The Strange Properties of

Left-handed Materials

C. M. SoukoulisAmes Lab. and Physics Dept. Iowa State University

andResearch Center of Crete, FORTH - Heraklion, Crete

Outline of Talk• Historical review left-handed materials

• Results of the transfer matrix method

• Determination of the effective refractive index

• Negative n and FDTD results in PBGs (ENE & SF)

• New left-handed structures• Experiments on negative refraction and

superlenses (Ekmel Ozbay, Bilkent)

• Applications/Closing Remarks

Peter Markos, E. N. Economou & S. Foteinopoulou

Rabia Moussa, Lei Zhang & Gary Tuttle (ISU) M. Kafesaki & T. Koschny (Crete)

A composite or structured material that exhibits properties not found in naturally occurring materials or compounds.

Left-handed materials have electromagnetic properties that are distinct from any known material, and hence are examples of metamaterials.

What is an Electromagnetic Metamaterial?

Veselago

We are interested in how waves propagate through various media, so we consider solutions to the wave equation.

(+,+)(-,+)

(-,-) (+,-)

space

€

k = ω εμ

€

∇2E = εμ∂2E∂t2 n = ± εμ

Sov. Phys. Usp. 10, 509 (1968)

Left-Handed Waves

• If then is a right set of vectors:

• If then is a left set of vectors:

0,0 >> ( )kHErrr

,,

0,0 << ( )kHErrr

,,

Er

kr

Hr

• Energy flux (Pointing vector):

– Conventional (right-handed) medium

– Left-handed medium

Energy flux in plane waves

Er

kr

Hr

Er

kr

Hr S

r

Frequency dispersion of LH medium

• Energy density in the dispersive medium

• Energy density W must be positive and this requires

• LH medium is always dispersive

• According to the Kramers-Kronig relations – it is always dissipative

( )0>

∂

∂

ω

μω

( ) ( ) 22 HEWω

μω

ω

εω

∂

∂+

∂

∂=

( );0>

∂

∂

ω

εω

“Reversal” of Snell’s Law

1

2

12

PIMRHM

PIMRHM

PIMRHM

NIMLHM

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

kr

Sr k

r

Sr

n=-1n=1 n=1

RH RHLHRH RHRH

n=1.3n=1 n=1

Focusing in a Left-Handed Medium

Left-handedLeft-handed Right-handedRight-handed

SourceSource SourceSource

n=-1 n=1,52

n=1

n=1

n=1

n=1

M. Kafesaki

           Objections to the left-handed ideas

S1 S2

A BO΄Μ

ΟCausality is violated

Parallel momentum is not conserved

Fermat’s Principle ndl minimum (?)

Superlensing is not possible

Reply to the objections

• Photonic crystals have practically zero absorption

• Momentum conservation is not violated

• Fermat’s principle is OK

• Causality is not violated

• Superlensing possible but limited to a cutoff kc or 1/L

€

ndl extremum∫

guS υrr

=

Sr

kr

ωα

d

nndl+≡1

0>αn

0, kn

cp

pg

rrr

r== υ

αυ

υ

€

rp =

εμc 2

r S +

r k

8π∂ε∂ω

r E 2 +

∂μ∂ω

r H 2

⎡ ⎣ ⎢

⎤ ⎦ ⎥ k

up

rrω

=

gυr kr

guS υrr

=

Sr

ωα

d

nndl+≡1

ωkc

n

r

=

0,

0,,0

<−>+

±=⇒>αα

α nnn

0, kn

cp

pg

rrr

r== υ

αυ

υ

Materials with < 0 and <0 Photonic Crystals

opposite to

opposite to

opposite to kr

gυr opposite to kr

,

Super lenses

€

ω2 = c 2 k||2 + k⊥

2( )⇒ if k|| > ω/c⇒ k⊥

€

e ik⊥r⊥ ~ e− k⊥ r⊥is imaginary

Wave components with decay, i.e. are lost , then max

If n < 0, phase changes sign

€

e ik⊥r⊥ ~ e k⊥ r⊥

ck /|| ω>

if imaginary ⊥k

⊥k

k

€

k||

ARE NOT LOST !!!

thus

Resonant response

-5

0

5

10

0 0.5 1 1.5 2 2.5 3

ω/ω0

qE

p

q

E p

qEp

Where are material resonances?

Most electric resonances are THz or higher.

For many metals, ωp occurs in the UV

Magnetic systems typically have resonances through the GHz (FMR, AFR; e.g., Fe, permalloy, YIG)

Some magnetic systems have resonances up to THz frequencies (e.g., MnF2, FeF2)

Metals such as Ag and Au have regions where <0, relatively low loss

Negative materials

<0 at optical wavelengths leads to important new optical phenomena.

<0 is possible in many resonant magnetic systems.

What about <0 and <0?

Unfortunately, electric and magnetic resonances do not overlap in existing materials.

This restriction doesn’t exist for artificial materials!

Obtaining electric response

-5

-4

-3

-2

-1

0

1

0 1 2 3

ω/ωp

0

0.5

1

1.5

2

ω/ω

p

k

Drude Model

-E-

-

-

-

Gap

€

(ω) = 1−ω p

2

ω2

€

ω =c

εk

€

ωp2 =

2πc 2

d2 ln(d / r)=

1d2Lε 0

Obtaining electric response (Cut wires)

Drude-Lorentz

E 0

0.5

1

1.5

2

ω/ω

p

k-10

-5

0

5

10

0 1 2 3

ω/ωp

- -

-- -

Gap

€

(ω) = 1−ω p

2

ω 2 − ω02

€

ω =c

μk

Obtaining magnetic response

To obtain a magnetic response from conductors, we need to induce solenoidal currents with a time-varying magnetic field

A metal disk is weakly diamagnetic

A metal ring is also weakly diamagnetic

Introducing a gap into the ring creates a resonance to enhance the response

H

+

+

-

-

Obtaining magnetic response

-3

-2

-1

0

1

2

3

0 1 2

ω/ωmp

0

0.5

1

1.5

2

ω/ω

mp

k

Gap

€

ω =c

μk

€

(ω) = 1−Fω p

2

ω 2 − ω02

Metamaterials Resonance Properties

€

ω( ) =1−ω p

2

ω2

€

ω( )=1−ω p

2

ω 2 −ω02

J. B. Pendry

First Left-Handed Test Structure

UCSD, PRL 84, 4184 (2000)

Wires alone

<0

Wires alone

Split rings alone

Transmission Measurements

4.5 7.05.0 5.5 6.0 6.5

Frequency (GHz)

Tra

nsm

itte

d P

ower

(dB

m)

>0<0>0

>0<0>0 <0<0

<0

UCSD, PRL 84, 4184 (2000)

Best LH peak observed in left-handed materials

Bilkent, ISU & FORTHw

t

dr1

r2

Single SRR Parameters: r1 = 2.5 mm r2 = 3.6 mm d = w = 0.2 mm t = 0.9 mm

A 2-D Isotropic Structure

UCSD, APL 78, 489 (2001)

Measurement of Refractive Index

UCSD, Science 292, 77 2001

Measurement of Refractive Index

UCSD, Science 292, 77 2001

Measurement of Refractive Index

UCSD, Science 292, 77 2001

Boeing free space measurements for negative refraction

PRL 90, 107401 (2003) & APL 82, 2535 (2003)

n

Transfer matrix is able to find:

• Transmission (p--->p, p--->s,…) p polarization

• Reflection (p--->p, p--->s,…) s polarization

• Both amplitude and phase• Absorption

Some technical details:

• Discretization: unit cell Nx x Ny x Nz : up to 24 x 24 x 24

• Length of the sample: up to 300 unit cells• Periodic boundaries in the transverse direction• Can treat 2d and 3d systems • Can treat oblique angles• Weak point: Technique requires uniform discretization

Structure of the unit cell

Periodic boundary conditionsare used in transverse directions

Polarization: p wave: E parallel to y s wave: E parallel to x

For the p wave, the resonance frequencyinterval exists, where with Re eff <0, Re eff<0 and Re np <0.For the s wave, the refraction index ns = 1.

Typical permittivity of the metallic components: metal = (-3+5.88 i) x 105

EM wave propagates in the z -direction

Typical size of the unit cell: 3.3 x 3.67 x 3.67 mm

Generic LH related Metamaterials

€

′ ω p

€

′ ω p

€

ωm

€

ωm

€

′ ω m

€

′ ω m

€

ωa/c

€

ωa/c

€

ωa/c

€

€

Resonance and anti-resonance

Typical LHM behavior

f (GHz)

T

30 GHz FORTH structure with 600 x 500 x 500 m3

SubstrateGaAsb=12.3

LHM Design used by UCSD, Bilkent and ISU

LHMSRRClosed LHM

T and R of a Metamaterial

€

ts =exp(−ikd)

cos nkd( )−1

2z +

1

z ⎛ ⎝

⎞ ⎠sin nkd( )

€

z =με

€

n = με

€

ω( )=1−ωmp

2

ω 2 −ω m02 + iΓm0

€

ω( ) =1−ωep

2

ω2 −ω e02 + iΓe 0

UCSD and ISU, PRB, 65, 195103 (2002)

€

rs = − ts exp(+ikd)i(z −1 / z)sin(nkd) / 2

d

z, n

Inversion of S-parameters

d

UCSD and ISU, PRB, 65, 195103 (2002)

€

e ik

€

teik

€

re− ik

€

=nz

€

=nz

€

n =1kd

cos−1 12 ′ t

1− r 2 − ′ t 2( )[ ]

⎛ ⎝

⎞ ⎠ +

2πmkd

€

z = ±1 + r( )

2− ′ t 2

1− r( )2 − ′ t 2

Effective permittivity ω and permeability ω of wires and SRRs

UCSD and ISU, PRB, 65, 195103 (2002)

€

ω( ) =1−ω p

2

ω2

€

ω( )=1−ωm

2

ω 2 − ω02 + iΓω

UCSD and ISU, PRB, 65, 195103 (2002)

Effective permittivity ω and permeability ω of LHM

UCSD and ISU, PRB, 65, 195103 (2002)

Effective refractive index nω of LHM

b=4.4

New designs for left-handed materials

Bilkent and ISU, APL 81, 120 (2002)

Bilkent & FORTH

Photonic Crystals with negative refraction.

Triangular lattice of rods with =12.96 and radius r, r/a=0.35 in air. H (TE) polarization.

Same structure as in Notomi, PRB 62,10696 (2000)

CASE 1

CASE 2

PRL 90, 107402 (2003)

Photonic Crystals with negative refraction.

υg

υg

Equal Frequency Surfaces (EFS)

Schematics for Refraction at the PC interface

EFS plot of frequency a/ = 0.58

Experimental verification of negative refraction

a

Lattice constant a=4.794 mmDielectric constant=9.61R/a=0.329Frequency=13.698 GHzsquare lattice E(TM) polarization

Bilkent & ISU

Band structure, negative refraction and experimental set up

Bilkent & ISU

Negative refraction is achievable in this frequency

range for certain angles of incidence.

Frequency = 13.7 GHz = 21.9 mm

17 layers in the x-direction and 21 layers in the y-direction

Superlensing in photonic crystals

FWHM = 0.21

Image Plane

Distance of the source from the PC interface is 0.7 mm (/30)

Subwavelength Resolution in PC based Superlens

The separation between the two point sources is /3

Subwavelength Resolution in PC based Superlens

The separation between the two point sources is /3 !

Power distribution along the image plane

Controversial issues raised for negative refraction

PIM NIMAmong othersAmong others

1) What are the allowed signs for the phase index np and group index ng ?

2) Signal front should move causally from AB to AO to AB’; i.e. point B reaches B’ in infinite speed.

Does negative refraction violate causalityviolate causality and the speed of light limitspeed of light limit ?

Valanju et. al., PRL 88, 187401 (2002)

Photonic Crystals with negative refraction.

FDTD simulations were used to study the time evolution of an EM wave as it hits the interface vacuum/photonic crystal.Photonic crystal consists of an hexagonal lattice of dielectric rods with =12.96. The radius of rods is r=0.35a. a is the lattice constant.

PhotonicCrystal

vacuum

QuickTime™ and aBMP decompressor

are needed to see this picture.

We use the PC system of case1 to address the controversial issue raised

Time evolution of negative refraction shows:

The wave is trapped initially at the interface.

Gradually reorganizes itself.

Eventually propagates in negative direction

Causality and speed of light limit not violated

S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, PRL 90, 107402 (2003)

Photonic Crystals: negative refraction

The EM wave is trapped temporarily at the interface and after a long time,the wave front moves eventually in the negative direction. Negative refraction was observed for wavelength of the EM wave = 1.64 – 1.75 a (a is the lattice constant of PC)

Conclusions

• Simulated various structures of SRRs & LHMs • Calculated transmission, reflection and absorption• Calculated eff and eff and refraction index (with UCSD)

• Suggested new designs for left-handed materials• Found negative refraction in photonic crystals • A transient time is needed for the wave to move along the - direction• Causality and speed of light is not violated.• Existence of negative refraction does not guarantee the existence of negative n and so LH behavior• Experimental demonstration of negative refraction and superlensing• Image of two points sources can be resolved by a distance of /3!!!

$$$ DOE, DARPA, NSF, NATO, EU

Publications:

P. Markos and C. M. Soukoulis, Phys. Rev. B 65, 033401 (2002) P. Markos and C. M. Soukoulis, Phys. Rev. E 65, 036622 (2002) D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002) M. Bayindir, K. Aydin, E. Ozbay, P. Markos and C. M. Soukoulis, APL 81, 120 (2002) P. Markos, I. Rousochatzakis and C. M. Soukoulis, Phys. Rev. E 66, 045601 (R) (2002) S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, PRL 90, 107402 (2003) S. Foteinopoulou and C. M. Soukoulis, Phys. Rev. B 67, 235107 (2003) P. Markos and C. M. Soukoulis, Opt. Lett. 28, 846 (2003) E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and CMS, Nature 423, 604 (2003) P. Markos and C. M. Soukoulis, Optics Express 11, 649 (2003) E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and CMS, PRL 91, 207401 (2003) T. Koshny, P. Markos, D. R. Smith and C. M. Soukoulis, PR E 68, 065602(R) (2003)

PBGs as Negative Index Materials (NIM)

Veselago : Materials (if any) with < 0 and < 0

> 0 Propagation k, E, H Left Handed (LHM) S=c(E x H)/4 opposite to k

   Snell’s law with < 0 (NIM)

   υg opposite to k

   Flat lenses

   Super lenses

€

n = − εμ

xy

z

t

wt»w

t=0.5 or 1 mmw=0.01 mm

l=9

cm

3 mm

3 mm

0.33 mm

0.33 mm

Periodicity:ax=5 or 6.5 mmay=3.63 mmaz=5 mm

Number of SRRNx=20Ny=25Nz=25

ax

0.33 mm

Polarization: TM

E

B

y

x

ax=6.5 mmt= 0.5 mm

Bilkent & ISU APL 81, 120 (2002)