Analysis of Signal Propagation in Optical Fiber Based on Finite-Difference Method, PhD presentation

1. University of Zagreb
Faculty of Electrical Engineering and Computing
Analysis of Signal Propagation in
Optical Fiber Based on
Finite - Difference Method
Sonja Zentner Pilinsky
Doctoral Thesis, Zagreb, 2003

2. 2
Contents:
1. Introduction
2. Pulse propagation in optical fiber
3. Numerical model and its accuracy
4. Selected simulation results
5. Conclusions

3. 3
1. Introduction
♦Motivation and goal
♦What we model
♦How we model
♦Additional devices needed

4. 4
Motivation
- need for accurate program with all effects included:
new sophisticated optical links
upgrade of existing fiber links
- expensive experiments
why to model optical link
- linear fiber communications at the edge (bit rates, capacity)
- optical transmission very sensitive to:
dispersion (cromatic, polarization mode)
loss
nonlinear effects
noise

5. 5
Goal - to model modern optical links
dispersion map

6. 6
What we model
Nonlinear Schrödinger equation (NLSE)
0
2
2 2
ω=ω
 β
β =  
ω 
d
d
nonlinear coeff.
describing SPM
fiber loss in dB/km
time
1
1
gv
β =
distance along fiber
optical pulse
complex envelope
2 0ω
γ =
eff
n
cA

7. 7
How we model
- FDM → Cranck - Nicholson
- pseudospectral → SSFM
- testing accuracy on canonical problems
- comparison with OptiSystem 2.0.
Models for additional devices
- EDFA model (G, ASE noise)
- Optical filter model (transfer function)

8. 8
2. Pulse propagation in optical fiber
♦Propagation equation
♦Fiber loss
♦Group velocity dispersion
♦Self - phase modulation
♦Polarization dispersion

9. 9
Maxwell equations
0
f
t
t
∂
∇ × = −
∂
∂
∇ × = +
∂
∇ ⋅ = ρ
∇ ⋅ =
f
B
E
D
H J
D
B
0
0
= ε +
= µ +
D E P
B H M
Optical fiber:
-no sources Jf ,ρf = 0
-nonmag.mat M = 0
( ) ( )
( ) ( )1
0, ,L t t t t dt
∝
−∝
′ ′ ′= ε χ − ⋅∫P r E r
( ) ( )
( ) ( ) ( ) ( )3
0 1 2 3 1 2 3 1 2 3, , , , , ,NL ijklt t t t t t t t t t dt dt dt
∝ ∝ ∝
−∝ −∝ −∝
= ε χ − − −∫ ∫ ∫P r E r E r E r

10. 10
Assumption and approximation:
( ) ( ) 2
/∇ ∇ ⋅ = ∇ ⋅∇ε ε << ∇E E E
( ) 2 2
∇ × ∇ × = ∇ ∇ ⋅ − ∇ ≈ −∇E E E E
WGA ∆ ≈ (ncore – ncladding)/ ncore << 1
- the EM field maintains its polarization along the fiber
- Weakly guiding approximation

11. 11
- PNL is treated as a small perturbation to PL
- nonlinear effects: Kerr and Raman (neglected for T0 > 1ps)
instantaneous
response
( )
( ) ( )
( ) ( ) ( )3 3
1 2 3 1 2 3, ,ijkl t t t t t t t t t t t tχ − − − = χ δ − δ − δ −
- SVEA - slowly varying envelope approximation
- envelope is slowly varying in z and t
- removes backscattered part of the envelope
( )0 , 1 L NLk
c
ω
= ε ω = + ε + ε
( ) ( ) ( )
( )
23
0
3
, , , ,
4
NL NL NLP t E t E t= ε ε ε = χr r r

12. 12
( )
( )
( )
2
2
32 0 0
4
,
2 3
,
8
,
I
xxxx eff
eff eff
F x y dxdy
n k Z
A
A n cA
F x y dxdy
∝ ∝
−∝ −∝
∝ ∝
−∝ −∝
 
 
ω  γ = = χ =
∫ ∫
∫ ∫
Propagation equation for pulse complex envelope:
SVEA assumption: ° ( ) ( ) ° ( ) 0
0 0, , , ej z
E F x y A z β
ω − ω = ω − ωr
HE11 mode ( )
( )
( ) ( )
0
0
,
,
,
a
J a
F x y a
J a e a
−γ ρ−
 κρ ρ ≤

= 
κ ρ ≥
ρ

13. 13
- α [dB/km]=-10log-α[1/km]
- absorption (intrinsic and extrinsic)
- scattering - linear: Rayleigh and Mie
- nonlinear: Raman and Brillouin
Fiber loss:

14. 14
- caused by material and waveguide dispersion
- mathematically described by
( ) ( ) ( ) ( ) ( )
2 3
0 1 0 2 0 3 0
1 1
..
2 3!
.
ω
β ω = ω = β +β ω− ω + β ω− ω + β ω− ω +n
c
1
1 ps
km
g
g
n
c v
 
β = =  
 
( ) ( )
0 0
2 23 2
0
2 2 2 2
ps
2
d n d n
c d c d kmω=ω λ=λ
ω λ  ω λ
β ≈ ≈  
ω π λ  
( ) ( )
0 0
2 3 3
3 02 3
1 ps
3
km
d n d n
c d dω=ω ω=ω
 ω ω  
 β = + ω  
ω ω    
2
0
2
D
T
L =
β
3
0
3
D
T
L′ =
β
Group velocity
dispersion:

15. 15
( ) ( )
2
2
2,
2
I
E
n E n n
Z
ω = ω +Kerr effect
SPM
XPM
FWM
2 0 2 0 1
Wkm
I I
eff eff
n k n
A cA
ω  
γ = =  
 
SPM without dispersion:
2
00
,
1
,
z
NL
NL
U j
e U U
z L
A
U L
PP
−α∂
=
∂
= =
γ
Self Phase
Modulation:
( ) ( ) ( )0 2
2 2 I
n z n I t z
π π 
Φ = ω + Φ + ω ⋅ 
λ λ 
( ) ( ) ( )
( ) ( )
,
2
, 0,
1
, 0, ,
NLi z T
z
eff
NL eff
NL
U z T U T e
z e
z T U T z
L
Φ
−α
=
−
Φ = =
α

16. 16
Polarization mode dispersion
- caused by circular asymmetries in the fiber
- locally birefringence
2
x y
x y
n
c c
n n
ω ω π
∆β = β − β = − = ∆
λ 1 1
2
B
x y
L
n
λ π
= =
∆ β − β
- measure of pulse splitting in biref. fiber - DGD
g
L d n d n
L L
v d c c d
∆β ∆ ω ∆ 
∆τ = = = + 
∆ ω ω 

17. 17
2
22
2 2
2
2 2
2 2
2
2 2 2 3
2
2 2 2 3
x x x
x x y x
y y y
y y x y
A A Aj
A j A A A
z t t
A A Aj
A j A A A
z t t
∂ ∂ ∂∆β α  
+ + β + = γ +  
∂ ∂ ∂  

∂ ∂ ∂∆β α   − + β + = γ +  ∂ ∂ ∂  
PMD (cont.)
- alternative method for linear optical element
( ) ( )
( ) ( )
,out in
a b
b a∗ ∗
 ω ω
= ⋅ =  
 − ω ω 
J A J A
0
00
2 2
0 0 0
1
x
y
j
xx
j
y y
x y
a eE
EE a e
E E E
φ
φ
  
= =        
= +
J
DGD ( ) ( ) ( )
2 2
2 a b′ ′∆τ ω = ω + ω a’(ω) ≈ [a(ω+∆ω) – a(ω)] / ∆ω
- globally - birefringence combined with random
polarization mode coupling:

18. 18
3. Numerical model and its accuracy
♦FDM or SSFM ?
♦Accuracy check and comparison with
OptiSystem 2.0
♦EDFA model and filter model

19. 19
- FDMs: Crank - Nicholson scheme
- pseudospectral method: SSFM
Nonlinear PDE modeling:
Criterion for selected FDM model:
- accuracy
- stability

20. 20
- solving numerical scheme to prescribed initial values
and boundary conditions
- errors: modeling, truncation, round-off
FDM steps
- dividing solution region into a grid of nodes
- PDE → finite difference equivalent (numerical stability!!)

21. 21
Derivative Finite difference approximation Type Error
1i i
t
+ψ −ψ
∆
FD O(∆t)
1i i
t
+ψ −ψ
∆
BD O(∆t)
1 1
2
i i
t
+ −ψ −ψ
∆
CD O(∆t2
)
2 14 3
2
i i i
t
+ +−ψ + ψ − ψ
∆
FD O(∆t2
)
1 23 4
2
i i i
t
− −ψ − ψ +ψ
∆
BD O(∆t2
)
t
′ψ
2 1 1 28 8
12
i i i i
t
+ + − −−ψ + ψ − ψ +ψ
∆
CD O(∆t4
)
( )
2 1
2
2i i i
t
+ +ψ − ψ +ψ
∆
FD O(∆t2
)
( )
1 2
2
2i i i
t
− −ψ − ψ +ψ
∆
BD O(∆t2
)
( )
2 1
2
2i i i
t
+ −ψ − ψ + ψ
∆
CD O(∆t2
)
tt
′′ψ
( )
2 1 1 2
2
16 30 16i i i i i
t
+ + − −−ψ + ψ − ψ + ψ −ψ
∆
CD O(∆t4
)
Accuracy

22. 22
2
2
.
A A
const
z t
∂ ∂
=
∂ ∂
First order (Euler)
( )
1 1
1
2
2
. i i
n n nn n
ii i
const
z t
+ −
+
ψ − ψ + ψψ − ψ
=
∆ ∆
- one step, explicit, unstable
( ) ( )
11 1 11 1
1 1 11
2 2
2 2.
2
i i i i
n n n n n nn n
i ii i const
z t t
+ − + −
+ + ++  ψ − ψ + ψ ψ − ψ + ψψ − ψ
= + 
∆ ∆ ∆  
Crank-Nicholson
- one step, implicit, accurate (1 in z, 2 in t), uncond. stable
( )
1 1
1 1
2
2
.
2
i i
n n nn n
ii i
const
z t
+ −
+ −
ψ − ψ + ψψ − ψ
=
∆ ∆
Leapfrog
- two step, explicit, accurate (2 in z, 2 in t), always unstable
Dufort-Frankel
( )
1 1
1 11 1
2
.
2
i i
n n n nn n
i ii i
const
z t
+ −
+ −+ − ψ − ψ − ψ + ψψ − ψ
=
∆ ∆
- two step, explicit, accurate (2 in z, 2 in t), uncond. stable
Various FDM schemes for eq.

23. 23
Accuracy
1. comparison with analytic solutions for simple problems
2. Comparison with simulations obtained by OptiSystem 2.0
( )
( ) ( )
1
1 1
NM
NM ex
i i
i
NM NM
NM NM ex ex
i i i i
i i
AKC a jb
∗
=
∗ ∗
= =
ψ ψ
= = +
ψ ψ ⋅ ψ ψ
∑
∑ ∑
1
1 NMAX
ex ZMAX
i i
i
ER
NMAX =
= ψ − ψ∑
2
1
1 NM
ex NM
i i
i
SER
NM =
= ψ − ψ∑
Mean error Mean square error
Correlation coefficient
( )
2 2
arg
AKC a b
b
AKC arctg
a
= +
=
Measure of accuracy:

24. 24
M E A N T I M E E R R O R = 2 . 7 1 8 7 0 2 5 7 4 3 9 6 3 5 9 E - 0 0 5
S Q U A R E M T E = 3 . 1 8 8 2 5 1 6 8 9 5 9 1 3 0 2 E - 0 0 9
A U T O C O R R E L A T I O N = 0 . 9 9 9 9 9 7 6 1 3 9 7 3 6 2 8 - j 1 . 3 3 4 9 1 6 9 4 5 0 6 8 7 2 8 E - 0 0 5
| A K C | = 0 . 9 9 9 9 9 7 6 1 4 0 6 2 7 2 9 a r g ( A K C ) = - 7 . 6 4 8 5 2 8 9 4 4 4 3 0 9 6 7 E - 0 0 4
Gaussian pulse
( )
2
2
02
00,
T
T
A T A e
−
=
( ) ( )
2
0
2
0 220
0 2
0 2
,
T
T j zT
A z T A e
T j z
−
− β
=
− β
Analytic solution:
Input pulse:
FDM:
ER = 1.77E-004
SER = 1.36E-007
1-|AKC| = 3.8E-005
arg (AKC)= 7E-004°
SSFM:
ER = 1.88E-004
SER = 1.56E-007
1-|AKC| = 4.4E-005
arg (AKC)= 2.75E-003°
0
0.0005
0.001
0.0015
0.002
0.0025
0.003
1
26
51
76
101
126
151
176
201
226
251
276
301
326
351
376
401
426
451
476
501
number of points in time window
pulsepower[W]
analytic solution
OptiSystem
FiberProp
Input
pulse
fiber
A0 = 0.01 W1/2
α= 0
T0 = 40 ps D = 16 ps/kmnm
λ0 = 1550 nm γ = 0

25. 25
Hyperbolic secant pulse:
2
2
2 2
2 2
A j A
A j A A
z T
∂ ∂ α
+ β + = γ
∂ ∂
2
2 0 0
0 20
, , , D
D NL
PTLA z T
U N
L T LP
γ
= ξ = τ = = =
β
2
22
2 2
sgn
2
U j U
jN U U
∂ ∂
+ β =
∂ξ ∂τ
( ) ( )0, sechu Nτ = τ ( ) ( ) 2
, sech
j
u e
ξ
ξ τ = τ
input pulse analytic solution
input pulse fiber
P0 = 22.6 mW α= 0
T0 = 2.7 ps β2 = -0.243 ps2
/km
λ0 = 1552 nm γ= 1.475 W-1
km-1
-0.0004
-0.0003
-0.0002
-0.0001
0
0.0001
0.0002
0.0003
0.0004
0.0005
0 100 200 300 400 500
number of points in time window
analyticvalue-computersimul.
FiberProp
OptiSystem
normalization:

26. 26
-3.00E-03
-2.00E-03
-1.00E-03
0.00E+00
1.00E-03
2.00E-03
3.00E-03
4.00E-03
0 100 200 300 400 500 600
number of points in time window
analyticsolution-progr.simulation
OptiSystem
FiberProp
Second order soliton pulse: input pulse
analytic solution
( ) ( )0, sechu Nτ = τ
( )
( ) ( )
( ) ( ) ( )
4
2
2cosh 3 6cosh
, 2
cosh 4 4cosh 2 3cos 4
j
j T T e
U T e
T T
ξξ
+
ξ =
+ + ξ
input pulse fiber
P0 = 90.4 mW α= 0
T0 = 2.7 ps β2 = -0.243 ps2
/km
λ0 = 1552 nm γ= 1.475 W-1
km-1
2
0
0
22 2
D
T
z L
π π
= =
β
L = 2z0 = 94.25 km

27. 27
-8.00E-03
-6.00E-03
-4.00E-03
-2.00E-03
0.00E+00
2.00E-03
4.00E-03
6.00E-03
8.00E-03
0 100 200 300 400 500
number of points in time window
analyticalsolution-progr.simulation
OptiSystem
FiberProp
Third order soliton pulse: input pulse
analytic
solution
( ) ( )0, sechu Nτ = τ
L = 5z0 = 235.67 km
( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
4
2
12 8 8 16
2cosh 8 32cosh 2 36cosh 4 16cosh 6
, 3
cosh 9 9cosh 7 64cosh 3 36cosh
20cosh 4 80cosh 2 5 45 20
36cosh 5 cos 4 20cosh 3 cos 12 90cosh cos 8
j
j
j j j j
T T T T e
U T e
T T T T
T T e e e e
T T T
ξξ
ξ −− ξ ξ ξ
 + + + + ξ =
+ + + +
 + + + + 
ξ + ξ + ξ
N = 3
P0 = 203.4 mW

28. 28
4th order soliton pulse - NO analytic solution:
N = 4
P0 = 361.56 mW
L = 2z0 = 94.25 km
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
1 33 65 97 129 161 193 225 257 289 321 353 385 417 449 481
number of points in time window
pulsepower[W]
FiberProp
OptiSystem

29. 29
EDFA model
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
1
26
51
76
101
126
151
176
201
226
251
276
301
326
351
376
401
426
451
476
501
number of points in time window
opticalpower[W]
FiberProp
OptiSystem
FIBER:
α = 0.1 dB/km
β2 = -0.243 ps2
/km
γ = 1.475 1/Wkm
INPUT PULSE:
N = 1
P0 = 32.54 mW
T0 = 2.7 ps
λ0 = 1552 nm
EDFA: ∆λ = 30 nm
λ0 = 1552 nm
G = 4.98 dB
nsp = 1.23

30. 30
Filter model
( )
2
0
1 2
1
2
f f
B
 
− − 
 
( )0
1
1 cos
2
f f
B
 π 
+ −  
  
( ) ( )
2
0
2
exp ln 2 f f
B
   
− −  
   
( )
2
0
1
2
1 f f
B
 
+ −  
Parabolic – shape characteristic
Cosine – shape characteristic
Lorentzian – shape characteristic
Gaussian – shape characteristic
Fabry-Perot
filter !!!

31. 31
B
B S
ω 0 f r e q u e n c y
f i l t e r t r a n s f e r f u n c t i o n
s o l i t o n s p e c t r u m
Why are filters used in nonlinear optical links?
• compensation of Gordon-Haus effect
• filtering at the receivers end
Filter model (cont.)

32. 32
4. Selected simulation results
♦FiberProp and its abilities
♦High bit rates soliton systems
♦Gordon-Haus effect and its compensation
♦Dispersion-compensated and
dispersion-managed systems
♦Polarization dispersion
♦Dispersion compensated system

33. 33
Simple FiberProp scheme:

34. 34
fiber param. EDFA param. input pulse
α= 0.1 dB/km ∆λ= 30 nm P0 = 31.48 mW
β2 = -0.243 ps2
/km G = 2.497 dB T0 = 1.543 ps
γ= 3.28 W-1
km-1
nsp = 1.5 λ0 = 1550 nm
40 Gb/s soliton transmission
PRBS
01111010
La = 25 km
L = 500 km

35. 35
80 Gb/s soliton transmission
fiber param. EDFA param. input pulse
α= 0 ∆λ= 30 nm P0 = 38 mW
β2 = -0.243 ps2
/km G = 2.497 dB T0 = 1.543 ps
γ= 3.28 W-1
km-1
nsp = 1.5 λ0 = 1550 nm
PRBS
0011110011100110
La = 25 km
L = 350 km

36. 36
0
0.005
0.01
0.015
0.02
0.025
0.03
1
20
39
58
77
96
115
134
153
172
191
210
229
248
267
286
305
324
343
362
381
400
419
438
457
476
495
number of grid points
power[W]
fiber param. EDFA param. input pulse
α= 0.1 dB/km ∆λ= 30 nm P0 = 14.83 mW
β2 = -0.243 ps2
/km λ0 = 1550 nm T0 = 5 ps
γ= 1.475 W-1
km-1
G = 5 dB N = 1.5
La = 50 km nsp = 2.2 λ0 = 1552 nm
L = 2500 km
tW = 1.5 T0
max 2972.59 kmTL ≤
Gordon-Haus transmission distance limitation:
1 a
a
L
L
Q
e−α⋅
α ⋅
=
−
22
2 c
D
π
= − β
λ

37. 37
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
1
20
39
58
77
96
115
134
153
172
191
210
229
248
267
286
305
324
343
362
381
400
419
438
457
476
495
number of grid points
power[W]
Inserted Fabry-Perot filter
before receiver:
( )
2
0
1
2
1 f f
B
 
+ −  
f0 = 193.414 THz
(λ0 = 1550 nm)
B = 100 GHz
Gordon-Haus transmission distance limitation (cont.):

38. 38
0
0.005
0.01
0.015
0.02
0.025
0.03
1
23
45
67
89
111
133
155
177
199
221
243
265
287
309
331
353
375
397
419
441
463
485
507
number of grid points
power[W]
L = 4000 km
tW = 3 T0 → Ltmax = 4718.7 km
( )
2
3
max
2
0,1372
1
FWHM w eff a
T
sp
T t A L Q
L
n n Dh G
≤
−
Gordon-Haus transmission distance limitation (cont.):

39. 39
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
1
20
39
58
77
96
115
134
153
172
191
210
229
248
267
286
305
324
343
362
381
400
419
438
457
476
495
number of grid points
power[W]
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
1
20
39
58
77
96
115
134
153
172
191
210
229
248
267
286
305
324
343
362
381
400
419
438
457
476
495
number of grid points
power[W]
Fabry - Perot filter
after each EDFA:
f0 = 193.414 THz
B = 100 GHz
f0 = 193.414 THz
B = 360 GHz
EDFA: G = 5.1081 dB
Inserted Fabry-Perot
filter before receiver:

40. 40
Fiber parameters
Two level disp. map Four level disp.map
α[dB/km] 0.078 0.15
β1″ [ps2
/km] 0.396 0.1275
β2″ [ps2
/km] -0.294 0.051
β3″ [ps2
/km] - -0.0765
β4″ [ps2
/km] - -0.306
βave [ps2
/km] -0.064 -0.051
L1 [km] 30.00 12.5
L2 [km] 60.00 12.5
L3 [km] - 12.5
L4 [km - 12.5
γ [W-1
km-1
] 2.65 2.65
Input pulse shape
sech sech
T0 [ps] 8.56 2.7
N 1.4 1.4
q0 2.92 4.63
Amplifier parameters
La [km] 90 50
G [dB] 7 7.383
λ0 [nm] 1550 1550
∆λ[nm] 30 30
nsp 1 1
DM soliton system:
20 Gb/s
40 Gb/s PRBS
01101110
β2
LL
L2
L1
β21
β22

41. 41
Polarization mode dispersion:
0.E+00
5.E-03
1.E-02
2.E-02
2.E-02
3.E-02
3.E-02
4.E-02
4.E-02
1
22
43
64
85
106
127
148
169
190
211
232
253
274
295
316
337
358
379
400
421
442
463
484
505
number of grid points
power[W]
Fiber param.:
α, β, γ, = 0
∆n = 4.2⋅10-5
h = 4.16 m
L = 64h = 266.24 m
input pulse: Gauss
T0 = 3 ps
P0 = 40 mW
λ0 = 1550 nm

42. 42
N=0.9
N=1.0
N=1.1
N=1.2
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
0 1 2 3 4 5 6 7
soliton periods z/z0
dislocationofpulsest/T0
Fiber parameters:
α = 0
β = -20 ps2
/km
γ = 1.475 1/Wkm
∆n = 4.2⋅10-5
input pulse:
lin. pol. at
45° at fiber axes
T0 = 5 ps
λ0 = 1550 nm
Polarization mode dispersion:
soliton prop. in birefringent fiber:
2
0
0
2
7 7 7
2 2
D
T
L z L
π π
= = =
β

43. 430 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0
0
5
1 0
1 5
2 0
2 5
3 0
3 5
4 0
4 5
5 0
Polarization mode dispersion -interaction compensation:
0 .0 0 1 0 . 0 0 2 0 .0 0 3 0 . 0 0 4 0 . 0 0
tim e [p s ]
0 .0 0
1 0 .0 0
2 0 .0 0
3 0 .0 0
4 0 .0 0
5 0 .0 0
6 0 .0 0
7 0 .0 0
8 0 .0 0
distance[km]
L = 85.76 km
Fiber:
α = 0
β2 = -20 ps2
/km
γ = 1.3 1/Wkm
Input pulse:
N = 1
P0 = 32.54 mW
T0 = 2 ps
λ0 = 1552 nm
q 0 = 4
L = 51.46 km

44. 44
Dispersion comp. system:
0 . 0 0 2 0 0 . 0 0 4 0 0 . 0 0 6 0 0 . 0 0
0 . 0 0
5 0 0 . 0 0
1 0 0 0 . 0 0
1 5 0 0 . 0 0
2 0 0 0 . 0 0
( )
2
0
1
2
00,
m
T
T
A T A e
 
−  
 
= Super Gauss (m=3)
T0 = 50 ps
P0 = 0.66 mW
λ0 = 1557.6 nm
FIBER PARAMETERS:
L1 = 89.8 km L2 = 16.2 km
L1 = 23⋅(L1 + L2) = 2438 km
α1 = 0.05231 1/km α2 = 0.13 1/km
β21 = -22.23 ps/km β22 = 119.48 ps/km
γ1 = γ2 = 1.35 1/Wkm
EDFA:
∆λ = 10 nm
λ0 = 1557.6 nm
G = 9.8 dB
nsp = 2.78
FABRY-PEROT FILTER:
f0 = 192.6 THz
B = 370 GHz

45. 45
Conclusions:
♦The derivation of optical pulse propagation equation is given
in details. All important effects influencing pulse propagation
in optical fiber are analyzed: fiber loss, cromatic dispersion,
polarization mode dispersion, nonlinear effects (especially
self-phase modulation)
♦Several numerical models are analyzed and the most
accurate one chosen for propagation equation modeling.
The accuracy is tested on simple canonical problems and
later on compared with commercially available software.
♦EDFA model strictly in time domain is developed, with
special attention given to ASE noise model. EDFA model and
optical filter model are included in computer program
FiberProp

46. 46
♦ The new approach to Gordon-Haus limitation derivation is
given. Timing jitter due to Gordon-Haus effect and its
suppression was analyzed with the FiberProp computer
program.
♦ Numerous examples of soliton and dispersion-managed
soliton transmission systems are analyzed and guidelines
for their design are given.
Conclusions (cont.):