Simulation and Understandingof Metamaterials

Th. Koschny, J. Zhou, C. M. Soukoulis

Ames Laboratory and Department of Physics, Iowa State University.

Th. Koschny, MURI NIMs Review May 2007, Purdue

Outline

1. Retrieval2. Breaking of Scaling3. Cut-wire pairs4. Diamagnetic response of SRR5. Anisotropic & Chiral metamaterials

Homogeneous Effective Medium Retrieval

z, n

d

€

e ik

€

teik

€

re− ik

PRB, 65, 195104 (2002),Opt. Exp. 11, 649 (2003).

Effective medium: Periodicity ArtifactsResonance/Anti-resonance “coupling”

“cut-off” deformationsnegative imaginary part

PRE, 68, 065602(R) (2003),PRL 95, 203901 (2005).

Curves are for our 200THz SRR,315nm x 330nm x 185nm unit cell

2 2 2( ) | | | | 2 | | ( ) ( )Q E H H n zω ε μ ω ω ω′′ ′′ ′′ ′= + =Energy loss is positive for causal branch Im(n) > 0 Re(z) > 0

ν

Periodic Effective medium description

PRB 71, 245105 (2005),PRE 71, 036617 (2005).

Dashed lines: Underlying physical resonancesSolid lines: Effective response due to periodicity

anti-resonance pseudo-resonance

“cut-off” at Brillouin zone edge

intermediateband gap

“cut-off” & shift

generic SRR

anti-pseudo-resonance

Outline

1. Retrieval2. Breaking of Scaling3. Cut-wire pairs4. Diamagnetic response of SRR5. Anisotropic & Chiral metamaterials

Breaking of Scaling

Metals are near-perfect conductors,the effective LC-resonator

depends on geometry only

Going to THz frequenciesIdea: geometric scaling

Scale: lenght S length time S time× ∧ ×Such that speed of light invariant and 0S →

0 rel

A

dC ε ε≈

2

0 00

8log 2

R RR

l rL

πμ μ⎛ ⎞

≈ ≈ −⎜ ⎟⎝ ⎠

densely stacked rings sparse rings

1 1mC S L S

SLCω∝ ∧ ∝ ⇒ = ∝

linearscaling

PRL 95, 223902 (2005),Opt. Lett. 31, 1259-1261 (2006).

Upper frequency limit of the SRRs?55 nm

Theory:Experiment:

2

2

1( )

21

2

e e e e

e

E n V m v

L I

=

=

21

2=m mE L I

ee

Iv

S e n=

2 2

1~e

ee

m VL

n e S a=

mL a∝ C a∝

Why saturation of ωm?

Key point: Kinetic energy of the electrons becomes comparable to magnetic energy in

small scale structures

1m

mL C=ω

1/m a∝ω (a: unit cell size)

V: wire effective volumeS: wire effective cross-sectionne: e- number density

Charge-carriers have non-zero mass !!

2

1 1

( ) .m

m eL L C a constω = ∝

+ +

Effective permeabilityCan be obtained by effective medium retrieval procedure from transmission & reflection

ordirectly via the magnetic moment of the SRR

1 11 , , 1

2 ( )metal

M r jM dV j i D

H Vμ ω

ε ω⎛ ⎞×

= + = =− −⎜ ⎟⎝ ⎠

∫

Limits of simple LC picture

“magnetic”modes

circularcurrent

(anti-symmetric)

“electric”modes

linearcurrent

(symmetric)

Magneticcoupling

or Electriccoupling

Electriccoupling

current density (arrows) & charge density (color)

~ / 2λ ~ 3 / 2λ ~ 5 / 2λ

2 ~ λ×2 ~ / 2λ× 2 ~ 2λ×

Outline

1. Retrieval2. Breaking of Scaling3. Cut-wire pairs4. Diamagnetic response of SRR5. Anisotropic & Chiral metamaterials

Electric mode

of coupled electric resonancesMagnetic mode

of coupled electric resonances

Electric resonance

Periodic Short-wire Pair arrays

Lagarkov & Sarychev, PRB 53, 6318 (1996);Panina et al., PRB 66, 155411 (2002);Shalaev et al., Opt. Lett. 30, 3356 (2005).

Opt. Lett. 31, 3620 (2006),Opt. Lett. 30, 3198 (2005).

With periodicity:

12 13 14 15 16

-8

-6

-4

-2

0

2

4

10 11 12 13 14 15

1/101/10

Frequency (GHz)

a b

14 15 16 17 18

-6

-4

-2

0 (b)

Frequency (GHz)

-2

0

2

4

6

(a)

APL 88, 221103 (2006)

14 15 16 17 18

-2

0

2

4

Real

Frequency (GHz)

Imaginary

ε < 0 and μ < 0

1.01 1.02 1.03 1.04

12.0

12.5

13.0

13.5

14.0

14.5

15.0

a

ay/l

fe

fm

b

Lm LmCe

Cm

Ce

Cm

(b)

(c) (d)

L

C1 1

2 2

Le LeCe

Cm

Ce

Cm

(a)

2

1

b

ay

ax

l

magnetic resonance electric resonance

Opt. Lett. 31, 3620 (2006)

The cross-over of themagnetic and electric resonance frequenciesis difficult to achieve!

2

1 1e m m

m e e

L C

L C

ωω

⎛ ⎞ ⎛ ⎞= + <⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

1e

e eL Cω =

( )1

m

m m eL C Cω =

+

Lm LmCe

Cm

Ce

Cm

(b)

(c) (d)

L

C1 1

2 2

Le LeCe

Cm

Ce

Cm

(a)

2

1

b

ay

ax

l

“Fishnet” structure

Zhang et al., PRL 95, 137404 (2005).

With periodicity:

Opt. Lett. 31, 1800 (2006).

Realization n<0 at 1.5Realization n<0 at 1.5μμm, Karlsruhe & ISUm, Karlsruhe & ISU

Since the first demonstration of an artificial LHM in 2000, there has been rapid development of metamaterials over a broad range of frequencies.

A Brief History of Left-handed Metamaterials

Iowa State University involved in designing, fabrication and testingof LHMs from GHz to optical frequencies [4,6,7,10,11,13,14].

Open symbol: µ<0Solid symbol: n<0

n<0 for 1.5 µm (ISU & Karlsruhe)Science 312, 892 (2006)

n<0 for 780 nm (ISU & Karlsruhe)Opt. Lett. 32, 53 (2007)

µ<0 for 6 THz(ISU & Crete)Opt. Lett. 30, 1348 (2005)

n<0 for 4 GHz(ISU & Bilkent )Opt. Lett. 29, 2623 (2004)

Science 315, 47 (2007)

Outline

1. Retrieval2. Breaking of Scaling3. Cut-wire pairs4. Diamagnetic response of SRR5. Anisotropic & Chiral metamaterials

Magnetic moment around resonance

2

2 2( ) 1

m

F

i

ωμ ωω ω γω

= +− +

according to

μ(ω) should return to unity below and above the resonance?

Two types of diamagnetic response

0B = 0B =

below resonanceB eliminated from area of ring metal

above resonanceB eliminated from all enclosed area

at resonance

Diamagnetic & Resonant currents

below resonance at resonance(note: scale is 10x larger)

L=10μmf=300GHz

L=10μmf=3.2THz

1( ) 1 ( )

( )metal

j i Dω ω ωε ω

⎛ ⎞= −⎜ ⎟

⎝ ⎠

r urwe describe metal by Drude model permittivitythen current density is available as:

Skin-depth

goodconductor

lossy negative“dielectric”Im

Re

Metals at THz frequenciesDrude model permittivity qualitatively good description for Au, Ag, Cu up to optical frequencies

Aluminum

Copper

Gold

SilverSkin-depth saturatesat optical frequencies !

RatioSkin-depth/structure size

becomes larger !!

first ~ω1/2 then ~o(1)

Drude model parameters from Experimental data:

Johnson & Christy, PRB 6, 4370 (1972);El-Kady et al., PRB 62, 15299 (2000).

1/ 2

S

cl

μωσ⎛ ⎞

≈⎜ ⎟⎝ ⎠

1/ 22

2

1,

ImSl qq c

με ω⎛ ⎞≈ = ⎜ ⎟

⎝ ⎠

for f < 1THz

Diamagnetic response of open and closed SRR ring

dependence on the ring width

L=10μmf~3THz

L=100nmf~70THz

Outline

1. Retrieval2. Breaking of Scaling3. Cut-wire pairs4. Diamagnetic response of SRR5. Anisotropic & Chiral metamaterials

200 250 300 350 400 450 500 550 600 650

-10

0

10

20

30

40

50 Re(ε) Im(ε) Re(μ) Im(μ)

Permittivity

,Permeability

Frequency (THz)

ωp

200 250 300 350 400 450 500 550 600 650-10

-8

-6

-4

-2

0

2

4

Re(ε) Im(ε) Re(μ) Im(μ)

Permittivity

,Permeability

Frequency (THz)

ωp

Short wires: radius=30nm, length=300nm, Drude-model Gold: F=11%

Continuous wires: radius=30nm, Drude-model Gold, (130nm)2 unit cell: F=16%

Anisotropic Arrays of Continuous or Short Nanowires

500 550 600 650 700 750 800-1.0

-0.5

0.0

0.5

1.0

1.5

Re(ε) Im(ε) Re(μ) Im(μ)

Permittivity

,Permeability

Frequency (THz)

ωp

wiresEur

P

500 550 600 650 700 750 800

0

1

2

3

4

Permittivity

,Permeability

Frequency (THz)

Re(ε) Im(ε) Re(μ) Im(μ)

wiresHuur

P

Beware:Periodicityartifacts

1, ( 1, 0.5)+ + −

anisotropicnegativerefraction

1, ( 1, 1)+ + −

1, ( 1, 1)− − −

left-handednegativerefraction

Note that the hyperbolic dispersion supports propagatingmodes for arbitrarily high parallel momenta (which would be evanescent in air).

•Bilayer chiral metamaterialsexhibits strong gyrotropyat optical frequencies.

•Specific rotatory power:Wavelength λ (nm) 660, 980, 1310Optical activity (°/mm) 600, 670, 2500

Eigenmodes in chiral medium:right circularly polarized (RCP, +) andleft circularly polarized (LCP, -), whose wavenumbers and effective indices are:

0

0

( ),

/ ( )

k k n

n k k n

χχ

±

± ±

= ±= = ±

nχ >

0, 0,k n− −< <

If the chirality parameter is very large,

the refractive index for the LCP eigenmode becomes negative.

0 0

0 0

j

j

ε χ μ ε

μ χ μ ε

= −

= +

D E H

B H E

then

Constitutive relations

V. A. Fedotov, CLEO Europe 2007

50nm Al50nm dielectric

Chiral Metamaterials: large gyrotropy & negative index

Experimental results

2 2| | | |s st t++ −−Δ = −arg( ) arg( )s st tδ ++ −−= −

LCP

RCP

5.25 GHzAν =

6.50 GHzBν =

5.25 GHzAν =

Frequency (GHz)

Tra

nsm

issi

on

(d

B)

6.58 GHzBν =

Frequency (GHz)

Δ (

dB

)

Frequency (GHz)

δ (d

egre

e)

A.V. Rogacheva, et al., PRL 97, 177401 (2006)

Simulations, J. Dong et al.

DElectron

Elasticcoupling

Electron

I nductiveCo u pl ing

D_

Svirko-Zheludev-OsipovMetamaterial (APL 78, 498 (2001))

Circular Dichroism: Experiment & Simulation

Thanks for your attention