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Physics4150Optics,Fall2017M. Goldman,Instructor

HW-1(totalvalue=40pts)dueTues,Sept.12,inclass

1) Sphericalwaves(20pts)

a) Findthegeneralwaveformofasphericallysymmetricscalarwave,A(r,t)withradialwavenumber,kr,whichisasol'ntothewaveeqn,v2∇2 −∂t

2( )A r,t( ) = 0 .(8pts)YourexpressionforA(r,t)shouldberealanddependonv,kr,initialphaseφ(atr=0andt=0)andrealamplitude.

InsphericalcoordinatestheLaplacianis

∇2A r,t( ) = 1r2∂r r

2 ∂r A( )+ 1r2 sinθ

∂θ sinθ ∂θ A( )+ 1r2 sin2θ

∂φ A .Sincethewaveis

sphericallysymmetric,A(r,t)=A(r,t)isindependentofθandφ,soonlythefirst

termisnon-zero.Hencewemustsolve v2

r2∂r r

2 ∂r A r,t( )( )−∂t2A r,t( ) = 0 .To

solve,firstletA(r,t)=a(r,t)/r,sothatr2∂rA=-a(r,t)+r[∂ra(r,t)]and∂r(r2∂rA)=r∂r2a.Thisgivesthefollowingeqnfora: v2 ∂r

2−∂t2( )a r,t( ) = 0 .Tryasolutionof

forma(r,t)=a0(t)Exp[ikrr],byanalogywithhowwesolvedasimilarwaveeqnforplanewaves.Nowthewaveeqn.becomes, kr

2v2 +∂t2( )a0 t( ) = 0 Thisisa

harmonicoscillatoreqnwithsolution,a0(t)=sExp[-iωt]wheresisacomplexnumberofforms0·Exp[iφ]andω=krcistheangularfrequencyofthewave.Hencethefullsolution(waveform)is

Re A r,t( )⎡⎣ ⎤⎦=s0rRe ei krr−krvt+φ( )⎡⎣

⎤⎦= s0

cos krr − krvt +φ[ ]r

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b) Plotthiswaveformatt=0aboutr=0asa3Dsurfaceoverthex-yplane

(fromx,y=-40to40)usingamathprocessorsuchasMathematicaforwavenumber,kr=1,andwithinitalphase(atr=0,t=0),φ=-π/2.(4pts)

cos[krr-π/2]=coskrrcosπ/2+sinkrrsinπ/2=sin(krr).Hencethewaveformissin r( )r

.From

Mathematicawecandisplaythisasa3Dsurfacec) Plotascon–

tourmapthespatialwaveformofthesumofthissphericalwaveaddedtoanothersphericalwavewithsameA0,kr,andinitialphasecenteredadistance10awayfromzeroonthex-axisattimet=0.Onceagaintheplotshouldbeasliceofthesphericalwaveformatz=0,goingfromx,y=-40to40.(6pts)

Dark Dark

DarkDark

BrightBrightBright

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d) Indicatewhereonascreenaty=40therewillbe"darkfringes"andwheretherewillbe"brightfringes."andexplainwhy.(2pts)

Thereisacentralbrightfringeat{x=0,y=40},darkfringesat{x=-10,y=40}andat{x=20,y=40}.Furtherouttherearebrightfringesanddarkfringescanbeperceivedmovingoutjustbeyondthescreenoneitherside.Thedarkfringesareduetodestructiveinterferenceandthebrightfringesareduetoconstructiveinterference(SeeMathematicFileshowinginterferenceoftwoplanewavesonclassWebsite)

2) Wavemagneticfield(10pts)a) Inclasswefoundthefollowingsolutionfortheelectricfieldofalightwave

withwavenumberkandfrequencyωk=ck,travellinginthedirectionofthewavevectork: ET r,t( ) =Ake

i k ·r−ωkt+ϕk( ), Ak , ϕ k real, Ak ·k = 0, ωk = ck .

UseFaraday'slawincgsunitstofindthewavemagneticfieldB(r,t).(7pts)

Faraday'slawsaysthat∇×E = −1c∂tB (cgs)

TheactionofthecurlonthevectorandspatialpartofETisgivenby∇× Ake

ik ·r( )whichisevaluatedbythefollowingdeterminant:

x y z∂x ∂y ∂z

Axeik ·r Aye

ik ·r Azeik ·r

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

= x ∂y Azeik ·r −∂z Aye

ik ·r( )− y ∂x Azeik ·r −∂z Axeik ·r( )+ z ∂x Aye

ik ·r −∂y Axeik ·r( )

= x ikyAzeik ·r − ikzAye

ik ·r( )− y ikxAzeik ·r − ikzAxeik ·r( )+ z ikxAye

ik ·r − ikyAxeik ·r( ) = ik× Aeik ·r( )

Hence, ik×E = −∂tB / c ,soB r,t( ) = ckωk

×Akei k ·r−ωkt+ϕk( ) = k×Ak( )ei k ·r−ωkt+ϕk( )

b) AfterexplaininghowtofindthedirectionofBintermsofkingeneral

answerthefollowing.(3pts)ThegeneralresultisthatthedirectionofBisgivenbykcrossedintoE.Ifkisinthex-directionandAkisiny-direction,whatisthedirectionofB?x× y = z soBisinthez-direction,orthogonaltobothxandy.AreBandEinphase?YesDoBandEhavethesameamplitude?Yes(incgsunits),butonlyinavacuum

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3) Fourieruncertaintyprincipleforcoherentwaves(10pts)

a) Thek-spectrumofthespatialwaveformoftheelectricfieldofalightwaveat

t=0travelinginthex-directionisaGaussiancenteredaboutk0with

Gaussianhalf-widthofΔk.ET k,t = 0( ) = 12πΔk

e− k−k0( )2 /2 Δk( )2 TheGaussian

half-widthisdeterminedbythevalueof|k-k0|=Δk.FindtheproductofΔkandΔx,thespatialGaussianhalf-widthoftherealspacewaveform.(6pts)

ThespatialwaveformistheinverseFouriertransform.Thiscanbecalculatedbyhandusingthemethodof"completingthesquare"intheexponential(aftershiftingtok-k0astheintegrationvariable),orbyusingMathematica,asfollows:

Thisisaplanewavewithwavenumberk0andspatialhalf-widthΔx=1/Δk,sotheproductΔxΔk=1,b) Interpretwhatthismeans.(4pts)Thismeansthatanarrowspectrumcorrespondstoabroadspatialwaveformandabroadspectrumcorrespondstoaarrowspatialwaveform.YoucanverifythisbyplottingvariouscombinationsinMathematica.