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Introduction to
Optical Networks
Chapter 2
Propagation of Signals in
Optical Fiber
1
2.Propagation of Signals in Optical Fiber
Advantages
• Low loss ~0.2dB/km at 1550nm
• Enormous bandwidth at least 25THz
• Light weight
• Flexible
• Immunity to interferences
• Low cost
Disadvantages and Impairments
• Difficult to handle
• Chromatic dispersion
• Nonlinear Effects
2
Cladding SiO2 , refractive index ≈1.45
SiO2
core 8~10μm, 50μm, 62.5μm doped
2.1 Light Propagation in Optical Fiber
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4
2.1.1Geometrical Optical Approach (Ray Theory)
This approach is only applicable to multimode fibers.
1 : incident angle (入射角)
 2 : refraction angle (折射角)
1r : reflection angle (反射角)
1r  1
Snell’s Law
n1 sin1  n2 sin2
n1, n2 : refractive indices
5
2f
n1  n2 and when 2   / 2
n2
,
n1
=>Critical angle c  Sin
When 1  c , total internal reflection occurs.
1
let 0 = air refractive index
0max= acceptance angle (total reflection will
6
occur at core/cladding interface)
n0 sin0max  n1 sin1max
n2
sinc  ,
n1
c 
sin(

2

2

max
1

max
1
n2
)
n1
n1 cos 1max  n2
sin 1max 
n22
1 2
n1
n0 sin  0max 
 0max  sin1
n12  n22
n12  n22
n0
(2.2)
7
n1  n2
Denote 
n1
n12  n22  (n1  n2 )(n1  n2 )
 ( n1  n2 )n1
If Δ is small (less than 0.01)
n12  n22  n1

max
0
 sin
For n1  1.5
1
n1
2
2
n0
(multimode)
  0.01
max

n0  1 0  12
max
n
sin

 n1 2
Numerical Aperture NA= 0
0
Because different modes have different lengths of
paths, intermodal dispersion occurs.
8
Infermode dispersion will cause digital pulse spreading
Let L be the length of the fiber
The ray travels along the center of the core
T f  Ln1 / C
The ray is incident at
c (slow ray)
Ln1
Ts 
c cos1max
Ln12
n2
max

cos1 
cn2
n1
 T  Ts  T f
Ln12
Ln1


cn2
c
Ln1( n1  n2 )

cn2
Ln12 

cn2
9
Assume that the bit rate = Bb/s
1
Bit duration T 
B
T
1
T 

2
2B
Ln12 
1

cn2
2B
The capacity is measured by BL (ignore loss)
n2c
c
BL 

2
2n1  2n1 
Foe example, if   0.01, n1  1.5 BL  (10mb / s )  km
10
For optimum graded-index fibers, δT is shorter than
that in the step-index fibers, because the ray travels
along the center slows down (n is larger) and the ray
traveling longer paths travels faster (n is small)
11
The time difference is given by (For Optical graded-index profile)
and
If
Ln1 2
T 
8c
1
2B
4c
BL 
n1 2
T 
(single mode
BL 
  0.01, n1  1.5 BL  8 (Gb / s )  km
c
2n1
)
Long haul systems use single-mode fibers
12
2.1.2 Wave Theory Approach
Maxwell’s equations
 D  
 B  0
B
t
D
 H  J 
t
  
D.1
D.2
D.3
D.4
ρ : the charge density, J: the current density
D : the electric flux density, B: the magnetic flux density
: the electric field, H : the magnetic field
13
Because the field are function of time and location in the
space, we denote them by
E ( r, t ) and H( r, t ) , where r and t are position vector and
time.
Assume the space is linear and time-invariant the Fourier
transform of E( r, t ) is

2.4
E(r, w)= E(r, t )exp(iwt )dt

-
w  2 f
let P be the induced electric polarization
D 0 E  P
0 : the permittivity of vacuum
B  0 ( H  M )
M : the magnetic polarization
 : the permeability of vacuum
注意有些書Fourier transform定義為

E ( r, w)=  E( r, t ) exp(  jwt )dt
-
2.5
2.6
0



-

E( r, t ) exp(  j 2 ft )dt
E ( r, t )=  E( r, w) exp( j 2 ft )dt
-
14
Locality of Response: P and E related to
dispersion and nonlinearities
If the response to the applied electric field is local
P(r1 ) depends only on E(r1 )
not on other values of E(r1 ), r  r1
This property holds in the 0.5~2μm wavelength
Isotropy: The electromagnetic properties are the
same for all directions in the medium
Birefringence: The refraction indexes along two
different directions are different (lithium niobate,
LiNbO , modulator, isolator, tunable filter)
3
15
Linearity：
P(r, t )  0



x(r, t  t ' )E(r,t ' )dt ' (Convolution Integral) 2.7
x( r, t ) : linear susceptibility
The Fourier transform of P( r, t ) is
P(r, w)  0 x(r, w)E(r, w)
2.8
Where x( r, w) is the Fourier transform of E( r, t )
( x( r, t ) is similar to the impulse response)
x( r, w) is function of frequency
=> Chromatic dispersion
16
Homogeneity: A homogeneous medium has the
same electromagnetic properties at all points
x( r, t )  x(t )
The core of a graded-index fiber is
inhomogeneous
Losslessness： No loss in the medium
At first we will only consider the core and cladding
regions of the fiber are locally responsive,
isotropic, linear, homogeneous, and lossless.
The refractive index is defined as
def
2
2.9
n ( w)  1  x( w)
n  1.5
For silica fibers x  1.25
17
From Appendix D
 D  
 B 0
B
t
D
 H  J 
t
  
For   0 (zero charge)
  0 (zero conductivity, dielectric material)
J E 0
D 0 E  P
B  0 ( H  M )
For nonmagnetic material M  0
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
 E  
 B
t

( 0   H )
t

2 D
 
( 0
)
2
t
t
2
 
0 (0 E  P )
2
t
2
2
  0 0
E  0
P
2
2
t
t
 
Assume linear and homogenence
E( r, w) 



E( r, t ) exp(iwt )dt
1 
E( r, t ) 
E( r, t ) exp( iwt )dw

2 

iw
t
19
2
2
    E( r, t )   0 0 2 E( r, t )  0 2 P( r, t )
t 
t
Take Fourier transform ( t iw)
2
2
 E(r, w)  0 0 w E(r, w)  0 w P(r, w)
Recall
P(r, w) 0 x(r, w)E(r, w)
2.8
    E(r, w)  0 0 w2 E(r, w)  0 0 w2 x(r, w) E(r, w)
Denote
c
1
0 0
c: speed of light
n( w)  1 x( w)
(Locally response, isotropic, linear,
homogeneous, lossless)
    E(r, w)  0 0 w2 (1 x(r, w))E(r, w)
w2 n2
 2 E( r, w)
c
2.9
20
    E ( r, w)  ( E( r, w))   E( r, w)
2
2
2
w n
 E ( r, w)  2 E ( r, w)  ( E ( r, w))
c
2
  palacian operation
2
 E ( r, t )    0,
2
2
w n ( w)
  E ( r, w ) 
E ( r, w )  0
2
c
2
2
2
or  E ( r, w)  n ( w)K 0 E ( r, w)  0
2
where K 0  w
c
 2

2.10
(free space wave number)
21
For Cartesian coordinates
2
2
2
2 





x2 y 2 z 2
For Cylindrical coordinatesρ. φ and z
 2 Ez 1 Ez 1  2 Ez  2 Ez

 2
 2  n2 k02 Ez  0
2
2

   
z
n1
a
n:{ n
 a
2
a: radius of the core
2 2
w
n ( w)
2

H
(
r
,
w
)

H ( r, w)  0
2
Similarly
c
Boundary conditions   0 E is finite
  , E  0, and continuity of field at ρ=a
2.11
References:
G.P. Agrawal “Fiber-Optical Communication System” Chapter 2
John Senior “Optical Fiber Communications, Principles and practice”
John Gowar “Optical Communication Systems”
注意有些書在 time domain運算
有些書在frequency domain運算
22
Fiber Modes
cladding
core
x
z
y
23
E core , E cladding , H core , and H cladding must satisfy
2.10, 2.11 and the boundary conditions.
let E(r, w)  Ex e x  E y e y  Ez e z
Where e x , e y , and e z are unit vectors
For the fundamental mode, the longitudinal
component is
Ez  2 J e ( x, y ) exp(i  z )
  wn
2 fn
2 n


c
c

 : the propagation constant
24
J ( x, y ) : Bessel functions
The transverse components ( Ex and E y )
Ex  2 J t ( x, y )exp(i  z )
For cylindrical symmetry of the fiber
J ( x, y ) and J t ( x, y ) depend only on x  y  
2
2
In general, we can write
E( r, w)  2 J ( x, y )exp(i  ( w) z )e( x, y )
(Appendix E)
25
Where
J ( x, y)  J ( x, y)  J ( x, y)
2
2
t
The multimode fiber can support many
modes. A single mode fiber only supports
the fundamental mode.
Different modes have different β,
such that they propagate at different
speeds.=>mode dispersion
(We can think of a “mode” as one possible
path that a guided ray can take)
26
For a fiber with core n1and cladding n2, if a wave
propagating purely in the core, then the propagation
constant is
wn1
2 n1
1 

c
  kn1
λ: free space wavelength
The wave number k  2 
Similarly if the wave propagating purely in the cladding,
then
 2  kn2
The fiber modes propagate partly in the cladding and partly
in the core,
so kn2    kn1

Define the effective index neff 
k
n n n
2
eff
1
The speed of the wave in the fiber= c n
eff
27
For a fiber with core radius a , the cutoff
condition is
def
2
V
a n12  n22  2.405

V : normalized wave number
n1  n2
Recall   n
1
V↓ when a↓
and △ ↓
For a single mode fiber, the typical values
are a=4μm and △=0.003
V  2 at 1550nm
28
The light energy is distributed in the core
and the cladding.
29
30
Since Δ is small, a significant portion of the light
energy can propagate in the cladding, the
modes are weakly guided.
The energy distribution of the core and the
cladding depends on wavelength.
kn1    kn2
n 
k
n2  neff  n1
eff
It causes waveguide dispersion (different from
material dispersion)
( Appendix E )
For longer wave, it has more energy in the
cladding and vice versa.
31
A multimode fiber has a large value of V
2
V
The number of modes  2
For example a=25μm, Δ=0.005
V=28 at 0.8μm
Define the normalized propagation constant (or
normalized effective index)
def
b
 k n
2
2
2
2
k n k n
2
2
1
2
2
2

2
eff
2
1
n
n
2
2
2
2
n n
b(V )  (1.1428  0.9960 / V )
2
HE11 mode
( H z  Ez )
b(V ) is used to investigate the wave
propagation in fibers
32
Polarization
Two fundamental modes exist for all λ. Others only
exist for λ< λcutoff,
E( r, t )  E x e x  E y e y  E z e z
E z : longitudinal component
E x , E y : transverse components
Linearly polarized field : Its direction is constant.
For the fundamental mode in a single-mode fiber
E x , E y  E z
33
34
35
Fibers are not perfectly circularly symmetric.
The two orthogonally polarized
fundamental modes have different β
=>Polarization-mode dispersion (PMD)
Differential group delay (DGD)
ps km
Δτ=Δβ/w ～typical value Δτ=0.5
100 km => 50 ps
Practically PMD varies randomly along the fiber and
may be cancelled from an segment to another
segment.
ps km
Empirically, Δτ ～0.1-1
Some elements such as isolators, circulators, filters
may have polarization-dependent loss (PDL).
36
2.2 Loss and Bandwidth
Pout  Pin e L
L : length of fiber
 : fiber loss in dB km
Pout
10 log10
  dB
Pin
 dB  (10 log10 e)  4.343
Two main loss mechanisms : material absorption
and Rayleigh scattering
The material absorption is negligible in 0.8  m ~ 1.6  m
37
38
c
Recall f   2 
Take the bandwidth over which the loss in dB/km is
with a factor of 2 of its minimum.
80nm at 1.3μm, 180nm at 1.55μm
=>BW=35 THz
39
Erbium-Doped Fiber Amplifiers (EDFA) operate in
the c and L bands, Fiber Raman Amplifiers (FRA)
operate in the S band.
All Wave fiber eliminates the absorption peaks due
to water.
40
41
2.2.1 Bending loss
A bend with r = 4cm, loss < 0.01dB
r↓
loss↑
2.2.3 Chromatic Dispersion
Different spectral components travel at different velocities.
a. Material dispersion n(w)
b. Waveguide dispersion, different wavelengths have
different energy distributions in core and cladding
=>different β, kn2< β < kn1
1  dB dw , 1 : group velocity
1
2
2  d B
: group velocity dispersion (GVD ) parameter
dw2
If  2  0 zero  dispersion
 2  0, the dispersion is normal
 2  0, the dispersion is anomalous
42
2.3.1 Chirped Gaussian Pulses
Chirped: frequency of the pulse changes
with time.
Cause of chirp: direct modulation, nonlinear
effects, generated on purpose. (soliton)
43
Appendix E, or Govind P. Agrawal “ Fiber- Optic
Communication Systems” 2nd Edition, John Wiley
& Sons. Inc. PP47~51
A chirped Gaussian pulse at z=0 is given by
1 ik  t



2 
 T0

G(t )  R A0 e


 A0 e
 A0 e
1 t
 
2
 T0




2
1 t

2
 T0




2





2
e
 i0 t




2

k  t  
cos  0t 

 

2  T0  


cos  (t )
kt 2
 (t )  0 t 
2T0
The instantaneous angular frequency
d  (t )
k
 0 
t
dt
T0
T0  Pulse width
44
k = The chirp factor
Define: The linearly chirped pulse: the instantaneous
angular frequency increases or decreases with time,
(k=constant)
 i0t


G
(
t
)

R
A
o
,
t
e


Note


A
A i
A2
Solve  1  2 2  o with the
initial
2
z
t 2 t
1 ik  t 
condition A(o, t )  A0 e
A0T0
We get A( z, t ) 




2 
 T0 
(E.7)

(1  ik )(t  1z )2 

exp  
2
2
 2 T0  i  2 z(1  ik ) 
T0  i  2 z(1  ik )


2




t  1z
1  ik

 
 Az exp  

2  T02  i  2 z(1  ik )   (E.8)

 



A(z,t) is also Gaussian pulse
45
Tz  T0   2 kz  i  2 z

Tz

T0
T0
  2 kz     2 z 
2
2
2

 2 z 
 2 kz 
1






2
T
T
0


 0 
2
Broadening of chirped Gaussian pulses
They have the same of broadening length.
Note a  b distance  2LD , c  d distance  0.4 LD
46
In Fig 2.9, 2  0, it is true for standard fibers at 1.55μm
LD 
Let
T02
2
If z
LD , dispersion can be neglected
T02
z
z 2
If
be the dispersion length
2
T (2.13) 
2
0
T02
Tz

T0
2 z
2
1
Tz
T0
1
and k  0 (unchirped pulse)
z  LD
For 2.5 Gb / s systems at 1.55  m ( return to zero pulse )
T
let T0   0.2ns ( half pulse duration ) LD  1800km
2
For 10 Gb / s, T0  0.05ns LD  115km
For NRZ ( return to zero )  LD  600km for 2.5 Gb / s
47
For kβ2 < 0
Tz

T0
2

 2 z 
k 2 z 

1


2 
T02 

 T0 
↓decreases
2
increases ↑
for certain z
For β2 > 0, high frequency travels faster
=> the tail travels fasters => compression (Fig 2.10)
=> make βk < 0, LD increases
48
Note β2> 0 , k< 0 β2k<0
49
2.3.2 Controlling the Dispersion Profile
Def:
Chromatic dispersion parameter
D =  2 c  2  2 in ps / nm  km
D = DM + D w
The standard single mode fiber has small chromatic dispersion at 1.3
μm but large at 1.55 μm


50
At 1.55μm loss is low, and EDFA is well developed.
Dispersion becomes an issue
We have not much control over DM, but Dw can be
controlled by carefully designed refractive index
profile.
Dispersion shifted fibers, which have zero dispersion
in 1.55μm band
51
52
2.4 Nonlinear Effects
For bit rate ≦2.5 Gb/s, power a few mw
Linear Assumption is valid
Nonlinear effect appears for high power or high bit
rate ≧ 10 Gb/s and WDM systems
The first category relates to the interaction of
lightwave with phonons (molecular vibrations)
- Rayleigh Scattering
- Stimulated Brillouin Scattering (SBS)
- Stimulated Raman Scattering (SRS)
53
The second category is due to the dependence of
the refractive index on the intensity
- self-phase modulation (SPM)
- four-wave mixing (FWM)
SBS and SRS transfer energy from short λ (pump)
to long λ(stokes wave)
Scattering gain coefficient, g, is measured in
meter/watt and Δf.
SPM induces chirping
In a WDM system, variation of n depending on the
intensity of all channels.
=>Yields Cross-phase modulation (CPM)
=>interchannel crosstalk
54
●
FWM, f1, f 2 , ... f n
fi , f j , f k  ( fi  f j  f k ), e.g 2 fi  f j , fi  f j  f k
55
2.4.1 Effective Length and Area
The nonlinear effect
depends on fiber length and cross-section.

P( z )  P0 e 
P0 Le 

L
z 0
z
P( z )dz
when Le : effective length
Le 
1  e  L

Typically   0.22 dB km at 1.55  m L
1
 (for long link)
Le  20 km
56
In addition nonlinear effect  intensity
Ae  effective cross  sectional area
2

F
(
r
,

)
rdrd 



  r 
4
  F ( r, ) rdrd
r
2

F ( r, ) : Fundamental mode intensity
Ie  P
 effective intensity
Ae
SMF Ae ~ 85  m 2 ,
DSF , Ae ~ 50  m 2
由 power point 50, DSF n1 大 n2
dispersion compensating fiber ( DCF ) n1 及 n2 差最多
Ae 更小, nonlinear effect 更嚴重
57
2.4.1 Stimulated Brillouin Scattering (SBS)
The scattering interaction occurs with acoustic
phonons over Δf =15 MHz, at 1.55μm, stokes and
pump waves propagate in opposite directions.
If spacing > 20 MHz => no effects on different channels
Ps(0)
Pp(L)
SBS
pumping
58
g B  4  10 11 m
w
independent of 
dI s
  gB I p Is   Is
dz
dI p
  gB I p Is   I p
dz
I s : Intensity of stokes, Ps  I s Ae
(2.14)
(2.15)
I p : Intensity of pump, Pp  I p Ae
Ae : effective area
Assuming I s is small, g B I p I s
dI p
Ip
  I p  I p ( z )  I p (0)e  z
dz
Pp ( L )  Pp (0)e  L
L : length
g B Pp (0) Le
Ps (0)  Ps ( L )e  L e
Le =
1-e  L
(2.16)
(P.78)

Pp ( L )  Pp (0)e
Ae
 L
59
2.4.3 Stimulated Roman Scattering (SRS)
SRS will deplete short wave power and amplifier long wave.
60
2.4.4 Propagation in a Nonlinear Medium
In a nonlinear medium, Fourier Transfer is not applicable.
When the electrical field has only one component, we can
write E ( r, t ) and P( r, t ) as the scalar functions E ( r, t ) ,
and P( r, t ) .
Appendix F, P( r, t ) contains higher order terms
P( r, t ) 0
 0
 0

t

x(1) ( r, t  t1 )E( r, t1 )dt1
t
 
t
 
t
t
x(2) (t  t1, t  t2 )E( r, t1 )E( r, t2 )dt1dt2
  
t
  
x(3) (t  t1, t  t2 , t  t3 )E( r, t1 )E( r, t2 )E( r, t3 )dt1dt2 dt3 ( F .1)
x(1) ( r, t ) : the linear susceptibility
x( i ) ( r, t ) : higher order nonlinear susceptibilities
P( r, t )  PL ( r, t )  PNL ( r, t )
linear polarization
nonlinear polarization
61
Because of symmetry x(2) ( r, t )  0 , and x(i )  0 i  4. 5...
t
t
t
PNL (r, t ) 0    x(3) (t  t1, t  t2, t  t3 )E(r,t1)E(r, t2 )E(r, t3 )dt1dt2dt3 ( F.2)
  
The nonlinear response occurs less than 100x10-15sec.
If the bit rate is less than 100 Gb/s, then
x(3) (t  t1, t  t2 , t  t3 )  x(3) (t  t1 ) ( t  t2 ) (t  t3 )
PNL ( r, t ) 0 x(3) E 3 (r, t )  E 3
(2.19)
x(3): the third-order nonlinear susceptibility independent of time
For simplicity, assume that the signals are monochromatic
plane waves
E(r,t )  E( z,t )  E cos( w0t  i z )
E is constant in the plane perpendicular to the dispersion of
propagation
In WDM systems with n wavelengths at the angular
frequencies w1, w2 ... wn .,( 1, 2 ... n ) 0
n
E( r, t )  E( z, t )   Ei cos( wi t  i z  i )
i 1
62
2.4.5 Self-phase Modulation (SPM)
Because n is intensity – dependent
=>induces phase shift proportional to the intensity
=>creates chirping => pulse broadening
It is significant for high power systems.
Consider a single channel case
E( z, t )  E cos( w0t  0 z )
PNL ( r, t ) 0 x(3) E 3 cos3 ( w0t  0 z )
1
3

0 x(3) E 3  cos( w0t  0 z )  cos(3 w0t  3 0 z ) 
4
4

3 w0 
0
3
(2.20)
shorter wavelength, the last term is negligible
3
PNL ( r, t )=( 0 x(3) E 2 )E cos( w0t  0 z )
4
E( z, t )
(2.21)
63
(1)
Recall n ( w)  1  x for linear medium
2
n
Now, we have to modify ( w) as
2
3 (3) 2
n ( w)  1  x  x E
4
(1)
2
We get
w
0  0
c
let
1 x
n  1 x
2
(1)
3 (3) 2
 x E
4
(1)
w0 n
3 (3) 2
0 
1
x E
2
c
4n
x(3) is very small
w
3 (3) 2
0  0 ( n 
x E )
c
8n
(2.22)
propagation constant changes with E 2
=> Phase changes with E
2
 intensity
64
E( z, t )  E cos( w0t  0 z ) ,whose phase changes as E z2 ,
this phenomenon is referal as self- phase modulation (SPM)
The intensity of the electrical field
1
I  0 cnE 2
in w 2
m
2
The intensity-dependent refractive index is
n( E )  n  nI
(2.23)
The nonlinear index coefficient
n 
2
3
x( 3 )
0 cn 8 n
2
2.2  3.4  10 8  m
in silica fiber
We take n  3.2  10 8  m w for example
Because a pulse has its finite temporal extent
=>The phase shift is different in different parts of the pulse
The leading edges have positive frequency shift
The tailing edges have negative frequency shift
=> SPM causes positive chirping
n
w
2
65
2.4.6 SPM-induced chirp for Gaussian Pulses
Consider an unchirped pulse with envelope

U (0, )  e 2 which has unit peak amplitude and
2
-width T0=1, and the peak power P0=1
Define the nonlinear
length as
e
1
e
LNL 
A
2 nP0
If link length ≧
LNL => nonlinear effect is severe
66
From Appendix E, (E.18)
U ( z, )  U (0, z )e
 U (0, z )e
iz U (0, )
2
LNL
iz
E.18
LNL  2
e
After propagation L distance,
The SPM-induced phase change is
' L
LNL
e
 2
The instantaneous frequency is given by
w( )  w0 
2 L  2
 e , w0 : central freg.
LNL
and the chirp factor is
2 L  2
k SPM ( ) 
e (1  2 2 )
LNL
References: Appendix E, and (Arg97)
67
k SPM ( ) 
2 L  2 (1 2 2 )
e
LNL
Recall Le 
<
1  e  L

increases with L
effective length
(2.25)
1

At the center of the pulse   0
 Ae
2
k SPM 
,
L

NL
LNL
2 nP0
At 1.55 m,   0.22 dB
For P0  1mw
P0  10 mw
LNL
km
 384km
LNL  38 km
negligible
significant
68
2.4.7 Cross-Phase Modulation
In WDM systems, the intensity-dependent nonlinear effects
(phase shift) are enhanced by other signals, this effect is
referred to as cross-phase modulation (CPM)
Consider two channels
E(r,t )  E1 cos( w1t  1z )  E2 cos( w2t  2 z )
(3) 3
P
(
r
,
t
)

x
E ( r, t )
Recall NL
0
0 x
(3)
(2.19)
 E1 cos( w1t  1z )  E2 cos( w2t  2 z )
3
69
2w1+w2, 2w2+w1, 3w1and 3w2 can be neglected
2w1-w2, 2w2-w1, are part of FWM.
Consider the w1 channel, the CPM term is
3
0 x(3) ( E12  2 E22 )E1 cos( w1t  1z )
4
(2.27)
CPM
If E1=E2 SPM
Apparently CPM effect is twice of SPM.
In practice, β1 and β2 are different
=> The pulses corresponding to individual channel
walk away from each other.
=> can not interact further
=> CPM is negligible for standard fibers
Note for DSF, they travel at same velocity, CPM is
significant
70
2.4.8 Four-Wave Mixing (FWM)
71
wi, wj ,wk (three waves) generate
wi ± wj ± wk (fourth wave)
For example, channel spacing Δw
w2 = w1 + Δw, w3 = w1 + 2Δw
w1- w2+ w3 = w2, 2w2-w1 = w3
72
Define wijk  wi  w j  wk , i, j  k
The degeneracy factor
d i. j .k 

i j
i j
3
6
( eq. 2.30)
( eq. 2.33)
The normalized Pijk(z,t) is given by
0 x(3)
Pijk ( z, t ) 
dijk Ei E j Ek cos ( wi  w j  wk ) t  ( i   j  k )z
4


(2.36)
If we assume that the optical signals
propagate as plane waves over Ae and
distance L, then the power is
 wijk dijk x
Pijk  
 8 Ae neff c

(3)
2

2
P
L
 PP
i j k

(using Fig 2.15 and 2.36)
Pi . Pj and Pk are powers at wi w j wk
73
For example
Pi  Pj  Pk  1mw ,
Ae  50  m 2
wi  w j , d ijk  6
n  3.0  10
8
 m2
w
L  20 km
Pijk  9.5  w
about 20dB below Pi  1mw
If another channel at wijk
Then FWM will interfere the wijk channel.
Practical FWM lacks of phase matching
=> No significant influence (in normal fibers)
74
2.4.9 New Optical Fiber Types
A. DSF is not suitable for WDM due to nonlinear effect.
To reduce nonlinear effect (different group
velocities lack phase matching)
=>to develop nonzero-dispersion fibers
(NZ-DSF)
a chromatic dispersion 1~6 ps/nm-km
or -1 ~ -6 ps/nm-km
NZ-DSF has most advantage of DSF (in c-band)
75
76
Large Effective Area Fiber (LEAF)
1
nonlinear effect  A for fix power
e
77
78
Positive and Negative Dispersion Fibers For
Chromatic dispersion compensation
79
80