Equalization of Modal Dispersion in Multimode Fiber Using Spatial Light Modulators

1. Equalization of Modal Dispersion in Multimode Fiber Using Spatial Light Modulators Elad Alon, Vladimir Stojanović, Joseph M. Kahn, Stephen Boyd, and Mark Horowitz

2. Multimode Fiber • Multimode is dominant type of fiber installed in current LANs • Existence of many modes is biggest advantage, and also biggest weakness • Easier to couple light into the fiber • Modes have different group velocities: modal dispersion

3. Modal Dispersion • Often characterize fiber optic systems by bit-rate·distance • Modal dispersion limits multimode fiber to ~1-3 Gb/s/km • Lots of previous work on compensation schemes • Electrical equalization, pre-generated optical masks • Noise enhancement, can’t be adapted

4. Lens Spatial light modulator Equalization Using Spatial Light Modulators • Control modes by shaping E-field at fiber input • Lens performs Fourier Transform, SLM performs filtering in spatial frequency • Refocus light into desired modes – no SNR penalty • Best SLM settings fiber and time dependent • Find efficient ways to optimize

5. Multimode Fiber Model (I) • Modes ideally orthogonal, but imperfections cause mode-mixing • Can still find orthogonal principal modes1 • This work: ideal, LP modes of weakly guiding step-index multimode fiber • Analysis relies only on mode orthogonality • Short fiber: negligible chromatic and polarization mode dispersion 1 S. Fan, J. M. Kahn, “Principal modes in multi-mode waveguides,” to be published in Optics Letters

6. Multimode Fiber Model (II) • Fiber represented as a matrix: • Each column one mode in sampled spatial frequency # of pixels # of modes

7. Communication System Model • SLM sets field projected onto fiber • Vector in spatial frequency: • Light field in basis of fiber modes: • Receiver detects intensity • Signal is quadratic function of SLM settings • Impulse response: Received signal Diagonal matrixwith mode delays

8. SLM Constraints • Passivity: • Usually can’t affect both magnitude & phase • Magnitude only: • Phase only: • Often limited precision • Binary phase-only:

9. Objective Function: Peak Signal-to-Interference Ratio • Standard SINR is ratio of 4th order polynomials, hard to optimize • Try peak SIR instead: • Non-convex, but has global solution • Extremely inefficient when scaled to meet passivity constraint • Lose global solution if add noise Pulse response

10. Maximize minimum distance instead: Objective Function: Minimum Distance dmin • With constraints: • Easy to show that optimal achieved when each pixel modifies only phase (use all energy) • But P is indefinite, so non-convex problem • Need heuristics

11. Dual Problems Dual Problem: Dual of the Dual: • Formulate dual problems to find heuristics and get performance bounds • Both problems are semidefinite programs1 (SDPs) with global solutions 1 S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge University Press, 2003

12. SDP-based Heuristic Dual of the Dual: • Dual of the dual same as for two-way partitioning • Same as our problem, but with • Well-known heuristic: • Solve SDP to find X • Use X as a covariance matrix to randomly generate set of vectors xfrom a zero-mean, normal dist. • Take sign of x and pick best solution • Easy to extend to our complex variables • Use X to independently generate real and imaginary components, and keep only resulting phase

13. Single-Coordinate Ascent • Motivated by simple hardware implementation • Algorithm: • Perturb one pixel by a small amount, measure dmin • Choose pixel setting with highest dmin • Move to next pixel, repeat • Interesting initial conditions for SCA: • Setting generated by SDP-based heuristic • Default setting - light focused on center of fiber

14. Example System • 1 km of 64 μm core, 0.2% step-index profile • 850nm laser @ 10 Gb/s • 88 modes (ignoring polarization) Fiber: SLM: • 20x20 phase-only • 1º phase resolution

15. Optimized Pulse Responses • Energy literally refocused • All 3 optimized results within 7% of upper bound

16. Optimized Field Patterns Default Setting SCA on Default Setting • Pulse responses similar, but fields different • Algorithms choose linear combination of modes that arrive at same symbol time • Lots of flexibility in choices • Probably why SCA works so well (a) (b) SDP-based Heuristic SCA on SDP-based Result

17. Performance Tradeoffs:Number of Pixels • SLM with 1° phase resolution • Diminishing returns above 400 pixels

18. Performance Tradeoffs:Phase Resolution • 20x20 pixels, binary phase control looks attractive • Less than 10% performance loss from 1° resolution 400 pixels

19. MIMO Over Multimode Fiber • Modal dispersion analogous to multi-path in wireless communications • Use orthogonality of modes to increase capacity with multiple channels

20. Example 2x2 Optimization Results • Results from using SCA to maximize worst dmin are promising • Negligible ISI and ICI • Slight loss in amplitude from extra filter

21. Conclusions • Light field processing with SLMs looks very promising to combat modal dispersion • Minimum distance cost function is non-convex, but can be effectively optimized to refocus signal energy • Both SDP-based heuristic and simple SCA perform extremely well • Can exploit available modal orthogonality to increase capacity with MIMO techniques