group velocity and pulse dispersion

7/30/2019 Group Velocity and Pulse Dispersion

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GroupVelocity and

PulseDispersion137

u

Atg


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zt

-v

= 50 ps(i.e.,at the front

end of the

pulse)Dw


3/80

12220

2( 5 0 1 0) (1)

p p-

- t +


4/80

12122

2 0 . 0 1 1 50 1 0 ( 1 0 0

1 0 )--

+

= +1.1


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108

HzThus at the

leading edgeof the pulse,

the


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frequenciesare

slightlyhigherwhich is

usuallyreferred to as

blue-shifted


7/80

. Notice0Dww

9

10

8


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Att

=gzv

,

Dw


9/80

= 0and atgzt

-v

= +50ps(i.e.,at the trailing


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edge of thepulse)

Dw

1.1


11/80

108

HzThus, atthe trailing

edge of thepulse, the

frequencies


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areslightlylower which

is usuallyreferred to as

red-shifted.


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FromExample

10.6, we canconclude the

following:F

or positive


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dispersion(i.e.,

negativevalue ofg

),p


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andk

will also benegative,

implying

that the


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instantaneous fre-quency

(within thepulse)

decreases

with time


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(we areof course

assumingz

> 0); this is

known as a


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down-chirped

pulse

in which theleading edge

of the pulse(t


19/80


20/80

(i.e., it hasfrequency

higher thanw0

) and thetrailing edge


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of the pulse(

t>z

/v

g


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) isred-shifted

(i.e., ithasfrequenc

y lower than

w0


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).This isshown in

Fig. 10.7where att


24/80

= 0 we haveanunchirped

pulse. Asthe pulse

propagates

farther, it


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willgetfurther

broadenedand also get

further

down-chirped


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.From Eq.(61) it

can bereadily seen

that at


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negativevalues of

z,p

(and

therefore


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k) will be

positive andtheleading edge

of the pulse (t


29/80


30/80

(i.e., it willhave

frequencylower thanw0

) and the


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trailing edge

of the pulse (

t>

z/

v

g


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) willbeblue-shifted

(i.e., it willhave

frequency

higher thanw


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0

).This

implies thatwe will

have an up-

chirpedpulse.


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Thusif anup-chirped

pulse ispassed

through a

medium


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charac-terized by

positivedispersion,

it will get

compressed


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untilitbecomes

unchirped,and then it

will

broaden


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againwithopposit

echirp.Simila

rly we can

discuss the


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case ofnegative

dispersion(implying a

positive

value of


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g). If a down-

chirpedpulse

ispassed

through a


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mediumcharacterize

d bynegative

disper-sion,

it will get


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compresseduntil it

becomesunchirped,

andthen it

will broaden


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again withopposite

chirp (seeFig. 10.8).10.4SELF

PHASE


43/80

MODULA

TION

As a pulsepropagates

through a

dispersive


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medium, thefre-quency

spectrumremains the

samei.e.,

no new


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frequenciesaregenerated.

Differentfrequencies

superpose

with


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differentphasesto

distort thetemporal

shape of the

pulse (see


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Prob. 10.10).Newfrequen

cies aregenerated

when the

medium is


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nonlinear we briefly

discuss thishere.The

refractive

index of any


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material is aconstant

onlyforsmall

intensities of

the


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propagatinglaser beam.

If the inten-sities are

large, the

refractive


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indexvariation is

approximatelygiven byn


52/80

~

n0

+n

2


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I

(63)

wheren

2is a constant

and


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I

represents

the intensityof the beam.

For


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example, forfused silica,

n0

~

1.47 and


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n2

~

3.2

10


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20

m2

W

1. Further, if

the effective


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area of thelightbeam is

Aeff

, then theintensity is

given by


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I

=

effP A

(64)where

P


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is the powerassociated

with thelight beam.

Now ina

single mode


61/80

fiber, thespot size

w0

of the beamis about5m


62/80

m (seeExamples

29.8 and29.9). Thus

the effective

3


63/80

cross-sectional

area of thebeam,

Aeff


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pw02


65/80

50m

m2

. For a 5mWlaser

beam propa


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gatingthrough

such a fiber,the resultant

intensity is

given by


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I

=

effP A


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3122

5 1 0 W

5 0 1 0 m--

= 108

W m


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2

(65)

Thus thechange in

refractive

index isgiven by


70/80

Dn

=n

2I


71/80

~3.2

10

12

(66)


72/80

Althoughthis is very

small, butwhen the

beam propa-

gates over


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an opticalfiber over

longdistances (a

fewhundred

to a few


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thousandkilometers),

theaccumulated

non-linear

effects can


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besignificant.

That is thegreat

advantageof

the optical


76/80

fiberthebeam

remainsconfined to

a verysmall

area for long


77/80

distances!We consider

a laser pulse(of

frequency

w0


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)propagatingt

hrough anoptical fiber;

the effective


79/80

propagationconstant3Values

adapted fromRef. 2.


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group velocity and pulse dispersion

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