Macroscopic Maxwell Equations

Home , Vacuum permeability, Vacuum permittivity

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

- 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:

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 푣𝑔푟 = 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



=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 von Idee

(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

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,  high cost of mesh generation.

Cooperations with C.Plessl/PC2, BMBF project HighPerMeshes Research topics in my group J. Förstner

complexity of material description hybrid/dynamic - - surfaces e -e intersubband meta materials bulk propagation scattering transitions

ph quantum dots nonlinear plasmonics Nano antennas, SBE (HF) Geometry optimization

coupled quantum wells spin currents quantum dots in Ensembles, photonic resonators Drude quantum dot ensemble pulse shaping Microdisks (bi-)chiral level system in liquid crystals structures complexity - 1d 2d 3d of optics description Higher Harmonic Generation J. Förstner Simulated emission using symmetrized grids:

THG

SHG

• SHG only in symmetry-broken direction (y) near • (Semi-classical) Fermi pressure negligible field • SHG mainly generated at edges

• Advection and charge shift counteracting, still larger than 퐽Ԧ × 퐵 nonlinearity. • Third harmonic generation (THG) in excitation direction hybrid plasmonic/dielectric nanoantennas J. Förstner related structure, single particle ("nanoantenna") experiment:

Model roughness, near fields at rough surface: Smooth

This explained the 1e 1e experimentally 11 13 observed strong SHG Rough signal.

5e 5e 12 15 [Light: Science & Applications 2016] Application of TD-DG to dust particles J. Förstner

Scattering of microwaves at larger particles (풓 ≪ 흀), e.g. at interplanetary dust and atmospheric ice particles:

e.g. used to determine size distribution of cometary dust from radar measurements

Large particles and rough surfaces are numerically very demanding → "Discontinuous Galerkin method" (lecture) Application of TD-DG to dust particles J. Förstner X=60:

TD-DG & GO

main result: size important, shape not so much

X=200:

TD-DG & DDA

good agreement, differences for imaginary part, roughness only has some influence, but small Biological photonic crystals J. Förstner

pronounced reflection band (with rotation Biomimetic (i.e. related to nature, but of the circular polarization by multiple technologically easier to realize) structure shows interference) ⇒ polarization filter same behaviour

we also investigate artificial photonic crystals: Cooperation with Xia Wu (UPB NW-C) Bi-chiral photonic crystals J. Förstner

Triple interveaved helix array (H. Giessen/Stuttgart): Experiment

Theory (Discontinuous Garlerkin): Ergebnisse: Theory

left circular polarized (LCP) and right circular polarized (RCP) light transmitted very differently ⇒ ultra thin polarizer HF circuits and antenna simulations J. Förstner

Simulation of a bluetooth antenna in a car radio/infotainment system Consider electronics and housing ⇒ optimize radiation and EMC (Electromagnetic compatibility)

with Continental Automotive

EMC simulation of SEPIC (DC-DC) Combination of Spice+Maxwell (with CST Studio)

• Verification of simulation method by measuring several designs • Reduction of interference to fulfill EMC requirements

with Phoenix Contact Thanks to J. Förstner My group

HPC: Christian Plessl and his team for acquiring and maintaining the PC2 systems, and cooperation/support on HPC programming

funding: And YOU for your attention!