Computational Methods for Optoelectronics

Transcription

1 IPT 526 Computational Methods for Optoelectronics Ray-Kuang Lee Department of Electrical Engineering and Institute of Photonics Technologies National Tsing-Hua University, Hsinchu, Taiwan Course info: rklee IPT-526, Spring 26 p./3

2 Computational Optoelectronics 2> ω a > Frequency (ωd/2πc) x j M Γ X M -5 5 x i IPT-526, Spring 26 p.2/3

3 IPT 526 Time: T3T4Tn (:-3: AM, Tuesday) Course Description: 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:. Two Homeworks, 7%; 2. One Project, 3%. IPT-526, Spring 26 p.3/3

4 Reference Books W. H. Press et al., "Numerical Recipes (in C, C++, or Fortran)," Cambridge University Press (992). W. Y. Yang et al., "Applied Numerical Methods Using MATLAB," Wiley (25). ftp://ftp.wiley.com/public/sci_tech_med/applied_numerical/ IPT-526, Spring 26 p.4/3

5 Related courses Data Structures Algorithm Numerical Analysis 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-526, Spring 26 p.5/3

6 , Linear Algebraic Equations A x = b, where A = a a 2 a N a 2 a 22 a 2N a M a M2 a MN, x = x x 2. x M, and b = b b 2.. b M. Gauss-Jordan elimination LU decomposition Jacobi iteration Gauss-Seidel iteration IPT-526, Spring 26 p.6/3

7 Multiple scattering method for PBG Helmholtz equation: 2 E + k 2 (M)E = a =.45 2r =.5 d =.45 l =. n c = 3.5 with k 2 (M) = k 2 ǫ = k 2 ǫ j if M C j (j =, 2,..., N) k 2 if M C j (j =, 2,..., N) write down the total field in decomposition, E(P) = + m= m= a l,m J m [kr l (P)]e imθ l(p) b l,m H () m [kr l (P)]e imθ l(p), Transmittance [db] # of layers = Normalized Frequency ωa/2πc then all we need is to solve 2.5 ˆb l j l S l T l,j ˆbj = S l Q l IPT-526, Spring 26 p.7/3

8 2, Interpolation, Curve Fitting, and Integration.5 equispaced points.5 Chebyshev points max error = max error = Polynomial interpolation and extrapolation Rational function interpolation Chebyshev approximation Padé approximation Fast Fourier Transform and Discrete Fourier Transform Trapezoidal and Simpson method for integration IPT-526, Spring 26 p.8/3

9 Finite difference approximation Second-order FD approximation for u (x j ) u (x j ) u j+ u j 2h in the matrix-vector form (with periodic boundary) u... u j... u N = h u... u j... u N IPT-526, Spring 26 p.9/3

10 3, Ordinary Differential Equations d 2 y dx 2 + q(x)dy dx = r(x), Euler s method Runge-Kutta method Adaptive stepsize control Predictor-corrector method Boundary value problems Relaxation method IPT-526, Spring 26 p./3

11 ODE with B. C.: D Bragg reflector coupled-mode equation: de + (z) dz de (z) dz with the Boundary Condition: = iδe + (z) + iκe (z) = iδe (z) + iκe + (z) A B A B A B A... B... E + (z = ) = E (z = L) = Transmission δ (/cm) Reflection δ (/cm) IPT-526, Spring 26 p./3

12 4, Partial Differential Equations A(x,y) 2 u x + B(x,y) 2 u 2 x y + u u C(x,y) 2 = f(x,y,u, y2 x, 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 IPT-526, Spring 26 p.2/3

13 Mach-Zehnder structure by BeamProp IPT-526, Spring 26 p.3/3

14 Metallic Waveguide by ToyFDTD IPT-526, Spring 26 p.4/3

15 5, Nonlinear Equations and Nonlinear PDE Nonlinear equation: f(x) = x 3 + x 2 = 5, Iterative method Newton-Raphson method Secant Method Nonlinear Schrödinger equation: i U t + 2 U x 2 + U 2 U = Crank-Nicholson method Slit-Step Fast Fourier Transform Pseudospectral method IPT-526, Spring 26 p.5/3

16 Propagation of solitons Nonlinear Schrödinger equation: i U t + 2 U x 2 + U 2 U = supports soliton solutions, U(t =,x) = sech(x) t x simulated by Fourier spectral + 4th-order explicit Runge-Kutta methods, N x = 28, N t = 5, error = 6, indep. of N t ; nlse.m. IPT-526, Spring 26 p.6/3

17 Bound solitons in CGLE Complex Ginzburg-Landau Equation: iu z + D 2 U tt + U 2 U = iδu + iǫ U 2 U + iβu tt + iµ U 4 U v U 4 U, seek for bound-state solutions by propagation method. N U(z,t) = U (z,t + ρ j )e iθ j j= R.-K. Lee, Y. Lai, and B. A. Malomed, Phys. Rev. A 7, 6387 (24). IPT-526, Spring 26 p.7/3

18 Vector bound solitons Coupled Nonlinear Schrödinger Equations: i U z + 2 i V z U t 2 + A U 2 U + B V 2 U = 2 V t 2 + A V 2 V + B U 2 V = where A = /3, B = 2/3; and U, V are circular polarization fields. U, V Time M. Haelterman, A. P. Sheppard, and A. W. Snyder, Opt. Lett. 8, 46 (993). IPT-526, Spring 26 p.8/3

19 6, Eigenvalues and Eigenvectors A x = λx, Similarity transformation and Diagonalization Jacobi method The QR algorithm Mathieu equation: u xx + 2q cos(2x)u = λx, λ q IPT-526, Spring 26 p.9/3

20 Band diagram and Density of States ω2 { E(r)} = ǫ(r) c E(r), Frequency (ωd/2πc) M Γ X M DOS (A.U.) x 4 IPT-526, Spring 26 p.2/3

21 Laplacian equation in a disk Eigenmodes of Laplacian equations, [ 2 x x 2 ]u(x, y) = λf(x, y). Mode λ =. Mode 3 λ = Mode 6 λ = Mode λ = L. N. Trefethen, Spectral methods in Matlab, SIAM (2). IPT-526, Spring 26 p.2/3

22 7, Finite Element Method Elements Mesh generation Element assembly Boundary condition IPT-526, Spring 26 p.22/3

23 Photonic Crystals IPT-526, Spring 26 p.23/3

24 8, Monte Carlo Method Random numbers with uniform deviates Transformation method Rejection method Random bits Monte Carlo methods IPT-526, Spring 26 p.24/3

25 9, Optimization x Simulated annealing Genetic algorithm Penalty function Optimal control method Matlab built-in routines IPT-526, Spring 26 p.25/3

26 , 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-526, Spring 26 p.26/3

27 Eigenfunction of Nonlinear PDF Gap solitons in optical lattices, i Φ (t, x) = 2 2 x 2 Φ (t, x) + V (x)φ (t, x) + g D φ (t, x) 2 φ (t, x) which has gap soliton solutions. 2 2 (a) 8 c - (b) P 6 4 b - (c) 2 a µ X by FS + NK methods, N x = 52, no. of iterations < ; gpeol.m. R.-K. Lee, E. A. Ostrovskaya, Yu. S. Kivshar, and Y. Lai, Phys. Rev. A 72, 3367 (25). IPT-526, Spring 26 p.27/3

28 Optical lithography IPT-526, Spring 26 p.28/3

29 Syllabus. Linear Algebraic Equations (Mar. 7) 2. Interpolation, Curve Fitting, and Integration (Mar. 4) 3. Ordinary Differential Equations (Mar. 2 and Mar. 28) Homework # : two-weeks to finish, deadline Apr.. 4. Partial Differential Equations (Apr. and Apr. 8) 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 6) 8. Monte Carlo Method (May 23) 9. Optimization (May 3) Project: one-month to finish.. Case studies (Jun. 6 and Jun 3) IPT-526, Spring 26 p.29/3

30 Evaluation. Two Homeworks (assigned), 7% HW, Ordinary Differential Equations HW2, Nonlinear Partial Differential Equations 2. One Project (chosen one that is related to your research), 3% Two/Three-dimensional PDE Nonlinear/Coupled ODE/PDE Finite Element Method Monte Carlo Method Optimization other suggestions IPT-526, Spring 26 p.3/3