Fresnel and Far-field Diffraction of a Truncated Elliptical Gaussian Beam Due to an Elliptical Aperture

Abstract

Using the Hopkins algorithm, expressions are derived for the intensity patterns, in both the Fresnel and far-field regions, associated with the diffraction of a plane-wave elliptical Gaussian beam truncated by an elliptical aperture. This is accomplished by evaluating the diffraction integral subject to the Fresnel approximation. Numerical results are presented that indicate how the truncation parameter affects the side-lobe level in the Fresnel and far-field regions.     

Propagation of decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems

A hard-edged elliptical aperture is described approximately by a tensor form, which can be expanded as a finite sum of complex Gaussian functions. An analytical propagation expression for a decentered elliptical Gaussian beam (DEGB) through an axially nonsymmetrical optical system with an elliptical aperture is derived by using vector integration. The approximate analytical results are compared with numerically integral ones, and it is shown that this method can significantly improve the efficiency of numerical calculation. Some numerical simulations are illustrated for the propagation properties of DEGBs through apertured and nonsymmetrical optical transforming systems.     

Propagation of elliptical Gaussian beams modulated by an elliptical annular aperture

An analytical propagation expression for a generalized astigmatic elliptical Gaussian beam (EGB) modulated by an elliptical annular aperture and passing through an axially nonsymmetrical optical system is obtained by the use of vector integration. The derived analytical results provide more convenience for studying the propagation and transformation of EGBs than the usual method of using the diffraction integral directly, and the efficiency of the numerical calculation is significantly improved. Some numerical simulations are illustrated for the propagation properties of EGBs passing through a free space with an elliptical annular aperture, an elliptical screen, or an elliptical aperture. Further extensions are also pointed out.     

Propagation of elliptical Gaussian beams in apertured and misaligned optical systems

On the basis of the fact that a hard-edged elliptical aperture can be expanded approximately as a finite sum of complex Gaussian functions in tensor form, an analytical propagation expression for an elliptical Gaussian beam (EGB) through a misaligned optical system with an elliptical aperture is derived by use of vector integration. The approximate analytical results provide more convenience for studying the propagation and transformation of EGBs than the usual way by using a diffraction integral directly, and the efficiency of numerical calculation is improved. Some numerical simulations are illustrated for the propagation properties of EGBs through apertured optical transforming systems with misaligned thin lenses.     

Focusing Properties of the Gaussian Beam Generated by Optical Resonators with Gaussian Reflectivity Mirrors

Abstract

This paper presents a parametric study concerning the focusing properties of the light beam generated by optical resonators with Gaussian reflectivity mirrors. The diffraction integral has been evaluated and analytical expressions are derived for intensity patterns near the focus in systems of different Fresnel numbers. Universal curves have been plotted against dimensionless parameters to show the variations in the intensity patterns due to the change in focusing geometry and the incident beam intensity profile. Modification to the well known diffraction formula are considered in order to extend the conventional Gaussian formalism to the case of Gaussian mirror resonators.     

Flat-topped light beams with non-circular cross-sections

Abstract

A set of flat-topped light beams with non-circular cross-sections is introduced. This set includes beams with rectangular, elliptical, pyramidal and one-dimensional Gaussian cross-sections. The one-dimensional Gaussian denotes a beam with Gaussian distribution in one direction and a long flattened distribution in the perpendicular direction. Field distributions of this set of beams can be expressed as sums of the lowest order Gaussian modes (TEM₀₀) having different parameters. The main features of the beam are investigated throughout the space without using the orthogonal polynomials.     

Fractional Fourier transform of truncated elliptical Gaussian beams

Based on the fact that a hard-edged elliptical aperture can be expanded approximately as a finite sum of complex Gaussian functions in tensor form, an analytical expression for an elliptical Gaussian beam (EGB) truncated by an elliptical aperture and passing through a fractional Fourier transform system is derived by use of vector integration. The approximate analytical results provide more convenience for studying the propagation and transformation of truncated EGBs than the usual way by using the integral formula directly, and the efficiency of numerical calculation is significantly improved.     

The Intensity Distribution of a Gaussian Schell-model Beam Focused by an Aperture Lens

Abstract

Starting from the generalized Huygens-Fresnel diffraction integral, we study the intensity distribution of a Gaussian Schell-model (GSM) beam diffracted at an aperture lens. A great number of numerical calculations have been performed to illustrate the focused field characteristics. Isophote diagrams are given for systems of different Fresnel numbers, which focus GSM beams, and the related analysis is presented.     

Laser-near-field-based Atomic Lens: Quantum Wave Optics Consideration

Abstract

This paper examines the focusing of an atomic beam by the near field of laser radiation diffracted by a small aperture. The wave properties of the atom are taken into consideration within the framework of an approximate Schrödinger equation. The possibility is demonstrated of focusing an atomic beam into a spot some 3 nm across, the atomic and laser beam characteristics being quite acceptable. The main contribution to the spot size is shown to be from spherical aberrations and diffraction effects, the chromatic aberrations of the atomic beam with a 5% velocity distribution broadening having a lesser effect.     

Transverse Mode Selection in Resonator with a Super-Gaussian Aperture

Abstract

It is shown that the use of a super-Gaussian aperture of order five to ten for fundamental mode selection in plano-concave cavity gives better results (1) than for a Gaussian aperture for its lower TEM₀₀ mode losses and, (2) than a hard aperture for its smoothing edge effects yielding more regular TEM₀₀ mode profiles. Explicit results are given in hard and soft aperture cases for the transverse mode discrimination and fundamental mode losses.     

Focal shifts in focused beams with an elliptical diffracting screen

The approximate analytical formula of the focal shift has been derived for an elliptical diffracting screen that consists of a circular lens and an elliptical aperture. It is found that the focal shift of the beam focused by this elliptical diffraction screen depends not only on the product of the Fresnel number of the focusing geometry and the standard deviation of a mapped function, but also on the ellipticity (the ratio between the minor and the major axes) of the elliptical aperture. The focal shift increases as the decrease in the ellipticity of elliptical aperture. By using an approximate analytical formula and the diffraction integral formula, some numerical simulation comparisons are done. The presented analyses show that the actual analytical expression for the focal shift of a rotationally asymmetric diffraction screen cannot generally be obtained by using the azimuthal average of the screen amplitude transmittance.     

Propagation characteristics of Gaussian beams through a paraxial ABCD optical system with an annular aperture

By introducing a hard aperture function into a finite sum of complex Gaussian functions, an approximate analytical expression for a Gaussian beam passing through a paraxial ABCD optical system with an annular aperture has been derived. The results could be reduced to the case of circular black screen or circular aperture. Some numerical simulations are also performed and illustrated for the propagation characteristics of a Gaussian beam through a paraxial ABCD optical system with an annular aperture, a circular black screen or a circular aperture.     

Approximate propagation equations of flattened gaussian beams passing through a paraxial ABCD system with hard-edge aperture

Abstract

The propagation of flattened Gaussian beams through a paraxial optical ABCD system with hard-edge aperture is studied. Approximate closed-form equations of FGBs for the apertured case are derived by means of the expansion of the circ function into a finite sum of complex Gaussian functions. The advantages of the method as compared with the straightforward integration of the Collins formula are discussed and illustrated with numerical examples.     

Diffractive optical elements with modulated zone sizes

Abstract

The standard design for phase-only diffractive optical elements comprises a transformation of the continuous phase function into a surface relief by means of wrapping the phase into regular intervals of M2π. This results in a structure with diffractive zones aligned in a horizontal plane. We present an alternative design concept with modulated zone sizes leading to non-periodic boundary positions and non-aligned surface structures. The diffractive properties are compared to those of conventional diffractive optical elements. It can be shown that they are fully equivalent for the design wavelength, but exhibit a different spectral behaviour for deviating wavelengths. These properties are exploited for the improvement of the optical performance of blazed gratings and diffractive lenses under conditions of deviating wavelengths. Special emphasis is put on the optimization of the ratio between diffractive efficiencies of the design order and other orders for blazed gratings and focusing diffractive lenses, as well as the suppression of interference effects within Gaussian beams collimated with diffractive lenses.     

The focus of light—linear polarization breaks the rotational symmetry of the focal spot

Abstract

We experimentally demonstrate for the first time that a linearly polarized beam is focused to an asymmetric spot when using a high-numerical aperture focusing system. This asymmetry was predicted by Richards and Wolf in 1959 and can only be measured when a polarization-insensitive sensor is placed in the focal region. We used a specially modified photodiode in a knife-edge-type set-up to obtain highly resolved images of the total electric energy density distribution at the focus. The results are in good agreement with the predictions of a vectorial focusing theory.     

Propagation of anisotropic Bessel-Gaussian beams: sidelobe control, mode selection, and field depth

Theoretical topics encountered in engineering analysis of diffraction-free beams are addressed. These topics are the following: synthesizing a class of diffraction-free beams with noncircular transverse irradiance distributions, e.g., elliptical beams; the effect of Gaussian apodization on sidelobe control; and mode selection that allows one to synthesize a nearly diffraction-free beam with a certain depth of field. The focusing properties of Bessel-Gaussian beams will be investigated the future.     

Reconfigurable imaging systems using elliptical nanowires

Materials that have subwavelength structure can add degrees of freedom to optical system design that are not possible with bulk materials. We demonstrate two lenses that are composed out of lithographically patterned arrays of elliptical cross-section silicon nanowires, which can dynamically reconfigure their imaging properties in response to the polarization of the illumination. In each element, two different focusing functions are polarization encoded into a single lens. The first nanowire lens has a different focal length for each linear polarization state, thereby realizing the front end of a nonmechanical zoom imaging system. The second nanowire lens has a different optical axis for each linear polarization state, demonstrating stereoscopic image capture from a single physical aperture.     

Focusing of axially symmetric flattened Gaussian beams

Abstract

We study the three-dimensional field distribution of a focused axially symmetric flattened Gaussian beam. In particular, exact closed-form expressions for the intensity along the optical axis and at the focal plane are provided, together with a comparison between our results and those pertinent to the case of a converging spherical wave diffracted by a hard-edge circular aperture. Some hints for future investigations are also given.     

Fresnel Diffraction of Gaussian Beams by a Circular Aperture in Gradient-index Media

Abstract

This work presents a method of evaluating the intensity distribution of a Gaussian beam in a weakly inhomogeneous medium when it has been truncated by a circular aperture. An analytical expression for the diffracted field is obtained in terms of Bessel functions.     

Internal waves in an elliptical channel

Abstract

Internal waves in open channels of various depths are studied, in this paper, and a wetted cross section with an elliptical bottom is considered. The frequencies of the first three sloshing modes of internal waves in two superposed fluids contained in an elliptical channel are calculated for various ratios of the depths of the two layers. Numerical solutions converge to analytical solutions based on the shallow water theory as the depth of the thin lower layer approaches zero. Also, solutions for the frequencies of the longitudinal modes of progressive internal waves in two superposed fluid layers contained in an elliptical channel are calculated for various ratios of the depths of the two layers and for two different wave numbers k.