Mie scattering in the time domain. Part II. The role of diffraction

doi:10.1364/JOSAA.28.001096

Mie scattering in the time domain. Part II. The role of diffraction

James A. Lock and Philip Laven »View Author Affiliations

JOSA A, Vol. 28, Issue 6, pp. 1096-1106 (2011)
http://dx.doi.org/10.1364/JOSAA.28.001096

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Abstract

The $p = 0$ term of the Mie–Debye scattering amplitude contains the effects of external reflection and diffraction. We computed the reflected intensity in the time domain as a function of the scattering angle and delay time for a short electromagnetic pulse incident on a spherical particle and compared it to the predicted behavior in the forward-focusing region, the specular reflection region, and the glory region. We examined the physical consequences of three different approaches to the exact diffraction amplitude, and determined the signature of diffraction in the time domain. The external reflection surface wave amplitude gradually replaces the diffraction amplitude in the angular transition region between forward-focusing and the region of specular reflection. The details of this replacement were studied in the time domain.

OCIS Codes
(260.1960) Physical optics : Diffraction theory
(290.4020) Scattering : Mie theory
(320.2250) Ultrafast optics : Femtosecond phenomena

ToC Category:
Scattering

History
Original Manuscript: February 15, 2011
Manuscript Accepted: March 16, 2011
Published: May 18, 2011

Virtual Issues
Vol. 6, Iss. 7 Virtual Journal for Biomedical Optics

Citation
James A. Lock and Philip Laven, "Mie scattering in the time domain. Part II. The role of diffraction," J. Opt. Soc. Am. A 28, 1096-1106 (2011)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=josaa-28-6-1096

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References

D. Halliday, R. Resnick, and J. Walker, “Diffraction,” in Fundamentals of Physics, 7th ed. (Wiley, 2005), p. 963.
E. Hecht, “Huygens principle,” in Optics, 2nd ed. (Addison-Wesley, 1987), p. 80.
J. W. Goodman, “The Kirchhoff formula of diffraction by a plane screen,” in Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 37–42.
M. Born and E. Wolf, “Two-dimensional diffraction of a plane wave by a half-plane,” in Principles of Optics, 6th ed.(Cambridge University, 1980), pp. 565–578.
M. Born and E. Wolf, “The nature of the solution,” in Principles of Optics, 6th ed. (Cambridge University, 1980), p. 571.
M. Born and E. Wolf, “The nature of the solution,” in Principles of Optics, 6th ed. (Cambridge University, 1980), p. 572.
M. Born and E. Wolf, “Expression of the solution in ters of Fresnel integrals,” in Principles of Optics, 6th ed. (Cambridge University, 1980), p. 569.
H. C. van de Hulst, “Rigorous scattering theory for spheres of arbitrary size,” in Light Scattering by Small Particles (Dover, 1957), pp. 114–130.
M. Kerker, “Scattering by a sphere,” in The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), pp. 27–96.
C. F. Bohren and D. R. Huffman, “Absorption and scattering by a sphere,” in Absorption and Scattering of Light by Small Particles (Wiley, 1983), pp. 82–129.
B. Van der Pol and H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 825–864 (1937).
H. C. van de Hulst, “The diffraction part,” in Light Scattering by Small Particles (Dover, 1957), p. 209.
H. M. Nussenzveig, “Uniform approximation in scattering by spheres,” J. Phys. A 21, 81–109 (1988), Section 4.2. [CrossRef]
J. A. Lock and E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993). [CrossRef]
H. M. Nussenzveig, “The Debye expansion,” in Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992), p. 96.
E. P. Wigner, “Lower limit for the energy derivative of the scattered phase shift,” Phys. Rev. 98, 145–147 (1955). [CrossRef]
H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. 34, 23–95 (1965). [CrossRef]
H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969). [CrossRef]
V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. dissertation (University of Rochester, 1976), pp. 89–109.
M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 366 Eq. (9.3.4), p. 448 Eq. (10.4.59), p. 449 Eq. (10.4.63).
M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 366 Eq. (9.3.4), p. 448 Eq. (10.4.60), p. 449 Eq. (10.4.64).
M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 366 Eq. (9.3.3).
H. C. van de Hulst, “The reflected and refracted light,” in Light Scattering by Small Particles (Dover, 1957), p. 212.
K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. 7, 259–286 (1959). [CrossRef]
D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Time dependence of internal intensity of a dielectric sphere on or near resonance,” J. Opt. Soc. Am. A 9, 1364–1373 (1992). [CrossRef]
E. E. M. Khaled, D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Internal and scattered time-dependent intensity of a dielectric sphere illuminated with a pulsed Gaussian beam,” J. Opt. Soc. Am. A 11, 2065–2071 (1994). [CrossRef]
K. S. Shifrin and I. G. Zolotov, “Quasi-stationary scattering of electromagnetic pulses by spherical particles,” Appl. Opt. 33, 7798–7804 (1994). [CrossRef] [PubMed]
P. Laven, “Separating diffraction from scattering: the million-dollar challenge,” J. Nanophoton. 4, 041593 (2010). [CrossRef]
J. A. Lock and P. Laven, “Mie scattering in the time domain. Part I. The role of surface waves,” J. Opt. Soc. Am. A 28, 1086–1095 (2011). [CrossRef]
J. A. Lock, “Observability of atmospheric glories and supernumerary rainbows,” J. Opt. Soc. Am. A 6, 1924–1930(1989). [CrossRef]
M.Abramowitz and I.A.Stegun, eds., “Bessel functions of fractional order,” in Handbook of Mathematical Functions(National Bureau of Standards, 1964), p. 478, Table 10.13.
H. C. van de Hulst, “Theory based on Mie’s formulae,” in Light Scattering by Small Particles (Dover, 1957), p. 253.
G. Arfken, “Recurrence relations and special properties,” in Mathematical Methods for Physicists, 3rd. ed. (Academic, 1985), p. 648 Eq. (12.28).
J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, “Legendre functions,” in Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, 1969), p. 76 Eqs. (1.367, 1.369).
H. M. Nussenzveig and W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991). [CrossRef] [PubMed]
J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, “Legendre functions,” in Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, 1969), p. 72 Eq. (1.335).
G. Arfken, “Asymptotic expansions,” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 620 Eq. (11.137).
J. B. Keller, “A geometrical theory of diffraction,” in Calculus of Variations and Its Applications, Proceedings of Symposia in Applied Mathematics, L.M.Graves, ed. McGraw-Hill, 1958), Vol. 3, pp. 27–52.
J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957). [CrossRef]
P. M. Morse and H. Feshbach, “Poisson sum formula,” in Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 466–467.
E. Hecht, “Babinet’s principle,” in Optics, 2nd ed. (Addison-Wesley, 1987), pp. 458–459.
W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980). [CrossRef] [PubMed]

Am. J. Phys.

J. A. Lock and E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993). [CrossRef]

Ann. Phys.

H. M. Nussenzveig, “High-frequency scattering by an impenetrable sphere,” Ann. Phys. 34, 23–95 (1965). [CrossRef]
K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. 7, 259–286 (1959). [CrossRef]

Appl. Opt.

J. Appl. Phys.

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957). [CrossRef]

J. Math. Phys.

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969). [CrossRef]

J. Nanophoton.

P. Laven, “Separating diffraction from scattering: the million-dollar challenge,” J. Nanophoton. 4, 041593 (2010). [CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

H. M. Nussenzveig, “Uniform approximation in scattering by spheres,” J. Phys. A 21, 81–109 (1988), Section 4.2. [CrossRef]

Philos. Mag.

B. Van der Pol and H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 825–864 (1937).

Phys. Rev.

E. P. Wigner, “Lower limit for the energy derivative of the scattered phase shift,” Phys. Rev. 98, 145–147 (1955). [CrossRef]

Phys. Rev. A

H. M. Nussenzveig and W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991). [CrossRef] [PubMed]

Other

J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, “Legendre functions,” in Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, 1969), p. 72 Eq. (1.335).
G. Arfken, “Asymptotic expansions,” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 620 Eq. (11.137).
J. B. Keller, “A geometrical theory of diffraction,” in Calculus of Variations and Its Applications, Proceedings of Symposia in Applied Mathematics, L.M.Graves, ed. McGraw-Hill, 1958), Vol. 3, pp. 27–52.
M.Abramowitz and I.A.Stegun, eds., “Bessel functions of fractional order,” in Handbook of Mathematical Functions(National Bureau of Standards, 1964), p. 478, Table 10.13.
H. C. van de Hulst, “Theory based on Mie’s formulae,” in Light Scattering by Small Particles (Dover, 1957), p. 253.
G. Arfken, “Recurrence relations and special properties,” in Mathematical Methods for Physicists, 3rd. ed. (Academic, 1985), p. 648 Eq. (12.28).
J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, “Legendre functions,” in Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, 1969), p. 76 Eqs. (1.367, 1.369).
H. C. van de Hulst, “The diffraction part,” in Light Scattering by Small Particles (Dover, 1957), p. 209.
H. M. Nussenzveig, “The Debye expansion,” in Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992), p. 96.
V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. dissertation (University of Rochester, 1976), pp. 89–109.
M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 366 Eq. (9.3.4), p. 448 Eq. (10.4.59), p. 449 Eq. (10.4.63).
M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 366 Eq. (9.3.4), p. 448 Eq. (10.4.60), p. 449 Eq. (10.4.64).
M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 366 Eq. (9.3.3).
H. C. van de Hulst, “The reflected and refracted light,” in Light Scattering by Small Particles (Dover, 1957), p. 212.
D. Halliday, R. Resnick, and J. Walker, “Diffraction,” in Fundamentals of Physics, 7th ed. (Wiley, 2005), p. 963.
E. Hecht, “Huygens principle,” in Optics, 2nd ed. (Addison-Wesley, 1987), p. 80.
J. W. Goodman, “The Kirchhoff formula of diffraction by a plane screen,” in Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 37–42.
M. Born and E. Wolf, “Two-dimensional diffraction of a plane wave by a half-plane,” in Principles of Optics, 6th ed.(Cambridge University, 1980), pp. 565–578.
M. Born and E. Wolf, “The nature of the solution,” in Principles of Optics, 6th ed. (Cambridge University, 1980), p. 571.
M. Born and E. Wolf, “The nature of the solution,” in Principles of Optics, 6th ed. (Cambridge University, 1980), p. 572.
M. Born and E. Wolf, “Expression of the solution in ters of Fresnel integrals,” in Principles of Optics, 6th ed. (Cambridge University, 1980), p. 569.
H. C. van de Hulst, “Rigorous scattering theory for spheres of arbitrary size,” in Light Scattering by Small Particles (Dover, 1957), pp. 114–130.
M. Kerker, “Scattering by a sphere,” in The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), pp. 27–96.
C. F. Bohren and D. R. Huffman, “Absorption and scattering by a sphere,” in Absorption and Scattering of Light by Small Particles (Wiley, 1983), pp. 82–129.
P. M. Morse and H. Feshbach, “Poisson sum formula,” in Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 466–467.
E. Hecht, “Babinet’s principle,” in Optics, 2nd ed. (Addison-Wesley, 1987), pp. 458–459.

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[1] D. Halliday, R. Resnick, and J. Walker, “Diffraction,” in Fundamentals of Physics, 7th ed. (Wiley, 2005), p. 963.

[2] E. Hecht, “Huygens principle,” in Optics, 2nd ed. (Addison-Wesley, 1987), p. 80.

[3] J. W. Goodman, “The Kirchhoff formula of diffraction by a plane screen,” in Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 37–42.

[4] M. Born and E. Wolf, “Two-dimensional diffraction of a plane wave by a half-plane,” in Principles of Optics, 6th ed.(Cambridge University, 1980), pp. 565–578.

[5] M. Born and E. Wolf, “The nature of the solution,” in Principles of Optics, 6th ed. (Cambridge University, 1980), p. 571.

[6] M. Born and E. Wolf, “The nature of the solution,” in Principles of Optics, 6th ed. (Cambridge University, 1980), p. 572.

[7] M. Born and E. Wolf, “Expression of the solution in ters of Fresnel integrals,” in Principles of Optics, 6th ed. (Cambridge University, 1980), p. 569.

[8] H. C. van de Hulst, “Rigorous scattering theory for spheres of arbitrary size,” in Light Scattering by Small Particles (Dover, 1957), pp. 114–130.

[9] M. Kerker, “Scattering by a sphere,” in The Scattering of Light and Other Electromagnetic Radiation (Academic, 1969), pp. 27–96.

[10] C. F. Bohren and D. R. Huffman, “Absorption and scattering by a sphere,” in Absorption and Scattering of Light by Small Particles (Wiley, 1983), pp. 82–129.

[11] B. Van der Pol and H. Bremmer, “The diffraction of electromagnetic waves from an electrical point source round a finitely conducting sphere, with applications to radiotelegraphy and the theory of the rainbow,” Philos. Mag. 24, 825–864 (1937).

[12] H. C. van de Hulst, “The diffraction part,” in Light Scattering by Small Particles (Dover, 1957), p. 209.

[13] H. M. Nussenzveig, “Uniform approximation in scattering by spheres,” J. Phys. A 21, 81–109 (1988), Section 4.2. [CrossRef]

[14] J. A. Lock and E. A. Hovenac, “Diffraction of a Gaussian beam by a spherical obstacle,” Am. J. Phys. 61, 698–707 (1993). [CrossRef]

[15] H. M. Nussenzveig, “The Debye expansion,” in Diffraction Effects in Semiclassical Scattering (Cambridge University, 1992), p. 96.

[16] E. P. Wigner, “Lower limit for the energy derivative of the scattered phase shift,” Phys. Rev. 98, 145–147 (1955). [CrossRef]

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

[18] H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969). [CrossRef]

[19] V. Khare, “Short-wavelength scattering of electromagnetic waves by a homogeneous dielectric sphere,” Ph.D. dissertation (University of Rochester, 1976), pp. 89–109.

[20] M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 366 Eq. (9.3.4), p. 448 Eq. (10.4.59), p. 449 Eq. (10.4.63).

[21] M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 366 Eq. (9.3.4), p. 448 Eq. (10.4.60), p. 449 Eq. (10.4.64).

[22] M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions (National Bureau of Standards, 1964), p. 366 Eq. (9.3.3).

[23] H. C. van de Hulst, “The reflected and refracted light,” in Light Scattering by Small Particles (Dover, 1957), p. 212.

[24] K. W. Ford and J. A. Wheeler, “Semiclassical description of scattering,” Ann. Phys. 7, 259–286 (1959). [CrossRef]

[25] D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Time dependence of internal intensity of a dielectric sphere on or near resonance,” J. Opt. Soc. Am. A 9, 1364–1373 (1992). [CrossRef]

[26] E. E. M. Khaled, D. Q. Chowdhury, S. C. Hill, and P. W. Barber, “Internal and scattered time-dependent intensity of a dielectric sphere illuminated with a pulsed Gaussian beam,” J. Opt. Soc. Am. A 11, 2065–2071 (1994). [CrossRef]

[27] K. S. Shifrin and I. G. Zolotov, “Quasi-stationary scattering of electromagnetic pulses by spherical particles,” Appl. Opt. 33, 7798–7804 (1994). [CrossRef] [PubMed]

[28] P. Laven, “Separating diffraction from scattering: the million-dollar challenge,” J. Nanophoton. 4, 041593 (2010). [CrossRef]

[29] J. A. Lock and P. Laven, “Mie scattering in the time domain. Part I. The role of surface waves,” J. Opt. Soc. Am. A 28, 1086–1095 (2011). [CrossRef]

[30] J. A. Lock, “Observability of atmospheric glories and supernumerary rainbows,” J. Opt. Soc. Am. A 6, 1924–1930(1989). [CrossRef]

[31] M.Abramowitz and I.A.Stegun, eds., “Bessel functions of fractional order,” in Handbook of Mathematical Functions(National Bureau of Standards, 1964), p. 478, Table 10.13.

[32] H. C. van de Hulst, “Theory based on Mie’s formulae,” in Light Scattering by Small Particles (Dover, 1957), p. 253.

[33] G. Arfken, “Recurrence relations and special properties,” in Mathematical Methods for Physicists, 3rd. ed. (Academic, 1985), p. 648 Eq. (12.28).

[34] J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, “Legendre functions,” in Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, 1969), p. 76 Eqs. (1.367, 1.369).

[35] H. M. Nussenzveig and W. J. Wiscombe, “Complex angular momentum approximation to hard-core scattering,” Phys. Rev. A 43, 2093–2112 (1991). [CrossRef] [PubMed]

[36] J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, “Legendre functions,” in Electromagnetic and Acoustic Scattering by Simple Shapes (Hemisphere, 1969), p. 72 Eq. (1.335).

[37] G. Arfken, “Asymptotic expansions,” in Mathematical Methods for Physicists, 3rd ed. (Academic, 1985), p. 620 Eq. (11.137).

[38] J. B. Keller, “A geometrical theory of diffraction,” in Calculus of Variations and Its Applications, Proceedings of Symposia in Applied Mathematics, L.M.Graves, ed. McGraw-Hill, 1958), Vol. 3, pp. 27–52.

[39] J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957). [CrossRef]

[40] P. M. Morse and H. Feshbach, “Poisson sum formula,” in Methods of Theoretical Physics (McGraw-Hill, 1953), pp. 466–467.

[41] E. Hecht, “Babinet’s principle,” in Optics, 2nd ed. (Addison-Wesley, 1987), pp. 458–459.

[42] W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980). [CrossRef] [PubMed]

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Mie scattering in the time domain. Part II. The role of diffraction