Chapter 5. Photonic Crystals, Plasmonics, and Metamaterials

Chapter 5. Photonic Crystals, Plasmonics, and Metamaterials Reading: Saleh and Teich Chapter 7 Novotny and Hecht Chapter 11 and 12

1. Photonic Crystals Periodic photonic structures 1D 2D 3D Period a ~ λ https://alexandramjurgens.wordpress.com/tag/ens-cachan/ http://optoelectronics.eecs.berkeley.edu/photonic_crystals.html

Natural Photonic Crystals http://optoelectronics.eecs.berkeley.edu/photonic_crystals.html

Photonic Crystal Optical Fibers Photonic Bandgap Fibers for Precision Surgery and Cancer Therapy http://optoelectronics.eecs.berkeley.edu/photonic_crystals.html

Photonic Integrated Circuits Photonic Crystal Waveguide Photonic Crystal Enhanced LED http://spie.org/x104683.xml http://optoelectronics.eecs.berkeley.edu/photonic_crystals.html

Wave Equation in a Photonic Crystal A photonic crystal of dielectric media has a periodic lattice structure whose constituent media have distinctive dielectric constants: ε r r = ε r r + u 3D for all Bravais lattice vectors, u = n 1 a 1 + n 2 a 2 + n 3 a 3 The electromagnetic modes in the photonic crystal take the form, H r, t = H r e iωt, E r, t = i ωε 0 ε r r H r, t where the spatial mode function H r is determined by the wave equation Eigenmodes H k r 1 ε r r ω2 H r = H r c2 Eigenvalues ω k

Incident light e ikz r T e ikz Bragg Reflection r d The total reflection coefficient r from a semi infinite structure: r T = r + re 2ikd + re 4ikd + re 6ikd r + = 1 e 2ikd Diverges if e 2ikd = 1 k = π d Constructive interference Bragg condition Light cannot propagate in a crystal, when the frequency of the incident light satisfies the Bragg condition. Origin of the photonic bandgap

Any 1d Periodic System has a Gap Treat it as artificially periodic w sin a x cos a x e 1 a e(x) = e(x+a) 0 π/a x = 0

Any 1d Periodic System has a Gap w Add a small real periodicity ε 2 = ε 1 + Δε sin a x cos a x a e(x) = e(x+a) e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 0 π/a x = 0

Any 1d Periodic System has a Gap w Add a small real periodicity ε 2 = ε 1 + Δε band gap sin a x cos a x Splitting of degeneracy: state concentrated in higher index (ε 2 ) has lower frequency a e(x) = e(x+a) e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 e 1 e 2 0 π/a x = 0

1D Photonic Crystal n 1 n 2 n z = n z + Λ d 1 d 2 Λ H z = H k z e ikz Dispersion relation: cos 2π k g = Re 1 t ω g = 2π Λ cos 2π k g = 1 t 12 t 21 cos π ω ω B r 12 2 cos πζ ω ω B t 12 t 21 = 4n 1n 2 n 1 + n 2 r 2 12 = n 2 n 1 2 n 1 + n 2 2 2 ζ = n 1d 1 n 2 d 2 n 1 d 1 + n 2 d 2 Bragg frequency ω B = cπ nλ n = n 1d 1 + n 2 d 2 Λ

Band structure: ω cos 2π k g = 1 t 12 t 21 cos π ω ω B r 12 2 cos πζ ω ω B 2ω B photonic bangap ω B photonic bangap ω k g 2 k g 2

Photonic Nanocavities Photonic cavities strongly confine light. Applications Coherent electron photon interactions Ultra-small optical filters Low-threshold lasers Photonic chips Nonlinear optics and quantum information processing Y. Akahane et. al. Nature 425, 944 (2003)

PC Waveguide: High transmission through sharp bends Highly efficient transmission of light around sharp corners in photonic bandgap waveguides A. Mekis et al, PRL, 77, 3786 (1996)

Tunneling through localized resonant state Complete transfer can occur between the continuums by creating resonant states of different symmetry, and by forcing an accidental degeneracy between them. S. Fan et. al., PRL 80, 960 (1998).

2. Surface Plasmons K. Yao and Y. Liu, Nanotech. Rev. 3, 177 (2014)

Electromagnetic Waves in Conductors Recall the inhomogeneous wave equation: 2 E z 2 1 2 E c 2 t 2 = μ 0 2 P t 2 Polarization P when there are free electrons: The equation of motion based on the forced oscillator model is d 2 x t dt 2 From this, we found the polarization: = γ dx t dt resistive force by scattering + ee 0 m e e iωt force due to the incident light field P t = Nex t = Ne2 /m e ω 2 + iωγ E(t)

Plasma Frequency Polarization in conductor: P t = Ne2 /m e ω 2 + iωγ E(t) Define a new constant, the plasma frequency ω p : ω p 2 = Ne2 ε 0 m e Thus P t = ε 0 ω p 2 ω 2 + iωγ E(t) We now plug this in for the polarization term in the wave equation.

Back to the wave equation 2 E z 2 1 c 2 2 E t 2 = μ 0 2 P t 2 = μ 0ε 0 2 E z 2 1 c 2 1 ω p ω 2 + iωγ 2 E z 2 n2 c 2 2 E t 2 = 0 ω2 p ω 2 + iωγ 2 E t 2 So this must be the (complex) refractive index for a metal. 2 2 E t 2 = 0 This is the wave equation for a wave propagating in a uniform medium, if we define the refractive index of the medium as: n 2 ω = ε r ω = 1 ω p 2 ω 2 + iωγ

Optical properties in the low-frequency limit, ω γ Dielectric function: ε ω = ε 0 1 ω p 2 ω 2 + iωγ ε 0 1 + i ω p ωγ 2 In the Drude model, J = σ 0 E σ 0 = Ne2 τ where m e = Ne2 m e γ ε ω = ε 0 1 + i σ 0 ε 0 ω In the high-frequency limit, ω γ ε ω ε 0 1 ω p 2 From Drude theory, that τ~10 14 sec, so γ = 1/τ ~10 14 Hz. (corresponding to the frequency of infrared light) For a typical metal, ω p is 100 or even 1000 times larger. (corresponding to the frequency of ultraviolet light) ω 2

Dielectric function of metals: ε ω = ε 0 1 ω p 2 ω 2 + iωγ A plot of Re ε and Im ε for illustrative values: ε ω /ε 0 5 4 3 2 1 0-1 -2-3 -4-5 Im ε ω p linear scale Re ε ω p = 4000 cm -1 γ = 40 cm -1 0 2000 4000 6000 8000 10000 Frequency (cm -1 ) ε ω /ε 0 10 5 10 3 10 1 10-1 10-3 10-5 log scale Re ε Im ε ω p 10 0 10 1 10 2 10 3 10 4 Frequency (cm -1 ) Re ε 10 5 Imaginary part gets very small for high frequencies Real part has a zero crossing at the plasma frequency

Drude theory at optical frequencies ε ω ε 0 ω p 2 = 1 ω 2 + iωγ 1 ω p 2 ω 2 + i γ ω ω p > ω γ ω p 2 ω 2 Re{ε} large and negative (below ω p ) ε 0 small and positive Im{ε} ε 0 IR visible

High frequency optical properties In the regime where ω > γ, n ω = 2 ε ω = 1 ω p ε 0 ω 2 For frequencies below the plasma frequency, n is complex, so the wave is attenuated and does not propagate very far into the metal. For high frequencies above the plasma frequency, n is real. The metal becomes transparent! It behaves like a nonabsorbing dielectric medium. Reflectivity drops abruptly at the plasma frequency. This is why x-rays can pass through metal objects.

Radio Waves in Ionosphere The uppermost part of the atmosphere, where many of the atoms are ionized. There are a lot of free electrons floating around here. For N~10 12 m 3, the plasma frequency is: ω p = Ne2 ε 0 m e = 2π 9 MHz Radiation above 9 MHz is transmitted, while radiation at lower frequencies is reflected back to earth. That s why AM radio broadcasts can be heard very far away.

Dielectric Function of Real Metals For real metals, there is a very broad range of frequencies for which Im ε ~0 and Re ε < 0. e(w)/e 0 20 16 12 8 4 0-4 -8-12 -16-20 Im ε Re ε 0 4000 8000 12000 16000 20000 Frequency (cm -1 ) ω p linear scale This has interesting implications!!! ω p = 15000 cm 1 γ = 40 cm 1

Waves trapped at an interface x z ε 1 ε 2 Consider a wave at the interface between two semi-infinite nonmagnetic media (μ 1 = μ 2 = μ 0 ). z = 0 Is there a solution to Maxwell s equations describing a wave that propagates along the surface? We can guess a solution of the form for media 1 and 2: E m = B m = E 1x, 0, E 1z e κ m z ei kx ωt 0, B 1y, 0 e κ m z i kx ωt e This propagates along the interface, and decays exponentially into both media. m = 1,2 (Note: this is not a transverse wave, but that s OK.)

Interface Waves In order to exist, the wave must satisfy Maxwell s equations: ω iκ 1 B 1y = ε 1 c 2 E 1x B = με E t ω iκ 2 B 2y = ε 2 c 2 E 2x and also the continuity boundary conditions at z = 0: B 1y z = 0 = B 2y z = 0 E 1x z = 0 = E 2x z = 0 It is easy to show that these conditions can only be satisfied if: ε 1 + ε 2 = 0 κ 1 κ 2 Since κ 1 and κ 2 are always positive, this shows that interface waves only exist if ε 1 and ε 2 have opposite signs. In a metal, ε < 0 for frequencies less than ω p.

Surface Plasmon Polaritons Surface Plasmon Polariton (SPP) - a surface wave moving along the interface between a metal and a dielectric (e.g., air) The electrons in the metal oscillate in conjunction with the surface wave, at the same frequency. In fact, an SPP is both an electromagnetic wave and a collective oscillation of the electrons.

SPP Electric field: κ 1 ε 1 + κ 2 ε 2 = 0 Dispersion relation: SPP Dispersion Relation E = E 0 e κ z i kx ωt e k 2 + κ 2 ω 2 and i = ε i i = 1,2 c k = ω c ε 1 ε 2 ε 1 + ε 2 1/2 For ε 1 ω = 1 ω p 2 ω 2 ω sp = ω p 1 + ε 2

Surface plasmon sensors Surface plasmons are very sensitive to molecules on the metal surface.

Surface plasmons on small objects Instead of considering a semi-infinite piece of metal, what if the metal object is small? e.g., a metal nanosphere We can still excite a plasmon, but in this case it does not propagate! The electrons just collectively slosh back and forth. excess negative charge excess positive charge There is a restoring force on the electron cloud! Once again, we encounter something like a mass on a spring, with a resonance

Surface plasmon resonance The sloshing electrons interact with light most strongly at the resonant frequency of their oscillation. 2.8 nm copper nanoparticles Pedersen et al., J Phys Chem C (2007) gold nanoparticles give rise to the red colors in stained glass windows

Controlling the surface plasmon resonance The frequency of the plasmon resonance can be tuned by changing the geometry of the metal nano-object. Halas group, Rice U. gold nano-shells

3. Metamaterials Negative refractive index K. Yao and Y. Liu, Nanotech. Rev. 3, 177 (2014) Metadevices L. Billings, Nature 500, 138 (2013) N. I. Zheludev and Y. S. Kivshar, Nature Mat. 11, 917 (2012)

Is an invisibility cloak magic or reality?

Invisibility Skin Cloak for Visible Light AFM Cloak on Cloak off Ni et. al. Science 349, 1310 (2015).

Electromagnetic Metamaterial Negative refraction Invisibility Physics Today Jun 2004 Physics Today Feb 2007 λ a, b a b Meta-atom

Material Parameter Space by ε and μ n 2 = ε r μ r n = ± ε r μ r S k S k k E = μωh k H = εωe ε = ε 0 ε r, μ = μ 0 μ r S = E H Y. Liu and X. Zhang, Chem. Soc. Rev. 40, 2494 (2011)

Basic metamaterial structures to implement artificial electric and magnetic Responses Periodic Wires Split Ring Resonators (SRR) Y. Liu and X. Zhang, Chem. Soc. Rev. 40, 2494 (2011)

First Negative Index Material Smith et. al., PRL14, 234 (2000)

Negative Refraction and Perfect Focusing sin θ = n sin θ r n = 1 n > 1 n = 1 n < 1 θ RHM θ r > 0 θ LHM θ r < 0 n = 1 n = 1 n = 1 Shelby et. al. Science 292, 77 (2001) point source Internal focus evanescent waves image Fang et. al. Science 308, 534 (2005) FIB 40nm AFM with superlens AFM w/o superlens

Invisibility Cloak Y. Liu and X. Zhang, Chem. Soc. Rev. 40, 2494 (2011)