Recent progress in light scattering has been the theoryof interaction between an incident arbitrary-shapedbeam and a homogeneous sphere @generalized Lorenz–Mie theory ~GLMT!; Refs. 1–4 and references therein#,later extended to the case in which the scattering par-ticle is a multilayered sphere,5 leading to many appli-cations ~Ref. 4, for example!. Although the GLMTallows us to deal with arbitrary-shaped beams, themost extensive effort has been devoted to Gaussianbeams, for obvious practical reasons, insofar as laserbeams used in the laboratory for experiments are, atleast approximately, Gaussian. Nevertheless, otherkinds of beams, such as laser sheets6–9 and top-hatbeams,10,11 have been considered too.
Similarly, I recently attacked the problem of inter-action between a Gaussian beam and an infinite cyl-inder and solved it in several steps up to the rathergeneral case in which the cylinder is arbitrarily lo-cated and arbitrarily oriented in Gaussianbeams,12,13 as described within the so-called Davisframework.11–14 The case of Gaussian beams forhigher-order descriptions within the Davis frame-work has also been considered ~no formulation hasyet been published, but is available upon request!.These formulations have recently been used to pro-duce the scattering diagrams in Ref. 15. In the case
The author is with the Laboratoire d’Energetique des Systemes etProcedes, Institut National des Sciences Appliquee de Rouen, Unitede Recherche Associee au Centre National de la Recherche Scienti-fique 230, Complexe de Recherche Interdisciplinaire en Aerother-mochimie, P.O. Box 08, 76131 Mont Saint Aignan Cedex, France.
Received 1 May 1996; revised manuscript received 7 October1996.
4292 APPLIED OPTICS y Vol. 36, No. 18 y 20 June 1997
of the GLMT for spheres, the Davis framework ~usedfor the cylinder in Refs. 12, 13, and 15! provided themost appealing way of describing Gaussian beams.However, as for the GLMT for spheres, it is used tostudy situations in which the cylinder is illuminatedby Gaussian beams with descriptions other than theDavis one, and more generally by arbitrary-shapedbeams. Therefore this paper is devoted to a finalgeneralization in which the cylinder is illuminated byan arbitrary-shaped beam. As far as the cylinder isconcerned, in a classical way, it is embedded in anonabsorbing medium and its material is homoge-nous, isotropic, and nonmagnetic.
When the theory for Gaussian beams within theDavis framework was developed, unexpected difficul-ties were uncovered, as discussed in Refs. 16 and 17.The underlying origin of these difficulties is the sta-tus of the separability theorem18 in cylindrical coor-dinates. Indeed, the usual functions do not allow usto write all solutions of physical interest as a linearcombination of basic generating functions,19 in par-ticular, the case of Gaussian beams described withinthe Davis framework. It was later established thatthe most general framework to be used to write allsolutions of physical interest is the theory of distri-butions.20 In particular, the theory of distributionsis indeed used for the case of Gaussian beams withinthe Davis framework,12,13,15 and, not surprisingly, itmust also be used in the case of arbitrary-shapedbeams considered in this paper.
Therefore an essential ingredient of the generaltheory in this paper is the use of the theory of distri-butions. Experts in light scattering are usually notfamiliar with this theory, which is rather recent.This fact may give a somewhat abstract character tothis paper. However, students nowadays are oftentrained in distributions very early. Therefore, what
can appear to be abstract for senior experts in lightscattering might very well in the future become nat-ural to future researchers. To help the reader who isnot necessarily familiar with distributions, a discus-sion of them is given in Appendix A. This discussionshould be sufficient to allow the reader to understandwhat is going on.
A few comments should be added here. Distribu-tions may be viewed as generalized functions. Exam-ples are provided by the Dirac distribution ~veryfamiliar!, derivatives of the Dirac distribution ~less fa-miliar!, the Heaviside step distribution, and many oth-ers. The theory of distributions provides themathematically precise way to handle such general-ized functions. Final results, however, are expressedin terms of functions, allowing classical implementa-tion in computer programs, as confirmed by the resultsin Ref. 15. Furthermore, I do not claim that the in-teraction between light and cylinders must always beexpressed in terms of distributions. The set of func-tions is actually a subset of the set of distributions,and, for some special incident beams ~like a planewave! or special descriptions of arbitrary-shapedbeams ~like a Gaussian beam described by a plane-wave spectrum!, the theory may be expressed in termsof functions. In such cases, my theory in terms ofdistributions may be simplified to a theory in terms offunctions ~see the discussion of the plane-wave casebelow!. Also, any theory in terms of functions mayconversely be expressed in terms of distributions ~sucha link will be published elsewhere for the case of aplane-wave spectrum description!. But for distribu-tions that provide the most general framework, it isnecessary to use them in the case of arbitrary-shapedbeams, as discussed here.
This paper is organized as follows. The descrip-tion of arbitrary-shaped beams in terms of electricand magnetic fields is given in Section 2. In Section3 Bromwich scalar potentials ~BSP’s! are introducedand a description of the beam is given in terms ofbeam-shape distributions ~BSD’s!, depending on thebeam-shape coefficients ~BSC’s!. Also, the incidentpotentials are expressed in terms of the usual func-tions. The potentials that describe the scatteredwave and the wave inside the cylinder and all fieldexpansions associated with the potentials are intro-duced in Section 4. In Section 5 boundary condi-tions at the surface of the cylinder are considered,allowing us to complete the theory. Section 6 is theconclusion, followed by a brief appendix.
2. Electromagnetic Description of Arbitrary-ShapedBeams
A. Original Cartesian System
We consider a Cartesian coordinate system ~Ouvw!~Fig. 1! and an electromagnetic arbitrary-shapedbeam propagating toward positive w’s in a nonab-sorbing medium. The beam is monochromatic witha time-dependent reading of exp~ivt!, where v is thepulsation of the wave. This time-dependent term isomitted from all formulas below, as is the normal
practice. It is convenient to introduce rescaled coor-dinates:
~U, V, W! 5 ~ku, kv, kw! (1)
where k is the wave number.Although the Cartesian coordinate system ~Ouvw!
may be, in principle, arbitrarily chosen, it is highlyrecommended to choose it in such a way that it inher-ently matches the biggest amount of symmetry presentin the beam in order to obtain a beam description ofmaximum simplicity. For example, let us assumethat the beam is a Gaussian beam. Then the axis Owshould identify with the symmetry axis of the beam~the so-called on-axis case!, and, still better, the originO should also identify with the beam-waist center ofthe beam ~so-called beam-waist-center case!.
Now, the electric-field Ei and the magnetic-field Hicomponents may always ~as discussed below! be writ-ten as
~Eu, Ev, Ew! 5 E0 exp~2iW! (p50
`
(q50
`
(l50
`
3 ~Epqlu , Epql
v , Epqlw !UpVqWl, (2)
~Hu, Hv, Hw! 5 H0 exp~2iW! (p50
`
(q50
`
(l50
`
3 ~Hpqlu , Hpql
v , Hpqlw !UpVqWl, (3)
where E0 and H0 designate the electric and the mag-netic strengths, respectively, exp~2iW! is the propa-gation term associated with the time-dependent termexp~ivt!, and Epql
u , . . . , Hpqlw are amplitude coeffi-
cients. Their values define the beam shaping, i.e.,any shaping may be obtained by specification of theamplitude coefficients.
The amplitude coefficients, however, are not inde-pendent because the field components must satisfyMaxwell’s equations in free space. This conditionimplies eight relations that link the amplitude coef-
Fig. 1. Original and translated Cartesian coordinate systems.OpO 5 ~U0, V0, W0! with rescaled coordinates.
20 June 1997 y Vol. 36, No. 18 y APPLIED OPTICS 4293
ficients21,22:
~q 1 1!Epq11lw 2 ~l 1 1!Epql11
v 1 i~Epqlv 1 Hpql
u ! 5 0, (4)
~l 1 1!Epql11u 2 ~p 1 1!Ep11ql
w 1 i~Hpqlv 2 Epql
u ! 5 0, (5)
~p 1 1!Ep11qlv 2 ~q 1 1!Epq11l
u 1 iHpqlw 5 0, (6)
~p 1 1!Hp11qlu 1 ~q 1 1!Hpq11l
v
1 ~l 1 1!Hpql11w 2 iHpql
w 5 0, (7)
~q 1 1!Hpq11lw 2 ~l 1 1!Hpql11
v 1 i~Hpqlv 2 Epql
u ! 5 0, (8)
~l 1 1!Hpql11u 2 ~p 1 1!Hp11ql
w 2 i~Hpqlu 1 Epql
v ! 5 0, (9)
~p 1 1!Hp11qlv 2 ~q 1 1!Hpq11l
u 2 iEpqlw 5 0, (10)
~p 1 1!Ep11qlu 1 ~q 1 1!Epq11l
v
1 ~l 1 1!Epql11w 2 iEpql
w 5 0. (11)
The reader who is familiar with the problem ofdescribing shaped beams may be surprised by the useof polynomial expansions such as those given in Eqs.~2! and ~3!. I comment on this choice, both frommathematical and physical points of view.
Indeed, amplitude functions in general are notpolynomials. See the example of Gaussian beams inthe Davis formulation.14 Mathematically, the factthat the field components may always be written inthe form of Eqs. ~2! and ~3! greatly relies on theconvergence Weierstrass theorem.23 According tothis theorem, if a function f ~x! is continuous, it ispossible to approximate it by a polynomial and tomake the error of approximation arbitrarily small byincreasing the degree of the approximating polyno-mial. As a result, it is emphasized that Eqs. ~2! and~3! provide a general framework of description asrequired for arbitrary-shaped beams.
The choice of such a description is also physicallymotivated by the history of the GLMT for spheres.Within this framework, the incident beam may be ex-pressed by partial-wave expansions weighted by so-called beam-shape coefficients.1,3 These coefficientsare determined from the mathematical expressions forthe radial electric-field Er and magnetic-field Hr com-ponents in spherical coordinates. Therefore, within ageneral framework, we may apparently be satisfied ifwe just state that the incident arbitrary-shaped beamis defined by Er and Hr, leaving actual expressions forthese fields unspecified. Similarly, if we adopt thesame attitude in the present case, we might just statethat the incident beam is described by the longitudinalelectric-field Ez and magnetic-field Hz components incylindrical coordinates, which are seemingly sufficientto express beam-shape coefficients, again leaving ac-tual expressions for these fields unspecified16 ~see alsoSubsection 3.B!.
To show that this attitude is actually not the mostcorrect one, let us consider the GLMT for sphereswith incident Gaussian beams described within theDavis framework. This framework actually pro-vides successive approximations to Gaussian beams;none of these approximations exactly satisfy Max-well’s equations.11,14,24 We then state that the de-scriptions are not Maxwellian. Therefore the fields
4294 APPLIED OPTICS y Vol. 36, No. 18 y 20 June 1997
Er and Hr also are not Maxwellian. It then happensthat the beam-shape coefficients evaluated from Erand Hr take the form of constant complex numbers asthey should, but are supplemented by additionalterms that retain a functional dependence.25–27
These additional terms are therefore artifacts pro-duced by the departures of the beam descriptionsfrom Maxwellianity. When these artifacts are re-moved, the Maxwellian contributions embodied in Erand Hr, once converted to Cartesian coordinates, in-deed take the form of Eqs. ~2! and ~3!, as extensivelydescribed in Ref. 28. This observation leads us tothe introduction of the arbitrary-shaped beam de-scription of Eqs. ~2! and ~3!, as discussed in Ref. 22.In particular, accounting for a difference of notations,the amplitude coefficients in Ref. 28 indeed satisfyrelations ~4!–~11!.
This discussion shows that, for the GLMT forspheres, it is somewhat illusory to leave Er and Hrunspecified insofar as the incident-beam descriptionused may be not Maxwellian. In many cases, thisattitude actually hides a potential difficulty. Simi-larly, in the present problem, it would be illusory toleave Ez and Hz unspecified; furthermore, when doingso in cylindrical coordinates, the need for the use ofdistributions required for obtaining a general frame-work is not clearly apparent.16 Therefore the arbi-trary beam description used in this paper ismathematically and physically motivated, with theadvantage of being automatically Maxwellian. Inparticular, it is suitable to handle the case of Gauss-ian beams described within a Davis framework.
Besides the example of Gaussian beams within theDavis framework discussed above, we also have thetrivial ~but interesting! example of the plane wavethat is electrically polarized in the u direction, inwhich all field components are 0 except
~Eu, Hv! 5 ~E0, H0!exp~2iW!, (12)
for which all amplitude coefficients are 0 but
E000u 5 H000
v 5 1. (13)
This is a case in which the scattering theory mayeasily be expressed in terms of functions.29,30 Whenexpressed in terms of distributions, the plane-wavedistributions used appear to be proportional to theDirac distribution @see, for example, relation ~77! inRef. 12#, i.e., they incorporate what can be consideredas the most trivial distribution. I claim that anycase that may be expressed in terms of functions mayalso be expressed in terms of trivial distributions, i.e.,distributions proportional to the Dirac distribution.From Ref. 12 ~but see also Ref. 16!, it appears that theplane-wave theory is a special case of the generaltheory developed in this paper.
From these plane-wave-case statements, thereader may correctly infer that a description of theincident beam in terms of a plane-wave spectrumshould also match the general description developedin this paper and that the light-scattering theory soobtained could also reduce to a more classical theory
in terms of functions. Such a link between the gen-eral framework discussed here and a plane-wave-spectrum approach ~as also used in Ref. 15! will begiven elsewhere. I just mention that distributionsappearing in the plane-wave-spectrum case again re-duce to trivial distributions, as expected. Indeed, inthe case of arbitrary-shaped beams, plane-wave-spectrum approaches may be found to be appealing~although they are not free of difficulties!. For thecase of Gaussian beams, see the recent works byLock.31,32 For two-dimensional Gaussian beams ~la-ser sheets!, see Refs. 33–38. It is, however, desir-able to possess a formulation that is general enoughto handle Gaussian beams in the Davis description,which has been so remarkably successful for spheri-cal scatterers: hence the need for distributions andfor the general framework described in this paper,from which all special cases, including plane-wave-spectrum approaches, may be recovered.
B. Translated Cartesian Description
We consider a point Op located on the axis of the infi-nite cylinder. In order to deal with an arbitrary loca-tion of the scatterer, we then introduce a secondCartesian coordinate system ~Opu9v9w9!, which istranslated with respect to the original system ~Ouvw!.With rescaled coordinates, the translation is defined by
OpO 5 ~U0, V0, W0!. (14)
Field components Eu9, . . . , Hw9, in ~Opu9v9w9! withrescaled coordinates U9, V9, W9 are then obtainedfrom field components Eu, . . . , Hw in ~Ouvw! bychanging ~U, V, W! to ~U9 2 U0, V9 2 V0, W9 2 W0!,leading to
~Eu9, Ev9, Ew9! 5 E0 exp@2i~W9 2 W0!#
3 (p50
`
(q50
`
(l50
`
~Epqlu , Epql
v , Epqlw !~U9 2 U0!
p
3 ~V9 2 V0!q~W9 2 W0!
l, (15)
~Hu9, Hv9, Hw9! 5 H0 exp@2i~W9 2 W0!#
3 (p50
`
(q50
`
(l50
`
~Hpqlu , Hpql
v , Hpqlw !~U9 2 U0!
p
3 ~V9 2 V0!q~W9 2 W0!
l. (16)
C. Rotated and Translated Cartesian Description
In order to deal also with arbitrary orientation of thescatterer, we now introduce a Cartesian coordinatesystem ~OPxyz! that is rotated with respect to thetranslated system ~OPu9v9w9!, as sketched in Fig. 2.The unit vector ei that defines the direction of prop-agation of the beam ~Fig. 1! is located in the plane~xOpz!. Axis z is oriented in such a way that theprojection of the unit vector ei on axis z is negative.Similarly, axis x is oriented in such a way that theprojection of the unit vector ei on axis x is negativetoo. We define the oriented angle G as
G 5 ~Op z, 2ei! (17)
With unit vectors ex and ez on ~Opx! and ~Opz!,respectively, we then have
ei 5 2Sex 2 Cez, (18)
where
S 5 sin G, C 5 cos G. (19)
The rotation between ~Opxyz! and ~OPu9v9w9! isthen completely defined if we specify the direction ofthe electric component Eu9. The general situation issplit into two cases, which are ~1! the parallel case inwhich Eu9 is located in the plane ~xOpz! and ~2! theperpendicular case in which Eu9 is located perpendic-ularly to the plane ~xOpz!. Relations for the paralleland perpendicular cases are similar13 in such a waythat it is sufficient to present the results for one caseonly: Let us consider the perpendicular case.
Cartesian field components Ex, . . . , Hz then are
Xx 5 Xv9C 2 Xw9S, (20)
Xy 5 Xu9, (21)
Xz 5 2Xv9S 2 Xw9C, (22)
where X stands for E or H.
D. Description in Cylindrical Coordinates
The cylindrical coordinates ~z, r, w!, as sketched inFig. 3, are introduced here. We then have
Xr 5 XxCw 1 XySw, (23)
Fig. 2. Rotated Cartesian coordinate system ~OPxyz!.
Fig. 3. Cylindrical coordinates. The axis ~Opz! is the cylinderaxis.
20 June 1997 y Vol. 36, No. 18 y APPLIED OPTICS 4295
Xw 5 2XxSw 1 XyCw, (24)
Xz 5 Xz, (25)
where, again, X stands for E or H and
Sw 5 sin w, Cw 5 cos w. (26)
Furthermore, we also have
U9 5 RSw, (27)
V9 5 RCCw 2 SZ, (28)
W9 5 2RSCw 2 CZ, (29)
in which we introduced rescaled coordinates:
~Z, R! 5 ~kz, kr!. (30)
We then obtain the field components in cylindricalcoordinates associated with the rotated and trans-lated Cartesian coordinate system:
Er 5 E0 A (p50
`
(q50
`
(l50
`
~CCw Epqlv 2 SCw Epql
w
1 Sw Epqlu !Apql, (31)
Ew 5 E0 A (p50
`
(q50
`
(l50
`
@SSw Epqlw 2 CSw Epql
v
1 CwEpqlu #Apql, (32)
Ez 5 E0 A (p50
`
(q50
`
(l50
`
~2SEpqlv 2 CEpql
w !Apql (33)
Hr 5 H0A (p50
`
(q50
`
(l50
`
~CCw Hpqlv 2 SCw Hpql
w
1 Sw Hpqlu !Apql, (34)
Hw 5 H0 A (p50
`
(q50
`
(l50
`
~SSw Hpqlw 2 CSw Hpql
v
1 Cw Hpqlu !Apql, (35)
Hz 5 H0 A (p50
`
(q50
`
(l50
`
~2SHpqlv 2 CHpql
w !Apql, (36)
where
A 5 exp@i~RSCw 1 CZ 1 W0!#, (37)
Apql 5 ~RSw 2 U0!p~RCCw 2 SZ 2 V0!
q
3 ~2RSCw 2 CZ 2 W0!l. (38)
3. Incident-Beam Description in Terms of Potentials
A. Bromwich Scalar Potentials
We first briefly recall the theory of Bromwich scalarpotentials ~BSP’s! in cylindrical coordinate systems,which is more extensively discussed in Ref. 16. Alldetails on the easy derivation of this theory are avail-able from a textbook not yet published but availableby request from the author. In the cylindrical coor-dinate system ~z, r, w!, the Pythagorean theorem is
ds2 5 ~e1!2dz2 1 ~e2!
2dr2 1 ~e3!2dw2, (39)
4296 APPLIED OPTICS y Vol. 36, No. 18 y 20 June 1997
where ds is the infinitesimal distance between twopoints P and ~P 1 dP!, where therefore
e1 5 e2 5 1, e3 5 r, (40)
which lead to
e1 5 1,(41)
]
]z Se2
e3D 5 0.
Because Eqs. ~41! are satisfied, the wave may bedescribed in terms of BSP’s. Any solution to Max-well’s equations ~for linear, local, isotropic, and ho-mogeneous media, with other assumptions explicitlydescribed in the above-mentioned textbook! is thesummation of two special solutions, the TM wave~transverse magnetic wave! and the TE wave ~trans-verse electric wave!. The special solutions may befound by first solving a partial derivative equation forTM and TE BSP’s, UTM and UTE, respectively. Thisequation, which is valid for both UTM and UTE, is
]2U]z2 1 k2U 1
1r
]
]rr
]U]r
11r2
]2U]w2 5 0, (42)
where k is the wave number in the considered materialand U stands for either UTM or UTE. Once UTM andUTE are determined, all TM- and TE-field componentsmay be evaluated with the following set of equations:
Ez, TM 5]2UTM
]z2 1 k2UTM, (43)
Er, TM 5]2UTM
]z]r, (44)
Ew, TM 51r
]2UTM
]z]w, (45)
Hz, TM 5 0, (46)
Hr, TM 5ive
r
]UTM
]w, (47)
Hw, TM 5 2ive]UTM
]r, (48)
Ez, TE 5 0, (49)
Er, TE 5 2ivm
r
]UTE
]w, (50)
Ew, TE 5 ivm]UTE
]r, (51)
Hz, TE 5]2UTE
]z2 1 k2UTE, (52)
Hr, TE 5]2UTE
]z]r, (53)
Hw, TE 51r
]2UTE
]z]w, (54)
where v is the wave angular frequency, and m and eare the magnetic and the electric capacities of theconsidered medium, respectively.
The general solution of Eq. ~42! is a linear combi-nation of generating functions G~z, r, w!:
G~z, r, w! 5 1Jm~r!Ym~r!Hm
~1!~r!Hm
~2!~r!2exp~imw!exp~ikgz!, (55)
where m [ Z, ~kg! [ R, Jm~r! and Ym~r! are theBessel functions of the first and the second kind,respectively, Hm
~1!~r! and Hm~2!~r! are the Hankel func-
tions that may be expressed in terms of Jm~r! andYm~r!, respectively, and
r 5 kr~1 2 g2!1y2. (56)
B. Introduction to Beam-Shape Distributions
The incident BSP’s UTMi and UTE
i therefore are a linearcombination of generating functions. They must in-volve Bessel functions of the first kind because, amongJm~r!, Ym~r!, Hm
~1!~r!, and Hm~2!~r!, they are the only ones
that do not diverge at r 5 0. Furthermore, it must beemphasized that the linear combinations have to bewritten in terms of distributions, leading to
UTMi 5
E0
k2 (m52`
1`
~2i!m exp~imw!^ImTM~g!,
3 Jm@R~1 2 g2!1y2#exp~igZ!&, (57)
UTEi 5
H0
k2 (m52`
1`
~2i!m exp~imw!^ImTE~g!,
3 Jm@R~1 2 g2!1y2#exp~igZ!&, (58)
where ImTM~g! and ImTE~g! are the beam-shape dis-tributions ~BSD’s!.
To some extent, expressions ~57! and ~58!, whichcontain distributions, may be a surprise to the readerwho is not necessarily familiar enough with distribu-tions and corresponding notation. This again re-quires a few extra comments and complementaryinformation.
For gaining an extensive enough background todistributions, which may be viewed as generalizedfunctions or as operators working on test functions,the reader may go to Refs. 39–41. Reference 39 ismotivated by the fact that Schwartz is considered thefounder of the theory of distributions. Reference 40provides pedagogic and comprehensive access forphysicists. Beside these books, which are written inFrench, the reader is also directed to Ref. 41. Theneed for distributions has been extensively com-mented on in the introduction and in Subsection 2.A.For the reader to avoid deeply entering the theory ofdistributions, a heuristic discussion is provided inAppendix A. This heuristic discussion, comple-mented by a few statements and expressions given inthe body of this paper, should allow the reader toproceed on an easy enough path.
Then, from relations ~43!, ~49!, ~57! and ~46!, ~52!,~58!, it is found that the BSD’s are determined by thelongitudinal components Ez
i and Hzi of the incident
electric and magnetic fields, according to
^ImTM~g!, ~1 2 g2!Jm@R~1 2 g2!1y2#exp~igZ!&
51
2p~2i!m *0
2p SEzi
E0Dexp~2imw!dw, (59)
^ImTE~g!, ~1 2 g2!Jm@R~1 2 g2!1y2#exp~igZ!&
51
2p~2i!m *0
2p SHzi
H0Dexp~2imw!dw, (60)
where superscript i is reintroduced for the word in-cident ~see comments in Appendix A!.
C. Evaluation of Beam-Shape Distributions
We first consider the TM BSP given by Eq. ~57!, inwhich the TM BSD is given by Eq. ~59!. Because Ez
i
is proportional to exp~iCZ!, as seen in Eq. ~37!, thesupport of the distribution ImTM~g! must be g 5 $C% insuch a way that the left-hand side ~lhs! of Eq. ~59!generates an exp~iCZ! term too. Such a distribu-tion, which takes the form of a single peak with sup-port at g 5 $C%, must be a linear combination of theDirac distribution and its derivatives, with supportsat g 5 $C% too. Therefore we have
ImTM~g! 5 (n50
`
ImTMn d~n!~g 2 C!, (61)
where InTMn ’s are constant complex numbers called
beam-shape coefficients ~BSC’s!. To evaluate theBSC’s, we successively consider the right-hand side~rhs! and the lhs of Eq. ~59!, denoted as Ar and Al,respectively.
Inserting expression ~33! for Ezi into Ar, we obtain
Ar 521
2p~2i!m exp~iCZ!exp~iW0! (p50
`
(q50
`
(l50
`
Epql
3 *0
2p
exp~iRSCw!exp~2imw!~RSw 2 U0!p
3 ~RCCw 2 SZ 2 V0!q~2RSCw 2 CZ 2 W0!
ldw,
(62)where
Epql 5 SEpqlv 1 CEpql
w . (63)
To evaluate the integral, denoted I in Eq. ~67!, wewrite
~RSw 2 U0!p 5 (
r50
p SprD~RSw!
r~2U0!p2r, (64)
~RCCw 2 SZ 2 V0!q 5 (
s50
q
(t50
s
~21!q2tSqsDSs
tD3 ~RCCw!
t~SZ!s2tV 0q2s, (65)
20 June 1997 y Vol. 36, No. 18 y APPLIED OPTICS 4297
~2RSCw 2 CZ 2 W0!l 5 (
u50
l
(v50
u
~21!lS luDSu
vD3 ~RSCw!
v~CZ!u2vW 0l2u, (66)
which lead to
I 5 (r50
p
(s50
q
(t50
s
(u50
l
(v50
u
~21!p2r1q2t1lSprDSq
sDSstDS l
uDSuvD
3 Rr1t1vZs2t1u2vCt1u2vSs2t1vU 0p2rV0
q2sW 0l2uJrtv~RS!,
(67)
where Jrtv~RS! designates an integral:
Jrtv~RS! 5 *0
2p
exp~iRSCw!exp~2imw!Cwt1vSw
rdw. (68)
This integral may be evaluated starting from12,13
I0 5 *0
2p
exp~iRSCw!exp~2imw!dw 52p
~2i!m Jm~RS!,
(69)
I1 5 *0
2p
exp~iRSCw!exp~2imw!Swdw
5 22pm~2i!m
Jm~RS!
RS, (70)
which lead to
Jrtv 52p
~2i!m
1it1v (
w50
ry2 Sry2w DJ m
~t1v12w!~RS!, r even,
(71)
Jrtv 5 22pm~2i!m
1it1v (
w50
~r21!y2 F~r 2 1!y2w G
3FJm~RS!
RS G~t1v12w!
, r odd. (72)
Many of these integrals are explicitly given in Ref.13. It is convenient to introduce the functionsGrtv~RS!, for even or odd r, given by
Jrtv~RS! 52p
~2i!m
1it1v Grtv~RS!. (73)
Inserting Eq. ~73! into Eq. ~67! and then rewriting
4298 APPLIED OPTICS y Vol. 36, No. 18 y 20 June 1997
Ar @Eq. ~62!#, we obtain
Ar 5 2~21!m exp~iCZ!exp~iW0!
3 (p50
`
(q50
`
(l50
`
Epql (r50
p
(s50
q
(t50
s
(u50
l
(v50
u
3~21!p2r1q2t1l
it1v SprDSq
sDSstDS l
uD3 Su
vDRr1t1vZs2t1u2vCt1u2v
3 Ss2t1vU0p2rV0
q2sW0l2uGrtv~RS! (74)
Using
(s50
q
(t50
s
5 (s50
q
(a50
s
, (75)
(u50
l
(v50
u
5 (u50
l
(b50
u
, (76)
we can collect Ar with respect to Z in the form
Ar 5 ~21!m11 exp~iCZ!exp~iW0!
3 (p50
`
(q50
`
(l50
`
(r50
p
(s50
q
(a50
s
(u50
l
(b50
u
ApqlrsaubZa1b, (77)
where
Apqlrsaub 5 ~2U0!p~2V0!
q~2W0!lEpqlS2R
U0DrSp
rDSqsD
3 S ss 2 aD~iR!s2a Cs2aSa
V0s S l
uDS uu 2 bD
3 ~2iR!u2b CbSu2b
W0u Gr, s2a, u2b~RS!. (78)
Relation ~77! is better rewritten as
Ar 5 ~21!m11 exp~iCZ!exp~iW0! (j50
`
AjZj, (79)
where the functions Aj may be readily extracted fromEq. ~77!. For example, A0 is extracted by collectingterms such as ~a 1 b! 5 0, A1 by collecting terms suchas ~a 1 b! 5 1, . . . , etc., which lead to
A0 5 (p50
`
(q50
`
(l50
`
(r50
p
(s50
q
(u50
l
Apqlrs0u0 (80)
A1 5 (p50
`
(q50
`
(l50
`
(r50
p
(s51
q
(u50
l
Apqlrs1u0
1 (p50
`
(q50
`
(l50
`
(r50
p
(s50
q
(u51
l
Apqlrs0u1. (81)
In practice, however, we deal with truncated expres-sions of Ez
i in Eq. ~33!, i.e., expressions in which p, q,and l do not range up to infinity. Such is obviouslythe situation for the trivial plane-wave case, but it isalso the case for Gaussian beams within the Davisframework.12,13 More generally, relying on theWeierstrass theorem, we may truncate the expressions
when we possess a good enough approximation to thebeam description. It then appears that the order oftruncation to be used depends on the actual descrip-tion of the incident beam. In special cases studiedpreviously ~plane wave, plane-wave spectrum, Davisformulation up to the so-called fifth order!, the order oftruncation obtained is small enough to be manageable,although the use of a mathematical assistant like aMaple symbolic computation software may be recom-mended to speed up algebraic manipulations ~forhigher-order Davis beams, for example!.
Then let us assume that q ranges up to Q and l upto L. As we see below, we do not need any trunca-tion with respect to p. Then, from Eq. ~77!, a rangesup to Q and b up to L, in such a way that the degreeof Eq. ~77! with respect to Z ranges up to N 5 Q 1 L.Relation ~79! now is
Ar 5 ~21!m11 exp~iCZ!exp~iW0! (j50
N
AjZj. (82)
We now study the lhs of Eq. ~59!, which, when Eq.~61! is used, becomes
Al 5 (n50
`
ImTMn ^d~n!~g 2 C!, ~1 2 g2!
3 Jm@R~1 2 g2!1y2#exp~igZ!&. (83)
But the theory of distributions tells us that39,40
^d~n!~g 2 C!, f ~g!& 5 ~21!n@ f ~g!#g5C~n! , (84)
which leads to
Al 5 (n50
`
ImTMn ~21!n$~1 2 g2!
3 Jm@R~1 2 g2!1y2#exp~igZ!%g5C~n! . (85)
We set
g~g! 5 ~1 2 g2!Jm@R~1 2 g2!1y2# (86)
to find
Al 5 exp~iCZ! (n50
`
ImTMn ~21!n (
r50
n SnrD~iZ!r dn2r
dgn2r g~g!g5C,
(87)
which, again truncated at the O~ZN! term included, is
Al 5 exp~iCZ! (j50
N
Bj~ImTMj , . . . , ImTM
N !Zj, (88)
where the functions Bj also depend on BSC’s that areexplicitly written as arguments.
Writing the equality between Ar and Al, whichmust be valid at any order with respect to Z, weobtain a set of equations:
BN~ImTMN ! 5 ~21!m11 exp~iW0!AN, (89)
BN21~ImTMN21 , ImTM
N ! 5 ~21!m11
3 exp~iW0!AN21, . . . , (90)
B0~ImTM0 , . . . , ImTM
N ! 5 ~21!m11 exp~iW0!A0. (91)
Solving the equations successively from Eq. ~89!down to Eq. ~91! we then evaluate all the BSC’s indescending order, from ImTM
N down to ImTM0 . This
downward procedure in particular assures us thatthe set of equations is not ill posed.
At this stage, however, the BSC’s are still R depen-dent because Ai’s and Bi’s are R dependent. How-ever, the internal coherence of the theory and aboveall the fact that the beam description exactly satisfiesMaxwell’s equations assure us that this dependenceis only apparent. It can actually be removed by re-peated applications of42
J9m~x! 512 @Jm21~x! 2 Jm11~x!#, (92)
Jm21~x! 1 Jm11~x! 52mx
Jm~x!, (93)
as exemplified in previous works on Gaussianbeams.12,13
For the TE distributions, we start from Eq. ~60! andset
ImTE~g! 5 (n50
`
ImTEn d~n!~g 2 C!. (94)
The TE BSC’s ImTEn are then evaluated quite sim-
ilarly as for the TM BSC’s.
D. Incident Potentials in Terms of Functions
We start with the TM BSP UTMi given by Eq. ~57! in
which we use a truncated BSD:
ImTM~g! 5 (n50
N
ImTMn d~n!~g 2 C!. (95)
From Eqs. ~57!, ~95!, and ~84!, UTMi becomes
UTMi 5
E0
k2 (m52`
1`
~2i!m exp~imw! (n50
N
ImTMn ~21!n
3 $Jm@R~1 2 g2!1y2#exp~igZ!%g5C~n! . (96)
We introduce
jmn ~R! 5
dn
dgn Jm@R~1 2 g2!1y2#g5C, (97)
and, in order to collect with respect to Z, we use
(n50
N
(l50
n
5 (l50
N
(n5l
N
, (98)
yielding
UTMi 5
E0
k2 exp~iCZ! (m52`
1`
~2i!m exp~imw!
3 (l50
N
Zl (n5l
N
ImTMn ~21!nSn
lDiljmn2l~R!. (99)
Similarly, we have
UTEi 5
H0
k2 exp~iCZ! (m52`
1`
~2i!m exp~imw!
3 (l50
N
Zl (n5l
N
ImTEn ~21!nSn
lDiljmn2l~R!. (100)
20 June 1997 y Vol. 36, No. 18 y APPLIED OPTICS 4299
4. Field Expansions
To deal later with boundary conditions at the surfaceof the cylinder, field expansions derived from BSP’sare needed for the incident, scattered, and internal~or cylinder! waves.
A. Incident Fields
By adding respective TM and TE solutions in the set ofEqs. ~43!–~54! specified for the case of a propagation infree space and by using rescaled coordinates, we findthat the incident-field expansions are given by
Ezi 5 k2S]2UTM
i
]Z2 1 UTMi D , (101)
Eri 5 k2S]2UTM
i
]Z]R2
iR
E0
H0
]UTEi
]w D , (102)
Ewi 5 k2S1
R]2UTM
i
]Z]w1 i
E0
H0
]UTEi
]R D , (103)
Hzi 5 k2S]2UTE
i
]Z2 1 UTEi D , (104)
Hri 5 k2S i
RH0
E0
]UTMi
]w1
]2UTEi
]Z]RD , (105)
Hwi 5 k2S1
iH0
E0
]UTMi
]R1
1R
]2UTEi
]Z]wD , (106)
where the potentials UTMi and UTE
i are given by re-lations ~99! and ~100!, respectively and in which weused the relation
vSemD 5 kSH0yE0
E0yH0D . (107)
It is then found that the incident-field componentsare
Xai 5 X0 exp~iCZ! (
m52`
1`
~2i!m exp~imw! (l50
N
Xamil ~R!Zl,
(108)
where X stands for E or H and a for z, r, or w, and thefunctions Xam
ij ~R! are given by
Ezmil ~R! 5 ImTM
l ~2i!lS2jm0 ~R! 1 ImTM
l11 ~2i!l~l 1 1!@2Cjm0 ~R!
2 S2jm1 ~R!# 1 (
n5l12
N
ImTMn ~21!nilFS2Sn
lD jmn2l~R!
2 2C~l 1 1!S nl 1 1D jm
n2l21~R! 2 ~l 1 1!~l 1 2!
3 S nl 1 2D jm
n2l22~R!G, l 5 0 · · · N 2 2;
(109)
EzmiN21~R! 5 2iCNImTM
N ~2i!Njm0 ~R!
1 S2 (n5N21
N
ImTMn ~21!nS n
N 2 1DiN21jmn2N11~R!;
(110)
4300 APPLIED OPTICS y Vol. 36, No. 18 y 20 June 1997
EzmiN ~R! 5 S2ImTM
N ~2i!Njm0 ~R!; (111)
Ermil ~R! 5 i~2i!lFCImTM
l djm0 ~R!
dR2 imImTE
l j m0 ~R!
R G1 (
n5l11
N
~21!nilFiCImTMn Sn
lD d jmn2l~R!
dR
1 ~l 1 1!ImTMn iS n
l 1 1D djmn2l21~R!
dR
2iR
imImTEn Sn
lD jmn2l~R!G, l 5 0 · · · N 2 1;
(112)
ErmiN ~R! 5 ~2i!NFiCImTM
N djm0 ~R!
dR2
iR
imImTEN jm
0 ~R!G ;
(113)
Ewmil 5 i~2i!lFCim
RImTM
l jm0 ~R! 1 ImTE
l d jm0 ~R!
dR G1 (
n5l11
N
i~21!nilFCimR
ImTMn Sn
lD jmn2l~R! 1
imR
3 ~l 1 1!ImTMn S n
l 1 1D jmn2l21~R!
1 ImTEn Sn
lD d jmn2l~R!
dR G , l 5 0 · · · N 2 1;
(114)
EwmiN 5 i~2i!NFCim
RImTM
N jm0 ~R! 1 ImTE
N d jm0 ~R!
dR G ;
(115)
Hrmil 5 i~2i!lFim
RImTM
l jm0 ~R! 1 CImTE
l djm0 ~R!
dR G1 (
n5l11
N
i~21!nilFimR
I mTMn Sn
lD jmn2l~R!
1 CImTEn Sn
lD d jmn2l~R!
dR1 ~l 1 1!ImTE
n
3 S nl 1 1D djm
n2l21~R!
dR G , l 5 0 · · · N 2 1;
(116)
HrmiN 5 i~2i!NFim
RImTM
N jm0 ~R! 1 CImTE
N d jm0 ~R!
dR G ;
(117)
Hwmil 5 i~2i!lF2ImTM
l djm0 ~R!
dR1
CR
imImTEl jm
0 ~R!G1 (
n5l11
N
i~21!nilF2ImTMn Sn
lD djmn2l~R!
dR
1CR
imImTEn Sn
lD jmn2l~R! 1
imR
~l 1 1!ImTEn
3 S nl 1 1D jm
n2l21~R!G , l 5 0 · · · N 2 1;(118)
HwmiN 5 i~2i!NF2ImTM
N d jm0 ~R!
dR1
CR
imImTEN jm
0 ~R!G .
(119)
For Hzi , we may compare Eqs. ~101!–~104! and ~99!
and ~100! and see that the functions Hzmil ~R! are ob-
tained from the functions Ezmil ~R! by changing ImTM
n ’sto ImTE
n ’s.
B. Scattered Fields
The BSP’s for scattered fields are written as12,13
UTMs 5 2
E0
k2 (m52`
1`
~2i!m exp~imw!^SmTM~g!,
Hm@R~1 2 g2!1y2#exp~igZ!& (120)
UTEs 5
H0
k2 (m52`
1`
~2i!m exp~imw!^SmTE~g!,
Hm@R~1 2 g2!1y2#exp~igZ!& , (121)
where SmTM~g! and SmTE~g! are distributions andHm’s are the Hankel functions Hm
~2!. Prefactors inthese BSP’s ~and in the others! are chosen in orderto recover the classical plane-wave case with theusual notations when the amplitude coefficients ofthe incident beam are accordingly specified.16
Next, because the incident BSP’s are truncated at0~ZN! inclusive, such must also be the case for thescattered BSP’s and also later on for the cylinderBSP’s. Hence the scattering distributions mustalso be truncated at the level of the Nth-order de-rivative of the Dirac distribution. Therefore, as forEq. ~95!, we set
SmTM~g! 5 (n50
N
SmTMn d~n!~g 2 C!, (122)
SmTE~g! 5 (n50
N
SmTEn d~n!~g 2 C!. (123)
Therefore we obtain
UTMs 5 2
E0
k2 (m52`
1`
~2i!m exp~imw!
3 (n50
N
SmTMn ~21!n$Hm@R~1 2 g2!1y2#
3 exp~igZ!%g5C~n! , (124)
UTEs 5
H0
k2 (m52`
1`
~2i!m exp~imw!
3 (n50
N
SmTEn ~21!n$Hm@R~1 2 g2!1y2#
3 exp~igZ!%g5C~n! , (125)
which can be also derived from UTMi , UTE
i by changingImTMn to 2SmTM
n , ImTEn to SmTE
n , and Jm to Hm.
We introduce, as for Eq. ~97!,
hmn ~R! 5
dn
dgn Hm@R~1 2 g2!1y2#g5C, (126)
and, from Eqs. ~99! and ~100!, by using substitutionswe immediately obtain
UTMs 5 2
E0
k2 exp~iCZ! (m52`
1`
~2i!m exp~imw! (l50
N
Zl
3 (n5l
N
SmTMn ~21!nSn
lDilhmn2l~R!, (127)
UTEs 5
H0
k2 exp~iCZ! (m52`
1`
~2i!m exp~imw! (l50
N
Zl
3 (n5l
N
SmTEn ~21!nSn
lDilhmn2l~R!. (128)
Now, because the scattered wave propagates infree space, as does the incident wave, the scattered-field components may be derived from the scat-tered BSP’s again by use of the set of Eqs. ~101!–~106!, with UTM
i , UTEi changed to UTM
s , UTEs , respec-
tively. We immediately discover, without anycomputation, that the scattered-field componentsare
X as 5 X0 exp~iCZ! (
m52`
1`
~2i!m exp~imw! (l50
N
X amsl ~R!Zl,
(129)
where again X stands for E or H and a for z, r, or w.Furthermore, we deduce the functions X am
sj ~R! fromthe functions Xam
ij ~R! by changing ImTMn to 2SmTM
n ,ImTEn to SmTE
n , and jmn to hm
n .
C. Internal Fields
The infinite cylinder of axis z, of cross-sectional ra-dius a, possesses a complex refractive indexM with respect to the surrounding medium,where
M 5 kcyk, (130)
kc is the wave number in the cylinder material, andk is the wave number outside the cylinder. Theelectric capacity e in the cylinder is denoted by ec,where
ec 5 M2e (131)
and e is the electric capacity outside the cylinder.Furthermore, the cylinder material is assumed to benonmagnetic:
mc 5 m. (132)
The cylinder BSP’s then are12,13
UTMc 5
E0
k2M (m52`
1`
~2i!m exp~imw!^CmTM~g!,
Jm@MR~1 2 g2!1y2#exp~igZ!& , (133)
20 June 1997 y Vol. 36, No. 18 y APPLIED OPTICS 4301
UTEc 5
iH0
k2 (m52`
1`
~2i!m exp~imw!^CmTE~g!,
Jm@MR~1 2 g2!1y2#exp~igZ!& , (134)
where, as for Eqs. ~122! and ~123!, we set
CmTM~g! 5 (n50
N
CmTMn d~n!~g 2 C!, (135)
CmTE~g! 5 (n50
N
CmTEn d~n!~g 2 C!. (136)
We proceed as above and introduce
j mn ~R! 5
dn
dgn Jm@MR~1 2 g2!1y2#g5C; (137)
the cylinder BSP’s become
UTMc 5
E0 exp~iCZ!
Mk2 (m52`
1`
~2i!m exp~imw! (l50
N
Zl
3 (n5l
N
CmTMn ~21!nSn
lDil j mn2l~R!, (138)
UTEc 5
iH0 exp~iCZ!
k2 (m52`
1`
~2i!m exp~imw! (l50
N
Zl
3 (n5l
N
CmTEn ~21!nSn
lDil j mn2l~R!. (139)
We note that the cylinder BSP’s may be obtainedfrom the incident BSP’s by changing ImTM
n toCmTM
n yM, ImTEn to iCmTE
n , and Jm~RS! to Jm~MRS! or,more conveniently, jm
n ~R! to #jmn ~R!.
Nevertheless, because the internal fields do notpropagate in free space, the set of Eqs. ~101!–~106!cannot be used any more. Instead, starting againfrom the set of Eqs. ~43!–~54! and specifying physicalconstants for the cylinder material, we now obtain
Ezc 5 k2S]2UTM
c
]Z2 1 M2UTMc D , (140)
Erc 5 k2S]2UTM
c
]Z]R2
iR
E0
H0
]UTEc
]w D , (141)
Ewc 5 k2S1
R]2UTM
c
]Z]w1 i
E0
H0
]UTEc
]R D , (142)
Hzc 5 k2S]2UTE
c
]Z2 1 M2UTEc D , (143)
Hrc 5 k2SiM2
RH0
E0
]UTMc
]w1
]2UTEc
]Z]RD , (144)
Hwc 5 k2S1
iM2 H0
E0
]UTMc
]R1
1R
]2UTEc
]Z]w D . (145)
We then find that the cylinder field componentsagain take a form similar to Eqs. ~108! and ~129!:
Xac 5 X0 exp~iCZ! (
m52`
1`
~2i!m exp~imw! (l50
N
Xamcl ~R!Zl.
(146)
4302 APPLIED OPTICS y Vol. 36, No. 18 y 20 June 1997
The functions Ezmcl ~R! may be found by applying Eq.
~140! to Eq. ~138!, yielding
Ezmcl ~R! 5
CmTMl
M~2i!l~M2 2 C2!jm
0 ~R! 1CmTM
l11
M~2i!l
3 ~l 1 1!@2Cjm0 ~R! 2 M2jm
1 ~R! 1 C2jm1 ~R!#
1 (n5l12
N CmTMn
M~21!nilF~M2 2 C2!Sn
lDj mn2l~R!
2 2C~l 1 1!S nl 1 1Dj m
n2l21~R! 2 ~l 1 1!~l
1 2!S nl 1 2Dj m
n2l22~R!G l 5 0 · · · N 2 2;
(147)
EzmCN21~R! 5 2iCN
CmTMN
M~2i!Njm
0 ~R! 1 ~M2 2 C2!
3 (n5N21
N CmTMn
M~21!nS n
N 2 1DiN21 j mn2N11~R!;
(148)
EzmCN~R! 5 ~M2 2 C2!
CmTMN
M~2i!Njm
0 ~R!. (149)
We note that the functions Ezmil ~R! may be recov-
ered from the functions Ezmcl ~R! by setting M 5 1 and
changing CmTMn to MImTM
n 5 ImTMn , CmTE
n to 2iImTEn ,
and jmn ~R! to jm
n ~R!. The functions Ezmsl ~R! may also
be recovered from the functions Ezmcl ~R! by use of sim-
ilar substitutions. Also, conversely, the functionsEzm
cl ~R! can be obtained either from Ezmil ~R! or Ezm
sl ~R!by use of substitutions, but this inverse process re-quires much care.
For Ermcl ~R! and Ewm
cl ~R!, however, comparing ex-pressions for the incident and the cylinder waves, wefind that they can readily be found from Erm
il ~R! andEwm
il ~R!, respectively, by changing ImTMn to CmTM
n yM,ImTEn to iCmTE
n , and jmn to jm
n .Next the functions Hzm
cl ~R! are obtained from the func-tions Ezm
cl ~R! by changing CmTMn to iMCmTE
n . Finally thefunctions Hrm
cl ~R! and Hwmcl ~R! are obtained from the func-
tions Hrmil ~R! and Hwm
il ~R!, respectively, by changing ImTMn
to MCmTMn , ImTE
n to iCmTEn , and jm
n to jmn .
5. Boundary Conditions
At the surface of the cylinder ~R 5 ka!, we have fourboundary conditions that express the continuity ofthe electric and the magnetic tangential fields:
Ezc 2 Ez
s 5 Ezi , Ew
c 2 Ews 5 Ew
i , (150)
Hzc 2 Hz
s 5 Hzi , Hw
c 2 Hws 5 Hw
i ; (151)
the lhs’s contain the unknown distribution coeffi-cients while the rhs’s are known when the nature ofthe incident beam is specified. Using Eqs. ~108!,~129!, and ~146! for the fields involved in Eqs. ~150!and ~151! and noting that equalities must be sepa-rately satisfied at each order 0~Zn!, n 5 0, . . . N, weobtain a set of boundary equations:
Ezmcj 2 Ezm
sj 5 Ezmij , Ewm
cj 2 Ewmsj 5 Ewm
ij , (152)
Hzmcj 2 Hzm
sj 5 Hzmij , Hwm
cj 2 Hwmsj 5 Hwm
ij , (153)
where j ranges from 0 to N. The functions Xamij ~R!,
Xamsj ~R!, and Xam
cj ~R! that are involved in the set of Eqs.~152! and ~153! are specified for R 5 ka. Being constantnumbers, they are written without any argument.
We are then faced with a linear set of 4~N 1 1!equations for 4~N 1 1! unknown distribution coeffi-cients. For a specified problem, this set may besolved by conventional methods. Therefore the the-ory is completed.
6. Conclusion
The theory of interaction between an infinite cylinderand an arbitrary-shaped beam was presented. Thistheory is called the GLMT for cylinders. From amathematical point of view, it possesses a somewhatunexpected structure insofar as it requires the use ofthe theory of distributions, in contrast with usuallight-scattering theories. Loosely speaking, this re-quirement may be viewed in part as the result of aconflict of symmetry in cylindrical coordinates be-tween the cylinder extending to 6` along its axis andthe incident shaped beam that is spatially confined.More correctly, it is the result of the status of one ofthe separation constants produced by the applicationof the separability theorem to the problem understudy. From a practical point of view, it opens theway to new investigations, such as in the field ofparticle characterization, similar to the GLMT forspheres and multilayered spheres, which lead tomany practical applications. Examples of scatter-ing diagrams are provided in Ref. 15 in which Gauss-ian incident beams are described by ~1! a Davisframework, ~2! a plane-wave-spectrum approach, and~3! a new localized approximation specific to cylindri-cal coordinates. More recently, phase-diameter re-lationships have been obtained, with the incidentbeam described by a first-order Davis beam, showingthat cylinder diameters may be measured withphase-Doppler instruments.
Appendix A.
To some extent, the most difficult part of my theory isin Subsection 3.B, in which BSD’s are introduced.Most experts in light-scattering theory do not masterthe theory of distributions and, as far as we are con-cerned, we have to learn it to solve the problem ad-dressed in this paper. Therefore, in this Appendix,some information from which the reader could atleast intuitively understand what is going on is in-troduced. The approach below is heuristic. For arigorous treatment, see Refs. 39 and 41.
In terms of functions, Eq. 57 that expresses UTMi as
a linear combination involving the Bessel functions ofthe first kind would be
UTMi 5
E0
k2 (m52`
1`
~2i!m exp~imw! * ImTM~g!
3 Jm@R~1 2 g2!1y2#exp~igZ!dg, (A1)
where ImTM~g! is a function that allows us to intro-duce a spectrum attached to the separability constantg. Equation ~A1! provides a linear combination, interms of functions, to generate functions given in Eq.~55!. Therefore it gives the most general structureof physical interest when the theory is expressed interms of the usual functions. Such an expressionwill produce oscillatory terms with respect to Z, butcannot produce polynomial terms with respect to Z ascontained in the field expressions discussed in Sub-section 2.4. Therefore the general theory cannot bereadily built in terms of functions.
Now, because the field expressions are proportional toexp~iCZ! @Eq. ~37!#, we see that actually g 5 C. In termsof functions, this would imply an oscillatory behaviorexp~iCZ! with respect to Z, which again does not matchthe field expressions that also contain polynomials withrespect to Z. Furthermore, because g is a constant, theintegral in Eq. ~A1! has no meaning any more. Equa-tion ~57! is the generalization of Eq. ~A1! from functionsto distributions. To understand how this generalizationworks, let us consider the following expression:
* d~x! f ~x!dx 5 f ~0!, (A2)
where d~x! is the Dirac distribution. Although Eq.~A2! is often written by physicists, in particular inquantum mechanics, it has no meaning because theDirac distribution d~x! is 0 outside of $0% and thereforewe are not allowed to integrate over the variable x.Within the framework of the theory of distributions,Eq. ~A2! is correctly written as
^d, f & 5 f ~0!. (A3)
The jump from Eq. ~A2! to Eq. ~A3! is similar to thejump from Eq. ~A1! to Eq. ~57!. We say in Eqs. ~A2!and ~A3! that the support of the distribution is $0%.Similarly, because g 5 C in terms of functions, we findfor our problem that the support of the distributions tobe used is g 5 $C%. Such distributions that possess asingle peak are known to be linear combinations of theDirac distribution and its derivatives. Because of theuse of these derivatives, we cannot conveniently usethe meaningless but usual form of Eq. ~A2!; it is betterto introduce the correct notation of Eq. ~A3!. Further-more, these derivatives will indeed generate the poly-nomial terms that we were looking for.
I also comment on the function f introduced in ~A3!.Such a function is called a test function. From thatpoint of view, a distribution may be considered as anoperator that is defined when we know how it works ontest functions. Distributions are usually defined by testfunctions that form the set of functions w~x! [ C, x [ Rn,infinitely smooth with a bounded support, which is avectorial space, denoted as D. Test functions in Eq. ~57!do not pertain to D. This is allowed, however, becausewe use distributions that have a support at g 5 $C%.Clearly, for such distributions, we need only test func-tions defined at the support location.
Our discussion for UTEi @Eq. ~58!# would be similar
20 June 1997 y Vol. 36, No. 18 y APPLIED OPTICS 4303
to that for UTMi . Finally, Eqs. ~59! and ~60! are ob-
tained with Eqs. ~43!, ~46!, ~49!, and ~52! togetherwith Eqs. ~57! and ~58!. Readers at this point willcertainly guess how it works, even if they do notmaster the theory of distributions, if they just try towrite down the intermediary steps.
References1. G. Gouesbet, B. Maheu, and G. Grehan, “Light scattering from
a sphere arbitrarily located in a Gaussian beam, using a Brom-wich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 ~1988!.
2. B. Maheu, G. Gouesbet, and G. Grehan, “A concise presenta-tion of the generalized Lorenz-Mie theory for arbitrary locationof the scatterer in an arbitrary incident profile,” J. Opt. ~Paris!,19, 59–67 ~1988!.
3. G. Gouesbet, G. Grehan, and B. Maheu, “Generalized Lorenz–Mie theory and applications to optical sizing,” in CombustionMeasurements N. Chigier, ed. ~Hemisphere, Washington, D.C.,1991!, pp. 339–384.
4. G. Gouesbet, “Generalized Lorenz–Mie theory and applica-tions,” Part. Part. Syst. Charact. 11, 22–34 ~1994!.
5. F. Onofri, G. Grehan, and G. Gouesbet, “Electromagnetic scat-tering from a multilayered sphere located in an arbitrarybeam,” Appl. Opt. 34, 7113–7124 ~1995!.
6. K. F. Ren, G. Grehan, and G. Gouesbet, “Laser sheet scattering byspherical particles,” Part. Part. Syst. Charact. 10, 146–151 ~1993!.
7. G. Grehan, K. F. Ren, G. Gouesbet, A. Naqwi, and F. Durst,“Evaluation of a particle sizing technique based on lasersheets,” Part. Part. Syst. Charact. 11, 101–106 ~1994!.
8. K. F. Ren, G. Grehan, and G. Gouesbet, “Evaluation of lasersheet beam shape coefficients in generalized Lorenz–Mie the-ory by using a localized approximation,” J. Opt. Soc. Am. A 11,2072–2079 ~1994!.
9. K. F. Ren, G. Grehan, and G. Gouesbet, “Electromagnetic fieldexpression of a laser sheet and the order of approximation,” J.Opt. ~Paris! 25, 165–176 ~1994!.
10. F. Corbin, G. Grehan, and G. Gouesbet, “Top-hat beam tech-nique: improvements and application to bubble measure-ments,” Part. Part. Syst. Charact. 8, 222–228 ~1991!.
11. G. Gouesbet, J. A. Lock, and G. Grehan, “Partial wave repre-sentations of laser beams for use in light-scattering calcula-tions,” Appl. Opt. 34, 2133–2143 ~1995!.
12. G. Gouesbet, “Interaction between Gaussian beams and infi-nite cylinders, by using the theory of distributions,” J. Opt.~Paris! 26, 225–239 ~1995!.
13. G. Gouesbet, “Scattering of a first-order Gaussian beam by aninfinite cylinder with arbitrary location and arbitrary orienta-tion,” Part. Part. Syst. Charact. 12, 242–256 ~1995!.
14. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A19, 1177–1179 ~1979!.
15. K. F. Ren, G. Gouesbet, and G. Grehan, “Scattering of a Gauss-ian beam by an infinite cylinder: numerical results in GLMT-framework,” presented at the Eighth InternationalSymposiumon Applications of Laser Techniques to Fluid Mechanics, Lis-bon, 8–11 July 1996.
16. G. Gouesbet and G. Grehan, “Interaction between shapedbeams and an infinite cylinder including a discussion of Gauss-ian beams,” Part. Part. Syst. Charact. 11, 299–308 ~1994!.
17. G. Gouesbet and G. Grehan, “Interaction between a Gaussianbeam and an infinite cylinder, using non-sigma-separable po-tentials,” J. Opt. Soc. Am. A 11, 3261–3273 ~1994!.
18. P. M. Morse and H. Feshback, Methods of Theoretical Physics~McGraw-Hill, New York, 1953!, Chap. 5, Part I.
19. G. Gouesbet, “The separability theorem revisited with applicationsto light scattering theory,” J. Opt. ~Paris! 26, 123–135 ~1995!.
4304 APPLIED OPTICS y Vol. 36, No. 18 y 20 June 1997
20. E. Lenglart and G. Gouesbet, “The separability theorem interms of distributions with discussion of electromagnetic scat-tering theory,” J. Math. Phys. 37, 9705–9710 ~1996!.
21. G. Gouesbet, “General description of arbitrary shaped beamsfor introduction in light scattering theories,” in Recent Re-search Developments in Applied Optics, S. M. Krishnan, ed.~Research Signpost, Trivandrum, India, edited by the Scien-tific Information Guild, 1996!.
22. G. Gouesbet, “Exact description of arbitrary shaped beams foruse in light scattering theories,” J. Opt. Soc. Am. A 13, 2434–2440 ~1996!.
23. J. F. Rice, The Approximation of Functions ~Addison-Wesley,Reading, Mass., 1969!, Vols. 1 and 2.
24. J. P. Barton and D. R. Alexander, “Fifth order corrected elec-tromagnetic field components for a fundamental Gaussianbeam,” J. Appl. Phys. 66, 2800–2802 ~1989!.
25. J. A. Lock and G. Gouesbet, “Rigorous justification of the lo-calized approximation to the beam shape coefficients in gener-alized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc.Am. A 11, 2503–2515 ~1994!.
26. G. Gouesbet and J. A. Lock, “Rigorous justification of the lo-calized approximation to the beam shape coefficients in gener-alized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc.Am. A 11, 2516–2525 ~1994!.
27. G. Gouesbet, C. Letellier, K. F. Ren, and G. Grehan, “Discus-sion of two quadrature methods of evaluating beam shapecoefficients in generalized Lorenz–Mie theory,” Appl. Opt. 35,1537–1542 ~1996!.
28. G. Gouesbet, “Higher-order description of Gaussian beams,” J.Opt. ~Paris! 27, 35–50 ~1996!.
29. M. Kerker, The Scattering of Light and Other ElectromagneticRadiation ~Academic, London, 1969!.
30. C. F. Bohren and D. R. Huffman, Absorption and Scattering ofLight by Small Particles ~Wiley, New York, 1983!.
31. J. A. Lock, “Scattering of a diagonally incident focused Gauss-ian beam by an infinitely long homogeneous circular cylinder,”J. Opt. Soc. Am. A 14, 640–652 ~1997!.
32. J. A. Lock, “Morphology-dependent resonances of an infinitelylong circular cylinder illuminated by a diagonally incidentplane wave or a focused Gaussian beam,” J. Opt. Soc. Am. A 14,653–661 ~1997!.
33. N. G. Alexopoulos and P. K. Park, “Scattering of waves withnormal amplitude distribution from cylinders,” IEEE Trans.Antennas Propag. AP-20, 216–217 ~1972!.
34. S. Kozaki, “Scattering of a Gaussian beam by a homogeneousdielectric cylinder,” J. Appl. Phys. 53, 7195–7200 ~1982!.
35. S. Kozaki, “A new expression for the scattering of a Gaussianbeam by a conducting cylinder,” IEEE Trans. AntennasPropag. AP-30, 881–887 ~1982!.
36. S. Kozaki, “Scattering of a Gaussian beam by an inhomogeneousdielectric cylinder,” J. Opt. Soc. Am. 72, 1470–1474 ~1982!.
37. T.-C. K. Rao, “Scattering by a radially inhomogeneous cylin-drical dielectric shell due to an incident Gaussian beam,” Can.J. Phys. 67, 471–475 ~1989!.
38. E. Zimmermann, R. Dandliker, N. Souli, and B. Krattiger,“Scattering of an off-axis Gaussian beam by a dielectric cylin-der compared with a rigorous electromagnetic approach,” J.Opt. Soc. Am. A 12, 398–403 ~1995!.
39. L. Schwartz, Methodes Mathematiques pour les Sciences Phy-siques ~Hermann, Paris, 1965!
40. F. Roddier, Distributions et Transformation de Fourier a l’Usagedes Physiciens et des Ingenieurs ~McGraw-Hill, Paris, 1982!.
41. E. Butkov, Mathematical Physics ~Addison-Wesley, Reading,Mass., 1968!.
42. G. Arfken, Mathematical Methods for Physicists ~Academic,New York, 1968!.
