Course detail

Fourier Methods in Optics

FSI-TFOAcad. year: 2025/2026

The course consists of three parts.
The first part is a mathematical one. The Fourier transform of two variables is transformed to polar coordinates and expressed in terms of Hankel's transforms. The Zernike polynomials are used for the description of wave aberrations.
The second part of the course deals with the wave description of an image formation by lenses. The problem is exposed by a direct application of the diffraction theory on one hand, and by the use of the formalism of linear systems (transfer function) on the other hand. The light distribution near the focus, the Abbe theory of image formation, the dark field method, the method of the phase contrast, schlieren method, the image processing by influencing the spectrum of spatial frequencies, and the principle of confocal microscopy are discussed.
The third part of the course provides an overview of the diffractive optics, of the image formation by zone plates and of optics of Gaussian beams. The course involves also the history of the Fourier optics as a whole.

Language of instruction

Czech

Number of ECTS credits

7

Mode of study

Not applicable.

Guarantor

prof. RNDr. Radim Chmelík, Ph.D.

Department

Institute of Physical Engineering (ÚFI)

Entry knowledge

Wave optics. Calculus of functions of several variables.

Rules for evaluation and completion of the course

Examination: Oral. The examined student has 90 minutes to prepare the solution of the problems and he/she may use books and notes.
Course-unit credit is conditional on active participation in lessons. The way of compensation for missed lessons is specified by the teacher.

Aims

The aim of the course is to provide students with basic ideas and history of Fourier optics.
Working knowledge of the Bessel functions, Lommel functions of two variables, Hankel transforms, Zernike polynomials and their applications for calculation in wave optics. A grasp of the Fourier optics.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Born M., Wolf E.: Principles of Optics. 7th ed., kap. 8, 9, Appendix VII, Cambridge University Press 1999.
Bracewell R. N.: The Fourier Transform and its Applications. 2nd ed., McGraw-Hill Book Co., New York 1986.
Goodman J. W.: Introduction to Fourier Optics. 2nd ed., McGraw-Hill Co., New York 1996.
Komrska J.: Fourierovské metody v teorii difrakce a ve strukturní analýze, VUTIUM, Brno 2001.
Papoulis A.: Systems and Transforms with Applications in Optics., McGraw-Hill Co., New York 1968.

Recommended reading

Iizuka K.: Engineering Optics. 2nd ed., Springer Verlag, Berlin 1987.
Saleh B. E. A., Teich C.: Základy fotoniky 1, Matfyzpress, Praha 1994.

Classification of course in study plans

  • Programme N-FIN-P Master's 1 year of study, summer semester, compulsory-optional
  • Programme N-PMO-P Master's 1 year of study, summer semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Radim Chmelík, Ph.D.

Syllabus

1. The Fourier series.

2. The Dirac distribution, its definition, properties, and expressions in various coordinate systems. Examples.

3. The Fourier transform, definition, fundamental theorem. Examples. The diffraction of plane wave by a three-dimensional structure. The Ewald spherical surface.

4. The Fraunhofer diffraction as the Fourier transform of the transmission function. Meanings of variables in the Fourier transform. Spatial frequencies.

5. 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.Convolution and the Fourier transform of convolution.

6. The Fourier transform in computer.

7. The Bessel functions. The intensity distribution near the focus.

8. The Fourier transform in polar coordinates. The Hankel transforms.

9. The Fourier transform in spherical coordinates.

10. The wave description of the image formation by a lens.

11. Linear systems. The transfer function.

12. Image formation by the zone plates. Diffraction optics.

Exercise

39 hod., compulsory

Teacher / Lecturer

prof. RNDr. Radim Chmelík, Ph.D.

Syllabus

Discussion, calculations and/or laboratory demonstrations of the topics specified during the lectures.