Dispersion Effects

8/6/2019 Dispersion Effects

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A study towards dispersion-effects using

slab-waveguide


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ERRORSENCOUNTERED

&

STAGESOFDEVELOPMENT


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Eg= exp((-4*(cos(theta))^2/(BW)^2*(y)^2)+imag(2*pi/lamda*y*sin(theta)))

theta=45 deg

BW= 2e-9 nm

Eg=exp(-(y^2)/(2*0.001^2))/sqrt(2*pi*0.001^2)

Eg= E0*sin(omega1*x)*sin(theta)

Gaussian Beam (normal incidence)

COMSOLOUTPUT : Working

Gaussian Beam (angular incidence)

COMSOLOUTPUT : Not Working


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HINTFORSIMULATING THE

FRINGEPATTERN


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FRINGEPATTERN GENERATION &

MATHEMATICALTREATMENT


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Time : 0sec


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Time : 1.569782e-4 sec


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Time : 2.162206e-4 sec


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Time : 2.196788e-4 sec


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ERROR: TIME-STEPTOOSMALL

TOEVALUATE

PROPOSED FIX: NONDIMENSIONALIZATIONOF

MAXWELLSEQUATION


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Introduction ofMATLAB for numerical

computations

The time co-ordinate is stretched to a[0,1] scale & the time-step for the

original time_array is used for time-

mapping


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clc;clear all;

%%%%%%%%%%%% Constants for the geometry, in COMSOL%%%%%%%%%%%%%%%%%%%%%%%%%%%mu_Si=1.0;epsilon_Si=11.8;

n_Si=3.4255;res_Si=640;sigma_Si=1000;epsilon_PMMA=2.9;

mu_PMMA=0.866;n_PMMA=1.4914;res_PMMA=1e-19;sigma_PMMA=1e-4;

epsilon_air=1;mu_air=1;n_air=1;c_light=3e+8;

res_air=1;sigma_air=3e-15;

pi=3.414;

%%%%%%%%%%%% Parameters which modulate the Diffraction amount %%%%%%%%%%%%%%

BW=2e-9; %%%%%%Bandwidth of incident beam%%%%%%%%%%%%%E0=1; %%%%%%Maximum Amplitude of incident beam%%%%%lamda=632e-9; %%%%%%Wavelength of incident beam%%%%%%%%%%%height=3.8e-6; %%%%%%height of the layer in Geometry%%%%%%%theta=30; %%%%%%angle of incidence %%%%%%%

%%%%%%%%% number of Time-steps (based on time of propagation) %%%%%%%

omega1=2*pi*c_light/lamda;freq=2*pi/omega1;

delT=lamda/c_light;time_propagation=abs(2*height/(c_light*cos(theta))); %%%%%%%Actual time: 1.6423e-13%%%%%%%%%%

Ntime_step=ceil(time_propagation/delT); %%%%%%%Number of time-steps %%%%%%%%

%%%%%%%%%%% co-ordinate stretching (time-axis) %%%%%%%%%%%%%%%%%%%%%%%%%%%time_array=linspace(0,time_propagation,Ntime_step);N=linspace(0,1,length(time_array));

increment=N(12)-N(11);sprintf('%6f',increment)


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ConceptRe-defined

Followed the procedures for aPhotonicCrystal (COMSOL->Modal

Library)

Understood the concept ofPML (perfectly matched layer) for a slab

waveguide

Electric field excitation is achieved by applying a Gaussian beam on the

boundary of the top layer

Introduced the weak-terms and other important co-efficients (h, q, r) in

theModel

Applied it for a Stationary Analysis.

It has to be verified for Time-variant and Wave-propagation in any

multi-layered slab waveguides

Understood thatCOMSOL was not able to calculate due to memory

problem but due to the wrongly assumed BoundaryConditions &improper meshing


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ConceptRe-defined

1. Tried to export a different module in the geometry.

2. Combined the PDE co-efficient form with the Structural Mechanicsmodule and simulated the combined modules together.

3. Understood that GardE gives the effective flux, outward direction.

4. Came back to the original PDE form and tried to incorporate the

User ModelsEM Wave propagationDiffraction pattern

5. The above module was an example of Stationary analysis. Hence,

simulated the module for a time-dependent analysis and simulated

the diffraction patterns. (Double slit fringe diffraction)

6. Re- calculated the simulation parameters and the other solver-timebased on the new method for calculation.

7. Reduced the geometry to non-dimensional form and found that the

solver-step-time which I was making 1, (delT/t_prop) is absurd.

Thus, the correct numerals [0:delT:N*(steps)] and simulated the

geometry for the Counter fringe patterns.


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CONTOUR VARIATIONS (in [sec])

0 6.32e-6 4.425e-5

0.00000632 1.9592e-4 0.00080264

0.00632 0.082160.014587

0.7584 1.43464 1.61792


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7.268

6.99624

2.04768 2.99568 3.99424

5.996784.99912


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Speed of light in vaccum (c1): 3e+8 [m/s]

Wavelength of incident beam (lamda) : 632 nm

Height of the layer in Geometry(h1) : 3.16e-4 m

Angle of incidence (theta) : 30 [deg]

Frequency of incident beam (f): 4.746835e+14[Hz]

Angular Frequency of incident beam (w): 2.982525e+15 [rad]

Time period of the incident wave (T): 2.106667e-15[s]

Minimum step-time required(T): 2.10667e-16[s]

Time of propagation (Ttotal) : 2.432584e-12[s]

No. of steps required(N): 11547

Non-dimentionalization

x->x/L t->c*t/L

y->y/L

L= 10 m

Ttotal= 2.432584e-17 [s]

delT = 0.00632

Number of time step for simulation (N) = 1154

Tcomsol

= 7.29328 [s]


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A new perspective

1. Even thought the calculations were exact, the desired

diffraction effect was far from achievement.

2. Re-visited the concepts of SPP (surface plasmon

polaritons) for the metallo-di electric slab waveguides.

3. Found that every parameter (mu, sigma, n, omega..) are

spatially dependent terms and not constant. This led to a

significant change in the Maxwells equation and the

corresponding boundary conditions in the geometry.

4. Instead of sinusoidal excitation, a Gaussian beam

excitation with a fixed wavelength is theoretically

calculated for the model.

5. The surface propagation constant (ksp) and other

necessary parameters were adjusted, and the geometry wassimulated again, based on the procedures followed for

obtaining the SPP diffraction effects at the metal-

dielectric interface.

6. Discussed the possibilities with Sir and the solution is

yet to be verified.


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9.48e-5

0.014089

0

2.92616

5.89024

6.32e-6

7.38808

8.0264e-4

CONTOUR


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Sine/Gauss spatial contour (Ey)

Sine/Gauss temporal contour (Ey)

Solver : GMRES

Static Analysis of PMMA layerexcluded slab waveguide

Normal Incidence

DropTolerance : .0001


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Solver : GMRES

Static Analysis of PMMA layerincluded slab waveguideNormal Incidence

Gauss-spatial contour (Ey)

Solver : GMRES

Sine-temporal contour (Ey)

Solver : GMRES

Gauss-temporal contour (Ey)Solver : GMRESSine-spatial contour (Ey)



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____Sine Pulse____Gaussian Pulse

Transient Analysis of PMMA layerexcluded slab waveguide

Gaussian Pulse

2.50272 s

0 s

2.5596 s

1.00498 s


Solver : GMRES


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4.00056 s

5.03702 s

6.29472 s

7.0468 s

Contd..

Solver : GMRES


____Sine Pulse

____Gaussian Pulse


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Transient Analysis of PMMA layerexcluded slab waveguide

Sine Pulse

0 s

9.48e-5 s

0.632 s

1.96552

s

Solver : GMRES



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4.97384 s

Contd..

7.0468 s

Solver : GMRES


Normal Incidence


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Transient Analysis of PMMA layerincluded slab waveguide

Gaussian PulseGaussian excitation

6.32e-6s

2.212 s3.58344 s


Solver : GMRES


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Transient Analysis of PMMA layerincluded slab waveguide

Sine Pulse

6.32e-6

1.34616 s2.54064 s

Solver : GMRES



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Contd..

6.32e-6 s

1.34616 s2.54064 s


Solver : GMRES


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Contd..

3.8236 s 5.0876 s

6.58544 s7.0468 s


Solver : GMRES


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Completely Lost the track.

1. The assumptions were all incorrect.

2. Surface plasmon and high related high-concept physics hasnothing to do with this model.

3. Verified the mathematics and the physics concept behind

the scene.

4. Went back to the original PDE co-efficient form.

5. Did a static analysis with a Neumann boundary condition.

6. Did a cross verification different kind of solver

settings.

7. Drop Tolerence to 0.0001 while using GMRES solver for

both time-dependent and frequency dependent analysis.

8. Got a hint that the mesh-parameters play a pivotal role

in FE-analysis.

9. Tried to use a chirped signal but wasnt successful.

10.The modified results are shown below:


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SOLVER: Direct (UMFPACK) STATIONARY ANALYSIS

Case II: electric field exists at the silicon bottom layer.

incident boundary electric field has both Ex and Ey

silicon boundary (last boundary) Ex 0 & Ey =0


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Case I: zero electric field at the silicon bottom layer.

incident boundary electric field has both Ex and Ey

silicon boundary (last boundary) Ex & Ey =0

Case II: electric field exists at the silicon bottom layer. incident boundary electric field has both Ex and Ey

silicon boundary (last boundary) Ex 0 & Ey =0


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Coming back to the right path.

1. Got salary for 2-months and bought new books.

2. Bought a new book for FEA for electro-magneticsimulation.

3. Revisited and re-learned the PDE in various form.

4. Learnt that various of Maxwells equation (integral,

differential etc.) essentially represent the same thing

and that it can be used for any kind of wave-propagationand not only for EM-wave simulations.

5. Got mathematical definition of mesh-parameter

adjustment and how it is related for the correct

simulation of a geometry.

6. Scraped every model (previously built) and started thesimulation freshly.

7. Verified the results for Si-Air layered slab and then

finally did a static analysis of the 3-layered geometry.


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AN UNBELIEVEBALE MISCONCEPTION

FINALLY

LED TO THE UNDERLYING

PHYSICS


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Waveguide ModelsWaveguide Models

Static AnalysisStatic Analysis


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TRANSIENT ANALYSIS

MAXWELLS EQUATION:

SCALAR HELMHOLTZ

VECTORIAL REPRESENTATION

TRANSIENT EQUATION

Which can be effectively derived from the transientWhich can be effectively derived from the transient

form we used in our solutionform we used in our solution

Thus we have to somehow retrieve and use the

value ofeigenvalue/propagation constanteigenvalue/propagation constant and

use it in the solution for our geometry.


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COMSOL TREATMENT FOR TRANSIENT ANALYSIS

HOW COMSOL IS DOING A TRANSIENT ANALYSIS


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EXPLANATION


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COMPUTATIONAL WINDOW

DETAILED ANALYSIS


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DETAILED ANALYSIS


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FOCUS :CO-EFFICENTS OF THE GENERALIZED PDE


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MISCONCEPTION FINALLY CLEARED

UNDERSTOOD THE BASICS OF BOTH

SLAB WAVEGUIDE&

DIELECTRIC INTERFACE


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STATIONARY ANALYSIS

2D 2-LAYER DIELECTRIC INTERFACE


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TRANSIENT ANALYSIS


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TANGENTIAL ELECTRIC FIELD

COMPONENT


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DOUBTS FINALLY CLEARED


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1. UNDERSTOOD WHY THE INCIDENT ELECTRIC FIELD IS IN TM-MODE & THE

DIELECTRIC -PLANE OF WAVE PROPAGATION IS IN TE MODE

2. HOW WE ARE USING THE GAUSSIAN PULSE/WAVE AS A DIFFRACTION LIMITED WAVE

FOR THE STUDY OF DISPERSION AND INTERFERENCE IN A DIELECTRIC MEDIUM

3. GOT THE KEY CONCEPT THAT THE DIRECTION OF ELECTRIC FIELD (LIGHT-WAVE)

PROPAGATION IS ALWAYS ORTHOGONAL TO THE PLANE OF LIGHT PROPAGATION

4. UNDERSTOOD THE BASIC PRINCIPLES OF MODE-DECOMPOSITON AND THE RELATIVE

IMPORTANCE OF IT IN CEM, FOR FIELD APPROXIMATIONS(FAR-FIELD & NEAR

FIELD)

5. RE-DERIVED THE GOVERNING EQUATIONS (MAXWELLS VECTORIAL FORM) AND

APPLIED THE PRINCIPLE OF SUPERPOSITION, TO CALCULATE THE TRANSVERSE

WAVES IN TERMS OF TWO SIMILAR FIELD (H/E) USING A SCALAR HELMHOLTZ

EQUATION

6. UNDERLYING MATHEMATICS OF TIME-HARMONIC FUNCTIONS, UNIFORM-HOMOGENEOUS

PALNE WAVE, INTRINSIC IMPEDENCE, COMPLEX PERMITTIVITY, TENSORIAL

REPRESENETATION

7. HOW TO RELATE THE NYQUIST-CRITERION, COURANTS CONDITION, STEP-TIME IN

TIME MARCHING, SATISFYING A STABILITY CRITERION IN THE NEUMERICAL

ANALYSIS


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Dispersion Effects