Abstract

The closed-form expression of the angular spectrum of multipole fields, both scalar and vectorial, of any order and degree, evaluated across a plane orthogonal to an arbitrary (fixed) direction, is provided. Such a result has been obtained by starting from the Weyl representation of multipole fields and using suitable transformation rules. Moreover, as far as the vectorial case is concerned, knowledge of the (vectorial) transverse angular spectrum allows one to gain some insight into the polarization structure of the multipole fields evaluated across a typical plane. Such information could be useful, for instance, in those problems dealing with the interaction between planar partially reflecting surfaces and waves.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. Weyl, “Ausbreitung elektromagnetischer Wallen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
    [CrossRef]
  2. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  3. A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
    [CrossRef]
  4. P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
    [CrossRef]
  5. T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
    [CrossRef]
  6. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, Oxford, UK, 1966).
  7. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 13.2.1.
  8. A. T. Friberg, E. Wolf, “Angular spectrum representation of scattered electromagnetic fields,” J. Opt. Soc. Am. 73, 26–32 (1983).
    [CrossRef]
  9. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 3.2.4.
  10. A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966), Sec. 2.13.
  11. E. Wolf, M. Nieto-Vesperinas, “Analiticity of the angular spectrum amplitude of scattered fields and some of its consequences,” J. Opt. Soc. Am. A 2, 886–890 (1985).
    [CrossRef]
  12. Note that the definition given in Eq. (7) is intended to be valid when m>0.Spherical harmonics evaluated for negative values of mare given by (38)Yl,m(α, β)=(-1)mYl,-m*(α, β),with the asterisk denoting the complex conjugate.
  13. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, New York, 2000), Sec. 8.81.
  14. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998).
  15. C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, New York, 1977).

1998 (1)

T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

1986 (1)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

1985 (1)

1983 (1)

1974 (1)

A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[CrossRef]

1919 (1)

H. Weyl, “Ausbreitung elektromagnetischer Wallen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
[CrossRef]

Baños, A.

A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966), Sec. 2.13.

Bobbert, P. A.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 13.2.1.

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, Oxford, UK, 1966).

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, New York, 1977).

Devaney, A. J.

A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[CrossRef]

Diu, B.

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, New York, 1977).

Doicu, A.

T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Friberg, A. T.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, New York, 2000), Sec. 8.81.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998).

Laloë, F.

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, New York, 1977).

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 3.2.4.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Nieto-Vesperinas, M.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, New York, 2000), Sec. 8.81.

Vlieger, J.

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

Weyl, H.

H. Weyl, “Ausbreitung elektromagnetischer Wallen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
[CrossRef]

Wolf, E.

E. Wolf, M. Nieto-Vesperinas, “Analiticity of the angular spectrum amplitude of scattered fields and some of its consequences,” J. Opt. Soc. Am. A 2, 886–890 (1985).
[CrossRef]

A. T. Friberg, E. Wolf, “Angular spectrum representation of scattered electromagnetic fields,” J. Opt. Soc. Am. 73, 26–32 (1983).
[CrossRef]

A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 13.2.1.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 3.2.4.

Wriedt, T.

T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

Ann. Phys. (Leipzig) (1)

H. Weyl, “Ausbreitung elektromagnetischer Wallen über einem ebenen Leiter,” Ann. Phys. (Leipzig) 60, 481–500 (1919).
[CrossRef]

J. Math. Phys. (1)

A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representations of the electromagnetic field,” J. Math. Phys. 15, 234–244 (1974).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

T. Wriedt, A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

Physica A (1)

P. A. Bobbert, J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

Other (9)

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, Oxford, UK, 1966).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Sec. 13.2.1.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Sec. 3.2.4.

A. Baños, Dipole Radiation in the Presence of a Conducting Half-Space (Pergamon, New York, 1966), Sec. 2.13.

Note that the definition given in Eq. (7) is intended to be valid when m>0.Spherical harmonics evaluated for negative values of mare given by (38)Yl,m(α, β)=(-1)mYl,-m*(α, β),with the asterisk denoting the complex conjugate.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 6th ed. (Academic, New York, 2000), Sec. 8.81.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998).

C. Cohen-Tannoudji, B. Diu, F. Laloë, Quantum Mechanics (Wiley, New York, 1977).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Complex integration paths for obtaining Weyl expansion of spherical waves.

Fig. 2
Fig. 2

Behavior of normalized modulus of Φl,m(p; z) given by Eq. (12) as a function of λ|p|, evaluated across the plane z=λ, for l=0 and m=0.

Fig. 3
Fig. 3

The same as in Fig. 2, but for l=1, m=0 (solid curve), and m=1 (circles).

Fig. 4
Fig. 4

The same as in Fig. 2, but for l=2, m=0 (solid curve), m=1 (circles), and m=2 (triangles).

Equations (42)

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

exp(ikr)r=iλ-ππdβC±dαsinα expi2π sˆλr,
exp(ikr)r=R2d2pexp(i2πpr)Φ(p; z),
Φ(p; z)=i(1/λ2-p2)1/2exp[i2π(1/λ2-p2)1/2|z|].
px=sinαcosβλ,py=sinαsinβλ,
dβdαsinα=λd2p(1/λ2-p2)1/2.
Πl,m(r)=hl(kr)Yl,m(θ, ϕ),
Yl,m(θ, ϕ)=2l+14π(l-m)!(l+m)!1/2exp(imϕ)Pl(m)(cosθ),
exp(ikr)r=ikΠ0,0(r).
Πl,m(r)=(-i)l×i2π-ππdβC±dαsinαYl,m(α, β)×expi2π sˆλr.
exp(imβ)=px+ipy|p|m.
Πl,m(r)=R2d2pexp(i2πpr)Φl,m(p; z),
Φl,m(p; z)=Fl,m(p)exp[i2π(1/λ2-p2)1/2|z|],
Fl,m(p)=i-lk2l+14π(l-m)!(l+m)!1/2×px+ipy|p|mPl(m)(1-|λp|2)1/2(1/λ2-p2)1/2,
Fl,m(p)=(-1)mi-lk2l+14π(l-|m|)!(l+|m|)!1/2×px-ipy|p||m|Pl(|m|)(1-|λp|2)1/2(1/λ2-p2)1/2.
Nl,m(r)=ik×[rΠl,m(r)],
Ml,m(r)=××[rΠl,m(r)],
Nl,m(r)=(-i)l×ik2π-ππdβC±dαsinαYl,m(α, β)×expi2π sˆλr,
Yl,m(α, β)=LYl,m(α, β),
L=1ip×p.
L=L+u-+L-u++Lzzˆ,
L±Yl,m(α, β)=aYl,m±1(α, β),
a±=[(l±m)(lm+1)]1/2.
Nl,m(r)=R2d2pexp(i2πpr)Φl,m(p; z),
Φl,m(p; z)=k exp[i2π(1/λ2-p2)1/2|z|]×{12[a-Fl,m+1(p)+a+Fl,m-1(p)]xˆ+12i[a-Fl,m+1(p)-a+Fl,m-1(p)]yˆ+mFl,m(p)zˆ}.
Ml,m(r)=(-i)l×ik2π-ππdβC±dα×sinα[sˆ×Yl,m(α, β)]expi2π sˆλr.
Ml,m(r)=R2d2pexp(i2πpr)Ψl,m(p; z),
Ψl,m(p; z)=λp×Φl,m(p; z).
Φl,0(p; z)
=(-i)l2l+14π1/2expi2π1λ2-p21/2|z|×Pl(1)(1-|λp|2)1/2(1/λ2-p2)1/2py|p|xˆ-px|p|yˆ,
Ψl,0(p; z)
=(-i)l2l+14π1/2expi2π1λ2-p21/2|z|×Pl(1)(1-|λp|2)1/2(1/λ2-p2)1/2×(1-|λp|2)1/2p|p|-|λp|zˆ,
(l+m)(l-m+1)Pl(m-1)(λpz)=-Pl(m+1)(λpz)-2m pz|p| Pl(m)(λpz),
Φl,m(p; z)=i-lk2l+14π(l-m)!(l+m)!1/2×exp[i2π(1/λ2-p2)1/2|z|](1/λ2-p2)1/2×[i exp(imβ)Pl(m+1)(λpz)p^×zˆ-2mpz|p|exp(imβ)Pl(m)(λpz)u^+exp(-iβ)+m exp(imβ)Pl(m)(λpz)zˆ],
02πexp(imβ)cosβ exp[iβcos(β-ϕ)]dβ=πim+1exp(imϕ)[exp(iϕ)Jm+1(β)-exp(-iϕ)Jm-1(β)],
02πexp(imβ)sinβ exp[iβcos(β-ϕ)]dβ=πimexp(imϕ)[exp(iϕ)Jm+1(β)+exp(-iϕ)Jm-1(β)],
Nl,m(r, z)=exp(imϕ)f(|r|, z)r^+exp(imϕ)g(|r|, z)zˆ×r^+m exp[i(m-1)ϕ]h(|r|, z)u^++m exp(imϕ)l(|r|, z)zˆ,
p×(zˆ×p)= |p|2(zˆ+p^),
p×u^+=-ipzu^++i(px-ipy)zˆ,
p×zˆ=pyxˆ-pxyˆ,
Ψl,m(p)=exp(imβ)S(|p|; z)p^+m exp(imβ)T(|p|; z)zˆ×p^+m exp[i(m-1)β]U(|p|; z)u++{exp(imβ)V(|p|; z)+m exp[i(m-2)β]W(|p|; z)}zˆ,
Ml,m(r, z)=exp(imϕ)s(|r|, z)r^+exp(imϕ)t(|r|, z)zˆ×r^+m exp[i(m-1)ϕ]u(|r|, z)u^++{exp(imϕ)v(|r|; z)+m exp[i(m-2)ϕ]w(|r|, z)}zˆ,
Yl,m(α, β)=(-1)mYl,-m*(α, β),

Metrics