Abstract

In Generalized Lorenz-Mie theories, (GLMTs), the most difficult task concerns the description of the illuminating beam. We provide an approach for expansions of the incident arbitrary shaped beam in spherical and spheroidal coordinates in the general case of oblique illumination. The representations for shaped beam coefficients are derived by using addition theorem for spherical vector wave functions under coordinate rotations. For

© 2007 Optical Society of America

1. Introduction

There is a rather large literature on the interaction of the incident shaped beam with some kinds of regular scatterers which has been of interest in such areas as particle sizing, optical tweezers, nonlinear optics, laser beam aerosol penetration, and so on. There are a number of mathematical theories for arbitrary beam scattering. Each of these theories relies on the decomposition of the incident beam into an infinite series of elementary constituents, such as partial waves or plane waves, with amplitudes and phases given by a set of beam-shape coefficients [1–10]. The generalized Lorenz-Mie theory (GLMT) developed by Gouesbet et al, one fundamental problem in it concerning the expansion of the incident shaped beam as a series of spherical vector wave functions, is effective to describe the electromagnetic scattering of a shaped beam by a spherical particle, in which the beam shape coefficients g m n are obtained when the shaped beam propagates parallel to the coordinate axis [1–3]. It has been extended to the case of multilayered spheres and of infinitely long circular cylinders [4,5]. Barton et al have calculated the intensity distributions internal and external to a sphere illuminated with a focused TEM00 mode laser beam by using spherical harmonic expansions of the scalar potential for the electric and magnetic fields and by matching the electromagnetic fields at the surface [7]. Such a procedure has been applied to the study of arbitrary beam scattering by a spheroidal particle, but Barton didn’t give the expressions of the beam shape coefficients of the shaped beam [8]. An approach presented by Khaled et al uses the angular spectrum of plane waves to model the shaped beam and the T-matrix method (TMM) or extended boundary condition method (EBCM) to compute the fields inside and outside a sphere [9]. Some numerical methods, such as Fourier Lorenz-Mie theory, mutiple multiple method (MMP), discrete dipole approximation (DDA), etc., have been used to study scattering of Gaussian beam by particles [11, 12, 13].

A strong effort has recently been devoted by us to the study of arbitrary beam scattering by spheroids, due to the fact that a large number of real objects can be modeled by spheroids with appropriate axial ratios and to the possibility of subsequent applications of exact analytical methods. In our previous papers, we studied the scattering of a spheroidal particle illuminated by a Gaussian beam propagating along the axis, and the expansion coefficients of the shaped beam in terms of spheroidal vector wave functions are determined [14–17]. In this paper, the approach to expand an arbitrary shaped beam in spherical and spheroidal coordinates in the general case of oblique illumination is presented. Once the beam-shape coefficients are determined, the solution of scattering for arbitrary shaped beam in oblique illumination by a spherical and spheroidal particle can be obtained by means of the method of separation of variables.

The paper is organized as follows. Section 2 provides a procedure to expand the incident arbitrary shaped beam in oblique illumination, and the beam shape coefficients ms gmsn, msn and Gmn,mn, corresponding to spherical and spheroidal coordinates respectively, are given. In section 3, for particle center located on Gaussian beam axis and plane wave, the simplified representations for beam shape coefficients are obtained. The convergence of the beam shape coefficients is discussed in section 4. Section 5 is a conclusion.

2. Expansion of incident shaped beam with respect to spherical and spheroidal coordinates

2.1. Shaped beam coefficients in spherical coordinates

Figures 1 and 2 show the geometry of the scatterer. The shaped beam propagates in free space and from the negative z´ to the positive z´ of O´x´y´z´, with the middle of beam waist located at origin O´. The system Ox´´y´´z´´ is parallel to O´x´y´z´ and the Cartesian coordinates of O´ in Ox´´y´´z´´ are x 0, y 0, z 0. The center of the scatterer is located at the point O of the Cartesian coordinate system Oxyz which is obtained by rotating the system Ox´´y´´z´´ through Euler 23]. For spheroidal particle, the major axe of the spheroid is along the z axis. In this paper, we assume that the time-dependent part of the electromagnetic fields is e-iωt.

 

Fig. 1. The center of an arbitrarily oriented scatterer is located at origin O of the Cartesian coordinate systems Oxyz and Ox´´y´´z´´. (For spheroid, the major axe along the z axis of Oxyz ). The xyz axes are obtained by a rigid-body rotation of the x´´y´´z´´ axes through Euler angles α, β, γ. The scatterer is illuminated by a shaped beam propagating along the z´ axis with the middle of its waist located at origin O´. Theoz´´ is parallel to ´oz´ , and with similar conditions for the other axes. The Cartesian coordinates of O´ in the system Ox´´y´´z´´ are (x 0,y 0,z 0).

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Fig. 2. Geometry of the scattering description for a scatterer in the Cartesian coordinate system Oxyz. The rotation axis of the spheroid is the z axis and its orientation in space is specified by the Euler angles α,β,gamma; of the xyz axes with respect to the x´´y´´z´´ axes.

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The incident shaped beam can be expanded in terms of spherical vector wave functions mmnoer(1)(kr,θ´´,ϕ´´) and nmnoer(1)(kr,θ´´,ϕ´´) natural to the system Ox´´y´´z´´ [1, 14] (TE Mode)

Ei=E0m=0n=m[gn,TEm¯memnr(1)(kr,θ´´,ϕ´´)+g'n,TEm¯momnr(1)(kr,θ´´,ϕ´´)+ig'n,TMm¯nemnr(1)(kr,θ´´,ϕ´´)+ign,TMm¯nomnr(1)(kr,θ´´,ϕ´´)]

where gn,TEm¯, g'n,TEm¯, gn,TMm¯ g'n,TMm¯, z´ axis. They can be 1–3], and by an approach presented by A. Doicu using the translational addition theorem for spherical vector wave functions [6] (see Appendix).

We have found it convenient to describe the addition theorem for spherical scalar wave functions under coordinate rotations by referring to the common forms in quantum mechanics presented by Edmonds [23].

Pnm(cosθ´´)eimθ´´=s=nnρ(m,s,n)Pns(cosθ)eisϕ

The coordinates θ,ϕ are the angle coordinates of a point in space defined in the usual manner with respect to the xyz axes, and θ ´´,ϕ´´ to the x´´y´´z´´ axes. Here the associated Legendre functions Pmn(cosθ) 27].

The coefficients ρ(m,s,n) are given by

ρ(m,s,n)=(1)s+meisγ[(n+m)!(ns)!(nm)!(n+s)!]12usm(n)(β)eimα
usm(n)(β)=[(n+s)!(ns)!(n+m)!(nm)!]12σn+mnsσnmσ(1)nsσ(cosβ2)2σ+s+m(sinβ2)2n2σsm

Let ρ(m,s,n) = ρ1(m,s,n) + iρ2(m,s,n). ρ1(m,s,n) and ρ2(m,s,n) are the real and imaginary parts of ρ(m,s,n) respectively.

Substituting ρ(m,s,n)into Eq. (2) and having the real and imaginary parts on both sides of the equality be equal respectively, we can obtain

Pnm(cosθ'')cos(mϕ'')sin(mϕ'')=s=nn[ρ1(m,s,n)Pns(cosθ)cos(sϕ)sin(sϕ)
+11ρ2(m,s,n)Pns(cosθ)sin(sϕ)cos(sϕ)]

Since the vector operator ∇ is invariant to a transformation of the coordinate system and r is the same position vector for both due to the common origin O of Cartesian coordinate systems Oxyz and Ox´´y´´z´´, by using Eq. (5) multiplied by the spherical Bessel functions jn(kr) and the definitions of mmnoer(1)(kr,θ,ϕ) nmnoer(1)(kr,θ,ϕ)[27], we have

wmnoer(1)(kr,θ'',ϕ'')=s=nnρ1(m,s,n)wsnoer(1)(kr,θ,ϕ)s=nnρ2(m,s,n)wsneor(1)(kr,θ,ϕ)

where w stands for the spherical vector wave functions m or n,

Taking into account the following relation

Pnm(cosθ)=(1)m(nm)!(n+m)!Pnm(cosθ)m>1

one can easily obtain from Eq. (5)

w-mnoer(i)(kr,θ,ϕ)=±(1)m(nm)!(n+m)!wmnoer(i)(kr,θ,ϕ)m>0

Substituting Eq. (6) into Eq. (1) and using Eq. (8), we can rewrite Eq. (1) or expand the incident beam in terms of the spherical vector wave functions mmnoer(1)(kr,θ,ϕ) and nmnoer(1)(kr,θ,ϕ) natural to the system Oxyz , as follows (TE Mode):

Ei=E0m=0n=ms=0n[gn,TEmsmesnr(1)(kr,θ,ϕ)g'n,TEmsmosnr(1)(kr,θ,ϕ)
+ig'n,TMmsnesnr(1)(kr,θ,ϕ)+ign,TMmsnosnr(1)(kr,θ,ϕ)]

where gn,TEms,gn,TE'ms,gn,TMms     and     gn,TM'ms are the expansion coefficients for arbitrary shaped beam in the spherical coordinates at oblique incidence

gn,TEmsg'n,TEmsgn,TMmsg'n,TMms=gn,TEm¯gn,TEm¯gn,TMm¯gn,TMm¯[ρ1(m,s,n)ρ2(m,s,n)ρ1(m,s,n)ρ2(m,s,n)+1111(1δ0s)(1)s(ns)!(n+s)!ρ1(m,s,n)ρ2(m,s,n)ρ1(m,s,n)ρ2(m,s,n)]
+1111g'n,TEm¯g'n,TEm¯g'n,TMm¯g'n,TMm¯[ρ2(m,s,n)ρ1(m,s,n)ρ2(m,s,n)ρ1(m,s,n)+1111(1δ0s)(1)s(ns)!(n+s)!ρ2(m,s,n)ρ1(m,s,n)ρ2(m,s,n)ρ1(m,s,n)]

2.2. Shaped beam coefficients in spheroidal coordinates

The following derivations are for the prolate spheroidal coordinate system, since the 26]

cic,ζiζ

The spherical wave functions can be expanded in terms of spheroidal ones[26].

Pnm(cosθ)jn(kr)=2(n+m)!(2n+1)(nm)!'l=m,m+1ilnNmldnmml(c)Sml(c,η)Rml(1)(c,ζ)

As similar derivation for Eq. (6) from Eq. (5), it follows immediately from Eq. (12) that the spherical and spheroidal vector wave functions are related by

wmnoer(1)(kr,θ,ϕ)=l=m=m+l'2(n+m)!(2n+1)(nm)!ilnNmldnmml(c)wmloer(1)(c,ζ,η,ϕ)

in which W stands for the spheroidal vector wave functions M or N.

Substituting Eq. (13) into Eq. (9) and considering that the summation of the series n=0m=0namnWmnr(i)is equal to that of the series m=0n=mamnWmnr(i), with amn representing the expansion coefficients, we can obtain

Ei=E0s=0l=sn=s,s+1m=0n2(n+s)!(2n+1)(ns)!ilnNsldnssl(c)[gn,TEmsMeslr(1)(c,ζ,η,ϕ)
g'n,TEmsMoslr(1)(c,ζ,η,ϕ)+ig'n,TMmsNeslr(1)(c,ζ,η,ϕ)+ign,TMmsNoslr(1)(c,ζ,η,ϕ)]

Let n - s = r. One replaces s by m,l by n and m by s , then Eq. (14) or the expansion of the incident beam in terms of spheroidal vector wave functions attached to an arbitrarily oriented spheroid can be written as follows (TE Mode):

Ei=E0m=0n=m[Gn,TEmMemnr(1)(c,ζ,η,ϕ)G'n,TEmMomnr(1)(c,ζ,η,ϕ)
+iGn,TMmNomnr(1)(c,ζ,η,ϕ)+iG'n,TMmNemnr(1)c,ζ,η,ϕ)]

where Gn,TEms,Gn,TE'ms,Gn,TMms         and      Gn,TM'ms are the expansion coefficients for arbitrary shaped beam at oblique incidence in the spheroidal coordinates.

[Gn,TEmG'n,TEmGn,TMmG'n,TMm]='r=0,1s=0r+m2(r+2m)!(2r+2m+1)r!irmNmndrmn(c)[gr+msm,TEg'r+msm,TEgr+msm,TMg'r+msm,TM]

Once the beam-shape coefficients in oblique illumination are determined in spherical and spheroidal coordinates, the incident, scattered and internal fields can be expanded in terms of spherical and spheroidal vector wave functions. For spheroidal particle, as an example, incident fields are given in Eq. (14) for TE mode, and the internal and scattered fields can be expressed as following [17]:

Ew=E0m=0n=min[δmnMemnr(1)(c,ζ,η,ϕ)+mnNomnr(1)(c,ζ,η,ϕ)
+δ'mnMomnr(1)(c,ζ,η,ϕ)+iγ'mnNemnr(1)(c,ζ,η,ϕ)]
Es=E0m=0n=min[βmnMemnr(3)(c,ζ,η,ϕβ'mnMomnr(3)(c,ζ,η,ϕ)
+iαmnNomnr(3)(c,ζ,η,ϕ)+iα'mnNemnr(3)(c,ζ,η,ϕ)]

The corresponding magnetic fields can be obtained with the following relations:

H=1iwμ×EMmnoe=1kNmnoeNmnoe=1kMmnoe

The unknown coefficients (γmn, δmn, γmn',δmn' , αmn, βmn, αmn',βmn' ) are determined by applying the boundary conditions of continuity of the tangential electromagnetic fields over the surface of the particle. Thus, the solution of scattering for arbitrary shaped beam by a homogeneous spheroidal particle can be obtained

3. Expressions of beam shape coefficients for particle center located on Gaussian beam axis or plane wave illumination

Equations (10) and (15) enable us to compute beam shape coefficients for sphere and spheroid at oblique illumination in all case. It may be interesting to calculate these coefficients for the special cases.

For particle center located on Gaussian beam axis (x 0 = 0, y 0 = 0) or plane wave illumination (x 0 = 0, y 0 = 0,z 0, w 2 → ∞, we have [1] [6]

gn,TEm¯=gn,TEm¯=gn,g'n,TEm¯=g'n,TMm¯=0m=1gn,TEm¯=g'n,TEm¯=gn,TMm¯=g'n,TMm¯=0m1}
gn,TE1mg'n,TE1mgn,TM1mg'n,TM1m=(1)m1(nm)!(n+m)!gn[(2δm0)dPnm(cosβ)dβcos(mγ)cosαsin(mγ)cosαsin(mγ)sinαcos(mγ)sinα
+2mPnm(cosβ)sinβsin(mγ)sinαcos(mγ)sinαcos(mγ)cosαsin(mγ)cosα]

By substituting Eq. (21) into Eq. (15), we obtain the simpler form of the beam shape coefficients for particle center on beam axis and plane wave (taking w 0 → ∞ in gn) as follows

Gn,TEmG'n,TEmGn,TMmG'n,TMm=2Nmnr=0,1/irm2r+2m+1drmn(c)gr+m(1)m1
×[(2δm0)dPr+mmdβcos(mγ)cosαsin(mγ)cosαsin(mγ)sinαcos(mγ)sinα+2mPr+mm(cosβ)sinβsin(mγ)sinαcos(mγ)sinαcos(mγ)cosαsin(mγ)cosα]

When α =γ = 0 , consider gr+m,=ir+m2r+2m+1(r+m)(r+m+1) for an incident plane wave, Eq. (15) and Eq. (22) can be reduced to

êyE0eikr(sinθcosφsinβ+cosθcosβ)=E0m=0n=min[Gn,TEmMemnr(1)(c,ζ,η,ϕ)+iGn,TMmNomnr(1)(c,ζ,η,ϕ]

where

Gn,TEmGn,TMm=2Nmn(1)m1r=0,1/drmn(c)(r+m)(r+m+1)(2δm0)dPr+mm(cosβ)2mPr+mm(cosβ)sinβ

Eq. (24) agrees with Eqs. (9.2.18) and (9.2.1) for plane wave in Flammer’s book [26, p72],with which to validate expressions for beam shape coefficients given in Section 2.

4. Numerical results

The expansion coefficients of a shaped beam in oblique illumination are given by Eqs. (10),(16), and those for particle center on beam axis and plane wave by Eq. (22). Numerical computations of the beam shape coefficients of a Gaussian beam are performed on the b of a/b=2 , and the size parameter 2πa/λ=5.

Figures 3 and 4 respectively show the convergence with n of the beam shape coefficients Gn,TMm,Gn,TEm , of particle center located on Gaussian beam axis with 2λ w 0=, given α=π/4,β=π/6,γ=0, x 0=0, y 0=0, z 0=0. Computational results indicate that Gn,TMm,Gn,TEm converge rapidly for increasing n and m.

 

Fig. 3. The convergence of the beam shape coefficients Gmn,TM

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Fig. 4. The convergence of the beam shape coefficients Gmn,TE

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5. Conclusion

A description of arbitrary shaped beam in the general case of oblique illumination is given by using addition theorem for spherical vector wave functions under coordinate rotations and relations between spheroidal vector wave functions and spherical ones. The representations for shaped beam coefficients are derived in spherical and spheroidal coordinates. For the special case of the plane wave, the simplified expressions are given, which agree with that in Flammer’s book. As a result, this approach provides a useful practical tool for calculating scattering of an arbitrary shaped beam by particles.

Appendix

At order L of approximation in the description of the Gaussian beam, the field components in the spherical coordinates (r,θ ´´,ϕ´´) with respect to Ox´´y´´z´´ are obtained as follows [1][25]

Er=E0ψ0[cosϕ′′sinθ′′(12Qlrcosθ′′)+2Qlx0cosθ′′]exp(K)
Eθ=E0ψ0[cosϕ′′(cosθ′′+2Qlrsin2θ′′)2Qlx0sinθ′′]exp(K)
Eϕ=E0ψ0sinϕ′′exp(K)
Hr,TE=H0ψ0[sinϕ′′sinθ′′(12Qlrcosθ′′)+2Qly0cosθ′′]exp(K)
Hϕ=H0ψ0[sinϕ′′(cosθ+2Qlrsin2θ′′)2Qly0sinθ′′]exp(K)
Hϕ=H0ψ0cosϕ′′exp(K)

in which

ψ0=iQexp[2iQw02rsinθ′′(x0cosϕ′′+ysinϕ′′)×exp(iQr2sin2θ′′w02)exp(iQx02+y02w02)
K=ik(rcosθz0)
Q=1i+(zz0)l

where l=kw02 , and w 0 is the beam’s electric-field half-width in the focal plane.

E 0 and H0 are linked by

E0H0=(με)1/2

When we replace E by - H, H by E, ε by μ, μ by ε , from Eqs. (A1)-(A6)order L of approximation can also be expressed by

Er,TE=E0ψ0[sinϕ′′sinθ′′(12Qlrcosθ′′)+2Qly0cosθ′′]exp(K)
Eθ=E0ψ0[sinϕ′′(cosθ′′+2Qlrsin2θ′′)2Qly0sinθ′′]exp(K)
Eϕ=E0ψ0cosϕ′′exp(K)
Hr=H0ψ0[cosϕ′′sinθ′′(12Qlrcosθ′′)+2Qlx0cosθ′′]exp(K)
Hθ=H0ψ0[cosϕ′′(cosθ′′+2Qlrsin2θ′′)2Qlx0sinθ′′]exp(K)
Hϕ=H0ψ0sinϕ′′exp(K)

The description of the Gaussian beam by Eqs. (A1-A6) is TM mode, and that of Eqs.

(A11-A16) is TE mode

The incident shaped beam can be expanded in terms of spherical vector wave functions mmnr(1)(kr,θ",φ") and nmnr(1)(kr,θ",φ") as in [1] [6]

Ei=E0n=1m=nnCnm[ign,TEmmmnr(1)(kr,θ′′,ϕ′′)+gn,TMmnmnr(1)(kr,θ′′,ϕ′′)]

in which the Cnm coefficients are normalized factors for negative values of the index m

Cnm={Cnm0(1)m(n+m)!(nm)!Cnm<0
(Cn=in12n+1n(n+1))

and

mmnr(1)(kr,θ′′,ϕ′′)nmnr(1)(kr,θ′′,ϕ′′)=memnr(1)(kr,θ′′,ϕ′′)nemnr(1)(kr,θ′′,ϕ′′)+imomnr(1)(kr,θ′′,ϕ′′)nomnr(1)(kr,θ′′,ϕ′′)

By substituting Eq. (A20) into Eq. (A17) and by considering Eq. (8), we can obtain Eq.(1), in which

(gn,TEm¯gn,TE'm¯gn,TMm¯gn,TM'm¯)=in2n+1n(n+1)1(1+δ0m)1ii1gn,TEm+gn,TEmgn,TEmgn,TEmgn,TMmgn,TMmgn,TMm+gn,TMm

Acknowledgments

This work is supported by NCET-04-0949 of China and by Nation Nature Science Foundation

References and links

1. G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam,using a Bromwich formulation,” J. Opt. Soc. Am. A 5,1427–1443 (1988). [CrossRef]  

2. G. Gouesbet, G. Gréhan, and B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. 27,4874–4883 (1988). [CrossRef]   [PubMed]  

3. G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients m ,” J. Opt. Soc. Am. A 7,998–1003 (1990). [CrossRef]  

4. K. F. Ren, G. Gréhan, and G. Gouesbet, “Scattering of a Gaussian beam by an infinite cylinder in the ,” J. Opt. Soc. Am. A 14, 3014–3025 (1997). [CrossRef]  

5. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, “Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres,” Appl. Opt. 36,5188–5198 (1997). [CrossRef]   [PubMed]  

6. A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. 36,2971–2978 (1997). [CrossRef]   [PubMed]  

7. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64,1632–1639 (1988). [CrossRef]  

8. J. P. Barton, “Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination,” Appl. Opt. 34,5542–5551 (1995). [CrossRef]   [PubMed]  

9. E. E. M. Khaled, S. C. Hill, and P. W. Barber, “Scattered and internal intensity of a sphere illuminated with a Gaussian beam,” IEEE Trans. Antennas Propagat. 41,259–303 (1993). [CrossRef]  

10. F. M. Schulz, K. Stamnes, and J. J. Stamnes, “Scattering of electromagnetic wave by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37,7875–7896 (1998). [CrossRef]  

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13. A. Doicu and T. Wriedt, “Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions,” J. Modern. Opt. 44,785–801 (1997). [CrossRef]  

14. Y. Han and Z. Wu, “Scattering of a spheroidal particle illuminated by a Gaussian beam,” Appl. Opt. 40,2501–2509 (2001). [CrossRef]  

15. Y. Han and Z. Wu, “The expansion coefficients of a Spheroidal Particle illuminated by Gaussian Beam,” IEEE Trans. Antennas Propagat. 49,615–620 (2001). [CrossRef]  

16. H. Y. Zhang and Y. P. Han, “Scattering by a confocal multilayered spheroidal particle illuminated by an axial Gaussian beam,” IEEE Trans. Antennas Propagat. 53,1514–1518 (2005). [CrossRef]  

17. Y. Han, H. Zhang, and X. Sun, “Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries,” Appl. Phys. B. 84,485–492 (2006). [CrossRef]  

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19. S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19,15–24 (1961).

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23. A. R. Edmonds, Angular momentum in quantum mechanics, (Princeton University Press, Princeton, N. J, 1957), Chap.4.

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References

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  1. G. Gouesbet, B. Maheu, and G. Gréhan, "Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation," J. Opt. Soc. Am. A 5, 1427-1443 (1988).
    [CrossRef]
  2. G. Gouesbet, G. Gréhan, and B. Maheu, "Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods," Appl. Opt. 27, 4874-4883 (1988).
    [CrossRef] [PubMed]
  3. G. Gouesbet, G. Gréhan, and B. Maheu, "Localized interpretation to compute all the coefficients gnm in the generalized Lorenz-Mie theory," J. Opt. Soc. Am. A 7, 998-1003 (1990).
    [CrossRef]
  4. K. F. Ren, G. Gréhan, and G. Gouesbet, "Scattering of a Gaussian beam by an infinite cylinder in the framework of generalized Lorenz-Mie theory formulation and numerical results," J. Opt. Soc. Am. A 14, 3014-3025 (1997).
    [CrossRef]
  5. Z. S. Wu, L. X. Guo, K. F. Ren, G. Gouesbet, and G. Gréhan, "Improved algorithm for electromagnetic scattering of plane waves and shaped beams by multilayered spheres," Appl. Opt. 36, 5188-5198 (1997).
    [CrossRef] [PubMed]
  6. A. Doicu and T. Wriedt, "Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions," Appl. Opt. 36, 2971-2978 (1997).
    [CrossRef] [PubMed]
  7. J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
    [CrossRef]
  8. J. P. Barton, "Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination," Appl. Opt. 34, 5542-5551 (1995).
    [CrossRef] [PubMed]
  9. E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propagat. 41, 259-303 (1993).
    [CrossRef]
  10. F. M. Schulz, K. Stamnes, and J. J. Stamnes, "Scattering of electromagnetic wave by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates," Appl. Opt. 37, 7875-7896 (1998).
    [CrossRef]
  11. H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999).
    [CrossRef]
  12. T. Evers, H. Dahl, and T. Wriedt, "Extension of the program 3D MMP with a fifth order Gaussian beam," Electron. Lett. 32, 1356-1357 (1996).
    [CrossRef]
  13. A. Doicu and T. Wriedt, "Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions," J. Modern. Opt. 44, 785-801 (1997).
    [CrossRef]
  14. Y. Han and Z. Wu, "Scattering of a spheroidal particle illuminated by a Gaussian beam," Appl. Opt. 40, 2501-2509 (2001).
    [CrossRef]
  15. Y. Han and Z. Wu, "The expansion coefficients of a Spheroidal Particle illuminated by Gaussian Beam," IEEE Trans. Antennas Propagat. 49, 615-620 (2001).
    [CrossRef]
  16. H. Y. Zhang and Y. P. Han, "Scattering by a confocal multilayered spheroidal particle illuminated by an axial Gaussian beam," IEEE Trans. Antennas Propagat. 53, 1514-1518 (2005).
    [CrossRef]
  17. Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B. 84, 485-492 (2006).
    [CrossRef]
  18. B. Friedman and J. Russek, "Addition theorems for spherical waves," Q. Appl. Math. 12, 13-23 (1954).
  19. S. Stein, "Addition theorems for spherical wave functions," Q. Appl. Math. 19, 15-24 (1961).
  20. O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).
  21. S. Asano and G. Yamamoto, "Light scattering by a spheroid particle," Appl. Opt. 14, 29-49 (1975).
    [PubMed]
  22. J. Dalmas and R. Deleuil, "Translational addition theorems for prolate spheroidal vector wave functions Mr and Nr ," Q. Appl. Math. 38,143-158 (1980).
  23. A. R. Edmonds, Angular momentum in quantum mechanics, (Princeton University Press, Princeton, N. J, 1957), Chap.4.
  24. B. P. Sinha and R. H. Macphie, "Translational addition theorems for spheroidal scalar and vector wave functions," Q. Appl. Math. 44,213-222 (1986).
  25. L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
    [CrossRef]
  26. C. Flammer, Spheroidal wave functions, (Stanford University Press, Stanford, California, 1957).
  27. J. A. Stratton, Electromagnetic Theory, (New York: McGraw-Hill, 1941).

2006

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B. 84, 485-492 (2006).
[CrossRef]

2005

H. Y. Zhang and Y. P. Han, "Scattering by a confocal multilayered spheroidal particle illuminated by an axial Gaussian beam," IEEE Trans. Antennas Propagat. 53, 1514-1518 (2005).
[CrossRef]

2001

Y. Han and Z. Wu, "The expansion coefficients of a Spheroidal Particle illuminated by Gaussian Beam," IEEE Trans. Antennas Propagat. 49, 615-620 (2001).
[CrossRef]

Y. Han and Z. Wu, "Scattering of a spheroidal particle illuminated by a Gaussian beam," Appl. Opt. 40, 2501-2509 (2001).
[CrossRef]

1999

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999).
[CrossRef]

1998

1997

1996

T. Evers, H. Dahl, and T. Wriedt, "Extension of the program 3D MMP with a fifth order Gaussian beam," Electron. Lett. 32, 1356-1357 (1996).
[CrossRef]

1995

1993

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propagat. 41, 259-303 (1993).
[CrossRef]

1990

1988

1986

B. P. Sinha and R. H. Macphie, "Translational addition theorems for spheroidal scalar and vector wave functions," Q. Appl. Math. 44,213-222 (1986).

1980

J. Dalmas and R. Deleuil, "Translational addition theorems for prolate spheroidal vector wave functions Mr and Nr ," Q. Appl. Math. 38,143-158 (1980).

1979

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

1975

1962

O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).

1961

S. Stein, "Addition theorems for spherical wave functions," Q. Appl. Math. 19, 15-24 (1961).

1954

B. Friedman and J. Russek, "Addition theorems for spherical waves," Q. Appl. Math. 12, 13-23 (1954).

Albrecht, H.-E.

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Asano, S.

Barber, P. W.

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propagat. 41, 259-303 (1993).
[CrossRef]

Barton, J. P.

J. P. Barton, "Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination," Appl. Opt. 34, 5542-5551 (1995).
[CrossRef] [PubMed]

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Borys, M.

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999).
[CrossRef]

Cruzan, O. R.

O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).

Dahl, H.

T. Evers, H. Dahl, and T. Wriedt, "Extension of the program 3D MMP with a fifth order Gaussian beam," Electron. Lett. 32, 1356-1357 (1996).
[CrossRef]

Dalmas, J.

J. Dalmas and R. Deleuil, "Translational addition theorems for prolate spheroidal vector wave functions Mr and Nr ," Q. Appl. Math. 38,143-158 (1980).

Damaschke, N.

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999).
[CrossRef]

Davis, L. W.

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Deleuil, R.

J. Dalmas and R. Deleuil, "Translational addition theorems for prolate spheroidal vector wave functions Mr and Nr ," Q. Appl. Math. 38,143-158 (1980).

Doicu, A.

A. Doicu and T. Wriedt, "Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions," J. Modern. Opt. 44, 785-801 (1997).
[CrossRef]

A. Doicu and T. Wriedt, "Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions," Appl. Opt. 36, 2971-2978 (1997).
[CrossRef] [PubMed]

Evers, T.

T. Evers, H. Dahl, and T. Wriedt, "Extension of the program 3D MMP with a fifth order Gaussian beam," Electron. Lett. 32, 1356-1357 (1996).
[CrossRef]

Friedman, B.

B. Friedman and J. Russek, "Addition theorems for spherical waves," Q. Appl. Math. 12, 13-23 (1954).

Gouesbet, G.

Gréhan, G.

Guo, L. X.

Han, Y.

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B. 84, 485-492 (2006).
[CrossRef]

Y. Han and Z. Wu, "Scattering of a spheroidal particle illuminated by a Gaussian beam," Appl. Opt. 40, 2501-2509 (2001).
[CrossRef]

Y. Han and Z. Wu, "The expansion coefficients of a Spheroidal Particle illuminated by Gaussian Beam," IEEE Trans. Antennas Propagat. 49, 615-620 (2001).
[CrossRef]

Han, Y. P.

H. Y. Zhang and Y. P. Han, "Scattering by a confocal multilayered spheroidal particle illuminated by an axial Gaussian beam," IEEE Trans. Antennas Propagat. 53, 1514-1518 (2005).
[CrossRef]

Hill, S. C.

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propagat. 41, 259-303 (1993).
[CrossRef]

Khaled, E. E. M.

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propagat. 41, 259-303 (1993).
[CrossRef]

Macphie, R. H.

B. P. Sinha and R. H. Macphie, "Translational addition theorems for spheroidal scalar and vector wave functions," Q. Appl. Math. 44,213-222 (1986).

Maheu, B.

Ren, K. F.

Russek, J.

B. Friedman and J. Russek, "Addition theorems for spherical waves," Q. Appl. Math. 12, 13-23 (1954).

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

Schulz, F. M.

Sinha, B. P.

B. P. Sinha and R. H. Macphie, "Translational addition theorems for spheroidal scalar and vector wave functions," Q. Appl. Math. 44,213-222 (1986).

Stamnes, J. J.

Stamnes, K.

Stein, S.

S. Stein, "Addition theorems for spherical wave functions," Q. Appl. Math. 19, 15-24 (1961).

Sun, X.

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B. 84, 485-492 (2006).
[CrossRef]

Tropea, C.

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999).
[CrossRef]

Wriedt, T.

A. Doicu and T. Wriedt, "Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions," J. Modern. Opt. 44, 785-801 (1997).
[CrossRef]

A. Doicu and T. Wriedt, "Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions," Appl. Opt. 36, 2971-2978 (1997).
[CrossRef] [PubMed]

T. Evers, H. Dahl, and T. Wriedt, "Extension of the program 3D MMP with a fifth order Gaussian beam," Electron. Lett. 32, 1356-1357 (1996).
[CrossRef]

Wu, Z.

Y. Han and Z. Wu, "The expansion coefficients of a Spheroidal Particle illuminated by Gaussian Beam," IEEE Trans. Antennas Propagat. 49, 615-620 (2001).
[CrossRef]

Y. Han and Z. Wu, "Scattering of a spheroidal particle illuminated by a Gaussian beam," Appl. Opt. 40, 2501-2509 (2001).
[CrossRef]

Wu, Z. S.

Yamamoto, G.

Zhang, H.

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B. 84, 485-492 (2006).
[CrossRef]

Zhang, H. Y.

H. Y. Zhang and Y. P. Han, "Scattering by a confocal multilayered spheroidal particle illuminated by an axial Gaussian beam," IEEE Trans. Antennas Propagat. 53, 1514-1518 (2005).
[CrossRef]

Appl. Opt.

Appl. Phys. B.

Y. Han, H. Zhang, and X. Sun, "Scattering of shaped beam by an arbitrarily oriented spheroid having layers with non-confocal boundaries," Appl. Phys. B. 84, 485-492 (2006).
[CrossRef]

Electron. Lett.

T. Evers, H. Dahl, and T. Wriedt, "Extension of the program 3D MMP with a fifth order Gaussian beam," Electron. Lett. 32, 1356-1357 (1996).
[CrossRef]

IEEE Trans. Antennas Propagat.

Y. Han and Z. Wu, "The expansion coefficients of a Spheroidal Particle illuminated by Gaussian Beam," IEEE Trans. Antennas Propagat. 49, 615-620 (2001).
[CrossRef]

H. Y. Zhang and Y. P. Han, "Scattering by a confocal multilayered spheroidal particle illuminated by an axial Gaussian beam," IEEE Trans. Antennas Propagat. 53, 1514-1518 (2005).
[CrossRef]

E. E. M. Khaled, S. C. Hill, and P. W. Barber, "Scattered and internal intensity of a sphere illuminated with a Gaussian beam," IEEE Trans. Antennas Propagat. 41, 259-303 (1993).
[CrossRef]

J. Appl. Phys.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam," J. Appl. Phys. 64, 1632-1639 (1988).
[CrossRef]

J. Modern. Opt.

A. Doicu and T. Wriedt, "Formulations of the extended boundary condition method for incident Gaussian beams using multiple-multipole expansions," J. Modern. Opt. 44, 785-801 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Meas. Sci. Technol.

H.-E. Albrecht, M. Borys, N. Damaschke, and C. Tropea, "The imaging properties of scattering particles in laser beams," Meas. Sci. Technol. 10,564-574 (1999).
[CrossRef]

Phys. Rev. A

L. W. Davis, "Theory of electromagnetic beam," Phys. Rev. A 19, 1177-1179 (1979).
[CrossRef]

Q. Appl. Math.

B. Friedman and J. Russek, "Addition theorems for spherical waves," Q. Appl. Math. 12, 13-23 (1954).

S. Stein, "Addition theorems for spherical wave functions," Q. Appl. Math. 19, 15-24 (1961).

O. R. Cruzan, "Translational addition theorems for spherical vector wave functions," Q. Appl. Math. 20, 33-40 (1962).

J. Dalmas and R. Deleuil, "Translational addition theorems for prolate spheroidal vector wave functions Mr and Nr ," Q. Appl. Math. 38,143-158 (1980).

B. P. Sinha and R. H. Macphie, "Translational addition theorems for spheroidal scalar and vector wave functions," Q. Appl. Math. 44,213-222 (1986).

Other

A. R. Edmonds, Angular momentum in quantum mechanics, (Princeton University Press, Princeton, N. J, 1957), Chap.4.

C. Flammer, Spheroidal wave functions, (Stanford University Press, Stanford, California, 1957).

J. A. Stratton, Electromagnetic Theory, (New York: McGraw-Hill, 1941).

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Figures (4)

Fig. 1.
Fig. 1.

The center of an arbitrarily oriented scatterer is located at origin O of the Cartesian coordinate systems Oxyz and Ox´´y´´z´´. (For spheroid, the major axe along the z axis of Oxyz ). The xyz axes are obtained by a rigid-body rotation of the x´´y´´z´´ axes through Euler angles α, β, γ. The scatterer is illuminated by a shaped beam propagating along the z´ axis with the middle of its waist located at origin O´. Theoz´´ is parallel to ´oz´ , and with similar conditions for the other axes. The Cartesian coordinates of O´ in the system Ox´´y´´z´´ are (x 0,y 0,z 0).

Fig. 2.
Fig. 2.

Geometry of the scattering description for a scatterer in the Cartesian coordinate system Oxyz . The rotation axis of the spheroid is the z axis and its orientation in space is specified by the Euler angles α,β,gamma; of the xyz axes with respect to the x´´y´´z´´ axes.

Fig. 3.
Fig. 3.

The convergence of the beam shape coefficients Gm n,TM

Fig. 4.
Fig. 4.

The convergence of the beam shape coefficients Gm n,TE

Equations (54)

Equations on this page are rendered with MathJax. Learn more.

E i = E 0 m = 0 n = m [ g n , T E m ¯ m emn r ( 1 ) ( kr , θ ´ ´ , ϕ ´ ´ ) + g ' n , TE m ¯ m omn r ( 1 ) ( kr , θ ´ ´ , ϕ´´ ) + i g ' n , TM m ¯ n emn r ( 1 ) ( kr , θ ´ ´ , ϕ ´ ´ ) + i g n , TM m ¯ n omn r ( 1 ) ( k r , θ ´ ´ , ϕ ´ ´ ) ]
P n m ( cos θ ´ ´ ) e imθ ´ ´ = s = n n ρ ( m , s , n ) P n s ( cos θ ) e i s ϕ
ρ ( m , s , n ) = ( 1 ) s + m e i s γ [ ( n + m ) ! ( n s ) ! ( n m ) ! ( n + s ) ! ] 1 2 u sm ( n ) ( β ) e i m α
u sm ( n ) ( β ) = [ ( n + s ) ! ( n s ) ! ( n + m ) ! ( n m ) ! ] 1 2 σ n + m n s σ n m σ ( 1 ) n s σ ( cos β 2 ) 2 σ + s + m ( sin β 2 ) 2 n 2 σ s m
P n m ( cos θ ' ' ) cos ( m ϕ ' ' ) sin ( m ϕ ' ' ) = s = n n [ ρ 1 ( m , s , n ) P n s ( cos θ ) cos ( s ϕ ) sin ( s ϕ )
+ 1 1 ρ 2 ( m , s , n ) P n s ( cos θ ) sin ( s ϕ ) cos ( s ϕ ) ]
w mn o e r ( 1 ) ( kr , θ ' ' , ϕ ' ' ) = s = n n ρ 1 ( m , s , n ) w sn o e r ( 1 ) ( kr , θ , ϕ ) s = n n ρ 2 ( m , s , n ) w sn e o r ( 1 ) ( k r , θ , ϕ )
P n m ( cos θ ) = ( 1 ) m ( n m ) ! ( n + m ) ! P n m ( cos θ ) m > 1
w -mn o e r ( i ) ( kr , θ , ϕ ) = ± ( 1 ) m ( n m ) ! ( n + m ) ! w mn o e r ( i ) ( kr , θ , ϕ ) m > 0
E i = E 0 m = 0 n = m s = 0 n [ g n , TE ms m esn r ( 1 ) ( kr , θ , ϕ ) g ' n , TE ms m osn r ( 1 ) ( kr , θ , ϕ )
+ ig ' n , TM ms n esn r ( 1 ) ( kr , θ , ϕ ) + ig n , TM ms n osn r ( 1 ) ( kr , θ , ϕ ) ]
g n , TE ms g ' n , TE ms g n , TM ms g ' n , TM ms = g n , T E m ¯ g n , T E m ¯ g n , T M m ¯ g n , T M m ¯ [ ρ 1 ( m , s , n ) ρ 2 ( m , s , n ) ρ 1 ( m , s , n ) ρ 2 ( m , s , n ) + 1 1 1 1 ( 1 δ 0 s ) ( 1 ) s ( n s ) ! ( n + s ) ! ρ 1 ( m , s , n ) ρ 2 ( m , s , n ) ρ 1 ( m , s , n ) ρ 2 ( m , s , n ) ]
+ 1 1 1 1 g' n , T E m ¯ g' n , T E m ¯ g' n , T M m ¯ g' n , T M m ¯ [ ρ 2 ( m , s , n ) ρ 1 ( m , s , n ) ρ 2 ( m , s , n ) ρ 1 ( m , s , n ) + 1 1 1 1 ( 1 δ 0 s ) ( 1 ) s ( n s ) ! ( n + s ) ! ρ 2 ( m , s , n ) ρ 1 ( m , s , n ) ρ 2 ( m , s , n ) ρ 1 ( m , s , n ) ]
c i c , ζ i ζ
P n m ( cos θ ) j n ( kr ) = 2 ( n + m ) ! ( 2 n + 1 ) ( n m ) ! ' l = m , m + 1 i l n N ml d n m ml ( c ) S ml ( c , η ) R ml ( 1 ) ( c , ζ )
w mn o e r ( 1 ) ( kr , θ , ϕ ) = l = m = m + l ' 2 ( n + m ) ! ( 2 n + 1 ) ( n m ) ! i l n N ml d n m ml ( c ) w ml o e r ( 1 ) ( c , ζ , η , ϕ )
E i = E 0 s = 0 l = s n = s , s + 1 m = 0 n 2 ( n + s ) ! ( 2 n + 1 ) ( n s ) ! i l n N sl d n s sl ( c ) [ g n , TE ms M esl r ( 1 ) ( c , ζ , η , ϕ )
g ' n , TE ms M osl r ( 1 ) ( c , ζ , η , ϕ ) + ig ' n , TM ms N esl r ( 1 ) ( c , ζ , η , ϕ ) + ig n , TM ms N osl r ( 1 ) ( c , ζ , η , ϕ ) ]
E i = E 0 m = 0 n = m [ G n , TE m M emn r ( 1 ) ( c , ζ , η , ϕ ) G ' n , TE m M o mn r ( 1 ) ( c , ζ , η , ϕ )
+ iG n , TM m N omn r ( 1 ) ( c , ζ , η , ϕ ) + iG ' n , TM m N emn r ( 1 ) c , ζ , η , ϕ ) ]
[ G n , TE m G ' n , TE m G n , TM m G ' n , TM m ] = ' r = 0,1 s = 0 r + m 2 ( r + 2 m ) ! ( 2 r + 2 m + 1 ) r ! i r m N mn d r mn ( c ) [ g r + m sm , TE g ' r + m sm , TE g r + m sm , TM g ' r + m sm , TM ]
E w = E 0 m = 0 n = m i n [ δ mn M emn r ( 1 ) ( c , ζ , η , ϕ ) + mn N omn r ( 1 ) ( c , ζ , η , ϕ )
+ δ ' mn M omn r ( 1 ) ( c , ζ , η , ϕ ) + i γ ' mn N emn r ( 1 ) ( c , ζ , η , ϕ ) ]
E s = E 0 m = 0 n = m i n [ β mn M emn r ( 3 ) ( c , ζ , η , ϕ β ' mn M omn r ( 3 ) ( c , ζ , η , ϕ )
+ i α mn N omn r ( 3 ) ( c , ζ , η , ϕ ) + i α ' mn N emn r ( 3 ) ( c , ζ , η , ϕ ) ]
H = 1 i w μ × E M mn o e = 1 k N mn o e N mn o e = 1 k M mn o e
g n , TE m ¯ = g n , TE m ¯ = g n , g ' n , TE m ¯ = g ' n , TM m ¯ = 0 m = 1 g n , TE m ¯ = g ' n , TE m ¯ = g n , TM m ¯ = g ' n , TM m ¯ = 0 m 1 }
g n , TE 1 m g ' n , TE 1 m g n , TM 1 m g ' n , TM 1 m = ( 1 ) m 1 ( n m ) ! ( n + m ) ! g n [ ( 2 δ m 0 ) dP n m ( cos β ) d β cos ( m γ ) cos α sin ( m γ ) cos α sin ( m γ ) sin α cos ( m γ ) sin α
+ 2 m P n m ( cos β ) sin β sin ( m γ ) sin α cos ( m γ ) sin α cos ( m γ ) cos α sin ( m γ ) cos α ]
G n , TE m G ' n , TE m G n , TM m G ' n , TM m = 2 N mn r = 0,1 / i r m 2 r + 2 m + 1 d r mn ( c ) g r + m ( 1 ) m 1
× [ ( 2 δ m 0 ) d P r + m m d β cos ( m γ ) cos α sin ( m γ ) cos α sin ( m γ ) sin α cos ( m γ ) sin α + 2 m P r + m m ( cos β ) sin β sin ( m γ ) sin α cos ( m γ ) sin α cos ( m γ ) cos α sin ( m γ ) cos α ]
ê y E 0 e ikr ( sin θ cos φ sin β + cos θ cos β ) = E 0 m = 0 n = m i n [ G n , TE m M emn r ( 1 ) ( c , ζ , η , ϕ ) + iG n , TM m N omn r ( 1 ) ( c , ζ , η , ϕ ]
G n , TE m G n , TM m = 2 N mn ( 1 ) m 1 r = 0,1 / d r mn ( c ) ( r + m ) ( r + m + 1 ) ( 2 δ m 0 ) dP r + m m ( cos β ) 2 m P r + m m ( cos β ) sin β
E r = E 0 ψ 0 [ cos ϕ′′ sin θ′′ ( 1 2 Q l r cos θ′′ ) + 2 Q l x 0 cos θ′′ ] exp ( K )
E θ = E 0 ψ 0 [ cos ϕ′′ ( cos θ′′ + 2 Q l r sin 2 θ′′ ) 2 Q l x 0 sin θ′′ ] exp ( K )
E ϕ = E 0 ψ 0 sin ϕ′′ exp ( K )
H r , TE = H 0 ψ 0 [ sin ϕ′′ sin θ′′ ( 1 2 Q l r cos θ′′ ) + 2 Q l y 0 cos θ′′ ] exp ( K )
H ϕ = H 0 ψ 0 [ sin ϕ′′ ( cos θ + 2 Q l r sin 2 θ′′ ) 2 Q l y 0 sin θ′′ ] exp ( K )
H ϕ = H 0 ψ 0 cos ϕ′′ exp ( K )
ψ 0 = iQ exp [ 2 iQ w 0 2 r sin θ′′ ( x 0 cos ϕ′′ + y sin ϕ′′ ) × exp ( iQ r 2 sin 2 θ′′ w 0 2 ) exp ( iQ x 0 2 + y 0 2 w 0 2 )
K = ik ( r cos θ z 0 )
Q = 1 i + ( z z 0 ) l
E 0 H 0 = ( μ ε ) 1 / 2
E r , TE = E 0 ψ 0 [ sin ϕ′′ sin θ′′ ( 1 2 Q l r cos θ′′ ) + 2 Q l y 0 cos θ′′ ] exp ( K )
E θ = E 0 ψ 0 [ sin ϕ′′ ( cos θ′′ + 2 Q l r sin 2 θ′′ ) 2 Q l y 0 sin θ′′ ] exp ( K )
E ϕ = E 0 ψ 0 cos ϕ′′ exp ( K )
H r = H 0 ψ 0 [ cos ϕ′′ sin θ′′ ( 1 2 Q l r cos θ′′ ) + 2 Q l x 0 cos θ′′ ] exp ( K )
H θ = H 0 ψ 0 [ cos ϕ′′ ( cos θ′′ + 2 Q l r sin 2 θ′′ ) 2 Q l x 0 sin θ′′ ] exp ( K )
H ϕ = H 0 ψ 0 sin ϕ′′ exp ( K )
E i = E 0 n = 1 m = n n C nm [ ig n , TE m m mn r ( 1 ) ( kr , θ′′ , ϕ′′ ) + g n , TM m n mn r ( 1 ) ( kr , θ′′ , ϕ′′ ) ]
C nm = { C n m 0 ( 1 ) m ( n + m ) ! ( n m ) ! C n m < 0
( C n = i n 1 2 n + 1 n ( n + 1 ) )
m mn r ( 1 ) ( kr , θ′′ , ϕ′′ ) n mn r ( 1 ) ( kr , θ′′ , ϕ′′ ) = m emn r ( 1 ) ( kr , θ′′ , ϕ′′ ) n emn r ( 1 ) ( kr , θ′′ , ϕ′′ ) + i m omn r ( 1 ) ( kr , θ′′ , ϕ′′ ) n omn r ( 1 ) ( kr , θ′′ , ϕ′′ )
( g n , T E m ¯ g n , T E ' m ¯ g n , T M m ¯ g n , T M ' m ¯ ) = i n 2 n + 1 n ( n + 1 ) 1 ( 1 + δ 0 m ) 1 i i 1 g n , TE m + g n , TE m g n , TE m g n , TE m g n , TM m g n , TM m g n , TM m + g n , TM m

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