Course detail

# Fourier Methods in Optics and in Structure Analysis

The course deals with the Fourier transform of functions of several variables and its use in diffraction theory and in structure analysis. The introductory parts are focused on the definition of the Fourier transform, spatial frequencies, spectrum of spatial frequencies and on the relevance of the Fourier transform to the diffraction theory. Then, the properties of the Fourier transform are presented via mathematical theorems and are illustrated by the Fraunhofer diffraction patterns. In this way a view of the general properties of the diffraction phenomena of this type is obtained. At the end the kinematical theory of diffraction by crystals is presented as an application of the Fourier transform of three-dimensional lattices. Analytical evaluation of the Fourier transform is exercised in tutorials and the Fraunhofer diffraction phenomena are registered by an optical diffractometer in the laboratory, and are interpreted in detail. The course puts accent on the common features of the diffraction in optics (two-dimensional objects) and in structure analysis (three-dimensional objects). Also emphasised are the common features of various methods of structure analysis (X-rays, neutrons, LEED, HEED). This way, the course provides a basis for activities in the Fourier optics and in various branches of structure analysis.

Learning outcomes of the course unit

The ability to calculate the Fourier transform. Knowledge of kinematical theory of diffraction in structure analysis. Ability to interpret the Fraunhofer diffraction phenomena in optics.

Prerequisites

Basic course of physics. Linear algebra. Calculus of functions of several variables.

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Bracewell R. N.: The Fourier Transform and its Applications. 2nd ed., McGraw-Hill Book Co., New York 1986.
Komrska J.: Fourierovské metody v teorii difrakce a ve strukturní analýze, Akademické nakladatelství CERM, s.r.o., Brno 2007.
Komrska J.: Matematické základy kinematické teorie difrakce: Fourierova transformace mřížky, In: Metody analýzy povrchů. Elektronová mikroskopie a difrakce. Ed. L. Eckertová, L. Frank. Academia, Praha 1996.
Papoulis A.: Systems and Transforms with Applications in Optics, McGraw-Hill Book Co., New York 1968.
Saleh B. E., Teich C.: Základy fotoniky 1., Matfyzpress, Praha 1996.
James J. F.: A students guide to Fourier transforms, Cambridge University Press, Cambridge 1996.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

Examination: Oral. Both practical and theoretical knowledge of the course is checked in detail. The examined student has 90 minutes to prepare the solution of the problems and he/she may use books and notes.

Language of instruction

Czech

Work placements

Not applicable.

Aims

The aim of the course is to provide students with the formalism of the Fourier transform of functions of several variables as well as understanding of kinematical theory of diffraction in structure analysis and of the Fraunhofer diffraction in optics.

Specification of controlled education, way of implementation and compensation for absences

Course-unit credit is conditional on active participation in seminars. The way of compensation of missed lessons is specified by the teacher.

Classification of course in study plans

• Programme B3A-P Bachelor's

branch B-FIN , 3. year of study, summer semester, 4 credits, optional (voluntary)

• Programme M2A-P Master's

branch M-PMO , 1. year of study, summer semester, 4 credits, compulsory

• Programme N-FIN-P Master's, 1. year of study, summer semester, 4 credits, optional

#### Type of course unit

Lecture

13 hours, optionally

Teacher / Lecturer

Syllabus

1. The Dirac distribution, its definition, properties and expressions in various coordinate systems. Examples.
2. The Fourier transform, definition, fundamental theorem. Examples. The diffraction of plane wave by a three-dimensional structure. The Ewald spherical surface.
3. The Fraunhofer diffraction as the Fourier transform of the transmission function. Meanings of variables in the Fourier transform. Spatial frequencies.
4. Linearity of the Fourier transform and the Babinet theorem. Examples. Rayleigh-Parseval theorem. Examples. Symmetry properties of the Fourier transform. Central symmetry, mirror symmetry, places of zero amplitude. The Friedel law.
5. The Fourier transform of functions related by regular linear transform of variables. Shift, rotation.
6. Convolution and the Fourier transform of convolution.
7. The Fourier transform of the function characterising a system of identical objects of the same orientation. Sampling theorem.
8. The Fourier transform of projection. The Abbe transform and the Abbe theorem.
9. Infinite lattice of points and its Fourier transform. The reciprocal lattice.
10. Infinite crystal lattice and its Fourier transform. The structure amplitude.
11. Finite lattice and its Fourier transform. Lattice amplitude and shape amplitude. Lattice amplitude as a sum of shape amplitudes.
12. Conditions of the principal diffraction maxima at the diffraction by lattices. The Laue equations and the Bragg equation.
13. Basic methods of structure analysis: X-ray diffraction, electron diffraction (LEED, HEED), neutron diffraction.

Exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

1. Examples of functions leading in the limit to the Dirac distribution.
2. Calculations of the Fourier transform.
3. Demonstration of the Fraunhofer diffraction phenomena in the laboratory.
4. The Fourier transform of the characteristic function of a parallelogram.
5. The function [sin(knaX/2)]/[sin(kaX/2)].
6. Demonstration of the Fraunhofer diffraction phenomena in the laboratory. The interpretation of the Fraunhofer diffraction phenomena.
7. The Fraunhofer diffraction by polygonal apertures.
8. The Fourier transform of the Fourier series.
9. The Fourier transform of the two-dimensional lattice. Interpretation of the Fraunhofer diffraction.
10. Calculation of the structure factors of body-centred, face-centred, diamond and h.c.p. structures.
11. Interpretation of the Debyegram of diamond.
12. Shape amplitudes of polyhedra, shapes of the diffraction spots at the diffraction by finite lattices, an illustration of the sampling theorem.

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