Abstract

We describe an efficient algorithm for the sizing of single microspheres having a known index of refraction as a function of wavelength. The algorithm employs a peak-detection routine that determines several resonant frequencies in the radiation scattered from the particle. These measured resonances are then compared with entries from a library of stored resonance locations in order to determine a few neighborhoods within which the size of the particle is likely to lie. A final step finds a local minimum of the cost function within each neighborhood, and the size estimate is determined by selecting the smallest of these local minima. The algorithm has modest computational and memory requirements, and it requires no analysis of complicated features of the resonance spectrum that would call for human intervention. Hence it could be automated for nearly real-time operation using a microprocessor. When applied to the measured resonance spectrum of a fluorescent polystyrene sphere, the algorithm finds the radius with an accuracy limited only by such factors as surface roughness, asphericity, and imperfect knowledge of the refractive index. The algorithm is currently limited to use with first-order resonances.

© 1984 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Chylek, “Partial-wave resonances and the ripple structure in the Mie normalized cross section,”J. Opt. Soc. Am. 66, 285–287 (1976).
    [CrossRef]
  2. P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
    [CrossRef]
  3. G. J. Rosasco, H. S. Bennett, “Internal field resonance structure: implications for optical absorption and scattering by microspheric particles,”J. Opt. Soc. Am. 68, 1242–1250 (1978).
    [CrossRef]
  4. P. Chylek, J. T. Kiehl, M. K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978).
    [CrossRef] [PubMed]
  5. R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phy. Rev. Let. 44, 475–478 (1980).
    [CrossRef]
  6. P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984).
    [CrossRef]
  7. P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,”IEEE Trans. Antennas Propag. AP-30, 168–172 (1982).
    [CrossRef]
  8. P. Chylek, V. Ramaswamy, A. Ashkin, J. M. Dziedzic, “Simultaneous determination of refractive index and size of spherical dielectric particles from light scattering data,” Appl. Opt. 22, 2302–2307 (1983).
    [CrossRef] [PubMed]
  9. A. J. Wilson, C. I. Franks, I. L. Freeston, “Algorithms for the detection of breaths from respiratory waveform recordings of infants,” Med. Biol. Eng. Comput. 20, 286–292 (1982).
    [CrossRef] [PubMed]
  10. K. S. Fu, Syntactic Pattern Recognition and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1982), pp. 411–444.
  11. J. T. Tou, ed., Software Engineering: Proceedings of the Third Symposium on Computer and Information Sciences (Academic, New York, 1971), pp. 183–202.
  12. T. Pavlidis, Structural Pattern Recognition (Springer-Verlag, New York, 1977), pp. 13–15.
  13. S. C. Hill, R. E. Benner, C. K. Rushforth, P. R. Conwell, “Structural resonances observed in the fluorescence emission from small spheres on substrates,” Appl. Opt. 23, 1680–1683 (1984).
    [CrossRef] [PubMed]

1984 (2)

1983 (1)

1982 (2)

A. J. Wilson, C. I. Franks, I. L. Freeston, “Algorithms for the detection of breaths from respiratory waveform recordings of infants,” Med. Biol. Eng. Comput. 20, 286–292 (1982).
[CrossRef] [PubMed]

P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,”IEEE Trans. Antennas Propag. AP-30, 168–172 (1982).
[CrossRef]

1980 (1)

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phy. Rev. Let. 44, 475–478 (1980).
[CrossRef]

1978 (3)

1976 (1)

Ashkin, A.

Barber, P. W.

P. R. Conwell, P. W. Barber, C. K. Rushforth, “Resonant spectra of dielectric spheres,” J. Opt. Soc. Am. A 1, 62–67 (1984).
[CrossRef]

P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,”IEEE Trans. Antennas Propag. AP-30, 168–172 (1982).
[CrossRef]

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phy. Rev. Let. 44, 475–478 (1980).
[CrossRef]

Benner, R. E.

S. C. Hill, R. E. Benner, C. K. Rushforth, P. R. Conwell, “Structural resonances observed in the fluorescence emission from small spheres on substrates,” Appl. Opt. 23, 1680–1683 (1984).
[CrossRef] [PubMed]

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phy. Rev. Let. 44, 475–478 (1980).
[CrossRef]

Bennett, H. S.

Chang, R. K.

P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,”IEEE Trans. Antennas Propag. AP-30, 168–172 (1982).
[CrossRef]

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phy. Rev. Let. 44, 475–478 (1980).
[CrossRef]

Chylek, P.

Conwell, P. R.

Dziedzic, J. M.

Franks, C. I.

A. J. Wilson, C. I. Franks, I. L. Freeston, “Algorithms for the detection of breaths from respiratory waveform recordings of infants,” Med. Biol. Eng. Comput. 20, 286–292 (1982).
[CrossRef] [PubMed]

Freeston, I. L.

A. J. Wilson, C. I. Franks, I. L. Freeston, “Algorithms for the detection of breaths from respiratory waveform recordings of infants,” Med. Biol. Eng. Comput. 20, 286–292 (1982).
[CrossRef] [PubMed]

Fu, K. S.

K. S. Fu, Syntactic Pattern Recognition and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1982), pp. 411–444.

Hill, S. C.

Kiehl, J. T.

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978).
[CrossRef] [PubMed]

Ko, M. K. W.

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Narrow resonance structure in the Mie scattering characteristics,” Appl. Opt. 17, 3019–3021 (1978).
[CrossRef] [PubMed]

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

Owen, J. F.

P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,”IEEE Trans. Antennas Propag. AP-30, 168–172 (1982).
[CrossRef]

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phy. Rev. Let. 44, 475–478 (1980).
[CrossRef]

Pavlidis, T.

T. Pavlidis, Structural Pattern Recognition (Springer-Verlag, New York, 1977), pp. 13–15.

Ramaswamy, V.

Rosasco, G. J.

Rushforth, C. K.

Wilson, A. J.

A. J. Wilson, C. I. Franks, I. L. Freeston, “Algorithms for the detection of breaths from respiratory waveform recordings of infants,” Med. Biol. Eng. Comput. 20, 286–292 (1982).
[CrossRef] [PubMed]

Appl. Opt. (3)

IEEE Trans. Antennas Propag. (1)

P. W. Barber, J. F. Owen, R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,”IEEE Trans. Antennas Propag. AP-30, 168–172 (1982).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Med. Biol. Eng. Comput. (1)

A. J. Wilson, C. I. Franks, I. L. Freeston, “Algorithms for the detection of breaths from respiratory waveform recordings of infants,” Med. Biol. Eng. Comput. 20, 286–292 (1982).
[CrossRef] [PubMed]

Phy. Rev. Let. (1)

R. E. Benner, P. W. Barber, J. F. Owen, R. K. Chang, “Observation of structure resonances in the fluorescence spectra from microspheres,” Phy. Rev. Let. 44, 475–478 (1980).
[CrossRef]

Phys. Rev. A (1)

P. Chylek, J. T. Kiehl, M. K. W. Ko, “Optical levitation and partial-wave resonances,” Phys. Rev. A 18, 2229–2233 (1978).
[CrossRef]

Other (3)

K. S. Fu, Syntactic Pattern Recognition and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1982), pp. 411–444.

J. T. Tou, ed., Software Engineering: Proceedings of the Third Symposium on Computer and Information Sciences (Academic, New York, 1971), pp. 183–202.

T. Pavlidis, Structural Pattern Recognition (Springer-Verlag, New York, 1977), pp. 13–15.

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 (6)

Fig. 1
Fig. 1

Calculated scattering efficiency versus size parameter for a dielectric sphere with m = 1.195. For size parameters ranging from 50 to 80, the sharp resonances seen are all first order. For size parameters ranging from 80 to 90, the shorter, wider peaks seen are a result of the merging of second-order (l = 2) electric and magnetic resonances. For size parameters 90 to 100, these second-order resonances become sufficiently narrow to be resolved independently.

Fig. 2
Fig. 2

An expanded view of a portion of Fig. 1. The resonances are shown with their attendant poles. The subscripts indicate mode and order numbers, respectively, of each resonance. For this case, only first-order resonances are observed. The smaller the imaginary part of the pole, the sharper the resonance. The imaginary part tends to get smaller as the mode is increased for the same-order poles.

Fig. 3
Fig. 3

Shown are two large peaks broken into long left and long right sides. These, in turn, are broken into a string of positive-sloping line segments, or primitives. Notice that the primitives are not contiguous line segments. A long. left side is defined as a string of primitives (starting with the bottom of a primitive and ending with a top) whose net vertical gain exceeds some preset threshold. A long right side is defined as a string whose net vertical loss (starting with the top of a primitive and ending with a bottom) exceeds the same threshold.

Fig. 4
Fig. 4

Plot of the mean-squared error incurred when a set of experimental and theoretical resonance locations is matched while the radius of the theoretical locations is varied. The local minima and maxima occur as the two resonance sets come into and out of partial registration, respectively.

Fig. 5
Fig. 5

Probability of detecting the wrong minimum as a function of number of resolved resonance peaks.

Fig. 6
Fig. 6

Fluorescence spectrum of a dye-impregnated polystyrene sphere having a nominal diameter of 4.72 μm. The spectrum was measured with an optical multichannel analyzer (top). Calculated elastic scattering efficiency of a polystyrene sphere having a radius of 4.719 μm (bottom). The dispersion in the refractive index has been taken into account. The mode numbers corresponding to the observed resonances are indicated.

Equations (15)

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

Q s = 2 x 2 n = 1 ( 2 n + 1 ) a n 2 + b n 2 ,
a n = j n ( x ) [ m x j n ( m x ) ] - m 2 j n ( m x ) [ x j n ( x ) ] h n ( 2 ) ( x ) [ m x j n ( m x ) ] - m 2 j n ( m x ) [ x h n ( 2 ) ( x ) ]
b n = j n ( x ) [ m x j n ( m x ) ] - j n ( m x ) [ x j n ( x ) ] h n ( 2 ) ( x ) [ m x j n ( m x ) ] - j n ( m x ) [ x h n ( 2 ) ( x ) ] ,
x ( m ) = α 0 + α 1 ( m - m 0 ) + α 2 ( m - m 0 ) ( m - m 1 ) + α 3 ( m - m 0 ) ( m - m 1 ) ( m - m 2 ) .
λ = 2 π r f ( λ ) / x ,
λ = K 1 r / x - K 2 r
r 1 = D 1 L M K 1 + K 2 D 1 .
Q i ( r 1 ) = K 1 r 1 L M - i + 1 - K 2 r 1 ,             i = 1 , 2 , , N .
r 2 = D 1 L M - 1 K 1 + K 2 D 1 ,
Q i ( r 2 ) = K 1 r 2 L ( M - 1 ) - i + 1 - K 2 r 2 ,             i = 1 , 2 , , N .
Q i ( r j ) = K 1 r j L ( M - j ) - i + 1 - K 2 r j ,             i = 1 , 2 , , N ,
r j = D 1 L M - j + 1 K 1 + K 2 D 1 .
E ( r j ) = i = 1 N [ D i - Q i ( r j ) ] 2 .
Q ( j ) ( r ) = K 1 r L ( M - j ) - i + 1 - K 2 r .
E ( r ) = i = 1 N [ D i - K 1 r L ( M - j ) - i + 1 - K 2 r ] 2 .

Metrics