Abstract

Basis vectors of a spherical system are shown to be linear combinations of functions commonly used in light scattering. These expressions are used to obtain an expansion for the unit dyadic as a linear combination of products of these functions, and this expansion is used to motivate the use of matrix orthonormality to obtain a complete set of sum rules for products of scalar light scattering functions. As an example demonstrating the utility of these expressions, sum rules are obtained for dyadic, component, and inner products of vector spherical harmonics used in the theory of light scattering.

© 2001 Optical Society of America

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References

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  1. J. D. Pendleton, D. L. Rosen, “Light scattering from an optically active sphere into a circular aperture,” Appl. Opt. 37, 7897–7905 (1998).
    [CrossRef]
  2. J. D. Pendleton, S. C. Hill, “Collection of emission from an oscillating dipole inside a sphere: analytical integration over a circular aperture,” Appl. Opt. 36, 8729–8737.
  3. G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), pp. 108, 118, 738.
  4. L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), p. 173.
  5. J. D. Pendleton, “Mie scattering into solid angles,” J. Opt. Soc. Am. 72, 1029–1033 (1982), Eq. (36).
    [CrossRef]
  6. M. E. Rose, Elementary Theory of Angular Momentum (Dover, Mineola, N.Y., 1995), pp. 54, 60, 103.
  7. D. M. Brink, G. R. Satchelor, Angular Momentum, 3rd. ed. (Clarendon, Oxford, UK, 1994), pp. 21, 22, 66, 67.
  8. M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), p. 124.
  9. T. B. Drew, Handbook of Vector and Polyadic Analysis (Reinhold, New York, 1961), pp. 19, 22.
  10. M. I. Mischenko, “Extinction of light by randomly oriented nonspherical grains,” Astrophys. Space Sci. 164, 1–13 (1990), Eqs. (36)–(38).
    [CrossRef]
  11. D. A. Varsholovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

1998 (1)

1990 (1)

M. I. Mischenko, “Extinction of light by randomly oriented nonspherical grains,” Astrophys. Space Sci. 164, 1–13 (1990), Eqs. (36)–(38).
[CrossRef]

1982 (1)

Arfken, G. B.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), pp. 108, 118, 738.

Brink, D. M.

D. M. Brink, G. R. Satchelor, Angular Momentum, 3rd. ed. (Clarendon, Oxford, UK, 1994), pp. 21, 22, 66, 67.

Drew, T. B.

T. B. Drew, Handbook of Vector and Polyadic Analysis (Reinhold, New York, 1961), pp. 19, 22.

Hill, S. C.

Khersonskii, V. K.

D. A. Varsholovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

Kong, J. A.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), p. 173.

Mischenko, M. I.

M. I. Mischenko, “Extinction of light by randomly oriented nonspherical grains,” Astrophys. Space Sci. 164, 1–13 (1990), Eqs. (36)–(38).
[CrossRef]

Moskalev, A. N.

D. A. Varsholovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

Pendleton, J. D.

Rose, M. E.

M. E. Rose, Elementary Theory of Angular Momentum (Dover, Mineola, N.Y., 1995), pp. 54, 60, 103.

Rosen, D. L.

Satchelor, G. R.

D. M. Brink, G. R. Satchelor, Angular Momentum, 3rd. ed. (Clarendon, Oxford, UK, 1994), pp. 21, 22, 66, 67.

Shin, R. T.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), p. 173.

Spiegel, M. R.

M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), p. 124.

Tsang, L.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), p. 173.

Varsholovich, D. A.

D. A. Varsholovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

Weber, H. J.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), pp. 108, 118, 738.

Appl. Opt. (2)

Astrophys. Space Sci. (1)

M. I. Mischenko, “Extinction of light by randomly oriented nonspherical grains,” Astrophys. Space Sci. 164, 1–13 (1990), Eqs. (36)–(38).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (7)

M. E. Rose, Elementary Theory of Angular Momentum (Dover, Mineola, N.Y., 1995), pp. 54, 60, 103.

D. M. Brink, G. R. Satchelor, Angular Momentum, 3rd. ed. (Clarendon, Oxford, UK, 1994), pp. 21, 22, 66, 67.

M. R. Spiegel, Mathematical Handbook of Formulas and Tables (McGraw-Hill, New York, 1968), p. 124.

T. B. Drew, Handbook of Vector and Polyadic Analysis (Reinhold, New York, 1961), pp. 19, 22.

G. B. Arfken, H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, San Diego, Calif., 1995), pp. 108, 118, 738.

L. Tsang, J. A. Kong, R. T. Shin, Theory of Microwave Remote Sensing (Wiley, New York, 1985), p. 173.

D. A. Varsholovich, A. N. Moskalev, V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).

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Equations (54)

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P¯nm(cos θ)(2n+1)(n-m)!2(n+m)!1/2Pnm(cos θ),
Pnm(μ)12nn! (1-μ2)m/2dn+mdμn+m (μ2-1)n
Ynm(θ, ϕ)=(-1)mP¯nm(cos θ)exp(imϕ)/2π,
Ynm(θ, ϕ)=(-1)mrexp(imϕ)2π [θˆτ¯nm(cos θ)+ϕˆiπ¯nm(cos θ)],
π¯nm(cos θ)mP¯nm(cos θ)sin θ,
τ¯nm(cos θ)dP¯nm(cos θ)dθ.
ξ^1-(xˆ+iyˆ)/2,
ξ^0zˆ,
ξ^-1(xˆ-iyˆ)/2,
r=xˆx+yˆy+zˆz,=r(xˆ sin θ cos ϕ+yˆ sin θ sin ϕ+zˆ cos ϕ),
r=rm=-1,0,14π31/2(-1)mY1,-m(θ, ϕ)ξ^m=rm=-1,0,14π31/2[Y1,m(θ, ϕ)]*ξ^m,
rˆ=r/r=m=-1,0,1231/2(-1)mP¯1,m(cos θ)exp(-imϕ)ξ^m,
rˆr/r|r/r|,
θˆr/θ|r/θ|=rˆ/θ,
ϕˆr/ϕ|r/ϕ|=(rˆ/ϕ)/sin θ.
rˆ=xˆ sin θ cos ϕ+yˆ sin θ sin ϕ+zˆ cos θ,
θˆ=xˆ cos θ cos ϕ+yˆ cos θ sin ϕ-zˆ sin θ,
ϕˆ=-xˆ sin ϕ+yˆ cos ϕ,
θˆ=m=-1,0,1231/2(-1)mdP¯1,m(cos θ)dθexp(-imϕ)ξ^m=m=-1,0,1231/2(-1)mτ¯1,m(cos θ)exp(-imϕ)ξ^m,
ϕˆ=m=-1,0,1231/2(-1)m-imP¯1,m(cos θ)sin θ×exp(-imϕ)ξ^m=-im=-1,0,1231/2(-1)mπ¯1,m(cos θ)exp(-imϕ)ξ^m.
{f(ϕ)}12πϕ=02πdϕf(ϕ),
Idxˆxˆ+yˆyˆ+zzˆ=rˆrˆ+θˆθˆ+ϕˆϕˆ=rˆr^*+θˆθ^*+ϕˆϕ^*,
Id={rˆr^*+θˆθ^*+ϕˆϕ^*}.
Id=23m=-1,0,1[(P¯1,m)2+(τ¯1,m)2+(π¯1,m)2]ξ^mξ^m*.
Id=m=-1,0,1ξ^mξ^m*
(P¯1,m)2+(τ¯1,m)2+(π¯1,m)2=32
M=-nndMmn(θ)dMmn(θ)=δmm,
M=-1,0,1[dMm1(θ)]2=1.
d±1,mn(θ)=2n(n+1)(2n+1)1/2(π¯nm±τ¯nm)
d0,mn(θ)=2(2n+1)1/2P¯nm.
m=-nndMmn(θ)dMmn(θ)=δMM,
m=-nn(π¯nm)2=m=-nn(τ¯nm)2=n(n+1)(2n+1)4,
m=-nnπ¯nmτ¯nm=0,
m=-nn(P¯nm)2=(2n+1)2,
m=-nnP¯nmπ¯nm=m=-nnP¯nmτ¯nm=0.
M¯nm(1)(kr)=ijn(ρ)exp(imϕ)(θˆπ¯nm+iϕˆτ¯nm),
N¯nm(1)(kr)=exp(imϕ)rˆn(n+1) jn(ρ)ρ P¯nm+ψn(ρ)ρ (θˆτ¯nm+iϕˆπ¯nm),
M¯nm(1)*(kr)=(-1)mM¯n,-m(1)(kr),
N¯nm(1)*(kr)=(-1)mN¯n,-m(1)(kr),
m=-nnM¯nm(1)(kr)[M¯nm(1)(kr)]*n(n+1)
=(θˆθˆ+ϕˆϕˆ) 2n+14 [jn(ρ)]2,
m=-nnN¯nm(1)(kr)[N¯nm(1)(kr)]*n(n+1)
=rˆrˆ n(n+1)(2n+1)2[jn(ρ)]2ρ2+(θˆθˆ+ϕˆϕˆ) 2n+14[ψn(ρ)]2ρ2.
n=0m=-nnM¯nm(1)(kr)[M¯nm(1)(kr)]*+N¯nm(1)(kr)[N¯nm(1)(kr)]*n(n+1)
=13 Id.
n=0m=-nnM¯nm(1)(kr)[N¯nm(1)(kr)]*+N¯nm(1)(kr)[M¯nm(1)(kr)]*n(n+1)
=0.
AB  CD=(AC)(BD),
Id=k=13e^k*e^k,
Ide^je^j*=δjj,
n=0m=-nn[M¯nm(1)(kr)e^j][M¯nm(1)(kr)e^j]*+[N¯nm(1)(kr)e^j][N¯nm(1)(kr)e^j]*n(n+1)=13 δjj,
n=0m=-nn[M¯nm(1)(kr)e^j][N¯nm(1)(kr)e^j]*+[N¯nm(1)(kr)e^j][M¯nm(1)(kr)e^j]*n(n+1)=0.
n=0m=-nn[M¯nm(1)(kr)][M¯nm(1)(kr)]*+[N¯nm(1)(kr)][N¯nm(1)(kr)]*n(n+1)=1,
n=0m=-nn[M¯nm(1)(kr)][N¯nm(1)(kr)]*+[N¯nm(1)(kr)][M¯nm(1)(kr)]*n(n+1)=0.

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