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Macroscopic Maxwell Equations

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Macroscopic Maxwell Equations J. Förstner How HPC helps exploring electromagnetic near fields Jens Förstner Theoretical Electrical Engineering Outline J. Förstner • Maxwell equations • some analytical solutions – homogeneous media – point-like sources • challenges for wavelength-sized structures • examples from the TET group Maxwell equations J. Förstner Starting point of this talk are the macroscopic Maxwell equations: div 퐷(푟Ԧ, 푡) = 휌(푟Ԧ, 푡) Gauss's law (Electric charges are the source of electro-static fields) Gauss's law for magnetism div 퐵(푟Ԧ, 푡) = 0 (There are no free magnetic charges/monopoles) curl 퐸(푟Ԧ, 푡) = −휕푡퐵(푟Ԧ, 푡) Faraday's law of induction (changes in the magnetic flux electric ring fields) curl 퐻(푟Ԧ, 푡) = 휕 퐷(푟Ԧ, 푡) + 퐽Ԧ (푟Ԧ, 푡) Ampere's law with Maxwell's addition 푡 (currents and changes in the electric flux density magnetic ring fields) 퐸 electric field strength 퐻 magnetic field strength 퐷 electric flux density 퐵 magnetic flux density 푑 휕푡 ≔ 푃Ԧ macroscopic polarization 푃Ԧ magnetic dipole density 푑푡 휌 free electric charge density 퐽Ԧ free electric current density 퐶2 푁 휀 = 8.85 ⋅ 10−12 vacuum permittivity 휇 = 4휋10−7 vacuum permeability 0 푁푚2 0 퐴2 Maxwell theory J. Förstner - magnetism (earth, compass) DC - binding force between electrons & nucleus => atoms pg - binding between atoms => molecules and solids l.j AC circuit - <1 kHz: electricity, LF electronics - antennas, radation: radio, satellites, cell phones, radar - metallic waveguides: TV, land-line communication, power transmission, HF electronics - lasers, LEDs, optical fibers Full range of effects are described by the - medical applications same theory: Maxwell equations However the material response depends - X-Ray scanning strongly on the frequency. - astronomy http://imagine.gsfc.nasa.gov/Images/science/EM_spectrum_full.jpg, http://imagine.gsfc.nasa.gov/Images/science/EM_spectrum_ful Maxwell equations J. Förstner Starting point of this talk are the macroscopic Maxwell equations: div 퐷(푟Ԧ, 푡) = 휌(푟Ԧ, 푡) Gauss's law (Electric charges are the source of electro-static fields) Gauss's law for magnetism div 퐵(푟Ԧ, 푡) = 0 (There are no free magnetic charges/monopoles) curl 퐸(푟Ԧ, 푡) = −휕푡퐵(푟Ԧ, 푡) Faraday's law of induction (changes in the magnetic flux electric ring fields) curl 퐻(푟Ԧ, 푡) = 휕 퐷(푟Ԧ, 푡) + 퐽Ԧ (푟Ԧ, 푡) Ampere's law with Maxwell's addition 푡 (currents and changes in the electric flux density magnetic ring fields) Together with the constitutive/material relations: 1 퐷 = 휀0퐸 + 푃 퐻 = 퐵 − 푀 휇0 퐸 electric field strength 퐻 magnetic field strength 퐷 electric flux density 퐵 magnetic flux density 푑 휕푡 ≔ 푃Ԧ macroscopic polarization 푃Ԧ magnetic dipole density 푑푡 휌 free electric charge density 퐽Ԧ free electric current density 퐶2 푁 휀 = 8.85 ⋅ 10−12 vacuum permittivity 휇 = 4휋10−7 vacuum permeability 0 푁푚2 0 퐴2 Material models J. Förstner The e.m. fields originate from free charges (휌, 퐽Ԧ) and bound charges (푃, 푀). The charges, however, feel a force via the electromagnetic fields: 퐹Ԧ = 푞퐸 + 푞푣Ԧ × 퐵 Coulomb Lorentz force force https://de.wikipedia.org/wiki/Datei:Polarisiertes_Atom.svg This force accelerates the charges leading to changes in 휌, 퐽Ԧ, 푃, and 푀: changes fields via MW eq. material state e.m. fields MW 휌, 퐽Ԧ, 푃Ԧ, 푀 퐸, 퐵, 퐷, 퐻 eq. changes material via forces ⇒ The material quantities are functionals of the fields, i.e. they may depend on the fields at all other points in space in time. ⇒ complex spatio-temporal coupled dynamics! Material models J. Förstner With a few assumptions (linearity, locality, causality, achirality, time invariance), the material relation for 퐷 can be written in frequency space as simple proportionality: 퐷 푟Ԧ, 휔 = 휀Ԧ 푟Ԧ, 휔 퐸(푟Ԧ, 휔) In non-conducting dielectric materials the restoring force on bound charges often scales mostly linear with the external force (Hooke's law, linear spring). This leads to a (damped) harmonic oscillator called Lorentz model. 푠 ሷ ሶ 2 + - 푢 + 훾푢 + 휔0푢 = 훼퐸 ⇒ 휀 휔 = 휀0 + 2 2 휔 − 휔0 − 푗휔훾 In solids there are many types of oscillations (electronic, atomic, dipolar, ionic) of different frequencies which superpose, i.e. sum up: real part 휀′ → dispersion imaginary part 휀′′ → damping /wiki/Permittivity en.wikipedia.org https:// The wave equation J. Förstner Assuming a spatial homogeneous material, i.e. spatially constant 휀ǁ & 휇, and no free charges one can derive the wave equation, in frequency domain called Helmholtz equation: 2 Δ 퐸 푟Ԧ, 휔 + 휔 휀ǁ 휔 휇 휔 퐸 푟Ԧ, 휔 = 푗휔휇 휔 퐽Ԧ푒(푟Ԧ, 휔) 푗휔푡−푗푘⋅푟Ԧ One set of solutions are plane waves (for 퐽푒 = 0): 푒 The (circular) frequency 휔 and wave number 푘 are linked via a dispersion relation: 푘2 = 휔2휀ǁ 휔 휇(휔). The real part 훽 = Re 푘 determines the wavelength 휆 = 2휋/훽 (i.e. spatial period), the speed of light in a medium 푣푝ℎ = 휔/훽, and it's derivative the group velocity 푣푟 = 휕휔/휕훽 The imaginary part 훼 = Im 푘 determines damping effects. Superpositions lead to more complex field patterns (interference). Interfaces J. Förstner Things get interesting at interfaces between homogeneous media: Refraction: 휀1 < 휀2 ⇒ towards normal 휀1 > 휀2 ⇒ away from normal https://en.wikipedia.org/wiki/Total_internal_re flection#/media/File:Total_internal_reflection_ of_Chelonia_mydas.jpg For 휀1 > 휀2 total reflection can occur above a critical angle: (100% reflection, evanescent decaying field in media 2) https://www.flickr.com/photos/jtbss/9393445794 This is the basis for wave guiding in dielectrics ⇒ fibre optics, integrated photonics https://www.photonics.com/Articles/Integrated_Photonics_A_Tale_of_Two_Materials/a60862 Dispersion J. Förstner The material parameter 휀(휔) depends on frequency ⇒ strength of refraction & speed of light differs for spectral components examples: rainbow material dispersion in fibers chromatic abberation of lenses https://www.opternus.de/anwendungsgebiete/optische- http://avax.news/touching/Simply_Some_Photos_Rainbow_04-12-2014.html messtechnik/cd-chromatische-dispersion prism https://rivel.com/the-prism-a-full-spectrum-of-color-on-governance-issues/ http://pixxel-blog.de/was-ist-eigentlich-chromatische-aberration/ Tiny particles J. Förstner Homogeneous media & simple boundaries ⇒ analytical solutions ⇒ no need for HPC. How about tiny particles (much smaller than the wavelength), look at point-like emitter: 2 퐸 z =F3SXmgm2uxM watch?v / 0 x 푆 휗 ∝ sin2 휗 www.youtube.com near field far field https:// 푝 1 2푗 electric field: 퐸 = 푒−푗푘푟 푘2 − sin 휗 휗 4휋휀 푘2푟2 푘푟 far angular dependence spherical near field via 푘 ∝ 휔 field (radiation pattern) waves frequency (small for dependence large r) Raleigh scattering J. Förstner This also explains how e.m. fields scatter off tiny particles (Rayleigh scattering): 1 푃푠푐푎푡푡푒푟푒푑 휔 ∝ 4 푃푖푛(휔) 휆 light !) scale Rayleigh_scattering / scattered wiki / of Some consequences: power (linear power (1) Blue sky en.wikipedia.org Percentage https:// sky_blue.html / white-yellow light directly from the sun blue sky from scattered light www.sciencemadesimple.com http:// Rayleigh Streuung J. Förstner (2) Sky pale/whiter near horizon (3) sunsets are red molecules scatter and dust reflects light from the sunlight sky near the sun appears white-yellow light red directly from the sun blue sky blue light scatters light directly blue light removed by from the sun additional scattering appears red http://www.sciencemadesimple.com/sky_blue.html Idee von Ideevon (3) scattering in "milk opal" losses in fibres " Rayleigh+UV+IR Photonik Jahns " Jahns s/0/0b/Why_is_the_sky_blue.jpg https://upload.wikimedia.org/wikipedia/common Tiny particles J. Förstner One tiny particle ⇒ no need for HPC. How about the mesoscopic e.m. "Mie" regime, i.e. particle size ≈ wavelength? Only few analytical solutions for high symmetry: Spherical: Mie solutions, spherical harmonic functions https://de.wikipedia.org/wiki/Kugelfl%C3%A4chenfunktionen Planar symmetries Cylindrical Everything more complex ⇒ numerical simulation One example J. Förstner Simulation of the Second Harmonic Generation (SHG) in arrays of gold split ring resonators. Question for the theory: Where and how is SHG signal generated? • Surface? • Bulk? • Substrate? • Depositions? Challenge: strong variation of fields J. Förstner P) - © Matthias Reichelt (UPB, NW (UPB, Reichelt Matthias © Electromagnetic fields are strongly enhanced and vary on extremely short scales Simulation of nanostructures J. Förstner -> challenges for theory: • Strong near field enhancement and extreme field variation, • Complex optical response of materials (dielectrics and metals): nonlinearites, nonlocality, anisotropy, decoherence, • Nontrivial short- and long distance coupling. Requires: • Advanced nonlinear/nonlocal/anisotropic material models, • Adaptive mesh time domain PDE solver, • Efficient parallel implementations. ⇒ All tested available tools failed Our numerical method of choice J. Förstner Nodal Discontinuous Galerkin Time-Domain Method (DGTD) (unstructured grid, related to FEM&FVM) Spatial distribution of interpolation nodes in an element The field components for 퐸 and 퐻 are expanded locally in each cell. There Maxwell and material equations are solved: Then exchange of e.m flux. Hesthaven, Warburton, Springer Book (2007) ∆E, ∆H Busch et al, Laser & Photonics Reviews (2011) Properties of the DGTD method J. Förstner The Discontinuous Galerkin Time Domain (DGTD) method: ☺ Unstructured, adaptive mesh -> multiscale, multiphysics, ☺ full geometrical flexibility (substrate, materials, etc) ☺ direct incorporation of nonlinear material equations in TD, ☺ stability can be proven, even for some nonlinearities, ☺ excellent parallel scaling, HPC! complex method, effort of implementation,
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