Abstract

Efficient methods for the calculation of light scattering intensity functions for concentrically coated spheres (~10-μ diam) are discussed. This model represents many types of biological cells whose nuclei have a low refractive index (~1.1) and cytoplasms with a slightly lower refractive index. Studies are made on the relationships between the scattering coefficients for nonabsorbing, spherically symmetric scatterers. The physical origin of these coefficients is examined for absorbingscatterers. A comparison of the angular half-width of the scattered intensity functions for the coated sphere and an equivalent homogeneous sphere shows that diffraction dominates the small angle scattering in both cases. At larger angles, the coated sphere scattering pattern is more structured and quite sensitive to core sphere size, suggesting a possible method of distinguishing types of biological cells that are similar in gross size but different in internal detail.

© 1972 Optical Society of America

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References

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  1. P. J. Wyatt, Appl. Opt. 7, 1879 (1968).
    [CrossRef] [PubMed]
  2. G. Mie, Ann. Phys. 25, 377 (1908).
    [CrossRef]
  3. P. F. Mullaney, P. N. Dean, Biophys. J. 10, 764 (1970).
    [CrossRef] [PubMed]
  4. P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, P. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
    [CrossRef] [PubMed]
  5. A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
    [CrossRef]
  6. A. L. Aden, J. Appl. Phys. 22, 601 (1951).
    [CrossRef]
  7. J. J. Stephens, J. R. Gerhardt, J. Meteorol. 18, 818 (1961).
    [CrossRef]
  8. J. J. Stephens, J. R. Gerhardt, J. Meteorol. 18, 819 (1961).
  9. G. W. Kattawar, G. N. Plass, Appl. Opt. 6, 1377 (1967).
    [CrossRef] [PubMed]
  10. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  11. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Nat. Bur. Stand., Washington, D.C., 1964) p. 445.
  12. D. Diermendjian, R. J. Clasen, Light Scattering on Partially Absorbing Homogeneous Spheres of Finite Size, Rep. R-393-PR (Rand Corp., Santa Monica, Calif., 1962), p. 36.
  13. British Association for the Advancement of Science, Bessel Functions—Part II, Mathematical Tables (Cambridge U. P., Cambridge, 1952), Vol. 10, p. xvii.
  14. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 135.
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  16. W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, J. Colloid Sci. 29, 501 (1965).
    [CrossRef]
  17. M. Kerker, E. Matijevic, J. Opt. Soc. Am. 51, 87 (1961).
    [CrossRef]
  18. D. Diermendjian, Tables of Mie Scattering Cross Sections and Amplitudes, Rep. R-407-PR (Rand Corp., Santa Monica, Calif., 1963).
  19. J. V. Dave, Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere, Rep. 320.3237 (IBM Scientific Center, Palo Alto, Calif., 1968).
  20. R. H. Boll, J. A. Leacock, G. C. Clark, S. W. Churchill, Tables of Light Scattering Functions (University of Michigan Press, Ann Arbor, 1958).
  21. Ref. 11, p. 344ff.
  22. R. O. Gumprecht, C. M. Sliepcevich, Ricatti-Bessel Functions for Large Arguments and Orders (Eng. Res. Inst., Univ. of Michigan, Ann Arbor, 1951).
  23. R. O. Gumprecht, C. M. Sliepcevich, Functions of First and Second Derivatives of Legendre Polynomials (Eng. Res. Inst., Univ. of Michigan, Ann Arbor, 1951).
  24. P. J. Wyatt, Phys. Res. 127, 1837 (1962).
    [CrossRef]
  25. P. J. Wyatt, Phys. Rev. 134, AB1 (1964).
    [CrossRef]
  26. P. J. Wyatt, J. Appl. Phys. 35, 1966 (1964).
    [CrossRef]
  27. P. J. Wyatt, J. Appl. Phys. 36, 3875 (1965).
    [CrossRef]
  28. Ref. 10, p. 223.
  29. P. J. Wyatt, Science Spectrum, Box 3003, Santa Barbara, Calif. 93105, private communication (3June1971).
  30. Ref. 14, p. 208.
  31. P. F. Mullaney, P. N. Dean, Appl. Opt. 8, 2361 (1969).
    [CrossRef] [PubMed]

1970

P. F. Mullaney, P. N. Dean, Biophys. J. 10, 764 (1970).
[CrossRef] [PubMed]

1969

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, P. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[CrossRef] [PubMed]

P. F. Mullaney, P. N. Dean, Appl. Opt. 8, 2361 (1969).
[CrossRef] [PubMed]

1968

1967

1965

P. J. Wyatt, J. Appl. Phys. 36, 3875 (1965).
[CrossRef]

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, J. Colloid Sci. 29, 501 (1965).
[CrossRef]

1964

P. J. Wyatt, Phys. Rev. 134, AB1 (1964).
[CrossRef]

P. J. Wyatt, J. Appl. Phys. 35, 1966 (1964).
[CrossRef]

1962

P. J. Wyatt, Phys. Res. 127, 1837 (1962).
[CrossRef]

1961

J. J. Stephens, J. R. Gerhardt, J. Meteorol. 18, 818 (1961).
[CrossRef]

J. J. Stephens, J. R. Gerhardt, J. Meteorol. 18, 819 (1961).

M. Kerker, E. Matijevic, J. Opt. Soc. Am. 51, 87 (1961).
[CrossRef]

1951

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

A. L. Aden, J. Appl. Phys. 22, 601 (1951).
[CrossRef]

1908

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Aden, A. L.

A. L. Aden, J. Appl. Phys. 22, 601 (1951).
[CrossRef]

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

Boll, R. H.

R. H. Boll, J. A. Leacock, G. C. Clark, S. W. Churchill, Tables of Light Scattering Functions (University of Michigan Press, Ann Arbor, 1958).

Churchill, S. W.

R. H. Boll, J. A. Leacock, G. C. Clark, S. W. Churchill, Tables of Light Scattering Functions (University of Michigan Press, Ann Arbor, 1958).

Clark, G. C.

R. H. Boll, J. A. Leacock, G. C. Clark, S. W. Churchill, Tables of Light Scattering Functions (University of Michigan Press, Ann Arbor, 1958).

Clasen, R. J.

D. Diermendjian, R. J. Clasen, Light Scattering on Partially Absorbing Homogeneous Spheres of Finite Size, Rep. R-393-PR (Rand Corp., Santa Monica, Calif., 1962), p. 36.

Coulter, J. R.

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, P. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[CrossRef] [PubMed]

Dave, J. V.

J. V. Dave, Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere, Rep. 320.3237 (IBM Scientific Center, Palo Alto, Calif., 1968).

Dean, P. N.

P. F. Mullaney, P. N. Dean, Biophys. J. 10, 764 (1970).
[CrossRef] [PubMed]

P. F. Mullaney, P. N. Dean, Appl. Opt. 8, 2361 (1969).
[CrossRef] [PubMed]

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, P. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[CrossRef] [PubMed]

Diermendjian, D.

D. Diermendjian, R. J. Clasen, Light Scattering on Partially Absorbing Homogeneous Spheres of Finite Size, Rep. R-393-PR (Rand Corp., Santa Monica, Calif., 1962), p. 36.

D. Diermendjian, Tables of Mie Scattering Cross Sections and Amplitudes, Rep. R-407-PR (Rand Corp., Santa Monica, Calif., 1963).

Espenscheid, W. F.

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, J. Colloid Sci. 29, 501 (1965).
[CrossRef]

Gerhardt, J. R.

J. J. Stephens, J. R. Gerhardt, J. Meteorol. 18, 819 (1961).

J. J. Stephens, J. R. Gerhardt, J. Meteorol. 18, 818 (1961).
[CrossRef]

Gumprecht, R. O.

R. O. Gumprecht, C. M. Sliepcevich, Ricatti-Bessel Functions for Large Arguments and Orders (Eng. Res. Inst., Univ. of Michigan, Ann Arbor, 1951).

R. O. Gumprecht, C. M. Sliepcevich, Functions of First and Second Derivatives of Legendre Polynomials (Eng. Res. Inst., Univ. of Michigan, Ann Arbor, 1951).

Kattawar, G. W.

Kerker, M.

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, J. Colloid Sci. 29, 501 (1965).
[CrossRef]

M. Kerker, E. Matijevic, J. Opt. Soc. Am. 51, 87 (1961).
[CrossRef]

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

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

Leacock, J. A.

R. H. Boll, J. A. Leacock, G. C. Clark, S. W. Churchill, Tables of Light Scattering Functions (University of Michigan Press, Ann Arbor, 1958).

Matijevic, E.

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, J. Colloid Sci. 29, 501 (1965).
[CrossRef]

M. Kerker, E. Matijevic, J. Opt. Soc. Am. 51, 87 (1961).
[CrossRef]

Mie, G.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Mullaney, P. F.

P. F. Mullaney, P. N. Dean, Biophys. J. 10, 764 (1970).
[CrossRef] [PubMed]

P. F. Mullaney, P. N. Dean, Appl. Opt. 8, 2361 (1969).
[CrossRef] [PubMed]

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, P. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[CrossRef] [PubMed]

Plass, G. N.

Sliepcevich, C. M.

R. O. Gumprecht, C. M. Sliepcevich, Ricatti-Bessel Functions for Large Arguments and Orders (Eng. Res. Inst., Univ. of Michigan, Ann Arbor, 1951).

R. O. Gumprecht, C. M. Sliepcevich, Functions of First and Second Derivatives of Legendre Polynomials (Eng. Res. Inst., Univ. of Michigan, Ann Arbor, 1951).

Stephens, J. J.

J. J. Stephens, J. R. Gerhardt, J. Meteorol. 18, 818 (1961).
[CrossRef]

J. J. Stephens, J. R. Gerhardt, J. Meteorol. 18, 819 (1961).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 135.

Van Dilla, M. A.

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, P. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[CrossRef] [PubMed]

Willis, E.

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, J. Colloid Sci. 29, 501 (1965).
[CrossRef]

Wyatt, P. J.

P. J. Wyatt, Appl. Opt. 7, 1879 (1968).
[CrossRef] [PubMed]

P. J. Wyatt, J. Appl. Phys. 36, 3875 (1965).
[CrossRef]

P. J. Wyatt, Phys. Rev. 134, AB1 (1964).
[CrossRef]

P. J. Wyatt, J. Appl. Phys. 35, 1966 (1964).
[CrossRef]

P. J. Wyatt, Phys. Res. 127, 1837 (1962).
[CrossRef]

P. J. Wyatt, Science Spectrum, Box 3003, Santa Barbara, Calif. 93105, private communication (3June1971).

Ann. Phys.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Appl. Opt.

Biophys. J.

P. F. Mullaney, P. N. Dean, Biophys. J. 10, 764 (1970).
[CrossRef] [PubMed]

J. Appl. Phys.

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[CrossRef]

A. L. Aden, J. Appl. Phys. 22, 601 (1951).
[CrossRef]

P. J. Wyatt, J. Appl. Phys. 35, 1966 (1964).
[CrossRef]

P. J. Wyatt, J. Appl. Phys. 36, 3875 (1965).
[CrossRef]

J. Colloid Sci.

W. F. Espenscheid, E. Willis, E. Matijevic, M. Kerker, J. Colloid Sci. 29, 501 (1965).
[CrossRef]

J. Meteorol.

J. J. Stephens, J. R. Gerhardt, J. Meteorol. 18, 818 (1961).
[CrossRef]

J. J. Stephens, J. R. Gerhardt, J. Meteorol. 18, 819 (1961).

J. Opt. Soc. Am.

Phys. Res.

P. J. Wyatt, Phys. Res. 127, 1837 (1962).
[CrossRef]

Phys. Rev.

P. J. Wyatt, Phys. Rev. 134, AB1 (1964).
[CrossRef]

Rev. Sci. Instrum.

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, P. N. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[CrossRef] [PubMed]

Other

D. Diermendjian, Tables of Mie Scattering Cross Sections and Amplitudes, Rep. R-407-PR (Rand Corp., Santa Monica, Calif., 1963).

J. V. Dave, Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere, Rep. 320.3237 (IBM Scientific Center, Palo Alto, Calif., 1968).

R. H. Boll, J. A. Leacock, G. C. Clark, S. W. Churchill, Tables of Light Scattering Functions (University of Michigan Press, Ann Arbor, 1958).

Ref. 11, p. 344ff.

R. O. Gumprecht, C. M. Sliepcevich, Ricatti-Bessel Functions for Large Arguments and Orders (Eng. Res. Inst., Univ. of Michigan, Ann Arbor, 1951).

R. O. Gumprecht, C. M. Sliepcevich, Functions of First and Second Derivatives of Legendre Polynomials (Eng. Res. Inst., Univ. of Michigan, Ann Arbor, 1951).

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

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Nat. Bur. Stand., Washington, D.C., 1964) p. 445.

D. Diermendjian, R. J. Clasen, Light Scattering on Partially Absorbing Homogeneous Spheres of Finite Size, Rep. R-393-PR (Rand Corp., Santa Monica, Calif., 1962), p. 36.

British Association for the Advancement of Science, Bessel Functions—Part II, Mathematical Tables (Cambridge U. P., Cambridge, 1952), Vol. 10, p. xvii.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), p. 135.

Ref. 10, p. 49ff.

Ref. 10, p. 223.

P. J. Wyatt, Science Spectrum, Box 3003, Santa Barbara, Calif. 93105, private communication (3June1971).

Ref. 14, p. 208.

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Figures (7)

Fig. 1
Fig. 1

The nonabsorbing circle. The an and bn coefficients fall on this circle of radius 0.5 in the complex plane for nonabsorbing, spherically symmetric scatters; αn is the phase angle for this an value (see van de Hulst14).

Fig. 2
Fig. 2

Semilog plot of |s| vs n for an absorbing core and nonabsorbing coating: ν = 86, m1 = 1.05–0.005i, and m2 = 1.03; α = 40 (——), 60 (-----), 80 (—–—).

Fig. 3
Fig. 3

Semilog plot of |s| vs n for a nonabsorbing core and absorbing coating: ν = 86, m1 = 1.05, and m2 = 1.03–0.005i; α = 40 (——), 60 (-----), 80 (—–—).

Fig. 4
Fig. 4

Semilog plot of |s| vs n for an absorbing core and coating: ν = 86, m1 = 1.05–0.0030i, and m2 = 1.03–0.0015i; α = 40 (——), 60 (-----), 80(—–—).

Fig. 5
Fig. 5

The localization principle. A ray passing a distance λn/2π from the center of a concentrically coated absorbing sphere is absorbed along u.

Fig. 6
Fig. 6

Log-log plot of the half-width of the intensity function vs the whole particle size parameter ν[α = (2/3)ν]. The values of m1/m2 are 1.05/1.02 (▼), 1.04/1.02 (■), 1.07/1.05(•), and 1.10/1.07 (▲). The straight line is the result of Fraunhofer diffraction.

Fig. 7
Fig. 7

Logarithm of the relative intensity function i2(θ) vs θ for a homogeneous sphere with α = ν = 85, m1 = 1.036, m2 = 1.0 (top curve), and a coated sphere with α = 2ν/3 = 56.67, ν = 85, m1 = 1.05, m2 = 1.03 (bottom curve). ICS and IHS are i2(θ) for the coated and homogeneous spheres, respectively. The plotting increment is 0.25°.

Equations (17)

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S 1 = n = 1 2 n + 1 n ( n + 1 ) [ a n π n ( cos θ ) + b n τ n ( cos θ ) ] , S 2 = n = 1 2 n + 1 n ( n + 1 ) [ a n τ n ( cos θ ) + b n π n ( cos θ ) ] ,
π n ( cos θ ) = [ P n ( 1 ) ( cos θ ) ] / sin θ
τ n ( cos θ ) = ( d / d θ ) P n ( 1 ) ( cos θ ) ,
( χ ψ ) n , ν = χ n ( m 2 ν ) ψ n ( ν ) m 2 χ n ( m 2 ν ) ψ n ( ν ) , ( χ ψ ) n , ν = m 2 χ n ( m 2 ν ) ψ n ( ν ) χ n ( m 2 ν ) ψ n ( ν ) , ( χ ψ ) n , α = m 1 χ n ( m 2 α ) ψ n ( m 1 α ) m 2 χ n ( m 2 α ) ψ n ( m 1 α ) , ( χ ψ ) n , α = m 2 χ n ( m 2 α ) ψ n ( m 1 α ) m 1 χ n ( m 2 α ) ψ n ( m 1 α ) .
a n = ( ψ ψ ) n , α · ( χ ψ ) n , ν ( χ ψ ) n , α · ( ψ ψ ) n , ν ( ψ ψ ) n , α · ( χ ζ ) n , ν ( χ ψ ) n , α · ( ψ ζ ) n , ν , b n = ( ψ ψ ) n , α · ( χ ψ ) n , ν ( χ ψ ) n , α · ( ψ ψ ) n , ν ( ψ ψ ) n , α · ( χ ζ ) n , ν ( χ ψ ) n , α · ( ψ ζ ) n , ν ;
q = ( υ w ) / υ ,
s = { [ 0.5 Re ( a n ) ] 2 + [ Im ( a n ) ] 2 } 1 2 0.5
[ Re ( S 12 ) + Im ( S 12 ) ] [ Re ( S 12 ) + Im ( S 12 ) ] <
Re ( S 12 ) + Im ( S 12 ) < ,
S 12 = | d S 1 d n | + | d S 2 d n |
S 12 = | S 1 | + | S 2 | .
a n = ψ n ( α ) W n ( 1 ) ( α ) m 2 W n ( 1 ) ( α ) ψ n ( α ) ζ n ( α ) W n ( 1 ) ( α ) m 2 W n ( 1 ) ( α ) ζ n ( α ) , b n = ψ n ( α ) G n ( 1 ) ( α ) G n ( 1 ) ( α ) ψ n ( α ) ζ n ( α ) G n ( 1 ) ( α ) G n ( 1 ) ( α ) ζ n ( α ) ,
tan α n = ψ n ( α ) W n ( 1 ) ( α ) m 2 W n ( 1 ) ( α ) ψ n ( α ) χ n ( α ) W n ( 1 ) ( α ) m 2 W n ( 1 ) ( α ) χ n ( α ) , tan β n = ψ n ( α ) G n ( 1 ) ( α ) G n ( 1 ) ( α ) ψ n ( α ) χ n ( α ) G n ( 1 ) ( α ) G n ( 1 ) ( α ) χ n ( α ) ·
a n = tan α n / ( tan α n i ) , b n = tan β n / ( tan β n i ) .
a n = | ψ n ( m 1 α 1 ) ψ n ( m 2 α 1 ) χ n ( m 2 α 1 ) 0 0 0 0 0 m 1 ψ m ( m 1 α 1 ) m 2 ψ n ( m 2 α 1 ) m 2 χ n ( m 2 α 1 ) 0 0 0 0 0 0 ψ n ( m 2 α 2 ) χ n ( m 2 α 2 ) ψ n ( m 3 α 2 ) χ n ( m 3 α 2 ) 0 0 0 0 m 2 ψ n ( m 2 α 2 ) m 2 χ n ( m 2 α 2 ) m 3 ψ n ( m 3 α 2 ) m 3 χ n ( m 3 α 2 ) 0 0 0 0 0 0 ψ n ( m 3 α 3 ) χ n ( m 3 α 3 ) 0 0 0 0 0 0 m 3 ψ n ( m 3 α 3 ) m 3 χ n ( m 3 α 3 ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ψ n ( m l α l ) χ n ( m l α l ) ψ n ( α l ) 0 0 0 0 0 m 1 ψ n ( m l α l ) m l χ n ( m l α l ) ψ n ( α l ) same as above except that ψ n ( α l ) and ψ n ( α l ) are replaced by ζ n ( α l ) and ζ n ( α l ) |
tan α n = ψ n ( α l ) C 2 l 1,2 l + ψ n ( α l ) C 2 l , 2 l ζ n ( α l ) C 2 l 1,2 l + ζ n ( α l ) C 2 l .2 l ·
a n = tan α n / ( tan α n i ) .

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