Slab Waveguides Fundamentals - users.ntua.grusers.ntua.gr/eglytsis/IO/Slab_Waveguides_p.pdf · Slab Waveguides Fundamentals 07/12/2016 ... G. Lifante, Integrated Photonics Fundamentals,

Slab Waveguides Fundamentals

07/12/2016

Integrated Optics Prof. Elias N. Glytsis

School of Electrical & Computer Engineering National Technical University of Athens

SLAB WAVEGUIDES

Step-index

Graded-index

G. Lifante, Integrated Photonics Fundamentals, Wiley 2003

Prof. Elias N. Glytsis, School of ECE, NTUA

x

z

h

nc

nf

ns

cover

substrate

film

θ θ θ θ

S1 S0

S2

SLAB WAVEGUIDE GEOMETRY


SLAB WAVEGUIDE SELF-CONSISTENCY CONDITION (waveguiding condition)

x

z

h

nc

nf

ns

cover

substrate

film θ θ

Α

Β

D O

C


SLAB WAVEGUIDE SELF-CONSISTENCY CONDITION (graphical interpretation)


SLAB WAVEGUIDE ELECTROMAGNETIC APPROACH

Helmholtz Equation:





TE Modes

TM Modes


SLAB WAVEGUIDE ELECTROMAGNETIC APPROACH TE Modes


SLAB WAVEGUIDE ELECTROMAGNETIC APPROACH TE Modes

Dispersion Equation


SLAB WAVEGUIDE ELECTROMAGNETIC APPROACH TM Modes


SLAB WAVEGUIDE ELECTROMAGNETIC APPROACH TM Modes

Dispersion Equation


SLAB WAVEGUIDE ELECTROMAGNETIC APPROACH TE, TM Modes Graphical Solution

Neff for TE Modes: 2.1700, 2.0783, 1.9190, 1.6831

Neff for TM Modes: 2.1642, 2.0542, 1.8636, 1.5968


SLAB WAVEGUIDE ELECTROMAGNETIC APPROACH TE Modes Example (nc= 1, nf =2.2, ns=1.5, h=1.2μm, λ0=1.0μm


SLAB WAVEGUIDE ELECTROMAGNETIC APPROACH TM Modes Example (nc= 1, nf =2.2, ns=1.5, h=1.2μm, λ0=1.0μm


SLAB WAVEGUIDES MODES

TE0

TE1



TE0 TE1

H.A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall 1984



TM0

TM1

H.A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall 1984


SLAB WAVEGUIDES SUBSTRATE MODES

TE Substrate Modes

TM Substrate Modes


SLAB WAVEGUIDES SUBSTRATE MODES TE/TM Substrate Modes Example (nc= 1, nf =2.2, ns=1.5, h=1.2μm, λ0=1.0μm


SLAB WAVEGUIDES RADIATION MODES

TE Radiation Modes

TM Radiation Modes


SLAB WAVEGUIDES RADIATION MODES TE/TM Radiation Modes Example (nc= 1, nf =2.2, ns=1.5, h=1.2μm, λ0=1.0μm


SLAB WAVEGUIDES UNPHYSICAL MODES

TE Unphysical Modes

TM Unphysical Modes


SLAB WAVEGUIDES UNPHYSICAL MODES TE/TM Radiation Modes Example (nc= 1, nf =2.2, ns=1.5, h=1.2μm, λ0=1.0μm


SLAB WAVEGUIDES MODE CLASSIFICATION



SLAB WAVEGUIDES RAY PICTURE



SLAB WAVEGUIDES SUBSTRATE AND RADIATION MODES



SLAB WAVEGUIDES EVANESCENT MODES

TE Forward Evanescent Modes

TM Forward Evanescent Modes


SLAB WAVEGUIDES EVANESCENT MODES TE/TM Forward Evanescent Modes Example (nc= 1, nf =2.2, ns=1.5, h=1.2μm, λ0=1.0μm)


× × × × × ×

βi /k0

βr /k0

nc

-nc

ns

-ns

nf

-nf

Forward Leaky Modes

Backward Leaky Modes

×

×

×

×

Forward Unphysical Modes

Backward Unphysical Modes

Forward Region Backward Region

Forward Evanescent

Modes

Backward Evanescent

Modes Forward Guided Modes

Backward Guided Modes

Forward Substrate

Modes

Radiation Modes

Backward Substrate

Modes

Continuum Spectrum

× × Discrete Spectrum

×

×

nc

nf

ns z

SLAB WAVEGUIDE MODE DIAGRAM


SLAB WAVEGUIDE CUTOFF CONDITIONS


SLAB WAVEGUIDE CUTOFF CONDITIONS


SLAB WAVEGUIDE NORMALIZED PARAMETERS

Dispersion for TE Guided Modes

Dispersion for TM Guided Modes


SLAB WAVEGUIDE NORMALIZED DIAGRAM


SLAB WAVEGUIDE NORMALIZED DIAGRAM


POWER CONSIDERATIONS FOR SLAB WAVEGUIDES

TE Modes



TM Modes






MODE ORTHOGONALITY & COMPLETENESS IN DIELECTRIC WAVEGUIDES



Z-Reversal Symmetry



Time-Reversal Symmetry





Guided Modes

Radiation Modes



Power Considerations


MODE ORTHOGONALITY IN SLAB WAVEGUIDES

TE Modes

TE Field Expansion


MODE ORTHOGONALITY IN SLAB WAVEGUIDES

TM Modes

TM Field Expansion


EXAMPLE DISCONTINUITY IN A SLAB WAVEGUIDE



Boundary Conditions



Boundary Conditions (neglecting radiation/substrate modes)

Formulation 1



Power Considerations (Formulation 1)



Boundary Conditions (neglecting radiation/substrate modes) Formulation 2



Power Considerations (Formulation 2)



Formulation 2 with Substrate & Radiation Modes




Boundary Conditions




Boundary Conditions after Projections



Convergence (Formulation 2) with Substrate and Radiation Modes






Formulation 2 with Substrate and Radiation Modes (100 modes)




Boundary Conditions after Projections











MULTILAYERED SLAB WAVEGUIDES

Range of propagation constant β possible values for guided modes



TE Modes



TE Modes

Boundary Conditions


MULTILAYERED SLAB WAVEGUIDES TE Modes


TE Guided Modes Example nc= 1.45, n1 =1.56, n2 =1.45, n3 =1.56 ns=1.45,

h1=0.75 μm, h2=0.50, h3=0.75 μm λ0=1.0μm




TM Modes



TM Modes

Boundary Conditions

Dispersion Equation


TM Guided Modes Example nc= 1.45, n1 =1.56, n2 =1.45, n3 =1.56 ns=1.45,

h1=0.75 μm, h2=0.50, h3=0.75 μm λ0=1.0μm



FINITE-DIFFERENCE FREQUENCY-DOMAIN (FDFD) ANALYSIS OF SLAB WAVEGUIDES

TE Modes

Helmholtz Equations

TM Modes



TE Modes

Normalized Distance:



TE Modes



TM Modes


Example: nc= 1.0, nf =3.4, ns=3.1, h =1.0 μm, λ0=1.3μm



FINITE-DIFFERENCE FREQUENCY-DOMAIN (FDFD) ANALYSIS OF SLAB WAVEGUIDES USING YEE’s CELL

(Solving Maxwell’s Equations)

TE Modes

Normalized Magnetic Field

R. C. Rumpf, PIERS B, vol. 36, 221-248 (2012)




TE Modes




TE Modes – Permittivity Averaging












TM Modes




TM Modes



(Solving Maxwell’s Equations) TM Modes










GRADED-INDEX SLAB WAVEGUIDES

Example of Graded-index Slab Waveguide



WKB Method (Wentzel-Kramers-Brillouin) Ey = ψ(x)

For TE Modes



WKB solution in general

WKB for the profile shown







WKB - Dispersion Equation for Graded Index Slab Waveguide


GRADED-INDEX SLAB WAVEGUIDES Example Case 1: εs=4.80, Δε=0.045, x0 = 8μm, w0 = 2μm, λ0 = 0.6328μm



Example Case 1: εs=4.80, Δε=0.045, x0 = 8μm, w0 = 2μm, λ0 = 0.6328μm



Example Case 1: εs=4.80, Δε=0.045, x0 = 8μm, w0 = 2μm, λ0 = 0.6328μm



Example Case 2: εs=4.80, Δε=0.045, w0 = 4μm, λ0 = 0.6328μm




WKB Modified Dispersion Equation




Finite-Difference Frequency-Domain Results




Multilayer Approximation Results (N=40)


Documents

Slab Waveguides Fundamentals - users.ntua.grusers.ntua.gr/eglytsis/IO/Slab_Waveguides_p.pdf · Slab Waveguides Fundamentals 07/12/2016 ... G. Lifante, Integrated Photonics Fundamentals,