Layered media and photonic crystals Cord Arnold / Anne L Huillier
Definition A photonic crystal is a periodic arrangement of a dielectric material that exhibits strong interaction with light Variation of refractive index on the scale of the wavelength λ Photonic Crystals; J.D. Joannopoulos
Artificial photonic crsytals 1D: Bragg Reflector 2D: Si pillar crystal J. Serbin LZH 3D: Wood pile diameter of the rods 200 nm, spacing in-between 250 nm 3D: Colloidal crystal
Natural Examples Natural opals http://www.viewsfromscience.com/documents/webpages/natural_photonics_p1.html
Multilayer Optics
How to analyze a multilayer system Wave-Transfer Matrix Scattering Matrix Conversion Analysis of multilayer optical system S i M i M Σ S Σ
Cascaded system - Airy formulas
Examples: Homogeneous medium
Examples: Partially reflective mirror (beamsplitter)
Examples: Single dielectric boundary
Off-axis waves in layered media Reflections become angle-dependent Reflections become polarisation dependent The phase is obtained from a projection on the optical axis The angle is different in different refractive index media
Single boundary Off Axis Waves in Layered Media
Fabry-Perot Etalon (Mirror) Free spectral range
Dielectric Slab as a Fabry-Perot Etalon
Off-Axis Transmittance of FPE
Bragg grating
Dielectric Bragg grating Condition ( nd + nd ) mπ ϕ1+ ϕ2 = k 0 1 1 2 2 = The accumulated phase shift must add to π, λ/2
Off-axis high-reflection mirrors
Examples for 1-dimensional multilayer structures Dielectric multi-layer mirror: - Alternating stack of high and low refractive index materials (n 1 >n 2 ) with optical thickness of lambda/4 in each layer. - Reflectivity >99.999% for narrow bands, >99% for very broad bands n 1 n 2 Typical coating materials: - Magnesium fluoride n~1.3 - Silicone dioxide n~1.45 - Tantalum pentoxide n~2 - Zinc sulfide n~2.3 - Titanium dioxide n~2.4 Image source: Wikipedia
Examples for 1-dimensional multilayer structures Chirped dielectric mirror: - The layer thickness changes as function of depth into the mirror. Blue wave lengths are reflected at the surface, red in the depth. - Chirped mirrors are used to cancel dispersion in ultrashort pulse oscillators. - They form the basis of today s femtosecond laser technology. - Reflectivity >99% for 500-1000nm.
Examples for 1-dimensional multilayer structures Fiber Bragg grating (FBG): - Wavelength filtering and multiplexing - Single frequency fiber lasers - Stretching, compression, dispersion compensation with chirped FBGs - Sensing, e.g. temperature and pressure Image source: Wikipedia
A short introduction into photonic crystals
Dispersion relation K(ω) and wave localization for a 1d photonic crystal made from alternating dielectric layers
Dispersion relation K(ω) for a 1d photonic crystal made from alternating dielectric layers
Phase and group velocity Slow region Phase velocity Group velocity c n c 0 vp = eff 0 vg = Neff
Photonic band gaps at off-axis propagation Internal off-axis waves External off-axis propagation coupled into the structure
Rectangular lattice Two-dimensional photonic crystals Triangular lattice
Photonic Crystal in Two Dimensions Band Gap for TM modes
2D PhC, Complete (!) 2D Band Gap Triangular lattice of holes!, a, r ~λ
3D Photonic Crystals: Yablonovite Historically 1 st 3D Band Gap Crystal
3D PhC Woodpile structure
Defect modes
Defect modes By introducing a defect in the highly periodic environment as for example removing one rod or simply changing the radius of one or a few roods, the periodicity is broken and defect modes can exist in the forbidden gap! This e.g. allows for laser cavities with dimensions of the order of one wave length!
Bending light in 2D PhC
PhC Add-Drop Splitter λ1 >λ2 >λ3 λ1 >λ3 λ2 NB: Add-drop filter with wavelength selectivity of ca 1% having dimensions of ca. one wavelength
End Of Lecture