DispersionSyed Abdul Rehman Rizvi

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X

Intermodel dispersion (Multimode dispersion)

XX Φc

L

L

X= L/SinΦc

SinΦc= n2/n1

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=

− ⎜ − 1 =⎟ ⎜ ⎟=

⎣ 2 ⎦

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 θ i = 0, and is just equal to the fiber lenght 'L'.T he longestpath occurs for θ i show n previously and has a lenght 'L/sin Φ c .

v = c / n1 , the tim e delay is given by ; ∆ T = TM ax − TM in

=sv=

x − Lv

=Ln1

n 2

cn1

− L=Ln1 n1 − n 2

cn 2 n1n1

Ln12 ∆cn 2

W hen << ∆ 1

under this condition =∆ n1 − n 2

n 2

m ay also be true

∆ T = Ln12 Ln1

cn 2 cLn1 ⎛ n1

c ⎝ n 2

⎞ Ln1 ⎛ n1 − n 2 ⎞ ⎠ c ⎝ n 2 ⎠

=Ln1 ∆c

=Ln12 2 ∆2 n1 c=

L ( N A ) 2

2 n1 c⎡Q (N A ) 2 = 2 n1 ∆ ⎤

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The time delay between the two rays taking the shortest and longestpaths is a measure of broadening experienced by an impulse launchedat the fiber input.

We can relate ∆T to the information-carrying capacity of the fibermeasured through the bit rate B. Although a precise relation betweenB and ∆T depends on many details,

Requirement for minimal inter symbol interference:B ∆t < 1

whereB = bit rate

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Dispersion in single mode fiberIntermodal dispersion in multimode fibers leads toconsiderable broadening of short optical pulses (~ 10ns/km).In geometrical optics description this is attributed todifferent paths followed by different rays.In the modal description it is related to the different modeindices or group velocities associated with different modes.The main advantage of single mode fiber is that intermodaldispersion is absent but that doesn’t mean that dispersionhas vanished altogether.The group velocity associated with the fundamental mode isfrequency dependent because of chromatic dispersion.The result is that different spectral components of the pulsetravel at slightly different group velocities and thephenomena is referred as group-velocity dispersion (GVD),intramodal dispersion or simply fiber dispersion.

Intramodal dispersion has two contributions, material dispersionand waveguide dispersion, such that

D = DM + DW

Where DM is material dispersion and DWis

waveguide dispersion.

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= −1

Group-Velocity DispersionConsider a single-mode fiber of length L. A specific spectralcomponent at the frequency ω would arrive at the output end ofthe fiber after a time delay

T = Lvg

Where vg is the group velocity defined as

vg =dω 1dβ dβ dω

= (dβ dω )

Remember at the same time, phase velocity is defined as

v p =ω

βIn nondispersive medium the phase velocity isindependent of the wave frequency and the groupvelocity and phase velocity are the same. So in suchcase

v p = vg

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ng

By using β = n k0 = n ω c

where n is mode index or effective index, β is propagatio n constant,

k0 is free space propagtion constant or free space wave number

k0 = ω c = 2π λ

and ω is angular frequency.

At the same time vg = c

Where ng is the group index given by

ng = n + ω (dn dω )

by

⎜ ⎟⎝d 2 βdω

-1

The frerquency dependence of group velocity leads to pulse broadening becausedifferent spectral components of the pulse disperse during propagation and don'tarrive simultaneously at the fiber output. If ∆ω is the spectral width of the pulse,the extent of pulse broadening for a fiber of length L is governeddb

∆T = dTdω

∆ω = d ⎛ Ldω ⎜ vg

⎞ ∆⎟ ω

⎠( Q T = L vg )

= L 2 ∆ω

= Lβ2 ∆ω

( Q vg = ( dβ dω) )

The parameter β2 = d 2 β dω 2 is known as the GVD parameter.It determines how much an optical pulse would broaden on propagationinside the fiber.

⎜ ⎞⎟ ⎜ ⎞

⎟⎝ ⎝

⎜λ⎝

In some optical communicat ion systems, the frequency spread ∆ωis determined by the range of waveleng ths ∆λ in place of ∆ω.By using ω = 2πc λ and ∆ω = (− 2πc λ2 )∆λ

Q ∆T =d ⎛ L

dω ⎜ v g

∆⎟ ω⎠

∴ ∆T = d ⎛ Ldλ ⎜ vg

∆⎟ ω = DL ∆λ⎠

Where D = d ⎛ 1dλ ⎜ vg

⎞ 2πc ⎟ = − 2 β 2

⎠D is called the dispersion parameter and is expressed in unitsof 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 ec rite rio n B ∆ T < 1 . B y p u ttin g th e v a lu e o f ∆ T = D L ∆ λ th is c o n d itio nbecom es

BL D ∆ λ < 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 r1 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 ∆ λ 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 ne 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 llyo 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 lem o d e s e m ic o n d u c t o r la s e rs w ith ∆ λ b e lo w 1 n m .

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⎜ ⎝ g ⎠

2π c d 1 2⎛ ⎞ π ⎜ v ⎟λ d ω λ

+ ω ⎜ 2 ⎟

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 = dd λ

⎛ 1 ⎞ ⎜ v ⎟

∴ D = − 2 = − ⎜ ⎟ 2 ⎝ g ⎠

u sin g th e relati o n n g = n

⎛ dn d 2 n ⎞ ⎝ d ω d ω 2 ⎠

+ ω ( dn d ω )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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DM = − 2 =

2

DW = − 2 ⎢ + ⎥2 ⎣ ⎦

2π dn 2g 1 dn 2g

λ dω c dλ

2π∆ ⎡ n 2g Vd2 ( Vb ) dn 2g d ( Vb ) ⎤λ ⎢ n 2ω dV dω dV ⎥

where n 2g is group index of material and V isnormalized frequency and b is normalized propagation constantas 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 dispersionMaterial dispersion occurs because the refractive index of silica,the material used for fiber fabrication, changes with the opticalfrequency ω.On a fundamental level, the origin of material dispersion isrelated to the characteristic resonance frequencies at which thematerial absorbs the electromagnetic radiation.

Far from the medium resonances, the refractive index n(ω) iswell approximated by the Sellmeier equation .

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where ωj is the resonance frequency and B j is theoscillator strength. Here n stands for n1 or n2,depending on whether the dispersive properties ofthe core or the cladding are considered.

In the case of optical fibers, the parameters Bj andωj are obtained empirically by fitting the measureddispersion 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 µmfor 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 isreferreddtto as the zero-dispersion wavelength λZD, since DM = 0 at λ= λZD.

The dispersion parameter DM is negative below λZD and becomes positive abovethat. In the wavelength range 1.25–1.66 µ m it can be approximated by an empiricalrelation.

Lemda zero may be extended to1550µm

Lowering the normalised freqIncreasing the relative refractive indexdifference ∆Suitable doping of the silica withgermenium.

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Waveguide DispersionThe contribution of waveguide dispersion DW to the dispersion parameter D isgiven 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 becomespositive above that.

D = DM +DW, for a typical single-mode fiber.

The main effect of waveguide dispersion is to shift λZD by an amount 30–40nm 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 µm 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, sincethe 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 DWdepends on fiber parameters such as thecore radius a and the index difference ∆,it is possible to design the fiber such thatλZD is shifted into the vicinity of 1.55µm. Such fibers are called dispersionshifted fibers.It is also possible to tailor the waveguidecontribution such that the total dispersionD is relatively small over a widewavelength range extending from 1.3 to1.6 µm . Such fibers are calleddispersion-flattened fibers.

The design of dispersion modified fibersinvolves the use of multiple claddinglayers and a tailoring of the refractive-index profile . Waveguide dispersion canbe used to produce dispersion-decreasing fibers in which GVD decreasesalong the fiber length because of axialvariations in the core radius. In anotherkind of fibers, known as the dispersioncompensating fibers, GVD is madenormal and has a relatively largemagnitude.

Higher order dispersionBL product of a single-mode fiber can be increased indefinitelyby operating at the zero-dispersion wavelength λZD where D =0.The dispersive effects, however, do not disappear completelyat λ = λZD.Optical pulses still experience broadening because of higher-order dispersive effects.This feature can be understood by noting that D cannot bemade zero at all wavelengths contained within the pulsespectrum centered at λZD.Clearly, the wavelength dependence of D will play a role inpulse broadening.Higher-order dispersive effects are governed by the dispersionslope S = dD/dλ. The parameter S is also called a differential-dispersion parameter.

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where β3 = dβ2/dω≡ d3 β/dω3 is the third-order dispersion parameter. Atλ= λZD, β2 = 0, and S is proportional to β3.

The numerical value of the dispersion slope S plays an important role inthe design of modern WDM systems.

It may appear from following Eq. that the limiting bit rate of a channeloperating at λ = λZD will be infinitely large. However, this is not the casesince 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 spectralwidth ∆ λ, the effective value of dispersion parameter becomes

D = S∆ λ

. The limiting bit rate–distance product can now be obtained by using thisvalue of D. The resulting condition becomes

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