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The closed trajectories of a Gaussian beam in plano–spherical degenerate and nondegenerate resonators with two Gaussian apertures (with two pump regions with the Gaussian distribution of the gain) are described analytically and the existence conditions are found for such trajectories. It was assumed that Gaussian apertures (gain regions) were located in the plane of a flat mirror at equal distances on both sides of the resonator axis and the size of apertures restricted their action by the corresponding half-plane.

Several optical schemes performing the complex-order fractional Fourier transform are considered. It is shown that these schemes, containing only Gaussian apertures or their combination with lenses, have eigenbeams represented by Hermite—Gaussian modes with transverse indices m, n≤1 and Laguerre—Gaussian modes with p=0 and l=1. The wave front of the eigenbeams is, as a rule, spherical.

The evolution of the cross-spectral density of an initially spatially incoherent beam propagating through an iterated sequence of spherical lenses and Gaussian apertures is studied. The global degree of spatial coherence, the spot size and the radius of wavefront curvature are calculated as functions of the number of transits. Steady-state values of the beam parameters are also given in a closed form. The dependence of the results on the aperture width and the focal length is discussed.

The transformation of simple Gaussian light beams by lenses with a complex focal length is investigated in an approximation of paraxial optics. The results obtained are compared with the corresponding results of geometric optics into which they go over in limiting cases.

In his works on diffraction [Proc. R. Soc. London Ser. A204, 533 (1951)], E. Wolf introduced the Q2n function, which enters his expressions for the encircled energy. This quantity specifies the fraction of the total energy within various rings in receiving planes parallel to the geometrical focal plane. In addition to the Q2n function, another special function, called the Yn function, was used in his formulation, which had been introduced by H. H. Hopkins [Proc. Phys. Soc. London Sect. B62, 22 (1949)]. The purpose of this study is to generalize both the Q2n and Yn functions for evaluating the encircled energy in systems of focused truncated Gaussian beams by apertures of different Fresnel numbers and different levels of beam truncation. The generalized Q and Y functions are functions of more than one variable and are applicable to all nonnegative integers m; they may therefore be called the Qm and the Ym functions. Computed results are shown graphically in the form of contour lines of the encircled energy. Part II of this study [J. Opt. Soc. Am. A24, 2033 (2007)] contains an analysis of maximizing beam energy concentration on a target.

Beam quality behavior of Hermite-Gauss modes and Gauss Schell-model fields propagating through super-Gaussian apertures are analyzed.

The beam quality behavior of Hermite-Gauss laser modes propagating through Gaussian apertures is analyzed.

Beam quality behavior of Gaussian Schell-model beams that propagate through Gaussian apertures is analyzed.

In Part I of this study [J. Opt. Soc. Am. A24, 2023 (2007)] the Q(2n) functions of E. Wolf and the Y(n) functions of H. H. Hopkins have been generalized for evaluating the fraction of the total energy in systems with focused truncated Gaussian beams by apertures of different Fresnel numbers and different ratios of aperture radius to beam radius. The generalized special functions provide a mathematical basis for a rigorous study of maximizing beam energy concentration on a target. This subject is addressed under two subtitles: (1) active focusing of a Gaussian beam onto a distant target and (2) optimizing photodetection in a focused field.

It is demonstrated that an approximation to the focus-wave-mode field can be generated from a dynamic Gaussian aperture. A source of this type is characterized by the time variation of its effective radius. The performance of such an aperture is studied in detail; it is demonstrated that the dynamic aperture shows a great enhancement over the corresponding static one. The types of source investigated provide an efficient scheme to launch narrow Gaussian pulses from extended apertures.

Diffraction of focused truncated Gaussian beams at a circular aperture is investigated experimentally. The beams are obtained from a uniformly illuminated Gaussian transmittance filter. The irradiance along the axis is measured point by point in order to demonstrate the existence of the focal shifts. Diffraction patterns in various planes normal to the axis are presented by photographs. Measured results are compared with theoretical predictions. Close agreement between them has been obtained.

The influence of super-Gaussian apertures on the beam power, beam size, beam quality factor and kurtosis parameter of Hermite-Gauss mode beams and Gauss Schell-model fields is investigated in detail.

Near-field irradiance patterns have been calculated for plane-wave gaussian beams truncated by circular apertures. The Kirchhoff diffraction integral was integrated numerically, using the small-angle Fresnel approximation. The effect of truncation was important when the aperture radius was less than twice the spot size of the gaussian beam.

This paper investigates Fresnel diffraction from plane-wave gaussian beams truncated by circular apertures. Kirchhoff’s integral has been evaluated analytically and (analytical) expressions are derived, in terms of infinite series, for the irradiance and total power of the diffracted field. Some graphs are presented for Fraunhofer conditions. The results obtained greatly reduce the labor of numerically computing the irradiance and the power.

A fast and accurate algorithm is developed for the numerical computation of the family of complex linear canonical transforms (CLCTs), which represent the input-output relationship of complex quadratic-phase systems. Allowing the linear canonical transform parameters to be complex numbers makes it possible to represent paraxial optical systems that involve complex parameters. These include lossy systems such as Gaussian apertures, Gaussian ducts, or complex graded-index media, as well as lossless thin lenses and sections of free space and any arbitrary combinations of them. Complex-ordered fractional Fourier transforms (CFRTs) are a special case of CLCTs, and therefore a fast and accurate algorithm to compute CFRTs is included as a special case of the presented algorithm. The algorithm is based on decomposition of an arbitrary CLCT matrix into real and complex chirp multiplications and Fourier transforms. The samples of the output are obtained from the samples of the input in approximately N log N time, where N is the number of input samples. A space-bandwidth product tracking formalism is developed to ensure that the number of samples is information-theoretically sufficient to reconstruct the continuous transform, but not unnecessarily redundant.

Many operations of vision require image regions to be isolated and inter-related. This is challenging when they are different in detail and extent. Practical methods of Computer Vision approach this through the tools of downsampling, pyramids, cropping and patches. In this paper we develop an ideal geometric structure for this, compatible with the existing scale space model of image measurement. Its elements are apertures which view the image like fuzzy-edged portholes of frosted glass. We establish containment and cause/effect relations between apertures, and show that these link them into cross-scale atlases. Atlases formed of Gaussian apertures are shown to be a continuous version of the image pyramid used in Computer Vision, and allow various types of image description to naturally be expressed within their framework. We show that views through Gaussian apertures are approximately equivalent to the jets of derivative of Gaussian filter responses that form part of standard Scale Space theory. This supports a view of the simple cells of mammalian V1 as implementing a system of local views of the retinal image of varying extent and resolution. As a worked example we develop a keypoint descriptor scheme that outperforms previous schemes that do not make use of learning.

A novel method to reduce the diffraction crosstalk for micro-lens based free space optical interconnects is presented. Instead of using microlenses with uniform transmittance apertures, the use of microlenses with non-uniform transmittance apertures is proposed. It is shown that the diffraction crosstalk which exists in the free space interconnects systems that use microlenses with uniform transmittance apertures can be substantially reduced by using microlenses with Gaussian transmittance. The optical field at the detectors array using both the uniform and Gaussian apertures have been derived and used to calculate the crosstalk. Numerical results have been introduced to show the improvement of the signal-to-crosstalk ratio when using the Gaussian transmittance for the microlens.

A theoretical and experimental study of a Hartmann sensor intended for measurements of light beam wavefronts and implemented on the basis of multielement amplitude masks with apodized apertures is presented. We perform a comparative analysis of the optical performance of the amplitude masks with hard-edge circular apertures and half-tone masks (HTMs) with apodized apertures forming smooth fast decaying optical responses on a photosensor. A technique for fabricating chromium HTMs is developed, which is based on laser thermochemical method of recording. This technique includes two main stages: first – the exposure of a chromium film by a focused laser beam, whose power is varied in a smooth and nonlinear manner versus the required density of HTM’s structure, and second – the developing of the film in a selective etcher. Specimens of HTMs consisting of 64×64 Gaussian apertures were fabricated and applied in a modified Hartmann sensor to measure the wavefront of powerful laser systems.

Multiple-input multiple-output (MIMO) techniques are employed in free-space optical (FSO) links to mitigate the degrading effects of atmospheric turbulence. We consider a MIMO FSO system, which consists of a radial laser array with partially coherent Gaussian beams at the transmitter and a detector array with Gaussian apertures at the receiver. The average power and the power correlation function at the finite-sized receiver apertures are formulated by using the extended Huygens-Fresnel principle in weak atmospheric turbulence. This let us further quantify the performance metrics such as the power scintillation index, the aperture averaging factor, and the average bit error rate (BER) as functions of system parameters. The derived power scintillation equation correctly reduces to the existing coherent and partially coherent Gaussian beam scintillation indices in the limiting cases. Using the performance metrics, we analyze the effect of various practical system parameters on the performance of a MIMO FSO system. Practical system parameters include the transmitter and receiver ring radius, number of beamlets, number of finite-aperture receivers, source size, degree of source coherence, receiver aperture radius, link distance, and the structure constant of atmosphere. (C) 2016 Society of Photo-Optical Instrumentation Engineers (SPIE)

Summary Based on the ABCD matrix formalism, it is shown that by introducing two thin lenses and/or two Gaussian apertures with appropriate geometric parameters into the input and output planes of a general complex ABCD system, the resulting system can be expressed in terms of the complex-order Fourier transform. Detailed analytical results are given and confirmed with illustrative examples.

Coherent optical systems are widely used in modern non-contact metrology [1]. To understand and design these systems appropriately, it is of utmost importance to be able to relate the optical field at the input of the system to that at the output of the system. Many different models exist for achieving this goal. Here however, we assume that the paraxial approximation is valid for the analysis. The paraxial model generally strikes a reasonable balance between calculating an accurate solution while maintaining a relatively simple description of the physical process. For general speckle metrology systems, consisting of many lenses, Gaussian apertures, and sections of free space, the ABCD-matrix-diffraction approach by Yura et. al. [2] provides an excellent design framework [3].

An extension of the ray-pulse matrix formalism is presented that includes self-phase-modulation, self-focusing, bandwidth limitations, and Gaussian apertures. With this more complete set of matrices it is shown how self-mode-locked lasers can be modeled and how the different pulse-shaping mechanisms interact in order to provide a stable femtosecond output.

Despite significant amount of research on automatic classification of facial expressions, recognizing a facial expression remains a complex task to be achieved by a computer vision system. Our approach is based on a close look at the mechanisms of the human visual system, the best automatic facial expression recognition system yet. The proposed model is made for the classification of the six basic facial expressions plus Neutral on static frames based on the permanent facial features deformations using the Transferable Belief Model. The aim of the proposed work is to understand how the model behaves in the same experimental conditions as the human observer, to compare their results and to identify the missing informations so as to enhance the model performances. To do this we have given our TBM based model the ability to deal with partially occluded stimuli and have compared the behavior of this model with that of humans in a recent experiment, in which human participants had to classify the studied expressions that were randomly sampled using Gaussian apertures. Simulations show first the suitability of the TBM to deal with partially occluded facial parts and its ability to optimize the available information to take the best possible decision. Second they show the similarities of the human and model observers performances. Finally, we reveal important differences between the use of facial information in the human and model observers, which open promising perspectives for future developments of automatic systems.

A novel 3 × 3 transfer-matrix method is developed to propagate off-axis Gaussian beams in astigmatic optical systems that may include tilted, displaced, or curved optical elements. Unlike in a previous generalized ray matrix formalism, optical elements that possess gain or loss such as Gaussian apertures, complex lenslike media, and amplifiers are included; and a new beam transformation is found. In addition, a novel exponential variable-reflectivity mirror, which displaces a Gaussian beam without changing its spot size, and a complex prismlike medium are introduced.

It is demonstrated that the definition of a fractional-order Fourier transform can be extended into the complexorder regime. A complex-order Fourier transform deals with the imaginary part as well as the real part of the exponential function in the integral. As a result, while the optical implementation of fractional-order Fourier transform involves gradient-index media or spherical lenses, the optical interpretation of complex-order Fourier transform is practically based on the convolution operation and Gaussian apertures. The beam width of a Gaussian beam subjected to the complex-order Fourier transform is obtained analytically with the ABCD matrix approach.

In this communication we have studied the imaging properties of the recently fabricated Selfoc lens. The theoretical results have been compared with the available experimental data. A brief discussion of diffraction by gaussian apertures has also been included.

A modified matrix model is proposed for stable cavities with intracavity Gaussian apertures, which approximate real apertures. This model differs from the known ones by a better grounded approach to selection of the Gaussian aperture parameters. Analytic relationships are derived for calculating the energy divergence angle of the output radiation and of the beam diameter at the output mirror position under multimode emission conditions. It is shown that the spatial characteristics of the beam are determined by the aperture diameter and by the complex magnification, which is an analogue of the geometric magnification used in the description of unstable cavities. The calculated results are in good agreement with experimental data.