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OPTICAL PROPERTIES OF OPTICAL PROPERTIES OF METALLIC NANOPARTICLES, METALLIC NANOPARTICLES,

MOLECULES AND POLYMERSMOLECULES AND POLYMERS

Dr. Mica GrujicicDr. Mica Grujicic

April, 2004April, 2004Department of Mechanical EngineeringDepartment of Mechanical Engineering

Mie Theory Mie Theory --Dilute Colloidal Dilute Colloidal Solution LimitSolution Limit

Spherical ParticlesSpherical Particles

Ref: Ref: C. F.C. F.Bohren Bohren and D. R. Huffman, Absorption and D. R. Huffman, Absorption and Scattering of Light by Small Particles, and Scattering of Light by Small Particles,

Wiley: New York, 1983Wiley: New York, 1983..

Extinction CrossExtinction Cross--section of section of Spherical ParticlesSpherical Particles

( ) 22

2/332

224

pmp

pmpext

RC

εεεε

λεπ

′′++′

′′=

εεmm –– Dielectric Function of the MediumDielectric Function of the Medium

RRpp -- Particle Radius Particle Radius λλ -- Incident WavelengthIncident Wavelength

εε’’pp –– Real Part of Dielectric Function of ParticlesReal Part of Dielectric Function of Particles

εε’’’’pp –– Imaginary Part of Dielectric Function of ParticlesImaginary Part of Dielectric Function of Particles

ppp iεεε ′′+′=

Dielectric Function of the NanoparticlesDielectric Function of the Nanoparticles

Complex Dielectric Function Complex Dielectric Function For Bulk MaterialFor Bulk Material

( ) ( ) ( ) ( )εεεεωω

ωωε bulkbulk

e

Pbulk i

im′′+′=

Γ+−= 2

2

1

ωω –– Excitation Angular FrequencyExcitation Angular Frequency

mmee –– Mass of ElectronMass of Electron

τ1

==Γe

F

lv

vvFF –– FermiFermi VelocityVelocity

02 εω eP mne=

nn –– Density of Free ElectronsDensity of Free Electrons

ee –– Electron ChargeElectron Charge

εε00 –– Permittivity of Free SpacePermittivity of Free Space

llee –– Mean Free PathMean Free Path

ττ –– Relaxation TimeRelaxation Time

Damping Damping FrequencyFrequency

Bulk Bulk PlasmonPlasmon FrequencyFrequency

( ) 22

2

1Γ+

−=′ωω

ωε Pbulk

Real Part of Dielectric Function of Bulk MaterialReal Part of Dielectric Function of Bulk Material

( ) ( )22

2

Γ+Γ

=′′ωωω

ωε Pbulk

Imaginary Part of Dielectric Function of Bulk MaterialImaginary Part of Dielectric Function of Bulk Material

( ) 22

2

Γ+−=′ ∞ ω

ωεωε P

bulk

Corrected Real Part of Dielectric Function of Bulk MaterialCorrected Real Part of Dielectric Function of Bulk Material

εε∞∞–– High Frequency Dielectric ConstantHigh Frequency Dielectric Constant

peeff Rll111

+=

Particle Size Effective MeanParticle Size Effective Mean--Free PathFree Path

eff

Feff l

v=Γ

Effective Damping FrequencyEffective Damping Frequency

Effect of the Small Particle SizeEffect of the Small Particle Size

( ) 22

2

eff

Pp Γ+

−=′ ∞ ωω

εωε

( ) ( )22

2

eff

effPp Γ+

Γ=′′

ωω

ωωε

FreeFree--electron Real Part of the electron Real Part of the Dielectric Function of Spherical ParticlesDielectric Function of Spherical Particles

FreeFree--electron Imaginary Part of the electron Imaginary Part of the Dielectric Function of Spherical ParticlesDielectric Function of Spherical Particles

Effect of Effect of Intrabound Intrabound TransitionsTransitions

( ) ( ) ( ) ( )pbulkierbandi

freeii ,int =+= ωεωεωε

Total Complex Dielectric FunctionTotal Complex Dielectric Function

( ) ( ) ( ) ( )ωεωεωεωε freebulkbulk

freepp −+=

Dielectric Constant in Metallic NanoparticlesDielectric Constant in Metallic Nanoparticles

Nextddl C ρα =

DiluteDilute--dispersion Limit dispersion Limit Adsorption CoefficientAdsorption Coefficient

ρρNN –– Number Density of ParticlesNumber Density of Particles

NonNon--spherical Particlesspherical Particles

Ref: R. Ref: R. GansGans, Ann. Phys., 47 (1915) 270, Ann. Phys., 47 (1915) 270

Extinction Cross Section of Extinction Cross Section of NonNon--Spherical ParticlesSpherical Particles

( )cbaj

PP

PRCj

pmj

jp

pjmext ,,

1

13

8

2

2

22332

=

′′+

−+′

′′= ∑

εεε

ελεπ

21

;111ln

211

2

2a

cbaP

PPrr

rrrP

−==

−

−+−

=

( )21 abr −=

Depolarization Vector for Nanorod (a>b=c)Depolarization Vector for Nanorod (a>b=c)

wherewhere

Gold

SilverGold

Silver

Input: Real and Imaginary Parts of the Dielectric Constants For Input: Real and Imaginary Parts of the Dielectric Constants For Gold and Silver as a Function of the Photon WavelengthGold and Silver as a Function of the Photon Wavelength 43704370

Ref: P. B. Johnson and R. W. Christy, Phys. Rev. B, 6 (1972) 4370

Wavelength, nm

Extin

ctio

nC

oeffi

cien

t,M

-1cm

-1

300 400 500 600 7000

1000

2000

3000

4000

5000

Calculated Absorption Spectra of Au Particles in Water

n = 1.334

ResultsResults

Dielectric Constant of

Water

Spherical Spherical ParticlesParticles

Wavelength, nm

Nor

mal

ized

Abso

rban

ce

500 525 550 575 6000

0.5

1

1.5

2

Calculated Absorption Spectra of Au Particles Media with DifferentDielectric Constant

n=1.602

1.3341.376

1.421

1.471

H2O (n=1.334)Cyclohexane (n=1.376)

Dodecane (n=1.421)Decalin (n=1.471)

CS2 (n=1.602)

Medium Dielectric Constant


n=1.334n=1.334 n=1.407n=1.407 n=1.481n=1.481 n=1.525n=1.525 n=1.583n=1.583

Ref: S. Underwood and P. Mulvaney,

Langmuir, 10 (1994) 3427-3430

15 nm Au Spherical Particles in Water and in

Mixtures of Butyl Acetate and

Carbon Disulfide

Spherical Gold ParticlesEffect of Dielectric Constant of the Medium

Mie Mie Theory Theory Transmission Transmission

ColorsColors


ExperimentExperiment

TheoryTheory

Elongated Ellipsoidal ParticlesElongated Ellipsoidal Particles

Wavelength, nm

Abso

rban

ce,a

.u

400 500 600 700 800 9000

50

100

150

200

250

300

350

400

450

Particle Aspect Ratio

3.63.33.1

2.9

2.6Longitudinal Longitudinal

PlasmonsPlasmons, Red Shift, Red ShiftTransverse Transverse PlasmonsPlasmons, Blue Shift, Blue Shift

Medium Dielectric Constant = 4.0

Wavelength, nm

Abso

rban

ce,a

.u

400 500 600 700 800 9000

50

100

150

200

250

300

350

400

450

Elongated Ellipsoidal ParticlesElongated Ellipsoidal Particles

Medium Dielectric Constant

3.0

2.5 Longitudinal Longitudinal PlasmonsPlasmons, Red Shift, Red ShiftTransverse Transverse

PlasmonsPlasmons, Red Shift, Red Shift

3.54.0 4.5

Aspect Ratio = 3.3

Regression Analysis of the Regression Analysis of the Wavelength at the Wavelength at the

LongitudinalLongitudinal PlasmonPlasmon PeakPeak

( ) 31.47231.4634.33max +−= mR ελ

Maxwell Garnett TheoryMaxwell Garnett TheoryNonNon--Dilute Colloidal SolutionsDilute Colloidal Solutions

Ref: J. C. Maxwell Garnett, Ref: J. C. Maxwell Garnett, PhilosPhilos. Trans. R. Soc. . Trans. R. Soc. London,203 (1904) 385. London,203 (1904) 385.

Au Core

SiO2 Shell

(a)(a) (b)(b)

(a) Silica Coated Gold Particle; (b) Ideal Packing of Silica Coated Gold Particles in the Film to Form FCC Lattice with Volume Fraction 0.74.

Average Electric Field in Composite MaterialAverage Electric Field in Composite Material

( ) pmav EEE φφ +−= 1

Particle Volume FractionParticle Volume Fraction

RRAuAu –– Radius of the Gold CoreRadius of the Gold Core

( )33

2

74.0

SiOAu

Au

RRR

+=φ

RRSiO2 SiO2 –– Thickness of SiOThickness of SiO22 ShellShell

EEmm –– Electrical Field in the Matrix MaterialElectrical Field in the Matrix Material

EEpp –– Electrical Field in the ParticleElectrical Field in the Particle

( )( ) ( ) ( ) avavppmmav EEEP 000 1111 εεεεφεεφ −=−+−−=

Average Polarization in Composite MaterialAverage Polarization in Composite Material

mmp

mp EE

εεε

23+

=

Electric Field Inside the ParticlesElectric Field Inside the Particles(Lorentz Cavity Field)(Lorentz Cavity Field)

εεmm –– Dielectric Function of the Matrix MaterialDielectric Function of the Matrix Material

Final Form of the EquationsFinal Form of the Equations

( ) mmp

mmav EEE

εεφε

φ2

31

++−=

Average Electric Field in Composite MaterialAverage Electric Field in Composite Material

Average Dielectric Function in Composite MaterialAverage Dielectric Function in Composite Material

( ) ( )( ) ( )φεφε

φεφεεε

++−

−++=

211221

mp

mpmav

( )λπεωα av

av

avav

knc

4Im==

Average Absorption Coefficient in Composite MaterialAverage Absorption Coefficient in Composite Material

Complex Dielectric FunctionComplex Dielectric Function

( ) ( )avddliikni iiiii ,2 =+=′′+′= εεε

( )avddlin iiii ,

2

21

22

=

′+′′+′=

εεε

( )avddlik iiii .

2

21

22

=

′−′′+′=

εεε

Complex Refractory IndexComplex Refractory Index

Optical ReflectanceOptical Reflectance

( )( )

( )avddliknkn

Rii

ii ,11

22

22

=++

+−=

( )( ) ( ) ( )

( )avddliRhhR

RRTii

,2cos2expexp

sin412

22

=+−+−

+−=

ψξααψ

Optical TransmittanceOptical Transmittance

h h –– Thickness of the Thickness of the Au@SiOAu@SiO22 FilmFilm

Optical ReflectanceOptical Reflectance

( )( )

( )avddliknkn

Rii

ii ,11

22

22

=++

+−=

( )( ) ( ) ( )

( )avddliRhhR

RRTii

,2cos2expexp

sin412

22

=+−+−

+−=

ψξααψ

Optical TransmittanceOptical Transmittance

Functions in the Above EquationFunctions in the Above Equation

( )avddlihni ,

4==

λπ

ξ ( )avddliknk

ii

i ,01

2tan 22

1 =≤≤

−+= − πψψ

h h –– Thickness of the Thickness of the Au@SiOAu@SiO22 FilmFilm

Wavelength, nm

Nor

mal

ised

Abso

rban

ce

200 300 400 500 600 7000

2

4

6

8

10

12

14

Effect of Film Thickness on the Calculated Absorption Spectra of Au@SiO2 Films at the Particle Volume Fraction φ = 0.05.

5nm

20nm

40nm

60nm

80nm

100nm

FilmThickness

ResultsResults

Wavelength, nm

Nor

mal

ised

Abso

rban

ce

400 450 500 550 600 650 7000

0.2

0.4

0.6

0.8

1

1.2

Calculated Absorption Spectra of Au Particles With Different Particle Volume Fractions.

0.100.20

0.30

0.400.50

0.60

Particle VolumeFraction

Volume Fraction of Au

Surfa

cePl

asm

onPe

akPo

sitio

n,nm

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7500

525

550

575

600

625

650

675

700

Effect of the Particle Volume Fraction on the Calculated Peak Positions of the Coupled Plasmon

Bands in Au@SiO2 Films

MG TheoryMG Theory

ExperimentalExperimental

t=17.5nmt=17.5nm t=12.5nmt=12.5nm t=4.6nmt=4.6nm t=2.9nmt=2.9nm t=1.5nmt=1.5nm

Ref: T. Ung, L. M. Liz-Marzan and P. Mulvaney, J.

Phys. Chem., B105 (2001) 3441-3452

15 nm Gold Spherical Particles Coated with Silica Shells of Various

Thickness


MaxwellMaxwell--Garnett Garnett Theory Theory

Transmission Transmission ColorsColors


TheoryTheory

t=17.5nmt=17.5nm t=12.5nmt=12.5nm t=4.6nmt=4.6nm t=2.9nmt=2.9nm t=1.5nmt=1.5nm

Ref: T. Ung, L. M. Liz-Marzan and P. Mulvaney, J.

Phys. Chem., B105 (2001) 3441-3452

15 nm Gold Spherical Particles Coated with Silica Shells of Various

Thickness


MaxwellMaxwell--Garnett Garnett Theory Theory

Reflection Reflection ColorsColors


TheoryTheory

Discrete Dipole Discrete Dipole ApproximationApproximation

Ref: Ref: J. J. Goodman, B. T.J. J. Goodman, B. T. DraineDraine, and P. J., and P. J.FlateauFlateau, Opt., Opt. LettLett. 16 (1991) 1198.. 16 (1991) 1198.

iii EαP ⋅=

Polarization of Each DipolePolarization of Each Dipole

ααii –– PolarizabilityPolarizability of the Dipole at of the Dipole at rrii

Total Total ElectriclElectricl Field at Position Field at Position rrii

iselfiinci ,, EEE +=

j

N

ijijiself PAE ⋅−= ∑

≠,

Electric Field From Other DipolesElectric Field From Other Dipoles

( )tii iiinc ω−⋅= rkEE exp0,

Electric Field of Incident Plain WaveElectric Field of Incident Plain Wave

k k –– Wave VectorWave Vector

EEoo –– The Amplitude of the Incident Electric Field The Amplitude of the Incident Electric Field

tt –– TimeTime ωω –– FrequencyFrequency

Final Equation for PolarizationFinal Equation for Polarization

( ) iinc

N

ijjijii ,

1 EPAPα =⋅+ ∑≠

−

( ) ( ) ( ) ( )[ ]

⋅−−

+××=⋅ jijijjijij

ijjijij

ij

ijjij r

rikr

krrki

PrrPPrrPA 31exp 2

22

3

Dyadic Green’s Function ApproachDyadic Green’s Function Approach

=

≠≡′− ji

jiijji 0

AA

Matrix A’Matrix A’

jj

jij

N

j

N

j

N

jjii

x

x

y

y

z

z

PAPAY ⋅′≡⋅′= ∑∑∑∑ −= = =

−

2

0

2

0

2

0

ConvolutionConvolution

∑

++≡

i z

zz

y

yy

x

xxin N

inNin

Nin

i222

expˆ YY

Discrete Fourier TransformDiscrete Fourier Transform

Extinction Cross SectionExtinction Cross Section

( )∑=

∗ ⋅=N

iiiext

kC1

2 Im4 PEE0

π

( )[ ]∑=

∗∗−

−⋅=

N

iiiiiabs k

EkC

1

2312

032Im4 PPP απ

Absorption Cross SectionAbsorption Cross Section

Scattering Cross SectionScattering Cross Section

absextsca CCC −=

EEii** –– Complex Conjugate of Total Electric Field at Complex Conjugate of Total Electric Field at rrii

Reflectivity Reflectivity

( ) ( )[ ]( ) ( )[ ]2

2

coscoscoscos

ri

ri

mm

Rθθθθ

+

−=

iknm +=

Complex Refractory IndexComplex Refractory Index

θθrr ––Refractive AngleRefractive Angle

θθii –– Incident AngleIncident Angle

( ) 2122 nmk −=

( ) ( )rin θθ sinsin=

Imaginary Part of Refractory IndexImaginary Part of Refractory Index

Real Part of Refractory IndexReal Part of Refractory Index

Wavelength, nm

Nor

mal

ised

Abso

rban

ce

300 400 500 600 700 8000

0.2

0.4

0.6

0.8

1

Particle Particle Volume Volume

Fractions Fractions 0.050.05

MG

DDA

Comparison of the Calculated Results from DDA and MG Effective Medium Method

UV Spectra of UV Spectra of MoleculesMolecules

Ref: Ref: Accelrys Accelrys VAMP TutorialVAMP Tutorial

Cinnamate Cinnamate MoleculeMolecule

Although self-consistent field calculations are adequate for the vast majority of ‘normal’ molecules, biradicals and excited states require a more sophisticated treatment.

This is often achieved using configuration interaction methods (CI). In CI calculations, the molecular orbitals for the ground state are calculated and then used unchanged to construct a series of further electronic configurations (microstates) that are mixed to form new electronic states.

CI calculations give not only the ground state, but also the excited states that result from mixing the microstates used. They can therefore be used for the calculation of UV/vis spectra, optimization of excited states, second order hyperpolarizabilities (sum-over-states method) etc.

CI calculations are available only for RHFwavefunctions. Any spin state (single, doublet, etc.) can be requested.

Configuration Interaction ResultsConfiguration Interaction Results

OscOsc. . StrStr..|r||r|

Dipole Length, ADipole Length, ADelDelMuMu

EnergiesEnergies

0.0220.0220.1830.1830.0000.000--0.1160.116--0.1420.1423.553.55168.9168.97.3397.339202011

0.0230.0230.1920.192--0.0190.0190.1910.191--0.0050.0051.721.72173.9173.97.1287.128191911

0.1820.1820.5600.560--0.0200.0200.2060.2060.5200.5206.536.53186.3186.36.6536.653161611

0.0700.0700.3490.3490.0500.0500.1060.106--0.3290.3294.774.77190.2190.26.5186.518151511

0.4610.4610.9500.950--0.0250.0250.9440.944--0.1010.1010.890.89212.5212.55.8345.834111111

0.3270.3270.8240.8240.0100.0100.2740.2740.7760.7763.603.60224.7224.75.5185.518101011

0.1800.1800.6320.6320.0310.031--0.5810.5810.2470.2475.315.31240.9240.95.1475.1479911

0.3940.3941.0471.0470.0480.048--1.0071.0070.2820.2822.852.85301.6301.64.1114.1116611

0.0690.0690.4410.441--0.0180.0180.3880.388--0.2100.2100.380.38305.3305.34.0604.0605511|z||z||y||y||x||x|nmnmevev

Excited Excited StateState

GroundGroundStateState

AccelrysAccelrys’’ VAMPVAMP

Wavelength, nm

Abso

rban

ce

200 250 300 350 400

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

CalculatedCalculated


Adsorption Spectrum forAdsorption Spectrum for CinnamateCinnamate

IR Spectra of IR Spectra of PolymersPolymers

Ref: A. Ref: A. Soldera Soldera and J.and J.--P. P. DognonDognon, , ““Optical Coefficients Optical Coefficients of Polymers Versus Wavelength Calculated From of Polymers Versus Wavelength Calculated From Classical Molecular SimulationsClassical Molecular Simulations””, ACS Division of , ACS Division of Polymeric Materials, Science and Engineering, 75 (1996) Polymeric Materials, Science and Engineering, 75 (1996) 227227--228. 228.

MethylMethyl--MethaMetha--AcrylateAcrylate(MMA)(MMA)

PolyPoly--MethylMethyl--MethaMetha--AcrylateAcrylate(PMMA)(PMMA)

0.13830.138315.999315.999399

0.08480.084817.264417.26441010

0.22850.228515.638315.638388

0.33910.339118.110618.11061111

Intensity Intensity km/molkm/mol

0.15070.150718.615418.61541212

0.29830.298314.076514.076577

0.02510.025113.778113.778166

0.19370.193712.088912.088955

0.13460.134611.093311.093344

0.45030.450310.504710.504733

0.00000.00000.00020.000222

0.00000.00000.00000.000011

0.00000.00000.00000.000000

Frequency Frequency 1/cm1/cmModeMode

Normal Mode AnalysisNormal Mode Analysis

AccelrysAccelrys’’DiscoverDiscover

Infra Red Absorption CoefficientInfra Red Absorption Coefficient(Ramsay Function)

( )( )∑

∆+−∆=

i i

i

m

SV

K2

21221 4

1303.221

νννν

πν

(Ramsay Function)

ννii –– WavenumberWavenumber

VVmm –– Molar VolumeMolar Volume

νν1/21/2 –– Half WidthHalf Width

SSii –– Integrated IntensityIntegrated Intensity

Frequency, cm-1

Abso

rptio

nC

oeffi

cien

t,a.

u.

10002000300040000

5

10

15

20

25



Infrared Absorption Spectra of PMMAInfrared Absorption Spectra of PMMA

Wavelength, microns

Rea

lPar

tofR

efra

ctiv

eIn

dex

3 4 5 6 71.35

1.375

1.4

1.425

1.45

1.475

1.5

1.525

1.55



Real Part of Refractive Index of PMMAReal Part of Refractive Index of PMMA

PPolymer Colloidal olymer Colloidal Crystal Photonic Crystal Photonic

BandgapBandgap StructureStructure

S.H.S.H. FoulgerFoulger, , D.W. Smith, Jr. and J.D.W. Smith, Jr. and J. BallatoBallatoClemson University Clemson University

A.L. Reynolds A.L. Reynolds –– ““TranslightTranslight”: A”: A Transfer Matrix CodeTransfer Matrix Codehttp://www.elec.http://www.elec.glagla.ac..ac.ukuk/groups//groups/optoopto//photoniccrystphotoniccryst

alal//PhotonicsPhotonics//photonicsmainphotonicsmain..htm htm

Polymerized State of the Aboven = 1.368

198.393PolystyrenePCCA

Water + Poly(ethylene glycol)methacrylate (PEG-MA) +

Poly(ethylene glycol) dimethacrylate (PEG-DMA)

2,2-diethoxyacetophenone(DENP)n = 1.367

198.993PolystyreneCCA/PEG

Water, n = 1.344185.293PolystyreneCCA

Capping MediumParticle Distance

(nm)

Particle Diameter

(nm)ParticlesType

Structural and Optical Parameters of Polymer Encapsulated FCC Crystalline Colloidal Arrays

Ref: S. H. Foulger, et. al., Langmuir, 17 (2001) 6023

Wavelength, nm

Ref

lect

ance

,a.u

.

450 500 550 600 650



Mechanochromic Mechanochromic Response of PCCA CompositeResponse of PCCA Composite

10% Compressed10% Compressed Stress FreeStress Free

Poly(ethylene glycol) +Poly(2-methoxyethyl methacrylate

(MOEM)) 203109PolystyreneMOEM

Poly(ethylene glycol) +Poly(2-methoxyethyl acrylate)-co-poly(2-methoxyethyl methacrylate)

203109PolystyreneMOEA+MOEM(50:50)

Poly(ethylene glycol) +Poly(2-methoxyethyl acrylate

(MOEA)) ,(nc = 1.489)203109PolystyreneMOEA

Capping MediumParticle Distance

(nm)

Particle Diameter

(nm)ParticlesType

Structural and Optical Parameters of Polymer Encapsulated FCC Crystalline Colloidal Arrays

Ref: S. H. Foulger, et. al., Adv. Mater., 15 (2003) 685

Wavelength, nm

Nor

mal

ized

Ref

lect

ance

,a.u

.

350 400 450 500 550 600 6500

0.2

0.4

0.6

0.8

1

1.2

Calculated/Measeured Reflectance Spectra for Different Compressive Stress

30% Compression

0% Compression

48% Compression

Experimental

Calculated

A Comparison of the Measured and Calculated Reflected Colors

0% Compression 30% Compression 48% Compression

TheoryTheory


OneOne--DimensionalDimensionalPhotonic Photonic BandgapBandgap

StructuresStructures

A.L. Reynolds A.L. Reynolds –– ““TranslightTranslight”: A”: A Transfer Matrix CodeTransfer Matrix Codehttp://www.elec.http://www.elec.glagla.ac..ac.ukuk/groups//groups/optoopto//photoniccrystphotoniccryst

alal//PhotonicsPhotonics//photonicsmainphotonicsmain..htm htm

Transfer Matrix MethodTransfer Matrix Method

OneOne--dimension Planar dimension Planar Periodic StructurePeriodic Structure

kk11 kk22

z

x

y

(i) Two-layer Planar Structure(ii) Periodic Two-layer Planar Structure

a b

N × d

Wave Vectors: kWave Vectors: k11, k, k22Period: d = a + bPeriod: d = a + b

(i)

(ii)

Definition of the Problem

K1 K2

a b

kk11

a

z

x

y

Z0

X

kk22kk11

φ= [φ1 (z<0), φ2 (0<z<b), φ3 (z>b)]

B1B1 B2B2

B: BoundaryB: Boundary

Maxwell Electromagnetic EquationsMaxwell Electromagnetic Equations

E – Electrical Field

( )HE ci ω=×∇ ( ) ( )ErH εω ci−=×∇

H – Magnetic Field

c – Light Speed

ω – Angular Frequency

ε(r) – Dielectric Constant

OneOne--Dimensional CaseDimensional Case

( )0,0,)( E=rE ( )0,,0)( H=rH

Hci

zE ω

−=∂∂

Ezci

zH )(εω

−=∂∂

−

( )HE ci ω=×∇

( ) ( )ErH εω ci−=×∇

0)( 2

2

2

2

=+∂∂ E

cz

zE ωε

Boundary ConditionsBoundary Conditions

At Boundary B1:

21 φφ =

=

BA

MDC

1

At Boundary B2:

32 φφ =

=

DC

MGF

2

zz ∂∂=∂∂ 21 φφ

zz ∂∂=∂∂ 32 φφ

Solution Form Solution Form -- Wave FunctionsWave Functions

H]E,[=φ

zikzik eBeA 111

−⋅+⋅=φ

zikzik eDeC 222

−⋅+⋅=φ

zikzik eGeF 113

−⋅+⋅=φ

E, H – Electrical, Magnetic Fields

k1, k2 – Wave Vector in Materials 1 and 2

A, C, F – Incident Waves Magnitude

B, D, G – Reflected Waves Magnitude

Wave VectorsWave Vectors

After Substitution of φ1 and φ2 into the Governing Equation one Obtains:

)/( 221

21 ck ωε ⋅=

)/( 222

22 ck ωε ⋅=

Transfer MatrixTransfer Matrix

=

BA

TGF

212221

1211 TTTTTT

T ⋅=

=

=

−

1122 k-k11

k-k11

T1

1

⋅⋅

⋅⋅

×

⋅⋅

⋅⋅=

−

))))

)))) 1

2

aexp(-ikk-aexp(ikkaexp(-ikaexp(ik

aexp(-ikk-aexp(ikkaexp(-ikaexp(ik

T

2222

22

1111

11

Bloch TheoremBloch Theorem

( )0

exp==

⋅⋅=zdz

diK φφ

⋅=

BA

diKGF

)exp(

⋅=

BA

diKBA

T )exp()(ω EigevalueProblem

Band Structure CalculationBand Structure Calculation

0)exp()(det =⋅⋅− IdiKT ω

)][cos(1 dKFunc ⋅= −ω

)()cos( ωFuncdK =⋅

Func-1 – A Multi-valued Function

Transmission CoefficientTransmission Coefficient

2

AFCtrans =

2111 )(Re −= TalCtrans

Band Structure CalculationBand Structure Calculation

Normalized Reciprocal Vector

Rad

ialF

requ

ency

,rad

/s

0 0.2 0.4 0.6 0.8 10

1E+15

2E+15

3E+15

4E+15

5E+15

6E+15

7E+15

Transmission SpectrumTransmission Spectrum

Wavelength, nm

Tran

smis

sion

Coe

ffici

ent

300 400 500 600 700 800 900 10000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1