光在材料中的傳播

Theequationsofopticsare

Maxwell’sequations.whereistheelectricfield,isthemagneticfield,risthechargedensity,eisthepermittivity,andmisthepermeabilityofthemedium.Maxwell’sequationssimplifytothewaveequationfortheelectricfield.whichhasasimplesine-wavesolution:LightisanElectromagneticWaveElectric(E)andmagnetic(B)fieldsareinphase.Theelectricfield,themagneticfield,andthepropagationdirectionareallperpendicular.

Whatisawave?Awaveisanythingthatmoves.Todisplaceanyfunctionf(x)totheright,justchangeitsargumentfromxtox-a,whereaisapositivenumber.Ifweleta=vt,wherevispositiveandtistime,thenthedisplacementwillincreasewithtime.Sorepresentsarightward,orforward,propagatingwave.Similarly,representsaleftward,orbackward,propagatingwave.v

willbethevelocityofthewave.f(x)f(x-3)f(x-2)f(x-1)x0123f(x-vt)f(x+vt)Theone-dimensionalwaveequationWe’llderivethewaveequationfromMaxwell’sequations.Hereitisinitsone-dimensionalformforscalar(i.e.,non-vector)functions,f:Lightwaves(actuallytheelectricfieldsoflightwaves)willbeasolutiontothisequation.Andvwillbethevelocityoflight.Thesolutiontotheone-dimensionalwaveequationwheref(u)canbeanytwice-differentiablefunction.Thewaveequationhasthesimplesolution:Proofthatf

(x

±

vt)solvesthewaveequationWritef

(x

±

vt)asf

(u),whereu=x±vt.Soand

Now,usethechainrule:

SoTandT

Substitutingintothewaveequation:QEDThe1DwaveequationforlightwavesWe’llusecosine-andsine-wavesolutions:

or

where:whereEisthelightelectricfieldAsimplerequationforaharmonicwave:

E(x,t)=Acos[(kx–wt)–q]Usethetrigonometricidentity:

cos(z–y)=cos(z)cos(y)+sin(z)sin(y)wherez=k

x–w

tandy=qtoobtain:

E(x,t)=Acos(kx–wt)cos(q)+Asin(kx–wt)sin(q)whichisthesameresultasbefore,

aslongas:

A

cos(q)=BandAsin(q)=CForsimplicity,we’lljustusetheforward-propagatingwave.Definitions:AmplitudeandAbsolutephase

E(x,t)=A

cos[(kx–wt)–q]

A=Amplitudeq=Absolutephase(orinitialphase)DefinitionsSpatialquantities:Temporalquantities:ThePhaseVelocityHowfastisthewavetraveling?Velocityisareferencedistancedividedbyareferencetime.Thephasevelocityisthewavelength/period:

v=l/tIntermsofthek-vector,k=2p/l,andtheangularfrequency,w=2p/t,thisis: v=w/k

ThePhaseofaWaveThisformulaisusefulwhenthewaveisreallycomplicated.Thephaseiseverythinginsidethecosine.

E(t)=A

cos(j),wherej=kx–wt–qIntermsofthephase,

w=–?j/?t

k=?j/?xAnd

–?j/?t

v=–––––––

?j/?xComplexnumbersSo,insteadofusinganorderedpair,(x,y),wewrite:

P=x+i

y

=A

cos(j)+iAsin(j)wherei=(-1)1/2

Considerapoint,P

=(x,y),ona2DCartesiangrid.Letthex-coordinatebetherealpartandthey-coordinatetheimaginarypartofacomplexnumber.Euler'sFormula

exp(ij)=cos(j)+isin(j)sothepoint,

P=Acos(j)+iAsin(j),canbewritten:

P=Aexp(ij)where

A=Amplitude

j=PhaseProofofEuler'sFormulaUseTaylorSeries:exp(ij)=cos(j)+isin(j)Ifwesubstitute

x=ijintoexp(x),then:ComplexnumbertheoremsMorecomplexnumbertheoremsAnycomplexnumber,z,canbewritten:

z=Re{z}+iIm{z}So

Re{z}=1/2(z+z*)and

Im{z}

=1/2i(z–z*)where

z*isthecomplexconjugateofz(i?–i)The"magnitude,"|z|,ofacomplexnumberis:

|z|2=zz*=Re{z}2+Im{z}2Toconvertzintopolarform,Aexp(ij):

A2=Re{z}2+Im{z}2

tan(j)=Im{z}/Re{z}Wecanalsodifferentiateexp(ikx)asiftheargumentwerereal.WavesusingcomplexnumbersTheelectricfieldofalightwavecanbewritten:

E(x,t)=Acos(kx–wt–q)Sinceexp(ij)=cos(j)+isin(j),

E(x,t)canalsobewritten:

E(x,t)=Re{Aexp[i(kx–wt–q)]}or

E(x,t)=1/2Aexp[i(kx–wt–q)]+c.c.where"+c.c."means"plusthecomplexconjugateofeverythingbeforetheplussign."Weoftenwritetheseexpressionswithoutthe?,Re,or+c.c.WavesusingcomplexamplitudesWecanlettheamplitudebecomplex:

wherewe'veseparatedtheconstantstufffromtherapidlychangingstuff.Theresulting"complexamplitude"is:

So:HowdoyouknowifE0isrealorcomplex? Sometimespeopleusethe"~",butnotalways. Soalwaysassumeit'scomplex.Addingwavesofthesamefrequency,butdifferentinitialphase,yieldsawaveofthesamefrequency.Thisisn'tsoobvioususingtrigonometricfunctions,butit'seasywithcomplexexponentials:whereallinitialphasesarelumpedintoE1,E2,andE3.The3Dwaveequationfortheelectricfieldanditssolution!or

whichhasthesolution:

where

andAlightwavecanpropagateinanydirectioninspace.Sowemustallowthespacederivativetobe3D:iscalleda“planewave.”Aplanewave'swave-frontsareequa