Photonic Wannier Functions: Generation and...

Institut für Theoretische Festkörperphysik – Photonics Group

photonics.tfp.uni-karlsruhe.de

Photonic Wannier Functions: Generation and Usage

Kurt Busch

Institut für Theoretische Festkörperphysik and

DFG-Center for Functional Nanostructures, KIT, Germany


Dr. Sabine Essig

Dr. Regine Frank

Dr. Patrick Mack

Dr. Jens Niegemann

Richard Diehl

Jens Küchenmeister

Michael König

Paolo Longo

Christian Wolff

Thomas Zebrowski

Ferdinand Kaplan

Martin Köhl*

Timo Köllner

Benjamin Lutz

Christoph Martens

Matthias Moeferdt

Petru Tighineanu

Michael Walz

Acknowledgments

Prof. Dr. Ralf Wehrspohn (Fraunhofer Institute &

University Halle)

Prof. Dr. Martin Wegener (KIT)

Prof. Dr. Georg von Freymann (University of Kaiserslautern)

Prof. Dr. Stefan Linden (University of Bonn)

Prof. Dr. Wolfram Hergert (University of Halle)

Dr. Antonio Garcia-Martin (now @ IMM-CSIC, Madrid)

Dr. Matthias Schillinger (now @ D-Fine)

Dr. Daniel Hermann (now @ 1&1)

Christian Blum (now @ HU Berlin)


Motivation

Wannier Functions: Generation

Wannier Functions: Usage

Conclusions

Outline


Motivation: Photonic Crystals

I. Staude et al., Opt. Lett. 35, 1094 (2010)

M. Deubel et al., Nature Materials 3, 444 (2004)


I. Staude et al., Opt. Lett. 36, 67 (2011)



Aequous HF

Wa

fer

cro

ss s

ectio

n

IlluminationDonator ion (e.g., P)

Hole

SCR

A. Birner et al., Adv. Mater. 13, 377 (2001)



P. W. Nolte et al., Adv. Mater. 22, 4731 (2010)




Numerically Large Systems even for Simple Constituent Materials

Complex Materials: Multi-Component Systems, Optical Anistropy ….

Design / Optimization of Photonic-Crystal-based Circuitry

Novel Functionalities

Disorder

Photonic Wannier Function Approach

K. Busch et al., J. Phys.: Cond. Mater. 15, R1233 (2003)

K. Busch et al., J. Mod. Opt. 58, 365 (2011)


Motivation



Conclusions

Outline


·@x

1

²(r)@x + @y

1

²(r)@y ¡

!2

c2

¸H(r; !) = 0

Hnk(r+R) = exp(ikR)Hnk(r)

WnR(r) =

r1

VBZ

Z

BZ

d2k exp(¡ikR)Hnk(r)

Photonic Wannier Functions

²(r) = ²(r+R) 8R 2R

2D H-Polarization: Wave Equation

Ideal Photonic Crystal: Bloch Functions

Localized Basis: Wannier Functions


Integration? Alright, but Where‘s the Band?


Band 26

Wannier Functions: Naive Integration


Band 26

Wannier Functions: “Correct“ Processing


WnR(r) =

r1

VBZ

Z

BZ

d2k exp(¡ikR)X

n0

Un0n(k)Hn0k(r)

Generalized Wannier Functions

Free Parameters (“Gauge Transformation“)

Un0n(k)

Minimize Spread Functional

[fUnn0(k)g] =X

n

Zd2rW¤

nR(r)¡r ¡ hri2n

¢2W¤

nR(r)

hrin =X

n

Zd2rW¤

nR(r)rW¤nR(r)


r = 0.45 a

Macroporous Silicon: Triangular Lattice


Souza-Marzari-Vanderbilt Algorithm

N. Marzari, D. Vanderbilt, Phys. Rev. B 56, 12847 (1997)

I. Souza et al., Phys. Rev. B 65, 035109 (2001)


Possible Wannier Function Symmetries for Various Wannier Centers

Wannier Function Symmetry from Bloch Function Symmetry („Band Symmetry“)

Grouping of Bands for Souza-Marzari-Vanderbilt: Selection Rules

E. Krüger, phys. stat. sol. (b) 52, 215 (1972)

E. Krüger, phys. stat. sol. (b) 52, 519 (1972)

Extended to Photonic Wannier Functions:

P. Mack, K. Busch, in preparation

Group-Theoretical Preprocessing



Selection Rules: 2D Triangular Lattice, H-Polarization



3D Photonic Wannier Functions: Woodpile

Band 1

Band 3

Band 15

Band 25



Band Structure Reconstruction from Wannier Functions



Motivation



Conclusions

Outline


·@x

1

²(r)@x + @y

1

²(r)@y ¡

!2

c2

¸H(r; !) = 0

2D H-Polarization: Wave Equation

H(r; !) =X

n

X

R

CnRWnR(r)

The Master Equation

²(r) = ²per(r) + ±²(r)


The Master Equation

µCnn0;RR0 +Dnn0;RR0 ¡ !2

c2Ann0;RR0

¶cn0R0 = snR

Ann0;RR0 = hWnRjWn0R0i

Cnn0;RR0 = hWnRj @x1

²per(R)@x + @y

1

²per(R)@y jWn0R0i

Dnn0;RR0 = hWnRj @x¡±²(r)

²(r)²per(r)@x + @y

¡±²(r)²(r)²per(r)

@yjWn0R0i


Types of Defects and Reusability of Matrices

Dnn0;RR0 =X

Rdef

Bnn0;(R¡Rdef) (R0¡Rdef)


Types of Defects and Reusability of Matrices


Convergence



Convergence: Auxiliary Wannier Functions



Convergence

Wannier Functions Auxiliary Wannier Functions



Convergence



Tunable Waveguides



Tunable Splitter



D. Hermann et al., J. Opt. Soc. Am. B. 25, 202 (2008)

Photonic-Crystal Circuit Theory


Photonic-Crystal Circuit Theory

D. Hermann et al., J. Opt. Soc. Am. B. 25, 202 (2008)


Green‘s Function in Wannier Representation

·@2x + @2y +

!2

c2²(r)

¸E(r; !) = 0

r = 0.18 a


G(r;rs;!) = ²per(rs)X

®;¯

W¤®(rs)G®¯(!)W¯(r)!)

G®¯(!) =X

m

Z

BZ

d2kUnm(k)U¤

n0m(k)

!2 ¡ !2mkexp (ik ¢ (R0 ¡R))

Green‘s Function in Wannier Representation


Density of States and Local Density of States



Disorder: Cluster CPA

M. Köhl, C. Wolff, K. Busch, submitted

5% Silicon Rods in Square Array of ZnO rods


Photonic Wannier Functions: Powerful Tool for Photonic Crystals

“Atomic Orbitals for Light”

Generation

Usage

Functional Elements

Photonic Crystal Circuit Theory

Disorder: Cluster CPA

Time-Domain Version

Conclusions

C. Blum, C. Wolff, K. Busch, Opt. Lett. 36, 307 (2011)

Photonic Wannier Functions: Generation and...

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