Transcript Photonics and Metamaterials

Class 3. Photonics Crystals and Metamaterials: from
Superlensing to Cloaking (Pendry’s dream)
Dr. Marc Madou
Chancellor’s Professor UC Irvine, 2012
1887
1-D
periodicin
onedirection
1987 2-D
periodicin
twodirections
3-D
periodicin
threedirections
Looking Ahead
 Photonics involves the use of radiant energy, and uses photons the same way that
electronic applications use electrons.
 With photons traveling much faster than electrons, photonics, potentially, offers an
effective solution to faster circuits by implementing optical communication systems
based on optical fibers and photonic circuits.
 Unfortunately traditional optical waveguides involves structures that cannot be made much
smaller than λ/2 because of diffraction.
 Furthermore, waveguiding along bent guides is very lossy . The latter problem is overcome in
photonic bandgap structures. But the dimensions of the structures are still limited by the
wavelength of light.
 The hot new photonic topic that might solve the size and bending problem and that is
plasmonics .
Problems with photonic devices . Bending and diffraction losses.
Looking Ahead
 Surface plasmon–based circuits: Waveguiding of plasmonic excitation in
closely placed metal nanoparticles ( today way too lossy!).
 This class:
 We start this class by categorizing natural crystals and metamaterials,
photonic crystals and geometric optics (a vs. ).
 We then consider in more detail photonic crystals and metamaterials and
explore applications for both.
 We also present a comparison table of electrons with photons.
Contents
From crystals and metamaterials to photonic
crystals and geometric optics: a matter of
repeat size a vs. .
Photonic Crystals.
Photonic Crystal Applications.
Photons vs. Electrons.
Metamaterials.
Metamaterial Applications:
Superlenses
Cloaking.
From Crystals and
Metamaterials to Photonic
Crystals and Geometric Optics
 Material properties are determined by the properties of their
sub-units with their spatial distribution. Electromagnetic
properties as a function of the ratio a the “lattice constant” of
the material/structure and λ the wavelength of the incoming
light (a/) can be organized in three large groups:
1. Natural crystals (violet Tanzanite in the figure further below) and
metamaterials have lattice constants much smaller than the light
wavelengths: a << λ. These materials are treated as homogeneous
media with parameters ε and μ (which are tensors in anisotropic
crystals). They either have a positive refractive index: n > 1, and show
no magnetic response at optical wavelengths (μ = 1) in natural crystals
or they have a negative refractive index in metamaterials (manmade).
From Crystals and
Metamaterials to Photonic
Crystals and Geometric Optics
2. When a is in the same range of the wavelength of the incoming light
one defines a photonic crystal; a meso-scale material with subunits
bigger than atoms but smaller than the EM wavelength. In photonic
crystals a is the distance between repeat units with a different
dielectric constant, so we can still call it “lattice constant”. Photonic
crystals have properties governed by the diffraction of the periodic
structures and may exhibit a bandgap for photons. They typically
are not described well using effective parameters ε and μ and may
be artificial or natural (in the figure that follows we show an artificial
3D photonic crystal and peacock feathers as an example of natural
photonic crystals).
3. When considering macroscopic optic components with a critical
dimension a >>>λ, geometric optics do apply. In this case a might
be the repeat unit of a grating, the aperture of a lens or the length of
a side of an optical triangular prism.
Photonics Crystals
 A breakthrough in photonics in 1987
was the proposal of photonic crystals,
with periodicity of the refractive index n
in two or three dimensions (2D and 3D)
(Sajeev John, Eli Yablonovitch). A 1D
“photonic crystal” had been known since
1887 (a Bragg mirror).
 The dispersion character of light in a
bulk medium with a uniform refractive
index, n was analyzed in class 2 (see also
figure).
 The dispersion curve of a 1D “photonic
crystal” deviates from the straight-line
dispersion curve of a uniform bulk
medium.The region of kz between –π/a
and π/a is called the First Brillouin zone.
'Yablonovite',
Eli Yablonovitch's
first photonic
band-gap structure
Photonics Crystals
 The principle of the 1D “photonic crystal” can be understood as
follows. Each interface between two consecutive materials
contributes a Fresnel reflection.
 For the design wavelength, the optical path length difference
between reflections from subsequent interfaces is half the
wavelength; in addition, the reflection coefficients for the
interfaces have alternating signs. Therefore, all reflected
components from the interfaces interfere constructively, which
results in a strong reflection. The reflectivity achieved is
determined by the number of layer pairs and by the refractive
index contrast between the layer materials.
 A semiconductor cannot support electrons of energy lying in the
electronic band gap. Similarly, a photonic crystal cannot
support photons lying in the photonic band gap. By preventing
or allowing light to propagate through a crystal, light processing
can be performed.
Photonic Crystals
 Before 1987, one-dimensional photonic
crystals in the form of periodic multi-layers
dielectric stacks (such as the Bragg mirror)
were studied extensively. Lord Rayleigh
started their study in 1887, by showing that
such systems have a one-dimensional
photonic band-gap, a spectral range of
large reflectivity, known as a stop-band.
 The 1987 definition of a “photonic crystal”
has two requirements: high index contrast
and a 2D or 3D periodic structure (1D is
often excluded).
Photonic Crystals
 Compared to 1D “photonic crystal”, it is relatively more difficult
to deduce the dispersion curves for 2D and 3D photonic
crystals. The software for doing this is publicly available at:
http://www.elec.gla.ac.uk/groups/opto/photoniccrystal/Software/
SoftwareMain.htm.
 Complete photonic band gaps can only be obtained under
favorable circumstances, i.e., with the right structures and
sufficient (threshold) refractive index contrast. The threshold
refractive index contrast for complete band gaps in 3-D
photonic crystals depends on the exact structure, for diamond,
for example, it is 1.87 and for an inverse opal structure it is
2.80.
 Similar to the periodic electron-crystal lattice, one can fabricate
photonic-crystal lattices. The refractive index varies with a
much larger period of around 200 nm.
Photonic Crystals:The First 3d
Bandgap Structure (theoretical)
frequency (c/a)
Overlapping Si spheres
11% gap
X

W
U L
K
for gap at  = 1.55µm,
sphere diameter ~ 330nm
K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
Photonic Crystals
 In spite of successes in the
microwave range (Yablonovite), it
took over a decade to fabricate
photonic crystals that work in the
near-infrared (780-3000 nm)
and visible (450-750 nm)
regions of the spectrum.
 The main challenges were to
find suitable materials and
processing techniques to
fabricate structures that are
about a thousandth the size of
the microwave crystals.That is
where MEMS came in (e.g.,
porous Si)
Photonic Crystals
 Nature is way ahead of the scientists,
the iridescence of various colorful
living creatures, from beetles to
peacock feathers to butterflies, are
due to photonic crystals.
 Unlike pigments, which absorb or
reflect certain frequencies of light as
a result of their chemical composition,
the way that photonic crystals reflect
light is a function of their physical
structure.
 These natural photonic crystals can
lead to shifting shades of iridescent
color that may help some animals
attract mates or establish territories.
waveguide
cavity
Photonic Crystals
 Most proposals for devices that make
use of photonic crystals do not use the
properties of the crystal directly but make
use of defect modes.
 Such a defect is made when the lattice is
changed locally. As a result, light with a
frequency inside the bandgap can now
propagate locally in the crystal, i.e. at the
position of the defect. It is however still
impossible to propagate in the
surrounding photonic crystal material.
 100% transmission at sharp bends is
possible.
Photonic Crystals
 A photonic band gap (PBG)
crystal is a structure that could
manipulate beams of light in the
same way semiconductors
control electric currents.
 In photonic crystals photons with
energy in the energy bandgap
can still move through lattice
defects regions
 Devices remain large though, for
small photonic devices
plasmonics are better. The latter,
at least today, unfortunately
come with very large losses.
Photonic Crystal Applications
 Super refraction: The
prism effect refers to
separation of colors by
refraction through a prism.
 The dispersion or variation
of refractive index with
wavelength, i.e., the
derivative (dn/dλ) of the
refractive index with
wavelength is low.
 In a normal bulk medium
like glass, dispersion is
small, but dn/dλ can be
made unusually large in
photonic crystals.
Photonic Crystal Applications
 Photonic crystal optic fibers are a special
class of 2D photonic crystals where the
dimension of the medium perpendicular to
the crystal plane could be 100s of meters
long. These are known by several names:
photonic crystal fiber, photonic bandgap
fiber, holey fiber, microstructured fiber and
Bragg fiber.
 The central defect (missing hole in the
middle-5 µm across) acts as the fiber's core.
The fiber is about 40 microns across.
 Photonic band gap fiber: guiding light in air.
Bragg fiber using perfect cylindrical dielectric
mirrors (the Omniguide fiber).
Photonic Crystal Applications
 Applications of photonic crystals:
 Better lasers
 Optical insulators
 Perfect dielectric mirrors
 Better LEDs
 Polarizers
 Better optical filters
 Micron size optical benches
 Photonic diodes and transistors
 Jewellery
 Negative refractive index (see also
metamaterials)
 Superlensing
Photonic Crystal
Applications:PBG Laser
 The smallest defect mode laser is
shown ( Axel Scherer, California
Institute of Technology).
 Periodic air holes in high index
material forms a 2D photonic
crystal.
 The center air hole is removed
and forms a resonant cavity. Light
is confined in the cavity.
 Spontaneous emission in the
band gap is prohibitted, but for the
defect mode is enhanced.
 This produces a microlaser with
PBG Defect Laser
Photons vs Electrons
Equation
Applicable particles
distribution
Solution
Generic
Wavelength
Momentum
Electrons
Schrš
d inger
Electrons Fermion s and follow Fermi-Dirac
statistics
Describes allowed energies of electrons
Free space
   0 e ik.rt  e ik.r t  : a scalar
In a medium
wavefunction
Propagation of Electrons affected by
Coulomb Potential
Equation

h
p
 = 
c
mv

Remark
Electron mo mentum p so the wavelength of
electrons is generally < 1 nm.
Equation

p = (h/2Ή)k or k
Remark
Same equation


Photons
Maxw ellΥs equations for light
Photons are bosons and follow BoseEin stein statistics
Describes the allowed frequencies of
light
E  E m e ik.rt  e ik.r t : a
vector field described by E and B
Propagation of Light affected by the
dielectric me dium (refractive index)
 =
h c

p 
Photon momentum p of light is
smaller so wavelength of light is
generally < 1 µm
p = (h/2Ή)k or k
Same equation

Energy
Equation
k
E=
E = pc =(h/2Ή)kc= kc
Remark
Parabolic
Lin ear
The velocity of an electron, v, can be
any value small er than c.
m=9.1x10 -31 kg
Fixed =
2
2me
Speed

Mass

c for electromagnetic
radiation is always 3 108 m/s
m=0
Propagation through
classically forbidden zones
Confinements
0D (no s ize
confinement)
1D
Electron wavefunctions decay exponentially
in forbidden zones
Single crystal-Bulk material
Photons tunnel through classically
forbidden zones. E and B fields
decay exponentially and k-vector is
ima ginary.
Photonic crystals-Bulk m aterial
2D
3D
Band gap-Forbidden energies
Possible states
Charge
Electronic band gap
Spin up and spin down
q=-1.6x 10 -19 C
Photonc band gap
Polarization
q=0
Metamaterials
Metamaterials
 We now delve into the
fascinating new topic of
metamaterials,
manmade structures with a
negative refractive index
 This could make for
perfect
lenses
and
cloaking
and
might
change the photonics
field forever.
Pendry
Metamaterials
 In vacuum and for most materials the
“right-hand rule” relates E, H, and k
because normally μ > 0 and ε > 0. Thy
are right-handed materials (RHM),
 But in 1968, Victor G. Veselago
postulated that if the permittivity ε and
permeability μ of materials are
negative simultaneously, there is no
violation of any fundamental physical
rules and the materials will have
special optical properties. For a
material with µ and negative, the
vectors E and H and k form a left set
of vectors and one refers to them as
Left-Handed Materials (LHMs)
 See also our earlier statements about S
and k.
Metamaterials
 As no naturally occurring material
or compound has ever been
demonstrated with both ε and μ
negative , Veselago (1968)
wondered whether this apparent
asymmetry in material properties
was just happenstance, or

perhaps had a more fundamental
origin.
 From 1996 on strong theoretical
and experimental evidence started
emerging (Pendry) that LeftHanded Materials, now called
metamaterials, could be fabricated
(at least in the rf range but not yet
in the optical range).
n( )    which for  < 0 and  < 0 is
n( )   
Most materials do come with ε > 1 (e.g.,
ε = 12 for Si), but this is not without
exceptions; we know that materials with
a negative ε at optical frequencies
(visible, IR) include metals such as Au,
Ag and Al.
Most materials have µ ≈ 1 and they are
non-magnetic but if µ ≠ 1, one deals
with a magnetic material. With µ >1 we
have a paramagnetic material at hand
and with µ <1, it is diamagnetic and
finally with µ>>1 we define a
ferromagnetic material. Materials with a
negative µ are again the exception,
they include resonant ferromagnetic or
antiferromagnetic systems at
microwave frequencies.
Metamaterials
Metamaterials
 Refraction: The direction of
the refracted light bends
away from the normal to the
interface between two media,
rather than toward the
normal, as in Snell's Law (see
also meta water above)
 First microwave were shown
to exhibit negative refractive
index (at around 10.5 GHz)
on a metamaterial with wire
and spit ring resonators on
the same substrate. Today
we are close to the optical
range.
Superlenses
 Conventional lenses need a wide
aperture NA for good resolution
but even so they are limited in
resolution by the wavelength
employed.
 The contribution to the image
from the far field are limited by
the free space wavelength λ0.
 From θ = 90° (see Abbe ‘limit),
we get a maximum value of the
wavevector kx = k0= ω/c0=2π/λ0
− the shortest wavelength
component of the image. Hence
the resolution R is no better than:
R

2 2c 0

 0
k0

 Contributions of the near field to the
image come from large values of kx
responsible for the finest details in
the source. But in the near field the
familiar ray diagram do not work
since:
 ‘Near field’ light decays
exponentially with distance
away from the source. The
missing components of a
traditional far field image are
thus contained in the near field
and this field decays
exponentially and cannot be
focused in the normal way.

Superlenses
traditional : k z   2c -20  k 2x when since  2c -20  k 2x - -k z is imaginary :
evanescent decay
Pendry lens : with n = -1, transport of energy in the z - direction requires
the z - component of the wavevector to have the opposite sign :
k z    2c -20  k 2x for large angular frequency the evanescent wave now grows
Superlenses
 It is here that Pendry, again, had an amazing insight : he suggested that a
super lens can be made from a flat slab of negative refractive index material
which not only brings rays to a focus but has the capacity to amplify the near
field so that it can contribute to the image thus removing the wavelength
limitation.
 Let us compare imaging with a traditional and a “Vesalago” lens. A negative
refractive index medium (LH) bends light to a negative angle relative to the
surface normal. Light formerly diverging from a point source is set in reverse
and converges back to a point. Released from the medium the light reaches
a focus for a second time. The new Pendry lens based on negative
refraction has unlimited resolution provided that the condition n = −1 is met
exactly. This can happen only at one frequency. So the secret of the new
lens is that it can focus the near field and to do this it must amplify the highly
localised near field to reproduce the correct amplitude at the image. This can
be understood from the Fermat principle that light takes the shortest optical
path between two points as illustrated further below. For a traditional lens
the shortest optical distance between object and image is:
n1d1+n2d2+n1d3=n1d'1+n2d'2+n1d'3
Superlenses
RH
RH
RH
n=1
n=1.3
n=1
 Light enters n > 0 material
 deflection
 Light enters n < 0 material
focusing
 For a perfect lens the
shortest optical distance
between object and image
is zero:
n1d1+n2d2+n1d3=0=
n1d'1+n2d'2+n1d'3
RH
LH RH
n=1
n=-1 n=1
Superlenses
 Both paths converge at the
same point because both
correspond to a minimum. In
the Pendry lens, n2 is negative
and the ray transverses
negative optical space.
 A super-lens prevents image
degradation and beats the
diffraction limit established by
Abbe! However the resonant
nature of the amplification
places sever demands on
materials: they must be very
low loss!
Superlenses
 In other words for a perfect
lens the image is the
object. The evanescent
waves are re-grown in a
negative refractive index
slab and fully recovered at
the image plane as
illustrated in this figure.
Superlenses
 A poor’s man near-field superlens (<1
and =1) has already been demonstrated.
In 2003, Zhang's group at UC Berkeley,
showed that optical evanescent waves
could indeed be enhanced as they passed
through a silver superlens.
 They took this work one step further and
imaged objects as small as 40-nm across
with blue light at 365 nm with their
superlens, which is just 35-nm thick.
With the superlens, using 365 nm
illumination features of a few tens of 10
nm were imaged, clearly breaking Abbe’s
diffraction limit.
Cloaking
 Pendry proposed to build a special material
that wraps around an object and which would
'grab' light heading towards it and make it flow
smoothly around the object rather than strike
it. To an observer the light would appear to
have behaved as if there was nothing there.
 Thus no light strikes the object, nor does the
object cast any shadow. It is completely
invisible.
 Diagram created by Dr. David Schurig. The
cloak deflects microwave beams so they flow
around a "hidden" object inside with little
distortion, making it appear almost as if
nothing were there at all.
Metamaterials
 Today cloak only works in
2D and only with
microwaves of a specific
wavelength.
 Making something invisible
to the human eye would
present a much greater
challenge.
 It is not very likely that this
technology would work in
the visible.
Photonics Micropolis MIT