mode-division multiplexing systems: propagation effects...
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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
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