DIFFRACTION INTEGRAL FORMULA FOR MISALIGNED OPTICAL SYSTEMS

The focusing properties of a so-called reflaxicon (a combination of a diverging and a converging axicon) are studied both theoretically and experimentally. Calculations of intensity distributions produced by this system are made by evaluating the Kirchhoff–Fresnel diffraction integral, first by means of an approximate technique, the stationary phase method, then by a more exact numerical method. The calculations are presented for various planes along the axis of the axicons. The effects of the presence of the supporting mount of the axicons and of some important misalignments of the system on the distributions is also investigated. Experimental results of actual intensity distributions produced by focusing a near-fundamental Gaussian beam by such a system are also presented and are seen to be in fair agreement with numerical calculations. Such calculations would be valuable in many applications for predicting important characteristics (e.g., peak intensity, length of the focal line, degree of asymmetry) of the intensity distributions formed by optical systems containing an axicon pair as the focusing component.

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A vectorial ray-based diffraction integral for optical systems

10.1117/12.2190816 ◽

2015 ◽

Author(s):

Birk Andreas

Keyword(s):

Optical Systems ◽

Diffraction Integral

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Generalized Huygens–Fresnel diffraction integral for misaligned asymmetric first-order optical systems and decentered anisotropic Gaussian Schell-model beams

Journal of the Optical Society of America A ◽

10.1364/josaa.19.000485 ◽

2002 ◽

Vol 19 (3) ◽

pp. 485 ◽

Cited By ~ 4

Author(s):

Guilin Ding ◽

Baida Lü

Keyword(s):

Fresnel Diffraction ◽

Optical Systems ◽

First Order ◽

Diffraction Integral

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Analysis of Phase Distribution of Focused Light in High Numerical Aperture Systems

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.437.616 ◽

2010 ◽

Vol 437 ◽

pp. 616-620

Author(s):

Alexander Normatov ◽

Boris Spektor ◽

Joseph Shamir

Keyword(s):

Phase Distribution ◽

Numerical Aperture ◽

Mathematical Expression ◽

Optical Systems ◽

Entrance Pupil ◽

High Numerical Aperture ◽

Diffraction Integral ◽

Quadratic Phase ◽

The Difference ◽

Kirchhoff Integral

High numerical aperture focusing is becoming increasingly important for nanotechnology related applications. Rigorous, vector evaluation of the focused field, in such cases, is usually performed using the Richards-Wolf method which is based on the Debye approach. The resulting field is known to have a piecewise quasi planar phase. A corresponding result, produced by a Fresnel-Kirchhoff integral for aplanatic optical systems of medium and low numerical apertures, leads to the well known physical fact that a quadratic phase exists when the entrance pupil is not located at the front focal plane. Yet, the amplitudes produced in both ways are in a good agreement. In this work we investigated the difference, presented above, in a 2D system with the help of the Stratton-Chu diffraction integral. The amplitude obtained by the Stratton-Chu integral was quite similar to the classic results while the phase exhibited a quadratic behavior, with the quadratic coefficient depending on the numerical aperture of the optical system. For lower numerical apertures it approached the result obtained by the Fresnel-Kirchhoff integral while for higher numerical apertures it was approaching the Richards-Wolf result. A mathematical expression for the quadratic coefficient was derived and verified for various values of numerical aperture.

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Generalized diffraction integral formulae for misaligned optical systems described by fractional Fourier transforms

Optik ◽

10.1078/0030-4026-00126 ◽

2002 ◽

Vol 113 (2) ◽

pp. 63-66 ◽

Cited By ~ 1

Author(s):

Daomu Zhao ◽

F.a.n. Ge ◽

Shaomin Wang

Keyword(s):

Fourier Transforms ◽

Optical Systems ◽

Diffraction Integral ◽

Fractional Fourier Transforms ◽

Integral Formulae

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Atomic Resolution in Crystal Aperture Stem

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100179932 ◽

1990 ◽

Vol 48 (1) ◽

pp. 236-237

Author(s):

J T Fourie

Keyword(s):

Optical System ◽

Spherical Aberration ◽

Three Dimensional ◽

Transmission Function ◽

Optical Systems ◽

Bragg Angle ◽

Electron Optical System ◽

Intimate Contact ◽

Diffraction Effects ◽

Design Aspect

The attempts at improvement of electron optical systems to date, have largely been directed towards the design aspect of magnetic lenses and towards the establishment of ideal lens combinations. In the present work the emphasis has been placed on the utilization of a unique three-dimensional crystal objective aperture within a standard electron optical system with the aim to reduce the spherical aberration without introducing diffraction effects. A brief summary of this work together with a description of results obtained recently, will be given.The concept of utilizing a crystal as aperture in an electron optical system was introduced by Fourie who employed a {111} crystal foil as a collector aperture, by mounting the sample directly on top of the foil and in intimate contact with the foil. In the present work the sample was mounted on the bottom of the foil so that the crystal would function as an objective or probe forming aperture. The transmission function of such a crystal aperture depends on the thickness, t, and the orientation of the foil. The expression for calculating the transmission function was derived by Hashimoto, Howie and Whelan on the basis of the electron equivalent of the Borrmann anomalous absorption effect in crystals. In Fig. 1 the functions for a g220 diffraction vector and t = 0.53 and 1.0 μm are shown. Here n= Θ‒ΘB, where Θ is the angle between the incident ray and the (hkl) planes, and ΘB is the Bragg angle.

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