20.500.12556/DKUM-90811 Prepovedani pasovi v fotonskih kristalih Band gaps in photonic crystals V magistrski nalogi obravnavamo fotonske kristale in se osredotočimo na izračun prepovedanih pasov. Fotonski kristali so materiali, v katerih se dielektrična konstanta s krajem periodično spreminja. V takšnih materialih lahko obstajajo prepovedani pasovi oziroma območje frekvenc, pri katerih se elektromagnetno valovanje ne more razširjati skozi material, ampak se odbije. Prepovedani pas, ki obstaja pri vseh možnih smereh razširjanja elektromagnetnega valovanja skozi material, se imenuje popolni prepovedani pas. Za izračun prepovedanih pasov v dvodimenzionalnih in tridimenzionalnih fotonskih kristalih uporabimo metodo prenosne matrike v realnem prostoru. Izpeljemo disperzijsko zvezo, ki opisuje odvisnost frekvence od valovnega števila. Zvezo uporabimo v numeričnih izračunih, ki jih izvedemo z računalniškim programom Wolfram Mathematica. Pri določeni smeri razširjanja elektromagnetnega valovanja skozi material prikažemo rezultate numeričnih izračunov na grafu frekvence v odvisnosti od valovnega števila, iz katerega so razvidni prepovedani pasovi. Ugotovimo, da metoda na osebnem računalniku v programu Wolfram Mathematica omogoča izračun prepovedanih pasov samo za dvodimenzionalne fotonske kristale. Za izračun prepovedanih pasov v tridimenzionalnih fotonskih kristalih uporabimo programska paketa MEEP in MPB. Programski paket MEEP omogoča shranjevanje geometrijskih lastnosti struktur, numerično reševanje Maxwellovih enačb in različne simulacije elektromagnetnih pojavov. Programski paket MPB je integriran v paket MEEP in je posebej zasnovan za izračun prepovedanih pasov v fotonskih kristalih za različne geometrijske lastnosti. Osredotočimo se na modri tekočekristalni fazi BPI in BPII. V osnovni celici modrih faz BPI in BPII se molekule uredijo v dvojno zvite cilindre. Predpostavimo, da dielektrična konstanta znotraj cilindrov ni odvisna od smeri, zato namesto dielektričnega tenzorja upoštevamo dielektrično konstanto. Ugotovimo, da v modrih fazah BPI in BPII nastanejo prepovedani pasovi, ki pa niso popolni. Za modro fazo BPI upoštevamo, da je velikost osnovne celice enaka 240 nm in izračunamo, da v vidnem območju elektromagnetnega spektra obstajajo prepovedani pasovi. Za modro fazo BPII pa upoštevamo, da je velikost osnovne celice enaka 150 nm in izračunamo, da v vidnem območju elektromagnetnega spektra prav tako obstajajo prepovedani pasovi. In the master’s thesis, we study photonic crystals and focus on the calculation of band gaps. Photonic crystals are materials in which the dielectric constant changes periodically with position. In such materials, band gaps can exist, which are frequency ranges where electromagnetic waves cannot propagate through the material but are instead reflected. A band gap that exists for all possible directions of electromagnetic wave propagation through the material is called a complete band gap. To calculate the band gaps in two-dimensional and three-dimensional photonic crystals, we use the transfer matrix method in real space. We derive the dispersion relation, which describes the dependence of frequency on wave number. This relation is used in numerical calculations performed with the Wolfram Mathematica software. For a specific direction of electromagnetic wave propagation through the material, we present the results of the numerical calculations in a graph of frequency as a function of wave number, showing the band gaps. We find that on a personal computer, the method in Wolfram Mathematica allows the calculation of band gaps only for two-dimensional photonic crystals. For the calculation of band gaps in three-dimensional photonic crystals, we use the software packages MEEP and MPB. The MEEP package enables the storage of geometric properties of structures, the numerical solution of Maxwell’s equations, and various simulations of electromagnetic phenomena. The MPB package is integrated into MEEP and is specifically designed to calculate band gaps in photonic crystals with different geometric properties. We focus on the blue liquid crystal phases BPI and BPII. In the unit cell of the blue phases BPI and BPII, molecules arrange into double-twisted cylinders. We assume that the dielectric constant within the cylinders is isotropic, so instead of the dielectric tensor, we consider the dielectric constant. We find that in blue phases BPI and BPII, band gaps occur but are not complete. For the blue phase BPI, we consider that the size of the unit cell is 240 nm and calculate that there are photonic band gaps in the visible range of the electromagnetic spectrum. For the blue phase BPII, we consider that the size of the unit cell is 150 nm and calculate that there are also photonic band gaps in the visible range of the electromagnetic spectrum. fotonski kristali fotonski prepovedani pasovi dielektrična konstanta disperzijska zveza metoda prenosne matrike v realnem prostoru modre faze MEEP in MPB photonic crystals photonic band gaps dielectric constant dispersion relation transfer matrix method in real space blue phases MEEP and MPB true true false [A. Ficko] Slovenski jezik Angleški jezik Magistrsko delo/naloga 2024-09-24 18:20:47 2024-10-18 14:54:08 2024-10-19 03:11:38 0000-00-00 00:00:00 2024 0 Maribor 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 538.9(043.2) 211710723 MAG_Ficko_Anica_2024.pdf MAG_Ficko_Anica_2024.pdf 1 AC7090B929BDDD4F62AFBEE95242D23F bde79afe5cbd7b651963c912d07b9993ab7c30a792cb71a38be50b7676bfa2b4 0571aa05-7f34-11ef-8e23-00155d000105 https://dk.um.si/Dokument.php?lang=slv&id=186057 Fakulteta za naravoslovje in matematiko 0 0 0