Computational Methods for Optoelectronics

IPT 5260
Computational Methods for Optoelectronics
Ray-Kuang Lee †
Department of Electrical Engineering
and Institute of Photonics Technologies
National Tsing-Hua University, Hsinchu, Taiwan
Course info: http://mx.nthu.edu.tw/∼rklee
†e-mail: rklee@ee.nthu.edu.tw
IPT-5260, Spring 2006 – p.1/30

Frequency (ωd/2πc)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
ω a
|2>
|1>
0
M Γ X M
Computational Optoelectronics
x j
50
0
-50
-50 0 50
xi IPT-5260, Spring 2006 – p.2/30

Time: T3T4Tn (10:10-13:00 AM, Tuesday)
Course Description:
IPT 52600
Fundamental numerical simulation techniques for solving Optoelectronics related
problems.
Taking this course you are asked to program by yourself.
Although this course is given primarily for the first year graduate students, those
who are undergraduates or senior graduates are encouraged to take this course.
Background: No required but you must learn how to program in C/C++, Fortran,
Matlab, Mathematica, or Mapple (at least one of these programming languages).
Remark: This course is more focused on numerical methods than numerical analysis.
Teaching Method: in-class lectures with examples and projects studies.
Evaluation:
1. Two Homeworks, 70%;
2. One Project, 30%.
IPT-5260, Spring 2006 – p.3/30

Reference Books
W. H. Press et al., "Numerical Recipes (in C, C++, or
Fortran)," Cambridge University Press (1992).
W. Y. Yang et al., "Applied Numerical Methods Using
MATLAB," Wiley (2005).
http://www.nr.com/
ftp://ftp.wiley.com/public/sci_tech_med/applied_numerical/
IPT-5260, Spring 2006 – p.4/30

Data Structures
Algorithm
Numerical Analysis
Related courses
Numerical Partial Differential Equations
Scientific Computation
Computational Physics
Computational Fluid Dynamics, CFD
First Principle calculation, ab-initio
Finite Element Method, FEM
Monte Carlo/Molecular Dynamics calculation, MC/MD
Computer-Aided Design, CAD
IPT-5260, Spring 2006 – p.5/30

where
A =
⎡
⎢
⎣
1, Linear Algebraic Equations
a11 a12 · · · a1N
a21 a22 · · · a2N
· · ·
aM1 aM2 · · · aMN
Gauss-Jordan elimination
LU decomposition
Jacobi iteration
Gauss-Seidel iteration
A · x = b,
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ , x = ⎢
⎥ ⎢
⎦ ⎢
⎣
x1
x2
.
xM
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥ , and b = ⎢
⎥ ⎢
⎦ ⎣
b1
b2
.
bM
⎤
⎥ .
⎥
⎦
IPT-5260, Spring 2006 – p.6/30

Multiple scattering method for PBG
Helmholtz equation:
with
˜k 2 (M) = k 2 ˜ǫ =
⎧
⎨
⎩
▽ 2 E + ˜ k 2 (M)E = 0
write down the total field in decomposition,
E(P) =
then all we need is to solve
+
k 2 ǫj if M ∈ Cj(j = 1, 2, . . . , N)
k 2 if M ∈ Cj(j = 1, 2, . . . , N)
∞
m=−∞
∞
m=−∞
al,mJm[krl(P)]e imθ l(P)
bl,mH (1)
m [krl(P)]e imθ l(P) ,
ˆbl −
SlTl,j ˆ bj = SlQl.
j=l
Transmittance [dB]
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
10
0
-10
-20
-30
-40
a = 0.45
2r = 0.15
d = 0.45
l = 1.0
n c = 3.5
-2 -1 0 1 2
# of layers
= 2 345
-50
0.2 0.3 0.4 0.5 0.6 0.7
Normalized Frequency ωa/2πc
2.5
2
1.5
1
0.5
0
-0.5
-1
-2 -1 0 1 2
IPT-5260, Spring 2006 – p.7/30

2, Interpolation, Curve Fitting, and Integration
1.5
1
0.5
0
−0.5
equispaced points
max error = 5.9001
−1
−1 −0.5 0 0.5 1
−0.5
Polynomial interpolation and extrapolation
Rational function interpolation
Chebyshev approximation
Padé approximation
1.5
1
0.5
0
Chebyshev points
max error = 0.017523
−1
−1 −0.5 0 0.5 1
Fast Fourier Transform and Discrete Fourier Transform
Trapezoidal and Simpson method for integration
IPT-5260, Spring 2006 – p.8/30

Finite difference approximation
Second-order FD approximation for u ′ (xj)
u ′ (xj) ≈ uj+1 − uj−1
2h
in the matrix-vector form (with periodic boundary)
⎛
⎜
⎝
u ′ 1
.
u ′ j
.
u ′ N
⎞
⎛
⎟ = h
⎟
⎠
−1
⎜
⎝
1
0 2
− 1
0 2
1
2
. ..
− 1
2 0
. ..
1
2
0
− 1
2
− 1
2
1
2
0
⎞ ⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
u1
.
uj
.
uN
⎞
⎟
⎠
IPT-5260, Spring 2006 – p.9/30

Euler’s method
3, Ordinary Differential Equations
Runge-Kutta method
Adaptive stepsize control
Predictor-corrector method
Boundary value problems
Relaxation method
d 2 y
+ q(x)dy
dx2 dx
= r(x),
IPT-5260, Spring 2006 – p.10/30

ODE with B. C.: 1D Bragg reflector
coupled-mode equation:
dE+(z)
dz
dE−(z)
dz
with the Boundary Condition:
Transmission
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-20 -10 0 10 20
δ (1/cm)
= iδE+(z) + iκE−(z)
= iδE−(z) + iκE+(z)
E+(z = 0) = 1
E−(z = L) = 0
Reflection
A B A B A B A
.... B ....
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-20 -10 0 10 20
δ (1/cm)
IPT-5260, Spring 2006 – p.11/30

4, Partial Differential Equations
A(x,y) ∂2u ∂x2 + B(x,y) ∂2u ∂x∂y + C(x,y)∂2 u
∂y
Euler method
Crank-Nicholson method
Beam Propagation Method
Slit-Step Fast Fourier Transform
Finite Difference Time Domain method
Absorption Boundary Condition
Periodic Boundary Condition
Perfect Matching Layers
∂u
= f(x,y,u, 2 ∂x
, ∂u
∂y ),
IPT-5260, Spring 2006 – p.12/30

Mach-Zehnder structure
by BeamProp
IPT-5260, Spring 2006 – p.13/30

Metallic Waveguide
by ToyFDTD
IPT-5260, Spring 2006 – p.14/30

5, Nonlinear Equations and Nonlinear PDE
Nonlinear equation:
Iterative method
Newton-Raphson method
f(x) = x 3 + x 2 = 5,
Secant Method
Nonlinear Schrödinger equation:
Crank-Nicholson method
Slit-Step Fast Fourier Transform
Pseudospectral method
i ∂U
∂t + ∂2 U
∂x 2 + |U|2 U = 0
IPT-5260, Spring 2006 – p.15/30

Propagation of solitons
Nonlinear Schrödinger equation:
i ∂U
∂t + ∂2 U
∂x 2 + |U|2 U = 0
supports soliton solutions, U(t = 0,x) = sech(x).
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−30
−20
−10
0
10
x
20
simulated by Fourier spectral + 4th-order explicit Runge-Kutta methods,
Nx = 128, Nt = 50, error = 10 −6 , indep. of Nt; nlse.m.
30
0
5
10
15
t
20
25
30
35
IPT-5260, Spring 2006 – p.16/30

Bound solitons in CGLE
Complex Ginzburg-Landau Equation:
iUz + D
2 Utt + |U| 2 U = iδU + iǫ|U| 2 U + iβUtt
+ iµ|U| 4 U − v|U| 4 U,
seek for bound-state solutions by propagation method.
N
U(z,t) = U0(z,t + ρj)e iθj
j=1
R.-K. Lee, Y. Lai, and B. A. Malomed, Phys. Rev. A 70, 063817 (2004).
IPT-5260, Spring 2006 – p.17/30

Vector bound solitons
Coupled Nonlinear Schrödinger Equations:
i ∂U
∂z
i ∂V
∂z
+ 1
2
+ 1
2
∂ 2 U
∂t 2 + A|U|2 U + B|V | 2 U = 0
∂ 2 V
∂t 2 + A|V |2 V + B|U| 2 V = 0
where A = 1/3, B = 2/3; and U, V are circular polarization fields. Time
U, V
1.5
1
0.5
0
-0.5
-1
-5 0 5
M. Haelterman, A. P. Sheppard, and A. W. Snyder, Opt. Lett. 18, 1406 (1993).
IPT-5260, Spring 2006 – p.18/30

6, Eigenvalues and Eigenvectors
A · x = λx,
Similarity transformation and Diagonalization
Jacobi method
The QR algorithm
Mathieu equation: −uxx + 2q cos(2x)u = λx,
λ
32
28
24
20
16
12
8
4
0
−4
−8
−12
−16
−20
−24
0 5 10 15
q
IPT-5260, Spring 2006 – p.19/30

Frequency (ωd/2πc)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Band diagram and Density of States
1
ǫ(r)
M Γ X M
ω2
∇ × {∇ × E(r)} = E(r),
c2 0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0 0.5 1 1.5 2 2.5 3
x 10 4
0
DOS (A.U.)
IPT-5260, Spring 2006 – p.20/30

Laplacian equation in a disk
Eigenmodes of Laplacian equations, [ ∂2
∂x 2 + ∂2
∂x 2 ]u(x, y) = −λf(x, y).
Mode 1
λ = 1.0000000000
Mode 6
λ = 2.2954172674
Mode 3
λ = 1.5933405057
Mode 10
λ = 2.9172954551
L. N. Trefethen, ”Spectral methods in Matlab,” SIAM (2000).
IPT-5260, Spring 2006 – p.21/30

Elements
Mesh generation
Element assembly
Boundary condition
7, Finite Element Method
IPT-5260, Spring 2006 – p.22/30

3.5
3
2.5
2
1.5
1
0.5
0
−0.5
−1
Photonic Crystals
−1.5
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
3.5
3
2.5
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5
IPT-5260, Spring 2006 – p.23/30

8, Monte Carlo Method
Random numbers with uniform deviates
Transformation method
Rejection method
Random bits
Monte Carlo methods
IPT-5260, Spring 2006 – p.24/30

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1.545 1.55 1.555
x 10 −3
0
Simulated annealing
Genetic algorithm
Penalty function
Optimal control method
Matlab built-in routines
9, Optimization
6
5
4
3
2
1
0
−1
−2
−3
0 2 4 6 8 10
IPT-5260, Spring 2006 – p.25/30

10, Case studies
Newton-Kantorovich method
2D Gross-Pitaevskii equation
Moment method
... your project
... your project
Numerical Libraries
Matlab
IMSL
Programming in Unix
GNU Make and Concurrent Version System
Parallel programming
IPT-5260, Spring 2006 – p.26/30

Eigenfunction of Nonlinear PDF
Gap solitons in optical lattices,
i ∂
∂ Φ0(t, x) = − 1
2
∂ 2
which has gap soliton solutions.
P
12
10
8
6
4
2
a
∂x 2 Φ0(t, x) + V (x)Φ0(t, x) + g1D|φ0(t, x)| 2 φ0(t, x)
0
1 2 3 4
µ
b
c
2
1
0
-1
1
0
-1
1
0
X
(a)
(b)
(c)
-1
-50 -25 0 25 50
by FS + NK methods, Nx = 512, no. of iterations < 10; gpeol.m.
R.-K. Lee, E. A. Ostrovskaya, Yu. S. Kivshar, and Y. Lai, Phys. Rev. A 72, 033607 (2005).
IPT-5260, Spring 2006 – p.27/30

Optical lithography
IPT-5260, Spring 2006 – p.28/30

1. Linear Algebraic Equations (Mar. 7)
Syllabus
2. Interpolation, Curve Fitting, and Integration (Mar. 14)
3. Ordinary Differential Equations (Mar. 21 and Mar. 28)
Homework # 1: two-weeks to finish, deadline Apr. 11.
4. Partial Differential Equations (Apr. 11 and Apr. 18)
5. Nonlinear Equations and Nonlinear PDE (Apr. 25 and May 2)
Homework # 2: two-weeks to finish.
6. Eigenvalues and Eigenvectors (May 9)
7. Finite Element Method (May 16)
8. Monte Carlo Method (May 23)
9. Optimization (May 30)
Project: one-month to finish.
10. Case studies (Jun. 6 and Jun 13)
IPT-5260, Spring 2006 – p.29/30

1. Two Homeworks (assigned), 70%
Evaluation
HW1, Ordinary Differential Equations
HW2, Nonlinear Partial Differential Equations
2. One Project (chosen one that is related to your research), 30%
Two/Three-dimensional PDE
Nonlinear/Coupled ODE/PDE
Finite Element Method
Monte Carlo Method
Optimization
· · ·
other suggestions
IPT-5260, Spring 2006 – p.30/30