1. Beam Propagation Method Devang Parekh 3/2/04 EE290F

2. Outline What is it? FFT FDM Conclusion

3. Beam Propagation Method Used to investigate linear and nonlinear phenomena in lightwave propagation Helmholtz’s Equation

4. BPM (cont.) Separating variables Substituting back in

5. BPM (cont.) Nonlinear Schrödinger Equation Optical pulse envelope Switch to moving reference frame

6. BPM (cont.) Substituting again First two-linear; last-nonlinear

7. Fast Fourier Transform (FFTBPM) Use operators to simplify Solution

8. Fast Fourier Transform (FFTBPM) A represents linear propagation Switch to frequency domain

9. Fast Fourier Transform (FFTBPM) Solving back for the time domain Plug in at h/2

10. Fast Fourier Transform (FFTBPM) Similarly for B(nonlinear) Using this we can find the envelope at z+h

11. Fast Fourier Transform (FFTBPM) Three step process 1. Linear propagation through h/2 2. Nonlinear over h 3. Linear propagation through h/2

12. Fast Fourier Transform (FFTBPM) Numerically solving Discrete Fourier Transform Fast Fourier Transform Divide and conquer method

13. Fast Fourier Transform (FFTBPM) Cool Pictures

14. Fast Fourier Transform (FFTBPM)

15. Finite Difference Method (FDMBPM) Represent as differential equation Apply Finite Difference Method

16. Finite Difference Method (FDMBPM)

17. Cool Pictures

18. Finite Difference Method (FDMBPM)

19. Conclusion Can be used for linear and nonlinear propagation Either method depending on computational complexity can be used Generates nice graphs of light propagation

20. Reference Okamoto K. 2000 Fundamentals of Optical Waveguides (San Diego, CA: Academic)