mode-division multiplexing systems: propagation effects...

Mode-Division Multiplexing Systems: Propagation Effects, Performance

and Complexity

Joseph M. Kahn1

Keang-Po Ho2

1 E. L. Ginzton Laboratory, Stanford University2 Silicon Image, Inc., Sunnyvale, CA

OFC 2013 ● Anaheim, CA ● March 21, 2013

2

Outline

Mode coupling Sources, effects and models

Modal dispersion Principal modes Statistics of group delays MDM system complexity

Modal gains and losses Statistics of gains/losses and noise MDM system performance Frequency diversity

Discussion

Research funded byNSF ECCS-1101905, ECCS-0700899, and Corning, Inc.

3

Outline

Mode coupling Sources, effects and models

Modal dispersion Principal modes Statistics of group delays MDM system complexity

Modal gains and losses Statistics of gains/losses and noise MDM system performance Frequency diversity

Discussion

4

Modes and Mode Coupling

Terminology Number of modes D includes spatial and polarization degrees of freedom. Fiber types

Single-mode: D = 2 Few-mode or multi-mode: D = 6, 10, 12, 16, 20, 24, 30, … Coupled multi-core: D = 2Ncore (single-mode cores, large core spacing)

Unintentional coupling: often lowpass power spectrum |F()|2 4 to 8

Index profile variations Bends or twists Offset connectors Crosstalk in modal (de)multiplexers

Intentional coupling: power spectrum can be tailored Mode couplers or scramblers Interconnection of different fiber types Fiber perturbations: analogous to “spinning” used to reduce PMD in SMF

5

Effects of Mode Coupling

Bad Good

Direct detection

Induces modal dispersion.

Complicates mode-division multiplexing.

Reduces group delay spread in plastic fiber (although it increases loss).

Coherent mode-

division multiplexing

Necessitates full-rank signal processing.

Reduces group delay spread, reducing signal processing complexity.

Mitigates mode-dependent gain/loss, increasing capacity and reducing outage probability.

6

Multi-Section Field Propagation Model

A field is described by a vector .

Total propagation matrix (D × D):

Propagation matrix in kth section:

Unitary matrices describing mode coupling in kth section:

Matrix describing gain/loss and modal dispersion in kth section (ignoring CD):

D

iii yxAyx

1

,, EE TDAA ,,1 A

inA outA 1M 2M … 1KM KM… kM

121t MMMMM KK

Hkkkk UΛVM

kk VU ,

)()(21

)(1

)(12

1

ω

ω

0

0

kD

kD

kk

jg

jg

k

e

e

i uncoupled group delay

ige uncoupled power gain/loss

7

Regimes of Mode Coupling

Coupling Regime

Correlation length of

modal fields Described in

multi-section model Group delay or

gain/loss Eigenmodes of group delay or

gain/loss

Weak ~ total fiber length L

Small K or large K with U, V describing partial or spatially correlated coupling

Accumulate linearly with K or L

Superpositions of a few uncoupled modes

Strong << total fiber length L

Large K with U, V describing full, random coupling

Accumulate as √K or √L

Superpositions of many uncoupled modes

8

Key Effects in Long-Haul Mode-Division Multiplexing

Downconv.

DataIn …

Mod.

Mod.

Mod.

Mod.

…D

1

…

TxLaser

FixedModalMux Multi-Mode

FiberMulti-ModeAmplifier

Multi-ModeFiber

Multi-ModeAmplifier

… …

Downconv.

Downconv.

Downconv.…LO

Laser

FixedModalDemux

AdaptiveMIMO

Equalizer

D

1

…

DataOut

2D

1

2D

1

(WDM not shown)

9

Modal dispersion Different modes propagate with different group velocities in transmission fibers.

Analogous to multipath delay spread in wireless systems.

Does not fundamentally limit system performance. Affects MIMO signal processing complexity.

Essential for frequency diversity.

Key Effects in Long-Haul Mode-Division Multiplexing

Downconv.

DataIn …

Mod.

Mod.

Mod.

Mod.

…D

1

…

TxLaser

FixedModalMux Multi-Mode

FiberMulti-ModeAmplifier

Multi-ModeFiber

Multi-ModeAmplifier

… …

Downconv.

Downconv.

Downconv.…LO

Laser

FixedModalDemux

AdaptiveMIMO

Equalizer

D

1

…

DataOut

2D

1

2D

1

(WDM not shown)

10

Key Effects in Long-Haul Mode-Division Multiplexing

Downconv.

DataIn …

Mod.

Mod.

Mod.

Mod.

…D

1

…

TxLaser

FixedModalMux Multi-Mode

FiberMulti-ModeAmplifier

Multi-ModeFiber

Multi-ModeAmplifier

… …

Downconv.

Downconv.

Downconv.…LO

Laser

FixedModalDemux

AdaptiveMIMO

Equalizer

D

1

…

DataOut

2D

1

2D

1

(WDM not shown)

Mode-dependent loss and gain Can arise in transmission fibers or inline amplifiers.

Analogous to multipath fading in wireless systems.

Causes variations among SNRs of multiplexed signals.Reduces MIMO capacity and potentially causes outage.

Narrowband systems: diversity-multiplexing tradeoff.Wideband systems: frequency diversity can reduce outage probability.

11

Key Effects in Long-Haul Mode-Division Multiplexing

Downconv.

DataIn …

Mod.

Mod.

Mod.

Mod.

…D

1

…

TxLaser

FixedModalMux Multi-Mode

FiberMulti-ModeAmplifier

Multi-ModeFiber

Multi-ModeAmplifier

… …

Downconv.

Downconv.

Downconv.…LO

Laser

FixedModalDemux

AdaptiveMIMO

Equalizer

D

1

…

DataOut

2D

1

2D

1

(WDM not shown)

Strong mode coupling Reduces delay spread and signal processing complexity.

Mitigates mode-dependent loss and gain (in combination with modal dispersion).

12

Outline

Mode coupling Sources, effects and models

Modal dispersion Principal modes Statistics of group delays MDM system complexity

Modal gains and losses Statistics of gains/losses and noise MDM system performance Frequency diversity

Discussion

13

Principal Modes

Consider a D-mode fiber in weak- or strong-coupling regime. Assume mode-dependent loss is negligible. Neglect loss to simplify notation. The propagation operator is unitary: .

Define an input PM and the corresponding output PM:

such that if you fix and vary , is unchanged (to first order in ).

One can define a (Hermitian) group delay operator:

The D input PMs are eigenmodes of G with eigenvalues given by the coupled group delays:

The input and output PMs differ, since .

C. D. Poole and R. E. Wagner, Electron. Lett. 22, 1029 (1986).S. Fan and J. M. Kahn, Opt. Lett. 30, 135 (2005).

IMM tt H

in PM,iA

in PM,tout PM,ii AMA

in PM,iA

out PM,iA

Hj t

t

MMG

0MG t,

Diiii ,,1in PM,in PM, AGA

14

First-Order Modal Dispersion

Field pattern of each PM varies over frequency , and has a coherence bandwidth.

Very weak coupling: PMs similar to ideal modes, coherence bandwidth very large.

Strong coupling: coherence bandwidth of order 1/gd (coupled r.m.s. group delay).

For a signal occupying a small bandwidth near frequency , the overall propagation operator is:

Unitary matrices, independent of frequency (to first order in ):

The columns of U(t) and V(t) represent the input and output PMs.

Diagonal matrix describing propagation of PMs without crosstalk or modal dispersion:

First-order PMs form the basis for avoiding modal dispersion or mode-division multiplexing using direct detection.

Htttt UΛVM

tt , VU

)t(

)t(1

ω

ω

t

0

0

Dj

j

e

eΛ

15

Higher-Order Modal Dispersion

First-order: includes terms up to order in propagation operator M(t)().

Higher-order: includes terms of order 2 and higher.

Higher-order modal dispersion: Limits dispersion avoidance or spatial multiplexing using

frequency-independent devices.

and leads to:

Nonlinear relationship between input and output intensity waveforms. Filling-in and broadening of impulse response. Polarization- and spatial mode-dependent chromatic dispersion. Depolarization and spatial mode mixing of modulated signals.

Analogous to higher-order polarization-mode dispersion.

M. B. Shemirani and J. M. Kahn, J. Lightw. Technol. 27, 5461 (2009).

16

L = 1 kmD = 2 × 55

0.1 1 100

0.2

0.4

0.6

0.8

1.0

1.2

Gro

up d

elay

i(n

s)

Standard deviation of curvature (m1)

Principal Mode Group Delays in 50-m Graded-Index MMF with Spatial- and Polarization-Mode Coupling

M. B. Shemirani et al, J. Lightw. Technol. 27, 1248 (2009).

Weak coupling: GDs form degenerate groups. GD spread scales as L.

Medium coupling: GD degeneracies broken.

Strong coupling: GD spread reduced, scales as √L.

Very strong coupling: GDs diverge. Violation of assumptions made in analysis.

17

2468

10

12

14

16

18

20

L = 1 kmD = 2 × 55

0.1 1 100

0.2

0.4

0.6

0.8

1.0

1.2

Gro

up d

elay

i(n

s)

Standard deviation of curvature (m1)

Principal Mode Group Delays in 50-m Graded-Index MMF with Spatial- and Polarization-Mode Coupling

M. B. Shemirani et al, J. Lightw. Technol. 27, 1248 (2009).

Weak coupling: GDs form degenerate groups. GD spread scales as L.

Medium coupling: GD degeneracies broken.

Strong coupling: GD spread reduced, scales as √L.

Very strong coupling: GDs diverge. Violation of assumptions made in analysis.

18

L = 1 kmD = 2 × 55

0.1 1 100

0.2

0.4

0.6

0.8

1.0

1.2

Gro

up d

elay

i(n

s)

Standard deviation of curvature (m1)

Principal Mode Group Delays in 50-m Graded-Index MMF with Spatial- and Polarization-Mode Coupling

M. B. Shemirani et al, J. Lightw. Technol. 27, 1248 (2009).

Weak coupling: GDs form degenerate groups. GD spread scales as L.

Medium coupling: GD degeneracies broken.

Strong coupling: GD spread reduced, scales as √L.

Very strong coupling: GDs diverge. Violation of assumptions made in analysis.

19

L = 1 kmD = 2 × 55

0.1 1 100

0.2

0.4

0.6

0.8

1.0

1.2

Gro

up d

elay

i(n

s)

Standard deviation of curvature (m1)

Principal Mode Group Delays in 50-m Graded-Index MMF with Spatial- and Polarization-Mode Coupling

M. B. Shemirani et al, J. Lightw. Technol. 27, 1248 (2009).

Weak coupling: GDs form degenerate groups. GD spread scales as L.

Medium coupling: GD degeneracies broken.

Strong coupling: GD spread reduced, scales as √L.

Very strong coupling: GDs diverge. Violation of assumptions made in analysis.

20

L = 1 kmD = 2 × 55

0.1 1 100

0.2

0.4

0.6

0.8

1.0

1.2

Gro

up d

elay

i(n

s)

Standard deviation of curvature (m1)

Principal Mode Group Delays in 50-m Graded-Index MMF with Spatial- and Polarization-Mode Coupling

M. B. Shemirani et al, J. Lightw. Technol. 27, 1248 (2009).

Weak coupling: GDs form degenerate groups. GD spread scales as L.

Medium coupling: GD degeneracies broken.

Strong coupling: GD spread reduced, scales as √L.

Very strong coupling: GDs diverge. Violation of assumptions made in analysis.

21

Statistics of Principal Mode Group Delays

Assume: D modes (or can be a subset of D modes uncoupled to other modes)K >> 1 independent sections (strong coupling regime)Negligible mode-dependent loss or gain

Total propagation operator (D × D):

Propagation operator in kth section:

Random unitary matrices describing mode coupling:

Matrix describing uncoupled group delays (assume ):

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3119 (2011).

121t MMMMM KK

Hkkkk UΛVM

kk VU ,

)(

)(1

ω

ω

0

0

kD

k

j

j

k

e

e

Λ

01 kD

k

22

Statistics of Principal Mode Group Delays (2)

Total group delay operator:

By chain rule of differentiation, G is the sum of K i.i.d. random matrices having independent eigenvectors. By the Central Limit Theorem, G is a zero-trace Gaussian unitary ensemble. The statistical properties of its eigenvalues (thePM group delays) are known for 2 ≤ D < .

Assuming all K sections are statistically identical:

As , the p.d.f. approaches a semicircle, and the peak-to-peak group delay

spread approaches:

D

Kgd

coupledr.m.s. GD

K sections

uncoupledr.m.s. GD1 section

K44 gdminmax

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3119 (2011).

Hj t

t

MMG

23

p.d.

f.

Normalized Group Delay gd

-2 -1 0 1 20

0.1

0.2

0.3

0.4

0.5

0.6D = 2Analytical

Two-sided Maxwellian distribution.

G. J. Foschini and C. D. Poole, J. Lightw. Technol. 9, 1439 (1991).

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3119 (2011).

Probability Densities of Group Delays

24

Probability Densities of Group Delays

p.d.

f.

Normalized Group Delay gd

-2 -1 0 1 20

0.1

0.2

0.3

0.4

0.5

0.6D = 4AnalyticalSimulation

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3119 (2011).

Simulation: four uncoupled modes have only two delays , , , :p.d.f. of coupled GDs insensitive to p.d.f. of uncoupled GDs.

25

Probability Densities of Group Delays

p.d.

f.

Normalized Group Delay gd

-2 -1 0 1 20

0.1

0.2

0.3

0.4

0.5

0.6D = 8AnalyticalSemicircle (D→)

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3119 (2011).

As D increases, the p.d.f. approaches a semicircle.

26

Probability Densities of Group Delays

p.d.

f.

Normalized Group Delay gd

-2 -1 0 1 20

0.1

0.2

0.3

0.4

0.5

0.6D = 16SimulationSemicircle (D→)

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3119 (2011).

As D increases, the p.d.f. approaches a semicircle.

27

Probability Densities of Group Delays

p.d.

f.

Normalized Group Delay gd

-2 -1 0 1 20

0.1

0.2

0.3

0.4

0.5

0.6D = 64SimulationSemicircle (D→)

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3119 (2011).

As D increases, the p.d.f. approaches a semicircle.

28

Statistics of Group Delay Spread Group delay spread maxmin determines memory length required in MIMO equalizer.

Can compute distribution using Fredholm determinant (any D) or approximate it using Tracy-Widom distribution (large D).

K.-P. Ho and J. M. Kahn, Photon. Technol. Lett. 24, 1906 (2012).

0 1 2 3 4 5 6108

106

104

102

100

Fredholm Det.Tracy-WidomSimulation

x

D = 30

Pr((

max m

in)/

gd>

x)

12 620

29

Statistics of Group Delay Spread Group delay spread maxmin determines memory length required in MIMO equalizer.

Can compute distribution using Fredholm determinant (any D) or approximate it using Tracy-Widom distribution (large D).

Given D and a desired probability p, can compute:

K.-P. Ho and J. M. Kahn, Photon. Technol. Lett. 24, 1906 (2012).

pxxpuD gdminmax /)(Prsuch that

0 1 2 3 4 5 6108

106

104

102

100

Fredholm Det.Tracy-WidomSimulation

x

D = 30

Pr((

max m

in)/

gd>

x)

12 620

typical values:uD(p) ~ 4-5

30

Coherent MDM System Example

S. O. Arik, D. Askarov and J. M. Kahn, J. Lightw. Technol. 31, 423 (2013).

…1 secK ampK

ampL

…1 secK 1 …1 secK 2 …secL

Fiber Span

Amplifier Amplifier

Fiber Span

Amplifier

Fiber Span

Total length: Ltot = Lamp × Kamp Section length: Lsec = Lamp / Ksec

Mode-averaged CD: 2,av Uncoupled r.m.s. MD: 1,rms

Symbol rate: Rs Oversampling ratio: ros

Chromatic dispersion memory length (sampling intervals):

Modal dispersion memory length (sampling intervals):

2sostotav,2CD 2 RrLN

sosgdMD )( RrpuN D totsecrms,1gd LL

31

Memory Length of Fiber Dispersion

S. O. Arik, D. Askarov and J. M. Kahn, J. Lightw. Technol. 31, 423 (2013).

Ltot = Kamp × Lamp = 20 × 100 km

Rs = 32 Gsym/s

ros = 2

p = 105

D = 2: step-index, NA = 0.10

D = 6, 12, 20, 30: graded-indexdepressed cladding, NA = 0.15

D 2,av (ps2/km)

1,rms (ps/km)

2 22.5

6 28.0 277

28.4 383

28.6 415

28.7 451

110102103

104

Section length Lsec (m)

Number of sections per span Ksec

NC

Dor

NM

D(s

ampl

ing

inte

rval

s)

10 103

104

10510

2

103

104

105

D = 6D = 12D = 20D = 30

CD, D = 6,…,30

MD

105

102

1

CD, D = 2

-20 -10 0 10 201.435

1.44

1.445

1.45

1.455

Radial position (m)

Ref

ract

ive

inde

x

32

Memory Length of Fiber Dispersion

S. O. Arik, D. Askarov and J. M. Kahn, J. Lightw. Technol. 31, 423 (2013).

Ltot = Kamp × Lamp = 20 × 100 km

Rs = 32 Gsym/s

ros = 2

p = 105

D = 2: step-index, NA = 0.10

D = 6, 12, 20, 30: graded-indexdepressed cladding, NA = 0.15

D 2,av (ps2/km)

1,rms (ps/km)

2 22.5

6 28.0 277

28.4 383

28.6 415

28.7 451

110102103

104

Section length Lsec (m)

Number of sections per span Ksec

NC

Dor

NM

D(s

ampl

ing

inte

rval

s)

10 103

104

10510

2

103

104

105

D = 6D = 12D = 20D = 30

CD, D = 6,…,30

MD

105

102

1

CD, D = 2

33

Equalization Complexity

S. O. Arik, D. Askarov and J. M. Kahn, J. Lightw. Technol. 31, 423 (2013).

DetectedSymbols

(complex)

HomodyneDownconverter

Outputs(complex)

rate rosRs

MD(adaptive

D × D)

…1

D…1

D

CD (fixed)

CD (fixed)

1

log1

CDFFT

FFT2FFTos FDECD,

NNNNrCMCD, FDE

1

log

MDFFT

FFT2FFTos FDEMD,

NNNDNrCMMD, FDE

CDosTDE CD, NrCM CD, TDE

MDosTDE MD, DNrCM MD, TDE

Assume separate equalization of CD and MD for simplicity (not necessarily optimal).

NFFT FFT block length (to minimize complexity: NFFT ~ 10NCD, NFFT ~ 20NMD).

CM complex multiplications/symbol (for equalization, not adaptation).

34

Equalization Complexity (2)

S. O. Arik, D. Askarov and J. M. Kahn, J. Lightw. Technol. 31, 423 (2013).

Com

plex

mul

t. pe

r sym

bol

107

103

104

105

106

CD, D = 6,..,30

D = 6D = 12D = 20D = 30

MD

110102103

104

Section length Lsec (m) 10

5

Number of sections per span Ksec

10 103

104

105

102

1

CD, D = 2

TDE

35

Equalization Complexity (2)

S. O. Arik, D. Askarov and J. M. Kahn, J. Lightw. Technol. 31, 423 (2013).

Com

plex

mul

t. pe

r sym

bol

20

40

60

80

100

120

CD, D = 6,…,30

MD, D = 12

MD, D = 6

MD, D = 20

MD, D = 30

110102103

104

Section length Lsec (m) 10

5

Number of sections per span Ksec

10 103

104

105

102

10

CD, D = 2

log 2

(NFF

T,op

t)

10

12

14

16

18

20

22

D = 6D = 12D = 20D = 30

CD, D = 2,…,30

MD

110102103

104

Section length Lsec (m) 10

5

Number of sections per span Ksec

10 103

104

105

102

1

FDE FDE

36

Equalization Complexity (2)

Computational complexity CM TDE: prohibitive. FDE: reasonable, insensitive to Ksec or Lsec using optimal NFFT .

FFT block length NFFT

Exacerbates phase noise and frequency offsets. Slows adaptation.DSP hardware complexity scales faster than D·NFFT. Probably must limit to NFFT ≤ 216.

S. O. Arik, D. Askarov and J. M. Kahn, J. Lightw. Technol. 31, 423 (2013).

Com

plex

mul

t. pe

r sym

bol

20

40

60

80

100

120

CD, D = 6,…,30

MD, D = 12

MD, D = 6

MD, D = 20

MD, D = 30

110102103

104

Section length Lsec (m) 10

5

Number of sections per span Ksec

10 103

104

105

102

10

CD, D = 2

log 2

(NFF

T,op

t)

10

12

14

16

18

20

22

D = 6D = 12D = 20D = 30

CD, D = 2,…,30

MD

110102103

104

Section length Lsec (m) 10

5

Number of sections per span Ksec

10 103

104

105

102

1

FDE FDE

216

2000 m 250 m

37

Outline

Mode coupling Sources, effects and models

Modal dispersion Principal modes Statistics of group delays MDM system complexity

Modal gains and losses Statistics of gains/losses and noise MDM system performance Frequency diversity

Discussion

38

Statistics of Mode-Dependent Loss and Gain

Assume: D modes (or can be a subset of D modes uncoupled to other modes)K >> 1 independent sections (strong coupling regime)Include mode-dependent loss or gain

Total propagation operator (D × D):

Propagation operator in kth section:

Random unitary matrices describing mode coupling:

Matrix describing uncoupled gains and group delays (assume ):

:

K.-P. Ho and J. M. Kahn, Opt. Expr. 19,16612 (2011).

121t MMMMM KK

Hkkkk UΛVM

kk VU ,

kD

kD

kk

jg

jg

k

e

e

21

21

0

011

Λ

01 kD

k gg

39

Statistics of Mode-Dependent Loss and Gain (2)

At any single frequency , can perform singular-value decomposition.Suppressing frequency dependence in :

Columns of U(t) and V(t) define transmit and receive bases that diagonalizeM(t) into D uncoupled spatial subchannels.

Matrix describing gains of spatial subchannels:

Spatial subchannel gains

are logs of eigenvalues of (modal gain or round-trip propagation operator).

Statistics of gains govern system performance.

Htttt UΛVM

t

t1

21

21

t

0

0

Dg

g

e

eΛ

tt2

t1

tDggg g

Htt MM

Htttt ,, UΛVM and

K.-P. Ho and J. M. Kahn, Opt. Expr. 19,16612 (2011).

tt2

t1

tDggg g

40

Statistics of Mode-Dependent Loss and Gain (3)

Approximate results (proven in low-MDL limit, accurate to moderate MDL)

Distribution of gains (measured on log scale) sameas distribution of eigenvalues of a zero-trace Gaussian unitary ensemble.

Relationship between accumulated MDL and overall MDL:

Assuming all K sections are statistically identical:

Assuming large number of noise sources (amplifiers), noises in differentspatial subchannels have equal powers and are statistically independent.

K.-P. Ho and J. M. Kahn, Opt. Expr. 19,16612 (2011).

accumulated r.m.s. MDLK sections

gK

uncoupled r.m.s. MDL1 section

tt2

t1

tDggg g

overall r.m.s. MDLK sections

2121

mdl 1

accumulated r.m.s. MDLK sections

41

Exact Statistics of Mode-Dependent Loss and Gain

Distribution of gains (measured on log scale) is sameas distribution of eigenvalues of:

G: zero-trace Gaussian unitary ensemble.

F: matrix with random eigenvectors, deterministic eigenvalues uniform on [1, 1].

D = D/ 2 (1 +D) : constant between 1/3 and 1/2, depending on number of modes.

: accumulated r.m.s. MDL.

K.-P. Ho, J. Lightw. Technol. 30, 3603 (2012).

FG 2 D

tt2

t1

tDggg g

42

Coherent MDM System Example

Kamp = 20 amplifiers

g = 1.1 dB uncoupled r.m.s. MDG per amplifier

Accumulated r.m.s. mode-dependent gain:

Overall r.m.s. mode-dependent gain:

Overall mean spatial non-whiteness of noise: ~0.2 dB

…1 secK ampK

ampL

…1 secK 1 …1 secK 2 …secL

Fiber Span

Amplifier Amplifier

Fiber Span

Amplifier

Fiber Span

dB 0.5gamp K

dB 3.51 2121

mdl

43

D = 2(Maxwellian)

A. Mecozzi and M. Shtaif, Photon. Technol. Lett. 14, 313 (2002).K.-P. Ho and J. M. Kahn, Opt. Expr. 19,16612 (2011).

-3 -2 -1 0 1 2 3

= 1 dB = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 9 = 10 = 11 = 12 = 13 = 14 = 15 = 16 = 17 = 18 = 19 = 20

0 5 10 15 200

5

10

15

20

25

30

35

= K1/2 g (dB)

Ove

rall

MD

L m

dl(d

B)

0

10

20

30

40

50

60

70

Mea

n M

DL

Diff

eren

ce (d

B)

Sim. mdlEq. (8)Eq. (1)Sim. MDL Diff.

Normalized Overall MDL g /mdl (dB)

10 dB

5 dB

Probability Densities of Mode-Dependent Loss and Gain

44K.-P. Ho and J. M. Kahn, Opt. Expr. 19,16612 (2011).

5 10 15 200

5

10

15

20

25

30

35

Ove

rall

MD

L

mdl

(dB

)

00

20

40

60

80

100

Max

imum

MD

L D

iffer

ence

(dB

)

Sim. mdlEq. (1)Sim. MDL Diff.

-3 -2 -1 0 1 2 3

= 1 dB = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 9 = 10 = 11 = 12 = 13 = 14 = 15 = 16 = 17 = 18 = 19 = 20

Normalized Overall MDL g /mdl (dB) = K1/2 g (dB)

D = 4

15.3 dB

5 dB

Probability Densities of Mode-Dependent Loss and Gain

45

-3 -2 -1 0 1 2 3

= 1 dB = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 9 = 10 = 11 = 12 = 13 = 14 = 15 = 16 = 17 = 18 = 19 = 20

0 5 10 15 200

5

10

15

20

25

30

35

Ove

rall

MD

L

mdl

(dB

)

0

20

40

60

80

100

Max

imum

MD

L D

iffer

ence

(dB

)

Sim. mdlEq. (1)Sim. MDL Diff.

Normalized Overall MDL g /mdl (dB) = K1/2 g (dB)

K.-P. Ho and J. M. Kahn, Opt. Expr. 19,16612 (2011).

D = 8

15.3 dB

5 dB

Probability Densities of Mode-Dependent Loss and Gain

46

0 5 100

5

10

15

20

25

30

35

Ove

rall

MD

L

mdl

(dB

)

15 200

20

40

60

80

100

120

140

Max

imum

MD

L D

iffer

ence

(dB

)

Sim. mdlEq. (1)Sim. MDL Diff.

-3 -2 -1 0 1 2 3

= 1 dB = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 9 = 10 = 11 = 12 = 13 = 14 = 15 = 16 = 17 = 18 = 19 = 20

Normalized Overall MDL g /mdl (dB) = K1/2 g (dB)

K.-P. Ho and J. M. Kahn, Opt. Expr. 19,16612 (2011).

D = 64(approaches semicircle)

20 dB

5 dB

Probability Densities of Mode-Dependent Loss and Gain

47

Spatial Non-Whiteness of Noise

Output noise contribution from one noise source is spatially non-white.

As number of noise sources K increases, non-whiteness of noise (STD ofspatial spectral distribution) decreases as (law of large numbers).

K.-P. Ho and J. M. Kahn, Opt. Expr. 19,16612 (2011).

K/1

D = 8100 realizations each KMax. STDMean STD

0 0.1 0.2 0.33 0.50

0.5

1

1.5

2

2.5

K1/2

25610064 25 16 9 4 Number of Noise Sources, K

STD

of t

he S

patia

l Spe

ctra

l Dis

trib

utio

n (d

B)

= 10 dB

= 5 dB

48

Channel Capacity without Mode-Dependent Loss/Gain

Capacity without MDL:

(b/s/Hz).

Signal-to-noise ratio:

DDC t

2 1log

SNR, t (dB)-5 0 5 10 15 20

100

101

102

Cha

nnel

Cap

acity

(bit/

s/H

z)

D = 1

2

4

16

64

8

C

mode per power noisemodesall in power signal received

tD

49

Effect of Mode-Dependent Loss and Gainon Capacity at One Frequency

Channel state information is knowledge of:optimal transmit basis U(t) and spatial subchannel gains g(t)

Not available at transmitter in long-haul system (tens of ms round-trip delay).

Capacity, assuming no CSI:

Because g(t) is random, C is random.The p.d.f. of C close to Gaussian, but skewed.

Outage probability Pout and outage capacity Cout:

Cout decreases as Pout decreases.If C is Gaussian:

D

i

tig

DC

1

)(2 exp1log

mode per power noisemodesall in power dtransmitte D

outout Pr CCP

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3719 (2011).P. J. Winzer and G. J. Foschini, Opt. Expr. 19,16680 (2011).

mode per power noisemodesall in power signal received avg.

tD

D = 10t= 20 dB = 10 dBK = 256

14 16 18 20 2210

-6

10-5

10-4

10-3

10-2

10-1

100

Prob

abili

ty D

ensi

ty

Capacity C (b/s/Hz)

Avg. Cap.Cavg

Outage Cap.Cout

OutageProb.

Pout = 103

C

outavgout

CCQP

50

Effect of Mode-Dependent Loss and Gainon Capacity at One Frequency (2)

When CSI not available, mode-dependent loss/gain always decreases average capacity and outage capacity.

Can compute C by:Generating Gaussian unitary ensemble (theory)Using multi-section model (simulation)

K.-P. Ho and J. M. Kahn, Opt. Expr. 19,16612 (2011).

No CSIt = 10 dBPout = 103

K = 256 (sim.)

0 5 10 15 200

2

4

6

8

10

12

= K1/2 g (dB)

D = 2

4

8

16

5 dB

TheorySimulation

Out

age

Cap

acity

, Cou

t(b

it/s/

Hz)

No CSIt = 10 dBK = 256 (sim.)

TheorySimulation

0 5 10 15 200

5

10

15

= K1/2g (dB)

Aver

age

Cap

acity

, Cav

g(b

it/s/

Hz)

D = 512

16

4

2

64

8

5 dB

51

Reducing Outage Probabilityto Increase Outage Capacity

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3719 (2011).

Components of g(t)() correlated over , with coherence bandwidth of order 1/gd.

52

Reducing Outage Probabilityto Increase Outage Capacity

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3719 (2011).

Components of g(t)() correlated over , with coherence bandwidth of order 1/gd.

Narrowband regime Occurs when .

Performance limited by diversity-multiplexing tradeoff.

To reduce Pout, use either:

strong FEC coding

space-time coding (as in narrowband MIMO wireless).

Both can reduce throughput or increase complexity.

10~1~ mdgds NR

Sig

nal G

ain

(log

units

)

Frequency (arbitrary units)

0

ωg(t)1

ωg(t)2

2t122

12t112

1 ωMωMlog

2t222

12t212

1 ωMωMlog

Frequency (arbitrary units)

0

Sig

nal G

ain

(log

units

)

D = 2

Min./max. gains

Gains in reference modes

53

Reducing Outage Probabilityto Increase Outage Capacity

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3719 (2011).

Components of g(t)() correlated over , with coherence bandwidth of order 1/gd.

Wideband regime Occurs when .

Frequency diversity can reduce Pout, enabling:

(law of large numbers).

To exploit frequency diversity, use either:

single-carrier with linear equalizer

multi-carrier with FEC codewordsspread over all carriers (as inwideband wireless).

Neither reduces throughput, nor increases complexity.

101 mdgds NR

avgout CC

Frequency (arbitrary units)

ωg(t)1

ωg(t)2

0

Sig

nal G

ain

(log

units

)

Frequency (arbitrary units)

0

Sig

nal G

ain

(log

units

)

2t222

12t212

1 ωMωMlog

2t122

12t112

1 ωMωMlog

D = 2D = 2

Min./max. gains

Gains in reference modes

54

Frequency Diversity Example

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3719 (2011).

Normalized frequency separation:

2gd

0 4 8 12 16

-20

-10

0

10

20

Normalized Frequency Separation,

Mod

al G

ains

, g(t) i

(dB

)

D = 10 = 10 dBK = 256No CSI

55

Frequency Diversity Example

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3719 (2011).

Normalized frequency separation:

2gd

0 0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

Normalized Frequency Separation,

Cor

rela

tion

Coe

ffici

ent

Capacity

g(t)i

D = 10 = 10 dBK = 256No CSI

56

Frequency Diversity Example

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3719 (2011).

Normalized frequency separation:

Approximate diversity order:

2gd

gdsgdsig RBb

D = 10 = 10 dBK = 256No CSIPout = 103

0 5 10 15 2002468

1012141618

SNR (dB)

Cha

nnel

Cap

acity

(b/s

/Hz)

Cout, b = 0 (single freq.)

Cavg

Cout, b = 1Cout, b = 4Cout, b = 8

57

Signal bandwidth Diversity order Average capacity Capacity variance Outage capacity 

0sig B 1D F avgC 21,C 1out,C

0sig B 1D F avgCD

21,

2, /

DFCFC Dout,FC

12

1012

2101

210

cccccccccc

ccc

R

k

kF1

D k : eigenvalues of R

1 : largest eigenvalue

D1,outavg

,outavg 1D

FCCCC F

Rigorous Definition of Diversity Order

Form matrix of gain correlation coefficients sampled at different normalized frequency separations:

Define frequency diversity order as number of independent components of R:

Outage capacity reduction ratio found to decrease as 1/√FD:

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3719 (2011).

58

Signal bandwidth Diversity order Average capacity Capacity variance Outage capacity 

0sig B 1D F avgC 21,C 1out,C

0sig B 1D F avgCD

21,

2, /

DFCFC Dout,FC

12

1012

2101

210

cccccccccc

ccc

R

k

kF1

D k : eigenvalues of R

1 : largest eigenvalue

D1,outavg

,outavg 1D

FCCCC F

Rigorous Definition of Diversity Order

Form matrix of gain correlation coefficients sampled at different normalized frequency separations:

Define frequency diversity order as number of independent components of R:

Outage capacity reduction ratio found to decrease as 1/√FD:

K.-P. Ho and J. M. Kahn, J. Lightw. Technol. 29, 3719 (2011).

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

FD1/2

Out

age

Cap

acity

Red

uctio

n R

atio

SNR = 20 dBSNR = 10 dB

100 25 16 9 4 1Diversity Order, F

D

Dout,1avg

out,avg 1D

FCCCC F

D = 10 = 10 dBK = 256No CSI

59

Outline

Mode coupling Sources, effects and models

Modal dispersion Principal modes Statistics of group delays MDM system complexity

Modal gains and losses Statistics of gains/losses and noise MDM system performance Frequency diversity

Discussion

60

Long-Haul Mode-Division Multiplexing

Modal dispersion Coupled GD spread gd affects complexity of MIMO signal processing.

Computational complexity depends weakly on gd (and D), buthardware complexity scales faster than D · gd.

If D · gd is too large, mode-division multiplexing will not be feasible. Transmission fibers should have low uncoupled GD spread.It may be necessary to intentionally enhance mode coupling.

Mode-dependent loss and gain Fundamentally limits performance: reduces capacity and can cause outage.

Transmission fibers and inline amplifiers must have low uncoupled MDL/MDG. Strong mode coupling can further reduce MDL/MDG.

Frequency diversity reduces outage probability substantially. Coupled GDspread gd must be sufficient to achieve diversity order required.

61

Viability of Long-Haul Spatial Multiplexing

Considerations Transceivers: integration

Fibers: loss, nonlinear effects, splicing

Optical amplifiers: noise figure, mode-dependent gain, pumping efficiency

Signal processing: hardware and computational complexity

System performance: capacity, outage probability

Optical switches: scalability to future traffic

Economics: technology development, deployment in network

Comments We have yet to make a convincing case for multi-core or multi-mode long-haul

systems.

Integrated transceiver arrays with parallel single-mode fibers yield many of the potential benefits.

62

To Learn More

ee.stanford.edu/~jmk