plasmonics - amolf.nl · 1 plasmonics femius koenderink center for nanophotonics amolf, amsterdam...
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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..