Analysis of the resonant scattering of light by cylinders at oblique incidence

Analysis of the resonant scatteringof light by cylinders at oblique incidence

Luiz Gallisa Guimaraes and Jose Paulo Rodrigues Furtado de Mendonca

We study the resonant scattering of light at oblique incidence by dielectric uncoated and coated cylinders.We develop a stable algorithm that permits us to calculate the resonances of a single dielectric cylinderas the tilting angle varies. This algorithm is based on semiclassical formulas for the distance betweenresonances. Results show that the resonances and the resonant electromagnetic energy flux near andinternal to the cylindrical surface are highly sensitive to variations in the tilting angle. In addition, thecoating effects are studied for scattering of light at oblique incidence by an infinite, perfect cylindricalconductor coated by a dielectric layer. In this case the resonance calculations show a peculiar similaritybetween this light scattering and atomic-molecular scattering. A physical interpretation for theseeffects is given, based on an analogy of optics and quantum mechanics. © 1997 Optical Society ofAmerica

Key words: Resonances of a cylinder, light scattering, molecular predissociation.

1. Introduction

The properties of light scattered by cylindrical obsta-cles have been used as a tool in many areas in phys-ics. For instance, guided modes of optical fibershave been used to excite the whispering gallerymodes of dielectric spheres1,2 as well as to illuminatea sample in near-field scanning optical microscopy.3In these investigations it is fundamental to know theresonance location and properties of the evanescentelectromagnetic field.4 In the present paper westudy the scattering of plane waves by tilted single-material and coated cylinders. We have an interestin the role of the tilting angle on the resonance loca-tion and width as well as the electromagnetic energypropagation along the fiber. Wait5 was the first tosolve the problem of the scattering of the plane waveby a tilted dielectric cylinder. Scattering of light bya composite cylinder at normal and oblique incidencewas studied by many authors during the past threedecades.6–12 For review of the mathematical and

L. G. Guimaraes is with Departamento de Fısica Nuclear, Insti-tuto de Fısica, Universidade Federal do Rio de Janeiro, C.P. 68528,21945-970 Rio de Janeiro, Brazil; J. P. R. Furtado de Mendonca iswith the Departamento de Fısica, Universidade Federal de Juiz deFora, 36036-330 Juiz de Fora–MG Brazil.

Received 12 November 1996; revised manuscript received 4March 1997.

0003-6935y97y308010-10$10.00y0© 1997 Optical Society of America

8010 APPLIED OPTICS y Vol. 36, No. 30 y 20 October 1997

numerical methods and the state of art in this re-search, we refer the reader to the paper by Kai andD’Alessio.13 To introduce some simplifications wefirst study the resonance calculation for scattering ofa plane wave at oblique incidence by a single-material ~uncoated! cylinder. We then comparethese results with those for the effects of a coating ona cylindrical surface. This paper is organized in thefollowing manner. In Section 2 for oblique inci-dence, we introduce the cylindrical multipole expan-sion theory ~CMET! and use this method to obtain thetranscendental equation related to the resonances ofa single-material dielectric cylinder. In Section 3 wedevelop a semiclassical theory for resonant modes.This theory helps us to give a physical interpretationfor these modes and to understand the role of tiltingangle in the resonant scattering. In Section 4 weanalyze the effects of coating on resonant modes.We calculate the resonances as the tilting angle andthe coating vary. A physical interpretation for theseresonant modes is given, too. In Section 6 we sum-marize our main results and discuss some possibleapplications.

2. Cylindrical Multipole Expansion Theory forResonances of a Single Cylinder

In the geometry for the scattering of a plane wave bya dielectric cylinder at oblique incidence we assumethat the incident vector wave kinc makes an angle fwith a vector normal to the cylindrical surface. Thecylindrical coordinate ~r, u, z! is adopted, and the

cylinder axis is coincident with the z axis. The po-larization of the scattered fields is related to the po-larization of the incident field in the following twocases.14,15 In case I the incident electric field is inthe incident plane ~the plane that contains kinc andthe z axis!. In case II the incident magnetic fieldvibrates in the incident plane. With CMET, for anincident wave the electromagnetic fields are given byan infinite cylindrical multipole series.5,14 The polesof this multipole expansion are related to resonances.The cylindrical multipole coefficients5 are functions

Optical resonances are related to high l values; inthis case finding the solution of Eq. ~1! is a nontrivialcomputational task. In the limit of sharp reso-nances ~b .. w! we can derive two useful formulasfrom Eq. ~1!. The first is an approximation for res-onance position, a transcendental equation; the sec-ond is an explicit formula for the resonance width w.These formulas are written, respectively, as

Xl~b!Zl~b! 5 Fl sin~f!S1v2 2

1u2DG2

1 O~w2!, (4)

w 52b~Xl 1 Zl!

pYl2~v!@@l2 2 v2 2 v ln Yl~v! 2 ~v ln Yl~v!!2#~Xl 1 Zl! 1 8~1 2 v2yu2!#

1 O~w3!, (5)

whose denominator is proportional to the function J.Here we assume that the cylinder is nonmagnetic ~m5 1 in all space! and lies in vacuum, so that, for adielectric cylinder with radius a characterized by re-fractive index N and for an incident vector wave kinc5 k@cos~f!ex 1 sin~f!ez#, these poles are the complexzeros b# of J~b# ! 5 0, with the transcendental functionJ given by5

J~b! ; Xl~b!Zl~b! 2 Fl sin~f!S1v2 2

1u2DG2

, (1)

where we define the size parameters b 5 ka, v 5 bcos~f!, and u 5 b@N2 2 sin~f!2#1y2. The complexfunctions Xl~b! and Zl~b! are written as

Xl~b! ;ln9 Hl

~1!~v!

v2

ln9 Jl~u!

u, (2)

Zl~b! ;ln9Hl

~1!~v!

v2 N2 ln9 Jl~u!

u. (3)

Here ln9 F~x! denotes the logarithmic derivative withrespect to the argument, and Jl and Hl

~1! are thecylindrical Bessel and Hankel functions with integerorder l, respectively. For a given l and angle f thesolutions of Eq. ~1! are complex and can be written asb# 5 b 2 iw, where b is the resonance position and wis the resonance width. Note that for normal inci-dence ~f 5 0! the zeros of Eq. ~1! are the complexsolutions of Xl~b! 5 0 or zeros of Zl~b! 5 0, so in thisparticular case the resonances of a dielectric cylinderare similar to resonances of dielectric sphere ~Mieresonances!.

3. Semiclassical Theory for Resonances of TiltedCylinders

In geometrical optics, if R$N % . 1, then the incidentlight ray converges to the interior of the cylindricalsurface, and internal optical caustics can arise.16 Inthis paper we assume that the cylinder is transparentwith N . 1, so absorption effects are not included.

where we define the real functions

Xl ;ln9 Yl~v!

v2

ln9 Jl~u!

u, (6)

Zl ;ln9 Yl~v!

v2 N2 ln9 Jl~u!

u, (7)

with Yl being the cylindrical Neumman function.Note that Eqs. ~4! and ~5! are written by means of realfunctions with real arguments. To calculate andprovide a physical interpretation for resonances, weuse the well-known analogy of optics and quantummechanics.17–19 In this analogy the radial Hertz–Debye scalar potentials14,20,21 are interpreted as so-lutions of a time-independent Schrodinger-likeequation subject to an effective potential Ueff.16–19

For an incident “particle” having positive “energy” k2

and l being the angular momentum Lz eigenvalue,the effective potential Ueff ~in units such that \ 5

Fig. 1. For a given Lz, angular momentum eigenvalue l, andincidence angle f, the effective potential Ueff for cylinder refractionindex N . 1 and positive energy k2~f!. The external and internalclassical turning points are rext~f! and rint~f!, respectively. Inthis framework resonances are quasi-bound states of the light.

20 October 1997 y Vol. 36, No. 30 y APPLIED OPTICS 8011

2m 5 1! takes the form of an attractive square well ofdepth k2~N2 2 1! plus the repulsive centrifugal bar-rier ~l2yr2! ~see Fig. 1!. In this approach the reso-nances can be interpreted as quasi-bound states oflight. Initially for calculation of the resonance posi-tion and width we can use semiclassical methodssuch as the Bohr–Sommerfeld and Gamow formu-las,22 respectively. For fixed values of l and the an-gle f the resonance position bnj~l, f! and width wnj~l,f! satisfy the following semiclassical equations:

w~unj, l! ; *rint

a

dr~knj2 2 Ueff!

1y2

5 ~n 1 1y4!p 1 tg21$ejN2~f!

3@~l2 2 vnj2!y~unj

2 2 l2!#1y2%,

n 5 0, 1, . . . nmax 2 1. (8)

wnj 5 bnjejN2~f!

~l2 2 vnj2!1y2

~unj2 2 l2!

cos2@w~unj, l!

2 py4#expF22 *a

rext

dr~Ueff 2 knj2!1y2G , (9)

where N~f! [ @N2 2 sin2~f!#1y2ycos~f! and j is thepolarization index, so ej assumes the values e1 5 1~ j 5 1, related to case I! and e2 5 1yN2 ~ j 5 2, relatedto case II!. The integer n is the resonance order~family number!, and nmax is the maximum numberof allowed resonances inside the well. The internaland external classical turning points are rint~l, f! 5~lyunj!a and rext~l, f! 5 ~lyvnj!a. These semiclassi-cal formulas for resonance position and width do nothold in the limit when ul 2 vu # O~l1y3! and uu 2 lu #O~l1y3!. For resonances close to the top of the bar-rier ~broad resonances! a gross estimate for the solu-tion of Eq. ~8! is given by

bn~l, f! < ltg21@Mycos~f!#

M1

~n 1 1y4!

Mp; n < nmax,

(10)

where M [ ~N2 2 1!1y2 and nmax ' l$Mycos~f! 2 tg21

@Mycos~f!#% 2 ~py4!; note that this estimate for nmaxshows that, as f increases, the number of allowedresonances increases, too. For many theoretical andexperimental purposes the calculation of resonanceseparation is useful. For instance, the separationbetween resonances permits the identification ofmany nonlinear optical process such as stimulatedRaman emission and lasing in micrometer-sized cy-lindrical liquid jets.23 In this semiclassical approx-imation Eq. ~8! permits us to calculate some distancesbetween resonances. For instance, for a fixed anglef we can obtain the following separations:


• Separation between resonances with consecu-tive angular moments and the same order and polar-ization,

Dbl ; bnj~l 1 1, f! 2 bnj~l, f!

< bnj

tg21@~unj2 2 l2!1y2yl#

~unj2 2 l2!1y2 . (11)

• Separation between resonances in consecutiveorder and with the same angular moment and polar-ization,

Dbn ; bn11j~l, f! 2 bnj~l, f! <pbnj

Îunj2 2 l2

. (12)

• Separation between resonances with differentpolarizations and the same order and angular mo-ment,

Dbj ; bnj11~l, f! 2 bnj~l, f! < @bnjy~unj2 2 l2!1y2#

3 $tg21@N2~f!~l2 2 vnj2yunj

2 2 l2!1y2#

2 tg21@~N2~f!yN2!~l2 2 vnj2yunj

2 2 l2!1y2#% . (13)

It is interesting to observe that when the incidenceangle f is zero ~normal incidence!, these results re-produce the Mie scattering case.19,24 For the bestlocalization of the resonance position we need an ac-curacy greater than the resonance width; however,the above semiclassical approach does not have suf-ficient precision for this. In other words, the aboveresults as well as other results obtained for Mie scat-tering with Schobe asymptotic formulas for Besselfunctions25 do not have sufficient accuracy for calcu-lating the resonance position and width.19 To thisend we need to use Olver uniform asymptotic expan-sion ~UAE! formulas for Bessel functions.19,26,27

However, we can use the above semiclassical resultsas an initial guess for a more sophisticated numericalmethod to solve Eq. ~4! based on UAE’s. In thepresent study, for resonance calculation with UAE’swe develop an algorithm based on a downward recur-rence method applied to the resonance order n. Forgiven N, l, j, and angle f we must perform the fol-lowing steps:

~i! Start from relation ~10! with n 5 nmax.~ii! Use the above estimate for resonance position

as an initial guess in some numerical method to solvethe semiclassical transcendental equation ~8!.

~iii! Again, use this last semiclassical solution asthe best improved initial guess to solve Eq. ~4! nu-merically.

~iv! If the desired resonance is of order n differentfrom nmax, use the distance formula ~12! in a down-ward direction obtain an estimate for resonance po-sition with n 5 nmax 2 1 and return to solve thesemiclassical transcendental equation ~8! in step ~ii!.This loop continues until the resonance position withorder n is given as a solution of eq ~4!.

~v! Then we obtain the related resonance widthwith this last result in Eq. ~5!.

To avoid some numerical instability it is advisableto use a numerical code based on UAE’s for Besselfunctions in steps ~iii! and ~iv!. Using this algo-rithm, we calculated some resonances of a tilted cyl-inder as the angle f varies. In the same way othersimilar algorithms can be derived by use of any otherresonance distance formula above.

4. Behavior of the Resonances and theElectromagnetic Fields

For a dielectric tilted cylinder, Figs. 2–4 show theresonance position and width as the incidence angle f~in degrees! varies. In these figures the symbol jn

l

denotes a resonance related to angular momentum l,polarization j of the incident field ~case I or II!, andorder n. We can see from Figs. 2–4 that, for anyorder n and for both polarizations, as f increases, thevalue b of the resonance position increases, too.This is in accordance with the fact that the depth ofthe well k2~f!~N2 2 1! increases as f increases.However, from Figs. 3 and 4 we note that for bothpolarizations, as f varies, the width of the resonancedecreases rapidly as this angle increases. To under-stand this apparent contradiction, we need to studythe behavior of the classical turning points. We ob-serve in Fig. 5 that the internal turning point rint~l,f! is a function that varies slowly with f. However,the external turning point rext~l, f! varies quicklyas f assume values greater than a critical anglefc ' sin21@~N4 2 N2 1 1!1y2 2 ~N2 2 1!#1y2. Thisimplies that the size of the barrier to be tunneled in-

Fig. 2. For N 5 1.45 and l 5 65, the resonance position b as theincidence angle f varies. The symbol jn

l denotes a resonance re-lated with polarization j, Lz angular momentum l, and resonanceorder n. The dashed and solid curves, related to resonances I1

65

and II165, respectively, are obtained through solution of Eq. ~4! by

means of a numerical algorithm based on the between-resonancedistance formula ~see Section 3!. Crosses ~1! are the exact resultsobtained by solution of Eq. ~1!. Note that for both polarizationsthe value of the resonance position b increases as f increases.Because the angle f increases, the height of the effective barrierincreases.

creases rapidly for angles greater than fc, so that theresonance width decreases quickly, too. Observethat the angle fc is resonance independent; in otherwords, fc depends only on the refraction index N.Another physical interpretation of the critical anglefc can be given if we observe that the density ofresonant modes in a strip f and f 1 Df has a max-imum at fc ~see Fig. 6!. We show below that this

Fig. 3. For N 5 1.45, order n 5 1, and l 5 65, the resonance widthw for both polarizations I ~dashed curve! and II ~solid curve! as theincidence angle f varies. These results are obtained with Eq. ~5!.Note that the resonance width decreases very quickly as f in-creases ~inset!.

Fig. 4. For polarization I, N 5 1.45, l 5 61, and consecutiveorders n 5 0, . . . , 3, the behavior of the resonance position andwidth as the incidence angle f varies. The left axis ~dottedcurves! shows the resonance positions for consecutive orders; notethat in this case the distance between resonances with consecutiveorders is not highly sensitive to f variations. This agrees withdistance formula ~12! and is the goal in our algorithm for resonancecalculation. The right axis ~solid curves! shows ln~w!; note thatthe resonance width of any order collapses as f 3 90°.


fact implies some constraints on electromagnetic en-ergy flux along the fiber.

For both practical and theoretical reasons a quan-titative study of internal and near fields as the inci-dence angle varies is desired. To this end we begindefining the average source function S in the follow-ing suitable form28,29:

S~ j!~r! ; ~1y2piEinc~ j!i2!

3 S*0

2p

duuEr~ j!~r, u!u2 1 *

0

2p

duuEz~ j!~r, u!u2D

; S'~ j!~r! 1 Sz

~ j!~r!, (14)

where iEinc~ j!i2 is the intensity of the incident field

with polarization j. It is interesting to observe thatthe electromagnetic energy flux in the z direction isrelated to S'

~ j!, while the electromagnetic energy fluxin directions perpendicular to the cylinder axis isrelated to Sz

~ j!. For normal incidence, S~II! 5 Sz~II!;

thus we identify these modes as orbiting modes be-cause in this particular case we have an analogybetween the light scattering by cylinders andspheres. For f Þ 0 a fraction of the incident energyproportional to S'

~ j! is transferred for guided modesalong the z direction. For several resonance condi-tions, Fig. 7~a! shows the sources functions Sz and S',respectively, for normal incidence and f 5 2°. Onthe basis of an analogy of scattering by a cylinder atnormal incidence to Mie scattering, we believe thatfor small values of f the height of the peak in sourcefunctions should be inversely proportional to reso-nance width,16 so sharper resonance would concen-trate the highest electromagnetic fields close to thecylindrical surface. In general for a resonance with

Fig. 5. For resonance I361, the internal rint ~dashed line! and the

external rext ~solid curve! classical turning points as the incidenceangle f varies. Note that rint is very close to the cylinder radiusa, so that the internal electromagnetic field has a small region tooscillate. On the other hand, for angles f greater than a criticalangle fc ' sin21 @~N4 2 N2 1 1!1y2 2 ~N2 2 1!#1y2 the externalturning point rext is greater than the cylinder radius. In this casethe barrier to be tunnelled is very large. So extremely narrowresonances can be excited for angles f . fc.


order n, the source function has n 1 1 peaks insidethe effective well Ueff ~see Fig. 7!. These peaks cor-respond to regions of caustics inside the dielectriccylinder. The occurrence of these caustics is limitedto the strip @ayN~f!# # r # a. In the limit f3 py2,N 3 `, so caustics related to grazing rays can arisecloser to the cylinder axis than caustics related toresonance at normal incidence ~f 5 0°!. However,as the incidence angle increases, an energy transferbetween orbitinglike and guidedlike modes can occur.In this case both source functions S'

~ j! and Sz~ j! can

exhibit points of maximum. For instance, denotingthe source function maximum as S'~r'

max! [ S'max and

Sz~rzmax! [ Sz

max, we note that these peaks are sepa-rated by a distance of Dr [ rz

max 2 r'max ' ~printy2l!,

so that guidedlike modes are closest to the cylinderaxis, and orbitinglike modes are closest to the cylin-drical surface. For resonances related to a givenfixed l the distance Dr is not highly sensitive to fvariations ~see left axis in Fig. 8~a!, since the internalturning point rint is a slowly varying function of f ~seeFig. 5!. The right axis in Fig. 8~a! shows ~for reso-nant conditions! the ratio between S'

max and Szmax as

f varies, so we observe that as f increases, it leads anincreasing energy transfer from orbitinglike modes toguidedlike modes. However, the resonance widthdecreases as the incidence angle f increases ~see Fig.3!. Note that the values of the peaks of the maxi-mum source functions S'

max and Szmax increase as f

varies until the critical angle fc is reached ~see Figs.8~b! and 8~c!#, so that for f . fc it does not hold thatthe height of the peak in source functions is inverselyproportional to resonance width. Otherwise, for f .fc we see that a small perturbation in the value of theangle f introduces great variations into the value ofthe external turning point rext ~see Fig. 5!. Conse-quently, for f . fc the incident electromagnetic field~plane wave! is not very efficient in exciting the res-

Fig. 6. For resonances II361, II1

65 and I165, the derivative of the

resonance position b in relation to incidence angle f. Note that,independent of the angular momentum, resonance order, and po-larization, all these derivatives have a maximum at the samepoint, f 5 fc. The main dependence of fc is only on the refractiveindex N.

Fig. 7. For several resonances, the normalized source functions Sz and S' for regions internal to and near the cylinder surface. Notethat for a resonance with order n, the source function has n 1 1 peaks. Since the value of the resonance width increases as the order nincreases, these graphics suggest that for close to normal incidence the heights of the peaks in the source function increase as the resonancewidth decreases.

onant modes, since, in this circumstance, the width ofthe barrier to be tunneled increases very quickly. Inaddition, for a given l and on-resonance conditions,the external evanescent field can propagate in radialdirections limited to the strip a # r # rext 2 al22y3,so that the width of this external caustic region in-creases very quickly for f . fc, because, in this sit-

Fig. 8. ~a! The left axis shows ~for N 5 1.45 and on-resonance II020

conditions! the internal points r'max ~circles! and rz

max ~squares!,where S' and Sz have maximum values. Note that these pointsare not highly sensitive to f variations. The right scale shows theratio S'

maxySzmax ~dashed curve! between maximum values of both

components of the source function. Notice that this ratio in-creases as f increases. ~b! and ~c! S'

max and Szmax ~in logarithm

scale!, respectively, as the incidence angle f varies. Note that thevalues of these peaks decrease for f . fc.

uation, the external turning point rext is greater thanthe cylinder radius a ~see Fig. 5!. Now we study thecoating effects.

5. Resonances of a Tilted Coated Cylinder

We study the scattering of light at oblique incidenceby a perfectly metallic infinite cylinder coated with atransparent dielectric infinite cylindrical layer.6,8,11

We assume that the refraction index of the layer isN . 1, the inner radius of the metallic cylinder is b,and the outer radius related to dielectric layer is a.For oblique incidence and within the framework ofthe CMET the resonances of this metallic-coated cyl-inder are related to the complex zeros of the 6 3 6determinant.12 For a given l the transcendentalequation derived from this determinant can be suit-ably expanded as

P~b! 5 J~b! 1P~b!

T~b! FJl~g!

Yl~g!G 1 O@@Jl~g!yYl~g!#2#· · ·

5 0, (15)

where T and P are transcendental functions12 andwe define the size parameter g 5 ~bya!u. Noticethat in the above equation, as b ,, a the ratioJl~g!yYl~g! goes to zero, so that in this limit thistranscendental equation is mapped onto the tran-scendental equation related to the single cylindercase. In the geometrical optics approximation thecaustics region in the radial direction occurs forayN~f! , r , aN~f!, the ratio ay@bN~f!# playing animportant role in the scattering analysis. In addi-tion, we hope that as the ratio ayb decreases, the


thickness of the effective well will decrease, too. Inthe particular case of a metallic cylinder the reso-nant modes cannot propagate; so as the ratio aybgoes to 1, we expect the value of the resonant levelto increase. To better understand the behavior ofthese resonances, we use again the semiclassicalarguments. In this particular case of the metalliccylindrical core coated by a dielectric cylindricallayer the effective potential is analyzed in two dif-ferent situations ~see Fig. 9!. In the first situationthe ratio ayb . N @Fig. 9~a!#, so the inner metalliccylinder does not intersect the caustics regions, im-plying that the resonances are similar to the natu-ral modes of the single-material dielectric cylinderand that the size parameters satisfy the inequalityg , v , l , u. In the other case, for ayb , N @Fig.9~b!# there are two different possibilities to be an-alyzed. The first occurs for an impact parametersuch that reflection on an inner metallic cylinder be-comes important. This situation is related to sizeparameters that satisfy the inequality v , g , l , u.The second situation is more critical. In this casethe thickness of the coating is very small, so that onlysurface waves can propagate. This situation is char-acterized by size parameters v , l , g , u. In Fig.9~b! we define for a given l and f the transitionregion D as being the resonances for which ug 2 lu 'O~l1y3!. This region is peculiar because the bound-ary conditions and reflection on an inner metalliccylinder create some interesting resonant effects, aswe will see below. To calculate the resonances of acoated cylinder, we developed a numerical method,denoted the evaporation algorithm, that is based onuse of the resonances of a single-material cylinder.More explicitly, if b is a solution of J~b! 5 0, an

Fig. 9. Effective potential Ueff for a metallic cylinder of radius bcoated with a dielectric cylinder of radius a. ~a! when b , rint ,a , rext and the ratio ayb . N, the resonances are similar to theresonances of a single dielectric cylinder. ~b! when b 5 rint , a ,rext and the ratio ayb , N, reflections on the surface of the innermetallic cylinder play an important role, so that resonances in theD transition region are highly sensitive to variations in the layerthickness.


initial guess b~0! to solve Eq. ~15! numerically isgiven by

b~0! < b 1P~b!

J9~b!T~b! FJl~g!

Yl~g!G , (16)

where the prime denotes differentiation with respectto the argument. Using this algorithm, we calculatesome resonances for normal and oblique incidence asthe ratio ayb varies. In what follows we presentthese results. In Fig. 10, for N 5 1.45 and normalincidence, we calculated the resonance position in therange 52.5 , b , 57. It is interesting to note that,for both cases I and II, as the thickness of the dielec-tric layer decreases in the limit ayb3 1, the values ofthe resonance position increase because the width ofthe effective well decreases as the ratio ayb decreases@see Fig. ~9!#. In addition, Fig. 10 shows that severalresonances are crossing curves, and resonances re-lated to polarization I have a minimum value. Theinset in Fig. ~10! shows that this minimum valueoccurs for resonances in which the size parameter g isclose to l. This peculiar behavior for resonances asthe ratio ayb varies resembles the electronic spec-trum of a strongly polarized diatomic molecule.30–33

All of these features of the resonances as the coatingthickness varies suggest that resonance position ishighly sensitive to slow variations in the ratio ayb.This fact is confirmed in Fig. 11, where we plot theextinction efficiency factor Qext for a composite cylin-der34,35 as the size parameter b varies. Notice inthis figure that slow variations in the ratio ayb candrastically change the positions between resonancesI260 and II2

60. Other interesting features of reso-nances in a coated cylinder appear in other directions

Fig. 10. For a composite cylinder and normal incidence ~f 5 0!,several resonance positions in the range 52.5 , b , 56.5 as theratio ayb varies. Note that many resonances present crossoverpoints, and resonances related to I polarization have a minimumvalue. These resonances resemble the electronic spectrum of adiatomic molecule. On other words, resonances related to II po-larization are antibonding orbitallike, and resonances related to Ipolarization are bonding orbitallike. The inset shows that theminimum value in resonances related with I polarization occurs forsize parameters for which g ' l.

of the incident field. For instance, small variationsof the normal incidence can introduce changes in thebehavior of the resonances. For f 5 5° and refrac-tive index of the coating layer N 5 1.45, Fig. 12 showsthe position of the resonance I1

65 and II165 as the ratio

ayb varies. Note that in this case the resonances donot exhibit crossing curves. Now the resonance po-sitions are similar to the molecular predissociationspectrum of a diatomic molecule.36–38 To under-stand this peculiar feature of the resonance, we usethe analogy between optics and quantum mechanics.More precisely, in atomic scattering the molecularpredissociation states can occur as a perturbationsuch as nucleus vibration ~corrections to the Born–Oppenheimer approximation! or spin–orbit interac-tion, which creates a coupling between twodegenerate atomic eigenstates ~crossing curves!. In

Fig. 11. Extinction efficiency factor Qext as the size parameter bvaries for normal incidence ~f 5 0! and N 5 1.45. Note that Qext

is highly sensitive to variations in the layer thickness, so that oncrossover resonance conditions ~solid curve! the related peak issuperimposed.

Fig. 12. For nonnormal incidence ~f 5 5°! the resonance positionb for both polarizations I and II. Note that in this situation thecurves do not exhibit crossing, so that in this case the resonantspectrum resembles a predissociation electronic molecular spec-trum.

general this coupling can remove this degeneres-cence. In this optical scattering case the coupling isintroduced by a mixture of polarizations; in otherwords, for an incidence different from the normal ~fÞ 0! the internal and the scattered fields have bothcomponents of polarization, independent of the polar-ization state of the incident field, so that the reso-nance transcendental equation @see Eq. ~15!# canhave solutions that combine both polarizations.Note that this analogy of optical resonances and mo-lecular crossing or predissociation curves is not a co-incidence but is related to the similarity between theelectromagnetic wave equation and the Schrodingerequation. In both wave equations tunneling andboundary conditions play a fundamental role in thespectrum’s behavior. To understand this analogybetter, we can assume that the inner cylinder is not aperfect conductor but is a good conductor that has acomplex refractive index Ninner with a great imaginarypart I$Ninner%. When the real part R$Ninner% of therefractive index of the inner cylinder is greater thanthe refraction index N of the cylindrical dielectriclayer, the effective potential Ueff is a double-well po-tential that resembles a model of the electronic poten-tial of a diatomically polarized molecule.30,31,36 As theinner cylinder has a strong absorption ~I$Ninner% .. 1!,the two wells are not correlated, so that in this limitUeff is equivalent to the strongly polarized diatomicmolecular potential. Finally, we note that similar ef-fects in the electromagnetic spectrum can occur forlight scattering in other geometries. For instance,resonance shift and resonance split can arise on scat-tering of light by a dielectric sphere near a conductingplane39 and in light scattering by dielectric bi-spheres,40 respectively.

6. Conclusions

In this study, based on semiclassical distance formu-las between resonances @Eqs. ~11!–~13!#, we developan algorithm that allows us to calculate the reso-nance of the dielectric cylinders as the incidence an-gle varies. The resonance calculation shows thatthe resonance position is an increasing function of theincidence angle,41 whereas the resonance width is adecreasing function of this angle ~see Figs. 2–4!.This behavior is related to the changes in the shape ofthe effective potential as the tilt angle varies. As theangle f increases, the height and the width of theeffective barrier increase. Then the resonance posi-tion value b increases, whereas the resonance widthvalue w decreases as f increases. For instance, res-onant grazing rays incident upon a cylindrical sur-face can excite optical resonant modes inside thedielectric cylinder with a mean lifetime greater thanthe mean lifetime related to the equivalent resonantmode excited by normal incident rays. In addition,we showed that for incidence angles greater than acritical angle fc the resonance width is highly sensi-tive to small perturbations of the incidence angle ~seeright axis in Fig. 4! and the peaks of maximum in-tensity of the internal electromagnetic fields havemaximum value at f 5 fc ~see Fig. 7!. This suggests


that the best generation of the optical nonlinear pro-cess in cylindrical wave guides can occur inside thecaustic regions created by resonant modes related toan incidence angle of f 5 fc. Using the analogy ofquantum mechanics and optics, we show that thecoating effects become important for impact param-eters of the order O@ayN~f!# and for resonances re-lated to size parameters in the D transition region~see Fig. 9!. In this case the resonances show a pe-culiar behavior, similar to the electronic spectrum ofa diatomic molecule. For normal incidence resonantmodes related to I polarization are bonding orbital-like, while the resonant modes related to II polariza-tion are antibonding orbitallike ~see Fig. 10!. Fornonnormal incidence the analogy between resonantmodes of a coated, tilted cylinder and the molecularelectronic spectrum is maintained. However, in thissituation the resonances resemble the predissocia-tion molecular electronic spectrum. On the basis ofthese results, we believe that composite, multilay-ered, scattered centers such as coated spheres or cyl-inders can support photon localization,42 because inthis case the effective potential can simultaneouslysupport shape resonances and transmission reso-nances. Therefore we believe that these materialsare suitable for observing photonic band-gap effects.

This work is supported in part by Brazilian agen-cies ~Conselho Nacional de Desenvolvimento Cientı-fico e Tecnologico!, Financiadora de Estudos eProjetos!, Fundacao de Amparo a Pesquisa de MinasGerais!, and Programa Especial de Visitantes. Theauthors are grateful to James A. Lock for sending usRef. 41, 43, and 44.

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Analysis of the resonant scattering of light by cylinders at oblique incidence