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Statistical mechanicsIntroduction to MB, BE, FD distributions
Application of probability theory (largepopulations)hermodynamics as a natural result ofstatistics and mechanics (classical and!uantum) at the microscopic le"el
#hat is most li$ely to happen (no actualmotions or interactions )%"erall beha"ior of system properties ofparticles
&annot re"eal the history but can say for
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Statistical distributions
o determine the most probable ay in hich agi"en amount of energy Eis distributed among Nmembers of the system of particles in thermale!uilibrium at absolute temperature T
(no. of differentways in which
particles can be
arranged)
P W
Eg. Rolling two dices: Total 36 possibilities
To get 12: (6,6)
(12) 1 36P
=
( ) ( ) ( )To get !: (1, "), 2,3 , 3, 2 , ",1
(!) " 36P =
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eneral formula for WMost probable distribution is the one ha"ing ma*imum W
#$%ber of particles of energy
( ) ( ) ( )n g f
=
( ) & n$%ber of states of energy
& statistical weigth corresponding to energy
g
( ) & distrib$tion f$nction
& a'erage n$%ber of particles in each state of energy
& probability of occ$pancy of each state of energy
f
Three types of distrib$tion f$nctions eist
according to their properties of the particles
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Distribution function
Maxwell-Boltzmann
Bose-Einstein Fermi-Dirac
Applies tosystems of
Identical,distinguishableparticles
Identical,indistinguishableparticles that do not
obey +auli principle
Identical,indistinguishableparticles that obey
+auli principle
1
ep( )A kT
1
ep( ) 1)*F kT
1
ep( ) * 1)F kT +( )f
&ategoryof particles
&lassical Bosons Fermions
+ropertiesof particles
Any spin, particlesare far enoughapart so a"efunctions do not
o"erlap
Integral spin a"efunctions aresymmetric tointerchange of
particles
-alf.integral spina"e functions areantisymmetric tointerchange of
particlesE*amples
Molecules of a gas +hotos in a ca"ity,phonons in a solid,li!uid -e at lo
Free electrons in ametal, nucleons innucleus
+ropertiesofdistribution
/o limit to numberof particles perstate
/o limit to numberof particles perstate moreparticles per statethanf
MB
at lo
energies
/e"er more than 0particle per statefeer particles perstate than fMBat lo
energiesapproachesfMBat
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Fermi.Dirac distribution
( )
1( )
1FFD kT
fe
= +
( )
1+, ( )
1FF FD kT
T fe
= =
+
1
To appreciate the significance of the -er%i energy, let $s consider a syste%of fer%ions at T & + and in'estigate the occ$pancy of states whose energies
are less than Fand greater than F
1
1e=
+
1
+ 1=
+
1=
( )
1+, ( )
1FF FD kT
T fe
= = +2
1
1e=
+1
=
+=
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3ASE4 5 Basics
6ery nearlymonochromatic
-ighly coherent
Does not di"erge
E*tremely intense
Light Ampli7cation by Stimulated Emission ofRadiation
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approach
( )i j i ijN N B u v = j jiN A ( )j jiN B u v
( )*j i j ji jiN N A B u v = +.t e/$ilibri$%,
i j j iN N = ( ) ( )*i ij j ji jiN B u v N A B u v = +
0i'iding both sides by
and sol'ing for ( )
j jiN B
u v ( ) ( )ij jii
j ji ji
B AN
u v u vN B B
= +
( )
1
ji ji
iji
j ji
A Bu v
BN
N B
=
Njato%s
Niato%s
ti%$lated
absoption
pontaneo$s
e%ission
ti%$lated
e%ission
#o of ato%s that
absorb photons
oefficient of proportionality
Energy density
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Einstein9s approach (contd')
ep( ) ep( )i i j jN C E kT N C E kT = =
ato%s%olec$les in a gas follow 4 distrib$tion
ep ( ) *i i jj
NE E kT
N= ep( ) *j iE E kT= ep( )hv kT=
( )1
ji ji
iji
j ji
A B
u v BN
N B
=
ep( ) 1
ji ji
ij
ji
A B
Bhv kT
B
=
3
3
5
6lanc7 radiation law: ( ) ep( ) 1
h v dv
u v dv c hv kt
=
onsistency between abo'e two epressions de%and
4ij jiB=
3
3
5
and
ji
ji
A hv
B c
=
i i : i
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Einstein9s coe:cients
Stimulated emission does occur and itsprobability is e!ual to the probability forabsorption
he ratio beteen to probabilities "ariesith v3, so the relati"e li$elihood ofspontaneous emission increases rapidly
ith the energy di;erence beteen theto states
All e need to $no is one of theprobabilitiesAij, Bij, Bjito 7nd others
4ij ji
B=3
3
5and
ji
ji
A hv
B c
=
bl
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Metastable state< le"els are re!uired to achie"e population in"ersion
0 5 %rdinary e*cited state= 5 Metastable state< 5 Stable (ground) state
An atom can e*ist in a metastable state for alonger time before radiating than it can in anordinary energy le"el
+ i i l f < l l l
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+rinciple of
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4uby laser (
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Four le"el laser
ntinuous operation is possible
- li / l
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-elium./eon laser
f l
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ypes of lasersas lasers
&hemical lasers
E*cimer lasers
Solid.state lasers
Fiber.hosted lasers
+hotonic crystal lasers
Semiconductor lasers
Dyelasers
Free e.
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poer
&D>D6D 4># ti
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&D>D6D 5 4># operation
- l h
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-olography%rdinary photograph8 only intensity is recorded =D imageHologram8 Images are formed by interference,
ithout lenses
Interference of to beams allos the 7lm to recordboth intensity and relati"e phase
&oherency is crucial
- l h ( td )
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-olography (contd')After de"eloping such a 7lm and placing it in laser
light, e get a
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&ommunication "ia light
-uman e e
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-uman eye
he rods and cones ofthe eye pass opticalsignals
Fiberscope
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Fiberscope
Endoscope is a familiar e*ampleState.of.art 7berscope has 0?,???
7bers of bundle ith 0mm dia
&apable of resol"ing ob@ects ?m across
otal internal reection
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otal internal reection
1 1 2 2sin sin 8nell9s lawn n =
2 1-or total internal reflection, + , C = =
1 2sin Cn n =2
1
sin Cn
n = 1 2
1
sinC
n
n
=
he optical 7ber
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he optical 7ber
/umerical aperture
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/umerical aperture
2
1
-or total internal reflection, sin cos n
n = =
2
2 2
1
cos n
n
=
2
2 2
1sin 1
n
n
=
2
2
1sin 1
n
n
=
2 2
1 1 2sinn n n =
+ 1 +-or air;core interface, sin sin 8 1n i n n= =
2 2
1 2#$%erical apert$re sin mNA i n n = =
Some details
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Some details
/umber of Modes in a Fiber
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/umber of Modes in a Fiber
+.!
core dia%eter
operating wa'elength
n$%erical apert$re
m
D NAN
D
NA
=
ypes of 7bers
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ypes of 7bers
Ad"antages of Fiber %ptics
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Ad"antages of Fiber %pticsLess expensive. Se"eral miles of optical cable canbe made cheaper than e!ui"alent lengths of copper
ire' his sa"es your pro"ider (cable 6, Internet) andyou money'
Thinner. %ptical 7bers can be dran to smallerdiameters than copper ire'
Higher carring capacit. Because optical 7bers arethinner than copper ires, more 7bers can be bundledinto a gi"en.diameter cable than copper ires' hisallos more phone lines to go o"er the same cable or
more channels to come through the cable into yourcable 6 bo*'
Less signal !egra!ation. he loss of signal inoptical 7ber is less than in copper ire'
Light signals. nli$e electrical signals in copper
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(contd')
Low power. Because signals in optical 7bers degradeless, loer.poer transmitters can be used instead ofthe high."oltage electrical transmitters needed forcopper ires' Again, this sa"es your pro"ider and youmoney'
Digital signals. %ptical 7bers are ideally suited forcarrying digital information, hich is especially useful incomputer netor$s'
"on-#amma$le. Because no electricity is passedthrough optical 7bers, there is no 7re haard'
Lightweight. An optical cable eighs less than acomparable copper ire cable' Fiber.optic cables ta$eup less space in the ground'
Flexi$le
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Medium
E"olution of 7ber lin$s
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E"olution of 7ber lin$s
&urrent status
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&urrent status
State.of.the.art numbers8
E*perimental trial8 Simens demonstrated bi.directional bps'
/E& demonstrated 'G bps (G? o"er 0?H9s in ?.- spacing) o"er 0J $m
&ommercial systems can go up to =bps
Single H8 /ortel demonstrated J? transmission o"er GJ? $m
Se"eral abo"e.tera.bps lin$s are beinginstalled orldide
Future trend8 More than 0?? H9s ill
be used
4eferences
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4eferences
A' Beiser 5 KConcepts of Modern PhysicsL, Ed',ata Mcra.-ill (/e Delhi, =??