Beam propagation method for waveguide device simulation

Transcription

1 1/29 Beam propagation method for waveguide device simulation Chrisada Sookdhis Photonics Research Centre, Nanyang Technological University This is for III-V Group Internal Tutorial

2 Overview EM theory, scalar and vectorial wave equation Finite-difference BPM Optiwave BPM Examples 2/29

3 Beam propagation method (BPM) Beam propagation method is a mathematical procedure used to study the evolution of electromagnetic fields in arbitrary inhomogenious medium. BPM yields the response of a given device to an external optical sigal, similar to in an experiment. 3/29 Applications Devices that defy eigenmode study, e.g. tapers Avoids difficult approximations Cases where radiation is important. Behaviours of special devices.

4 Starting from Maxwell s Equations (CGS units) E = 1 H c t 4π M c t H = 1 E c t + 4π P c t + 4π c J E = 4π P + 4πρ H = 4π M (1) 4/29 Assumptions: Non-magnetic materials M = 0, no charge ρ = 0, zero current J = 0, linear isotropic media P = χ e E, e iωt time dependence. where ɛ = 1 + 4πχ e = n 2 E = ik 0 H H = ik 0 ɛe ɛe = 0 H = 0, (2)

5 From (2), we can eliminate the electric or magnetic field to arrive at the Helmholtz equations ( E) = k 2 0ɛE (3) 1 ɛ ( H) = k2 0 H (4) 5/29 E = E t + E z ẑ, H = H t + H z ẑ and = t + z

6 E formulation Equation (3) can be written as z-invariant media 2 E + k 2 0ɛE = ( E) ɛ z E z 0 2 E t + ɛk 2 0 E t = t ( t ln ɛ E t ) (5) or in terms of x and y components: 2 E x + ɛk 2 0 E x = ( ) ln ɛ x x E x ( ) ln ɛ x y E y 2 E y + ɛk 2 0 E y = ( ) ln ɛ y x E x ( ) ln ɛ y y E y 6/29

7 Slowly varying envelope approximation E t = Ê t e in 0k 0 z, 2 Ê t z 2 2n Ê t 0k 0 z 7/29 With above approximations, the electric field s evolution is given by Vectorial BPM i z where (for illustration) (Êx Ê y ) A xy Ê y = 1 2n 0 k 0 = ( x ( ) ) Axx A xy (Êx A yy Ê y A yx [ 1 ɛ ] (ɛê y ) y ) 2 Ê y x y A xx =, A yy =, A yx = (6)

8 Semi-vector BPM i Ê x z = A xxêx i Ê y z = A yyêy (7) 8/29 This taks into account the polarization (A xx A yy ), but neglects the coupling between E x and E y (A xy = A yx = 0). When variation of refractive index is small in transverse dimension, polarization dependency and coupling are weak and may be neglected. It is safe to treat the two polarizations as decoupled, as long as they are not synchronized with some mechanism within the device.

9 Scalar BPM i E z = A scalare (8) 9/29 where A scalar = 1 2n 0 k 0 ( ) 2 x y + (ɛ 2 n2 0)k 2 0 (8) governs the conventional scalar beam propagation method. This is useful where device is weakly-guiding and/or polarization dependence can be neglected.

10 Finite-difference scheme In beam propagation 10/29 we apply the propagator U to calculate z E t (z + z) = U( z)e t z+ z Propagator U can take many forms, depending on the chosen BPM technique, e.g.

11 Paraxial/Wide angle Vectorial/Scalar BPM Boundary conditions 11/29 Discretization x n-1 n n+1 2D FD-BPM m+1 m m-1 n-1 n n+1 y 3D FD-BPM

12 Finite-difference BPM x x m+1 m m-1 12/29 m+1 m m-1 n-1 n n+1 y z+ z n-1 n n+1 y z Discretize the device volume and replace differentiations in BPM propagator with difference operator. x = x m+1 x m f m (x) = f (x m ) f m x = f m+1 f m 1 2 x 2 f m x = f m+1 2f m + f m 1 2 x 2

13 For example A xy Ê y = 1 2n 0 k 0 ( x [ 1 ɛ ] (ɛê y ) y ) 2 Ê y x y becomes ([ ] 1 ɛ(m + 1, n + 1, l) A xy Ê y = 1 Ê y (m + 1, n + 1) 8n 0 k 0 x y ɛ(m + 1, n, l) [ ] ɛ(m + 1, n 1, l) 1 Ê y (m + 1, n 1) ɛ(m + 1, n, l) [ ] ɛ(m 1, n + 1, l) 1 Ê y (m 1, n + 1) ɛ(m 1, n, l) [ ] ) ɛ(m 1, n 1, l) 1 Ê y (m + 1, n 1) ɛ(m 1, n, l) All are known quantities. ɛ is from device definition, while Ê y is from the last calculation step. 13/29

14 So we replace i z (Êx Ê y ) = ( ) ) Axx A xy (Êx A yy Ê y A yx with 14/29 E t (z + z) = U FD-BPM ( z)e t (9) This has been a simplest discussion, the method entails much more finer points and best left to experts or commercial software developers. Things we have not consider: Iterative algorithms for solving the coupled equations Speed Boundary conditions Stability

15 Optiwave OptiBPM Device definition 15/29

16 Profile definition 16/29 This is a new implementation from version 5 of OptiBPM. The material and profile library can be called from many layout files.

17 17/29

18 2D vs. 3D FD-BPM Comparing the time complexity O of 2D- and 3D-BPM, O 3D O 2D n x (10) It is recommended to use 2D algorithm whenever possible. So when is it possible? Device is actually 2D. (slab waveguides) Using Effective index method (EIM) for Weakly guiding structures with low level of radiation Full-vectorial BPM is not required ε(x,y) ε(y) 18/29

19 19/29 After calculation, the electric field distribuion within the device is displayed. We can choose to open the Analyzer module for data analysis.

20 Examples of data presentation 20/29

21 21/29

22 Waveguide mode calculation 3 micron 22/29 3 micron Waveguide modes are solved using Alternate Direct Implicit (ADI) method. Users can specify The solver engine (Vectorial, scalar) Boundary conditions (Transparent, Neumann) Accuracy of the resultant mode field (e.g. to 1E-007) whether to start with the fundamental mode or any other mode.

23 23/29 Waveguide mode summary table

24 Star couplers 24/29

25 25/29

26 3dB couplers 26/29

27 Scanning script Const NumIterations = 7 d = 0 For x = 1 to NumIterations ParamMGR.SetParam "offset", CStr(d) ParamMgr.Simulate WGMgr.Sleep(5) d = d Next ParamMGR.SetParam "offset", 0 27/29

28 28/29 Scanning script results. Best performance is between iteration 3 and 4. So we know the separation should be 2.5 µm.

29 The End We have discussed 29/29 What is BPM The technical backgrounds behind BPM Optiwave BPM and its features Device definition Path monitor and output extractions Waveguide Mode solver Parameterization and scripting for optimization of device Q & A