# Finite element Fourier and Abbe transform methods for generalization of aperture function and geometry in Fraunhofer diffraction theory

## Abstract

This paper discusses methods for calculating Fraunhofer intensity fields resulting from diffraction through one- and two-dimensional apertures are presented. These methods are based on the geometric concept of finite elements and on Fourier and Abbe transforms. The geometry of the two-dimensional diffracting aperture(s) is based on biquadratic isoparametric elements, which are used to define aperture(s) of complex geometry. These elements are also used to build complex amplitude and phase functions across the aperture(s) which may be of continuous or discontinuous form. The transform integrals are accurately and efficiently integrated numerically using Gaussian quadrature. The power of these methods is most evident in two dimensions, where several examples are presented which include secondary obstructions, straight and curved secondary spider supports, multiple-mirror arrays, synthetic aperture arrays, segmented mirrors, apertures covered by screens, apodization, and phase plates. Typically, the finite element Abbe transform method results in significant gains in computational efficiency over the finite element Fourier transform method, but is also subject to some loss in generality.

- Authors:

- (Idaho National Engineering Lab., EG and G Idaho, Inc., Idaho Falls, ID (US))

- Publication Date:

- OSTI Identifier:
- 7114269

- DOE Contract Number:
- AC07-76ID01570

- Resource Type:
- Journal Article

- Journal Name:
- Optical Engineering; (United States)

- Additional Journal Information:
- Journal Volume: 30:8; Journal ID: ISSN 0091-3286

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 47 OTHER INSTRUMENTATION; DIFFRACTION; FRAUNHOFER LINES; APERTURES; CALCULATION METHODS; EFFICIENCY; FINITE ELEMENT METHOD; FOURIER TRANSFORMATION; GAUSS FUNCTION; INTEGRALS; NUMERICAL ANALYSIS; ONE-DIMENSIONAL CALCULATIONS; TWO-DIMENSIONAL CALCULATIONS; COHERENT SCATTERING; FUNCTIONS; INTEGRAL TRANSFORMATIONS; MATHEMATICS; NUMERICAL SOLUTION; OPENINGS; SCATTERING; TRANSFORMATIONS; 661300* - Other Aspects of Physical Science- (1992-); 440600 - Optical Instrumentation- (1990-)

### Citation Formats

```
Kraus, H.G.
```*Finite element Fourier and Abbe transform methods for generalization of aperture function and geometry in Fraunhofer diffraction theory*. United States: N. p., 1991.
Web. doi:10.1117/12.55900.

```
Kraus, H.G.
```*Finite element Fourier and Abbe transform methods for generalization of aperture function and geometry in Fraunhofer diffraction theory*. United States. doi:10.1117/12.55900.

```
Kraus, H.G. Thu .
"Finite element Fourier and Abbe transform methods for generalization of aperture function and geometry in Fraunhofer diffraction theory". United States. doi:10.1117/12.55900.
```

```
@article{osti_7114269,
```

title = {Finite element Fourier and Abbe transform methods for generalization of aperture function and geometry in Fraunhofer diffraction theory},

author = {Kraus, H.G.},

abstractNote = {This paper discusses methods for calculating Fraunhofer intensity fields resulting from diffraction through one- and two-dimensional apertures are presented. These methods are based on the geometric concept of finite elements and on Fourier and Abbe transforms. The geometry of the two-dimensional diffracting aperture(s) is based on biquadratic isoparametric elements, which are used to define aperture(s) of complex geometry. These elements are also used to build complex amplitude and phase functions across the aperture(s) which may be of continuous or discontinuous form. The transform integrals are accurately and efficiently integrated numerically using Gaussian quadrature. The power of these methods is most evident in two dimensions, where several examples are presented which include secondary obstructions, straight and curved secondary spider supports, multiple-mirror arrays, synthetic aperture arrays, segmented mirrors, apertures covered by screens, apodization, and phase plates. Typically, the finite element Abbe transform method results in significant gains in computational efficiency over the finite element Fourier transform method, but is also subject to some loss in generality.},

doi = {10.1117/12.55900},

journal = {Optical Engineering; (United States)},

issn = {0091-3286},

number = ,

volume = 30:8,

place = {United States},

year = {1991},

month = {8}

}