Wave optics class 12.....................

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1. WAVEOPTICS-|
1.ElectromagneticWave
2.Wavefront
3.Huygens'Principle
4.ReflectionofLightbasedonHuygens'Principle
5.RefractionofLightbasedonHuygens'Principle
6.BehaviourofWavefront inaMirror,LensandPrism
7.CoherentSources
8.Interference
9.Young'sDoubleSlitExperiment
10.Colours inThinFilms

2. ElectromagneticWave:
Bof..
1.Variationsinbothelectricandmagneticfieldsoccursimultaneously.
Therefore,theyattaintheirmaximaandminimaatthesameplaceandat
thesametime.
2.Thedirectionofelectricandmagneticfieldsaremutuallyperpendicular
toeachotherandaswellastothedirectionofpropagationofwave.
3.Thespeedofelectromagneticwavedependsentirelyontheelectricand
magneticpropertiesofthemedium,inwhichthewavetravelsandnoton
theamplitudesoftheirvariations.
WaveispropagatingalongX-axiswithspeedc=1/NPgE
FordiscussionofopticalpropertyofEMwave,moresignificance isgivento
ElectricField,E.Therefore,ElectricFieldiscalled'lightvector'.

3. Wavefront:
Awaveletisthepointofdisturbanceduetopropagationoflight.
Awavefrontisthelocusofpoints(wavelets)havingthesamephaseof
oscillations.
Alineperpendiculartoawavefrontiscalledaray'.
Spherical
Wavefront
fromapoint
source
Cylindrical
Wavefront
fromalinear
source
Plane
Wavefront
PinkDots-Wavelets
BlueEnvelope-Wavefront
RedLine-Ray

4. Huygens'ConstructionorHuygens'PrincipleofSecondary
Wavelets:
NewWavefront
(Spherical)
(Wavelets-Reddotsonthewavefront)
New
Wave
front
(Plane)
1.Eachpointonawavefrontactsasafreshsourceofdisturbanceoflight.
2.Thenewwavefrontatanytimelaterisobtainedbytakingtheforward
envelopeofallthesecondarywaveletsatthattime.
Note:Backwardwavefrontisrejected.Why?
Amplitudeofsecondarywaveletisproportionalto%(1+cos0).Obviously,
forthebackwardwavelet0=180°and(1+cose)is0.

5. LawsofReflectionataPlaneSurface(OnHuygens'Principle):
Ifcbethespeedoflight,t
bethetimetakenbylightto
gofromBtoCorAtoDor
N
EtoGthroughF,then
t=
FG
t=
AFsin i
C
FCsinr
C
F
nY
t=
ACsinr+AF(sini-sinr)
C
AB-Incidentwavefront
CD-Reflectedwavefront
XY-Reflectingsurface
Forraysoflightfromdifferentpartsontheincidentwavefront,thevaluesof
AFaredifferent.Butlightfromdifferentpointsoftheincidentwavefront
shouldtakethesametimetoreachthecorresponding pointsonthe
reflectedwavefront.
So,tshouldnotdependuponAF.Thisispossibleonlyifsin i-sinr=0.
i.e.sin i=sinror i=r

6. LawsofRefractionataPlaneSurface(OnHuygens'Principle):
Ifcbethespeedoflight,t
bethetimetakenbylightto
gofromBtoCorAtoDor
EtoGthrough F,then
N
B
N
Rarer
EF FG X
Denser
Y
t=
AFsini FCsinr
ACsinr
+AF(
sin isinr
AB-Incidentwavefrontt=
c CD-Refractedwavefront
XY-Refractingsurface
Forraysoflightfromdifferentpartsontheincidentwavefront,thevaluesof
AFaredifferent.Butlightfromdifferentpointsoftheincidentwavefront
shouldtakethesametimetoreachthecorresponding pointsonthe
refractedwavefront.
So,tshouldnotdependuponAF.Thisispossibleonly
if
sinisinr
=0
or
sin isinr
or
sin i C
C V C V
sinr

7. BehaviourofaPlaneWavefrontinaConcaveMirror,ConvexMirror,
ConvexLens,ConcaveLensandPrism:
C
B
ConcaveMirror
B
D
ConvexMirror
ConvexLens
B
ConcaveLens
AB-Incidentwavefront CD-Reflected/Refractedwavefront

8. Prism
AB-Incidentwavefront CD-Refractedwavefront
Prism
CoherentSources:
CoherentSourcesoflightarethosesourcesoflightwhichemitlightwavesof
samewavelength,samefrequencyandinsamephaseorhavingconstant
phasedifference.
Coherentsourcescanbeproducedbytwomethods:
1.Bydivisionofwavefront(Young'sDoubleSlitExperiment,Fresnel's
BiprismandLloyd'sMirror)
2.Bydivisionofamplitude(Partialreflectionorrefraction)

9. InterferenceofWaves:
E,+E,
i(f....
ConstructiveInterferenceE=E,+E,
E,
S
S
BrightBand
DarkBand
BrightBand
DarkBand
BrightBand
Crest
DestructiveInterferenceE=E -E, Trough
BrightBand
1stWave(E,)
DarkBand
2ndWave(E,)
ResultantWave
Thephenomenonofonewaveinterfering
ReferenceLine
withanotherandtheresulting
redistributionofenergyinthespace
aroundthetwosourcesofdisturbance is
calledinterferenceofwaves.

10. TheoryofInterferenceofWaves:
E,=asinwt
Thewavesarewithsamespeed,wavelength,frequency,
timeperiod,nearlyequalamplitudes,travellinginthe
E,=bsin(wt+o) samedirectionwithconstantphasedifferenceof0.
wistheangularfrequencyofthewaves,a,barethe
amplitudesandE,E,aretheinstantaneousvaluesof
Electricdisplacement.
Applyingsuperpositionprinciple,themagnitudeoftheresultantdisplacement
ofthewavesisEE,+E,
E=asinwt+bsin(wt+0)
E(a+bcos)sinwt+bsincoswt
Puttinga+bcos=Acos
(whereEisthe
resultant
Asin0
bsin= Asin9 displacement,A
bsin
Weget E=Asin(wt+0)
istheresultant
amplitudeand
0istheresultant
A=V(a²+b²+2abcos)
phasedifference) a
bcos
tan
bsin
a+bcos
Acos

11. A=(a?+b²+2abcos)
Intensity Iisproportionaltothesquareoftheamplitudeofthewave.
So,IaA? i.e.Ia(a?+b²+2abcos)
ConditionforConstructiveInterference ofWaves:
Forconstructiveinterference, Ishouldbemaximumwhichispossible
onlyifcos=+1.
i.e.=2nrwhere n=0,1,2,3,.....
Corresponding pathdifferenceisA=(A/2)x2n
A=nA
maxa(a+b)?
ConditionforDestructiveInterference ofWaves:
Fordestructiveinterference, Ishouldbeminimumwhichispossible
onlyifcos=-1.
i.e. =(2n+1)r wheren=0,1,2,3,.....
CorrespondingpathdifferenceisA=(A/2m)x(2n+1)T
A=(2n+1)A/2 |mint(a-b)?

12. Comparisonofintensitiesofmaximaandminima:
maxa(a+b)?
mina(a-b)?
Imax
Imin
(a+b)?
(a-b)²
(a/b+1)
(alb-1)2
Imax (r+12
=
Imin (r-12
wherer=alb (ratio oftheamplitudes)
RelationbetweenIntensity(),Amplitude(a)ofthewaveand
Width(w)oftheslit:
Iaa²
aavw
(a,)? W
(a)? W2

13. Young'sDoubleSlitExperiment:
S
SingleSlit DoubleSlit
V Screen
ThewavesfromS,andS,reachthepointPwith
somephasedifferenceandhencepathdifference
A=S,P-s,P
s,P2-s,p2=(D2+(y+(di2)}1-[D2+y-(d/2)}1
(S,P-S,P)(S,P+S,P)=2yd A(2D)=2yd A=yd/D

14. PositionsofBrightFringes: Positions ofDarkFringes:
ForabrightfringeatP,
A=yd/D=nÀ
wheren=0,1,2,3,...
y=nDAld
Forn=0, Yo=0
Forn=1, y,=DAId
Forn=2,
y,=2DA/d
For n=n, Y,=n DAld
ForadarkfringeatP,
A=yd/D=(2n+1)N2
wheren=0,1,2,3,...
y=(2n+1)DA/2d
Forn=0,
Y,'=DA2d
Forn=1, y,'=3DA/2d
Forn=2, Y'=5DA/2d
Forn=n,
y,'=(2n+1)DA/2d
ExpressionforDarkFringeWidth: Expression forBrightFringeWidth:
=nDAld-(n-1)DAld
=DAld
=(2n+1)DA/2d-(2(n-1+1)DA/2d
=DAd
Theexpressionsforfringewidthshowthatthefringesareequallyspacedon
thescreen.

15. DistributionofIntensity:
Intensity
Supposethetwointerferingwaves
havesameamplitudesay'a',then
Imara(a+a)² i.e.Imaxa4a?
Allthebrightfringeshavethissame
intensity.
Imin=0
0
Allthedarkfringeshavezero
intensity.
Conditionsforsustainedinterference:
1.Thetwosourcesproducinginterferencemustbecoherent.
2.Thetwointerferingwavetrainsmusthavethesameplaneof
polarisation.
3.Thetwosourcesmustbeveryclosetoeachotherandthepatternmust
beobservedatalargerdistancetohavesufficientwidthofthefringe.
(DAld)
4.Thesourcesmustbemonochromatic. Otherwise,thefringesofdifferent
colourswilloverlap.
5.Thetwowavesmustbehavingsameamplitudeforbettercontrast
betweenbrightanddarkfringes.

16. ColoursinThinFilms:
Itcanbeprovedthatthepath
differencebetweenthelightpartially
reflectedfromPQandthatfrom
partiallytransmittedandthen
reflectedfromRSis
P
A=2utcosr t
SincethereisareflectionatO,the
rayOAsuffersanadditionalphase
differenceofrandhencethe
corresponding pathdifferenceof
N2.
R
FortheraysOAandBCtointerfere
constructively (Brightfringe),the
pathdifferencemustbe(n+%)A
So, 2utcosr=(n+%)A
FortheraysOAandBCtointerfere
destructively(Darkfringe),thepath
differencemustbenà
So, 2utcosr=nA
Whenwhitelightfromthesunfallsonthinlayerofoilspreadoverwaterinthe
rainyseason,beautifulrainbowcoloursareformedduetointerferenceoflight.
EndofWaveOptics- I