modern optics introduction
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Modern Optics
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Modern Optics
Alexander QuandtInstitut fr Physik, Uni Greifswald
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A somewhat personal selection :
L. Novotny and B. Hecht, Principles of Nano-Optics, CambridgeUniversity Press (2006).
H. C. Van de Hulst, Light Scattering by Small Particles, Dover (1982).
B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, Wiley (2007).
J. D. Joannopoulos, S. G. Johnson, J. N. Winn and R. D. Meade,Photonic crystals (2nd ed.), Princeton University Press (2008).
M. Born and E. Wolf, Principles of Optics (7th ed.), Cambridge University
Press (1999).
M. A. Silverman, Waves and Grains, Princeton University Press (1998).
J. D. Jackson, Classical Electrodynamics (3rd ed.), Wiley (1999).
Literature
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Introduction
Photons, colors & butterflies
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Wanted- dead or alive !
The photon
Description:
-Zero mass.
-Spin 1, but only two directions of
polarization!
-Tends to disguise
, once as a particle,
once as a wave.
Is responsible for acts oflightand
massive electromagneticexchange.
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Witness 1:
According to the assumption to be contemplated here, when a light
rayis spreading from a point, the energyis not distributed
continuouslyover ever-increasing spaces, but consists of a finite
number of energy quanta that are localizedin points in space, move
without dividing, and can be absorbedorgenerated onlyas a whole.
(1905)
Would it not be possible to replace the hypothesis of light quanta by
another assumption that would also fitthe known phenomena? If it is
necessary to modifythe elements of the theory, would it not bepossible to retain at least the equations for thepropagation of
radiation and conceive onlythe elementaryprocessess ofemission
andabsorption differentlythan they have been until now?(1909)
(A. Einstein)
Witnesses (see: Mead)
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Witnesses (see: Mead)
Witness 2:
It is generallyassumedthat a radiating body emits light in every
direction, quite regardless of whether there are nearordistant objects
which may ultimatelyabsorb that light; in other words that itradiates
into space.
I am going to make the contrary assumption that an atom never emits
lightexceptto another atom
I propose to eliminate the idea ofmere emission of light and
substitute the idea oftransmission, or a process ofexchange of
energybetween two definite atoms both atoms must playcoordinate andsymmetricalparts in the process of exchange
(1926)
(G. N. Lewis, who coined the word photon)
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Photons finally caught ?
From : Nature 433, p. 230-238, Jan. 2005.
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A classical experiment (Taylor)See: Proc. Cambridge Phil. Soc. 15, 114-115 (1909)
Fundamental question:
Does a particle-likephoton still produceinterference patterns,
thus acting like a wave ?
Description ofexperiment:
Produce diffraction pattern of a needle using light from a gasflame, and record results on a photographic plate.
Gradually dim the gas light by inserting a series ofopaqueplates, until one single photon at a time should hit thephotographic plate, only.
Extreme case corresponds to standard candle more than onemile away, and an exposure time of roughly three months.
Result: no change in the interference patterns, even forlongest exposures !
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A modern experiment (Grangier, Roger, Aspect)
See: Europhys. Lett. 1, 173-179 (1986)
S1 T2
R
2
Sourceproduces two
correlatedphotons. 50/50beam
splitter.
Detects reflected
photon 2.
Detects transmitted
photon 2.
Detects a leftgoing
photon 1, and thus
signals the emission of a
rightgoing photon 2.
1. Semiclassical theory (quantized atoms and classical electromagneticfields): R2 and T2 should occasionally detect in coincidence, becausedetection probability is proportional to square of field amplitudes.
2. Quantum mechanical theory (quantized atoms and photons): R2 or T2measure position of photon 2. Thus they should neverdetected incoincidence. This was actually seen by Grangier, Roger and Aspect.
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De Broglie Waves
Electrons as matter waves:
(L. De Broglie, Nobel Lectures in Physics
1922-1941, p.239-259).
Louis De Broglie
Free electron theory:(Collective model characterized byFermi energyEFand
Fermi wavenumberkF)Fermi wavelengthlF amounts toseveral , only.
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Einsteins propsal : A similar theory of light
Photonic plane wave:
(Leads to properinterpretation ofPlancks theoryofblackbody radiation).
Cavity resonator(collective photon system):(Closed[rectangular]boxof mirrors containinglight. The latter comprises of a set of
modes, each of them containing an integral numbern ofphotons. Characteristics of
each mode are assignedto thephoton).
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Whispering Gallery Modes
Acoustic paradigms:
Photonic analogues:
Microdisk whisperinggallery modes:
Tamboli et al., Nature
Photonics 1, 61 (2007).
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Re-discovering Planck: Thermal light
The model:System ofphotons and atoms at thermal equilibrium. Interactionsthrough absorption, spontaneous emission and stimulated emission.
System contained within cavity at temperature T.
Atoms are sitting in the walls of the cavity. They are described by two-level systems, where the levels 1 and 2 are separated by an energy hn.
Blackbody system, able to absorb all of the incoming light (continuum).
T
nav
N1(t )
N2(t )
E1
E2
hn
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Spontaneous emission:
Absorption:
Stimulated emission:
Resulting rate equation (ignoring nonradiative processes) :
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Add missing thermodynamical ingredients:
Average energy of radiation mode:
Spectral energy density:
Atoms attemperature T are in thermal equilibrium, and their levelpopulations are thus Boltzmann distributed.
Blackbody radiation spectrum
r(n)
n
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Natural black body radiation
Oursun:
Cosmic microwave background:
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Radiative transfer between atoms (see: Mead)
Atoms as two-level systems:
The model:
E1
E2
Two atoms act like small dipole oscillators, and energy is radiativelytransferred between them.
Starting configuration:
Atom Iinitially in state 2, but slightlyperturbed:
Atom IIinitially in state 1, with matchingperturbation:
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Oscillating dipole moments:
Hellman-Feynman theorem:
Basic behaviourof this model system:Radiative couplingdecreases energy offirstatom (AI increases and BIdecreases). On the other hand, itincreases energy ofsecondatom (AIIdecreases and BII increases).
The rate of energy loss (or energy gain) is supposed to beproportional
to the square of the oscillating amplitude dAIBI(or dAIIBII).
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Time evolution of model system:
Solutions:
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Observations:
Comment:
Precondi t ions: At the beginning, someperturbationputs both atoms ina mixedstate with exactlythe same difference ofenergies and exactly
the right phase.
Self reinfo rced: transferred energyincreases minoritystate, thus
increasingthe dipole moment, thus reinforcingthe coupling ...
Cont inuou s exchangeof a photon! (see below)
Are there quantum jumps ? (E. Schrdinger)
There are no quantum jumps, norare there particles ! (Manifesto of H. D. Zeh)
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Semiclassical Model (see: Scully & Sargant, Physics Today March 1972)
General scheme:
Accounts for:
atom-fieldinteractions
stimulatedemission
resonancefluorescence
photoelectriceffect (!!!)
.
Shortcomings:
This is the original scan
Spontaneous emission:
No dipole associated with pure quantum state, thus no decay ofexcited states.
We need to add vacuum fluctuations (i.e. noise !!!).
M d O i
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For Heretics
W. E. Lamb, Jr., Anti-Photon, Appl. Phys. B 60, 77-84 (1995)
at the first of the 1960s Rochester Coherence Conferences, I
suggested that a license be required foruse of the word photon, and
offered to give such a license toproperly qualified people. My records
show thatnobodyworking in Rochester, and very few other people
elsewhere, ever took out a license to use the word photon
Professors
M d O ti
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Nano-Optics in a Nutshell (see: Novotny/Hecht)
About the resolution limit (i.e. Rayleight limit) :
Breaking the resolution limit:
M d O ti
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Problem:
Solution (Nano-Optics):
Introduce matter, such that unphysical solutions are sorted out, andreplaced by physical solutions, due to suitable boundary conditions.
Problematic vacuum case becomes irrelevant.
Optical antenna, see http://www.optics.rochester.edu/workgroups/novotny/antenna.html
M d O ti
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J. W. von Goethe & I. Newton
As to what I have done as apoet I take
no pride in it but that in my century I am
the only person who knows the truth in the
difficultscience ofcolours of that, I say, I
am not a little proud, and here I have a
consciousness of a superiorityto many.
Newtons color circlewith seven sections
proportial to diatonic
musical scale. Mixed
color zobtained through
azimuthal center of
gravitycalculation.
Radial positiondetermines saturation.
See:
N. Ribe and F. Steinle,
Physics Today July2002, p. 43.
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Sunlight and Colors
400
nm
450 500 550 600 650 700
eV3 2.75 2.5 2.25 2.0 1.77
blue
violet
green
yellow
red
Visible ranges:
Wavelength: 400-700 nm
Energy: 3.1-1.77 eV
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Rainbows and the deep blue sky
Rainbow
over
Newtonsbirthplace.
Sketches of a
rainbowfrom
Newtons Opticks.
Observations:
(1) Primary bow (violet inside, red outside).
(2) A faint and inverted secondary bow.
(3) Darkened area between bows (Alexanders band)
(4) Interference patterns.
(5) Polarization (use polarization filters).
(1)
(2)
(3)
(4)
???
???Why is the
sky blue, and
why are the
sunsets red???
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Exploratory experimentation
Experiment 1 (Prism, dispersive refraction):
Experiment 2 (Domestic rainbow):
Experiment 3 (Milk):
The rainbow and the blue sky must be caused by the dispersive
scattering ofdaylight on waterdroplets, dust, molecules .
Red
lightBlue
light
White
light
Rainbows can also be observed on a sunny day, for example while finally cleaning
your caroutside with apressurized watergun, or aroundgarden sprinklers, or
Take milkfresh from the cow,
anddilute it with water inside
a large glass container.
Liquid
appears
blueish.
And reddish
on direct
transmission.
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A geometrical water droplet model
Reflections anddispersive refractions:
Primary rainbow consistsofclass 3 rays thatundergo one internalreflection.
Secondary rainbowconsists ofclass 4 raysthat undergo two internalreflections.
Higher orders illuminatedarkened area betweenbows (black otherwise).
Impactparameter
Incident ray
Class 1
Class 2
Class 3(primary r.)
Class 4
(secondary r.)
Higher orders
Water droplet
a
Examine diffusion angle a as a function ofimpact parameterfor rays ofclass 3 and class 4 (explains origin ofdarkened area between bows):
- With increasing impact parameter, the diffusion angle ofclass 3 decreases from180, goes through a minimum around 138, and increases again.
- Similarly, class 4 increases from 0 and goes through a maximum at 130.
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Colors of the rainbow and interference effects
Generation ofcolors:
Interference (Young):
Droplet is uniformlyilluminated.
Maximum intensityin a region where
diffusion angle varies slowestwith
impact parameter (i.e. extrema).
The so-calledrainbow rayis a class 3ray that is scatteredat the minimum
diffuse angle (i.e. rainbow angle).
Dispersive refraction:
Rainbow angle 13758 for
redlight.
Rainbow angle 13943 forvioletlight.
Rainbow rayParallel rays
Rays ofclass 3 with impact parameters
slightly higherandlowerthan the rainbowray
may re-appear under the same diffuse angle.
According to Young, thoseparallelrays may
interfere.
Whenever thepath taken by both rays differs
byhalf a wavelength, interference will be
destructive.
Fringes may appear at angles higherthan the
rainbow angle (forsmallerdroplets < 1 mm).
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Polarization of rainbow
Mechanisms ofcomplete polarization (see Silverman):
s-polarization (in plane)
p-polarization (normalto plane)
Incident ray
Scattered ray
90- Rayleighscattering at amolecule.
Brewsterangle:Reflection atBrewstersangle QB.
Reflectionswithin waterdroplets are
close to QB !
QB
Incident ray Reflected ray
Refracted ray
n1
n2
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Mie and Rayleigh scattering (see Silverman)
A general scattering model:
Rayleigh scattering:
Electromagneticwaves offrequencywscatteredby a homogeneous
object have to obey the vector wave (Helmholtz) equation:
There exists an exact, butextremely complexsolution in the case of
scattering by a homogeneous sphere ofradius a, due to Mie andDebye:
Smallsphere immersedin light, no standing waves:
Scatteredintensities mainly from shorterwavelengths (blue
sky); red sunsets due to longer path through atmosphere.
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Atmospheric Optics (Halos, glories and spectres)
Gallery I:
Solar halo andsun dogs Lunar halo andmoon dogs Solar halo at South pole
Glorynear hot springs Gloryseen from an airplane Brocken spectre
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Halos (Ice crystals)
The phenomenon:
(Dispersive) refraction by ice crystals:
Sun dogs (hexagonalplates)
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Greifswald, SS 2010 Alex Quandt, Universitt Greifswald (D)
Ice crystals:
Essential preconditions: Myriads ofsimple crystals like colums and
plates in a cloud, some degree oforder.
Complex crystals have too many facets. They
diffuse light and produce weak halos.
Cloud can neitherbe too thin nortoo thick.
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Greifswald, SS 2010 Alex Quandt, Universitt Greifswald (D)
Glories & Spectres (Surface waves ???)
The phenomenon:
Desert view watch tower (Grand Canyon):
D. M. Black, The Brocken spectre of the
Desert View Watch Tower, Grand Canyon,
Arizona, Science 119, 164 (1954).
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Greifswald, SS 2010 Alex Quandt, Universitt Greifswald (D)
Explanation forglories (van de Hulst):
Rainbow
Glory
Back scattering (180) ofsunlight from small (< 35 mm) droplets of water.
Nussenzveig (2003):The gloryprovides direct
and visually stunning
experimental evidence of
the importance of
resonances andlight
tunnelingin clouds.
See also:
P. Laven, How are
glories formed ?, Appl.
Optics 44, 5675 (2005).
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Gallery II:
Beware:
The Grey Man,
the Brocken Gespenst
its real!
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The Brocken spectre at night:
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Basic mechanisms that generate colorsSee: K. Nassau, The Causes of Colors,SCIAM 243, 124-154 (1980)
Electronic excitations Flames, arc discharges, lasers.
Vibrations Water: Ocean blue, blue ice.
Transition metal compounds Pigments of painting colors, lasers.
Transition metal impurities Ruby, emerald, lasers.
Color centers Amethyst, smoky quartz.
Charge transfer Sapphire, magnetite.
Conjugate bonding Organic colors, dye laser.
Metals Copper, iron, gold, silver.
Pure semiconductors Silicon, diamond.
Doped semiconductors Blue and yellow diamond, sc lasers.
Dispersive refraction Rainbow, chromatic aberration.
Diffusion Blue sky, red mountains.
Interference Colors of insects, benzine on water.
Diffraction grids Opal, liquid crystals, colors of insects.
Electronic transitions in
free atoms; molecular
vibrations.
Crystal field splittings;
fluorescence effects.
Transitions between
molecular orbitals.
Transitions between
bands.
Geometrical optics,
physics.
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Interaction of light with atoms/solids
Atoms and molecules:
(a) (b) (c)
Basic mechanism:
-Electrons occupydiscrete levels.
-Photon may be absorbed, if it has just
the rightenergy to liftelectron into an
upperlevel, see (a),(b).
-Electron decays back to originallevel,
emittingaphoton with energyequal to
electroniclevel spacing, see (c).
Remember: visual range between 1.77 eV (red) and 3.1 eV (violet).
Unfortunately, most atoms and moleculesinteract with light in the ultraviolet range!
Absorption in semiconductors:
E
lectronenergies
E
lectronenergies
Absorption in metals:
Valence band
(full)
Conductance
band (empty)
Fermi levelGap Fermi level
Full
Empty
Photon energies.
0 eV
1.77 eV
3.1 eV
visible
black
no color
After excitation, energy isdissipated via phonons etc.
Color
depends
on gap
location
Highreflectance:
photons ofall
visible energies
are absorbed
and re-emitted
immediately.
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An oscillator model of light-matter interactionsSee: V. Weisskopf, How light interacts with matter, SCIAM 219, 60-71 (1968)
Damped oscillating particle interacting with external field:
Dielectric medium described by polarization density:
Resonant dielectric medium interacting with electric field:
Assume periodic excitations and considerreal parts:
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Real and imaginary parts ofsusceptibility:
nn0
c0
c(n)
nn0
-c(n)
Interpretation:
Realistic modelling (Jackson):
n> n0: response smalland180phase shift.
n= n0: maximum response, and90phase shift.
damped and driven
harmonic oscillator
Attenuation ofmonochromatic light:
Several electronsper atom. Determine w0, s, ... quantum mechanically.
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One minute of Zen - Basho (1644-1694)
Lady butterfly
perfumes her wings
by floating
Over the orchid.
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Beauty of Nature Butterflies, Birds and FishesSee: P. Vukusic and J. R. Sambles, Photonic structures in biology, Nature 424,853 (2003).
For a systemat icreverse eng ineeringof Moth er Nature read: A. R. Parker and H. E.
Townley, Biomimetics of photonic nanostructures, Nature Nanotechology 2, 348 (2007).
A showcase:
Corresponding microstructures:
Wing of butterfly Peacock Koi fish
Wing: Discrete multilayers
of chitin cuticle and air.
Feather: Melanin rods
and airholes inside
kreatin matrix.Scale: Multilayer stackingof
guanine crystals and cytoplasm.
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Beauty of Nature Opals
Anothershowcase:
The microstructures of opal:
Opal: hydrated silica.
Mineraloidgel.
Gems: fcc closed
packedmicrospheres.
Also finalstage of
fossilation.
Structure of a opalgem.Closeup oflattice made of 150-
300 nm spheres and airholes. Closeup ofnear gem opal.
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Reflections, Refractions (and Metamaterials)
Refraction at interfaces:
For metamaterials see: V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
aI
aR aT
aT
kTkR
kIkT
Components of wavefunctions at interface :
Most general matching conditions :Arguments of wavefunctions have to be equal atall times tand at every point r0of the interface.
nl nr
Asymmetry in materials properties ?
Refraction along black path corresponds to an interface between two materials with positiveindex of refraction. The red path proceeds through a material with negative index of refraction(metamaterial).
Spectacularproperties ofmetamaterials: near field amplification, optical antimatter ...
Realization ofmetamaterials: microwave arrays, photonic crystals
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X-ray Bragg diffractionSee: J. B. Pendry, Photonic Gap Materials, Current Science 76, 1311-1316 (1999).
d
Q
Planes ofatoms withina crystal are able to actlike mirrors and reflectX-rays.
Braggs law:
X-rays shine on (atomic) Braggplanes with mutualdistance d andangle of incidence Q.
Bragg condition: Rays from parallel planes are able to
interfere constructively, if the difference in opticalpathlength (red) is an integermultiple of the wavelength l:
As indicated, Braggs lawusually holds over a wholerange of angles 2df, depending on the details of the atomic
scatteringprocess (Bubble model). There are manypossible Braggplanes for a given crystal.
Once that crystal is rotated, those planes willglint, as soon
as the Bragg condition is met.
However, crystals rejectX-rays onlyfor a limitedrange of
angles of incidence (see illustration on the left).
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Colors due to Bragg diffractionSee: J. B. Pendry, Photonic Gap Materials, Current Science 76, 1311-1316 (1999).
Making colors:
In the visible range, the essentialprecondition for Bragg
scattering is that the crystalhas to be made ofcomponents on
the scale of the wavelength of light.
Photonic insulator(photonic crystal): scattering is so strong,
that the ranges ofrejection angles fordifferentBragg planeswilloverlap completely (see illustration on the left).
Photons within a forbidden bandof energies will be rejectedin
whatever direction they will enter the photonic crystal.
Sketch of basicBragg
scatteringprocess foropal.
Opalstructure as aparadigm for
an imperfectphotonic insulator.
Iridescence due to Braggscattered
photons ofdifferent wavelength.
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Case study a weevilSee: A. R. Parker et. al., Opal analogue discovered in a weevil, Nature 426, 787 (2003).
Description:
-Australian weevil pachyrhynchus argus (a) living in the forests of
Queensland.
- Visible from all directions due to 3Dphotonic structure analogous
to opal.
- Metallic colormediated byscales (b) of 0.1 mm diameter.
- Interiorofscales (c) consists of a hexagonal close packedarrays of
transparentmicrospheres with 250 nm diameterD, embedded into
chitinous exoskeleton matrix.
- Sub-wavelenth arrays of microspheres act as 3D diffraction
gratings. Light will be Bragg reflectedatlayers of microspheres.
- But first, lightcrosses airwith index of refraction n = 1. Then itenters a refractive media, made ofmicrospheres with ns= 1.56, and
an exoskeleton matrixwith nm= 1.33.
- Spectrum (d) measured forangle of incidence f= 20 normaltosurface. Wavelength of maximum reflectance lmax= 530 nm.
- Theoretical task: Explain these results, andpredictlmaxat different
angles of incidence f.
Metallicappearance due to
embedded3D photonic structures.
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Predicting the colors of a weevilSee: A. R. Parker et. al., Opal analogue discovered in a weevil, Nature 426, 787 (2003).
d
Q
f ni
neff
Braggs law including refraction
Pachyrhynchus argus
Substitute spheres by point scatterers, and substitute spheresand chitin matrix by an effective dielectric medium, f.e. :
Determine distance between diffracting hexagonal layers:
Bragg reflections and colors:
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Yablonovite: (Yablonovitch et. al., PRL 67, 2295-2298 and 3380-3383 (1991))
-A slab of material with refractive index 3.6or higher is covered by a maskconsisting of a triangulararray ofholes.
- Each hole is drilledthrough three times, at an angle 35.26 away from
normal, andspread out120 on the azimuth.
-The resultingcriss-cross of holes below the surface of the slab produces a
3D periodicfcc structure.
- System size of a fewmillimeters, photonicband gap between 13-16 GHz.
Photonic insulators and YablonoviteSee: E. Yablonovitch, Inhibited Spontaneous Emission in Solid-State Physics and Electronics, PRL 58, 2059
(1987).
Drilling a Yablonovite crystal
Microwave band gap
Whats the matter with photonic insulators ?
- The inside of an idealphotonic insulator would be extremelydark, such that
matches couldnotbe lit, atoms wouldnotdecay, and even zero-point
fluctuations would be suppressed.
-Thepractical utilityof such a material was originallyforeseen by Yablonovitch,
for example in reducingundesiredlosses due to spontaneous emission.- Use scaling laws to design a suitable photonic crystals: If a structure has aband gap atl, anotherstructure with twice its dimensions will have a band gapat 2l. Bulkiermodels may thus be testedin the microwave range etc.
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Concluding remarks
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Let us assume that photons are real.
Visible daylight is but a small range in the 6000 Kblackbody radiation spectrum generated by the sun.
Rainbows, glories, halos and the blue sky are caused
by a rather complex dispersive scattering ofdaylight(Mie scattering theory).
Electrons are usually absorbing daylight in theultraviolet range. But under a large variety offavourable circumstances, they are also able to
generate colors.
Photonic crystals are the reason behind thespectacularcoloring ofinsects, birds, fishes, andopals.
Concluding remarks
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