Abstract

The extinction paradox is examined by applying partial-wave analysis to a two-dimensional light beam interacting with a long transverse cylinder without absorption, assuming always short wavelengths. We show that the (conventional) power scattered, Psca, except for a very narrow beam hitting a transparent cylinder on axis, is always double the power directly intercepted by the scatterer, Pitc, including a zero result for Psca when the incident beam is basically off the material surface. This contradicts the interpretation that attributes one half of Psca to edge diffraction by the scatterer. Furthermore, we identify the shadow-forming wave (SFW) from the partial-wave sum in the forward direction and show that the actual power scattered or, equivalently, the power depleted from the incident beam is equal to one unit of Pitc for a narrow beam, gets larger for a broader beam, and approaches 2Pitc for a very broad beam. The larger value in the latter cases is due to the extent of divergence of the SFW beam out of the incident beam at distances well beyond the Rayleigh range.

© 2004 Optical Society of America

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References

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  1. J. A. Stratton, H. G. Houghton, “A theoretical investigation of the transmission of light through fog,” Phys. Rev. 38, 159–165 (1931).
    [CrossRef]
  2. D. Sinclair, “Light scattering by spherical particles,” J. Opt. Soc. Am. 37, 475–480 (1947).
    [CrossRef] [PubMed]
  3. L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).
    [CrossRef]
  4. J. M. Blatt, V. F. Weisskopf, Theoretical Nuclear Physics (Wiley, New York, 1952), pp. 324–325.
  5. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 2, pp. 1380–1381.
  6. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 107–108.
  7. K. Gottfried, Quantum Mechanics, Vol. 1, Fundamentals (Benjamin, Reading, Mass., 1966), pp. 106–108.
  8. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), pp. 104–108.
  9. J. B. Keller, “Quantum mechanical cross sections for small wavelengths,” Am. J. Phys. 40, 1035–1036 (1972).
    [CrossRef]
  10. R. Peierls, Surprises in Theoretical Physics (Princeton U. Press, Princeton, N.J., 1979), pp. 6–10 and references therein.
  11. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 107–111.
  12. J. A. Lock, “Interpretation of extinction in Gaussian-beam scattering,” J. Opt. Soc. Am. A 12, 929–938 (1995).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 722–723.
  14. J. J. Sakurai, Modern Quantum Mechanics, revised ed. (Addison-Wesley, Reading, Mass., 1994), pp. 406–410.
  15. R. O. Grumprecht, C. M. Sliepcevich, “Scattering of light by large spherical particles,” J. Phys. Chem. 57, 90–95 (1953).
    [CrossRef]
  16. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 495–502.
  17. G. Grehan, B. Maheu, G. Gouesbet, “Scattering of laser beams by Mie scatter centers: numerical results using a localized approximation,” Appl. Opt. 25, 3539–3548 (1986).
    [CrossRef] [PubMed]
  18. Ref. 6, p. 208.
  19. H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. (N.Y.) 34, 23–95 (1965).
    [CrossRef]
  20. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., prepared by Yu. V. Geronimus, M. Yu. Tseytlin and translated by A. Jeffrey (Academic, New York, 1965), p. 414.
  21. Ref. 6, p. 21.
  22. Ref. 13, p. 65, Eqs. (57) and (58).

1995 (1)

1986 (1)

1972 (1)

J. B. Keller, “Quantum mechanical cross sections for small wavelengths,” Am. J. Phys. 40, 1035–1036 (1972).
[CrossRef]

1965 (1)

H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. (N.Y.) 34, 23–95 (1965).
[CrossRef]

1953 (1)

R. O. Grumprecht, C. M. Sliepcevich, “Scattering of light by large spherical particles,” J. Phys. Chem. 57, 90–95 (1953).
[CrossRef]

1949 (1)

L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).
[CrossRef]

1947 (1)

1931 (1)

J. A. Stratton, H. G. Houghton, “A theoretical investigation of the transmission of light through fog,” Phys. Rev. 38, 159–165 (1931).
[CrossRef]

Blatt, J. M.

J. M. Blatt, V. F. Weisskopf, Theoretical Nuclear Physics (Wiley, New York, 1952), pp. 324–325.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 107–111.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 722–723.

Brillouin, L.

L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 2, pp. 1380–1381.

Geronimus, Yu. V.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., prepared by Yu. V. Geronimus, M. Yu. Tseytlin and translated by A. Jeffrey (Academic, New York, 1965), p. 414.

Gottfried, K.

K. Gottfried, Quantum Mechanics, Vol. 1, Fundamentals (Benjamin, Reading, Mass., 1966), pp. 106–108.

Gouesbet, G.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., prepared by Yu. V. Geronimus, M. Yu. Tseytlin and translated by A. Jeffrey (Academic, New York, 1965), p. 414.

Grehan, G.

Grumprecht, R. O.

R. O. Grumprecht, C. M. Sliepcevich, “Scattering of light by large spherical particles,” J. Phys. Chem. 57, 90–95 (1953).
[CrossRef]

Houghton, H. G.

J. A. Stratton, H. G. Houghton, “A theoretical investigation of the transmission of light through fog,” Phys. Rev. 38, 159–165 (1931).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 107–111.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 495–502.

Keller, J. B.

J. B. Keller, “Quantum mechanical cross sections for small wavelengths,” Am. J. Phys. 40, 1035–1036 (1972).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), pp. 104–108.

Lock, J. A.

Maheu, B.

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 2, pp. 1380–1381.

Nussenzveig, H. M.

H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. (N.Y.) 34, 23–95 (1965).
[CrossRef]

Peierls, R.

R. Peierls, Surprises in Theoretical Physics (Princeton U. Press, Princeton, N.J., 1979), pp. 6–10 and references therein.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., prepared by Yu. V. Geronimus, M. Yu. Tseytlin and translated by A. Jeffrey (Academic, New York, 1965), p. 414.

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics, revised ed. (Addison-Wesley, Reading, Mass., 1994), pp. 406–410.

Sinclair, D.

Sliepcevich, C. M.

R. O. Grumprecht, C. M. Sliepcevich, “Scattering of light by large spherical particles,” J. Phys. Chem. 57, 90–95 (1953).
[CrossRef]

Stratton, J. A.

J. A. Stratton, H. G. Houghton, “A theoretical investigation of the transmission of light through fog,” Phys. Rev. 38, 159–165 (1931).
[CrossRef]

Tseytlin, M. Yu.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., prepared by Yu. V. Geronimus, M. Yu. Tseytlin and translated by A. Jeffrey (Academic, New York, 1965), p. 414.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 107–108.

Weisskopf, V. F.

J. M. Blatt, V. F. Weisskopf, Theoretical Nuclear Physics (Wiley, New York, 1952), pp. 324–325.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 722–723.

Am. J. Phys. (1)

J. B. Keller, “Quantum mechanical cross sections for small wavelengths,” Am. J. Phys. 40, 1035–1036 (1972).
[CrossRef]

Ann. Phys. (N.Y.) (1)

H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. (N.Y.) 34, 23–95 (1965).
[CrossRef]

Appl. Opt. (1)

J. Appl. Phys. (1)

L. Brillouin, “The scattering cross section of spheres for electromagnetic waves,” J. Appl. Phys. 20, 1110–1125 (1949).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. Chem. (1)

R. O. Grumprecht, C. M. Sliepcevich, “Scattering of light by large spherical particles,” J. Phys. Chem. 57, 90–95 (1953).
[CrossRef]

Phys. Rev. (1)

J. A. Stratton, H. G. Houghton, “A theoretical investigation of the transmission of light through fog,” Phys. Rev. 38, 159–165 (1931).
[CrossRef]

Other (14)

R. Peierls, Surprises in Theoretical Physics (Princeton U. Press, Princeton, N.J., 1979), pp. 6–10 and references therein.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 107–111.

J. M. Blatt, V. F. Weisskopf, Theoretical Nuclear Physics (Wiley, New York, 1952), pp. 324–325.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 2, pp. 1380–1381.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 107–108.

K. Gottfried, Quantum Mechanics, Vol. 1, Fundamentals (Benjamin, Reading, Mass., 1966), pp. 106–108.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), pp. 104–108.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1999), pp. 495–502.

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999), pp. 722–723.

J. J. Sakurai, Modern Quantum Mechanics, revised ed. (Addison-Wesley, Reading, Mass., 1994), pp. 406–410.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed., prepared by Yu. V. Geronimus, M. Yu. Tseytlin and translated by A. Jeffrey (Academic, New York, 1965), p. 414.

Ref. 6, p. 21.

Ref. 13, p. 65, Eqs. (57) and (58).

Ref. 6, p. 208.

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Figures (8)

Fig. 1
Fig. 1

Sketch of the cross section of a (circular) cylinder and a 2D incident wave beam of width 2w focused at (x0, 0).

Fig. 2
Fig. 2

Three sets of graphs showing the scattering efficiency versus the focal position of a Gaussian beam of half-width w scanning a dielectric cylinder of radius a for s=10,000, 10,002, and 10,004, respectively, with w/a=0.05, 0.1, and 0.2 in each set. The bases for the three sets are appropriately chosen to be 0, 2, and 4, respectively.

Fig. 3
Fig. 3

Two sets of graphs showing the scattering efficiency versus the focal position of a Gaussian beam of half-width w scanning a perfectly conducting cylinder of radius a for s=104 and 103, respectively, with w/a=0.05, 0.1, and 0.2 in each set. The bases for the two sets are chosen to be 0 and 2, respectively.

Fig. 4
Fig. 4

The three solid curves give the scattering efficiency versus the focal position of a Gaussian beam of half-width w scanning a dielectric cylinder of radius a for w/a=1, 5, and 10, respectively, with s=103. The dashed curves give the corresponding 2Pitc/Pinc curves.

Fig. 5
Fig. 5

Sketch of a collimated yet narrow incident wave beam wholly striking the (circular) cylinder, the dashed curves showing the further path of the beam in the absence of the scatterer.

Fig. 6
Fig. 6

Sketch of the SFW beam, indicated by the dashed lines, in four regions behind the (circular) cylinder as a result of interaction with a very broad incident beam, indicated by the two parallel long solid lines (width not to scale). The bulk of the SFW expands completely out of the incident beam in the top region.

Fig. 7
Fig. 7

Actual scattering cross section versus w/a according to Eq. (33) for w/a1, where the dotted, dashed, and solid curves are for N=1, 2, and 3, respectively. The line from the origin to (1, 2a) for w/a1 is also drawn.

Fig. 8
Fig. 8

“Scattering efficiency” versus s for n=1.5. The dotted curve is from Eq. (A1), the dashed curve is the asymptotic result from Eq. (19), and the three solid curves are the numerical results from Eq. (14) for w/a=0.05 (curve 3), 0.02 (curve 2), and 0.005 (curve 1).

Equations (37)

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Einc(x, y)=-kkdkxE˜(kx; r0)exp(ikxx+ikyy),
sin γ=kx/k,
Einc(x, y)=mβmJm(kr)exp(imϕ)
βm=-kkdkxE˜(kx; r0)exp(imγ)
Esca(x, y)=imβm(sin δm)Hm(+)(kr)exp(imϕ+iδm),
tan δmJm(ns)Jm(s)-nJm(ns)Jm(s)Jm(ns)Ym(s)-nJm(ns)Ym(s)
tan δmJm(s)/Ym(s)
Psca=2μ0ωm|βm|2sin2 δm.
βm=Einc(m/k, 0)
Esq|m|kwJm(kr)exp(imϕ),
-dx|Einc(x, 0)|21km|Einc(m/k, 0)|2.
ηPsca/Pinc=4m|βm|2sin2 δmm|βm|2
σsca/2a=2smsin2 δm,
βm=exp-(m/k-x0)22w2
η=4πkwmsin2 δmexp-m-kx0kw2.
Pitc/Pinc-aadxc0|Einc|2/2Pinc,
η0for|x0|>a+w
η2for|x0|a
sin2 δm
{(n+1)sin[(n-1)s](n-1)cos[(n+1)s]}22[n2+1(n2-1)sin 2ns],
η2=1-2n[(n2+1)cos 2ns cos 2s+2n sin 2ns sin 2s](n2+1)2cos2 2ns+4n2sin2 2ns
η=21-(-1)p2nn2+1cospπn,
-dx0-kkdkxexp-kx2w22exp[ikx(x-x0)+ikyy],
Einc(x, y)=mβmHm(+)(kr)+Hm(-)(kr)2exp(imϕ),
Esca(x, y)=12mβm[exp(2iδm)-1]Hm(+)(kr)exp(imϕ)
Escaa(x, y)=12mβmHm(+)(kr)exp[i(mϕ+2δm)],
Pscaa=12μ0ωm|βm|2
Einc=|m|s+|m|>s×βmHm(+)(kr)+Hm(-)(kr)2exp(imϕ).
Esca=12|m|s[exp(2iδm)-1]Hm(+)(kr)exp(imϕ),
Esf=-122πkr1/2expikr-iπ4m=-ssexp(imψ),
Esf=-Aaexpikr-iπ4
Aa2πkr1/2sin(kaψ)ψ,
I(r, ψ)I0=2πkrsin2 kwψψ2+sin2 kaψψ2-2 sin kwψ sin kaψψ2,
PdepI0=2a-Nπa/wNπa/wdξ2 sinwξa-sin ξsin ξπξ2,
PdepI0=4aπ1+wasiNπ1+aw+1-wasiNπ1-aw-si2Nπaw+sin2(Nπa/w)Nπa/w,
si(x)0xsin ξξdξ.
η=2-2×2n[(n2+1)sin 2ns sin 2s+2n cos 2ns cos 2s](n2+1)2sin2 2ns+4n2cos2 2ns.

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