Abstract

In this paper, we develop an analytic theory for describing the photoacoustic wave generation from a spheroidal droplet and derive the first complete analytic solution. Our derivation is based on solving the photoacoustic Helmholtz equation in spheroidal coordinates with the separation-of-variables method. As the verification, besides carrying out the asymptotic analyses which recover the standard solutions for a sphere, an infinite cylinder and an infinite layer, we also confirm that the partial transmission and reflection model previously demonstrated for these three geometries still stands. We expect that this analytic solution will find broad practical uses in interpreting experiment results, considering that its building blocks, the spheroidal wave functions (SWFs), can be numerically calculated by the existing computer programs.

© 2014 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Light Scattering by a Spheroidal Particle

Shoji Asano and Giichi Yamamoto
Appl. Opt. 14(1) 29-49 (1975)

References

  • View by:
  • |
  • |
  • |

  1. G. J. Diebold, “Photoacoustic monopole radiation: waves from objects with symmetry in one, two, and three dimensions,” in Photoacoustic Imaging and Spectroscopy, L. V. Wang, ed. (Taylor and Francis, 2009).
  2. G. J. Diebold, M. I. Khan, and S. M. Park, “Photoacoustic “signatures” of particulate matter: optical production of acoustic monopole radiation,” Science 250(4977), 101–104 (1990).
    [Crossref] [PubMed]
  3. G. J. Diebold, T. Sun, and M. I. Khan, “Photoacoustic monopole radiation in one, two, and three dimensions,” Phys. Rev. Lett. 67(24), 3384–3387 (1991).
    [Crossref] [PubMed]
  4. L. V. Wang and S. Hu, “Photoacoustic tomography: in vivo imaging from organelles to organs,” Science 335(6075), 1458–1462 (2012).
    [Crossref] [PubMed]
  5. S. Hu and L. V. Wang, “Optical-resolution photoacoustic microscopy: auscultation of biological systems at the cellular level,” Biophys. J. 105(4), 841–847 (2013).
    [Crossref] [PubMed]
  6. E. I. Galanzha and V. P. Zharov, “Circulating tumor cell detection and capture by photoacoustic flow cytometry in vivo and ex vivo,” Cancers 5(4), 1691–1738 (2013).
    [Crossref] [PubMed]
  7. E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “Probing red blood cell morphology using high-frequency photoacoustics,” Biophys. J. 105(1), 59–67 (2013).
    [Crossref] [PubMed]
  8. E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “High frequency label-free photoacoustic microscopy of single cells,” Photoacoustics 1(3–4), 49–53 (2013).
    [Crossref]
  9. C. Flammer, Spheroidal Wave Functions (Stanford University, 1957).
  10. R. D. Spence and S. Granger, “The scattering of sound from a prolate spheroid,” J. Acoust. Soc. Am. 23(6), 701–706 (1951).
    [Crossref]
  11. S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975).
    [Crossref] [PubMed]
  12. J. P. Barton, N. L. Wolff, H. Zhang, and C. Tarawneh, “Near-field calculations for a rigid spheroid with an arbitrary incident acoustic field,” J. Acoust. Soc. Am. 113(3), 1216–1222 (2003).
    [Crossref] [PubMed]
  13. B. R. Rapids and G. C. Lauchle, “Vector intensity field scattered by a rigid prolate spheroid,” J. Acoust. Soc. Am. 120(1), 38–48 (2006).
    [Crossref]
  14. M. J. Mendes, I. Tobias, A. Marti, and A. Luque, “Near-field scattering by dielectric spheroidal particles with sizes on the order of the illuminating wavelength,” J. Opt. Soc. Am. B 27(6), 1221–1231 (2010).
    [Crossref]
  15. M. J. Mendes, I. Tobías, A. Martí, and A. Luque, “Light concentration in the near-field of dielectric spheroidal particles with mesoscopic sizes,” Opt. Express 19(17), 16207–16222 (2011).
    [Crossref] [PubMed]
  16. K. T. McDonald, “Gaussian laser beams via oblate spheroidal waves,” arXiv:physics/0312024v1 (2003).
  17. M. Zeppenfeld, “Solutions to Maxwell's equations using spheroidal coordinates,” New J. Phys. 11(7), 073007 (2009).
    [Crossref]
  18. M. Zeppenfeld and P. W. H. Pinkse, “Calculating the fine structure of a Fabry-Perot resonator using spheroidal wave functions,” Opt. Express 18(9), 9580–9591 (2010).
    [Crossref] [PubMed]
  19. L. V. Wang and H.-I. Wu, Biomedical Optics (Wiley, 2007), Chap. 12.
  20. V. E. Gusev and A. A. Karabutov, Laser Optoacoustics (American Institute of Physics Press, 1993), Chap. 2.
  21. A. D. Kotsis and J. A. Roumeliotis, “Acoustic scattering by a penetrable spheroid,” Acoust. Phys. 54(2), 153–167 (2008).
    [Crossref]
  22. W. M. John, “Asymptotic approximations for prolate spheroidal wave functions,” Stud. Appl. Math. 54, 315–349 (1975).
  23. M. I. Khan, T. Sun, and G. J. Diebold, “Photoacoustic waves generated by absorption of laser radiation in optically thin layers,” J. Acoust. Soc. Am. 93(3), 1417–1425 (1993).
    [Crossref]
  24. S. Zhang and J. Jin, Computation of Special Functions (Wiley, 1997).
  25. W. J. Thompson, “Spheroidal wave functions,” Comput. Sci. Eng. 1, 84–87 (1999).
  26. L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (Wiley, 2002).
  27. P. E. Falloon, P. C. Abbott, and J. B. Wang, “Theory and computation of spheroidal wave functions,” J. Phys. Math. Gen. 36(20), 5477–5495 (2003).
    [Crossref]

2013 (4)

S. Hu and L. V. Wang, “Optical-resolution photoacoustic microscopy: auscultation of biological systems at the cellular level,” Biophys. J. 105(4), 841–847 (2013).
[Crossref] [PubMed]

E. I. Galanzha and V. P. Zharov, “Circulating tumor cell detection and capture by photoacoustic flow cytometry in vivo and ex vivo,” Cancers 5(4), 1691–1738 (2013).
[Crossref] [PubMed]

E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “Probing red blood cell morphology using high-frequency photoacoustics,” Biophys. J. 105(1), 59–67 (2013).
[Crossref] [PubMed]

E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “High frequency label-free photoacoustic microscopy of single cells,” Photoacoustics 1(3–4), 49–53 (2013).
[Crossref]

2012 (1)

L. V. Wang and S. Hu, “Photoacoustic tomography: in vivo imaging from organelles to organs,” Science 335(6075), 1458–1462 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (2)

2009 (1)

M. Zeppenfeld, “Solutions to Maxwell's equations using spheroidal coordinates,” New J. Phys. 11(7), 073007 (2009).
[Crossref]

2008 (1)

A. D. Kotsis and J. A. Roumeliotis, “Acoustic scattering by a penetrable spheroid,” Acoust. Phys. 54(2), 153–167 (2008).
[Crossref]

2006 (1)

B. R. Rapids and G. C. Lauchle, “Vector intensity field scattered by a rigid prolate spheroid,” J. Acoust. Soc. Am. 120(1), 38–48 (2006).
[Crossref]

2003 (2)

J. P. Barton, N. L. Wolff, H. Zhang, and C. Tarawneh, “Near-field calculations for a rigid spheroid with an arbitrary incident acoustic field,” J. Acoust. Soc. Am. 113(3), 1216–1222 (2003).
[Crossref] [PubMed]

P. E. Falloon, P. C. Abbott, and J. B. Wang, “Theory and computation of spheroidal wave functions,” J. Phys. Math. Gen. 36(20), 5477–5495 (2003).
[Crossref]

1999 (1)

W. J. Thompson, “Spheroidal wave functions,” Comput. Sci. Eng. 1, 84–87 (1999).

1993 (1)

M. I. Khan, T. Sun, and G. J. Diebold, “Photoacoustic waves generated by absorption of laser radiation in optically thin layers,” J. Acoust. Soc. Am. 93(3), 1417–1425 (1993).
[Crossref]

1991 (1)

G. J. Diebold, T. Sun, and M. I. Khan, “Photoacoustic monopole radiation in one, two, and three dimensions,” Phys. Rev. Lett. 67(24), 3384–3387 (1991).
[Crossref] [PubMed]

1990 (1)

G. J. Diebold, M. I. Khan, and S. M. Park, “Photoacoustic “signatures” of particulate matter: optical production of acoustic monopole radiation,” Science 250(4977), 101–104 (1990).
[Crossref] [PubMed]

1975 (2)

S. Asano and G. Yamamoto, “Light scattering by a spheroidal particle,” Appl. Opt. 14(1), 29–49 (1975).
[Crossref] [PubMed]

W. M. John, “Asymptotic approximations for prolate spheroidal wave functions,” Stud. Appl. Math. 54, 315–349 (1975).

1951 (1)

R. D. Spence and S. Granger, “The scattering of sound from a prolate spheroid,” J. Acoust. Soc. Am. 23(6), 701–706 (1951).
[Crossref]

Abbott, P. C.

P. E. Falloon, P. C. Abbott, and J. B. Wang, “Theory and computation of spheroidal wave functions,” J. Phys. Math. Gen. 36(20), 5477–5495 (2003).
[Crossref]

Asano, S.

Barton, J. P.

J. P. Barton, N. L. Wolff, H. Zhang, and C. Tarawneh, “Near-field calculations for a rigid spheroid with an arbitrary incident acoustic field,” J. Acoust. Soc. Am. 113(3), 1216–1222 (2003).
[Crossref] [PubMed]

Berndl, E. S. L.

E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “High frequency label-free photoacoustic microscopy of single cells,” Photoacoustics 1(3–4), 49–53 (2013).
[Crossref]

E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “Probing red blood cell morphology using high-frequency photoacoustics,” Biophys. J. 105(1), 59–67 (2013).
[Crossref] [PubMed]

Diebold, G. J.

M. I. Khan, T. Sun, and G. J. Diebold, “Photoacoustic waves generated by absorption of laser radiation in optically thin layers,” J. Acoust. Soc. Am. 93(3), 1417–1425 (1993).
[Crossref]

G. J. Diebold, T. Sun, and M. I. Khan, “Photoacoustic monopole radiation in one, two, and three dimensions,” Phys. Rev. Lett. 67(24), 3384–3387 (1991).
[Crossref] [PubMed]

G. J. Diebold, M. I. Khan, and S. M. Park, “Photoacoustic “signatures” of particulate matter: optical production of acoustic monopole radiation,” Science 250(4977), 101–104 (1990).
[Crossref] [PubMed]

Falloon, P. E.

P. E. Falloon, P. C. Abbott, and J. B. Wang, “Theory and computation of spheroidal wave functions,” J. Phys. Math. Gen. 36(20), 5477–5495 (2003).
[Crossref]

Galanzha, E. I.

E. I. Galanzha and V. P. Zharov, “Circulating tumor cell detection and capture by photoacoustic flow cytometry in vivo and ex vivo,” Cancers 5(4), 1691–1738 (2013).
[Crossref] [PubMed]

Granger, S.

R. D. Spence and S. Granger, “The scattering of sound from a prolate spheroid,” J. Acoust. Soc. Am. 23(6), 701–706 (1951).
[Crossref]

Hu, S.

S. Hu and L. V. Wang, “Optical-resolution photoacoustic microscopy: auscultation of biological systems at the cellular level,” Biophys. J. 105(4), 841–847 (2013).
[Crossref] [PubMed]

L. V. Wang and S. Hu, “Photoacoustic tomography: in vivo imaging from organelles to organs,” Science 335(6075), 1458–1462 (2012).
[Crossref] [PubMed]

John, W. M.

W. M. John, “Asymptotic approximations for prolate spheroidal wave functions,” Stud. Appl. Math. 54, 315–349 (1975).

Khan, M. I.

M. I. Khan, T. Sun, and G. J. Diebold, “Photoacoustic waves generated by absorption of laser radiation in optically thin layers,” J. Acoust. Soc. Am. 93(3), 1417–1425 (1993).
[Crossref]

G. J. Diebold, T. Sun, and M. I. Khan, “Photoacoustic monopole radiation in one, two, and three dimensions,” Phys. Rev. Lett. 67(24), 3384–3387 (1991).
[Crossref] [PubMed]

G. J. Diebold, M. I. Khan, and S. M. Park, “Photoacoustic “signatures” of particulate matter: optical production of acoustic monopole radiation,” Science 250(4977), 101–104 (1990).
[Crossref] [PubMed]

Kolios, M. C.

E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “Probing red blood cell morphology using high-frequency photoacoustics,” Biophys. J. 105(1), 59–67 (2013).
[Crossref] [PubMed]

E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “High frequency label-free photoacoustic microscopy of single cells,” Photoacoustics 1(3–4), 49–53 (2013).
[Crossref]

Kotsis, A. D.

A. D. Kotsis and J. A. Roumeliotis, “Acoustic scattering by a penetrable spheroid,” Acoust. Phys. 54(2), 153–167 (2008).
[Crossref]

Lauchle, G. C.

B. R. Rapids and G. C. Lauchle, “Vector intensity field scattered by a rigid prolate spheroid,” J. Acoust. Soc. Am. 120(1), 38–48 (2006).
[Crossref]

Luque, A.

Marti, A.

Martí, A.

Mendes, M. J.

Park, S. M.

G. J. Diebold, M. I. Khan, and S. M. Park, “Photoacoustic “signatures” of particulate matter: optical production of acoustic monopole radiation,” Science 250(4977), 101–104 (1990).
[Crossref] [PubMed]

Pinkse, P. W. H.

Rapids, B. R.

B. R. Rapids and G. C. Lauchle, “Vector intensity field scattered by a rigid prolate spheroid,” J. Acoust. Soc. Am. 120(1), 38–48 (2006).
[Crossref]

Roumeliotis, J. A.

A. D. Kotsis and J. A. Roumeliotis, “Acoustic scattering by a penetrable spheroid,” Acoust. Phys. 54(2), 153–167 (2008).
[Crossref]

Spence, R. D.

R. D. Spence and S. Granger, “The scattering of sound from a prolate spheroid,” J. Acoust. Soc. Am. 23(6), 701–706 (1951).
[Crossref]

Strohm, E. M.

E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “Probing red blood cell morphology using high-frequency photoacoustics,” Biophys. J. 105(1), 59–67 (2013).
[Crossref] [PubMed]

E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “High frequency label-free photoacoustic microscopy of single cells,” Photoacoustics 1(3–4), 49–53 (2013).
[Crossref]

Sun, T.

M. I. Khan, T. Sun, and G. J. Diebold, “Photoacoustic waves generated by absorption of laser radiation in optically thin layers,” J. Acoust. Soc. Am. 93(3), 1417–1425 (1993).
[Crossref]

G. J. Diebold, T. Sun, and M. I. Khan, “Photoacoustic monopole radiation in one, two, and three dimensions,” Phys. Rev. Lett. 67(24), 3384–3387 (1991).
[Crossref] [PubMed]

Tarawneh, C.

J. P. Barton, N. L. Wolff, H. Zhang, and C. Tarawneh, “Near-field calculations for a rigid spheroid with an arbitrary incident acoustic field,” J. Acoust. Soc. Am. 113(3), 1216–1222 (2003).
[Crossref] [PubMed]

Thompson, W. J.

W. J. Thompson, “Spheroidal wave functions,” Comput. Sci. Eng. 1, 84–87 (1999).

Tobias, I.

Tobías, I.

Wang, J. B.

P. E. Falloon, P. C. Abbott, and J. B. Wang, “Theory and computation of spheroidal wave functions,” J. Phys. Math. Gen. 36(20), 5477–5495 (2003).
[Crossref]

Wang, L. V.

S. Hu and L. V. Wang, “Optical-resolution photoacoustic microscopy: auscultation of biological systems at the cellular level,” Biophys. J. 105(4), 841–847 (2013).
[Crossref] [PubMed]

L. V. Wang and S. Hu, “Photoacoustic tomography: in vivo imaging from organelles to organs,” Science 335(6075), 1458–1462 (2012).
[Crossref] [PubMed]

Wolff, N. L.

J. P. Barton, N. L. Wolff, H. Zhang, and C. Tarawneh, “Near-field calculations for a rigid spheroid with an arbitrary incident acoustic field,” J. Acoust. Soc. Am. 113(3), 1216–1222 (2003).
[Crossref] [PubMed]

Yamamoto, G.

Zeppenfeld, M.

Zhang, H.

J. P. Barton, N. L. Wolff, H. Zhang, and C. Tarawneh, “Near-field calculations for a rigid spheroid with an arbitrary incident acoustic field,” J. Acoust. Soc. Am. 113(3), 1216–1222 (2003).
[Crossref] [PubMed]

Zharov, V. P.

E. I. Galanzha and V. P. Zharov, “Circulating tumor cell detection and capture by photoacoustic flow cytometry in vivo and ex vivo,” Cancers 5(4), 1691–1738 (2013).
[Crossref] [PubMed]

Acoust. Phys. (1)

A. D. Kotsis and J. A. Roumeliotis, “Acoustic scattering by a penetrable spheroid,” Acoust. Phys. 54(2), 153–167 (2008).
[Crossref]

Appl. Opt. (1)

Biophys. J. (2)

S. Hu and L. V. Wang, “Optical-resolution photoacoustic microscopy: auscultation of biological systems at the cellular level,” Biophys. J. 105(4), 841–847 (2013).
[Crossref] [PubMed]

E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “Probing red blood cell morphology using high-frequency photoacoustics,” Biophys. J. 105(1), 59–67 (2013).
[Crossref] [PubMed]

Cancers (1)

E. I. Galanzha and V. P. Zharov, “Circulating tumor cell detection and capture by photoacoustic flow cytometry in vivo and ex vivo,” Cancers 5(4), 1691–1738 (2013).
[Crossref] [PubMed]

Comput. Sci. Eng. (1)

W. J. Thompson, “Spheroidal wave functions,” Comput. Sci. Eng. 1, 84–87 (1999).

J. Acoust. Soc. Am. (4)

M. I. Khan, T. Sun, and G. J. Diebold, “Photoacoustic waves generated by absorption of laser radiation in optically thin layers,” J. Acoust. Soc. Am. 93(3), 1417–1425 (1993).
[Crossref]

R. D. Spence and S. Granger, “The scattering of sound from a prolate spheroid,” J. Acoust. Soc. Am. 23(6), 701–706 (1951).
[Crossref]

J. P. Barton, N. L. Wolff, H. Zhang, and C. Tarawneh, “Near-field calculations for a rigid spheroid with an arbitrary incident acoustic field,” J. Acoust. Soc. Am. 113(3), 1216–1222 (2003).
[Crossref] [PubMed]

B. R. Rapids and G. C. Lauchle, “Vector intensity field scattered by a rigid prolate spheroid,” J. Acoust. Soc. Am. 120(1), 38–48 (2006).
[Crossref]

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

J. Phys. Math. Gen. (1)

P. E. Falloon, P. C. Abbott, and J. B. Wang, “Theory and computation of spheroidal wave functions,” J. Phys. Math. Gen. 36(20), 5477–5495 (2003).
[Crossref]

New J. Phys. (1)

M. Zeppenfeld, “Solutions to Maxwell's equations using spheroidal coordinates,” New J. Phys. 11(7), 073007 (2009).
[Crossref]

Opt. Express (2)

Photoacoustics (1)

E. M. Strohm, E. S. L. Berndl, and M. C. Kolios, “High frequency label-free photoacoustic microscopy of single cells,” Photoacoustics 1(3–4), 49–53 (2013).
[Crossref]

Phys. Rev. Lett. (1)

G. J. Diebold, T. Sun, and M. I. Khan, “Photoacoustic monopole radiation in one, two, and three dimensions,” Phys. Rev. Lett. 67(24), 3384–3387 (1991).
[Crossref] [PubMed]

Science (2)

L. V. Wang and S. Hu, “Photoacoustic tomography: in vivo imaging from organelles to organs,” Science 335(6075), 1458–1462 (2012).
[Crossref] [PubMed]

G. J. Diebold, M. I. Khan, and S. M. Park, “Photoacoustic “signatures” of particulate matter: optical production of acoustic monopole radiation,” Science 250(4977), 101–104 (1990).
[Crossref] [PubMed]

Stud. Appl. Math. (1)

W. M. John, “Asymptotic approximations for prolate spheroidal wave functions,” Stud. Appl. Math. 54, 315–349 (1975).

Other (7)

L. V. Wang and H.-I. Wu, Biomedical Optics (Wiley, 2007), Chap. 12.

V. E. Gusev and A. A. Karabutov, Laser Optoacoustics (American Institute of Physics Press, 1993), Chap. 2.

S. Zhang and J. Jin, Computation of Special Functions (Wiley, 1997).

L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (Wiley, 2002).

G. J. Diebold, “Photoacoustic monopole radiation: waves from objects with symmetry in one, two, and three dimensions,” in Photoacoustic Imaging and Spectroscopy, L. V. Wang, ed. (Taylor and Francis, 2009).

C. Flammer, Spheroidal Wave Functions (Stanford University, 1957).

K. T. McDonald, “Gaussian laser beams via oblate spheroidal waves,” arXiv:physics/0312024v1 (2003).

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 (2)

Fig. 1
Fig. 1 Illustration of photoacoustic (PA) wave generation of a prolate spheroidal droplet or an oblate spheroidal droplet. In the left panel, on the top a prolate spheroid along with its prolate spheroidal coordinate system is plotted and at the bottom an oblate spheroid along with its oblate spheroidal coordinate system is plotted, where the two black dots represent the foci and the z-axis represents the revolution axis for each system. In the right panel. the three geometries, infinitely long cylinder, sphere, and infinitely large layer, represent respectively the asymptotic spheroids under the extreme conditions of infinite or zero interfocal distance. For these three geometries, the analytic solutions of the photoacoustic waves are well-known [13].
Fig. 2
Fig. 2 Illustration for the model of outgoing photoacoustic wave partial transmission and reflection at the boundary of the spheroidal droplet. The left side figure illustrates the case for a prolate spheroidal droplet, and the right side figure illustrates the case for an oblate spheroidal droplet. In each figure, the orange contour plots the droplet boundary, and the black curves plot the spheroidal coordinate system where the thick black line corresponds to the area with the minimum ξ . The red arrows inside the boundary represent the outgoing photoacoustic waves, the red arrows outside the boundary represent the partially transmitted photoacoustic waves, and the green arrows represent the partially reflected photoacoustic waves. All of these photoacoustic waves propagate along the direction normal to the ξ surfaces.

Equations (94)

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

t T= κ ρ C P 2 T+ H ρ C P ,
2 p 1 v 2 2 t 2 p= α v 2 2 t 2 T,
2 p 1 v 2 2 t 2 p= β C p t H,
{ 2 p s 1 v s 2 2 t 2 p s = β C p H t (insidethedroplet) 2 p f 1 v f 2 2 t 2 p f =0(outsidethedroplet)
p s = p s1 + p s2 ,
1 v s 2 2 t 2 p s1 = β C p H t
2 p s2 1 v s 2 2 t 2 p s2 =0.
H= ε th μ a I 0 e iωt
p s1 (ω) p 0 =i ε th μ a β I 0 v s 2 ω C p .
p s2 (ω,ξ,η,ϕ)= p 0 m=0 nm p mn s S mn ({ c s i c s ,η) R mn (1) ({ c s ,ξ i c s ,iξ ) e ±imϕ ,
p f (ω,ξ,η,ϕ)= p 0 m=0 nm p mn f S mn ({ c f i c f ,η) R mn (3) ({ c f ,ξ i c f ,iξ ) e ±imϕ .
c s = ω v s d 2 = k ^ s d 2 ,and c f = ω v f d 2 = k ^ f d 2
ξ 0 = a d/2 1(prolatespheroid),
ξ 0 = b d/2 0(oblatespheroid).
x= d 2 (1 η 2 )( ξ 2 1) cosϕ,y= d 2 (1 η 2 )( ξ 2 1) sinϕ,z= d 2 ηξ(prolatespheroid),
x= d 2 (1 η 2 )( ξ 2 +1) cosϕ,y= d 2 (1 η 2 )( ξ 2 +1) sinϕ,z= d 2 ηξ(oblatespheroid).
p s (ω, ξ 0 ,η,ϕ)= p f (ω, ξ 0 ,η,ϕ), p s (ω,ξ,η,ϕ) ρ s ξ | ξ= ξ 0 = p f (ω,ξ,η,ϕ) ρ f ξ | ξ= ξ 0 .
whenξ,ηcosθ,
m=0,n=2k.
p s2 ϕ =0, p f ϕ =0,
p s2 (η)= p s2 (η), p f (η)= p f (η)
S 0n ({ c ic ,η)= l=0,1 ' d l 0n ({ c ic ) P l (η)
p s2 (ω,ξ,η,ϕ)= p 0 n=0 p 0n s S 0n ({ c s i c s ,η) R 0n (1) ({ c s ,ξ i c s ,iξ ) ,n=2k,
p f (ω,ξ,η,ϕ)= p 0 n=0 p 0n f S 0n ({ c f i c f ,η) R 0n (3) ({ c f ,ξ i c f ,iξ ) ,n=2k.
{ δ 0,l + n=0 p 0n s d l 0n ({ c s i c s ) R 0n (1) ({ c s , ξ 0 i c s ,i ξ 0 ) = n=0 p 0n f d l 0n ({ c f i c f ) R 0n (3) ({ c f , ξ 0 i c f ,i ξ 0 ) n=0 p 0n s d l 0n ({ c s i c s ) R 0n (1) ({ c s , ξ 0 i c s ,i ξ 0 ) = ρ s ρ f n=0 p 0n f d l 0n ({ c f i c f ) R 0n (3) ({ c f , ξ 0 i c f ,i ξ 0 ) , n=2k,l=2 k 0,
{ A+ D s R 1s P s = D f R 3f P f D s R 1s P s = ρ s ρ f D f R 3f P f .
A i = δ i,1 .
(D s ) ij = d 2i2 0(2j2) ({ c s i c s ), (D f ) ij = d 2i2 0(2j2) ({ c f i c f ),
(R 1s ) ij = δ i,j R 0(2i2) (1) ({ c s , ξ 0 i c s ,i ξ 0 ), ( R 1s ) ij = δ i,j R 0(2i2) (1) ({ c s , ξ 0 i c s ,i ξ 0 ),
(R 3f ) ij = δ i,j R 0(2i2) (3) ({ c f , ξ 0 i c f ,i ξ 0 ), ( R 3f ) ij = δ i,j R 0(2i2) (3) ({ c f , ξ 0 i c f ,i ξ 0 ).
P s = [ ρ f ρ s D f R 3f ( R 3f ) 1 (D f ) 1 D s R 1s D s R 1s ] 1 A,
P f = [ D f R 3f ρ s ρ f D s R 1s ( R 1s ) 1 (D s ) 1 D f R 3f ] 1 A.
I(t)= 1 2π + I ˜ (ω) e iωt dω .
p s (t,ξ,η,ϕ)= 1 2π + I ˜ (ω) I 0 p 0 e iωt [1+ n=0 p 0n s S 0n ({ c s i c s ,η) R 0n (1) ({ c s ,ξ i c s ,iξ )] dω ,n=2k,
p f (t,ξ,η,ϕ)= 1 2π + I ˜ (ω) I 0 p 0 e iωt n=0 p 0n f S 0n ({ c f i c f ,η) R 0n (3) ({ c f ,ξ i c f ,iξ ) dω ,n=2k,
I(t)= I 0 δ(t) I ˜ (ω) I 0 =1,
c s 0, c f 0.
S 0n ({ c ic ,η)~ P n (η),n=2k,forc0,
d 2i 0(2j) ({ c ic )~ δ i,j ,forc0.
D s E, D f E.
( P f ) i ~ δ i,1 × 1 (R 3f ) 11 ρ s ρ f (R 1s ) 11 ( R 3f ) 11 ( R 1s ) 11 ,
R 0n (1) ({ c,ξ ic,iξ )~ j n ( k ^ r),n=2k,forc0,
R 0n (3) ({ c,ξ ic,iξ )~ h n (1) ( k ^ r),n=2k,forc0,
cξ k ^ r,forc0
( P f ) i ~ δ i,1 × 1 h 0 ( k ^ f a) ρ s ρ f k ^ f h 1 (1) ( k ^ f a) k ^ s j 1 ( k ^ s a) j 0 ( k ^ s a) .
p f (ω,ξ,η,ϕ)~ p 0 j 1 ( k ^ s a) h 0 (1) ( k ^ f r) j 1 ( k ^ s a) h 0 ( k ^ f a) ρ s v s ρ f v f h 1 (1) ( k ^ f a) j 0 ( k ^ s a) .
p f (ω,ξ,η,ϕ)~ p 0 r/a e i q ^ τ × (sin q ^ q ^ cos q ^ )/ q ^ [(1 ρ s ρ f )(sin q ^ / q ^ )cos q ^ +i ρ s v s ρ f v f sin q ^ ] ,
c s , c f .
D s ~D f ,
P f ~ [ R 3f ρ s ρ f R 1s ( R 1s ) 1 R 3f ] 1 (D f ) 1 A.
R 0n (1) (c,ξ)~ π 2c J 0 (c ξ 2 1 ),n=2k,forc,
R 0n (3) (c,ξ)~ π 2c H 0 (1) (c ξ 2 1 ),n=2k,forc,
S 0n (c,η)~ h 0 n 2 n/2 e c η 2 /2 H n ( c η),n=2k,forc,
η0,
c ξ 2 1 k ^ r.
P f ~ J 1 ( k ^ s b) π 2 c f H 0 (1) ( k ^ f b) J 1 ( k ^ s b) ρ s v s ρ f v f π 2 c f J 0 ( k ^ s b) H 1 (1) ( k ^ f b) (D f ) 1 A.
p f (ξ,η)~ p 0 J 1 ( k ^ s b) H 0 (1) ( k ^ f r) H 0 (1) ( k ^ f b) J 1 ( k ^ s b) ρ s v s ρ f v f J 0 ( k ^ s b) H 1 (1) ( k ^ f b) k=0 [ ( D f ) 1 ] (k+1)1 S 0(2k) ( c f ,η) .
k=0 [ ( D f ) 1 ] (k+1)1 S 0(2k) ( c f ,η) = k=0 [ ( D f ) 1 ] (k+1)1 k =0 d 2 k 0(2k) P 2 k (η) = k =0 k=0 [ ( D f ) 1 ] (k+1)1 ( D f ) ( k +1)(k+1) P 2 k (η) = δ k ,0 P 2 k (η)=1,
p f (ω,ξ,η,ϕ)~ p 0 J 1 ( k ^ s b) H 0 (1) ( k ^ f r) H 0 (1) ( k ^ f b) J 1 ( k ^ s b) ρ s v s ρ f v f J 0 ( k ^ s b) H 1 (1) ( k ^ f b) .
c s , c f .
R 0n (1) (ic,iξ)~ cos(cξ) c × e i n 2 π ,n=2k,forc,
R 0n (3) (ic,iξ)~ e icξ c × e i n 2 π ,n=2k,forc.
S 0n (c,η)~ A 0 0n { e c(1η) L n/2 [2c(1η)]+ e c(1+η) L n/2 [2c(1+η)]},n=2k,forc,
η±1.
cξ k ^ |z|.
P f ~ c f sin( k ^ s b) e i k ^ f b sin( k ^ s b)+i ρ s v s ρ f v f cos( k ^ s b) E ¯ (D f ) 1 A,
p f (ω,ξ,η,ϕ)~ p 0 sin( k ^ s b) e i k ^ f (|z|b) sin( k ^ s b)+i ρ s v s ρ f v f cos( k ^ s b) .
p f (t,ξ,η,ϕ)= 1 2π + p 0 e iωt n=0 [(E++ 2 +) P f_i ] (n/2+1) S 0n ({ c f i c f ,η) R 0n (3) ({ c f ,ξ i c f ,iξ ) dω ,n=2k,
p s (t,ξ,η,ϕ)= 1 2π + p 0 e iωt dω + 1 2π + p 0 e iωt n=0 [ P s_i ] (n/2+1) S 0n ({ c s i c s ,η) R 0n (1) ({ c s ,ξ i c s ,iξ ) dω + 1 2π + p 0 e iωt n=0 [2(E++ 2 +) P f_i ] (n/2+1) S 0n ({ c s i c s ,η) R 0n (1) ({ c s ,ξ i c s ,iξ ) dω ,n=2k.
P f_i = [ R 3s ( R 1s ) 1 R 1s R 3s ] 1 (D s ) 1 A,
P s_i = [ R 3s ( R 3s ) 1 R 1s R 1s ] 1 (D s ) 1 A.
2 R 0n (1) = R 0n (3) + R 0n (4) ,
p I (ω,ξ,η,ϕ)= p 0 n=0 B (n/2+1) S 0n ({ c s i c s ,η) R 0n (3) ({ c s ,ξ i c s ,iξ ) , B (n/2+1) 1,n=2k
{ p T (ω,ξ,η,ϕ)= p 0 n=0 T (n/2+1) S 0n ({ c f i c f ,η) R 0n (3) ({ c f ,ξ i c f ,iξ ) p R (ω,ξ,η,ϕ)= p 0 n=0 R (n/2+1) S 0n ({ c s i c s ,η) R 0n (4) ({ c s ,ξ i c s ,iξ ) ,n=2k
{ p I (ω, ξ 0 ,η,ϕ)+ p R (ω, ξ 0 ,η,ϕ)= p T (ω, ξ 0 ,η,ϕ) [ p I (ω,ξ,η,ϕ)+ p R (ω,ξ,η,ϕ)] ρ s ξ | ξ= ξ 0 = p T (ω,ξ,η,ϕ) ρ f ξ | ξ= ξ 0 .
R 0n (3) = R 0n (1) +i R 0n (2) , R 0n (4) = R 0n (1) i R 0n (2) ,
{ D s R 3s B +D s R 4s R =D f R 3f T D s R 3s B +D s R 4s R= ρ s ρ f D f R 3f T ,
(R 3s ) ij = δ i,j R 0(2i2) (3) ({ c s , ξ 0 i c s ,i ξ 0 ), ( R 3s ) ij = δ i,j R 0(2i2) (3) ({ c s , ξ 0 i c s ,i ξ 0 ),
(R 4s ) ij = δ i,j R 0(2i2) (4) ({ c s , ξ 0 i c s ,i ξ 0 ), ( R 4s ) ij = δ i,j R 0(2i2) (4) ({ c s , ξ 0 i c s ,i ξ 0 ).
{ T=B R=B ,
{ = [ D f R 3f ρ s ρ f D s R 4s ( R 4s ) 1 (D s ) 1 D f R 3f ] 1 [ D s R 3s D s R 4s ( R 4s ) 1 R 3s ] = [ ρ f ρ s D f R 3f ( R 3f ) 1 (D f ) 1 D s R 4s D s R 4s ] 1 [ D s R 3s ρ f ρ s D f R 3f ( R 3f ) 1 (D f ) 1 D s R 3s ] .
(E) 1 = 1 2 [ ρ f ρ s D f R 3f ( R 3f ) 1 (D f ) 1 D s R 1s D s R 1s ] 1 [ ρ f ρ s D f R 3f ( R 3f ) 1 (D f ) 1 D s R 4s D s R 4s ].
R 0n (1) R 0n (2) R 0n (1) R 0n (2) ={ 1 c( ξ 2 1) (forprolatespheroid) 1 c( ξ 2 +1) (foroblatespheroid) ,
= [ R 4s (D s ) 1 D f R 3f ρ s ρ f R 4s (D s ) 1 D f R 3f ] 1 ×{ 2i c s ( ξ 2 1) 2i c s ( ξ 2 +1) .
[ R 4s (D s ) 1 D f R 3f ρ s ρ f R 4s (D s ) 1 D f R 3f ] 1 [ ρ f ρ s D f R 3f ( R 3f ) 1 (D f ) 1 D s R 1s D s R 1s ] 1 = [ R 1s (D s ) 1 D f R 3f ρ s ρ f R 1s (D s ) 1 D f R 3f ] 1 [ ρ f ρ s D f R 3f ( R 3f ) 1 (D f ) 1 D s R 4s D s R 4s ] 1 ,
(E) 1 = [ R 1s (D s ) 1 D f R 3f ρ s ρ f R 1s (D s ) 1 D f R 3f ] 1 ×{ i c s ( ξ 2 1) i c s ( ξ 2 +1) .
P f = (E) 1 R 1s (D s ) 1 A×{ i c s ( ξ 2 1) i c s ( ξ 2 +1) .
P f_i = R 1s (D s ) 1 A×{ i c s ( ξ 2 1) i c s ( ξ 2 +1) ,
P f = (E) 1 P f_i .
2 (E) 1 = [ ρ f ρ s D f R 3f ( R 3f ) 1 (D f ) 1 D s R 1s D s R 1s ] 1 [ D s R 3s ρ f ρ s D f R 3f ( R 3f ) 1 (D f ) 1 D s R 3s ].
R 3s = ( R 1s ) 1 R 1s R 3s ( R 1s ) 1 ×{ i c s ( ξ 2 1) i c s ( ξ 2 +1)
2 (E) 1 = R 3s ( R 1s ) 1 [ ρ f ρ s D f R 3f ( R 3f ) 1 (D f ) 1 D s R 1s D s R 1s ] 1 D s ( R 1s ) 1 ×{ i c s ( ξ 2 1) i c s ( ξ 2 +1) .
P s_i = R 3s (D s ) 1 A×{ i c s ( ξ 2 1) i c s ( ξ 2 +1) ,
P s =2 (E) 1 P f_i + P s_i .

Metrics