Abstract

The Airy theory of the rainbow is extended to polarized light. For both polarization directions a simple analytic expression is obtained for the intensity distribution as a function of the scattering angle in terms of the Airy function and its derivative. This approach is valid at least down to droplet diameters of 0.3 mm in visible light. The degree of polarization of the rainbow is less than expected from geometrical optics; it increases with droplet size. For a droplet diameter >1 mm the locations of the supernumerary rainbows are equal for both polarization directions, but for a diameter <1 mm the supernumerary rainbows of the weaker polarization component are located between those in the strong component.

© 1979 Optical Society of America

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References

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  1. G. B. Airy, see any textbook on meteorological optics.
  2. V. Khare, H. M. Nussenzveig, Phys. Rev. Lett. 33, 976 (1974); H. M. Nussenzveig, Sci. Am. 16 (April1977).
    [CrossRef]
  3. C. Boyer, The Rainbow, from Myth to MathematicsYoseloff, New York, 1959).
  4. J. Bricard, Ann. Phys. 14, 148 (1940).
  5. H. C. van de Hulst, Light-Scattering by Small Particles (Wiley, New York, 1957), pp. 249et seq.
  6. W. J. Humphreys, Physics of the Air (McGraw-Hill, New York, 1940), pp. 476et seq.
  7. S. Rosch, Appl. Opt. 7, 233 (1968).
    [CrossRef] [PubMed]

1974 (1)

V. Khare, H. M. Nussenzveig, Phys. Rev. Lett. 33, 976 (1974); H. M. Nussenzveig, Sci. Am. 16 (April1977).
[CrossRef]

1968 (1)

1940 (1)

J. Bricard, Ann. Phys. 14, 148 (1940).

Airy, G. B.

G. B. Airy, see any textbook on meteorological optics.

Boyer, C.

C. Boyer, The Rainbow, from Myth to MathematicsYoseloff, New York, 1959).

Bricard, J.

J. Bricard, Ann. Phys. 14, 148 (1940).

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (McGraw-Hill, New York, 1940), pp. 476et seq.

Khare, V.

V. Khare, H. M. Nussenzveig, Phys. Rev. Lett. 33, 976 (1974); H. M. Nussenzveig, Sci. Am. 16 (April1977).
[CrossRef]

Nussenzveig, H. M.

V. Khare, H. M. Nussenzveig, Phys. Rev. Lett. 33, 976 (1974); H. M. Nussenzveig, Sci. Am. 16 (April1977).
[CrossRef]

Rosch, S.

van de Hulst, H. C.

H. C. van de Hulst, Light-Scattering by Small Particles (Wiley, New York, 1957), pp. 249et seq.

Ann. Phys. (1)

J. Bricard, Ann. Phys. 14, 148 (1940).

Appl. Opt. (1)

Phys. Rev. Lett. (1)

V. Khare, H. M. Nussenzveig, Phys. Rev. Lett. 33, 976 (1974); H. M. Nussenzveig, Sci. Am. 16 (April1977).
[CrossRef]

Other (4)

C. Boyer, The Rainbow, from Myth to MathematicsYoseloff, New York, 1959).

H. C. van de Hulst, Light-Scattering by Small Particles (Wiley, New York, 1957), pp. 249et seq.

W. J. Humphreys, Physics of the Air (McGraw-Hill, New York, 1940), pp. 476et seq.

G. B. Airy, see any textbook on meteorological optics.

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of the origin of supernumerary rainbows. Two light rays with a different impact on a rain drop may lead to scattering in the same direction. Interference between these rays arises from differences in their optical path lengths.

Fig. 2
Fig. 2

Schematic view of the phase jump of a light wave by internal reflection inside a droplet for both polarization directions. IB denotes the Brewster angle, i is the angle of incidence. Interference between two rays with i < IB and i > IB leads to a different interference pattern for the two polarization directions, due to the additional phase jump of 180° for i < IB in the lower part of the figure. Such interference plays an important role in rainbow formation. Apart from the phase jump illustrated here, a weakening of the waves occurs at reflection. This is not shown in the diagram.

Fig. 3
Fig. 3

The intensity distribution as a function of angle for the rainbow at both polarization directions, for 2πa = 1500λ. This corresponds to a droplet radius of about 0.14 mm in visible light. The differences in locations of the supernumerary rainbows can be clearly seen from the graphs. Curve (a) represents the common rainbow, curve (b) the polarized rainbow, corresponding to the E vector tangential and radial with respect to the rainbow, respectively. The formulas used are given in the figure. The intensity scales are comparable. θR denotes the scattering angle; θR the rainbow angle.

Equations (27)

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y = [ h / ( 3 a 2 ) ] x 3 ,
x = a α cos I R .
V ( t ) = - k ( x ) sin ( ω t - δ ) d x ,
V ( t ) = - k ( x ) cos δ d x sin ω t - - k ( x ) sin δ d x cos ω t A sin ω t - B cos ω t .
I n t = A 2 + B 2 .
δ = 2 π λ ( h x 3 3 a 2 - x θ d ) .
A = - k ( x ) cos 2 π λ ( h x 3 3 a 2 - x θ d ) d x , B = - k ( x ) sin 2 π λ ( h x 3 3 a 2 - x θ d ) d x } ,
A m 1 = - [ 1 - sin 2 ( i - r ) sin 2 ( i + r ) ] sin ( i - r ) sin ( i + r ) A m 2 = [ 1 - tan 2 ( i - r ) tan 2 ( i + r ) ] tan ( i - r ) tan ( i + r ) } .
γ = i - I B ,             = r - R B .
1 n cos I B cos R B γ = ;             = 1 n 2 γ = 9 16 γ for water ( n = 4 / 3 ) .
A m 2 / A m 1 ( γ + ) / cos 3 ( I B - R B ) ,
γ = α + I R - I B .
k ( x ) = A m 2 / A m 1 = ( γ + ) / cos 3 ( I B - R B ) = ( 1 + 1 / n 2 ) γ cos 3 ( I B - R B ) = 1.77 γ
k ( x ) = 1.77 a cos I R ( x 0 + x ) ,
x 0 = a ( I R - I B ) cos I R .
- 2 π x θ d λ u z ,             2 π h x 3 3 a 2 λ / 3 1 u 3 ,
A = 2 0 cos ( / 3 1 u 3 + z u ) d u ( a 2 λ 2 π h ) 1 / 3 2 π ( a 2 λ 2 π h ) 1 / 3 A i ( z ) B = 0 } ,
A = 2 · 1.77 a cos I R a cos I R ( I R - I B ) 0 cos ( / 3 1 u 3 + z u ) d u ( a 2 λ ) 1 / 3 2 π h = 2 π · 1.77 ( I R - I B ) ( a 2 λ 2 π h ) 1 / 3 A i ( z ) B = 2 · 1.77 a cos I R 0 u sin ( / 3 1 u 3 + z u ) d u ( a 2 λ 2 π h ) 2 / 3 = - 2 π 1.77 a cos I R ( a 2 λ 2 π h ) 2 / 3 A i ( z ) } ,
z = - ( 4 π 2 a 2 h λ 2 ) 1 / 3 θ d .
I n t 1 = [ A i ( z ) ] 2 ,
I n t 2 = [ 1.77 ( I R - I B ) A i ( z ) ] 2 + ( a 2 λ 2 π h ) 2 / 3 [ 1.77 a cos I R A i ( z ) ] 2 ,
I n t 1 = [ A i ( z ) ] 2 I n t 2 = 0.0376 [ A i 2 ( z ) ] + 0.232 a - 2 / 3 [ A i ( z ) ] 2 } ,
A i ( - z ) π - 1 / 2 z - 1 / 4 cos ( / 3 2 z 3 / 2 - π / 4 ) A i ( - z ) π - 1 / 2 z 1 / 4 sin ( / 3 2 z 3 / 2 - π / 4 ) }
I n t 2 = 0.0376 π - 1 z - 1 / 2 [ 1 + ( 0.232 a - 2 / 3 z - 1 ) sin · 2 ( / 3 2 z 3 / 2 - π / 4 ) ] .
0.232 a - 2 / 3 z - 1 > 0 ,
R = 0.0376 + 0.00465 a - 2 / 3 ,
R = 0.0087 a - 2 / 3 [ A i ( - 4.08 ) / A i ( - 3.25 ) ] 2 = 0.032 a - 2 / 3 ,

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