Post on 10-Jul-2018
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
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
Spherical Spherical ParticlesParticles
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
Spherical Spherical ParticlesParticles
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
Spherical Gold ParticlesEffect of Dielectric Constant of the Medium
MaxwellMaxwell--Garnett Garnett Theory Theory
Transmission Transmission ColorsColors
ExperimentExperiment
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
Spherical Gold ParticlesEffect of Dielectric Constant of the Medium
MaxwellMaxwell--Garnett Garnett Theory Theory
Reflection Reflection ColorsColors
ExperimentExperiment
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
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
ExperimentalExperimental
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.
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
CalculatedCalculated
ExperimentalExperimental
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
CalculatedCalculated
ExperimentalExperimental
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
CalculatedCalculated
ExperimentalExperimental
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
ExperimentExperiment
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
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
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