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Plasmonics

Femius KoenderinkCenter for Nanophotonics

AMOLF, Amsterdam

Plasmon: elementary excitation of a

plasma (gas of free charges)

Plasmonics: nano-scale optics done with

plasmons at metal interfaces

Flavours

M-I-MTaper and wire

Hybrid V-groove Wedge/hybrid

Dispersion

Dispersion relation with loss - very large k, flat dispersion

- also note superluminal part

1. Can a material response attain arbitrary values?

No – Kramers-Kronig bound mean that Re e brings Im e

2. Extreme confinement appears nice for optical circuits

But pulsed signals disperse – phase, group & energy velocity

Kramers-Kronig

2013 / 5

Can a material have arbitrary real and imaginary e ?

No: material response functions are constrained by

causality

Frequency domain

Time domainNote how convolution turns to product

Physics: no response before cause

Kramers Kronig

Either you have non-dispersive vacuum e=1, i.e., c=0, or

- A window of real c implies a window of absorption

- Real c(w) >0 means c(w) < 0 at other w (to avoid gain)

- “No dispersion” but a refractive index not 1 is impossible

Considerations hold for any physical response function

Typical solids

Absorption bands close to

intrinsic resonances

Real n to the red

also outside absorption

Most materials have ’normal

dispersion’, i.e.,

goes up with energy

is higher towards the blue

is higher towards short

Until you go through an absorption resonance

1. Can a material response attain arbitrary values?

No – Kramers-Kronig bound mean that Re e brings Im e

2. Extreme confinement appears nice for optical circuits

But pulsed signals disperse – phase, group & energy velocity

Dispersion relation in a Lorentzian gas

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Near resonance

(g ~ 0.05 w0)

- Strongly dispersive

- Very low vg

- Superluminal phase velocity

- Negative & superluminal vg

Phase velocity

A front of constant phase follows

Phase velocity

Suppose now I have a wave packet

e.g.

Note that at x=0, this is just a pulse

Group velocity

Maybe you have dispersion

The pulse envelope has a velocity

known as the group velocity

Indices

Group index [blue curve]

very different

from phase index [orange]

Negative vg

Superluminal vg, vf

Region of strong absorption – Kramers-Kronig in action

Movie – dispersive propagation

Strongly dispersive,

weakly absorbing

wc~ 0.85 w0

(here loss length > 50l)

Pulse envelope tracks vg

(Note phase fronts walk faster)

Strongly dispersive,

weakly absorbing

(here loss length > 50l)

Pulse envelope tracks vg

Superluminal velocities

Strongly dispersive,

strongly absorbing

wc~ 1.03 w0

vf=1.25c, vgroup=1.7c

(at carrier frequency)

Note how the packet

- barely moves

- vg >c not apparent

- package break up

Loss length ~ 3 l

Slow group v, superluminal phase v

Superluminal phase

Slow group velocity

Weak abosrption

wc~ 1.1 w0

vf=1.15c, vgroup=c/2

(at carrier frequency)

Note how the group

velocity has regained

meaning

Loss length ~ 20 l

Take home messages

• Phase velocity describes phase front propagation

You get it from the ratio of w/c and |k|

• For weak absorption, group velocity describes Gaussian envelope

• Strong dispersion also entails strong absorption

• Superluminal phase and group velocities are common

• These are irrelevant to describe pulse-energy propagation

• Pulse break up (group velocity dispersion)

• Strong attenuation

Plasmonics is strongly dispersive and at superluminality paradox

Confinement, dispersion and absorption are all linked

Pros and cons

Signals suffer from:

Absorption

Pulse dispersion

Stopped/slow light:

Enhanced |E|2

Longer time to excite matter

Pros and cons

Signals suffer from:

Absorption

Pulse dispersion

Light-matter interaction

Longer time to excite matter

Enhanced |E|2

Metamaterials

Femius Koenderink

Center for NanophotonicsFOM Institute AMOLFAmsterdam

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Thought experiment

Damped solutions Propagating waves

Damped solutionsPropagating waves

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What is special about e<0, m<0

Veselago (1968, Russian only) / Pendry (2000)

Conventional choice:

If e<0, m<0, one should choose:

Propagating waves with `Negative index of refraction’

Snell’s law with negative index

Does ‘negative index’ mean negative refraction of rays ?

S1S2

Povray raytrace of Snells law

Energy flow and k

If both e = m = -1, plane wave:

(1) k, E, B

E

B

Phase frontsto the right

(2) Poynting vector sets energy flow S

k

HSEnergy flowto the left

means

Snell’s law

Exactly what does negativerefraction mean ??

(1) k|| is conserved

(2) Energy flows away from the interface

(3) Phase advances towards the interface

(4) Snell’s law for rays holds withnegative refractive index

kin

k|| k

Energy flux

n=1 n=-1

Refraction movies

Positive refraction

n=1 n=2

Negative refraction

n=1 n=-1

W.J. Schaich, Indiania

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Negative index lens

NIM slab

A flat n=-1Negative index slabfocuses light

The image is uprightThe lens position is irrelevantObject-image spacing is 2d

Perfect lens

Claim: The negative index slab creates a perfect image by ‘amplifying’ the evanescent field via surface modes

Surface modes

Does amplification violateenergy conservation ?

1). Evanescent wave has noflux along z

2) n=-1 is only possible as a resonant effectneeds time to build up

More bizarre optics

`Transformation optics’ - bend rays in space smoothly

Coordinate distortion is equivalent to transforming e & m

Maxwell equationsmap onto Maxwell whencoordinates are stretched

Transformation + itsderivatives set new

e, m tensor

Again: Pendry (Science, 2005)

Conformal mapping

A transformation that locally preserves geometry, in particular angles

Area / volume is not conservedDeformation metric yields e and m

A more relaxed version:“quasi conformal”

Cloak

Merit of the idea - a mathematical receipe to convertyour desired field into a required e(r) & m(r)

On paper: perfect cloaks in space, perfect cloaks in time, perfect lenses, perfect …

Problems

• None of this works without magnetism

• The receipe provides receipes for graded anisotropic e and m - impossible to make

• Fundamentally the ideas are narrowband

• Fundamentally absorption is strong

How e,m come about

Conventional material `Meta material’

Artificial ‘atoms’Magnetic polarizabilityForm effective medium

The idea of an effective mediumA complicated heterogeneous system can have effective homogeneous medium properties.Electrical resistivity, thermal conductivity, mechanical strength, diffusivity, viscosity, sound velocity, refractive index, e, m

inclusions 2 networks

Mixing rule depends ongeometry, topology,coupling parameters, …

‘Bruggeman’

‘Lorentz’

Length scales for waves

l/a1 Photonic crystals Gratings

& diffraction

10 Metamaterials

Conventional materials

1000

Geometrical ray optics

Classical optics & microscopy

0.1

`Homogenizable media’

200 x 200 nm gold, 30 nm highReported m = -0.25

cm-sized printed circuit boardmicrowave negative m

l

How does a single SRR work ?

Faraday: flux change sets up a voltage over a loop

Ohm’s law: current depending on impedance

Resonance when |Z| is minimum (or 0)

Circulating current I has a magnetic dipole moment

(pointing out of the loop)

Pioneering metamaterial

Copper SRR, 0.7 cm size1 cm pitch lattice, l=2.5 cm

cm-sized printed circuit boardmicrowave negative m

First demonstration of negative refraction

Idea: beam deflection by a negative index wedge has ‘wrong’ sign

Measurement for microwaves(10.2 GHz, or 3 cm wavelength)Shelby, Smith, Schultz, Science 2001

“Carpet cloak”

Relax the number of viewing directions for which the

cloak should work

Zentgraf & Zhang

Nature Nanotechn

Wegener carpet cloak

Object: goldbump

Cloak:

3D printed graded polymer

Tested in amplitude and

Interferometrically

Wegener / Karlsruhe

Cloaking solar cell contacts

Karlsruhe group - 2016

Thermal cloak

Heat cloak [copper & PDMS]

Light, elastostatics, elastodynamics

fluid dynamics, diffusion, etc.

From metamaterial to metasurface

Metamaterial:

Designer e and m

perform a function

Metasurface

Designed sheet

Performs a function

Individually tailored scatterers,

controlled amplitude, polarization and phase

Example

A lens provides:

1. Unit transmission over its

surface

2. A phase advance that

increases with r as

f ~ (r/f )2

Na

Metasurface minicamera - Caltech600 nm tall posts

CMOS-flat-lens f=0.7 mm, NA= 0.5570% focusing efficiency

Achromatic [here 850 ± 40 nm]Large F.O.V. and angle tolerance Nat Comm 7 13682 (2016)

Take home messages

• Metals allow very strong confinement

• Kramers-Kronig: non-trivial Re e implies loss

• Very strong confinement, means very strong dispersion

Phase, group and energy velocity

• Maxwell-equations allow negative refraction and cloaks

• Transformation optics – derive required e and m from function

• Effective media can “spoof” unavailable e and m

• Metamaterial idea extends to

mechanics, sound, heat, current,…

• Why spoof a volume, if even a metasurface provides function..