Abstract

Photometric immersion refractometry is a technique for determining the refractive index of particulate material. In this technique, the attenuation of light by a suspension of particles is measured as a function of the refractive index of the immersion medium. A minimum attenuation occurs at the refractive index of the medium equal to that of the particles. This technique can serve as a benchmark method for the refractive index determination because it is independent of assumptions invoked by other techniques, such as those based on the inversion of the spectral attenuation data. We present a rigorous model of the photometric immersion refractometry based on the anomalous diffraction approximation for the attenuation efficiency of particles. This model permits one to determine the average value of the real part of the refractive index of the particles, its variance, and the average imaginary part of the refractive index of the particles. In addition, the fourth moment of the particle size distribution can be determined if the concentration and shape of the particles are known. We analyze the sensitivity of this model to experimental errors and discuss the applicability of photometric immersion refractometry to marine microbial particles. We also present experimental results of this technique as applied to heterotrophic marine bacteria. The results indicate that the refractive index of these bacteria was narrowly distributed about the average value of 1.3886.

© 1997 Optical Society of America

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References

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  1. K. L. Carder, P. R. Betzer, D. W. Eggiman, “Physical, chemical and optical measures of suspended particle concentrations: their intercomparison and application to the West African shelf,” in Suspended Solids in Water, J. R. Gibbs, ed. (Plenum, New York, 1974), pp. 221–227.
  2. E. Aas, The Refractive Index of Phytoplankton (Institute of Geophysics, Oslo University, Oslo, Norway, 1981), .
  3. K. F. A. Ross, E. Billing, “The water and solid content of living bacterial spores and vegetative cells as indicated by refractive index measurements,” J. Gen. Microbiol. 16, 418–425 (1957).
    [CrossRef] [PubMed]
  4. R. Barer, S. Joseph, “Refractometry of living cells. Part I: Basics principles,” Q. J. Micros. Sci. 95, 399–423 (1954).
  5. D. Stramski, R. A. Reynolds, “Diel variations in the optical properties of a marine diatom,” Limnol. Oceanogr. 38, 1347–1364 (1993).
    [CrossRef]
  6. D. Stramski, A. Morel, “Optical properties of photosynthetic picoplankton in different physiological states as affected by growth irradiance,” Deep-Sea Res. 37, 245–266 (1990).
    [CrossRef]
  7. S. G. Ackleson, R. W. Spinrad, C. M. Yentsch, J. Brown, W. Korjeff-Bellows, “Phytoplankton optical properties: flow-cytometric examinations of dilution-induced effects,” Appl. Opt. 27, 1262–1269 (1988).
    [CrossRef] [PubMed]
  8. A. Morel, Y.-H. Ahn, “Optics of heterotrophic nanoflagellates and cilliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 1–26 (1991).
    [CrossRef]
  9. D. Stramski, D. A. Kiefer, “Lightscattering by microorganisms in the open ocean,” Prog. Oceanog. 28, 343–383 (1991).
    [CrossRef]
  10. Ch. Waltham, J. Boyle, B. Ramey, J. Smit, “Light scattering and absorption caused by bacterial activity in water,” Appl. Opt. 33, 7536–7541 (1994).
    [CrossRef] [PubMed]
  11. P. Gerhardt, T. C. Beaman, T. R. Corner, J. T. Greenamyre, L. S. Tisa, “Photometric immersion refractometry of bacterial spores,” J. Bacteriol. 150, 643–648 (1982).
    [PubMed]
  12. R. E. Marquis, “Immersion refractometry of isolated bacterial cell walls,” J. Bacteriol. 116, 1273–1279 (1973).
    [PubMed]
  13. J. B. Bateman, J. Wagman, E. L. Carstensen, “Refraction and absorption of light in bacterial suspensions,” Kolloid Z. Z. Polym. 8, 44–58 (1966).
    [CrossRef]
  14. P. Latimer, “Influence of selective light scattering on measurements of absorption spectra of Chlorella,” Plant Physiol. 34, 193–199 (1959).
    [CrossRef] [PubMed]
  15. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1957).
  16. C. D. Mobley, Light and Water. Radiative Transfer in Natural Waters (Academic, San Diego, 1994).
  17. W. Heller, R. M. Tabibian, “Experimental investigations on the light scattering of colloidal spheres. 2. Sources of error in turbidity measurements,” J. Colloid Sci. 12, 5–39 (1957).
    [CrossRef]
  18. J. D. Klett, R. A. Sutherland, “Approximate methods for modeling the scattering properties of nonspherical particles: evaluation of the Wentzel-Kramers-Brillouin method,” Appl. Opt. 31, 373–386 (1992).
    [CrossRef] [PubMed]
  19. C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
    [CrossRef]
  20. D. Hudson, Statistics (CERN, Geneva, 1964).
  21. Y.-H. Ahn, A. Bricaud, A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res. 39, 1835–1855 (1992).
    [CrossRef]
  22. J. E. Hobbie, R. J. Daley, S. Jasper, “Use of Nuclepore filters for counting bacteria by fluorescent microscopy,” Appl. Environ. Microbiol. 33, 1225–1228 (1977).
    [PubMed]
  23. V. Vouk, “Projected area of convex bodies,” Nature (London) 162, 330–331 (1948).
    [CrossRef]
  24. A. Morel, A. Bricaud, “Inherent properties of algal cells including picoplankton: theoretical and experimental results,” Can. Bull. Fish. Aqua. Sci. 214, 521–559 (1986).
  25. R. Barer, “Spectrophotometry of clarified cell suspensions,” Science 121, 709–715 (1955).
    [CrossRef] [PubMed]
  26. D. Stramski, D. A. Kiefer, “Optical properties of marine bacteria,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 250–268 (1990).
    [CrossRef]
  27. M. Jonasz, G. Fournier, “Approximation of the size distributions of marine particles by a sum of log-normal functions,” Limnol. Oceanogr. 41, 744–754 (1996).
    [CrossRef]

1996 (1)

M. Jonasz, G. Fournier, “Approximation of the size distributions of marine particles by a sum of log-normal functions,” Limnol. Oceanogr. 41, 744–754 (1996).
[CrossRef]

1994 (1)

1993 (1)

D. Stramski, R. A. Reynolds, “Diel variations in the optical properties of a marine diatom,” Limnol. Oceanogr. 38, 1347–1364 (1993).
[CrossRef]

1992 (2)

1991 (2)

A. Morel, Y.-H. Ahn, “Optics of heterotrophic nanoflagellates and cilliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 1–26 (1991).
[CrossRef]

D. Stramski, D. A. Kiefer, “Lightscattering by microorganisms in the open ocean,” Prog. Oceanog. 28, 343–383 (1991).
[CrossRef]

1990 (1)

D. Stramski, A. Morel, “Optical properties of photosynthetic picoplankton in different physiological states as affected by growth irradiance,” Deep-Sea Res. 37, 245–266 (1990).
[CrossRef]

1988 (1)

1987 (1)

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

1986 (1)

A. Morel, A. Bricaud, “Inherent properties of algal cells including picoplankton: theoretical and experimental results,” Can. Bull. Fish. Aqua. Sci. 214, 521–559 (1986).

1982 (1)

P. Gerhardt, T. C. Beaman, T. R. Corner, J. T. Greenamyre, L. S. Tisa, “Photometric immersion refractometry of bacterial spores,” J. Bacteriol. 150, 643–648 (1982).
[PubMed]

1977 (1)

J. E. Hobbie, R. J. Daley, S. Jasper, “Use of Nuclepore filters for counting bacteria by fluorescent microscopy,” Appl. Environ. Microbiol. 33, 1225–1228 (1977).
[PubMed]

1973 (1)

R. E. Marquis, “Immersion refractometry of isolated bacterial cell walls,” J. Bacteriol. 116, 1273–1279 (1973).
[PubMed]

1966 (1)

J. B. Bateman, J. Wagman, E. L. Carstensen, “Refraction and absorption of light in bacterial suspensions,” Kolloid Z. Z. Polym. 8, 44–58 (1966).
[CrossRef]

1959 (1)

P. Latimer, “Influence of selective light scattering on measurements of absorption spectra of Chlorella,” Plant Physiol. 34, 193–199 (1959).
[CrossRef] [PubMed]

1957 (2)

W. Heller, R. M. Tabibian, “Experimental investigations on the light scattering of colloidal spheres. 2. Sources of error in turbidity measurements,” J. Colloid Sci. 12, 5–39 (1957).
[CrossRef]

K. F. A. Ross, E. Billing, “The water and solid content of living bacterial spores and vegetative cells as indicated by refractive index measurements,” J. Gen. Microbiol. 16, 418–425 (1957).
[CrossRef] [PubMed]

1955 (1)

R. Barer, “Spectrophotometry of clarified cell suspensions,” Science 121, 709–715 (1955).
[CrossRef] [PubMed]

1954 (1)

R. Barer, S. Joseph, “Refractometry of living cells. Part I: Basics principles,” Q. J. Micros. Sci. 95, 399–423 (1954).

1948 (1)

V. Vouk, “Projected area of convex bodies,” Nature (London) 162, 330–331 (1948).
[CrossRef]

Aas, E.

E. Aas, The Refractive Index of Phytoplankton (Institute of Geophysics, Oslo University, Oslo, Norway, 1981), .

Ackleson, S. G.

Ahn, Y.-H.

Y.-H. Ahn, A. Bricaud, A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res. 39, 1835–1855 (1992).
[CrossRef]

A. Morel, Y.-H. Ahn, “Optics of heterotrophic nanoflagellates and cilliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 1–26 (1991).
[CrossRef]

Barer, R.

R. Barer, “Spectrophotometry of clarified cell suspensions,” Science 121, 709–715 (1955).
[CrossRef] [PubMed]

R. Barer, S. Joseph, “Refractometry of living cells. Part I: Basics principles,” Q. J. Micros. Sci. 95, 399–423 (1954).

Bateman, J. B.

J. B. Bateman, J. Wagman, E. L. Carstensen, “Refraction and absorption of light in bacterial suspensions,” Kolloid Z. Z. Polym. 8, 44–58 (1966).
[CrossRef]

Beaman, T. C.

P. Gerhardt, T. C. Beaman, T. R. Corner, J. T. Greenamyre, L. S. Tisa, “Photometric immersion refractometry of bacterial spores,” J. Bacteriol. 150, 643–648 (1982).
[PubMed]

Betzer, P. R.

K. L. Carder, P. R. Betzer, D. W. Eggiman, “Physical, chemical and optical measures of suspended particle concentrations: their intercomparison and application to the West African shelf,” in Suspended Solids in Water, J. R. Gibbs, ed. (Plenum, New York, 1974), pp. 221–227.

Billing, E.

K. F. A. Ross, E. Billing, “The water and solid content of living bacterial spores and vegetative cells as indicated by refractive index measurements,” J. Gen. Microbiol. 16, 418–425 (1957).
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

Boyle, J.

Bricaud, A.

Y.-H. Ahn, A. Bricaud, A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res. 39, 1835–1855 (1992).
[CrossRef]

A. Morel, A. Bricaud, “Inherent properties of algal cells including picoplankton: theoretical and experimental results,” Can. Bull. Fish. Aqua. Sci. 214, 521–559 (1986).

Brown, J.

Carder, K. L.

K. L. Carder, P. R. Betzer, D. W. Eggiman, “Physical, chemical and optical measures of suspended particle concentrations: their intercomparison and application to the West African shelf,” in Suspended Solids in Water, J. R. Gibbs, ed. (Plenum, New York, 1974), pp. 221–227.

Carstensen, E. L.

J. B. Bateman, J. Wagman, E. L. Carstensen, “Refraction and absorption of light in bacterial suspensions,” Kolloid Z. Z. Polym. 8, 44–58 (1966).
[CrossRef]

Corner, T. R.

P. Gerhardt, T. C. Beaman, T. R. Corner, J. T. Greenamyre, L. S. Tisa, “Photometric immersion refractometry of bacterial spores,” J. Bacteriol. 150, 643–648 (1982).
[PubMed]

Daley, R. J.

J. E. Hobbie, R. J. Daley, S. Jasper, “Use of Nuclepore filters for counting bacteria by fluorescent microscopy,” Appl. Environ. Microbiol. 33, 1225–1228 (1977).
[PubMed]

Eggiman, D. W.

K. L. Carder, P. R. Betzer, D. W. Eggiman, “Physical, chemical and optical measures of suspended particle concentrations: their intercomparison and application to the West African shelf,” in Suspended Solids in Water, J. R. Gibbs, ed. (Plenum, New York, 1974), pp. 221–227.

Fournier, G.

M. Jonasz, G. Fournier, “Approximation of the size distributions of marine particles by a sum of log-normal functions,” Limnol. Oceanogr. 41, 744–754 (1996).
[CrossRef]

Gerhardt, P.

P. Gerhardt, T. C. Beaman, T. R. Corner, J. T. Greenamyre, L. S. Tisa, “Photometric immersion refractometry of bacterial spores,” J. Bacteriol. 150, 643–648 (1982).
[PubMed]

Greenamyre, J. T.

P. Gerhardt, T. C. Beaman, T. R. Corner, J. T. Greenamyre, L. S. Tisa, “Photometric immersion refractometry of bacterial spores,” J. Bacteriol. 150, 643–648 (1982).
[PubMed]

Heller, W.

W. Heller, R. M. Tabibian, “Experimental investigations on the light scattering of colloidal spheres. 2. Sources of error in turbidity measurements,” J. Colloid Sci. 12, 5–39 (1957).
[CrossRef]

Hobbie, J. E.

J. E. Hobbie, R. J. Daley, S. Jasper, “Use of Nuclepore filters for counting bacteria by fluorescent microscopy,” Appl. Environ. Microbiol. 33, 1225–1228 (1977).
[PubMed]

Hudson, D.

D. Hudson, Statistics (CERN, Geneva, 1964).

Jasper, S.

J. E. Hobbie, R. J. Daley, S. Jasper, “Use of Nuclepore filters for counting bacteria by fluorescent microscopy,” Appl. Environ. Microbiol. 33, 1225–1228 (1977).
[PubMed]

Jonasz, M.

M. Jonasz, G. Fournier, “Approximation of the size distributions of marine particles by a sum of log-normal functions,” Limnol. Oceanogr. 41, 744–754 (1996).
[CrossRef]

Joseph, S.

R. Barer, S. Joseph, “Refractometry of living cells. Part I: Basics principles,” Q. J. Micros. Sci. 95, 399–423 (1954).

Kiefer, D. A.

D. Stramski, D. A. Kiefer, “Lightscattering by microorganisms in the open ocean,” Prog. Oceanog. 28, 343–383 (1991).
[CrossRef]

D. Stramski, D. A. Kiefer, “Optical properties of marine bacteria,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 250–268 (1990).
[CrossRef]

Klett, J. D.

Korjeff-Bellows, W.

Latimer, P.

P. Latimer, “Influence of selective light scattering on measurements of absorption spectra of Chlorella,” Plant Physiol. 34, 193–199 (1959).
[CrossRef] [PubMed]

Marquis, R. E.

R. E. Marquis, “Immersion refractometry of isolated bacterial cell walls,” J. Bacteriol. 116, 1273–1279 (1973).
[PubMed]

Mobley, C. D.

C. D. Mobley, Light and Water. Radiative Transfer in Natural Waters (Academic, San Diego, 1994).

Morel, A.

Y.-H. Ahn, A. Bricaud, A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res. 39, 1835–1855 (1992).
[CrossRef]

A. Morel, Y.-H. Ahn, “Optics of heterotrophic nanoflagellates and cilliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 1–26 (1991).
[CrossRef]

D. Stramski, A. Morel, “Optical properties of photosynthetic picoplankton in different physiological states as affected by growth irradiance,” Deep-Sea Res. 37, 245–266 (1990).
[CrossRef]

A. Morel, A. Bricaud, “Inherent properties of algal cells including picoplankton: theoretical and experimental results,” Can. Bull. Fish. Aqua. Sci. 214, 521–559 (1986).

Ramey, B.

Reynolds, R. A.

D. Stramski, R. A. Reynolds, “Diel variations in the optical properties of a marine diatom,” Limnol. Oceanogr. 38, 1347–1364 (1993).
[CrossRef]

Ross, K. F. A.

K. F. A. Ross, E. Billing, “The water and solid content of living bacterial spores and vegetative cells as indicated by refractive index measurements,” J. Gen. Microbiol. 16, 418–425 (1957).
[CrossRef] [PubMed]

Smit, J.

Spinrad, R. W.

Stramski, D.

D. Stramski, R. A. Reynolds, “Diel variations in the optical properties of a marine diatom,” Limnol. Oceanogr. 38, 1347–1364 (1993).
[CrossRef]

D. Stramski, D. A. Kiefer, “Lightscattering by microorganisms in the open ocean,” Prog. Oceanog. 28, 343–383 (1991).
[CrossRef]

D. Stramski, A. Morel, “Optical properties of photosynthetic picoplankton in different physiological states as affected by growth irradiance,” Deep-Sea Res. 37, 245–266 (1990).
[CrossRef]

D. Stramski, D. A. Kiefer, “Optical properties of marine bacteria,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 250–268 (1990).
[CrossRef]

Sutherland, R. A.

Tabibian, R. M.

W. Heller, R. M. Tabibian, “Experimental investigations on the light scattering of colloidal spheres. 2. Sources of error in turbidity measurements,” J. Colloid Sci. 12, 5–39 (1957).
[CrossRef]

Tisa, L. S.

P. Gerhardt, T. C. Beaman, T. R. Corner, J. T. Greenamyre, L. S. Tisa, “Photometric immersion refractometry of bacterial spores,” J. Bacteriol. 150, 643–648 (1982).
[PubMed]

van de Hulst, H. C.

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

Vouk, V.

V. Vouk, “Projected area of convex bodies,” Nature (London) 162, 330–331 (1948).
[CrossRef]

Wagman, J.

J. B. Bateman, J. Wagman, E. L. Carstensen, “Refraction and absorption of light in bacterial suspensions,” Kolloid Z. Z. Polym. 8, 44–58 (1966).
[CrossRef]

Waltham, Ch.

Yentsch, C. M.

Am. J. Phys. (1)

C. F. Bohren, “Multiple scattering of light and some of its observable consequences,” Am. J. Phys. 55, 524–533 (1987).
[CrossRef]

Appl. Environ. Microbiol. (1)

J. E. Hobbie, R. J. Daley, S. Jasper, “Use of Nuclepore filters for counting bacteria by fluorescent microscopy,” Appl. Environ. Microbiol. 33, 1225–1228 (1977).
[PubMed]

Appl. Opt. (3)

Can. Bull. Fish. Aqua. Sci. (1)

A. Morel, A. Bricaud, “Inherent properties of algal cells including picoplankton: theoretical and experimental results,” Can. Bull. Fish. Aqua. Sci. 214, 521–559 (1986).

Deep-Sea Res. (2)

Y.-H. Ahn, A. Bricaud, A. Morel, “Light backscattering efficiency and related properties of some phytoplankters,” Deep-Sea Res. 39, 1835–1855 (1992).
[CrossRef]

D. Stramski, A. Morel, “Optical properties of photosynthetic picoplankton in different physiological states as affected by growth irradiance,” Deep-Sea Res. 37, 245–266 (1990).
[CrossRef]

J. Bacteriol. (2)

P. Gerhardt, T. C. Beaman, T. R. Corner, J. T. Greenamyre, L. S. Tisa, “Photometric immersion refractometry of bacterial spores,” J. Bacteriol. 150, 643–648 (1982).
[PubMed]

R. E. Marquis, “Immersion refractometry of isolated bacterial cell walls,” J. Bacteriol. 116, 1273–1279 (1973).
[PubMed]

J. Colloid Sci. (1)

W. Heller, R. M. Tabibian, “Experimental investigations on the light scattering of colloidal spheres. 2. Sources of error in turbidity measurements,” J. Colloid Sci. 12, 5–39 (1957).
[CrossRef]

J. Gen. Microbiol. (1)

K. F. A. Ross, E. Billing, “The water and solid content of living bacterial spores and vegetative cells as indicated by refractive index measurements,” J. Gen. Microbiol. 16, 418–425 (1957).
[CrossRef] [PubMed]

J. Mar. Res. (1)

A. Morel, Y.-H. Ahn, “Optics of heterotrophic nanoflagellates and cilliates: a tentative assessment of their scattering role in oceanic waters compared to those of bacterial and algal cells,” J. Mar. Res. 49, 1–26 (1991).
[CrossRef]

Kolloid Z. Z. Polym. (1)

J. B. Bateman, J. Wagman, E. L. Carstensen, “Refraction and absorption of light in bacterial suspensions,” Kolloid Z. Z. Polym. 8, 44–58 (1966).
[CrossRef]

Limnol. Oceanogr. (2)

D. Stramski, R. A. Reynolds, “Diel variations in the optical properties of a marine diatom,” Limnol. Oceanogr. 38, 1347–1364 (1993).
[CrossRef]

M. Jonasz, G. Fournier, “Approximation of the size distributions of marine particles by a sum of log-normal functions,” Limnol. Oceanogr. 41, 744–754 (1996).
[CrossRef]

Nature (London) (1)

V. Vouk, “Projected area of convex bodies,” Nature (London) 162, 330–331 (1948).
[CrossRef]

Plant Physiol. (1)

P. Latimer, “Influence of selective light scattering on measurements of absorption spectra of Chlorella,” Plant Physiol. 34, 193–199 (1959).
[CrossRef] [PubMed]

Prog. Oceanog. (1)

D. Stramski, D. A. Kiefer, “Lightscattering by microorganisms in the open ocean,” Prog. Oceanog. 28, 343–383 (1991).
[CrossRef]

Q. J. Micros. Sci. (1)

R. Barer, S. Joseph, “Refractometry of living cells. Part I: Basics principles,” Q. J. Micros. Sci. 95, 399–423 (1954).

Science (1)

R. Barer, “Spectrophotometry of clarified cell suspensions,” Science 121, 709–715 (1955).
[CrossRef] [PubMed]

Other (6)

D. Stramski, D. A. Kiefer, “Optical properties of marine bacteria,” in Ocean Optics X, R. W. Spinrad, ed., Proc. SPIE1302, 250–268 (1990).
[CrossRef]

K. L. Carder, P. R. Betzer, D. W. Eggiman, “Physical, chemical and optical measures of suspended particle concentrations: their intercomparison and application to the West African shelf,” in Suspended Solids in Water, J. R. Gibbs, ed. (Plenum, New York, 1974), pp. 221–227.

E. Aas, The Refractive Index of Phytoplankton (Institute of Geophysics, Oslo University, Oslo, Norway, 1981), .

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

C. D. Mobley, Light and Water. Radiative Transfer in Natural Waters (Academic, San Diego, 1994).

D. Hudson, Statistics (CERN, Geneva, 1964).

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Figures (7)

Fig. 1
Fig. 1

Log-normal particle size distribution (left axis) used in the simulation of the photometric immersion refractometry (Table 1 and Fig. 2): g(D) = FD max exp[-(lnD - lnD peak)2/(2σD 2)], where FD max = 4 × 108 cm-3 µm-1, Dpeak = 0.4 µm, and σ D = 0.57. The number concentration of the particles is 2.7 × 108 cm-3. The contributions of the various particle size classes to the total optical density (OD, right axis) were calculated with the anomalous diffraction approximation to Q at(n s) for spheres at a wavelength of light of 0.55 µm in vacuum and at the refractive index of 1.36 of the immersion medium. The refractive index of particles in each size class was sampled from a uniform probability distribution with a mean of 1.39 and a spread of ±0.0025. The contributions to the OD obtained with the parabolic approximation to Q at(n s) are also shown (total OD error ≈ +5%).

Fig. 2
Fig. 2

Optical density OD of a particle suspension (size distribution in Fig. 1) as a function of the refractive index of the immersion n s . The OD was calculated with the anomalous diffraction approximation (ADA) for spheres. By using the central 7 points (|〈n〉 - n s | < 0.02) one obtains much better fit to the parabolic approximation to ADA than by using all data points (|〈n〉 - n s | < 0.06).

Fig. 3
Fig. 3

Wavelength spectrum (solid squares) of the optical density OD of a suspension of particles (size distribution in Fig. 1) calculated with the anomalous diffraction approximation (ADA). The refractive index of the immersion medium equals 1.332. The power-law OD spectrum (const × λ-s, line) is characteristic of our experiment with marine bacteria.

Fig. 4
Fig. 4

Marine bacteria used in the immersion refractometry experiment. Bacteria were stained with acridine orange and viewed with an epifluorescent microscope.

Fig. 5
Fig. 5

Aggregates of marine bacteria from the immersion refractometry experiment. Bacteria were stained with acridine orange and viewed with an epifluorescent microscope.

Fig. 6
Fig. 6

Single-cell size distribution of marine bacteria in our experiment. The particle diameter is the diameter of a volume-equivalent sphere. A log-normal function g(D) = FD max exp[-(lnD - ln〈D peak〉) 2/(2σ D 2)] was fitted to the data with the least squares (LSQ). The fit parameters are FD max = 1.85 × 108 cm-3 µm-1, D peak = 0.34 µm, and σ D = 0.4. The number concentration of cells is 1.3 × 108 cm-3. We calculated the LSQ fit by letting the weight of a data point be equal to its y value, as follows from the Poisson probability distribution of the number concentration of bacteria.

Fig. 7
Fig. 7

Optical density OD of a suspension of marine bacteria as measured in solutions of bovine serum albumin with various refractive indices n s . Only seven data points, roughly centered about the ODmin, are shown. The horizontal error bars represent a ±0.001 n s measurement error. The vertical bars represent a ±0.002 OD measurement error. See Table 2 for the OD values, Table 3 for retrieved characteristics of the bacteria, and Fig. 6 for the single-cell size distribution of bacteria.

Tables (3)

Tables Icon

Table 1 Summary of the Simulation of the Photometric Immersion Refractometry

Tables Icon

Table 2 Measured OD (at a wavelength of 0.589 µm in vacuum) of a Suspension of Free-Living Marine Bacteria as a Function of the Refractive Index n s of the Immersion Medium

Tables Icon

Table 3 Summary of a Photometric Immersion Refractometry Experiment at a Wavelength of 0.589 µm (in vacuum) with Heterotrophic Marine Bacteria in Bovine Serum Solutions

Equations (53)

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

OD=cL log e,
N=FDM D, mdD dm,
D=4/πP1/2,
FDMD, m=gDhm,
hmhnhk.
c= CatD, m, λFDMD, mdD dm,
Cat=QatD, m, λPD
Qat=2-4/ρexp-ρ tan βcos β sinρ-β-4/ρ2exp-ρ tan βcos2 β cosρ-2β+4/ρ2cos2 β cos 2β,
ρ=2πD/λn-ns,
tan β=k/n-ns,
m=n-ik,
Qat2π/λ2D2n-ns2-k2+8/3π/λDk.
QatCs4π2/λ2D2n-ns2-k2+Cs16/3×π/λDk,
ODCsA D4gDdD n-ns2-k2hmdm+4/3λ/π  D3gDdD  khmdm,
A=π3 log eLλ-2.
xr= xr fxdx fxdx
OD=CsAND4n-ns2-D4k2+4/3λ/πD3k,
x2=σx2+x2,
ODCsAND4ns2-CsAND42nns+CsAND4σn2+n2+CsAN4/3λ/πD3k,
ODminCsAND4σn2+4/3λ/πD3k.
ODminCsAN4/3λ/πD3k.
OD=a2ns2+a1ns+a0.
a2=CsAND4,
a1=-CsAND42n,
a0=CsAND4σn2+n2+4/3λ/πD3k.
n=-a1/2a2.
D4=a2/CsAN.
kODmin/CsAN4/3λ/πD3.
D3=D43/4 exp-3/2σD2,
kGa0/a23/4-a12/4a27/4,
G=3π exp3/2σD2/4λCsAN1/4.
σn=1/2a22σ2a1+a1/2a222σ2a2-a1/2a23cova1, a21/2,
σk=G1/a23/42σ2a0+a1/2a27/42σ2a1+-3/4a0/a27/4+7/16a12/a211/42σ2a2-a1/a210/4cova0, a1+-3/2a0/a210/4+7/8a12/a214/4cova0, a2+3/4a0a1/a214/4+7/16a13/a218/4cova1, a21/2,
σD41/4=1/4AN-1/4a2-3/42σ2a21/2.
ODminCsAND4σn2.
a0=CsAND4σn2+n2.
σn2=a0/a2-a1/2a22.
σσn2=1/a22σ2a0+a1/2a222σ2a1+-a0/a22+a12/2a232σ2a2-a1/a23cova0, a1+2-a0/a23+a12/2a24cova0, a2-2-a0a1/2a24+a134a25cova1, a21/2,
σσn=1/4σn2σ2σn21/2.
nλ1-nsλ1/nλ2-nsλ2=ODλ1/ODλ21/2λ1/λ2.
σn2=KnSn2λ,
k=KkSkλ,
ODminλCsAND4KnSn2λ+4/3λ/π×D3KkSkλ.
Pavg=1/4πDcylLcell,
OD=const λ-s.
ODCsπ3/λ2L log e  D4n-ns2FDMD, ndD dn,  Csπ3/λ2L log e  D4n2-2D4n ns+D4ns2FDMD, ndD dn.
ODCsAND4n2-2CsAND4nns+CsAND4ns2.
a0=CsAND4n2,
a1=-2CsAND4n,
a2=CsAND4.
ns min=-a1/2a2,  =--2CsAND2n/(2CsAND4,  =D4n/D4.
gD=FDmax exp-lnD-lnDpeak2/2σD2,
D41/4=Dexp3/2σD2.

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