Scattering of an Electromagnetic Plane Wave by a Luneburg Lens. II. Wave Theory

12 Pages·2016·0.42 MB·English

by  James A. Lock

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Physics Faculty Publications Physics Department
12-1-2008
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James A. Lock
Cleveland State University, [email protected]
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OOrriiggiinnaall CCiittaattiioonn
Lock, James A. "Scattering of an Electromagnetic Plane Wave by a Luneburg Lens. II. Wave Theory."
Journal of the Optical Society of America A: Optics Image Science and Vision 25 (2008): 2980-2990.
Repository Citation
Lock, James A., "Scattering of an Electromagnetic Plane Wave by a Luneburg Lens. II. Wave Theory" (2008).
Physics Faculty Publications. 31.
https://engagedscholarship.csuohio.edu/sciphysics_facpub/31
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2980 J.Opt.Soc.Am.A/Vol.25,No.12/December2008 JamesA.Lock
Scattering of an electromagnetic plane wave by a
Luneburg lens. II. Wave theory
JamesA.Lock
DepartmentofPhysics,ClevelandStateUniversity,Cleveland,Ohio44115,USA([email protected])
ReceivedJuly30,2008;acceptedSeptember11,2008;
postedSeptember29,2008(Doc.ID99458);publishedNovember12,2008
Thepartialwavescatteringandinterioramplitudesfortheinteractionofanelectromagneticplanewavewith
amodifiedLuneburglensarederivedintermsoftheexteriorandinteriorradialfunctionsofthescalarradia-
tion potentials evaluated at the lens surface.ADebye series decomposition of these amplitudes is also per-
formedanddiscussed.Theeffectivepotentialinsidethelensforthetransverseelectricpolarizationisquali-
tatively examined, and the approximate lens size parameters of morphology-dependent resonances are
determined. Finally, the physical optics model is used to calculate wave scattering in the vicinity of the ray
theoryorbitingconditioninordertodemonstratethesmoothingofraytheorydiscontinuitiesbythediffraction
ofscatteredwaves. © 2008OpticalSocietyofAmerica
OCIScodes: 050.1960,080.2710,290.4020.
1. INTRODUCTION tential approach borrowed from quantum mechanics [4].
It is found that for f=1, scattering of the grazing inci-
This is the second in a series of papers whose purpose is
denceraycorrespondstothesemiclassicalphenomenonof
to examine electromagnetic scattering by a modified
orbiting[5,6].Forf(cid:1)1,theeffectivepotentialhasawide,
Luneburglensasaprototypeexampleforunderstanding
shallow well inside the modified Luneburg lens that can
the scattering phenomena of a radially inhomogeneous
support a series of morphology-dependent resonances
sphere and to compare and contrast them to the way in
(MDRs). The approximate size parameter of the reso-
whichtheyoccurforscatteringbyahomogeneoussphere.
nances is derived as a function of partial wave number
The first paper [1] considered ray theory transmission
andradialmodenumber.InSection4,thephysicaloptics
through a sphere of radius a whose dielectric constant
N2(cid:1)r(cid:2)variesparabolicallyinrinsuchawaythatthefocal model for scattering of the grazing incident ray is devel-
opedforf=1,andthesmoothingoftheraytheorydiscon-
point of all the transmitted rays is fa. Such a sphere is
tinuities at the orbiting condition by wave diffraction is
called a modified Luneburg lens. A discontinuity was
demonstratedandphysicallyinterpreted.Finally,Appen-
found as a function of f in the trajectory of the ray with
grazingincidenceonthesphere.Forf(cid:1)1,thegrazingin- dix A gives the details of orbiting for classical scattering
cidenceraywasscatteredthrough(cid:2)=180°,forf=1itwas ofaparticlebyaforcecenter.AppendixBderivestheDe-
scattered through (cid:2)=90°, and for f(cid:3)1 it was scattered bye series expansion of the partial wave scattering and
through(cid:2)=0°.Similarly,therewasadiscontinuityinthe interior amplitudes for a sphere with an arbitrary radi-
scattered intensity of the grazing incident ray. For f(cid:3)1 ally inhomogeneous refractive index profile. This Debye
the intensity diverged as f→1 due to the transmission serieshasaslightlydifferentphysicalinterpretationthan
rainbowapproaching(cid:2)=90°.Butforf=1,theintensityat itdoesforahomogeneoussphere,whichiscommentedon
(cid:2)=90° vanished. Both of these discontinuities signal a inAppendix B as well. The results obtained here and in
[1]arenumericallytestedin[7]whereamodifiedLuneb-
nonuniform convergence of ray theory in the vicinity of
urglensisapproximatedbyafinelystratifiedmultilayer
thegrazingincidenceray.
sphere. In [7] also, a new algorithm for computing scat-
In this paper the ray theory discontinuities are
teringbyamultilayerspherebasedonananalogytothe
smoothed by diffractive effects when scattering by light
successivedoublingstrategyofthefastFouriertransform
wavesisconsidered.InSection2,anincidentplanewave
(FFT)algorithmisdevelopedandimplemented.
is decomposed into partial waves. The scattering of each
partialwavebyaspherehavinganarbitraryradiallyin-
homogeneous refractive index profile is determined in
terms of the solutions of the second-order differential
2. FORMAL SOLUTION TO THE WAVE
equationfortheradialfunctionsofthetransverseelectric
SCATTERING PROBLEM
(TE) and transverse magnetic (TM) scalar radiation po-
tentials. For TE scattering by a modified Luneburg lens, A.RadialWaveEquations
the radial functions are related to Whittaker functions In analogy to the procedure followed in Mie theory for
[2,3]. In order to obtain an intuitive understanding of a scatteringofaplanewavebyahomogeneoussphere,the
numberoftheresultingscatteringphenomena,inSection TEandTMfieldsinsideaspherehavingthearbitraryre-
3thedifferentialequationfortheTEpartialwaveradial fractiveindexprofileN(cid:1)r(cid:2) areobtainedbytakingvarious
function is qualitatively analyzed using the effective po- vector derivatives of the scalar radiation potentials. For
1084-7529/08/122980-11/$15.00 ©2008OpticalSocietyofAmerica
JamesA.Lock Vol.25,No.12/December2008/J.Opt.Soc.Am.A 2981
the TE polarization, a partial wave scalar radiation po- B.BoundaryConditionMatching
tentialinsidethesphereisoftheform Consider a plane wave with electric field strength E ,
0
traveling in the positive z direction in the external me-
(cid:4) (cid:1)kr,(cid:2),(cid:5)(cid:2)=(cid:3)F (cid:1)kr(cid:2)/kr(cid:4)P m(cid:3)cos(cid:1)(cid:2)(cid:2)(cid:4)exp(cid:1)im(cid:5)(cid:2), (cid:1)1(cid:2)
n,m n n dium and polarized in the x direction. It is incident on a
radially inhomogeneous sphere of radius a and arbitrary
where(cid:6)isthewavelengthoftheincidentelectromagnetic
refractive index profile N(cid:1)r(cid:2) whose center is at the origin
waveinvacuum,thewavenumberisk=2(cid:7)/(cid:6),thepartial
ofcoordinates.Thecompletescalarradiationpotential(cid:12)
wavenumberisn,theazimuthalmodenumberism,and
oftheTEandTMcomponentsoftheplanewavecontains
P m are associated Legendre polynomials. Equation (1)
n onlythem=±1 azimuthalmodesandis
satisfiesthewaveequation
(cid:13)
(cid:1)2(cid:4)n,m+N2(cid:1)r(cid:2)k2(cid:4)n,m(cid:1)kr(cid:2)=0, (cid:1)2(cid:2) (cid:12) (cid:1)kr,(cid:2),(cid:5)(cid:2)=(cid:6)(cid:5)in(cid:1)2n+1(cid:2)/(cid:3)n(cid:1)n
inc
n=1
whichreducesto[8–11]
+1(cid:2)(cid:4)(cid:7)(cid:4)(cid:1)kr(cid:2)P 1(cid:1)cos(cid:2)(cid:2)(cid:14)(cid:1)(cid:5)(cid:2), (cid:1)9(cid:2)
d2F /d(cid:1)kr(cid:2)2+(cid:5)N2(cid:1)r(cid:2)−n(cid:1)n+1(cid:2)/(cid:1)kr(cid:2)2(cid:4)F (cid:1)kr(cid:2)=0 (cid:1)3(cid:2) n n
n n
where
for the partial wave radial function F (cid:1)kr(cid:2). If the refrac-
tiveindexNisconstant,F (cid:1)kr(cid:2)isalinnearcombinationof (cid:14)(cid:1)(cid:5)(cid:2)=cos(cid:1)(cid:5)(cid:2) forTM,
n
Riccati–Bessel functions (cid:4)(cid:1)Nkr(cid:2)=Nkrj (cid:1)Nkr(cid:2) and
n n
Riccati–Neumannfunctions(cid:8)(cid:1)Nkr(cid:2)=Nkrn (cid:1)Nkr(cid:2),where =sin(cid:1)(cid:5)(cid:2) forTE. (cid:1)10(cid:2)
n n
j and n are spherical Bessel functions and spherical
n n
Similarly, the complete scalar radiation potential of the
Neumann functions, respectively. The partial wave TE
scatteredwaveintheexteriormediumis
electricandmagneticfieldsassociatedwiththisradiation
potentialare (cid:13)
(cid:6)
(cid:12) (cid:1)kr,(cid:2),(cid:5)(cid:2)=− (cid:5)in(cid:1)2n+1(cid:2)/(cid:3)n(cid:1)n+1(cid:2)(cid:4)(cid:7)
E TE(cid:1)r,(cid:2),(cid:5)(cid:2)=−r(cid:9)(cid:1)(cid:4) , (cid:1)4a(cid:2) scat
n,m n,m n=1
(cid:9)(cid:15)(cid:1)1(cid:2)(cid:1)kr(cid:2)P 1(cid:1)cos(cid:2)(cid:2)(cid:16)(cid:1)(cid:5)(cid:2), (cid:1)11(cid:2)
B TE(cid:1)r,(cid:2),(cid:5)(cid:2)=(cid:1)i/(cid:10)(cid:2)(cid:1) (cid:9)(cid:1)r(cid:9)(cid:1)(cid:4) (cid:2), (cid:1)4b(cid:2) n n n
n,m n,m
where (cid:15)(cid:1)1(cid:2)(cid:1)kr(cid:2)=krh (cid:1)1(cid:2)(cid:1)kr(cid:2) are radially outgoing Riccati–
wherethespeedoflightinvacuumisc and(cid:10)=ck. n n
Hankelfunctions,
For the TM polarization, a partial wave scalar radia-
tionpotentialisoftheform (cid:16)(cid:1)(cid:5)(cid:2)=a cos(cid:1)(cid:5)(cid:2) forTM,
n n
(cid:11) (cid:1)kr,(cid:2),(cid:5)(cid:2)=N(cid:1)r(cid:2)(cid:4) (cid:1)kr,(cid:2),(cid:5)(cid:2)
n,m n,m =b sin(cid:1)(cid:5)(cid:2) forTE, (cid:1)12(cid:2)
n
=(cid:3)G (cid:1)kr(cid:2)/kr(cid:4)P m(cid:1)cos(cid:2)(cid:2)exp(cid:1)im(cid:5)(cid:2) (cid:1)5(cid:2)
n n
anda andb arethepartialwavescatteringamplitudes.
n n
andsatisfiesthewaveequation Thecompleteinteriorscalarradiationpotentialis
(cid:1)2(cid:11)n,m−(cid:3)2(cid:1)dN/dr(cid:2)/N(cid:4)(cid:2)(cid:11)n,m/(cid:2)r (cid:12) (cid:1)kr,(cid:2),(cid:5)(cid:2)=(cid:6)(cid:13) (cid:5)in(cid:1)2n+1(cid:2)/(cid:3)n(cid:1)n+1(cid:2)(cid:4)(cid:7)(cid:17)(cid:1)kr,(cid:5)(cid:2)P 1(cid:1)cos(cid:2)(cid:2),
int n n
−(cid:3)2(cid:1)dN/dr(cid:2)/(cid:1)Nr(cid:2)(cid:4)(cid:11)n,m+N2(cid:1)r(cid:2)k2(cid:11)n,m=0. (cid:1)6(cid:2) n=1
(cid:1)13(cid:2)
Itreducesto[8–11]
where
d2G /d(cid:1)kr(cid:2)2−2(cid:1)dN/dr(cid:2)(cid:3)dG /d(cid:1)kr(cid:2)(cid:4)/Nk
n n
(cid:17)(cid:1)kr,(cid:5)(cid:2)=G (cid:1)kr(cid:2)c cos(cid:1)(cid:5)(cid:2) forTM,
+(cid:3)N2(cid:1)r(cid:2)−n(cid:1)n+1(cid:2)/(cid:1)kr(cid:2)2(cid:4)G (cid:1)kr(cid:2)=0 (cid:1)7(cid:2) n n n
n
for the partial wave radial function Gn(cid:1)kr(cid:2). If the refrac- =Fn(cid:1)kr(cid:2)dnsin(cid:1)(cid:5)(cid:2) forTE, (cid:1)14(cid:2)
tive index N is constant, G (cid:1)kr(cid:2) is again a linear combi-
nationof(cid:4)(cid:1)Nkr(cid:2)and(cid:8)(cid:1)Nknr(cid:2).ThepartialwaveTMelec- whereGn(cid:1)kr(cid:2) andFn(cid:1)kr(cid:2) arethesolutionsofEqs.(3)and
n n (7), respectively, that vanish at the origin and c and d
tric and magnetic fields associated with this radiation n n
arethepartialwaveinterioramplitudes.Thencontinuity
potentialare
ofD ,E ,andB atr=a gives[8–11]
rad tan tan
En,mTM(cid:1)r,(cid:2),(cid:5)(cid:2)=(cid:1)ic/N2(cid:10)(cid:2)(cid:1) (cid:9)(cid:1)r(cid:9)(cid:1)(cid:11)n,m(cid:2), (cid:1)8a(cid:2) a =(cid:3)G (cid:1)(cid:1)ka(cid:2)(cid:4)(cid:1)ka(cid:2)−N2(cid:1)a(cid:2)G (cid:1)ka(cid:2)(cid:4)(cid:1)(cid:1)ka(cid:2)(cid:4)/(cid:3)G (cid:1)(cid:1)ka(cid:2)
n n n n n n
Bn,mTM(cid:1)r,(cid:2),(cid:5)(cid:2)=(cid:1)1/c(cid:2)r(cid:9)(cid:1)(cid:11)n,m. (cid:1)8b(cid:2) (cid:9)(cid:15)n(cid:1)1(cid:2)(cid:1)ka(cid:2)−N2(cid:1)a(cid:2)Gn(cid:1)ka(cid:2)(cid:15)n(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4), (cid:1)15a(cid:2)
The structure of the differential equation for the TE ra- b =(cid:3)F (cid:1)(cid:1)ka(cid:2)(cid:4)(cid:1)ka(cid:2)−F (cid:1)ka(cid:2)(cid:4)(cid:1)(cid:1)ka(cid:2)(cid:4)/(cid:3)F (cid:1)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)ka(cid:2)
dialfunctionF ofEq.(3)isexactlyasitwasforahomo- n n n n n n n
n
geneoussphere,exceptthatnowtherefractiveindexisa −F (cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4), (cid:1)15b(cid:2)
n n
function of r. The differential equation for the TM radial
functionGnofEq.(7)containsafirstderivativetermpro- c =−iN2(cid:1)a(cid:2)/(cid:3)G (cid:1)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)ka(cid:2)−N2(cid:1)a(cid:2)G (cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4),
portionaltodN/dr,whichgreatlycomplicatesitsanalyti- n n n n n
calsolution. (cid:1)15c(cid:2)
2982 J.Opt.Soc.Am.A/Vol.25,No.12/December2008 JamesA.Lock
d =−i/(cid:3)F (cid:1)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)ka(cid:2)−F (cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4). (cid:1)15d(cid:2) tentialapproachisdiscussedforTEscattering.Thefinely
n n n n n
stratified sphere model is implemented and discussed in
In Eqs. (15a)–(15d) the prime symbol indicates a deriva- [7].
tive with respect to kr, and N(cid:1)a(cid:2) is evaluated inside the The differential equation of Eq. (17) for F (cid:1)kr(cid:2) for the
n
sphere as r→a. These partial wave scattering and inte- modifiedLuneburglenscanberewrittenas
rior amplitudes are then substituted into Eqs. (11) and
−d2F /d(cid:1)kr(cid:2)2+U (cid:1)kr(cid:2)F (cid:1)kr(cid:2)=F (cid:1)kr(cid:2), (cid:1)20(cid:2)
(13), from which the scattered and interior fields can be n eff n n
determined. This constitutes the formal solution to the wheretheeffectivepotentialis
electromagneticboundaryvalueproblemofaplanewave
scattered by a radially inhomogeneous sphere. The solu- Ueff(cid:1)kr(cid:2)=n(cid:1)n+1(cid:2)/(cid:1)kr(cid:2)2+(cid:3)−1+(cid:1)kr(cid:2)2/(cid:1)ka(cid:2)2(cid:4)/f2 forr(cid:19)a,
tion is formal in the sense that the radial functions F
n
and G still need to be determined for a given refractive =n(cid:1)n+1(cid:2)/(cid:1)kr(cid:2)2 forr(cid:3)a. (cid:1)21(cid:2)
n
index profile, and the infinite series of partial waves
Equations (20) and (21) are analogous to a one-
needs to be summed in order to identify and understand
dimensional quantum mechanical Schrodinger equation
variousfeaturesofthescatteredintensity.
[4]withwavefunctionF (cid:1)kr(cid:2),withunitenergy,andwith
n
the potential well U (cid:1)kr(cid:2). The effective potential for
C.WaveScatteringbyaModifiedLuneburgLens eff
r(cid:19)a is the sum of the centrifugal potential proportional
AmodifiedLuneburglenshastherefractiveindexprofile
to1/r2andaharmonicoscillatorpotentialproportionalto
[1]
r2. Only the centrifugal potential occurs outside the
N(cid:1)r(cid:2)=(cid:3)1+f2−(cid:1)r/a(cid:2)2(cid:4)1/2/f. (cid:1)16(cid:2) sphere,andU iscontinuousatr=a.Asthepartialwave
eff
number n increases for constant ka, the effective energy
Equation (3) for the TE scalar radiation potential Fn(cid:1)kr(cid:2) remains constant and the centrifugal barrier becomes
thenbecomes higher. The radial interval for which U (cid:1)1 corresponds
eff
toaclassicallyallowedregionwhereF (cid:1)kr(cid:2) isoscillatory,
d2F /d(cid:1)kr(cid:2)2+(cid:3)−n(cid:1)n+1(cid:2)/(cid:1)kr(cid:2)2+(cid:1)f2+1(cid:2)/f2−(cid:1)kr(cid:2)2/(cid:1)fka(cid:2)2(cid:4)F n
n n andtheradialintervalforwhichU (cid:3)1correspondstoa
eff
=0, (cid:1)17(cid:2) classically forbidden region where F (cid:1)kr(cid:2) is damped. The
n
second derivative of F (cid:1)kr(cid:2) vanishes at the boundary be-
n
whose solution, after a change of variables, is the Whit- tween these two regions and is called a classical turning
takerfunction[2,3] point.Fortheremainderofthissection,thepartialwave
F (cid:1)kr(cid:2)=(cid:3)(cid:1)kr(cid:2)2/fka(cid:4)(cid:1)n+1(cid:2)/2exp(cid:3)−(cid:1)kr(cid:2)2/2fka(cid:4)(cid:9)M(cid:3)(cid:1)2n+3(cid:2)/4 numbern isparameterizedbyX,where
n
X(cid:8)n(cid:1)n+1(cid:2)/(cid:1)ka(cid:2)2. (cid:1)22(cid:2)
−(cid:1)f2+1(cid:2)ka/4f, (cid:1)2n+3(cid:2)/2; (cid:1)kr(cid:2)2/fka(cid:4), (cid:1)18(cid:2)
Theeffectivepotentialapproachcannotbeappliedtothe
where M(cid:1)a,c;w(cid:2) is a confluent hypergeometric function.
TM partial wave radial function because of the presence
In the context of quantum mechanics, Eqs. (17) and (18)
ofthefirstderivativeterminEq.(19).
are identical to the radial Schrodinger equation for the
three-dimensionalharmonicoscillatorpotentialinspheri-
B.ModifiedLuneburgLenswithf(cid:3)1
cal coordinates and its solution, with the replacement TheshapeofU (cid:1)kr(cid:2) fortheTEpolarizationisshownfor
(cid:1)f2+1(cid:2)(cid:1)ka(cid:2)/4f→(cid:1)E/2(cid:18)(cid:2)(cid:1)m/k(cid:2)1/2. For the TM polarization, a number of vaeluffes of the partial wave X for f(cid:3)1 in Fig.
Eq.(7)forthescalarradiationpotentialGn(cid:1)kr(cid:2) becomes 1(a), for f=1 in Fig. 1(b), and for f(cid:1)1 in Fig. 1(c). For
d2G /d(cid:1)kr(cid:2)2+(cid:3)2kr/(cid:1)fka(cid:2)2(cid:4)(cid:3)(cid:1)f2+1(cid:2)/f2 f(cid:3)1 and low partial waves corresponding to X(cid:1)1, the
n
classically allowed region inside the modified Luneburg
−(cid:1)kr/fka(cid:2)2(cid:4)−1dGn/d(cid:1)kr(cid:2)+(cid:3)−n(cid:1)n+1(cid:2)/(cid:1)kr(cid:2)2+(cid:1)f2+1(cid:2)/f2 lensis
−(cid:1)kr(cid:2)2/(cid:1)fka(cid:2)2(cid:4)G =0. (cid:1)19(cid:2) (cid:5)(cid:1)f2+1(cid:2)−(cid:3)(cid:1)f2+1(cid:2)2−4f2X(cid:4)1/2(cid:7)/2(cid:19)(cid:1)kr/ka(cid:2)2(cid:19)1. (cid:1)23(cid:2)
n
The analytical solution of this equation [12] is consider- Apartialwavewithn(cid:1)ka isincidentonthelenssurface
ablymorecomplicatedthanthatofEq.(17). and penetrates into it until the centrifugal barrier’s in-
creasing strength converts the wave from oscillatory to
evanescent, after which it tunnels to the origin. As the
3. EFFECTIVE POTENTIAL APPROACH FOR partial wave penetrates into the lens for some distance,
TE SCATTERING its oscillatory shape is distorted and its phase is shifted
by the refractive index variation, and it thus contributes
A.GeneralConsiderations
Since the radial functions F (cid:1)kr(cid:2) and G (cid:1)kr(cid:2) of Eqs. (17) tothescatteredintensity.Thelocalizationprincipleofvan
n n
and (19) for a modified Luneburg lens are both compli- de Hulst associates a partial wave with the impact pa-
cated and relatively unfamiliar, two procedures may be rameterofanincidentray,sin(cid:1)(cid:16)(cid:2),andiswrittenas
used to gain intuition concerning the behavior of these X(cid:9)sin2(cid:1)(cid:16)(cid:2) (cid:1)24(cid:2)
functions: (i) analyzing the sphere’s effective potential
well and (ii) approximating the radially inhomogeneous in the notation of Eq. (22) and [1]. Substituting Eq. (24)
refractive index profile by a finely stratified multilayer intotheleftsideofEq.(23),theclassicalturningpointof
sphere and then numerically solving the multilayer thepartialwaveradialfunctioninsidethelensisfoundto
spherescatteringproblem.Inthissectiontheeffectivepo- beidenticaltothedistanceofclosestapproachofthecor-
JamesA.Lock Vol.25,No.12/December2008/J.Opt.Soc.Am.A 2983
sidethesphereisnegativeasr→a,andthepartialwave
radial function is evanescent for the entire lens interior.
This is shown in Fig. 1(a). This higher partial wave can-
not effectively probe the details of the lens interior, and
thus its contribution to the scattered intensity is small.
When the partial wave number further increases to X
(cid:3)1, the contribution to the scattered field rapidly de-
creases as the centrifugal barrier becomes increasingly
highandthedampingrateoftheevanescentwaveinside
thespherebecomesprogressivelyfaster.
C.ClassicalOrbitinginaLuneburgLenswithf=1
Whenf=1,theeffectivepotentialis
U (cid:1)kr(cid:2)=X(cid:1)ka/kr(cid:2)2+(cid:1)kr/ka(cid:2)2−1 forr(cid:19)a,
eff
=X(cid:1)ka/kr(cid:2)2 forr(cid:3)a. (cid:1)25(cid:2)
For low partial waves with X(cid:1)1, the classically allowed
region of the lens interior is again given by Eq. (23), and
for high partial waves with X(cid:3)1 the interior is a classi-
cally forbidden region that produces minimal scattering.
For the partial wave corresponding to X=1, the classical
turning point is at r=a, and the first derivative of U
eff
vanishesinsidethesphereasr→a.ThisisshowninFig.
1(b).However,thederivativedoesnotvanishoutsidethe
sphere as r→a, since only the monotonically decreasing
centrifugal potential contributes to U there. As is de-
eff
scribed in Appendix A, the phenomenon of orbiting [5,6]
inclassicalscatteringoccursforaparticleinanattractive
potentialwhenthefirstderivativeofU vanishesatthe
eff
classicalturningpoint,e.g.,whenU islocallyquadratic
eff
and the turning point occurs at the relative maximum of
U . When this condition is met, a classical particle ap-
eff
proaching the force center will not be scattered by it but
willbecapturedbytheforcecenterandorbititforever.It
shouldbenotedthattheparticulargeometryofaLuneb-
urg lens permits the orbiting condition to be met at the
relative minimum of the parabolic potential, as in Fig.
1(b), instead of the relative maximum. Although the
Luneburg lens effective potential for X=1 does not pre-
cisely correspond to the condition for classical orbiting,
since the first derivative of U at the classical turning
eff
pointvanishesonlyasrincreasestowarda,itisaboutas
close to it as a relatively simple optical scattering geom-
etry can come. If a classical particle is incident on the
force center with slightly different initial conditions, it
will be temporarily captured by the force center, orbit it
for a number of cycles, and then eventually escape. The
Fig.1. EffectiveradialpotentialofEq.(21)asafunctionofr/a
optical analog of temporary capture is closely approxi-
for ka=50.5, partial waves n=45, 50, and 55, and (a) f=1.2, (b)
f=1.0, and (c) f=0.8. The effective energy of the size parameter matedbyanMDR[13].
ka=50.5isthehorizontallineU =2550.Forf=0.8,aninternal Aswasseenin[1],thegeometricalraycorrespondingto
eff
well potential is formed by partial waves with n slightly larger the X=1 partial wave strikes the lens surface with graz-
than50.Forf=1,thepartialwaven=50isattheconditionfor ing incidence and travels in a circular arc on the surface
orbiting.
for a quarter cycle before breaking free and scattering at
respondingraytotheoriginobtainedinEq.24ofRef.[1], 90°. But intuitively, once the initially grazing ray is tra-
furtherreinforcingthecorrespondencebetweentheradial versing the lens surface, there appears to be no special
propagation of partial waves and the curved trajectories physical constraint that would limit the ray to orbit for
ofrays. only a quarter cycle. It could equally well traverse the
When the partial wave number increases to X=1, the lens surface for any number of orbits, thus being the op-
classicallyallowedregionendsatr=a,theslopeofU in- tical analog of an orbiting particle in classical scattering
eff
2984 J.Opt.Soc.Am.A/Vol.25,No.12/December2008 JamesA.Lock
[14]. In Section 4 this phenomenon is analyzed more X1/2=(cid:3)(cid:1)f2+1(cid:2)/2f(cid:4)−(cid:3)(cid:1)2S+1(cid:2)/ka(cid:4), (cid:1)31(cid:2)
quantitativelyusingthephysicalopticsmodel.
whereS=0,1,2,3...,analogoustotheenergyeigenvalues
D.Morphology-DependentResonancesofaModified oftheSchrodingerequation,thedifferentialequationhas
LuneburgLenswithf(cid:1)1 aboundstatesolution.Usingtheapproximation
Forf(cid:1)1 andlowpartialwaveswithX(cid:1)1,theclassically
n(cid:1)n+1(cid:2)(cid:9)(cid:1)n+1/2(cid:2)2 (cid:1)32(cid:2)
allowedregioninsidethemodifiedLuneburglensisagain
given by Eq. (23). For X=1, the slope of Ueff inside the forlargepartialwaves,Eq.(31)becomes
sphere is positive as r→a, signaling the presence of a
relativeminimumofU forr(cid:1)a.ThisisapparentinFig. n+1/2=(cid:3)(cid:1)f2+1(cid:2)ka/2f(cid:4)−(cid:1)2S+1(cid:2) (cid:1)33(cid:2)
eff
1(c).ForpartialwavesX(cid:20)1 with
or
1(cid:19)X(cid:19)(cid:3)(cid:1)f2+1(cid:2)/2f(cid:4)2 (cid:1)26(cid:2)
ka=(cid:1)n+2S+3/2(cid:2)(cid:3)2f/(cid:1)f2+1(cid:2)(cid:4). (cid:1)34(cid:2)
corresponding to geometrical rays that just miss striking
Interestingly, Eq. (34) is also the condition for which the
the lens surface, a classically allowed well region is
exactconfluenthypergeometricradialfunction[15]inEq.
formedinsidethelensthatextendsfrom
(18) becomes a polynomial of degree S, again giving a
(cid:5)(cid:1)f2+1(cid:2)−(cid:3)(cid:1)f2+1(cid:2)2−4f2X(cid:4)1/2(cid:7)/2 bound state solution to Eq. (17) that damps to zero as r
→(cid:13).Apparently,thetwoapproximationsofEqs.(30)and
(cid:19)(cid:1)kr/ka(cid:2)2(cid:19)(cid:5)(cid:1)f2+1(cid:2)+(cid:3)(cid:1)f2+1(cid:2)2−4f2X(cid:4)1/2(cid:7)/2. (32) cancel each other. Considering only the condition for
(cid:1)27(cid:2) theexistenceofaboundstateinthelocallyparabolicwell
andneglectingsmallcorrectionstokaduetotheneedfor
In order to get to this interior well region, the partial matching boundary conditions at both ends of the cen-
wave must tunnel through the centrifugal barrier, which trifugal barrier, Eq. (34) is the approximate size param-
extendsfrom eter for the formation of a TE-polarized MDR in the par-
tial wave n of radial order S. In [7] this prediction is
(cid:5)(cid:1)f2+1(cid:2)+(cid:3)(cid:1)f2+1(cid:2)2−4f2X(cid:4)1/2(cid:7)/2(cid:19)(cid:1)kr/ka(cid:2)2(cid:19)X. (cid:1)28(cid:2)
numericallytested,andtheTMresonantsizeparameters
areobtained.
Equation(26)definestheso-callededgeregionforscatter-
ing by a modified Luneburg lens with f(cid:1)1, and MDRs Bywayofcomparison,forscatteringbyahomogeneous
spherewithn slightlylargerthanka,theradialeffective
shouldoccurwhenpartialwavesinthisintervalarereso-
potentialconsistsofacentrifugalbarrieroutsidethepar-
nantlycapturedbytheinteriorwell.UsingEq.(15b),this
ticle and an approximately linearly decreasing well, i.e.,
occurswhen
anAirywell,justinsidetheparticlesurfacethatalsosup-
F (cid:1)(cid:1)ka(cid:2)/F (cid:1)ka(cid:2)=(cid:8)(cid:1)(cid:1)ka(cid:2)/(cid:8)(cid:1)ka(cid:2). (cid:1)29(cid:2) ports TE and TM resonances. The size parameter of the
n n n n
homogeneoussphereresonancesisgiventofirstorderby
Asimple approximation for the values of the resonant
[16–18]
sizeparameterforagivenpartialwaveintheedgeregion
can be derived as follows. When Ueff is Taylor series ex- ka(cid:9)(cid:1)n+1/2(cid:2)/N+(cid:1)n+1/2(cid:2)1/3wS/(cid:1)21/3N(cid:2)−P/(cid:1)N2−1(cid:2)1/2
pandedaboutitsrelativeminimum,theresultingwell
+ ... , (cid:1)35(cid:2)
U (cid:1)kr(cid:2)(cid:9)(cid:3)(cid:1)2X1/2/f(cid:2)−(cid:1)1/f2(cid:2)(cid:4)+4(cid:3)kr−(cid:1)ka(cid:2)f1/2X1/4(cid:4)2/(cid:1)kaf(cid:2)2
eff where
(cid:1)30(cid:2)
P=1 forTE,
islocallyparabolic.ThefirstterminEq.(30)isaconstant
baseline,andthesecondtermisaharmonicoscillatorpo- =1/N2 forTM, (cid:1)36(cid:2)
tential centered on the position r=af1/2X1/4. When the
partial wave is at the high end of the edge region, X is where Ai(cid:1)−w (cid:2)=0 and S=1,2,3.... The modified Luneb-
S
slightlylessthan(cid:3)(cid:1)f2+1(cid:2)/2f(cid:4)2inEq.(26),thewellisvery urglensMDRsareexpectedtohavesomewhatofadiffer-
shallow,anditscenterisapproximatelyhalfwaybetween entbehaviorthanthehomogeneoussphereMDRs.Thelo-
thetwoclassicalturningpointsofEq.(27).Whenthepar- cally parabolic effective potential of Eqs. (21) and (30) is
tialwaveisatthelowendoftheedgeregion,Xisslightly relatively wide and shallow and thus supports a bound
greaterthan1inEq.(26),andthewellbecomessubstan- statequiteclosetothebottomofthewell,whereastheen-
tiallywiderandsomewhatdeeper.Ifthisinteriorwellbe- ergy of the first bound state in an Airy well lies higher
comesdeepenough,itcansupportoneormoreharmonic above the bottom of the well. As a result, the modified
oscillator bound states that damp to zero as r→(cid:13) if Eq. Luneburg lens MDRs for a given partial wavenumber
(30) were valid all the way out to infinity. Instead, these shouldoccuratalowervalueofXthandotheMDRsofa
states slowly decay as they leak through the centrifugal homogeneous sphere. Also, since the deepest part of the
barrierofEq.(28)totheclassicallyallowedregionoutside well for a modified Luneburg lens lies further inside the
the lens. These metastable states are the MDRs. Substi- sphere than does the deepest part of the Airy well, the
tuting Eq. (30) into Eq. (20), the resulting differential Luneburg lens MDRs should also lie deeper inside the
equation is identical to a one-dimensional Schrodinger sphere, whereas the homogeneous sphere MDRs lie just
equation with a harmonic oscillator potential. If X takes beneath the sphere surface. These issues are treated
onthespecialvalues morequantitativelyandfullyin[7].
JamesA.Lock Vol.25,No.12/December2008/J.Opt.Soc.Am.A 2985
4. PHYSICAL OPTICS MODEL OF agreeswiththeraytheoryscatteredfieldofEq.18of[1].
SCATTERING OF THE ORBITING RAY BY A Finally,Eq.(38d)wasderivedin[1].
LUNEBURG LENS WITH f=1 This analysis encounters difficulties at the nongeneric
angle (cid:2)=(cid:7)/2 for two reasons. First, the density of outgo-
InraytheorytheamplitudeforscatteringbyaLuneburg
ingraysinthehorizontaldirectiongoestozerothere.Sec-
lens with f=1 is proportional to (cid:1)cos(cid:2)(cid:2)1/2 for 0(cid:19)(cid:2)(cid:19)(cid:7)/2 ond, ray theory predicts that no rays are scattered for (cid:2)
and vanishes for larger angles. The slope of this ampli- (cid:3)(cid:7)/2, so there are no transmitted rays to integrate over
tudehasaninfinitediscontinuityat(cid:2)=(cid:7)/2andisknown forthe−(cid:13)(cid:1)x(cid:1)(cid:1)0 portionoftheexitplanecorresponding
as a weak caustic [19]. This divergence is softened in
to
wave theory, as is shown here using the physical optics
model. Consider a plane wave incident on a spherical (cid:2)=(cid:1)(cid:7)/2(cid:2)+(cid:22), (cid:1)39(cid:2)
scatterer.Thespatialdensityoftraysassociatedwiththe
plane wave is E 2, and the wavefront at the entrance with(cid:22)(cid:3)0.Rayscatteringoccursonlyfor(cid:2)(cid:1)(cid:7)/2,i.e.,for
0 (cid:22)(cid:1)0,andthesetransmittedrayscrosstheexitplanefor
planetangenttothespheresurfaceisflat.Forscattering
at what is called a generic angle (cid:2), the scattered wave- 0(cid:1)x(cid:1)(cid:1)(cid:13).Bothdifficultiesareremediedbythesamepre-
scription. The ray density factor A is slowly varying at
front in the exit plane tangent to the sphere surface cen- h
tered on the ray exiting in the (cid:2)direction is approxi- angles far from (cid:7)/2 and rapidly decreases only as (cid:2)
→(cid:7)/2 due to its infinite slope at (cid:2)=(cid:7)/2.As a result, one
mately parabolic in both the horizontal and vertical
canapproximateA bysomeconstantaveragevalueA ave
directions, having the radii of curvature R and R , re- h h
h v for 0(cid:19)(cid:2)(cid:19)(cid:7)/2 corresponding to rays that cross the exit
spectively. As one follows the incident flux tube through
plane for 0(cid:19)x(cid:1)(cid:1)(cid:13), and by zero for (cid:7)/2(cid:19)(cid:2)(cid:19)(cid:7) corre-
thesphere,thedensityofoutgoingraysinthehorizontal
sponding to the absence of transmitted rays crossing the
and vertical directions in the exit plane is taken to be
exitplanefor−(cid:13)(cid:1)x(cid:1)(cid:1)0.Onethenobtains
A E andA E ,respectively.Lettheexitplane’shorizon-
h 0 v 0
taldirectionbex(cid:1)anditsverticaldirectionbey(cid:1).Letb(cid:2)be E(cid:3)(cid:1)(cid:7)/2(cid:2)+(cid:22)(cid:4)=(cid:3)−ikE exp(cid:1)ikr(cid:2)/2(cid:7)r(cid:4)A aveA exp(cid:3)i(cid:21)(cid:1)(cid:7)/2(cid:2)(cid:4)
0 h v
theimpactparameteroftheincidentraythatistransmit- (cid:10)
ted through the Luneburg lens and exits at the angle (cid:2). (cid:13)
(cid:9) dx(cid:1)exp(cid:1)ikx(cid:1)2/2R (cid:2)exp(cid:1)ikx(cid:1)(cid:22)(cid:2)
Then for f=1, rays with incident impact parameters b h
(cid:1)b(cid:2)are scattered through smaller angles and cross the (cid:10)−(cid:13)
exitplaneatx(cid:1)(cid:3)0.Incidentrayswithimpactparameters (cid:13)
b(cid:3)b(cid:2)are scattered through larger angles and cross the (cid:9) dy(cid:1)exp(cid:1)iky(cid:1)2/2Rv(cid:2)
exitplaneatx(cid:1)(cid:1)0. −(cid:13)
Inthephysicalopticsmodel[20],thefar-zonescattered =(cid:1)1/21/2r(cid:2)(cid:3)E aA aveexp(cid:1)ikr+i(cid:21)(cid:1)(cid:7)/2(cid:2)−i(cid:7)/4(cid:2)(cid:4)
electric field at the scattering angle (cid:2)is obtained by 0 h
Fraunhofer diffracting the electric field in the scatterer’s (cid:9)exp(cid:1)−ika(cid:22)2/2(cid:2)(cid:5)F(cid:1)(cid:13)(cid:2)−F(cid:3)(cid:22)(cid:1)ka/(cid:7)(cid:2)1/2(cid:4)(cid:7),
exitplane (cid:1)40(cid:2)
E(cid:1)(cid:2)(cid:2)=(cid:3)−ikE0exp(cid:1)ikr(cid:2)/2(cid:7)r(cid:4)AhAvexp(cid:3)i(cid:21)(cid:1)(cid:2)(cid:2)(cid:4) whereF(cid:1)w(cid:2) istheFresnelintegral
(cid:10) (cid:10) (cid:10)
(cid:13) (cid:13) W
(cid:9) dx(cid:1)exp(cid:1)ikx(cid:1)2/2Rh(cid:2) dy(cid:1)exp(cid:1)iky(cid:1)2/2Rv(cid:2) F(cid:1)w(cid:2)= dvexp(cid:1)i(cid:7)v2/2(cid:2). (cid:1)41(cid:2)
−(cid:13) −(cid:13) 0
=(cid:3)E0exp(cid:1)ikr(cid:2)/r(cid:4)AhAv(cid:1)RhRv(cid:2)1/2exp(cid:3)i(cid:21)(cid:1)(cid:2)(cid:2)(cid:4), (cid:1)37(cid:2) In the ray theory illuminated region, (cid:22) is negative and
the Fresnel straight-edge pattern F(cid:1)(cid:13)(cid:2)−F(cid:3)(cid:22)(cid:1)ka/(cid:7)(cid:2)1/2(cid:4) is
where (cid:21)(cid:1)(cid:2)(cid:2) is the phase of the wavefront at the center of oscillatory. In the ray theory shadowed region, (cid:22) is posi-
theexitplane.ForraytransmissionthroughaLuneburg tive and F(cid:1)(cid:13)(cid:2)−F(cid:3)(cid:22)(cid:1)ka/(cid:7)(cid:2)1/2(cid:4) monotonically decreases.
lens with f=1 and assuming 100% transmission through
Thissmoothingofthetransitionfromtheilluminatedre-
thelensforboththeTEandTMpolarizations,araytrac- gion to the shadowed region in the vicinity of (cid:2)=(cid:7)/2 is
inganalysisgives,aftermuchalgebra,
showninFig.5of[11]forka=60.0,andinFig.2herefor
R =R =a(cid:3)1−cos(cid:1)(cid:2)(cid:2)(cid:4), (cid:1)38a(cid:2) ka=350.0.
v h
These results have a pleasing physical interpretation.
Inraytheorytheorbitingraytravelsinacirculararcon
Ah=(cid:5)cos(cid:2)/(cid:3)1−cos(cid:1)(cid:2)(cid:2)(cid:4)(cid:7)1/2, (cid:1)38b(cid:2) the surface of the lens and exits only at (cid:2)=(cid:7)/2. In the
physical optics model the orbiting ray travels along the
A =(cid:5)1/(cid:3)1−cos(cid:1)(cid:2)(cid:2)(cid:4)(cid:7)1/2, (cid:1)38c(cid:2) surfaceofthelensforever,continuallysheddingradiation
v
tangentially. The radiation it sheds for (cid:2)(cid:1)(cid:7)/2 interferes
(cid:21)(cid:1)(cid:2)(cid:2)=ka(cid:3)(cid:1)(cid:7)/2(cid:2)−cos(cid:1)(cid:2)(cid:2)(cid:4)−(cid:7). (cid:1)38d(cid:2) withthelighttransmittedthroughthespherebyincident
rayswithsmallerimpactparameters,producinganinter-
As a check of Eqs. (38a)–(38d), the radii of curvature of ferencepattern.For(cid:2)(cid:3)(cid:7)/2,theshedradiationistheonly
the phase fronts at the exit plane are equal to the dis- contribution to the scattered light. As the amplitude of
tance along the ray exiting in the (cid:2)direction from the the orbiting ray decreases, the amount of radiation it
pointfocusonthebackoftheLuneburglens,sincethefo- sheds farther on in its trajectory decreases as well. This
calpointisthesourceofthephasefronts.Further,when behavior is qualitatively similar to that of the radiation
Eqs. (38a)–(38c) are substituted into Eq. (37), the result shedbyelectromagneticsurfacewavesintotheshadowed
2986 J.Opt.Soc.Am.A/Vol.25,No.12/December2008 JamesA.Lock
terpropagating orbiting radiation shed by the ray with
grazing incidence at the bottom of the sphere.All the ef-
fects qualitatively described here are treated quantita-
tively in [7] using a finely stratified multilayer sphere to
modelthemodifiedLuneburglens.
APPENDIX A: ORBITING IN CLASSICAL
SCATTERING
In the context of classical mechanics, consider a spheri-
cally symmetric attractive potential V(cid:1)r(cid:2) surrounding a
forcecenterlocatedattheoriginofcoordinates.Aparticle
ofmassmisincidentonthepotentialfrominfinity.Ithas
the impact parameter b with respect to the force center,
energyE=mv2(cid:1)(cid:13)(cid:2)/2,andangularmomentumL=mv(cid:1)(cid:13)(cid:2)b,
where v(cid:1)(cid:13)(cid:2) is the magnitude of the particle’s velocity at
infinity. The particle is deflected through an angle (cid:2)by
the potential. The particle moves in the effective radial
Fig. 2. TE scattered intensity as a function of the scattering
angle(cid:2)forf=1.0,a=28.40(cid:23)m,(cid:6)=0.51(cid:23)m,andka=350.0com- potential
putedbythemethoddescribedin[7].For30°(cid:1)(cid:2)(cid:1)100°,thein- U (cid:1)r(cid:2)=(cid:1)L2/2mr2(cid:2)+V(cid:1)r(cid:2), (cid:1)A1(cid:2)
tensity resembles that of a Fresnel straight-edge pattern corre- eff
spondingtoEq.(40).
where the first term is the centrifugal potential and the
second term is the attractive potential surrounding the
region at a Fock transition [21]. This analogy is only force center. During the deflection of the particle, its dis-
qualitative,however,sincetheangulardependenceofthe tanceofclosestapproachtotheoriginr occurswhenthe
0
electromagnetic surface wave radiation damps exponen- radialkineticenergyvanishesand
tially in (cid:22), whereas deep in the classically shadowed re-
gion here the falloff of Eq. (40) is proportional to 1/(cid:22). In E=Ueff(cid:1)r0(cid:2). (cid:1)A2(cid:2)
Fig. 2, the falloff of the intensity in the shadowed region
Thedeflectionangleoftheparticleis[22]
wasfoundtobebetterfitbyastraightlineusingalog–log
(cid:10)
graph rather than a semilog graph, indicating a power (cid:13)
law behavior. In addition, if one considers f=1 scattering (cid:2)=(cid:7)−2 (cid:1)Ldr/r2(cid:2)/(cid:5)2m(cid:3)E−U (cid:1)r(cid:2)(cid:4)(cid:7)1/2. (cid:1)A3(cid:2)
eff
asthelimitoff(cid:3)1scattering,thef(cid:3)1rainbowangleap- r0
proaches(cid:2)=(cid:7)/2andtherainbowelectricfieldshiftsfrom
For the generic situation where the particle arrives at r
anAiry integral to the derivative of anAiry integral [1], 0
andapositiveradialforceactsonitattemptingtopushit
thus suppressing the main rainbow peak. The supernu-
backout,onehas
merary interference pattern of the f(cid:3)1 rainbow, which
occurs at smaller scattering angles than the main rain- (cid:11)(cid:1)dU /dr(cid:2)(cid:11) (cid:3)0. (cid:1)A4(cid:2)
eff r0
bow peak, evolves into the ripple pattern of the Fresnel
straight-edge field of Eq. (40). The complex ray of the The integral in Eq. (A3) is convergent, the deflection
f(cid:3)1rainbow,whichoccursatangleslargerthanthemain angleisfinite,andtheparticlespendsafiniteamountof
rainbowpeak,evolvesintothesmoothfalloffoftheradia- timeinthevicinityoftheforcecenter.Forthenongeneric
tion shed by the orbiting ray in the classically shadowed situationwheretheparticlegetstor andthereisnora-
0
region.AllthesefeaturesarequalitativelyevidentinFig. dialforceactingonittopushitbackout,onehas
2.ThephysicalopticsmodelofEq.(40),however,provides
(cid:11)(cid:1)dU /dr(cid:2)(cid:11) =0. (cid:1)A5(cid:2)
only an approximation to the scattered field in the tran- eff r0
sition region. For ka=350, the main peak of the Fresnel
The deflection angle of Eq. (A3) diverges logarithmically
straight-edge pattern should occur at (cid:2)=86.6° according
atthelowerlimitofintegration[5,6].Theparticleiscap-
to Eq. (40). But in Fig. 2 it actually occurs at (cid:2)=68.8°.
turedbytheforcecenterinanorbitthatslowlydecaysto-
Similarly, the periodicity of the first number of oscilla-
wardr=r ,anditspendsaninfiniteamountoftimeinthe
tions observed in Fig. 2 for (cid:2)(cid:1)68.8° is a factor of 1.48 0
vicinity of the force center. Equation (A5) is the orbiting
greater than the periodicity predicted from Eq. (40). As
condition,anditcorrespondstoaparticlewhoseenergyis
the sphere size parameter further increases toward the
suchthatU islocallyquadraticinthevicinityofr and
geometrical optics limit, these differences should slowly eff 0
hasarelativemaximumthere.Finally,ifthepotentialis
decrease.
moreslowlyvaryinginthevicinityofr sothat
Finally,inFig.2thediffractionstructureisevidentfor 0
(cid:2)(cid:1)10°, and the radiation shed by waves orbiting in all (cid:11)(cid:1)dU /dr(cid:2)(cid:11) =(cid:11)(cid:1)d2U /dr2(cid:2)(cid:11) =0, (cid:1)A6(cid:2)
planes of incidence constructively interferes to form a eff r0 eff r0
broad glory enhancement in the transmitted intensity at thedivergenceofthedeflectionangleatthelowerlimitof
(cid:2)(cid:9)(cid:7). The fine oscillatory structure for (cid:2)(cid:3)100° is due to integrationinEq.(A3)isstrongerandthedecayoftheor-
interference of orbiting radiation shed by the ray with bittowardr=r isslower.Equation(A6)correspondstoa
0
grazing incidence at the top of the sphere and the coun- particle whose energy is such that the effective potential
JamesA.Lock Vol.25,No.12/December2008/J.Opt.Soc.Am.A 2987
is locally cubic in the vicinity of r0 and has its inflection lim Un(cid:1)kr(cid:2)(cid:12)sin(cid:3)N(cid:1)(cid:13)(cid:2)kr−(cid:5)n(cid:4). (cid:1)B2(cid:2)
pointthere. kr→(cid:13)
Thesefunctionsareradialstandingwaves.Let
APPENDIX B. DEBYE SERIES FOR A Xn(cid:1)1/2(cid:2)(cid:1)kr(cid:2)=Fn(cid:1)kr(cid:2)±iUn(cid:1)kr(cid:2) (cid:1)B3(cid:2)
SCATTERING BY A RADIALLY
be the radially outgoing (1) and incoming (2) waves
INHOMOGENEOUS SPHERE
formed from these standing waves. Similarly, let the lin-
In this appendix the Debye series for scattering by a ra- early independent solutions of the second-order differen-
dially inhomogeneous sphere of radius a with the arbi- tial equation of Eq. (7) for the TM polarization that are
traryrefractiveindexprofileN(cid:1)r(cid:2) inanexternalmedium well behaved and singular at the origin be G (cid:1)kr(cid:2) and
n
with refractive index N=1 is derived and compared with V (cid:1)kr(cid:2),where
n
two different Debye series decompositions for a
multilayersphereapproximatingN(cid:1)r(cid:2).Theinteractionof lim Gn(cid:1)kr(cid:2)(cid:12)(cid:1)kr(cid:2)n+1,
each incident partial wave with the radially inhomoge- kr→0
neoussphereisseenheretobedecomposedintoasumof
lim G (cid:1)kr(cid:2)(cid:12)cos(cid:3)N(cid:1)(cid:13)(cid:2)kr−(cid:5)(cid:4), (cid:1)B4(cid:2)
diffraction, external reflection from the sphere surface, n n
kr→(cid:13)
and transmission through the sphere surface following p
−1 internalreflectionsfromthesurfacewithp=1,2,3....
lim V (cid:1)kr(cid:2)(cid:12)(cid:1)kr(cid:2)−n,
This decomposition for a radially inhomogeneous sphere n
kr→0
merits a number of comments. Whenever a wave is inci-
dentonaninterfacebetweentwomediawithdifferentre- lim V (cid:1)kr(cid:2)(cid:12)sin(cid:3)N(cid:1)(cid:13)(cid:2)kr−(cid:5)(cid:4), (cid:1)B5(cid:2)
n n
fractive indices, part of the wave is transmitted through kr→(cid:13)
the interface and part is reflected by it. Thus as an inci-
and where we have additionally assumed that (cid:1)dN/dr(cid:2)
dent wave penetrates into a radially inhomogeneous 0
=0.Againlet
sphere,itcontinuallyshedsnewreflectedwavesatevery
point along its trajectory. A portion of these reflected Z (cid:1)1,2(cid:2)(cid:1)kr(cid:2)=G (cid:1)kr(cid:2)±iV (cid:1)kr(cid:2) (cid:1)B6(cid:2)
waves are in turn reflected over and over again as they n n n
encounter further refractive index changes while propa- be the radially outgoing (1) and incoming (2) traveling
gatingthroughthesphere.Theentiremultiple-scattering waves formed from these standing waves. For scattering
infinite series of all the transmissions and reflections a byahomogeneoussphereofrefractiveindexN,thefunc-
partial wave can make before finally exiting the sphere tionsF andG ,U andV ,andX (cid:1)1,2(cid:2)andZ (cid:1)1,2(cid:2)become
results in the radial functions Fn(cid:1)kr(cid:2) and Gn(cid:1)kr(cid:2) of Eqs. Riccati–nBessel,n Rinccati–Nneumannn, and Ricncati–Hankel
(3) and (7) for the TE and TM polarizations. For scatter- functionsofNkr,respectively.
ing by a finely stratified multilayer sphere that approxi- In analogy to the treatment of [23], let the amplitudes
mates the refractive index profile N(cid:1)r(cid:2), the individual N ,D ,P ,andQ fortheTEpolarizationbedefinedas
n n n n
terms of the multiple-scattering series described above
wereenumeratedandstudiedin[23–25].IntheDebyese- N =F (cid:1)(cid:1)ka(cid:2)(cid:4)(cid:1)ka(cid:2)−F (cid:1)ka(cid:2)(cid:4)(cid:1)(cid:1)ka(cid:2), (cid:1)B7a(cid:2)
n n n n n
ries decomposition for an inhomogeneous sphere pre-
sentedinthisappendix,allthemultiple-scatteringinter- D =F (cid:1)(cid:1)ka(cid:2)(cid:8)(cid:1)ka(cid:2)−F (cid:1)ka(cid:2)(cid:8)(cid:1)(cid:1)ka(cid:2), (cid:1)B7b(cid:2)
n n n n n
actions a partial wave makes within the sphere are
summed implicitly, while only reflections and transmis-
P =U (cid:1)(cid:1)ka(cid:2)(cid:4)(cid:1)ka(cid:2)−U (cid:1)ka(cid:2)(cid:4)(cid:1)(cid:1)ka(cid:2), (cid:1)B7c(cid:2)
sionsatthesurfaceoftheradiallyinhomogeneoussphere n n n n n
arecountedexplicitly.Thederivationproceedsasfollows.
Consider the partial wave radial functions for the TE Q =U (cid:1)(cid:1)ka(cid:2)(cid:8)(cid:1)ka(cid:2)−U (cid:1)ka(cid:2)(cid:8)(cid:1)(cid:1)ka(cid:2). (cid:1)B7d(cid:2)
n n n n n
polarization. Since Eq. (3) is a second-order differential
equation, it has two linearly independent solutions. If ThecorrespondingamplitudesfortheTMpolarizationare
N(cid:1)r(cid:2) is finite both at the origin and at infinity, the well- obtained by replacing Fn(cid:1)ka(cid:2) and Un(cid:1)ka(cid:2) with
abseyhmavpetdotsioclubetihoanvaiotrtheoriginiscalledFn(cid:1)kr(cid:2)andhasthe aNn2d(cid:1)a(cid:2)UGnn(cid:1)(cid:1)(cid:1)kkaa(cid:2)(cid:2) awnidthNG2(cid:1)na(cid:1)(cid:2)(cid:1)Vkan(cid:1)(cid:2)kaan(cid:2),daVndn(cid:1)(cid:1)bkya(cid:2)r,epwlhaecirnegNF2n(cid:1)(cid:1)a(cid:1)(cid:2)kais(cid:2)
evaluated inside the sphere as r→a. The Wronskian of
lim F (cid:1)kr(cid:2)(cid:12)(cid:1)kr(cid:2)n+1, X (cid:1)1(cid:2) andX (cid:1)2(cid:2) andofZ (cid:1)1(cid:2) andZ (cid:1)2(cid:2) isdefinedas
n n n n n
kr→0
W TE(cid:1)ka(cid:2)=i(cid:3)X (cid:1)1(cid:2)(cid:1)ka(cid:2)X (cid:1)2(cid:2)(cid:1)(cid:1)ka(cid:2)−X (cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)X (cid:1)2(cid:2)(cid:1)ka(cid:2)(cid:4)/2,
n n n n n
lim F (cid:1)kr(cid:2)(cid:12)cos(cid:3)N(cid:1)(cid:13)(cid:2)kr−(cid:5)(cid:4), (cid:1)B1(cid:2) (cid:1)B8a(cid:2)
n n
kr→(cid:13)
where(cid:5) isasuitablychosenphasefactor.Theotherlin- WnTM(cid:1)ka(cid:2)=i(cid:3)Zn(cid:1)1(cid:2)(cid:1)ka(cid:2)Zn(cid:1)2(cid:2)(cid:1)(cid:1)ka(cid:2)−Zn(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)Zn(cid:1)2(cid:2)(cid:1)ka(cid:2)(cid:4)/2.
n
earlyindependentsolutionUn(cid:1)kr(cid:2)issingularattheorigin (cid:1)B8b(cid:2)
andhastheasymptoticbehavior
TheseWronskiansarefoundtosatisfy
lim U (cid:1)kr(cid:2)(cid:12)(cid:1)kr(cid:2)−n,
n W TE=N Q −D P , (cid:1)B9a(cid:2)
kr→0 n n n n n
2988 J.Opt.Soc.Am.A/Vol.25,No.12/December2008 JamesA.Lock
N2(cid:1)a(cid:2)W TM=N Q −D P . (cid:1)B9b(cid:2) R 11=−(cid:3)X (cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)ka(cid:2)−X (cid:1)1(cid:2)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4)/
n n n n n n n n n n
Consider the interior of the radially inhomogeneous (cid:3)X (cid:1)2(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)ka(cid:2)−X (cid:1)2(cid:2)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4) (cid:1)B13b(cid:2)
n n n n
sphere to be medium 1 and the exterior to be medium 2.
The partial wave transmission and reflection coefficients fortheTEpolarizationand
at the surface of the radially inhomogeneous sphere for
T 12=−2iW TM/(cid:3)Z (cid:1)2(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)ka(cid:2)−N2(cid:1)a(cid:2)
bothpolarizationsarederivedasfollows.Whenaradially n n n n
incomingpartialwaveinmedium2withtheradialfunc- (cid:9)Z (cid:1)2(cid:2)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4), (cid:1)B14a(cid:2)
tion (cid:15)(cid:1)2(cid:2)(cid:1)kr(cid:2) is incident on the surface of the radially in- n n
n
homogeneoussphere,aportionT 21istransmittedatthe
surfaceofthespherewiththeradnialfunctionX (cid:1)2(cid:2)(cid:1)kr(cid:2) or Rn11=−(cid:3)Zn(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:15)n(cid:1)1(cid:2)(cid:1)ka(cid:2)−N2(cid:1)a(cid:2)Zn(cid:1)1(cid:2)(cid:1)ka(cid:2)(cid:15)n(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4)/
n
Z (cid:1)2(cid:2)(cid:1)kr(cid:2), and a portion R 22 is reflected from the surface (cid:3)Z (cid:1)2(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)ka(cid:2)−N2(cid:1)a(cid:2)Z (cid:1)2(cid:2)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4)
n n n n n n
back into medium 2 with the radial function (cid:15)(cid:1)1(cid:2)(cid:1)kr(cid:2).
n (cid:1)B14b(cid:2)
Matching the boundary conditions of the various compo-
nentsofE andB atthesurfaceofthespheregives fortheTMpolarization.Forbothpolarizations,thetrans-
missionandreflectioncoefficientsareoftheform
T 21=−2i/(cid:3)X (cid:1)2(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)ka(cid:2)−X (cid:1)2(cid:2)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4),
n n n n n T 12=−2iW /(cid:3)(cid:1)N +Q (cid:2)+i(cid:1)D −P (cid:2)(cid:4), (cid:1)B15a(cid:2)
(cid:1)B10a(cid:2) n n n n n n
R 11=(cid:3)(cid:1)−N +Q (cid:2)−i(cid:1)D +P (cid:2)(cid:4)/(cid:3)(cid:1)N +Q (cid:2)
n n n n n n n
R 22=−(cid:3)X (cid:1)2(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:15)(cid:1)2(cid:2)(cid:1)ka(cid:2)−X (cid:1)2(cid:2)(cid:1)ka(cid:2)(cid:15)(cid:1)2(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4)/
n n n n n +i(cid:1)D −P (cid:2)(cid:4). (cid:1)B15b(cid:2)
n n
(cid:3)X (cid:1)2(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)ka(cid:2)−X (cid:1)2(cid:2)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4) (cid:1)B10b(cid:2)
n n n n The partial wave scattering amplitudes can then be
shown,afterareasonableamountofalgebra,tobeofthe
fortheTEpolarizationand
form (cid:13) (cid:14)
T 21=−2iN2(cid:1)a(cid:2)/(cid:3)Z (cid:1)2(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)ka(cid:2)−N2(cid:1)a(cid:2) a
n n n n =N /(cid:1)N +iD (cid:2)
(cid:9)Z (cid:1)2(cid:2)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4), (cid:1)B11a(cid:2) bn (cid:15)n(cid:16) n n
n n
1
= (cid:3)1−R 22−T 21T 12/(cid:1)1−R 11(cid:2)(cid:4)
Rn22=−(cid:3)Zn(cid:1)2(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:15)n(cid:1)2(cid:2)(cid:1)ka(cid:2)−N2(cid:1)a(cid:2)Zn(cid:1)2(cid:2)(cid:1)ka(cid:2)(cid:15)n(cid:1)2(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4)/ (cid:15)2(cid:16)(cid:17) n n n n (cid:18)
(cid:3)Z (cid:1)2(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)ka(cid:2)−N2(cid:1)a(cid:2)Z (cid:1)2(cid:2)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4) 1 (cid:6)(cid:13)
n n n n = 1−R 22− T 21(cid:1)R 11(cid:2)p−1T 12 .
(cid:1)B11b(cid:2) 2 n p=1 n n n
(cid:1)B16(cid:2)
fortheTMpolarization.Forbothpolarizations,thetrans-
missionandreflectioncoefficientsareoftheform ThisistheDebyeseriesexpansionforaradiallyinhomo-
geneous sphere. The first term on the right-hand side of
T 21=−2i(cid:24)/(cid:3)(cid:1)N +Q (cid:2)+i(cid:1)D −P (cid:2)(cid:4), (cid:1)B12a(cid:2) Eq. (B16) represents diffraction of the plane wave and is
n n n n n
independent of the composition of the sphere. Thus, this
term is identical for both a homogeneous sphere and a
R 22=(cid:3)(cid:1)−N +Q (cid:2)+i(cid:1)D +P (cid:2)(cid:4)/(cid:3)(cid:1)N +Q (cid:2)
n n n n n n n modified Luneburg lens. The second term is external re-
+i(cid:1)D −P (cid:2)(cid:4), (cid:1)B12b(cid:2) flection from the surface of the radially inhomogeneous
n n sphere. Whenever a wave reflects from a surface, the re-
flection amplitude depends on the details of the material
(cid:24)=1 forTE, the wave is reflecting from, even though the reflected
wave never enters the material. This dependence is de-
scribedbytheinteriorfunctionsX (cid:1)2(cid:2)andZ (cid:1)2(cid:2)evaluated
=N2(cid:1)a(cid:2) forTM. (cid:1)B12c(cid:2) n n
at the surface of the sphere in Eqs. (B10b) and (B11b).
Similarly,whenaradiallyoutgoingpartialwaveinme- The last term in Eq. (B16) represents transmission
dium1withtheradialfunctionX (cid:1)1(cid:2)(cid:1)kr(cid:2)orZ (cid:1)1(cid:2)(cid:1)kr(cid:2)isin- through the sphere following p−1 internal reflections at
cident on the surface of the rnadially inhnomogeneous the surface. For example, the term Tn21Tn12 corresponds
sphere,aportionT 12istransmittedatthesurfacetothe to a partial wave crossing the sphere surface, making an
n
outside with the radial function (cid:15)(cid:1)1(cid:2)(cid:1)kr(cid:2), and a portion infiniteseriesoftransmissionsandreflectionsinside,and
R 11isreflectedfromthesurfacebanckinsidewiththera- when encountering the sphere surface for the second
dinal function X (cid:1)2(cid:2)(cid:1)kr(cid:2) or Z (cid:1)2(cid:2)(cid:1)kr(cid:2). Boundary condition time,istransmittedout.Similarly,thetermTn21Rn11Tn12
n n in Eq. (B16) corresponds to a partial wave crossing the
matchingatthesurfaceofthespheregives
sphere surface, making an infinite series of reflections
T 12=−2iW TE/(cid:3)X (cid:1)2(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)ka(cid:2)−X (cid:1)2(cid:2)(cid:1)ka(cid:2)(cid:15)(cid:1)1(cid:2)(cid:1)(cid:1)ka(cid:2)(cid:4), and transmissions inside the sphere, eventually return-
n n n n n n ing to the surface where it is reflected back inside, mak-
(cid:1)B13a(cid:2) ing another infinite series of transmissions and reflec-