Abstract

The complete scattering matrix S of spheres was measured with a flow cytometer. The experimental equipment allows simultaneous detection of two scattering-matrix elements for every sphere in the distribution. Two-parameter scatterplots with x and y coordinates determined by the S 11 + S ij and S 11S ij values are measured. Samples of spheres with very narrow size distributions (< 1%) were analyzed with a FlowCytometer, and they produced unexpected two-parameter scatterplots. Instead of compact distributions we observed Lissajous-like loops. Simulation of the scatterplots, using Lorenz–Mie theory, shows that these loops are due not to experimental errors but to true Lorenz–Mie scattering. It is shown that the loops originate from the sensitivity of the scattered field on the radius of the spheres. This paper demonstrates that the interpretation of rare events and hidden features in flow cytometry needs reconsideration.

© 1994 Optical Society of America

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References

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1991 (2)

1990 (2)

1989 (1)

1988 (1)

1987 (2)

B. G. Grooth, L. W. M. M. Terstappen, G. J. Puppels, J. Greve, “Light-scattering polarization measurements as a new parameter in flow cytometry,” Cytometry 8, 539–544 (1987).
[CrossRef] [PubMed]

E. Gulari, “Latex particle size distributions from multiwavelength turbidity spectra,” Part. Characteriz. 4, 96–101 (1987).
[CrossRef]

1986 (1)

1985 (2)

1984 (1)

1979 (1)

1908 (1)

G. Mie, “Considerations on the optics of turbid media, especially colloidal metal sols,” Ann. Phys. 25, 377–442 (1908).
[CrossRef]

1890 (1)

L. V. Lorenz, “Upon the light reflected and refracted by a transparent sphere,” Vidensk. Sel’sk. Shrifter 6, 1–62 (1890).

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Chang, R. K.

Dean, P. N.

M. A. van Dilla, P. N. Dean, O. D. Laerum, M. R. Melamed, Flow Cytometry: Instrumentation and Data Analysis (Academic, San Diego, Calif., 1985), Chap. 1.

Figdor, C. G.

Gouesbet, G.

Grehan, G.

Gréhan, G.

Greve, J.

B. G. Grooth, L. W. M. M. Terstappen, G. J. Puppels, J. Greve, “Light-scattering polarization measurements as a new parameter in flow cytometry,” Cytometry 8, 539–544 (1987).
[CrossRef] [PubMed]

Grooth, B. G.

B. G. Grooth, L. W. M. M. Terstappen, G. J. Puppels, J. Greve, “Light-scattering polarization measurements as a new parameter in flow cytometry,” Cytometry 8, 539–544 (1987).
[CrossRef] [PubMed]

Gulari, E.

E. Gulari, “Latex particle size distributions from multiwavelength turbidity spectra,” Part. Characteriz. 4, 96–101 (1987).
[CrossRef]

Hoekstra, A. G.

P. M. A. Sloot, A. G. Hoekstra, H. van der Liet, C. G. Figdor, “Scattering matrix elements of biological particles measured in a flow through system: theory and practice,” Appl. Opt. 28, 1752–1762 (1989).
[CrossRef] [PubMed]

P. M. A. Sloot, A. G. Hoekstra, “Arbitrarily-shaped particles measured in flow through systems,” in Proceedings of the Second International Congress on Optical Particle Sizing, E. D. Hirleman, ed. (Arizona State University, Tempe, Ariz., 1990), p. 605.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Kiefer, W.

Laerum, O. D.

M. A. van Dilla, P. N. Dean, O. D. Laerum, M. R. Melamed, Flow Cytometry: Instrumentation and Data Analysis (Academic, San Diego, Calif., 1985), Chap. 1.

Lock, J. A.

Long, M. B.

Lorenz, L. V.

L. V. Lorenz, “Upon the light reflected and refracted by a transparent sphere,” Vidensk. Sel’sk. Shrifter 6, 1–62 (1890).

Maheu, B.

Melamed, M. R.

M. A. van Dilla, P. N. Dean, O. D. Laerum, M. R. Melamed, Flow Cytometry: Instrumentation and Data Analysis (Academic, San Diego, Calif., 1985), Chap. 1.

Mie, G.

G. Mie, “Considerations on the optics of turbid media, especially colloidal metal sols,” Ann. Phys. 25, 377–442 (1908).
[CrossRef]

Nussenzveig, H. M.

Puppels, G. J.

B. G. Grooth, L. W. M. M. Terstappen, G. J. Puppels, J. Greve, “Light-scattering polarization measurements as a new parameter in flow cytometry,” Cytometry 8, 539–544 (1987).
[CrossRef] [PubMed]

Qian, S. X.

Show, J. B.

Sloot, P. M. A.

P. M. A. Sloot, A. G. Hoekstra, H. van der Liet, C. G. Figdor, “Scattering matrix elements of biological particles measured in a flow through system: theory and practice,” Appl. Opt. 28, 1752–1762 (1989).
[CrossRef] [PubMed]

P. M. A. Sloot, A. G. Hoekstra, “Arbitrarily-shaped particles measured in flow through systems,” in Proceedings of the Second International Congress on Optical Particle Sizing, E. D. Hirleman, ed. (Arizona State University, Tempe, Ariz., 1990), p. 605.

Terstappen, L. W. M. M.

B. G. Grooth, L. W. M. M. Terstappen, G. J. Puppels, J. Greve, “Light-scattering polarization measurements as a new parameter in flow cytometry,” Cytometry 8, 539–544 (1987).
[CrossRef] [PubMed]

Thurn, R.

Tzeng, H. M.

van de Hulst, H. C.

van der Liet, H.

van Dilla, M. A.

M. A. van Dilla, P. N. Dean, O. D. Laerum, M. R. Melamed, Flow Cytometry: Instrumentation and Data Analysis (Academic, San Diego, Calif., 1985), Chap. 1.

Wall, K. F.

Wang, R. T.

Wangand, R. T.

Ann. Phys. (1)

G. Mie, “Considerations on the optics of turbid media, especially colloidal metal sols,” Ann. Phys. 25, 377–442 (1908).
[CrossRef]

Appl. Opt. (6)

Cytometry (1)

B. G. Grooth, L. W. M. M. Terstappen, G. J. Puppels, J. Greve, “Light-scattering polarization measurements as a new parameter in flow cytometry,” Cytometry 8, 539–544 (1987).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Lett. (2)

Part. Characteriz. (1)

E. Gulari, “Latex particle size distributions from multiwavelength turbidity spectra,” Part. Characteriz. 4, 96–101 (1987).
[CrossRef]

Vidensk. Sel’sk. Shrifter (1)

L. V. Lorenz, “Upon the light reflected and refracted by a transparent sphere,” Vidensk. Sel’sk. Shrifter 6, 1–62 (1890).

Other (4)

P. M. A. Sloot, A. G. Hoekstra, “Arbitrarily-shaped particles measured in flow through systems,” in Proceedings of the Second International Congress on Optical Particle Sizing, E. D. Hirleman, ed. (Arizona State University, Tempe, Ariz., 1990), p. 605.

M. A. van Dilla, P. N. Dean, O. D. Laerum, M. R. Melamed, Flow Cytometry: Instrumentation and Data Analysis (Academic, San Diego, Calif., 1985), Chap. 1.

E. D. Hirleman, ed., Proceedings of the Second International Congress on Optical Particle Sizing (Arizona State University, Tempe, Ariz., 1990), p. 1; Appl. Opt.30, 688–699 (1991).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

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Figures (9)

Fig. 1
Fig. 1

Schematic drawing of the optical system of the FlowCytometer. The laser beam is focused by lens I1 on a cell flowing through the cuvet. The incident beam is polarized by means of the polarizer, P. The intensity of the forward-scattered light is measured by detector d1, and the sideward-scattered light is focused by lens I2 on detectors d2 and d3. The side-scattering signals are passed through two different analyzers, A1 and A2. B, beam splitter.

Fig. 2
Fig. 2

Example of a two-parameter scatterplot for a large number of polystyrene spheres with a mean diameter of 1.98 μm, measured with a FlowCytometer. The wavelength was 0.6328 μm. Every dot represents a single sphere; the value of the x coordinate is the intensity of the forward-scattered (FS) light, and the value of the y coordinate is the intensity of the sideward-scattered (SS) light. The intensities are in arbitrary units.

Fig. 3
Fig. 3

Normalized experimental and theoretical (s 11 + s 12, s 11s 12) scatterplot for polystyrene spheres with a mean diameter of 7.04 μm. The horizontal axis is the s 11 + s 12 signal in the sideward direction, and the vertical axis is the s 11s 12 signal in the sideward direction. The inset shows the theoretical curve only, without scaling of the dot diameter. The arrow represents the starting point (d i = d mean − 4σ d ) and the loop direction, as d i increases, of the theoretical curve.

Fig. 4
Fig. 4

Same as in Fig. 3 but for the (s 11 + s 33, s 11s 33) scatterplot.

Fig. 5
Fig. 5

Same as in Fig. 3 but for the (s 11 + s 34, s 11s 34) scatterplot.

Fig. 6
Fig. 6

Normalized experimental and theoretical (s 11 + s 12, s 11s 12) scatterplot for polystyrene spheres with a mean diameter of 1.98 μm. The horizontal axis is the s 11 + s 12 signal in the sideward direction, and the vertical axis is the s 11s 12 signal in the sideward direction.

Fig. 7
Fig. 7

Integrated-scattering-matrix elements, as a function of the diameter, d, of the sphere (in micrometers). The solid curve is s 11, the dotted curve is s 12, the dashed curve is s 33, and the dashed-dotted curve is s 34. a.u., arbitrary units.

Fig. 8
Fig. 8

S 11 element as a function of the scattering angle θ for d = 1.98 μm. The shaded area denotes the field of view of the side-scattering detectors.

Fig. 9
Fig. 9

S 11 element as a function of the scattering angle θ for d = 7.04 μm. The shaded bar denotes the field of view of the side-scattering detectors.

Equations (10)

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n water = n 0 + n 2 / λ 2 + n 4 / λ 4 .
n rel = n polysty / n water = 1.192 ,
I A = I 0 C ( S 11 ± S ij ) ,             ij = 12 , 33 , 34 ,
I det = d Ω I A d ω = I 0 C ( d Ω S 11 d ω ± d Ω S ij d ω ) I 0 C ( s 11 ± s ij ) ,
d mean - 4 σ d d d mean + 4 σ d ,
r dot = r max exp [ - ( d i - d m ) 2 2 σ d 2 ] ,
scale ex = 1 2 p i = 1 p ( I A1 i + I A 2 i ) = 1 p I 0 C i = 1 p s 11 i ,
( 1 p i = 1 p s 11 i ) - 1 ( s 11 ± s ij ) .
1 p i = 1 p s 11 i
s 11 ¯ = 0 1 σ d 2 π exp [ - ( d - d mean ) 2 2 σ d 2 ] s 11 ( d ) δ d .

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