Beam Propagation Method Solution to the Seminar Tasks

Transcription

1 Beam Propagation Method Solution to the Seminar Tasks Matthias Zilk The task was to implement a 1D beam propagation method (BPM) that solves the equation z v(xz) = i 2 [ 2k x 2 + (x) k 2 ik2 v(x, z) = Lv(x, z) (1) 2k ] using a centered finite-difference approximation of the x-derivative and two different finite-difference schemes along the propagation direction z. Voluntarily, the Crank-Nicolson could be implemented as a third solution method. The implementations had to be tested with a Gaussian beam v(x) = exp ( x2 w 2 ) that is slightly too large to fit into a step index waveguide n n(x) = core for x < x b /2; { n cladding otherwise. (2) (3) The parameters of the test geometry are given in table 1. 1 Task I: The Explicit FTCS Scheme The first task was the implementation of the explicit FTCS scheme, where the transverse second derivative is discretized by a central difference while the z-derivative is discretized by a forward difference. The field update of this scheme is given by v n+1 = (I + ΔzL)v n = Av n (4) where I is the identity matrix and L is the discretized beam propagation operator of equation (1). The vector v n = (v n 1,, vn i,, vn N x ) T contains the discretized field at the lateral positions x = x a /2 + (i 1)Δx and at z = (n 1)Δz. Width of computational domain x a = 5 µm Waveguide width x b = 2 µm Cladding refractive index n cladding = 1.45 Core refractive index n core = 1.46 Beam width w = 4 µm Vacuum wavelength λ = 1 µm Propagation distance z end = 1 µm Table 1: Parameters of the test geometry. 1

2 1.1 Implementation For the implementation of the explicit scheme the matrix of the discretized beam propagation operator L is required. As this matrix is also required by the other schemes, it it useful to implement it in a separate function. This function is called bpm_operator: bpm_operator.m 1 function L = bpm_operator (dx,n,nd, lambda ) 2 % L = bpm_operator (lambda,n,nd,dx) 3 % Calculates the finite difference discretization of the 4 % BPM differential operator L = 1i /(2* kbar )*( d ^2/ dx ^2 + k ^2 - kbar ^2). 5 % All lengths have to be specified in µm. 6 % Arguments : 7 % dx : Lateral step size ( scalar ) 8 % n : Refractive index distribution ( vector ) 9 % nd : Reference refractive index ( scalar ) 1 % lambda : Wavelength ( scalar ) 11 % Returns : 12 % L : Beam propagation operator ( sparse matrix ) Nx = numel (n); 15 k = 2* pi/ lambda *n (:); 16 kbar = 2* pi/ lambda *nd; 17 alpha = 1/(( dx ^2)* 2* kbar ); % abbreviation for prefactor 18 Wi = ( k.^2 - kbar ^2)/( 2* kbar ); % abbreviation for last term 19 2 main_ diag = 1i*( Wi - 2* alpha ); 21 aux_diag = 1i* alpha * ones (Nx, 1); 22 L = spdiags ([ aux_diag, main_diag, aux_diag ], [-1,, 1], Nx, Nx ); The implementation is a literal translation of the formulas from the seminar slides. The discretized beam propagation operator is a symmetric tridiagonal matrix. The main and the secondary diagonals are set up in line 2 and 21. In line 22 the matrix is created using the spdiags function. This produces a sparse matrix that stores only the non-zero entries. As L has only 3 non-zero diagonals, using sparse matrix saves quite a lot of memory. Also numerical operations with sparse matrices are computationally more efficient than operation with dense matrices that contain mostly zeros. With the matrix L we can now implement the explicit scheme: beamprop_explicit.m 1 function [ v_out, z] = beamprop_ explicit ( v_in, lambda, dx, n, nd,... 2 z_end, dz, output_ step ) 3 % [ v_out, z] = beamprop_ explicit ( v_in, lambda, dx, n, nd,... 4 % z_end, dz, output_ step ) 5 % Propagates an initial field over a given distance based on the 6 % solution of the paraxial wave equation in an inhomogeneous 7 % refractive index distribution using the explicit 8 % forward Euler scheme. All lengths have to be specified in µm. 9 % Arguments : 1 % v_in : Initial field ( vector ) 11 % lambda : Wavelength ( scalar ) 12 % dx : Transverse step size ( scalar ) 13 % n : Refractive index distribution ( vector ) 14 % nd : Reference refractive index ( scalar ) 15 % z_end : Propagation distance ( scalar ) 2

3 16 % dz : Step size in propagation direction ( scalar ) 17 % output_ step : Number of steps between field outputs ( integer ) 18 % Returns : 19 % v_out : Propagated field ( matrix ) 2 % z : z- coordinates of field output ( vector ) % calculate grid size 23 Nx = length (n); 24 Nz = round ( z_end /( output_ step * dz )) + 1; % +1 for initial field at z = 25 % calculate required number of iterations 26 Niter = ( Nz - 1)* output_ step ; % generate the output grid 29 v_out = zeros ( Nx, Nz ); 3 z = ( : (Nz - 1))* dz* output_step ; % generate iteration matrix A = I + dz* L 33 A = speye (Nx) + dz* bpm_operator (dx,n,nd, lambda ); % iteration 36 curr_ slice = 1; % current postion in output matrix 37 v_ curr = v_in (:); % current field 38 v_out (:, curr_ slice ) = v_ curr ; % store input field as first output 39 for iter = 1 : Niter 4 % advance field : multiply current field with iteration matrix 41 v_curr = A* v_curr ; 42 % check if current iteration has to be stored 43 if mod( iter, output_ step ) == 44 curr_ slice = curr_ slice + 1; % move to next output slot 45 v_out (:, curr_ slice ) = v_ curr ; % store current solution 46 end 47 end The function consists of three parts: In line 23 to 3 the output matrices are set up and the required number of iterations is calculated. In line 33 the iteration matrix is created using the auxiliary function bpm_operator. This line is a literal translation of equation (4). As we are dealing with sparse matrices, the function speye has to be used to create the identity matrix. Finally, in line 36 to 47 the field is propagated using a for loop and is stored at the requested intervals. At each iteration the field is advanced by Δz by multiplying the current field with the iteration matrix. As the iteration matrix is tridiagonal this operation has a linear complexity of O(N x ). The if condition in line 43 uses a modulo operation to decide if the current field has to be stored in the output. 1.2 Results The implementation was tested with a step index waveguide and a Gaussian initial filed. The parameters of the test case are given in table 1. For the lateral discretization N x = 251 points were used, resulting in a lateral step size of Δx = x a /(N x 1) =.2 µm. As the explicit scheme is unstable, a very small propagation step size of Δz = 2.5 nm had to be used to achieve a propagation distance of z end = 1 µm. For the subtraction of the carrier a reference refractive index of n d = was used. The refractive index distribution, the initial field and the fundamental waveguide mode of the step index waveguide (the waveguide supports only a single mode) are shown in figure 1. The waveguide 3

4 lateral position [µm] real part of field real part of field afer 1 µm input field fundamental waveguide mode index difference n(x) - n cladding lateral position [µm] Figure 1: Refractive index distribution, initial field and fundamental waveguide mode of the test geometry. The initial field and the waveguide mode have been normalized to the same power. explicit FTCS schme dz = 2.5nm, Nx=251 dz = 5.nm, Nx=251 dz = 2.5nm, Nx= propagation distance [µm] lateral position [µm] Figure 2: Evolution of the field propagated with the explicit scheme using a propagation step size of Δz = 2.5 nm and a lateral step size of Δx = x a /(N x 1) =.2 µm. Figure 3: Influence of the parameters Δz and N x on the stability of the explicit scheme. The plots show the real part of the field after a propagation distance of 1 µm for various discretization parameters. The individual curves have been offset vertically for better distinction. mode has been calculated with the finite-difference mode solver of seminar 3. It is apparent that the initial field is slightly larger than the waveguide mode. The effective index of the waveguide mode is n eff = which is very close to the chosen reference index. The evolution of the field propagated with the explicit scheme is shown in figure 2. Most of the initial field is coupled to the waveguide mode and propagates in the waveguide. However, as the overlap between the initial field and the waveguide mode is not perfect, parts of the initial field are scattered and create oscillating sidelobes that spread laterally with increasing propagation distance. The field in the waveguide shows almost no phase variation. This is because the BPM calculates only the slowly varying envelop of the actual field. The oscillating carrier is removed by the choice of k = 2πn d /λ. As k was chosen very close to the propagation constant of the waveguide mode, the envelop of the field in the waveguide exhibits almost no phase evolution upon propagation. The field shown figure 2 looks well but we know that the explicit scheme is unstable. So what happens when we change the propagation step size or the lateral resolution? In the seminar it has been shown that the squared magnitude of the eigenvalues of the propagation matrix in absence of 4

5 a refractive index modulation (i.e. k(x) = k) is given by g(κ) 2 = 1 + (Δz)2 k 2 (Δx) [cos(κδx) 4 1]2 (5) whereby κ is the transverse wave number of the eigenvector of the beam propagation operator. The eigenvalues are larger than unity for all κ > making the explicit scheme unstable. The eigenvalues are largest for high frequency components with κ close to π/δx. The magnitude of the eigenvalues increases both when Δz is increased or when Δx is decreased. From equation (5) we can expect that both increasing Δz and N x will increase the amplification of numerical errors which will in turn lead to the appearance of high frequency noise. This is exactly what can be seen in figure 3. With the original parameters of Δz = 2.5 nm and N x = 251 there are already some small high frequency distortions visible on the field after a propagation distance of 1 µm. When Δz or N x is increased, these distortions grow strongly and the solution becomes completely unusable. 2 Task II: The Implicit BTCS Scheme The stability problem of the explicit scheme is circumvented in the implicit BTCS scheme by using a backward difference for the approximation for the derivative in propagation direction. The field update of this scheme is given by v n+1 = (I ΔzL) 1 v n = A 1 v n (6) and requires the solution of a linear system of equations. However, as the matrix A is tridiagonal, this can still be done with a complexity of O(N x ). 2.1 Implementation The implementation of the implicit scheme is very similar to the implementation of the explicit scheme. Actually only two lines have to be changed: beamprop_implicit.m 32 % generate implicit iteration matrix A = I - dz* L 33 A = speye (Nx) - dz* bpm_operator (dx,n,nd, lambda ); and 39 for iter = 1 : Niter beamprop_implicit.m 4 % advance field : solve tridiagonal system of equations 41 v_curr = A\ v_curr ; 42 % check if current iteration has to be stored 43 if mod( iter, output_ step ) == 44 curr_ slice = curr_ slice + 1; % move to next output slot 45 v_out (:, curr_ slice ) = v_ curr ; % store current solution 46 end 47 end In line 33 the new iteration matrix is assembled according to equation (6). In line 41 the multiplication of the field vector with the iteration matrix is replaced by the solution of a linear system of equations using Matlab s backslash operator. This selects an efficient algorithm automatically. 5

6 lateral position [µm] real part of field lateral position [µm] real part of field real part of field -2-1 implicit BTCS scheme initial field propagated field after 1µm waveguide mode (scaled) propagation distance [µm] lateral position [µm] Figure 4: Evolution of the field propagated with the implicit scheme using a propagation step size of Δz =.1 µm and a lateral step size of Δx = x a /(N x 1) =.2 µm. Figure 5: Plot of the field of figure 4 at z = 1 µm. implicit BTCS scheme propagation distance [µm] Figure 6: Evolution of the field over a longer propagation distance. The parameters are the same as in figure Results The implicit scheme is unconditionally stable. Hence, a larger propagation step size of Δz =.1 µm can be used for testing the implementation. The evolution of the propagated field is shown in figure 4. Despite the much larger step size it looks very similar to the result of the explicit method. Figure 5 shows the field after a propagation distance of 1 µm. The high frequency distortions of the explicit method are gone. The central part of the field distribution strongly resembles the waveguide mode while the oscillating sidelobes originate from the parts of the initial field that are not coupled to the waveguide. The dominance of the waveguide mode justifies the selection of the reference index of n d = which almost matches the effective index of the waveguide mode of n eff = Figure 6 shows the evolution of the field over a longer propagation distance. It reveals a fundamental flaw of our simple BPM implementation: It cannot simulate an infinite computational domain. When the scattered field that is not coupled to the waveguide reaches the boundaries, it is reflected at the perfectly conducting walls. It propagates back and forth in the computational domain and interferes with the waveguide mode. In most applications this behavior is undesirable. Hence, more sophisticated BPM implementations provide some kind of open boundaries, for example in 6

7 form of perfectly matched layers (PMLs), that do not reflect impinging waves but absorb them. 3 Task III: The Crank-Nicolson Scheme The implicit scheme is stable but the backward difference approximation is only first order accurate. The implicit scheme is also not energy conserving in the absence of absorption. These shortcomings are mended by the Crank-Nicolson (CN) scheme which uses a centered finite-difference approximation of the derivative in propagation direction. The field update of this scheme is given by v n+1 = ( I 1 2 ΔzL 1 ) ( I ΔzL ) vn = A 1 Bv n (7) which is a combination of the explicit and the implicit scheme. 3.1 Implementation The implementation of the CN scheme requires only very few changes with respect to the implementation of the implicit scheme. Instead of a single iteration matrix now two matrices A and B have to be assembled: beamprop_cn.m 32 % generate iteration matrices 33 % A = I -.5* dz*l and B = I +.5* dz*l 34 B =.5* dz* bpm_operator (dx,n,nd, lambda ); 35 A = speye ( Nx) - B; 36 B = speye ( Nx) + B; The update of the fields in the iteration loop has to be adapted to equation (7) as well: 42 for iter = 1 : Niter beamprop_cn.m 43 % advance field : solve tridiagonal system of equations 44 % parentheses around ( B* v_ curr ) are very important for efficiency 45 v_curr = A\(B* v_curr ); 46 % check if current iteration has to be stored 47 if mod( iter, output_ step ) == 48 curr_ slice = curr_ slice + 1; % move to next output slot 49 v_out (:, curr_ slice ) = v_ curr ; % store current solution 5 end 51 end The parentheses around the expression (B*v_curr) in line 45 are very important! Without them, Matlab left-divides B by A before multiplying with v_curr. This means solving N x linear systems and results in more densely populated matrix that has to be multiplied with v_curr. With parentheses, v_curr is multiplied with B first and a single tridiagonal system is solved with the resulting vector afterwards. This is much more efficient. 7

8 lateral position [µm] real part of field Crank-Nicolson scheme propagation distance [µm] Figure 7: Evolution of the field propagated with the Crank-Nicolson scheme using a propagation step size of Δz =.5 µm and a lateral step size of Δx = x a /(N x 1) =.2 µm. 3.2 Results The evolution of the field in the test geometry upon porpagation with the CN scheme is shown in figure 7. As the CN scheme is more accurate than the implicit scheme, the propagation step size was increased to Δz =.5 µm. The results are nearly identical to the result of the implicit scheme shown in figure 4. 4 Comparison of the Solution Schemes In the following sections we will take a closer look at the various aspects of the different schemes 4.1 Conservation of Energy First, we have a look sum of the squared magnitude of the field N x E(z) = v i (z) 2. (8) i=1 The test geometry does not contain absorbing materials or open boundaries. Hence, E(z) should be conserved upon propagation. Figure 8 shows the evolution of E(z) along the propagation distance for the different solution schemes and various propagation step sizes. Only the Crank-Nicolson scheme is energy conserving. The explicit scheme exhibits an exponential growth due to the instability that causes an amplification of high frequency noise. The amplification is stronger for larger propagation step sizes as it is indicated by equation (5). Also the implicit method does not conserve the field energy. Instead the solution is damped upon propagation. The damping is stronger for larger propagation step sizes. 8

9 sum of squared magnitude explicit scheme, dz=2.5nm explicit scheme, dz=1.nm implicit scheme, dz=1nm implicit scheme, dz=1nm Crank-Nicolson scheme, dz=5nm propagation distance [µm] Figure 8: Comparison of the total power of the field within the computational domain along the propagation distance for the different solution schemes and different propagation step sizes. 4.2 Covergence Finally, we have a look of the influence of the discretization onto the accuracy of the obtained solution. As the explicit method is unstable it does not make sense to investigate its accuracy. Only the implicit scheme and the Crank-Nicolson scheme are considered in the following. First, we will have a look at the influence of the propagation step size. To avoid any impact of the geometry we replace the discontinuous step index waveguide with a smooth Gaussian index distribution n(x) = exp ( x2 w 2 n ) (9) with w n = 2 µm. To avoid boundary effects, the width of the computational domain is chosen larger than the propagation length (x a = 5 µm, z end = 2 µm). The relative error e is given by the square root of sum of the squared magnitude of the difference between the propagated field v(z end ) and the true result v (z end ): N x e = v i (z end ) v i (z end ) 2. (1) i=1 The initial field was normalized to have a sum of the squared magnitude equal to unity. The true result was approximated by the result of the Crank-Nicolson scheme at the smallest step size. The lateral step size was set to Δx = 5 nm. Figure 9 shows the convergence of the two methods with respect to the propagation steps size together with according power law fits. As expected from the order of the finite-difference approximation of the z-derivative, the implicit scheme exhibits only a linear convergence while the Crank-Nicolson scheme converges quadratically. At very small steps sizes the Crank-Nicolson scheme is limited by the lateral resolution. Next, we investigate the influence of the lateral resolution. As the number of grid points is not constant in this experiment, the relative error cannot be calculated with equation (1). Instead it is 9

10 relative error relative error implicit scheme power law fit, k=.97 Crank-Nicolson scheme power law fit, k= implicit scheme Crank-Nicolson scheme power law fit, k= propagation step size / lateral step size lateral step size / propagation step size Figure 9: Convergence of the implicit and the Crank-Nicolson scheme in dependence of the propagation step size Δz. A Gaussian refractive index distribution with w n = 2 µm was assumed instead of the step index distribution of the original test geometry. The lateral step size was set to Δx = 5 nm. Figure 1: Convergence of the implicit and the Crank-Nicolson scheme in dependence of the lateral step size Δx. A Gaussian refractive index distribution with w n = 2 µm was assumed instead of the step index distribution of the original test geometry. The propagation step size was set to Δz = 5 nm. approximated by the value of the field in the center of the waveguide: e = v (N x 1)/2+1(z end ) v (N x 1)/2+1 (z end). (11) v (N x 1)/2+1 (z end ) For both methods the propagation step size was set to Δz = 5 nm. The results are shown in figure 1. Both schemes use the same discretization of the second derivative with respect to x, hence both schemes exhibit the expected quadratic convergence. However, the accuracy of implicit scheme is limited much earlier by the propagation step size than the accuracy of the Crank-Nicolson scheme. How does the convergence change when we use the original step index waveguide instead of the smooth Gaussian index distribution? The answer can be found in figures 11 and 12. The convergence with respect to the propagation step size is slowed for both methods but the Crank-Nicolson scheme still converges faster than the implicit scheme. The convergence with respect to the lateral resolution is also slowed and becomes very irregular as the actual width of the waveguide cannot be represented correctly at all resolutions. This example shows, that the true convergence of a numerical method is not only influenced by its mathematical properties but also by the actual geometry of the problem. Hence, convergence should be tested for each geometry individually. 1

11 relative error relative error implicit scheme power law fit, k=.63 Crank-Nicolson scheme power law fit, k= implicit scheme Crank-Nicolson scheme power law fit, k= propagation step size / lateral step size lateral step size / propagation step size Figure 11: Convergence of the implicit and the Crank-Nicolson scheme in dependence of the propagation step size Δz using the step index distribution of the original test geometry. The lateral step size was set to Δx = 5 nm. Figure 12: Convergence of the implicit and the Crank-Nicolson scheme in dependence of the lateral step size Δx using the step index distribution of the original test geometry. The propagation step size was set to Δz = 5 nm. 11