presentation on metamaterials

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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Outline

IntroductionAims and MotivationWhat are Metamaterials?

BackgroundEarly History plotModern developmentsThe superlens

The projectMethodologySummary



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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.



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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

=



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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)



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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.



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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)



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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.



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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



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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



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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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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



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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



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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



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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 (!)



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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.



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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



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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.



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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)



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MethodologySummary

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.)



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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



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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?



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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?



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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.



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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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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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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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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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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



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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)



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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.


IntroductionAims and MotivationWhat are Metamaterials?

BackgroundEarly History- plotModern developmentsThe superlens

The projectMethodologySummary

presentation on metamaterials

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