Basic Optics. Principles and Concepts

985 Pages·2016·11.25 MB·English

by  Avijit Lahiri

Basic Optics
Basic Optics
Principles and Concepts
Avijit Lahiri
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Dedication
I dedicate this book to the memory of
Jogendra Nath Maitra
and Bijali Bhusan Lahiri
—two great men in my life whom I deeply miss.
Acknowledgments
IacknowledgeaheartfulofthankstomywifeAnita,tomydaughterAnindita,andtomyson
Indrajit.Purba,youdeservespecialthanks.
ParikshitMannacreatedallthefiguresinthisbookforme,andhisfineeffortsarehereforall
tosee.
SankhasubhraNagneverrefusedtobebymysidewheneverIneededhim,andhishelpand
supportpermeatethisbook.
LittleAditya,youhavebeenaconstantsourceofloveandjoy.Attheageofthree,youhave
beenwiseenoughtomercilesslypenthroughmostofmynotes.Your“criticism”didmea
worldofgood.
RajkumarMoitra,BrahmanandaDasgupta,RanjanBhattacharya,andDebashis
Mukherjee—thisbookwouldnothaveseenthelightofdaybutforyourunqualifiedsupport.
Thanks.
MymotherSovanaLahirihasforeverbeenafountainofinspiration.AsIwritethesefew
lines,withthebookalmostreadytogotopress,she,allofasudden,isnomore.Neverknew
howtothankher.
Kolkata,July2016.
xxv
CHAPTER 1
Electromagnetic Theory
and Optics
1.1 Introduction
Opticalphenomenaareofanimmensediversity.Yet,amazingly,theexplanationofallofthose
canbetracedbacktoaveryfewbasicprinciples.Thisisnottosaythat,oncethesebasic
principlesareknown,onecanarriveatapreciseexplanationofeachandeveryoptical
phenomenonorataprecisesolutionforeachandeveryprobleminoptics.Inreality,optical
phenomenacanbegroupedintoclasseswhereeachclassofphenomenahascertain
characteristicfeaturesincommon,andanadequateexplanationofeachclassofphenomenais
achallengeinitself,requiringappropriateapproximationschemes.Butwhatever
approximationsonehastomake,thesewillbefoundtoinvolvenoprinciplesmore
fundamentalthan,orindependentof,thebasicones.
What,then,arethesebasicprinciplesofoptics?Asfaraspresent-dayknowledgegoes,the
mostbasicprincipleunderlyingtheexplanationofopticalphenomena,asindeedofall
physicalphenomena,istobefoundinquantumtheory.However,amoreusefulandconcrete
wayofputtingthingswouldbetosaythatthetheoreticalbasisofopticsisprovidedby
electromagnetictheory,which,inturn,isbasedentirelyonMaxwell’sequations.
ThequestionthenarisesastowhetherMaxwell’sequationsandelectromagnetictheoryareto
beconsideredfromthepointofviewofclassicalphysicsorofquantumtheory.
Ofcourse,oneknowsthatthesetwopointsofviewarenotindependentofeachother.Ina
sense,classicalexplanationsareapproximationstothemorecomplete,quantumtheoretic
descriptions.Butonceagaintheseapproximationsare,inasense,necessaryingredientsinthe
explanationofalargebodyofobservedphenomena.Inotherwords,whileagreatdealis
knownaboutthewayclassicalphysicsisrelatedtoquantumtheoryandwhileitcanbestated
thatthelatterisamorefundamentaltheoryofnature,itstillmakessensetosaythatthe
classicalandthequantumtheoriesaretwowaysofdescribingandexplainingobserved
phenomena,validintheirownrespectiverealms,wheretheformerrelatestothelatterina
certainlimitingsense.
BasicOptics
Copyright©2016AvijitLahiri.PublishedbyElsevierInc.Allrightsreserved. 1
2 Chapter1
ThishasbearingonthequestionIposedabove,theanswertowhichonemaystateasfollows:
Whilethequantumtheoryoftheelectromagneticfieldprovidestheultimatebasisofoptics,an
adequateexplanationofalargebodyofopticalphenomenacanbearrivedatfromthe
classicalelectromagnetictheorywithoutovertreferencetothequantumtheory.Theredo
remain,however,opticalphenomenathatcannotbeadequatelyexplainedwithoutinvoking
quantumprinciples.
Opticalphenomenaarerelatedtothebehaviorofelectromagneticfieldswherethetypical
frequenciesofvariationofthefieldcomponentsliewithinacertainrangeconstitutingthe
spectrumofvisiblelight,thoughthetheoreticalmethodsandprinciplesofopticsarerelevant
evenbeyondthisrange.
Withthisinmind,Iproposeinthisbooktolookattheprinciplesandbasicconceptsofoptics,
startingfromtheclassicalelectromagnetictheory.Atthesametime,Iproposetolookat
quantumopticsaswell,whereopticalphenomenaarelinkedtothequantumtheoryofthe
electromagneticfield.Intheprocesswewillberewardedwithabroadoverviewofthe
fascinatingsubjectofoptics.
Theapproachofexplainingopticalphenomenaonthebasisofclassicalelectromagnetic
theoryissometimesreferredtoas‘classicaloptics’soastodistinguishitfromquantum
optics.Buttheterm‘classicaloptics’ismorecommonlyusednowtorefertoacertain
traditionalwayoflookingatopticsandtodistinguishthisapproachfromwhatisknownas
‘modernoptics.’ThelatterincludesareassuchasFourieroptics,statisticaloptics,nonlinear
optics,and,aboveall,quantumoptics.Notalloftheseinvolvequantumtheory,somebeing
mostlybasedonclassicalelectromagnetictheoryalone.Thustheterm‘classicaloptics’has
twomeaningsattachedtoit—oneinthesenseofacertaintraditionalapproachinoptics,and
theotherinthesenseofanapproachbasedontheclassicalelectromagnetictheory.
Classicalelectromagnetictheoryisasubjectofvastdimensions.ThereisnowayIcaneven
sketchilysummarizeheretheprincipalresultsofthistheory.Instead,Iwillsimplystartfrom
Maxwell’sequations,whichconstitutethefoundationsofthetheory,andthenstateanumber
ofbasicresultsofrelevanceinoptics.Fortunately,formostofclassicalopticsoneneednot
delvedeeperintoelectromagnetictheory.Iwillnotpresentderivationsoftheresultsof
electromagnetictheorywewillneedinthisbook,forwhichyouwillhavetorefertostandard
textsonthesubject.
1.2 Maxwell’s Equations in Material Media and in Free Space
1.2.1 Electromagnetic FieldVariables
Thebasicideaunderlyingelectromagnetictheoryisthatspaceispermeatedwithelectricand
magneticfieldswhosespatialandtemporalvariationsarecoupledtooneanotherandare
relatedtosourcedensities(ie,distributionsofchargesandcurrents).
ElectromagneticTheoryandOptics 3
Theelectromagneticfield,moreover,isadynamicalsysteminitself,endowedwithenergy,
momentum,andangularmomentum,andcapableofexchangingthesewithbodiescarrying
chargeandcurrent.Thevariationsoftheelectricandmagneticfieldintensitiesaredescribed
byasetofpartialdifferentialequations—theMaxwellequations(commonlyreferredtoasthe
fieldequationsinthecontextofelectromagnetictheory).AsIhavealreadymentioned,the
behavioroftheelectromagneticfieldasadynamicalsystemcanbedescribedfromeitherthe
classicalorthequantumtheoreticpointofview.Thequantumpointofviewissubtler
thantheclassicalone,andwewillhaveatasteofitwhenItalkofquantumopticslater
inthisbook.
Maxwell’sequationsforamaterialmediuminvolvefourelectromagneticfield
variables—namely,theelectricfieldintensity(E),electricdisplacement(D),magneticfield
intensityorfluxdensity(B),andmagneticfieldstrength(H)—eachofthesebeingfunctionsof
spaceandtimevariablesrandt.Notallofthesefieldvariablesareindependentsincethe
electricvectorsDandEarerelatedtoeachotherthroughasetofconstitutiveequations
relatingtothematerialpropertiesofthemedium.Similarly,themagneticvariablesHandB
arerelatedthroughanothersetofconstitutiveequations.
1.2.1.1 Digression:Thenamingofthefieldvariables
Thefieldvectorsdonothaveuniversallyacceptednamesattachedtothem.ThusEisreferred
tovariouslyasthe‘electricfieldstrength,’the‘electricfieldintensity’(or‘electricintensity,’
inbrief),orsimply,the‘electricvector.’Agreaterdegreeofnonuniformityaffectsthenaming
ofBandH.Theformerisoftenreferredtoasthe‘magneticfluxdensity’orthe‘magnetic
induction,’whilethelatteriscommonlydescribedasthe‘magneticfieldstrength.’Inthis
bookIwillmostlyrefertoEandHasthe‘electricfieldstrength’andthe‘magneticfield
strength,’respectively,while,attimes,usingtheterm‘intensity’intheplaceof‘fieldstrength’
(theterm‘magneticintensity’willmorecommonlybeusedtodenotethevectorB).Theterm
‘intensity’hasanotheruseinelectromagnetictheory—namely,indescribingtherateofflow
ofelectromagneticfieldenergyperunitareaorientedperpendicularlytothedirectionof
energyflow.However,itwillalwaysbepossibletodistinguishouruseoftheterm‘intensity’
inconnectionwiththefieldvariablesEandBfromthisotherusageofthetermby
referencetothecontext.Attimestheterms‘electricfieldvector’and‘magneticfieldvector’
willbeusedtodenotethevectorsEandH,respectively.ThevectorDwillbenamedthe
‘electricdisplacement,’which,toagreaterdegree,isacommonlyacceptednamein
theliterature.
Nonspecifictermssuchas‘fieldvectors’and‘fieldvariables’aresometimesusedtodescribe
oneormoreofthevectorsnamedabove,orsomeoftheircomponents,especiallywhensome
commonfeaturesofthesevectorsarebeingreferredto.Onceagainthemeaningwillhaveto
bereadfromthecontext.
4 Chapter1
1.2.1.2 Digression:Thenamingofthefieldvariablesandtheirspace-timevariationsinoptics
Finally,inoptics,certaincharacteristicfeaturesofthespace-timevariationsofthefield
vectorsortheircomponentsareoftenreferredtobytermssuchas‘opticalfield,’‘optical
disturbance,’and‘opticalsignal.’Thusthetimevariationofanyofthefieldcomponentsata
pointoratvariouspointsinagivenregionofspaceissaidtoconstituteanopticaldisturbance
inthatregion.Thetimevariationofthefieldvariablesatanygivenpointinspaceisattimes
referredtoastheopticalsignalatthatpoint,andonecanthentalkofthepropagationofthe
opticalsignalfrompointtopoint,especiallyinthecontextofinformationbeingcarriedbythe
timevariationofthefieldvariables.
Inopticsitoftensufficestoconsiderthevariationsofascalarvariableratherthanthoseofthe
fieldvectors,wherethescalarvariablemaystandforanyofthecomponentsofafieldvector,
orevenforasurrogatevariablesimulatingthevariationsofthefieldvectors.Forinstance,
suchascalarvariablemaybeinvokedtoexplainthevariationofintensityatvariouspointsin
somegivenregionofspace,whereamoredetaileddescriptionintermsofthefieldvectors
themselvesmayinvolveunnecessarycomplexitieswithoutanyaddedbenefitsintermsof
conceptualclarity.
Suchscalarfieldswillprovetobeusefulinexplaininginterferenceanddiffraction
phenomena,inFourieroptics,andindescribinganumberofcoherencecharacteristicsof
opticaldisturbances.Thespace-timevariationsofsuchascalarvariablearealsoreferredtoas
an‘opticaldisturbance’oran‘opticalsignal,’andthescalarvariableitselfiscommonly
termeda‘fieldvariable.’Avectororscalarfieldvariable(identifiedfromthecontext)willalso
betermedawavefunctionsincesuchavariablecommonlysatisfiesawaveequationasin
acoustics.
Incidentally,thetemporalvariationofawavefunctionatanygivenpointinspaceisreferred
toasitswaveformatthatpoint.Itisoftenusefultothinkofawaveformasagraphofthewave
functionplottedagainsttime.
1.2.2 Maxwell’sEquations
Maxwell’sequations—fourinnumber—relatethespace-timedependenceofthefield
variablestothesourcedistributions—namely,thechargedensityfunctionρ(r,t)andthe
currentdensityfunctionj(r,t):
divD = ρ, (1.1a)
∂B
curlE = − , (1.1b)
∂t
divB = 0, (1.1c)
∂D
curlH = j+ . (1.1d)
∂t
ElectromagneticTheoryandOptics 5
Eqs.(1.1a)and(1.1d)areconsistentwiththeequationofcontinuity,
∂ρ
divj+ = 0. (1.1e)
∂t
Thisequationconstitutesthemathematicalstatementoftheprincipleofconservationof
charge.
Intheaboveequations,ρ andjaretobeinterpretedasthefreechargeandcurrentdensities
settinguptheelectromagneticfieldunderconsideration,wherethebound chargesand
currents,associatedwiththedielectricpolarizationandmagnetizationofthemediumunder
consideration,areexcluded.
1.2.3 Material MediaandtheConstitutiveRelations
1.2.3.1 Linearmedia
Theconstitutiveequationsarephenomenologicalrelationsdependingonthetypeofthe
mediumunderconsideration.Thereexistapproximatemicroscopictheoriesoftheserelations
forsometypesofmedia.Thefollowingrelationsholdforwhatareknownaslinearmedia:
D = [(cid:4)]E, (1.1f)
B = [μ]H. (1.1g)
Inthiscontextonehastodistinguishbetweenisotropicandanisotropicmedia.Foran
isotropicmediumthesymbols[(cid:4)]and[μ]intheaboveconstitutiveequationsstandforscalar
constants(tobedenotedby(cid:4) andμ,respectively)thatmay,ingeneral,befrequency
dependent(seelater).Forananisotropicmedium,ontheotherhand,thesymbols[(cid:4)]and[μ]
intheconstitutiverelationsstandforsecond-ranksymmetrictensorsrepresented,inanygiven
Cartesiancoordinatesystem,bysymmetricmatriceswithelements,say,(cid:4) andμ ,
ij ij
respectively(i,j = 1,2,3).
Digression:tensorsandtensorfields
Foragivenrandgivent,avectorsuchasE(r,t)isanelementofarealthree-dimensional
linearvectorspacewhichwedenoteasR(3).Atensorofrank2isthenanelementofanine-
dimensional vector space T that includes the direct product R(3) ×R(3) and, in addition,
containsallpossiblelinearcombinationsofdirectproductsofpairsofvectors.Ifnˆ ,nˆ ,nˆ
1 2 3
constituteanorthonormalbasisinR(3),thenanorthonormalbasisinT willbemadeupof
theobjectsnˆ nˆ (i,j=1,2,3),andatensorofrank2canbeexpressedasalinearcombination
(cid:2)i j
oftheform C nˆ nˆ .Thus,withreferencetothisbasis,thetensorunderconsiderationis
i,j ij i j
completelydescribedbythe3×3matrixwithelementsC .Thematrix(andalsothetensor)
ij
istermed‘symmetric’ifC =C (i,j=1,2,3).Thematrixissaidtobepositivedefiniteif
ij ji
allitseigenvaluesarepositive.
Now consider any of the above field vectors (say, E(r,t)) at a given time instant but at all
possiblepointsr.Thismeansavectorassociatedwitheverypointinsomespecifiedregionin
space.Thesetofallthesevectorsistermedavectorfieldintheregionunderconsideration.