Negative refraction and Left-handed behavior in Photonic Crystals:

FDTD and Transfer matrix method studies

Peter Markos, S. Foteinopoulou and C. M. Soukoulis

Outline of Talk• What are metamaterials?

• 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 refractions (Bilkent)

• Applications/Closing Remarks

E. N. Economou & S. Foteinopoulou

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?

Electromagnetic Metamaterials

Example: Metamaterials based on repeated cells…

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

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)

   g opposite to k

   Flat lenses

   Super lenses

€

n = − εμ

           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 rr=

Sr

kr

ωα

d

nndl+≡1

0>αn

0, kn

cp

pg

rrr

r==

α

⎥⎦

⎤⎢⎣

⎡∂∂

+∂∈∂

+∈

= 222 8

HEk

Sc

prr

rrr

ωμ

ωπμ

ku

prr

ω=

gr kr

guS rr

=

Sr

ωα

d

nndl+≡1

ωkc

n

r

=

0,

0,,0

<−>+

±=⇒>αα

α nnn

0, kn

cp

pg

rrr

r==

α

Materials with < 0 and <0 Photonic Crystals

opposite to

opposite to

opposite to kr

gr 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

Metamaterials Extend 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)

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

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: emetal = (-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

Structure of the unit cell:

SRR

LHM

EM waves propagate in the z-direction.Periodic boundary conditions are used in the xy-plane

Left-handed material: array of SRRs and wires

Resonance frequency as a function of

metallic permittivity

complex m Real m

Dependence of LHM peak on metallic permittivity

The length of the system is 10 unit cells

Dependence of LHM peak on metallic permittivity

PRB 65, 033401 (2002)

Example of Utility of 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

The transmission coefficient is an example of a quantity that can be determined simply and analytically, if the bulk material parameters are known.

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

€

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

Effective permittivity ω and permeability ω of wires and SRRs

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

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

Determination of effective parameters from transmission studies

From transmission and reflection data, the index of refraction n was calculated.

Frequency interval with Re n<0 and very small Im n was found.

Permittivity , Permeability and ImpedanceZ

Re μ < 0

Im μ > 0

Re ε < 0

Im ε < 0

Im Z < 0Re Ζ > 0

Energy losses Q are always positive in spite of the fact that Im ε is negative:

Q(ω)= 2ω/(2π) |H|2 Im (n) Re (Z) > 0

???

Another 1D left-handed structure:

Both SRR and wires are located on the same side of the dielectric board.Transmission depends on the orientation of SRR.

Bilkent & ISU APL 2002

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

b=4.4

New designs for left-handed materials

Bilkent and ISU, APL 81, 120 (2002)

ax=6.5 mmt= 0.5 mm

Bilkent & ISU APL 2002

ax=6.5 mmt= 1 mm

Bilkent & ISU APL 2002

Cut wires: Positive and negative n

•In both the LHM and PC literature there is still a lot of confusion regarding the phase refractive index np

and the group refractive index ng. How these

properties relate to “negative refraction”“negative refraction” and LH LH behavior has not yet been fully examined.

•There is controversy over the “negative refraction” phenomenon. There has been debate over the allowed signs (+ /-) for np and ng in the LH system.

Phase and group refractive index

DEFINING phase and group refractive index np and ng

In any general case:

The equifrequency surfaces (EFS) (i.e. contours of constant frequency in 2D k-space) in air and in the PC are needed to find the refracted wavevector kf (see figure).

vphase=c/|np| and vgroup= c/|ng|

Where c is the velocity of light

So from k// momentum conservation: |np|=c kf () /ω.

=∇ ωk

Remarks In the PC system vgroup=venergy so |ng|>1. Indeed this holds !

np <1 in many cases, i.e. the phase velocity is larger than c in many cases.

np can be used in Snell’s formula to determine the angle of the propagating wavevector. In general this angle is not the propagation angle of the signal. This angle is the propagation angle of the signal only when dispersion is linear (normal), i.e. the EFS in the PC is circular (i.e. kf independent of theta).

ng can never be used in a Snell-like formula to determine the signals propagation angle.

Index of refraction of photonic crystals

– The wavelength is comparable with the period of the photonic crystal

– An effective medium approximation is not valid

Equifrequency surfaces

ky

k x

Eff

ectiv

e in

dex

ω

Ref

ract

ion

angl

e

Incident angle

Photonic Crystals with negative refraction.

Photonic Crystals with negative refraction.

S. Foteinopoulou, E. N. Economou and C. M. Soukoulis

EFS plot of frequency a/ = 0.58

Schematics for Refraction at the PC interface

Schematics for Refraction at the PC interface

EFS plot of frequency a/ = 0.535

Negative refraction and left-handed behavior for a/ = 0.58

Negative refraction but NO left-handed behavior for a/ = 0.535

Superlensing in 2D Photonic Crystals

Lattice constant=4.794 mmDielectric constant=9.73r/a=0.34, square lattice

Experiment by Ozbay’s group

Negative Refraction in a 2d Photonic Crystal

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

Superlensing in photonic crystals

Subwavelength Resolution in PC based Superlens

The separation between the two point sources is /3

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

Photonic Crystals with negative refraction.

t0=1.5TT=/c

Photonic Crystals with negative refraction.

Photonic Crystals with negative refraction.

Photonic Crystals with negative refraction.

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, Appl. Phys. Lett. (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, accepted (2003) S. Foteinopoulou and C. M. Soukoulis, submitted Phys. Rev. B (2002) P. Markos and C. M. Soukoulis, submitted to Opt. Lett. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and C. M. Soukoulis, submitted to Nature P. Markos and C. M. Soukoulis, submitted to Optics Express

The keen interest to the topic

• Left-Handed Medium (LH)• Metamaterial• Backward Medium (BW)• Double Negative Medium (DNG)• Negative Phase Velocity (NPV)• Materials with Negative Refraction (MNR)

0

1020

3040

1968 1980 1990 1996 1999 2001

• Terminology

Collaboration between Crete, Greece and Bilkent University, Turkey

Crete

GM

Image Plane

Menu

Display

LHM

Cal

Store

Network Analyzer

Experimental Setup

Scanned Power Distribution at the Image Plane

Dependence of LHM peak on L and Im m

Dependence on the incident angle

Transmission peak does not depend on the angle of incidence !

Transition peak strongly dependson the angle of incidence.

This structure has an additionalxz - plane of symmetry

Transmission depends on the orientation of SRR

Lower transmissionNarrower resonance intervalLower resonance frequency

Higher transmissionBroader resonance intervalHigher resonance frequency

Transmission properties depend on the orientation of the SRR:

Dependence of the LHM T peak on the Im Board

Losses in the dielectric board are crucial for the transmission properties of the LH structures.

In our simulations, we have:

Periodic boundary condition,therefore no losses due toscattering into another direction.

Very high Im metal thereforevery small losses in the metalliccomponents.

New / Alternate Designs

Superprism Phenomena in Photonic Crystals Experiment

– H.Kosaka, T.Kawashima et. al. Superprism phenomena in photonic crystals, Phys. Rev. B 58, 10096 (1998)

Scattering of the photonic crystalHexagonal 2D photonic crystal

Vanishingly small index modulation Small index modulation

-M.Natomi, Phys. Rev. B M.Natomi, Phys. Rev. B 6262, 10696 (2000), 10696 (2000)

Using an equifrequency surface (EFS) plotsUsing an equifrequency surface (EFS) plots

Photonic crystal as a perfect lens C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. PendryC. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. PendryPhys. Rev. B, Phys. Rev. B, 6565, 201104 (2002), 201104 (2002)

Resolution limit 0.67