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.

© 2011 Optical Society of America

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References

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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).
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  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]

2011

2010

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

1994

1993

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

1992

1991

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

1989

1988

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

1980

1969

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

1965

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

1959

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

1957

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

1955

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

1937

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).

Arfken, G.

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

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

Barber, P. W.

Bohren, C. F.

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.

Born, M.

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.

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.

Bowman, J. J.

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).

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).

Bremmer, H.

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).

Chowdhury, D. Q.

Feshbach, H.

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

Ford, K. W.

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

Goodman, J. W.

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

Halliday, D.

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

Hecht, E.

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

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

Hill, S. C.

Hovenac, E. A.

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

Huffman, D. R.

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.

Keller, J. B.

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

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.

Kerker, M.

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

Khaled, E. E. M.

Khare, V.

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

Laven, P.

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]

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

Lock, J. A.

Morse, P. M.

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

Nussenzveig, H. M.

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

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

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

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

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

Resnick, R.

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

Senior, T. B. A.

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).

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).

Shifrin, K. S.

Uslenghi, P. L. E.

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).

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).

van de Hulst, H. C.

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

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

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

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

Van der Pol, B.

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).

Walker, J.

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

Wheeler, J. A.

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

Wigner, E. P.

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

Wiscombe, W. J.

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

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

Wolf, E.

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, “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, “Expression of the solution in ters of Fresnel integrals,” in Principles of Optics, 6th ed. (Cambridge University, 1980), p. 569.

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

Zolotov, I. G.

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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Figures (6)

Fig. 1
Fig. 1

Intensity of the p = 0 Mie–Debye term as a function of the scattering angle and delay time for the unpolarized raised cosine pulse of [29] with λ 0 = 0.65 μm . and τ = 5 fs incident on a spherical particle of radius a = 10 μm and refractive index N + i K = 1.3326 + i ( 1.67 × 10 8 ) .

Fig. 2
Fig. 2

Diffracted rays a and b are assumed to arrive at the entrance plane A A with the same phase. Then ray a advances its phase by k a θ ( 1 + 1 / k a ) between the entrance plane and the exit plane B B for scattering at the angle θ. The diffracted ray b retards its phase by k a θ ( 1 + 1 / k a ) between the entrance and exit planes.

Fig. 3
Fig. 3

Diffracted intensity corresponding to Eq. (33) as a function of the scattering angle and delay time for the unpolarized raised cosine pulse and spherical particle of Fig. 1.

Fig. 4
Fig. 4

Intensity of the p = 0 Mie–Debye term as a function of scattering angle and delay time in the near-forward direction for the unpolarized raised cosine pulse and spherical particle of Fig. 1. The fundamental sampling interval is 0.135 fs .

Fig. 5
Fig. 5

Diffracted intensity corresponding to Eq. (4) as a function of the scattering angle and delay time for the unpolarized raised cosine pulse of [29] with λ 0 = 0.65 μm and τ = 10 fs incident on a spherical particle of radius a = 8 μm and refractive index N + i K = 1.3326 + i ( 1.67 × 10 8 ) .

Fig. 6
Fig. 6

Intensity of the p = 0 Mie–Debye term as a function of the scattering angle and delay time for the unpolarized raised cosine pulse of Fig. 5 incident on the spherical particle of Fig. 1. The upper limit of the Mie sum is (a)  0.4 k a , (b)  0.6 k a , (c)  0.8 k a , and (d)  1.0 k a .

Equations (58)

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E scatt ( θ , φ ) = i S 2 ( θ ) cos ( φ ) u θ i S 1 ( θ ) sin ( φ ) u φ ,
S 1 ( θ ) = n = 1 { ( 2 n + 1 ) / [ n ( n + 1 ) ] } [ a n π n ( θ ) + b n τ n ( θ ) ] ,
S 2 ( θ ) = n = 1 { ( 2 n + 1 ) / [ n ( n + 1 ) ] } [ a n τ n ( θ ) + b n π n ( θ ) ] .
a n = ( 1 / 2 ) ( 1 R n TM ) ,
b n = ( 1 / 2 ) ( 1 R n TE ) ,
S 1 diff ( θ ) = S 2 diff ( θ ) = ( 1 / 2 ) n = 1 k A { ( 2 n + 1 ) / [ n ( n + 1 ) ] } [ π n ( θ ) + τ n ( θ ) ] .
S 1 ref ( θ ) = ( 1 / 2 ) n = 1 [ k a ε max ( k a ) 1 / 3 ] 1 { ( 2 n + 1 ) / [ n ( n + 1 ) ] } [ R n TM π n ( θ ) + R n TE τ n ( θ ) ] ,
S 2 ref ( θ ) = ( 1 / 2 ) n = 1 [ k a ε max ( k a ) 1 / 3 ] 1 { ( 2 n + 1 ) / [ n ( n + 1 ) ] } [ R n TM τ n ( θ ) + R n TE π n ( θ ) ] ,
S 1 graz + tunn ( θ ) = ( 1 / 2 ) n = [ k a ε max ( k a ) 1 / 3 ] k A { ( 2 n + 1 ) / [ n ( n + 1 ) ] } [ R n TM π n ( θ ) + R n TE τ n ( θ ) ] + ( 1 / 2 ) n = k A + 1 [ k a + ε max ( k a ) 1 / 3 ] { ( 2 n + 1 ) / [ n ( n + 1 ) ] } [ ( 1 R n TM ) π n ( θ ) + ( 1 R n TE ) τ n ( θ ) ] ,
S 2 graz + tunn ( θ ) = ( 1 / 2 ) n = [ k a ε max ( k a ) 1 / 3 ] k A { ( 2 n + 1 ) / [ n ( n + 1 ) ] } [ R n TM τ n ( θ ) + R n TE π n ( θ ) ] + ( 1 / 2 ) n = k A + 1 [ k a + ε max ( k a ) 1 / 3 ] { ( 2 n + 1 ) / [ n ( n + 1 ) ] } [ ( 1 R n TM ) τ n ( θ ) + ( 1 R n TE ) π n ( θ ) ] .
R n TE = R n TM 1 i exp ( 2 5 / 2 ε 3 / 2 / 3 ) + O [ 1 / ( k a ) 1 / 3 ]
1 R n TE = 1 R n TM i exp ( 2 5 / 2 ε 3 / 2 / 3 ) + O [ 1 / ( k a ) 1 / 3 ] .
R n TE = R n TM i exp ( i 2 5 / 2 | ε | 3 / 2 / 3 ) + O [ 1 / ( k a ) 1 / 3 ] .
S 1 ref ( θ ) = i ( k a / 2 ) r TE ( θ ) exp [ 2 i k a sin ( θ / 2 ) ] ,
S 2 ref ( θ ) = i ( k a / 2 ) r TM ( θ ) exp [ 2 i k a sin ( θ / 2 ) ] ,
r TE ( θ ) = { [ N 2 cos 2 ( θ / 2 ) ] 1 / 2 sin ( θ / 2 ) } / { [ N 2 cos 2 ( θ / 2 ) ] 1 / 2 + sin ( θ / 2 ) } ,
r TM ( θ ) = { [ N 2 cos 2 ( θ / 2 ) ] 1 / 2 N 2 sin ( θ / 2 ) } / { [ N 2 cos 2 ( θ / 2 ) ] 1 / 2 + N 2 sin ( θ / 2 ) } .
E ref ( t , θ , φ ) = [ r TM ( θ ) cos ( φ ) u θ + r TE ( θ ) sin ( φ ) u φ ] × ( d k / 2 π ) A ( k ) ( k a / 2 ) exp [ 2 i k a sin ( θ / 2 ) i c k t ] .
A ( k ) = σ ( π ) 1 / 2 exp [ σ 2 ( k k 0 ) 2 / 4 ] ,
I ref ( t , θ , φ ) = [ ( k 0 a ) 2 / 4 ] { [ r TE ( θ ) ] 2 cos 2 ( φ ) + [ r TM ( θ ) ] 2 sin 2 ( φ ) } × exp { 2 [ c t + 2 a sin ( θ / 2 ) ] 2 / σ 2 } ,
t = ( 2 a / c ) sin ( θ / 2 ) a θ / c
E graz + tunn ( θ , φ ) = { ( k a ) 4 / 3 exp ( 5 π i / 6 ) + ( k a ) ( N 2 + 1 ) / [ 2 ( N 2 1 ) 1 / 2 ] } J 0 ( k a θ ) u x [ ( k a ) ( N 2 1 ) 1 / 2 J 2 ( k a θ ) / 2 ] [ cos ( 2 φ ) u x + sin ( 2 φ ) u y ] .
E graz + tunn ( θ , φ ) = ( k a ) 5 / 6 T ( θ ) B ( k , θ ) exp [ i k a Φ ( k , θ ) ] × { [ 1 N 2 θ / ( N 2 1 ) 1 / 2 ] cos ( φ ) u θ [ 1 θ / ( N 2 1 ) 1 / 2 ] sin ( φ ) u φ } ,
T ( θ ) = exp ( 7 i π / 12 ) / { 2 5 / 6 [ π sin ( θ ) ] 1 / 2 [ A i ( X ) ] 2 } ,
B ( k , θ ) = exp { θ [ 3 1 / 2 X ( k a ) 1 / 3 / 2 4 / 3 ( 1 / 2 ) ( N 2 + 1 ) / ( N 2 1 ) 1 / 2 ] } ,
Φ ( k , θ ) = θ { 1 + X / [ 2 4 / 3 ( k a ) 2 / 3 ] } .
I graz + tunn ( t , θ , φ ) = ( k 0 a ) 5 / 3 [ T ( θ ) B ( k 0 , θ ) ] 2 exp { 2 [ c t a Φ ( k 0 , θ ) ] 2 / σ 2 } .
t = ( a θ / c ) { 1 + X / [ 2 4 / 3 ( k 0 a ) 2 / 3 ] } .
E graz + tunn ( θ , φ ) = ( k a ) 4 / 3 { exp ( 4 i π / 3 ) B ( k , π ) exp [ i k a Φ ( k , π ) ] / 2 1 / 3 [ A i ( X ) ] 2 } × { [ π ( N 2 1 ) 1 / 2 / 2 ] J 0 ( k a δ ) u x + [ 1 + π ( N 2 + 1 ) / 2 ( N 2 1 ) 1 / 2 ] J 2 ( k a δ ) [ cos ( 2 φ ) u x + sin ( 2 φ ) u y ] } .
S em diff ( θ ) = n = 1 k A ( n + 1 / 2 ) P n [ cos ( θ ) ] + [ 1 cos ( θ ) ] n = 1 k A { ( n + 1 / 2 ) / [ n ( n + 1 ) ] } d P n [ cos ( θ ) ] / d cos ( θ ) .
P n [ cos ( θ ) ] = [ θ / sin ( θ ) ] 1 / 2 { J 0 [ ( n + 1 / 2 ) θ ] + J 1 [ ( n + 1 / 2 ) θ ] [ cot ( θ ) ( 1 / θ ) ] / [ 8 ( n + 1 / 2 ) ] + O [ 1 / ( n + 1 / 2 ) 2 ] } ,
S em diff ( θ ) [ θ / sin ( θ ) ] 1 / 2 { ( k A ) 2 J 1 ( k A θ ) / ( k A θ ) + [ 1 / sin ( θ ) ( 1 / 8 θ ) 7 cot ( θ ) / 8 ] [ 1 J 0 ( k A θ ) ] + O ( 1 / k A ) } ,
S scalar diff ( θ ) [ θ / sin ( θ ) ] 1 / 2 { ( k A ) 2 J 1 ( k A θ ) / ( k A θ ) + [ ( 1 / 8 θ ) + cot ( θ ) / 8 ] [ 1 J 0 ( k A θ ) ] + O ( 1 / k A ) } .
S diff ( θ ) ( k a ) 2 J 1 ( k a θ ) / ( k a θ )
P n [ cos ( θ ) ] = 1 n ( n + 1 ) θ 2 / 4 + [ n ( n + 1 ) / 48 + ( n 1 ) n ( n + 1 ) ( n + 2 ) / 64 ] θ 4 + O ( θ 6 ) .
S em diff ( θ ) = [ k A ( k A + 2 ) / 2 ] { 1 ( θ 2 / 8 ) [ ( k A ) 2 + 2 k A 1 ] + O ( θ 4 ) } .
S em diff ( θ ) ( k a + 1 ) 2 J 1 [ ( k a + 1 ) θ ] / [ ( k a + 1 ) θ ] .
S scalar diff ( θ ) = [ ( k A + 1 ) 2 / 2 ] [ 1 k A ( k A + 2 ) θ 2 / 8 + O ( θ 4 ) ] ,
S diff ( θ ) { ( k a + 1 ) / [ 2 π sin ( θ ) ] } 1 / 2 ( 1 / θ ) × { exp [ i k a θ ( 1 + 1 / k a ) 3 i / 4 ] + exp [ i k a θ ( 1 + 1 / k a ) + 3 i / 4 ] } .
E diff ( θ , φ ) [ E 0 ( k a + 1 ) 1 / 2 / k r ] { exp [ i k r + i k a θ ( 1 + 1 / k a ) i ω t ] i exp [ i k r i k a θ ( 1 + 1 / k a ) i ω t ] } [ cos ( φ ) u θ sin ( φ ) u φ ] .
r = a ( 1 + 1 / k a ) .
I diff ( t , θ ) = { k 0 a / [ 2 π θ 2 sin ( θ ) ] } { exp [ 2 ( c t a θ ) 2 / σ 2 ] + exp [ 2 ( c t + a θ ) 2 / σ 2 ] 2 sin ( 2 k 0 a θ ) exp [ ( c t a θ ) 2 / σ 2 ] exp [ ( c t + a θ ) 2 / σ 2 ] } .
t = ± a θ / c ,
P n [ cos ( π δ ) ] = ( 1 ) n P n [ cos ( δ ) ] ,
S em diff ( θ ) = exp ( i π k A ) [ k A ( k A + 1 ) ( k A + 2 ) δ 2 / 16 + O ( δ 4 ) ] .
S em diff ( θ ) exp ( i π k a ) [ ( k a + 1 ) / 2 ] J 2 [ ( k a + 1 ) δ ] .
S scalar diff ( θ ) = exp ( i π k A ) [ ( k A + 1 ) / 2 ] [ 1 δ 2 k A ( k A + 2 ) / 4 + O ( δ 4 ) ] ,
exp ( i π k a ) [ ( k a + 1 ) / 2 ] J 0 [ ( k a + 1 ) δ ] .
n = 0 k A f ( x = n + 1 / 2 ) = m = ( 1 ) m 0 k A + 1 d x f ( x ) exp ( 2 π i m x ) .
f ( x ) = x P x 1 / 2 [ cos ( θ ) ] .
S diff ( θ ) T 0 ( θ ) + B 0 ( θ ) + m = 1 ( 1 ) m [ T m + ( θ ) + B m + ( θ ) + T m ( θ ) + B m ( θ ) ] ,
T 0 ( θ ) = ( D / θ ) exp [ i ( k a + 1 ) θ 3 π i / 4 ] ,
B 0 ( θ ) = ( D / θ ) exp [ i ( k a + 1 ) θ + 3 π i / 4 ] ,
T m + ( θ ) = [ D / ( 2 π m + θ ) ] exp [ i ( k a + 1 ) ( 2 π m + θ ) 3 π i / 4 ] ,
B m + ( θ ) = [ D / ( 2 π m + θ ) ] exp [ i ( k a + 1 ) ( 2 π m + θ ) + 3 π i / 4 ] ,
T m ( θ ) = [ D / ( 2 π m θ ) ] exp [ i ( k a + 1 ) ( 2 π m θ ) + i π / 4 ] ,
B m ( θ ) = [ D / ( 2 π m θ ) ] exp [ i ( k a + 1 ) ( 2 π m θ ) i π / 4 ] ,
D = { ( k a + 1 ) / [ 2 π sin ( θ ) ] } 1 / 2 .

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