Abstract

The modal expansion of the field diffracted by a spherical object is transformed to an angular spectrum representation. Outside the object the angular spectrum representation yields the same diffracted field as that from the modal expansion. By neglecting the inhomogeneous plane waves in this representation, we also obtain the virtual, or backpropagated, fields for planes within or behind the object. These virtual fields are the fields that, to an external observer, appear to exist in free space within or behind the object. Thus these virtual fields are different from the actual fields existing in these regions. The image field model is obtained by including aperture limitations to the angular spectrum representation. These image fields could have been obtained by the more cumbersome approach of first computing the field in the entrance pupil of the imaging system by using the modal expansion and then propagating it back to a plane within or behind the object. The physical properties of images of the center plane of spheres are derived for the scalar case. The conditions for representing a sphere with an equivalent thin-object model are derived along with the criteria for directly representing this thin-object model by the modal coefficients. The usefulness of the image field model for numerical field calculations is illustrated by several examples. Diffraction effects that are present only in imaging situations are presented along with field calculations in the Fresnel region, where both the image field model and the modal expansion yield the same results.

© 1998 Optical Society of America

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References

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  1. N. E. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
    [CrossRef]
  2. A. Clebsh, “Über die Refexion an einer Kugelfläche,” J. Math. 61, 195–262 (1863).
  3. G. Mie, “Beitrage zur Optik trüber Medien, speziell kollodialer Metallösungen,” Ann. Phys. (Leipzig) 25, 377–442 (1908).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Exeter, UK, 1981).
  5. J. J. Bowman, T. B. A. Senior, P. L. E. Ushlenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).
  6. J. B. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 116–130 (1962).
    [CrossRef] [PubMed]
  7. A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representation of the electromagnetic field,” J. Math. Phys. (N.Y.) 15, 183–201 (1995).
  8. G. C. Gaunaurd, H. Überall, “Acoustics of finite beams,” J. Acoust. Soc. Am. 63, 5–16 (1978).
    [CrossRef]
  9. P. Zinin, W. Weise, O. Lobkis, O. Kosolov, S. Boseck, “Fourier optics analysis of spherical particles image formation in reflection acoustic microscopy,” Optik (Stuttgart) 98, 45–60 (1994).
  10. W. Weise, P. Zinin, S. Boseck, “Angular spectrum approach for imaging of spherical particles in reflection and transmission SAM,” in Acoustical Imaging, L. Masotti, P. Tortoli, eds. (Plenum, New York, 1995), Vol. 22, pp. 707–712.
  11. W. Weise, P. Zinin, T. Wilson, D. A. D. Briggs, S. Boseck, “Imaging of spheres with the confocal scanning optical microscope,” Opt. Lett. 21, 1800–1802 (1996).
    [CrossRef] [PubMed]
  12. P. Zinin, W. Weise, O. Lobkis, S. Boseck, “The theory of three dimensional imaging of strong scatterers in scanning acoustic microscopy,” Wave Motion 25, 213–236 (1997).
    [CrossRef]
  13. O. I. Lobkis, T. Kundu, P. V. Zinin, “A theoretical analysis of acoustic microscopy for spherical cavities,” Wave Motion 21, 183–201 (1995).
    [CrossRef]
  14. O. I. Lobkis, K. I. Maslov, T. Kundu, P. V. Zinin, “Spherical inclusion characterization by the acoustical microscope—axisymmetrical case,” J. Acoust. Soc. Am. 99, 33–45 (1996).
    [CrossRef]
  15. K. Kvien, “The validity of weak scattering models in forward two dimensional optical scattering,” Appl. Opt. 34, 8447–8459 (1995).
    [CrossRef] [PubMed]
  16. K. Kvien, “Shift of the shadow boundary in images of two-dimensional circular cylinders,” Opt. Commun. 141, 107–112 (1997).
    [CrossRef]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  18. J. R. Sherwell, E. Wolf, “Inverse scattering and a new reciprocity theorem,” J. Opt. Soc. Am. 58, 1596–1603 (1968).
    [CrossRef]
  19. G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1969).
    [CrossRef]
  20. A. J. Devaney, G. C. Sherman, “Plane wave representations for scalar wave fields,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 765–786 (1973).
  21. J. J. Stamnes, “Focusing of two-dimensional waves,” J. Opt. Soc. Am. 71, 15–31 (1981).
    [CrossRef]
  22. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  23. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1968).
  24. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  25. K. Rottmann, Mathematische Formelsammlung (Bibliographishes Institut, Mannheim, 1960).
  26. I. S. Gradsteyn, I. M. Ryshik, Tables of Series, Products and Integrals (Springer-Verlag, Berlin, 1981).
  27. M. Abramowitz, I. A. Stegun, Handbook for Mathematical Functions (Dover, New York, 1964).
  28. L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Nonrelativistic Theory) (Pergamon, Oxford, 1959).
  29. A. Erdêlyi, “Zur der Theorie der Kugelwellen,” Physica (Utrecht) 2, 107–120 (1937).
    [CrossRef]
  30. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
  31. W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974).
  32. D. A. Varshalowich, V. K. Khersonskii, A. N. Moskalev, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific, Singapore, 1988).
  33. J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986).

1997 (2)

P. Zinin, W. Weise, O. Lobkis, S. Boseck, “The theory of three dimensional imaging of strong scatterers in scanning acoustic microscopy,” Wave Motion 25, 213–236 (1997).
[CrossRef]

K. Kvien, “Shift of the shadow boundary in images of two-dimensional circular cylinders,” Opt. Commun. 141, 107–112 (1997).
[CrossRef]

1996 (2)

W. Weise, P. Zinin, T. Wilson, D. A. D. Briggs, S. Boseck, “Imaging of spheres with the confocal scanning optical microscope,” Opt. Lett. 21, 1800–1802 (1996).
[CrossRef] [PubMed]

O. I. Lobkis, K. I. Maslov, T. Kundu, P. V. Zinin, “Spherical inclusion characterization by the acoustical microscope—axisymmetrical case,” J. Acoust. Soc. Am. 99, 33–45 (1996).
[CrossRef]

1995 (3)

K. Kvien, “The validity of weak scattering models in forward two dimensional optical scattering,” Appl. Opt. 34, 8447–8459 (1995).
[CrossRef] [PubMed]

O. I. Lobkis, T. Kundu, P. V. Zinin, “A theoretical analysis of acoustic microscopy for spherical cavities,” Wave Motion 21, 183–201 (1995).
[CrossRef]

A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representation of the electromagnetic field,” J. Math. Phys. (N.Y.) 15, 183–201 (1995).

1994 (1)

P. Zinin, W. Weise, O. Lobkis, O. Kosolov, S. Boseck, “Fourier optics analysis of spherical particles image formation in reflection acoustic microscopy,” Optik (Stuttgart) 98, 45–60 (1994).

1981 (1)

1978 (1)

G. C. Gaunaurd, H. Überall, “Acoustics of finite beams,” J. Acoust. Soc. Am. 63, 5–16 (1978).
[CrossRef]

1973 (1)

A. J. Devaney, G. C. Sherman, “Plane wave representations for scalar wave fields,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 765–786 (1973).

1969 (1)

1968 (1)

1965 (1)

N. E. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
[CrossRef]

1962 (1)

1937 (1)

A. Erdêlyi, “Zur der Theorie der Kugelwellen,” Physica (Utrecht) 2, 107–120 (1937).
[CrossRef]

1908 (1)

G. Mie, “Beitrage zur Optik trüber Medien, speziell kollodialer Metallösungen,” Ann. Phys. (Leipzig) 25, 377–442 (1908).
[CrossRef]

1863 (1)

A. Clebsh, “Über die Refexion an einer Kugelfläche,” J. Math. 61, 195–262 (1863).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook for Mathematical Functions (Dover, New York, 1964).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Exeter, UK, 1981).

Boseck, S.

P. Zinin, W. Weise, O. Lobkis, S. Boseck, “The theory of three dimensional imaging of strong scatterers in scanning acoustic microscopy,” Wave Motion 25, 213–236 (1997).
[CrossRef]

W. Weise, P. Zinin, T. Wilson, D. A. D. Briggs, S. Boseck, “Imaging of spheres with the confocal scanning optical microscope,” Opt. Lett. 21, 1800–1802 (1996).
[CrossRef] [PubMed]

P. Zinin, W. Weise, O. Lobkis, O. Kosolov, S. Boseck, “Fourier optics analysis of spherical particles image formation in reflection acoustic microscopy,” Optik (Stuttgart) 98, 45–60 (1994).

W. Weise, P. Zinin, S. Boseck, “Angular spectrum approach for imaging of spherical particles in reflection and transmission SAM,” in Acoustical Imaging, L. Masotti, P. Tortoli, eds. (Plenum, New York, 1995), Vol. 22, pp. 707–712.

Bowman, J. J.

J. J. Bowman, T. B. A. Senior, P. L. E. Ushlenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Briggs, D. A. D.

Clebsh, A.

A. Clebsh, “Über die Refexion an einer Kugelfläche,” J. Math. 61, 195–262 (1863).

Devaney, A. J.

A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representation of the electromagnetic field,” J. Math. Phys. (N.Y.) 15, 183–201 (1995).

A. J. Devaney, G. C. Sherman, “Plane wave representations for scalar wave fields,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 765–786 (1973).

Erdêlyi, A.

A. Erdêlyi, “Zur der Theorie der Kugelwellen,” Physica (Utrecht) 2, 107–120 (1937).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1968).

Gaunaurd, G. C.

G. C. Gaunaurd, H. Überall, “Acoustics of finite beams,” J. Acoust. Soc. Am. 63, 5–16 (1978).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Gradsteyn, I. S.

I. S. Gradsteyn, I. M. Ryshik, Tables of Series, Products and Integrals (Springer-Verlag, Berlin, 1981).

Keller, J. B.

Khersonskii, V. K.

D. A. Varshalowich, V. K. Khersonskii, A. N. Moskalev, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific, Singapore, 1988).

Kosolov, O.

P. Zinin, W. Weise, O. Lobkis, O. Kosolov, S. Boseck, “Fourier optics analysis of spherical particles image formation in reflection acoustic microscopy,” Optik (Stuttgart) 98, 45–60 (1994).

Kundu, T.

O. I. Lobkis, K. I. Maslov, T. Kundu, P. V. Zinin, “Spherical inclusion characterization by the acoustical microscope—axisymmetrical case,” J. Acoust. Soc. Am. 99, 33–45 (1996).
[CrossRef]

O. I. Lobkis, T. Kundu, P. V. Zinin, “A theoretical analysis of acoustic microscopy for spherical cavities,” Wave Motion 21, 183–201 (1995).
[CrossRef]

Kvien, K.

K. Kvien, “Shift of the shadow boundary in images of two-dimensional circular cylinders,” Opt. Commun. 141, 107–112 (1997).
[CrossRef]

K. Kvien, “The validity of weak scattering models in forward two dimensional optical scattering,” Appl. Opt. 34, 8447–8459 (1995).
[CrossRef] [PubMed]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Nonrelativistic Theory) (Pergamon, Oxford, 1959).

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Nonrelativistic Theory) (Pergamon, Oxford, 1959).

Lobkis, O.

P. Zinin, W. Weise, O. Lobkis, S. Boseck, “The theory of three dimensional imaging of strong scatterers in scanning acoustic microscopy,” Wave Motion 25, 213–236 (1997).
[CrossRef]

P. Zinin, W. Weise, O. Lobkis, O. Kosolov, S. Boseck, “Fourier optics analysis of spherical particles image formation in reflection acoustic microscopy,” Optik (Stuttgart) 98, 45–60 (1994).

Lobkis, O. I.

O. I. Lobkis, K. I. Maslov, T. Kundu, P. V. Zinin, “Spherical inclusion characterization by the acoustical microscope—axisymmetrical case,” J. Acoust. Soc. Am. 99, 33–45 (1996).
[CrossRef]

O. I. Lobkis, T. Kundu, P. V. Zinin, “A theoretical analysis of acoustic microscopy for spherical cavities,” Wave Motion 21, 183–201 (1995).
[CrossRef]

Logan, N. E.

N. E. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).

Maslov, K. I.

O. I. Lobkis, K. I. Maslov, T. Kundu, P. V. Zinin, “Spherical inclusion characterization by the acoustical microscope—axisymmetrical case,” J. Acoust. Soc. Am. 99, 33–45 (1996).
[CrossRef]

Mie, G.

G. Mie, “Beitrage zur Optik trüber Medien, speziell kollodialer Metallösungen,” Ann. Phys. (Leipzig) 25, 377–442 (1908).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1968).

Moskalev, A. N.

D. A. Varshalowich, V. K. Khersonskii, A. N. Moskalev, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific, Singapore, 1988).

Rottmann, K.

K. Rottmann, Mathematische Formelsammlung (Bibliographishes Institut, Mannheim, 1960).

Ryshik, I. M.

I. S. Gradsteyn, I. M. Ryshik, Tables of Series, Products and Integrals (Springer-Verlag, Berlin, 1981).

Senior, T. B. A.

J. J. Bowman, T. B. A. Senior, P. L. E. Ushlenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

Sherman, G. C.

A. J. Devaney, G. C. Sherman, “Plane wave representations for scalar wave fields,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 765–786 (1973).

G. C. Sherman, “Diffracted wave fields expressible by plane-wave expansions containing only homogeneous waves,” J. Opt. Soc. Am. 59, 697–711 (1969).
[CrossRef]

Sherwell, J. R.

Stamnes, J. J.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook for Mathematical Functions (Dover, New York, 1964).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

Überall, H.

G. C. Gaunaurd, H. Überall, “Acoustics of finite beams,” J. Acoust. Soc. Am. 63, 5–16 (1978).
[CrossRef]

Ushlenghi, P. L. E.

J. J. Bowman, T. B. A. Senior, P. L. E. Ushlenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Varshalowich, D. A.

D. A. Varshalowich, V. K. Khersonskii, A. N. Moskalev, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific, Singapore, 1988).

Weise, W.

P. Zinin, W. Weise, O. Lobkis, S. Boseck, “The theory of three dimensional imaging of strong scatterers in scanning acoustic microscopy,” Wave Motion 25, 213–236 (1997).
[CrossRef]

W. Weise, P. Zinin, T. Wilson, D. A. D. Briggs, S. Boseck, “Imaging of spheres with the confocal scanning optical microscope,” Opt. Lett. 21, 1800–1802 (1996).
[CrossRef] [PubMed]

P. Zinin, W. Weise, O. Lobkis, O. Kosolov, S. Boseck, “Fourier optics analysis of spherical particles image formation in reflection acoustic microscopy,” Optik (Stuttgart) 98, 45–60 (1994).

W. Weise, P. Zinin, S. Boseck, “Angular spectrum approach for imaging of spherical particles in reflection and transmission SAM,” in Acoustical Imaging, L. Masotti, P. Tortoli, eds. (Plenum, New York, 1995), Vol. 22, pp. 707–712.

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974).

Wilson, T.

Wolf, E.

A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representation of the electromagnetic field,” J. Math. Phys. (N.Y.) 15, 183–201 (1995).

J. R. Sherwell, E. Wolf, “Inverse scattering and a new reciprocity theorem,” J. Opt. Soc. Am. 58, 1596–1603 (1968).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Exeter, UK, 1981).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).

Zinin, P.

P. Zinin, W. Weise, O. Lobkis, S. Boseck, “The theory of three dimensional imaging of strong scatterers in scanning acoustic microscopy,” Wave Motion 25, 213–236 (1997).
[CrossRef]

W. Weise, P. Zinin, T. Wilson, D. A. D. Briggs, S. Boseck, “Imaging of spheres with the confocal scanning optical microscope,” Opt. Lett. 21, 1800–1802 (1996).
[CrossRef] [PubMed]

P. Zinin, W. Weise, O. Lobkis, O. Kosolov, S. Boseck, “Fourier optics analysis of spherical particles image formation in reflection acoustic microscopy,” Optik (Stuttgart) 98, 45–60 (1994).

W. Weise, P. Zinin, S. Boseck, “Angular spectrum approach for imaging of spherical particles in reflection and transmission SAM,” in Acoustical Imaging, L. Masotti, P. Tortoli, eds. (Plenum, New York, 1995), Vol. 22, pp. 707–712.

Zinin, P. V.

O. I. Lobkis, K. I. Maslov, T. Kundu, P. V. Zinin, “Spherical inclusion characterization by the acoustical microscope—axisymmetrical case,” J. Acoust. Soc. Am. 99, 33–45 (1996).
[CrossRef]

O. I. Lobkis, T. Kundu, P. V. Zinin, “A theoretical analysis of acoustic microscopy for spherical cavities,” Wave Motion 21, 183–201 (1995).
[CrossRef]

Ann. Phys. (Leipzig) (1)

G. Mie, “Beitrage zur Optik trüber Medien, speziell kollodialer Metallösungen,” Ann. Phys. (Leipzig) 25, 377–442 (1908).
[CrossRef]

Appl. Opt. (1)

J. Acoust. Soc. Am. (2)

G. C. Gaunaurd, H. Überall, “Acoustics of finite beams,” J. Acoust. Soc. Am. 63, 5–16 (1978).
[CrossRef]

O. I. Lobkis, K. I. Maslov, T. Kundu, P. V. Zinin, “Spherical inclusion characterization by the acoustical microscope—axisymmetrical case,” J. Acoust. Soc. Am. 99, 33–45 (1996).
[CrossRef]

J. Math. (1)

A. Clebsh, “Über die Refexion an einer Kugelfläche,” J. Math. 61, 195–262 (1863).

J. Math. Phys. (N.Y.) (1)

A. J. Devaney, E. Wolf, “Multipole expansions and plane wave representation of the electromagnetic field,” J. Math. Phys. (N.Y.) 15, 183–201 (1995).

J. Opt. Soc. Am. (4)

Opt. Commun. (1)

K. Kvien, “Shift of the shadow boundary in images of two-dimensional circular cylinders,” Opt. Commun. 141, 107–112 (1997).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (1)

P. Zinin, W. Weise, O. Lobkis, O. Kosolov, S. Boseck, “Fourier optics analysis of spherical particles image formation in reflection acoustic microscopy,” Optik (Stuttgart) 98, 45–60 (1994).

Physica (Utrecht) (1)

A. Erdêlyi, “Zur der Theorie der Kugelwellen,” Physica (Utrecht) 2, 107–120 (1937).
[CrossRef]

Proc. IEEE (1)

N. E. Logan, “Survey of some early studies of the scattering of plane waves by a sphere,” Proc. IEEE 53, 773–785 (1965).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) Rev. (1)

A. J. Devaney, G. C. Sherman, “Plane wave representations for scalar wave fields,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 765–786 (1973).

Wave Motion (2)

P. Zinin, W. Weise, O. Lobkis, S. Boseck, “The theory of three dimensional imaging of strong scatterers in scanning acoustic microscopy,” Wave Motion 25, 213–236 (1997).
[CrossRef]

O. I. Lobkis, T. Kundu, P. V. Zinin, “A theoretical analysis of acoustic microscopy for spherical cavities,” Wave Motion 21, 183–201 (1995).
[CrossRef]

Other (15)

W. Weise, P. Zinin, S. Boseck, “Angular spectrum approach for imaging of spherical particles in reflection and transmission SAM,” in Acoustical Imaging, L. Masotti, P. Tortoli, eds. (Plenum, New York, 1995), Vol. 22, pp. 707–712.

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Exeter, UK, 1981).

J. J. Bowman, T. B. A. Senior, P. L. E. Ushlenghi, Electromagnetic and Acoustic Scattering by Simple Shapes (North-Holland, Amsterdam, 1969).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1968).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

K. Rottmann, Mathematische Formelsammlung (Bibliographishes Institut, Mannheim, 1960).

I. S. Gradsteyn, I. M. Ryshik, Tables of Series, Products and Integrals (Springer-Verlag, Berlin, 1981).

M. Abramowitz, I. A. Stegun, Handbook for Mathematical Functions (Dover, New York, 1964).

L. D. Landau, E. M. Lifshitz, Quantum Mechanics (Nonrelativistic Theory) (Pergamon, Oxford, 1959).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, London, 1974).

D. A. Varshalowich, V. K. Khersonskii, A. N. Moskalev, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols (World Scientific, Singapore, 1988).

J. J. Stamnes, Waves in Focal Regions (Hilger, London, 1986).

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Figures (14)

Fig. 1
Fig. 1

Scattering geometry for scalar waves.

Fig. 2
Fig. 2

Scattering geometry for vectorial waves.

Fig. 3
Fig. 3

Coordinate transformations for TM polarization.

Fig. 4
Fig. 4

Coordinate transformations for TE polarization.

Fig. 5
Fig. 5

Spherical and Cartesian unit vectors.

Fig. 6
Fig. 6

Modal contribution in the approximation by tangents.

Fig. 7
Fig. 7

Image at z=0 of a perfectly reflecting sphere of radius a=2λ. The dashed circle is the geometrical shadow boundary of the sphere. The incident field is linearly polarized with the electric-field vector parallel with the x axis, and the imaging aperture half-angle θa60°. Because of polarization effects the image of the sphere is elliptical.

Fig. 8
Fig. 8

Image fields along the x and y axes in Fig. 7 plotted with, respectively, a dashed curve and a solid curve. The geometrical shadow boundary is at η=a. The steps are the positions of the shadow boundary found for cylinders. The image field along the x and y axes corresponds to, respectively, the TM- and TE-polarization cases for cylinders.

Fig. 9
Fig. 9

Virtual focal region for an acoustically transparent sphere of radius a=10λ and index of refraction nˆ=0.9 relative to that of the background. The sphere acts like a concave lens with a virtual focus on the negative z axis. The dashed line is the position of the virtual focal plane found from paraxial optics.

Fig. 10
Fig. 10

Paraxial imaging of a soft sphere with ka=10: (a) cross section of the phase function of a flat circularly symmetric object, the phase function being equal to the scattering phase ψn for the sphere; (b) the dashed curve is the intensity in a paraxial image with aperture half-angle θa=20°, and the solid curve is the corresponding exact image intensity.

Fig. 11
Fig. 11

Point image D64(η) of the off-axis point η64 for several values of the aperture half-angle θa.

Fig. 12
Fig. 12

Image field amplitude at z=0 of the sphere in Fig. 9. The dashed and solid curves are, respectively, the exact and geometrical-optics fields.

Fig. 13
Fig. 13

Paraxial scattering case where the backpropagated field at z=0 is directly given by the mode coefficients. The scattering phase of the sphere in Fig. 9 is plotted with a dashed curve along with the field phase in the geometrical-optics approximation.

Fig. 14
Fig. 14

Scattered field amplitude at z=20λ of an acoustically soft sphere with radius a=2λ. The solid curve and the plusses are the field amplitudes computed with, respectively, the angular spectrum representation and the modal expansion.

Equations (137)

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

Ui(r)=exp(ikir)=exp(ikr cos γ0),
Ui=n=0in(2n+1)Pn(cos γ0)jn(kr),
Us(r)=n=0in(2n+1)anPn(cos γ0)hn(1)(kr),
Pn(cos γ0)=Pn[cos θ cos θ0+sin θ sin θ0 cos(ϕ-ϕ0)]=m=0nm (n-m)!(n+m)!Pnm(cos θ0)×Pnm(cos θ)cos[m(ϕ-ϕ0)],
jn(z)=π/2zJn+1/2(z),
hn(1)(z)=π/2zHn+1/2(1)(z).
an=-nˆjn(kanˆ)jn(ka)-jn(kanˆ)jn(ka)nˆjn(kanˆ)hn(1)(ka)-jn(kanˆ)hn(1)(ka),
an=-12[1+exp(iψn)],
ψn=-2 arg[nˆjn(kanˆ)hn(1)(ka)-jn(kanˆ)hn(1)(ka)]
ψn=-2 arg[hn(1)(ka)],
ψn=-2 arg[hn(1)(ka)].
Pn(cos θ)hn(1)(kr)=12πin02πdβ0π/2-idα×sin α×exp(ikr)Pn(cos α),
Pn(cos γ0)hn(1)(kr)=12πin02πdβ0π/2-idα×sin α exp(ikr)Pn(cos γα0),
Us(r)=12π02πdβ0π/2-idα sin α exp(ikr)S(cos γα0),
S(cos γα0)=n=0(2n+1)anPn(cos γα0).
hn(1)(z)1z(-i)n+1exp(iz),z.
Us(r)exp(ikr)ikrS(cos γ0),kr.
Ei(x, y, z)=ex exp(ikz),
Hi(x, y, z)=ey exp(ikz).
Ei(x, y, z)=n=0in 2n+1n(n+1)[mo1n(1)-ine1n(1)],
Hi(x, y, z)=-n=0in 2n+1n(n+1)[me1n(1)+ino1n(1)].
πn(cos θ)=Pn1(cos θ)sin θ,
τn(cos θ)=ddθPn1(cos θ),
mo1n(1)=πn(cos θ)jn(kr)cos ϕeθ-jn(kr)τn(cos θ)×sin ϕeϕ,
no1n(1)=n(n+1)krjn(kr)Pn1(cos θ)sin ϕer+1kr[krjn(kr)]τn(cos θ)sin ϕeθ+1kr[krjn(kr)]πn(cos θ)cos ϕeϕ.
me1n(1)=-πn(cos θ)jn(kr)sin ϕeθ-jn(kr)τn(cos θ)×cos ϕeϕ,
ne1n(1)=n(n+1)krjn(kr)Pn1(cos θ)cos ϕer+1kr[krjn(kr)]τn(cos θ)cos ϕeθ-1kr[krjn(kr)]πn(cos θ)sin ϕeϕ.
Es(r)=n=0in 2n+1n(n+1)[anmo1n(3)-ibnne1n(3)],
Hs(r)=-n=0in 2n+1n(n+1)[bnme1n(3)+ianno1n(3)],
an=-jn(kanˆ)[kajn(ka)]-[kanˆjn(kanˆ)]jn(ka)jn(kanˆ)[kahn(1)(ka)]-[kanˆjn(kanˆ)]hn(1)(ka),
bn=-[kanˆjn(kanˆ)]jn(ka)-nˆ2jn(ka)nˆ[kajn(ka)][kanˆjn(kanˆ)]hn(1)(ka)-nˆ2jn(kanˆ)[kahn(1)(ka)].
an=-12[1+exp(iψn)],
bn=-12[1+exp(iϕn)],
ψn=-2 arg(jn(kanˆ)[kahn(1)(ka)]-[kanˆjn(kanˆ)]hn(1)(ka)),
ϕn=-2 arg([kanˆjn(kanˆ)]hn(1)(ka)-nˆ2jn(kanˆ)[kahn(1)(ka)])
ψn=-2 arg[hn(1)(ka)],
ϕn=-2 arg([kahn(1)(ka)]).
Es(r)=n=1in 2n+1n(n+1)[anmo1n(3)(r)-ibnne1n(3)(r)],
Hs(r)=-n=1in 2n+1n(n+1)[bnme1n(3)(r)+ianno1n(3)(r)],
mσ1n(3)(r)
=12πin02πdβ0π/2-i exp(ikr)C1nσ(α, β)sin α dα,
nσ1n(3)(r)
=i2πin02πdβ0π/2-i exp(ikr)B1nσ(α, β)sin α dα,
C1ne(θ, ϕ)=-πn(cos θ)sin ϕeθ-τn(cos θ)cos ϕeϕ,
B1ne(θ, ϕ)=τn(cos θ)cos ϕeθ-πn(cos θ)sin ϕeϕ,
C1no(θ, ϕ)=πn(cos θ)cos ϕeθ-τn(cos θ)sin ϕeϕ,
B1no(θ, ϕ)=τn(cos θ)sin ϕeθ+πn(cos θ)cos ϕeϕ,
Es(r)=12π02πdβ0π/2-i exp(ikr)×n=1 2n+1n(n+1)[anC1no(α, β)+bnB1ne(α, β)]sin α dα,
Hs(r)=-12π02πdβ0π/2-i exp(ikr)×n=1 2n+1n(n+1)[bnC1ne(α, β)-anB1no(a, β)]sin α dα.
n=1 2n+1n(n+1)[anC1no(α, β)+bnB1ne(α, β)]
=S1(cos α)cos βeα-S2(cos α)sin βeβ,
S1(cos α)=n=0 2n+1n(n+1)[anπn(cos α)+bnτn(cos α)],
S2(cos α)=n=0 2n+1n(n+1)[anτn(cos α)+bnπn(cos α)].
n=1 2n+1n(n+1)[bnC1ne(α, β)-anB1no(α, β)]
=-S2(cos α)sin βeα-S1(cos α)cos βeβ.
Es(r)=12π02πdβ0π/2-i exp(ikr)[S1(cos α)cos βeα-S2(cos α)sin βeβ]sin α dα,
Hs(r)=12π02πdβ0π/2-i exp(ikr)[S2(cos α)sin βeα+S1(cos α)cos βeβ]sin α dα.
[zhn(1)(z)](-i)n exp(iz),z.
Es(r)exp(ikr)ikr[S1(cos θ)cos ϕeθ-S2(cos θ)×sin ϕeϕ],kr,
Hs(r)exp(ikr)ikr[S2(cos θ)sin ϕeθ+S1(cos θ)×cos ϕeϕ],kr,
Ei=ex exp(ikz),
Hi=ey exp(ikz).
ex=cos θ0ex-sin θ0ez,
ey=ey,
ez=sin θ0ex+cos θ0ez,
α=arccos(cos θ0 cos α+sin θ0 sin α cos β),
β=arcsinsin β sin αsin α,
ex=ex,
ey=cos θ0ey-sin θ0ez,
ez=sin θ0ey+cos θ0ez,
α=arccos(cos θ0 cos α+sin θ0 sin α sin β),
β=arccoscos β sin αsin α,
Es(r)=12π02πdβ0π/2-i exp(ikr)×[S1(cos α)cos βeα-S2(cos α)sin βeβ]β=β(α, β, θ0)α=α(α, β, θ0), ×sin α dα,
Hs(r)=12π02πdβ0π/2-i exp(ikr)×[S2(cos α)sin βeα+S1(cos α)cos βeα]β=β(α, β, θ0)α=α(α, β, θ0),×sin α dα,
eα=cos α cos βex+cos α sin βey-sin αez,
eβ=-sin βex+cos βey.
Us(r)=12π02πdβ0π/2S(cos γα0)exp(ikr)sin α dα,
Us(r)=12π02πdβ0π/2S(cos γα0)exp(ikr)H(α)sin α dα.
H(α)=1,αθa0,α>θa,
Us(r)=12π02πdβ0θaS(cos γα0)exp(ikr)sin α dα.
U(r)=Ui(r)+Us(r),θ0θaUs(r),θ0>θa.
|γα0|[0,θ0+θa],θ0θa[θ0-θa,θ0+θa],θa<θ0<π-θa[θ0-θa,π],π-θa<θ0π.
Es(r)=Ex(r)ex+Ey(r)ey+Ez(r)ez,
Hs(r)=Hx(r)ex+Hy(r)ey+Hz(r)ez,
Us(η, 0)=12π02πdβ0π/2S(cos α)×exp[ikη sin α×cos(ϕ-β)]sin α dα.
02πdβ exp[ikη sin α cos(ϕ-β)]=2πJ0(kη sin α),
Us(η, 0)=0π/2S(cos α)J0(kη sin α)sin α dα.
Us(η, 0)=12π02πdβ0θaS(cos α)×exp[ikη sin α cos(ϕ-β)]sin α dα,
Us(η, 0)=0θaS(cos α)J0(kη sin α)sin α dα.
Us(η, 0)=n=0 1k2anDn(η),
Dn(η)=kn+120θaPn(cos α)J0(kη sin α)sin α dα.
Pn(cos α)=2π0α cos[(n+12)t](cos t-cos α)1/2dt=2απ0π/2 cos τ[cos(α sin τ)-cos α]1/2×cos[(n+12)α sin τ]dτ.
Pn(cos α)J0((n+12)α).
D(ηn, η)k2ηn0θaJ0(kηnα)×J0(kηα)α dα,
01J0(at)J0(bt)t dt=aJ1(a)J0(b)-bJ1(b)J0(a)a2-b2,
D(ηn, η)k2nnθa×kηnJ1(kθaηn)J0(kθaη)-kηJ1(kθaη)J0(kθaηn)(kηn)2-(kη)2,
D(ηn, ηn)k2θa2ηn[J1(kθaηn)2+J0(kθaηn)2].
J0(b)2πb cosb-π4,b1,
J1(a)2πa sina-π4,a1,
D(ηn, η)kθaπηnη sin[kθa(ηn-η)]kθa(ηn-η),
kηn, kη1.
UI(x, y)=--U0(x2+y2)h(x-x, y-y)dxdy,
h(x, y)=1(2π)2--H(kx, ky)×exp[i(kxx+kyy)]dkxdky
h(η)=k2(2π)202πdβ0θadα sin α×exp[ikη sin α cos(ϕ-β)]=k22π0θadα(sin α)J0(kη sin α),
h(η)k22π0θaJ0(kηα)α dα.
UI(η)=0dη U0(η)h˜(η, η),
h˜(η, η)=η02πdϕ h((η2+η2-2ηη cos ϕ)1/2).
J0(mR)=J0(mr)J0(mρ)+n=1Jn(mr)Jn(mρ)cos(nϕ),
h˜(η, η)=k2η0θaJ0(kηα)J0(kηα)α dα
UI(η)=0 1kU0(ηn)h˜(ηn, η).
U0(η)=exp[iΨ(η)]-1.
Us(η, 0)=n=0 1k2anDn(η).
D(ηn, η)kηnη sin[k(ηn-η)Nπ]k(ηn-η)Nπ,N1.
Us(ηn, 0)=2an.
Us(r)=n=0 1k2anDn(r),
Dn(r)=kn+1202πdβ0θαdαsin α×Pn(cos α)exp(ikr).
D(ηn, r)=k2ηn0θαdα02πdβ0π/2dτ f(α, τ)×{exp[iΨ+(α, β, τ)]+exp[iΨ-(α, β, τ)]}.
f(α, τ)=α sin α22π2cos τ[cos(α sin τ)-cos α]1/2,
Ψ±(α, β, τ)=kr[cos α cos θ+sin α sin θ cos(β-ϕ)]±kηnα sin τ.
Ψ+(α, β, τ)Ψ+(αs, βs, τs)+122Ψ+α2α=αs,β=βs,τ=τs(α-αs)2+122Ψ+β2α=αs,β=βs,τ=τs(β-βs)2+122Ψ+τ2α=αs,β=βs,τ=τs(τ-τs)2,
f(αs, τs)(αs sin αs)1/22π2,
- expi B2β2dβ=2π|B| expi π4sgn(B),
-τs expi C2(τ-τs)2dτ=12- expi C2τ2dτ,
D(ηn, r)kηnη expikr-i π42πkr×[-i exp(ikηnθ2)+exp(ikηnθ1)],
Dn(r)=k(n+12)inhn(1)(kr)Pn(cos θ),
Dn(r)=kn+12expi π2nπ2kr Hn+1/2(1)(kr)×J0n+12α.
Hν(1)(z)=2πz2-ν21/2 expiz2-ν2-ν arccos νz-π4,zν.
D(ηn, r)kηnrθ expikr-i π42πkr[-i exp(ikηnθ2)+exp(ikηnθ1)],rηn.
Ψ+(α, β, τ)Ψ+αα=0+,β=βs1,τ=τs1α+163Ψ+α3α=0+,β=βs1,τ=τs1α3+123Ψ+β2αα=0+,β=βs1,τ=τs1α(β-βs1)2+123Ψ+τ2αα=0+,β=βs1,τ=τs1α(τ-τs1)2,
f(α, τs)(α sin α)1/22π2=12π2[α+O(α3)],
D(ηn, η)k2πηnη -θaθa ×expik(ηn-η)α+kη6α3dα,
ηnη.
D(ηn, η)kηnη 2kη1/3×Ai2kη1/3k(ηn-η),
ηnη,
Ai(z)=12π- expizα+13α3dα
kη<3π/θa3.
D(ηn, η)k2πηnη -θaθa exp[ik(ηn-η)α]dα=kθaπηnη sin[kθa(ηn-η)]kθa(ηn-η),

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