Electromagnetic Metamaterials

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ca ca Department of Electrical and Computer Engineering

1 Photonic Bandgap and Electromagnetic Metamaterials Andrew Kirk ca Department of Electrical and Computer Engineering McGill Institute for Advanced Materials A Kirk 11/24/2008 Photonic bandgap and metamaterials 1

2 References and further reading StevenJohnson (MIT), Tutorial slides and notes: initio.mit.edu/photons/tutorial/ J M Lourtioz, Photonic Crystals: Towards Nanoscale Devices, Springer, 2005 Maksim Skorobogatiy (Ecole Polytechnique), course in photonic crystals A Kirk 11/24/2008 Photonic bandgap and metamaterials 2

3 Contents Background: EM waves Photonic crystals Bandgap effects Resonant cavities Superprisms Negative index properties Metamaterials and optical cloaking Fabrication A Kirk 11/24/2008 Photonic bandgap and metamaterials 3

4 After attending this class, you should be able to: Explain what is meant by a photonic bandgap gp Explain what is meant by an electromagnetic metamaterial Estimate the critical dimensions for photonic bandgap structures, as a function of wavelength Describe the way that photonic bandgap structures can be used to modify the reflective, refractive or dispersive properties of materials Describe the common fabrication processes for photonic bandgap and metamaterials A Kirk 11/24/2008 Photonic bandgap and metamaterials 4

5 A Kirk 11/24/2008 Photonic bandgap and metamaterials 5

6 Harmonic Waves U 0 0 -U 0 /2 /2 z U z, t U 0 cos( t kz ) 2 Space 2 1 c f k representation f Time representation Wavelength : distance needed to recover the same phase. Wave period : time needed to recover the same phase. Phase of the wave: state at that point in space time Wave frequency : inverse oftheperiod period. Phase velocity: rate at which constant phase propagates A Kirk 11/24/2008 Photonic bandgap and metamaterials 6

7 An electromagnetism review y Lightis is anelectromagnetic wave: x Orthogonal electric (E) and magnetic (H) fields H Wave equation: E E 2 t v H H 2 t Phase velocity: which is 3x10 8 m/s in vacuum p E z Material properties: Permittivity describes electric field response Permeability describes magnetic field response n v Refractive index n is relative speed of light 1/ p A Kirk 11/24/2008 Photonic bandgap and metamaterials 7

8 Transverse EM waves (plane waves) E 0 Wavefronts: planes of constant phase Have infinite extent -E 0 0 k How do we describe a wave that does not travel on the z axis? Introducethe the wave vector k A Kirk 11/24/2008 Photonic bandgap and metamaterials 8

9 Transverse EM waves Write the wave as: xyzt,,, exp t E E k r 0 k a x k x a y k y a z k z k = Wave vector (m 1 1 ) P k k 2 2 k 2 2 x k 2 k 2 z 2 y u What is the plane for which kris k.r constant? a n 0 z R Plane of constant phase WAVEFRONT A Kirk 11/24/2008 k is always normal to wave front Photonic bandgap and metamaterials 9

10 Dispersion relation Relationship between wave number and frequency For an anisotropic medium, we need to consider the wave vector k Freque ency vk p Wave vector Group velocity is local lgradient of dispersion curve k c k v g d dk In free space, dispersion relation is linear A Kirk 11/24/2008 Photonic bandgap and metamaterials 10

11 Photonic crystals Also known as photonic bandgap materials Periodic dielectric and/or metallic structures Period is typically close to the wavelength of light Can be 1 D, 2 D, 3 D structures May contain defects, df heterostructures and other more complex inclusions 3 D 2 D 1 D n 1 n 2 a period Multilayer Hexagonal A Kirk 11/24/2008 Photonic bandgap and metamaterials Woodpile 11

12 Scattering from periodic structures Coherent illumination with wavelength 0 Each plane scatters waves which interfere, constructively ti or destructively Reflection maxima occur at angles given by Bragg s law: a 2 a cos Or equivalently m ak m A Kirk 11/24/2008 Photonic bandgap and metamaterials ( / ) 0 n eff 12

13 Floquet Bloch theorem Infinitely periodic structure Implies that EM fields will have same periodicity Fields can be expanded as a summation of Bloch waves (i.e. Fourier series) This is Floquet Bloch Theorem Structure has reciprocal lattice, with basis vectors G j Fields are written: u exp jkr. V G exp jkr. Where V derives from the Fourier series Bloch waves are conserved as they move through the lattice G A Kirk 11/24/2008 Photonic bandgap and metamaterials 13

14 Brillouin zones In periodic structure, all of dispersion curve can be mapped into the irreducible Brillouin zone n x n x a a period k is periodic k+2/a is equivalent to k Limit of zero modulation Frequenc cy Frequenc cy A Kirk 11/24/2008 Wavevector k Photonic bandgap and metamaterials ka 0.5 Wavevector 2 14

15 Brillouin zones In periodic structure, all of dispersion curve can be mapped into the irreducible Brillouin zone n x n x a a period k is periodic k+2/a is equivalent to k Finite index difference Frequenc cy Frequenc cy Bandgap A Kirk 11/24/2008 Wavevector k Photonic bandgap and metamaterials ka 0.5 Wavevector 2 15

16 Degeneracy splitting at band edge EM waves d m a period n x n x a Bandgapwidth is approximately Finite index difference Bandgap ka 2 Freque ency Wavevector d 0 m State concentrated in higher indexhaslower frequency A Kirk 11/24/2008 Photonic bandgap and metamaterials 16

17 Bandgap Bandgap: Frequency band for which propagation is forbidden All 1 D photonic crystals have a bandgap Only some 2 D and 3 D crystals have complete bandgaps Freque ency Finite index difference Bandgap ka Wavevector A Kirk 11/24/2008 Photonic bandgap and metamaterials 17

18 a Square 2 D lattice (From Johnson) Rods in air 1 a / uency (2πc/a) = Photonic Band Gap =12:1 freq TM bands 0 irreducible Brillouin zone X M M E gap for k X TM n > ~1.75:1 H A Kirk 11/24/2008 Photonic bandgap and metamaterials 18

19 Electric field distribution (from Johnson) E z (+ 90 rotated version) Photonic Band Gap TM bands E z 0 X M E gap for + TM n > ~1.75:1 H A Kirk 11/24/2008 Photonic bandgap and metamaterials 19

20 3 D photonic crystal: complete gap, =12:1 I. II % gap I: rod layer II: hole layer L' U' X U'' W K' U W' L K UÕ L X W K z gap for n > ~4:1 [ S. G. Johnson et al., Appl. Phys. Lett. 77, 3490 (2000) ] A Kirk 11/24/2008 Photonic bandgap and metamaterials 20

21 Properties of Bulk Crystals (from Johnson) band diagram (dispersion relation) (cartoon) ncy conserv ved freque photonic band gap synthetic medium for propagation backwards slope: negative refraction d/dk 0: slow light (e.g. DFB lasers) strong curvature: super prisms, (+ negative refraction) conserved wavevector k A Kirk 11/24/2008 Photonic bandgap and metamaterials 21

22 Properties of Bulk Crystals (from Johnson) band diagram (dispersion relation) (cartoon) ncy conserv ved freque photonic band gap synthetic medium for propagation backwards slope: negative refraction d/dk 0: slow light (e.g. DFB lasers) strong curvature: super prisms, (+ negative refraction) conserved wavevector k A Kirk 11/24/2008 Photonic bandgap and metamaterials 22

23 Bandgap effects Make use of bandgap to confine light within `defect` waveguide Hexagonal crystal structure typically used Vertical confinement achieved via total internal reflection (i.e. Conventional guiding) Advantage over conventional index guiding is small size However scattering loss is higher (4 db/cm reported) Operating bandwidth is typically <30 nm Watanabe 2007 Standard bend Optimized bend A Kirk 11/24/2008 Photonic bandgap and metamaterials 23

24 Thermo optic optic Mach Zender modulator Camargo 2006 A Kirk 11/24/2008 Photonic bandgap and metamaterials 24

25 Resonant cavity structures Photonic crystal cavity with displaced holes (marked by letters A, B and C), fabricated in SOI [Asano 2006] 2 D bandgaps can be used to define very small cavities Applications include single photon lasers Require high Q factor and small volume (i.e. large Q/V) A Kirk 11/24/2008 Photonic bandgap and metamaterials 25

26 Heterostructure cavities Asano 2006 A Kirk 11/24/2008 Photonic bandgap and metamaterials 26

27 Properties of Bulk Crystals (from Johnson) band diagram (dispersion relation) (cartoon) ncy conserv ved freque photonic band gap synthetic medium for propagation backwards slope: negative refraction d/dk 0: slow light (e.g. DFB lasers) strong curvature: super prisms, (+ negativerefraction) conserved wavevector k A Kirk 11/24/2008 Photonic bandgap and metamaterials 27

28 Photonic crystal superprism Photonic oo ccysaope crystal operated ed at wavelength above bandgap Periodic structure significantly modifies dispersion Can be used to form optical multiplexer/demultiplexer p Potential for very compact devices A Kirk 11/24/2008 Photonic bandgap and metamaterials Kosaka

29 S vector and k vector effects 1 D photonic crystal n x =k x /k 0 1 x n 1 n 2 k r S r ( 1 ) k i S r k i k r ( 1 ) k r ( 2 ) Effective index diagram S r ( 2 ) k 2 z z n =k /k z z 0 S vector: Group velocity dispersion Beam is directed along normal to dispersion surface k vector: Phase velocity dispersion Wavefront is refracted according to effective index Which effect is most useful? Which lattice is best A Kirk 11/24/ D, 2 D, hexagonal, square? Photonic bandgap and metamaterials 29

30 S vector and k vector superprisms S vector device Spatial separation within photonic crystal k vector device Angular dispersion Can make use of non parallel edges Design objectives: High spectral resolution Small device area A Kirk 11/24/2008 Photonic bandgap and metamaterials 30

31 Material system All designs were carried out for silicon on insulator (SOI) Design wavelength 1550 nm Plane wave expansion method analysis (3 D) Air Si n=3.45 n m SiO 2 Si n= m A Kirk 11/24/2008 Photonic bandgap and metamaterials 31

32 k vector scales to DWDM Results k-vector designs (DWDM 32 channels x 100 GHz) 1-D 2-D square 2-D hexagonal Period (nm) Hole size (nm) Prism area (mm 2 ) Lattices equal Only n is important S-vector designs (CWDM 4 channels x 20 nm) 1-D 2-D square 2-D hexagonal Period (nm) Hole size (nm) Prism area (mm 2 ) D significantly more compact due to lower band curvature

33 Layout for a 3X32 DWDM k vector multiplexer A Kirk 11/24/2008 Photonic bandgap and metamaterials 33

34 Properties of Bulk Crystals (from Johnson) band diagram (dispersion relation) (cartoon) ncy conserv ved freque photonic band gap synthetic medium for propagation backwards slope: negative refraction d/dk 0: slow light (e.g. DFB lasers) strong curvature: super prisms, (+ negative refraction) conserved wavevector k A Kirk 11/24/2008 Photonic bandgap and metamaterials 34

35 Negative refractive index materials OrtwinHess, Optics: Farewell to Flatland, Nature 455, (18 September 2008) A Kirk 11/24/2008 Photonic bandgap and metamaterials 35

36 Incident ray i Strange properties p of negative index materials Positive index material n i n Air sin n sin i i t t Negative index material Incident ray i ni Air t n t Transmitted ray t n t Transmitted ray Flat lens (originally proposed by Pendry) A Kirk 11/24/2008 Photonic bandgap and metamaterials 36

37 Negative index region Periodic arrays of air holes in dielectrics can also have n= 1 Imaging properties of dielectric photonic crystal slabs for large object distances Guilin Sun, Aju S. Jugessur, and Andrew G. Kirk A Kirk 11/24/2008 Photonic bandgap and metamaterials 37 Optics Express, Vol. 14, Issue 15, pp

38 Electromagnetic metamaterials So far we have discussed periodic dielectric structures We have modulated but not What happens if we modulate both? Result ltis a metamaterial: t Can have properties not found in nature Typically the modulation is on a period smaller than the wavelength of light, so effect is not due to interference A Kirk 11/24/2008 Photonic bandgap and metamaterials 38

2006 Science paper he works out what electromagnetic properties this

39 Example: Optical cloaking If we could bend light around an object, we could make it invisible This is called Optical cloaking In Pendry s 2006 Science paper he works out what electromagnetic properties this cloak should have A Kirk 11/24/2008 Photonic bandgap and metamaterials 39

r r Permitt tivity, perme ability 5 4 3 2 r z, z 1 r, r z 2 1 Natural materials do not usually

40 Material requirements Pendry showed that the material needs to anisotropic The permittivitity and permeability (for a cylinder) must be given by: r r r r r r R r 1 2 r R r R r z z R 1 R2 R1 r r Permitt tivity, perme ability r z, z 1 r, r z 2 1 Natural materials do not usually have = Radius, r A Kirk 11/24/2008 Photonic bandgap and metamaterials 40

41 Engineering a microwave cloak Only 5 months after Pendry s paper, pp the first experimental demonstration of electromagnetic cloaking was published The research was led by D.R.Smith at Duke University A Kirk 11/24/2008 Photonic bandgap and metamaterials 41

5 mm wavelength) In order to simplify the

42 How it was done Cloak is made of many split ring resonators Curved conductors with a precisely calculated inductance This is a metamaterial for microwaves (3.5 mm wavelength) In order to simplify the experiment, the structure guided EM waves in the correct direction but did not provide the full impedance matching Therefore it still reflected some radiation A Kirk 11/24/2008 Photonic bandgap and metamaterials 42

43 Measurement system Probe Cloak Microwave source (8.5 GHz, wavelength is3 3.5 mm) A Kirk 11/24/2008 Photonic bandgap and metamaterials 43

44 Results: Electric field patterns Simulation: ideal materials Simulation: actual materials Experiment: uncloaked copper cylinder Experiment: cloaked copper cylinder A Kirk 11/24/2008 Photonic bandgap and metamaterials 44

45 Electromagnetic cloaking for light In April 2007, in a letter in Nature, Vladimir Shalaev at Purdue showed (by simulation) that optical cloaking should possible by using metallic wires in a dielectric i medium A Kirk 11/24/2008 Photonic bandgap and metamaterials 45

46 Simulated results (at 632 nm) Cloaked Uncloaked As before, this does not provide impedance matching, so reflections still occur A Kirk 11/24/2008 Photonic bandgap and metamaterials 46

47 Demonstration of negative index optical materials il An experimental demonstration is probably not far off The Purdue group have already demonstrated negative refractive index optical materials Silver gratings separated by 38 nm of alumina A Kirk 11/24/2008 Photonic bandgap and metamaterials 47

48 Results A Kirk 11/24/2008 Photonic bandgap and metamaterials 48

$3 D negative refractive index materials Researchers at Berkeley have recently$ 2008): Silver and magnesium fluoride fishnet structure Layers are 80 nm thick

2008): Silver and magnesium fluoride fishnet structure Layers are 80 nm thick

49 3 D negative refractive index materials Researchers at Berkeley have recently demonstrated a 3 D negative index optical meatmaterial (Nature 455 September 2008): Silver and magnesium fluoride fishnet structure Layers are 80 nm thick and period is 860 nm A Kirk 11/24/2008 Photonic bandgap and metamaterials 49

50 Results J Valentine et al. Nature 000, 1 4 (2008) doi: /nature07247 A Kirk 11/24/2008 Photonic bandgap and metamaterials 50

51 Fabrication Fabrication of photonic crystal structures is typically achieved via nanolithography Electron beam lithography is often employed Deep UV optical lithography is also suitable (and more efficient i for mass production) Focused ion beam etching is also used Typical materials are silicon on insulator (SOI), silicon and III V semiconductors A Kirk 11/24/2008 Photonic bandgap and metamaterials 51

52 Summary Photonic crystals are materials with periodically modulated permittivity (on a scale of the wavelength) This modifies the reflective, dispersive and refractive properties Metamaterials typically have modulated permeability, in addition i to permittivity, i i and do not operate via interference A Kirk 11/24/2008 Photonic bandgap and metamaterials 52

Photonic crystals. Semi-conductor crystals for light. The smallest dielectric lossless structures to control whereto and how fast light flows

Photonic crystals. Semi-conductor crystals for light. The smallest dielectric lossless structures to control whereto and how fast light flows Photonic crystals Semi-conductor crystals for light The smallest dielectric lossless structures to control whereto and how fast light flows Femius Koenderink Center for Nanophotonics AMOLF, Amsterdam f.koenderink@amolf.nl