Abstract

The angular distribution of intensity of light scattered from a collimated beam incident upon a spherical air bubble in water is determined for any bubble with radius greater than a few wavelengths of the incident light. The computations are for wavelength 5893 A and n=1.3334, the relative index of refraction of water at 15°C. One external reflection, five internal reflections, and six refractions are considered. A general equation for the geometric attenuation factor is developed, with special forms for external reflection and for zero angle of incidence. The limits of accuracy of the equations for angles of scattering in the neighborhood of 0° and 180° are evaluated. The effects of diffraction and interference are assumed to be negligible. The fractions of the incident light scattered from each of the six “surfaces” (successive points of incidence of any ray on the spherical surface) are shown by accurate curves for all angles of deviation. The corresponding geometric attenutation factors are similarly shown. Final intensity values are tabulated and are plotted both with rectangular and with polar coordinates.

© 1955 Optical Society of America

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References

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  1. R. Clausius, Ann. Physik u. Chemie 76, 161–188 (1849).
    [CrossRef]
  2. (a) Rayleigh, Phil. Mag. Ser. 4, 41, 107–120, 274–279, 447–454 (1871); (b)Scie. Papers1, 87–110; (c)Phil. Mag. Ser. 5,  12, 81–101 (1881); (d)Scie. Papers 1, 518–536; (e)Phil. Mag. Ser. 5,  47, 375–384 (1899); (f)Scie. Papers4, 397–405; (g)Proc. Roy. Soc. (London) A84, 25–46 (1910); (h)Scie. Papers5, 547–568; (i)Proc. Roy. Soc. (London) A90, 219–225 (1914); (j)Scie. Papers6, 220–226; (k)Proc. Roy. Soc. (London) A94, 296–300 (1918); (l)Scie. Papers6, 518–522.
  3. G. Mie, Ann. Physik 25, 377–445 (1908).
    [CrossRef]
  4. F. Mierdel, Beitr. Phys. fr. Atm. 8, 95 (1919).
  5. W. Shoulejkin, Phil. Mag. Ser. 6,  48, 307–320 (1924).
    [CrossRef]
  6. (a)G. Jobst, Ann. Physik Ser. 4,  76, 863–888 (1925); (b)Ann. Physik Ser. 4,  78, 157–166 (1925).
  7. (a)H. Blumer, Z. Physik 32, 119–134 (1925); (b)Z. Physik 38, 304–328 (1926); (c)Z. Physik 38, 920–947 (1926); (d)Z. Physik 39, 195–214 (1926).
  8. F. Roth, Meteorol. Z. 2, 52–62 (1885).
  9. C. Wiener, Abhandl. Kaiser-Leopoldinisch-Carolinische deut. Akad. Naturforsch.731907 (first part) and Abhandl. Kaiser-Leopoldinisch-Carolinische deut. Akad. Naturforsch. 911909 (second part).
  10. F. Richarz, Meteorol. Z. 25, 19 (1908); Sitzber. Ges. Beförder. ges. Naturw. Marburg, No.  1, 1 (1912).
  11. R. Mecke, Ann. Physik, Ser. 4,  61, 471–500 (1920); Ann. Physik, Ser. 4,  62, 623–648 (1920); Ann. Physik, Ser. 4,  65, 257–273 (1921).
    [CrossRef]

1925 (2)

(a)G. Jobst, Ann. Physik Ser. 4,  76, 863–888 (1925); (b)Ann. Physik Ser. 4,  78, 157–166 (1925).

(a)H. Blumer, Z. Physik 32, 119–134 (1925); (b)Z. Physik 38, 304–328 (1926); (c)Z. Physik 38, 920–947 (1926); (d)Z. Physik 39, 195–214 (1926).

1924 (1)

W. Shoulejkin, Phil. Mag. Ser. 6,  48, 307–320 (1924).
[CrossRef]

1920 (1)

R. Mecke, Ann. Physik, Ser. 4,  61, 471–500 (1920); Ann. Physik, Ser. 4,  62, 623–648 (1920); Ann. Physik, Ser. 4,  65, 257–273 (1921).
[CrossRef]

1919 (1)

F. Mierdel, Beitr. Phys. fr. Atm. 8, 95 (1919).

1908 (2)

G. Mie, Ann. Physik 25, 377–445 (1908).
[CrossRef]

F. Richarz, Meteorol. Z. 25, 19 (1908); Sitzber. Ges. Beförder. ges. Naturw. Marburg, No.  1, 1 (1912).

1907 (1)

C. Wiener, Abhandl. Kaiser-Leopoldinisch-Carolinische deut. Akad. Naturforsch.731907 (first part) and Abhandl. Kaiser-Leopoldinisch-Carolinische deut. Akad. Naturforsch. 911909 (second part).

1885 (1)

F. Roth, Meteorol. Z. 2, 52–62 (1885).

1871 (1)

(a) Rayleigh, Phil. Mag. Ser. 4, 41, 107–120, 274–279, 447–454 (1871); (b)Scie. Papers1, 87–110; (c)Phil. Mag. Ser. 5,  12, 81–101 (1881); (d)Scie. Papers 1, 518–536; (e)Phil. Mag. Ser. 5,  47, 375–384 (1899); (f)Scie. Papers4, 397–405; (g)Proc. Roy. Soc. (London) A84, 25–46 (1910); (h)Scie. Papers5, 547–568; (i)Proc. Roy. Soc. (London) A90, 219–225 (1914); (j)Scie. Papers6, 220–226; (k)Proc. Roy. Soc. (London) A94, 296–300 (1918); (l)Scie. Papers6, 518–522.

1849 (1)

R. Clausius, Ann. Physik u. Chemie 76, 161–188 (1849).
[CrossRef]

Blumer, H.

(a)H. Blumer, Z. Physik 32, 119–134 (1925); (b)Z. Physik 38, 304–328 (1926); (c)Z. Physik 38, 920–947 (1926); (d)Z. Physik 39, 195–214 (1926).

Clausius, R.

R. Clausius, Ann. Physik u. Chemie 76, 161–188 (1849).
[CrossRef]

Jobst, G.

(a)G. Jobst, Ann. Physik Ser. 4,  76, 863–888 (1925); (b)Ann. Physik Ser. 4,  78, 157–166 (1925).

Mecke, R.

R. Mecke, Ann. Physik, Ser. 4,  61, 471–500 (1920); Ann. Physik, Ser. 4,  62, 623–648 (1920); Ann. Physik, Ser. 4,  65, 257–273 (1921).
[CrossRef]

Mie, G.

G. Mie, Ann. Physik 25, 377–445 (1908).
[CrossRef]

Mierdel, F.

F. Mierdel, Beitr. Phys. fr. Atm. 8, 95 (1919).

Rayleigh,

(a) Rayleigh, Phil. Mag. Ser. 4, 41, 107–120, 274–279, 447–454 (1871); (b)Scie. Papers1, 87–110; (c)Phil. Mag. Ser. 5,  12, 81–101 (1881); (d)Scie. Papers 1, 518–536; (e)Phil. Mag. Ser. 5,  47, 375–384 (1899); (f)Scie. Papers4, 397–405; (g)Proc. Roy. Soc. (London) A84, 25–46 (1910); (h)Scie. Papers5, 547–568; (i)Proc. Roy. Soc. (London) A90, 219–225 (1914); (j)Scie. Papers6, 220–226; (k)Proc. Roy. Soc. (London) A94, 296–300 (1918); (l)Scie. Papers6, 518–522.

Richarz, F.

F. Richarz, Meteorol. Z. 25, 19 (1908); Sitzber. Ges. Beförder. ges. Naturw. Marburg, No.  1, 1 (1912).

Roth, F.

F. Roth, Meteorol. Z. 2, 52–62 (1885).

Shoulejkin, W.

W. Shoulejkin, Phil. Mag. Ser. 6,  48, 307–320 (1924).
[CrossRef]

Wiener, C.

C. Wiener, Abhandl. Kaiser-Leopoldinisch-Carolinische deut. Akad. Naturforsch.731907 (first part) and Abhandl. Kaiser-Leopoldinisch-Carolinische deut. Akad. Naturforsch. 911909 (second part).

Abhandl. Kaiser-Leopoldinisch-Carolinische deut. Akad. Naturforsch. (1)

C. Wiener, Abhandl. Kaiser-Leopoldinisch-Carolinische deut. Akad. Naturforsch.731907 (first part) and Abhandl. Kaiser-Leopoldinisch-Carolinische deut. Akad. Naturforsch. 911909 (second part).

Ann. Physik (2)

G. Mie, Ann. Physik 25, 377–445 (1908).
[CrossRef]

R. Mecke, Ann. Physik, Ser. 4,  61, 471–500 (1920); Ann. Physik, Ser. 4,  62, 623–648 (1920); Ann. Physik, Ser. 4,  65, 257–273 (1921).
[CrossRef]

Ann. Physik Ser. 4 (1)

(a)G. Jobst, Ann. Physik Ser. 4,  76, 863–888 (1925); (b)Ann. Physik Ser. 4,  78, 157–166 (1925).

Ann. Physik u. Chemie (1)

R. Clausius, Ann. Physik u. Chemie 76, 161–188 (1849).
[CrossRef]

Beitr. Phys. fr. Atm. (1)

F. Mierdel, Beitr. Phys. fr. Atm. 8, 95 (1919).

Meteorol. Z. (2)

F. Richarz, Meteorol. Z. 25, 19 (1908); Sitzber. Ges. Beförder. ges. Naturw. Marburg, No.  1, 1 (1912).

F. Roth, Meteorol. Z. 2, 52–62 (1885).

Phil. Mag. Ser. (1)

(a) Rayleigh, Phil. Mag. Ser. 4, 41, 107–120, 274–279, 447–454 (1871); (b)Scie. Papers1, 87–110; (c)Phil. Mag. Ser. 5,  12, 81–101 (1881); (d)Scie. Papers 1, 518–536; (e)Phil. Mag. Ser. 5,  47, 375–384 (1899); (f)Scie. Papers4, 397–405; (g)Proc. Roy. Soc. (London) A84, 25–46 (1910); (h)Scie. Papers5, 547–568; (i)Proc. Roy. Soc. (London) A90, 219–225 (1914); (j)Scie. Papers6, 220–226; (k)Proc. Roy. Soc. (London) A94, 296–300 (1918); (l)Scie. Papers6, 518–522.

Phil. Mag. Ser. 6 (1)

W. Shoulejkin, Phil. Mag. Ser. 6,  48, 307–320 (1924).
[CrossRef]

Z. Physik (1)

(a)H. Blumer, Z. Physik 32, 119–134 (1925); (b)Z. Physik 38, 304–328 (1926); (c)Z. Physik 38, 920–947 (1926); (d)Z. Physik 39, 195–214 (1926).

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Figures (19)

Fig. 1
Fig. 1

Reflection and refraction of components of an incident ray.

Fig. 2
Fig. 2

Traces of some of the rays which emerge parallel to the optic axis (angle of deviation, DN=0° or a multiple of −π). Drawn for n=1.3334. The numbers 3 to 6 indicate the ordinal numbers of the “surfaces” from which the rays emerge.

Fig. 3
Fig. 3

Reflection and refraction of components of a tubular beam of infinitesimal thickness, coaxial with the bubble.

Fig. 4
Fig. 4

Angle of incidence ≠0°, emergent beam parallel to optic axis.

Fig. 5
Fig. 5

Angle of incidence ≠0°, emergent beam approaching optic axis at small angle.

Fig. 6
Fig. 6

Rays shown emerging from surface N, after N−2 internal reflections (not shown). The angle of deviation, DN, ranges from 82.8° to −720°, for surfaces 2 to 6, inclusive. Oa is drawn parallel to bd.

Fig. 7
Fig. 7

Reflection from 1st surface of air bubble in water.

Fig. 8
Fig. 8

Transmission through 2nd surface of air bubble in water.

Fig. 9
Fig. 9

Transmission through 3rd surface of air bubble in water.

Fig. 10
Fig. 10

Transmission through 4th surface of air bubble in water.

Fig. 11
Fig. 11

Transmission through 5th surface of air bubble in water.

Fig. 12
Fig. 12

Transmission through 6th surface of air bubble in water.

Fig. 13
Fig. 13

Attenuation due to deviation and divergence, for rays emerging from 2nd surface of air bubble in water.

Fig. 14
Fig. 14

Attenuation due to deviation and divergence, for rays emerging from 3rd surface of air bubble in water.

Fig. 15
Fig. 15

Attenuation due to deviation and divergence, for rays emerging from 4th surface of air bubble in water.

Fig. 16
Fig. 16

Attenuation due to deviation and divergence, for rays emerging from 5th surface of air bubble in water.

Fig. 17
Fig. 17

Attenuation due to deviation and divergence, for rays emerging from 6th surface of air bubble in water.

Fig. 18
Fig. 18

Angular distribution of intensity of light scattered by an air bubble in water. I0, intensity of incident beam; R′, radius of bubble; R, distance to point of observation.

Fig. 19
Fig. 19

Angular distribution of intensity of light scattered by an air bubble in water. I0, intensity of incident beam; R′, radius of bubble; R, distance to point of observation.

Tables (4)

Tables Icon

Table I Errors in calculated intensities, for values of i not near 0°, emergent beam nearly parallel to optic axis. Errors due to approximation in assuming H=R sinDN in deriving Eq. (9) for the geometric attenuation factor. Angles DN are actual or equivalent. For example, 1° represents actual deviations of 1°, −1°, −359° and −361°; 179° represents deviations of 179°, −179°, −181°, −539° and −541°.

Tables Icon

Table II Sample calculation. Intensity, I, of scattered light at distance R, at angle of deviation DN=10°. Initial intensity, I0. I=FG. Each value of I shown in Table III was similarly calculated, except values taken from curves.

Tables Icon

Table III Angular distribution of intensity of light scattered by bubble of air in water.a Incident beam collimated, initial intensity I0, index of refraction relative to air, 1.3334, R, distance to point of observation, and R′, radius of bubble. Angle of deviation measured from line and direction of travel of incident rays.

Tables Icon

Table IV Surfaces which together contribute 95 percent or more of the intensity, in order of decreasing magnitudes of their contributions.

Equations (13)

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d A 1 = 8 π R 2 sin i cos i d i .
d E = I 0 d A = I 0 · 2 π R 2 sin i cos i d i .
I = d E / d A 1 = I 0 R 2 / 4 R 2 ,
G = ( I / I 0 ) ( R / R ) 2 .
d A = 2 π H d w = 2 π R 2 sin D 2 d D 2 .
I = d E d A = I 0 · R 2 sin i cos i R 2 sin D N · d i d D N ,
D N = ( 2 N - 2 ) r - 2 i - ( N - 2 ) π .
D N = ( 2 N - 2 ) sin - 1 ( n sin i ) - 2 i - ( N - 2 ) π .
G = ( 1 - n 2 sin 2 i ) 1 2 sin i cos i [ ( 2 N - 2 ) n cos i - 2 ( 1 - n 2 sin 2 i ) 1 2 ] sin D N .
G = 1 4 [ n ( N - 1 ) - 1 ] 2 ,
h = [ 2 n ( N - 1 ) - 1 ] R i .
f 1 = sin 2 ( i - r ) sin 2 ( i + r ) ,
f 2 = tan 2 ( i - r ) tan 2 ( i + r ) ,