Abstract

The geometrical optics approach is used to derive i1(θ) = |S1(θ)|2 and i2(θ) = |S2(θ)|2, the angular intensity functions for light scattered by a spherical water droplet of a radius comparable with or larger than the wavelength of light. In contrast to previously published results, these functions are obtained in closed form and as functions of the scattering angle θ, which greatly enhance their usefulness in numerical work and in the reduction of large sphere scattering data. The range of validity of these expressions is investigated by graphical comparison of calculated angular intensity patterns with those obtained from rigorous Mie theory. Our main objective is to study the feasibility of using the geometrical optics expressions as a basis for practical laser water droplet sizing work. A criterion is established for the range of applicability of the relationship I(θ,R) = K(θ)R2, which relates the scattering intensity at a particular angle θ to the radius R of the droplet. Accuracy of the laser water droplet sizing technique is thus quantitatively established.

© 1981 Optical Society of America

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References

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  1. F. Shofner, Y. Watanabe, T. Carlson, ISA Trans. 12, 56 (1973).
  2. J. Chan, M. Golay, Atmos. Environ. 2, 775 (1977).
  3. I. Landa, E. Tebay, IEEE Trans. Instrum. Meas. IM-21, 59 (Feb.1971).
  4. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1950).
  5. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  6. G. Mie, Ann. Phys. 25, 377 (1908).
    [CrossRef]
  7. P. Debye, Ann. Phys. 30, 57 (1909).
    [CrossRef]
  8. C. Wiener, Abh. Kaiser Leopold. Carol. Dtsch. Akad. Naturforsch.73 (1907);Abh. Kaiser Leopold. Carol. Dtsch. Akad. Naturforsch.91 (1909).
  9. R. Mecke, Ann. Phys. 62, 623 (1920).
    [CrossRef]
  10. W. Shoulejkin, Philos. Mag. (6), 38, 307 (1924).
  11. H. Blumer, Z. Phys. 38, 920 (1926).
    [CrossRef]
  12. J. Bricard, J. Phys. 4, 57 (1943).
  13. H. C. van de Hulst, Rech. Astron. Obs. Utrecht XI, Part 1 (1946).
  14. G. E. Davis, J. Opt. Soc. Am. 45, 572 (1955).
    [CrossRef]
  15. J. V. Dave, Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere, IBM Data Processing Division, Palo Alto Scientific Center (1968).
  16. P. Chylek, J. Opt. Soc. Am. 63, 1467 (1973);W. G. Wiscombe, P. Chylek, J. Opt. Soc. Am. 67, 572 (1977).
    [CrossRef]
  17. A. Cohen, C. Acquista, J. A. Cooney, Appl. Opt. 19, 2264 (1980).
    [CrossRef] [PubMed]

1980 (1)

1977 (1)

J. Chan, M. Golay, Atmos. Environ. 2, 775 (1977).

1973 (2)

1971 (1)

I. Landa, E. Tebay, IEEE Trans. Instrum. Meas. IM-21, 59 (Feb.1971).

1955 (1)

1946 (1)

H. C. van de Hulst, Rech. Astron. Obs. Utrecht XI, Part 1 (1946).

1943 (1)

J. Bricard, J. Phys. 4, 57 (1943).

1926 (1)

H. Blumer, Z. Phys. 38, 920 (1926).
[CrossRef]

1924 (1)

W. Shoulejkin, Philos. Mag. (6), 38, 307 (1924).

1920 (1)

R. Mecke, Ann. Phys. 62, 623 (1920).
[CrossRef]

1909 (1)

P. Debye, Ann. Phys. 30, 57 (1909).
[CrossRef]

1908 (1)

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

1907 (1)

C. Wiener, Abh. Kaiser Leopold. Carol. Dtsch. Akad. Naturforsch.73 (1907);Abh. Kaiser Leopold. Carol. Dtsch. Akad. Naturforsch.91 (1909).

Acquista, C.

Blumer, H.

H. Blumer, Z. Phys. 38, 920 (1926).
[CrossRef]

Bricard, J.

J. Bricard, J. Phys. 4, 57 (1943).

Carlson, T.

F. Shofner, Y. Watanabe, T. Carlson, ISA Trans. 12, 56 (1973).

Chan, J.

J. Chan, M. Golay, Atmos. Environ. 2, 775 (1977).

Chylek, P.

Cohen, A.

Cooney, J. A.

Dave, J. V.

J. V. Dave, Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere, IBM Data Processing Division, Palo Alto Scientific Center (1968).

Davis, G. E.

Debye, P.

P. Debye, Ann. Phys. 30, 57 (1909).
[CrossRef]

Golay, M.

J. Chan, M. Golay, Atmos. Environ. 2, 775 (1977).

Kerker, M.

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

Landa, I.

I. Landa, E. Tebay, IEEE Trans. Instrum. Meas. IM-21, 59 (Feb.1971).

Mecke, R.

R. Mecke, Ann. Phys. 62, 623 (1920).
[CrossRef]

Mie, G.

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

Shofner, F.

F. Shofner, Y. Watanabe, T. Carlson, ISA Trans. 12, 56 (1973).

Shoulejkin, W.

W. Shoulejkin, Philos. Mag. (6), 38, 307 (1924).

Tebay, E.

I. Landa, E. Tebay, IEEE Trans. Instrum. Meas. IM-21, 59 (Feb.1971).

van de Hulst, H. C.

H. C. van de Hulst, Rech. Astron. Obs. Utrecht XI, Part 1 (1946).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1950).

Watanabe, Y.

F. Shofner, Y. Watanabe, T. Carlson, ISA Trans. 12, 56 (1973).

Wiener, C.

C. Wiener, Abh. Kaiser Leopold. Carol. Dtsch. Akad. Naturforsch.73 (1907);Abh. Kaiser Leopold. Carol. Dtsch. Akad. Naturforsch.91 (1909).

Abh. Kaiser Leopold. Carol. Dtsch. Akad. Naturforsch. (1)

C. Wiener, Abh. Kaiser Leopold. Carol. Dtsch. Akad. Naturforsch.73 (1907);Abh. Kaiser Leopold. Carol. Dtsch. Akad. Naturforsch.91 (1909).

Ann. Phys. (3)

R. Mecke, Ann. Phys. 62, 623 (1920).
[CrossRef]

G. Mie, Ann. Phys. 25, 377 (1908).
[CrossRef]

P. Debye, Ann. Phys. 30, 57 (1909).
[CrossRef]

Appl. Opt. (1)

Atmos. Environ. (1)

J. Chan, M. Golay, Atmos. Environ. 2, 775 (1977).

IEEE Trans. Instrum. Meas. (1)

I. Landa, E. Tebay, IEEE Trans. Instrum. Meas. IM-21, 59 (Feb.1971).

ISA Trans. (1)

F. Shofner, Y. Watanabe, T. Carlson, ISA Trans. 12, 56 (1973).

J. Opt. Soc. Am. (2)

J. Phys. (1)

J. Bricard, J. Phys. 4, 57 (1943).

Philos. Mag. (1)

W. Shoulejkin, Philos. Mag. (6), 38, 307 (1924).

Rech. Astron. Obs. Utrecht (1)

H. C. van de Hulst, Rech. Astron. Obs. Utrecht XI, Part 1 (1946).

Z. Phys. (1)

H. Blumer, Z. Phys. 38, 920 (1926).
[CrossRef]

Other (3)

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1950).

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

J. V. Dave, Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere, IBM Data Processing Division, Palo Alto Scientific Center (1968).

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Figures (7)

Fig. 1
Fig. 1

Reflection and refraction of an incident ray with intensity Iinc(θi) by successive air–water and water–air interfaces. Only the first three emerging rays are shown.

Fig. 2
Fig. 2

Angular scattering diagrams for 5-μm droplets as obtained by geometrical optics and Mie approaches are shown for different angular ranges. Differential intensities i(θ) as defined by Eq. (22) are plotted as functions of scattering angle θ.

Fig. 3
Fig. 3

Same as Fig. 2 except diagrams are for 10-μm droplets. Since the angular increment between successive extrema in the scattering pattern goes as 1/x, smaller angular ranges were chosen in this case to still resolve successive maxima and minima to the same degree as was accomplished in Fig. 2.

Fig. 4
Fig. 4

Same as Figs. 2 and 3 except that in this case the graphs are for 50-μm droplets. Because of a further increased radius, the angular increments are taken still smaller.

Fig. 5
Fig. 5

Integrated and differential intensities i(R,α) and i ̅ ( R , α ) as obtained from Mie theory are presented as functions of droplet radius R. The scattering angle is 50°, and to calculate i ̅ ( R , α ) we chose Δα = 1.6°. Clearly, the effect of integrating or optically averaging the scattering pattern is to linearize i ̅ ( R , α ) as a function of R on a log–log plot. Note that for small droplets the difference between i(R,α) and i ̅ ( R , α ) vanishes because insufficient averaging in this case causes i ̅ ( R , α ) to approach i(R,α).

Fig. 6
Fig. 6

Integrated intensity i ̅ ( R , α ) as a function of R for Δα = 1.6° is shown for three different angles. These results were obtained using our geometrical optics method. Again for small radii the log–log linearity is seen to break down.

Fig. 7
Fig. 7

Here we demonstrate graphically the effect of increasing the averaging angle Δα keeping everything else constant. Clearly the linearity of i ̅ ( R , α ) as a function of R on these log–log plots is best for the case of Δα = 10°.

Equations (40)

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S 1 ( x , m , θ ) = n = 1 2 n + 1 n ( n + 1 ) [ a n ( x , m ) π n ( cos θ ) + b n ( x , m ) τ n ( cos θ ) ] ,
S 2 ( x , m , θ ) = n = 1 2 n + 1 n ( n + 1 ) [ a n ( x , m ) τ n ( cos θ ) + b n ( x , m ) π n ( cos θ ) ] .
θ N ( θ i ) = 2 ( N 1 ) θ r 2 θ i ( N 2 ) π ,
I scat , j ( N ) ( I 0 , R / r , m , θ i ) = I 0 ( R r ) 2 α j ( N ) ( m , θ i ) G ( N ) ( m , θ i ) ,
G N ( m , θ i ) = sin θ i cos θ i sin θ N ( θ i ) × | m 2 sin 2 θ i 2 ( N 1 ) cos θ i m 2 sin 2 θ i | .
α 1 ( 1 ) = r 1 2 α 2 ( 1 ) = r 2 2 } for N = 1 ,
α 1 ( N ) = [ ( 1 r 1 2 ) ( r 1 ) ( N 2 ) ] 2 α 2 ( N ) = [ ( 1 r 2 2 ) ( r 2 ) ( N 2 ) ] 2 } for N 2 ,
r 1 = cos θ i m cos θ r cos θ i + m cos θ r r 2 = m cos θ i cos θ r m cos θ i + cos θ r .
i j ( N ) ( x , m , θ i ) = x 2 α j ( N ) ( m , θ i ) G ( N ) ( m , θ i ) j = 1 , 2 .
S j ( N ) ( x , m , θ i ) = i j ( N ) ( x , m , θ i ) exp [ i σ N ( x , m , θ i ) ] j = 1 , 2 ,
σ 1 ( x , m , θ i ) = π 2 + 2 x cos θ i σ 2 ( x , m , θ i ) = 3 π 2 + 2 x ( cos θ i m cos θ r ) } .
i j ( x , m , θ ) = θ N ( θ i ) = θ i j ( N ) ( x , m , θ i ) j = 1 , 2 ,
S j ( x , m , θ ) = θ N ( θ i ) = θ S j ( N ) ( x , m , θ i ) j = 1 , 2 .
S 1 ( 1 ) ( x , m , θ ) = x sin θ 2 m 2 cos 2 θ 2 sin θ 2 + m 2 cos 2 θ 2 1 2 exp [ i ( π 2 + 2 x sin θ 2 ) ] ,
S 2 ( 1 ) ( x , m , θ ) = x m 2 sin θ 2 m 2 cos 2 θ 2 m 2 sin θ 2 + m 2 cos 2 θ 2 1 2 exp [ i ( π 2 + 2 x sin θ 2 ) ] .
sin θ i = m sin θ 2 1 + m 2 2 m cos θ 2 ,
cos θ i = m cos θ 2 1 1 + m 2 2 m cos θ 2 .
S 1 ( 2 ) ( x , m , θ ) = x [ 1 ( 1 + m 2 2 m cos θ 2 1 m 2 ) 2 ] m 2 sin θ 2 ( m cos θ 2 1 ) ( m cos θ 2 ) 2 sin θ ( 1 + m 2 2 m cos θ 2 ) 2 × exp [ i ( 3 π 2 2 x 1 + m 2 2 m cos θ 2 ) ] ,
S 2 ( 2 ) ( x , m , θ ) = x [ 1 ( ( 1 + m 2 ) cos θ 2 2 m ( m 2 1 ) cos θ 2 ) 2 ] m 2 sin θ 2 ( m cos θ 2 1 ) ( m cos θ 2 ) 2 sin θ ( 1 + m 2 2 m cos θ 2 ) 2 × exp [ i ( 3 π 2 2 x 1 + m 2 2 m cos θ 2 ) ] .
S diff ( x , m , θ ) = x 2 [ J 1 ( x sin θ ) x sin θ ] .
S j ( x , m , θ ) = S diff ( x , m , θ ) + S j ( 1 ) ( x , m , θ ) + S j ( 2 ) ( x , m , θ ) j = 1 , 2 ,
i 1 ( x , m , θ ) = | S 1 ( x , m , θ ) | 2 = x 2 { x 2 [ J 1 ( x sin θ ) x sin θ ] 2 + 2 x [ J 1 ( x sin θ ) x sin θ ] × { 1 2 ( sin θ 2 m 2 cos 2 θ 2 sin θ 2 + m 2 cos 2 θ 2 ) cos ( π 2 + 2 x sin θ 2 ) + [ 1 ( 1 + m 2 cos θ 2 1 m 2 ) 2 ] m 2 sin θ 2 ( m cos θ 2 1 ) ( m cos θ 2 ) 2 sin θ ( 1 + m 2 2 m cos θ 2 ) 2 × cos ( 3 π 2 2 x 1 + m 2 2 m cos θ 2 ) } + 1 4 ( sin θ 2 m 2 cos 2 θ 2 sin θ 2 + m 2 cos 2 θ 2 ) 2 + [ 1 ( 1 + m 2 2 m cos θ 2 1 m 2 ) 2 ] 2 m 2 sin θ 2 ( m cos θ 2 1 ) ( m cos θ 2 ) 2 sin θ ( m 2 + 1 2 m cos θ 2 ) 2 ( sin θ 2 m 2 cos 2 θ 2 sin θ 2 m 2 cos 2 θ 2 ) [ 1 ( 1 + m 2 2 m cos θ 2 1 m 2 ) 2 ] × m 2 sin θ 2 ( m cos θ 2 1 ) ( m cos θ 2 ) 2 sin θ ( 1 + m 2 2 m cos θ 2 ) 2 cos [ 2 x ( sin θ 2 + 1 + m 2 2 m cos θ 2 ) ] } ,
i 2 ( x , m , θ ) = | S 2 ( x , m , θ ) | 2 = x 2 { x 2 [ J 1 ( x sin θ ) x sin θ ] 2 + 2 x [ J 1 ( x sin θ ) x sin θ ] × { 1 2 m 2 sin θ 2 m 2 cos 2 θ 2 m 2 sin θ 2 + m 2 cos 2 θ 2 cos ( π 2 + 2 x sin θ 2 ) + [ 1 ( ( 1 + m 2 ) cos θ 2 2 m ( m 2 1 ) cos θ 2 ) 2 ] m 2 sin θ 2 ( m cos θ 2 1 ) ( m cos θ 2 ) 2 sin θ ( m 2 + 1 2 m cos θ 2 ) 2 × cos ( 3 π 2 2 x 1 + m 2 2 m cos θ 2 ) } + 1 4 ( m 2 sin θ 2 m 2 cos 2 θ 2 m 2 sin θ 2 + m 2 cos 2 θ 2 ) 2 + [ 1 ( ( 1 + m 2 ) cos θ 2 2 m ( m 2 1 ) cos θ 2 ) 2 ] 2 m 2 sin θ 2 ( m cos θ 2 1 ) ( m cos θ 2 ) 2 sin θ ( 1 + m 2 2 m cos θ 2 ) 2 ( m 2 sin θ 2 m 2 cos 2 θ 2 m 2 sin θ 2 + m 2 cos 2 θ 2 ) [ 1 ( ( 1 + m 2 ) cos θ 2 2 m ( m 2 1 ) cos θ 2 ) 2 ] × m 2 sin θ 2 ( m cos θ 2 1 ) ( m cos θ 2 ) 2 sin θ ( 1 + m 2 2 m cos θ 2 ) 2 cos [ 2 x ( sin θ 2 + 1 + m 2 2 m cos θ 2 ) ] } .
i ( θ ) = i 1 ( θ ) + i 2 ( θ ) 2 ,
i 1 ( x , m , θ ) = | S 1 ( x , m , θ ) | 2 = x 2 { 1 4 ( sin θ 2 m 2 cos 2 θ 2 sin θ 2 + m 2 cos 2 θ 2 ) 2 + [ 1 ( 1 + m 2 2 m cos θ 2 1 m 2 ) 2 ] 2 m 2 sin θ 2 ( m cos θ 2 1 ) ( m cos θ 2 ) 2 sin θ ( m 2 + 1 2 m cos θ 2 ) 2 ( sin θ 2 m 2 cos 2 θ 2 sin θ 2 + m 2 cos 2 θ 2 ) [ 1 ( 1 + m 2 2 m cos θ 2 1 m 2 ) 2 ] × m 2 sin θ 2 ( m cos θ 2 1 ) ( m cos θ 2 ) 2 sin θ ( 1 + m 2 2 m cos θ 2 ) 2 cos [ 2 x ( sin θ 2 + m 2 + 1 2 m cos θ 2 ) ] } .
I 2 ( x , m , θ ) = | S 2 ( x , m , θ ) | 2 = x 2 { 1 4 ( m 2 sin θ 2 m 2 cos 2 θ 2 m 2 sin θ 2 + m 2 cos 2 θ 2 ) 2 + [ 1 ( ( 1 + m 2 ) cos θ 2 2 m ( m 2 1 ) cos θ 2 ) 2 ] 2 m 2 sin θ 2 ( m cos θ 2 1 ) ( m cos θ 2 ) 2 sin θ ( 1 + m 2 2 m cos θ 2 ) 2 ( m 2 sin θ 2 m 2 cos 2 θ 2 m 2 sin θ 2 + m 2 cos 2 θ 2 ) [ 1 ( ( 1 + m 2 ) cos θ 2 2 m ( m 2 1 ) cos θ 2 ) 2 ] m 2 sin θ 2 ( m cos θ 2 1 ) ( m cos θ 2 ) 2 sin θ ( 1 + m 2 2 m cos θ 2 ) 2 cos [ 2 x ( sin θ 2 + m 2 + 1 2 m cos θ 2 ) ] } .
f ( x , m , θ ) = 2 x ( sin θ 2 + 1 + m 2 2 m cos θ 1 ) .
Δ θ 2 π x ( cos θ 2 + m sin θ 2 / 1 + m 2 2 m cos θ 2 ) 1 ,
Δ x π ( sin θ 2 + 1 + m 2 2 m cos θ 2 ) 1 .
I scat ( R , m , θ ) = K ( m , θ ) R 2 ,
I ̅ scat ( R , m , α ) = ( 2 π λ ) 2 i ̅ ( x , m , α ) r 2 = ( 2 π λ ) 2 i ̅ 1 ( x , m , α ) + i ̅ 2 ( x , m , α ) 2 r 2 ,
i ̅ j ( x , m , α ) = 1 Δ α α 2 Δ α α + 2 Δ α i j ( x , m , θ ) d θ j = 1 , 2 , = 1 Δ α { Δ α [ | S j ( 1 ) ( x , m , θ ) | 2 + | S j ( 2 ) ( x , m , θ ) | 2 ] d θ 2 Δ α | S j ( 1 ) ( x , m , θ ) | | S j ( 2 ) ( x , m , θ ) | cos [ f ( x , m , θ ) ] d θ } .
f ( x , m , θ ) = f ( x , m , α ) + f ( x , m , α ) ( θ α ) ,
i ̅ j ( x , m , α ) = 1 Δ α { [ | S j ( 1 ) ( x , m , α ) | 2 + | S j ( 2 ) ( x , m , α ) | 2 ] Δ α d θ 2 | S j ( 1 ) ( x , m , α ) | | S j ( 2 ) ( x , m , α ) | Δ α cos [ f ( x , m , α ) + f ( x , m , α ) × ( θ α ) ] d θ } j = 1 , 2 .
i ̅ j ( x , m , α ) = [ | S j ( 1 ) ( x , m , α ) | 2 + | S j ( 2 ) ( x , m , α ) | 2 ] ( 1 k ) j = 1 , 2 ,
k = 2 | S j ( 1 ) ( x , m , α ) | | S j ( 2 ) ( x , m , α ) | | S j ( 1 ) ( x , m , α ) | 2 + | S j ( 2 ) ( x , m , α ) 2 | cos [ f ( x , m , α ) ] × sin [ f ( x , m , α ) Δ α 2 ] f ( x , m , α ) Δ α 2 .
k 1 f ( x , m , α ) Δ α 2 ,
x Δ α 4 π ; Δ α in radius
Q ext = 4 x 2 Re [ S ( 0 ) ] .
lim x Q ext = 2

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