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).