applied mathematics, photonics

Exposing the student to various mathematical concepts often used in photonics. The aim is to make the student acquainted with the basic principles and references, in order to allow him to independently further research these concepts.

• 1: Complex analysis: wave problems as problems from complex analysis, complex functions, analytic functions, derivatives, line integrals, poles, zeros, branch cuts, residue calculus, limit theorems, Cauchy principal value, Kramers-Kronig dispersion relation, conformal transformations, bend losses in optical waveguides
• 2: Special functions: modes of an optical fibre, Bessel and Neuman functions, generating functions, recursion relations, integrals, orthogonality, series expansion, higher order solutions of the paraxial wave euqation, Hermite polynomials, generating function, recurrence relation, differential equation, orthogonality, series expansion
• 3: Numerical techniques: finite elements, finite differences, variational methods, eigenmode expansion, method of weighted residuals
• 4: Periodicity and symmetry in photonic systems: using symmetries to classify modes, Bloch theorem, band diagrams, photonic crystals
• 5: Dynamical systems: origins of non-linearity in optical systems, stability, fixed points, the logistic map, sadlle points, bifurcations, chaos, period doubling, Lyaponuc exponent, stable and unstable manifold, phase portrait

mathematics from the bachelor program

being able to apply complex analysis to photonic problems; being able to apply special functions and orthogonal polynomials to photonic problems; getting a basic insight in the effects of symmetry on photonic systems; getting a basic insight into numerical techniques for photonics; being able to study the dynamics of a photonic system; being able to study a new mathematical topic in an independent and critical manner and apply it in a creative way.

full lecture notes

Classroom lectures; Classroom problem solving sessions

Evaluation during examination period

During examination period: written open-book exam - problems; oral open-book exam, written preparation