MIMO and Mode Division Multiplexing in Multimode Fibers

Transcription

1 MIMO and Mode Division Multiplexing in Multimode Fibers Kumar Appaiah Department of Electrical Engineering Indian Institute of Technology Bombay Tutorial: National Conference on Communications 2017, Chennai 1

2 Prerequisites Basic EE, physics Maxwell s equations Linear system theory, communication theory Optics, reflection, refraction etc. 2

3 Outline Introduction Basic optical fiber concepts Propagation characteristics MIMO, MDM in multimode fibers Recent trends, adoption, future Summary 3

4 Introduction Optical fibers: waveguides that carry EM waves High bandwidths, long distance carrying abilities Challenges: alignment, dispersion, nonlinearities Insert picture here 4

5 Electromagnetic spectrum Frequency (Hz) Gamma-rays Wavelength 0.1 Å X-rays 1Å 0.1 nm 1 nm 400 nm 10 nm Ultraviolet Visible Near IR 100 nm 1000 nm 1 µm 500 nm 600 nm Infra-red 10 µm MHz Thermal IR Far IR 100 µm 1000 µm 1 mm 700 nm 500 MHz UHF Microwaves Radar 1 cm 10 cm VHF Radio, TV 1 m 100 MHz 50 MHz FM VHF AM 10 m 100 m Long-waves 1000 m 5

6 Optical fibre Guided medium High bandwidth Large bandwidth-length product Examples: Long undersea links Medium haul links Short links: data centres, vehicles 6

7 Maxwell s equations E = B t H = D t (1) (2) D = 0 (3) B = 0 (4) where D = ϵe, B = μh. 7

8 Maxwell s equations Take curl of equation (1) Identity: ( E) = μ t ( H) = μϵ 2 E t 2 ( E) = ( E) 2 E Using E = 0, we get the wave equations (repeated for H): 2 E = μϵ 2 E t 2 2 H = μϵ 2 H t 2 8

9 Waveguide equations Using cylindrical coordinate system: E(r, ϕ, z) = E 0 (r, ϕ)e j(ωt βz) H(r, ϕ, z) = H 0 (r, ϕ)e j(ωt βz) β: z-component of propagation vector, determined by the boundary conditions. Substitute above in the curl equations. Use in cylindrical coordinates: 1 1 rêr ê ϕ rêz A = det r ϕ z A r A ϕ A z 9

10 Waveguide equations 1 E z r ( ϕ + jrβe ϕ ) = jωμh r jβe r + E z r = jωμh ϕ 1 r ( r (re ϕ) E r ϕ ) = jμωh z 1 H z r ( ϕ + jrβh ϕ ) = jωμe r jβh r + H z r = jωμe ϕ 1 r ( r (rh ϕ) H r ϕ ) = jμωe z Six fields are coupled, eliminate to keep only H z and E z. 10

11 Waveguide equations E r = j q 2 ( β E z r + μω r H z ϕ ) E ϕ = j β E z q 2 ( r ϕ μω H z r ) H r = j q 2 ( β H z r ϵω r E z ϕ ) H ϕ = j β H z q 2 ( r ϕ + ϵω E z r ) We now substitute the above into 2 E = μϵ 2 E t 2, to get 2 E z r r E z r + 1 r 2 2 E z ϕ 2 + q2 E z = 0 where q 2 = ω 2 μϵ β 2. 11

12 Waveguide equations Separation of variables: let s see if this works: E z = AF 1 (r)f 2 (ϕ)f 3 (z)f 4 (t) where F 3 (z)f 4 (t) = e j(ωt βz) by prior assumption. By symmetry, we also have F 2 (ϕ) = e jνϕ. Finally, for F 1 (z), we have: 2 E z r r E z r + 1 r 2 2 E z ϕ 2 + q2 E z = 0 2 F 1 r r F 1 r + ( q2 ν2 r 2 ) F 1 = 0 Solution of above differential equation: Bessel functions (Please refer) 12

13 Boundary conditions Boundary conditions lead to Bessel functions J ν, K ν. 1. The field can t be at the centre. 2. The field has to die far away from the centre (cladding). Solution with boundary conditions: E z (r < a) = AJ ν (ur)e jνϕ e j(ωt βz) H z (r < a) = BJ ν (ur)e jνϕ e j(ωt βz) E z (r > a) = CK ν (wr)e jνϕ e j(ωt βz) H z (r > a) = DK ν (wr)e jνϕ e j(ωt βz) where u 2 = k 2 1 β 2, w 2 = β 2 k 2 2. We need F 1 to be real everywhere. So, u 2 0, v 2 0 k 2 β k 1. 13

14 Boundary conditions At the core-cladding interface, E ϕ, E z, H ϕ, H z must be continuous. AJ ν (ua) = CK ν (ua) BJ ν (ua) = DK ν (wa) Similarly, E ϕ, H ϕ can be found from E z, H z, and the continuity condition may be obtained as 1 u 2 [ Ajνβ a J ν(ua) BωμuJ ν(ua) ] = 1 w 2 [ Cjνβ a K ν(wa) DωμwK ν(wa) ] 1 u 2 [ Bjνβ a J ν(ua) + Aωϵ 1 uj ν(ua) ] = 1 w 2 [ Djνβ a K ν(wa) + Cωϵ 2 wk ν(wa) ] 14

15 Boundary conditions Consolidating above equation, we get det J ν (ua) 0 K ν (wa) 0 βν au J jωμ βν ν(ua) 2 u J ν(ua) aw K jωμ ν(wa) 2 w K ν(wa) 0 J ν (ua) 0 K ν (wa) jωϵ 1 u J ν(ua) βν au 2 J ν(ua) jωϵ 2 w K ν(wa) βν aw 2 K ν(wa) = 0 J ν(ua) ( uj ν (ua) + K ν(wa) k 2 1J ν(ua) wk ν (wa)) ( uj ν (ua) + k2 2K ν(wa) wk ν (wa) ) = ( 2 βν a ) Solving above for β gives us the values of β that correspond to the propagating modes of the fibre. 1 ( u w 2 ) 2 15

16 Fibre modes Inspecting the equation for ν = 0, we get J 0(ua) ( uj 0 (ua) + K 0(wa) k 2 1J 0(ua) wk 0 (wa)) ( uj 0 (ua) + k2 2K 0(wa) wk 0 (wa) ) = 0 which implies that J 0(ua) ( uj 0 (ua) + K 0(wa) wk 0 (wa)) = 0 TE modes k or 2 1J 0(ua) ( uj 0 (ua) + k2 2K 0(wa) wk 0 (wa) ) TM modes = 0 Proof that these are TE/TM: homework. For ν 1, solution is more complex. 16

17 Fibre modes Convenience substitutions: Normalized frequency or V-number, defined by: 2 2 2πa 2πa V 2 = (u 2 + w 2 )a 2 = (n ( λ 2 1 n 2 2) = NA 2 ) ( λ ) Normalized propagation constant b, defined as: b = a2 w 2 V 2 = (β/k)2 n 2 2 n 2 1 n

18 Linearly polarized modes In the weakly guiding case (i.e. Δ << 1 or n 1 n 2 << 1), several mode pairs are almost degenerate. That is, HE ν+1,m and EH ν 1,m are very similar. Similar is the case for TE 0m, TM 0m and HE 2m. So, only four field components need to be considered, instead of six. Linearly polarized modes: Simplification 1. LP 0m is derived from HE 1m mode 2. LP 1m is derived from HE 2m, TE 0m and TM 0m modes 3. LP νm for ν 2 is derived from an HE ν+1,m and an EH ν 1,m mode 18

19 Fibre modes

20 Ideal modes Ideal modes have specific field distributions Different modes may have different propagation constants Orthonormal: overlap integral zero Provide spatial degrees of freedom 20

21 Ideal modes are orthogonal E p q (x, y)e pq (x, y)dxdy = 0 if p p or q q = 1 if p = p and q = q Any arbitrary pattern propagating in the fiber can be represented as the linear combination of ideal modes. Since each mode can be in x polarized or in y polarized, pattern E(x, y) propagating inside fiber (supporting M spatial modes) can be written as E 2M 1 = [a x 1 a x 2 a x M a y 1 a y 2 a y M] T 21

22 Types of fiber Criterion Single-mode Few-mode Multimode Core diameter < 10 μm < 14 μm 50 μm Alignment tolerance 0.5 μm 0.5 μm >3 μm Data rate limits 100 Gb/s-km > 200 Gb/s-km 2 Gb/s-km Length scales km km 10 m - a few km Materials Silica Silica Silica, plastic 22

23 Comparison of SMF and MMF Single Mode Fiber Multimode Fiber SMF: no modal dispersion MMF: easier to couple Tradeoff: bandwidth-length product 23

24 Modal Dispersion Detector Ideal MMF t Laser t 24

25 Modal Dispersion Different modes travel at different speeds Different have different spatial distributions Modes detected at the photodetector retain individual propagation delays Pulse spreading: dispersion, limits data rates Conventional MMFs: BW-length product below 1 Gb/s km 25

26 Selective excitation of modes Detector SLM IM Pattern Ideal MMF t Laser t Computer 26

27 Intermodal coupling SLM IM Pattern Imperfect MMF Detector t Laser t Computer 27

28 Multiplexing using fiber modes Idea: Shape inputs to match each mode shape Shape outputs to match each mode shape Ideal fiber: spatial multiplexing! Issue: Intermodal coupling mixes the signal 28

29 Handling mode mixing Mode mixing: linear or nonlinear? Coherent or incoherent? Depends on what? Fiber length? What else? Complexity of implementation? 29

30 Next topics Incoherent multiplexing techniques Coherent multiplexing techniques Mode generation, shaping DSP implementation issues 30

31 Incoherent systems Incoherent in laser Like amplitude modulation 31

32 Incoherent multiplexing Stuart: dispersive multiplexing Offset coupling based solutions Mode group diversity multiplexing Issues and limitations 32

33 Incoherent multiplexing Laser Source 1 Intensity Modulator 1 Detector 1 Detector 2 Laser Source 2 Intensity Modulator 2 Mode Multiplexer Mode De-Multiplexer Laser Source M Intensity Modulator M Detector M 33

34 Mode group diversity multiplexing Idea: Modes with close β values travel together, so use them to multiplex Intermodal mixing: linear methods help reduce cross-talk 34

35 Coherent multiplexing Coherent: laser + polarization recovered at receiver Requires PLL (optical/electrical) or homodyne Improves receiver sensitivity Price: requires balanced detection Modulator Modulator POL MUX POL DEMUX Quadrature Optical Hybrid Balanced Detectors Digital Signal Processor Laser Local Oscillator 35

36 PSK and DPSK

37 PSK: Phase noise

38 Photonic Lantern Isolated SMFs New Cladding FMF The SMF ends act as spatial filter Different locations of SMFs causes coupling to different modes Post-processing required to separate modes 38

39 Free space coupling SMF Phase Plate Beam Splitters SMF FMF Mirror Phase Plate Lens 1 Lens 2 SMF 39

40 Free space coupling Mirror 1 SMF Periscope FMF SMF Mirror 2 Lens 1 Lens 2 SMF 40

41 Latest trends NEC Labs: 1.05 Petabit/s with multicore fiber Infinera: Advanced modulation, multiplexing Bell Labs: Several SDM experiments New fibers: OM4 (100GBASE over 150 metres) OM5 + SWDM: Further increase in data rates 41