Dispersion

Syed Abdul Rehman Rizvi

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Intermodel dispersion (Multimode dispersion)

X L L X= L/SinΦ c SinΦc= n 2 /n 1 Φ c

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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

/

n

1 , the tim e delay is given by ;

T

=

T M ax T M in

=

s v

=

x v L

=

L n

1

n

2

c L n

1 =

Ln

1

n

1

n

2

n

1

cn

2

n

1 =

Ln

12

cn

2 W hen  1 under this condition =

n

1

n

2

n

2 m ay also be true =

T

=

Ln

1

c Ln Ln

1

cn

2

c

=

Ln

2 12

n

2 1

c

=

Ln

1

c



n n

1 2 1  = 

Ln

1

c



n

1

n n

2 2   =

L

( 2

N A n

1

c

) 2  2 

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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

=

D M

+

D W

Where

D M

waveguide dispersion .

is material dispersion and

D W

is

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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

T

=

L v g

Where

v g

is the group velocity defined as

v g

=

d

ω 1

d

β

d

β

d

ω = (

d

β

d

ω ) 1

Remember at the same time, phase velocity is defined as

v p

= ω 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

=

v g

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By using

β

=

n k

0 =

n



c

where

n

is mode index or effective index,  is propagatio n constant,

k

0 is free space propagtion constant or free space wave number

k

0 = ω

c

= 2

π λ

and  is angular frequency.

At the same time

v g

=

c n g

Where

n g

is the group index given by

n g

=

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

by

T

=

dT d

ω ∆ ω =

d d



L

ω ⎜

v g

⎞ ⎟ ⎠ ∆ ω ( Q

T

=

L v g

) =

d d

2 ω β 2 =

L

β 2 ∆ ω ω ( Q

v g

= (

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 By using

ω

= 2 

c

 and  = ( 2 

c

 2 ) 

λ

in place of

ω

.

Q

T

=

d d



L

ω ⎜

v g

  ⎠ ∆ ω 

T

=

d d

λ  ⎜

L v g

  ⎠ ∆ ω =

DL

λ

 Where

D

=

d d

λ

 ⎜ 1

v g

⎞ ⎠ 2  =

π

c

2

β

2

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 d

λ  1

v g

  

D

= 2 λ

d

ω 2 π 

c d g

 ⎛ 2 λ 1  

d

π + ω ω 2

d n

ω   2 ⎠ u sin g th e relati o n

n g

=

n

+ ω (

dn d

ω )

D

can b e w ritten as su m o f tw o term s

D

=

D M

+

D W

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

D W

are g iven b y

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D

M

=

π 2

dn

2g

1 dn

2g

2π∆ D

W

=

2

 λ

⎢ ⎡

n

2g

Vd

2

n

2 2 (

Vb

)

dn

2g

d ω dV dω dV

(

 Vb

)

 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 Vd 2 (Vb)/dV 2 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

D M

=

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,

D M

is negative for wavelengths below λ ZD and becomes positive above that.

D = D

M

+D

W

, 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

D M

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 core radius

a

µm. Such fibers are called

DW

depends on fiber parameters such as the and the index difference ∆, it is possible to design the fiber such that λZD is shifted into the vicinity of 1.55

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 refractive index profile . Waveguide dispersion can be used to produce

dispersion decreasing

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

magnitude.

fibers, GVD is made normal and has a relatively large

Higher order dispersion

BL

product of a single-mode fiber can be increased indefinitely by operating at the zero-dispersion wavelength λ ZD where

D

= 0.

The dispersive effects, however, do not disappear completely at λ = λZD.

Optical pulses still experience broadening because of higher order 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

differential dispersion

parameter.

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where β 3 = dβ 2 /dω≡ d 3 β/dω 3 is the third-order dispersion parameter. At λ= λ ZD , 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 = λ ZD 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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