Modern Optics Introduction

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Modern Optics

Greifswald, SS 2010 Alex Quandt, Universitt Greifswald (D)

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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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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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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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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G if ld SS 2010 Al Q d U i i G if ld (D)


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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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G if ld SS 2010 Al Q dt U i itt G if ld (D)


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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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Greifswald SS 2010 Alex Quandt Universitt Greifswald (D)


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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

Documents

Modern Optics Introduction