Abstract

The critical scattering angle at 82.8° from an air bubble in water locates the transition from partial to total reflection from elementary geometrical optics. The irradiance scattered into a narrow angular region near the critical scattering is a monotonically increasing function of bubble radius a provided a ≫ λ, and the weak contributions from rays reflected internally from the far side of the bubble are neglected. The asymptotic series for critical angle scattering derived here leads to a simple approximation for the irradiance. It also describes the breakdown of elementary geometrical optics for reflection at the critical angle from a curved surface. The leading correction to the scattering amplitude relative to the perfect reflection amplitude is found to be O(β−1/4), where β = 2πa/λ is the size parameter and λ is the wavelength of light in water. The series is confirmed by comparison (as a function of β) with smoothed Mie computations. The leading correction is significant for β as large as 20,000, and it is larger when the light is polarized with the E field parallel to the scattering plane rather than perpendicular to it. The dependence on β−1/4 is also shown from an average of the reflection coefficient over a Fresnel zone. Applications to optical bubble sizing are noted, and the nature of approximations in previous physical-optics models of critical angle scattering is clarified.

© 1991 Optical Society of America

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References

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  1. N. Fiedler-Ferrari, “Espalhamento de Mie na vizinhança do ângulo crítico,” Ph.D. dissertation (University of São Paulo, São Paulo, Brazil, 1983);see also N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
    [Crossref] [PubMed]
  2. H. M. Nussenzveig, “Recent developments in high-frequency scattering,” Rev. Bras. Fiz. (Braz. Rev. Phys., special issue)302–320 (1984).
  3. H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
    [Crossref]
  4. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 83–104, 111–114.
  5. P. L. Marston, D. L. Kingsbury, “Scattering by a bubble in water near the critical angle: interference effects,” J. Opt. Soc. Am. 71, 192–196 (1981);errata 71, 917 (1981).
    [Crossref]
  6. P. L. Marston, “Critical angle scattering by a bubble: physical-optics approximation and observations,” J. Opt. Soc. Am. 69, 1205–1211 (1979);errata 70, 353 (1980).
    [Crossref] [PubMed]
  7. C. E. Dean, “Analysis of scattered light: I. Asymptotic series for critical angle scattering from bubbles,” part 1 of Ph.D. dissertation (Washington State University, Pullman, Wash., 1989).
  8. D. L. Kingsbury, P. L. Marston, “Mie scattering near the crtiical angle of bubbles in water,” J. Opt. Soc. Am. 71, 358–361 (1981).
    [Crossref]
  9. D. S. Langley, P. L. Marston, “Critical-angle scattering of laser light from bubbles in water: measurements, models, and application to sizing in bubbles,” Appl. Opt. 23, 1044–1054 (1984).
    [Crossref] [PubMed]
  10. P. L. Marston, J. L. Johnson, S. P. Love, B. L. Brim, “Critical angle scattering of white light from a cylindrical bubble in glass: photographs of colors and computations,” J. Opt. Soc. Am. 73, 1658–1664 + plate X (1983).
    [Crossref]
  11. G. M. Hansen, “Mie scattering as a technique for the sizing of air bubbles,” Appl. Opt. 24, 3214–3220 (1985).
    [Crossref] [PubMed]
  12. S. Baldy, M. Bourguel, “Measurements of bubbles in a stationary field of breaking waves by a laser-based single-particle scattering technique,” J. Geophys. Res. 90, 1037–1047 (1985).
    [Crossref]
  13. D. L. Kingsbury, P. L. Marston, “Scattering by bubbles in glass: Mie theory and physical optics approximation,” Appl. Opt. 20, 2348–2350 (1981).
    [Crossref] [PubMed]
  14. R. G. Holt, “Experimental observation of the nonlinear response of single bubbles to an applied acoustic field,” Ph.D. dissertation, University of Mississippi, University, Miss., 1988);R. G. Holt, L. A. Crum, “Mie scattering used to determine spherical bubble oscillations,” Appl. Opt. 29, 4182–4191 (1990).
    [Crossref] [PubMed]
  15. W. P. Arnott, P. L. Marston, “Optical glory of small freely-rising gas bubbles in water: observed and computed cross-polarized backscattering patterns,” J. Opt. Soc. Am. A 5, 496–506 (1988).
    [Crossref]
  16. P. L. Marston, D. S. Langley, “Bubbles in liquid 4He and 3He: Mie and physical-optics models of light scattering, and quantum tunneling and spinodal models of nucleation,” in Near Zero: New Frontiers in Physics, J. D. Fairbank, A. S. Deaver, C. W. F. Everitt, P. F. Michelson, eds. (Freeman, San Francisco, Calif., 1988), pp. 127–140.
  17. P. L. Marston, “Light scattering from bubbles in water,” in Oceans'89, Publication 89CH2780-5 (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 1186–1193.
  18. W. J. Wiscombe, “Improved Mie scattering algorithms,” Appl. Opt. 19, 1505–1509 (1980).
    [Crossref] [PubMed]
  19. S. W. Lee, “Uniform asymptotic theory of electromagnetic edge diffraction: a review,” in Electromagnetic Scattering, P. L. E. Uslenghi, ed. (Academic, New York, 1978), pp. 69–71.
  20. S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986), pp. 49–53.
  21. A. W. Snyder, J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975).
    [Crossref]
  22. Y. A. Kravtsov, “Rays and caustics as physical objects,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam, 1988), Vol. 26, Chap. 4, pp. 277–280.
    [Crossref]
  23. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 448–449.
  24. T. Pearcey, “The structure of an electromagnetic field in the neighborhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).
  25. I. S. Gradshteyn, I. M. Ryzhik, in Table of integrals, Series, and Products, A. Jeffrey, ed. (Academic, New York, 1980), pp. 399–400.
  26. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), pp. 119–121.
  27. Ref. 25, integrals (3.761.4) and (3.761.8), pp. 420 and 421.
  28. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, San Francisco, Calif., 1986), p. 420.
  29. A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2, p. 221.

1988 (1)

1985 (2)

G. M. Hansen, “Mie scattering as a technique for the sizing of air bubbles,” Appl. Opt. 24, 3214–3220 (1985).
[Crossref] [PubMed]

S. Baldy, M. Bourguel, “Measurements of bubbles in a stationary field of breaking waves by a laser-based single-particle scattering technique,” J. Geophys. Res. 90, 1037–1047 (1985).
[Crossref]

1984 (2)

D. S. Langley, P. L. Marston, “Critical-angle scattering of laser light from bubbles in water: measurements, models, and application to sizing in bubbles,” Appl. Opt. 23, 1044–1054 (1984).
[Crossref] [PubMed]

H. M. Nussenzveig, “Recent developments in high-frequency scattering,” Rev. Bras. Fiz. (Braz. Rev. Phys., special issue)302–320 (1984).

1983 (1)

1981 (3)

1980 (1)

1979 (2)

1975 (1)

A. W. Snyder, J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975).
[Crossref]

1946 (1)

T. Pearcey, “The structure of an electromagnetic field in the neighborhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

Arnott, W. P.

Baldy, S.

S. Baldy, M. Bourguel, “Measurements of bubbles in a stationary field of breaking waves by a laser-based single-particle scattering technique,” J. Geophys. Res. 90, 1037–1047 (1985).
[Crossref]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 83–104, 111–114.

Bourguel, M.

S. Baldy, M. Bourguel, “Measurements of bubbles in a stationary field of breaking waves by a laser-based single-particle scattering technique,” J. Geophys. Res. 90, 1037–1047 (1985).
[Crossref]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, San Francisco, Calif., 1986), p. 420.

Brim, B. L.

Crosignani, B.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986), pp. 49–53.

Dean, C. E.

C. E. Dean, “Analysis of scattered light: I. Asymptotic series for critical angle scattering from bubbles,” part 1 of Ph.D. dissertation (Washington State University, Pullman, Wash., 1989).

DiPorto, P.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986), pp. 49–53.

Erdélyi, A.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2, p. 221.

Fiedler-Ferrari, N.

N. Fiedler-Ferrari, “Espalhamento de Mie na vizinhança do ângulo crítico,” Ph.D. dissertation (University of São Paulo, São Paulo, Brazil, 1983);see also N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
[Crossref] [PubMed]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, in Table of integrals, Series, and Products, A. Jeffrey, ed. (Academic, New York, 1980), pp. 399–400.

Hansen, G. M.

Holt, R. G.

R. G. Holt, “Experimental observation of the nonlinear response of single bubbles to an applied acoustic field,” Ph.D. dissertation, University of Mississippi, University, Miss., 1988);R. G. Holt, L. A. Crum, “Mie scattering used to determine spherical bubble oscillations,” Appl. Opt. 29, 4182–4191 (1990).
[Crossref] [PubMed]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 83–104, 111–114.

Johnson, J. L.

Kingsbury, D. L.

Kravtsov, Y. A.

Y. A. Kravtsov, “Rays and caustics as physical objects,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam, 1988), Vol. 26, Chap. 4, pp. 277–280.
[Crossref]

Langley, D. S.

D. S. Langley, P. L. Marston, “Critical-angle scattering of laser light from bubbles in water: measurements, models, and application to sizing in bubbles,” Appl. Opt. 23, 1044–1054 (1984).
[Crossref] [PubMed]

P. L. Marston, D. S. Langley, “Bubbles in liquid 4He and 3He: Mie and physical-optics models of light scattering, and quantum tunneling and spinodal models of nucleation,” in Near Zero: New Frontiers in Physics, J. D. Fairbank, A. S. Deaver, C. W. F. Everitt, P. F. Michelson, eds. (Freeman, San Francisco, Calif., 1988), pp. 127–140.

Lee, S. W.

S. W. Lee, “Uniform asymptotic theory of electromagnetic edge diffraction: a review,” in Electromagnetic Scattering, P. L. E. Uslenghi, ed. (Academic, New York, 1978), pp. 69–71.

Love, J. D.

A. W. Snyder, J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975).
[Crossref]

Love, S. P.

Magnus, W.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2, p. 221.

Marston, P. L.

W. P. Arnott, P. L. Marston, “Optical glory of small freely-rising gas bubbles in water: observed and computed cross-polarized backscattering patterns,” J. Opt. Soc. Am. A 5, 496–506 (1988).
[Crossref]

D. S. Langley, P. L. Marston, “Critical-angle scattering of laser light from bubbles in water: measurements, models, and application to sizing in bubbles,” Appl. Opt. 23, 1044–1054 (1984).
[Crossref] [PubMed]

P. L. Marston, J. L. Johnson, S. P. Love, B. L. Brim, “Critical angle scattering of white light from a cylindrical bubble in glass: photographs of colors and computations,” J. Opt. Soc. Am. 73, 1658–1664 + plate X (1983).
[Crossref]

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

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

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

P. L. Marston, D. S. Langley, “Bubbles in liquid 4He and 3He: Mie and physical-optics models of light scattering, and quantum tunneling and spinodal models of nucleation,” in Near Zero: New Frontiers in Physics, J. D. Fairbank, A. S. Deaver, C. W. F. Everitt, P. F. Michelson, eds. (Freeman, San Francisco, Calif., 1988), pp. 127–140.

P. L. Marston, “Light scattering from bubbles in water,” in Oceans'89, Publication 89CH2780-5 (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 1186–1193.

Nussenzveig, H. M.

H. M. Nussenzveig, “Recent developments in high-frequency scattering,” Rev. Bras. Fiz. (Braz. Rev. Phys., special issue)302–320 (1984).

H. M. Nussenzveig, “Complex angular momentum theory of the rainbow and the glory,” J. Opt. Soc. Am. 69, 1068–1079 (1979).
[Crossref]

Oberhettinger, F.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2, p. 221.

Pearcey, T.

T. Pearcey, “The structure of an electromagnetic field in the neighborhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, in Table of integrals, Series, and Products, A. Jeffrey, ed. (Academic, New York, 1980), pp. 399–400.

Snyder, A. W.

A. W. Snyder, J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975).
[Crossref]

Solimeno, S.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986), pp. 49–53.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), pp. 119–121.

Tricomi, F. G.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2, p. 221.

Wiscombe, W. J.

Appl. Opt. (4)

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microwave Theory Tech. MTT-23, 134–141 (1975).
[Crossref]

J. Geophys. Res. (1)

S. Baldy, M. Bourguel, “Measurements of bubbles in a stationary field of breaking waves by a laser-based single-particle scattering technique,” J. Geophys. Res. 90, 1037–1047 (1985).
[Crossref]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (1)

Philos. Mag. (1)

T. Pearcey, “The structure of an electromagnetic field in the neighborhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

Rev. Bras. Fiz. (Braz. Rev. Phys., special issue) (1)

H. M. Nussenzveig, “Recent developments in high-frequency scattering,” Rev. Bras. Fiz. (Braz. Rev. Phys., special issue)302–320 (1984).

Other (15)

N. Fiedler-Ferrari, “Espalhamento de Mie na vizinhança do ângulo crítico,” Ph.D. dissertation (University of São Paulo, São Paulo, Brazil, 1983);see also N. Fiedler-Ferrari, H. M. Nussenzveig, W. J. Wiscombe, “Theory of near-critical-angle scattering from a curved interface,” Phys. Rev. A 43, 1005–1038 (1991).
[Crossref] [PubMed]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 83–104, 111–114.

C. E. Dean, “Analysis of scattered light: I. Asymptotic series for critical angle scattering from bubbles,” part 1 of Ph.D. dissertation (Washington State University, Pullman, Wash., 1989).

P. L. Marston, D. S. Langley, “Bubbles in liquid 4He and 3He: Mie and physical-optics models of light scattering, and quantum tunneling and spinodal models of nucleation,” in Near Zero: New Frontiers in Physics, J. D. Fairbank, A. S. Deaver, C. W. F. Everitt, P. F. Michelson, eds. (Freeman, San Francisco, Calif., 1988), pp. 127–140.

P. L. Marston, “Light scattering from bubbles in water,” in Oceans'89, Publication 89CH2780-5 (Institute of Electrical and Electronics Engineers, New York, 1989), pp. 1186–1193.

S. W. Lee, “Uniform asymptotic theory of electromagnetic edge diffraction: a review,” in Electromagnetic Scattering, P. L. E. Uslenghi, ed. (Academic, New York, 1978), pp. 69–71.

S. Solimeno, B. Crosignani, P. DiPorto, Guiding, Diffraction, and Confinement of Optical Radiation (Academic, New York, 1986), pp. 49–53.

R. G. Holt, “Experimental observation of the nonlinear response of single bubbles to an applied acoustic field,” Ph.D. dissertation, University of Mississippi, University, Miss., 1988);R. G. Holt, L. A. Crum, “Mie scattering used to determine spherical bubble oscillations,” Appl. Opt. 29, 4182–4191 (1990).
[Crossref] [PubMed]

I. S. Gradshteyn, I. M. Ryzhik, in Table of integrals, Series, and Products, A. Jeffrey, ed. (Academic, New York, 1980), pp. 399–400.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), pp. 119–121.

Ref. 25, integrals (3.761.4) and (3.761.8), pp. 420 and 421.

R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, San Francisco, Calif., 1986), p. 420.

A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954), Vol. 2, p. 221.

Y. A. Kravtsov, “Rays and caustics as physical objects,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam, 1988), Vol. 26, Chap. 4, pp. 277–280.
[Crossref]

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 448–449.

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Figures (11)

Fig. 1
Fig. 1

Ray diagram showing rays directed to the critical scattering angle θc from a bubble of air in water with a relative index of refraction N = 3/4. The index l is the number of times a ray crosses the optic axis. Only rays with l = 0 will be designated as near side rays in this paper, since such rays are confined to the same side of the bubble as the reflected ray. Near side rays in addition to the p = 0 reflected ray shown are directed to the region with θ < θc.

Fig. 2
Fig. 2

Plots of the dimensionless irradiance i′1 (θc, β) vs size parameter β for perpendicular polarization. The fine solid line is from smoothed Mie computations. The dashed line is the O(β−1/2) asymptotic series expansion derived in this paper. (These curves are almost indistinguishable.) The dot-dashed line represents the irradiance for a perfectly reflecting sphere, that is, the geometrical optics limit. Smoothing for the Mie curve is by Ferrari's method (as explained in Section II) followed by a moving average of I1 over a range of 20 in β, which corresponds to 21 samples.

Fig. 3
Fig. 3

Plots of the dimensionless irradiance i′j vs size parameter β computed as in Fig. 2 but with j = 2 (parallel polarization).

Fig. 4
Fig. 4

Comparison of I2 versus size parameter β for smoothed and unsmoothed Mie theory calculations. The averaging was done by the method of Ferrari, as explained in Section II. The periodicity of the unsmoothed curve is aliased because of a relatively coarse sampling of the unsmoothed values at each decade in size parameter. This sampling period was intentionally chosen so as to allow presentation of the two curves without the smoothed data being masked by the direct Mie theory results. The actual ripple period Δβ of the unsmoothed Mie computations is 3.21.

Fig. 5
Fig. 5

Mie theory for scattering from a bubble in water with N = ¾ and β = 300. This corresponds to a bubble radius of 22 μm in red light having a wavelength in air of 633 nm. The normalized irradiance is shown where geometric reflection from a mirrorlike sphere gives log Ij = 0. The dashed and solid curves correspond to light polarized perpendicular to or parallel to the scattering plane, j = 1 and 2, respectively. The finely spaced structure is ordinarily associated with contributions to the scattering of widely spaced rays. The Ij terms generally increase as the scattering angle θ falls below θc ≈ 82.8°. In the region below θc there is a coarse structure discussed in Refs. 5 and 6. To avoid this structure, it is convenient to consider the scattering at θ = θc, and there is general dependence there on β, which is calculated in the present paper. Note that the fine structure in I2 is suppressed near the Brewster scattering angle of 106°.

Fig. 6
Fig. 6

Comparison of the two-angle method of averaging with Ferrari's method of subtracting off the far side contribution. The Ferrari method values (solid curve) are slightly lower than the two-angle method values (dashed curve) as explained in the text. Each curve displays a moving average over a β range of 40, which corresponds to five samples.

Fig. 7
Fig. 7

Plot of perpendicularly polarized normalized irradiance I1 versus size parameter β. The solid curve is smoothed Mie theory. The dashed line is the O(β−1/4) asymptotic series expansion; that is, Q = 1. There is also a dotted–dashed curve representing the O(β−1/2) expansion (Q = 2), but it is almost entirely on top of the smoothed Mie theory curve. Smoothing of the Mie theory is by Ferrari's method followed by a moving average over a range of 160 in β, which corresponds to 17 samples.

Fig. 8
Fig. 8

Plot of parallel polarized normalized irradiance I2 versus size parameter β. The solid curve is smoothed Mie theory. The dashed curve is the O(β−1/4) asymptotic series expansion, that is, Q = 1. The dotted–dashed curve represents the O(β−1/2) expansion (Q = 2). Smoothing is by Ferrari's method followed by a moving average over a range of 40 in β, which corresponds to five samples.

Fig. 9
Fig. 9

Plot of perpendicularly polarized normalized irradiance I1 vs size parameter β. The solid curve is smoothed Mie theory. The dashed curve is the O(β−1/4) asymptotic series expansion; that is, Q = 1. There is also a dotted–dashed curve representing the O(β−1/2) expansion (Q = 2), but it is almost entirely on top of the smoothed Mie theory curve except when β is small. Smoothing is by Ferrari's method followed by a moving average over a range of 20 in β, which corresponds to 21 samples.

Fig. 10
Fig. 10

Plot of parallel polarized normalized irradiance I2 versus size parameter β. The solid curve is smoothed Mie theory. The dashed curve is the O(β−1/4) asymptotic series expansion; that is, Q = 1. The dotted–dashed curve represents the O(β−1/2) expansion (Q = 2). Smoothing is by Ferrari's method followed by a moving average over a range of 4 in β, which corresponds to five samples.

Fig. 11
Fig. 11

Integration contour used to convert the form of special integrals considered in Appendix D. In each case a semi-Fourier transform is converted to a Laplace transform. Branch cuts coincide with the real and imaginary axes, and poles are outside the contour.

Tables (1)

Tables Icon

Table I Asymptotic Series Coefficients from Eqs. (12) for N = 3/4a

Equations (97)

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

I j ( θ , β ) = | S j ( θ , β ) | 2 ( 2 / β ) 2 ,
i j = i inc I j a 2 / 4 R 2 ,
i j ( θ , β ) = ( i j / i inc ) 4 k 2 R 2 = β 2 I j ,
S j ( θ , β ) = S j 0 + p 2 S jp f . s . + p 1 S jp n . s . ,
I j ( θ c , β ) | q = 0 Q C j , q β q / 4 | 2 ,
C j , q = A j , q + B j , q ,
I j ( θ , β ) = | S j S j 2 f . s . | 2 ( 2 / β ) 2 .
Ī 2 ( θ c , β ) = ½ [ I 2 ( θ + , β ) + I 2 ( θ , β ) ] ,
θ ± = θ c ± δ θ ,
δ θ = ¼ τ ,
τ = 2 π b β ,
S j 0 ( θ c , β ) = K [ P F ( 0 , y ) + F F ] ,
P F ( 0 , y ) + F F = P F 1 + F F 1 + ( P F r + F F r ) ,
P F ( 0 , y ) + F F = π 1 / 2 exp ( i π / 4 ) q = 0 Q C j , q β q / 4 ,
A j , 0 = ½ ,
A j , 1 = exp ( i π / 8 ) ( 2 N / π ) 1 / 2 M 3 / 4 η j Γ ( 3 / 4 ) ,
B j , 0 = ½ ,
B j , 1 = exp ( i 5 π / 8 ) ( 2 N / π ) 1 / 2 M 3 / 4 η j Γ ( 3 / 4 ) ,
C j , 2 = exp ( i π / 4 ) π 1 / 2 ( N / 2 ) M 3 / 2 η j ,
Φ j = arg [ S j 0 ( θ c , β ) ] arg [ S j ( θ c , β ) S j 2 f . s . ( θ c , β ) ] ,
S j 0 ( θ , β ) κ ( θ , β ) r j ( μ ) exp [ i k ( α c μ 2 ɛ μ ) ] d μ ,
Δ j B j μ 1 / 2 , μ a ,
B j = η j ( 8 N / a M 2 ) 1 / 2 ,
S j > ( θ , β ) κ 0 exp [ i ( k ɛ μ k α c μ 2 + B j μ 1 / 2 ) ] d μ .
S j > ( a M / k ) 1 / 2 κ 0 exp [ i ( u 2 + x u + η j y u 1 / 2 ) ] d u ,
1 r j B j | μ | 1 / 2 , | μ | a ,
S j 0 < ( θ c , β ) κ ( θ c , β ) 0 ( 1 B j | μ | 1 / 2 ) exp ( i k α c μ 2 ) d μ .
a f = ( 2 π ρ / k ) 1 / 2 ,
1 2 a f a f 0 B j | μ | 1 / 2 d μ = 1 3 B j a f 1 / 2 η j N 1 / 2 M 3 / 4 β 1 / 4 ,
S j 0 = S j 0 < + S j > ,
K = exp ( i π / 4 ) ( M N 2 π sin θ c ) 1 / 2 β exp ( i 2 β M ) ,
P F ( x , y ) = 0 exp { i [ u 2 x u ( y / 2 1 / 2 ) 4 / 3 η j ln Ai ( 2 4 / 3 u / y 2 / 3 ) ] } d u ,
ln Ai ( z ) z 1 / 2 , | z | 1 ,
P F ( x , y ) 2 0 exp [ i ( t 4 x t 2 + η j y t ) ] t d t ,
P F ( x , y ) 0 exp [ i ( u 2 x u + η j y u 1 / 2 ) ] d u .
F F = 0 1 + g ( υ ) 1 g ( υ ) exp ( i υ 2 ) d υ ,
g ( υ ) = exp ( i π / 6 ) ( N γ η j / M ) ln Ai [ exp ( 2 π / 3 ) z ] ,
S j 2 f . s . i β ( s j / sin θ c ) 1 / 2 ( 2 N c 1 c 2 ) 3 / 2 / ( 2 c 1 N c 2 ) 1 / 2 R j × exp [ 2 i β ( 2 N c 2 c 1 ) ] [ 1 + 0 ( β 1 ) ] ,
R j = ( N c 2 c 1 ) / ( N c 2 + c 1 ) 3 , j = 1 , = ( c 2 N c 1 ) / ( c 2 + N c 1 ) 3 , j = 2 ,
I P ( Y ) = 2 0 exp [ i ( t 4 + Y t ) ] t d t ,
P F ( x = 0 , y ) I P ( Y ) = 2 Y { 0 exp [ i ( t 4 + Y t ) d t ] } .
0 exp [ i ( t 4 + Y t ) ] d t = 1 4 k = 0 ( Y ) k k ! Γ ( k + 1 4 ) exp ( i 1 3 k 8 π ) ,
I P ( Y ) = 2 i Y [ 1 4 k = 0 ( Y ) k k ! Γ ( k + 1 4 ) exp ( i 1 3 k 8 π ) ] .
I P ( Y ) = i 2 k = 1 ( Y ) k 1 ( k 1 ) ! Γ ( k + 1 4 ) exp ( i 1 3 k 8 π ) ,
P F ( 0 , y ) P F 1 = 1 2 π 1 / 2 exp ( i π / 4 ) + y 2 η j Γ ( 3 / 4 ) × exp ( i 7 π / 8 ) + i y 2 4 η j 2 ,
A = 2 1 / 3 N 2 / 3 M 1 η j β 1 / 3 ,
B = 2 1 / 6 N 1 / 6 M 1 / 4 β 1 / 12 ,
F F = 0 1 + A exp ( i π / 6 ) ln Ai [ exp ( i 2 π / 3 ) B 2 z ] 1 A exp ( i π / 6 ) ln Ai [ exp ( i 2 π / 3 ) B 2 z ] exp ( i υ 2 ) d υ .
ln Ai [ exp ( i 2 π / 3 ) B 2 x ] iBx 1 / 2 exp ( i 2 π / 3 ) ,
F F I F ( β ) = 0 1 ABx 1 / 2 1 + ABx 1 / 2 exp ( i x 2 ) d x .
1 1 + ABx 1 / 2 1 ABx 1 / 2 + A 2 B 2 x + .
I F 0 ( 1 2 ABx 1 / 2 + 2 A 2 B 2 x ) exp ( i x 2 ) d x .
I F F F 1 = I 2 ABJ + 2 A 2 B 2 K ,
I = 0 exp ( i x 2 ) d x ,
J = 0 x 1 / 2 exp ( i x 2 ) d x ,
K = 0 x exp ( i x 2 ) d x .
I = ½ π 1 / 2 exp ( i π / 4 ) ,
J = ½ Γ ( 3 / 4 ) exp ( i 3 π / 8 ) ,
K = i 2 .
F F ½ π 1 / 2 exp ( i π / 4 ) exp ( i 3 π / 8 ) 2 1 / 2 N 1 / 2 M 3 / 4 η j Γ ( 3 / 4 ) β 1 / 4 ,
P F r = P F 0 P F 1 ,
F F r = F F 0 F F 1 ,
ln Ai ( z ) A i ( z ) Ai ( z ) ,
ln Ai [ x exp ( i 2 π / 3 ) ] i x 1 / 2 exp ( i 2 π / 3 ) 1 ( 7 i / 48 ) x 3 / 2 1 + ( 5 i / 48 ) x 3 / 2 .
F F 0 = 0 1 [ D x 1 / 2 ( 1 + i δ x 3 / 2 ) / ( 1 + i γ x 3 / 2 ) ] 1 + [ D x 1 / 2 ( 1 + i δ x 3 / 2 ) / ( 1 + i γ x 3 / 2 ) ] exp ( i x 2 ) d x ,
δ = 7 48 ( N 2 ) 1 / 2 M 3 / 4 β 1 / 4 ,
γ = 5 48 ( N 2 ) 1 / 2 M 3 / 4 β 1 / 4 ,
D = ( 2 N ) 1 / 2 M 3 / 4 η j β 1 / 4 .
F F 0 0 d x [ 1 D x 1 / 2 ( 1 + i δ x 3 / 2 1 + i γ x 3 / 2 ) ] exp ( i x 2 ) × [ 1 D x 1 / 2 ( 1 + i δ x 3 / 2 1 + i γ x 3 / 2 ) + D 2 x ( 1 + i δ x 3 / 2 1 + i γ x 3 / 2 ) 2 ] .
F F 0 0 d x [ 1 2 D x 1 / 2 ( 1 + i δ x 3 / 2 1 + i γ x 3 / 2 ) + 2 D 2 x ( 1 + i δ x 3 / 2 1 + i γ x 3 / 2 ) 2 ] exp ( i x 2 ) ,
F F 1 0 [ 1 2 D x 1 / 2 + 2 D 2 x ] exp ( i x 2 ) d x ,
F F r = 0 [ 1 2 D x 1 / 2 ( 1 + i δ x 3 / 2 1 + i γ x 3 / 2 ) + 2 D 2 x ( 1 + i δ x 3 / 2 1 + i γ x 3 / 2 ) 2 1 + 2 D x 1 / 2 2 D 2 x ] exp ( i x 2 ) d x .
F F r L = 2 D 0 x 1 / 2 [ 1 1 + i δ x 3 / 2 1 + i γ x 3 / 2 ] exp ( i x 2 ) d x .
L = i D ( γ δ ) 0 y 1 / 4 exp ( i λ y ) y 3 / 4 + i d y .
I ( α , λ ) = 0 x 1 / 4 exp ( λ x ) d x 1 + exp ( i α x 3 / 4 ) .
P F 0 = P F ( 0 , y ) = 0 exp { i [ u 2 2 A ln Ai ( B 2 u ) ] } d u ,
ln Ai ( B 2 u ) B u 1 / 2 ( 1 δ u 3 / 2 1 γ u 3 / 2 ) ,
P F 0 2 0 t exp ( i t 4 ) exp [ i 2 Dt ( 1 δ t 3 1 γ t 3 ) ] d t ,
P F r = 2 0 t exp ( i t 4 ) { exp [ i 2 Dt ( 1 δ t 3 1 γ t 3 ) ] exp ( i 2 Dt ) } d t .
P F r 2 0 t exp ( i t 4 ) [ i 2 Dt ( 1 δ t 3 1 γ t 3 ) + i 2 Dt ] d t ,
P F r i 4 D ( γ δ ) 0 t 2 exp ( i t 4 ) d t t 3 γ .
P F r D ( γ δ ) exp ( i π / 8 ) I ( 5 π / 8 , λ ) ,
F F r D ( γ δ ) exp ( i 3 π / 8 ) I ( 7 π / 8 , λ ) ,
I ( α , λ ) = 0 1 x 1 / 4 exp ( λ x ) d x 1 + exp ( i α ) x 3 / 4 + exp ( i α ) 1 x 1 exp ( λ x ) d x 1 + exp ( i α ) x 3 / 4 .
I ( α , λ ) = 0 1 d x x 1 / 4 exp ( λ x ) j = 0 [ exp ( i α ) x 3 / 4 ] j + exp ( i α ) 1 d x x 1 exp ( λ x ) j = 0 [ exp ( i α ) x 3 / 4 ] j .
I ( α , λ ) = j = 0 [ exp ( i α ) ] j 0 1 d x x ( 3 j 1 ) / 4 exp ( λ x ) + exp ( i α ) × j = 0 [ exp ( i α ) ] j 1 d x x ( 3 j 4 ) / 4 exp ( λ x ) .
0 1 exp ( λ x ) x μ d x = λ μ 1 γ ( μ + 1 , λ ) ,
1 exp ( λ x ) x μ d x = λ μ 1 Γ ( μ + 1 , λ ) ,
I ( α , λ ) = j = 0 [ exp ( i α ) ] j 4 3 j + 3 γ ( 3 j + 1 4 , λ ) j = 0 [ exp ( i α ) ] j + 1 s j ,
s 0 = E 1 ( λ ) = λ x 1 exp ( x ) d x C ln λ , λ 1 ,
s j = λ 3 j / 4 Γ ( 3 j 4 , λ ) λ 3 j / 4 Γ ( 3 j 4 ) + 4 3 j ,
s j = ( 1 ) m + 1 ( λ ) m E 1 ( λ ) m ! + exp ( λ ) k = 0 m 1 ( 1 ) k λ k j ( j 1 ) ( j k ) .
I ( α , λ ) 4 3 j = 0 [ exp ( i α ) ] j 1 j + 1 + exp ( i α ) [ E 1 ( λ ) ] 4 3 j = 0 [ exp ( i α ) ] j + 1 1 j ,
I ( α , λ ) exp ( i α ) ( E 1 ( λ ) + 4 / 3 { ln [ 1 + exp ( i α ) ] ln [ 1 + exp ( i α ) ] } ) ,
I ( α , λ ) exp ( i α ) [ E 1 ( λ ) + 4 / 3 i α ] .
F F r + P F r D ( γ δ ) [ i E 1 ( λ ) + 7 π / 6 i E 1 ( λ ) + 5 π / 6 ]
F F r + P F r D ( γ δ ) 2 π ,

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