Abstract

A novel effect in the scattering of white light from bubbles consists of colored bands that appear near the critical scattering angle. The bands were photographed in the far-zone scattering from a cylindrical bubble in glass. Their existence is associated with the coarse structure present in the exact scattered intensity. A digital Butterworth filter was developed to remove (from computed intensities) fine structures that are lost when the optical bandwidth is large. The colors are found to be due to the combined effects of interference and diffraction (near the critical scattering angle) and dispersion of the refractive index. Coarse structures were previously modeled in the monochromatic scattering from spherical air bubbles in water. Colors are also to be expected in the appearance of clouds of bubbles in water. Such colors were reported [ C. Pulfrich, Ann. Phys. Chem. (Leipzig) 33, 209 ( 1888)]. Some implications for the optical measurement of bubble size and surface quality are noted.

© 1983 Optical Society of America

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References

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  1. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  3. P. L. Marston, “Critical angle scattering by a bubble: physical-optics approximation and observations,” J. Opt. Soc. Am. 69, 1205–1211 (1979); J. Opt. Soc. Am. 70, 353 (E) (1980).
    [Crossref]
  4. P. L. Marston and D. L. Kingsbury, “Scattering by a bubble in water near the critical angle: interference effects,” J. Opt. Soc. Am. 71, 192–196 (1981); J. Opt. Soc. Am. 71, 917 (E) (1981).
    [Crossref]
  5. D. L. Kingsbury and P. L. Marston, “Mie scattering near the critical angle of bubbles in water,” J. Opt. Soc. Am. 71, 358–361 (1981).
    [Crossref]
  6. D. L. Kingsbury and P. L. Marston, “Scattering by bubbles in glass: Mie theory and physical optics approximation,” Appl. Opt. 20, 2348–2350 (1981).
    [Crossref] [PubMed]
  7. P. L. Marston, D. S. Langley, and D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).
    [Crossref]
  8. D. S. Langley and P. L. Marston, “Critical-angle scattering by a bubble in water: experimental results,” J. Opt. Soc. Am. 72, 1826 (1982).
  9. D. S. Langley and P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981); P. L. Marston, “Light scattering by bubbles in liquids: comments and application of results to circularly polarized incident light,” Appl. Sci. Res. 40, 3–5 (1983).
    [Crossref]
  10. C. Pulfrich, “Ueber eine dem Regenbogen verwandte Erscheinung der Totalreflexion,” Ann. Phys. Chem. (Leipzig) 33, 209–212 (1888).
  11. F. Volz, “Optik der Tropfen,” in Handbuch der Geophysik, F. Linke and F. Moller, eds. (Gebruder Borntraeger, Berlin, 1961), Band 8, pp. 943–1026.
  12. P. L. Marston and D. L. Kingsbury, “Acoustic scattering from fluid spheres: diffraction and interference near the critical scattering angle,” J. Acoust. Soc. Am. 70, 1488–1495 (1981).
    [Crossref]
  13. D. Ludwig, “Diffraction by a circular cavity,” J. Math. Phys. 11, 1617–1630 (1970).
    [Crossref]
  14. Rayleigh (J. W. Strutt), “On the Electromagnetic theory of light,” Phil. Mag. 12, 81–101 (1881).
  15. W. A. Farone and M. Kerker, “Light scattering from long sub-micron glass cylinders at normal incidence,” J. Opt. Soc. Am. 56, 481–487 (1966).
    [Crossref]
  16. P. W. Barber, J. F. Owen, and R. K. Chang, “Resonant scattering for characterization of axisymmetric dielectric objects,” IEEE Trans. Antennas Propag. AP-30, 168–172 (1982).
    [Crossref]
  17. P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908); G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 2nd ed. (Institution of Electrical Engineers, London, 1980), Sec. 3.4.1, Eq. (138).
  18. A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Sec. 5.2.1.
  19. N. J. Kreidl and J. L. Rood, “Optical materials,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, New York, 1965), Vol. 1, pp. 153–200.
  20. S. J. Williamson and H. Z. Cummins, Light and Color in Nature and Art (Wiley, New York, 1983), Plate 10.
  21. C. Pulfrich, “Ein experimenteller Beitrag zur Theorie des Regenbogens und der uberzahligen Bogen,” Ann. Phys. Chem. (Leipzig) 33, 194–208 (1888).
  22. S. Gowing and S. C. Ling, “Measurements of micro-bubbles in a water tunnel,” (David W. Taylor Naval Ship R & D Center, Bethesda, Md., 1981).
  23. D. A. Glaser, “The bubble chamber,” in Handbuch der Physik, (Springer-Verlag, Berlin, 1958), Band 45, pp. 314–341.
    [Crossref]
  24. P. L. Marston, J. L. Johnson, S. P. Love, and B. L. Brim, “Scattering of white light from a cylindrical bubble: observations of colors near the critical scattering angle,” in Digest of the Topical Meeting on Meteorological Optics (Optical Society of America, Washington, D.C.1983), pp. ThA3-1–ThA3-4.
  25. H. J. Orchard, “The phase and envelope delay of Butterworth and Tchebycheff filters,” IRE Trans. Circuit Theory CT-7, 180–181 (1960).
    [Crossref]

1982 (3)

P. L. Marston, D. S. Langley, and D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).
[Crossref]

D. S. Langley and P. L. Marston, “Critical-angle scattering by a bubble in water: experimental results,” J. Opt. Soc. Am. 72, 1826 (1982).

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

1981 (5)

D. S. Langley and P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981); P. L. Marston, “Light scattering by bubbles in liquids: comments and application of results to circularly polarized incident light,” Appl. Sci. Res. 40, 3–5 (1983).
[Crossref]

P. L. Marston and D. L. Kingsbury, “Scattering by a bubble in water near the critical angle: interference effects,” J. Opt. Soc. Am. 71, 192–196 (1981); J. Opt. Soc. Am. 71, 917 (E) (1981).
[Crossref]

D. L. Kingsbury and P. L. Marston, “Mie scattering near the critical angle of bubbles in water,” J. Opt. Soc. Am. 71, 358–361 (1981).
[Crossref]

D. L. Kingsbury and P. L. Marston, “Scattering by bubbles in glass: Mie theory and physical optics approximation,” Appl. Opt. 20, 2348–2350 (1981).
[Crossref] [PubMed]

P. L. Marston and D. L. Kingsbury, “Acoustic scattering from fluid spheres: diffraction and interference near the critical scattering angle,” J. Acoust. Soc. Am. 70, 1488–1495 (1981).
[Crossref]

1979 (1)

1970 (1)

D. Ludwig, “Diffraction by a circular cavity,” J. Math. Phys. 11, 1617–1630 (1970).
[Crossref]

1966 (1)

1960 (1)

H. J. Orchard, “The phase and envelope delay of Butterworth and Tchebycheff filters,” IRE Trans. Circuit Theory CT-7, 180–181 (1960).
[Crossref]

1908 (1)

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908); G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 2nd ed. (Institution of Electrical Engineers, London, 1980), Sec. 3.4.1, Eq. (138).

1888 (2)

C. Pulfrich, “Ein experimenteller Beitrag zur Theorie des Regenbogens und der uberzahligen Bogen,” Ann. Phys. Chem. (Leipzig) 33, 194–208 (1888).

C. Pulfrich, “Ueber eine dem Regenbogen verwandte Erscheinung der Totalreflexion,” Ann. Phys. Chem. (Leipzig) 33, 209–212 (1888).

1881 (1)

Rayleigh (J. W. Strutt), “On the Electromagnetic theory of light,” Phil. Mag. 12, 81–101 (1881).

Barber, P. W.

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

Brim, B. L.

P. L. Marston, J. L. Johnson, S. P. Love, and B. L. Brim, “Scattering of white light from a cylindrical bubble: observations of colors near the critical scattering angle,” in Digest of the Topical Meeting on Meteorological Optics (Optical Society of America, Washington, D.C.1983), pp. ThA3-1–ThA3-4.

Chang, R. K.

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

Cummins, H. Z.

S. J. Williamson and H. Z. Cummins, Light and Color in Nature and Art (Wiley, New York, 1983), Plate 10.

Debye, P.

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908); G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 2nd ed. (Institution of Electrical Engineers, London, 1980), Sec. 3.4.1, Eq. (138).

Farone, W. A.

Glaser, D. A.

D. A. Glaser, “The bubble chamber,” in Handbuch der Physik, (Springer-Verlag, Berlin, 1958), Band 45, pp. 314–341.
[Crossref]

Gowing, S.

S. Gowing and S. C. Ling, “Measurements of micro-bubbles in a water tunnel,” (David W. Taylor Naval Ship R & D Center, Bethesda, Md., 1981).

Johnson, J. L.

P. L. Marston, J. L. Johnson, S. P. Love, and B. L. Brim, “Scattering of white light from a cylindrical bubble: observations of colors near the critical scattering angle,” in Digest of the Topical Meeting on Meteorological Optics (Optical Society of America, Washington, D.C.1983), pp. ThA3-1–ThA3-4.

Kerker, M.

Kingsbury, D. L.

P. L. Marston, D. S. Langley, and D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).
[Crossref]

P. L. Marston and D. L. Kingsbury, “Scattering by a bubble in water near the critical angle: interference effects,” J. Opt. Soc. Am. 71, 192–196 (1981); J. Opt. Soc. Am. 71, 917 (E) (1981).
[Crossref]

D. L. Kingsbury and P. L. Marston, “Mie scattering near the critical angle of bubbles in water,” J. Opt. Soc. Am. 71, 358–361 (1981).
[Crossref]

D. L. Kingsbury and P. L. Marston, “Scattering by bubbles in glass: Mie theory and physical optics approximation,” Appl. Opt. 20, 2348–2350 (1981).
[Crossref] [PubMed]

P. L. Marston and D. L. Kingsbury, “Acoustic scattering from fluid spheres: diffraction and interference near the critical scattering angle,” J. Acoust. Soc. Am. 70, 1488–1495 (1981).
[Crossref]

Kreidl, N. J.

N. J. Kreidl and J. L. Rood, “Optical materials,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, New York, 1965), Vol. 1, pp. 153–200.

Langley, D. S.

D. S. Langley and P. L. Marston, “Critical-angle scattering by a bubble in water: experimental results,” J. Opt. Soc. Am. 72, 1826 (1982).

P. L. Marston, D. S. Langley, and D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).
[Crossref]

D. S. Langley and P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981); P. L. Marston, “Light scattering by bubbles in liquids: comments and application of results to circularly polarized incident light,” Appl. Sci. Res. 40, 3–5 (1983).
[Crossref]

Ling, S. C.

S. Gowing and S. C. Ling, “Measurements of micro-bubbles in a water tunnel,” (David W. Taylor Naval Ship R & D Center, Bethesda, Md., 1981).

Love, S. P.

P. L. Marston, J. L. Johnson, S. P. Love, and B. L. Brim, “Scattering of white light from a cylindrical bubble: observations of colors near the critical scattering angle,” in Digest of the Topical Meeting on Meteorological Optics (Optical Society of America, Washington, D.C.1983), pp. ThA3-1–ThA3-4.

Ludwig, D.

D. Ludwig, “Diffraction by a circular cavity,” J. Math. Phys. 11, 1617–1630 (1970).
[Crossref]

Marston, P. L.

P. L. Marston, D. S. Langley, and D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).
[Crossref]

D. S. Langley and P. L. Marston, “Critical-angle scattering by a bubble in water: experimental results,” J. Opt. Soc. Am. 72, 1826 (1982).

D. S. Langley and P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981); P. L. Marston, “Light scattering by bubbles in liquids: comments and application of results to circularly polarized incident light,” Appl. Sci. Res. 40, 3–5 (1983).
[Crossref]

D. L. Kingsbury and P. L. Marston, “Scattering by bubbles in glass: Mie theory and physical optics approximation,” Appl. Opt. 20, 2348–2350 (1981).
[Crossref] [PubMed]

D. L. Kingsbury and P. L. Marston, “Mie scattering near the critical angle of bubbles in water,” J. Opt. Soc. Am. 71, 358–361 (1981).
[Crossref]

P. L. Marston and D. L. Kingsbury, “Scattering by a bubble in water near the critical angle: interference effects,” J. Opt. Soc. Am. 71, 192–196 (1981); J. Opt. Soc. Am. 71, 917 (E) (1981).
[Crossref]

P. L. Marston and D. L. Kingsbury, “Acoustic scattering from fluid spheres: diffraction and interference near the critical scattering angle,” J. Acoust. Soc. Am. 70, 1488–1495 (1981).
[Crossref]

P. L. Marston, “Critical angle scattering by a bubble: physical-optics approximation and observations,” J. Opt. Soc. Am. 69, 1205–1211 (1979); J. Opt. Soc. Am. 70, 353 (E) (1980).
[Crossref]

P. L. Marston, J. L. Johnson, S. P. Love, and B. L. Brim, “Scattering of white light from a cylindrical bubble: observations of colors near the critical scattering angle,” in Digest of the Topical Meeting on Meteorological Optics (Optical Society of America, Washington, D.C.1983), pp. ThA3-1–ThA3-4.

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Sec. 5.2.1.

Orchard, H. J.

H. J. Orchard, “The phase and envelope delay of Butterworth and Tchebycheff filters,” IRE Trans. Circuit Theory CT-7, 180–181 (1960).
[Crossref]

Owen, J. F.

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

Pulfrich, C.

C. Pulfrich, “Ein experimenteller Beitrag zur Theorie des Regenbogens und der uberzahligen Bogen,” Ann. Phys. Chem. (Leipzig) 33, 194–208 (1888).

C. Pulfrich, “Ueber eine dem Regenbogen verwandte Erscheinung der Totalreflexion,” Ann. Phys. Chem. (Leipzig) 33, 209–212 (1888).

Rayleigh,

Rayleigh (J. W. Strutt), “On the Electromagnetic theory of light,” Phil. Mag. 12, 81–101 (1881).

Rood, J. L.

N. J. Kreidl and J. L. Rood, “Optical materials,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, New York, 1965), Vol. 1, pp. 153–200.

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Sec. 5.2.1.

van de Hulst, H. C.

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

Volz, F.

F. Volz, “Optik der Tropfen,” in Handbuch der Geophysik, F. Linke and F. Moller, eds. (Gebruder Borntraeger, Berlin, 1961), Band 8, pp. 943–1026.

Williamson, S. J.

S. J. Williamson and H. Z. Cummins, Light and Color in Nature and Art (Wiley, New York, 1983), Plate 10.

Ann. Phys. Chem. (Leipzig) (2)

C. Pulfrich, “Ueber eine dem Regenbogen verwandte Erscheinung der Totalreflexion,” Ann. Phys. Chem. (Leipzig) 33, 209–212 (1888).

C. Pulfrich, “Ein experimenteller Beitrag zur Theorie des Regenbogens und der uberzahligen Bogen,” Ann. Phys. Chem. (Leipzig) 33, 194–208 (1888).

Appl. Opt. (1)

Appl. Sci. Res. (1)

P. L. Marston, D. S. Langley, and D. L. Kingsbury, “Light scattering by bubbles in liquids: Mie theory, physical-optics approximations, and experiments,” Appl. Sci. Res. 38, 373–383 (1982).
[Crossref]

IEEE Trans. Antennas Propag. (1)

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

IRE Trans. Circuit Theory (1)

H. J. Orchard, “The phase and envelope delay of Butterworth and Tchebycheff filters,” IRE Trans. Circuit Theory CT-7, 180–181 (1960).
[Crossref]

J. Acoust. Soc. Am. (1)

P. L. Marston and D. L. Kingsbury, “Acoustic scattering from fluid spheres: diffraction and interference near the critical scattering angle,” J. Acoust. Soc. Am. 70, 1488–1495 (1981).
[Crossref]

J. Math. Phys. (1)

D. Ludwig, “Diffraction by a circular cavity,” J. Math. Phys. 11, 1617–1630 (1970).
[Crossref]

J. Opt. Soc. Am. (5)

Phil. Mag. (1)

Rayleigh (J. W. Strutt), “On the Electromagnetic theory of light,” Phil. Mag. 12, 81–101 (1881).

Phys. Rev. Lett. (1)

D. S. Langley and P. L. Marston, “Glory in optical backscattering from air bubbles,” Phys. Rev. Lett. 47, 913–916 (1981); P. L. Marston, “Light scattering by bubbles in liquids: comments and application of results to circularly polarized incident light,” Appl. Sci. Res. 40, 3–5 (1983).
[Crossref]

Phys. Z. (1)

P. Debye, “Das elektromagnetische Feld um einen Zylinder und die Theorie des Regenbogens,” Phys. Z. 9, 775–778 (1908); G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, 2nd ed. (Institution of Electrical Engineers, London, 1980), Sec. 3.4.1, Eq. (138).

Other (9)

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Sec. 5.2.1.

N. J. Kreidl and J. L. Rood, “Optical materials,” in Applied Optics and Optical Engineering, R. Kingslake, ed. (Academic, New York, 1965), Vol. 1, pp. 153–200.

S. J. Williamson and H. Z. Cummins, Light and Color in Nature and Art (Wiley, New York, 1983), Plate 10.

F. Volz, “Optik der Tropfen,” in Handbuch der Geophysik, F. Linke and F. Moller, eds. (Gebruder Borntraeger, Berlin, 1961), Band 8, pp. 943–1026.

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

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

S. Gowing and S. C. Ling, “Measurements of micro-bubbles in a water tunnel,” (David W. Taylor Naval Ship R & D Center, Bethesda, Md., 1981).

D. A. Glaser, “The bubble chamber,” in Handbuch der Physik, (Springer-Verlag, Berlin, 1958), Band 45, pp. 314–341.
[Crossref]

P. L. Marston, J. L. Johnson, S. P. Love, and B. L. Brim, “Scattering of white light from a cylindrical bubble: observations of colors near the critical scattering angle,” in Digest of the Topical Meeting on Meteorological Optics (Optical Society of America, Washington, D.C.1983), pp. ThA3-1–ThA3-4.

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Figures (6)

Fig. 1
Fig. 1

Rays having near-critical angles of incidence leading to a common scattering angle ϕ. Point C lies on the cylinder’s axis.

Fig. 2
Fig. 2

Computations of the normalized scattering intensity near ϕc ≃ 94.5° for x = 5014. The incident and scattered E fields lie in the scattering plane. The finely structured curve is the exact result from Eq. (3). The smooth curve has been low-pass filtered.

Fig. 3
Fig. 3

Low-pass-filtered intensities for blue light (with and without dispersion of the refractive index) and, as in Fig. 2, for red light. The curves illustrate the relative importances of dispersion, interference, and diffraction in the colored scattering. The digital filter is described in Appendix A.

Fig. 4
Fig. 4

View of the apparatus in the plane of scattering, which was horizontal in the laboratory. The bottom of the tube was plugged to prevent liquid from entering its central hole. The relative size of the tube is enlarged in this figure.

Fig. 5
Fig. 5

Photographs of near-critical-angle scattering of white incident light obtained with black-and-white film. The scattering angle increases from left to right. The bubble radius for (a) and (b) is 343 μm and for (c) and (d) it is 1588 μm. The incident light is unpolarized in (b) and (d), whereas in (a) and (c) both the incident and the scattered E fields lie in the scattering plane.

Plate X
Plate X

(Marston et al., p. 1658). Photographs of colored bands adjacent to the critical scattering angle that occur when white light is incident on a bubble. The scattering angle increases from left to right. Polarization and bubble radius for each figure correspond to those in Fig. 5.

Equations (9)

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

i s j = ( a / 2 R ) i j I j ( m , x , ϕ ) ,
I j = sin ( ϕ / 2 ) .
I j = ( 4 / π x ) T j ( m , x , ϕ ) 2 .
T j = n = 0 n a j , n cos ( n ϕ ) .
a 1 , n = m J n ( y ) J n ( x ) - J n ( y ) J n ( x ) m J n ( y ) H n ( x ) - J n ( y ) H n ( x ) ,
a 2 , n = J n ( y ) J n ( x ) - m J n ( y ) J n ( x ) J n ( y ) H n ( x ) - m J n ( y ) H n ( x ) ,
H ( f ) = [ 1 + ( f / f B ) 2 N ] - 1 / 2 ,
Δ ϕ a m - 1 Δ ϕ
h n = k = 1 N α k h n - k + k = 0 N - 1 β k g n - k .