engineering optical space with metamaterials: from ...shalaev/rome_2007/rome_tutorial.pdf · by...

$: Engineering Optical Space with Metamaterials: from ...shalaev/Rome_2007/Rome_Tutorial.pdf · by diffraction and interference a>> ... using geometrical optics and ray tracing Example:$
Birck Nanotechnology Center

Vladimir M. ShalaevSchool of Electrical & Computer Engineering

Purdue University, USA

2007.10.26

Engineering Optical Space with Metamaterials: from Metamagnetics to Negative-Index and

Cloaking

Metamaterials’2007, RomeEuropean Doctoral School on Metamaterials

V. M. Shalaev, Oct. 26, 2007 2


Invisibility: An Ancient Dream

Tarnhelm of invisibility(Norse mythology)

Perseus’ helmet(Greek mythology)

Ring of Gyges(“The Republic”, Plato)

The 12 Dancing Princesses(Brothers Grimm, Germany)

Cloaking devices(Star Trek, USA)

Harry Potter’s cloak(J. K. Rowling, UK)



Invisibility in Nature: Chameleon Camouflage



Invisibility by Transformation of Time-Space

Black hole



Invisibility to Radar: Stealth Technology

Stealth technique:Radar cross-section reductions by absorbing paint / non-metallic frame / shape effect…



Active Camouflage: Real time capture and re-display

Optical Camouflage, Tachi Lab, U. of Tokyo, Japan

Illustrating the concept: active capture and re-display, creates an "illusory transparency",



Cloaking ≠ Invisibility

• Cloaking is more than invisibility or camouflage– Camouflage: an adaptation to the surrounding environment.– Cloaking: No need to adapt to a particular environment, with the

ultimate goal of transparency — no scattering; no shade.

• Criteria for an ideal cloak– Macroscopic, not limit to subwavelength size or near field region.– Independent to the object to be cloaked.– Minimized absorption and scattering.– Passive– Broadband– …



Ideal Cloak: from fiction to fact?

The Invisible Man by H. G. Wells (1897)

“The invisible woman” in The Fantastic 4 by Lee & Kirby (1961)

Examples with scientific elements:



Ideal Cloak: from fiction to fact?

The Invisible Man by H. G. Wells (1897)

“The invisible woman” in The Fantastic 4 by Lee & Kirby (1961)

Examples with scientific elements:

"... it was an idea ... to lower the refractive index of a substance, solid or liquid, to that of air — so far as all practical purposes are concerned.” -- Chapter 19 "Certain First Principles"

"... she achieves these feats by bending all wavelengths of light in the vicinity around herself ... without causing any visible distortion.” -- Introduction from Wikipedia

Pendry et al.; Leonhard, Science, 2006(Earlier work: cloak of thermal conductivity by Greenleaf et al., 2003)

V. M. Shalaev, Oct. 26, 2007 10


Transformation of Maxwell’s equations

Straight field line in Cartesian coordinate

Distorted field line in distorted coordinate

Spatial profile of ε & μ tensors determines the distortion of coordinate

Seeking for profile of ε & μ to make light avoid particular region in space — optical cloaking

Pendry et al., Science, 2006

V. M. Shalaev, Oct. 26, 2007 11


Pendry et al., 2006

The bending of light due to the gradient in refractive index in a desert mirage

A similarity in Mother Nature

V. M. Shalaev, Oct. 26, 2007 12


Cloaking in spherical system

The transformation in spherical system:

0 r b< < 'a r b< <

' ' 'b ar r ab

θ θ φ φ−= + = =

2

2

( )r r

b r ab a rbb abb a

θ θ

φ φ

ε μ

ε μ

ε μ

⎧ −= =⎪ −⎪

⎪ = =⎨ −⎪⎪ = =⎪ −⎩

Pendry et al., Science, 2006

Similar idea was proposed by Leonhardt, Science, 2006

a b

V. M. Shalaev, Oct. 26, 2007 13


Route to Cloaking: Tailor the (ε,μ) distribution

ε, μ diagram:

V. M. Shalaev, Oct. 26, 2007 14


Natural Optical Materials

Semiconductors

Crystals

WaterAir

metals

V. M. Shalaev, Oct. 26, 2007 15


Metamaterials: Artificial media with designed (ε,μ)

--

Metamaterial is an arrangement of artificial structural elements, designed to achieve advantageous and unusual electromagnetic properties. ---Metamorphose

μετα = meta = beyond (Greek)

A natural material with its atoms

A metamatestructured “atoms”

rial with artificially

V. M. Shalaev, Oct. 26, 2007 16


Electromagnetic properties v.s. characteristic sizes

0 1 aa<< .

Effective mediumdescription using

Maxwell equations with, , n, Z

a~Structure dominates.

Properties determinedby diffraction and

interference

a>>Properties described

using geometrical opticsand ray tracing

Example:Optical crystalsMetamaterials

Example:Photonics crystalsPhased array radarX-ray diffraction optics

Example:Lens systemShadows

V. M. Shalaev, Oct. 26, 2007 17


Natural Crystals

... have lattice constants much smaller

than light wavelengths: a <<λ

… are treated as homogeneous media

with parameters ε, μ, n, Z (tensors in

anisotropic crystals)

… have a positive refractive index: n > 1

… show no magnetic response at optical

wavelengths: μ =1

V. M. Shalaev, Oct. 26, 2007 18


Photonic crystals

... have lattice constants comparable

to light wavelengths: a ~ λ

… can be artificial or natural

… have properties governed by the diffraction of the periodic structures

… may exhibit a bandgap for photons

… typically are not well described

using effective parameters ε, μ, n, Z

… often behave like but they are nottrue metamaterials

V. M. Shalaev, Oct. 26, 2007 19


Noble metal: ε < 1 in nature

500 1000 1500 2000-250

-200

-150

-100

-50

0

50

Wavelength (nm)

Per

mitt

ivity

of S

ilver

Re(ε), experimentIm(ε), experimentRe(ε), DrudeIm(ε), Drude

2

0( )( )

p

iω

ε ω εω ω

= −+ Γ

0 5.09.216

0.0212p eV

eV

εω

==

Γ =

Drude model for permittivity: Silver parameters:

Experimental data from Johnson & Christy, PRB, 1972

V. M. Shalaev, Oct. 26, 2007 20


Electrical metamaterials:metal wires arrays with tunable plasma frequency

A periodic array of thin metal wires with r<<a<<λ acts as a low frequency plasma

The effective ε is described with modified ωp

Plasma frequency depends on geometry rather than on material properties

2

2 2 20

' " 1( / )

p

p

ii a r

ωε ε ε

ω ω ε ω π σ= + = −

+

Pendry, PRL (1996)

22

2

2ln( / )pc

a a rπω =

V. M. Shalaev, Oct. 26, 2007 21


Metal-Dielectric Composites and Mixing Rules

( )1 1 2 2

1 2 1 2 2 1

c c

c c

ε ε ε

ε ε ε ε ε⊥

= +⎧⎪⎨

= +⎪⎩

( ) ( )( ) ( )

( ) ( )( ) ( )ωεωε

ωεωεωεωε

ωεωε

hi

hi

hMG

hMG f22 +

−=

+−

[ ]2

' "( 1) ( 1 )1

2( 1) ( 1) ( 1 ) 4( 1)

e e e

m d

m d m d

idp d dp

d dp d dp d

ε ε εε ε

ε ε ε ε

= +

− + − −⎧ ⎫⎪ ⎪= ⎨ ⎬− ± − + − − + −⎪ ⎪⎩ ⎭

V. M. Shalaev, Oct. 26, 2007 22


Electromagnetic properties of metal wires

2 3/ 2 2 1/ 2 2 1/ 20 2( ) ( ) ( )i j k

ii j k

a a a dsq

s a s a s a∞

=+ + +∫

(1 ) /q qκ = −

(1 ) 0m eff d eff

m eff d eff

f fε ε ε ε

ε κε ε κε− −

+ − =+ +

{ }21 42eff m dε ε ε κε εκ

= ± + [( 1) 1] [ ( 1) ]m df fε κ ε κ κ ε= + − + − +

10-2

10-1

100

101

1020

0.2

0.4

0.6

0.8

1

Aspect ratio, α:1:1

Dep

olar

izat

ion

fact

or, p

Lorentz depolarization factor for a spheroid with aspect ratio α:1:1

p(1:1:1)=1/3

Depolarization factor:

Screening factor:

Clausius-Mossotti yields shape-dependent EMT:

V. M. Shalaev, Oct. 26, 2007 23


Absence of Optical Magnetism in Nature

“… the magnetic permeability µ(ω) ceases to have any physical meaning at relatively low frequencies…there is certainly no meaning in using the magnetic susceptibility from optical frequencies onwards, and in discussion of such phenomena we must put µ=1.”

Landau and Lifshitz, ECM, Chapter 79.

Magnetic coupling to an atom: ~ 0/ 2B ee m c eaμ α= =

0eaElectric coupling to an atom: ~

(Bohr magneton)

Magnetic effect / electric effect ≈ α2 ≈ (1/137)2 < 10 -4

V. M. Shalaev, Oct. 26, 2007 24


SRR: the first magnetic metamaterials

Theory: Pendry et al., 1999.

H

A bulk metal has no magnetism in optics

Make it a ring: loop

current induced by H

A metal ring: weak magnetic response

Cut the ring to

introduce resonance

A split ring: magnetic resonance

Double

the

ring to

stre

ngthen

the

reso

nance

Double SRR: enhanced magnetic resonance

Split-ring resonator (SRR)

Experiment: Smith et al., 2000.

V. M. Shalaev, Oct. 26, 2007 25


Towards Optical Magnetic Metamaterials

Terahertz magnetism

A) Yen, et al. ~ 1THz (2-SRR) – 2004 Katsarakis, et al (SRR – 5 layers) - 2005

b) Zhang et al ~50THz (SRR+mirror) - 2005c) Linden, et al. 100THz (1-SRR) -2004d) Enkrich, et al. 200THz (u-shaped)-2005

2004-2007 years: from 10 GHz to 500 THZ

V. M. Shalaev, Oct. 26, 2007 26


Limits of size scaling in SRRs

Direct scaling-down the SRR dimensions doesn’t help much…

total coil kineticL L L= +

Zhou et al, PRL (2005); Klein, et al., OL (2006)

coilL size∝1

kineticLsize

∝

totalC size∝

2

1 1 1( / ) ( ) .

restotal totalL C A size B size C size size const

ω ∝ = ∝× ⋅ + ⋅ ⋅ +

Saturation

Loss in metal gives kinetic inductance

V. M. Shalaev, Oct. 26, 2007 27


Magnetic Metamaterial: Nanorod to Nanostrip

E

H

k

Dielectric

Metal

Nanorod pair Nanorod pair array Nanostrip pair

• Nanostrip pair has a much stronger magnetic response

Lagar’kov, Sarychev PRB (1996) - µ > 0 Podolskiy, Sarychev & Shalaev, JNOPM (2002) - µ < 0 & n < 0Kildishev et al, JOSA B (2006); Shvets et al JOSA (2006) – strip pairsSvirko, et al, APL (2001) - “crossed” rods for chirality

V. M. Shalaev, Oct. 26, 2007 28


Visible magnetism: structure and geometries

35 40 2 bt nm d nm p w= = ≈

Purdue groupYuan, et al., Opt. Expr., 2007 – red lightCai, et al., Opt. Expr., 2007 – all the visible

Width varies from 50 nm to 127 nm

V. M. Shalaev, Oct. 26, 2007 29


Negative Magnetic Response

EH

k

V. M. Shalaev, Oct. 26, 2007 30


Width tunes resonance

Resonant TMTransmission

Non-resonant TETransmission

Resonant TMReflection

Non-resonant TEReflection

Sample # A B C D E F

Width w (nm) 50 69 83 98 118 127

Cai, et al., Opt. Expr., 15, 3333 (2007)

400 500 600 700 800 9000.00.10.20.30.40.50.60.70.80.9

Tran

smis

sion

Wavelength (nm)

A B C D E F

400 500 600 700 800 9000.00.10.20.30.40.50.60.70.80.9

Wavelength (nm)

A B C D E F

400 500 600 700 800 9000.00.10.20.30.40.50.60.70.80.9

Ref

lect

ion

Wavelength (nm)

A B C D E F

400 500 600 700 800 9000.00.10.20.30.40.50.60.70.80.9

Wavelength (nm)

A B C D E F

160 μm

V. M. Shalaev, Oct. 26, 2007 31


Dependence of λm and μ′ on w

λm as a function of strip width “w”: experiment vs. theory

Negligible saturation effect on size-scaling (as opposed to SRRs)

50 60 70 80 90 100 110 120 130450

500

550

600

650

700

750

800 Experimental Analytical Permeability

Strip width, w (nm)

λ m (n

m)

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0P

ermeability (μ')

V. M. Shalaev, Oct. 26, 2007 32


Negative refractive index: A historical review

Sir Arthur Schuster Sir Horace Lamb

L. I. Mandel’stam

V. G. Veselago

Sir John Pendry

… energy can be carried forward at the group velocity but in a direction that is anti-parallel to the phase velocity…

Schuster, 1904

Negative refraction and backward propagation of waves

Mandel’stam, 1945

Left-handed materials: the electrodynamics of substances with simultaneously negative values of ε and μ

Veselago, 1968

Pendry, the one who whipped up the recent boom of NIM researches

Perfect lens (2000)EM cloaking (2006)

V. M. Shalaev, Oct. 26, 2007 33


Metamaterials with Negative Refraction

Single-negative:n<0 when ε′ < 0 whereas μ′ > 0 (F is low)

Double-negative:n<0 with both ε′ < 0 and μ′ < 0 (F can be large)

n < 0, if ε′|μ| + μ′|ε| < 0

εμn

εμn

±=

=2

Refraction:

Figure of meritF = |n’|/n”

PIM

NIM

V. M. Shalaev, Oct. 26, 2007 34


Negative index metamaterials

Negative electrical metamaterial + Negative magnetic metamaterial= Negative index metamaterial

+ =

Smith, et al., UCSD, PRL (2000)

V. M. Shalaev, Oct. 26, 2007 35


The first NIM in microwave

Smith, et al., UCSD, PRL (2000)

Transmission properties of the structure

E,H ~ exp[in(ω/c)z]n = ±√(εμ)

V. M. Shalaev, Oct. 26, 2007 36


Negative-index Material in Microwave

Smith, et al., UCSD, Science (2001)

2D waveguide 3D free space

The Boeing Cube

Parazzoli, et al., Boeing, PRL (2003)

NIM Teflon

V. M. Shalaev, Oct. 26, 2007 37


Negative Index Design for Optical Frequencies

E

H

k

Dielectric

Metal

Nanostrip pair (TM)

μ < 0 (resonant)

Nanostrip pair (TE)

ε < 0 (non-resonant)

Fishnet

ε and μ < 0

S. Zhang, et al., PRL (2005)

V. M. Shalaev, Oct. 26, 2007 38


A Negative-index Material close to the Visible

• E-beam lithography

• Period = 300 nm along both axis

• Average width of strips along H = 130 nm Average width of strips along E = 95 nm

H

E

Alumina

Silver

ITO300 nm

300 nm

Stacking:

10 nm of Al2O333 nm of Ag38 nm of Al2O333 nm of Ag10 nm of Al2O3

SEM image and primary polarization

V. M. Shalaev, Oct. 26, 2007 39


Solid line : Experimental

Spectra for Primary Polarization

• Magnetic resonance around λ = 800 nm

• Electric resonance around λ = 600 nm

U. K. Chettiar, et al., Optics Letters 32, 1671 (2007)

• Finite Elements

Dashed line: Simulated

V. M. Shalaev, Oct. 26, 2007 40


Field Maps for Primary Polarization

Electrical Resonance, λ = 625 nm Magnetic Resonance, λ = 815 nm

E

H

k

V. M. Shalaev, Oct. 26, 2007 41


Double Negative NIM (n’=-1.0, FOM=1.3, at 810 nm) Single Negative NIM (n’=-0.9, FOM=0.7, at 770 nm)

wavelength (nm)wavelength (nm)

pe

rme

ab

ility

pe

rmitt

ivity

500 600 700 800 900-1

0

1

2

-4-2

0

2

4μ'

ε'

500 600 700 800 900

-1

0

1-n'/n"

-n'E

H

pe

rme

ab

ility

pe

rmitt

ivity

μ'

ε'

wavelength (nm) wavelength (nm)

-n'/n"

-n'E

H

700 800 900-1

0

1

700 800 900

0.5

1

1.5

-6

-4

-2

0

Chettiar et alOL (2007)

V. M. Shalaev, Oct. 26, 2007 42


Materials ready…

ε, μ diagram:

V. M. Shalaev, Oct. 26, 2007 43


Gradient Metamaterial: A critical step

Negative index gradient lens

n = -2.7 on edge

n = -1.0 on center

D. R. Smith Group, PRE 2005; APL 2006

V. M. Shalaev, Oct. 26, 2007 44


Back to Cloak: cylindrical system

The transformation in cylindrical system:

0 r b< < 'a r b< <

' ' 'b ar r a z zb

θ θ−= + = =

2

r r

z z

r arrr ab r ab a r

θ θ

ε μ

ε μ

ε μ

⎧ −= =⎪

⎪⎪

= =⎨ −⎪⎪ −⎛ ⎞= =⎪ ⎜ ⎟−⎝ ⎠⎩

Smith et al., arXiv, 2006

V. M. Shalaev, Oct. 26, 2007 45


The transformation in cylindrical system:

0 r b< < 'a r b< <

' ' 'b ar r a z za

θ θ−= + = =

2

r r

z z

r arrr ab r ab a r

θ θ

ε μ

ε μ

ε μ

⎧ −= =⎪

⎪⎪

= =⎨ −⎪⎪ −⎛ ⎞= =⎪ ⎜ ⎟−⎝ ⎠⎩

2

z

r

b r ab a rrr ar ar

θ

ε

μ

μ

⎧ −⎛ ⎞=⎪ ⎜ ⎟−⎝ ⎠⎪⎪⎪ =⎨ −⎪

−⎪ =⎪⎪⎩

2

2

1

z

r

bb a

r ar

θ

ε

μ

μ

⎧ ⎛ ⎞=⎪ ⎜ ⎟−⎝ ⎠⎪⎪ =⎨⎪

−⎛ ⎞⎪ = ⎜ ⎟⎪ ⎝ ⎠⎩

TE incidence

To maintain the dispersion relation only

z

r z

constantconstant

θμ εμ ε

=⎧⎨ =⎩

Towards an experimental demonstration

Schurig et al., Science, 2006

V. M. Shalaev, Oct. 26, 2007 46


The First Metamaterial Cloak

Duke group, 2006

V. M. Shalaev, Oct. 26, 2007 47


Ideal case

Reduced parameter

Experimental data

Structure of the cloak

Experimental demonstration at microwave frequency

Schurig et al., Science, 2006

V. M. Shalaev, Oct. 26, 2007 48


Movie: How it works

News release, Duke Univ., Oct. 2006

V. M. Shalaev, Oct. 26, 2007 49


Nature Photonics (to be published)

Optical Cloaking with Metamaterials:Can Objects be Invisible in the Visible?

Cover article of Nature Photonics (April, 2007)

V. M. Shalaev, Oct. 26, 2007 50


How about optical frequencies?

Scaling the microwave cloak design?Intrinsic limits to the scaling of SRR size

High loss in resonant structures

TM incidence

To maintain the dispersion relation

z

z r

constantconstant

θμ εμ ε

=⎧⎨ =⎩

2

, ,r r z zr a r b r ar r a b a rθ θε μ ε μ ε μ− −⎛ ⎞= = = = = = ⎜ ⎟− −⎝ ⎠

2

z

r

b r ab a rrr ar ar

θ

μ

ε

ε

⎧ −⎛ ⎞=⎪ ⎜ ⎟−⎝ ⎠⎪⎪⎪ =⎨ −⎪

−⎪ =⎪⎪⎩

2

2 2

1z

r

bb a

b r ab a r

θ

μ

ε

ε

⎧⎪ =⎪⎪⎪ ⎛ ⎞=⎨ ⎜ ⎟−⎝ ⎠⎪⎪ −⎛ ⎞ ⎛ ⎞⎪ = ⎜ ⎟ ⎜ ⎟−⎪ ⎝ ⎠ ⎝ ⎠⎩

A constant permittivity of a dielectric; 1θ

No magnetism required!

ε >

(for in-plane k)

Gradient in r direction only; εr changing from 0 to 1.

Cai, et al., Nature Photonics, 1, 224 (2007)

HE

k

V. M. Shalaev, Oct. 26, 2007 51


Structure of the cloak: “Round brush”

metal needles embedded in dielectric host

Unit cell:

Flexible control of εr ;

Negligible perturbation in εθ

Cai, et al., Nature Photonics, 1, 224 (2007)

V. M. Shalaev, Oct. 26, 2007 52


Cloaking performance: Field mapping movies

Example:Non-magnetic cloak @ 632.8nm

Cloak ONCloak OFF

V. M. Shalaev, Oct. 26, 2007 53


Scattering issue in a non-magnetic cloak

Non-magnetic (linear) Cloak

Ideal Cloak

1zr b

r b

Zθ

με=

=

= =

Perfectly matched impedance results in zero scattering

1zr b

r b

aZ bθ

με=

=

= = −

Detrimental scattering due to impedance mismatch

V. M. Shalaev, Oct. 26, 2007 54


Normalized scattered field

V. M. Shalaev, Oct. 26, 2007 55


High-order transformations minimize scattering

Linear transformation

b ar r ab− ′= +

Cai, et al., App. Phys. Lett, 91, 111105 (2007)

Jacobian Matrix

Nonlinear transformation

( )r g r′= (0) ; ( ) ; ( ) 0g a g b b g r r′ ′= = ∂ ∂ >

ε And μ tensors for ideal cloak

( )( )( )

1

1

( )

( )

( )

r r

z z

r r g r r

r r g r r

r r g r r

θ θ

ε μ

ε μ

ε μ

−

−

⎧⎪ ′ ′ ′= = ∂ ∂⎪⎪⎪⎪⎪ ⎡ ⎤′ ′ ′= = ∂ ∂⎨ ⎢ ⎥⎣ ⎦⎪⎪⎪⎪ ⎡ ⎤′ ′ ′= = ∂ ∂⎪ ⎢ ⎥⎪ ⎣ ⎦⎩

Corresponding non-magnetic parameters ( )2 2

; ( ) ; 1r zr r g r rθε ε μ−⎡ ⎤′ ′ ′= = ∂ ∂ =⎢ ⎥⎣ ⎦

Set Z=1 at r=b to fix g(r′) ( ) 1r b zr br b

Z g r rθμ ε===

′ ′= = ∂ ∂ =

HE

k

V. M. Shalaev, Oct. 26, 2007 56


Example: Optimized quadratic transformation

( ) 2( ) 1r g r a b p r b r a with p a b⎡ ⎤′ ′ ′= = − + − + =⎢ ⎥⎣ ⎦

A second-order transformation for non-magnetic cloak with minimized scattering

V. M. Shalaev, Oct. 26, 2007 57


Transformation and impedance

0 1 2 3 40

1

2

3

4

r/a, old coordinates

r/a, n

ew c

oord

inat

es

0 1 2 3 40

0.5

1

1.5

r/a, new coordinates

impe

danc

e at

r=b

linearquadratic

b b

V. M. Shalaev, Oct. 26, 2007 58


Reduced scattering from nonlinear cloak

Normalized scattered field

V. M. Shalaev, Oct. 26, 2007 59


Suppression in both magnitude and directivity

Scattering radiation pattern

270°

90°

180° 0°

0.0 0.5 1.0

Ei

k

scatterer

linearquadratic


Engineering Optical Meta-Space:Controlling & Manipulating Light

via design and fabrication of (ε,μ)-distribution

Light-concentrating slab (as an example):

V. M. Shalaev, Oct. 26, 2007 61


Example of Engineering Optical Meta-Space:Optical concentrator

Electrical field distribution Power flow distribution

Rahm et al. (Duke U.), arXiv 0706.2452

V. M. Shalaev, Oct. 26, 2007 62


Example of Engineering Optical Meta-Space:Square cloak

Rahm et al. (Duke U.), arXiv 0706.2452

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