8/6/2019 Dispersion Effects
1/72
A study towards dispersion-effects using
slab-waveguide
8/6/2019 Dispersion Effects
2/72
ERRORSENCOUNTERED
&
STAGESOFDEVELOPMENT
8/6/2019 Dispersion Effects
3/72
8/6/2019 Dispersion Effects
4/72
8/6/2019 Dispersion Effects
5/72
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
8/6/2019 Dispersion Effects
6/72
HINTFORSIMULATING THE
FRINGEPATTERN
8/6/2019 Dispersion Effects
7/72
8/6/2019 Dispersion Effects
8/72
FRINGEPATTERN GENERATION &
MATHEMATICALTREATMENT
8/6/2019 Dispersion Effects
9/72
Time : 0sec
8/6/2019 Dispersion Effects
10/72
Time : 1.569782e-4 sec
8/6/2019 Dispersion Effects
11/72
Time : 2.162206e-4 sec
8/6/2019 Dispersion Effects
12/72
Time : 2.196788e-4 sec
8/6/2019 Dispersion Effects
13/72
ERROR: TIME-STEPTOOSMALL
TOEVALUATE
PROPOSED FIX: NONDIMENSIONALIZATIONOF
MAXWELLSEQUATION
8/6/2019 Dispersion Effects
14/72
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
8/6/2019 Dispersion Effects
15/72
8/6/2019 Dispersion Effects
16/72
8/6/2019 Dispersion Effects
17/72
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)
8/6/2019 Dispersion Effects
18/72
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
8/6/2019 Dispersion Effects
19/72
8/6/2019 Dispersion Effects
20/72
8/6/2019 Dispersion Effects
21/72
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.
8/6/2019 Dispersion Effects
22/72
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
8/6/2019 Dispersion Effects
23/72
7.268
6.99624
2.04768 2.99568 3.99424
5.996784.99912
8/6/2019 Dispersion Effects
24/72
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]
8/6/2019 Dispersion Effects
25/72
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.
8/6/2019 Dispersion Effects
26/72
8/6/2019 Dispersion Effects
27/72
9.48e-5
0.014089
0
2.92616
5.89024
6.32e-6
7.38808
8.0264e-4
CONTOUR
8/6/2019 Dispersion Effects
28/72
Sine/Gauss spatial contour (Ey)
Sine/Gauss temporal contour (Ey)
Solver : GMRES
Static Analysis of PMMA layerexcluded slab waveguide
Normal Incidence
DropTolerance : .0001
8/6/2019 Dispersion Effects
29/72
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)
DropTolerance : .0001
8/6/2019 Dispersion Effects
30/72
____Sine Pulse____Gaussian Pulse
Transient Analysis of PMMA layerexcluded slab waveguide
Gaussian Pulse
2.50272 s
0 s
2.5596 s
1.00498 s
DropTolerance : .0001
Solver : GMRES
8/6/2019 Dispersion Effects
31/72
4.00056 s
5.03702 s
6.29472 s
7.0468 s
Contd..
Solver : GMRES
DropTolerance : .0001
____Sine Pulse
____Gaussian Pulse
8/6/2019 Dispersion Effects
32/72
Transient Analysis of PMMA layerexcluded slab waveguide
Sine Pulse
0 s
9.48e-5 s
0.632 s
1.96552
s
Solver : GMRES
DropTolerance : .0001
8/6/2019 Dispersion Effects
33/72
4.97384 s
Contd..
7.0468 s
Solver : GMRES
DropTolerance : .0001
Normal Incidence
8/6/2019 Dispersion Effects
34/72
Transient Analysis of PMMA layerincluded slab waveguide
Gaussian PulseGaussian excitation
6.32e-6s
2.212 s3.58344 s
DropTolerance : .0001
Solver : GMRES
8/6/2019 Dispersion Effects
35/72
8/6/2019 Dispersion Effects
36/72
Transient Analysis of PMMA layerincluded slab waveguide
Sine Pulse
6.32e-6
1.34616 s2.54064 s
Solver : GMRES
DropTolerance : .0001
8/6/2019 Dispersion Effects
37/72
Contd..
6.32e-6 s
1.34616 s2.54064 s
DropTolerance : .0001
Solver : GMRES
8/6/2019 Dispersion Effects
38/72
Contd..
3.8236 s 5.0876 s
6.58544 s7.0468 s
DropTolerance : .0001
Solver : GMRES
8/6/2019 Dispersion Effects
39/72
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:
8/6/2019 Dispersion Effects
40/72
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
8/6/2019 Dispersion Effects
41/72
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
8/6/2019 Dispersion Effects
42/72
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.
8/6/2019 Dispersion Effects
43/72
8/6/2019 Dispersion Effects
44/72
8/6/2019 Dispersion Effects
45/72
AN UNBELIEVEBALE MISCONCEPTION
FINALLY
LED TO THE UNDERLYING
PHYSICS
8/6/2019 Dispersion Effects
46/72
Waveguide ModelsWaveguide Models
Static AnalysisStatic Analysis
8/6/2019 Dispersion Effects
47/72
8/6/2019 Dispersion Effects
48/72
8/6/2019 Dispersion Effects
49/72
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.
8/6/2019 Dispersion Effects
50/72
COMSOL TREATMENT FOR TRANSIENT ANALYSIS
HOW COMSOL IS DOING A TRANSIENT ANALYSIS
8/6/2019 Dispersion Effects
51/72
EXPLANATION
8/6/2019 Dispersion Effects
52/72
COMPUTATIONAL WINDOW
DETAILED ANALYSIS
8/6/2019 Dispersion Effects
53/72
DETAILED ANALYSIS
8/6/2019 Dispersion Effects
54/72
FOCUS :CO-EFFICENTS OF THE GENERALIZED PDE
8/6/2019 Dispersion Effects
55/72
8/6/2019 Dispersion Effects
56/72
8/6/2019 Dispersion Effects
57/72
MISCONCEPTION FINALLY CLEARED
UNDERSTOOD THE BASICS OF BOTH
SLAB WAVEGUIDE&
DIELECTRIC INTERFACE
8/6/2019 Dispersion Effects
58/72
STATIONARY ANALYSIS
2D 2-LAYER DIELECTRIC INTERFACE
8/6/2019 Dispersion Effects
59/72
8/6/2019 Dispersion Effects
60/72
2D 2-LAYER DIELECTRIC INTERFACE
TRANSIENT ANALYSIS
8/6/2019 Dispersion Effects
61/72
8/6/2019 Dispersion Effects
62/72
8/6/2019 Dispersion Effects
63/72
8/6/2019 Dispersion Effects
64/72
8/6/2019 Dispersion Effects
65/72
8/6/2019 Dispersion Effects
66/72
8/6/2019 Dispersion Effects
67/72
8/6/2019 Dispersion Effects
68/72
1D 2-LAYER DIELECTRIC INTERFACE
TANGENTIAL ELECTRIC FIELD
COMPONENT
8/6/2019 Dispersion Effects
69/72
DOUBTS FINALLY CLEARED
8/6/2019 Dispersion Effects
70/72
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
8/6/2019 Dispersion Effects
71/72
8/6/2019 Dispersion Effects
72/72