Dispersion Compensation Using Spatial Light Moulator

1. Dispersion Compensation Using Spatial Light Moulator Optical Communication Group Mahdieh Bagher Shemirani Spring 2008

2. Outline Introduction System configuration Review of the problem Previous work Looking for better solutions Simulation issues Results Discussion and summery

3. Introduction: • Modal dispersion introduces ISI • Dominant limitation to the bit rate-distance product • Electrical equalization can be used to mitigate ISI • Leads to noise amplification and a worse BER • It has been shown that a pulse launched in one principal mode gets to the other end of fiber as a single pulse • Using adaptive optics via an SLM we can shape the spatial profile of light into one of these principal modes • Advantageous to electrical equalization in that no noise is amplified

4. System configuration

5. Problem definition:I. Impulse response in terms of input signal • Projecting the input field into the 2M input principal modes: • The impulse response can be written as the delayed version of the power of principal modes • Which gives us the impulse response operator as: • Impulse response is derived as: Summarized from: Panicker et al. “Compensation of Multimode Fiber Dispersion using Adaptive Optics via Convex Optimization”,…

6. 32 32 Problem definition:II. SLM Characteristics • Light in the form of 0th order Hermite-Gaussian coming from a SMF is incident on SLM, • SLM is a passive device with a 2D complex reflectance function written in terms of N independent basis functions • So light entered MMF can be written as: • Where is the element wise product of SLM function with the input Gaussian beam

7. Problem definition:I. Impulse response in terms of input signal

8. p(t) h(t) r(t) g(nT,t0) t0+nT ISI and eye opening • Maximizing the eye opening is equivalent to minimizing ISI • It can be shown that in a practical system where polarization mode dispersion is weak Summarized from: Panicker et al. “Compensation of Multimode Fiber Dispersion using Adaptive Optics via Convex Optimization”,…

9. Previous work: Problem 0 • Maximizing signal to interference ratio • not convex, • generalized rayleigh quotient, • has global minimum, • not very efficient, • adding noise term doesn’t guarantee an optimal solution • Maximizing the minimum distance between 0 and 1 • not convex, • dual problem always convex SDP, • gives an upper bound on the achievable minimum distance, • optimal SLM pattern would just change the phase and not the magnitude Drawbacks: • This is a heuristic approach which doesn’t model all the physics of the system rather dealing with random matrices • the optimization problem has very high computational intensity • It is argued that although this problem has a global solution it has many other local optimal points • There is no garauntee for the algorithms to converge Alon et al. “Equalization fo Modal Dispersion in Multimode Fiber using Spatial Light Modulators”,…

10. Previous work: Problem 1 • Maximizing the eye opening • Equivalent to minimizing the ISI • Considering the physics of the system theproblem can be changed to a SOCP • Efficient • The algorithm used to solve the problem using the measurements of eye opening is SCA Some Issues: • Although the physics of the system has been carefully taken into account no way has been proposed to calculate the principal modes of the system, instead they are derived by random coupling of the ideal modes • Each basis function is defined to be 1 in a block consisting of a set of pixels and zero in the rest. • The accuracy depends on the number of blocks (maximum number=128*128) • Increasing the number of blocks increases the number of constraints, longer execution time to find the solution and also for algorithms to converge • Block by block adaptation algorithms are slow since each block is adapted at one time • Choosing block by block basis function to approximate the principal modes which are a combination of Hermite-Gaussian ideal modes may not be the best approach Summarized from: Panicker et al. “Compensation of Multimode Fiber Dispersion using Adaptive Optics via Convex Optimization”,…

11. results • SLM is divided into 8*8 blocks • The P matrix is calculated by random coupling • Optimal SLM pattern is calculated via convex optimization N=64

12. Looking for better approaches • To the first order PMs are a linear combination of ideal Hermite-Gaussian modes • Using these modes as our SLM basis function may help to get better results with less number of modes

13. Redefining the convex optimization problem: Problem 2 • The problem has infinite number of constraints • Can be transformed to infinitely constrained SOCP • Yet we can solve it for finite number of sampled constraints and check that the constraints hold for all points • We choose one sampling point in each 4*4 block => L=32*32

14. P matrix and normalization • The P matrix is different from the previous case in that the one dimensional SLM basis functions are Hermite-Gaussians instead of spatially distributed block functions • Also we need to normalize the P matrix so that with a blank SLM, total power in the modes is 1 • The problem is that unlike to the previous problem our Hermite-Gaussian functions are not enough to define a blank SLM (the set is not complete) • So we always lose some energy by projecting the blank SLM on our basis functions

15. Checking the constraints • At the end of simulation we need to check if the constraints are valid for all the points on SLM, unfortunately this never happens • At some points that the Hermite-Gaussian functions have maximum magnitude we can get constraint values from 1.003 to 1.6 depending on the mode radius of the basis functions • By changing the mode radius we can get better performance while enhancing the constraints

16. SLM with Block by block basis function N=36 N=64

17. SLM with hermite-Gaussian basis functions N=36 N=64

18. N=12*12

19. N=11*11

20. N=10*10

21. N=9*9

22. N=8*8

23. N=7*7

24. N=6*6

25. N=5*5

26. N=4*4

27. N=3*3

28. N=2*2

29. N=7*7

30. N=7*7

33. Effect of power loss in normalization of a blank SLM

34. Summery of results • Depending on N, the optimal value of eye opening happens at different mode radius. • For very large N the optimum happens at mode radius a little more than that of MMF, also optimal eye opening for problem 2 converges to that of problem 1 • As we decrease N the mode radius required for maximizing the eye pattern decreases, at some point problem 2 gives a better optimal value than problem 1 • Due to loss of energy for normalization, reducing N further results in worse performance for problem 2 • for very small N the mode radius is less than that of SMF, the performance of problem 2 gets better for very small values of N

35. Future work • Resolving the normalization problem in order to have comparable results. • Resolving the issue of change in mode radius due to the fact that input signal is gaussian itself with a different mode radius than that of MMF and SLM is acting as a matching device between the two fibers • Resolving the problem with constraints not met for all points on SLM • Using different functions than Hermite-Gaussians

36. Acknowledgment: • Professor Joseph M. Kahn • Rahul Panicker • Alan Lau