Transcript Slide 1

Dispersion
Syed Abdul Rehman Rizvi
Optical Communication
Intermodel dispersion (Multimode dispersion)
Φc
X
X
X
L
L
X= L/SinΦc
SinΦc= n2/n1
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Intermodel dispersion (Multimode dispersion)
T he extent of pulse broadening can be estim ated by considering the longest and shortest ray paths.
T he shortest path occurs for  i =0, and is just equal to the fiber lenght 'L'.T he longest
path occurs for  i show n previously and has a lenght 'L/sin c .
v =c / n1 , the tim e delay is given by ;
T =TM ax TM in
L n1 L
x L
s
= =
= n2
= Ln1 n1 n 2 n1
v
c
v
cn 2
n1
n1
= Ln12 
cn 2
W hen 1
under this condition =
n1 n 2
m ay also be true
n2
1
Ln1 n1 n 2 
T = Ln12  Ln1 = Ln1 n
 1 =


cn 2
c
c n 2
c n 2 
L(NA)2
= Ln1  = Ln12 2  =
c
2 n1 c
2 n1 c
Q
(N A ) 2 =2 2n1 


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The time delay between the two rays taking the shortest and longest
paths is a measure of broadening experienced by an impulse launched
at the fiber input.
We can relate ∆T to the information-carrying capacity of the fiber
measured through the bit rate B. Although a precise relation between
B and ∆T depends on many details,
Requirement for minimal inter symbol interference:
B ∆t < 1
where
B = bit rate
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Dispersion in single mode fiber
Intermodal dispersion in multimode fibers leads to
considerable broadening of short optical pulses (~ 10
ns/km).
In geometrical optics description this is attributed to
different paths followed by different rays.
In the modal description it is related to the different mode
indices or group velocities associated with different modes.
The main advantage of single mode fiber is that intermodal
dispersion is absent but that doesn’t mean that dispersion
has vanished altogether.
The group velocity associated with the fundamental mode is
frequency dependent because of chromatic dispersion.
The result is that different spectral components of the pulse
travel at slightly different group velocities and the
phenomena is referred as group-velocity dispersion (GVD),
intramodal dispersion or simply fiber dispersion.
Intramodal dispersion has two contributions, material dispersion
and waveguide dispersion, such that
D =DM +DW
Where
DM
is material dispersion and
waveguide dispersion.
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DW
is
Group-Velocity Dispersion
Consider a single-mode fiber of length L. A specific spectral
component at the frequency ω would arrive at the output end of
the fiber after a time delay
L
T=
vg
Where
vg
is the group velocity defined as
dω
1
1
=
(
d
β
d
ω
)
=
vg =
dβ dβ dω
Remember at the same time, phase velocity is defined as
ω
vp=
β
In nondispersive medium the phase velocity is
independent of the wave frequency and the group
velocity and phase velocity are the same. So in such
case
v p =vg
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By using β =n k0 =n  c
where n is mode index or effective index, is propagatio n constant,
k0 is free space propagtion constant or free space wave number
k0 = ω c = 2π λ
and  is angular frequency.
At the same time vg =c
ng
Where ng is the group index given by
ng =n + (dn d )
The frerquency dependence of group velocity leads to pulse broadening because
different spectral components of the pulse disperse during propagation and don't
arrive simultaneously at the fiber output. If  is the spectral width of the pulse,
the extent of pulse broadening for a fiber of length L is governeddb
by
d L ⎞
dT
T =
∆ω =
 ⎟
∆ω

dω
dω 
⎜ vg⎠
d 2β
=L 2 ω
dω
= Lβ2 ∆ω
( Q T =L vg )
( Q vg =(dβ dω) )
-1
The parameter 2 =d 2 d 2 is known as the GVD parameter.
It determines how much an optical pulse would broaden on propagation
inside the fiber.
In some optical communicat ion systems, the frequency spread 
is determined by the range of waveleng ths λ in place of ω.
By using ω =2c  and  =(2c  2 )
d L 
d L 

 ⎟∆ω = DL λ
⎟∆
ω

T
=
Q T =

dω 
⎜ v
g
d
λ
⎜
v
g
 ⎠
⎠
Where
d 1 ⎞
2πc
 =2 β 2
D=
λ
dλ ⎜
 vg⎠
D is called the dispersion parameter and is expressed in units
of ps/(km - nm).
T h e e ffe c t o f d is p e rs io n o n th e b it ra te B c a n b e e s tim a te d b y u s in g th e
c rite rio n B T 1 . B y p u ttin g th e v a lu e o f T =D L  th is c o n d itio n
becom es
BL D  1
F o r s ta n d a rd fib e r s D is re la tiv e ly s m a ll in th e w a v e le n g h t re g io n n e a r
1 3 0 0 n m [ D ~ 1 p s /(k m -n m )]. F o r a s e m ic o n d u c to r la s e r th e s p e c tra l
w id th  is 2 -4 n m . T h e B L p ro d u c t o f s u c h lig h t w a v e s y s te m s c a n
e x c e e d 1 0 0 (G b /s ) -k m . T e le c o m m s y s te m s w o rk in g a t 1 3 0 0 n m ty p ic a lly
o p e ra te a t a b it ra te o f 2 G b /s w ith a re p e a te r s p a c in g o f 4 0 -5 0 k m .
B L p ro d u c t o f s in g le m o d e fib e r c a n e x c e e d 1 (T b /s )-k m w h e n s in g le
m o d e s e m ic o n d u c t o r la s e rs w ith  b e lo w 1 n m .
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D d ep en d s u p o n o p eratin g w avelen g th b ecau se o f freq u en cy d ep en d en ce
o f m o d e in d ex n . W e alread y k n o w th at
d
D=
dλ
1 

v
g 
2π c d ⎛ 1 ⎞
dn 2π d 2 n 
+ω
=2 2


λ d ω v
λ
d
ω
d
ω
2 ⎠
g 
u sin g th e relati o n
n g =n + ω ( dn d ω )
D =2
D can b e w ritten as su m o f tw o term s
D =D M +DW
W h ere th e m aterial d isp ersio n D M an d th e w aveg u id e d isp ersio n DW
are g iven b y
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2π dn 2g 1 dn 2g
=
DM =2
λ dω c dλ
2π∆ ⎡ n 22g Vd2 (Vb )dn 2g d (Vb )
+

DW =2 
2
λ ⎢
n 2ω dV
dω dV 
where n 2g is group index of material and V is
normalized frequency and b is normalized propagation constant
as already defined.
A/z
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ZMD
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• The single-mode values of interest are from V = 2.0 to 2.4, as shown in Fig.
•The value of Vd2(Vb)/dV2 decreases monotonically from 0.64 down to 0.25.
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Material dispersion
Material dispersion occurs because the refractive index of silica,
the material used for fiber fabrication, changes with the optical
frequency ω.
On a fundamental level, the origin of material dispersion is
related to the characteristic resonance frequencies at which the
material absorbs the electromagnetic radiation.
Far from the medium resonances, the refractive index n(ω) is
well approximated by the Sellmeier equation .
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where ωj is the resonance frequency and B j is the
oscillator strength. Here n stands for n1 or n2,
depending on whether the dispersive properties of
the core or the cladding are considered.
In the case of optical fibers, the parameters Bj and
ωj are obtained empirically by fitting the measured
dispersion curves.
They depend on the amount of dopants (Boron,
arsenic and Antimony etc).
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Figure shows the wavelength dependence of n and n g in the range 0.5–1.6 µm
for fused silica. Material dispersion DM is related to the slope of ng by the relation DM
= c−1(dng/d λ). It turns out that dng/d λ= 0 at λ= 1.276 µ m. This wavelength is
referreddtto as the zero-dispersion wavelength λZD, since DM = 0 at λ= λZD.
The dispersion parameter DM is negative below λZD and becomes positive above
that. In the wavelength range 1.25–1.66 µ m it can be approximated by an empirical
relation.
Lemda zero may be extended to
1550µm
Lowering the normalised freq
Increasing the relative refractive index
difference ∆
Suitable doping of the silica with
germenium.
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Waveguide Dispersion
The contribution of waveguide dispersion DW to the dispersion parameter D is
given by following Eq.
DW is negative in the entire wavelength range 0–1.6 µm.
On the other hand, DM is negative for wavelengths below λZD and becomes
positive above that.
D = DM +DW,
for a typical single-mode fiber.
The main effect of waveguide dispersion is to shift λZD by an amount 30–40
nm so that the total dispersion is zero near 1.31 µ m.
It also reduces D from its material value DM in the wavelength range 1.3–1.6 µ
m that is of interest for optical communication systems.
Typical values of D are in the range 15–18 ps/(km-nm) near 1.55 µ m.
This wavelength region is of considerable interest for lightwave systems, since
the fiber loss is minimum near 1.55 µ m.
High values of D limit the performance of 1.55- µ m lightwave systems.
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Since the waveguide contribution DW
depends on fiber parameters such as the
core radius a and the index difference ∆,
it is possible to design the fiber such that
λZD is shifted into the vicinity of 1.55
µm. Such fibers are called dispersion
shifted fibers.
It is also possible to tailor the waveguide
contribution such that the total dispersion
D is relatively small over a wide
wavelength range extending from 1.3 to
1.6 µm . Such fibers are called
dispersion-flattened fibers.
The design of dispersion modified fibers
involves the use of multiple cladding
layers and a tailoring of the refractiveindex profile . Waveguide dispersion can
be used to produce dispersiondecreasing fibers in which GVD decreases
along the fiber length because of axial
variations in the core radius. In another
kind of fibers, known as the dispersion
compensating fibers, GVD is made
normal and has a relatively large
magnitude.
Higher order dispersion
BL product of a single-mode fiber can be increased indefinitely
by operating at the zero-dispersion wavelength λZD where D =
0.
The dispersive effects, however, do not disappear completely
at λ = λZD.
Optical pulses still experience broadening because of higherorder dispersive effects.
This feature can be understood by noting that D cannot be
made zero at all wavelengths contained within the pulse
spectrum centered at λZD.
Clearly, the wavelength dependence of D will play a role in
pulse broadening.
Higher-order dispersive effects are governed by the dispersion
slope S = dD/dλ. The parameter S is also called a differentialdispersion parameter.
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where β3 = dβ2/dω≡ d3 β/dω3 is the third-order dispersion parameter. At
λ= λZD, β2 = 0, and S is proportional to β3.
The numerical value of the dispersion slope S plays an important role in
the design of modern WDM systems.
It may appear from following Eq. that the limiting bit rate of a channel
operating at λ = λZD will be infinitely large. However, this is not the case
since S or β3 becomes the limiting factor in that case.
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We can estimate the limiting bit rate by noting that for a source of spectral
width ∆ λ, the effective value of dispersion parameter becomes
D = S∆ λ
. The limiting bit rate–distance product can now be obtained by using this
value of D. The resulting condition becomes
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