Resolution Enhancement of a LeftResolution Enhancement of a Left--Handed Handed Material Material SuperlensSuperlens

C. K. C. K. OngOng

Centre Centre for for Superconducting Superconducting and Magnetic Materials (CSMM) and and Magnetic Materials (CSMM) and Department of Physics, National University of SingaporeDepartment of Physics, National University of Singapore

X. S. X. S. RaoRao

Temasek Temasek Laboratories, National University of SingaporeLaboratories, National University of Singapore

(ICMAT 2003, Singapore, December 2003)

Outline

OutlineOutline

1.1. IntroductionIntroduction

2.2. The FDTD SimulationThe FDTD Simulation

3.3. Results and DiscussionsResults and Discussions

4.4. SummarySummary

An insight ahead of the time

02

2

002 =

∂∂

−∇tEEr

rµµεε

,)( )(0

rktjeEErrrr⋅−= ωω

0

22

ck ωεµ=r

0,0 << µε

Veselago’s results

Fundamental physical laws do not rule out the possibility of siFundamental physical laws do not rule out the possibility of simultaneous multaneous negative negative εε and and µµ..

EM properties of media with simultaneous negative EM properties of media with simultaneous negative εε and and µµ are very are very different from those of the normal materials with positive different from those of the normal materials with positive εε and and µµ..

leftleft--handedhanded

negative indexnegative index--ofof--refractionrefraction

reversed Doppler effectreversed Doppler effect

reversed reversed CerenkovCerenkov radiationradiation

focusing through a slab of such a mediumfocusing through a slab of such a medium

The rise of left-handed metamaterials

UCSD,UCSD,

PRL PRL 8484,4184 (2000); ,4184 (2000); ScienceScience 292292, 77 (2001)., 77 (2001).

PRL PRL 7676,4773 (1996);,4773 (1996);

IEEE MTTIEEE MTT 4747, 2075 (1999)., 2075 (1999).

Imaging by a LHM slab

Propagating wave : Phase compensation (Propagating wave : Phase compensation (Veselago Veselago 1964)1964)

Evanescent wave : Amplitude reconstruction (Evanescent wave : Amplitude reconstruction (Pendry Pendry 2000)2000)

“Perfect” lens“Perfect” lens 1)()( −== ωµωε rr

All geometric details of the source can be reconstructed at the All geometric details of the source can be reconstructed at the image!image!

Amplification of evanescent waves ?

No !No ! UnphysicallyUnphysically large field at the 2large field at the 2ndnd interfaceinterface

Breaking square Breaking square integrability integrability of EM fields if of EM fields if t>dt>d

Loss transforms amplified wave into decaying oneLoss transforms amplified wave into decaying one

… …… …

Methodology

full wavefull wave

straightstraight--forwardforward

causality guaranteedcausality guaranteed

dynamicaldynamical

FiniteFinite--difference timedifference time--domain (FDTD)domain (FDTD)

Diverse FDTD results

ZiolkovskiZiolkovski & & HeymanHeyman [[PREPRE 6464, 056625 (2001)] , 056625 (2001)]

No stable image observedNo stable image observed

Loschialpo Loschialpo et al.et al. [[PREPRE 6767, 025602 (2003)] , 025602 (2003)]

stable image but no resolution enhancement observedstable image but no resolution enhancement observed

CummerCummer [[APLAPL 8282, 1503 (2003)] , 1503 (2003)]

image contains image contains subwavelength subwavelength components, enhanced resolutioncomponents, enhanced resolution

KarkkainenKarkkainen [[condcond--mat/0302407 (2003)] mat/0302407 (2003)]

amplification of evanescent modes in a amplification of evanescent modes in a lossless lossless LHM slabLHM slab

Generate pure evanescent wave

Normal FDTD simulation involves both propagating and Normal FDTD simulation involves both propagating and evanescent waves. evanescent waves.

Difficult to differentiate effects from propagating and evanesceDifficult to differentiate effects from propagating and evanescent wave. nt wave.

Difficult to study the dependence of the behavior of evanescentDifficult to study the dependence of the behavior of evanescent wave on wave on different parameters. different parameters.

How to generate How to generate purepure evanescent wave ?evanescent wave ?

Total internal reflectionTotal internal reflection

Guided mode of a planar dielectric Guided mode of a planar dielectric waveguildwaveguild

“Periodic” boundary condition“Periodic” boundary condition

The sample system

x-axis

air airLHM

L/2 L/2L

plasmonicplasmonic dispersiondispersion

zz--polarized wave propagates in polarized wave propagates in the the xx--directiondirection

periodic boundary conditions are applied in the transverse periodic boundary conditions are applied in the transverse yy--direction direction

( for ( for evanescentevanescent waves)waves)

absorbing boundary conditions are applied at both ends of the absorbing boundary conditions are applied at both ends of the xx--directiondirection

FDTD details

Yee cell with leapfrog staggered Yee cell with leapfrog staggered E and and H sublatticesublattice

spatial gridspatial grid

total simulation spacetotal simulation space

time steptime step

monochromatic EM source [monochromatic EM source [ZiolkovskiZiolkovski & & HeymanHeyman, Phys. Rev. E 64, 056625 (2001), Phys. Rev. E 64, 056625 (2001)]]

The The dispersive dispersive ε and and µ are handled in time domain using the are handled in time domain using the piecewisepiecewise--linear recursive convolution (PLRC) methodlinear recursive convolution (PLRC) method

Dynamic features

0 30 60 90 120 150 1800.0

0.5

1.0

L = 80 ∆x ~ 0.14 λ0, ky = 3k0

γ = 0.005 γ = 0.01 γ = 0.02 γ = 0.05 source

Ampl

itude

of E

z at t

he im

age

plan

e

t / T

small absorption (or large transverse k) small absorption (or large transverse k) long relaxation time to stable statelong relaxation time to stable state

Amplification of evanescent waves

1961 2001 2041 2081 2121

1.0

2.0

0.5

(a)� = 0.001

� = 0.005

� = 0.01

� = 0.05

� = 0.1

� = 0.2

Am

plit

ude

ofE

z

x / �x

1E-3 0.01 0.1

0.5

1.0

1.5

2.0(b)

1st surface

2nd surface

Am

plit

ude

ofE

z

�

1921 2001 2081 2161 22410.1

1.0

(a)

� = 0.001

� = 0.005

� = 0.01

� = 0.05

� = 0.1

� = 0.2

Am

plit

ude

ofE

z

x / �x

1E-3 0.01 0.10.0

1.0

2.0

3.0

4.0 (b)

1st surface

2nd surface

Am

plit

ude

ofE

z

�

Surface polariton

)(2 ωε

)(2 ωµ1ε

1µ

RHMRHM LHMLHM

)(2 ωε

)(2 ωµ

)(2 ωε

)(2 ωµ1ε

1µ1ε

1µ

RHMRHM LHMLHM RHMRHM LHMLHM RHMRHMRHMRHM LHMLHM RHMRHM

Evanescent wave from nearEvanescent wave from near--field sourcefield source

Surface Surface polariton polariton at the 2at the 2ndnd interfaceinterface

Surface Surface polariton polariton at the 1at the 1stst interfaceinterface

The 2The 2ndnd SP is stronger than the 1SP is stronger than the 1stst SP SP Amplification !Amplification !

Physical model

Forced vibration and resonance of coupled oscillatorsForced vibration and resonance of coupled oscillators

Physical model vs. numerical results

1961 2001 2041 2081 2121

1.0

2.0

0.5

(a)� = 0.001

� = 0.005

� = 0.01

� = 0.05

� = 0.1

� = 0.2

Am

plit

ude

ofE

z

x / �x

1E-3 0.01 0.1

0.5

1.0

1.5

2.0(b)

1st surface

2nd surface

Am

plit

ude

ofE

z

�

1921 2001 2081 2161 22410.1

1.0

(a)

� = 0.001

� = 0.005

� = 0.01

� = 0.05

� = 0.1

� = 0.2A

mplit

ude

ofE

z

x / �x

1E-3 0.01 0.10.0

1.0

2.0

3.0

4.0 (b)

1st surface

2nd surface

Am

plit

ude

ofE

z

�

Resolution enhancement

1 2 3 4 5 6

0.1

1

L = 80�x

� = 0.005

� = 0.01

� = 0.02

� = 0.05

|�|

ky/ k

0

Dependence on Dependence on γ

2 4 6 8 10

0.1

1

� = 0.01

L = 40�x

L = 80�x

L = 160�x

|�|

ky/ k

0

Dependence on Dependence on L

Settle the discrepancies

ZiolkovskiZiolkovski & & HeymanHeyman [[PREPRE 6464, 056625 (2001)] , 056625 (2001)]

No stable image observed No stable image observed ((dipersive dipersive nature of LHM?)nature of LHM?)

1.1. LosslessLossless

2.2. γγ ~ 0.001~ 0.001

Loschialpo Loschialpo et al.et al. [[PREPRE 6767, 025602 (2003)] , 025602 (2003)]

stable image but no resolution enhancement observedstable image but no resolution enhancement observed

L ~ 3.2 L ~ 3.2 λλ00

Summary

Amplification of evanescent waves can be realized in LHM slab, Amplification of evanescent waves can be realized in LHM slab, through excitation of coupled surface through excitation of coupled surface polaritonspolaritons..

Stringent constraints apply for the amplification of evanescentStringent constraints apply for the amplification of evanescent waves. waves. Only evanescent waves with limited transverse wave numbers can bOnly evanescent waves with limited transverse wave numbers can be e amplified in amplified in lossylossy LHM slabs of finite width.LHM slabs of finite width.

Enhanced resolution can be achieved by a LHM Enhanced resolution can be achieved by a LHM superlenssuperlens. . The The enhancement is also limited by absorption and finite width of thenhancement is also limited by absorption and finite width of the LHM e LHM slab.slab.

Stable image can’t be obtained in the ideal Stable image can’t be obtained in the ideal losslesslossless case, so that case, so that “perfect” lens is not realizable.“perfect” lens is not realizable.

X.S. Rao and C.K. Ong, Phys. Rev. B 68, 113103 (2003); Phys. Rev. E 68, 0676xx (2003).