Mathematics of Wave Optics (FSI-9MAV)

Academic year 2020/2021
Supervisor: prof. RNDr. Jiří Petráček, Dr.  
Supervising institute: ÚFI all courses guaranted by this institute
Teaching language: Czech or English
Aims of the course unit:
To gain an overview of mathematics used in wave optics.
Learning outcomes and competences:
An overview of special functions.
Applications of special functions in wave optics.
Prerequisites:
The exposition is kept in frames of functions of real variables and applications are emphasized.
Course contents:
Special functions are frequently used in monographs and papers dealing with wave optics. In spite of that they are not involved in the curricula (e. g. the Lommel functions of two variables or the Fresnel integrals). In some cases a mathematical literature does not exist at all (e. g. the Zernike polynomials). Most of the graduates from the technological universities never studied special functions, not even the standard ones (e. g. the Bessel functions). Therefore, the post-graduate students of optical engineering have troubles with the study of books and papers, with mathematical treatment of their own results, and with numerical calculations. The present course offers an overview of mathematics used in wave optics. The exposition is kept in frames of functions of real variables and applications are emphasized.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline.
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.
Controlled participation in lessons:
The presence of students at practice is obligatory and is monitored by a tutor. The way how to compensate missed practice lessons will be decided by a tutor depending on the range and content of the missed lessons.
Type of course unit:
    Lecture  10 × 2 hrs. optionally                  
Course curriculum:
    Lecture 1. Elementary functions.
2. Gamma and digamma functions.
3. Sine and cosine integrals.
4. The Fresnel integrals.
5. The Dirac distribution.
6. Orthogonal systems of functions. The Gramm-Schmidt orthogonalization process.
7. Hypergeometric functions.
8. The Bessel functions.
9. The Fourier transform.
10. The Hankel transforms.
11. The Jacobi polynomials.
12. The Gegenbauer polynomials.
13. The Chebyshev polynomials.
14. The Zernike polynomials.
Literature - fundamental:
1. Andrews L. C.: Special functions of mathematics for engineers. 2nd ed.. McGraw-Hill Inc., New York 1992.
2. Sneddon I. N.: Special functions of mathematical physics and chemistry. Oliver and Boyd, Edinburgh 1966.
3. Temne N. M.: Special Functions. John Wiley & Sons, Inc., New York 1996.
Literature - recommended:
1. Whittaker E. T., Watson G. N.: A Course of Modern Analysis. Cambridge University Press, Cambridge 1965.
2. Lebeděv N. N.: Speciální funkce a jejich použití. SNTL, Praha 1956.
3. Watson G. N.: A Treatise on the Theory of Bessel Functions. 2nd ed.. Cambridge University Press, Cambridge 1966.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
D4F-P full-time study D-FMI Physical and Materials Engineering F Physical Engineering DrEx 0 Recommended course 3 1 W