Metamaterials & magnetic
scatterers
Femius Koenderink
Center for Nanophotonics
FOM Institute AMOLF
Amsterdam
Contents
1. Metamaterials -what are they & what’s the hype ?
• Maxwell with negative ε and µ
• Perfect lens, negative refraction
• Transformation optics & cloaks
• Effective media
2. Research on magnetism in metamaterials
• Split ring as magnetic polarizable object
• Evidence for magnetism, superlensing,….
• Metasurfaces
Optical response of natural materials
Damped solutions
Free electron plasma
Drude model
Propagating waves
Localized polarizabilities
Lorentz oscillator model
Thought experiment
Damped solutions
Propagating waves
Propagating waves
Damped solutions
What is special about ε<0, µ<0
Veselago (1968, Russian only) / Pendry (2000)
Conventional choice:
If ε<0, µ<0, one should choose:
Propagating waves with `Negative index of refraction’
Snell’s law with negative index
Povray raytrace of Snells law
S1
S2
Does ‘negative index’ mean negative refraction of rays ?
Refraction
k||
n1 n2
n1ω/c
n2ω/c
Generic solution steps:
1) Plane waves in each medium to be matched at boundary
2) Use k|| conservation to find allowed refracted wave vectors
3) Use causality to keep only outgoing waves
4) Match field continuity at boundary to find r and t
Energy flow and k
If both ε = µ = -1, plane wave:
(1) k, E, B
means
(2) Poynting vector sets energy flow S
Energy flow
to the left
S
E H
B
k
Phase fronts
to the right
Snell’s law
n=-1
n=1
Energy flux
kin
k||
k
Exactly what does negative
refraction 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 with
negative refractive index
Refraction movies
Positive refraction
n=1
W.J. Schaich, Indiania
n=2
Negative refraction
n=1
n=-1
Negative index lens
A flat n=-1
Negative index slab
focuses light
NIM slab
The image is upright
The lens position is irrelevant
Object-image spacing is 2d
Conventional lenses
Ray optics:
Image is flipped & sharp
Sharp features (large
Exact wave optics:
Image sharpness limited to λ/2
) don’t reach the lens since
Reflection /transmission of a slab
1
2
3
Reflection /transmission of a slab
n=3.5, 400 nm thick
Typical thin film:
`Etalon’ resonances
Phase increment integer λ/2
Reflection /transmission of a slab
Air slab in air:
- Unit T + phase delay for real θ
- Exponential damping beyond
Reflection /transmission of a slab
Negative index slab
- Negative phase delay
- Amplification beyond sin θ =1
Perfect lens
Claim: The negative index slab creates a perfect image
by ‘amplifying’ the evanescent field via surface modes
Does amplification violate
energy conservation ?
1). Evanescent wave has no
flux along z
Surface modes
2) n=-1 is only possible as
a resonant effect
needs time to build up
Kramers Kronig
Either you have non-dispersive vacuum ε=1, i.e., χ=0, or
- A window of real χ implies a window of absorption
- Real χ(ω) >0 means χ(ω) < 0 at other ω (to avoid gain)
- “No dispersion” but a refractive index not 1 is impossible
Considerations hold for any physical response function
More bizarre optics
`Transformation optics’ - bend rays in space smoothly
Coordinate distortion is equivalent to transforming ε & µ
Maxwell equations
map onto Maxwell when
coordinates are stretched
Transformation + its
derivatives set new
ε, µ tensor
Again: Pendry (Science, 2005)
Conformal mapping
A transformation that locally preserves
geometry, in particular angles
Area / volume is not conserved
Deformation metric yields ε and µ
A more relaxed version:
“quasi conformal”
Cloak
Merit of the idea - a mathematical receipe to convert
your desired field into a required ε(r) & µ(r)
On paper: perfect cloaks in space, perfect cloaks in time,
perfect lenses, perfect …
Perfect cloaking
A perfect cloak
- keeps external radiation out, and internal radiation inside the cloak
- works for any incident wave field
- cloaks the object in near and far field
- leaves no imprint on the phase of scattered light
Min Qiu, KTH Stockholm
Problems
• None of this works without magnetism
• The receipe provides receipes for graded
anisotropic ε and µ - impossible to make
• Fundamentally the ideas are narrowband
• Fundamentally absorption is strong
Narrowband
Superluminal rays
violate causality?
Amplification via surface wave
Resonance
Not if single ω only
Kramers-Kronig implies resonances, and absorption
To date: loss lengths in the visible are < 3 wavelengths
Metamaterials
effective medium, include spoofing magnetism
What became of this idea ?
1. Plasmonic scatterers, that spoof magnetism
2. New ideas for all-dielectric gradient index optics
3. Metasurfaces
Artificially nano-structured 2D films that mimic 3D optical components
by controlling the phase by which E and H are scatterered
How ε,µ come about
Conventional material
`Meta material’
Artificial ‘atoms’
Magnetic polarizability
Form effective medium
The idea of an effective medium
A complicated heterogeneous system can have effective homogeneous
medium properties.
Electrical resistivity, thermal conductivity, mechanical strength, diffusivity,
viscosity, sound velocity, refractive index, ε, µ
Mixing rule depends on
geometry, topology,
coupling parameters, …
‘Bruggeman’
inclusions
2 networks
‘Lorentz’
Length scales for waves
0.1 Geometrical ray optics
Classical optics & microscopy
λ/a
1
Photonic crystals Gratings
& diffraction
10 Metamaterials
1000 Conventional
materials
`Homogenizable media’
λ
Science 2001
Shelby, Smith Schultz
First shrunk to λ=1.5 µm (2005)
Linden Wegener, Giessen
cm-sized printed circuit board
microwave negative µ
200 x 200 nm gold, 30 nm high
Reported µ = -0.25
This sample – Ivana Sersic (AMOLF)
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 size
1 cm pitch lattice, λ=2.5 cm
Science 2001
Shelby, Smith Schultz
Calculation Pendry et al, ‘99
cm-sized printed circuit board
microwave negative µ
Questions
• What about the superlens ?
• What about cloaking ?
• Practical challenges for negative ε and µ
• Conceptual challenges
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
Superlens
Poor mans superlens: plasmon slab (ε<0 only)
Surface modes
Amplify evanscent field
Berkeley: image `Nano’ through 35 nm silver slab in photoresist
Superlens
Object (mask)
2 um scale
AFM of resist
with superlens
AFM of resist
Ag replaced by
PMMA
Atomic Force Microscope to detect sub-λ features in the image
Result: the opaque 35 nm Ag slab makes the image sharper !
Cloaking
2-dimensional experiment at microwave frequencies (λ=3cm)
Cloaked object: metal cylinder
No cloak
Cloak
Schurig et al., Science 2006
“Carpet cloak”
Relax the number of viewing directions for which the
cloak should work
Zentgraf & Zhang
Nature Nanotechn
Thermal cloak
Heat cloak
Light, elastostatics, elastodynamics
fluid dynamics, diffusion, etc.
Cloaking solar cell contacts
Practical challenges
1. Absorption & dispersion
2. Anisotropy
A. Planar arrays
B. Out-of-plane
response
Spatial inhomogeneity
Vector anisotropy
Question:
Can we make 3D isotropic NIM’s ?
Negative µ implies absorption
Current 1/e decay length ~ 4 λ
Split ring spectrum
Higher order resonance
LC – frequency
LC circuit
dI
Q
+ RI + =
−iωµ0 A H z + Ex d
L
dt
C
LC resonator (resonance λ ~ 10 x size, down to 750 nm)
 p   αE
  =  −iα
m 
C
iα C   E 
α H   H 
strong electric dipole
strong magnetic dipole
strong “magneto-electric coupling”
Femius Koenderink – FOM AMOLF
42
Optical activity
Most biomolecules are weakly chiral
Weak magnetic m induced by E
Weak perturbative effect
αC = 10-3 αE and αH = 10-6 αE
 p   αE
  = 
 m   − iα C
iα C  E 
 
α H  H 
Magnetic meta-atoms are hugely cross coupled αC = αE =αH ∼ λ3
Huge optical activity
Huge optical activity
1. Huge optical activity (though no geometrical chirality)
2. Symmetry: an angle a split ring looks like one handed helix turn
3. LC circuit: : iωΗzΑ + Εx d represents handed driving
4. LC bewegingsvergelijking is chiraal
True importance: handedness per building block has 100% contrast
any planar magnetic scatterer is hugely chiral
Sersic, in press PRL 2012
44
Possible 3D materials
Wegener group: split ring bars
Extremely difficult to make
Giessen group: split ring stacks
3D but anisotropic
Metasurfaces
Many optical components use phase from path differences
in 3D structures
- Lenses
- Waveplates use bulk birefringence
- Spiral phase plates to create structured beams,
donut beams, angular momentum beams,...
- Spatial light modulators, adaptive optics, ....
Metasurface - purely 2D technology
Metasurfaces – example 1
Generalization from `grating’ to `arbitrary phase plate’
Flat lens
60-nm thick film acts as lens at λ=1.55 µm
demonstrated r=0.5 mm, f=3cm, NA=0.015 –
designs up to f=0.4 mm, NA=0.77
Metasurfaces – example 2
Spiral phase plate:
Pass a Gaussian beam through a spiral plate
Azimuthal phase increase of a multiple of 2π
Vortex beam, orbital angular momentum beam
Optical traps, information processing
Metasurfaces – example 2
intensity
Pro’s: beam shaping through arbitrary phase
beam steering & splitting through
generalized reflection, refraction, diffraction
polarization control
planar fabrication
Capasso group, Harvard
Con’s: feature sizes
< λ / 100
Operating
λ=5-15 µm
Interferogram
very difficult to scale to visible λ
Pass a Gaussian beam through a spiral plate
Conceptual link to plasmon antenna arrays, etc.
Azimuthal phase increase of a multiple of 2π
Vortex beam, orbital angular momentum beam
Conclusions
Pendry’s metamaterials:
- Effective media with spoof magnetic response
- Transformation optics philosphy – cloaks, lenses
- Fundamentally narrowband, lossy, anisotropic ε,µ
Current developments
- Magnetic, and superchiral scatterers join plasmonics
- New design ideas for GRIN optics in 2D, and 3D
- Metasurfaces as arbitrary 2D shapers to replace 3D optics