Effective Medium Modelling of Plasmonic MetaMaterials

transcript

Gennady Shvets, The University of Texas at Austin

Metamorphose School’09: “Current research and recent developments on opticalMetamorphose School 09: Current research and recent developments on optical metamaterials”, Crete, Greece, June 13, 2009

Outline of the talk

• Introduction to optical metamaterials

• Attempts to answer puzzling questions that I would have had if I was entering the field:1. Why is the optical magnetism difficult to achieve (a.k.a why are

the microwave experts looking down on us?)

2 A l i ff t d b d f ki ti l2. Are plasmonic effects good or bad for making optical metamaterials?

3. How does one analyze periodic plasmonic structures?3. How does one analyze periodic plasmonic structures?

4. Is it possible to to homogenize plasmonic nanostructures?

5 Why is everybody talking about spatial dispersion and how5. Why is everybody talking about spatial dispersion and how important is it?

• Perspectives on the optical metamaterials field: where the• Perspectives on the optical metamaterials field: where the promising applications are?

Who suggested particle clusters for ti ?magnetic response?

Shvets & Urzhumov, JOPA 7, S23 (2005);

PRL 93, 243902 (2004)

What is a MetaMaterial?

O i i t f G k d " ft /b d"Originates from a Greek word : "after/beyond"

Example: metaphysics ("beyond nature"): a branch of philosophy concerned with giving a general and fundamental account of the way the world is. (Wilkepidia)

•Metamaterials are artificially engineered materials i ti ( h i l ti l l t i l)possessing properties (e.g., mechanical, optical, electrical)

that are not encountered in naturally occurring materials.

•The emphasis of this talk in on unusual electromagnetic properties such as dielectric permittivity , magnetic permeability and refractive index npermeability , and refractive index n.

What is a MetaMaterial?

Material properties are determined by theMaterial properties are determined by the properties of the sub-units plus their spatial distribution. p

For a << effective medium theory.

F h i ffFor a ~ photonic effects.

What about meso-scale materials: bigger than atom butbigger than atom but smaller than the wavelength??g

Microscopes: the Engines of Discovery

V L h k (1676)Van Leeuwenhooke (1676): discovered bacteria, blood cells

)sin(222.1

What if the imaged specimen is aspecimen is a sub-wavelengthgrating?

Difficult to resolve sub- features

L >> d/2

Small features (or large wavenumbers) of the object are lost because of the exponential j pevanescence of short-wavelength waves

Evanescence of large-k waves

•Fields in the object plane:

yikekAdkyx )( ),0(Hz

Spectral width k feature size y = k-1

•Optical system propagates A(k) through glass, vacuum

xkkiekA22

For smooth features with k < k0 = 0/c

xkkekA20

For sharp features with k > k0 = 0/c

•How to overcome decay? Go near-field!

(a) get very close (L < d/2) (b) lifdecay? Go near field! (b) amplify evanescent waves

Negative Index Materials to the Rescue: = -1 = -1 n = -1 = -1, = -1 n = -1

L/2 L L/2

•Super-lens is a near-field concept amplifies evanescent waves to restore image

•Near-field superlens possible with just < 0 (Pendry, 2000)

Negative Index Materials for -waves

2/12 2 c 2F2 log(D/r)

pp 0 1 22

Smith et.al., PRL'2000

•Basic Elements of a NIM: (a) Split ring resonator: just a well designed inductor ( ) p g j gresonating at << c/D gives < 0(b) Metal wires (continuous or cut): r << D to ensure that < 0 for << c/D

Sub- NIM Using Microwave Tricks•In microwave range: use “perfectly” conducting components to simulate < 0 and < 0 Smith et al (2000)components to simulate < 0 and < 0, Smith et.al., (2000)

Metal poles: = 1 – 2/2 < 0Metal poles: 1 p / < 0

Split-ring resonators, Pendry’99: “geometric” resonance at “geometric” resonance at M

0 1 22

Mi t i k bl i i t i ti ( << /L)

•Microwave tricks enable miniaturization (p, M << c/L): (a) very thin wires still good conductors(b) split ring design high capacitance and inductance(b) split ring design high capacitance and inductance

Why is it difficult to make a ymagnetically-active metamaterial

?in the optical domain?

Making a sub- NIM: from -waves to plasmonics by downsizing!to plasmonics by downsizing!

Microwave SRR, =1.57cm Plasmonic SRR, =3.44m

W=0.8mm, Gap = 80m (filled with d =4 dielectric)

Downsize by about x5,000 W=160nm, Gap = 16nm (filled with d 4 dielectric)

Magnetic cutoff = 0 at = 20*Period how??

pMagnetic cutoff = 0 at = 20*Period (i) how? Plasmonics effects! (ii) Can we

Narrow gap + high- dielectric high capacitance

Plasmonics effects! (ii) Can we move to shorter wavelength by further downsizing? No!

Disappearance of magnetism for optical frequencies ( < 3m)optical frequencies ( < 3m)

Energy content of microwave structure:Electric energy

,22 BdVUEdVU

BEBEtot

8dVUdVU BE

Magnetic energy In EE language:

CQU BE

Compare with the plasmonic structure:

Disappearance of magnetism for optical frequencies ( < 3m)optical frequencies ( < 3m)

Energy content of Electron kinetic energy2

1 pEnergy content of

plasmonic structure:

UUUU kinBEtot

Electron kinetic energy

22 BdVUEdVU BE

kinBEtot

Magnetic energylect

2 EdVU pkin

Energy balance at = 3.44m:

U 0 5U U 0 32U

Plasmonic parameter at= 3.44m (silver): 7.1 kin

UE = 0.5Utot, U kin = 0.32U tot, UB = 0.18 U tot

Electrostatic resonance at =-300 (or = 3 m for Ag): Tp = ∞

Y.A.Urzhumov and G.Shvets, Solid State Comm. 146, 208 (2008).

What happens when you downsize an ideal (lossless) plasmonic structure?ideal (lossless) plasmonic structure?

•Regime I: Very large unit cell (appropriate for -wave frequencies) metals behave as PEC's, miniaturization (if any) accomplished by

increasing capacitance

•Regime II: Intermediate unit cell shrinking th ll th ti l th ( f /the cell causes the operating wavelength (e.g., for magnetic resonance) to shrink proportionally: = C*Period

•Regime III: Small unit cell approaches the electrostatic resonance wavelength res unit

cell shrinks faster than the wavelength: > C*Period

•Regime IV: No magnetic activity for < res. Sorry…

What if > 3 m is not enough: simplify the structure!simplify the structure!

res = -330 res =3 m res = -82 res =1.5 m (or res=2.25 m for glass spacer)2.25 m for glass spacer)

silver strips

Strip Pair-One Film (SPOF) Lomakin, Fainman, U h G S O t Strip Pair One Film (SPOF)

res = -8.8 res = 0.5 m (or res = 0.7 m with glass spacer)

Urzhumov, G.S., Opt. Exp. 14, 11164 (2006); Urzhumov & G.S., Solid State Comm 146 si

lvSolid State Comm. 146,208 (2008).

Effect of plasmonic losses on developing sub wavelength NIMssub-wavelength NIMs

•Why not operate very close to the electrostic plasmonic resonance (extreme minimization)? Losses destroy the NIM band!

•Total energy increases as the plasmonic number Tpi th l t t ti

Bptot UTU 12increases near the electrostatic resonance:

•Group velocity decreases as Utot increases:mg

p y tottotUc

•NIM propagation band is destroyed if the 1 P•NIM propagation band is destroyed if the

transit time across a single cell is longer than the decay time -1 :

Three-dimensional example: is my fishnet better than yours?my fishnet better than yours?

Electrostatic resonances teach us about relative advantages/disadvantages of various structures!

Novel Plasmonic Double-Fishnet Structures

•Traditional double fishnet has never become a sub-negative index metamaterial

•Still room for innovation: look for

lt ti !

Planar fishnets are easy to •Too “simple” to be sub-

alternatives!

•Planar fishnets are easy to fabricate & align (e.g. nanoimprint lithography)All d

Too simple to be sub without plasmonics: P > /2n Kafesaki et.al., PRB’07: P = 1 cm = 2cm n = 2•All to-date structures are not

sub-wavelength: P > /2n Brueck et.al., PRL’05:

P = 1 cm, = 2cm, n = 2•Novel plasmonic design sub-wavelength!P f t i d,

P = 0.8 m, = 2 m, n=1.5 •Perfect impedance matching: zero reflection!

Modified vs regular fishnet Minimum wavelength for magnetic activity from ES simulations

Modified double-fishnet: "Regular" double-fishnet:min = 1.2 m (m/d = -62) at = 1.5 m can still be sub-(close to plasmonic resonance!)

min = 0.73 m (m/d = -8.3)

To be be sub-, must operate at < 900nm, or Period < 300nm

Impedance-matched NIM at = 1550nmcontinuous silvercontinuous silver wires control eff

impedance matching

20 nm50 nm “cut”

silver wires

320 nm80 nm

control eff

320 nm250 nm

•This structure is sub-l h P i d /5wavelength: Period = /5

•It operates in the desirable = = -1 regime re = re = -1

•Applications: detection (SEIRA, et.)

Effect of plasmonic losses

How to chartacterize periodic plasmonic structures?plasmonic structures?

•Standard approach: compute frequency vs wavenumber k

•This is not so easy to accomplish: premittivity of metals itselfThis is not so easy to accomplish: premittivity of metals itself depends on the frequency

•There is a way to proceed: compute k vs !•There is a way to proceed: compute k vs !

•This is even better: we can find complex k’s which give us the t f ti l d i id th b d f i di t t !rate of spatial decay inside the band gap for periodic structures!

Calculating k vs omegag g

Solve using the weak formulation of the PDE:

Calculating k vs omegag g

Assume Bloch-periodic solutions:

Substitute into:

Obtain:

Now discretize and solve for k as an eigenvalueg

Array of nanorods

SPOF structure: propagation bands

Is it possible to homogenize an optical metamaterial?optical metamaterial?

•Homogenization involves introducing a set of effective parameters (epsilon, mu, bi-anisotropy coefficients) by postulating some type of field averaging procedure and/or fit to Fresnel coefficients (transmission/reflection)

•Preferably, these parameters should have a clear and intuitive physical meaning (e.g., conform with our understanding of

it i d t t )capacitance, inductance, etc.)

•The situation can be further complicated by strong non-local dependence of D and B on E and H (spatial dispersion)

Example: effective permittivityp p y

extE QQAQAQC

)()( Eeff

)()(effeff

ffperiodic

-1V +1V

(a) Impose constant voltage drop sidewalls of a single unit cell: nothing more to do for dielectric non dispersive MMs

periodic

nothing more to do for dielectric non-dispersive MMs

(b) For plasmonic MMs: Scan frequency and recover Q() t ( )compute eff()

(c) Peaks of E-field (or Q) correspond to ES resonances

Example: effective permittivity tensorp p y

A l t l fi ld

Apply external field:

Solve Poisson’s equation:

Set up boundary conditions:p y

Extracting the permittivity tensorg p y

A l t l fi ld

Apply external field:

What if you have a dispersive material with () ?with () ?

1 S th f d

1. Scan the frequency and repeat the capacitance measurementy measurement

2. Find a better way: Generalized EigenvalueaxGeneralized Eigenvalue Differential Equation

Generalized Eigenvalue Equation

periodic

0 xE00

Define

periodic

B & S d’92 S k l ’01Bergman & Stroud’92; Stockman et.al.’01; Shvets & Urzhumov, PRL’04 & JOPA’05

Generalized Eigenvalue Equationperiodic

Define an eigenvalue equation:

periodic

)()()( 2

And use the eigenfunctions to solve:

)()()( 012

Examples of dipole-active electrostatic resonancesresonances

1)( s() = [1 - )]-1

iqs ssd 1

2 )(1)(

si resonance “frequency”

Fi oscillator strength

s = 2/ 2 = 0 14 ( = - 6) s = 2/ 2 = 0 40 ( = 1 5)

Two strongest ES hybridized dipole

s1 = /p = 0.14 ( = - 6) s2 = 2/p2 = 0.40 ( = -1.5)

y presonances.

Note: red-shifting of strongestof strongest resonance due to particle-particle interactioninteraction

Solutions of the driven equationq

*),(, yxdxdywhere

Now use a simple definition of the permittivity tensor:

k )(where jk

kj EE 0)(

Effective Permittivity Tensor!y

Note: f0 does not vanish only if the plasmonic inclusion forms a continuous phase!continuous phase!

Reminder:dm

Consequence: Drude conductivity for small frequencies!

Summary: effective permittivity of plasmonic MMsplasmonic MMs

•Assume a fully symmetric (C4v) structure: ssu e u y sy e c (C4v) s uc u e:scalar dielectric permittivity:

)()(1)(

i iq sss 1 )()(

•This term is only present when the metal crosses the cell’s boundary continuous

d i i f l f i

conductivity for low frequencies

Example: strip pair-single film (SPOF)extE +Q-Q

(a) Electrostatic simulation: scan frequency (m) obtain qs()magnetic

(b) Compare quasi-static (intuitive!)

qs with the electromagnetically

gresonance, band edge

extracted eff

(c) Sub- (/7) NIM structure qs ≈eff

Does this result in a true homogenization?

Multi-scale expansion theory for equations with oscillating periodic coefficients

X: large scale ll l

Expand potential: )(),(),()( 210 OyXyXx

X: large scale y: small scale

Derive effective permittivity for

yXX ),()( 0

j y )()(

ijijeff y

)()()()(

1Generalization of Bergman-Stockman-Stroud permittivity:

0))(()( )(1 j

Small-scale potential functions:

Physical meaning: Match dipole moment with that of homogeneous medium

complicated field profile in metamaterial

uniformly polarizedhomogeneous medium

Generalized quasi-static permittivity tensor for nanostructures with a continuous plasmonic phase:

ijijeff ss

0Ed eff

Generalized sum rule:Permittivity determined by strength of dipole-active resonances:

1)(2 p

Note: ES permittivity of every structure with a continuous metal phase (e.g. fishnet) has a ...1)( 2

pcontinuous metal phase (e.g. fishnet) has a

metal-like permittivity pole at =0

Unexpected connection: effective pepsilon and Extraordinary Optical Transmission

Experiment vs Theory: Transmission, Absorption, Reflection through perforated filmsReflection through perforated films

•Reflection drops, 1.0

n Reflection: p ,transmission and absorption spike in the vicinity of SPP(0,1)

rptio FTIR

Theory

vicinity of SPP(0,1) resonance

•Resonant effects 0.4

Transmission: FTIR Theory

Fano spike

observed for both S and P-polarizations, for all sample orientations0.2

ran Theory

•Resonances sensitive to polarization and sample

i i10 10.5 11 11.5 12 12.5

Absorption

orientation10 10.5 11 11.5 12 12.5

Wavelength (microns)

Radius: 1 m P i d 7 Experiments and simulationsPeriod: 7 m Wavelength: 11.8m Incidence angle: 21deg

Experiments and simulations agree: but what is the physics??

Quasi-static resonances of a periodically perforated filmperforated film

iFF01)(s() = [1 - )]-1

si resonance “frequency”

qs sss 1 )()(1)(

si resonance frequency

Fi oscillator strength

Bergman & Stroud’92; Stockman et.al.’01; P t l f ti fil Shvets & Urzhumov, PRL’04 & JOPA’05•Present only for continuous films

•Responsible for enhanced transmission

LSP 0 55 SPP(1 0) 7LSP at = -0.55 period independent

SPP(1,0) at = -7 depends on period Note:

•This technique is reminiscent of the Fano resonances approach

•No parameter fitting to EM simulations is required

How does the effective explain extra transmission and absorbance?

8Real(ε

Imag(εeff

) Absorption on

Korobkin et. al., Appl.Phys.A 88, 605 (2007)

Imag(εeff

)Real(ε

Imag(εSiC

Absorption maximum

Transmission maximum D = 2 m P = 7 m

Transmission minimum red: absorption

blue: transmission

•Enhanced transmission is due to correction to Real( ff )

10.4 10.6 10.8 11 11.2 11.4−10

λ, microns

EEnhanced transmission is due to correction to Real(eff )

•Enhanced absorption is due to enhanced Imag(eff) predicted & measured 40% absorbance!

•In metamaterials with continuous plasmonic phase resonances result in enhanced transmission!

How did the effective medium description do?

0.4Absorption (EM)Transmission (EM)

Quasi-Electrostatic (ES) R d hif

0.35 Transmission (EM)Absorption (ES)Transmission (ES)

Q ( )approach:

Compute electric dipole ( t th Fso

Red shift: ~(P/)2,

D = 2 m P = 7 m

0.2 resonances (strengths Fiand “frequencies” si)

C t ( )mis

−0.05

0.05 Compute eff()

Fresnel formulas transmission/absorptiontr

10 10.5 11 11.5 12 12.5

−0.1transmission/absorption

Wavelength (microns)

ieff ss

0 )(1)(1)(

( ) [1 )] 1

F0 = 0.88, s0 = 0 ( = -∞)F1 = 0.04, s1 = 0.12 ( = -7)

s() = [1 - )]-1

si resonance “frequency” Fi oscillator strength

1 1F2 = 0.005, s2 = 0.2 ( = -4)

Subtle effects: spatial dispersionp p

Spatial dispersion in “artificial plasma”

P d t l PRL ’96 •Pendry et.al., PRL ’96 thin-wire arrays have effective = 1 – 2/2effective 1 p /

•Early work: W. Rothman: “Plasma simulation byPlasma simulation by artificial dielectrics”, ’62

)8/log(16

22p rdd

c Plasma frequency is detemined

by wire spacing d and radius r00)g( 0 y p g 00

Is this plasma-like really correct? important for the i f f d b li i !existence of surface waves and sub- applications!

If it supports plasma waves, it must be plasma!

Wire width w = d/5z

Wire width w d/5

Cutoff pd/2c = 0.42

Wavenumber k d = x

Wavenumber kzd =

Freq.: d/2c = 0.45y

“Photon”: E k0

Bulk Plasmon: E ||k

d/2c = 0.56

−1 0 1

−0.5

−1 0 1

−0.5d/2c = 0.45

−1 0 1y

Shvets et.al., SPIE Proc. v. 5218, 156 (2003)

Bulk and Surface Modes in 3-D Wire Mesh

wave wave

d=0.58cm w=0.1cm

wwwww|| wave

ddddddsurface wave

Surface waves exist but their dispersion differs from that of plasmonic materials manifestation of spatial dispersion

Shapiro, Shvets, Sirigiri, Temkin., Opt. Lett. 31, 156 (2006)

What is the spatial dispersion: start with 2D

Propagation of TM waves in periodic 2D structure of thin rods: origin of the cutoff

Simulate/analyze Ez in single cell: (d2/dx2 + d2/dy2) Ez = (kz

2 - 2/c2) Ez

Boundary conditions: Ez = 0 at wires & E ( d/2 ) E ( d/2 ) ik dEz(x=d/2,y) = Ez(x=-d/2,y) eik

Ez(x,y=d/2) = Ez(x,y=-d/2) eiky

Electric/magnetic fields in 2D

•Non-vanishing fields: Ez B Er (only forEz, B, Er (only for finite kz)

C l E C•Color: Ez, Contours: magnetic field lines

•Plotted: cutoff point

•EM waves run along gthe wires for finite kz

)22/log(8

p Numerical tool: COMSOL )22/log( 0rddNumerical tool: COMSOL

Numerical simulations and effective

Parameters: d = 2, r0/d = 0.05, p = 1

• Thin-wire structure is isotropic

• Dispersion relation is

2 = p2 + kz

2c2 + 2(kx2 + ky

2)c2p y

where = 0.953

Fields can be described using cell-averaged quantities: <E> and <B> related by diagonal dielectric tensor ij: xx = yy =1

|| 09.01ck

ckck zz

Spatial dispersion of dielectric tensor

•Consider simple example of kz2 > 0 and kx = ky = 0

<Ex> = <Ey> = 0 but <Ez> = 1 what is Poynting flux <Pz > ?

BEecP zz

does not vanish after cell-averaging

BEcP jiij .

Landau & Lifshits, “Electrodynamics of C ti M di ”k84 Continuous Media”

vanishes does not vanish

|| 1),(ck

Strong dependence on kz

G. Shvets, AIP Conf. Proc., v. 647, 371 (2002); P. Belov et. al., Phys. Rev. B 67, 113103 (2003).

Even for 3-D meshes the spatial dispersion is still very strong!is still very strong!

)()()( mlkkck 20 )()(),(

ijlmijij k

(from Agranovich, Ginzburg, “Crystal Optics with Spatial Dispersion”)(from Agranovich, Ginzburg, Crystal Optics with Spatial Dispersion )

pFor cubic symmetry the tensor is characterized by three independents

Example of a wave with Hz, Ex, Ey field components (TM):

y pconstants only:

Example of a wave with Hz, Ex, Ey field components (TM):

222 20)( kkkk

)(00)(

),(xyyx

yxyxij kkkk

kkkkck

How do we extract these parameters?

For cubic symmetry the dispresion of waves near Gamma-point is:

(from Agranovich, Ginzburg, “Crystal Optics with Spatial Dispersion”)

Dependence of the spatial anisotropy ffi i t ( ’ ) th /d ticoefficients (’s) on the w/d ratio

•Only 2 decays as a power law

α2er a power law (2 = 1.3 w/d)

•Other s decay

00 0.02 0.04 0.06 0.08 0.1

•Other -s decay, at best, logarithmically)

g y)-0.4

α1-0.4

Note: in many cases, only 2 matters!

Spatial dispersion does not always show

Simplest geometry: wave normally incident on a vacuum/mesh interface

xzy ekkeHHeEE 000 ,,

vacuum/mesh interface

Dispersion relation:

dw3.12

from Smith, Schultz, Markos, Soukoulis, PRE’02

…but sometimes it does, with vengeance!

•Vacuum/mesh interface at z=0: BEcP

wwwww assume a surface wave with Hy, Ex, Ez components

dddddd

•Require the continuity of energy flux:

ccEEk jiij .

Outline of the talk

• Introduction to optical metamaterials

• Attempts to answer puzzling questions that I would have had if I was entering the field:1. Why is the optical magnetism difficult to achieve (a.k.a why are

the microwave experts looking down on us?)

2 A l i ff t d b d f ki ti l2. Are plasmonic effects good or bad for making optical metamaterials?

3. How does one analyze periodic plasmonic structures?3. How does one analyze periodic plasmonic structures?

4. Is it possible to to homogenize plasmonic nanostructures?

5 Why is everybody talking about spatial dispersion and how5. Why is everybody talking about spatial dispersion and how important is it?

• Perspectives on the optical metamaterials field: where the• Perspectives on the optical metamaterials field: where the promising applications are?

Effective Medium Modelling of Plasmonic MetaMaterials

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