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IntroductionBackgroundThe project
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Metamaterial SuperlensesFinite Size Effects
A. McMurray
Electromagnetic Materials Group,University of Exeter
Supervisor: Dr Stavroula Foteinopoulou
November 7, 2011
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Outline
IntroductionAims and MotivationWhat are Metamaterials?
BackgroundEarly History plotModern developmentsThe superlens
The projectMethodologySummary
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
Backup slides
Aims and MotivationWhat are Metamaterials?
Aims and MotivationWhat and why?
I Ideal superlens is infinite in lateral extent (width).
I In practice, any superlens must have a finite lateral
extent.
I This reduces attainable resolution
I We intend to quantify the effect of the finite sizeupon
resolution via analytical calculations.
I Superlenses optical lithographynanoelectronics hard drives,
chemicaldetectors.
Anti-reflective moth-eye coating forsolar cells,
University of Southampton.
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Aims and MotivationWhat are Metamaterials?
Definitions
I The refractive index, n, of a medium is the ratio of thespeed
of light in vacuum to its speed in the medium:
n =cv
=
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Aims and MotivationWhat are Metamaterials?
Definitions
I Poyntings vector, S = 10 (E B)I Derived from Poyntings
theorem:
dWdt
= ddt
V
12
(0E2 +10
B2)d S
10
(E B) dA
I S points in the direction of energy propagation. (i.e.
thedirection of wave propagation)
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Aims and MotivationWhat are Metamaterials?
Definitions
I The wavevector, k is defined as:
|k| = 2pi
k points in the direction normal to the surfaces of
constantphase (i.e. the wavefronts) which is not always the sameas
the direction of wave propagation.
I k S < 0 backward waveI k S > 0 forward wave.
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Aims and MotivationWhat are Metamaterials?
What are Metamaterials?
I Metamaterial artificial material with properties not foundin
nature.
I We are interested in metamaterials with NegativeRefractive
Index.
I Also called NRI materials/ Negative Index Materials (NIM)
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Aims and MotivationWhat are Metamaterials?
What are Metamaterials?I A NRI is achieved by having and be
simultaneously
negative. (i.e. both negative for the same frequency range)I
When , < 0, n = ||.
Figure: CG Images of water in a a) empty glass, b) glass of
water(n = 1.3), c) glass of NRI water (n = 1.3)1
1Dolling G. et al (2006), "Photorealistic images of objects in
effective negative-index materials",
Optics Express 14.
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Aims and MotivationWhat are Metamaterials?
What are Metamaterials?
I Meta-atoms: discrete cells,size < .
I Periodic arrays used to createbulk NRI medium.
I Their small size relative to means the light wave
interactswith them collectively - not asindividual components
Figure: An SRR/wire lattice. Each
resonator is ~1cm.1
1Shadrivov I., 2008, Nonlinear metamaterials: a new degree of
freedom, SPIE
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Aims and MotivationWhat are Metamaterials?
Invisibility cloak
I In 2006, SRRs were used to construct an invisibility
cloak1:
I 2D region, GHz frequencies.
1http://people.ee.duke.edu/~drsmith/gallery.html, Research Group
of David R. Smith, Duke
University
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Early History plotModern developmentsThe superlens
Early HistoryI Victor Veselago, 1967 Left-Handed
Materials(LHMs)I Simultaneously negative and I E,H and k form a
left-handed vector set.
E
k
H
E
kH
RHM: k (E H*) = kS>0 LHM: k (E H*) = kS
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IntroductionBackgroundThe project
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Early History plotModern developmentsThe superlens
Early historyI Although LHMs do not exist in nature, Veselago
was able
to predict many of their properties including:
I Negative refractive indexI Negative group velocityI Reversed
Doppler Effect
(receding sources areblue-shifted)
I Reversed Cherenkov radiation(backward Cherenkov radiationcone)
Cherenkov Radiation,
American Chemical Society
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Early History plotModern developmentsThe superlens
Definitions
I Propagating waves: A eikx ,extend to far field,
conventionalmicroscopy
I Evanescent waves: A ex ,confined to near field (~), carryno
energy, carry minusculedetails of object, total internalreflection
fluorescencemicroscopy.
A
x
A
x
A. McMurray Metamaterial Superlenses
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Early History plotModern developmentsThe superlens
Most dielectrics
Double Negative (DNG)
Metamaterials
Noble Metals(e.g. Ag, Au)
in IR/vis. region
Mu-negative (MNG)
materials
Propagating waves
Propagating waves Evanescent Waves
Evanescent Waves
Quadrant IQuadrant II
Quadrant III Quadrant IV
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Early History plotModern developmentsThe superlens
Modern developments
I Metamaterials largely abandoned for three decades.I 1996, J.B.
Pendry: parallel wires via self-inductanceme mN (factor 104
increase)
I p =
ne20me
I () = 1 2p2
I < p < 0I Without shift, would be large and negative ~GHz
large
R from Fresnel (!)
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Early History plotModern developmentsThe superlens
Modern developments
I In 1999, Pendry also suggested that using SRRs one couldtune
the value of for microwave frequencies.
I Combine the two ideas a lattice of straight wires andSRRs -ve
, at microwave frequencies.
I Demonstrated by D.R. Smith in 2000 and a negative n
wasdirectly confirmed in 2001 by R. A. Shelby.
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Early History plotModern developmentsThe superlens
The ideal superlens
O
I
A B
P
DC
ds
d
di
Amplitude of Evanescent Wave
A0 2A I Ideal superlens has = = 1
I Infinite in lateral extent.
I All rays focused to image point,I.
I All rays pass through point P.
I Perfect lens formula: ds + di = d
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Early History plotModern developmentsThe superlens
The ideal superlens
O
I
A B
P
DC
ds
d
di
Amplitude of Evanescent Wave
A0 2A I Propagating componentspreserve phase.
I Evanescent componentspreserve amplitude.
I Preservation of all components perfect image
reconstruction.
A. McMurray Metamaterial Superlenses
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MethodologySummary
Aims and Motivation RecapWhat and why?
I All practical superlenses must have a finite lateral
extent.
I Affects the possible resolution as smaller extent lesswave
components are focused reduced image quality.
I Also have losses in the medium lower quality.I We intend to
quantify the effect of lateral size upon
resolution via analytical calculations and perhaps alsoconsider
losses.
I This will allow for the optimisation of superlenses
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MethodologySummary
MethodologyHow do we do this?
1. Decompose incident light wave, Ei(r) (Gaussianwaveform), into
separate k-components, Fn(k), via Fouriertransforms.
2. Calculate the transmission functions, tn(k)
3. Calculate the resultant k-components, gn(k) = tn(k)Fn(k)
A. McMurray Metamaterial Superlenses
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MethodologyHow do we do this?
4. Recombine the resultant k-components to obtain thetransmitted
wave, Et(r) via inverse Fourier transforms.Resolution obtained from
FWHM.
5. Repeat for different configurations (e.g. different
lateralsizes, included components etc.)
A. McMurray Metamaterial Superlenses
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MethodologySummary
Summary
I Now possible to construct non-ideal superlenses.
I Images are imperfect finite extent (ratio of ds to
lateralextent), losses in the medium etc.
I Quantify the effect of finite size upon resolution
viaanalytical calculations.
I Begin with only far-field components, non-dispersive
(!),lossless media. Improve as the project develops
I Results should allow superlens design to be optimised,leading
to improvements in many devices
A. McMurray Metamaterial Superlenses
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MethodologySummary
End of talk
I would like to thank my supervisor,Dr. Stavroula
Foteinopoulou,
and my colleague, Alun Daley,for their help and advice.
Any questions?
A. McMurray Metamaterial Superlenses
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IntroductionBackgroundThe project
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Negative refraction (mathematical reasoning)
Negative refraction (graphical reasoning)
Perfect Lens Equation geometrical proof
Requirements for propagating waves
How does the SRR work?
Conservation of phase
Modifications to Fresnels equations
Hyperlens
Why must NIMs be dispersive?
Why dont evanescent waves carry energy?
A. McMurray Metamaterial Superlenses
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Negative refraction (mathematical reasoning)
I An EM-wave propagating in z direction will havecomponents:
I E = E0eikzit x
I H = H0eikzit y
I From wave equation: k = nc =c
I k and hence n must have positive imaginary parts to bephysical
(otherwise amplitude exponentially increases)
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Negative refraction (mathematical reasoning)I n2 = ||||ei(e+m)I
n =
||||e i(e+m)2 +impiIm
Re
Root 1, m=0
Root 2, m=1
I So to get +ve imag. part of n, must have -ve real part.
A. McMurray Metamaterial Superlenses
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Negative refraction (mathematical reasoning)
I This is a simplification as wave may propagate in
-zdirection
I Therefore consider time-averaged Poynting vector1:S = 12Re(E
H) zeIm(k)z
I Therefore Im(k) must be positive for the solution to
bephysical.
I So as before the real part of n must be negative.
1Ziolkowski R. & Heyman E., 2001, Wave propagation in media
having negative permittivity and
permeability, Physical Review E, 64, 056625
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Negative refraction (graphical reasoning)
k-components parallel to the interface are conserved.
LHM
Air
Magnitude of parallel k component
EFS of dielectric
EFS of air
kinc
kref
S
A
B
y
Dr. Stavroula Foteinopoulou
I In refraction, frequencyconserved
I incident k-vector lies onequifrequency surface(EFS) inair
(radius c )
I Refracted k-vector lies on EFSin dielectric (radius |kref | =
|n|c )
I kref could lie on A or B (kmust be conserved)
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Negative refraction (graphical reasoning)
LHM
Air
Magnitude of parallel k component
EFS of dielectric
EFS of air
kinc
kref
S
A
B
y
Dr. Stavroula Foteinopoulou
I LHM k S < 0I Energy flow (direction of S) must
be in causal direction away fromsource (y-direction)
I kref must lie on point A
I direction of energy (and thuswave) propagation S is onopposite
side of normal
I Negative refraction
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Perfect Lens Equation geometrical proof
O
I
A B
P
DC
ds
d
di
AB = OB tan
BP = ABtan
BP = OB, independent of andthus the same for all rays
emittedfrom O.
OP = OB + PB so all rays passthrough point P
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Perfect Lens Equation geometrical proof
O
I
A B
P
DC
ds
d
di
PC = DCtan
DC = IC tan
PC = IE
PC + BP = OB + IC
d = ds + di
The Perfect Lens Formula
A. McMurray Metamaterial Superlenses
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Requirements for propagating wavesI From the EM Wave Equation:
k2E 2c2 E = 0
I k2 = 2c2
I If , both have the same sign then k2 is +ve and k is real
I If their signs differ then k is imaginary
I E = E0eikxeit
I So wave is evanescent ( ekx ) if signs differ (Quadrants IIand
IV)
I If the signs are the same it is a propagating wave ( eikx
)(Quadrants I and III)
A. McMurray Metamaterial Superlenses
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How does the SRR work?
I Splits in rings force currents to oscillate in ringcirculating
currents store magnetic energyinductor.
I Large capacitance between smaller oppositelyoriented ring and
outer one.
I Analogous to L-C circuit.I Resonant frequency: 0 = 1LCI = 1
2
220I < 0 for > 0
I. Shadrivov, ANU
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Conservation of phase
O
I
A B
P
DC
ds
d
di
I Perfect lens formula: ds + di = dI ~OA = ~DI = ~AP = ~PDI k =
ncI Optical path for ~OA and ~DI =
(+|n|c ~OA) + (+|n|c ~DI) = 2|n|c ~OAI Optical path for
~AD = |n|c ~AD = 2|n|c ~OAI Total optical path = 0, phase
conserved.
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Fresnel modificationsI NIM: n =
I r = n1 cosn2 cosn1 cos+n2 cos
I Perfect lens (1st interface): n1 = 1, n2 = 1, = I r = 2
cos0
I Exact expression1: r = z2 cosz1 cosz2 cos+z1 cos
I z =
, in air/vacuum , = 1, in NIM , = 1
I 1st interface: r = 02 cos = 0 as expected.
1Veselago V., 2006, Negative Refractive Index Materials, J.
Comput. Theor. Nanosci. 3, 1-30
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Future possibilites - Hyperlens
I The hyperlens enhances the evanescent waves bycoupling them
into propagating waves.
I Near-field evanescent components far field
I obtain sub- resolution without needing to form theimage in the
near-field of the lens
I Applications in microscopy, cellular imaging.
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Future possibilites - Hyperlens
Figure: a)Conventional lens. b) Near-field superlens. c)
Far-fieldsuperlens. d) Hyperlens. The wavy curves are propagating
waves.Smooth curves are evanescent waves.1
1Zhang X. & Liu Z., 2008, Superlenses to overcome the
diffraction limit, Nature Materials 7, 435-441
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Why must NIMs be dispersive?
I The energy density in non-dispersive media is given by:
U =12
( Re(EE) + Re(HH))
I U < 0 if , < 0 unphysicalI If the media is dispersive,
i.e. = (), = (), then:
U =12
(()
Re(EE) +
()
Re(HH)
)I U can be > 0 only if the NIM is dispersive.
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Why dont evanescent waves carry energy?
I In an evanescent wave the E and H components are 90 orpi2 out
of phase
I the time-average of the Poynting vector (i.e.
consideringenergy transfer over a whole cycle) is zero.
S =12
Re(E H) = 0I No energy is transferred.
A. McMurray Metamaterial Superlenses
IntroductionAims and MotivationWhat are Metamaterials?
BackgroundEarly History- plotModern developmentsThe
superlens
The projectMethodologySummary