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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, n1 and Laguerre—Gaussian modes with p=0 and l=1. The wave front of the eigenbeams is, as a rule, spherical.

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.

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.

In his works on diffraction [Proc. R. Soc. London Ser. A204, 533 (1951)], E. Wolf introduced the Q_2n 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 Q_2n function, another special function, called the Y_n 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 Q_2n and Y_n 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 Q_m and the Y_m 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.

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.

Abstract: 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.

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.

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

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.

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

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.

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 ∼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.

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.

The evolution of the main parameters (spot size, radius of curvature, and far-field divergence) of the beam provided by a high-gain, short-pulse Xe–Cl laser equipped with a plane-parallel Gaussian resonator has been investigated both theoretically and experimentally as a function of the number of round trips. It is shown that a theoretical analysis, formerly developed to study the growth of coherence of a Schell–Gauss model beam propagating through a periodic sequence of Gaussian apertures, provides a satisfactory description of the temporal evolution of the Xe–Cl laser beam. A good agreement between theoretical and experimental data has been found.

The evolution of relevant beam parameters of a light beam propagating through an iterated sequence of spherical lenses and Gaussian apertures immersed in an active medium is investigated. This structure is a classical model for an optical resonator. The effective size of the field distribution, the beam quality factor and the radius of wave front curvature are calculated as functions of the number of transits. Steady-state values of these parameters are also determined and the dependence of the results on the gain of the medium, the aperture width and the focal length is illustrated. A comparison with the case without active medium is also discussed.

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.

It is demonstrated that the definition of a fractional-order Fourier transform can be extended into the complex-order 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.

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.

A model that characterizes the effects of beams and waveforms on the measurement of ultrasonic scattering is described and analyzed. The model includes arbitrary emitter beam patterns and detector apertures as angular spectra of plane waves. Arbitrary emitted pulses and detector time gates are incorporated through frequency spectra of temporal harmonics. A transformation of variables is employed to express the spectrum of the measured pressure as a product in wave space of a system function and the Fourier transform of the medium variations. The analysis shows the influence of beam patterns and a finite detector gate, relates the current model to a prior narrowband analysis, provides asymptotic expressions for beam pattern effects produced by radially symmetric Gaussian apertures, and treats the backscattering configuration for arbitrary apertures and correlated scattering. Numerical computations illustrate the effect of Gaussian, exponential, and uniform circularly symmetric spatial aperture functions on the measurement of the power spectrum of medium variations. The results may be used to design scattering experiments in which measurements yield scaled values of intrinsic parameters of scattering.