photoreceptors on the basis of waveguide theory that includes diffraction between the eye pupil and the photoreceptor apertures, and we show that valuable insight can be gained from a Gaussian approximation to the mode field. We apply this knowledge to a detailed study of the relationship between the Stiles-Crawford effect and its optical counterpart.     

Generalized Jinc functions and their application to focusing and diffraction of circular apertures

A family of generalized Jinc functions is defined and analyzed. The zero-order one is just the traditional Jinc function. In terms of these functions, series-form expressions are presented for the Fresnel diffraction of a circular aperture illuminated by converging spherical waves or plane waves. The leading term is nothing but the Airy formula for the Fraunhofer diffraction of circular apertures, and those high-order terms are directly related to those high-order Jinc functions. The truncation error of the retained terms is also analytically investigated. We show that, for the illumination of a converging spherical wave, the first 19 terms are sufficient for describing the three-dimensional field distribution in the whole focal region.     

The Structure of Fraunhofer Diffraction Patterns of Apertures with N-fold Rotational Symmetry

The moment theorem is used to show that the innermost part of the Fraunhofer diffraction pattern of any real aperture with higher than two-fold rotational symmetry is rotationally invariant. Then a formalism is presented in which aperture transmission-functions are represented by series of Zernike circle polynomials and diffracted field-amplitudes by series of Bessel functions, from which it is easily shown that the diffraction patterns of such apertures consist of regions, contained between well-defined values of the radius, whose rotational symmetries are integral multiples of that of the aperture. The central region, extending from = 0 to , N ( measures the diffraction angle, and N is the degree of rotational symmetry of the aperture) is rotationally invariant, and successive circumjacent regions have progressively higher rotational symmetries. The diffraction patterns of sectoral apertures and of rings of pinholes are derived and shown to exemplify these general conclusions. Finally it is shown how the diffraction patterns of some apertures (‘chiral apertures’) with rotational symmetries but no mirror symmetry can be deduced from the diffraction pattern of a related aperture with mirror symmetries, to which a chiral perturbation is applied.     

Le Congrès sur les Progrès Récents en Optique Physique Paris 2-7 Mai 1966

A vector Kirchhoff diffraction theory is used to obtain the Fraunhofer diffraction patterns and total power transmissions of circular apertures illuminated by partially coherent light, characterized only by the mutual intensity function in the aperture. The results are shown to agree with experimental results previously reported in the literature, and to be valid in and beyond the paraxial region. The results for both complete coherence and complete incoherence are recovered by using the appropriate limits for the mutual intensity function.     

Diffraction effects in the propagation of radiation in polarizable layers

The very low transmission of light through holes smaller than the wavelength has been found to be enhanced for subwavelength apertures in metallic surfaces with periodic corrugations. This effect has been attributed to the interaction of light with surface plasmons. Similar effects obtained subsequently for non-metallic surfaces have been attributed to evanescent waves on the surface produced by the diffracted Bloch waves from different points in the array. We present an exact solution of Maxwell's equations in the discrete dipole approximation (DDA) for a periodic array of polarizable point dipoles in a layer. Metallic as well as non metallic layers are described. When the wavelength is smaller than the lattice period there is a Bragg's scattered wave, while for subwavelength conditions an evanescent wave on the surface appears. The transmission/reflection coefficients are found to oscillate as a function of frequency, with resonances occurring in a broad range of frequencies depending on the polarizability, at which the evanescent field is enhanced. A detailed study is presented for nanostructured arrays. We find that this model agrees with features observed in experiments through hole arrays supporting the role played by diffraction during light transmission through such arrays without invoking surface plasmons and providing a base to analyze more complex geometries.     

Light intensity distributions of the Gaussian laser beam through an aperture team

Based on the propagating theory of the laser beam, the propagating characteristics of the Gaussian beam through an aperture team that comprises two apertures and a convergent lens, are studied. The approximate expressions for the field distribution are derived by the diffracted integral equation in detail under the condition of approximations. In comparison with the approximate expression and the precise expression, we know that there are the approximate same results for the two expressions if the radius of the second aperture is not too large. The numerical examples are given to confirm the correctness of our calculated results.     

Optical Diffraction in Close Proximity to Plane Apertures. I. Boundary-Value Solutions for Circular Apertures and Slits

In this paper the classical Rayleigh-Sommerfeld and Kirchhoff boundary-value diffraction integrals are solved in closed form for circular apertures and slits illuminated by normally incident plane waves. The mathematical expressions obtained involve no simplifying approximations and are free of singularities, except in the aperture plane itself. Their use for numerical computations was straightforward and provided new insight into the nature of diffraction in the near zone where the Fresnel approximation does not apply. The Rayleigh-Sommerfeld integrals were found to be very similar to each other, so that polarization effects appear to be negligibly small. On the other hand, they differ substantially at sub-wavelength differences from the aperture plane and do not correctly describe the diffracted field as an analytical continuation of the incident geometrical field.     

Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes through complex ABCD optical systems

The change of coherence and polarization of an electromagnetic beam modulated by a random anisotropic phase screen passing through any optical system is found within the framework of complex ABCD-matrix theory This means that the formalism can treat imaging and Fourier transform and free-space optical systems, as well as fractional Fourier transform systems, with finite-size limiting apertures of Gaussian transmission shape. Thus, the current paper shall be considered as a continuation, extension, and generalization of a previous work by Shirai and Wolf [J. Opt. Soc. Am. A21, 1907 (2004)]. It will be shown that the inclusion of apertures in the optical system strongly influences not only the propagation of spatial coherence but also the degree of polarization of a propagating field. Analytical expressions of coherence and polarization propagation will be given in terms of the matrix elements for any complex optical system.     

Measurement of seeing and the atmospheric time constant by differential scintillations

A simple differential analysis of stellar scintillations measured simultaneously with two apertures opens the possibility to estimate seeing. Moreover, some information on the vertical turbulence distribution can be obtained. A general expression for the differential scintillation index for apertures of arbitrary shape and for finite exposure time is derived, and its applications are studied. Correction for exposure time bias by use of the ratio of scintillation indices with and without time binning is studied. A bandpass-filtered scintillation in a small aperture (computed as the differential-exposure index) provides a reasonably good estimate of the atmospheric time constant for adaptive optics.     

Simulating diffraction limiting in a pinhole camera

Abstract

Images synthesized by the simulation of light transport are typically generated with the premiss of rays passing through a theoretical or ideal pinhole, the implication being that an infinitely small hole will produce a perfectly sharp image. However, this is not the case. Ray and particle based models of light transport cannot reproduce the effect of diffraction limiting found of small aperture systems.

Here we consider a more appropriate wave based model. While computationally expensive, and practical for only very small apertures, it is capable of reproducing the results of a real pinhole camera. In simulating such a system, we observe that superposition of waves from multiple photons is unlikely to occur in real systems, and that Nyquist theory can be used to explain diffraction limiting as a sampling artefact.