Integral localized approximation in generalized Lorenz–Mie theory Kuan Fang Ren, Ge´rard Gouesbet, and Ge´rard Gre´han

The generalized Lorenz–Mie theory deals with the interaction between spheres and arbitrarily shaped illuminating beams. An efficient use of the theory requires efficient evaluation of the so-called beamshape coefficients involved in the description of the illuminating beam. A less time-consuming method of evaluation relies on the localized approximation. However, it lacks flexibility when the description of the illuminating beam is modified. We present a new version of this method, called the integral localized approximation, that exhibits the desired property of flexibility. © 1998 Optical Society of America OCIS codes: 290.4020, 290.0290.

1. Introduction

A generalized Lorenz–Mie theory ~GLMT!, which describes the interaction between homogeneous spheres and arbitrarily shaped beams, has been developed during the past two decades and applied to many practical situations ~see Refs. 1 and 2 and references therein, among others!. In this theory we have to evaluate so-called beamshape coefficients ~BSC’s!. The possibility of using the GLMT for extensive applications depends much on our ability to evaluate BSC’s efficiently. Essentially, the word “efficiency” here designates two desirable properties, speed and flexibility. Speed means the amount of work left to the computer and is satisfied when numerical computations are not time consuming in terms of the CPU. Flexibility is related the amount of work left to the human brain and is satisfied when a change in the description of the illuminating beam does not require a heavy extraanalytical work. Historically, the first method used to evaluate BSC’s was that of using quadratures. A review of this approach is available in Ref. 3. This method is flexible because only kernels have to be changed The authors are with Laboratoire d’Energe´tique des Syste`mes et Proce´de´s, Unite´ Mixte de Recherche 6614yComplexe de Recherche Interprofessionnel en Aerotherochimie, Centre National de la Recherche Scientifique, Universite´ et Institut National des Sciences Applique´s de Rouen, B.P. 08, 76131-Mont-Saint-Aignan Cedex, France. Received 27 May 1997; revised manuscript received 2 December 1997. 0003-6935y98y194218-08$15.00y0 © 1998 Optical Society of America 4218

APPLIED OPTICS y Vol. 37, No. 19 y 1 July 1998

when the description of the illuminating beam is changed. However, it is computer-time consuming. Later, the finite-series method appeared.4 This method is much faster than the quadrature method and it meets the requirement of speed necessary for extensive applications of the GLMT. Unfortunately, it is not flexible. A less time-consuming method, however, is the localized approximation. It originally appeared for on-axis Gaussian beams5 and was afterward extended to the cases of off-axis Gaussian beams6 and of laser sheets.7 It also allows one to handle top-hat beams.8 Originally the localized approximation relied on an analogy to the van de Hulst localization principle9 but without any firm mathematical basis. A first attempt ~only partially successful! to provide a rigorous justification of the localized approximation was given in Ref. 10. Afterward, a rigorous justification of the localized approximation was provided for Gaussian beams, both in the on-axis and in the off-axis cases.11,12 This technique to evaluate BSC’s has good computational speed but is not flexible. This paper is therefore devoted to another method ~which satisfies both speed and flexibility!, called the integral localized approximation, that can be viewed as a hybrid of the quadrature method ~it still uses a quadrature! and the localized approximation ~it still uses prescriptions from the localized approximation!. The paper is organized as follows. The integral localized approximation is described in Section 2. Section 3 is devoted to the application of the method to the case of a fifth-order Gaussian beam with validations by comparison with one of the other established methods. Section 4 is devoted to laser sheets and demonstrates that the integral localized approx-

u!exp~imw! in Hm Here k designates the wave numr . ber of the electromagnetic wave. After this process, m m m Em r and Hr lead to Fr and Ir , respectively. m m ˆ ~iii! Apply to Fr and Ir a localization operator G 1 that changes R 5 kr to ~n 1 ⁄2! and u to py2. Note that these prescriptions are not exactly the ones given in Ref. 6, but they are equivalent and simpler ~see Ref. 8!. ~iv! Multiply the resultant expressions normalization factors that read as6

Fig. 1. Coordinate systems in generalized Lorenz–Mie theory. Op xyz is attached to the scattering sphere, and OG uvw to the Gaussian beam.

Z0n 5

2n~n 1 1!i , 2n 1 1

Zm n 5

S D 22i 2n 1 1

(1)

umu21

,

m Þ 0.

(2)

The BSC’s in the localized approximation framework then read as imation exhibits another desirable property, stability. In Section 5 we discuss the case of laser beams in the mode TEM01*, a situation for which we are not able to evaluate the BSC’s by quadratures or finite series. Section 6 discusses computational times. Section 7 is a conclusion. 2. Integral Localized Approximation

The GLMT uses two Cartesian coordinate systems, OG uvw and OP xyz, with parallel axes, as indicated in Fig. 1, and a spherical coordinate system ~r, u, w! based on OP xyz. The coordinate system OP xyz is attached to the spherical scattering particle whose center is located at OP. The coordinate system OG uvw is attached to the illuminating beam. For a Gaussian beam, OG is conveniently taken to be the beam waist center, and the beam propagates toward positive values of w. The coordinates of OG with respect to OP xyz are denoted ~x0, y0, z0!. The illuminating beam is described by use of expansions into partial waves expressed in the spherical coordinate system ~r, u, w!. The partial wave m m coefficients are BSC’s and are denoted gn,TM and gn,TE . . . ~n 5 1, . . . , `; m 5 2n 1n; TM, transverse magnetic; TE, transverse electric!. They can be determined from the mathematical expressions of the m radial electric field Er ~for gn,TM ! and of the radial m magnetic field Hr ~for gn,TE!, by use of the quadrature of the finite-series method. The procedure to obtain the expressions of the BSC’s in the framework of the localized approximation is given in Ref. 6 for Gaussian beams and can be summarized as follows: ~i! Expand the radial electric Er~r, u, w! and the radial magnetic Hr~r, u, w! field components into azm imuthal modes Em r and Hr , respectively; each azimuthal mode has a superscript m that is proportional to exp~imw!. ~ii! Remove from these azimuthal modes the plane-wave contributions, namely, E0 sin u exp~2ikr cos u!exp~imw! in Em r and H0 sin u exp~2ikr cos

m ˆ m gn,TM 5 Zm n G~Fr !,

(3)

m ˆ m 5 Zm gn,TE n G~Ir !.

(4)

This procedure has been rigorously justified in Ref. 12 for off-axis Gaussian beams and the simpler case of on-axis Gaussian beams was rigorously justified in Ref. 11. The rigorous justification of the localized approximation induces a slight modification of the ˆ , leading to a modified locallocalization operator G ized approximation. This modification is small enough to be irrelevant and is not included in this paper. Let us now consider the expansions of radial fields into azimuthal modes, reading as6

SD

1`

S D

Er Em r 5 m . Hr H r m52`

(

(5)

Extracting the azimuthal modes from the radial fields according to Eq. ~5! is an analytical process that can be tedious, as exemplified in Ref. 6 for off-axis Gaussian beams and in Ref. 7 for laser sheets. Therefore we have a gain of flexibility if this step is m avoided. Because the azimuthal modes Em r and Hr are proportional to exp~imw! and because

*

2p

[email protected]~m 2 m9!w#dw 5 2pdmm9,

(6)

0

we can achieve our aim by evaluating the azimuthal modes according to Em r 5

Hm r 5

1 exp~imw! 2p

*

1 exp~imw! 2p

*

2p

Er~r, u, w9!exp~2imw9!dw9,

(7)

Hr~r, u, w9!exp~2imw9!dw9.

(8)

0

2p

0

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From Eqs. ~3!, ~4!, ~7!, and ~8! and using the relam m m tions between Em r and Fr , Hr and Ir , we then find that the BSC’s now read as m 5 gn,TM

m gn,TE 5

Zm n 2pE0 Zm n 2pH0

* *

2p

ˆ @Er~r, u, w!#exp~2imw!dw, G

(9)

ˆ @Hr~r, u, w!#exp~2imw!dw. G

(10)

0 2p

0

Equations ~9! and ~10! define the integral localized approximation. If the description of the beam is modified, we need only to change the kernel, leading to a flexible scheme. The price to pay is that quadratures over w must be numerically evaluated, possibly leading to a loss of speed. However, this loss cannot be big because quadratures over w are not time consuming to carry out ~see Section 6 below!. For later use we end this section with a remark: ˆ changes u to py2, Inasmuch as localization operator G it changes z 5 r cos u to 0. Then the BSC’s in the localized approximation ~and also in the integral localized approximation! depend only on the radial fields expressed in the plane ~z 5 0!.

own right, are called localized beams. Furthermore, it has been shown that, although it is based on a first-order beam, the localized approximation nearly perfectly anticipates the information embodied in higher-order beam descriptions.8,11,12 Therefore, localized beams based on a first-order Davis description provide Maxwellian descriptions that are close to Gaussian beams as described by infinite-order Davis beams. B.

Fifth-Order Davis Beam

Assume that you are given a fifth-order ~symmetrized! Davis beam whose radial fields read as14 Er 5 E0c0 exp~2ikr cos u 1 ikz0!sin u 1 $cos w 1 [email protected]~1 2 ir2Q!cos w 2 2Q2j~j cos w 1 h sin w!# 1 [email protected]~2 2 3ir2Q 2 1⁄2r4Q2!cos w 1 r2Q4~8 2 2ir2Q!j~j cos w 1 h sin w!#% 1 E0c0 exp~2ikr cos u 1 ikz0!j cos [email protected]~22Q! 1 s3r2Q3~6 2 2ir2Q! 1 s5r4Q5~220 1 10ir2Q 1 r4Q2!#,

(11)

Hr 5 H0c0 exp~2ikr cos u 1 ikz0!sin u$sin w

3. Gaussian Beams

1 [email protected]~1 2 ir2Q!sin w 2 2Q2h~j cos w

A.

1 h sin w!# 1 [email protected]~2 2 3ir2Q 2 1⁄2r4Q2!sin w

Generalities

The most celebrated description of Gaussian beams is one provided by Davis,13 later symmetrized by Barton and Alexander.14 A review of this scheme of description can be found in Ref. 8. In this framework, Gaussian beams are described by a sequence of successive approximations, here called kth-order Davis beams ~k 5 1, 3, 5, . . .!, it being understood that we refer in this paper to use the symmetrized version. None of these beams satisfies Maxwell’s equations, except in the limit k 3 `. We therefore say that these descriptions are non-Maxwellian. However, a kth-order Davis beam comprises a Maxwellian contribution supplemented by a nonMaxwellian contribution that can be viewed as an artifact resulting from the imperfect description of the beam.11,12 The localized approximation is based on the firstorder Davis beam5,6; i.e., the radial fields Er and Hr implemented in Eqs. ~3! and ~4! are taken to be those given by the first-order Davis beam expressions. Although the first-order Davis beam is not Maxwellian, the BSC’s produced by Eqs. ~3! and ~4! in the localized approximation ~and equivalently in the integral localized approximation! are constant complex numbers. Therefore the partial wave exm pansions of Er and Hr, where coefficients gn,TM and m gn,TE are constant complex numbers, automatically provide a Maxwellian description. We then can say that the localized approximation induces a remodeling of the beam in which, starting from a non-Maxwellian description, we arrive at a Maxwellian description.11,12 These remodeled beams, which can be viewed as Maxwellian beams in their 4220

APPLIED OPTICS y Vol. 37, No. 19 y 1 July 1998

1 r2Q4~8 2 2ir2Q!h~h sin w 1 j cos w!#% 1 H0c0 exp~2ikr cos u 1 ikz0!h cos [email protected]~22Q! 1 s3r2Q3~6 2 2ir2Q! 1 s5r4Q5~220 1 10ir2Q 1 r4Q2!#,

(12)

in which

S

c0 5 iQ exp 2iQ sin2 u

D

r2 , w02

(13)

1 , i 1 2s k~r cos u 2 z0!

(14)

j5

x0 r cos w sin u 2 , w0 w0

(15)

h5

r y0 sin w sin u 2 , w0 w0

(16)

Q5

2

r2 5 j2 1 h2,

(17)

and s is the beam-confinement parameter, which reads as s5

1 , kw0

(18)

in which w0 is the beam waist radius. Also, assume that you are not given any information concerning the structure of the Davis scheme of description. It would then be inconvenient to apply the localized approximation stricto sensu, inasmuch

m Table 1. Comparison of BSC’s Tgn,TM Calculated by the Integral Localized Approximation and by the Quadrature Method for l 5 0.5145 mm, w0 5 1 mm, and x0 5 y0 5 z0 5 0.5 mma

n 1 1 5 5 5 10 10 10

m 0 1 0 1 5 0 1 10

Integral Localized Approximation 21

Quadrature

21

0.148002 1021, 0.476673 1021 0.297132, 20.671027 1021 0.187199, 0.620954 0.264935, 20.718648 1021 0.108205 1026, 0.651323 1027 0.417063, 0.151725 1011 0.189987, 20.711338 1021 20.347003 10217, 0.464010 10218

0.142575 10 , 0.462098 10 0.294466, 20.675677 1021 0.181435, 0.604687 0.263098, 20.715839 1021 20.916280 1027, 0.540129 1027 0.411071, 0.149534 1011 0.189731, 20.704528 1021 20.157495 10218, 20.188324 10217

a

We have chosen to present data with six digits.

as this localized approximation is based on a firstorder Davis beam that you are not supposed to know. Although this first-order description is actually embedded as a contribution in the fifth-order description above, it would be difficult to extract in the absence of any other information. We can then decide to apply Eqs. ~3! and ~4! with Er and Hr given by Eqs. ~11! and ~12!, which requires us to extract analytically the azimuthal modes Em r and ConHm r and Er and Hr, a tedious task indeed. versely, we can apply the integral localized approximation with Eqs. ~9! and ~10!. In practice, the loss of speed implied by this procedure is found to be insignificant. The result of the procedure is to provide us with constant BSC’s, i.e., with a remodeled Maxwellian localized beam, although the original fifth-order beam is not Maxwellian. This localized beam based on a fifth-order Davis beam is an approximation to a perfect Gaussian beam ~defined as an infinite-order Davis beam!, which is, however, not so good as a localized beam obtained from a first-order Davis beam.11 Nevertheless, the difference between localized beams based on a first-order beam and on a fifth-order beam is necessarily small when the beamconfinement parameter is small enough. Significant differences are expected only when s becomes big enough, approaching its theoretical limit s ; 0.25 ~as defined and discussed in Ref. 8!. Also, in the laboratory an actual beam may depart from its ideal theoretical description, making the difference between the two above kinds of localized beam rather irrelevant in most cases. Examples of results are provided in Table 1, which m evaluated by the integral locompares BSC’s gn,TM calized approximation and by a quadrature method.3 Here we use l 5 0.5145 mm, w0 5 1 mm, and x0 5 y0 5 z0 5 0.5 mm. Significant differences appear only when the modulus of the BSC’s becomes very small ~n 5 10, m 5 10!. The corresponding partial waves would, however, be of little influence in the scattering process. For the other cases the agreement is satisfactory, although the beam-confinement parameter is rather big ~s ; 0.1! because the chosen beam is highly focused ~w0 ; 2l!. For smaller degrees of focusing the differences between the two methods would obviously decrease a great deal.

4. Laser Sheets

The structure of a laser sheet, which can be viewed as an elliptical Gaussian, is sketched in Fig. 2. The electromagnetic description of such a beam is discussed in Ref. 15. In particular, the radial fields Er and Hr read as

F

Er 5 E0csh 0 cos f sin u 2

G

2Qx cos u~r cos f sin u 2 x0! lx

3 exp~2ikr cos u 1 ikz0!,

F

Hr 5 H0csh 0 cos f sin u 2

(19)

G

2Qy cos u~r sin f sin u 2 y0! ly

3 exp~2ikr cos u 1 ikz0!, where

F

csh 0 5 Î2Q x Q y exp 2 2

iQx ~r cos f sin u 2 x0!2 w0x2

G

iQy ~r sin f sin u 2 y0!2 , w0y2 1

Qx 5 i1 Qy 5

(20)

2 lx~r cos u 2 z0! 1

2 i1 ly~r cos u 2 z0!

(21)

,

(22)

,

(23)

in which lx 5 kw0x2,

(24)

ly 5 kw0y2,

(25)

in which w0x and w0y can be called beam waist radii along x and y, respectively, as shown in Fig. 2. If we take w0x 5 w0y 5 w0,

(26)

we recover the case of a Gaussian beam. Also, the above description of the radial fields in Eqs. ~19! and ~20! identifies with the first-order Davis beam description of a Gaussian beam. Recall that a localized ap1 July 1998 y Vol. 37, No. 19 y APPLIED OPTICS

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Fig. 4. g1n computed by the localized approximation for w0x 5 3 mm, w0y 5 2 mm, l 5 0.5145 mm, x0 5 0, y0 5 16.5 mm, and z0 5 0.2 mm. Fig. 2. Structure of a laser sheet beam defined by two radii, w0x and w0y, with associated spherical coordinates ~r, u, w!.

proximation must ideally be based on a first-order description; Eqs. ~19! and ~20! then permit the development of a localized approximation in the same spirit as for Gaussian beams. Such an approximation was presented in Ref. 7. It was afterward observed that the approximation lacks stability when n becomes great. This statement is exemplified in Fig. 3, exhib1 1 iting gn,TM and gn,TE versus n for w0x 5 3 mm, w0y 5 2 mm, l 5 0.5145 mm, x0 5 y0 5 0, and z0 5 0.2 mm, and in Fig. 4, again for w0x 5 3 mm, w0y 5 2 mm, and l 5 0.5145 mm but with x0 5 0, y0 5 16.5 mm, and z0 5 0.2 mm. One of these examples corresponds to what we can call the on-axis case ~x0 5 y0 5 0! and the other to what we can call an off-axis case ~x0 5 0, y0 Þ 0! by analogy to on-axis and off-axis Gaussian beams. Conversely, this lack of stability no longer appears if the BSC’s are evaluated by the integral localized approximation instead of by the localized approximation. Compare Fig. 5 with Fig. 3 and Fig. 6 with Fig. 4.

Fig. 3. g1n computed by the localized approximation for w0x 5 3 mm, w0y 5 2 mm, l 5 0.5145 mm, x0 5 y0 5 0, and z0 5 0.2 mm. 4222

APPLIED OPTICS y Vol. 37, No. 19 y 1 July 1998

Fig. 5. g1n computed by the integral localized approximation for w0x 5 3 mm, w0y 5 2 mm, l 5 0.5145 mm, x0 5 y0 5 0, and z0 5 0.2 mm.

Fig. 6. g1n computed by the integral localized approximation for w0x 5 3 mm, w0y 5 2 mm, l 5 0.5145 mm, x0 5 0, y0 5 16.5 mm, and z0 5 0.2 mm.

Outside the instability ranges versus n the localized approximation and the integral localized approximation agree, as they should, because the two techniques are analytically equivalent. This equivalence also points out the fact that the instabilities observed with the localized approximation for laser sheets have a numerical origin. The corresponding numerical problems are automatically avoided by use of the integral localized approximation. 5. TEM01* Laser Beam

We now apply the localized approximation to the case of a beam that does not share the Gaussian properties ~along x and along y! exhibited in the two previous cases, namely, to a TEM01* laser beam. We use the description available from Roosen and Imbert16 in which, in the plane z 5 0, Ex 5

S D 8m0 cF pw2

1y2

S D

r2 r exp 2 2 , w w

Ey 5 0,

(27) (28)

in which w is a width parameter of the beam. Let us insist on the fact that the description is not given everywhere in space but only at z 5 0. This condition prevents us from using the quadrature or the finite-series technique to evaluate the BSC’s but perfectly matches the possibilities of the localized approximations ~end of Section 2!. The radial component Er of the electric field Ei then reads as

S D

r r2 Er 5 E0 cos w exp 2 2 . w w

(29)

* F S DG 2p

0

r r2 ˆ G exp 2 2 cos w exp~2imw!dw. w w (30)

We then recall that ˆ ~r! 5 G

1 ~n 1 1⁄2! 5 rn, k

m n,TM

g

5 S D

rn r2n exp 2 2 5 2w w 0

umu 5 1

.

(32)

umu Þ 1

Next, a TEM01* laser beam possesses a symmetry around its axis of propagation, so it can be viewed as an axisymmetric laser beam in the sense defined in Ref. 17. With Er proportional to cos w, it is furthermore what we call a special axisymmetric beam, again in the sense defined in Ref. 17. From Eq. ~29!, 1 21 in Ref. 17, and noting that gn,TM 5 gn,TM @Eq. ~32!#, we can then define a set of special beam-shape coefficients gn that read as 1 21 gn 5 2gn,TM 5 2gn,TM 5

S D

r2 rn exp 2 2 . w w

(31)

(33)

These special BSC’s do not distinguish between the TM and TE contributions to the electromagnetic field and therefore also automatically incorporate the description of the magnetic field that is not originally given by Eqs. ~27! and ~28!. Once the special BSC’s are known, we can write down partial wave expansions of the electric and magnetic field components. In particular, the expansions for the electric field components read as17–19 Er 5

` E0 cos f ~2i!n11~2n 1 1! R2 n51

(

3 gncn~R! Pn1~cos u!, Eu 5

(34)

` E0 2n 1 1 cos f ~2i!n11 R n~n 1 1! n51

(

3 [email protected]~R!tn~cos u! 1 icn~R!pn~cos u!#,

We then also note that ]Ery]r Þ 0 at r 5 0, an unpleasant fact that is enough to permit us to claim that the model provided by Eqs. ~27! and ~28! is not a perfect description of the physics of a TEM01* laser beam. Nevertheless, the localized approximations will be able to handle this model correctly. In the present case, however, in which Er is proportional to cos w, the decomposition of Er into azimuthal waves Erm is easy. We can therefore decide to use either the localized approximation or the integral localized approximation, and, in both cases, the BSC’s will be obtained under the form of ~equivalent! analytical expressions. In particular, if we insert Eq. ~29! into Eq. ~9! we obtain Zm n m gn,TM 5 2p

so we readily obtain

Ef 5 2

(35)

` E0 2n 1 1 ~2i!n11 sin f R n~n 1 1! n51

(

3 [email protected]~R!pn~cos u! 2 icn~R!tn~cos u!#,

(36)

in which R 5 kr,

(37)

cn~R! 5 Rjn~R!,

(38)

pn~cos u! 5

Pn1~cos u! , sin u

(39)

tn~cos u! 5

d Pn1~cos u!, du

(40)

in which jn~R! are the spherical Bessel functions and Pn1~cos u! are the associated Legendre polynomials. Because the BSC’s evaluated by the localized approximations cannot be compared with BSC’s evaluated by other techniques, we compare directly the modulus of the electric field provided by Eqs. ~27! and ~28!, called the original field, and the modulus of the electric field deduced from Eqs. ~34!–~36!, called the 1 July 1998 y Vol. 37, No. 19 y APPLIED OPTICS

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Fig. 7. Comparison of the original and the reconstructed electric fields for a TEM01* laser beam model with w 5 5 mm and l 5 0.6328 mm.

reconstructed field. Such a comparison is shown in Fig. 7. The perfect agreement between the two kinds of field exemplifies once more the validity of the localized approximations. 6. Computational Times

In this section we discuss computational times and refine our statements concerning the difference in speed between the localized approximation and the integral approximation. It happens that computational times for the integral localized approximation cannot be much larger than for the localized approximation, although numerical quadratures are involved. Actually, depending on the beam under study, it may even happen that the integral localized approximation is less time consuming. As an example, the localized approximation leads to the evaluation of single summations for Gaussian beams6 and to double summations for laser sheets,7 whereas, in both cases, the integral localized approximation requires the evaluation of single quadratures. It then happens that, for laser sheets, the integral localized Table 2. Computational time ~CPU! in secondsa

nmax y0 5 0, N 5 32 200 400 600 800 900

LA 0.9 2.7 6.2 11.9 17.5

y0 5 16.5, N 5 64 100 384 200 1,046 300 4,254 400 13,900 500 38,636

ILA 2.4 4.8 7.2 9.8 10.9 2.3 4.6 6.9 9.2 11.4

LA, localized approximation; ILA, integral localized approximation. 4224

APPLIED OPTICS y Vol. 37, No. 19 y 1 July 1998

approximation may be less time consuming than the localized approximation. We exemplify this statement by comparing computational times ~on a workstation! for the same cases 1 as in Figs. 3– 6 ~Table 2!. The BSC’s gn,TM are computed for n # nmax, with N integration points in the integral localized approximation. The portion of Table 2 in which y0 5 0 and N 5 32 corresponds to the cases in Figs. 3 and 5, i.e., to an axis situation for laser sheets. The localized approximation is less time consuming than the integral approximation for nmax up to 600, but, for larger values of nmax, it becomes more time consuming. Conversely, the bottom half of Table 2, which corresponds to the off-axis situation in Figs. 4 and 6, the integral localized approximation is much less time consuming than the localized approximation. In such a case the integral localized approximation is superior in flexibility, stability, and also speed to the localized approximation and therefore provides the most efficient scheme available to evaluate BSC’s. 7. Conclusion

In this paper we have introduced the integral localized approximation to evaluate beam-shape coefficients in generalized Lorenz–Mie theory. This method exhibits good properties of speed, flexibility, and stability. In particular, the gain in flexibility is impressive and the loss of speed is insignificant. Also, in some cases, the integral localized approximation may even be much less time consuming than the localized approximation. We have exemplified this technique by considering the cases of Gaussian beams and laser sheets. We also considered the case of a model of a TEM01* laser beam for which both the nonintegral and the integral localized approximations led to analytical expressions for the beam-shape coefficients. This last example demonstrated once more the efficiency of the localized approximations in modeling shaped beams. References 1. G. Gouesbet, B. Maheu, and G. Gre´han, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 ~1988!. 2. G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 ~1994!. 3. G. Gouesbet, C. Letellier, K. F. Ren, and G. Gre´han, “Discussion of two quadrature methods to evaluate beam shape coefficients in generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542 ~1996!. 4. G. Gouesbet, G. Gre´han, and B. Maheu, “Expressions to compute the coefficients gm n in the generalized Lorenz–Mie theory using finite series,” J. Opt. ~Paris! 19, 35– 48 ~1988!. 5. G. Gre´han, B. Maheu, and G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539 –3548 ~1986!. 6. G. Gouesbet, G. Gre´han, and B. Maheu, “A localized interpretation to compute all the coefficients gm n in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998 –1007 ~1990!. 7. K. F. Ren, G. Gre´han, and G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 ~1994!.

8. G. Gouesbet, J. A. Lock, and G. Gre´han, “Partial wave representation of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 ~1995!. 9. H. C. van de Hulst, Light Scattering by Small Particles ~Wiley, New York, 1957!. 10. J. A. Lock, “The contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 ~1994!. 11. J. A. Lock and G. Gouesbet, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 ~1994!. 12. G. Gouesbet and J. A. Lock, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516 –2525 ~1994!. 13. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 ~1979!. 14. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800 –2802 ~1989!.

15. K. F. Ren, G. Gre´han, and G. Gouesbet, “Electromagnetic field expression of a laser sheet and the order of approximation,” J. Opt. ~Paris! 25, 165–176 ~1994!. 16. G. Roosen and C. Imbert, “The TEM01* mode laser beam—a powerful tool for optical levitation of various types of spheres,” Opt. Commun. 26, 432– 436 ~1978!. 17. G. Gouesbet, “Partial wave expansions and properties of axisymmetric light beams,” Appl. Opt. 35, 1543–1555 ~1996!. 18. G. Gouesbet, G. Gre´han, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center, using a Bromwich formulation,” J. Opt. ~Paris! 16, 83– 83 ~1985!; republished in Selected Papers on Light Scattering, M. Kerker, ed., Vol. 951 of SPIE Milestone Series ~SPIE, Bellingham, Wash., 1988!, Part I, pp. 361–371. 19. G. Gouesbet, G. Gre´han, and B. Maheu, “The order of approximation in a theory of the scattering of a Gaussian beam by Mie scatter center,” J. Opt. ~Paris! 16, 239 –247 ~1985!, republished in Selected Papers on Light Scattering, M. Kerker, ed., Vol. 951 of SPIE Milestone Series ~SPIE, Bellingham, Wash., 1988!, Part I, pp. 352–360.

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The generalized Lorenz–Mie theory deals with the interaction between spheres and arbitrarily shaped illuminating beams. An efficient use of the theory requires efficient evaluation of the so-called beamshape coefficients involved in the description of the illuminating beam. A less time-consuming method of evaluation relies on the localized approximation. However, it lacks flexibility when the description of the illuminating beam is modified. We present a new version of this method, called the integral localized approximation, that exhibits the desired property of flexibility. © 1998 Optical Society of America OCIS codes: 290.4020, 290.0290.

1. Introduction

A generalized Lorenz–Mie theory ~GLMT!, which describes the interaction between homogeneous spheres and arbitrarily shaped beams, has been developed during the past two decades and applied to many practical situations ~see Refs. 1 and 2 and references therein, among others!. In this theory we have to evaluate so-called beamshape coefficients ~BSC’s!. The possibility of using the GLMT for extensive applications depends much on our ability to evaluate BSC’s efficiently. Essentially, the word “efficiency” here designates two desirable properties, speed and flexibility. Speed means the amount of work left to the computer and is satisfied when numerical computations are not time consuming in terms of the CPU. Flexibility is related the amount of work left to the human brain and is satisfied when a change in the description of the illuminating beam does not require a heavy extraanalytical work. Historically, the first method used to evaluate BSC’s was that of using quadratures. A review of this approach is available in Ref. 3. This method is flexible because only kernels have to be changed The authors are with Laboratoire d’Energe´tique des Syste`mes et Proce´de´s, Unite´ Mixte de Recherche 6614yComplexe de Recherche Interprofessionnel en Aerotherochimie, Centre National de la Recherche Scientifique, Universite´ et Institut National des Sciences Applique´s de Rouen, B.P. 08, 76131-Mont-Saint-Aignan Cedex, France. Received 27 May 1997; revised manuscript received 2 December 1997. 0003-6935y98y194218-08$15.00y0 © 1998 Optical Society of America 4218

APPLIED OPTICS y Vol. 37, No. 19 y 1 July 1998

when the description of the illuminating beam is changed. However, it is computer-time consuming. Later, the finite-series method appeared.4 This method is much faster than the quadrature method and it meets the requirement of speed necessary for extensive applications of the GLMT. Unfortunately, it is not flexible. A less time-consuming method, however, is the localized approximation. It originally appeared for on-axis Gaussian beams5 and was afterward extended to the cases of off-axis Gaussian beams6 and of laser sheets.7 It also allows one to handle top-hat beams.8 Originally the localized approximation relied on an analogy to the van de Hulst localization principle9 but without any firm mathematical basis. A first attempt ~only partially successful! to provide a rigorous justification of the localized approximation was given in Ref. 10. Afterward, a rigorous justification of the localized approximation was provided for Gaussian beams, both in the on-axis and in the off-axis cases.11,12 This technique to evaluate BSC’s has good computational speed but is not flexible. This paper is therefore devoted to another method ~which satisfies both speed and flexibility!, called the integral localized approximation, that can be viewed as a hybrid of the quadrature method ~it still uses a quadrature! and the localized approximation ~it still uses prescriptions from the localized approximation!. The paper is organized as follows. The integral localized approximation is described in Section 2. Section 3 is devoted to the application of the method to the case of a fifth-order Gaussian beam with validations by comparison with one of the other established methods. Section 4 is devoted to laser sheets and demonstrates that the integral localized approx-

u!exp~imw! in Hm Here k designates the wave numr . ber of the electromagnetic wave. After this process, m m m Em r and Hr lead to Fr and Ir , respectively. m m ˆ ~iii! Apply to Fr and Ir a localization operator G 1 that changes R 5 kr to ~n 1 ⁄2! and u to py2. Note that these prescriptions are not exactly the ones given in Ref. 6, but they are equivalent and simpler ~see Ref. 8!. ~iv! Multiply the resultant expressions normalization factors that read as6

Fig. 1. Coordinate systems in generalized Lorenz–Mie theory. Op xyz is attached to the scattering sphere, and OG uvw to the Gaussian beam.

Z0n 5

2n~n 1 1!i , 2n 1 1

Zm n 5

S D 22i 2n 1 1

(1)

umu21

,

m Þ 0.

(2)

The BSC’s in the localized approximation framework then read as imation exhibits another desirable property, stability. In Section 5 we discuss the case of laser beams in the mode TEM01*, a situation for which we are not able to evaluate the BSC’s by quadratures or finite series. Section 6 discusses computational times. Section 7 is a conclusion. 2. Integral Localized Approximation

The GLMT uses two Cartesian coordinate systems, OG uvw and OP xyz, with parallel axes, as indicated in Fig. 1, and a spherical coordinate system ~r, u, w! based on OP xyz. The coordinate system OP xyz is attached to the spherical scattering particle whose center is located at OP. The coordinate system OG uvw is attached to the illuminating beam. For a Gaussian beam, OG is conveniently taken to be the beam waist center, and the beam propagates toward positive values of w. The coordinates of OG with respect to OP xyz are denoted ~x0, y0, z0!. The illuminating beam is described by use of expansions into partial waves expressed in the spherical coordinate system ~r, u, w!. The partial wave m m coefficients are BSC’s and are denoted gn,TM and gn,TE . . . ~n 5 1, . . . , `; m 5 2n 1n; TM, transverse magnetic; TE, transverse electric!. They can be determined from the mathematical expressions of the m radial electric field Er ~for gn,TM ! and of the radial m magnetic field Hr ~for gn,TE!, by use of the quadrature of the finite-series method. The procedure to obtain the expressions of the BSC’s in the framework of the localized approximation is given in Ref. 6 for Gaussian beams and can be summarized as follows: ~i! Expand the radial electric Er~r, u, w! and the radial magnetic Hr~r, u, w! field components into azm imuthal modes Em r and Hr , respectively; each azimuthal mode has a superscript m that is proportional to exp~imw!. ~ii! Remove from these azimuthal modes the plane-wave contributions, namely, E0 sin u exp~2ikr cos u!exp~imw! in Em r and H0 sin u exp~2ikr cos

m ˆ m gn,TM 5 Zm n G~Fr !,

(3)

m ˆ m 5 Zm gn,TE n G~Ir !.

(4)

This procedure has been rigorously justified in Ref. 12 for off-axis Gaussian beams and the simpler case of on-axis Gaussian beams was rigorously justified in Ref. 11. The rigorous justification of the localized approximation induces a slight modification of the ˆ , leading to a modified locallocalization operator G ized approximation. This modification is small enough to be irrelevant and is not included in this paper. Let us now consider the expansions of radial fields into azimuthal modes, reading as6

SD

1`

S D

Er Em r 5 m . Hr H r m52`

(

(5)

Extracting the azimuthal modes from the radial fields according to Eq. ~5! is an analytical process that can be tedious, as exemplified in Ref. 6 for off-axis Gaussian beams and in Ref. 7 for laser sheets. Therefore we have a gain of flexibility if this step is m avoided. Because the azimuthal modes Em r and Hr are proportional to exp~imw! and because

*

2p

[email protected]~m 2 m9!w#dw 5 2pdmm9,

(6)

0

we can achieve our aim by evaluating the azimuthal modes according to Em r 5

Hm r 5

1 exp~imw! 2p

*

1 exp~imw! 2p

*

2p

Er~r, u, w9!exp~2imw9!dw9,

(7)

Hr~r, u, w9!exp~2imw9!dw9.

(8)

0

2p

0

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From Eqs. ~3!, ~4!, ~7!, and ~8! and using the relam m m tions between Em r and Fr , Hr and Ir , we then find that the BSC’s now read as m 5 gn,TM

m gn,TE 5

Zm n 2pE0 Zm n 2pH0

* *

2p

ˆ @Er~r, u, w!#exp~2imw!dw, G

(9)

ˆ @Hr~r, u, w!#exp~2imw!dw. G

(10)

0 2p

0

Equations ~9! and ~10! define the integral localized approximation. If the description of the beam is modified, we need only to change the kernel, leading to a flexible scheme. The price to pay is that quadratures over w must be numerically evaluated, possibly leading to a loss of speed. However, this loss cannot be big because quadratures over w are not time consuming to carry out ~see Section 6 below!. For later use we end this section with a remark: ˆ changes u to py2, Inasmuch as localization operator G it changes z 5 r cos u to 0. Then the BSC’s in the localized approximation ~and also in the integral localized approximation! depend only on the radial fields expressed in the plane ~z 5 0!.

own right, are called localized beams. Furthermore, it has been shown that, although it is based on a first-order beam, the localized approximation nearly perfectly anticipates the information embodied in higher-order beam descriptions.8,11,12 Therefore, localized beams based on a first-order Davis description provide Maxwellian descriptions that are close to Gaussian beams as described by infinite-order Davis beams. B.

Fifth-Order Davis Beam

Assume that you are given a fifth-order ~symmetrized! Davis beam whose radial fields read as14 Er 5 E0c0 exp~2ikr cos u 1 ikz0!sin u 1 $cos w 1 [email protected]~1 2 ir2Q!cos w 2 2Q2j~j cos w 1 h sin w!# 1 [email protected]~2 2 3ir2Q 2 1⁄2r4Q2!cos w 1 r2Q4~8 2 2ir2Q!j~j cos w 1 h sin w!#% 1 E0c0 exp~2ikr cos u 1 ikz0!j cos [email protected]~22Q! 1 s3r2Q3~6 2 2ir2Q! 1 s5r4Q5~220 1 10ir2Q 1 r4Q2!#,

(11)

Hr 5 H0c0 exp~2ikr cos u 1 ikz0!sin u$sin w

3. Gaussian Beams

1 [email protected]~1 2 ir2Q!sin w 2 2Q2h~j cos w

A.

1 h sin w!# 1 [email protected]~2 2 3ir2Q 2 1⁄2r4Q2!sin w

Generalities

The most celebrated description of Gaussian beams is one provided by Davis,13 later symmetrized by Barton and Alexander.14 A review of this scheme of description can be found in Ref. 8. In this framework, Gaussian beams are described by a sequence of successive approximations, here called kth-order Davis beams ~k 5 1, 3, 5, . . .!, it being understood that we refer in this paper to use the symmetrized version. None of these beams satisfies Maxwell’s equations, except in the limit k 3 `. We therefore say that these descriptions are non-Maxwellian. However, a kth-order Davis beam comprises a Maxwellian contribution supplemented by a nonMaxwellian contribution that can be viewed as an artifact resulting from the imperfect description of the beam.11,12 The localized approximation is based on the firstorder Davis beam5,6; i.e., the radial fields Er and Hr implemented in Eqs. ~3! and ~4! are taken to be those given by the first-order Davis beam expressions. Although the first-order Davis beam is not Maxwellian, the BSC’s produced by Eqs. ~3! and ~4! in the localized approximation ~and equivalently in the integral localized approximation! are constant complex numbers. Therefore the partial wave exm pansions of Er and Hr, where coefficients gn,TM and m gn,TE are constant complex numbers, automatically provide a Maxwellian description. We then can say that the localized approximation induces a remodeling of the beam in which, starting from a non-Maxwellian description, we arrive at a Maxwellian description.11,12 These remodeled beams, which can be viewed as Maxwellian beams in their 4220

APPLIED OPTICS y Vol. 37, No. 19 y 1 July 1998

1 r2Q4~8 2 2ir2Q!h~h sin w 1 j cos w!#% 1 H0c0 exp~2ikr cos u 1 ikz0!h cos [email protected]~22Q! 1 s3r2Q3~6 2 2ir2Q! 1 s5r4Q5~220 1 10ir2Q 1 r4Q2!#,

(12)

in which

S

c0 5 iQ exp 2iQ sin2 u

D

r2 , w02

(13)

1 , i 1 2s k~r cos u 2 z0!

(14)

j5

x0 r cos w sin u 2 , w0 w0

(15)

h5

r y0 sin w sin u 2 , w0 w0

(16)

Q5

2

r2 5 j2 1 h2,

(17)

and s is the beam-confinement parameter, which reads as s5

1 , kw0

(18)

in which w0 is the beam waist radius. Also, assume that you are not given any information concerning the structure of the Davis scheme of description. It would then be inconvenient to apply the localized approximation stricto sensu, inasmuch

m Table 1. Comparison of BSC’s Tgn,TM Calculated by the Integral Localized Approximation and by the Quadrature Method for l 5 0.5145 mm, w0 5 1 mm, and x0 5 y0 5 z0 5 0.5 mma

n 1 1 5 5 5 10 10 10

m 0 1 0 1 5 0 1 10

Integral Localized Approximation 21

Quadrature

21

0.148002 1021, 0.476673 1021 0.297132, 20.671027 1021 0.187199, 0.620954 0.264935, 20.718648 1021 0.108205 1026, 0.651323 1027 0.417063, 0.151725 1011 0.189987, 20.711338 1021 20.347003 10217, 0.464010 10218

0.142575 10 , 0.462098 10 0.294466, 20.675677 1021 0.181435, 0.604687 0.263098, 20.715839 1021 20.916280 1027, 0.540129 1027 0.411071, 0.149534 1011 0.189731, 20.704528 1021 20.157495 10218, 20.188324 10217

a

We have chosen to present data with six digits.

as this localized approximation is based on a firstorder Davis beam that you are not supposed to know. Although this first-order description is actually embedded as a contribution in the fifth-order description above, it would be difficult to extract in the absence of any other information. We can then decide to apply Eqs. ~3! and ~4! with Er and Hr given by Eqs. ~11! and ~12!, which requires us to extract analytically the azimuthal modes Em r and ConHm r and Er and Hr, a tedious task indeed. versely, we can apply the integral localized approximation with Eqs. ~9! and ~10!. In practice, the loss of speed implied by this procedure is found to be insignificant. The result of the procedure is to provide us with constant BSC’s, i.e., with a remodeled Maxwellian localized beam, although the original fifth-order beam is not Maxwellian. This localized beam based on a fifth-order Davis beam is an approximation to a perfect Gaussian beam ~defined as an infinite-order Davis beam!, which is, however, not so good as a localized beam obtained from a first-order Davis beam.11 Nevertheless, the difference between localized beams based on a first-order beam and on a fifth-order beam is necessarily small when the beamconfinement parameter is small enough. Significant differences are expected only when s becomes big enough, approaching its theoretical limit s ; 0.25 ~as defined and discussed in Ref. 8!. Also, in the laboratory an actual beam may depart from its ideal theoretical description, making the difference between the two above kinds of localized beam rather irrelevant in most cases. Examples of results are provided in Table 1, which m evaluated by the integral locompares BSC’s gn,TM calized approximation and by a quadrature method.3 Here we use l 5 0.5145 mm, w0 5 1 mm, and x0 5 y0 5 z0 5 0.5 mm. Significant differences appear only when the modulus of the BSC’s becomes very small ~n 5 10, m 5 10!. The corresponding partial waves would, however, be of little influence in the scattering process. For the other cases the agreement is satisfactory, although the beam-confinement parameter is rather big ~s ; 0.1! because the chosen beam is highly focused ~w0 ; 2l!. For smaller degrees of focusing the differences between the two methods would obviously decrease a great deal.

4. Laser Sheets

The structure of a laser sheet, which can be viewed as an elliptical Gaussian, is sketched in Fig. 2. The electromagnetic description of such a beam is discussed in Ref. 15. In particular, the radial fields Er and Hr read as

F

Er 5 E0csh 0 cos f sin u 2

G

2Qx cos u~r cos f sin u 2 x0! lx

3 exp~2ikr cos u 1 ikz0!,

F

Hr 5 H0csh 0 cos f sin u 2

(19)

G

2Qy cos u~r sin f sin u 2 y0! ly

3 exp~2ikr cos u 1 ikz0!, where

F

csh 0 5 Î2Q x Q y exp 2 2

iQx ~r cos f sin u 2 x0!2 w0x2

G

iQy ~r sin f sin u 2 y0!2 , w0y2 1

Qx 5 i1 Qy 5

(20)

2 lx~r cos u 2 z0! 1

2 i1 ly~r cos u 2 z0!

(21)

,

(22)

,

(23)

in which lx 5 kw0x2,

(24)

ly 5 kw0y2,

(25)

in which w0x and w0y can be called beam waist radii along x and y, respectively, as shown in Fig. 2. If we take w0x 5 w0y 5 w0,

(26)

we recover the case of a Gaussian beam. Also, the above description of the radial fields in Eqs. ~19! and ~20! identifies with the first-order Davis beam description of a Gaussian beam. Recall that a localized ap1 July 1998 y Vol. 37, No. 19 y APPLIED OPTICS

4221

Fig. 4. g1n computed by the localized approximation for w0x 5 3 mm, w0y 5 2 mm, l 5 0.5145 mm, x0 5 0, y0 5 16.5 mm, and z0 5 0.2 mm. Fig. 2. Structure of a laser sheet beam defined by two radii, w0x and w0y, with associated spherical coordinates ~r, u, w!.

proximation must ideally be based on a first-order description; Eqs. ~19! and ~20! then permit the development of a localized approximation in the same spirit as for Gaussian beams. Such an approximation was presented in Ref. 7. It was afterward observed that the approximation lacks stability when n becomes great. This statement is exemplified in Fig. 3, exhib1 1 iting gn,TM and gn,TE versus n for w0x 5 3 mm, w0y 5 2 mm, l 5 0.5145 mm, x0 5 y0 5 0, and z0 5 0.2 mm, and in Fig. 4, again for w0x 5 3 mm, w0y 5 2 mm, and l 5 0.5145 mm but with x0 5 0, y0 5 16.5 mm, and z0 5 0.2 mm. One of these examples corresponds to what we can call the on-axis case ~x0 5 y0 5 0! and the other to what we can call an off-axis case ~x0 5 0, y0 Þ 0! by analogy to on-axis and off-axis Gaussian beams. Conversely, this lack of stability no longer appears if the BSC’s are evaluated by the integral localized approximation instead of by the localized approximation. Compare Fig. 5 with Fig. 3 and Fig. 6 with Fig. 4.

Fig. 3. g1n computed by the localized approximation for w0x 5 3 mm, w0y 5 2 mm, l 5 0.5145 mm, x0 5 y0 5 0, and z0 5 0.2 mm. 4222

APPLIED OPTICS y Vol. 37, No. 19 y 1 July 1998

Fig. 5. g1n computed by the integral localized approximation for w0x 5 3 mm, w0y 5 2 mm, l 5 0.5145 mm, x0 5 y0 5 0, and z0 5 0.2 mm.

Fig. 6. g1n computed by the integral localized approximation for w0x 5 3 mm, w0y 5 2 mm, l 5 0.5145 mm, x0 5 0, y0 5 16.5 mm, and z0 5 0.2 mm.

Outside the instability ranges versus n the localized approximation and the integral localized approximation agree, as they should, because the two techniques are analytically equivalent. This equivalence also points out the fact that the instabilities observed with the localized approximation for laser sheets have a numerical origin. The corresponding numerical problems are automatically avoided by use of the integral localized approximation. 5. TEM01* Laser Beam

We now apply the localized approximation to the case of a beam that does not share the Gaussian properties ~along x and along y! exhibited in the two previous cases, namely, to a TEM01* laser beam. We use the description available from Roosen and Imbert16 in which, in the plane z 5 0, Ex 5

S D 8m0 cF pw2

1y2

S D

r2 r exp 2 2 , w w

Ey 5 0,

(27) (28)

in which w is a width parameter of the beam. Let us insist on the fact that the description is not given everywhere in space but only at z 5 0. This condition prevents us from using the quadrature or the finite-series technique to evaluate the BSC’s but perfectly matches the possibilities of the localized approximations ~end of Section 2!. The radial component Er of the electric field Ei then reads as

S D

r r2 Er 5 E0 cos w exp 2 2 . w w

(29)

* F S DG 2p

0

r r2 ˆ G exp 2 2 cos w exp~2imw!dw. w w (30)

We then recall that ˆ ~r! 5 G

1 ~n 1 1⁄2! 5 rn, k

m n,TM

g

5 S D

rn r2n exp 2 2 5 2w w 0

umu 5 1

.

(32)

umu Þ 1

Next, a TEM01* laser beam possesses a symmetry around its axis of propagation, so it can be viewed as an axisymmetric laser beam in the sense defined in Ref. 17. With Er proportional to cos w, it is furthermore what we call a special axisymmetric beam, again in the sense defined in Ref. 17. From Eq. ~29!, 1 21 in Ref. 17, and noting that gn,TM 5 gn,TM @Eq. ~32!#, we can then define a set of special beam-shape coefficients gn that read as 1 21 gn 5 2gn,TM 5 2gn,TM 5

S D

r2 rn exp 2 2 . w w

(31)

(33)

These special BSC’s do not distinguish between the TM and TE contributions to the electromagnetic field and therefore also automatically incorporate the description of the magnetic field that is not originally given by Eqs. ~27! and ~28!. Once the special BSC’s are known, we can write down partial wave expansions of the electric and magnetic field components. In particular, the expansions for the electric field components read as17–19 Er 5

` E0 cos f ~2i!n11~2n 1 1! R2 n51

(

3 gncn~R! Pn1~cos u!, Eu 5

(34)

` E0 2n 1 1 cos f ~2i!n11 R n~n 1 1! n51

(

3 [email protected]~R!tn~cos u! 1 icn~R!pn~cos u!#,

We then also note that ]Ery]r Þ 0 at r 5 0, an unpleasant fact that is enough to permit us to claim that the model provided by Eqs. ~27! and ~28! is not a perfect description of the physics of a TEM01* laser beam. Nevertheless, the localized approximations will be able to handle this model correctly. In the present case, however, in which Er is proportional to cos w, the decomposition of Er into azimuthal waves Erm is easy. We can therefore decide to use either the localized approximation or the integral localized approximation, and, in both cases, the BSC’s will be obtained under the form of ~equivalent! analytical expressions. In particular, if we insert Eq. ~29! into Eq. ~9! we obtain Zm n m gn,TM 5 2p

so we readily obtain

Ef 5 2

(35)

` E0 2n 1 1 ~2i!n11 sin f R n~n 1 1! n51

(

3 [email protected]~R!pn~cos u! 2 icn~R!tn~cos u!#,

(36)

in which R 5 kr,

(37)

cn~R! 5 Rjn~R!,

(38)

pn~cos u! 5

Pn1~cos u! , sin u

(39)

tn~cos u! 5

d Pn1~cos u!, du

(40)

in which jn~R! are the spherical Bessel functions and Pn1~cos u! are the associated Legendre polynomials. Because the BSC’s evaluated by the localized approximations cannot be compared with BSC’s evaluated by other techniques, we compare directly the modulus of the electric field provided by Eqs. ~27! and ~28!, called the original field, and the modulus of the electric field deduced from Eqs. ~34!–~36!, called the 1 July 1998 y Vol. 37, No. 19 y APPLIED OPTICS

4223

Fig. 7. Comparison of the original and the reconstructed electric fields for a TEM01* laser beam model with w 5 5 mm and l 5 0.6328 mm.

reconstructed field. Such a comparison is shown in Fig. 7. The perfect agreement between the two kinds of field exemplifies once more the validity of the localized approximations. 6. Computational Times

In this section we discuss computational times and refine our statements concerning the difference in speed between the localized approximation and the integral approximation. It happens that computational times for the integral localized approximation cannot be much larger than for the localized approximation, although numerical quadratures are involved. Actually, depending on the beam under study, it may even happen that the integral localized approximation is less time consuming. As an example, the localized approximation leads to the evaluation of single summations for Gaussian beams6 and to double summations for laser sheets,7 whereas, in both cases, the integral localized approximation requires the evaluation of single quadratures. It then happens that, for laser sheets, the integral localized Table 2. Computational time ~CPU! in secondsa

nmax y0 5 0, N 5 32 200 400 600 800 900

LA 0.9 2.7 6.2 11.9 17.5

y0 5 16.5, N 5 64 100 384 200 1,046 300 4,254 400 13,900 500 38,636

ILA 2.4 4.8 7.2 9.8 10.9 2.3 4.6 6.9 9.2 11.4

LA, localized approximation; ILA, integral localized approximation. 4224

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approximation may be less time consuming than the localized approximation. We exemplify this statement by comparing computational times ~on a workstation! for the same cases 1 as in Figs. 3– 6 ~Table 2!. The BSC’s gn,TM are computed for n # nmax, with N integration points in the integral localized approximation. The portion of Table 2 in which y0 5 0 and N 5 32 corresponds to the cases in Figs. 3 and 5, i.e., to an axis situation for laser sheets. The localized approximation is less time consuming than the integral approximation for nmax up to 600, but, for larger values of nmax, it becomes more time consuming. Conversely, the bottom half of Table 2, which corresponds to the off-axis situation in Figs. 4 and 6, the integral localized approximation is much less time consuming than the localized approximation. In such a case the integral localized approximation is superior in flexibility, stability, and also speed to the localized approximation and therefore provides the most efficient scheme available to evaluate BSC’s. 7. Conclusion

In this paper we have introduced the integral localized approximation to evaluate beam-shape coefficients in generalized Lorenz–Mie theory. This method exhibits good properties of speed, flexibility, and stability. In particular, the gain in flexibility is impressive and the loss of speed is insignificant. Also, in some cases, the integral localized approximation may even be much less time consuming than the localized approximation. We have exemplified this technique by considering the cases of Gaussian beams and laser sheets. We also considered the case of a model of a TEM01* laser beam for which both the nonintegral and the integral localized approximations led to analytical expressions for the beam-shape coefficients. This last example demonstrated once more the efficiency of the localized approximations in modeling shaped beams. References 1. G. Gouesbet, B. Maheu, and G. Gre´han, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 ~1988!. 2. G. Gouesbet, “Generalized Lorenz–Mie theory and applications,” Part. Part. Syst. Charact. 11, 22–34 ~1994!. 3. G. Gouesbet, C. Letellier, K. F. Ren, and G. Gre´han, “Discussion of two quadrature methods to evaluate beam shape coefficients in generalized Lorenz–Mie theory,” Appl. Opt. 35, 1537–1542 ~1996!. 4. G. Gouesbet, G. Gre´han, and B. Maheu, “Expressions to compute the coefficients gm n in the generalized Lorenz–Mie theory using finite series,” J. Opt. ~Paris! 19, 35– 48 ~1988!. 5. G. Gre´han, B. Maheu, and G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539 –3548 ~1986!. 6. G. Gouesbet, G. Gre´han, and B. Maheu, “A localized interpretation to compute all the coefficients gm n in the generalized Lorenz–Mie theory,” J. Opt. Soc. Am. A 7, 998 –1007 ~1990!. 7. K. F. Ren, G. Gre´han, and G. Gouesbet, “Evaluation of laser sheet beam shape coefficients in generalized Lorenz–Mie theory by using a localized approximation,” J. Opt. Soc. Am. A 11, 2072–2079 ~1994!.

8. G. Gouesbet, J. A. Lock, and G. Gre´han, “Partial wave representation of laser beams for use in light scattering calculations,” Appl. Opt. 34, 2133–2143 ~1995!. 9. H. C. van de Hulst, Light Scattering by Small Particles ~Wiley, New York, 1957!. 10. J. A. Lock, “The contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 ~1994!. 11. J. A. Lock and G. Gouesbet, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A 11, 2503–2515 ~1994!. 12. G. Gouesbet and J. A. Lock, “A rigorous justification of the localized approximation to the beam shape coefficients in the generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516 –2525 ~1994!. 13. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 ~1979!. 14. J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800 –2802 ~1989!.

15. K. F. Ren, G. Gre´han, and G. Gouesbet, “Electromagnetic field expression of a laser sheet and the order of approximation,” J. Opt. ~Paris! 25, 165–176 ~1994!. 16. G. Roosen and C. Imbert, “The TEM01* mode laser beam—a powerful tool for optical levitation of various types of spheres,” Opt. Commun. 26, 432– 436 ~1978!. 17. G. Gouesbet, “Partial wave expansions and properties of axisymmetric light beams,” Appl. Opt. 35, 1543–1555 ~1996!. 18. G. Gouesbet, G. Gre´han, and B. Maheu, “Scattering of a Gaussian beam by a Mie scatter center, using a Bromwich formulation,” J. Opt. ~Paris! 16, 83– 83 ~1985!; republished in Selected Papers on Light Scattering, M. Kerker, ed., Vol. 951 of SPIE Milestone Series ~SPIE, Bellingham, Wash., 1988!, Part I, pp. 361–371. 19. G. Gouesbet, G. Gre´han, and B. Maheu, “The order of approximation in a theory of the scattering of a Gaussian beam by Mie scatter center,” J. Opt. ~Paris! 16, 239 –247 ~1985!, republished in Selected Papers on Light Scattering, M. Kerker, ed., Vol. 951 of SPIE Milestone Series ~SPIE, Bellingham, Wash., 1988!, Part I, pp. 352–360.

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