Download - 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
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12220
2( 5 0 1 0) (1)
p p-
- t +
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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
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. Notice0Dww
9
10
8
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Att
=gzv
,
Dw
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= 0and atgzt
-v
= +50ps(i.e.,at the trailing
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edge of thepulse)
Dw
1.1
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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
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(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
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= 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
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(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
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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
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~
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
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fiber, thespot size
w0
of the beamis about5m
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m (seeExamples
29.8 and29.9). Thus
the effective
3
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cross-sectional
area of thebeam,
Aeff
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pw02
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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
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Dn
=n
2I
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~3.2
10
12
(66)
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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
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fiberthebeam
remainsconfined to
a verysmall
area for long
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distances!We consider
a laser pulse(of
frequency
w0
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)propagatingt
hrough anoptical fiber;
the effective
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propagationconstant3Values
adapted fromRef. 2.
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