# 03/11/13, Eikonal Equations, Superposition of EM Waves Lecture Note (Nick Fang)

```03/11/13,EikonalEquations,SuperpositionofEMWaves
LectureNote(NickFang)
Outline: ‐
‐
‐
Connection of EM wave to geometric optics Path of Light in an Inhomogeneous Medium Superposition of waves, coherence A. HighFrequencyLimit,connectiontoGeometricOptics:
Slowly varying envelope
HowcanweobtainGeometricopticspicture
E0(r)
suchasraytracingfromwaveequations?Now
let’sgobacktorealspaceandfrequency
domain(inanisotropicmediumbutwith
spatiallyvaryingpermittivity , ,for
example).
E=E0 (r)exp(ik0)
,
0
ExampleofdecompositionofE
fieldintotheproductofslowly
varyingenvelopeandafast
oscillatingphaseexp(ik Φ Toseetheconnectiontogeometricoptics,we
decomposethefieldE(r,)intotwoforms:a
fastoscillatingcomponentexp(ik0),
k
/ andaslowlyvaryingenvelopeE0(r)asillustratedinthetextbox.
Furthermore,iftheenvelopeoffieldvariesslowly
≪ 1,
withwavelength(e.g.
≪ 1)
E0
thenwecanconvertwaveequationstothewell‐
knownEikonalequation:
,
,
H0
1
Observation(notproof):
Theaboveequationyields:| Φ|
,or| Φ|
2
1
3
Geometricalrelationship
ofE,H,and Φ
.
ThisisequivalenttotheFermat’sPrincipleonopticalpathlength(OPL):
| Φ|
Φ
.
ofphasecontour Φ(avector).Wewilluseittodeterminethepathoflightina
generalinhomogeneousmedium.
B. PathofLightinanInhomogeneousMedium
‐
Example 1: 1D problems (Gradient index waveguides, Mirage Effects) ThebestknownexampleofthiskindisprobablytheMirageeffectindesertor
nearaseashore,andweheardoftheexplanationsuchastherefractiveindex
increaseswithdensity(andhencedecreaseswithtemperatureatagivenaltitude).
Withthepictureinmind,nowcanwepredictmoreaccuratelytheraypathand
imageformingprocesses?
Image of Mirage effect removed due to copyright restrictions.
StartingfromtheEikonalequationandweassume
thenwefind:
Φ
,
Φ
isonlyafunctionofx,
Sincethereistheindexinindependentofz,wemayassumetheslopeofphase
changeinzdirectionislinear:
=C(const)
Thisallowsustofind
Φ
2
phasefront:
Φ
z‐direction:
x‐direction:
C
2
2
Therefore,thelightpath(x,z)isdeterminedby:
C
Hence
C
1
C
2
0
1
Tofindtheintegralexplicitlywemaytakethefollowingtransformationofthe
variablex:
Therefore,
C
√
C
√
3
Ormorecommonly,
sinusoidpattern!Theperiodicityisdeterminedbyaconstant
√
Indexof
refraction
n x .
x
dx
dz
z
Observation:theconstantCisrelatedtotheoriginal“launching”angle ofthe
opticalray.Tocheckthatwestartby: C
IfweassumeC=
‐
,then
cosβ
Other popular examples: Luneberg Lens TheLuneberglensisinhomogeneousspherethatbringsacollimatedbeamoflighttoa
2
4
,
Suchlenswasmathemateicallyconceivedduringthe2ndworldwarbyR.K.Luneberg,
(see:R.K.Luneberg,MathematicalTheoryofOptics(BrownUniversity,Providence,
RhodeIsland,1944),pp.189‐213.)TheapplicationsofsuchLuneberglenswasquickly
demonstratedinmicrowavefrequencies,andlaterforopticalcommunicationsaswell
asinacoustics.Recently,suchdevicegainednewinterestsininphasedarray
communications,inilluminationsystems,aswellasconcentratorsinsolarenergy
harvestingandinimagingobjectives.
Image of Luneberg lens removed
Left:PictureofanOpticalLunebergLens(aglassball60mmindiameter)usedas
sphericalretro‐reflectoronMeteor‐3Mspacecraft.(Nasa.gov)
parallelrays(redsolidcurves)comingfromtheleft‐handsideoftheLuneberglenswill
focustoapointontheedgeofthesphere.
C. Superposition of Waves ‐ Waves in complex numbers Forexample,theelectricfieldoflightfieldcanbeexpressedas:
,
Since exp
cos
sin ,
Or
1
. .
,
2
‐ Complex numbers simply optics! ‐ Coherence 5
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Spring 2014